Бидний тухай
Багш ажилтан
We investigate model spaces that are invariant under composition operators in the context of holomorphic self-maps of the open unit disc in the complex plane. The composition operator, defined as Cφ ◦ f = f ◦ φ for f ∈ H2 (D), where H2 (D), denotes the Hardy space on D, gives rise to a bounded linear operator. Model spaces, which are backward shift-invariant closed subspaces of H2 (D), are examined with a particular focus on their connection with inner functions. The research emphasizes the study of finite-dimensional model spaces, affine transformations, and linear fractional transformations. The findings contribute to the understanding of the behavior and properties of these model spaces under composition operators.
We study a class of time-fractional diffusion-wave equations with variable coefficients using Lie symmetry analysis. We obtain not only in infinitesimal symmetries but also a complete group classifi cation and a classifi cation of group invariant solutions of this class of equations. Group invariant solutions are given explicitly corresponding to every element in an optimal system of Lie algebras generated by in nitesimal symmetries of equations in the class. We express the solutions in terms of Mittag-Lefer functions, generalized Wright functions, and Fox H-functions. These solutions contain previously known solutions for particular cases.
МУИС нь анхлан 1942 онд Мал эмнэлэг-зоотехник, Хүн эмнэлэг, Багш нарын факультет гэсэн гурван салбартайгаар хичээллэж эхлэхэд байгуулагдсан долоон тэнхимийн нэг нь Математикийн тэнхим юм. 1962 онд Математикийн тэнхим нь Математик анализын тэнихим, алгебрын тэнхим гэж хоёр тэнхим болж хуваагдсанаар Монгол улсад энэ салбар ухаанаар нэрлэгдсэн биеэ даасан нэгж бий болсон түүхтэй. Математик анализын тэнхим нь 2014 он хүртэл энэ нэрээ ямар нэг байдлаар хадгалсаар ирсэн байна. Энэхүү өгүүллээр МУИС-ийн математик анализын салбарын түүхэн хөгжил, бүрэлдэхүүн, сургалт, судалгааны голлох үр дүнгүүдийн тухай товч танилцуулсан болно.
We find an explicit expression of the determinantal polynomials of a weighted shift matrix with palindromic geometric weights.
Эллиптик тоон мужын тухай теоремыг алгебрын муруйн чанар болон Киппенганы теоремыг ашиглан хэрхэн баталсаныг авч үзнэ.
Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ and let $H^2(\mathbb{D})$ be the Hardy space over $\mathbb{D}$. For $\varphi$ a holomorphic self-map of $\mathbb{D}$, define the composition operator $C_\varphi$ on $H^2(\mathbb{D})$ by \begin{equation} C_\varphi f=f \circ \varphi \quad (f \in H^2(\mathbb{D})). \end{equation} It is well known that $C_\varphi \in B(H^2(\mathbb{D}) )$. Here $B(H^2(\mathbb{D}) )$ denote the set of all bounded linear operators on $H^2(\mathbb{D})$.\\ The theory of composition operators is highly interdisciplinary with its natural connections to complex analysis, linear dynamics, complex geometry, and functional analysis. \begin{conj}[\cite{Mut}] Consider the finite Blaschke product $\theta= \prod_{i=1}^m b_{\alpha_i}^{n_i}$ corresponding to $\alpha_1, \dots, \alpha_n \in \mathbb{D}$. Characterize holomorphic self maps of $\mathbb{D}$ such that $$C_\varphi Q_\theta \subseteq Q_\theta.$$ \end{conj} \begin{center} Partial answers to the conjecture. \end{center} \begin{theorem} Let $\varphi$ be a holomorphic self map of $\mathbb{D}$, $\alpha, \beta \in \mathbb{D}\setminus \{0\}, \alpha \not =\beta $, and suppose $ \displaystyle \theta(z)= \frac{z-\alpha}{1-\overline{\alpha}z}\frac{z-\beta}{1-\overline{\beta}z}$. Then $Q_\theta$ is invariant under $C_\varphi$ if and only if $$\varphi(z)=\begin{cases} z, & \text{ if } \alpha \not =-\beta\\ z \text{ or } -z, & \text{ if } \alpha =-\beta \end{cases}$$ . \end{theorem} \begin{theorem} Let $\varphi$ be a holomorphic self map of $\mathbb{D}$, $\alpha, \beta \in \mathbb{D}\setminus \{0\}, \alpha \not =\beta $, and suppose $\displaystyle \theta(z)= z \frac{z-\alpha}{1-\overline{\alpha}z}\frac{z-\beta}{1-\overline{\beta}z}$. \begin{itemize} \item[i)] For $\alpha+\beta=0$, $Q_\theta$ is invariant under $C_\varphi$ if and only if \begin{enumerate} \item $\varphi $ is constant or \item $\varphi =z$ or \item $\varphi =-z$ \end{enumerate} \item[ii)] For $\alpha+\beta\not =0$, $Q_\theta$ is invariant under $C_\varphi$ if and only if \begin{enumerate} \item $\varphi $ is constant or \item $\varphi =z$ or \end{enumerate} \end{itemize} \end{theorem}
Let $n$ be a product of the prime numbers whose positive integer powers are of the form $a^2+Db^2$ where $D> 4$ is a square-free number and $a, b$ positive integers. For $n\leq 3072$, we obtained a refinement of the upper bound of the radius of a circular points set which was previously given in Tables 1 and 2 of \cite{Ganbileg}. In order to prove this, we showed that there are points on the circle with the radius $R=\dfrac{n\sqrt{D}}{2D}$ such that mutual distances between these points are all integers. Consequently, if $n$ is a product of the prime numbers whose squares are of the form $a^2+Db^2$, then we showed that there are $\tau(n)$ points on the circle with the radius $R=\dfrac{n\sqrt{D}}{2D}$ such that mutual distances between these points are all integer numbers, where $\tau(n)$ is the number of all positive divisors of $n$.
We generalize the results in \cite{Ganbileg} when $n$ is a product of the prime numbers whose positive integer powers are of the form $a^2+Db^2$ with $D> 4$ is a square-free number and $a, b$ positive integers. As in an application, we improved the results in \cite{Ganbileg} about the upper bound for the minimum radius of the circle with $n$ integral point set for $n \leq 3072$, see Table 1 and Table 2. For both tables, we denote by $R_n$ the minimum value of the radius of the circle of a circular $n$ integral points set. And denote by $R(n; D)$ the minimum value of the radius $R=\frac{n\sqrt{D}}{2D}$ of the circle of a circular $n$ integral points set.
Палиндром геометр жинтэй шилжилт матрицын тодорхойлогч олон гишүүнтийг олж тогтоосон.
A new type of hypergeometric differential equations was introduced and studied by H. Sekiguchi in 1996. The investigated system of partial differential equations generalizes the Gauss-Aomoto-Gelfand system. Gauss-Aomoto-Gelfand systems can be expressed as the determinants of $2\times 2$ matrices of derivations with respect to certain variables. The Gauss-Aomoto-Gelfand hypergeometric system arises in numerous problems of algebraic geometry, partial differential equations, the theory of special functions and combinatorics. H. Sekiguchi generalized this construction by looking at determinants of $l\times l$ matrices of derivations with respect to certain variables. In this talk we studied the dimension of global solutions to the generalized systems of Gauss-Aomoto-Gelfand hypergeometric systems.
A set Ω is a spectral set for an operator T if the spectrum of T is contained in Ω, and von Neumann's inequality holds for T with respect to the algebra R(Ω) of rational functions with poles off of Ω‾. It is a complete spectral set if for all r∈N, the same is true for Mr(C)⊗R(Ω). The rational dilation problem asks, if Ω is a spectral set for T, is it a complete spectral set for T? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (eg, over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is naturally associated to a constrained subalgebra of the disk algebra C+z2A(D). Here it is shown that such a result is generic for a large class of varieties associated to constrained algebras. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler–Pick interpolation theorem is given and it is proved that there exist Kaijser–Varopoulos style examples of non-contractive unital representations where the generators are contractions.
