The article is devoted to the studies of the existence of solutions of an integro-differential equation in the case of the double scale anomalous diffusion with the sum of the two negative Laplacians raised to two distinct fractional powers in ({mathbb R}^{d}, d=4, 5). The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for the non-Fredholm elliptic operators in unbounded domains are used.
In this paper, we study those mappings in unit ball satisfying the Dirichlet problem of the following differential operators
$$begin{aligned} Delta _{gamma }=big (1-|x|^{2}big )cdot left[ frac{1-|x|^{2}}{4}cdot sum _{i}frac{partial ^{2}}{partial x_{i}^{2}}+gamma sum _{i}x_{i}cdot frac{partial }{partial x_{i}}+gamma left( frac{n}{2}-1-gamma right) right] . end{aligned}$$Our aim is to establish the Schwarz type inequality, Heinz-Schwarz type inequality and boundary Schwarz inequality for those mappings.
We investigate the generalized tree properties and guessing model properties introduced by Weiß and Viale, as well as natural weakenings thereof, studying the relationships among these properties and between these properties and other prominent combinatorial principles. We introduce a weakening of Viale and Weiß’s Guessing Model Property, which we call the Almost Guessing Property, and prove that it provides an alternate formulation of the slender tree property in the same way that the Guessing Model Property provides and alternate formulation of the ineffable slender tree property. We show that instances of the Almost Guessing Property have sufficient strength to imply, for example, failures of square or the nonexistence of weak Kurepa trees. We show that these instances of the Almost Guessing Property hold in the Mitchell model starting from a strongly compact cardinal and prove a number of other consistency results showing that certain implications between the principles under consideration are in general not reversible. In the process, we provide a new answer to a question of Viale by constructing a model in which, for all regular (theta ge omega _2), there are stationarily many (omega _2)-guessing models (M in {mathscr {P}}_{omega _2} H(theta )) that are not (omega _1)-guessing models.
We explore the evaluation of infinite products involving the automatic sequence ((d_{n}: n in mathbb {N}_{0})) known as the period-doubling sequence, inspired by the work of Allouche, Riasat, and Shallit on the evaluation of infinite products involving the Thue–Morse or Golay–Shapiro sequences. Our methods allow for the application of integral operators that result in new product expansions for expressions involving the dilogarithm function, resulting in new formulas involving Catalan’s constant G, such as the formula
$$begin{aligned} prod _{n=1}^{infty } left( left( frac{n+2}{n}right) ^{n+1} left( frac{4 n + 3}{4 n+5}right) ^{4 n+4}right) ^{d_{n}} = frac{e^{frac{2 G}{pi }}}{sqrt{2}} end{aligned}$$introduced in this article. More generally, the evaluation of infinite products of the form ( prod _{n=1}^{infty } e(n)^{d_{n}} ) for an elementary function e(n) is the main purpose of our article. Past work on infinite products involving automatic sequences has mainly concerned products of the form ( prod _{n=1}^{infty } R(n)^{a(n)} ) for an automatic sequence a(n) and a rational function R(n), in contrast to our results as in above displayed product evaluation. Our methods also allow us to obtain new evaluations involving (frac{zeta (3)}{pi ^2}) for infinite products involving the period-doubling sequence.
We investigate the rational cohomology of the quotient of (generalized) braid groups by the commutator subgroup of the pure braid groups. We provide a combinatorial description of it using isomorphism classes of certain families of graphs. We establish Poincaré dualities for them and prove a stabilization property for the infinite series of reflection groups.
The notion of conditional coproduct of a family of abelian pro-Lie groups in the category of abelian pro-Lie groups is introduced. It is shown that the cartesian product of an arbitrary family of abelian pro-Lie groups can be characterized by the universal property of the conditional coproduct.
In 1957, Aigner (Monatsh Math 61:147–150, 1957) showed that the equations (x^6+y^6=z^6) and (x^9+y^9=z^9) have no solutions in any quadratic number field with (xyzne 0). We show that Aigner’s result holds for all equations (x^{3n}+y^{3n}=z^{3n}), where (nge 2) is a positive integer. The proof combines Aigner’s idea with deep results on Fermat’s equation and its variants.
We prove a Voronoi summation formula for non-holomorphic half-integral weight Maass forms on (Gamma _0(4)) without any restrictions on the denominator of a fraction in the exponential function. As an application we obtain a Voronoi summation formula for the values of Zagier L-series.
Let T be an (ntimes n) truncation of an ((n+alpha )times (n+alpha )) Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of T. We prove that as (nrightarrow + infty ) with (alpha ) fixed, the associated moment generating function enjoys asymptotics of the form
$$begin{aligned} exp big ( C_{1} n + C_{2} + o(1) big ), end{aligned}$$where the constants (C_{1}) and (C_{2}) are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.
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