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arxiv-676801 | math/0602059 | The Matrix of Maximum Out Forests of a Digraph and Its Applications | <|reference_start|>The Matrix of Maximum Out Forests of a Digraph and Its Applications: We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph G is a spanning subgraph of G that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains any out arborescences, then maximum out forests coincide with them. We provide a new proof to the Markov chain tree theorem saying that the matrix of Ces`aro limiting probabilities of an arbitrary stationary finite Markov chain coincides with the normalized matrix of maximum out forests of the weighted digraph that corresponds to the Markov chain. We discuss the applications of the matrix of maximum out forests and its transposition, the matrix of limiting accessibilities of a digraph, to the problems of preference aggregation, measuring the vertex proximity, and uncovering the structure of a digraph.<|reference_end|> | arxiv | @article{agaev2006the,
title={The Matrix of Maximum Out Forests of a Digraph and Its Applications},
author={Rafig Agaev and Pavel Chebotarev},
journal={Automation and Remote Control 61 (2000) 1424--1450},
year={2006},
archivePrefix={arXiv},
eprint={math/0602059},
primaryClass={math.CO cs.DS math.AG}
} | agaev2006the |
arxiv-676802 | math/0602061 | Spanning Forests of a Digraph and Their Applications | <|reference_start|>Spanning Forests of a Digraph and Their Applications: We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expression are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff (Laplacian) matrix. These expressions involve the matrix of maximum out forest of the digraph. Every matrix of out forests with a fixed number of arcs and the normalized matrix of out forests are represented as polynomials in the Kirchhoff matrix; with the help of these identities new proofs are given for the matrix-forest theorem and some other statements. A connection is specified between the forest dimension of a digraph and the degree of an annihilating polynomial for the Kirchhoff (Laplacian) matrix. Some accessibility measures for digraph vertices are considered. These are based on the enumeration of spanning forests.<|reference_end|> | arxiv | @article{agaev2006spanning,
title={Spanning Forests of a Digraph and Their Applications},
author={Rafig Agaev and Pavel Chebotarev},
journal={Automation and Remote Control 62 (2001) No.3 443-466},
year={2006},
doi={10.1023/A:1002862312617},
archivePrefix={arXiv},
eprint={math/0602061},
primaryClass={math.CO cs.DM math.RA}
} | agaev2006spanning |
arxiv-676803 | math/0602070 | The Matrix-Forest Theorem and Measuring Relations in Small Social Groups | <|reference_start|>The Matrix-Forest Theorem and Measuring Relations in Small Social Groups: We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure "dissociation," "solitariness," and "provinciality" of vertices are also considered. A nonstandard metric on the set of vertices is introduced, which is determined by their connectivity. The application of these indices in sociometry is discussed.<|reference_end|> | arxiv | @article{chebotarev2006the,
title={The Matrix-Forest Theorem and Measuring Relations in Small Social Groups},
author={Pavel Chebotarev and Elena Shamis},
journal={Automation and Remote Control 58 (1997) No. 9 1505-1514},
year={2006},
archivePrefix={arXiv},
eprint={math/0602070},
primaryClass={math.CO cs.IR math.AG}
} | chebotarev2006the |
arxiv-676804 | math/0602073 | On Proximity Measures for Graph Vertices | <|reference_start|>On Proximity Measures for Graph Vertices: We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted structures and take into account not only the shortest, but also all other connections, which is desirable in many applications. To apply these proximity measures to unweighted structures, every edge should be assigned the same weight which determines the proportion of taking account of two routes, from which one is one edge longer than the other. Among the proximity measures we consider path accessibility, route accessibility, relative forest accessibility along with its components, accessibility via dense forests, and connection reliability. A number of characteristic conditions is introduced and employed to characterize the proximity measures. A topological interpretation is obtained for the Moore-Penrose generalized inverse of the Laplacian matrix of a weighted multigraph.<|reference_end|> | arxiv | @article{chebotarev2006on,
title={On Proximity Measures for Graph Vertices},
author={Pavel Chebotarev and Elena Shamis},
journal={Automation and Remote Control 59 (1998), No. 10, Part 2 1443-1459.
Erratum: 60 (1999), No. 2, Part 2 297},
year={2006},
archivePrefix={arXiv},
eprint={math/0602073},
primaryClass={math.CO cs.DS cs.NI math.MG}
} | chebotarev2006on |
arxiv-676805 | math/0602154 | What is the Inverse of Repeated Square and Multiply Algorithm? | <|reference_start|>What is the Inverse of Repeated Square and Multiply Algorithm?: It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm efficiently. The technical hitch is in fixing the right sign of the square root and this is the heart of the discrete logarithm problem over finite fields of characteristic not equal to 2. In this paper a couple of probabilistic algorithms to compute the discrete logarithm over finite fields are given by bypassing this difficulty. One of the algorithms was inspired by the famous 3x+1 problem.<|reference_end|> | arxiv | @article{gadiyar2006what,
title={What is the Inverse of Repeated Square and Multiply Algorithm?},
author={H. Gopalkrishna Gadiyar, K M Sangeeta Maini, R. Padma and Mario Romsy},
journal={Colloq. Math. 116 (2009) 1-14},
year={2006},
archivePrefix={arXiv},
eprint={math/0602154},
primaryClass={math.NT cs.CR}
} | gadiyar2006what |
arxiv-676806 | math/0602171 | Preference fusion when the number of alternatives exceeds two: indirect scoring procedures | <|reference_start|>Preference fusion when the number of alternatives exceeds two: indirect scoring procedures: We consider the problem of aggregation of incomplete preferences represented by arbitrary binary relations or incomplete paired comparison matrices. For a number of indirect scoring procedures we examine whether or not they satisfy the axiom of self-consistent monotonicity. The class of {\em win-loss combining scoring procedures} is introduced which contains a majority of known scoring procedures. Two main results are established. According to the first one, every win-loss combining scoring procedure breaks self-consistent monotonicity. The second result provides a sufficient condition of satisfying self-consistent monotonicity.<|reference_end|> | arxiv | @article{chebotarev2006preference,
title={Preference fusion when the number of alternatives exceeds two: indirect
scoring procedures},
author={Pavel Chebotarev and Elena Shamis},
journal={Journal of the Franklin Institute, 336 (1999), No.2, 205-226.
Erratum: 336 (1999) No.4, 747-748
(http://dx.doi.org/10.1016/S0016-0032(99)00004-6)},
year={2006},
doi={10.1016/S0016-0032(98)00017-9},
archivePrefix={arXiv},
eprint={math/0602171},
primaryClass={math.OC cs.MA math.CO}
} | chebotarev2006preference |
arxiv-676807 | math/0602282 | The Diffie-Hellman Key Exchange Protocol and non-abelian nilpotent groups | <|reference_start|>The Diffie-Hellman Key Exchange Protocol and non-abelian nilpotent groups: In this paper we study a key exchange protocol similar to Diffie-Hellman key exchange protocol using abelian subgroups of the automorphism group of a non-abelian nilpotent group. We also generalize group no.92 of Hall-Senior table \cite{halltable}, for arbitrary prime $p$ and show that for those groups, the group of central automorphisms commute. We use these for the key exchange we are studying.<|reference_end|> | arxiv | @article{mahalanobis2006the,
title={The Diffie-Hellman Key Exchange Protocol and non-abelian nilpotent
groups},
author={Ayan Mahalanobis},
journal={arXiv preprint arXiv:math/0602282},
year={2006},
archivePrefix={arXiv},
eprint={math/0602282},
primaryClass={math.GR cs.CR}
} | mahalanobis2006the |
arxiv-676808 | math/0602493 | The dimension of a variety | <|reference_start|>The dimension of a variety: We invent the notion of a {\it dimension of a variety} $V$ as the cardinality of all its proper {\it derived} subvarieties (of the same type). The dimensions of varieties of lattices, varieties of regular bands and other general algebraic structures are determined.<|reference_end|> | arxiv | @article{graczyńska2006the,
title={The dimension of a variety},
author={Ewa Graczy'nska and Dietmar Schweigert},
journal={published in Discussiones Mathematicae, General Algebra and
Applications, 27, 2007, pp. 35--47},
year={2006},
archivePrefix={arXiv},
eprint={math/0602493},
primaryClass={math.LO cs.LO math.GM}
} | graczyńska2006the |
arxiv-676809 | math/0602505 | MDL Convergence Speed for Bernoulli Sequences | <|reference_start|>MDL Convergence Speed for Bernoulli Sequences: The Minimum Description Length principle for online sequence estimation/prediction in a proper learning setup is studied. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is finitely bounded, implying convergence with probability one, and (b) it additionally specifies the convergence speed. For MDL, in general one can only have loss bounds which are finite but exponentially larger than those for Bayes mixtures. We show that this is even the case if the model class contains only Bernoulli distributions. We derive a new upper bound on the prediction error for countable Bernoulli classes. This implies a small bound (comparable to the one for Bayes mixtures) for certain important model classes. We discuss the application to Machine Learning tasks such as classification and hypothesis testing, and generalization to countable classes of i.i.d. models.<|reference_end|> | arxiv | @article{poland2006mdl,
title={MDL Convergence Speed for Bernoulli Sequences},
author={Jan Poland and Marcus Hutter},
journal={Statistics and Computing, 16 (2006) pages 161-175},
year={2006},
doi={10.1007/s11222-006-6746-3},
number={IDSIA-04-06},
archivePrefix={arXiv},
eprint={math/0602505},
primaryClass={math.ST cs.IT cs.LG math.IT math.PR stat.TH}
} | poland2006mdl |
arxiv-676810 | math/0602522 | Characterizations of scoring methods for preference aggregation | <|reference_start|>Characterizations of scoring methods for preference aggregation: The paper surveys more than forty characterizations of scoring methods for preference aggregation and contains one new result. A general scoring operator is {\it self-consistent} if alternative $i$ is assigned a greater score than $j$ whenever $i$ gets no worse (better) results of comparisons and its `opponents' are assigned respectively greater (no smaller) scores than those of $j$. We prove that self-consistency is satisfied if and only if the application of a scoring operator reduces to the solution of a homogeneous system of algebraic equations with a monotone function on the left-hand side.<|reference_end|> | arxiv | @article{chebotarev2006characterizations,
title={Characterizations of scoring methods for preference aggregation},
author={Pavel Chebotarev and Elena Shamis},
journal={Annals of Operations Research, 1998. V. 80. P.299-332},
year={2006},
archivePrefix={arXiv},
eprint={math/0602522},
primaryClass={math.OC cs.MA math.FA}
} | chebotarev2006characterizations |
arxiv-676811 | math/0602552 | From Incomplete Preferences to Ranking via Optimization | <|reference_start|>From Incomplete Preferences to Ranking via Optimization: We consider methods for aggregating preferences that are based on the resolution of discrete optimization problems. The preferences are represented by arbitrary binary relations (possibly weighted) or incomplete paired comparison matrices. This incomplete case remains practically unexplored so far. We examine the properties of several known methods and propose one new method. In particular, we test whether these methods obey a new axiom referred to as {\it Self-Consistent Monotonicity}. Some results are established that characterize solutions of the related optimization problems.<|reference_end|> | arxiv | @article{chebotarev2006from,
title={From Incomplete Preferences to Ranking via Optimization},
author={Pavel Chebotarev and Elena Shamis},
journal={A version of this paper was published as: P.Yu.Chebotarev,
E.V.Shamis. Constructing an objective function for aggregating incomplete
preferences, In: A.Tangian and J.Gruber, eds. Econometric Decision Models:
Constructing Scalar-Valued Objective Functions. Lecture Notes in Economics
and Mathematical Systems, Springer-Verlag, 1997, P.100-124},
year={2006},
archivePrefix={arXiv},
eprint={math/0602552},
primaryClass={math.OC cs.MA math.CO}
} | chebotarev2006from |
arxiv-676812 | math/0602573 | The Forest Metrics for Graph Vertices | <|reference_start|>The Forest Metrics for Graph Vertices: We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs, it takes into account all paths between vertices. Formally, it is defined as d(i,j)=q_{ii}+q_{jj}-q_{ij}-q_{ji}, where q_{ij} is the (i,j)-entry of the {\em relative forest accessibility matrix} Q(\epsilon)=(I+\epsilon L)^{-1}, L is the Laplacian matrix of the (weighted) (multi)graph, and \epsilon is a positive parameter. By the matrix-forest theorem, the (i,j)-entry of the relative forest accessibility matrix of a graph provides the specific number of spanning rooted forests such that i and j belong to the same tree rooted at i. Extremely simple formulas express the modification of the proposed distance under the basic graph transformations. We give a topological interpretation of d(i,j) in terms of the probability of unsuccessful linking i and j in a model of random links. The properties of this metric are compared with those of some other graph metrics. An application of this metric is related to clustering procedures such as "centered partition." In another procedure, the relative forest accessibility and the corresponding distance serve to choose the centers of the clusters and to assign a cluster to each non-central vertex. The notion of cumulative weight of connections between two vertices is proposed. The reasoning involves a reciprocity principle for weighted multigraphs. Connections between the resistance distance and the forest distance are established.<|reference_end|> | arxiv | @article{chebotarev2006the,
title={The Forest Metrics for Graph Vertices},
author={Pavel Chebotarev and Elena Shamis},
journal={Electronic Notes in Discrete Mathematics 11 (July 2002) 98-107},
year={2006},
doi={10.1016/S1571-0653(04)00058-7},
archivePrefix={arXiv},
eprint={math/0602573},
primaryClass={math.CO cs.NI math.MG}
} | chebotarev2006the |
arxiv-676813 | math/0602575 | Matrix-Forest Theorems | <|reference_start|>Matrix-Forest Theorems: The Laplacian matrix of a graph $G$ is $L(G)=D(G)-A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees. According to the Matrix-Tree Theorem, the number of spanning trees in $G$ is equal to any cofactor of an entry of $L(G)$. A rooted forest is a union of disjoint rooted trees. We consider the matrix $W(G)=I+L(G)$ and prove that the $(i,j)$-cofactor of $W(G)$ is equal to the number of spanning rooted forests of $G$, in which the vertices $i$ and $j$ belong to the same tree rooted at $i$. The determinant of $W(G)$ equals the total number of spanning rooted forests, therefore the $(i,j)$-entry of the matrix $W^{-1}(G)$ can be considered as a measure of relative ''forest-accessibility'' of vertex $i$ from $j$ (or $j$ from $i$). These results follow from somewhat more general theorems we prove, which concern weighted multigraphs. The analogous theorems for (multi)digraphs are also established. These results provide a graph-theoretic interpretation for the adjugate to the Laplacian characteristic matrix.<|reference_end|> | arxiv | @article{chebotarev2006matrix-forest,
title={Matrix-Forest Theorems},
author={Pavel Chebotarev and Elena Shamis},
journal={arXiv preprint arXiv:math/0602575},
year={2006},
archivePrefix={arXiv},
eprint={math/0602575},
primaryClass={math.CO cs.DM math.RA}
} | chebotarev2006matrix-forest |
arxiv-676814 | math/0603024 | Towards a better list of citation superstars: compiling a multidisciplinary list of highly cited researchers | <|reference_start|>Towards a better list of citation superstars: compiling a multidisciplinary list of highly cited researchers: A new approach to producing multidisciplinary lists of highly cited researchers is described and used for compiling a multidisciplinary list of highly cited researchers. This approach is essentially related to the recently discovered law of the constant ratios (Podlubny, 2004) and gives a better-balanced representation of different scientific fields.<|reference_end|> | arxiv | @article{podlubny2006towards,
title={Towards a better list of citation superstars: compiling a
multidisciplinary list of highly cited researchers},
author={Igor Podlubny, Katarina Kassayova},
journal={Research Evaluation, vol. 15, no. 3, December 2006, pp. 154-162},
year={2006},
archivePrefix={arXiv},
eprint={math/0603024},
primaryClass={math.ST cs.GL physics.soc-ph stat.TH}
} | podlubny2006towards |
arxiv-676815 | math/0603155 | Vers une commande multivariable sans mod\`ele | <|reference_start|>Vers une commande multivariable sans mod\`ele: A control strategy without any precise mathematical model is derived for linear or nonlinear systems which are assumed to be finite-dimensional. Two convincing numerical simulations are provided.<|reference_end|> | arxiv | @article{fliess2006vers,
title={Vers une commande multivariable sans mod\`ele},
author={Michel Fliess (INRIA Futurs, LIX), C'edric Join (INRIA Futurs, CRAN),
Mamadou Mboup (INRIA Futurs), Hebertt Sira-Ramirez},
journal={Dans Conf\'erence internationale francophone d'automatique (CIFA
2006) (2006)},
year={2006},
archivePrefix={arXiv},
eprint={math/0603155},
primaryClass={math.OC cs.CE cs.RO physics.class-ph}
} | fliess2006vers |
arxiv-676816 | math/0603207 | Fast matrix multiplication is stable | <|reference_start|>Fast matrix multiplication is stable: We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63--72]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438--449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379--388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.<|reference_end|> | arxiv | @article{demmel2006fast,
title={Fast matrix multiplication is stable},
author={James Demmel, Ioana Dumitriu, Olga Holtz, Robert Kleinberg},
journal={Numer. Math. 106 (2007), no. 2, 199-224},
year={2006},
doi={10.1007/s00211-007-0061-6},
archivePrefix={arXiv},
eprint={math/0603207},
primaryClass={math.NA cs.CC cs.DS math.GR}
} | demmel2006fast |
arxiv-676817 | math/0603248 | Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets | <|reference_start|>Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets: In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results.<|reference_end|> | arxiv | @article{basu2006computing,
title={Computing the First Betti Numberand Describing the Connected Components
of Semi-algebraic Sets},
author={Saugata Basu, Richard Pollack, Marie-Francoise Roy},
journal={arXiv preprint arXiv:math/0603248},
year={2006},
archivePrefix={arXiv},
eprint={math/0603248},
primaryClass={math.AG cs.SC math.AT}
} | basu2006computing |
arxiv-676818 | math/0603256 | An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions | <|reference_start|>An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions: We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of $s$ polynomials in $\R[X_1,...,X_k]$ whose degrees are at most $d$ is bounded by \[ \frac{(2d)^k}{k!}s^k + O(s^{k-1}). \] This improves the best upper bound known previously which was \[ {1/2}\frac{(8d)^k}{k!}s^k + O(s^{k-1}). \] The new bound matches asymptotically the lower bound obtained for families of polynomials each of which is a product of generic polynomials of degree one.<|reference_end|> | arxiv | @article{basu2006an,
title={An asymptotically tight bound on the number of semi-algebraically
connected components of realizable sign conditions},
author={Saugata Basu, Richard Pollack, Marie-Francoise Roy},
journal={arXiv preprint arXiv:math/0603256},
year={2006},
archivePrefix={arXiv},
eprint={math/0603256},
primaryClass={math.CO cs.CG math.AG}
} | basu2006an |
arxiv-676819 | math/0603262 | Computing the Top Betti Numbers of Semi-algebraic Sets Defined by Quadratic Inequalities in Polynomial Time | <|reference_start|>Computing the Top Betti Numbers of Semi-algebraic Sets Defined by Quadratic Inequalities in Polynomial Time: For any $\ell > 0$, we present an algorithm which takes as input a semi-algebraic set, $S$, defined by $P_1 \leq 0,...,P_s \leq 0$, where each $P_i \in \R[X_1,...,X_k]$ has degree $\leq 2,$ and computes the top $\ell$ Betti numbers of $S$, $b_{k-1}(S), ..., b_{k-\ell}(S),$ in polynomial time. The complexity of the algorithm, stated more precisely, is $ \sum_{i=0}^{\ell+2} {s \choose i} k^{2^{O(\min(\ell,s))}}. $ For fixed $\ell$, the complexity of the algorithm can be expressed as $s^{\ell+2} k^{2^{O(\ell)}},$ which is polynomial in the input parameters $s$ and $k$. To our knowledge this is the first polynomial time algorithm for computing non-trivial topological invariants of semi-algebraic sets in $\R^k$ defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed $s$, we obtain by letting $\ell = k$, an algorithm for computing all the Betti numbers of $S$ whose complexity is $k^{2^{O(s)}}$.<|reference_end|> | arxiv | @article{basu2006computing,
title={Computing the Top Betti Numbers of Semi-algebraic Sets Defined by
Quadratic Inequalities in Polynomial Time},
author={Saugata Basu},
journal={arXiv preprint arXiv:math/0603262},
year={2006},
doi={10.1007/s10208-005-0208-8},
archivePrefix={arXiv},
eprint={math/0603262},
primaryClass={math.AG cs.CC math.LO}
} | basu2006computing |
arxiv-676820 | math/0603263 | Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time | <|reference_start|>Computing the First Few Betti Numbers of Semi-algebraic Sets in Single Exponential Time: In this paper we describe an algorithm that takes as input a description of a semi-algebraic set $S \subset \R^k$, defined by a Boolean formula with atoms of the form $P > 0, P < 0, P=0$ for $P \in {\mathcal P} \subset \R[X_1,...,X_k],$ and outputs the first $\ell+1$ Betti numbers of $S$, $b_0(S),...,b_\ell(S).$ The complexity of the algorithm is $(sd)^{k^{O(\ell)}},$ where where $s = #({\mathcal P})$ and $d = \max_{P\in {\mathcal P}}{\rm deg}(P),$ which is singly exponential in $k$ for $\ell$ any fixed constant. Previously, singly exponential time algorithms were known only for computing the Euler-Poincar\'e characteristic, the zero-th and the first Betti numbers.<|reference_end|> | arxiv | @article{basu2006computing,
title={Computing the First Few Betti Numbers of Semi-algebraic Sets in Single
Exponential Time},
author={Saugata Basu},
journal={arXiv preprint arXiv:math/0603263},
year={2006},
archivePrefix={arXiv},
eprint={math/0603263},
primaryClass={math.AG cs.SC}
} | basu2006computing |
arxiv-676821 | math/0603606 | Lanczos $\tau$-method optimal algorithm in APS for approximating the mathematical functions | <|reference_start|>Lanczos $\tau$-method optimal algorithm in APS for approximating the mathematical functions: A new procedure is constructed by means of APS in APLAN language. The procedure solves the initial-value problem for linear differential equations of order $k$ with polynomial coefficients and regular singularity in the initialization point in the interval $[a, b]$ and computes the algebraic polynomial $y_n$ of given order $n$. A new algorithm of Lanczos $\tau$-method is built for this procedure, the solution existence $y_n$ of the initial-value problem proved on this algorithm and also is proved the optimality by precision of order $k$ derivative of the initial-value problem solution.<|reference_end|> | arxiv | @article{denisenko2006lanczos,
title={Lanczos $\tau$-method optimal algorithm in APS for approximating the
mathematical functions},
author={P.N. Denisenko},
journal={arXiv preprint arXiv:math/0603606},
year={2006},
archivePrefix={arXiv},
eprint={math/0603606},
primaryClass={math.NA cs.MS math.CA}
} | denisenko2006lanczos |
arxiv-676822 | math/0603727 | Spectral Analysis of Pollard Rho Collisions | <|reference_start|>Spectral Analysis of Pollard Rho Collisions: We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho graph, and is close to the conjectured optimal bound of O(sqrt(n)). The result is derived by showing that the mixing time for the random walk on this graph is O((log n)^3); without the squaring step in the Pollard rho algorithm, the mixing time would be exponential in log n. The technique involves a spectral analysis of directed graphs, which captures the effect of the squaring step.<|reference_end|> | arxiv | @article{miller2006spectral,
title={Spectral Analysis of Pollard Rho Collisions},
author={Stephen D. Miller, Ramarathnam Venkatesan},
journal={arXiv preprint arXiv:math/0603727},
year={2006},
archivePrefix={arXiv},
eprint={math/0603727},
primaryClass={math.NT cs.CR cs.DM math.CO}
} | miller2006spectral |
arxiv-676823 | math/0604226 | A Dynamic View of Circular Colorings | <|reference_start|>A Dynamic View of Circular Colorings: The main contributions of this paper are three-fold. First, we use a dynamic approach based on Reiter's pioneering work on Karp-Miller computation graphs to give a new and short proof of Mohar's Minty-type Theorem. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni's results on circular chromatic number can be generalized to edge-weighted symmetric directed graphs. Third, we use the above-mentioned dynamic view of circular colorings to construct new improved lower bounds on the circular chromatic number of a graph. We show as an example that the circular chromatic number of the line graph of the Petersen graph can be determined very easily by using these bounds.<|reference_end|> | arxiv | @article{yeh2006a,
title={A Dynamic View of Circular Colorings},
author={Hong-Gwa Yeh},
journal={arXiv preprint arXiv:math/0604226},
year={2006},
archivePrefix={arXiv},
eprint={math/0604226},
primaryClass={math.CO cs.DC cs.DM}
} | yeh2006a |
arxiv-676824 | math/0604233 | Generalization error bounds in semi-supervised classification under the cluster assumption | <|reference_start|>Generalization error bounds in semi-supervised classification under the cluster assumption: We consider semi-supervised classification when part of the available data is unlabeled. These unlabeled data can be useful for the classification problem when we make an assumption relating the behavior of the regression function to that of the marginal distribution. Seeger (2000) proposed the well-known "cluster assumption" as a reasonable one. We propose a mathematical formulation of this assumption and a method based on density level sets estimation that takes advantage of it to achieve fast rates of convergence both in the number of unlabeled examples and the number of labeled examples.<|reference_end|> | arxiv | @article{rigollet2006generalization,
title={Generalization error bounds in semi-supervised classification under the
cluster assumption},
author={Philippe Rigollet (PMA)},
journal={arXiv preprint arXiv:math/0604233},
year={2006},
archivePrefix={arXiv},
eprint={math/0604233},
primaryClass={math.ST cs.LG stat.TH}
} | rigollet2006generalization |
arxiv-676825 | math/0604331 | On the reduction of a random basis | <|reference_start|>On the reduction of a random basis: For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra & Lov\'asz}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r\_j^{(n)}-1)\_j$ tends in distribution in some sense to an explicit process $({\mathcal R}\_j -1)\_j$; some properties of this latter are provided.<|reference_end|> | arxiv | @article{akhavi2006on,
title={On the reduction of a random basis},
author={Ali Akhavi (LIAFA), Jean-Franc{c}ois Marckert (LaBRI), Alain Rouault
(LM-Versailles)},
journal={arXiv preprint arXiv:math/0604331},
year={2006},
archivePrefix={arXiv},
eprint={math/0604331},
primaryClass={math.PR cs.DS}
} | akhavi2006on |
arxiv-676826 | math/0604366 | The Kesten-Stigum Reconstruction Bound Is Tight for Roughly Symmetric Binary Channels | <|reference_start|>The Kesten-Stigum Reconstruction Bound Is Tight for Roughly Symmetric Binary Channels: We establish the exact threshold for the reconstruction problem for a binary asymmetric channel on the b-ary tree, provided that the asymmetry is sufficiently small. This is the first exact reconstruction threshold obtained in roughly a decade. We discuss the implications of our result for Glauber dynamics, phylogenetic reconstruction, and so-called ``replica symmetry breaking'' in spin glasses and random satisfiability problems.<|reference_end|> | arxiv | @article{borgs2006the,
title={The Kesten-Stigum Reconstruction Bound Is Tight for Roughly Symmetric
Binary Channels},
author={Christian Borgs, Jennifer Chayes, Elchanan Mossel, Sebastien Roch},
journal={arXiv preprint arXiv:math/0604366},
year={2006},
archivePrefix={arXiv},
eprint={math/0604366},
primaryClass={math.PR cs.CC q-bio.PE}
} | borgs2006the |
arxiv-676827 | math/0604367 | Network Delay Inference from Additive Metrics | <|reference_start|>Network Delay Inference from Additive Metrics: We demonstrate the use of computational phylogenetic techniques to solve a central problem in inferential network monitoring. More precisely, we design a novel algorithm for multicast-based delay inference, i.e. the problem of reconstructing the topology and delay characteristics of a network from end-to-end delay measurements on network paths. Our inference algorithm is based on additive metric techniques widely used in phylogenetics. It runs in polynomial time and requires a sample of size only $\poly(\log n)$.<|reference_end|> | arxiv | @article{bhamidi2006network,
title={Network Delay Inference from Additive Metrics},
author={Shankar Bhamidi, Ram Rajagopal, Sebastien Roch},
journal={arXiv preprint arXiv:math/0604367},
year={2006},
archivePrefix={arXiv},
eprint={math/0604367},
primaryClass={math.PR cs.DS cs.NI math.ST stat.TH}
} | bhamidi2006network |
arxiv-676828 | math/0604371 | Constructing Non-Computable Julia Sets | <|reference_start|>Constructing Non-Computable Julia Sets: We completely characterize the conformal radii of Siegel disks in the family $$P_\theta(z)=e^{2\pi i\theta}z+z^2,$$ corresponding to {\bf computable} parameters $\theta$. As a consequence, we constructively produce quadratic polynomials with {\bf non-computable} Julia sets.<|reference_end|> | arxiv | @article{braverman2006constructing,
title={Constructing Non-Computable Julia Sets},
author={Mark Braverman, Michael Yampolsky},
journal={arXiv preprint arXiv:math/0604371},
year={2006},
archivePrefix={arXiv},
eprint={math/0604371},
primaryClass={math.DS cs.CC}
} | braverman2006constructing |
arxiv-676829 | math/0604584 | Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find | <|reference_start|>Enumeration of non-orientable 3-manifolds using face pairing graphs and union-find: Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P^2-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed non-orientable P^2-irreducible 3-manifolds that can be formed from at most ten tetrahedra.<|reference_end|> | arxiv | @article{burton2006enumeration,
title={Enumeration of non-orientable 3-manifolds using face pairing graphs and
union-find},
author={Benjamin A. Burton},
journal={Discrete and Computational Geometry 38 (2007), no. 3, 527-571},
year={2006},
doi={10.1007/s00454-007-1307-x},
archivePrefix={arXiv},
eprint={math/0604584},
primaryClass={math.GT cs.CG math.CO}
} | burton2006enumeration |
arxiv-676830 | math/0604611 | Variational optimization of probability measure spaces resolves the chain store paradox | <|reference_start|>Variational optimization of probability measure spaces resolves the chain store paradox: In game theory, players have continuous expected payoff functions and can use fixed point theorems to locate equilibria. This optimization method requires that players adopt a particular type of probability measure space. Here, we introduce alternate probability measure spaces altering the dimensionality, continuity, and differentiability properties of what are now the game's expected payoff functionals. Optimizing such functionals requires generalized variational and functional optimization methods to locate novel equilibria. These variational methods can reconcile game theoretic prediction and observed human behaviours, as we illustrate by resolving the chain store paradox. Our generalized optimization analysis has significant implications for economics, artificial intelligence, complex system theory, neurobiology, and biological evolution and development.<|reference_end|> | arxiv | @article{gagen2006variational,
title={Variational optimization of probability measure spaces resolves the
chain store paradox},
author={Michael J. Gagen and Kae Nemoto},
journal={arXiv preprint arXiv:math/0604611},
year={2006},
archivePrefix={arXiv},
eprint={math/0604611},
primaryClass={math.OC cond-mat.stat-mech cs.GT}
} | gagen2006variational |
arxiv-676831 | math/0605232 | Borne sur le degr\'e des polyn\^omes presque parfaitement non-lin\'eaires | <|reference_start|>Borne sur le degr\'e des polyn\^omes presque parfaitement non-lin\'eaires: The vectorial Boolean functions are employed in cryptography to build block coding algorithms. An important criterion on these functions is their resistance to the differential cryptanalysis. Nyberg defined the notion of almost perfect non-linearity (APN) to study resistance to the differential attacks. Up to now, the study of functions APN was especially devoted to power functions. Recently, Budaghyan and al. showed that certain quadratic polynomials were APN. Here, we will give a criterion so that a function is not almost perfectly non-linear. H. Janwa showed, by using Weil's bound, that certain cyclic codes could not correct two errors. A. Canteaut showed by using the same method that the functions powers were not APN for a too large value of the exponent. We use Lang and Weil's bound and a result of P. Deligne on the Weil's conjectures (or more exactly improvements given by Ghorpade and Lachaud) about surfaces on finite fields to generalize this result to all the polynomials. We show therefore that a polynomial cannot be APN if its degree is too large.<|reference_end|> | arxiv | @article{rodier2006borne,
title={Borne sur le degr\'e des polyn\^omes presque parfaitement
non-lin\'eaires},
author={Franc{c}ois Rodier (IML)},
journal={arXiv preprint arXiv:math/0605232},
year={2006},
archivePrefix={arXiv},
eprint={math/0605232},
primaryClass={math.AG cs.CR}
} | rodier2006borne |
arxiv-676832 | math/0605242 | N-Fold Integer Programming | <|reference_start|>N-Fold Integer Programming: In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed {\em n-fold integer programming problems} are polynomial time solvable. Our proof involves two heavy ingredients discovered recently: the equivalence of linear optimization and so-called directed augmentation, and the stabilization of certain Graver bases. We discuss several applications of our algorithm to multiway transportation problems and to packing problems. One important consequence of our results is a polynomial time algorithm for the $d$-dimensional integer transportation problem for long multiway tables. Another interesting application is a new algorithm for the classical cutting stock problem.<|reference_end|> | arxiv | @article{de loera2006n-fold,
title={N-Fold Integer Programming},
author={Jes'us A. De Loera, Raymond Hemmecke, Shmuel Onn, Robert Weismantel},
journal={Discrete Optimization, 5:231--241, 2008},
year={2006},
archivePrefix={arXiv},
eprint={math/0605242},
primaryClass={math.OC cs.CC cs.DM math.CO}
} | de loera2006n-fold |
arxiv-676833 | math/0605334 | Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations | <|reference_start|>Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations: In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gr\"obner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gr\"obner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.<|reference_end|> | arxiv | @article{gerdt2006gr\"obner,
title={Gr\"obner Bases and Generation of Difference Schemes for Partial
Differential Equations},
author={Vladimir P. Gerdt, Yuri A. Blinkov and Vladimir V. Mozzhilkin},
journal={SIGMA 2 (2006), 051, 26 pages},
year={2006},
doi={10.3842/SIGMA.2006.051},
archivePrefix={arXiv},
eprint={math/0605334},
primaryClass={math.RA cs.NA cs.SC math.NA}
} | gerdt2006gr\"obner |
arxiv-676834 | math/0605472 | An algebraic approach to Polya processes | <|reference_start|>An algebraic approach to Polya processes: P\'olya processes are natural generalization of P\'olya-Eggenberger urn models. This article presents a new approach of their asymptotic behaviour {\it via} moments, based on the spectral decomposition of a suitable finite difference operator on polynomial functions. Especially, it provides new results for {\it large} processes (a P\'olya process is called {\it small} when 1 is simple eigenvalue of its replacement matrix and when any other eigenvalue has a real part $\leq 1/2$; otherwise, it is called large).<|reference_end|> | arxiv | @article{pouyanne2006an,
title={An algebraic approach to Polya processes},
author={Nicolas Pouyanne (LM-Versailles)},
journal={Annales de l'IHP - Probabilit\'es et Statistiques (2008) Vol. 44,
No. 2, 293-323},
year={2006},
doi={10.1214/07-AIHP130},
archivePrefix={arXiv},
eprint={math/0605472},
primaryClass={math.CO cs.DM cs.DS math.PR}
} | pouyanne2006an |
arxiv-676835 | math/0605498 | Cross-Entropic Learning of a Machine for the Decision in a Partially Observable Universe | <|reference_start|>Cross-Entropic Learning of a Machine for the Decision in a Partially Observable Universe: Revision of the paper previously entitled "Learning a Machine for the Decision in a Partially Observable Markov Universe" In this paper, we are interested in optimal decisions in a partially observable universe. Our approach is to directly approximate an optimal strategic tree depending on the observation. This approximation is made by means of a parameterized probabilistic law. A particular family of hidden Markov models, with input \emph{and} output, is considered as a model of policy. A method for optimizing the parameters of these HMMs is proposed and applied. This optimization is based on the cross-entropic principle for rare events simulation developed by Rubinstein.<|reference_end|> | arxiv | @article{dambreville2006cross-entropic,
title={Cross-Entropic Learning of a Machine for the Decision in a Partially
Observable Universe},
author={Frederic Dambreville (DGA/CTA/DT/GIP)},
journal={arXiv preprint arXiv:math/0605498},
year={2006},
archivePrefix={arXiv},
eprint={math/0605498},
primaryClass={math.OC cs.AI cs.LG cs.NE cs.RO math.ST stat.TH}
} | dambreville2006cross-entropic |
arxiv-676836 | math/0605610 | Nonlinear Bipartite Matching | <|reference_start|>Nonlinear Bipartite Matching: We study the problem of optimizing nonlinear objective functions over bipartite matchings. While the problem is generally intractable, we provide several efficient algorithms for it, including a deterministic algorithm for maximizing convex objectives, approximative algorithms for norm minimization and maximization, and a randomized algorithm for optimizing arbitrary objectives.<|reference_end|> | arxiv | @article{berstein2006nonlinear,
title={Nonlinear Bipartite Matching},
author={Yael Berstein and Shmuel Onn},
journal={Discrete Optimization, 5:53--65, 2008},
year={2006},
archivePrefix={arXiv},
eprint={math/0605610},
primaryClass={math.OC cs.CC cs.DM math.CO}
} | berstein2006nonlinear |
arxiv-676837 | math/0605740 | Sharp thresholds for high-dimensional and noisy recovery of sparsity | <|reference_start|>Sharp thresholds for high-dimensional and noisy recovery of sparsity: The problem of consistently estimating the sparsity pattern of a vector $\betastar \in \real^\mdim$ based on observations contaminated by noise arises in various contexts, including subset selection in regression, structure estimation in graphical models, sparse approximation, and signal denoising. We analyze the behavior of $\ell_1$-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish a sharp relation between the problem dimension $\mdim$, the number $\spindex$ of non-zero elements in $\betastar$, and the number of observations $\numobs$ that are required for reliable recovery. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we establish existence and compute explicit values of thresholds $\ThreshLow$ and $\ThreshUp$ with the following properties: for any $\epsilon > 0$, if $\numobs > 2 (\ThreshUp + \epsilon) \log (\mdim - \spindex) + \spindex + 1$, then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for $\numobs < 2 (\ThreshLow - \epsilon) \log (\mdim - \spindex) + \spindex + 1$, then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble, we show that $\ThreshLow = \ThreshUp = 1$, so that the threshold is sharp and exactly determined.<|reference_end|> | arxiv | @article{wainwright2006sharp,
title={Sharp thresholds for high-dimensional and noisy recovery of sparsity},
author={Martin J. Wainwright},
journal={arXiv preprint arXiv:math/0605740},
year={2006},
archivePrefix={arXiv},
eprint={math/0605740},
primaryClass={math.ST cs.IT math.IT stat.TH}
} | wainwright2006sharp |
arxiv-676838 | math/0606022 | Imprimitive permutations groups generated by the round functions of key-alternating block ciphers and truncated differential cryptanalysis | <|reference_start|>Imprimitive permutations groups generated by the round functions of key-alternating block ciphers and truncated differential cryptanalysis: We answer a question of Paterson, showing that all block systems for the group generated by the round functions of a key-alternating block cipher are the translates of a linear subspace. Following up remarks of Paterson and Shamir, we exhibit a connection to truncated differential cryptanalysis. We also give a condition that guarantees that the group generated by the round functions of a key-alternating block cipher is primitive. This applies in particular to AES.<|reference_end|> | arxiv | @article{caranti2006imprimitive,
title={Imprimitive permutations groups generated by the round functions of
key-alternating block ciphers and truncated differential cryptanalysis},
author={A. Caranti, F. Dalla Volta, M. Sala, F. Villani},
journal={arXiv preprint arXiv:math/0606022},
year={2006},
archivePrefix={arXiv},
eprint={math/0606022},
primaryClass={math.GR cs.CR}
} | caranti2006imprimitive |
arxiv-676839 | math/0606122 | Diagonal Peg Solitaire | <|reference_start|>Diagonal Peg Solitaire: We study the classical game of peg solitaire when diagonal jumps are allowed. We prove that on many boards, one can begin from a full board with one peg missing, and finish with one peg anywhere on the board. We then consider the problem of finding solutions that minimize the number of moves (where a move is one or more jumps by the same peg), and find the shortest solution to the "central game", which begins and ends at the center. In some cases we can prove analytically that our solutions are the shortest possible, in other cases we apply A* or bidirectional search heuristics.<|reference_end|> | arxiv | @article{bell2006diagonal,
title={Diagonal Peg Solitaire},
author={George I. Bell},
journal={INTEGERS: Electronic Journal of Combinatorial Number Theory 7
(2007) #G01},
year={2006},
archivePrefix={arXiv},
eprint={math/0606122},
primaryClass={math.CO cs.DM cs.DS}
} | bell2006diagonal |
arxiv-676840 | math/0606315 | Bayesian Regression of Piecewise Constant Functions | <|reference_start|>Bayesian Regression of Piecewise Constant Functions: We derive an exact and efficient Bayesian regression algorithm for piecewise constant functions of unknown segment number, boundary location, and levels. It works for any noise and segment level prior, e.g. Cauchy which can handle outliers. We derive simple but good estimates for the in-segment variance. We also propose a Bayesian regression curve as a better way of smoothing data without blurring boundaries. The Bayesian approach also allows straightforward determination of the evidence, break probabilities and error estimates, useful for model selection and significance and robustness studies. We discuss the performance on synthetic and real-world examples. Many possible extensions will be discussed.<|reference_end|> | arxiv | @article{hutter2006bayesian,
title={Bayesian Regression of Piecewise Constant Functions},
author={Marcus Hutter},
journal={arXiv preprint arXiv:math/0606315},
year={2006},
number={IDSIA-14-05},
archivePrefix={arXiv},
eprint={math/0606315},
primaryClass={math.ST cs.LG math.PR stat.TH}
} | hutter2006bayesian |
arxiv-676841 | math/0606643 | Entropy And Vision | <|reference_start|>Entropy And Vision: In vector quantization the number of vectors used to construct the codebook is always an undefined problem, there is always a compromise between the number of vectors and the quantity of information lost during the compression. In this text we present a minimum of Entropy principle that gives solution to this compromise and represents an Entropy point of view of signal compression in general. Also we present a new adaptive Object Quantization technique that is the same for the compression and the perception.<|reference_end|> | arxiv | @article{kanhouche2006entropy,
title={Entropy And Vision},
author={Rami Kanhouche (CMLA)},
journal={arXiv preprint arXiv:math/0606643},
year={2006},
archivePrefix={arXiv},
eprint={math/0606643},
primaryClass={math.PR cs.CV cs.DB cs.DM cs.LG math.CO}
} | kanhouche2006entropy |
arxiv-676842 | math/0606734 | Codes in spherical caps | <|reference_start|>Codes in spherical caps: We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps. Several new upper bounds on codes in caps are derived. Applications of these bounds to estimates of the kissing numbers and one-sided kissing numbers are considered. It is proved that the maximum size of codes in spherical caps for large dimensions is determined by the maximum size of spherical codes, so these problems are asymptotically equivalent.<|reference_end|> | arxiv | @article{barg2006codes,
title={Codes in spherical caps},
author={Alexander Barg and Oleg R. Musin},
journal={Advances in Mathematics of Communications, vol. 1, no. 1, 2007,
131-149},
year={2006},
archivePrefix={arXiv},
eprint={math/0606734},
primaryClass={math.MG cs.IT math.IT}
} | barg2006codes |
arxiv-676843 | math/0606771 | Hard Instances of the Constrained Discrete Logarithm Problem | <|reference_start|>Hard Instances of the Constrained Discrete Logarithm Problem: The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds.<|reference_end|> | arxiv | @article{mironov2006hard,
title={Hard Instances of the Constrained Discrete Logarithm Problem},
author={Ilya Mironov, Anton Mityagin, Kobbi Nissim},
journal={In proceedings of 7th Algorithmic Number Theory Symposium (ANTS
VII), pages 582--598, 2006},
year={2006},
doi={10.1007/11792086_41},
archivePrefix={arXiv},
eprint={math/0606771},
primaryClass={math.NT cs.CR}
} | mironov2006hard |
arxiv-676844 | math/0607051 | Discrete differential geometry of triangle tiles and algebra of closed trajectories | <|reference_start|>Discrete differential geometry of triangle tiles and algebra of closed trajectories: This paper proposes a new mathematical framework that can be applied to biological problems such as analysis of the structures of proteins and protein complexes. In particular, it gives a new method for encoding the three-dimensional structure of a protein into a binary sequence, where proteins are approximated by a folded tetrahedron sequence. It also gives a new algebraic framework for describing molecular complexes and their interactions. For simplicity, we shall explain the framework in the case of two-dimensional objects. Then, the binary code of a plane curve is obtained as the ``second derivative'' of the curve and ``fusion and fission'' of closed trajectories is described algebraically.<|reference_end|> | arxiv | @article{morikawa2006discrete,
title={Discrete differential geometry of triangle tiles and algebra of closed
trajectories},
author={Naoto Morikawa},
journal={arXiv preprint arXiv:math/0607051},
year={2006},
archivePrefix={arXiv},
eprint={math/0607051},
primaryClass={math.CO cs.DM math.MG}
} | morikawa2006discrete |
arxiv-676845 | math/0607071 | NP-completeness of 4-incidence colorability of semi-cubic graphs | <|reference_start|>NP-completeness of 4-incidence colorability of semi-cubic graphs: The incidence coloring conjecture, proposed by Brualdi and Massey in 1993, states that the incidence coloring number of every graph is at most ${\it \Delta}+2$, where ${\it \Delta}$ is the maximum degree of a graph. The conjecture was shown to be false in general by Guiduli in 1997, following the work of Algor and Alon. However, in 2005 Maydanskiy proved that the conjecture holds for any graph with ${\it \Delta}\leq 3$. It is easily deduced that the incidence coloring number of a semi-cubic graph is 4 or 5. In this paper, we show that it is already NP-complete to determine if a semi-cubic graph is 4-incidence colorable, and therefore it is NP-complete to determine if a general graph is $k$-incidence colorable.<|reference_end|> | arxiv | @article{li2006np-completeness,
title={NP-completeness of 4-incidence colorability of semi-cubic graphs},
author={Xueliang Li, Jianhua Tu},
journal={arXiv preprint arXiv:math/0607071},
year={2006},
archivePrefix={arXiv},
eprint={math/0607071},
primaryClass={math.CO cs.CC}
} | li2006np-completeness |
arxiv-676846 | math/0607088 | Odd minimum cut sets and b-matchings revisited | <|reference_start|>Odd minimum cut sets and b-matchings revisited: The famous Padberg-Rao separation algorithm for b-matching polyhedra can be implemented to run in O(n^2m log(n^2/m)) time in the uncapacitated case, and in O(nm^2 log(n^2/m)) time in the capacitated case (where n is the number of vertices and m is the number of edges of the underlying graph). We give a new and simple algorithm for the capacitated case which can be implemented to run in O(n^2m log(n^2/m)) time.<|reference_end|> | arxiv | @article{letchford2006odd,
title={Odd minimum cut sets and b-matchings revisited},
author={Adam N. Letchford and Dirk Oliver Theis},
journal={arXiv preprint arXiv:math/0607088},
year={2006},
archivePrefix={arXiv},
eprint={math/0607088},
primaryClass={math.OC cs.DM}
} | letchford2006odd |
arxiv-676847 | math/0607243 | An active curve approach for tomographic reconstruction of binary radially symmetric objects | <|reference_start|>An active curve approach for tomographic reconstruction of binary radially symmetric objects: This paper deals with a method of tomographic reconstruction of radially symmetric objects from a single radiograph, in order to study the behavior of shocked material. The usual tomographic reconstruction algorithms such as generalized inverse or filtered back-projection cannot be applied here because data are very noisy and the inverse problem associated to single view tomographic reconstruction is highly unstable. In order to improve the reconstruction, we propose here to add some a priori assumptions on the looked after object. One of these assumptions is that the object is binary and consequently, the object may be described by the curves that separate the two materials. We present a model that lives in BV space and leads to a non local Hamilton-Jacobi equation, via a level set strategy. Numerical experiments are performed (using level sets methods) on synthetic objects.<|reference_end|> | arxiv | @article{abraham2006an,
title={An active curve approach for tomographic reconstruction of binary
radially symmetric objects},
author={Isabelle Abraham (DCRE), Romain Abraham (MAPMO), Maitine Bergounioux
(MAPMO)},
journal={arXiv preprint arXiv:math/0607243},
year={2006},
archivePrefix={arXiv},
eprint={math/0607243},
primaryClass={math.OC cs.CV}
} | abraham2006an |
arxiv-676848 | math/0607368 | The Newton Polytope of the Implicit Equation | <|reference_start|>The Newton Polytope of the Implicit Equation: We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.<|reference_end|> | arxiv | @article{sturmfels2006the,
title={The Newton Polytope of the Implicit Equation},
author={Bernd Sturmfels, Jenia Tevelev, Josephine Yu},
journal={Moscow Mathematical Journal 7 (2007), no. 2, 327--346, 351},
year={2006},
archivePrefix={arXiv},
eprint={math/0607368},
primaryClass={math.CO cs.SC math.AG}
} | sturmfels2006the |
arxiv-676849 | math/0607411 | Extending the scalars of minimizations | <|reference_start|>Extending the scalars of minimizations: In the classical theory of formal languages, finite state automata allow to recognize the words of a rational subset of $\Sigma^*$ where $\Sigma$ is a set of symbols (or the alphabet). Now, given a semiring $(\K,+,.)$, one can construct $\K$-subsets of $\Sigma^*$ in the sense of Eilenberg, that are alternatively called noncommutative formal power series for which a framework very similar to language theory has been constructed Particular noncommutative formal power series, which are called rational series, are the behaviour of a family of weighted automata (or $\K$-automata). In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states. Minimization processes of $\K$-automata already exist for $\K$ being: {\bf a)} a field, {\bf b)} a noncommutative field, {\bf c)} a PID . When $\K$ is the bolean semiring, such a minimization process (with isomorphisms of minimal objects) is known within the category of deterministic automata. Minimal automata have been proved to be isomorphic in cases {\bf (a)} and {\bf (b)}. But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of $\K^n$. Here we give an independent algorithm which reproves this fact and an example of a pair of nonisomorphic minimal automata. Moreover, we examine the possibility of extending {\bf (c)}. To this end, we provide an {\em Effective Minimization Process} (or {\em EMP}) which can be used for more general sets of coefficients.<|reference_end|> | arxiv | @article{duchamp2006extending,
title={Extending the scalars of minimizations},
author={G'erard Duchamp (LIPN), Eric Laugerotte (LIFAR EA2655), Jean-Gabriel
Luque (IGM-LabInfo)},
journal={SCI, \'{E}tats-Unis d'Am\'{e}rique (2001)},
year={2006},
archivePrefix={arXiv},
eprint={math/0607411},
primaryClass={math.CO cs.DS cs.SC}
} | duchamp2006extending |
arxiv-676850 | math/0607412 | Direct and dual laws for automata with multiplicities | <|reference_start|>Direct and dual laws for automata with multiplicities: We present here theoretical results coming from the implementation of the package called AMULT (automata with multiplicities in several noncommutative variables). We show that classical formulas are ``almost every time'' optimal, characterize the dual laws preserving rationality and also relators that are compatible with these laws.<|reference_end|> | arxiv | @article{duchamp2006direct,
title={Direct and dual laws for automata with multiplicities},
author={G'erard Duchamp (LIPN), Marianne Flouret (LIH EA3219), Eric
Laugerotte (LIFAR EA2655), Jean-Gabriel Luque (IGM-LabInfo)},
journal={Theoretical Computer Science 267 (2001) 105-120},
year={2006},
archivePrefix={arXiv},
eprint={math/0607412},
primaryClass={math.CO cs.DM cs.SC}
} | duchamp2006direct |
arxiv-676851 | math/0607420 | Transitive factorizations of free partially commutative monoids and Lie algebras | <|reference_start|>Transitive factorizations of free partially commutative monoids and Lie algebras: Let $\M(A,\theta)$ be a free partially commutative monoid. We give here a necessary and sufficient condition on a subalphabet $B\subset A$ such that the right factor of a bisection $\M(A,\theta)=\M(B,\theta\_B).T$ be also partially commutative free. This extends strictly the (classical) elimination theory on partial commutations and allows to construct new factorizations of $\M(A,\theta)$ and associated bases of $L\_K(A,\theta)$.<|reference_end|> | arxiv | @article{luque2006transitive,
title={Transitive factorizations of free partially commutative monoids and Lie
algebras},
author={Jean-Gabriel Luque (IGM-LabInfo), G'erard Henry Edmond Duchamp (LIPN)},
journal={Discrete Mathematics 246, Issue 1-3 (2002) 83 - 97},
year={2006},
archivePrefix={arXiv},
eprint={math/0607420},
primaryClass={math.CO cs.DM cs.SC math.GM}
} | luque2006transitive |
arxiv-676852 | math/0607462 | De l'oprateur de trace dans les jeux de Conway | <|reference_start|>De l'oprateur de trace dans les jeux de Conway: In this report, we propose a game semantics model of intuitionistic linear logic with a notion of brackets and a trace operator. This model is a revised version of Conway games augmented with an algebraicly defined gain which enable to describe well bracketed strategies. We then show the existence of a free cocommutative comonoid in the category of Conway. To conclude, we propose a new model of an Algol-like language with higher-order using the presence of a trace operator in our model to describe the memorial aspect of the language.<|reference_end|> | arxiv | @article{tabareau2006de,
title={De l'oprateur de trace dans les jeux de Conway},
author={Nicolas Tabareau (PPS)},
journal={arXiv preprint arXiv:math/0607462},
year={2006},
archivePrefix={arXiv},
eprint={math/0607462},
primaryClass={math.CT cs.PL}
} | tabareau2006de |
arxiv-676853 | math/0607507 | In-Degree and PageRank of Web pages: Why do they follow similar power laws? | <|reference_start|>In-Degree and PageRank of Web pages: Why do they follow similar power laws?: The PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that the PageRank obeys a `power law' with the same exponent as the In-Degree. This paper presents a novel mathematical model that explains this phenomenon. The relation between the PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of the PageRank, and is analogous to the well-known distributional identity for the busy period in the M/G/1 queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail behavior of the PageRank and the In-Degree differ only by a multiplicative factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data.<|reference_end|> | arxiv | @article{litvak2006in-degree,
title={In-Degree and PageRank of Web pages: Why do they follow similar power
laws?},
author={N. Litvak, W.R.W. Scheinhardt and Y. Volkovich},
journal={arXiv preprint arXiv:math/0607507},
year={2006},
number={Memorandum 1807, Dept. of Applied. Mathematics, University of Twente},
archivePrefix={arXiv},
eprint={math/0607507},
primaryClass={math.PR cs.IR}
} | litvak2006in-degree |
arxiv-676854 | math/0607597 | A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies | <|reference_start|>A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies: We present an accurate Lagrangian method based on vortex particles, level-sets, and immersed boundary methods, for animating the interplay between two fluids and rigid solids. We show that a vortex method is a good choice for simulating bi-phase flow, such as liquid and gas, with a good level of realism. Vortex particles are localized at the interfaces between the two fluids and within the regions of high turbulence. We gain local precision and efficiency from the stable advection permitted by the vorticity formulation. Moreover, our numerical method straightforwardly solves the two-way coupling problem between the fluids and animated rigid solids. This new approach is validated through numerical comparisons with reference experiments from the computational fluid community. We also show that the visually appealing results obtained in the CG community can be reproduced with increased efficiency and an easier implementation.<|reference_end|> | arxiv | @article{coquerelle2006a,
title={A Vortex Method for Bi-phasic Fluids Interacting with Rigid Bodies},
author={Mathieu Coquerelle (LMC - Imag, Gravir - Imag), J'er'emie Allard
(GRAVIR - Imag), Georges-Henri Cottet (LMC - Imag), Marie-Paule Cani (GRAVIR
- Imag)},
journal={arXiv preprint arXiv:math/0607597},
year={2006},
archivePrefix={arXiv},
eprint={math/0607597},
primaryClass={math.NA cs.GR}
} | coquerelle2006a |
arxiv-676855 | math/0607648 | Singular Values and Eigenvalues of Tensors: A Variational Approach | <|reference_start|>Singular Values and Eigenvalues of Tensors: A Variational Approach: We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem.<|reference_end|> | arxiv | @article{lim2006singular,
title={Singular Values and Eigenvalues of Tensors: A Variational Approach},
author={Lek-Heng Lim},
journal={Proceedings of the IEEE International Workshop on Computational
Advances in Multi-Sensor Adaptive Processing (CAMSAP '05), Vol. 1 (2005), pp.
129--132},
year={2006},
number={SCCM Technical Report 05-10},
archivePrefix={arXiv},
eprint={math/0607648},
primaryClass={math.SP cs.IR cs.NA math.NA math.OC}
} | lim2006singular |
arxiv-676856 | math/0608116 | Conflict Free Rule for Combining Evidences | <|reference_start|>Conflict Free Rule for Combining Evidences: Recent works have investigated the problem of the conflict redistribution in the fusion rules of evidence theories. As a consequence of these works, many new rules have been proposed. Now, there is not a clear theoretical criterion for a choice of a rule instead another. The present chapter proposes a new theoretically grounded rule, based on a new concept of sensor independence. This new rule avoids the conflict redistribution, by an adaptive combination of the beliefs. Both the logical grounds and the algorithmic implementation are considered.<|reference_end|> | arxiv | @article{dambreville2006conflict,
title={Conflict Free Rule for Combining Evidences},
author={Frederic Dambreville (DGA/CEP/GIP/SRO)},
journal={arXiv preprint arXiv:math/0608116},
year={2006},
archivePrefix={arXiv},
eprint={math/0608116},
primaryClass={math.LO cs.LO math.ST stat.TH}
} | dambreville2006conflict |
arxiv-676857 | math/0608210 | Recognising the Suzuki groups in their natural representations | <|reference_start|>Recognising the Suzuki groups in their natural representations: Under the assumption of a certain conjecture, for which there exists strong experimental evidence, we produce an efficient algorithm for constructive membership testing in the Suzuki groups Sz(q), where q = 2^{2m + 1} for some m > 0, in their natural representations of degree 4. It is a Las Vegas algorithm with running time O{log(q)} field operations, and a preprocessing step with running time O{log(q) loglog(q)} field operations. The latter step needs an oracle for the discrete logarithm problem in GF(q). We also produce a recognition algorithm for Sz(q) = <X>. This is a Las Vegas algorithm with running time O{|X|^2} field operations. Finally, we give a Las Vegas algorithm that, given <X>^h = Sz(q) for some h in GL(4, q), finds some g such that <X>^g = Sz(q). The running time is O{log(q) loglog(q) + |X|} field operations. Implementations of the algorithms are available for the computer system MAGMA.<|reference_end|> | arxiv | @article{bäärnhielm2006recognising,
title={Recognising the Suzuki groups in their natural representations},
author={Henrik B"a"arnhielm},
journal={J. Algebra 300 (1), 171-198, 2006},
year={2006},
doi={10.1016/j.jalgebra.2006.02.010},
archivePrefix={arXiv},
eprint={math/0608210},
primaryClass={math.GR cs.DS}
} | bäärnhielm2006recognising |
arxiv-676858 | math/0608522 | Graph Laplacians and their convergence on random neighborhood graphs | <|reference_start|>Graph Laplacians and their convergence on random neighborhood graphs: Given a sample from a probability measure with support on a submanifold in Euclidean space one can construct a neighborhood graph which can be seen as an approximation of the submanifold. The graph Laplacian of such a graph is used in several machine learning methods like semi-supervised learning, dimensionality reduction and clustering. In this paper we determine the pointwise limit of three different graph Laplacians used in the literature as the sample size increases and the neighborhood size approaches zero. We show that for a uniform measure on the submanifold all graph Laplacians have the same limit up to constants. However in the case of a non-uniform measure on the submanifold only the so called random walk graph Laplacian converges to the weighted Laplace-Beltrami operator.<|reference_end|> | arxiv | @article{hein2006graph,
title={Graph Laplacians and their convergence on random neighborhood graphs},
author={Matthias Hein, Jean-Yves Audibert and Ulrike von Luxburg},
journal={arXiv preprint arXiv:math/0608522},
year={2006},
archivePrefix={arXiv},
eprint={math/0608522},
primaryClass={math.ST cs.LG stat.TH}
} | hein2006graph |
arxiv-676859 | math/0608556 | On optimal quantization rules for some problems in sequential decentralized detection | <|reference_start|>On optimal quantization rules for some problems in sequential decentralized detection: We consider the design of systems for sequential decentralized detection, a problem that entails several interdependent choices: the choice of a stopping rule (specifying the sample size), a global decision function (a choice between two competing hypotheses), and a set of quantization rules (the local decisions on the basis of which the global decision is made). This paper addresses an open problem of whether in the Bayesian formulation of sequential decentralized detection, optimal local decision functions can be found within the class of stationary rules. We develop an asymptotic approximation to the optimal cost of stationary quantization rules and exploit this approximation to show that stationary quantizers are not optimal in a broad class of settings. We also consider the class of blockwise stationary quantizers, and show that asymptotically optimal quantizers are likelihood-based threshold rules.<|reference_end|> | arxiv | @article{nguyen2006on,
title={On optimal quantization rules for some problems in sequential
decentralized detection},
author={XuanLong Nguyen, Martin J. Wainwright, Michael I. Jordan},
journal={arXiv preprint arXiv:math/0608556},
year={2006},
doi={10.1109/TIT.2008.924647},
archivePrefix={arXiv},
eprint={math/0608556},
primaryClass={math.ST cs.IT math.IT stat.TH}
} | nguyen2006on |
arxiv-676860 | math/0608571 | Intensional Models for the Theory of Types | <|reference_start|>Intensional Models for the Theory of Types: In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up.<|reference_end|> | arxiv | @article{muskens2006intensional,
title={Intensional Models for the Theory of Types},
author={Reinhard Muskens},
journal={arXiv preprint arXiv:math/0608571},
year={2006},
archivePrefix={arXiv},
eprint={math/0608571},
primaryClass={math.LO cs.AI}
} | muskens2006intensional |
arxiv-676861 | math/0608603 | Sequences with constant number of return words | <|reference_start|>Sequences with constant number of return words: An infinite word has the property $R_m$ if every factor has exactly $m$ return words. Vuillon showed that $R_2$ characterizes Sturmian words. We prove that a word satisfies $R_m$ if its complexity function is $(m-1)n+1$ and if it contains no weak bispecial factor. These conditions are necessary for $m=3$, whereas for $m=4$ the complexity function need not be $3n+1$. New examples of words satisfying $R_m$ are given by words related to digital expansions in real bases.<|reference_end|> | arxiv | @article{balkova2006sequences,
title={Sequences with constant number of return words},
author={Lubomira Balkova, Edita Pelantova, Wolfgang Steiner (LIAFA)},
journal={arXiv preprint arXiv:math/0608603},
year={2006},
archivePrefix={arXiv},
eprint={math/0608603},
primaryClass={math.CO cs.DM}
} | balkova2006sequences |
arxiv-676862 | math/0608713 | Occam's hammer: a link between randomized learning and multiple testing FDR control | <|reference_start|>Occam's hammer: a link between randomized learning and multiple testing FDR control: We establish a generic theoretical tool to construct probabilistic bounds for algorithms where the output is a subset of objects from an initial pool of candidates (or more generally, a probability distribution on said pool). This general device, dubbed "Occam's hammer'', acts as a meta layer when a probabilistic bound is already known on the objects of the pool taken individually, and aims at controlling the proportion of the objects in the set output not satisfying their individual bound. In this regard, it can be seen as a non-trivial generalization of the "union bound with a prior'' ("Occam's razor''), a familiar tool in learning theory. We give applications of this principle to randomized classifiers (providing an interesting alternative approach to PAC-Bayes bounds) and multiple testing (where it allows to retrieve exactly and extend the so-called Benjamini-Yekutieli testing procedure).<|reference_end|> | arxiv | @article{blanchard2006occam's,
title={Occam's hammer: a link between randomized learning and multiple testing
FDR control},
author={Gilles Blanchard (FHG FIRST.IDA), Franc{c}ois Fleuret (EPFL -- CVLAB)},
journal={arXiv preprint arXiv:math/0608713},
year={2006},
archivePrefix={arXiv},
eprint={math/0608713},
primaryClass={math.ST cs.LG stat.TH}
} | blanchard2006occam's |
arxiv-676863 | math/0608733 | Context for models of concurrency | <|reference_start|>Context for models of concurrency: Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computer-scientific information. That is, one wants to replace a given model with a simpler model with the same directed homotopy-type. Unfortunately, the obvious definition of directed homotopy equivalence is too coarse. This paper introduces the notion of context to refine this definition.<|reference_end|> | arxiv | @article{bubenik2006context,
title={Context for models of concurrency},
author={Peter Bubenik},
journal={Electron. Notes Theor. Comput. Sci. 230 (2009) 3-21},
year={2006},
doi={10.1016/j.entcs.2009.02.014},
archivePrefix={arXiv},
eprint={math/0608733},
primaryClass={math.AT cs.DC}
} | bubenik2006context |
arxiv-676864 | math/0608789 | One method for proving inequalities by computer | <|reference_start|>One method for proving inequalities by computer: In this article we consider a method for proving a class of analytical inequalities via minimax rational approximations. All numerical calculations in this paper are given by Maple computer program.<|reference_end|> | arxiv | @article{malesevic2006one,
title={One method for proving inequalities by computer},
author={Branko J. Malesevic},
journal={arXiv preprint arXiv:math/0608789},
year={2006},
archivePrefix={arXiv},
eprint={math/0608789},
primaryClass={math.CA cs.GR cs.MS cs.NA math.GM math.NA}
} | malesevic2006one |
arxiv-676865 | math/0609360 | Minimal Polynomials for the Coordinates of the Harborth Graph | <|reference_start|>Minimal Polynomials for the Coordinates of the Harborth Graph: The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furthermore some algebraic properties of these polynomials, and consequences to the structure of the Harborth graph are determined.<|reference_end|> | arxiv | @article{gerbracht2006minimal,
title={Minimal Polynomials for the Coordinates of the Harborth Graph},
author={Eberhard H.-A. Gerbracht},
journal={arXiv preprint arXiv:math/0609360},
year={2006},
archivePrefix={arXiv},
eprint={math/0609360},
primaryClass={math.CO cs.SC}
} | gerbracht2006minimal |
arxiv-676866 | math/0609461 | Cross-Entropy method: convergence issues for extended implementation | <|reference_start|>Cross-Entropy method: convergence issues for extended implementation: The cross-entropy method (CE) developed by R. Rubinstein is an elegant practical principle for simulating rare events. The method approximates the probability of the rare event by means of a family of probabilistic models. The method has been extended to optimization, by considering an optimal event as a rare event. CE works rather good when dealing with deterministic function optimization. Now, it appears that two conditions are needed for a good convergence of the method. First, it is necessary to have a family of models sufficiently flexible for discriminating the optimal events. Indirectly, it appears also that the function to be optimized should be deterministic. The purpose of this paper is to consider the case of partially discriminating model family, and of stochastic functions. It will be shown on simple examples that the CE could fail when relaxing these hypotheses. Alternative improvements of the CE method are investigated and compared on random examples in order to handle this issue.<|reference_end|> | arxiv | @article{dambreville2006cross-entropy,
title={Cross-Entropy method: convergence issues for extended implementation},
author={Frederic Dambreville (DGA/CTA/DT/GIP)},
journal={arXiv preprint arXiv:math/0609461},
year={2006},
archivePrefix={arXiv},
eprint={math/0609461},
primaryClass={math.OC cs.LG cs.NE math.ST stat.TH}
} | dambreville2006cross-entropy |
arxiv-676867 | math/0609562 | On quadratic residue codes and hyperelliptic curves | <|reference_start|>On quadratic residue codes and hyperelliptic curves: A long standing problem has been to develop "good" binary linear codes to be used for error-correction. This paper investigates in some detail an attack on this problem using a connection between quadratic residue codes and hyperelliptic curves. One question which coding theory is used to attack is: Does there exist a c<2 such that, for all sufficiently large $p$ and all subsets S of GF(p), we have |X_S(GF(p))| < cp?<|reference_end|> | arxiv | @article{joyner2006on,
title={On quadratic residue codes and hyperelliptic curves},
author={David Joyner},
journal={arXiv preprint arXiv:math/0609562},
year={2006},
archivePrefix={arXiv},
eprint={math/0609562},
primaryClass={math.CO cs.IT math.AG math.IT math.NT}
} | joyner2006on |
arxiv-676868 | math/0610067 | On the context-freeness of the set of words containing overlaps | <|reference_start|>On the context-freeness of the set of words containing overlaps: We show that the set of binary words containing overlaps is not unambiguously context-free and that the set of ternary words containing overlaps is not context-free. We also show that the set of binary words that are not subwords of the Thue-Morse word is not unambiguously context-free.<|reference_end|> | arxiv | @article{rampersad2006on,
title={On the context-freeness of the set of words containing overlaps},
author={Narad Rampersad},
journal={arXiv preprint arXiv:math/0610067},
year={2006},
archivePrefix={arXiv},
eprint={math/0610067},
primaryClass={math.CO cs.FL}
} | rampersad2006on |
arxiv-676869 | math/0610121 | Fast Jacobian group operations for C_3,4 curves over a large finite field | <|reference_start|>Fast Jacobian group operations for C_3,4 curves over a large finite field: Let C be an arbitrary smooth algebraic curve of genus g over a large finite field K. We revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi (math.NT/0409209, to appear in Math. Comp.). The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| >> g, and which asymptotically require O(g^{2.376}) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, we provide explicit formulae for performing the group law on Jacobians of C_{3,4} curves of genus 3. We show that, typically, the addition of two distinct elements in the Jacobian of a C_{3,4} curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.<|reference_end|> | arxiv | @article{salem2006fast,
title={Fast Jacobian group operations for C_{3,4} curves over a large finite
field},
author={Fatima K. Abu Salem (Computer Science Department, American University
of Beirut) and Kamal Khuri-Makdisi (Center for Advanced Mathematical
Sciences, American University of Beirut)},
journal={LMS J. Comput. Math. 10 (2007) 307-328, may be downloaded from
http://www.lms.ac.uk/jcm/10/lms2006-049/},
year={2006},
archivePrefix={arXiv},
eprint={math/0610121},
primaryClass={math.NT cs.SC math.AG}
} | salem2006fast |
arxiv-676870 | math/0610184 | Adaptive Poisson disorder problem | <|reference_start|>Adaptive Poisson disorder problem: We study the quickest detection problem of a sudden change in the arrival rate of a Poisson process from a known value to an unknown and unobservable value at an unknown and unobservable disorder time. Our objective is to design an alarm time which is adapted to the history of the arrival process and detects the disorder time as soon as possible. In previous solvable versions of the Poisson disorder problem, the arrival rate after the disorder has been assumed a known constant. In reality, however, we may at most have some prior information about the likely values of the new arrival rate before the disorder actually happens, and insufficient estimates of the new rate after the disorder happens. Consequently, we assume in this paper that the new arrival rate after the disorder is a random variable. The detection problem is shown to admit a finite-dimensional Markovian sufficient statistic, if the new rate has a discrete distribution with finitely many atoms. Furthermore, the detection problem is cast as a discounted optimal stopping problem with running cost for a finite-dimensional piecewise-deterministic Markov process. This optimal stopping problem is studied in detail in the special case where the new arrival rate has Bernoulli distribution. This is a nontrivial optimal stopping problem for a two-dimensional piecewise-deterministic Markov process driven by the same point process. Using a suitable single-jump operator, we solve it fully, describe the analytic properties of the value function and the stopping region, and present methods for their numerical calculation. We provide a concrete example where the value function does not satisfy the smooth-fit principle on a proper subset of the connected, continuously differentiable optimal stopping boundary, whereas it does on the complement of this set.<|reference_end|> | arxiv | @article{bayraktar2006adaptive,
title={Adaptive Poisson disorder problem},
author={Erhan Bayraktar, Savas Dayanik, Ioannis Karatzas},
journal={Annals of Applied Probability 2006, Vol. 16, No. 3, 1190-1261},
year={2006},
doi={10.1214/105051606000000312},
number={IMS-AAP-AAP0171},
archivePrefix={arXiv},
eprint={math/0610184},
primaryClass={math.PR cs.IT math.IT math.ST stat.TH}
} | bayraktar2006adaptive |
arxiv-676871 | math/0610680 | Gaussian limits for multidimensional random sequential packing at saturation (extended version) | <|reference_start|>Gaussian limits for multidimensional random sequential packing at saturation (extended version): Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume $\lambda$ is asymptotically normal as $\lambda \to \infty$. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.<|reference_end|> | arxiv | @article{schreiber2006gaussian,
title={Gaussian limits for multidimensional random sequential packing at
saturation (extended version)},
author={T. Schreiber, Mathew D. Penrose and J. E. Yukich},
journal={arXiv preprint arXiv:math/0610680},
year={2006},
doi={10.1007/s00220-007-0218-2},
archivePrefix={arXiv},
eprint={math/0610680},
primaryClass={math.PR cs.OH}
} | schreiber2006gaussian |
arxiv-676872 | math/0611194 | Directed animals in the gas | <|reference_start|>Directed animals in the gas: In this paper, we revisit the enumeration of directed animals using gas models. We show that there exists a natural construction of random directed animals on any directed graph together with a particle system that explains at the level of objects the formal link known between the density of the gas model and the generating function of directed animals counted according to the area. This provides some new methods to compute the generating function of directed animals counted according to area, and leads in the particular case of the square lattice to new combinatorial results and questions. A model of gas related to directed animals counted according to area and perimeter on any directed graph is also exhibited.<|reference_end|> | arxiv | @article{borgne2006directed,
title={Directed animals in the gas},
author={Yvan Le Borgne (LaBRI), Jean-Franc{c}ois Marckert (LaBRI)},
journal={arXiv preprint arXiv:math/0611194},
year={2006},
archivePrefix={arXiv},
eprint={math/0611194},
primaryClass={math.PR cs.DM}
} | borgne2006directed |
arxiv-676873 | math/0611422 | Cartes auto-organis\'ees pour l'analyse exploratoire de donn\'ees et la visualisation | <|reference_start|>Cartes auto-organis\'ees pour l'analyse exploratoire de donn\'ees et la visualisation: This paper shows how to use the Kohonen algorithm to represent multidimensional data, by exploiting the self-organizing property. It is possible to get such maps as well for quantitative variables as for qualitative ones, or for a mixing of both. The contents of the paper come from various works by SAMOS-MATISSE members, in particular by E. de Bodt, B. Girard, P. Letr\'{e}my, S. Ibbou, P. Rousset. Most of the examples have been studied with the computation routines written by Patrick Letr\'{e}my, with the language IML-SAS, which are available on the WEB page http://samos.univ-paris1.fr.<|reference_end|> | arxiv | @article{cottrell2006cartes,
title={Cartes auto-organis\'{e}es pour l'analyse exploratoire de donn\'{e}es et
la visualisation},
author={Marie Cottrell (MATISSE, Samos), Sma"Il Ibbou (MATISSE, Samos),
Patrick Letr'emy (MATISSE, Samos), Patrick Rousset (CEREQ)},
journal={Journal de la Soci\'{e}t\'{e} Fran\c{c}aise de Statistique 144
n°4 (2003) 67-106},
year={2006},
archivePrefix={arXiv},
eprint={math/0611422},
primaryClass={math.ST cs.NE nlin.AO stat.TH}
} | cottrell2006cartes |
arxiv-676874 | math/0611433 | Working times in atypical forms of employment: the special case of part-time work | <|reference_start|>Working times in atypical forms of employment: the special case of part-time work: In the present article, we attempt to devise a typology of forms of part-time employment by applying a widely used neuronal methodology called Kohonen maps. Starting out with data that we describe using category-specific variables, we show how it is possible to represent observations and the modalities of the variables that define them simultaneously, on a single map. This allows us to ascertain, and to try to describe, the main categories of part-time employment.<|reference_end|> | arxiv | @article{letrémy2006working,
title={Working times in atypical forms of employment: the special case of
part-time work},
author={Patrick Letr'emy (SAMOS), Marie Cottrell (SAMOS)},
journal={Connectionist Approaches in Economics and Management Sciences
Kluwer (Ed.) (2003) 111-129},
year={2006},
archivePrefix={arXiv},
eprint={math/0611433},
primaryClass={math.ST cs.NE stat.TH}
} | letrémy2006working |
arxiv-676875 | math/0611666 | Anomalous heat-kernel decay for random walk among bounded random conductances | <|reference_start|>Anomalous heat-kernel decay for random walk among bounded random conductances: We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$, driven by a field of bounded random conductances $\omega_{xy}\in[0,1]$. The conductance law is i.i.d. subject to the condition that the probability of $\omega_{xy}>0$ exceeds the threshold for bond percolation on $\Z^d$. For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the $2n$-step return probability $P_\omega^{2n}(0,0)$. We prove that $P_\omega^{2n}(0,0)$ is bounded by a random constant times $n^{-d/2}$ in $d=2,3$, while it is $o(n^{-2})$ in $d\ge5$ and $O(n^{-2}\log n)$ in $d=4$. By producing examples with anomalous heat-kernel decay approaching $1/n^2$ we prove that the $o(n^{-2})$ bound in $d\ge5$ is the best possible. We also construct natural $n$-dependent environments that exhibit the extra $\log n$ factor in $d=4$. See also math.PR/0701248.<|reference_end|> | arxiv | @article{berger2006anomalous,
title={Anomalous heat-kernel decay for random walk among bounded random
conductances},
author={Noam Berger, Marek Biskup, Christopher E. Hoffman, Gady Kozma},
journal={Ann. Inst. H. Poincare Probab. Statist. 274 (2008), no. 2, 374-392},
year={2006},
doi={10.1214/07-AIHP126},
archivePrefix={arXiv},
eprint={math/0611666},
primaryClass={math.PR cs.DM math-ph math.MP}
} | berger2006anomalous |
arxiv-676876 | math/0611679 | Longest Common Pattern between two Permutations | <|reference_start|>Longest Common Pattern between two Permutations: In this paper, we give a polynomial (O(n^8)) algorithm for finding a longest common pattern between two permutations of size n given that one is separable. We also give an algorithm for general permutations whose complexity depends on the length of the longest simple permutation involved in one of our permutations.<|reference_end|> | arxiv | @article{rossin2006longest,
title={Longest Common Pattern between two Permutations},
author={Dominique Rossin (LIAFA), Mathilde Bouvel (LIAFA)},
journal={Algebr. Geom. Topol. 7 (2007) 829-843},
year={2006},
doi={10.2140/agt.2007.7.829},
archivePrefix={arXiv},
eprint={math/0611679},
primaryClass={math.CO cs.DM cs.DS}
} | rossin2006longest |
arxiv-676877 | math/0611918 | Solving random equations in Garside groups using length functions | <|reference_start|>Solving random equations in Garside groups using length functions: We give a systematic exposition of memory-length algorithms for solving equations in noncommutative groups. This exposition clarifies some points untouched in earlier expositions. We then focus on the main ingredient in these attacks: Length functions. After a self-contained introduction to Garside groups, we describe length functions induced by the greedy normal form and by the rational normal form in these groups, and compare their worst-case performances. Our main concern is Artin's braid groups, with their two known Garside presentations, due to Artin and due to Birman-Ko-Lee (BKL). We show that in $B_3$ equipped with the BKL presentation, the (efficiently computable) rational normal form of each element is a geodesic, i.e., is a representative of minimal length for that element. (For Artin's presentation of $B_3$, Berger supplied in 1994 a method to obtain geodesic representatives in $B_3$.) For an arbitrary number of strands, finding the geodesic length of an element is NP-hard, by a 1991 result of by Paterson and Razborov. We show that a good estimation of the geodesic length of a braid in Artin's presentation is measuring the length of its rational form in the \emph{BKL} presentation. This is proved theoretically for the worst case, and experimental evidence is provided for the generic case.<|reference_end|> | arxiv | @article{hock2006solving,
title={Solving random equations in Garside groups using length functions},
author={Martin Hock and Boaz Tsaban},
journal={arXiv preprint arXiv:math/0611918},
year={2006},
archivePrefix={arXiv},
eprint={math/0611918},
primaryClass={math.GR cs.CR math.AG}
} | hock2006solving |
arxiv-676878 | math/0611937 | Remarks on Inheritance Systems | <|reference_start|>Remarks on Inheritance Systems: We try a conceptual analysis of inheritance diagrams, first in abstract terms, and then compare to "normality" and the "small/big sets" of preferential and related reasoning. The main ideas are about nodes as truth values and information sources, truth comparison by paths, accessibility or relevance of information by paths, relative normality, and prototypical reasoning.<|reference_end|> | arxiv | @article{schlechta2006remarks,
title={Remarks on Inheritance Systems},
author={Karl Schlechta (LIF)},
journal={arXiv preprint arXiv:math/0611937},
year={2006},
archivePrefix={arXiv},
eprint={math/0611937},
primaryClass={math.LO cs.AI}
} | schlechta2006remarks |
arxiv-676879 | math/0612046 | On the Submodularity of Influence in Social Networks | <|reference_start|>On the Submodularity of Influence in Social Networks: We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the spread of influence in social networks. A social network can be represented by a directed graph where the nodes are individuals and the edges indicate a form of social relationship. A simple way to model the diffusion of ideas, innovative behavior, or ``word-of-mouth'' effects on such a graph is to consider an increasing process of ``infected'' (or active) nodes: each node becomes infected once an activation function of the set of its infected neighbors crosses a certain threshold value. Such a model was introduced by KKT in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several natural assumptions: the threshold values are (uniformly) random; and the activation functions are monotone and submodular. For an initial set of active nodes $S$, let $\sigma(S)$ denote the expected number of active nodes at termination. Here we prove a conjecture of KKT: we show that the function $\sigma(S)$ is submodular under the assumptions above. We prove the same result for the expected value of any monotone, submodular function of the set of active nodes at termination.<|reference_end|> | arxiv | @article{mossel2006on,
title={On the Submodularity of Influence in Social Networks},
author={Elchanan Mossel, Sebastien Roch},
journal={arXiv preprint arXiv:math/0612046},
year={2006},
archivePrefix={arXiv},
eprint={math/0612046},
primaryClass={math.PR cs.GT cs.SI}
} | mossel2006on |
arxiv-676880 | math/0612083 | Termination orders for 3-dimensional rewriting | <|reference_start|>Termination orders for 3-dimensional rewriting: This paper studies 3-polygraphs as a framework for rewriting on two-dimensional words. A translation of term rewriting systems into 3-polygraphs with explicit resource management is given, and the respective computational properties of each system are studied. Finally, a convergent 3-polygraph for the (commutative) theory of Z/2Z-vector spaces is given. In order to prove these results, it is explained how to craft a class of termination orders for 3-polygraphs.<|reference_end|> | arxiv | @article{guiraud2006termination,
title={Termination orders for 3-dimensional rewriting},
author={Yves Guiraud},
journal={Journal of Pure and Applied Algebra, Volume 207, Issue 2, October
2006, Pages 341-371},
year={2006},
doi={10.1016/j.jpaa.2005.10.011},
archivePrefix={arXiv},
eprint={math/0612083},
primaryClass={math.CT cs.LO}
} | guiraud2006termination |
arxiv-676881 | math/0612084 | Termination orders for 3-polygraphs | <|reference_start|>Termination orders for 3-polygraphs: This note presents the first known class of termination orders for 3-polygraphs, together with an application.<|reference_end|> | arxiv | @article{guiraud2006termination,
title={Termination orders for 3-polygraphs},
author={Yves Guiraud},
journal={Comptes-Rendus de l'Academie des Sciences Serie I, Volume 342,
Issue 4, 15 February 2006, Pages 219-222},
year={2006},
doi={10.1016/j.crma.2005.12.019},
archivePrefix={arXiv},
eprint={math/0612084},
primaryClass={math.CT cs.LO}
} | guiraud2006termination |
arxiv-676882 | math/0612088 | Two polygraphic presentations of Petri nets | <|reference_start|>Two polygraphic presentations of Petri nets: This document gives an algebraic and two polygraphic translations of Petri nets, all three providing an easier way to describe reductions and to identify some of them. The first one sees places as generators of a commutative monoid and transitions as rewriting rules on it: this setting is totally equivalent to Petri nets, but lacks any graphical intuition. The second one considers places as 1-dimensional cells and transitions as 2-dimensional ones: this translation recovers a graphical meaning but raises many difficulties since it uses explicit permutations. Finally, the third translation sees places as degenerated 2-dimensional cells and transitions as 3-dimensional ones: this is a setting equivalent to Petri nets, equipped with a graphical interpretation.<|reference_end|> | arxiv | @article{guiraud2006two,
title={Two polygraphic presentations of Petri nets},
author={Yves Guiraud},
journal={Theoretical Computer Science, Volume 360, Issues 1-3, 21 August
2006, Pages 124-146},
year={2006},
doi={10.1016/j.tcs.2006.02.015},
archivePrefix={arXiv},
eprint={math/0612088},
primaryClass={math.CT cs.LO}
} | guiraud2006two |
arxiv-676883 | math/0612089 | The three dimensions of proofs | <|reference_start|>The three dimensions of proofs: In this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting.<|reference_end|> | arxiv | @article{guiraud2006the,
title={The three dimensions of proofs},
author={Yves Guiraud},
journal={Annals of Pure and Applied Logic, Volume 141, Issues 1-2, August
2006, Pages 266-295},
year={2006},
doi={10.1016/j.apal.2005.12.012},
archivePrefix={arXiv},
eprint={math/0612089},
primaryClass={math.CT cs.LO math.LO}
} | guiraud2006the |
arxiv-676884 | math/0612264 | Fast linear algebra is stable | <|reference_start|>Fast linear algebra is stable: In an earlier paper, we showed that a large class of fast recursive matrix multiplication algorithms is stable in a normwise sense, and that in fact if multiplication of $n$-by-$n$ matrices can be done by any algorithm in $O(n^{\omega + \eta})$ operations for any $\eta > 0$, then it can be done stably in $O(n^{\omega + \eta})$ operations for any $\eta > 0$. Here we extend this result to show that essentially all standard linear algebra operations, including LU decomposition, QR decomposition, linear equation solving, matrix inversion, solving least squares problems, (generalized) eigenvalue problems and the singular value decomposition can also be done stably (in a normwise sense) in $O(n^{\omega + \eta})$ operations.<|reference_end|> | arxiv | @article{demmel2006fast,
title={Fast linear algebra is stable},
author={James Demmel, Ioana Dumitriu, Olga Holtz},
journal={Numer. Math. 108 (2007), no. 1, 59-91},
year={2006},
doi={10.1007/s00211-007-0114-x},
archivePrefix={arXiv},
eprint={math/0612264},
primaryClass={math.NA cs.CC cs.DS}
} | demmel2006fast |
arxiv-676885 | math/0612612 | The minimum size required of a solitaire army | <|reference_start|>The minimum size required of a solitaire army: The solitaire army is a one-person peg jumping game where a player attempts to advance an "army" of pegs as far as possible into empty territory. The game was introduced by John Conway and is also known as "Conway's Soldiers". We consider various generalizations of this game in different 2D geometries, unify them under a common mathematical framework, and find the minimum size army capable of advancing a given number of steps.<|reference_end|> | arxiv | @article{bell2006the,
title={The minimum size required of a solitaire army},
author={George I. Bell, Daniel S. Hirschberg, Pablo Guerrero-Garcia},
journal={INTEGERS: Electronic Journal of Combinatorial Number Theory 7
(2007) #G07},
year={2006},
archivePrefix={arXiv},
eprint={math/0612612},
primaryClass={math.CO cs.DM}
} | bell2006the |
arxiv-676886 | math/0612682 | Convergence Speed in Distributed Consensus and Control | <|reference_start|>Convergence Speed in Distributed Consensus and Control: We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm.<|reference_end|> | arxiv | @article{olshevsky2006convergence,
title={Convergence Speed in Distributed Consensus and Control},
author={Alex Olshevsky, John N. Tsitsiklis},
journal={arXiv preprint arXiv:math/0612682},
year={2006},
archivePrefix={arXiv},
eprint={math/0612682},
primaryClass={math.OC cs.SY}
} | olshevsky2006convergence |
arxiv-676887 | math/0701020 | Some notes on a method for proving inequalities by computer | <|reference_start|>Some notes on a method for proving inequalities by computer: In this article we consider mathematical fundamentals of one method for proving inequalities by computer, based on the Remez algorithm. Using the well-known results of undecidability of the existence of zeros of real elementary functions, we demonstrate that the considered method generally in practice becomes one heuristic for the verification of inequalities. We give some improvements of the inequalities considered in the theorems for which the existing proofs have been based on the numerical verifications of Remez algorithm.<|reference_end|> | arxiv | @article{banjac2006some,
title={Some notes on a method for proving inequalities by computer},
author={Bojan D. Banjac, Milica D. Makragic, Branko J. Malesevic},
journal={arXiv preprint arXiv:math/0701020},
year={2006},
doi={10.1007/s00025-015-0485-8},
archivePrefix={arXiv},
eprint={math/0701020},
primaryClass={math.CA cs.MS math.NA}
} | banjac2006some |
arxiv-676888 | math/0701102 | Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits | <|reference_start|>Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits: It is well known that R^N has subspaces of dimension proportional to N on which the \ell_1 norm is equivalent to the \ell_2 norm; however, no explicit constructions are known. Extending earlier work by Artstein--Avidan and Milman, we prove that such a subspace can be generated using O(N) random bits.<|reference_end|> | arxiv | @article{lovett2007almost,
title={Almost Euclidean sections of the N-dimensional cross-polytope using O(N)
random bits},
author={Shachar Lovett and Sasha Sodin},
journal={Commun. Contemp. Math. 10 (2008), no. 4, 477--489.},
year={2007},
doi={10.1142/S0219199708002879},
archivePrefix={arXiv},
eprint={math/0701102},
primaryClass={math.FA cs.CC math.MG}
} | lovett2007almost |
arxiv-676889 | math/0701131 | Compressed Sensing and Redundant Dictionaries | <|reference_start|>Compressed Sensing and Redundant Dictionaries: This article extends the concept of compressed sensing to signals that are not sparse in an orthonormal basis but rather in a redundant dictionary. It is shown that a matrix, which is a composition of a random matrix of certain type and a deterministic dictionary, has small restricted isometry constants. Thus, signals that are sparse with respect to the dictionary can be recovered via Basis Pursuit from a small number of random measurements. Further, thresholding is investigated as recovery algorithm for compressed sensing and conditions are provided that guarantee reconstruction with high probability. The different schemes are compared by numerical experiments.<|reference_end|> | arxiv | @article{rauhut2007compressed,
title={Compressed Sensing and Redundant Dictionaries},
author={Holger Rauhut, Karin Schnass, Pierre Vandergheynst},
journal={IEEE Trans. Inform. Theory, 54(5):2210-2219, 2008},
year={2007},
doi={10.1109/TIT.2008.920190},
archivePrefix={arXiv},
eprint={math/0701131},
primaryClass={math.PR cs.IT math.IT}
} | rauhut2007compressed |
arxiv-676890 | math/0701142 | On the use of self-organizing maps to accelerate vector quantization | <|reference_start|>On the use of self-organizing maps to accelerate vector quantization: Self-organizing maps (SOM) are widely used for their topology preservation property: neighboring input vectors are quantified (or classified) either on the same location or on neighbor ones on a predefined grid. SOM are also widely used for their more classical vector quantization property. We show in this paper that using SOM instead of the more classical Simple Competitive Learning (SCL) algorithm drastically increases the speed of convergence of the vector quantization process. This fact is demonstrated through extensive simulations on artificial and real examples, with specific SOM (fixed and decreasing neighborhoods) and SCL algorithms.<|reference_end|> | arxiv | @article{de bodt2007on,
title={On the use of self-organizing maps to accelerate vector quantization},
author={Eric De Bodt (ESA, Iag-Fin, Dice), Marie Cottrell (MATISSE, Samos),
Patrick Letr'emy (MATISSE, Samos), Michel Verleysen (DICE)},
journal={Neurocomputing 56 (2004) 187-203},
year={2007},
doi={10.1016/j.neucom.2003.09.009},
archivePrefix={arXiv},
eprint={math/0701142},
primaryClass={math.ST cs.NE stat.TH}
} | de bodt2007on |
arxiv-676891 | math/0701144 | Statistical tools to assess the reliability of self-organizing maps | <|reference_start|>Statistical tools to assess the reliability of self-organizing maps: Results of neural network learning are always subject to some variability, due to the sensitivity to initial conditions, to convergence to local minima, and, sometimes more dramatically, to sampling variability. This paper presents a set of tools designed to assess the reliability of the results of Self-Organizing Maps (SOM), i.e. to test on a statistical basis the confidence we can have on the result of a specific SOM. The tools concern the quantization error in a SOM, and the neighborhood relations (both at the level of a specific pair of observations and globally on the map). As a by-product, these measures also allow to assess the adequacy of the number of units chosen in a map. The tools may also be used to measure objectively how the SOM are less sensitive to non-linear optimization problems (local minima, convergence, etc.) than other neural network models.<|reference_end|> | arxiv | @article{de bodt2007statistical,
title={Statistical tools to assess the reliability of self-organizing maps},
author={Eric De Bodt (ESA, Iag-Fin), Marie Cottrell (SAMOS, Matisse), Michel
Verleysen (DICE)},
journal={Neural Networks 15, 8-9 (2002) 967-978},
year={2007},
archivePrefix={arXiv},
eprint={math/0701144},
primaryClass={math.ST cs.NE stat.TH}
} | de bodt2007statistical |
arxiv-676892 | math/0701145 | Bootstrap for neural model selection | <|reference_start|>Bootstrap for neural model selection: Bootstrap techniques (also called resampling computation techniques) have introduced new advances in modeling and model evaluation. Using resampling methods to construct a series of new samples which are based on the original data set, allows to estimate the stability of the parameters. Properties such as convergence and asymptotic normality can be checked for any particular observed data set. In most cases, the statistics computed on the generated data sets give a good idea of the confidence regions of the estimates. In this paper, we debate on the contribution of such methods for model selection, in the case of feedforward neural networks. The method is described and compared with the leave-one-out resampling method. The effectiveness of the bootstrap method, versus the leave-one-out methode, is checked through a number of examples.<|reference_end|> | arxiv | @article{kallel2007bootstrap,
title={Bootstrap for neural model selection},
author={Riadh Kallel (MATISSE, Samos), Marie Cottrell (MATISSE, Samos),
Vincent Vigneron (MATISSE, Samos)},
journal={Neurocomputing 48 (2002) 175-183},
year={2007},
doi={10.1016/S0925-2312(01)00650-6},
archivePrefix={arXiv},
eprint={math/0701145},
primaryClass={math.ST cs.NE stat.TH}
} | kallel2007bootstrap |
arxiv-676893 | math/0701152 | Missing values : processing with the Kohonen algorithm | <|reference_start|>Missing values : processing with the Kohonen algorithm: The processing of data which contain missing values is a complicated and always awkward problem, when the data come from real-world contexts. In applications, we are very often in front of observations for which all the values are not available, and this can occur for many reasons: typing errors, fields left unanswered in surveys, etc. Most of the statistical software (as SAS for example) simply suppresses incomplete observations. It has no practical consequence when the data are very numerous. But if the number of remaining data is too small, it can remove all significance to the results. To avoid suppressing data in that way, it is possible to replace a missing value with the mean value of the corresponding variable, but this approximation can be very bad when the variable has a large variance. So it is very worthwhile seeing that the Kohonen algorithm (as well as the Forgy algorithm) perfectly deals with data with missing values, without having to estimate them beforehand. We are particularly interested in the Kohonen algorithm for its visualization properties.<|reference_end|> | arxiv | @article{cottrell2007missing,
title={Missing values : processing with the Kohonen algorithm},
author={Marie Cottrell (MATISSE, Samos), Patrick Letr'emy (MATISSE, Samos)},
journal={Dans ASMDA 2005 CD-ROM Proceedings - ASMDA 2005, Brest : France
(2005)},
year={2007},
archivePrefix={arXiv},
eprint={math/0701152},
primaryClass={math.ST cs.NE stat.TH}
} | cottrell2007missing |
arxiv-676894 | math/0701261 | Tracking Stopping Times Through Noisy Observations | <|reference_start|>Tracking Stopping Times Through Noisy Observations: A novel quickest detection setting is proposed which is a generalization of the well-known Bayesian change-point detection model. Suppose \{(X_i,Y_i)\}_{i\geq 1} is a sequence of pairs of random variables, and that S is a stopping time with respect to \{X_i\}_{i\geq 1}. The problem is to find a stopping time T with respect to \{Y_i\}_{i\geq 1} that optimally tracks S, in the sense that T minimizes the expected reaction delay E(T-S)^+, while keeping the false-alarm probability P(T<S) below a given threshold \alpha \in [0,1]. This problem formulation applies in several areas, such as in communication, detection, forecasting, and quality control. Our results relate to the situation where the X_i's and Y_i's take values in finite alphabets and where S is bounded by some positive integer \kappa. By using elementary methods based on the analysis of the tree structure of stopping times, we exhibit an algorithm that computes the optimal average reaction delays for all \alpha \in [0,1], and constructs the associated optimal stopping times T. Under certain conditions on \{(X_i,Y_i)\}_{i\geq 1} and S, the algorithm running time is polynomial in \kappa.<|reference_end|> | arxiv | @article{niesen2007tracking,
title={Tracking Stopping Times Through Noisy Observations},
author={Urs Niesen, Aslan Tchamkerten},
journal={IEEE Transactions on Information Theory, vol. 55, pp. 422-432,
January 2009},
year={2007},
doi={10.1109/TIT.2008.2008115},
archivePrefix={arXiv},
eprint={math/0701261},
primaryClass={math.ST cs.IT math.IT stat.TH}
} | niesen2007tracking |
arxiv-676895 | math/0701419 | Strategies for prediction under imperfect monitoring | <|reference_start|>Strategies for prediction under imperfect monitoring: We propose simple randomized strategies for sequential prediction under imperfect monitoring, that is, when the forecaster does not have access to the past outcomes but rather to a feedback signal. The proposed strategies are consistent in the sense that they achieve, asymptotically, the best possible average reward. It was Rustichini (1999) who first proved the existence of such consistent predictors. The forecasters presented here offer the first constructive proof of consistency. Moreover, the proposed algorithms are computationally efficient. We also establish upper bounds for the rates of convergence. In the case of deterministic feedback, these rates are optimal up to logarithmic terms.<|reference_end|> | arxiv | @article{lugosi2007strategies,
title={Strategies for prediction under imperfect monitoring},
author={Gabor Lugosi, Shie Mannor, Gilles Stoltz (DMA)},
journal={Mathematics of Operations Research (2008) \`a para\^itre},
year={2007},
archivePrefix={arXiv},
eprint={math/0701419},
primaryClass={math.ST cs.LG stat.TH}
} | lugosi2007strategies |
arxiv-676896 | math/0701647 | Counting non-isomorphic maximal independent sets of the n-cycle graph | <|reference_start|>Counting non-isomorphic maximal independent sets of the n-cycle graph: The number of maximal independent sets of the n-cycle graph C_n is known to be the nth term of the Perrin sequence. The action of the automorphism group of C_n on the family of these maximal independent sets partitions this family into disjoint orbits, which represent the non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets. We provide exact formulas for the total number of orbits and the number of orbits having a given number of isomorphic representatives. We also provide exact formulas for the total number of unlabeled (i.e., defined up to a rotation) maximal independent sets and the number of unlabeled maximal independent sets having a given number of isomorphic representatives. It turns out that these formulas involve both Perrin and Padovan sequences.<|reference_end|> | arxiv | @article{bisdorff2007counting,
title={Counting non-isomorphic maximal independent sets of the n-cycle graph},
author={Raymond Bisdorff, Jean-Luc Marichal},
journal={Journal of Integer Sequences 11 (5) (2008), Article 08.5.7},
year={2007},
archivePrefix={arXiv},
eprint={math/0701647},
primaryClass={math.CO cs.DM math.GR}
} | bisdorff2007counting |
arxiv-676897 | math/0701791 | Linear versus Non-linear Acquisition of Step-Functions | <|reference_start|>Linear versus Non-linear Acquisition of Step-Functions: We address in this paper the following two closely related problems: 1. How to represent functions with singularities (up to a prescribed accuracy) in a compact way? 2. How to reconstruct such functions from a small number of measurements? The stress is on a comparison of linear and non-linear approaches. As a model case we use piecewise-constant functions on [0,1], in particular, the Heaviside jump function. Considered as a curve in the Hilbert space, it is completely characterized by the fact that any two its disjoint chords are orthogonal. We reinterpret this fact in a context of step-functions in one or two variables. Next we study the limitations on representability and reconstruction of piecewise-constant functions by linear and semi-linear methods. Our main tools in this problem are Kolmogorov's n-width and entropy, as well as Temlyakov's (N,m)-width. On the positive side, we show that a very accurate non-linear reconstruction is possible. It goes through a solution of certain specific non-linear systems of algebraic equations. We discuss the form of these systems and methods of their solution, stressing their relation to Moment Theory and Complex Analysis. Finally, we informally discuss two problems in Computer Imaging which are parallel to the problems 1 and 2 above: compression of still images and video-sequences on one side, and image reconstruction from indirect measurement (for example, in Computer Tomography), on the other.<|reference_end|> | arxiv | @article{ettinger2007linear,
title={Linear versus Non-linear Acquisition of Step-Functions},
author={Boris Ettinger, Niv Sarig, Yosef Yomdin},
journal={arXiv preprint arXiv:math/0701791},
year={2007},
archivePrefix={arXiv},
eprint={math/0701791},
primaryClass={math.CA cs.CV}
} | ettinger2007linear |
arxiv-676898 | math/0701801 | Deterministic modal Bayesian Logic: derive the Bayesian inference within the modal logic T | <|reference_start|>Deterministic modal Bayesian Logic: derive the Bayesian inference within the modal logic T: In this paper a conditional logic is defined and studied. This conditional logic, DmBL, is constructed as a deterministic counterpart to the Bayesian conditional. The logic is unrestricted, so that any logical operations are allowed. A notion of logical independence is also defined within the logic itself. This logic is shown to be non-trivial and is not reduced to classical propositions. A model is constructed for the logic. Completeness results are proved. It is shown that any unconditioned probability can be extended to the whole logic DmBL. The Bayesian conditional is then recovered from the probabilistic DmBL. At last, it is shown why DmBL is compliant with Lewis' triviality.<|reference_end|> | arxiv | @article{dambreville2007deterministic,
title={Deterministic modal Bayesian Logic: derive the Bayesian inference within
the modal logic T},
author={Frederic Dambreville (DGA/CTA/DT/GIP)},
journal={arXiv preprint arXiv:math/0701801},
year={2007},
archivePrefix={arXiv},
eprint={math/0701801},
primaryClass={math.LO cs.LO math.PR}
} | dambreville2007deterministic |
arxiv-676899 | math/0701904 | Noncomputable Spectral Sets | <|reference_start|>Noncomputable Spectral Sets: It is possible to enumerate all computer programs. In particular, for every partial computable function, there is a shortest program which computes that function. f-MIN is the set of indices for shortest programs. In 1972, Meyer showed that f-MIN is Turing equivalent to 0'', the halting set with halting set oracle. This paper generalizes the notion of shortest programs, and we use various measures from computability theory to describe the complexity of the resulting "spectral sets." We show that under certain Godel numberings, the spectral sets are exactly the canonical sets 0', 0'', 0''', ... up to Turing equivalence. This is probably not true in general, however we show that spectral sets always contain some useful information. We show that immunity, or "thinness" is a useful characteristic for distinguishing between spectral sets. In the final chapter, we construct a set which neither contains nor is disjoint from any infinite arithmetic set, yet it is 0-majorized and contains a natural spectral set. Thus a pathological set becomes a bit more friendly. Finally, a number of interesting open problems are left for the inspired reader.<|reference_end|> | arxiv | @article{teutsch2007noncomputable,
title={Noncomputable Spectral Sets},
author={Jason Teutsch},
journal={arXiv preprint arXiv:math/0701904},
year={2007},
archivePrefix={arXiv},
eprint={math/0701904},
primaryClass={math.LO cs.LO}
} | teutsch2007noncomputable |
arxiv-676900 | math/0702109 | Longest Common Separable Pattern between Permutations | <|reference_start|>Longest Common Separable Pattern between Permutations: In this article, we study the problem of finding the longest common separable pattern between several permutations. We give a polynomial-time algorithm when the number of input permutations is fixed and show that the problem is NP-hard for an arbitrary number of input permutations even if these permutations are separable. On the other hand, we show that the NP-hard problem of finding the longest common pattern between two permutations cannot be approximated better than within a ratio of $sqrt{Opt}$ (where $Opt$ is the size of an optimal solution) when taking common patterns belonging to pattern-avoiding classes of permutations.<|reference_end|> | arxiv | @article{bouvel2007longest,
title={Longest Common Separable Pattern between Permutations},
author={Mathilde Bouvel (LIAFA), Dominique Rossin (LIAFA), Stephane Vialette
(LRI)},
journal={Combinatorial Pattern Matching (CPM) 2007 (2007) 00},
year={2007},
archivePrefix={arXiv},
eprint={math/0702109},
primaryClass={math.CO cs.CC}
} | bouvel2007longest |
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