We will see that this illustrates a case of Deligne’s conjecture. We’ll start with the following simple task: describe Artin motives (“motives of dimension 0”) over a number field K. We will see that a reasonable category of dimension-0 motives is equivalent to a category of finite-dimensional Galois representations. We will be interested in the determinant of the period matrix of such objects....

let $r$ be an associative ring and let $m$ be a left $r$-module.let $spec_{r}(m)$ be the collection of all prime submodules of $m$ (equipped with classical zariski topology). there is a conjecture which says that every irreducible closed subset of $spec_{r}(m)$ has a generic point. in this article we give an affirmative answer to this conjecture and show that if $m$ has a noetherian spectrum, t...

amara in jurisprudential sources is said to any evidence which indicate the fact. indication is not definitively in the sight of foghaha and osooliyeen, but due to the prevailing conjecture which can be achieved from that evidence. witnesses’ testimony (bayyinah) as valid evidence in the jurisprudence and law is known as an “amara” in the sight of jurists witch its validity is from legislator. ...

The crepant resolution conjecture states that the Gromov–Witten invariants of an orbifold X should be determined in a precise way by the Gromov–Witten invariants of a crepant resolution of its coarse moduli space. We compute the Gromov–Witten invariants of the stack symmetric square of P 2 and compare them with the Gromov– Witten invariants of its crepant resolution, Hilb 2 P 2 (which were comp...

A conjecture of Loebl, also known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an n-vertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi [Graph theory, combinatorics, and algorithms, Vol. 2, 1135–1146, 1995] proved an approximate version o...

Let $p=(q^4+q^3+q^2+q+1)/(5,q-1)$ be a prime number, where $q$ is a prime power. In this paper, we will show $Gcong mathrm{PSL}(5,q)$ if and only if $|G|=|mathrm{PSL}(5,q)|$, and $G$ has a conjugacy class size $frac{| mathrm{PSL}(5,q)|}{p}$. Further, the validity of a conjecture of J. G. Thompson is generalized to the groups under consideration by a new way.

A conjecture of Loebl, also known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an n-vertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an approximate version of this conjecture. We prove it exactly for sufficiently large n. This i...

Recently Jia [4] solved a conjecture of Long and showed that for any irrational number α > 0, there exist infinitely many primes not in the form 2n + 2[αn] + 1, where [x] denotes the largest integer not exceeding x. Subsequently, in [2] Banks and Shparlinski investigated the primes in the sequence {[αn + β] : n ≥ 1}. Motivated by the binary Goldbach conjecture and the twin primes conjecture, we...

Investigations of graph polynomials Mirkó Visontai Advisor: James Haglund This thesis consists of two parts. The first part is a brief introduction to graph polynomials. We define the matching, rook and hit polynomials, reveal the connection between them and show necessary conditions that imply that all roots of these polynomials are real. In the second part, we focus on the closely related Mon...

Felice Russo _\/tcr()J7 7ecil17()/o!!:1' 110/\' '.-1 re~~CIl7() (-1(1) IW(1' In this note v.e report the results regarding ,he check of the third Smarandache conjecture on primes [1 J.[2] for P ~ 2 and 2 ~ k ~ 10, In the range analysed the conjecture is true, n \loreover. according to experimental data obtained. it seems likely that the conjecture is true for all primes and for all positi\'e va...