Finite Projective Geometries and Classification of the Weight Hierarchies of Codes （Ⅰ）

The weight hierarchy of a binary linear [n,κ] code C is the sequence (d1,d2,...,dκ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ).     

既不含4-圈又不含6-圈的平面图的非正常染色

设d₁, d₂,..., d_k是k个非负整数. 若图G=(V,E)的顶点集V能被剖分成k个子集V₁, V₂,...,V_k, 使得对任意的i=1, 2,..., k, V_i的点导出子图G[V_i] 的最大度至多为d_i, 则称图G是(d₁, d₂,...,d_k)-可染的. 本文证明既不含4-圈又不含6-圈的平面图是(3, 0, 0)-和(1, 1, 0)-可染的.     

Estimate of a complete rational trigonometric sum

Supposef is a polynomial of degree n≥3 with integral coefficientsa ₀,a ₁,...,a _n; q is a natural number; (a ₁,...,a _n, q)=1,f(0) = 0. It is proved that $$\left| {\sum\nolimits_{x = 1}^q {e^{2\pi if(x)/q} } } \right|< e^{5n^2 /\ln n} q^{1 - 1/n} $$ .     

Cyclic Just-In-Time Sequences Are Optimal

Consider the Product Rate Variation problem. Given n products 1,...,i,...,n, and n positive integer demands d ₁,..., d_i,...,d_n. Find a sequence =₁,...,_T, T = _i=1 ⁿ d _i, of the products, where product i occurs exactly d _i times that always keeps the actual production level, equal the number of product i occurrences in the prefix ₁,..., _t, t=1,...,T, and the desired production level, equal r _i t, where r _i=d_i/T, of each product i as close to each other as possible. The problem is one of the most fundamental problems in sequencing flexible just-in-time production systems. We show that if is an optimal sequence for d ₁,...,d_i,...,d_n, then concatenation ^m of m copies of is an optimal sequence for md ₁,..., md_i,...,md_n.     

Regular sequences in Rees and symmetric algebras,II

In continuing [7] we study necessary and sufficient conditions for a system of elements b₁,...,b_s,a₁,...,a_t of a local Noetherian ring A such that the sequence b₁,...,b_s,a₁T,a₁-a₂T,...,a_t–1-a_tT,a_t in the Rees algebra A[a₁ T,...,a_t T], T is an indeterminate, constitutes a regular sequence.     

The Method of Extremal Metric in Extremal Decomposition Problems with Free Parameters

Let a₁,..., a_n be a system of distinct points on the z-sphere , and let be a system of all non-overlapping simply-connected domains D₁,..., D_n on such that a_k ∈ D_k, k = 1,..., n. Let M (D_k, a_k) be the reduced module of the domain D_k with respect to the point a_k ∈ Dk. In the present paper, we solve some problems concerning the maximum of weighted sums of the reduced modules M (D_k, a_k) in certain families of systems of domains {D_k} described above, where the systems of points {a_k} satisfy prescribed symmetry conditions. In each case, the proof is based on an explicit construction of an admissible metric of the module problem, which is equivalent to the extremal problem under consideration, from known extremal metrics of simpler module problems. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 52–67.     

A remark on the intersection arrays of distance regular graphs and the distance regular graphs of diameter d = 3i − 1 with b i = 1 and k > 2

Let be a distance regular graph with intersection array b ₀, b ₁,..., b _d–1; c ₁,..., c _d. It is shown that in some cases (c _i–1, a _i–1, b _i–1) = (c ₁, a ₁, b ₁)and (c _2i–1, a _2i–1, b _2i–1) imply k 2b _i + 1. As a corollary all distance regular graphs of diameter d = 3i – 1 with b _i = 1 and k > 2 are determined.     

Accuracy of several multidimensional refinable distributions

Compactly supported distributions f₁,..., f_r on ℝ^d are fefinable if each f_i is a finite linear combination of the rescaled and translated distributions f_j(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝^d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σ_k∈Λ c_k f(Ax−k), where Λ is a finite subset of Γ, the c_k are r×r matrices, and f=(f₁,...,f_r)^T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f₁,...,f_r along the lattice Γ. We determine the accuracy p from the matrices c_k. Moreover, we determine explicitly the coefficients y_α,i(k) such that x^α=Σ _i=1 ^r Σ_k∈Γy_α,i(k) fi(x+k). These coefficients are multivariate polynomials y_α,i(x) of degree |α| evaluated at lattice points k∈Γ.     

Optimal holey packings OHP4(2, 4, n ,3)'s

An optimal holey packing OHP_d(2, k, n, g) is equivalent to a maximal (g + 1)‐ary (n, k, d) constant weight code. In this paper, we provide some recursive constructions for OHP_d(2, k, n, g)'s and use them to investigate the existence of an OHP₄(2, 4, n, 3) for n ≡ 2, 3 (mod 4). Combining this with Wu's result ( 18 ), we prove that the necessary condition for the existence of an OHP₄(2, 4, n, 3), namely, n ≥ 5 is also sufficient, except for n ∈ {6, 7} and except possibly for n = 26. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 111–123, 2006     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains in a circle

Let a_k, k=1,2,3, be distinct points of the circle U={z:¦z¦<1}, a_3+k=1/¯a_k, k=1,2,3. Let D₁,...,D₆ be a system of nonoverlapping simply connected domains D₁,...,D₆ on,a_k D_k, k=1,...,6. Let R(D_k,a_k) be the conformal radius of the domain D_k with respect to the point a_k. One formulates the following theorem. For any points a_k U, k=1,2,3, and any system of the indicated domains one has the sharp inequality One points out all the cases when equality prevails in (1). One indicates the main steps of the proof of this theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 99–113, 1983.     

The Erdös-Jacobson-Lehel conjecture on potentiallyP k-graphic sequence is true

A variation in the classical Turan extrernal problem is studied. A simple graphG of ordern is said to have propertyPk if it contains a clique of sizek+1 as its subgraph. Ann-term nonincreasing nonnegative integer sequence π=(d₁, d₂,⋯, d₂) is said to be graphic if it is the degree sequence of a simple graphG of ordern and such a graphG is referred to as a realization of π. A graphic sequence π is said to be potentiallyP _k-graphic if it has a realizationG having propertyP _k . The problem: determine the smallest positive even number σ(k, n) such that everyn-term graphic sequence π=(d₁, d₂,…, d₂) without zero terms and with degree sum σ(π)=(d ₁+d ₂+ …+d ₂) at least σ(k,n) is potentially P_k-graphic has been proved positive. Project supported by the National Natural Science Foundation of China (Grant No. 19671077) and the Doctoral Program Foundation of National Education Department of China.     

Fractional dimension of partial orders

Given a partially ordered setP=(X, ), a collection of linear extensions {L ₁,L ₂,...,L _r } is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someL _i (andy<x in someL _j ). For a positive integerk, we call a multiset {L ₁,L ₂,...,L _t } ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theL _i 's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask. We prove various results about the fractional dimension of a poset.Research supported in part by the Office of Naval Research.     

TheO(n3) algorithm for a special case of the maximum cost-to-time ratio cycle problem and its coherence with an eigenproblem of a matrix

Let twon×n matrices be given, namely a real matrixA=(a_ij) and a (0, 1)-matrixT=(t_ij). For a cyclic permutation=(i ₁,i ₂,...,i _k) of a subset of N={1, 2, ..., n} we define _A;T(), the cost-to-time ratio weight of, as . This paper presents an O(n³) algorithm for finding (A;T)=max _A;T(), the maximum cost-to-time ratio weight of the matricesA andT. Moreover a generalised eigenproblem is proposed.     

On a More General Characterisation of Steiner Systems

We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let be a [k,2(k−t+1)]-sparse geometry of cardinality n, with k \> t \> 1$$" align="middle" border="0"> . If , then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let and link every block with the code word where c_i=1 if and only if the point p_i is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries).     

Complexity of solution of linear systems in rings of differential operators

Suppose given a k₁×k₂ system of linear equations over the Weyl algebraA _n = F[X₁,...X₁,D₄,...,D_n] or over the algebra of differential operatorsK _n = F[X₁,...X₁,D₄,...,D_n], where the degree of each coefficient of the system is less than d. It is proved that if the system is solvable overA _n, orK _n, respectively, then it has a solution of degree at most (k, d)²⁰⁽ⁿ⁾.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 192, pp. 47–59, 1991.     

Residue races

Given a prime p and distinct non-zero integers a₁, a₂,..., a_k (mod p), we investigate the number N = N(a₁, a₂,..., a_k; p) of residues n (mod p) for which (na₁)_p < (na₂)_p < ... < (na_k)_p, where (b)_p is the least non-negative residue of b (mod p). We give complete solutions to the problem when k = 2,3,4, and establish some general results corresponding to k≥5. The first author is a Presidential Faculty Fellow. He is also supported, in part, by the National Science Foundation. 2000 Mathematics Subject Classification Primary—11A07; Secondary—11F20     

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ? ⁿ ,n>1, is said to be diagonal if there existn functions,a ₁,...,a _n , on some intervals of ?, such that the covariance matrixV _F (m) ofF has diagonal (a ₁(m ₁),...,a _n (m _n )), for allm=(m ₁,...,m _n ) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ? ^k and ? ^n-k , for somek=1,...,n?1. This paper shows that there are only six types of irreducible diagonal NEFs in ? ⁿ , that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ? ⁿ , under what conditions is its projectionp(F) in ? ^k , underp(x ₁,...,x _n )∶=(x ₁,...,x _k ),k=1,...,n?1, still an NEF in ? ^k ? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofV _F (m ₁,...,m _n ) does not depend on (m _k+1,...,m _n ).     

Oscillatory solutions of nonhomogeneous linear differential equations

The complex oscillation of nonhomogeneous linear differential equations with transcendental coefficients is discussed. Results concerning the equation f ^(k)+a _k−1 f ^(k−1)+...+a ₀ f=F where a ₀,...,a _k−i and Fare entire functions, possessing an oscillatory solution subspace in which all solutions (with at most one exception) have infinite exponent of convergence of zeros are obtained. All solutions of the equation are also characterized when the coefficients a ₀,a ₁,...,a _k−1 are polynomials and F=h exp (p ₀), where p ₀ is a polynomial and h is an entire function. Author supported by Max-Planck-Gesellschaft and by NSFC.     

Open Caps and Cups in Planar Point Sets

Points p₁,p₂,...,p_k in the plane, ordered in the x-direction, form a k-cap (k-cup}, respectively) if they are in convex position and p₂,...,p_k-1 lie all above (below, respectively) the segment p₁p_k. We prove the following generalization of the Erdos-Szekeres theorem. For any k, any sufficiently large set P of points in general position contains k points, p₁1,p₂,...,p_k, that form either a k-cap or a k-cup, and there is no point of P vertically above the polygonal line p₁p₂···p_k. We give double-exponential lower and upper bounds on the minimal size of P. We also give several related results.     

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.