On some algebras diagonalized by M-bases of ℓ2

We study reflexive algebrasA whose invariant lattices LatA are generated by M-bases of ². Examples are given whereA differs from ( being the rank one subalgebra ofA), and where together with the identity I is not strongly dense inA. For M-bases in a special class, we characterize the cases when they are strong, and also when the identity I is the ultraweak limit of a sequence of contractions in . We show that this holds provided that I is approximable by compact operators inA at any two points of ². We show that the spaceA+^* (where is the annihilator of ) is ultraweakly dense in (²), and characterize the M-bases in this class for which the sum is direct. We give a class of automorphisms ofA which are strongly continuous but not spatial.     

On some properties of the Navier–Stokes system of equations

We discuss the initial and boundary value problems for the system of dimensionless Navier–Stokes equations describing the dynamics of a viscous incompressible fluid using the method of characteristics and the geometric method developed by the authors. Some properties of the formulation of these problems are considered. We study the effect of the Reynolds number on the flow of a viscous fluid near the surface of a body.     

On complemented copies of c 0 and ℓ ∞

We furnish examples of pairs of Banach spaces X, Y so that none of c ₀ and l _∞ live inside X ^? and Y, but they embed complementably into the space DP(X,Y) of the Dunford–Pettis operators from X into Y.     

On the directional bias of the ℓbp-norm

The weighted sum of order p, the ℓ_bp-norm, is a generalization of the well-known ℓ_p-norm used in predicting distances in a transportation network. The properties of the directional bias function and the unit balls for the ℓ_bp-norm are of theoretical and practical interest. We investigate these properties and compare them with the properties of the ℓ_p-norm's directional bias function and the unit balls. We find that the ℓ_bp-norm is better at capturing the nonlinearity in a transportation network than the weighted ℓ_p-norm. It is also shown that, in contrast to the weighted ℓ_p-norm, where the optimal parameter p value is confined to the interval (1,2), for the ℓ_bp-norm the parameter p can have an optimal value greater than 2.     

On some properties of the ε-core of games with coalition structure

Chang [1991] stated some inclusion relation about kernel, reasonable set, and ε-core of games with coalition structure. We give counterexamples and modify erroneous results. Received: May 1997/Revised version: January 1999     

On the Schatten–von Neumann properties of some pseudo-differential operators

We obtain a number of explicit estimates for quasi-norms of pseudo-differential operators in the Schatten–von Neumann classes _Sq${\mathfrak{S}}_q$ with 0<q?1$0<q?1$. The estimates are applied to derive semi-classical bounds for operators with smooth or non-smooth symbols.     

On some statistical properties of the “Book Stack” transformation

This paper is devoted to studying the statistical properties of the “Book Stack” transformation proposed by B.Ya. Ryabko (Probl. Inf. Transm., 1980, vol. 16, no. 4) as a data compression method. The same transformation has been used by Ryabko and A.I. Pestunov (Probl. Inf. Transm., 2004, vol. 40, no. 1) to construct the similarly named statistical test. This test is designed for the verification of the null hypothesis that an available input i.i.d. sample corresponds to a discrete uniform distribution with a known support. They propose to verify this hypothesis for a new sample obtained via the Book Stack transformation instead of the input sample. This gives rise to the natural problem of comparing the results given by the same statistical test in the application to input and output samples. If the null hypothesis is true, these procedures prove to be equivalent; however, this is actually not the case anymore when there are some violations of this hypothesis. The results of comparing the criteria surely depend on the class of the alternatives considered. This paper deals with the natural alternative consisting of the fact that the initial replicated sample corresponds to a discrete, albeit, nonuniform, distribution with a fixed support. It has been demonstrated that some standard criteria for the verification of the null hypothesis prove to be more powerful for an input sample in comparison with a transformed sample. In particular, this takes place for the likelihood ratio criterion and (with some formal constraints) the χ²-criterion.     

On the composition subalgebras of Bridgeland’s Hall algebras

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.     

On some Kähler manifolds with Norden metric

We investigate the Kähler manifolds with Norden metric whose curvature tensor can be expressed in the terms of Ricci tensors only and whose holomorphically conformal curvature tensor vanishes.     

On the Structure of ∞-Harmonic Maps

Let H ∈ C ²(? ^N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E _∞(u, Ω) = ‖H(Du)‖ _{L ^∞(Ω)} defined on maps u: Ω ? ? ⁿ  → ? ^N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? ^N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.     

On some homotopical and homological properties of monoid presentations

If a monoid S is given by some finite complete presentation ℘, we construct inductively a chain of CW-complexes

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator T _p [u] := tr _p u _xx on the set ${\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}$ and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in ${C^2(\bar{\Omega})}$ for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I _p [u; f] in ${C^2(\bar{\Omega})}$ and we derive some applications.     

On the metric structure of non-Kähler complex surfaces

We give a characterization of a locally conformally K?hler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric. Received: 23 July 1998 / Revised: 2 June 1999     

On the λ-means of some trigonometric series

For functions λ differentiable on (0, 1) there are examined the means (2) of some trigonometric series (1). Section 2 contains a few basic estimates. In Section 3 the author presents two main theorems, which correspond to Propositions 1, 2 of [3].     

On the λ-means of some trigonometric series

For functions λ differentiable on (0, 1) there are examined the means (2) of some trigonometric series (1). Section 2 contains a few basic estimates. In Section 3 the author presents two main theorems, which correspond to Propositions 1, 2 of [3].