[1710.01968] Memetic Multilevel Hypergraph Partitioning

november 2017 by Vaguery

Hypergraph partitioning has a wide range of important applications such as VLSI design or scientific computing. With focus on solution quality, we develop the first multilevel memetic algorithm to tackle the problem. Key components of our contribution are new effective multilevel recombination and mutation operations that provide a large amount of diversity. We perform a wide range of experiments on a benchmark set containing instances from application areas such VLSI, SAT solving, social networks, and scientific computing. Our new algorithm computes the best result on almost all instances.

graph-partitioning
graph-theory
algebra
optimization
combinatorics
nudge-targets
consider:looking-to-see
november 2017 by Vaguery

[1603.00051] Algebraic Method in Tilings

october 2017 by Vaguery

In this paper we introduce a new algebraic method in tilings. Combining this method with Hilbert's Nullstellensatz we obtain a necessary condition for tiling n-space by translates of a cluster of cubes. Further, the polynomial method will enable us to show that if there exists a tiling of n-space by translates of a cluster V of prime size then there is a lattice tiling by V as well. Finally, we provide supporting evidence for a conjecture that each tiling by translates of a prime size cluster V is lattice if V generates n-space.

polyominoes
algebra
representation
rather-interesting
nudge-targets
consider:representation
october 2017 by Vaguery

[1709.09022] On Integer Images of Max-plus Linear Mappings

october 2017 by Vaguery

Let us extend the pair of operations (max,+) over real numbers to matrices in the same way as in conventional linear algebra. We study integer images of max-plus linear mappings. The question whether Ax (in the max-plus algebra) is an integer vector for at least one x has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems. We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case.

representation
linear-algebra
out-of-the-box
mathematics
algebra
nudge-targets
consider:looking-to-see
consider:representation
october 2017 by Vaguery

[1610.01369] Self-referential Functions

october 2017 by Vaguery

We introduce the concept of fractels for functions and discuss their analytic and algebraic properties. We also consider the representation of polynomials and analytic functions using fractels, and the consequences of these representations in numerical analysis.

fractals
algebra
representation
dynamical-systems
to-understand
define-your-terms
to-write-about
nudge
consider:fractel-type-in-Klapaucius
october 2017 by Vaguery

[1608.08087] Equisectional equivalence of triangles

october 2017 by Vaguery

We study equivalence relation of the set of triangles generated by similarity and operation on a triangle to get a new one by joining division points of three edges with the same ratio. Using the moduli space of similarity classes of triangles introduced by Nakamura and Oguiso, we give characterization of equivalent triangles in terms of circles of Apollonius (or hyperbolic pencil of circles) and properties of special equivalent triangles. We also study rationality problem and constructibility problem.

plane-geometry
compass-and-straightedge
looking-to-see
rather-interesting
algebra
nudge-targets
consider:feature-discovery
october 2017 by Vaguery

[1709.06334] On the number of representations of $n=a+b$ with $ab$ a multiple of a polygonal number

september 2017 by Vaguery

In this paper, we study the number of representations of a positive integer n by two positive integers whose product is a multiple of a polygonal number.

number-theory
combinatorics
algebra
nudge-targets
consider:looking-to-see
consider:symbolic-regression
september 2017 by Vaguery

[math/0604097] A note on integral points on elliptic curves

september 2017 by Vaguery

We investigate a problem considered by Zagier and Elkies, of finding large integral points on elliptic curves. By writing down a generic polynomial solution and equating coefficients, we are led to suspect four extremal cases that still might have nondegenerate solutions. Each of these cases gives rise to a polynomial system of equations, the first being solved by Elkies in 1988 using the resultant methods of~\Macsyma, with there being a unique rational nondegenerate solution. For the second case we found that resultants and/or Gr\"obner bases were not very efficacious. Instead, at the suggestion of Elkies, we used multidimensional p-adic Newton iteration, and were able to find a nondegenerate solution, albeit over a quartic number field. Due to our methodology, we do not have much hope of proving that there are no other solutions. For the third case we found a solution in a nonic number field, but we were unable to make much progress with the fourth case. We make a few concluding comments and include an appendix from Elkies regarding his calculations and correspondence with Zagier.

number-theory
elliptic-equations
algebra
rather-interesting
consider:looking-to-see
nudge-targets
september 2017 by Vaguery

News on quadratic polynomials

september 2017 by Vaguery

Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning quadratic polynomials, which uses a bridge between algebra and analysis. We study the iterations of quadratic polynomials, obtained by computing the value of a polynomial for a given number and feeding the outcome into the exact same polynomial again. These iterations of polynomials have interesting applications, such as in fractal theory.

mathematics
number-theory
algebra
fractals
rather-interesting
dynamical-systems
to-write-about
mathematical-recreations
september 2017 by Vaguery

Safely Footed Spiderwebs – The Inner Frame

september 2017 by Vaguery

The first three are equiangular still, making angles of 10, 90 and 240 degrees at the corners, respectively. The spiderwebs are conformal images of polar coordinates on the disk, thus illustrating the Schwarz-Christoffel formula for circular polygons. The bat down below is a neat optical illusion, too: Would you think that the vertices are at the corners of an equilateral triangle?

mathematical-recreations
visualization
to-write-about
geometry
algebra
september 2017 by Vaguery

[1512.00906] Butcher series: A story of rooted trees and numerical methods for evolution equations

september 2017 by Vaguery

Butcher series appear when Runge-Kutta methods for ordinary differential equations are expanded in power series of the step size parameter. Each term in a Butcher series consists of a weighted elementary differential, and the set of all such differentials is isomorphic to the set of rooted trees, as noted by Cayley in the mid 19th century. A century later Butcher discovered that rooted trees can also be used to obtain the order conditions of Runge-Kutta methods, and he found a natural group structure, today known as the Butcher group. It is now known that many numerical methods also can be expanded in Butcher series; these are called B-series methods. A long-standing problem has been to characterize, in terms of qualitative features, all B-series methods. Here we tell the story of Butcher series, stretching from the early work of Cayley, to modern developments and connections to abstract algebra, and finally to the resolution of the characterization problem. This resolution introduces geometric tools and perspectives to an area traditionally explored using analysis and combinatorics.

numerical-methods
differential-equations
symbolic-methods
algebra
mathematics
approximation
rather-interesting
to-write-about
consider:rediscovery
september 2017 by Vaguery

[math/0411380] Cosine products, Fourier transforms, and random sums

may 2017 by Vaguery

We investigate several infinite product of cosines and find the closed form using the Fourier transform. The answers provide limiting distributions for some elementary probability experiments.

mathematical-recreations
history
trigonometry
algebra
nudge-targets
consider:looking-to-see
consider:novelty-search
may 2017 by Vaguery

[1704.07503] Learning of Human-like Algebraic Reasoning Using Deep Feedforward Neural Networks

may 2017 by Vaguery

There is a wide gap between symbolic reasoning and deep learning. In this research, we explore the possibility of using deep learning to improve symbolic reasoning. Briefly, in a reasoning system, a deep feedforward neural network is used to guide rewriting processes after learning from algebraic reasoning examples produced by humans. To enable the neural network to recognise patterns of algebraic expressions with non-deterministic sizes, reduced partial trees are used to represent the expressions. Also, to represent both top-down and bottom-up information of the expressions, a centralisation technique is used to improve the reduced partial trees. Besides, symbolic association vectors and rule application records are used to improve the rewriting processes. Experimental results reveal that the algebraic reasoning examples can be accurately learnt only if the feedforward neural network has enough hidden layers. Also, the centralisation technique, the symbolic association vectors and the rule application records can reduce error rates of reasoning. In particular, the above approaches have led to 4.6% error rate of reasoning on a dataset of linear equations, differentials and integrals.

rather-interesting
neural-networks
deep-learning
algebra
rewriting-systems
to-write-about
consider:neural-GP-crossovers
may 2017 by Vaguery

[math/0609127] Square Eulerian Quadruples

april 2017 by Vaguery

We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.

number-theory
algebra
constraint-satisfaction
rather-interesting
nudge-targets
consider:performance-measures
algorithms
april 2017 by Vaguery

[0909.1666] On Sets of Integers where Each Pair Sums to a Square

april 2017 by Vaguery

We discuss the problem of finding distinct integer sets {x1,x2,...,xn} where each sum xi+xj,i≠j is a square, and n≤7. We confirm minimal results of Lagrange and Nicolas for n=5 and for the related problem with triples. We provide new solution sets for n=6 to add to the single known set. This provides new information for problem D15 in Guy's {\it Unsolved Problems in Number Theory}

number-theory
algebra
constraint-satisfaction
open-problems
nudge-targets
consider:looking-to-see
april 2017 by Vaguery

[1006.3153] Perfect Quadrilateral Right Prisms

april 2017 by Vaguery

We consider right prisms with horizontal quadrilateral bases and tops, and vertical rectangular sides. We look for examples where all the edges, face diagonals and space diagonals are integers. We find examples when the base is an isosceles trapezium and a parallelogram, but no solution for a kite or rhombus.

number-theory
algebra
constraint-satisfaction
catalog
nudge-targets
consider:looking-to-see
april 2017 by Vaguery

[1109.2396] New Solutions of $d=2x^3+y^3+z^3$

april 2017 by Vaguery

We discuss finding large integer solutions of d=2x3+y3+z3 by using Elsenhans and Jahnel's adaptation of Elkies' LLL-reduction method. We find 28 first solutions for |d|<10000.

number-theory
algebra
polynomials
constraint-satisfaction
rather-interesting
to-write-about
nudge-targets
consider:looking-to-see
stamp-collecting
algorithms
april 2017 by Vaguery

[1312.1793] "Nice" Rational Functions

april 2017 by Vaguery

We consider simple rational functions Rmn(x)=Pm(x)/Qn(x), with Pm and Qn polynomials of degree m and n respectively. We look for "nice" functions, which we define to be ones where as many as possible of the roots, poles, critical points and (possibly) points of inflexion are integer or, at worst, rational.

algebra
number-theory
rather-interesting
constraint-satisfaction
stamp-collecting
nudge-targets
consider:looking-to-see
consider:feature-discovery
april 2017 by Vaguery

[1703.09097] An explicit formula for the pressure of box-like affine iterated function systems

april 2017 by Vaguery

In this article we investigate the pressure function and affinity dimension for iterated function systems associated to the "box-like" self-affine fractals investigated by D.-J. Feng, Y. Wang and J.M. Fraser. Combining previous results of V. Yu. Protasov, A. K\"aenm\"aki and the author we obtain an explicit formula for the pressure function which makes it straightforward to compute the affinity dimension of box-like self-affine sets. We also prove a variant of this formula which allows the computation of a modified singular value pressure function defined by J.M. Fraser. We give some explicit examples where the Hausdorff and packing dimensions of a box-like self-affine fractal may be easily computed.

geometry
algebra
algorithms
fractals
nudge-targets
consider:rediscovery
consider:representation
april 2017 by Vaguery

[1703.05164] Convergent and Divergent Series in Physics

april 2017 by Vaguery

The aim of this review, based on a series of four lectures held at the 22nd "Saalburg" Summer School (2016), is to cover selected topics in the theory of perturbation series and their summation. The first part is devoted to strategies for accelerating the rate of convergence of convergent series, namely Richardson extrapolation, and the Shanks transformations, and also covers a few techniques for accelerating the convergence of Fourier series. The second part focuses on divergent series, and on the tools allowing one to retrieve information from them. These techniques include Euler summation, Borel summation, generic summation, and the method of continued functions, including in particular the Pad\'e theory based on continued fractions. Finally, a brief discussion of Stieltjes series and functions is given in order to study the convergence properties of Pad\'e approximants.

mathematics
algebra
introduction
nudge-targets
consider:looking-to-see
april 2017 by Vaguery

[1604.01760] Solving Diophantine Equations

march 2017 by Vaguery

In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\eta}({\pi}(x)) = {\pi}({\eta}(x)), where {\eta} is the Smarandache function and {\pi} is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and then we check all possible situations, and of course we retain among them only those solutions that verify our equation. In other words, we say that the equation does not have solutions in the search domain, or the equation has n solutions in this domain. This mode of solving is called partial resolution. Partially solving a Diophantine equation may be a good start for a complete solving of the problem. The authors have identified 62 Diophantine equations that impose such approach and they partially solved them. For an efficient resolution it was necessarily that they have constructed many useful tools for partially solving the Diophantine equations into a reasonable time. The computer programs as tools were written in Mathcad, because this is a good mathematical software where many mathematical functions are implemented. Transposing the programs into another computer language is facile, and such algorithms can be turned to account on other calculation systems with various processors.

algebra
number-theory
rather-interesting
books
nudge-targets
consider:looking-to-see
algorithms
numerical-methods
march 2017 by Vaguery

[1202.4358] Natural Product Xn on matrices

march 2017 by Vaguery

This book has eight chapters. The first chapter is introductory in nature. Polynomials with matrix coefficients are introduced in chapter two. Algebraic structures on these polynomials with matrix coefficients is defined and described in chapter three. Chapter four introduces natural product on matrices. Natural product on super matrices is introduced in chapter five. Super matrix linear algebra is introduced in chapter six. Chapter seven claims only after this notion becomes popular we can find interesting applications of them. The final chapter suggests over 100 problems some of which are at research level.

matrices
algebra
open-questions
rather-interesting
book
nudge-targets
consider:looking-to-see
march 2017 by Vaguery

[1605.01750] Some results on the spectral radii of uniform hypergraphs

may 2016 by Vaguery

Let A(G) be the adjacency tensor (hypermatrix) of uniform hypergraph G. The maximum modulus of the eigenvalues of A(G) is called the spectral radius of G. In this paper, the conjecture of Fan et al. in [5] related to compare the spectral radii of some three uniform hypergraphs is solved. Moreover, some eigenvalues properties of a kind of uniform hypergraphs are obtained.

hypergraphs
tensors
representation
algebra
graph-theory
nudge-targets
proof
combinatorics
may 2016 by Vaguery

[1605.01721] $E_8$, the most exceptional group

may 2016 by Vaguery

The five exceptional simple Lie algebras over the complex number are included one within the other as G2⊂F4⊂E6⊂E7⊂E8. The biggest one, E8, is in many ways the most mysterious. This article surveys what is known about it including many recent results, focusing on the point of view of Lie algebras and algebraic groups over fields.

algebra
group-theory
review
to-understand
may 2016 by Vaguery

[1407.3254] Matrix Completion for the Independence Model

may 2016 by Vaguery

We investigate the problem of completing partial matrices to rank-one matrices in the standard simplex. The motivation for studying this problem comes from statistics: A lack of eligible completion can provide a falsification test for partial observations to come from the independence model. For each pattern of specified entries, we give equations and inequalities which are satisfied if and only if an eligible completion exists. We also describe the set of valid completions, and we optimize over this set.

matrices
inference
inverse-problems
number-theory
algebra
probability-theory
rather-interesting
nudge-targets
consider:looking-to-see
consider:feature-discovery
may 2016 by Vaguery

[1601.07838] The Hurwitz continued fraction expansion as applied to real numbers (expository paper)

april 2016 by Vaguery

Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the Hurwitz complex continued fraction algorithm is not "reducible" to another complex continued fraction algorithm, over the reals the story is different. In this note we make clear the relation between the restriction of Hurwitz's algorithm to the real numbers and the classical continued fraction algorithm. As an application we deduce a theorem which captures the spirit of the main result of Choudhuri and Dani (2015).

continued-fractions
number-theory
algebra
complex-numbers
representation
nudge-targets
consider:looking-to-see
april 2016 by Vaguery

[1209.4588] The Minkowski $?(x)$ function, a class of singular measures, quasi-modular and mean-modular forms. I

april 2016 by Vaguery

The Minkowski question mark function is a rich object which can be explored from the perspective of dynamical systems, complex dynamics, metric number theory, multifractal analysis, transfer operators, integral transforms, and as a function itself via analysis of continued fractions and convergents. Our permanent target, however, was to get arithmetic interpretation of the moments of ?(x) (which are relatives of periods of Maass wave forms) and to relate the function ?(x) to certain modular objects. In this paper we establish this link, embedding ?(x) not into the modular-world itself, but into a space of functions which are generalizations and which we call mean-modular forms. For this purpose we construct a wide class of measures, and also investigate modular forms for congruence subgroups which additionally satisfy the three term functional equation. From this perspective, the modular forms for the whole modular group as well as the Stieltjes transform of ?(x) (the dyadic period function) minus the Eisenstein series of weight 2 fall under the same uniform definition. The main result is the construction of the canonical isomorphism between the spaces of quasi-modular forms and mean-modular forms. This gives unexpected Minkowski question mark function-related interpretation of quasi-modular forms.

number-theory
fractals
continued-fractions
algebra
rather-interesting
mathematics
representation
nudge-targets
consider:representation
april 2016 by Vaguery

[1602.08028] Features of a high school olympiad problem

april 2016 by Vaguery

This paper is a supplement to a talk for mathematics teachers given at the 2016 LSU Mathematics Contest for High School Students. The paper covers more details and aspects than could be covered in the talk.

We start with an interesting problem from the 2009 Iberoamerican Math Olympiad concerning a particular sequence. We include a solution to the problem, but also relate it to several areas of mathematics. This problem demonstrates the countability of the rational numbers with a direct one-to-one correspondence. The problem also shows the one-to-one correspondence of finite continued fractions and rational numbers. The subsequence of odd indexed terms was constructed by Johannes Kepler and is discussed. We also show that the indices have an extension to the 2-adic integers giving a one-to-one correspondence between the positive real numbers and the 2-adic integers. Everything but the extension to the 2-adic integers is known to have appeared elsewhere.

continued-fractions
algebra
mathematical-recreations
number-theory
group-theory
nudge-targets
consider:looking-to-see
We start with an interesting problem from the 2009 Iberoamerican Math Olympiad concerning a particular sequence. We include a solution to the problem, but also relate it to several areas of mathematics. This problem demonstrates the countability of the rational numbers with a direct one-to-one correspondence. The problem also shows the one-to-one correspondence of finite continued fractions and rational numbers. The subsequence of odd indexed terms was constructed by Johannes Kepler and is discussed. We also show that the indices have an extension to the 2-adic integers giving a one-to-one correspondence between the positive real numbers and the 2-adic integers. Everything but the extension to the 2-adic integers is known to have appeared elsewhere.

april 2016 by Vaguery

[1505.00667] An Unusual Continued Fraction

april 2016 by Vaguery

We consider the real number σ with continued fraction expansion [a0,a1,a2,…]=[1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,…], where ai is the largest power of 2 dividing i+1. We compute the irrationality measure of σ2 and demonstrate that σ2 (and σ) are both transcendental numbers. We also show that certain partial quotients of σ2 grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.

continued-fractions
number-theory
algebra
mathematics
nudge-targets
consider:representation
consider:looking-to-see
april 2016 by Vaguery

[1509.05239] Stern Sequences for a Family of Multidimensional Continued Fractions: TRIP-Stern Sequences

april 2016 by Vaguery

The Stern diatomic sequence is closely linked to continued fractions via the Gauss map on the unit interval, which in turn can be understood via systematic subdivisions of the unit interval. Higher dimensional analogues of continued fractions, called multidimensional continued fractions, can be produced through various subdivisions of a triangle. We define triangle partition-Stern sequences (TRIP-Stern sequences for short), higher-dimensional generalizations of the Stern diatomic sequence, from the method of subdividing a triangle via various triangle partition algorithms. We then explore several combinatorial results about TRIP-Stern sequences, which may be used to give rise to certain well-known sequences. We finish by generalizing TRIP-Stern sequences and presenting analogous results for these generalizations.

continued-fraction
algebra
number-theory
mathematics
nudge-targets
consider:representation
april 2016 by Vaguery

Gaussian elimination - Wikipedia, the free encyclopedia

april 2016 by Vaguery

It strikes me that the broader suite of operations from which these few are drawn, and their application over arbitrary fields, are woefully under-discussed. What algorithms other than this one are there, which do things other than the goal this one has?

algebra
mathematics
nudge-targets
consider:looking-to-see
april 2016 by Vaguery

[1504.06482] Convergence properties of the classical and generalized Rogers-Ramanujan continued fraction

april 2016 by Vaguery

The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set of measure zero on which the Rogers-Ramanujan continued fraction R(x) diverges and which enlarges a set previously found by Bowman and Mc Laughlin. We further study the generalized Rogers-Ramanujan continued fractions Ra(x) for roots of unity a and give explicit convergence and divergence conditions. As such, we extend some work of Huang towards a question originally investigated by Ramanujan and some work of Schur on the convergence of R(x) at roots of unity. In the end, we state several conjectures and possible directions for generalizing Schur's result to all Rogers-Ramanujan continued fractions Ra(x).

number-theory
continued-fractions
Ramanujan
algebra
feature-construction
nudge-targets
consider:looking-to-see
april 2016 by Vaguery

[1209.4598] Pasting and Reversing Operations over some Vector Spaces

march 2016 by Vaguery

Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be applied over vector spaces. In particular we study Pasting and Reversing over vectors, matrices and we rewrite some properties for polynomials as vector space. Finally we study some properties of Reversing through linear transformations as for example an analysis of palindromic and antipalindromic vector subspaces.

oh-yeah
algebra
linear-algebra
out-of-the-box
nudge-targets
group-theory
consider:looking-to-see
march 2016 by Vaguery

[1010.1967] On Pasting and Reversing operations over some rings

march 2016 by Vaguery

In this paper we study two operations, Pasting and Reversing, defined from a natural way to be applied over some rings such as the ring of polynomials and the ring of linear differential operators, which is a differential ring. We obtain some properties of these operations over these rings, in particular over the set of natural numbers, in where we rewrite some properties incoming from recreational mathematics.

oh-yeah
group-theory
algebra
mathematics
rather-interesting
out-of-the-box
nudge-targets
march 2016 by Vaguery

[1603.07065] Pasting and Reversing Approach to Matrix Theory

march 2016 by Vaguery

The aim of this paper is to study some aspects of matrix theory through Pasting and Reversing. We start giving a summary of previous results concerning to Pasting and Reversing over vectors and matrices, after we rewrite such properties of Pasting and Reversing in matrix theory using linear mappings to finish with new properties and new sets in matrix theory involving Pasting and Reversing. In particular we introduce new linear mappings: Palindromicing and Antipalindromicing mappings, which allow us to obtain palindromic and antipalindromic vectors and matrices.

oh-yeah
matrices
out-of-the-box
group-theory
algebra
nudge-targets
consider:rediscovery
consider:doing-this-asap
march 2016 by Vaguery

[1507.08366] On the matrix square root via geometric optimization

march 2016 by Vaguery

This paper is triggered by the preprint "\emph{Computing Matrix Squareroot via Non Convex Local Search}" by Jain et al. (\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent for computing the square root of a positive definite matrix. Contrary to claims of~\citet{jain2015}, our experiments reveal that Newton-like methods compute matrix square roots rapidly and reliably, even for highly ill-conditioned matrices and without requiring commutativity. We observe that gradient-descent converges very slowly primarily due to tiny step-sizes and ill-conditioning. We derive an alternative first-order method based on geodesic convexity: our method admits a transparent convergence analysis (<1 page), attains linear rate, and displays reliable convergence even for rank deficient problems. Though superior to gradient-descent, ultimately our method is also outperformed by a well-known scaled Newton method. Nevertheless, the primary value of our work is its conceptual value: it shows that for deriving gradient based methods for the matrix square root, \emph{the manifold geometric view of positive definite matrices can be much more advantageous than the Euclidean view}.

matrices
algebra
algorithms
computational-complexity
nudge-targets
horse-races
consider:looking-to-see
march 2016 by Vaguery

[1502.01384] Strictly self-similar fractals composed of star-polygons that are attractors of Iterated Function Systems

january 2016 by Vaguery

In this paper are investigated strictly self-similar fractals that are composed of an infinite number of regular star-polygons, also known as Sierpinski n-gons, n-flakes or polyflakes. Construction scheme for Sierpinsky n-gon and n-flake is presented where the dimensions of the Sierpinsky ∞-gon and the ∞-flake are computed to be 1 and 2, respectively. These fractals are put in a general context and Iterated Function Systems are applied for the visualisation of the geometric iterations of the initial polygons, as well as the visualisation of sets of points that lie on the attractors of the IFS generated by random walks. Moreover, it is shown that well known fractals represent isolated cases of the presented generalisation. The IFS programming code is given, hence it can be used for further investigations.

IFS
fractals
self-similarity
nonlinear-dynamics
algebra
generalization
nudge-targets
consider:feature-discovery
consider:feature-construction
january 2016 by Vaguery

[1512.00908] Lattice Surfaces and smallest triangles

december 2015 by Vaguery

We calculate the area of the smallest triangle and the area of the smallest virtual triangle for many known lattice surfaces. We show that our list of the lattice surfaces for which the area of the smallest virtual triangle greater than .05 is complete. In particular, this means that there are no new lattice surfaces for which the area of the smallest virtual triangle is greater than .05. Our method follows an algorithm described by Smillie and Weiss and improves on it in certain respects.

tiling
algebra
integer-lattices
computational-geometry
looking-to-see
rather-interesting
nudge-targets
consider:feature-discovery
december 2015 by Vaguery

[1405.0219] Complete Duality for Quasiconvex and Convex Set-Valued Functions

december 2015 by Vaguery

This paper provides an unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set valued functions.

set-valued-functions
algebra
optimization
mathematical-programming
representation
to-understand
december 2015 by Vaguery

[1412.4365] Fast Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

november 2015 by Vaguery

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for the case of a low-dimensional projective space. In this study, we construct an efficient decoding algorithm for all PRM codes. For this purpose, we divide a projective space into a union of affine spaces. In addition, we evaluate the computational complexity and number of correctable errors of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.

information-theory
algorithms
representation
algebra
signal-processing
error-correcting-codes
nudge-targets
consider:performance-measures
consider:representation
polynomials
november 2015 by Vaguery

Graph product - Wikipedia, the free encyclopedia

november 2015 by Vaguery

The following table shows the most common graph products, with ; denoting “is connected by an edge to”, and denoting non-connection. The operator symbols listed here are by no means standard, especially in older papers.

graph-theory
algebra
operations
ReQ
consider:implementation
november 2015 by Vaguery

[1505.07144] When are the Cayley-Salmon lines conjugate?

november 2015 by Vaguery

Given six points on a conic, Pascal's theorem gives rise to a well-known configuration called the \emph{hexagrammum mysticum}. It consists of, amongst other things, twenty Steiner points and twenty Cayley-Salmon lines. It is a classical theorem due to von Staudt that the Steiner points fall into ten conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. It is shown in this paper that the C-S lines are pairwise conjugate precisely when the original sextuple is~\emph{tri-involutive}. The variety of tri-involutive sextuples turns out to be arithmetically Cohen-Macaulay of codimension two. We determine its SL2-equivariant minimal resolution.

geometry
algebra
mathematics
rather-interesting
nudge-targets
consider:exploration
november 2015 by Vaguery

[1409.5223] Why Local Search Excels in Expression Simplification

september 2015 by Vaguery

Simplifying expressions is important to make numerical integration of large expressions from High Energy Physics tractable. To this end, Horner's method can be used. Finding suitable Horner schemes is assumed to be hard, due to the lack of local heuristics. Recently, MCTS was reported to be able to find near optimal schemes. However, several parameters had to be fine-tuned manually. In this work, we investigate the state space properties of Horner schemes and find that the domain is relatively flat and contains only a few local minima. As a result, the Horner space is appropriate to be explored by Stochastic Local Search (SLS), which has only two parameters: the number of iterations (computation time) and the neighborhood structure. We found a suitable neighborhood structure, leaving only the allowed computation time as a parameter. We performed a range of experiments. The results obtained by SLS are similar or better than those obtained by MCTS. Furthermore, we show that SLS obtains the good results at least 10 times faster. Using SLS, we can speed up numerical integration of many real-world large expressions by at least a factor of 24. For High Energy Physics this means that numerical integrations that took weeks can now be done in hours.

rather-interesting
algebra
expression-simplification
metaheuristics
nudge-targets
local-search
september 2015 by Vaguery

[1307.3474] Kneading determinants of infinite order linear recurrences

september 2015 by Vaguery

Infinite order linear recurrences are studied via kneading matrices and kneading determinants. The concepts of kneading matrix and kneading determinant of an infinite order linear recurrence, introduced in this work, are defined in a purely linear algebraic context. These concepts extend the classical notions of Frobenius companion matrix to infinite order linear recurrences and to the associated discriminant of finite order linear recurrences. Asymptotic Binet formulas are deduced for general classes of infinite order linear recurrences as a consequence of the analytical properties of the generating functions obtained for the solutions of these infinite order linear recurrences.

rather-interesting
group-theory
algebra
generative-models
nudge-targets
consider:rediscovery
consider:performance-measures
september 2015 by Vaguery

[1502.03669] Nonsimple polyominoes and prime ideals

july 2015 by Vaguery

It is known that the polyomino ideal of a simple polyomino is a prime ideal. A new class of nonsimple polyominoes $\Pc$ for which the polyomino ideal $I_{\Pc}$ is a prime ideal will be presented.

algebra
polyominoes
combinatorics
to-understand
nudge-targets
(?)
july 2015 by Vaguery

[1403.6920] Gr"obner bases of balanced polyominoes

july 2015 by Vaguery

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gr\"obner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced.

combinatorics
algebra
to-understand
stamp-collecting
feature-construction
nudge-targets
hmm
july 2015 by Vaguery

[1203.0407] Ideals generated by 2-minors, collection of cells and stack polyominoes

july 2015 by Vaguery

In this paper we study ideals generated by quite general sets of 2-minors of an m×n-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given.

graph-theory
algebra
algorithms
to-understand
reference
nudge-targets
consider:classification
july 2015 by Vaguery

[1412.4453] Diagrams and rectangular extensions of planar semimodular lattices

april 2015 by Vaguery

In 2009, G. Gr\"atzer and E. Knapp proved that every planar semimodular lattice has a rectangular extension. We prove that, under reasonable additional conditions, this extension is unique. This theorem naturally leads to a hierarchy of special diagrams of planar semimodular lattices. Besides that these diagrams are unique in a strong sense, we explore many of their further properties. Finally, we demonstrate the power of our new diagrams in two ways. First, we prove a simplified version of our earlier Trajectory Coloring Theorem, which describes the inclusion con(p)\supseteq\con(q) for prime intervals p and q in slim rectangular lattices. Second, we prove G. Gr\"atzer's Swing Lemma for the same lattices, which describes the same inclusion more simply.

to-understand
group-theory
graph-theory
algebra
pretty-pitchers
april 2015 by Vaguery

[1408.1324] Unit balls of constant volume: which one has optimal representation?

november 2014 by Vaguery

In the family of unit balls with constant volume we look at the ones whose algebraic representation has some extremal property. We consider the family of nonnegative homogeneous polynomials of even degree d whose sublevel set $\G=\{\x: g(\x)\leq 1\}$ (a unit ball) has same fixed volume and want to find in this family the one that minimizes either the ℓ1-norm or the ℓ2-norm of its vector of coefficients. Equivalently, among all degree-d polynomials of constant ℓ1− or ℓ2-norm, which one minimizes the volume of its level set $\G$. We first show that in both cases this is a convex optimization problem with a unique optimal solution g∗1 and g∗2 respectively. We also show that g∗1 is the Lp-norm polynomial $\x\mapsto\sum_{i=1}^n x_i^{p}$, thus recovering a parsimony property of the Lp-norm via ℓ1-norm minimization. (Indeed n=∥g∗1∥0 is the minimum number of non-zero coefficient for $\G$ to have finite volume.) This once again illustrates the power and versatility of the ℓ1-norm relaxation strategy in optimization when one searches for an optimal solution with parsimony properties. Next we show that g∗2 is not sparse at all (and so differs from g∗1) but is still a sum of p-powers of linear forms. We also characterize the unique optimal solution of the same problem where one searches for an SOS homogeneous polynomial that minimizes the trace of its associated (psd) Gram matrix, hence aiming at finding a solution which is a sum of a few squares only. Finally, we also extend these results to generalized homogeneous polynomials, which includes Lp-norms when $0

geometry
algebra
sittin-in-a-tree
as-though-taken-directly-from-Pickering's-quaternion-paper
bridgehead
rather-interesting
creative-heuristics
november 2014 by Vaguery

[1211.3680] Ahlfors circle maps: historical ramblings

february 2013 by Vaguery

[This may be one of the best-written mathematical papers I've ever seen. More please.]

mathematics
academic-culture
history
nanohistory
review
topology
algebra
visualization
personal-accounts
narrative
february 2013 by Vaguery

Computer Algebra Systems

april 2010 by Vaguery

"In the spirit of sparse representation, we chose to use a syntax tree as the internal data structure for our symbolic calculator. A syntax tree is a kind of tree, which in turn is a kind of linked data structure. Briefly, a linked data structure is an object which contains references, or links, to other like objects. A simple example is a linked list, where each element contains the data for it list entry and a link to the next list element. A tree is a linked structure that starts with a single "root" node. One or more "child" nodes are referenced from the root node, and each of these child nodes may in turn have children of their own. This linking pattern produces a branching data structure, as seen in the following diagram; hence the name "tree"."

computer-algebra
mathematics
algebra
representation
programming
structures
Nudge
april 2010 by Vaguery

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