着色的琼斯函数(英文)

In this paper,we discuss the properties of the colored Jones function of knots.Particularly,we calculate the colored Jones function of some knots（3₁,4₁,5₁,5₂）.Furthermore,one can compute the Kashaev’s invariants and study some properties of the Kashaev’s conjecture.     

Rigidity of Minimizers in Nonlocal Phase Transitions Ⅱ

In this paper we extend the results of[12]to the borderline case s=We obtain the classification of global bounded solutions with asymptotically flat level sets for semilinear nonlocal equations of the type△^1/2u=W′(u)in R^n,where VV is a double well potential.     

Covering complete partite hypergraphs by monochromatic components

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson (1971)) states that $k$-partite intersecting hypergraphs have transversals of at most $k?1$ vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in Gyárfás (1977): if the edges of a complete graph $K$ are colored with $k$ colors then the vertex set of $K$ can be covered by at most $k?1$ sets, each forming a connected graph in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: it was proved in Király (2013) that in every $k$-coloring of the edges of the $r$-uniform complete hypergraph $K^r$ ($r\ge 3$), the vertex set of $K^r$ can be covered by at most $?k∕r?$ sets, each forming a connected hypergraph in some color.Here we investigate the analogue problem for complete $r$-uniform $r$-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of every color used in the coloring. We propose the following analogue of Ryser’s conjecture.In every spanning $(r+t)$-coloring of the edges of a complete $r$-uniform $r$-partite hypergraph, the vertex set can be covered by at most $t+1$ sets, each forming a connected hypergraph in some color.We show that the conjecture (if true) is best possible. Our main result is that the conjecture is true for $1\le t\le r?1$. We also prove a slightly weaker result for $t\ge r$, namely that $t+2$ sets, each forming a connected hypergraph in some color, are enough to cover the vertex set.To build a bridge between complete $r$-uniform and complete $r$-uniform $r$-partite hypergraphs, we introduce a new notion. A hypergraph is complete $r$-uniform $(r,?)$-partite if it has all $r$-sets that intersect each partite class in at most $?$ vertices (where $1\le ?\le r$).Extending our results achieved for $?=1$, we prove that for any $r\ge 3,2\le ?\le r,k\ge 1+r??$, in every spanning $k$-coloring of the edges of a complete $r$-uniform $(r,?)$-partite hypergraph, the vertex set can be covered by at most $1+?\frac{k?r+??1}{?}?$ sets, each forming a connected hypergraph in some color.     

Involutive automorphism of symmetric groups

Let (𝔖_n,S) be a Coxeter system of the symmetric group, we show that the set of automorphisms of 𝔖_n which are involutions and leave S stable is a finite group of order less than 3.     

Zero divisors and units with small supports in group algebras of torsion-free groups

Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽[G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽₂ is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.     

Connectivity keeping stars or double-stars in 2-connected graphs

In Mader (2010), Mader conjectured that for every positive integer $k$ and every finite tree $T$ with order $m$, every $k$-connected, finite graph $G$ with $\delta \left(G\right)\ge ?\frac{3}{2}k?+m?1$ contains a subtree $T^{\prime }$ isomorphic to $T$ such that $G?V\left(T^{\prime }\right)$ is $k$-connected. In the same paper, Mader proved that the conjecture is true when $T$ is a path. Diwan and Tholiya (2009) verified the conjecture when $k=1$. In this paper, we will prove that Mader’s conjecture is true when $T$ is a star or double-star and $k=2$.     

The Bateman–Horn conjecture: Heuristic,history, and applications

The Bateman–Horn conjecture is a far-reaching statement about the distribution of the prime numbers. It implies many known results, such as the prime number theorem and the Green–Tao theorem, along with many famous conjectures, such the twin prime conjecture and Landau’s conjecture. We discuss the Bateman–Horn conjecture, its applications, and its origins.     

Freeness for 13 lines arrangements is combinatorial

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane.     

Dade's Invariant Conjecture for the Symplectic Group Sp4(2 n ) and the Special Unitary Group SU4(22n ) in Defining Characteristic

In this article, we verify Dade's projective invariant conjecture for the symplectic group Sp₄(2 ⁿ ) and the special unitary group SU₄(2²ⁿ ) in the defining characteristic, that is, in characteristic 2. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) in the defining characteristic, that is, Sp₄(2 ⁿ ) and SU₄(2²ⁿ ) are good for the prime 2 in the sense of Isaacs, Malle, and Navarro.     

Improving the Clark–Suen bound on the domination number of the Cartesian product of graphs

A long-standing Vizing’s conjecture asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers; one of the most significant results related to the conjecture is the bound of Clark and Suen, $\gamma (G\square H)\ge \gamma \left(G\right)\gamma \left(H\right)∕2$, where $\gamma$ stands for the domination number, and $G\square H$ is the Cartesian product of graphs $G$ and $H$. In this note, we improve this bound by employing the 2-packing number $\rho \left(G\right)$ of a graph $G$ into the formula, asserting that $\gamma (G\square H)\ge (2\gamma \left(G\right)?\rho \left(G\right))\gamma \left(H\right)∕3$. The resulting bound is better than that of Clark and Suen whenever $G$ is a graph with $\rho \left(G\right)<\gamma \left(G\right)∕2$, and in the case $G$ has diameter 2 reads as $\gamma (G\square H)\ge (2\gamma \left(G\right)?1)\gamma \left(H\right)∕3$.     

Altermatic number of categorical product of graphs

In this paper, we prove some relaxations of Hedetniemi’s conjecture in terms of altermatic number and strong altermatic number of graphs, two combinatorial parameters introduced by the present authors Alishahi and Hajiabolhassan (2015) providing two sharp lower bounds for the chromatic number of graphs. In terms of these parameters, we also introduce some sharp lower bounds for the chromatic number of the categorical product of two graphs. Using these lower bounds, we present some new families of graphs supporting Hedetniemi’s conjecture.     

On a generalization of starlike functions

In this paper, we mainly study the order of $q$-starlikeness of the well-known basic hypergeometric function. In addition, we discuss the Bieberbach-type problem and the second order Hankel determinant for a generalized class of starlike functions.