Let \Omega \in L^{s}\left ( S^{n-1} \right ) for s\geqslant 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral μΩ and b is defined by \left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}. In this paper, the author proves the \left (L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ),L^{p\left ( \cdot \right )}\left ( \mathbb{R}^{n} \right ) \right )-boundedness of the Marcinkiewicz integral operator μΩ and its commutator [b, μΩ ] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about μΩ and [b, μΩ ] on Herz spaces with variable exponent., Hongbin Wang., and Obsahuje seznam literatury
Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$, and define a random vector $Z_n C[0,t]\to\mathbb R^{n+1}$ by Z_n(x)=\biggl(x(0)+a(0), \int_0^{t_1}h(s) {\rm d} x(s)+x(0)+a(t_1), \cdots,\int_0^{t_n}h(s) {\rm d} x(s)+x(0)+a(t_n)\biggr), where $a\in C[0,t]$, $h\in L_2[0,t]$, and $0<t_1 < \cdots< t_n\le t$ is a partition of $[0,t]$. Using simple formulas for generalized conditional Wiener integrals, given $Z_n$ we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions $F$ in a Banach algebra which corresponds to Cameron-Storvick's Banach algebra $\mathcal S$. Finally, we express the generalized analytic conditional Feynman integral of $F$ as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space $C[0,t]$., Byoung Soo Kim, Dong Hyun Cho., and Obsahuje bibliografické odkazy
We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL(2, 11). Additionally we prove that the Prime graph question is true for the automorphism group of the simple group PSL(2, 13)., Joe Gildea., and Obsahuje seznam literatury