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    <title>Structural Study of the Accelerated Collatz Map - Archive ouverte HAL</title>
    <dc:date>2026-05-24T17:17:59+00:00</dc:date>
    <link>https://hal.science/hal-05574765v1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this paper, we develop a structural analysis of a fully accelerated Collatz map on odd integers, which corresponds to the OEIS sequence A363270, and analyze possible cycles by linking exponential valuation growth, harmonic correction terms, and algebraic encoding. Using an affine logarithmic representation, we derive the recursive identity where the slack variable δ i has an explicit closed form, and is limited by 0 < δ i < 1. Furthermore, we evaluate periodic trajectories under the recursive representation, which yields the identity M N -αK N = ∆ N , expressing exponential imbalance as a cumulative harmonic defect, and show that the same quantity governs the determinant of an associated cyclic linear system. We also analyze convergent and extremal behaviors, congruence restrictions, and collapse configurations for the accelerated map. Combining the affine identity with existing computational bounds, we obtain tight Diophantine constraints that possible periodic orbits must satisfy. While not resolving the longstanding Collatz conjecture, this framework isolates structural mechanisms in the accelerated dynamics.

]]></description>
<dc:subject>dynamical-systems Collatz nonlinear-dynamics statistical-mechanics rather-interesting to-write-about to-simulate</dc:subject>
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<dc:identifier>https://pinboard.in/u:Vaguery/b:71e9416a54d4/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2012.03892">
    <title>[2012.03892] Three characterizations of a self-similar aperiodic 2-dimensional subshift</title>
    <dc:date>2026-05-24T11:57:00+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.03892</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The goal of this chapter is to illustrate a generalization of the Fibonacci word to the case of 2-dimensional configurations on ℤ2. More precisely, we consider a particular subshift of ℤ2 on the alphabet ={0,…,15} for which we give three characterizations: as the subshift Φ generated by a 2-dimensional morphism Φ defined on ; as the Wang shift Ω defined by a set  of 16 Wang tiles; as the symbolic dynamical system ,R representing the orbits under some ℤ2-action R defined by rotations on 𝕋2 and coded by some topological partition  of 𝕋2 into 16 polygonal atoms. We prove their equality Ω=Φ=,R by showing that they are self-similar with respect to the substitution Φ.
This chapter provides a transversal reading of results divided into four different articles obtained through the study of the Jeandel-Rao Wang shift. It gathers in one place the methods introduced to desubstitute Wang shifts and to desubstitute codings of ℤ2-actions by focussing on a simple 2-dimensional self-similar subshift. SageMath code to find marker tiles and compute the Rauzy induction of ℤ2-rotations is provided allowing to reproduce the computations. The chapter contains many exercises whose solutions are provided at the end.
]]></description>
<dc:subject>nonlinear-dynamics rewriting-systems dynamical-systems permutations research-maneuvers rather-interesting to-write-about to-simulate consider:L-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:fdc10ba71d50/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2203.13545">
    <title>[2203.13545] Automorphism groups of random substitution subshifts</title>
    <dc:date>2025-09-11T20:39:50+00:00</dc:date>
    <link>https://arxiv.org/abs/2203.13545</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.
]]></description>
<dc:subject>strings rewriting-systems automata stochastic-systems rather-interesting dynamical-systems group-theory to-understand to-simulate consider:computation quasicrystals aperiodic-tiling symbolic-dynamics</dc:subject>
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<item rdf:about="https://www.mdpi.com/2504-3110/9/8/528">
    <title>Design and Control of Fractional-Order Systems Based on Fractal Operators</title>
    <dc:date>2025-08-18T14:41:29+00:00</dc:date>
    <link>https://www.mdpi.com/2504-3110/9/8/528</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In recent years, we have abstracted physical fractal space from biological structures and movements within living organisms, revealing the profound intrinsic connections between fractional order time and fractional-dimensional space, and providing partial explanations for the sources and orders of fractional order. We have confirmed that the topological invariants of fractal cells, the order of physical components, and the mismatch of spatiotemporal order are important factors determining the fractional order of operators. This paper is a continuation of the previous work. Inspired by bone fractal operators, this article attempts to identify other factors that affect the order of operators. Specifically, the following contents are included: (1) originating from the bone fractal operators, we present the construction process of the “apparent half-order” system; (2) using the Schiessel–Blumen model as the comparative object, we analyze the origin and characteristics of the “γ-order” system; (3) using the continued fraction theory and operatorization thought as the link, we establish the design and control method for general fractional-order systems, and discuss the factors affecting the order of fractional-order operators.
]]></description>
<dc:subject>fractals continued-fractions dynamical-systems nonlinear-dynamics rather-interesting to-write-about to-simulate consider:recursion consider:representation diffy-Qs models-and-modes</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
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<item rdf:about="https://personal.math.ubc.ca/~gerg/index.shtml?abstract=UECFE">
    <title>The unreasonable effectualness of continued function expansions</title>
    <dc:date>2025-07-23T22:28:20+00:00</dc:date>
    <link>https://personal.math.ubc.ca/~gerg/index.shtml?abstract=UECFE</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behavior.
]]></description>
<dc:subject>continued-fractions mathematics number-theory representation oh-no-attractive-nuisance to-write-about to-simulate consider:genetic-programming consider:convergence consider:rewriting-systems dynamical-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
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<item rdf:about="https://arxiv.org/abs/2507.12662">
    <title>[2507.12662] Walking on Archimedean Lattices: Insights from Bloch Band Theory</title>
    <dc:date>2025-07-23T11:47:43+00:00</dc:date>
    <link>https://arxiv.org/abs/2507.12662</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after n steps. Determining the total number of returning walks of a given length n is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As a condensed matter physics application, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.
]]></description>
<dc:subject>random-walks tiling enumeration rather-interesting dynamical-systems number-theory group-theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:96e231b5884d/</dc:identifier>
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<item rdf:about="https://research-portal.uu.nl/en/publications/dynamics-of-number-expansions-and-translation-surfaces">
    <title>Dynamics of number expansions and translation surfaces - Utrecht University</title>
    <dc:date>2025-07-20T13:34:48+00:00</dc:date>
    <link>https://research-portal.uu.nl/en/publications/dynamics-of-number-expansions-and-translation-surfaces</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This thesis uses dynamical systems to study two types of objects: number expansions (Chapters 1 and 2) and translation surfaces (Chapter 3). Our first chapter builds a broad, unifying theory for a large class of continued fraction algorithms producing what we call contracted Farey expansions. These algorithms are based on three ideas: (i) contraction of generalised continued fractions, (ii) induced transformations, and (iii) the natural extension of the Farey tent map. Within this theory, we find several well-studied algorithms; a new subfamily of superoptimal continued fractions with arbitrarily good convergence and approximation properties; and a unifying framework to prove several old and new results in Diophantine approximation. Chapter 2 introduces a new, one-parameter family of functions called skewed symmetric golden maps. Using tools from ergodic theory, we study the relative frequencies of digits typically occurring in number expansions produced by these maps. The central tool for our analysis is a mysterious phenomenon of our functions called matching, which has been recently observed and exploited to understand several other families of functions generating number expansions. Our final chapter deals with translation surfaces, i.e., surfaces obtained by gluing pairs of parallel, equal-length, and oppositely oriented edges of planar polygons. The Veech group of a translation surface is the group of Jacobians of its orientation-preserving affine automorphisms. We develop a novel algorithm to construct translation surfaces with prescribed lattice Veech groups in given strata. Our ideas are also used to obtain obstructions for the realisability of certain Veech groups in certain strata. In particular, we show that the square torus is the only translation surface in any minimal stratum whose Veech group is the modular group.]]></description>
<dc:subject>number-theory dynamical-systems nonlinear-dynamics representation rather-interesting continued-fractions to-understand to-write-about consider:rewriting-systems consider:visualization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9bd39e53430f/</dc:identifier>
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<item rdf:about="https://arxiv.org/abs/2410.18316">
    <title>[2410.18316] Periodic orbits for square and rectangular billiards</title>
    <dc:date>2025-07-19T16:23:51+00:00</dc:date>
    <link>https://arxiv.org/abs/2410.18316</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterised by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We classify all possible periodic orbits on square and rectangular tables. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We also present a connection between the number of different classes of periodic orbits and Euler's totient function, which for any integer N counts how many integers smaller than N share no common divisor with N other than 1. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions, and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).
]]></description>
<dc:subject>dynamical-systems billiards topology review periodic-systems linear-systems representation to-write-about enumeration</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:01600c3bc99f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:review"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:periodic-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:linear-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:enumeration"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2403.19806">
    <title>[2403.19806] Feature-Based Echo-State Networks: A Step Towards Interpretability and Minimalism in Reservoir Computer</title>
    <dc:date>2024-10-30T12:37:11+00:00</dc:date>
    <link>https://arxiv.org/abs/2403.19806</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper proposes a novel and interpretable recurrent neural-network structure using the echo-state network (ESN) paradigm for time-series prediction. While the traditional ESNs perform well for dynamical systems prediction, it needs a large dynamic reservoir with increased computational complexity. It also lacks interpretability to discern contributions from different input combinations to the output. Here, a systematic reservoir architecture is developed using smaller parallel reservoirs driven by different input combinations, known as features, and then they are nonlinearly combined to produce the output. The resultant feature-based ESN (Feat-ESN) outperforms the traditional single-reservoir ESN with less reservoir nodes. The predictive capability of the proposed architecture is demonstrated on three systems: two synthetic datasets from chaotic dynamical systems and a set of real-time traffic data.
]]></description>
<dc:subject>reservoir-computing neural-networks time-series recurrent-networks dynamical-systems to-write-about to-simulate consider:discretization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:27ce316a19c0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:reservoir-computing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:time-series"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:recurrent-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:discretization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2303.13253">
    <title>[2303.13253] What is a degree of freedom? Configuration spaces and their topology</title>
    <dc:date>2024-09-28T15:37:42+00:00</dc:date>
    <link>https://arxiv.org/abs/2303.13253</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Understanding degrees of freedom in classical mechanics is fundamental to characterizing physical systems. Counting them is usually easy, especially if we can assign them a clear meaning. However, the precise definition of a degree of freedom is not usually presented in first-year physics courses since it requires mathematical knowledge only learned in more advanced courses. In this paper, we use a pedagogical approach motivated by simple but non-trivial mechanical examples to define degrees of freedom and configuration spaces. We highlight the role that topology plays in understanding these ideas.
]]></description>
<dc:subject>define-your-terms mathematics philosophy-of-science explanation physics statistics to-write-about rather-interesting topology dynamical-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:446c6a5e7624/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:define-your-terms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:philosophy-of-science"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:explanation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:physics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2210.12229">
    <title>[2210.12229] Deep Reinforcement Learning for Stabilization of Large-scale Probabilistic Boolean Networks</title>
    <dc:date>2023-10-10T10:26:46+00:00</dc:date>
    <link>https://arxiv.org/abs/2210.12229</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The ability to direct a Probabilistic Boolean Network (PBN) to a desired state is important to applications such as targeted therapeutics in cancer biology. Reinforcement Learning (RL) has been proposed as a framework that solves a discrete-time optimal control problem cast as a Markov Decision Process. We focus on an integrative framework powered by a model-free deep RL method that can address different flavours of the control problem (e.g., with or without control inputs; attractor state or a subset of the state space as the target domain). The method is agnostic to the distribution of probabilities for the next state, hence it does not use the probability transition matrix. The time complexity is linear on the time steps, or interactions between the agent (deep RL) and the environment (PBN), during training. Indeed, we explore the scalability of the deep RL approach to (set) stabilization of large-scale PBNs and demonstrate successful control on large networks, including a metastatic melanoma PBN with 200 nodes.
]]></description>
<dc:subject>boolean-networks dynamical-systems nonlinear-dynamics control-theory rather-interesting neural-networks all-the-things Kauffmania to-write-about to-visualize</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0cb34f5bc93a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:boolean-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:control-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:all-the-things"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:Kauffmania"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2207.01810">
    <title>[2207.01810] An additive framework for kirigami design</title>
    <dc:date>2023-09-28T12:36:36+00:00</dc:date>
    <link>https://arxiv.org/abs/2207.01810</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present an additive approach for the inverse design of kirigami-based mechanical metamaterials by focusing on the empty (negative) spaces instead of the solid tiles. By considering each negative space as a four-bar linkage, we identify a simple recursive relationship between adjacent linkages, yielding an efficient method for creating kirigami patterns. This allows us to solve the kirigami design problem using elementary linear algebra, with compatibility, reconfigurability and rigid-deployability encoded into an iterative procedure involving simple matrix multiplications. The resulting linear design strategy circumvents the solution of a non-convex global optimization problem and allows us to control the degrees of freedom in the deployment angle field, linkage offsets and boundary conditions. We demonstrate this by creating a large variety of rigid-deployable, compact, reconfigurable kirigami patterns. We then realize our kirigami designs physically using two simple but effective fabrication strategies with very different materials. Altogether, our additive approaches present routes for efficient mechanical metamaterial design and fabrication based on ori/kirigami art forms.
]]></description>
<dc:subject>kirigami engineering-design mechanism-design rather-interesting mathematical-recreations constraint-satisfaction dynamical-systems to-write-about to-simulate consider:looking-to-see consider:genetic-programming</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:fdbb6baa285b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:kirigami"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:engineering-design"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mechanism-design"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://cinquantesignes.blogspot.com/2022/07/does-this-iteration-end-sum-and-erase.html">
    <title>Does this iteration end? (Sum and erase)</title>
    <dc:date>2022-08-06T12:24:10+00:00</dc:date>
    <link>http://cinquantesignes.blogspot.com/2022/07/does-this-iteration-end-sum-and-erase.html</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Does this iteration end?

The procedure is easy to understand; we take any integer N (for example N = 1124), make the sum S of its digits (S = 8 here) and concatenate S at the end of N (we get 11248).

We iterate from there until the leftmost digit d of N appears in S: we then erase all d’s of the last concatenation – and start from there a new iteration.

]]></description>
<dc:subject>mathematical-recreations integer-sequences rather-interesting dynamical-systems to-write-about consider:learning-from-watching nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:650aff92cee3/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:integer-sequences"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:learning-from-watching"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1207.1067">
    <title>[1207.1067] Bounding differences in Jager Pairs</title>
    <dc:date>2022-01-01T13:22:30+00:00</dc:date>
    <link>https://arxiv.org/abs/1207.1067</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their difference. Results will also apply to the classical regular and backwards continued fractions expansions, which are realized as special cases.
]]></description>
<dc:subject>continued-fractions number-theory approximation rather-interesting dynamical-systems to-write-about to-simulate consider:stochastic-search</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:a28f68ffe2fe/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:stochastic-search"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1811.00384">
    <title>[1811.00384] Encoding and Visualization in the Collatz Conjecture</title>
    <dc:date>2021-11-09T11:34:17+00:00</dc:date>
    <link>https://arxiv.org/abs/1811.00384</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The Collatz conjecture is one of the easiest mathematical problems to state and yet it remains unsolved. For each n≥2 the Collatz iteration is mapped to a binary sequence and a corresponding unique integer which can recreate the iteration. The binary sequence is used to produce the Collatz curve, a 2-D visualization of the iteration on a grid, which, besides the aesthetics, provides a qualitative way for comparing iterations. Two variants of the curves are explored, the r-curves and on-change-turn-right curves. There is a scarcity of acyclic r-curves and only three r-curves were found having a cycle of minimum length greater than 4.
]]></description>
<dc:subject>Collatz-numbers visualization mathematical-recreations rather-interesting dynamical-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9aa55863bb04/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:Collatz-numbers"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2003.07798">
    <title>[2003.07798] Pressio: Enabling projection-based model reduction for large-scale nonlinear dynamical systems</title>
    <dc:date>2021-10-23T06:50:26+00:00</dc:date>
    <link>https://arxiv.org/abs/2003.07798</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This work introduces Pressio, an open-source project aimed at enabling leading-edge projection-based reduced order models (ROMs) for large-scale nonlinear dynamical systems in science and engineering. Pressio provides model-reduction methods that can reduce both the number of spatial and temporal degrees of freedom for any dynamical system expressible as a system of parameterized ordinary differential equations (ODEs). We leverage this simple, expressive mathematical framework as a pivotal design choice to enable a minimal application programming interface (API) that is natural to dynamical systems. The core component of Pressio is a C++11 header-only library that leverages generic programming to support applications with arbitrary data types and arbitrarily complex programming models. This is complemented with Python bindings to expose these C++ functionalities to Python users with negligible overhead and no user-required binding code. We discuss the distinguishing characteristics of Pressio relative to existing model-reduction libraries, outline its key design features, describe how the user interacts with it, and present two test cases -- including one with over 20 million degrees of freedom -- that highlight the performance results of Pressio and illustrate the breath of problems that can be addressed with it.
]]></description>
<dc:subject>nonlinear-dynamics approximation numerical-methods rather-interesting dynamical-systems to-understand system-identification</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:69ce0d534dcf/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:system-identification"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1907.10881">
    <title>[1907.10881] Quadratic Cyclic Sequences</title>
    <dc:date>2021-10-03T20:55:37+00:00</dc:date>
    <link>https://arxiv.org/abs/1907.10881</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We explore relations between cyclic sequences determined by a quadratic difference relation, cyclotomic polynomials, Eulerian digraphs and walks in the plane. These walks correspond to closed paths for which at each step one must turn either left or right through a fixed angle. In the case when this angle is 2π/n, then non-symmetric phenomena occurs for n≥12. Examples arise from algebraic numbers of modulus one which are not n'th roots of unity.
]]></description>
<dc:subject>number-theory dynamical-systems rewriting-systems rather-interesting to-understand algebra</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:144513bbdb7d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rewriting-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algebra"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1904.05980">
    <title>[1904.05980] Parallel algorithms development for programmable logic devices</title>
    <dc:date>2021-10-03T20:50:24+00:00</dc:date>
    <link>https://arxiv.org/abs/1904.05980</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Programmable Logic Devices (PLDs) continue to grow in size and currently contain several millions of gates. At the same time, research effort is going into higher-level hardware synthesis methodologies for reconfigurable computing that can exploit PLD technology. In this paper, we explore the effectiveness and extend one such formal methodology in the design of massively parallel algorithms. We take a step-wise refinement approach to the development of correct reconfigurable hardware circuits from formal specifications. A functional programming notation is used for specifying algorithms and for reasoning about them. The specifications are realised through the use of a combination of function decomposition strategies, data refinement techniques, and off-the-shelf refinements based upon higher-order functions. The off-the-shelf refinements are inspired by the operators of Communicating Sequential Processes (CSP) and map easily to programs in Handel-C (a hardware description language). The Handel-C descriptions are directly compiled into reconfigurable hardware. The practical realisation of this methodology is evidenced by a case studying the matrix multiplication algorithm as it is relatively simple and well known. In this paper, we obtain several hardware implementations with different performance characteristics by applying different refinements to the algorithm. The developed designs are compiled and tested under Celoxica's RC-1000 reconfigurable computer with its 2 million gates Virtex-E FPGA. Performance analysis and evaluation of these implementations are included.
]]></description>
<dc:subject>programming-language formal-languages to-understand dynamical-systems consider:ReQ</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4ebb4501ad5a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:programming-language"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:formal-languages"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ReQ"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1712.05340">
    <title>[1712.05340] Shifts of finite type and random substitutions</title>
    <dc:date>2021-07-28T16:14:55+00:00</dc:date>
    <link>https://arxiv.org/abs/1712.05340</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We prove that every topologically transitive shift of finite type in one dimension is topologically conjugate to a subshift arising from a primitive random substitution on a finite alphabet. As a result, we show that the set of values of topological entropy which can be attained by random substitution subshifts contains all Perron numbers and so is dense in the positive real numbers. We also provide an independent proof of this density statement using elementary methods.
]]></description>
<dc:subject>rewriting-systems formal-languages L-systems rather-interesting dynamical-systems to-write-about to-visualize</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:19d01cd4bbe0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rewriting-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:formal-languages"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:L-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1808.05934">
    <title>[1808.05934] Periodic points in random substitution subshifts</title>
    <dc:date>2021-07-21T21:26:57+00:00</dc:date>
    <link>https://arxiv.org/abs/1808.05934</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study various aspects of periodic points for random substitution subshifts. In order to do so, we introduce a new property for random substitutions called the disjoint images condition. We provide a procedure for determining the property for compatible random substitutions-random substitutions for which a well-defined abelianisation exists. We find some simple necessary criteria for primitive, compatible random substitutions to admit periodic points in their subshifts. In the case that the random substitution further has disjoint images and is of constant length, we provide a stronger criterion. A method is outlined for enumerating periodic points of any specified length in a random substitution subshift.
]]></description>
<dc:subject>rewriting-systems rather-interesting formal-languages probability-theory dynamical-systems to-write-about to-simulate consider:looking-to-see consider:performance-measures</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:583f2d50b1ff/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rewriting-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:formal-languages"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1507.05782">
    <title>[1507.05782] The random continued fraction transformation</title>
    <dc:date>2021-07-15T11:10:25+00:00</dc:date>
    <link>https://arxiv.org/abs/1507.05782</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce a random dynamical system related to continued fraction expansions. It uses random combination of the Gauss map and the Rényi (or backwards) continued fraction map. We explore the continued fraction expansions that this system produces as well as the dynamical properties of the system.
]]></description>
<dc:subject>continued-fractions dynamical-systems number-theory rather-interesting looking-to-see to-write-about to-visualize consider:distribution-drawing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4cbb471bf331/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-visualize"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:distribution-drawing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1512.02171">
    <title>[1512.02171] Right-jumps and pattern avoiding permutations</title>
    <dc:date>2021-07-12T10:30:23+00:00</dc:date>
    <link>https://arxiv.org/abs/1512.02171</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study the iteration of the process "a particle jumps to the right" in permutations. We prove that the set of permutations obtained in this model after a given number of iterations from the identity is a class of pattern avoiding permutations. We characterize the elements of the basis of this class and we enumerate these "forbidden minimal patterns" by giving their bivariate exponential generating function: we achieve this via a catalytic variable, the number of left-to-right maxima. We show that this generating function is a D-finite function satisfying a nice differential equation of order~2. We give some congruence properties for the coefficients of this generating function, and we show that their asymptotics involves a rather unusual algebraic exponent (the golden ratio (1+5‾√)/2) and some unusual closed-form constants. We end by proving a limit law: a forbidden pattern of length n has typically (lnn)/5‾√ left-to-right maxima, with Gaussian fluctuations.
]]></description>
<dc:subject>permutations pattern-avoiding dynamical-systems rewriting-systems rather-interesting to-understand to-simulate consider:probability-theory combinatorics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:779c5b3a0040/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:permutations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:pattern-avoiding"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rewriting-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:probability-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2103.01235">
    <title>[2103.01235] Statistical mechanics of dimers on quasiperiodic tilings</title>
    <dc:date>2021-07-04T11:08:45+00:00</dc:date>
    <link>https://arxiv.org/abs/2103.01235</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study classical dimers on two-dimensional quasiperiodic Ammann-Beenker (AB) tilings. Despite the lack of periodicity we prove that each infinite tiling admits 'perfect matchings' in which every vertex is touched by one dimer. We introduce an auxiliary 'AB∗' tiling obtained from the AB tiling by deleting all 8-fold coordinated vertices. The AB∗ tiling is again two-dimensional, infinite, and quasiperiodic. The AB∗ tiling has a single connected component, which admits perfect matchings. We find that in all perfect matchings, dimers on the AB∗ tiling lie along disjoint one-dimensional loops and ladders, separated by 'membranes', sets of edges where dimers are absent. As a result, the dimer partition function of the AB∗ tiling factorizes into the product of dimer partition functions along these structures. We compute the partition function and free energy per edge on the AB∗ tiling using an analytic transfer matrix approach. Returning to the AB tiling, we find that membranes in the AB∗ tiling become 'pseudomembranes', sets of edges which collectively host at most one dimer. This leads to a remarkable discrete scale-invariance in the matching problem. The structure suggests that the AB tiling should exhibit highly inhomogenous and slowly decaying connected dimer correlations. Using Monte Carlo simulations, we find evidence supporting this supposition in the form of connected dimer correlations consistent with power law behaviour. Within the set of perfect matchings we find quasiperiodic analogues to the staggered and columnar phases observed in periodic systems.]]></description>
<dc:subject>domino-tiling tiling dynamical-systems rather-interesting quasicrystals aperiodic-tiling to-write-about consider:animation symmetry constraint-satisfaction</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c0988dd4a678/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:domino-tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:quasicrystals"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:aperiodic-tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:animation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:symmetry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1605.06860">
    <title>[1605.06860] Controlling Chaotic Maps by Feedback Control Modulation</title>
    <dc:date>2021-06-28T11:04:04+00:00</dc:date>
    <link>https://arxiv.org/abs/1605.06860</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This paper deals whith the stabilization of any UPO of a chaotic map by modulation of a control parameter. It concentrates on proportional and delayed feedback control methods. Alternative types of these methods are proposed and their achievments are investigated analyticaly and numerically.
]]></description>
<dc:subject>nonlinear-dynamics control-theory dynamical-systems nudge-targets consider:performance-measures consider:representation rather-interesting consider:visualization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:806d780619ba/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:control-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.futilitycloset.com/2019/06/12/aliquot-sequences/">
    <title>Aliquot Sequences - Futility Closet</title>
    <dc:date>2021-06-25T09:39:09+00:00</dc:date>
    <link>https://www.futilitycloset.com/2019/06/12/aliquot-sequences/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Many of these sequences arrive at some resolution — they terminate in a constant, or an alternating pair, or some regular cycle. But it’s an open question whether all of them do this. The fate of the aliquot sequence of 276 is not known; by step 469 it’s reached 149384846598254844243905695992651412919855640, but possibly it reaches some apex and then descends again and finds some conclusion (the sequence for the number 138 reaches a peak of 179931895322 but eventually returns to 1). Do all numbers eventually reach a resolution? For now, no one knows.

]]></description>
<dc:subject>open-questions number-theory dynamical-systems rather-interesting to-simulate to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c7b41a72dfb6/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:open-questions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2003.13775">
    <title>[2003.13775] Coupled Dynamics on Hypergraphs: Master Stability of Steady States and Synchronization</title>
    <dc:date>2021-06-17T21:53:37+00:00</dc:date>
    <link>https://arxiv.org/abs/2003.13775</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In the study of dynamical systems on networks/graphs, a key theme is how the network topology influences stability for steady states or synchronized states. Ideally, one would like to derive conditions for stability or instability that instead of microscopic details of the individual nodes/vertices rather make the influence of the network coupling topology visible. The master stability function is an important such tool to achieve this goal. Here we generalize the master stability approach to hypergraphs. A hypergraph coupling structure is important as it allows us to take into account arbitrary higher-order interactions between nodes. As for instance in the theory of coupled map lattices, we study Laplace type interaction structures in detail. Since the spectral theory of Laplacians on hypergraphs is richer than on graphs, we see the possibility of new dynamical phenomena. More generally, our arguments provide a blueprint for how to generalize dynamical structures and results from graphs to hypergraphs.
]]></description>
<dc:subject>dynamical-systems hypergraphs nonlinear-dynamics coupled-oscillators rather-interesting consider:visualization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e8b7af76dde5/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:hypergraphs"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:coupled-oscillators"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1406.4386">
    <title>[1406.4386] Visualising rate of change: application to age-specific fertility</title>
    <dc:date>2021-06-16T11:09:05+00:00</dc:date>
    <link>https://arxiv.org/abs/1406.4386</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Visualisation methods help in the discovery of characteristics that might not have been apparent using mathematical models and summary statistics. However, visualisation methods have not received much attention in demography, with the exceptions of scatter plot and Lexis surface. We utilise a phase-plane plot to visualise the rate of change, obtained from derivatives of a continuous function. The phase-plane plot bears a resemblance to hysteresis loops, isogrowth curves, and solutions to differential equations. Using Australian and Chilean fertility, we present phase-plane plots. Similarly to the scatter plot and Lexis surface, the phase-plane plot identifies the age with maximum fertility rate and displays skewness of fertility distribution. Unlike the scatter plot and Lexis surface, the phase-plane plot identifies the age with maximum positive or negative velocity (i.e., trend), can compare the magnitude of the rate of change between any two years based on the size of the radius of circles. The phase-plane plot allows the visualisation of dynamic changes in fertility for a given age over the years and is potentially useful for visualising dynamic changes in birth-cohort fertility. Via the animate package in LaTeX, a dynamic phase-plane plot is also proposed to visualise changes in fertility over age or year.
]]></description>
<dc:subject>visualization dynamical-systems rather-interesting to-write-about consider:more-examples lexis-surface</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c8ab93df1816/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:more-examples"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:lexis-surface"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.biorxiv.org/content/10.1101/275008v1?rss=1">
    <title>Predicting improved protein conformations with a temporal deep recurrent neural network | bioRxiv</title>
    <dc:date>2021-06-04T11:19:21+00:00</dc:date>
    <link>https://www.biorxiv.org/content/10.1101/275008v1?rss=1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Accurate protein structure prediction from amino acid sequence is still an unsolved problem. The most reliable methods centre on template based modelling. However, the accuracy of these models entirely depends on the availability of experimentally resolved homologous template structures. In order to generate more accurate models, extensive physics based molecular dynamics (MD) refinement simulations are performed to sample many different conformations to find improved conformational states. In this study, we propose a deep recurrent network model, called DeepTrajectory, that is able to identify these improved conformational states, with high precision, from a variety of different MD based sampling protocols. The proposed model learns the temporal patterns of features computed from the MD trajectory data in order to classify whether each recorded simulation snapshot is an improved conformational state, decreased conformational state or a none perceivable change in state with respect to the starting conformation. The model is trained and tested on 904 trajectories from 42 different protein systems with a cumulative number of more than 1.7 million snapshots. We show that our model outperforms other state of the art machine-learning algorithms that do not consider temporal dependencies. To our knowledge, DeepTrajectory is the first implementation of a time-dependent deep-learning protocol that is re-trainable and able to adapt to any new MD based sampling procedure, thereby demonstrating how a neural network can be used to learn the latter part of the protein folding funnel.

]]></description>
<dc:subject>protein-folding structural-biology machine-learning recurrent-neural-network deep-learning rather-interesting dynamical-systems to-simulate to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:82ac88eb634f/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:protein-folding"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:structural-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:recurrent-neural-network"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:deep-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.05503">
    <title>[2012.05503] Geometric algorithms for sampling the flux space of metabolic networks</title>
    <dc:date>2021-05-22T21:50:23+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.05503</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Systems Biology is a fundamental field and paradigm that introduces a new era in Biology. The crux of its functionality and usefulness relies on metabolic networks that model the reactions occurring inside an organism and provide the means to understand the underlying mechanisms that govern biological systems. Even more, metabolic networks have a broader impact that ranges from resolution of ecosystems to personalized medicine.The analysis of metabolic networks is a computational geometry oriented field as one of the main operations they depend on is sampling uniformly points from polytopes; the latter provides a representation of the steady states of the metabolic networks. However, the polytopes that result from biological data are of very high dimension (to the order of thousands) and in most, if not all, the cases are considerably skinny. Therefore, to perform uniform random sampling efficiently in this setting, we need a novel algorithmic and computational framework specially tailored for the properties of metabolic networks.We present a complete software framework to handle sampling in metabolic networks. Its backbone is a Multiphase Monte Carlo Sampling (MMCS) algorithm that unifies rounding and sampling in one pass, obtaining both upon termination. It exploits an improved variant of the Billiard Walk that enjoys faster arithmetic complexity per step. We demonstrate the efficiency of our approach by performing extensive experiments on various metabolic networks. Notably, sampling on the most complicated human metabolic network accessible today, Recon3D, corresponding to a polytope of dimension 5 335 took less than 30 hours. To our knowledge, that is out of reach for existing software.
]]></description>
<dc:subject>dynamical-systems reaction-networks simulation algorithms approximation sampling rather-interesting to-write-about consider:general-case consider:ReQ systems-biology</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:59ee2e9eec9c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:reaction-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:sampling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:general-case"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ReQ"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:systems-biology"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2102.04906">
    <title>[2102.04906] Dynamic Neural Networks: A Survey</title>
    <dc:date>2021-05-19T10:56:00+00:00</dc:date>
    <link>https://arxiv.org/abs/2102.04906</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Dynamic neural network is an emerging research topic in deep learning. Compared to static models which have fixed computational graphs and parameters at the inference stage, dynamic networks can adapt their structures or parameters to different inputs, leading to notable advantages in terms of accuracy, computational efficiency, adaptiveness, etc. In this survey, we comprehensively review this rapidly developing area by dividing dynamic networks into three main categories: 1) instance-wise dynamic models that process each instance with data-dependent architectures or parameters; 2) spatial-wise dynamic networks that conduct adaptive computation with respect to different spatial locations of image data and 3) temporal-wise dynamic models that perform adaptive inference along the temporal dimension for sequential data such as videos and texts. The important research problems of dynamic networks, e.g., architecture design, decision making scheme, optimization technique and applications, are reviewed systematically. Finally, we discuss the open problems in this field together with interesting future research directions.
]]></description>
<dc:subject>deep-learning dynamical-systems machine-learning neural-networks representation to-understand consider:visualization</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2d0193bfb3b0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:deep-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2012.09229">
    <title>[2012.09229] Tag-based Genetic Regulation for Genetic Programming</title>
    <dc:date>2021-03-27T11:08:03+00:00</dc:date>
    <link>https://arxiv.org/abs/2012.09229</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce and experimentally demonstrate tag-based genetic regulation, a new genetic programming (GP) technique that allows evolving programs to dynamically adjust which code modules to express. Tags are evolvable labels that provide a flexible mechanism for referring to code modules. Tag-based genetic regulation extends existing tag-based naming schemes to allow programs to "promote" and "repress" code modules. This extension allows evolution to structure a program as a gene regulatory network where program modules are regulated based on instruction executions. We demonstrate the functionality of tag-based regulation on a range of program synthesis problems. We find that tag-based regulation improves problem-solving performance on context-dependent problems; that is, problems where programs must adjust how they respond to current inputs based on prior inputs (i.e., current context). We also observe that our implementation of tag-based genetic regulation can impede adaptive evolution when expected outputs are not context-dependent (i.e., the correct response to a particular input remains static over time). Tag-based genetic regulation broadens our repertoire of techniques for evolving more dynamic genetic programs and can easily be incorporated into existing tag-enabled GP systems.
]]></description>
<dc:subject>genetic-programming distributed-processing hey-I-know-this-guy dynamical-systems architecture to-write-about consider:ReQ-similarity</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1383bae9e8e8/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:distributed-processing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:hey-I-know-this-guy"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:architecture"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:ReQ-similarity"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://blog.tanyakhovanova.com/2020/03/a-game-with-the-devil/">
    <title>Tanya Khovanova's Math Blog » Blog Archive » A Game with the Devil</title>
    <dc:date>2021-02-10T20:28:16+00:00</dc:date>
    <link>https://blog.tanyakhovanova.com/2020/03/a-game-with-the-devil/</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Puzzle. You are playing a game with the Devil. There are n coins in a line, each showing either H (heads) or T (tails). Whenever the rightmost coin is H, you decide its new orientation and move it to the leftmost position. Whenever the rightmost coin is T, the Devil decides its new orientation and moves it to the leftmost position. This process repeats until all coins face the same way, at which point you win. What’s the winning strategy?

]]></description>
<dc:subject>game-theory puzzles mathematical-recreations rather-interesting dynamical-systems to-simulate to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:1dfe5f8679df/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:game-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:puzzles"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2011.06661">
    <title>[2011.06661] Stabilization of the fluidic pinball with gradient-enriched machine learning control</title>
    <dc:date>2021-02-02T21:29:12+00:00</dc:date>
    <link>https://arxiv.org/abs/2011.06661</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We stabilize the flow past a cluster of three rotating cylinders, the fluidic pinball, with automated gradient-enriched machine learning algorithms. The control laws command the rotation speed of each cylinder in an open- and closed-loop manner. These laws are optimized with respect to the average distance from the target steady solution in three successively richer search spaces. First, stabilization is pursued with steady symmetric forcing. Second, we allow for asymmetric steady forcing. And third, we determine an optimal feedback controller employing nine velocity probes downstream. As expected, the control performance increases with every generalization of the search space. Surprisingly, both open- and closed-loop optimal controllers include an asymmetric forcing, which surpasses symmetric forcing. Intriguingly, the best performance is achieved by a combination of phasor control and asymmetric steady forcing. We hypothesize that asymmetric forcing is typical for pitchfork bifurcated dynamics of nominally symmetric configurations. Key enablers are automated machine learning algorithms augmented with gradient search: explorative gradient method for the open-loop parameter optimization and a gradient-enriched machine learning control (gMLC) for the feedback optimization. gMLC learns the control law significantly faster than previously employed genetic programming control.
]]></description>
<dc:subject>control-theory machine-learning rather-interesting dynamical-systems optimization to-simulate to-understand consider:interactive genetic-programming fluid-dynamics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:470579b43f07/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:control-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:interactive"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fluid-dynamics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://bit-player.org/2019/my-god-its-full-of-dots">
    <title>My God, It’s Full of Dots! | bit-player</title>
    <dc:date>2020-12-09T12:24:29+00:00</dc:date>
    <link>http://bit-player.org/2019/my-god-its-full-of-dots</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This scheme for filling space with randomly placed objects is the invention of John Shier, a physicist who worked for many years in the semiconductor industry and who has also taught at Normandale Community College near Minneapolis. He explains the method and the mathematics behind it in a recent book, Fractalize That! A Visual Essay on Statistical Geometry. (For bibliographic details see the links and references at the end of this essay.) I learned of Shier’s work from my friend Barry Cipra.
Shier hints at the strangeness of these doings by imagining a set of 100 round tiles in graduated sizes, with a total area approaching one square meter. He would give the tiles to a craftsman with these instructions:
]]></description>
<dc:subject>nonlinear-dynamics mathematical-recreations fractals rather-interesting dynamical-systems to-write-about to-simulate consider:inverse-problem generative-art algorithms constant-finding</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:2c7d562b2cc2/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fractals"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:inverse-problem"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-art"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constant-finding"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1606.06325">
    <title>[1606.06325] Applications of Thin Orbits</title>
    <dc:date>2020-10-03T12:31:53+00:00</dc:date>
    <link>https://arxiv.org/abs/1606.06325</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[This text is based on a series of three expository lectures on a variety of topics related to "thin orbits," as delivered at Durham University's Easter School on "Dynamics and Analytic Number Theory" in April 2014. The first lecture reviews closed geodesics on the modular surface and the reduction theory of binary quadratic forms before discussing Duke's equidistribution theorem (for indefinite classes). The second lecture exposits three quite different but (it turns out) not unrelated problems, due to Einsiedler-Lindenstrauss-Michel-Venkatesh, McMullen, and Zaremba. The third lecture reformulates these in terms of the aforementioned thin orbits, and shows how all three would follow from a single "Local-Global" Conjecture of Bourgain and the author. We also describe some partial progress on the conjecture, which has lead to some results on the original problems.
]]></description>
<dc:subject>number-theory nonlinear-dynamics to-understand dynamical-systems to-simulate to-write-about consider:visualization consider:computation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:fc286a676a4a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:computation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="http://www.cleonis.nl/physics/phys256/coupling.php">
    <title>Rotational-vibrational coupling</title>
    <dc:date>2020-07-15T20:35:59+00:00</dc:date>
    <link>http://www.cleonis.nl/physics/phys256/coupling.php</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Rotational-vibrational coupling occurs when there is a 1:2 ratio of rotation frequency of an object and a natural internal vibration frequency. The animation on the right shows the simplest example of this phenomenon. The motion depicted in the animation is for the idealized situation that the force exerted by the spring is proportional to the amount of extension. Note that in this demonstration the spring isn't alternating between pulling and pushing, the spring is exerting a contracting force all the time; given the chance the idealized spring would contract all the way down to zero length. Also, since the animation keeps on looping, the animation depicts what would occur if there would not be any friction.

In molecular physics it is recognized that there is a coupling of rotational and vibrational energy-levels. In molecular physics rotational-vibrational coupling is also called rovibronic coupling and Coriolis coupling. The physics of actual diatomic molecules is more complicated than the example in the animation, but because of its simplicity the animation is useful for illustrating the basic principles.

]]></description>
<dc:subject>dynamical-systems simulation physics rather-interesting via:twitter to-write-about to-simulate consider:exponents consider:varying-constants</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c4a410a72bee/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:physics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:via:twitter"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:exponents"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:varying-constants"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.biorxiv.org/content/10.1101/223651v1?rss=1">
    <title>The feasibility and stability of large complex biological networks: a random matrix approach | bioRxiv</title>
    <dc:date>2020-06-17T13:49:54+00:00</dc:date>
    <link>https://www.biorxiv.org/content/10.1101/223651v1?rss=1</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In his theoretical work of the 70’s, Robert May introduced a Random Matrix Theory (RMT) approach for studying the stability of large complex biological systems. Unlike the established paradigm, May demonstrated that complexity leads to instability in generic models of biological networks. The RMT approach has since similarly been applied in many disciplines. Central to the approach is the famous “circular law” that describes the eigenvalue distribution of an important class of random matrices. However the “circular law” generally does not apply for ecological and biological systems in which density-dependence (DD) operates. Here we directly determine the far more complicated eigenvalue distributions of complex DD systems. A simple mathematical solution falls out, that allows us to explore the connection between feasible systems (i.e., having all equilibrium populations positive) and stability. In particular, for these RMT systems, almost all feasible systems are stable. The degree of stability, or resilience, is shown to depend on the minimum equilibrium population, and not directly on factors such as network topology.

]]></description>
<dc:subject>theoretical-biology systems-biology reaction-networks rather-interesting dynamical-systems looking-to-see to-write-about to-simulate consider:dynamics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:26149d19517e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:theoretical-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:systems-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:reaction-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:dynamics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2004.13935">
    <title>[2004.13935] An Averaging Processes on Hypergraphs</title>
    <dc:date>2020-06-14T11:09:51+00:00</dc:date>
    <link>https://arxiv.org/abs/2004.13935</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Consider the following iterated process on a hypergraph H. Each vertex v has an initial vertex weight. At each step, we uniformly at random select an edge F in H, and for each vertex v in F we replace the weight of v by the average value of the vertex weights over all vertices in F. This is a generalization of an interactive process on graphs, first proposed by Aldous and Lanoue. In this paper, we use the eigenvalues of a Laplacian for hypergraphs to bound the rate of convergence for the iterated averaging process.
]]></description>
<dc:subject>hypergraphs dynamical-systems stochastic-systems rather-interesting simulation to-write-about to-simulate consider:structure consider:extreme-values</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:bc1e179bbbe4/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:hypergraphs"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:stochastic-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:structure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:extreme-values"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2004.12497">
    <title>[2004.12497] Forty New Invariants of N-Periodics in the Elliptic Billiard</title>
    <dc:date>2020-06-14T11:04:35+00:00</dc:date>
    <link>https://arxiv.org/abs/2004.12497</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present some 40 newfound invariants displayed by N-periodics in the Elliptic Billiard, obtained through experimental exploration. These involve distances, areas, angles and centers of mass of N-periodics and associated polygons (inner, outer, pedal, antipedal). Some depend on the parity of N, others on other positional constraints. A few invariants have already been proven with elegant tools of Analytic and Algebraic Geometry. We welcome reader input to add to the list of proofs.
]]></description>
<dc:subject>billiards dynamical-systems rather-interesting numerical-methods invariants feature-construction looking-to-see to-write-about to-simulate consider:genec</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:6936f8f957f7/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:numerical-methods"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:invariants"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genec"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2004.14926">
    <title>[2004.14926] Tanaka-Ito $α$-continued fractions and matching</title>
    <dc:date>2020-05-23T11:53:53+00:00</dc:date>
    <link>https://arxiv.org/abs/2004.14926</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Two closely related families of α-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter α has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito α-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold, called bifurcation set, is a zero measure set (even if it has full Hausdorff dimension). This property is also shared by Nakada's α-continued fractions, and yet there also are some substantial differences: not only does the bifurcation set for Tanaka-Ito continued fractions contain infinitely many rational values, it also contains numbers with unbounded partial quotients.
]]></description>
<dc:subject>continued-fractions number-theory representation algorithms feature-construction rather-interesting to-simulate to-write-about consider:algorithms dynamical-systems consider:convergence-as-multiobjective</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0be98b96648b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:convergence-as-multiobjective"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/2004.13491">
    <title>[2004.13491] As Time Goes By: Reflections on Treewidth for Temporal Graphs</title>
    <dc:date>2020-05-23T11:50:19+00:00</dc:date>
    <link>https://arxiv.org/abs/2004.13491</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Treewidth is arguably the most important structural graph parameter leading to algorithmically beneficial graph decompositions. Triggered by a strongly growing interest in temporal networks (graphs where edge sets change over time), we discuss fresh algorithmic views on temporal tree decompositions and temporal treewidth. We review and explain some of the recent work together with some encountered pitfalls, and we point out challenges for future research.
]]></description>
<dc:subject>data-structures graph-theory dynamical-systems algorithms to-write-about to-simulate consider:looking-to-see consider:autocorrelation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:acf02c840c2a/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:data-structures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:autocorrelation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1702.02764">
    <title>[1702.02764] A new method to reduce the number of time delays in a network</title>
    <dc:date>2020-05-07T12:30:09+00:00</dc:date>
    <link>https://arxiv.org/abs/1702.02764</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Time delays may cause dramatic changes to the dynamics of interacting oscillators. Coupled networks of interacting dynamical systems can behave unexpectedly when the signal between the vertices are time delayed. It has been shown for a very general class of systems that the time delays can be rearranged as long as the total time delay over the constitutive loops of the network is conserved. This fact allows to reduce the number of time delays of the problem without loss of information. There is a theoretical lower bound for this number, but in many cases we can find a numerical solution that beats this limit. Here we propose a formulation of the problem and a numerical method to even further reduce the number of time delays in a network.
]]></description>
<dc:subject>coupled-oscillators complexology inference rather-interesting dynamical-systems collective-behavior to-write-about to-simulate consider:looking-to-see consider:heuristics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:28be316b5332/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:coupled-oscillators"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:complexology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inference"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:collective-behavior"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:heuristics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.nature.com/articles/s41467-019-12306-2">
    <title>Stable memory with unstable synapses | Nature Communications</title>
    <dc:date>2020-05-02T14:59:53+00:00</dc:date>
    <link>https://www.nature.com/articles/s41467-019-12306-2</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[What is the physiological basis of long-term memory? The prevailing view in Neuroscience attributes changes in synaptic efficacy to memory acquisition, implying that stable memories correspond to stable connectivity patterns. However, an increasing body of experimental evidence points to significant, activity-independent fluctuations in synaptic strengths. How memories can survive these fluctuations and the accompanying stabilizing homeostatic mechanisms is a fundamental open question. Here we explore the possibility of memory storage within a global component of network connectivity, while individual connections fluctuate. We find that homeostatic stabilization of fluctuations differentially affects different aspects of network connectivity. Specifically, memories stored as time-varying attractors of neural dynamics are more resilient to erosion than fixed-points. Such dynamic attractors can be learned by biologically plausible learning-rules and support associative retrieval. Our results suggest a link between the properties of learning-rules and those of network-level memory representations, and point at experimentally measurable signatures.

]]></description>
<dc:subject>representation neural-networks rather-interesting to-write-about ReQ dynamical-systems consider:feature-discovery to-simulate analog-computing</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:b8a27d00f959/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:neural-networks"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:ReQ"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:analog-computing"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.pnas.org/content/117/1/43">
    <title>Mechanics unlocks the morphogenetic puzzle of interlocking bivalved shells | PNAS</title>
    <dc:date>2020-05-02T14:51:20+00:00</dc:date>
    <link>https://www.pnas.org/content/117/1/43</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Brachiopods and mollusks are 2 shell-bearing phyla that diverged from a common shell-less ancestor more than 540 million years ago. Brachiopods and bivalve mollusks have also convergently evolved a bivalved shell that displays an apparently mundane, yet striking feature from a developmental point of view: When the shell is closed, the 2 valve edges meet each other in a commissure that forms a continuum with no gaps or overlaps despite the fact that each valve, secreted by 2 mantle lobes, may present antisymmetric ornamental patterns of varying regularity and size. Interlocking is maintained throughout the entirety of development, even when the shell edge exhibits significant irregularity due to injury or other environmental influences, which suggests a dynamic physical process of pattern formation that cannot be genetically specified. Here, we derive a mathematical framework, based on the physics of shell growth, to explain how this interlocking pattern is created and regulated by mechanical instabilities. By close consideration of the geometry and mechanics of 2 lobes of the mantle, constrained both by the rigid shell that they secrete and by each other, we uncover the mechanistic basis for the interlocking pattern. Our modeling framework recovers and explains a large diversity of shell forms and highlights how parametric variations in the growth process result in morphological variation. Beyond the basic interlocking mechanism, we also consider the intricate and striking multiscale-patterned edge in certain brachiopods. We show that this pattern can be explained as a secondary instability that matches morphological trends and data.

]]></description>
<dc:subject>developmental-biology evo-devo rather-interesting pattern-formation theoretical-biology dynamical-systems performance-measure to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4fb2b291258e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:developmental-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:evo-devo"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:pattern-formation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:theoretical-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:performance-measure"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://www.sciencedirect.com/science/article/abs/pii/S0196677498909889">
    <title>Data Structures for Mobile Data - ScienceDirect</title>
    <dc:date>2020-03-08T19:06:30+00:00</dc:date>
    <link>https://www.sciencedirect.com/science/article/abs/pii/S0196677498909889</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Akinetic data structure(KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a concentual framework for kinetic data structures, we propose a number of criteria for the quality of such structures, and we describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behave well according to the proposed quality criteria for KDSs.]]></description>
<dc:subject>data-structures rather-interesting to-simulate to-write-about consider:processing-sims modeling dynamical-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:71d465109b7d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:data-structures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:processing-sims"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:modeling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1803.09639">
    <title>[1803.09639] On the multipacking number of grid graphs</title>
    <dc:date>2020-01-26T14:04:10+00:00</dc:date>
    <link>https://arxiv.org/abs/1803.09639</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In 2001, Erwin introduced broadcast domination in graphs. It is a variant of classical domination where selected vertices may have different domination powers. The minimum cost of a dominating broadcast in a graph G is denoted γb(G). The dual of this problem is called multipacking: a multipacking is a set M of vertices such that for any vertex v and any positive integer r, the ball of radius r around v contains at most r vertices of M . The maximum size of a multipacking in a graph G is denoted mp(G). Naturally mp(G) ≤γb(G). Earlier results by Farber and by Lubiw show that broadcast and multipacking numbers are equal for strongly chordal graphs. In this paper, we show that all large grids (height at least 4 and width at least 7), which are far from being chordal, have their broadcast and multipacking numbers equal.
]]></description>
<dc:subject>graph-theory feature-construction rather-interesting dynamical-systems to-simulate to-write-about consider:generalizations consider:prediction</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:3ac74c815fcd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:generalizations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:prediction"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1809.09676">
    <title>[1809.09676] Chip-Firing and Fractional Bases</title>
    <dc:date>2020-01-19T18:03:44+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.09676</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study a particular chip-firing process on an infinite path graph. At any time when there are at least a+b chips at a vertex, a chips fire to the left and b chips fire to the right. We describe the final state of this process when we start with n chips at the origin.
]]></description>
<dc:subject>chip-firing dynamical-systems number-theory mathematical-recreations rather-interesting to-write-about to-simulate combinatorics</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9cc6c790c3dd/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:chip-firing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1802.09904">
    <title>[1802.09904] Algorithmic Causal Deconvolution of Intertwined Programs and Networks by Generative Mechanism</title>
    <dc:date>2019-11-25T23:47:41+00:00</dc:date>
    <link>https://arxiv.org/abs/1802.09904</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Complex data usually results from the interaction of objects produced by different generating mechanisms. Here we introduce a universal, unsupervised and parameter-free model-oriented approach, based upon the seminal concept of algorithmic probability, that decomposes an observation into its most likely algorithmic generative sources. Our approach uses a causal calculus to infer model representations. We demonstrate its ability to deconvolve interacting mechanisms regardless of whether the resultant objects are strings, space-time evolution diagrams, images or networks. While this is mostly a conceptual contribution and a novel framework, we provide numerical evidence evaluating the ability of our methods to separate data from observations produced by discrete dynamical systems such as cellular automata and complex networks. We think that these separating techniques can contribute to tackling the challenge of causation, thus complementing other statistically oriented approaches.
]]></description>
<dc:subject>machine-learning dynamical-systems inverse-problems rather-interesting system-identification to-simulate to-try to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:aa810e1dd392/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inverse-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:system-identification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-try"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1902.10834">
    <title>[1902.10834] An evolutionary model that satisfies detailed balance</title>
    <dc:date>2019-11-25T17:36:03+00:00</dc:date>
    <link>https://arxiv.org/abs/1902.10834</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We propose a class of evolutionary models that involves an arbitrary exchangeable process as the breeding process and different selection schemes. In those models, a new genome is born according to the breeding process, and then a genome is removed according to the selection scheme that involves fitness. Thus the population size remains constant. The process evolves according to a Markov chain, and, unlike in many other existing models, the stationary distribution -- so called mutation-selection equilibrium -- can be easily found and studied. The behaviour of the stationary distribution when the population size increases is our main object of interest. Several phase-transition theorems are proved.
]]></description>
<dc:subject>machine-learning statistical-mechanics Markov-models dynamical-systems looking-under-the-tractable-lamp-post to-simulate to-write-about consider:lexicase</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:01a8a5a8bd77/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:statistical-mechanics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:Markov-models"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-under-the-tractable-lamp-post"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:lexicase"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1506.01202">
    <title>[1506.01202] Introducing the Plaid Model</title>
    <dc:date>2019-11-24T23:39:44+00:00</dc:date>
    <link>https://arxiv.org/abs/1506.01202</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is closely related to outer billiards on kites, and also is related to corner percolation, to Hooper's Truchet tile system, to self-similar tilings, and to polyhedron exchange transformations.
]]></description>
<dc:subject>construction plane-geometry rather-interesting dynamical-systems tiling to-simulate to-write-about consider:feature-discovery consider:inverse-problem</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0f8f8585bfbf/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:inverse-problem"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1909.12501">
    <title>[1909.12501] Dynamics in a time-discrete food-chain model with strong pressure on preys</title>
    <dc:date>2019-10-11T12:57:22+00:00</dc:date>
    <link>https://arxiv.org/abs/1909.12501</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Ecological systems are complex dynamical systems. Modelling efforts on ecosystems' dynamical stability have revealed that population dynamics, being highly nonlinear, can be governed by complex fluctuations. Indeed, experimental and field research has provided mounting evidence of chaos in species' abundances, especially for discrete-time systems. Discrete-time dynamics, mainly arising in boreal and temperate ecosystems for species with non-overlapping generations, have been largely studied to understand the dynamical outcomes due to changes in relevant ecological parameters. The local and global dynamical behaviour of many of these models is difficult to investigate analytically in the parameter space and, typically, numerical approaches are employed when the dimension of the phase space is large. In this article we provide topological and dynamical results for a map modelling a discrete-time, three-species food chain with two predator species interacting on the same prey population. The domain where dynamics live is characterized, as well as the so-called escaping regions, for which the species go rapidly to extinction after surpassing the carrying capacity. We also provide a full description of the local stability of equilibria within a volume of the parameter space given by the prey's growth rate and the predation rates. We have found that the increase of the pressure of predators on the prey results in chaos. The entry into chaos is achieved via a supercritical Neimarck-Sacker bifurcation followed by period-doubling bifurcations of invariant curves. Interestingly, an increasing predation directly on preys can shift the extinction of top predators to their survival, allowing an unstable persistence of the three species by means of periodic and strange chaotic attractors.
]]></description>
<dc:subject>food-webs rather-interesting theoretical-biology dynamical-systems collective-behavior nonlinear-dynamics looking-to-see simulation to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e0e2adccb467/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:food-webs"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:theoretical-biology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:collective-behavior"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nonlinear-dynamics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:simulation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1809.02942">
    <title>[1809.02942] Cellular automata as convolutional neural networks</title>
    <dc:date>2019-09-22T12:07:37+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.02942</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Deep learning techniques have recently demonstrated broad success in predicting complex dynamical systems ranging from turbulence to human speech, motivating broader questions about how neural networks encode and represent dynamical rules. We explore this problem in the context of cellular automata (CA), simple dynamical systems that are intrinsically discrete and thus difficult to analyze using standard tools from dynamical systems theory. We show that any CA may readily be represented using a convolutional neural network with a network-in-network architecture. This motivates our development of a general convolutional multilayer perceptron architecture, which we find can learn the dynamical rules for arbitrary CA when given videos of the CA as training data. In the limit of large network widths, we find that training dynamics are strongly stereotyped across replicates, and that common patterns emerge in the structure of networks trained on different CA rulesets. We train ensembles of networks on randomly-sampled CA, and we probe how the trained networks internally represent the CA rules using an information-theoretic technique based on distributions of layer activation patterns. We find that CA with simpler rule tables produce trained networks with hierarchical structure and layer specialization, while more complex CA tend to produce shallower representations---illustrating how the underlying complexity of the CA's rules influences the specificity of these internal representations. Our results suggest how the entropy of a physical process can affect its representation when learned by neural networks.
]]></description>
<dc:subject>cellular-automata dynamical-systems rather-interesting to-write-about to-simulate machine-learning representation feature-construction information-theory consider:NN-complexity-as-constructed-features</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:184925fd8d9c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:cellular-automata"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:information-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:NN-complexity-as-constructed-features"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1907.01178">
    <title>[1907.01178] On a proof of the Tree conjecture for triangle tiling billiards</title>
    <dc:date>2019-09-21T11:18:36+00:00</dc:date>
    <link>https://arxiv.org/abs/1907.01178</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Tiling billiards model a movement of light in heterogeneous medium consisting of homogeneous cells in which the coefficient of refraction between two cells is equal to −1. The dynamics of such billiards depends strongly on the form of an underlying tiling. In this work we consider periodic tilings by triangles (and cyclic quadrilaterals), and define natural foliations associated to tiling billiards in these tilings. By studying these foliations we manage to prove the Tree Conjecture for triangle tiling billiards that was stated in the work by Baird-Smith, Davis, Fromm and Iyer, as well as its generalization that we call Density property.
]]></description>
<dc:subject>billiards dynamical-systems tiling rather-interesting to-understand to-simulate consider:feature-reduction consider:classification</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:8267e599550b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-reduction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:classification"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1804.00181">
    <title>[1804.00181] Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket</title>
    <dc:date>2019-09-12T10:33:35+00:00</dc:date>
    <link>https://arxiv.org/abs/1804.00181</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.
]]></description>
<dc:subject>dynamical-systems billiards tiling rather-interesting plane-geometry to-simulate to-understand emergent-design</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:e05b5e86b99e/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:emergent-design"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1809.08490">
    <title>[1809.08490] On 3-Inflatable Permutations</title>
    <dc:date>2019-09-09T11:05:16+00:00</dc:date>
    <link>https://arxiv.org/abs/1809.08490</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Call a permutation k-inflatable if it can be "blown up" into a convergent sequence of permutations by a uniform inflation construction, such that this sequence is symmetric with respect to densities of induced subpermutations of length k. We study properties of 3-inflatable permutations, finding a general formula for limit densities of pattern permutations in the uniform inflation of a given permutation. We also characterize and find examples of 3-inflatable permutations of various lengths, including the shortest examples with length 17.
]]></description>
<dc:subject>permutations strings rather-interesting mathematical-recreations dynamical-systems formal-languages constraint-satisfaction to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:614d8c6b8614/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:permutations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:strings"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:mathematical-recreations"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:formal-languages"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1301.6807">
    <title>[1301.6807] Modified Stern-Brocot Sequences</title>
    <dc:date>2019-09-08T23:06:30+00:00</dc:date>
    <link>https://arxiv.org/abs/1301.6807</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We present the classical Stern-Brocot tree and provide a new proof of the fact that every rational number between 0 and 1 appears in the tree. We then generalize theStern-Brocot tree to allow for arbitrary choice of starting terms, and prove that in all cases the tree maintains the property that every rational number between the two starting terms appears exactly once.
]]></description>
<dc:subject>number-theory continued-fractions dynamical-systems rather-interesting visualization to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f280512ebaa0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:visualization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1604.00222">
    <title>[1604.00222] Ordering of nested square roots of 2 according to Gray code</title>
    <dc:date>2019-08-06T22:36:05+00:00</dc:date>
    <link>https://arxiv.org/abs/1604.00222</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this paper we discuss some relations between zeros of Lucas-Lehmer polynomials and Gray code. We study nested square roots of 2 applying a "binary code" that associates bits 0 and 1 to ⊕ and ⊖ signs in the nested form. This gives the possibility to obtain an ordering for the zeros of Lucas-Lehmer polynomials, which assume the form of nested square roots of 2.
]]></description>
<dc:subject>number-theory approximation dynamical-systems rather-interesting Ramanujan representation to-simulate to-write-about</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:4bb59ec80d60/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:approximation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:Ramanujan"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1705.06314">
    <title>[1705.06314] Tire tracks and integrable curve evolution</title>
    <dc:date>2019-08-06T22:24:25+00:00</dc:date>
    <link>https://arxiv.org/abs/1705.06314</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study a simple model of bicycle motion: a segment of fixed length in multi-dimensional Euclidean space, moving so that the velocity of the rear end is always aligned with the segment. If the front track is prescribed, the trajectory of the rear wheel is uniquely determined via a certain first order differential equation -- the bicycle equation. The same model, in dimension two, describes another mechanical device, the hatchet planimeter. 
Here is a sampler of our results. We express the linearized flow of the bicycle equation in terms of the geometry of the rear track; in dimension three, for closed front and rear tracks, this is a version of the Berry phase formula. We show that in all dimensions a sufficiently long bicycle also serves as a planimeter: it measures, approximately, the area bivector defined by the closed front track. We prove that the bicycle equation also describes rolling, without slipping and twisting, of hyperbolic space along Euclidean space. We relate the bicycle problem with two completely integrable systems: the AKNS (Ablowitz, Kaup, Newell and Segur) system and the vortex filament equation. We show that "bicycle correspondence" of space curves (front tracks sharing a common back track) is a special case of a Darboux transformation associated with the AKNS system. We show that the filament hierarchy, encoded as a single generating equation, describes a 3-dimensional bike of imaginary length. We show that a series of examples of "ambiguous" closed bicycle curves (front tracks admitting self bicycle correspondence), found recently F. Wegner, are buckled rings, or solitons of the planar filament equation. As a case study, we give a detailed analysis of such curves, arising from bicycle correspondence with multiply traversed circles.
]]></description>
<dc:subject>plane-geometry rather-interesting topology constraint-satisfaction dynamical-systems algorithms consider:looking-to-see to-simulate to-write-about consider:rediscovery nudge-targets</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c1528efd65ac/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:constraint-satisfaction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:rediscovery"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:nudge-targets"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1902.01530">
    <title>[1902.01530] Flip cycles in plabic graphs</title>
    <dc:date>2019-08-06T10:10:32+00:00</dc:date>
    <link>https://arxiv.org/abs/1902.01530</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian Gr≥0(n,k). Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope Z(n,3). Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem.
]]></description>
<dc:subject>combinatorics domino-tiling graph-theory dynamical-systems rather-interesting representation to-write-about group-theory</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f115f438d3b0/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:domino-tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:group-theory"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1903.02748">
    <title>[1903.02748] (Re)constructing code loops</title>
    <dc:date>2019-08-06T09:40:13+00:00</dc:date>
    <link>https://arxiv.org/abs/1903.02748</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The Parker loop is a central extension of the extended binary Golay code. It is an example of a general class of non-associative structures known as \emph{code loops}, which have been studied from a number of different algebraic and combinatorial perspectives. This expository paper aims to also highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops' twisted cocycles satisfy. As a biproduct one can reconstruct the multiplication in the Parker loop from a mere fragment of its twisted cocycle, and we have found relatively large subspaces of the Golay code over which the Parker loop splits as a direct product.
]]></description>
<dc:subject>combinatorics coding-theory to-understand group-theory dynamical-systems generative-models</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f39d14698b1b/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:coding-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:group-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:generative-models"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1907.08226">
    <title>[1907.08226] Who is Afraid of Big Bad Minima? Analysis of Gradient-Flow in a Spiked Matrix-Tensor Model</title>
    <dc:date>2019-08-06T09:35:46+00:00</dc:date>
    <link>https://arxiv.org/abs/1907.08226</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Gradient-based algorithms are effective for many machine learning tasks, but despite ample recent effort and some progress, it often remains unclear why they work in practice in optimising high-dimensional non-convex functions and why they find good minima instead of being trapped in spurious ones. 
Here we present a quantitative theory explaining this behaviour in a spiked matrix-tensor model. 
Our framework is based on the Kac-Rice analysis of stationary points and a closed-form analysis of gradient-flow originating from statistical physics. We show that there is a well defined region of parameters where the gradient-flow algorithm finds a good global minimum despite the presence of exponentially many spurious local minima. 
We show that this is achieved by surfing on saddles that have strong negative direction towards the global minima, a phenomenon that is connected to a BBP-type threshold in the Hessian describing the critical points of the landscapes.
]]></description>
<dc:subject>machine-learning optimization fitness-landscapes metaheuristics representation to-understand topology dynamical-systems consider:performance-measures consider:better-faster</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:c6507c15b09d/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:optimization"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:fitness-landscapes"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:metaheuristics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:topology"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:performance-measures"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:better-faster"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1812.11224">
    <title>[1812.11224] Planar maps, random walks and circle packing</title>
    <dc:date>2019-08-03T11:26:20+00:00</dc:date>
    <link>https://arxiv.org/abs/1812.11224</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[These are lecture notes of the 48th Saint-Flour summer school, July 2018, on the topic of planar maps, random walks and the circle packing theorem.
]]></description>
<dc:subject>packing graph-theory plane-geometry representation algorithms combinatorics review to-read dynamical-systems</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:cdaaeeae3fdb/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:packing"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:review"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-read"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1109.0516">
    <title>[1109.0516] The bifurcation locus for numbers of bounded type</title>
    <dc:date>2019-08-02T11:06:09+00:00</dc:date>
    <link>https://arxiv.org/abs/1109.0516</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We define a family B(t) of compact subsets of the unit interval which generalizes the sets of numbers whose continued fraction expansion has bounded digits. We study how the set B(t) changes as one moves the parameter t, and see that the family undergoes period-doubling bifurcations and displays the same transition pattern from periodic to chaotic behavior as the usual family of quadratic polynomials. The set E of bifurcation parameters is a fractal set of measure zero and Hausdorff dimension 1. We also show that the Hausdorff dimension of B(t) varies continuously with the parameter, and the dimension of each individual set equals the dimension of a corresponding section of the bifurcation set E.]]></description>
<dc:subject>continued-fractions number-theory dynamical-systems representation rather-interesting to-understand to-simulate to-write-about consider:feature-discovery</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:0d1d4203e756/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:continued-fractions"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:number-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:feature-discovery"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1004.3025">
    <title>[1004.3025] Outer Billiards and the Pinwheel Map</title>
    <dc:date>2019-08-02T10:59:14+00:00</dc:date>
    <link>https://arxiv.org/abs/1004.3025</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer billiards system has unbounded orbits if and only if the pinwheel map has unbounded orbits. As the pinwheel map is much easier to analyze directly, we think that this bijection will be helpful in attacking some of the main questions about polyonal outer billiards.
]]></description>
<dc:subject>plane-geometry construction dynamical-systems algorithms iterated-systems to-write-about to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:99e12dd95fff/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:algorithms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:iterated-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1506.08415">
    <title>[1506.08415] PLG2: Multiperspective Processes Randomization and Simulation for Online and Offline Settings</title>
    <dc:date>2019-07-25T11:17:34+00:00</dc:date>
    <link>https://arxiv.org/abs/1506.08415</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Process mining represents an important field in BPM and data mining research. Recently, it has gained importance also for practitioners: more and more companies are creating business process intelligence solutions. The evaluation of process mining algorithms requires, as any other data mining task, the availability of large amount of real-world data. Despite the increasing availability of such datasets, they are affected by many limitations, in primis the absence of a "gold standard" (i.e., the reference model). 
This paper extends an approach, already available in the literature, for the generation of random processes. Novelties have been introduced throughout the work and, in particular, they involve the complete support for multiperspective models and logs (i.e., the control-flow perspective is enriched with time and data information) and for online settings (i.e., generation of multiperspective event streams and concept drifts). The proposed new framework is able to almost entirely cover the spectrum of possible scenarios that can be observed in the real-world. The proposed approach is implemented as a publicly available Java application, with a set of APIs for the programmatic execution of experiments.
]]></description>
<dc:subject>process-mining learning-by-watching machine-learning statistics modeling dynamical-systems what-gets-measured to-write-about representation discrete-event-simulators to-simulate time-series</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:122986d87d39/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:process-mining"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:learning-by-watching"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:machine-learning"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:statistics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:modeling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:what-gets-measured"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:representation"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:discrete-event-simulators"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:time-series"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1806.09644">
    <title>[1806.09644] How to hear the shape of a billiard table</title>
    <dc:date>2019-07-24T11:31:08+00:00</dc:date>
    <link>https://arxiv.org/abs/1806.09644</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The bounce spectrum of a polygonal billiard table is the collection of all bi-infinite sequences of edge labels corresponding to billiard trajectories on the table. We give methods for reconstructing from the bounce spectrum of a polygonal billiard table both the cyclic ordering of its edge labels and the sizes of its angles. We also show that it is impossible to reconstruct the exact shape of a polygonal billiard table from any finite collection of finite words from its bounce spectrum.
]]></description>
<dc:subject>billiards dynamical-systems spectra rather-interesting inverse-problems to-understand to-write-about to-simulate consider:looking-to-see consider:classification impossibility-proof consider:approximation</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:9a50eefee19c/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:spectra"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:inverse-problems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:classification"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:impossibility-proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:approximation"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/0911.1984">
    <title>[0911.1984] Perfect Retroreflectors and Billiard Dynamics</title>
    <dc:date>2019-07-24T11:18:16+00:00</dc:date>
    <link>https://arxiv.org/abs/0911.1984</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We construct semi-infinite billiard domains which reverse the direction of most incoming particles. We prove that almost all particles will leave the open billiard domain after a finite number of reflections. Moreover, with high probability the exit velocity is exactly opposite to the entrance velocity, and the particle's exit point is arbitrarily close to its initial position. The method is based on asymptotic analysis of statistics of entrance times to a small interval for irrational circle rotations. The rescaled entrance times have a limiting distribution in a limit when the number of iterates tends to infinity and the length of the interval vanishes. The proof of the main results follows from the study of related limiting distributions and their regularity properties.
]]></description>
<dc:subject>billiards plane-geometry dynamical-systems engineering-design rather-interesting existence-proof to-write-about consider:looking-to-see to-simulate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:d09333590656/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:engineering-design"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:existence-proof"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-simulate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1510.07742">
    <title>[1510.07742] Iterating evolutes and involutes</title>
    <dc:date>2019-07-24T11:06:28+00:00</dc:date>
    <link>https://arxiv.org/abs/1510.07742</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[We study iterations of two classical constructions, the evolutes and involutes of plane curves, and we describe the limiting behavior of both constructions on a class of smooth curves with singularities given by their support functions. 
Next we study two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices, or by the incenters of the triples of consecutive sides. The space of polygons is a vector bundle over the space of the side directions, and both kinds of evolutes define vector bundle morphisms. In both cases, we describe the linear maps of the fibers. In the first case, the induced map of the base is periodic, whereas, in the second case, it is an averaging transformation. We also study the dynamics of the related inverse constructions, the involutes of polygons. 
In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.
]]></description>
<dc:subject>plane-geometry dynamical-systems construction to-understand rather-interesting to-write-about consider:genetic-programming consider:closed-forms consider:classification</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:472c3da85f71/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:plane-geometry"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:genetic-programming"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:closed-forms"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:classification"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1006.2782">
    <title>[1006.2782] Outer Billiards, Arithmetic Graphs, and the Octagon</title>
    <dc:date>2019-07-14T12:53:57+00:00</dc:date>
    <link>https://arxiv.org/abs/1006.2782</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. 
When the shape is a polygon, the system has a combinatorial flavor. In the polygonal case, there is a natural acceleration of the map, a first return map to a certain strip in the plane. The arithmetic graph is a geometric encoding of the symbolic dynamics of this first return map. 
In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period tends to infinity. We show that the rescaled limit of essentially any sequence of these graphs converges to a fractal curve that simultaneously projects one way onto a variant of the Koch snowflake and another way onto a variant of the Sierpinski carpet. In a sense, this gives a complete description of the asymptotic behavior of the symbolic dynamics of the first return map. 
What makes all our proofs work is an efficient (and basically well known) renormalization scheme for the dynamics.
]]></description>
<dc:subject>billiards dynamical-systems rather-interesting consider:looking-to-see to-write-about to-animate</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:ac46d3107a40/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:billiards"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:consider:looking-to-see"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-write-about"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-animate"/>
</rdf:Bag></taxo:topics>
</item>
<item rdf:about="https://arxiv.org/abs/1808.07409">
    <title>[1808.07409] The domino shuffling height process and its hydrodynamic limit</title>
    <dc:date>2019-06-24T11:13:42+00:00</dc:date>
    <link>https://arxiv.org/abs/1808.07409</link>
    <dc:creator>Vaguery</dc:creator><description><![CDATA[The famous domino shuffling algorithm was invented to generate the domino tilings of the Aztec Diamond. Using the domino height function, we view the domino shuffling procedure as a discrete-time random height process on the plane. The hydrodynamic limit from an arbitrary continuous profile is deduced to be the unique viscosity solution of a Hamilton-Jacobi equation ut+H(ux)=0, where the determinant of the Hessian of H is negative everywhere. The proof involves interpolation of the discrete process and analysis of the limiting semigroup of the evolution. In order to identify the limit, we use the theories of dimer models as well as Hamilton-Jacobi equations. 
It seems that our result is the first example in d>1 where such a full hydrodynamic limit with a nonconvex Hamiltonian can be obtained for a discrete system. We also define the shuffling height process for more general periodic dimer models, where we expect similar results to hold.]]></description>
<dc:subject>combinatorics graph-theory feature-construction rather-interesting dynamical-systems domino-tiling to-understand</dc:subject>
<dc:source>https://pinboard.in/</dc:source>
<dc:identifier>https://pinboard.in/u:Vaguery/b:f99acfeabd08/</dc:identifier>
<taxo:topics><rdf:Bag>	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:combinatorics"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:graph-theory"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:feature-construction"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:rather-interesting"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:dynamical-systems"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:domino-tiling"/>
	<rdf:li rdf:resource="https://pinboard.in/u:Vaguery/t:to-understand"/>
</rdf:Bag></taxo:topics>
</item>
</rdf:RDF>