Conservation laws of second order for the Born-Infeld equation and other related equations

We describe a class of quasilinear partial differential equations of second order with two independent variables in the general case of mixed type for which we construct conservation laws of second order which are quadratic with respect to the second derivatives. As examples, we present similar conservation laws for the Born-Infeld equation, for the equations of minimal and maximal surfaces in Minkowski space, and for the classical equation of minimal surfaces.     

On the zeros of functions in Bergman spaces and in some other related classes of functions

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f?0 is a function which belongs to the Bergman space Ap (0<p<∞) and {zk} is a sequence of zeros of f which is contained in a Stolz angle, then {zk} satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.     

The chromatic number and other functions of the lexicographic product

The values of the chromatic and achromatic number, point- and line-connectivity, and point independence number of the lexicographic product of two graphs are examined in relation to the values of the respective parameters on the factor graphs.     

The blue and minque in Gauss-Markoff model with linear transformation of the observable variables

For a singular linear model A = (y, Xβ, σ2 V) and its transformed model AF = (Fy, FXβ, σ2FVF'), where V is nonnegative definite and X can be rank-deficient,the expressions for the differences of the estimates for the vector of FXβ and the variance factor σ2 are given. Moreover, the necessary and sufficient conditions for the equalities of the estimates for the vector of FXβ and the variance factor σ2 are also established. In the meantime, works in Baksalary and Kala (1981) are strengthened and consequences in Puntanen and Nurhonen (1992), and Puntanen (1996) are extended.     

The stability of the Rogers-Shephard inequality and of some related inequalities

An essentially optimal stability version of the Rogers-Shephard inequality and stability versions of some other related affine invariant inequalities are verified. The characteristic feature of all these inequalities is that simplices provide the equality case.     

The sojourn time and related characteristics of the age-dependent branching process

An age-dependent branching process where disasters occur as a renewal process leading to annihilation or survival ofall the cells, is considered. For such a process, the total mean sojourn time of all the cells in the system is analysed using the regeneration point technique. The mean number of cells which die in timet and its asymptotic behaviour are discussed. When the disasters arrival as a Poisson process and the lifetime of the cells follows exponential distribution, elegant inter-relationships are found among the means of (i) the total number of cells which die in timet (ii) the total sojourn time of all cells in the system upto timet and (iii) the number of living cells at timet. Some of the existing results are deduced as special cases for related processes.     

The Haruki-Rassias and related integral representations of the Bernoulli and Euler polynomials

Haruki and Rassias [H. Haruki, T.M. Rassias, New integral representations for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 175 (1993) 81-90] found the integral representations of the classical Bernoulli and Euler polynomials and proved them by making use of the properties of certain functional equation. In this sequel, we rederive, in a completely different way, the results of Haruki and Rassias and deduce related and new integral representations. Our proofs are quite simple and remarkably elementary.     

The Relative Sizes of the Terms in Chebyshev and other Ultraspherical Expansions

Lanczos (1952) claims that expansion of a function in a seriesof Chebyshev polynomials is usually superior to expansion ina series of ultraspherical polynomials for any possible non-zerovalue of the parameter . We look for conditions on the functionunder which this superiority can be proved and measured; weachieve some success on the assumption that is positive, butthe results for negative are less satisfactory.     

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.     

The propagation of non-Lefschetz type, the Gottlieb group and related questions

This is a brief note which indicates how the property of being non-Lefschetz may be propagated by equivariant symplectic maps. We also discuss some questions related to the Gottlieb group and nilpotency of symplectic manifolds. To Vladimir Arnold, who taught me symplectic geometry without knowing it     

The Demyanov metric and some other metrics in the family of convex sets

We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.     

The first variation of the total mass of log-concave functions and related inequalities

On the class of log-concave functions on Rⁿ${\mathbb{R}}^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such a first variation, which suggest to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski’s first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski’s problem for convex bodies, and prove some partial results towards its solution.     

The mapping class group of powers of the long ray and other non-metrisable spaces

We identify the mapping class group, i.e. the space of homeomorphisms modulo isotopy, of powers of the long ray and long line as well as generalisations of the long plane obtained by taking copies of the first octant of the long plane and identifying them along their boundaries. We show that every countable group is the mapping class group of such a space. We also consider homotopy classes of continuous functions between these spaces.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

The reactivity of the double bond in coumarins and related unsaturated carbonyl compounds

Proceedings - Mathematical Sciences - The addition of cyanoacetamide to the double bond of 7-methoxy and 7-hydroxycoumarins is very slow and it is attributed to the influence of the substituents...