On a theorem by Sohr for the Navier-Stokes equations

In this paper we study some criteria for the full (space-time) regularity of weak solutions to the Navier-Stokes equations. In particular, we generalize some classical and very recent criteria involving the velocity, or its derivatives. More specifically, we show with elementary tools that if a weak solution, or its vorticity, is small in appropriate Marcinkiewicz spaces, then it is regular.     

On the best constants in inequalities of the Markov and Wirtinger types for polynomials on the half-line

The main topic of the paper is best constants in Markov-type inequalities between the norms of higher derivatives of polynomials and the norms of the polynomials themselves. The norm is the L² norm with Laguerre weight. The leading term of the asymptotics of the constants is determined and tight bounds for the principal coefficient in this term, which is the operator norm of a Volterra operator, are given. For best constants in inequalities of the Wirtinger type, the limit is computed and an asymptotic formula for the error term is presented.     

On finsler spaces satisfying theT-condition

A Finsler space has been shown to satisfy theT-condition if the Finslerian metric tensor is quadratic in the unit tangent vectors. In the case where the curvature tensor of the indicatrix vanishes the converse statement is valid. The wide class of the Finslerian metric functions satisfying the condition of the quadratic dependence of the metric tensor on the unit tangent vectors, and hence theT-condition, has been found.     

On a result of Birkhoff

The research of the first two authors was supported by the NSERC of Canada.     

On distance preserving transformations of Euclidean-like planes over the rational field

Aequationes mathematicae -     

Variational principle for the Finslerian extension of general relativity

The Lagrangian density for formulating the Finslerian gravitational field equations is constructed by replacing the tangent vectors entering a direction-dependent density by the auxiliary vector field. The Lagrangian derivative is represented in terms of the tensor densities associated with an initial direction-dependent density. A particular case, where the direction-dependent density is chosen in the form of the contraction of the FinslerianK-tensor of curvature multiplied by the Jacobian, is treated in detail.     

Some new nonlinear inequalities and applications to boundary value problems

In this paper, we establish some new nonlinear integral inequalities of the Gronwall–Bellman–Ou-Iang-type in two variables. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations. We illustrate this by applying our new results to certain boundary value problem.     

On the topology of graph picture spaces

We study the space of pictures of a graph G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. For certain special graphs called orchards, the picture space is smooth and has the structure of an iterated projective bundle. We give a Borel presentation of the cohomology ring of the picture space of an orchard, and use this presentation to develop an analogue of the classical Schubert calculus.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

On the identric and logarithmic means

Summary Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b ^b /a ^a ) ^1/(b–a) , fora b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b – a)/(logb – loga), fora b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) = . In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.     

Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type

We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal Skyrmions.     

Generalized-Finslerian connection coefficients for the static spherically symmetric space

Summary The metric torsionless connection coefficients are found in an explicit way in the case of static spherically symmetric spaces defined in a generalized-Finslerian way. The connection coefficients are determined in terms of the metric tensor and its first derivatives.