A note on an open question on ω- and τ- Matrices

In a recent paper by Engel and Schneider, it was asked if, for every n ? 1, A ∈ τ_<n> implies (A+D) ∈ τ_<n> for every D = diag[d₁, d₂,… d_n] with d_i ? 0, 1 ? i ? n. We answer this question in the negative. More precisely, we show that for, any n ? 3, the set

(τ _{< n>}): = {D ∈C^n,n:(A+D)∈τ < n> for all A∈τ_<n>} is exactly given by

(Gt_<n>) = {γI_n:γ ? 0}.     

Monotonicity of permanents of direct sums of doubly stochastic matrices

Let Ωn be the set of all n × n doubly stochastic matrices, let J_n be the n × n matrix all of whose entries are 1/n and let σ _k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω _n and A ≠ J_J then g_A,k (θ) ? σ _k ((1 θ)J_n 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A ₁ ⊕ ⊕A_t (t ≥ 2) is an n × n matrix where A_i for i = 1,2, …,t, and if for each i g_Ai,ki (θ) is non-decreasing on [0.1] for k_t = 2,3,…,n_i , then g_A,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.     

On cross t-intersecting families of sets

For all p, t with 0<p<0.11 and 1?t?1/(2p), there exists n₀ such that for all n, k with n>n₀ and k/n=p the following holds: if A and B are k-uniform families on n vertices, and |A∩B|?t holds for all A∈A and B∈B, then .     

Hasse principle for classical groups over function fields of curves over number fields

In (Letter to J.-P. Serre, 12 June 1991) Colliot-Thélène conjectures the following: Let F be a function field in one variable over a number field, with field of constants k and G be a semisimple simply connected linear algebraic group defined over F. Then the map has trivial kernel, denoting the set of places of k.The conjecture is true if G is of type 1A∗, i.e., isomorphic to SL₁(A) for a central simple algebra A over F of square free index, as pointed out by Colliot-Thélène, being an immediate consequence of the theorems of Merkurjev-Suslin [S1] and Kato [K]. Gille [G] proves the conjecture if G is defined over k and F=k(t), the rational function field in one variable over k. We prove that the conjecture is true for groups G defined over k of the types 2A∗, B_n, C_n, D_n (D₄ nontrialitarian), G₂ or F₄; a group is said to be of type 2A∗, if it is isomorphic to SU(B,τ) for a central simple algebra B of square free index over a quadratic extension k′ of k with a unitary k′|k involution τ.     

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p₁,p₂,…,p_k are distinct odd primes and a_i>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph K_n,n up to isomorphism, where t=1 if a=0, t=2^k if a=1, t=2^k+1 if a=2, and t=3·2^k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of K_n,n up to isomorphism, where s=1 if a=0, s=2^k if a=1, s=2^k+1 if a=2, and s=2^k+2 if a?3.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

Invariant hypersurfaces for derivations in positive characteristic

Let A be an integral k-algebra of finite type over an algebraically closed field k of characteristic p>0. Given a collection D of k-derivations on A, that we interpret as algebraic vector fields on , we study the group spanned by the hypersurfaces V(f) of X invariant under D modulo the rational first integrals of D. We prove that this group is always a finite dimensional F_p-vector space, and we give an estimate for its dimension. This is to be related to the results of Jouanolou and others on the number of hypersurfaces invariant under a foliation of codimension 1. As a application, given a k-algebra B between A^p and A, we show that the kernel of the pull-back morphism is a finite F_p-vector space. In particular, if A is a UFD, then the Picard group of B is finite.     

Non-uniform Turán-type problems

Given positive integers n,k,t, with 2?k?n, and t<2^k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T_?r(n,k,t) and T_r(n,k,t), where T_?r(n,k,t) (T_r(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T_?r(n,k,t)=(1+o(1))T_r(n,k,t) for fixed r,k,t with and n→∞.     

Circle grids and bipartite graphs of distances

Fort fixed,n+t pointsA ₁,A ₂,...,A _n andB ₁,B ₂,...,B _t are constructed in the plane withO(n) distinct distancesd(A _i B _j ) As a by-product we show that the graph of thek largest distances can contain a complete subgraphK _{t, n} withn=(k ²), which settles a problem of Erds, Lovász and Vesztergombi.Research partially supported by the Hungarian National Science Fund (OTKA) # 2117.     

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A₁ ⊕ A₂ with M′ ? M, there is a decomposition M′ =M₁ ⊕ M₂ such that A = M′ ⊕ [A₁ ∩ (M₁ ⊕ B)] ⊕ [A₂ ∩ (M₂ ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.     

A subdeterminant inequality for normal matrices

Let A be an n × n normal matrix over $\text{C}$, and Q_m, _n be the set of strictly increasing integer sequences of length m chosen from 1,…,n. For α, β ? Q_m, _n denote by A[α|β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k ? {0, 1,…, m} we write |α∩β| = k if there exists a rearrangement of 1,…, m, say i₁,…, i_k, i_k+1,…, i_m, such that α(i_j) = β(i_j), i = 1,…, k, and {α(i_k+1),…, α(i_m) } ∩ {β(i_k+1),…, β(i_m) } = ?. A new bound for |detA[α|β ]| is obtained in terms of the eigenvalues of A when 2m = n and |α∩β| = 0.Let $\text{U}$_n be the group of n × n unitary matrices. Define the nonnegative number

where | α ∩ β| = k. It is proved that

Let A be semidefinite hermitian. We conjecture that ρ₀(A) ? ρ₁(A) ? ··· ? ρ_m(A). These inequalities have been tested by machine calculations.     

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {α_n} satisfies ▽~3(α_n~2)≠0for all n≥2.We prove that if A and B are two simple unilateral weighted shifts,then AI+IB is reducible if and only if A and B are unitarily equivalent.We also study the reducing subspaces of A~kI+IB~l and give some examples.As an application,we study the reducing subspaces of multiplication operators Mzk+αωl on function spaces.     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

Characterizations of Buchsbaum complexes

Letk be a field and an abstract simplicial complex with vertex set . In this article we study the structure of the Ext modules Ext _a ⁱ (A/m _(l ,k[]) of the Stanley-Reisner ringk[] whereA=k[x ₁,...,x _n ] andm _l =(x _l ¹ ,...,x _l ⁿ ). Using this structure theorem we give a characterization of Buchsbaumness ofk[] by means of the length of the modules Ext _A ⁱ (A/m _l ,k[]). That isk[] is Buchsbaum if and only if for allik[], the length of the modules Ext _A ⁱ (A/m _l ,k[]) is independent ofl.     

Diameters of uniform subset graphs

Let r,s be positive integers with r>s, k a nonnegative integer, and n=2r−s+k. A uniform subset graph G(n,r,s) is a graph with vertex set [n]^r and where two r-subsets A,B∈[n]^r are adjacent if and only if |A∩B|=s. Let denote the diameter of a graph G.In this paper, we prove the following results: (1) If k>0, then if r≥2s+k+2, 2 if k≥s and 2s≤r≤s+k, or k<s and s+k≤r≤2s, and 3 otherwise; (2) If k=0, then . This generalizes a result in [M. Valencia-Pabon, J.-C. Vera, On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Edge-isoperimetric inequalities in the grid

The grid graph is the graph on [k] ⁿ ={0,...,k–1} ⁿ in whichx=(x _i ) ₁ ⁿ is joined toy=(y _i ) ₁ ⁿ if for somei we have |x _i –y _i |=1 andx _j =y _j for allji. In this paper we give a lower bound for the number of edges between a subset of [k] ⁿ of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA[k] ⁿ satisfiesk ⁿ /4|A|3k ⁿ /4 then there are at leastk ^n–1 edges betweenA and its complement.Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.We also give a best possible upper bound for the number of edges spanned by a subset of [k] ⁿ of given cardinality. In particular, forr=1,...,k we show that ifA[k] ⁿ satisfies |A|r ⁿ then the subgraph of [k] ⁿ induced byA has average degree at most 2n(1–1/r).Research partially supported by NSF Grant DMS-8806097