Asymmetric Ramsey Properties of Random Graphs Involving Cliques

Consider the following problem: Forgiven graphs G and F(1),..., F(k), find a coloring of the edges of G with k colors such that G does not contain F; in color i. Rodl and Rucinski studied this problem for the random graph G,,, in the symmetric case when k is fixed and F(1) = ... = F(k) = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p <= bn(-beta) for some constants b = b(F,k) and beta = beta(F). This result is essentially best possible because for p >= Bn(-beta), where B = B(F, k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n(-beta(F1,..., Fk)) for arbitrary F(1), ..., F(k). In this article we address the case when F(1),..., F(k) are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G(n,p) with p <= bn(-beta) for some constant b = b(F(1),..., F(k)), where beta = beta(F(1),..., F(k)) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F,,..., Fk) such that for p >= Bn(-beta) the random graph G(n,p) a.a.s. does not have a valid k-edge-coloring provided the so-called KLR-conjecture holds. (C) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 34, 419-453, 2009

Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

A Conjecture Concerning Transformation Of A Supercooled Hard-Sphere Liquid To A Metastable Disordered Solid

It is conjectured that the hard sphere system has several distinct solid phases, all but one of which are metastable. The bifurcation theory analysis of freezing is extended to the description of the transition between a supercooled liquid and a disordered solid by defining a restricted phase space for the disordered solid. This approach leads to the prediction of a first order transition between a supercooled hard sphere fluid and a disordered metastable hard sphere solid. The results of the calculation are in qualitative agreement with the results of Woodcock's molecular dynamics computer simulations. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Curves on threefolds and a conjecture of Griffiths-Harris

We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

The Herzog-Vasconcelos conjecture for affine semigroup rings

Let S be a simplicial affine semigroup such that its semigroup ring A = k[S] is Buchsbaum. We prove for such A the Herzog-Vasconcelos conjecture: If the A-module Der(k)A of k-linear derivations of A has finite projective dimension then it is free and hence A is a polynomial ring by the well known graded case of the Zariski-Lipman conjecture.

OOCs, Partial Relative Difference Families and a Conjecture of Golomb

The cyclic difference sets constructed by Singer are also examples of perfect distinct difference sets (DDS). The Bose construction of distinct difference sets, leads to a relative difference set. In this paper we introduce the concept of partial relative DDS and prove that an optical orthogonal code (OOC) construction due to Moreno et. al., is a partial relative DDS. We generalize the concept of ideal matrices previously introduced by Kumar and relate it to the concepts of this paper. Another variation of ideal matrices is introduced in this paper: Welch ideal matrices of dimension n by (n - 1). We prove that Welch ideal matrices exist only for n prime. Finally, we recast an old conjecture of Golomb on the Welch construction of Costas arrays using the concepts of this paper. This connection suggests that our construction of partial relative difference sets is in a sense, unique

Ontheproof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

On the proof of the Poincare´ conjecture

In 2002, Perelman proved the Poincare conjecture, building on the work of Richard Hamilton on the Ricci flow. In this article, we sketch some of the arguments and attempt to place them in a broader dynamical context.

A unified approach to known and unknown cases of Berge's conjecture

Berge's elegant dipath partition conjecture from 1982 states that in a dipath partition P of the vertex set of a digraph minimizing , there exists a collection Ck of k disjoint independent sets, where each dipath P?P meets exactly min{|P|, k} of the independent sets in C. This conjecture extends Linial's conjecture, the GreeneKleitman Theorem and Dilworth's Theorem for all digraphs. The conjecture is known to be true for acyclic digraphs. For general digraphs, it is known for k=1 by the GallaiMilgram Theorem, for k?? (where ?is the number of vertices in the longest dipath in the graph), by the GallaiRoy Theorem, and when the optimal path partition P contains only dipaths P with |P|?k. Recently, it was proved (Eur J Combin (2007)) for k=2. There was no proof that covers all the known cases of Berge's conjecture. In this article, we give an algorithmic proof of a stronger version of the conjecture for acyclic digraphs, using network flows, which covers all the known cases, except the case k=2, and the new, unknown case, of k=?-1 for all digraphs. So far, there has been no proof that unified all these cases. This proof gives hope for finding a proof for all k.

On linear series and a conjecture of D. C. Butler

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H-0 (L) of dimension n+1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V circle times O-C -> L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

Beyond Schur's Conjecture

Schur 4] conjectured that the maximum length N of consecutive quadratic nonresidues modulo a prime p is less than root p if p is large enough. This was proved by Hummel in 2003. In this note, we outline a clear improvement over Hummel's bound for p > 23.

Beyond Schur's Conjecture

Schur 4] conjectured that the maximum length N of consecutive quadratic nonresidues modulo a prime p is less than root p if p is large enough. This was proved by Hummel in 2003. In this note, we outline a clear improvement over Hummel's bound for p > 23.

On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.