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Beck's Conjecture For Multiplicative Lattices [article]

Vinayak Joshi, Sachin Sarode
2013 arXiv   pre-print
Further, we prove that Beck's conjecture is true for reduced multiplicative lattice which extends the result of Behboodi and Rakeei[7], and Aalipour et. al.[1].  ...  In this paper, we introduce the zero divisor graph of a multiplicative lattice. We provide a counter example to Beck's conjecture for multiplicative lattices.  ...  Thus Beck's Conjecture is not true in the case of multiplicative lattices.  ... 
arXiv:1310.4594v1 fatcat:jgj5l4fr7rdodlzcb6lz2pecvm

The L 2 Discrepancy of Irrational Lattices [chapter]

Dmitriy Bilyk
2013 Monte Carlo and Quasi-Monte Carlo Methods 2012  
In this paper we extend the prior work to arbitrary values of α and N. We heavily rely on Beck's study of the behavior of the sums ∑ {kα} − 1 2 .  ...  However, it remained unclear whether this holds for the original lattices without any modifications.  ...  Nevertheless, it still remained a mystery whether these modifications are indeed necessary and whether the original lattices have asymptotically minimal L 2 discrepancy.  ... 
doi:10.1007/978-3-642-41095-6_11 fatcat:4z5xfoddnngrnhqywxzxrt6tay

Extremal properties of line arrangements in the complex projective plane [chapter]

Piotr Pokora
2019 Analytic and Algebraic Geometry 3  
Using Hirzebruch-type inequalites we provide some new results on r-rich lines, symplicial arrangements of lines, and the so-called free line arrangmenets. 2010 Mathematics Subject Classification. 14N10  ...  Taking into account that Bojanowski's inequality is more acurate, we can formulate the following conjecture as it was suggested by de Zeeuw [7, Conjecture 4.5].  ...  In this setting, C 3 is the lattice element0 and the rank one elements of L A are the planes. In this section we denote by S the polynomial ring C[x, y, z], Definition 4.1.  ... 
doi:10.18778/8142-814-9.14 fatcat:nei2dt772ng2fos2gvalqfhzai

Page 3337 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews  
with the conjecture is given.  ...  3337 counterexamples, that the conjecture cannot be generally valid.  ... 

Clique number and Chromatic number of a graph associated to a Commutative Ring with Unity [article]

T. Kavaskar
2020 arXiv   pre-print
In 1993 [2], Anderson and Naseer disproved the conjecture by giving a counter example of the conjecture (Note that, till date this is a only one counter example).  ...  In 1988 [8], Beck raised the conjecture that the chromatic number and clique number are same in a graph associated to any commutative ring with unity.  ...  That is, ω(R) = 5 and χ(R) = 6. Here we construct infinitely many counter examples of the Beck's conjecture. We now recall a result of Beck in [8] .  ... 
arXiv:1912.10199v2 fatcat:ozhnjrqgq5ex5p4wgkgo6zkjfm

Problems, problems, problems

William O.J. Moser
1991 Discrete Applied Mathematics  
Beck's Theorem.  ...  "You know", he replied, "addition, subtraction, multiplication and division." 0 LM 36 ( 036 1966). Given a linear point set on (0,l) of measure + (say).  ...  Here disto(x, y) and dist(x, y) denote respectively the distance between x and y in the graph G (i.e., the length of shortest path connecting them in G) and the Euclidean distance between x and y. (2)  ... 
doi:10.1016/0166-218x(91)90071-4 fatcat:ldhyppgesnapliwe62ovwmwkei

Topological and Geometric Combinatorics

Anders Björner, Gil Kalai, Isabella Novik, Günter Ziegler
2011 Oberwolfach Reports  
The following conjecture is coined three-permutations-conjecture or simply Beck's conjecture (see Problem 1.9 in [1]): Given any 3 permutations on n symbols, one can color the symbols with red and blue  ...  So far the best known bound on D perm 3 (n) is O(log n) and more generally D perm k (n) can be bounded by O(k log n) [2] and by O( Theorem 1 . 1 If Beck's conjecture holds, then the integrality gap of  ...  The coefficients c j are easily seen to lie in Z, and although wellstudied, interpretations and explicit formulas for them remain elusive.  ... 
doi:10.4171/owr/2011/08 fatcat:kcwryms27fecjjs4hciq6ixa2q

OLD AND NEW PROBLEMS IN COMBINATORIAL ANALYSIS AND GRAPH THEORY

P. Erdös
1979 Annals of the New York Academy of Sciences  
Brown, Simonovits, and I have some more general conjectures for multiple rgraphs, but here even the case r -2 presents enormous difficulties .  ...  The lattice points (x, y) 0 5 x, y < n -I show that if true this conjecture is the best possible .  ... 
doi:10.1111/j.1749-6632.1979.tb32789.x fatcat:kcvx4gls6besdlqtrvh3vo2v2m

The Surprising Accuracy of Benford's Law in Mathematics [article]

Zhaodong Cai, Matthew Faust, A.J. Hildebrand, Junxian Li, Yuan Zhang
2019 arXiv   pre-print
We prove results that explain many, but not all, of these surprising accuracies, and we relate the observed behavior to classical results in Diophantine approximation as well as recent deep conjectures  ...  Similar "perfect hits" can be observed in other instances, such as the digit 1 and 2 counts for the first billion powers of 3.  ...  We are grateful to the referees for their careful reading of the paper and helpful suggestions and comments.  ... 
arXiv:1907.08894v2 fatcat:7udffwu3cvce3nlusj6x65lm5m

Combinatorial Applications of the Subspace Theorem [article]

Ryan Schwartz, Jozsef Solymosi
2013 arXiv   pre-print
It has appeared in various forms and been adapted and improved over time.  ...  Other applications of the Subspace Theorem include linear recurrence sequences and finite automata.  ...  We are also thankful to the organizers of the workshop in Pisa, "Geometry, Structure and Randomness in Combinatorics", where the parts of this paper were presented.  ... 
arXiv:1311.3743v1 fatcat:y6deyg2x5na4jfpm2xu2m6ntle

Zero-divisor graphs of nilpotent-free semigroups

Neil Epstein, Peyman Nasehpour
2012 Journal of Algebraic Combinatorics  
We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph.  ...  under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.  ...  In investigating this question, we encountered connections with topology, semigroup theory, lattice theory, multiplicative ideal theory, and other areas.  ... 
doi:10.1007/s10801-012-0377-x fatcat:xgv7xkaa5nb6pc3gzkeww4iuoe

Representation Theory of Finite-Dimensional Algebras

Michael Ringel, Idun Reiten
2005 Oberwolfach Reports  
Methods and results from the representation theory of finite dimensional algebras have led to many interactions with other areas of mathematics.  ...  addition to stimulating progress in the representation theory of algebras, to further develop such interactions with commutative algebra, algebraic geometry, group representation theory, Lie-algebras and  ...  Applications include a proof of some cases of the strong no loops conjecture, and results relating Brauer algebras with various symmetric groups in the context of [2].  ... 
doi:10.4171/owr/2005/06 fatcat:nu7cke4ylfdkzf7wrx2m4efhlu

Disparity between dorsal and ventral networks in patients with obsessive-compulsive disorder: evidence revealed by graph theoretical analysis based on cortical thickness from MRI

Seung-Goo Kim, Wi Hoon Jung, Sung Nyun Kim, Joon Hwan Jang, Jun Soo Kwon
2013 Frontiers in Human Neuroscience  
As one of the most widely accepted neuroanatomical models on obsessive-compulsive disorder (OCD), it has been hypothesized that imbalance between an excitatory direct (ventral) pathway and an inhibitory  ...  More importantly, however, disparity between the dorsal and the ventral networks in the OCD patients was found in terms of graph theoretical measures, suggesting a positive evidence to the imbalance theory  ...  This work was supported by the National Research Foundation of Korea grant (2012-0005150) funded by the Ministry of Education, Science and Technology (MEST) of the Republic of Korea.  ... 
doi:10.3389/fnhum.2013.00302 pmid:23840184 pmcid:PMC3699763 fatcat:xyj6lvbpvnbl5jr3eatp3farbu

Verma-type Modules for Quantum Affine Lie Algebras [article]

Vyacheslav M. Futorny, Duncan J. Melville, Alexander N. Grishkov
2000 arXiv   pre-print
Let g be an untwisted affine Kac-Moody algebra and M_J(lambda) a Verma-type module for g with J-highest integral weight lambda.  ...  We construct quantum Verma-type modules M_J^q(lambda) over the quantum group U_q(g), investigate their properties and show that M_J^q(lambda) is a true quantum deformation of M_J(\l) in the sense that  ...  Extend the root latticeQ ofġ to a lattice Q =Q ⊕ Zδ, and extend the form (.|.) to Q by setting (q|δ) = 0 for all q ∈Q and (δ|δ) = 0.  ... 
arXiv:math/0010322v1 fatcat:2ouqj6ppbrh6pdfatbibrd6fye

A PBW basis for lusztig's form of untwisted affine quantum groups

Fabio Gavarini
1999 Communications in Algebra  
Let g be an untwisted affine Kac-Moody algebra over the field K , and let U_q(g) be the associated quantum enveloping algebra; let U_q(g) be the Lusztig's integer form of U_q(g) , generated by q -divided  ...  i the root lattice of g, and Q ∨ 0 := α∈Φ 0 Zα ∨ = ⊕ i∈I 0 Zα ∨ i the coroot lattice; W 0 the Weyl group of g.  ...  involved in the "quantized (5.1), (5.2)", and from these achieve the claim of Conjecture B.  ... 
doi:10.1080/00927879908826468 fatcat:w5u7rzs5wvhfhpohkflh7scj6y
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