RECONSTRUCTION OF BINARY RELATIONS FROM THEIR RESTRICTIONS OF CARDINALITY 2, 3, 4 and (n - 1) II

We shall prove here that any binary relation on a base E with cardinality n > 6 is reconstructible from its restrictions of cardinality 2, 3, 4 and (n - 1). This proof needs results of part I of this paper where we characterize any pair of relations R, R' which are 2-, 3- and 4-hypomorphic. As a corollary we obtain that any binary relation is (n - 4)-reconstructible (when n > 9).     

Construction of nonnegative matrices and the inverse eigenvalue problem

This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ₁,λ₂,λ₃,σ) from a given realizable list of n complex numbers (c ₁,c ₂,σ), where c ₁ is the Perron eigenvalue, c ₂ is a real number and σ is a list of n???2 complex numbers.     

Triangular embeddings of K((i-2) n,n,…,n)

For complete i-partite graphs of the form K(n₁, n, n, …, n) the largest value of n₁ that allows the graph to be triangularly-embedded into a surface is (i-2)n. In this paper the author constructs triangular embeddings into surfaces of some complete partite graphs of the form K((i-2)n, n, …, n). The embeddings are exhibited using embedding schemes but the surfaces into which K((i-2)n, n, …, n) are triangularly embedded can be seen to be particularly nice branched covers of a surface into which K(i-2, 1, 1,…,1) is triangularly embedded.     

Optimal (2, n) visual cryptographic schemes

In (2,n) visual cryptographic schemes, a secret image(text or picture) is encrypted into n shares, which are distributed among n participants. The image cannot be decoded from any single share but any two participants can together decode it visually, without using any complex decoding mechanism. In this paper, we introduce three meaningful optimality criteria for evaluating different schemes and show that some classes of combinatorial designs, such as BIB designs, PBIB designs and regular graph designs, can yield a large number of black and white (2,n) schemes that are optimal with respect to all these criteria. For a practically useful range of n, we also obtain optimal schemes with the smallest possible pixel expansion.     

Generating Sets of Finite Transformation Semigroups PK(n,r) and K (n,r)

Let P _n and T _n be the partial transformation and the full transformation semigroups on the set {1,…, n}, respectively. In this paper we find necessary and sufficient conditions for any set of partial transformations of height r in the subsemigroup PK(n, r) = {α ∈P _n : |im (α)| ≤r} of P _n to be a (minimal) generating set of PK(n, r); and similarly, for any set of full transformations of height r in the subsemigroup K(n, r) = {α ∈T _n : |im (α)| ≤r} of T _n to be a (minimal) generating set of K(n, r) for 2 ≤ r ≤ n ? 1.     

A universality theorem for the Riemann zeta-function with respect to Beatty sets

We prove that, for given positive numbers α and h, the Riemann zeta-function ζ can approximate any nonvanishing analytic function on a simply connected compact subset of the right open half of the critical strip by shifts of type ζ(s?+?ih?αn?).

     

The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX?+?BX ^??=?0, where A, B?∈?? ^m×n are two given matrices, X?∈?? ^n×n is an unknown matrix, and ? denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A?+?λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A?+?λB to the canonical form by strict equivalence.     

A note on marking lines in [k] n

In ${[k]^n=[k]{\times}[k]{\times}\cdots{\times}[k]}$ , a coordinate line consists of the collection of points where all but one coordinate is fixed and the unfixed coordinate varies over all possibilities. We consider the problem of marking (or designating) one point on each line in [k] ⁿ so that each point in [k] ⁿ is marked either a or b times, for some fixed a or b. This is equivalent to forming a strategy for a hat guessing game for n players with k different colors of hats where the number of correct guesses, regardless of hats placed, is either a or b. If we let s?≥?0 and t?≥?0 denote the number of vertices marked a and b times respectively, then we have the following obvious necessary conditions: s?+?t?=?k ⁿ (the number of points) and as?+?bt?=?nk ^n–1 (the number of lines). Our main result is to show for n?≤ 5, and k arbitrary, that these necessary conditions are also sufficient.     

The multiplier conjecture for the case n = 4n1

In this article we prove that in the case n = 4n₁ if (v,2 · 7 · 31) = 1 and v is not divisible by 15, then the Second Multiplier Theorem holds without the assumption n₁ > λ. This improves a result due to McFarland. © 1995 John Wiley & Sons, Inc.     

Embedding the Outer Automorphism Group Out(F n ) of a Free Group of Rank n in the Group Out(F m ) for m>n

It is proved that for every n 1, the group Out(F _n )is embedded in the group Out(F _m ) with m=1+(n-1)k ⁿ , where k is an arbitrary natural number coprime to n-1.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

RECONSTRUCTION OF BINARY RELATIONS FROM THEIR RESTRICTIONS OF CARDINALITY 2, 3, 4 and (n - 1) I

We prove that any binary relation with underlying set (base) E with cardinality n > 6 is reconstructible from its restrictions of cardinality 2, 3, 4 and (n - 1). In part I we characterize relations R and R' on the same base E such that R/X and R'/X are isomorphic for every subset X of E with cardinality 2, 3, 4. In part II we shall prove that R and R' are isomorphic as soon as n > 6 when R/X and R/X' are isomorphic for every subset X of E with cardinality 2, 3, 4 and (n - 1).     

Finite groups with the set of the number of subgroups of possible order containing exactly two elements

Let G be a finite group, and n(G) be the set of the number of subgroups of possible order of G. We investigate the structure of G satisfying that n(G)?=?{1, m} for any positive integer m?>?1. At first, we prove that the nilpotent length of G is less than 2. Secondly, we investigate nilpotent groups with m?=?p?+?1 or p ²?+?p?+?1 (p is a prime), and we get the classification of such kinds of groups. At last, we investigate non-nilpotent groups with m?=?p?+?1 and get the classification of the groups under consideration.     

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

Moebius geometry of submanifolds in ? n

In this paper we define a Moebius invariant metric and a Moebius invariant second fundamental form for submanifolds in ? ⁿ and show that in case of a hypersurface with n≥ 4 they determine the hypersurface up to Moebius transformations. Using these Moebius invariants we calculate the first variation of the moebius volume functional. We show that any minimal surface in ? ⁿ is also Moebius minimal and that the image in ? ⁿ of any minimal surface in ℝ ⁿ unter the inverse of a stereographic projection is also Moebius minimal. Finally we use the relations between Moebius invariants to classify all surfaces in ?³ with vanishing Moebius form. Received: 18 November 1997     

Construction of representation-finite selfinjective artin algebras of classB n andC n

In this paper, we discuss the representation-finite selfinjective artin algebras of classB _n andC _n and obtain the following main results: For any fieldk, let Λ be a representation-finite selfinjective artin algebras of classB _n orC _n overk.

3·2 r+1 -Cycle Systems of K n −C n

 In this paper we investigate the necessary and sufficient conditions for the existence of 3 · 2 ^{r +1}-cycle system of K _n – C _n , where C _n is any Hamilton cycle of K _n and r ≥ 0. Received: December 17, 1999 Final version received: July 25, 2000 Present address: 15 Guang Ming 10 St. Sec. 1, Chu-Bei, Hsin-Chu, 302, Taiwan, ROC     

On Liouville’s theorem for locally quasiregular mappings inR n

Letf:R ⁿ→Rⁿ be locally quasiregular in the sense that the restriction off to any ball |x|<r has finite inner dilatationK ₁(r). Suppose that the growth condition ∫^∞r^-1K₁(r)^1/(1-n) holds. Then Liouville’s theorem is valid:If f is bounded, f is a constant. An example shows that this growth condition is relatively sharp.