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• 1. Aaghabali, M.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Upper bounds on the number of perfect matchings and directed 2-factors in graphs with given number of vertices and edges2015In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 45, p. 132-144Article in journal (Refereed)

We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on 2n vertices. The upper bound is sharp for even n. For odd n we state a conjecture on a sharp upper bound.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
On a lower bound for the connectivity of the independence complex of a graph2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 21, p. 2566-2569Article in journal (Refereed)

Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.

• 3.
Blekinge Institute of Technology, School of Engineering, Department of Telecommunication Systems.
Security Issues in Wireless Systems2009Independent thesis Advanced level (degree of Master (One Year))Student thesis

ireless Communication is one of the fields of Telecommunications which is growing with the tremendous speed. With the passage of time wireless communication devices are becoming more and more common. It is not only the technology of business but now people are using it to perform their daily tasks, be it for calling, shopping, checking their emails or transfer their money. Wireless communication devices include cellular phones, cordless phones and satellite phones, smart phones like Personal Digital Assistants (PDA), two way pagers, and lots of their devices are on their way to improve this wireless world. In order to establish two way communications, a wireless link may be using radio waves or Infrared light. The Wireless communication technologies have become increasingly popular in our everyday life. The hand held devices like Personal Digital Assistants (PDA) allow the users to access calendars, mails, addresses, phone number lists and the internet. Personal digital assistants (PDA) and smart phones can store large amounts of data and connect to a broad spectrum of networks, making them as important and sensitive computing platforms as laptop PCs when it comes to an organization’s security plan. Today’s mobile devices offer many benefits to enterprises. Mobile phones, hand held computers and other wireless systems are becoming a tempting target for virus writers. Mobile devices are the new frontier for viruses, spam and other potential security threats. Most viruses, Trojans and worms have already been created that exploit vulnerabilities. With an increasing amount of information being sent through wireless channels, new threats are opening up. Viruses have been growing fast as handsets increasingly resemble small computers that connect with each other and the internet. Hackers have also discovered that many corporate wireless local area networks (WLAN) in major cities were not properly secured. Mobile phone operators say that it is only a matter of time before the wireless world is hit by the same sorts of viruses and worms that attack computer software.

• 4. Akbari, Saieed
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On 1-sum flows in undirected graphs2016In: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 31, p. 646-665Article in journal (Refereed)

Let G = (V, E) be a simple undirected graph. For a given set L subset of R, a function omega: E -> L is called an L-flow. Given a vector gamma is an element of R-V , omega is a gamma-L-flow if for each v is an element of V, the sum of the values on the edges incident to v is gamma(v). If gamma(v) = c, for all v is an element of V, then the gamma-L-flow is called a c-sum L-flow. In this paper, the existence of gamma-L-flows for various choices of sets L of real numbers is studied, with an emphasis on 1-sum flows. Let L be a subset of real numbers containing 0 and denote L* := L \ {0}. Answering a question from [S. Akbari, M. Kano, and S. Zare. A generalization of 0-sum flows in graphs. Linear Algebra Appl., 438:3629-3634, 2013.], the bipartite graphs which admit a 1-sum R* -flow or a 1-sum Z* -flow are characterized. It is also shown that every k-regular graph, with k either odd or congruent to 2 modulo 4, admits a 1-sum {-1, 0, 1}-flow.

• 5.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
LLT polynomials, chromatic quasisymmetric functions and graphs with cycles2018In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 341, no 12, p. 3453-3482Article in journal (Refereed)

We use a Dyck path model for unit-interval graphs to study the chromatic quasisymmetric functions introduced by Shareshian and Wachs, as well as unicellular LLT polynomials, revealing some parallel structure and phenomena regarding their e-positivity. The Dyck path model is also extended to circular arc digraphs to obtain larger families of polynomials, giving a new extension of LLT polynomials. Carrying over a lot of the noncircular combinatorics, we prove several statements regarding the e-coefficients of chromatic quasisymmetric functions and LLT polynomials, including a natural combinatorial interpretation for the e-coefficients for the line graph and the cycle graph for both families. We believe that certain e-positivity conjectures hold in all these families above. Furthermore, beyond the chromatic analogy, we study vertical-strip LLT polynomials, which are modified Hall-Littlewood polynomials.

• 6.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Combinatorics and zeros of multivariate polynomials2019Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

• 7.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Spectrahedrality of hyperbolicity cones of multivariate matching polynomials2018In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Refereed)

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

• 8.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Stable multivariate generalizations of matching polynomialsManuscript (preprint) (Other academic)

The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan d-coverings, Hall, Puder and Sawin introduced the d-matching polynomial of a graph G, defined as the uniform average of matching polynomials over the set of d-sheeted covering graphs of G. We prove that a natural multivariate version of the d-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the d-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.

• 9.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The cone of cyclic sieving phenomena2019In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 342, no 6, p. 1581-1601Article in journal (Refereed)

We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone corresponds to a non-negative integer matrix which jointly records the statistic and cyclic order distribution associated with the set of objects realizing the CSP. In particular we consider a universal subcone onto which every CSP matrix linearly projects such that the projection realizes a CSP with the same cyclic orbit structure, but via a universal statistic that has even distribution on the orbits.

Reiner et.al. showed that every cyclic action gives rise to a unique polynomial (mod q^n-1) complementing the action to a CSP. We give a necessary and sufficient criterion for the converse to hold. This characterization allows one to determine if a combinatorial set with a statistic gives rise (in principle) to a CSP without having a combinatorial realization of the cyclic action. We apply the criterion to conjecture a new CSP involving stretched Schur polynomials and prove our conjecture for certain rectangular tableaux. Finally we study some geometric properties of the CSP cone. We explicitly determine its half-space description and in the prime order case we determine its extreme rays.

• 10.
KTH, Centres, Nordic Institute for Theoretical Physics NORDITA.
Dublin Inst Adv Studies, Sch Theoret Phys, 10 Burlington Rd, Dublin 4, Ireland.. Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England.;Univ Oxford, Math Inst, Radcliffe Observ Quarter, Andrew Wiles Bldg,Woodstock Rd, Oxford OX2 6GG, England.. Museo Stor Fis, Via Panisperna 89A, I-00184 Rome, Italy.;Ctr Studi & Ric Enrico Fermi, Via Panisperna 89A, I-00184 Rome, Italy.;Univ Padua, Dipartimento Fis & Astron Galileo Galilei, Via Marzolo 8, I-35131 Padua, Italy.;INFN, Sez Padova, Via Marzolo 8, I-35131 Padua, Italy.. Ctr Astron & Particle Theory, Univ Pk, Nottingham NG7 2RD, England.. Imperial Coll London, Blackett Lab, Theoret Phys, London SW7 2AZ, England..
The mile high magic pyramid2019In: NONASSOCIATIVE MATHEMATICS AND ITS APPLICATIONS / [ed] Vojtechovsky, P Bremner, MR Carter, JS Evans, AB Huerta, J Kinyon, MK Moorhouse, GE Smith, JDH, AMER MATHEMATICAL SOC , 2019, p. 1-27Conference paper (Refereed)

Using a unified formulation of N = 1, 2, 4, 8, super Yang-Mills theories in D = 3 spacetime dimensions with fields valued respectively in R, C, H, O, it was shown that tensoring left and right multiplets yields a Freudenthal magic square of D = 3 supergravities. When tied in with the more familiar R, C, H, O description of super Yang-Mills in D = 3, 4, 6, 10 this results in a magic pyramid of supergravities: the known 4x4 magic square at the base in D = 3, a 3x3 square in D = 4, a 2x2 square in D = 6 and Type II supergravity at the apex in D = 10.

• 11.
Umeå University.
Umeå University. Umeå University.
Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, p. 230-234Article in journal (Refereed)
• 12.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding Arrays of Odd Order by Latin Squares2013In: Combinatorics, probability & computing, ISSN 0963-5483, E-ISSN 1469-2163, Vol. 22, no 2, p. 184-212Article in journal (Refereed)

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) x (2k + 1) array A on the symbols 1, ... , 2k + 1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k + 1) times in every row and column is avoidable; that is, there is a (2k + 1) x (2k + 1) Latin square S on the symbols 1, ... , 2k + 1 such that, for each i, j is an element of {1, ... , 2k + 1}, the symbol in position (i, j) of S does not appear in the corresponding cell in Lambda. This settles the last open case of a conjecture by Haggkvist. Using this result, we also show that there is a constant rho, such that, for any positive integer n, if each cell in an n x n array B is assigned a set of m <= rho n symbols, where each set is chosen independently and uniformly at random from {1, ... , n}, then the probability that B is avoidable tends to 1 as n -> infinity.

• 13.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On the Ising problem and some matrix operations2007Doctoral thesis, comprehensive summary (Other academic)

The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

• 14.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoidability by Latin squares of arrays of even orderManuscript (preprint) (Other academic)

We prove that for any k and any 2k × 2k array A such that no cell in A contains more than   k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.

• 15.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoidability of random arraysManuscript (preprint) (Other academic)

An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring $K_{n,n}$ with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.

• 16.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding (m, m, m)-arrays of order n = 2kManuscript (preprint) (Other academic)

An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.

• 17.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
On Latin squares and avoidable arrays2010Doctoral thesis, comprehensive summary (Other academic)

This thesis consists of the four papers listed below and a survey of the research area.

I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k

II Lina J. Andrén: Avoidability of random arrays

III Lina J. Andr´en: Avoidability by Latin squares of arrays with even order

IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares

Papers I, III and IV are all concerned with a conjecture by Häggkvist saying that there is a constant c such that for any positive integer n, if m ≤ cn, then for every n × n array A of subsets of {1, . . . , n} such that no cell contains a set of size greater than m, and none of the elements 1, . . . , n belongs to more than m of the sets in any row or any column of A, there is a Latin square L on the symbols 1, . . . , n such that there is no cell in L that contains a symbol that belongs to the set in the corresponding cell of A. Such a Latin square is said to avoid A. In Paper I, the conjecture is proved in the special case of order n = 2k . Paper III improves on the techniques of Paper I, expanding the proof to cover all arrays of even order. Finally, in Paper IV, similar methods are used together with a recoloring theorem to prove the conjecture for all orders. Paper II considers another aspect of the problem by asking to what extent way a deterministic result concerning the existence of Latin squares that avoid certain arrays can be used when the sets in the array are assigned randomly.

• 18.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Avoiding arrays of odd order by Latin squaresManuscript (preprint) (Other academic)

We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

• 19.
Department of Applied Mathematics and Statistics, Stony Brook University, USA .
Institute of Computer Science, Freie Universität Berlin, Germany . Institute of Computer Science, Universität Bayreuth, Germany . Department of Applied Mathematics and Statistics, Stony Brook University, USA . Helsinki Institute for Information Technology, CS Dept, University of Helsinki, Finland . Institute of Computer Science, Freie Universität Berlin, Germany . Department of Computer Science, Stony Brook University, USA .
Convex transversals2014In: Computational Geometry, ISSN 0925-7721, Vol. 47, no 2, p. 224-239Article in journal (Refereed)

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”

We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.

The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.

• 20.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of Southern Denmark, Denmark.
Solution of Vizings Problem on Interchanges for the case of Graphs with Maximum Degree 4 and Related Results2016In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 82, no 4, p. 350-373Article in journal (Refereed)

Let G be a Class 1 graph with maximum degree 4 and let t amp;gt;= 5 be an integer. We show that any proper t-edge coloring of G can be transformed to any proper 4-edge coloring of G using only transformations on 2-colored subgraphs (so-called interchanges). This settles the smallest previously unsolved case of a well-known problem of Vizing on interchanges, posed in 1965. Using our result we give an affirmative answer to a question of Mohar for two classes of graphs: we show that all proper 5-edge colorings of a Class 1 graph with maximum degree 4 are Kempe equivalent, that is, can be transformed to each other by interchanges, and that all proper 7-edge colorings of a Class 2 graph with maximum degree 5 are Kempe equivalent. (C) 2015 Wiley Periodicals, Inc.

• 21.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Yerevan State Univ, Armenia; Natl Acad Sci, Armenia.
Cyclic deficiency of graphs2019In: Discrete Applied Mathematics, ISSN 0166-218X, E-ISSN 1872-6771, Vol. 266, p. 171-185Article in journal (Refereed)

A proper edge coloring of a graph G with colors 1, 2, . . . , t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. In this paper we introduce and investigate a new notion, the cyclic deficiency of a graph G, defined as the minimum number of pendant edges whose attachment to G yields a graph admitting a cyclic interval coloring; this number can be considered as a measure of closeness of G of being cyclically interval colorable. We determine or bound the cyclic deficiency of several families of graphs. In particular, we present examples of graphs of bounded maximum degree with arbitrarily large cyclic deficiency, and graphs whose cyclic deficiency approaches the number of vertices. Finally, we conjecture that the cyclic deficiency of any graph does not exceed the number of vertices, and we present several results supporting this conjecture. (C) 2018 Elsevier B.V. All rights reserved.

• 22.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Yerevan State University, Armenia; National Academic Science, Armenia.
Some results on cyclic interval edge colorings of graphs2018In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 87, no 2, p. 239-252Article in journal (Refereed)

A proper edge coloring of a graph G with colors 1,2,,t is called a cyclic interval t-coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree (G)4 admits a cyclic interval (G)-coloring if for every vertex v the degree dG(v) satisfies either dG(v)(G)-2 or dG(v)2. We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for (a,b)-biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)-biregular graphs as well as all (2r-2,2r)-biregular (r2) graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.

• 23.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Southern Polytechnic State University, Marietta, Georgia. University of Illinois, Urbana, Illinois.
Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs2009In: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, no 2, p. 88-97Article in journal (Refereed)

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.

• 24.
Univ Politecn Cataluna, Dept Comp Sci, Barcelona, Spain..
Univ Politecn Cataluna, Dept Comp Sci, Barcelona, Spain.. KTH, School of Electrical Engineering and Computer Science (EECS). Sapienza Univ Roma, Dept Stat Sci, Rome, Italy.. KTH, School of Electrical Engineering and Computer Science (EECS). Univ Chicago, Chicago, IL 60637 USA.;Steklov Math Inst, Moscow, Russia..
Clique Is Hard on Average for Regular Resolution2018In: STOC'18: PROCEEDINGS OF THE 50TH ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING / [ed] Diakonikolas, I Kempe, D Henzinger, M, ASSOC COMPUTING MACHINERY , 2018, p. 866-877Conference paper (Refereed)

We prove that for k << (4)root n regular resolution requires length n(Omega(k)) to establish that an Erdos Renyi graph with appropriately chosen edge density does not contain a k-clique. This lower bound is optimal up to the multiplicative constant in the exponent, and also implies unconditional n(Omega(k)) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

• 25.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
Dense Subset Sum may be the hardest2016In: Leibniz International Proceedings in Informatics, LIPIcs, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2016Conference paper (Refereed)

The Subset Sum problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O∗(2n/2)-time algorithm for Subset Sum by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O∗(2(0.5-δ)n)-time algorithm, with some constant δ &gt; 0. Continuing an earlier work [STACS 2015], we study Subset Sum parameterized by the maximum bin size β, defined as the largest number of subsets of the n input integers that yield the same sum. For every ∈ &gt; 0 we give a truly faster algorithm for instances with β ≤ 2(0.5-∈)n, as well as instances with β ≥ 20.661n. Consequently, we also obtain a characterization in terms of the popular density parameter n/log2 t: if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for Subset Sum and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.

Stockholm University, Faculty of Science, Department of Mathematics.
Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 274, p. 118-166Article in journal (Refereed)

For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.

• 27. Ayyer, Arvind
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Some genelalized juggling processes (extended abstract)2015In: DMTCS proc. FPSAC'15, Nancy, France, 2015, p. 925-936Conference paper (Refereed)

We consider generalizations of juggling Markov chains introduced by Ayyer, Bouttier, Corteel and Nunzi. We first study multispecies generalizations of all the finite models therein, namely the MJMC, the add-drop and the annihilation models. We then consider the case of several jugglers exchanging balls. In all cases, we give explicit product formulas for the stationary probability and closed-form expressions for the normalization factor if known.

• 28.
University of St Andrews, Scotland.
University of St Andrews, Scotland. Mid Sweden University, Faculty of Science, Technology and Media, Department of Mathematics and Science Education.
Sesqui-arrays, a generalisation of triple arrays2018In: The Australasian Journal of Combinatorics, ISSN 1034-4942, Vol. 71, no 3, p. 427-451Article in journal (Refereed)

A triple array is a rectangular array containing letters, each letter occurring equally often with no repeats in rows or columns, such that the number of letters common to two rows, two columns, or a rowand a column are (possibly different) non-zero constants. Deleting the condition on the letters common to a row and a column gives a double array. We propose the term sesqui-array for such an array when only the condition on pairs of columns is deleted. In this paper we give three constructions for sesqui-arrays. Therst gives $(n + 1)\times n^2$ arrays on n(n + 1) letters for $n\geq 2$. (Suchan array for n = 2 was found by Bagchi.) This construction uses Latin squares. The second uses the Sylvester graph, a subgraph of the Hoffman--Singleton graph, to build a good block design for 36 treatments in 42 blocks of size 6, and then uses this in a 736 sesqui-array for 42 letters.We also give a construction for K(K-1)(K-2)/2 sesqui-arrays on K(K-1)/2 letters from biplanes. The construction starts with a block of a biplane and produces an array which satises the requirements for a sesqui-array except possibly that of having no repeated letters in a row or column. We show that this condition holds if and only if the Hussain chains for the selected block contain no 4-cycles. A sufficient condition for the construction to give a triple array is that each Hussain chain is a union of 3-cycles; but this condition is not necessary, and we give a few further examples. We also discuss the question of which of these arrays provide good designs for experiments.

• 29.
Universitat Politècnica de Catalunya, Barcelona, Spain.
The University of Western Australia, Crawley, WA, Australia. The University of Western Australia, Crawley, WA, Australia. Universitat Rovira i Virgili, Tarragona, Spain.
An alternative way to generalise the pentagon2013In: Journal of combinatorial designs (Print), ISSN 1063-8539, E-ISSN 1520-6610, Vol. 21, no 4, p. 163-179Article in journal (Refereed)

We introduce the concept of a pentagonal geometry as a generalization of the pentagon and the Desargues configuration, in the same vein that the generalized polygons share the fundamental properties of ordinary polygons. In short, a pentagonal geometry is a regular partial linear space in which for all points x, the points not collinear with the point x, form a line. We compute bounds on their parameters, give some constructions, obtain some nonexistence results for seemingly feasible parameters and suggest a cryptographic application related to identifying codes of partial linear spaces.

• 30.
Stockholm University, Faculty of Science, Department of Mathematics.
Classifications and volume bounds of lattice polytopes2017Licentiate thesis, monograph (Other academic)

In this licentiate thesis we study relations among invariants of lattice polytopes, with particular focus on bounds for the volume.In the first paper we give an upper bound on the volume vol(P^*) of a polytope P^* dual to a d-dimensional lattice polytope P with exactly one interiorlattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. In the second paper we classify the three-dimensional lattice polytopes with two lattice points in their strict interior. Up to unimodular equivalence thereare 22,673,449 such polytopes. This classification allows us to verify, for this case only, the sharp conjectural upper bound for the volume of a lattice polytope with interior points, and provides strong evidence for more general new inequalities on the coefficients of the h^*-polynomial in dimension three.

• 31.
Christian-Albrechts Universität Kiel.
Christian-Albrechts Universität Kiel. Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Christian-Albrechts Universität Kiel. Christian-Albrechts Universität Kiel.
Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection2015In: Journal of the Operational Research Society, ISSN 0160-5682, E-ISSN 1476-9360, Vol. 66, no 4, p. 615-626Article in journal (Refereed)

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.

• 32.
Christian-Albrechts Universität Kiel, Germany.
Christian-Albrechts Universität Kiel, Germany. Christian-Albrechts Universität Kiel, Germany.
Construction of Sparse Asymmetric Connectors2003In: Proceedings of European Conference on Combinatorics, Graph Theory and Applications (Eurocomb 2003), 2003Conference paper (Refereed)
• 33.
Christian-Albrechts-Universität Kiel, Germany.
Christian-Albrechts-Universität Kiel, Germany. Christian-Albrechts-Universität Kiel, Germany.
Constructions of Sparse Asymmetric Connectors2003In: Proceedings of 23rd Conference of Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2003) / [ed] P.K. Lodaya and J. Radhakrishnan, Berlin-Heidelberg: Springer Berlin/Heidelberg, 2003, p. 13-22Conference paper (Refereed)
• 34.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany. Mathematisches seminar, Christian-Albrechts-Universität zu Kiel, Germany.
Constructions of Sparse Asymmetric Connectors with Number Theoretic Methods2005In: Networks, ISSN 0028-3045, E-ISSN 1097-0037, Vol. 45, no 3, p. 119-124Article in journal (Refereed)

We consider the problem of connecting a set I of n inputs to a set O of N outputs (n ≤ N) by as few edges as possible such that for every injective mapping f : I → O there are n vertex disjoint paths from i to f(i) of length k for a given k $\in \mathbb{N}$. For k = Ω(log N + log$^{2}$n) Oruς (1994) gave the presently best (n,N)-connector with O(N+n·log n) edges. For k=2 and N the square of a prime, Richards and Hwang (1985) described a construction using $N\left[\sqrt{n+5/4-1/2}\right] + n\left[\sqrt{n+5/4-1/2} \right]\sqrt{N}$edges. We show by a probabilistic argument that an optimal (n,N)-connector has Θ (N) edges, if $n\leq N^{1/2-\varepsilon}$for some ε>0. Moreover, we give explicit constructions based on a new number theoretic approach that need at most $N\left[\sqrt{3n/4} \right]+2n\left[\sqrt{3n/4} \right]\left[\sqrt{N} \right]$edges for arbitrary choices of n and N. The improvement we achieve is based on applying a generalization of the Erdös-Heilbronn conjecture on the size of restricted sums.

• 35. Barak, B.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
Making the long code shorter2015In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 44, no 5, p. 1287-1324Article in journal (Refereed)

The long code is a central tool in hardness of approximation especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results including the following: (1) For any ε &gt; 0, we show the existence of an n-vertex graph G where every set of o(n) vertices has expansion 1-ε but G's adjacency matrix has more than exp(logδ n) eigenvalues larger than 1 - ε, where δ depends only on ε. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563-572] who asked whether one can improve over the noise graph on the Boolean hypercube that has poly(log n) such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo K into instances with alphabet k with a blowup of kpolylog(K) , improving over the previously known gadget with blowup of kω(K). (3) An n-variable integrality gap for Unique Games that survives exp(poly(log log n)) rounds of the semidefinite programming version of the Sherali-Adams hierarchy, improving on the previously known bound of poly(log log n). We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.

• 36.
Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE).
Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE), Halmstad Embedded and Intelligent Systems Research (EIS), MPE-lab.
Meshfree methods in option pricing2011Independent thesis Advanced level (degree of Master (One Year)), 10 credits / 15 HE creditsStudent thesis

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.

• 37.
Stockholm University, Faculty of Science, Department of Mathematics.
Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)

Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

• 38.
Stockholm University, Faculty of Science, Department of Mathematics.
Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)

Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

• 39. Bhattacharya, S.
KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
New deterministic approximation algorithms for fully dynamic matching2016In: STOC'16: PROCEEDINGS OF THE 48TH ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING, 2016, p. 398-411Conference paper (Refereed)

We present two deterministic dynamic algorithms for the maximum matching problem. (1) An algorithm that maintains a (2 + ϵ)-approximate maximum matching in general graphs with O(poly(log n, 1/ϵ)) update time. (2) An algorithm that maintains an αk approximation of the value of the maximum matching with O(n2/K) update time in bipartite graphs, for every sufficiently large constant positive integer K. Here, 1 ≤ αk ≤ 2 is a constant determined by the value of K. Result (1) is the first deterministic algorithm that can maintain an o(log n)-approximate maximum matching with polylogarithmic update time, improving the seminal result of Onak et al. [STOC 2010]. Its approximation guarantee almost matches the guarantee of the best randomized polylogarithmic update time algorithm [Baswana et al. FOCS 2011]. Result (2) achieves a better-than-two approximation with arbitrarily small polynomial update time on bipartite graphs. Previously the best update time for this problem was O(m1/4) [Bernstein et al. ICALP 2015], where m is the current number of edges in the graph.

• 40. Bierbrauer, Jürgen
Örebro University, Department of Natural Sciences.
Almost independent and weakly biased arrays: efficient constructions and cryptologic applications2000In: Advances in cryptology: CRYPTO 2000 / [ed] Mihir Bellare, Springer Berlin/Heidelberg, 2000, p. 533-543Conference paper (Other academic)
• 41.
Umeå University, Faculty of Science and Technology, Department of Computing Science.
Umeå University, Faculty of Science and Technology, Department of Computing Science. Department of Information Science, Stellenbosch University, Stellenbosch, South Africa.
Minimization of Finite State Automata Through Partition Aggregation2016In: Logical Aspects of Computational Linguistics: Celebrating 20 Years of LACL (1996–2016) / [ed] Amblard, M DeGroote, P Pogodalla, S Retore, C, SPRINGER-VERLAG BERLIN , 2016, p. 328-328Conference paper (Refereed)
• 42.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Graphical representations of Ising and Potts models: Stochastic geometry of the quantum Ising model and the space-time Potts model2009Doctoral thesis, monograph (Other academic)

HTML clipboard Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models.

In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter 2 we define the appropriate space–time random-cluster model and explore a range of useful probabilistic techniques for studying it. The space– time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded fk-cluster, and the resulting lower bound on the critical value in .

In Chapter 3 we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory— and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which gives the results.

In Chapter 4 we explore some consequences and possible extensions of the results established in Chapters 2 and 3. For example, we determine the critical point for the quantum Ising model in and in ‘star-like’ geometries.

• 43.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
On the connectivity of manifold graphs2015In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 143, no 10, p. 4123-4132Article in journal (Refereed)

This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold M is (2d - b_M)-connected. The parameter b_M has the property that b_M = 0 if the complex M is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d+1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of b_M involves the concept of banner triangulations of manifolds, a generalization of flag triangulations.

• 44.
Linköping University, Department of Computer and Information Science. Linköping University, The Institute of Technology.
Sport-sort: sorting algorithms and sport tournaments1987Licentiate thesis, monograph (Other academic)

Arrange a really short, thrilling and fair tournament! Execute parallel sorting in a machine of a new architecture! The author shows how these problems are connected. He designs several new tournament schemes, and analyses them both in theory and in extensive simulations. He uses only elementary mathematical and statistical methods. The results are much better than previous ones, and close to the theoretical limit. Now personal computers can be used to arrange tournaments which give the complete ranking list of several thousands of participants within only 20 - 30 rounds.

• 45. Bogart, Tristram
Stockholm University, Faculty of Science, Department of Mathematics.
Finitely many smooth d-polytopes with n lattice points2015In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 207, no 1, p. 301-329Article in journal (Refereed)

We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙ n that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.

• 46.
Uppsala University, Interfaculty Units, Centre for Image Analysis. Uppsala University, Interfaculty Units, Centre for Image Analysis. Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Information Technology, Computerized Image Analysis.
Kedjekod - ett sätt att beskriva former i digitala bilder2005In: Problemlösning är # 1, Liber, Stockholm , 2005, p. 38-42Chapter in book (Other scientific)
• 47.
Stockholm University, Faculty of Science, Department of Mathematics. -.
On Minimal Non-(2, 1)-Colorable Graphs2017Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

A graph is (2, 1)-colorable if it allows a partition of its vertices into two classes such that both induce graphs with maximum degree at most one. A non-(2, 1)-colorable graph is minimal if all proper subgraphs are (2, 1)-colorable. We prove that such graphs are 2-edge-connected and that every edge sits in an odd cycle. Furthermore, we show properties of edge cuts and particular graphs which are no induced subgraphs. We demonstrate that there are infinitely many minimal non-(2, 1)-colorable graphs, at least one of order n for all n ≥ 5. Moreover, we present all minimal non-(2, 1)- colorable graphs of order at most seven. We consider the maximum degree of minimal non-(2, 1)-colorable graphs and show that it is at least four but can be arbitrarily large. We prove that the average degree is greater than 8/3 and give sufficient properties for graphs with average degree greater than 14/5. We conjecture that all minimal non-(2, 1)-colorable graphs fulfill these properties.

• 48.
University of Skövde, School of Humanities and Informatics. University of Skövde, The Informatics Research Centre.
Maximizing the Area under the ROC Curve with Decision Lists and Rule Sets2007In: Proceedings of the 7th SIAM International Conference on Data Mining / [ed] C. Apte, B. Liu, S. Parthasarathy, D. Skillicorn, Society for Industrial and Applied Mathematics , 2007, p. 27-34Conference paper (Refereed)

Decision lists (or ordered rule sets) have two attractive properties compared to unordered rule sets: they require a simpler classi¯cation procedure and they allow for a more compact representation. However, it is an open question what effect these properties have on the area under the ROC curve (AUC). Two ways of forming decision lists are considered in this study: by generating a sequence of rules, with a default rule for one of the classes, and by imposing an order upon rules that have been generated for all classes. An empirical investigation shows that the latter method gives a significantly higher AUC than the former, demonstrating that the compactness obtained by using one of the classes as a default is indeed associated with a cost. Furthermore, by using all applicable rules rather than the first in an ordered set, an even further significant improvement in AUC is obtained, demonstrating that the simple classification procedure is also associated with a cost. The observed gains in AUC for unordered rule sets compared to decision lists can be explained by that learning rules for all classes as well as combining multiple rules allow for examples to be ranked according to a more fine-grained scale compared to when applying rules in a fixed order and providing a default rule for one of the classes.

• 49.
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria; Faculty of Engineering, South-Western University, Blagoevgrad, Bulgaria.
Linköping University, Department of Electrical Engineering, Communication Systems. Faculty of Mathematics and Informatics, Sofia University, Sofia, Bulgaria.
Refinements of Levenshtein Bounds in q-ary Hamming Spaces2018In: Problems of Information Transmission, ISSN 0032-9460, E-ISSN 1608-3253, Vol. 54, no 4, p. 329-342Article in journal (Refereed)

We develop refinements of the Levenshtein bound in q-ary Hamming spaces by taking into account the discrete nature of the distances versus the continuous behavior of certain parameters used by Levenshtein. We investigate the first relevant cases and present new bounds. In particular, we derive generalizations and q-ary analogs of the MacEliece bound. Furthermore, we provide evidence that our approach is as good as the complete linear programming and discuss how faster are our calculations. Finally, we present a table with parameters of codes which, if exist, would attain our bounds.

• 50.
Universitat Rovira i Virgili, Catalonia, Spain.
Universitat Rovira i Virgili, Catalonia, Spain. Universitat Rovira i Virgili, Catalonia, Spain.
Configuraciones combinatóricas y recuperación privada de información por pares2009Conference paper (Other academic)
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