Stability of some versions of the Prékopa–Leindler inequality

Two consequences of the stability version of the one dimensional Prékopa–Leindler inequality are presented. One is the stability version of the Blaschke–Santaló inequality, and the other is a stability version of the Prékopa– Leindler inequality for even functions in higher dimensions, where a recent stability version of the Brunn–Minkowski inequality is also used in an essential way.     

Stability of some versions of the Prékopa–Leindler inequality

We study the behavior of positive solutions of the following Dirichlet problem

$$\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}\right. $$

when s → p ^?. Here \({p >1 , s\,{\in}\,]1,p]}\) and q > p with \({q\leq\frac{Np}{N-p}}\) if N > p.

     

Approximation of Functions by the Szász-Mirakjan-Kantorovich Operator

We characterize the approximation of functions in the L_p-norm by the Szász-Mirakjan-Kantorovich operator. We prove a direct and a strong converse inequality of type B in terms of an appropriate K-functional.     

Quermassintegrals of quasi-concave functions and generalized Prékopa–Leindler inequalities

We extend to a functional setting the concept of quermassintegrals, well-known within the Minkowski theory of convex bodies. We work in the class of quasi-concave functions defined on the Euclidean space, and with the hierarchy of their subclasses given by α-concave functions. In this setting, we investigate the most relevant features of functional quermassintegrals, and we show they inherit the basic properties of their classical geometric counterpart. As a first main result, we prove a Steiner-type formula which holds true by choosing a suitable functional equivalent of the unit ball. Then we establish concavity inequalities for quermassintegrals and for other general hyperbolic functionals, which generalize the celebrated Prékopa–Leindler and Brascamp–Lieb inequalities. Further issues that we transpose to this functional setting are integral-geometric formulae of Cauchy–Kubota type, valuation property and isoperimetric/Urysohn-like inequalities.     

On Ivády’s bounds for the gamma function and related results

Periodica Mathematica Hungarica - We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all $$x\in [0,1]$$ , $$\beta \ge {\beta...     

Direct estimates of the weighted simultaneous approximation by the Szász–Mirakjan operator

Periodica Mathematica Hungarica - We establish direct estimates of the rate of weighted simultaneous approximation by the Szász–Mirakjan operator for smooth functions in the supremum...     

Approximation by the Complex form of a Link Operator Between the Phillips and the Szász–Mirakjan Operators

Results in Mathematics - The link operator $${P_{\alpha}^\rho(f,x)=\sum_{k=1}^\infty s_{\alpha,k}(x)\int_0^\infty \theta_{\alpha,k}^\rho(t)f(t)dt+e^{-\alpha x}f(0)}$$ , $${\alpha, \rho >...     

Saddlepoint approximations to the distribution of the total distance of the multivariate isotropic and von Mises–Fisher random walks

This article considers the random walk over R^p, with p ≥ 2, where the directions taken by the individual steps follow either the isotropic or the vonMises–Fisher distributions. Saddlepoint approximations to the density and to upper tail probabilities of the total distance covered by the random walk, i.e., of the length of the resultant, are derived. The saddlepoint approximations are onedimensional and simple to compute, even though the initial problem is p-dimensional. Numerical illustrations of the high accuracy of the proposed approximations are provided.     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

Some conjectures on the zeros of approximates to the Riemann ≡-function and incomplete gamma functions

Riemann conjectured that all the zeros of the Riemann ≡-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums ≡ _N (z) in Riemann’s uniformly convergent infinite series expansion of ≡(z) involving incomplete gamma functions. We conjecture that when the zeros of ≡ _N (z) in the first quadrant of the complex plane are listed by increasing real part, their imaginary parts are monotone nondecreasing. We show how this conjecture implies the RH, and discuss some computational evidence for this and other related conjectures.     

Solution of the linear and nonlinear advection–diffusion problems on a sphere

Various linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite-volume method of second-order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman-Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow-up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray-Scott reaction–diffusion model.     

Further results on the Moore–Penrose invertibility of projectors and its applications

The aim of this article is to investigate new results on the Moore–Penrose invertibility of the products and differences of projectors and generalized projectors. The range relations of projectors and the detailed representations for Moore–Penrose inverses are presented.     

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities:

$$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$

which is related to the Tzitzéica equation. Here \(h_1, h_2\) are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser–Trudinger inequality related to this problem. Finally, we give a general existence result.

     

Holomorphic Banach vector bundles on the maximal ideal space of H∞ and the operator corona problem of Sz.-Nagy

We establish triviality of some holomorphic Banach vector bundles on the maximal ideal space $M\left(H^{\infty }\right)$ of the Banach algebra $H^{\infty }$ of bounded holomorphic functions on the unit disc $\mathbb{D}\subset \mathbb{C}$ with pointwise multiplication and supremum norm. We apply the result to the study of the Sz.-Nagy operator corona problem.     

The effect of order and co‐ordination of the problem quantities on the difficulty of missing value proportion problems

One variable of the solution process of missing value proportion problems is the order co‐ordination of the missing value and the units of measure. A scheme is devised to analyse the syntactical structure of such problems in a standard self‐paced college arithmetic text. Student performance is assessed to determine the impact of this co‐ordination on problem difficulty.