Extensions of the Generalized Pólya Process

Methodology and Computing in Applied Probability - A new self-exciting counting process is here considered, which extends the generalized Pólya process introduced by Cha (Adv Appl Probab...     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

On the Generalized Bochner Intergral

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On the Generalized Virasoro-like Algebras

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On the shape of random Pólya structures

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Pólya trees: a uniform random Pólya tree of size $n$ consists of a conditioned critical Galton–Watson tree $C_n$ and many small forests, where with probability tending to one, as $n$ tends to infinity, any forest $F_n\left(v\right)$, that is attached to a node $v$ in $C_n$, is maximally of size $\left|F_n\left(v\right)\right|=O(logn)$. Their proof used the framework of a Boltzmann sampler and deviation inequalities.In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for $\left|F_n\left(v\right)\right|$, namely $\left|F_n\left(v\right)\right|=\Theta (logn)$. Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Pólya tree. Third, we derive the limit probability that for a random node $v$ the attached forest $F_n\left(v\right)$ is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Pólya structures.     

On the Mixed Pólya–Szeg? Principle

In this paper,the mixed Pólya-Szeg? principle is established.By the mixed Pólya-Szeg? principle,the mixed Morrey-Sobolev inequality and some new analytic inequalities are obtained.     

On the Existence of Generalized Pólya Frequency Functions Corresponding to Entire Functions with Zeros in Angular Sectors

Generalized Pólya frequency functions are introduced through inverse Mellin transformations of the reciprocals of real entire functions with all zeros in sectors A ^? and ? A ^? for 0 ≤ ? ≤ π/4, where A ^? := {z ? C||arg z| ≤ ?}. It is shown that the constant π/4 is best possible in this context.     

On Some Geometric Properties of Generalized Calderón-Lozanovskii Spaces

Strict monotonicity, lower and upper local uniform monotonicities, rotundity, uniform rotundity and uniform non-squareness of generalized Calderón-Lozanovskii spaces are studied.     

On Weighted Estimates of High-Order Riesz-Bessel Transformations Generated by the Generalized Shift Operator

In this paper, we establish sufficient conditions on weights which ensure that high-order Riesz-Bessel transformations generated by the generalized shift operator act boundedly from one weighted Lp-space into another.     

On the Pólya Conversion Problem for Permanents and Determinants

Let $ \mathbb{F} $ be a finite field of characteristics different from 2. We show that no bijective map transforms the permanent to the determinant when the cardinality of $ \mathbb{F} $ is sufficiently large. Also we determine Gibson barriers (the maximal and minimal numbers of nonzero elements) for convertible (0, 1)-matrices and solve several related problems in different matrix subspaces. Our results are illustrated by examples. This paper is based on the joint work with G. Dolinar, B. Kuzma, and M. Orel.     

On the Asymptotic Behavior of the Storage Process Fed by a Markov Modulated Brownian Motion

Consider a storage model fed by a Markov modulated Brownian motion. We prove that the stationary distribution of the model exits and that the running maximum of the storage process over the interval [0, t] grows asymptotically like log t as t→∞.     

On the Center of the Generalized Liénard System

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)₁

On the Generalized Upwind Scheme for the Neutron Transport Equation

Let Ω be a convex domain in the (x, y) plane with boundary . Denote by n=(n_x, n_y) the outward unit normal to. Consider the following problem (1) (2) where μ, v are real parameters and σ=σ(x, y) satisfies σ≥σ_0>0. Equation (1) arises in neutron transport theory. In this paper, we study theoretically the stability and convergence properties of a discontinuous Galerkin approximation to prob-     

On Approximation of Functions by Generalized Abel–Poisson Operators

We find the exact constant in the principal term of the deviation of a function in the Zygmund class from the generalized Abel-Poisson operators determined by the Fourier series over the trigonometric system with particular summation factors.     

On the Constant Metric Dimension of Generalized Petersen Graphs P（n, 4）

In this paper,we consider the family of generalized Petersen graphs P(n,4).We prove that the metric dimension of P(n,4) is 3 when n ≡ 0(mod 4),and is 4 when n = 4k + 3(k is even).For n ≡ 1,2(mod 4) and n = 4k + 3(k is odd),we prove that the metric dimension of P(n,4) is bounded above by 4.This shows that each graph of the family of generalized Petersen graphs P(n,4)has constant metric dimension.     

On the Generalized Szász–Mirakyan Operators

In this paper, we construct sequences of Szász–Mirakyan operators which are based on a function ρ. This function not only characterizes the operators but also characterizes the Korovkin set ${\left \{ 1,\rho ,\rho ^{2} \right \}}$ in a weighted function space. We give theorems about convergence of these operators to the identity operator on weighted spaces which are constructed using the function ρ and which are subspaces of the space of continuous functions on ${\mathbb{R} ^{+}}$ . We give quantitative type theorems in order to obtain the degree of weighted convergence with the help of a weighted modulus of continuity constructed using the function ρ. Further, we prove some shape-preserving properties of the operators such as the ρ-convexity and the monotonicity. Our results generalize the corresponding ones for the classical Szász operators.     

On One Generalization of the Hardy–Littlewood–Pólya Inequality

We generalize the Hardy–Littlewood–Pólya inequality for numerical sets to certain sets of vectors on a plane.