On a Problem of Fenchel

We give a necessary and sufficient condition for a given set of positive real numbers to be the dihedral angles of a hyperbolic n -simplex in this note. This answers a question of W. Fenchel raised in his book, Elementary Geometry in Hyperbolic Space, (De Gruyter, Berlin, 1989, p. 174) where he obtained some necessary conditions for which six numbers have to satisfy in order to be the dihedral angles of a hyperbolic tetrahedron. We also present a simple proof of the known necessary and sufficient condition for the dihedral angles of Euclidean n-simplexes.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Solving Chisini’s functional equation

We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x), . . . , G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results.     

Inverse Sturm–Liouville problem on equilateral regular tree

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0,?a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.     

Cycles and paths in edge‐colored graphs with given degrees

Sufficient degree conditions for the existence of properly edge‐colored cycles and paths in edge‐colored graphs, multigraphs and random graphs are investigated. In particular, we prove that an edge‐colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to ?(n+1)/2? has properly edge‐colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge‐colored paths and hamiltonian paths between given vertices are considered as well. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 63–86, 2010     

Some Remarks on Almost and Stable Almost Complex Manifolds

In the first part we give necessary and sufficient conditions for the existence of a stable almost complex structure on a 10-manifold M with H₁(M;?) = 0 and no 2-torsion in H₁(M;?) for i = 2,3. Using the Classification Theorem of Donaldson we give a reformulation of the conditions for a 4-manifold to be almost complex in terms of Betti numbers and the dimension of the ±-eigenspaces of the intersection form. In the second part we give general conditions for an almost complex manifold to admit infinitely many almost complex structures and apply these to symplectic manifolds, to homogeneous spaces and to complete intersections.     

On the fundamental group of real toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π₁(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forC _Δ to be aK (π, 1) space whereC _Δ is the complement of a real subspace arrangement associated to Δ.     

Vertex Angles of a Simplex in Hyperbolic Space H n

For an n-simplex in H ⁿ or S ⁿ , we give sine theorems and their results for kth-order vertex angles which are defined in this paper. The Gram and edge matrices, and polar simplex of n-simplex in H ⁿ or S ⁿ are used to prove the theorems and their results.     

Stability properties of superoptimal preconditioner from numerical range

A matrix is said to be stable if the real parts of all the eigenvalues are negative. In this paper, for any matrix A_n, we give some sufficient and necessary conditions for the stability of superoptimal preconditioner E_U(A_n) proposed by Tyrtyshnikov (SIAM J. Matrix Anal. Appl. 1992; 13 :459–473). Copyright © 2005 John Wiley & Sons, Ltd.     

On inverse spectrum problems for normal nonnegative matrices

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.     

RADICAL THEORY FOR SEMIRINGS

Abstract

In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ?_∞(S) and ?^ε(S) and two similar ideals β_∞ (S) and β^ε(S) is widely solved. We prove ?_∞(S) ? ?^ε(S) = β^∞(S) = β^ε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M_ε(U) is always semisimple, a result which is not true for the special class M_∞(U).     

An existence theorem for packing problems with implications for the computation of optimal machine schedules

Motivated by a job-shop problem we ask on which conditions n intervals v_i: with given lengths d_i , can be arranged non-overlapping on the real axis, so that every v_i is placed in a given frame [f_i s_i ].We prove a necessary and sufficient criterion analogous to the “marriage theorem” but with an additional monotony in the ages. Using this criterion we can accelerate branch and bound algorithms for job-shop scheduling by fixing partial sequences. For n = 2, the approved pair-combinatorics by Carlier and Pinson results. To show how the method works for other n. we derive the complete triple-combinatorics (n = 3: five sufficient criterions, which are necessary as a whole)     

n维双曲空间和n维球面空间中的正弦定理及应用

本文研究了n维双曲空间和n维球面空间中单形的正弦定理和相关几何不等式. 应用距离几何的理论和方法, 给出了n维双曲空间和n维球面空间中一种新形式的正弦定理, 利用建立的正弦定理获得了Hadamard 型和Veljan-Korchmaros型不等式. 另外, 建立了涉及两个n维双曲单形和n维球面单形的"度量加"的一些几何不等式.     

Properties of digraphs connected with some congruence relations

The paper extends the results given by M. Křížek and L. Somer, On a connection of number theory with graph theory, Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer n define a digraph Γ(n) whose set of vertices is the set H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a ³ ≡ b (mod n). The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph Γ(n) is proved. The formula for the number of fixed points of Γ(n) is established. Moreover, some connection of the length of cycles with the Carmichael λ-function is presented.      

Interpolating sequences in the ball ofC n

LetB be the unit ball ofC ⁿ , I give necessary conditions on sequenceS of points inB to beH ^∞(B) interpolating in term of aC ⁿ valued holomorphic function zero onS (a substitute for the interpolating Blaschke product). These conditions are sufficient to prove that the sequenceS is interpolating for ∩ _p>1 (B) and is also interpolating forH ^p (B) for 1≤p<∞.     

Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex. This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province (Grant Nos. 20070628 and Y200802999)     

Non-Zero Degree Maps Between <Emphasis Type="Italic">2n</Emphasis>-Manifolds

Abstract Thom–Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism ϕ : H ⁿ (L;Z) → H ⁿ (M;Z) to be realized by a map f : M → L of degree k for closed (n − 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M. Both authors are supported by MSTC, NSFC. The comments of F. Ding, J. Z. Pan, Y. Su and the referee enhance the quality of the paper     

Sufficient conditions for λ′‐optimality in graphs with girth g

For a connected graph the restricted edge‐connectivity λ′(G) is defined as the minimum cardinality of an edge‐cut over all edge‐cuts S such that there are no isolated vertices in G–S. A graph G is said to be λ′‐optimal if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G defined as ξ(G) = min{d(u) + d(v) ? 2:uv ∈ E(G)}, d(u) denoting the degree of a vertex u. A. Hellwig and L. Volkmann [Sufficient conditions for λ′‐optimality in graphs of diameter 2, Discrete Math 283 (2004), 113–120] gave a sufficient condition for λ′‐optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g ? 1, g being the girth of the graph, and show that a graph G with diameter at most g ? 2 is λ′‐optimal. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 73–86, 2006     

Matrix Transformations of the Maddox Vector-Valued Sequence Spaces

In this paper, we give the matrix characterizations from any normal vector-valued FK-space containing ø(X) into scalar-valued sequence space c(q) and by applying this result, we also obtain necessary and sufficient conditions for infinite matrices mapping the sequence spaces , and F_r(X,p) into the space c(q), where p = (p_k) and q = (q_k) are bounded sequences of positive real numbers and r ≥ 0.AMS Subject Classification (2000): 46A45.     

Highly edge‐connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z₊. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν₁,…,ν_r(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003