The dynamics of solutions for the plate equation without delay effects has been investigated by many authors. But there are few works on plate equation with variable delay on unbounded domain. Therefore, we firstly consider the existence of pullback attractor for the plate equation with variable delay on (mathbb{R}^{n}) by using the theory of multivalued dynamical system.
In the paper, we discuss the uniqueness problem of meromorphic function (f(z)) when (f^{prime}(z)) shares (a), (b) and (infty) CM with (Delta_{c}f(z)), where (a) and (b) are two distinct finite values. The obtained result improves the recent result of Qi et al. [6] by dropping the condition ‘‘order of growth of (f) is not an integer or infinite’’ by ‘‘(rho(f)<infty)’’. Also in the paper we give an affirmative answer of the question raised by Wei et al. [9].
In this paper, we study the existence of meromorphic solutions of hyperorder strictly less than 1 to functional equation (f(z)^{2}+f(z+c)^{3}=e^{P},f(z)^{2}+f(z+c)^{4}=e^{P}) and the solution of the difference analogue of Fermat-type equation of the form (f(z)^{3}+[c_{1}f(z+c)+c_{0}f(z)]^{3}=e^{P}), where (P) is a polynomial. These results generalize the results of Lü and Guo [Mediterr. J. Math. 2022] and Ahamed [J. Contemp. Math. Anal. 2021].
First, it will be shown that Banach spaces (V) of harmonic or holomorphic functions with (L^{p}) norm satisfy minimal norm property, i.e., in any set
$$V_{z,c}:={fin V>|>f(z)=c},$$if nonempty, there is exactly one element with minimal norm. Later, it will be proved that this element depends continuously on a deformation of a norm and on an increasing sequence of domains in a precisely defined sense.
An example in the article shows that the first derivative of (f(z)=frac{2}{1-e^{-2z}}) sharing (0) CM and (1,infty) IM with its shift (pi i) cannot obtain they are equal. In this paper, we study the uniqueness of meromorphic function sharing small functions with their shifts concerning its (k)th derivatives. We use a different method from Qi and Yang [1] to improves entire function to meromorphic function, the first derivative to the (k)th derivatives, and also finite values to small functions. As for (k=0), we obtain: Let (f(z)) be a transcendental meromorphic function of (rho_{2}(f)<1), let (c) be a nonzero finite value, and let (a(z)notequivinfty,b(z)notequivinftyinhat{S}(f)) be two distinct small functions of (f(z)) such that (a(z)) is a periodic function with period (c) and (b(z)) is any small function of (f(z)). If (f(z)) and (f(z+c)) share (a(z),infty) CM, and share (b(z)) IM, then either (f(z)equiv f(z+c)) or
$$e^{p(z)}equivfrac{f(z+c)-a(z+c)}{f(z)-a(z)}equivfrac{b(z+c)-a(z+c)}{b(z)-a(z)},$$where (p(z)) is a nonconstant entire function of (rho(p)<1) such that (e^{p(z+c)}equiv e^{p(z)}).
Value-at-risk (VaR) serves as a measure for assessing the risk associated with individual securities and portfolios. When calculating VaR for portfolios, the dimension of the covariance matrix increases as more securities are included. In this study, we present a solution to address the issue of dimensionality by directly computing the VaR of a portfolio using a single security, therefore requiring only one variance and one mean. Our results demonstrate that, under the assumption of Gaussian distribution, the deviation between the computed VaR and actual values is relatively small.
In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with its derivatives and obtain a result, which improve several previous results. Also in the paper we include some applications of our main result.
In this paper, we study the following fractional Kirchhoff-type problem with Liouville–Weyl fractional derivatives:
$$begin{cases}left[a+bleft(intlimits_{mathbb{R}}(|u|^{2}+|{{}_{-infty}}D_{x}^{beta}u|^{2})dxright)^{varrho-1}right]({{}_{x}}D_{infty}^{beta}({{}_{-infty}}D_{x}^{beta}u)+u)=|u|^{2^{*}_{beta}-2}u,in~mathbb{R}, uinmathbb{I}_{-}^{beta}(mathbb{R}),end{cases}$$where (betain(0,frac{1}{2})), (varrho>1), ({{}_{-infty}}D_{x}^{beta}u(cdot)), and ({{}_{x}}D_{infty}^{beta}u(cdot)) denote the left and right Liouville–Weyl fractional derivatives, (2_{beta}^{*}=frac{2}{1-2beta}) is fractional critical Sobolev exponent (ageq 0) and (b>0). Under suitable values of the parameters (varrho), (a) and (b), we obtain a nonexistence result of nontrivial solutions of infinitely many nontrivial solutions for the above problem.
The fourth-order properly elliptic equation with multiple root is considered in the elliptic domain. The conditions, necessary and sufficient for the unique solvability of the Dirichlet problem for this equation are found, and if these conditions fail the defect numbers of this problem are determined. The solution of the problem is found in explicit form.
In this paper, based on the discretization method, we construct a new 2-parameter regularly varying discrete distribution generated by Waring-type probability (2-RDWP). Some useful plots are displayed for the model. From the mathematical point of view, to suggest 2-RDWP as a new discrete probability distribution in bioinformatics, some statistical facts such as unimodality, skewness to the right, upward/downward convexity, regular variation at infinity and asymptotically constant slowly varying component are established for the model. We provide the conditions of coincidence of solution for the system of likelihood equations with the maximum likelihood estimators for the unknown parameters. Simulation studies are performed using the Monte Carlo method and Nelder–Mead optimization algorithm to obtain maximum likelihood estimations of the unknown parameters. Asymptotic expansion of the probability function with two terms is considered, and then the moment’s existence of integer orders is investigated. Finally, a real count data set is used to show the applicability of the new model compared to other models in bioinformatics.
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