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# Some Landau–Kolmogorov Type Inequalities for Differential Operators Generated by Polynomials

Studia Scientiarum Mathematicarum Hungarica
Authors:
Vu Nhat Huy
,
Nguyen Ngoc Huy
, and
Chu Van Tiep
In this paper, we establish some Landau–Kolmogorov type inequalities for differential operators generated by polynomials in the following form
$P ( D ) f p ≤ K 1 ( ε , P ) f q + K 2 ( ε , m ) D m ( P ( D ) f ) p$

for all $ε > 0$ , where 0 < gp ≤ ∞, and the differential operator P (D) is obtained from the polynomial P (x) by substituting $x → − i ∂ / ∂ x$ . Moreover, the explicit form of $K 1 ( ε , p )$ and $K 2 ( ε , m )$

are given.

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# A Unified Version of Weighted Weak Type Inequality for Martingale Maximal Operators

Studia Scientiarum Mathematicarum Hungarica
Authors:
Yanbo Ren
and
Congbian Ma
Let ɣ and Φ1 be nondecreasing and nonnegative functions defined on [0, ∞), and Φ2 is an N -function, u, v and w are weights. A unified version of weighted weak type inequality of the form
$Φ 1 ( λ ) ℙ u ( f * > λ ) ≤ C 𝔼 Φ 2 C f ∞ υ γ ( λ ) w$

for martingale maximal operators f is considered, some necessary and su@cient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.

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# Weak Solutions for Obstacle Problems with Weak Monotonicity

Studia Scientiarum Mathematicarum Hungarica
Authors:
and
Elhoussine Azroul

This paper is concerned with the existence of weak solutions for obstacle problems. By means of the Young measure theory and a theorem of Kinderlehrer and Stampacchia, we obtain the needed result.

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# Limiting distributions associated with moments of exponential Brownian functionals

Studia Scientiarum Mathematicarum Hungarica
Authors:
Y. Hariya
and
M. Yor

During the last decade, a number of explicit results about the distributions of exponential functionals of Brownian motion with drift: have been obtained, often originating with the works of D. Dufresne.

In the present paper, we rely extensively on these results to show the existence of limiting measures as $T → ∞$ , when the law of ${ B t + μ t , 0 ≤ _ t ≤ _ T }$ is perturbed by the Radon-Nikodym density consisting of either of the normalized functionals exp $( − α A T ( μ ) )$ or $1 / ( A T ( μ ) ) m$ . The results exhibit different regimes according to whether in the first case, and to a partition of the $( μ , m )$ -plane in the second case.

Although a large number of similar studies have been made for, say, one-dimensional diffusions, the present study, which focuses upon Brownian exponential functionals, appears to be new.

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# Automorphisms of Finite Order of Soluble Groups of Finite Rank

Studia Scientiarum Mathematicarum Hungarica
Author:
Bertram A. F. Wehrfritz

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

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# Borel Directions and the Uniqueness of Algebroid Functions Dealing with Multiple Values

Studia Scientiarum Mathematicarum Hungarica
Authors:
Yang Tan
and
Qingcai Zhang

In this paper, we investigate the uniqueness of algebroid functions in angular domain by the method of conformal mapping. We discuss the relations between the Borel directions and uniquenss with the multiple values of algebroid functions and obtain some results which extend some uniqueness results of meromorphic functions to that of algebroid functions.

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# Estimates of Functions with Transformed Double Fourier Series

Studia Scientiarum Mathematicarum Hungarica
Author:
Boris V. Simonov

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

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# Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

Studia Scientiarum Mathematicarum Hungarica
Authors:
Jinjiang Li
,
Min Zhang
, and
Haonan Zhao

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+ s ) exceptions, all even positive integers up to N can be represented in the form $p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 6 + p 6 6 ,$

where p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are prime numbers.

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# Existence Results for a Class of p(x)-Kirchhoff Problems

Studia Scientiarum Mathematicarum Hungarica
Author:
Mostafa Allaoui

This paper is concerned with the existence of solutions to a class of p(x)-Kirchhoff-type equations with Robin boundary data as follows:

$− M ∫ Ω 1 p ( x ) ∇ u p ( x ) d x + ∫ ∂ Ω β ( x ) p ( x ) ∇ u p ( x ) d σ div( ∇ u p(x)-2 ∇ u)=f(x,u)in Ω ,$

Where $β ∈ L ∞ ( ∂ Ω )$ and $f : Ω × ℝ → ℝ$ satisfies Carathéodory condition. By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions.

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# A Note on Weakly 𐒎 -Subgroups

Studia Scientiarum Mathematicarum Hungarica
Author:
Changwen Li

The major aim of the note is to give new brief proofs of the results in the paper “The influence of weakly H -subgroups on the structure of finite groups” [Studia Scientiarum Mathematicarum Hungarica, 51 (1), 27–40 (2014)].

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