A solution to one of Kano's conjectures concerningk-th maximal spanning trees

Let (G) be the collection of all spanning trees of a connected and weighted graphG, andF ₁,F ₂, ...,F _m the partition of (G) such thatF _n the set ofi-th maximal spanning trees ofG. Kano^[1] conjectured that for anyA F ₁ and every integerk, 1km, there existsT F _k such that |T/A|k–1. This paper gives the conjecture a very simple proof, and related results.     

Kato's square root problem in Banach spaces

Let L be an elliptic differential operator with bounded measurable coefficients, acting in Bochner spaces Lp(Rn;X) of X -valued functions on Rn. We characterize Kato's square root estimates and the H^∞-functional calculus of L in terms of R-boundedness properties of the resolvent of L, when X is a Banach function lattice with the UMD property, or a noncommutative Lp space. To do so, we develop various vector-valued analogues of classical objects in Harmonic Analysis, including a maximal function for Bochner spaces. In the special case X=C, we get a new approach to the Lp theory of square roots of elliptic operators, as well as an Lp version of Carleson's inequality.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

The equivalence of two conjectures of Berge and Fulkerson

Let G be a bridgeless cubic graph. Fulkerson conjectured that there exist six 1‐factors of G such that each edge of G is contained in exactly two of them. Berge conjectured that the edge‐set of G can be covered with at most five 1‐factors. We prove that the two conjectures are equivalent. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:125‐128, 2011     

On Kato's inequality for the Weyl quantized relativistic Hamiltonian

In this note we give a simpler proof of Kato's inequality for the Weyl quantized relativistic HamiltonianH _A ^m of a spinless particle with a more singular vector potentialA(x) than in a previous paper of one of the authors. This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990     

On two congruence conjectures

In this paper, we mainly prove a congruence conjecture of M. Apagodu [3] and a supercongruence conjecture of Z.-W. Sun [25].     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some pen problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations, Please submit your problems and conjectures with all relevant information to G. Ladas.     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjecutures with all relevant informations to G. Ladas     

On steiner ratio conjectures

LetM be a metric space andP a finite set of points inM. The Steiner ratio inM is defined to be(M)=inf{L _s(P)/L _m(P) |P M}, whereL _s(P) andL _m(P) are the lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. In this paper, we study various conjectures on(M). In particular, we show that forn-dimensional Euclidean space _n ,( _n )>0.615.Supported in part by the National Science Foundation of China.     

Open problems and conjectures

In this section we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relavant information to G.Ladas     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.