Direct Constructions of Hyperplanes of Dual Polar Spaces Arising from Embeddings

Let e be one of the following full projective embeddings of a finite dual polar space Δ of rank n ≥ 2: (i) The Grassmann-embedding of the symplectic dual polar space Δ ≅ DW(2n – 1, q); (ii) the Grassmann-embedding of the Hermitian dual polar space Δ ≅ DH(2n – 1, q ²); (iii) the spin-embedding of the orthogonal dual polar space Δ ≅ DQ(2n, q); (iv) the spin-embedding of the orthogonal dual polar space Δ ≅DQ ⁻(2n + 1, q). Let H_e{\mathcal{H}_{e}} denote the set of all hyperplanes of Δ arising from the embedding e. We give a method for constructing the hyperplanes of H_e{\mathcal{H}_{e}} without implementing the embedding e and discuss (possible) applications of the given construction.     

A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.     

Blocking sets in PG(2, q n ) from cones of PG(2n, q)

Let Ω and be a subset of Σ = PG(2n−1,q) and a subset of PG(2n,q) respectively, with Σ ⊂ PG(2n,q) and . Denote by K the cone of vertex Ω and base and consider the point set B defined by

On symplectic polar spaces over non-perfect fields of characteristic 2

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n ? 1, 𝕂) be the symplectic polar space defined in PG(2n ? 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n ? 1, 𝕂) ? Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n ? 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n ? 1, 𝕂). We show that, in spite of this fact, W(2n ? 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n ? 1, 𝕂) of W(2n ? 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).     

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

Distribution of a certain partition function modulo powers of primes

In this paper, we study a certain partition function a(n) defined by Σ _n≥0 a(n)q ⁿ := Π _n=1(1 − q ⁿ )⁻¹(1 − q ²ⁿ )⁻¹. We prove that given a positive integer j ≥ 1 and a prime m ≥ 5, there are infinitely many congruences of the type a(An + B) ≡ 0 (mod m ^j ). This work is inspired by Ono’s ground breaking result in the study of the distribution of the partition function p(n).     

The Valuations of the Near Octagon {\mathbb{H}}_4

Let , n  ≥ 2, be the near 2n-gon on the 2-factors of a complete graph with 2n + 2 vertices. In this paper, we classify the valuations of the near octagon . We use this classification to study isometric full embeddings of into DQ(8,2) and DH(7,4). We show that there is up to isomorphism a unique isometric full embedding of into each of these dual polar spaces. Further applications are expected in the classification of dense near polygons with lines of size 3.     

Index sets and parametric reductions

We investigate the index sets associated with the degree structures of computable sets under the parameterized reducibilities introduced by the authors. We solve a question of Peter Cholakand the first author by proving the fundamental index sets associated with a computable set A, {e : W _e ≤ _q ^u A} for q∈ {m, T} are Σ₄ ⁰ complete. We also show hat FPT(≤ _q ⁿ ), that is {e : W _e computable and ≡ _q ⁿ ?}, is Σ₄ ⁰ complete. We also look at computable presentability of these classes. Received: 13 July 1996 / Revised version: 14 April 2000 / Published online: 18 May 2001     

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation: Δ²(x(n) − p(n)x(n −τ)) =q(n)x(g(n)). (1) In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

零点有次线性项的椭圆问题的变号解

Abstract. It is proved that the semilinear elliptic problem with zero boundary value     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).     

On a Delay Reaction-Diffusion Difference Equation of Neutral Type

In this paper, we first consider a delay difference equation of neutral type of the form: Δ(y _n + py _n−k + q _n y _n−l = 0 for n∈ℤ⁺(0) (1*) and give a different condition from that of Yu and Wang (Funkcial Ekvac, 1994, 37(2): 241–248) to guarantee that every non-oscillatory solution of (1*) with p = 1 tends to zero as n→∞. Moreover, we consider a delay reaction-diffusion difference equation of neutral type of the form: Δ₁(u _n,m + pu _n−k,m ) + q _n,m u _n−l,m = a ²Δ² ₂ u _{n +1, m−1} for (n,m) ∈ℤ⁺ (0) ×Ω, (2*) study various cases of p in the neutral term and obtain that if p≥−1 then every non-oscillatory solution of (2*) tends uniformly in m∈Ω to zero as n→∞; if p = −1 then every solution of (2*) oscillates and if p < −1 then every non-oscillatory solution of (2*) goes uniformly in m∈Ω to infinity or minus infinity as n→∞ under some hypotheses. Received July 14, 1999, Revised August 10, 2000, Accepted September 30, 2000     

Nonexistence of Ground States of -△u = u^p - u^q

The authors use the method of moving spheres to prove the nonexistence of ground states of -△u = u^p - u^q for n≥3,-∞〈p〈（n＋2）/（n-2） and q〉max （1,p）,
In fact this conclusion is a special case of -△u =f（u） for n≥2.     

Degenerate principal series and local theta correspondence II

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω ^p,q and Ω ^p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].     

Sulla funzione di Hilbert di un sottoschema zero-dimensionale diP 3

Riassunto In questo lavoro diamo una caratterizzazione aritmetica della differenza prima ΔH(X,−) della funzione di Hilbert di un sottoschema chiuso 0-dimensionaleX diP ³. Il risultato principale viene applicato per dimostrare che seX è contenuto in una completa intersezione di tipo(a, b, c), a≦b≦c allora ΔH(X, n) è decrescente pern≧a+c−2.

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Summary In this paper we give an aritmetical characterization of the first difference ΔH(X,−) of the Hilbert function of a closed 0-dimensional subschemeX ofP ³. The main result is then applied to prove that ifX is contained in a complete intersection of type(a, b, c), a≦b≦c then ΔH(X, n) is decreasing forn≧a+c−2.

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Lavoro svolto con finanziamento MPI.     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

Chern-Simons invariant and conformal embedding of a 3-manifold

This note studies the Chern-Simons invariant of a closed oriented Riemannian 3-manifold M. The first achievement is to establish the formula CS（e） - CS（e） = degA, where e and e are two （global） frames of M, and A ： M → SO（3） is the ＂difference＂ map. An interesting phenomenon is that the ＂jumps＂ of the Chern-Simons integrals for various frames of many 3-manifolds are at least two, instead of one. The second purpose is to give an explicit representation of CS（e＋） and CS（e_）, where e＋ and e_ are the ＂left＂ and ＂right＂ quaternionic frames on M3 induced from an immersion M^3 → E^4, respectively. Consequently we find many metrics on S^3 （Berger spheres） so that they can not be conformally embedded in E^4.     

Large Monotone Paths in Graphs with Bounded Degree

 We prove that for every ε>0 and positive integer r, there exists Δ₀=Δ₀(ε) such that if Δ>Δ₀ and n>n(Δ,ε,r) then there exists a packing of K _n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn ² edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let α_Δ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥α_Δ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K _n ). Received: March 15, 1999?Final version received: October 22, 1999