Let & micro;(rho,alpha) be the Fourier transform of the self-similar measure & micro;(rho,alpha) defined by & micro;(rho,alpha) = 1/m Sigma(m)(j=1) & micro;(rho,alpha) o S- j(-1) , where Sj (x) = rho(-1)x + a(j), rho > 1, a(j +1)-a(j) = alpha for some alpha > 0. In this paper, we study the asymptotic behavior of & micro;(rho,alpha)(t) when rho = q is an integer greater than m. We prove that the upper limit of |& micro;(q,alpha)(t)| at infinity is | & micro;(q,alpha)(alpha(-1)) > 0. For integer q >= 2m, we give a sufficient and necessary condition for limsup|n|->infinity & micro;q,alpha(n) and limsup|t|->infinity |& micro;(q,alpha)(t)| to be equal. Moreover, we show that the exceptional set E-q is of zero Lebesgue measure and one Hausdorff dimension, where E-q = {alpha is an element of R+ : limsup(|n|->infinity) & micro;q,alpha(n) =/ limsup |t|(->infinity) & micro;(q,alpha)(t)}.
In this paper we consider 4-superlinear fourth order elliptic equations on RN involving u triangle(u2) and indefinite potentials. The potential V here is bounded so that the working space is H2(RN), which can not be compactly embedded into Lebesgue spaces. To overcome this difficulty, we study the nonlinearity with integrable weights. By a local linking argument and Morse theory, we obtain the existence of nontrivial solutions for this problem.
Let p(m, n) be the number of partitions of n with m more odd parts than even parts. We consider the first and second moment of p(m, n) and p(m, n)-p(-m, n) to further investigate parity bias in integer partitions. We give the generating functions for these moments and derive asymptotic formulas by employing Wright's circle method.
In this paper, we prove that any compact manifold equipped with an alpha-Bach-flat metric (as defined in (1)) that satisfies certain sectional curvature conditions, and also has either positive constant scalar curvature or positive constant sigma(2)(A(g)), is Einstein. In particular, we show that a compact manifold with divergence-free Cotton tensor, positive constant scalar curvature R, and sectional curvature KM satisfying K-M >= epsilon R > 0, is an Einstein manifold. Moreover, if the sectional curvature of such a manifold satisfies K-M >= 1/n(n+2) R > 0, then it is isometric to S-n or CPm when n = 2m.
Morse homology on a noncompact manifold generally depends on the choice of Morse functions. The escape of critical points or gradient flow lines along a homotopy of Morse functions violates the invariance property. More subtly, homology classes can escape along a homotopy even though critical points and gradient flow lines do not. In this note, we focus on the latter phenomenon, which also arises in Rabinowitz Floer theory. We present an abstract example illustrating the escape of homology classes and investigate conditions on homotopies that prevent this behavior. Applications to the invariance problem in Rabinowitz Floer homology are also discussed.
. Let Cent(R) denote the set of all distinct centralizers in a ring R. A ring R is said to be n-centralizer ring if Cent(R) = n, where n is an element of N. In this paper, we illustrate some connections between the centralizers and pairwise non-commuting elements of a finite ring. After that, we characterize all 10-centralizer finite rings. Also, we compute the commuting probability for all 10-centralizer finite rings.
In this article, we apply the Moser iteration technique and the Saloff-Coste's Sobolev inequality to study Cheng-Yau logarithmic gradient estimates for solutions to the following nonlinear elliptic equation Delta(phi)u + a|del u|(2)/u + b(x)u(p )(ln(u + c))(q) = 0 on smooth metric measure space (M-n, g, e(-phi )d nu), where c >= 1, a > -1, p and q are constants, and b(x) is allowed to change sign. Moreover, we also obtained the Liouville theorem and Harnack inequality when b(x) is constant. Our results generalize and improve many previous works.
Since RG is the group ring of a group G over an associative and unital ring R, a novel type of normalized units has been introduced as G-Jacobson. A unit u in normalized unit group V(RG) is defined as G-Jacobson if u = gw for some g is an element of G and w is an element of 1 + I(J(R)G; G) where J(R) is the jacobson radical of R and I(J(R)G; G) is a fundamental ideal of RG. Besides, some necessary and sufficient conditions for V(RG) = G & times; (1 + I(J(R)G; G)) concerning an abelian R-favorable torsion group G have been investigated using some prominent characteristics of Jacobson radicals. Results figure out an open problem defined in [4] extending it to Jacobson radicals.
The Riemann problem with available analytical solutions is concretely investigated for a hyperbolic system of conservation laws in the one-dimensional setting, which originates from the backward-forward model of mean-field games by taking the quadratic Hamiltonian term and the singular coupling term. The Riemann solution is constituted by using either a single delta shock wave or the association of two contact discontinuities according to the assigned initial data. Furthermore, the interactions of delta shock waves are carefully explored by taking the initial data in three intervals with each interval being in a constant state, where some fascinating nonlinear wave phenomena are observed. Finally, the displayed numerical tests are in well agreement with the Riemann solutions we obtained theoretically before.
This paper investigates the optimal robust reinsurance and investment problems for the insurer and reinsurer in a liquid financial market under a principal-agent framework based on the generalized variance premium principle. The surplus process of the insurer is assumed to follow a diffusion risk process (which is an approximation of the classical compound Poisson model). Both the insurer and the reinsurer are allowed to invest in a risk-free asset and a risky asset whose price process is described by the Heston model. The targets of the insurer and the reinsurer are supposed to maximize the expected exponential utility from their terminal wealth. The ambiguity-averse reinsurer has deterministic ambiguity aversion preferences against the diffusion risk caused by the financial market and the approximated diffusion risk which comes from the claims process. By employing stochastic optimal control approaches and constructing the HJB (or HJBI) equations, the optimal (robust) reinsurance-investment strategies and the corresponding value functions are obtained explicitly. Numerical examples and sensitivity analysis are presented. Some results are found, for example, within our model framework the reinsurer premium income is the least but the reinsurance payout is the most under the variance principle by comparing with the expected value principle and the generalized variance principle.
. This paper investigates the inverse problems for the photoacoustic imaging in attenuating media, where the wave propagation is described by an attenuated wave equation with some attenuation coefficient. We define the attenuated wave forward operator as the transform that assigns to a given function f the solution of the attenuated wave equation on the detector surface (where the detectors are located) with the initial function f. Here we study the injectivity of the attenuated wave forward operator, its normal derivative and their combination. This means that the initial function is uniquely determined by the knowledge of the solution of the attenuated wave equation on the unit sphere.
We consider the initial-boundary value problem for a class of p-Kirchhoff type parabolic equations with logarithmic nonlinearity. The existence of local solution is proved by using Galerkin's method. By employing the concavity method, we prove that solutions blow-up either in infinite time or finite time. Furthermore, by establishing first-order differential inequalities and combining analytical techniques, the estimates of upper and lower bounds for the blow-up time are obtained.
. In this work, we consider a class of nonlinear Schro & uml;dingerPoisson systems involving the fractional p-Laplacian and a combined Sobolev critical exponent and the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By using the constraint variational method and quantitative deformation lemma, we prove the existence of the least energy sign-changing solution for the system. Moreover, we show that the energy of the least energy sign-changing solution is strictly larger than twice that of the ground state energy.
In this paper, we mainly study the multichannel sampling and reconstruction of signals in function spaces associated with the special affine Fourier transform (SAFT). Three kinds of convolutions are used to construct function spaces for modeling non-bandlimited signals in the SAFT domain. Moreover, based on these convolution operators, multichannel linear systems with the modulation-time-invariant property are developed to produce samples with distinct sampling rate in each channel. We establish the sampling formula in each type of function space under some appropriate conditions. Furthermore, some examples are provided to verify the conditions in the corresponding sampling expansions.
A map f : X -> Y induces a homomorphism from 7rn(X) to 7rn(Y). The Gottlieb groups Gn(X) and Gn(Y) are subgroups of 7rn(X) and 7rn(Y), respectively. When f is a cyclic map, it induces a homomorphism between these subgroups. In this case, the homomorphism is known to preserve elements from Gn(X) to Gn(Y). This study aims to classify self-maps that preserve Gk(Sn) for k = n, n + 1. Moreover, we characterize self-maps that induce an isomorphism on Gk(Sn) fork = n, n + 1, providing new insights into their structure.
We study forward self-similar solutions of the p-harmonic map heat flow from R1+1 to the circle S-1. We consider three cases 1 < p < 2, p = 2 and 2 < p < infinity. Solution behaviors are different according to the range of p.
This paper aims to investigate the sectional curvature of a given poly-Norden semi-Riemannian manifold and check the weakly Einstein conditions for such manifolds. First it is shown that there exist no poly-Norden semi-Riemannian manifolds with constant (real) sectional curvature by analyzing the characteristics of the curvature tensor field. Next, it is shown that both holomorphic-like sectional curvature and holomorphic-like bi-sectional curvature do not work for poly-Norden manifolds. Therefore, a new sectional curvature called poly-Norden sectional curvature is introduced and an example is given. Also, in the case that this poly-Norden sectional curvature is constant, the expression of the curvature tensor field is obtained. Finally, if this poly-Norden sectional curvature is constant, the conditions for the poly-Norden manifold to be Einstein and weakly Einstein are investigated.
We obtain the Jackson q-integral analogues of the Euler cosine integral and the Raabe type formula for alternating Hurwitz zeta functions. Further, we present several series of representations of the Dirichlet lambda function.
. In this paper, we establish some results on Baum-Katz theorem for adapted sequences in noncommutative probability. More precisely, we prove the convergence rate given in (1.1) under certain conditions for adapted sequences of measurable operators and positive functions l, phi. As applications, we discuss the convergence rates for weighted sums in noncommutative Lorentz and Marcinkiewicz spaces.
Recently, Liu and Liu gave a q-supercongruence from Andrews' terminating q-analogue of Watson's formula. In this paper, employing the method of creative microscoping devised by the author and Zudilin in 2019, we deduce more q-supercongruences from Andrew's summation.
