A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar’s refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an Appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

     

On a Refined Stark Conjecture for Function Fields

We prove that a refinement of Stark's Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin's holds true.     

The Arc‐Weighted Version of the Second Neighborhood Conjecture

Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. Seymour's conjecture has been verified in several special cases, most notably for tournaments by Fisher  6 . One extension of the conjecture that has been used by several researchers is to consider vertex‐weighted digraphs. In this article we introduce a version of the conjecture for arc‐weighted digraphs. We prove the conjecture in the special case of arc‐weighted tournaments, strengthening Fisher's theorem. Our proof does not rely on Fisher's result, and thus can be seen as an alternate proof of said theorem.     

An Asymptotic Version of the Multigraph 1‐Factorization Conjecture

We give a self‐contained proof that for all positive integers r and all , there is an integer such that for all any regular multigraph of order 2n with multiplicity at most r and degree at least is 1‐factorizable. This generalizes results of Perkovi? and Reed (Discrete Math 165/166 (1997), 567–578) and Plantholt and Tipnis (J London Math Soc 44 (1991), 393–400).     

A Relaxed Version of the Erdős–Lovász Tihany Conjecture

The Erd?s–Lovász Tihany conjecture asserts that every graph G with ) contains two vertex disjoint subgraphs G ₁ and G ₂ such that and . Under the same assumption on G , we show that there are two vertex disjoint subgraphs G ₁ and G ₂ of G such that (a) and or (b) and . Here, is the chromatic number of is the clique number of G , and col(G ) is the coloring number of G .     

A Formulation of the Kepler Conjecture

This paper is the second in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. The top level structure of the proof is described. A compact topological space is described. Each point of this space can be described as a finite cluster of balls with additional combinatorial markings. A continuous function on this compact space is defined. It is proved that the Kepler conjecture will follow if the value of this function is never greater than a given explicit constant.     

A case of the Jacobian Conjecture

H. Bass, E.H. Connell and D. Wright settled a case for the Jacobian Conjecture. In the present paper we will give another proof of the case. Project supported by the National Natural Science Foundation of China.     

A Proof of the Molecular Conjecture

We prove the Molecular Conjecture posed by Tay and Whiteley. This implies that a graph G can be realized as an infinitesimally rigid panel-hinge framework in ℝ ^d by mapping each vertex to a rigid panel and each edge to a hinge if and only if (((d+1) || 2)-1)G\bigl({d+1 \choose 2}-1\bigr)G contains ((d+1) || 2){d+1\choose2} edge-disjoint spanning trees, where (((d+1) || 2)-1)G\bigl({d+1 \choose2}-1\bigr)G is the graph obtained from G by replacing each edge by (((d+1) || 2)-1)\bigl({d+1\choose2}-1\bigr) parallel edges.     

关于Hunt－Yorke猜想的证明

考虑具有ｎ个变时滞的泛函微分方程其中ｑ＿ｉ（ｔ），Ｔ＿ｉ（ｔ）∈Ｃ（［０，＋∞），Ｒ￣＋），ｉ＝１，２，…，ｎ。本文证明了Ｈｕｎｔ－Ｙｏｒｋｅ猜想；同时还得到了在Ｋｗｏｎｇ－Ｐａｔｕｌａ意义下泛函微分方程强振动的充分条件。     

A proof of the Gordon Conjecture

A combinatorial proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized.     

A Remark on the Rank Conjecture

We prove a result about the action of -operations on the homology of linear groups. We use this to give a sharper formulation of the rank conjecture as well as some shorter proofs of various known results. We formulate a conjecture about how the sharper formulation of the rank conjecture together with another conjecture could give rise to a different point of view on the isomorphism between and K_n^{(p)} (F)$ for an infinite field F, and we prove part of this new conjecture.     

A note on the Multiplier Conjecture

In order to identify multipliers of abelian (υ, k, λ)-difference sets the First and the Second Multiplier Theorem of Hall, Ryser and Chowla, resp. of Hall and Menon, need a divisor m of n = k − λ that is coprime to υ. Moreover, both theorems require that m > λ. The famous Multiplier Conjecture asserts that the restriction m > λ is not necessary. We present a generalization of the Second Multiplier Theorem where m is not necessarily coprime to υ. Here the requirement that m > λ generalizes to the condition m/(υ, m) > λ. This gives rise to a generalized Multiplier Conjecture which asserts that this condition is not necessary. We disprove this conjecture by showing that there exist counterexamples.     

A Note on the Monomial Conjecture

Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed.

     

A unitary test of the Ratios Conjecture

The Ratios Conjecture of Conrey, Farmer and Zirnbauer (2008) [CFZ1], (preprint) [CFZ2] predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal (Huynh and Miller (preprint) [HuyMil], Miller (2009) [Mil5], Miller and Montague (in press) [MilMo]) and symplectic (Miller (2008) [Mil3], Stopple (2009) [St]) families of L-functions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet L-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (−1,1), and for support up to (−2,2) we show agreement up to a power savings in the family's cardinality.     

A proof of the Pfister Factor Conjecture

It is shown that any split product of quaternion algebras with orthogonal involution is adjoint to a Pfister form. This settles the Pfister Factor Conjecture formulated by D.B. Shapiro. A more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quaternion algebra.     

A relaxation of the strong Bordeaux Conjecture

Let be k nonnegative integers. A graph G is ‐colorable if the vertex set can be partitioned into k sets , such that the subgraph , induced by , has maximum degree at most for . Let denote the family of plane graphs with neither adjacent 3‐cycles nor 5‐cycles. Borodin and Raspaud (2003) conjectured that each graph in is (0, 0, 0)‐colorable (which was disproved very recently). In this article, we prove that each graph in is (1, 1, 0)‐colorable, which improves the results by Xu (2009) and Liu‐Li‐Yu (2016).     

A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)^∗3, i.e. F(X)=(x₁+(a₁₁x₁+?+a_1nx_n)³,…,x_n+(a_n1x₁+?+a_nnx_n)³), satisfies TrJ((AX)^∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .     

A Helly Type Conjecture

A family of sets is Π ⁿ , or n -pierceable, if there exists a set of n points such that each member of the family contains at least one of them. It is Π _k ⁿ if every subfamily of size k or less is Π ⁿ . Helly's theorem is one of the fundamental results in Combinatorial Geometry. It asserts, in the special case of finite families of convex sets in the plane, that Π ₃ ¹ implies Π ¹ . However, there is no k such that Π _k ² implies 2 -pierceability for all finite families of convex sets in the plane. It is therefore natural to propose the following: Conjecture. There exists a k ₀ such that, for all planar finite families of convex sets , Π _k0 ² implies Π ³ . Proofs of this conjecture for restricted families of convex sets are discussed. Received October 8, 1996, and in revised form August 12, 1997.     

A Stein Conjecture for the Circle

We discuss the manner in which one might expect directional maximal functions to control the Fourier extension operator via L ² weighted inequalities. We prove a general inequality of this type for the extension operator restricted to circles in the plane.