A simple characterization of the set ofμ-entropy pairs and applications

We present simple characterizations of the setsE _μ andE _X of measure entropy pairs and topological entropy pairs of a topological dynamical system (X, T) with invariant probability measureμ. This characterization is used to show that the set of (measure) entropy pairs of a product system coincides with the product of the sets of (measure) entropy pairs of the component systems; in particular it follows that the product of u.p.e. systems (topological K-systems) is also u.p.e. Another application is to show that the proximal relationP forms a residual subset of the setE _X . Finally an example of a minimal point distal dynamical system is constructed for whichE _X ∩(X ₀×X ₀)≠ , whereX ₀ is the denseG _δ subset of distal points inX.     

Estimate of the norm ∥f(A1, A2)-f(B1, B2)∥ for pairs of self-adjoint commuting operators

In this note we announce the estimate for the difference mentioned in the title, where are the lengths of intervals containing the spectra of the operators A₁, B₂ and A₂, B₂; the function f is from the class Lip 1 (with constant 1). The outline of the proof of this estimate is given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 175–177, 1984.     

KSBA surfaces with elliptic quotient singularities, π1 = 1, pg = 0, and K2 = 1, 2

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π₁ = 1, p_g = 0, and K² = 1, 2 for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants π₁ = 1, p_g = 0, and K² = 1, 2, 3, 4.     

The moduli component of the space of semistable rank-2 sheaves on ℙ3 with singularities of mixed dimension

A new irreducible component of the Gieseker–Maruyama moduli scheme M(3) of semistable coherent sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 3, and c₃ = 0 on P³ such that its general point corresponds to a sheaf whose singular locus contains components of dimensions 0 and 1 is described. These sheaves are obtained by elementary transformations of stable reflexive sheaves of rank 2 with Chern classes c₁ = 0, c₂ = 2, and c₃ = 2 along the projective line. The constructed family of sheaves is the first example of an irreducible component of a Gieseker–Maruyama scheme whose general point corresponds to a sheaf with singularities of mixed dimension.

     

The image inH 2(Q 3;ℝ) of the set of closed 2-forms with preassigned kernel

If (P²ⁿ, ω) is a symplectic manifold and Q³ is its orientable closed submanifold such that ω/Q³≠0, then there arises a one-dimensional distribution ￡=(ω/Q³). We study the dependence of ω in a neighborhood of Q³ and of [ω] ɛH ²(Q ³,ℝ) on ￡. Bibliography: 13 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 125–136.     

Comments on “Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1+A2X2B2=C”

This note points out some technical problems in the proofs of some lemmas in the above-mentioned paper, and presents their corresponding corrections. Nevertheless, the main results of that paper are still true.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.     

On the Solvability of x∈Ax＋λBx in a General Cone with A and B Multivalued k－set－contractive

ＯｎｔｈｅＳｏｌｖａｂｉｌｉｔｙｏｆｘ∈Ａｘ＋λＢｘｉｎａＧｅｎｅｒａｌＣｏｎｅ　ｗｉｔｈＡａｎｄＢＭｕｌｔｉｖａｌｕｅｄ　ｋ－ｓｅｔ－ｃｏｎｔｒａｃｔｉｖｅＺｈａｎｇＱｉｎｇｙｏｎｇ（张庆雍）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，Ｓｉｃ...     

A generalization of König-Egervary graphs and heuristics for the maximum independent set problem with improved approximation ratios

We first study a generalization of the König-Egervary graphs, the class of the κ-KE graphs, and propose an exact polynomial time algorithm solving maximum independent set problem in this class. Next, we show how this result can be efficiently used to devise polynomial time approximation algorithms with improved approximation ratios for the maximum independent set problem in general graphs.     

Radius of locally convex subsets in Alexandrov spaces with curvature ≥1 and radius 〉 π/2

Let X be a complete Alexandrov space with curvature ≥1 and radius 〉 π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius 〉 π/2.     

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.     

A Remark on Lower Bounds for the Chromatic Numbers of Spaces of Small Dimension with Metrics ℓ1 and ℓ2

Mathematical Notes - A particular class of estimates related to the Nelson–Erd?s–Hadwiger problem is studied. For two types of spaces, Euclidean and spaces with metric ?1,...     

Generalized normal form and the formal equivalence of systems with zero approximation (x 2 3 , — x 1 3 )

We consider formal systems of differential equations of the form

A family of translation planes of order q 2m+1 with two orbits of length 2 and q 2m+1−1 on l ∞

A class of translation planes of order q ^2m+1, where q is an odd prime power and m1, is constructed. If m=1, then this class is contained in the class of order q ³ constructed by Hiramine [5]. These planes of order q ^2m+1 are of dimension 2m+1 over their kernels. If q ^2m+13³, then the linear translation complements of these planes have two orbits of length 2 and q ^2m+1–1 on l and this class contains many planes which are not generalized André planes. If q ^2m+1= 3³, then each plane of this class is isomorphic to the Hering plane of order 27.Dedicated to Professor Tuyosi Oyama on his 60th birthday     

Evaluation of the Assistant Function with Four Zeros and all Nonprincipal Characters but X1X2X3X4

To obtain a better Vinogradov's bound, assistant function with two or three zeros and all Dirichlet's characters has been evaluated. Assistant function g(X1,X2,X3,X4)with four zeros and all nonprincipal characters but X1X2X3X4 is constructed and evaluated in the paper. And the constant obtained is 8.395 511 841 994.     

On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

Let w and µ be respectively the conditional Wiener measure in C ₀([0, 1]) and the centered Gaussian measure in L ₂[0, 1] with the correlation operator (?d ²/dx ²)^?1. We prove the equivalence of these two measures in the following sense: for any Borel set A ? L ₂[0, 1] the set A ∩ C ₀([0, 1]) is a Borel subset of C ₀([0, 1]) and µ(A) = w(A∩C ₀([0, 1])).     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.