86.30 Number Theory from a Combinatorial Point of View

The German version of Riemann’s Collected Works is confined to a single volume of 690 pages. Even so, this volume has had an abiding and profound impact on modern mathematics and physics, as we shall see. In fifteen years of activity, from 1851, when he gained his doctorate at the University of Göttingen, to his death in 1866, two months short of his fortieth birthday, Riemann contributed to almost all areas of mathematics. He perceived mathematics from the analytic point of view and used analysis to illuminate subjects as diverse as number theory and geometry. Although regarded principally as a mathematician Riemann had an abiding interest in physics and researched significantly in the methods of mathematical physics, particularly in the area of partial differential equations.

Spectral statistics of permutation matrices

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2012.0508 ◽

2014 ◽

Vol 372 (2007) ◽

pp. 20120508 ◽

Cited By ~ 1

Author(s):

Idan Oren ◽

Uzy Smilansky

Keyword(s):

Number Theory ◽

Point Of View ◽

Spectral Form ◽

Permutation Matrices ◽

Divisor Functions ◽

Point Correlation ◽

Point Correlation Function ◽

The Subject ◽

Spectral Statistics ◽

The Mean

We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.

Graphs and complete intersection toric ideals

Journal of Algebra and Its Applications ◽

10.1142/s0219498815400113 ◽

2015 ◽

Vol 14 (09) ◽

pp. 1540011 ◽

Cited By ~ 5

Author(s):

I. Bermejo ◽

I. García-Marco ◽

E. Reyes

Keyword(s):

Complete Intersection ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Point Of View ◽

Toric Ideals ◽

Induced Subgraphs ◽

Toric Ideal ◽

Vertex Set ◽

Combinatorial Point ◽

Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

Möbius Inversion and Mechanical Properties of Ni3Al

MRS Proceedings ◽

10.1557/proc-364-357 ◽

1994 ◽

Vol 364 ◽

Author(s):

Nan-Xian Chen ◽

Mi Li ◽

Shao-Jun Liu ◽

Zhao-Duo Chen

Keyword(s):

Mechanical Properties ◽

Number Theory ◽

Point Of View ◽

Möbius Inversion ◽

Industrial Material

AbstractIn this work, a very abstract formula from number theory has been applied to solving the practical problems related to the important industrial material Ni3Al from the microscopic point of view.

Analytic number theory for 0-cycles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000767 ◽

2017 ◽

Vol 166 (1) ◽

pp. 123-146

Author(s):

WEIYAN CHEN

Keyword(s):

Number Theory ◽

Analytic Number Theory ◽

Point Of View ◽

Analytic Number ◽

Geometric Point ◽

Vast Literature ◽

Prime Factors ◽

Affine Line

AbstractThere is a well-known analogy between integers and polynomials over 𝔽q, and a vast literature on analytic number theory for polynomials. From a geometric point of view, polynomials are equivalent to effective 0-cycles on the affine line. This leads one to ask: Can the analogy between integers and polynomials be extended to 0-cycles on more general varieties? In this paper we study prime factorisation of effective 0-cycles on an arbitrary connected varietyVover 𝔽q, emphasizing the analogy between integers and 0-cycles. For example, inspired by the works of Granville and Rhoades, we prove that the prime factors of 0-cycles are typically Poisson distributed.

Complex surface singularities from the combinatorial point of view

Topology and its Applications ◽

10.1016/0166-8641(95)00038-i ◽

1995 ◽

Vol 66 (3) ◽

pp. 251-265 ◽

Cited By ~ 4

Author(s):

Francisco Larrión ◽

José Seade

Keyword(s):

Complex Surface ◽

Point Of View ◽

Surface Singularities ◽

Combinatorial Point

The combinatorics of minimal unsatisfiability: connecting to graph theory

10.23889/suthesis.57815 ◽

2021 ◽

Author(s):

◽

Hoda Abbasizanjani

Keyword(s):

Graph Theory ◽

Building Blocks ◽

Point Of View ◽

New Class ◽

Strongly Connected ◽

Combinatorial Point ◽

Minimal Unsatisfiability ◽

Full Classification ◽

New Framework

Minimally Unsatisﬁable CNFs (MUs) are unsatisﬁable CNFs where removing any clause destroys unsatisﬁability. MUs are the building blocks of unsatisﬁa-bility, and our understanding of them can be very helpful in answering various algorithmic and structural questions relating to unsatisﬁability. In this thesis we study MUs from a combinatorial point of view, with the aim of extending the understanding of the structure of MUs. We show that some important classes of MUs are very closely related to known classes of digraphs, and using arguments from logic and graph theory we characterise these MUs.Two main concepts in this thesis are isomorphism of CNFs and the implica-tion digraph of 2-CNFs (at most two literals per disjunction). Isomorphism of CNFs involves renaming the variables, and ﬂipping the literals. The implication digraph of a 2-CNF F has both arcs (¬a → b) and (¬b → a) for every binary clause (a ∨ b) in F .In the ﬁrst part we introduce a novel connection between MUs and Minimal Strong Digraphs (MSDs), strongly connected digraphs, where removing any arc destroys the strong connectedness. We introduce the new class DFM of special MUs, which are in close correspondence to MSDs. The known relation between 2-CNFs and implication digraphs is used, but in a simpler and more direct way, namely that we have a canonical choice of one of the two arcs. As an application of this new framework we provide short and intuitive new proofs for two im-portant but isolated characterisations for nonsingular MUs (every literal occurs at least twice), both with ingenious but complicated proofs: Characterising 2-MUs (minimally unsatisﬁable 2-CNFs), and characterising MUs with deﬁciency 2 (two more clauses than variables).In the second part, we provide a fundamental addition to the study of 2-CNFs which have eﬃcient algorithms for many interesting problems, namely that we provide a full classiﬁcation of 2-MUs and a polytime isomorphism de-cision of this class. We show that implication digraphs of 2-MUs are “Weak Double Cycles” (WDCs), big cycles of small cycles (with possible overlaps). Combining logical and graph-theoretical methods, we prove that WDCs have at most one skew-symmetry (a self-inverse ﬁxed-point free anti-symmetry, re-versing the direction of arcs). It follows that the isomorphisms between 2-MUs are exactly the isomorphisms between their implication digraphs (since digraphs with given skew-symmetry are the same as 2-CNFs). This reduces the classiﬁ-cation of 2-MUs to the classiﬁcation of a nice class of digraphs.Finally in the outlook we discuss further applications, including an alter-native framework for enumerating some special Minimally Unsatisﬁable Sub-clause-sets (MUSs).

Dual Pairs of Matroids with Coefficients in Finitary Fuzzy Rings of Arbitrary Rank

Mathematica Pannonica ◽

10.1556/314.2020.00006 ◽

2021 ◽

Vol 27_NS1 (1) ◽

pp. 48-60

Author(s):

Walter Wenzel

Keyword(s):

Point Of View ◽

The Other ◽

Finite Intersection ◽

Dual Pairs ◽

Arbitrary Rank ◽

Other Hand ◽

Combinatorial Point ◽

Fuzzy Ring

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.5820 ◽

2021 ◽

Vol vol. 22 no. 3, Computational... (Special issues) ◽

Author(s):

OLivier Bodini ◽

Matthieu Dien ◽

Antoine Genitrini ◽

Frédéric Peschanski

Keyword(s):

Integral Formula ◽

Explicit Construction ◽

Point Of View ◽

Combinatorial Explosion ◽

Control Graph ◽

International Audience ◽

Sampling Algorithms ◽

Generate Process ◽

Quantitative Results ◽

Combinatorial Point

International audience In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space.

Trace of Products in Finite Fields from a Combinatorial Point of View

SIAM Journal on Discrete Mathematics ◽

10.1137/19m1279903 ◽

2019 ◽

Vol 33 (4) ◽

pp. 2126-2139

Author(s):

Sam Mattheus

Keyword(s):

Finite Fields ◽

Point Of View ◽

Combinatorial Point