Symmetric Differential Operators Research Articles

Characterization of self-adjoint domains for regular odd order C-symmetric differential operators

We enlarge the class of regular odd order differential operators and find the self-adjoint boundary condition of every operator. In this paper, we give the characterization of self-adjoint domains for regular odd order C-symmetric differential operators with two-point boundary conditions. The previously known characterization of self-adjoint operators for regular odd order symmetric differential operators is the special case of this one.

A New Soft Set Operation: Soft Binary Piecewise Symmetric Difference Operation

Molodtsov tarafından geliştirilen esnek küme teorisi hem teorik hem de pratik olarak birçok alanda uygulanmıştır. Belirsizliği ele almak için yararlı bir matematiksel araçtır. Ortaya atıldığından bu yana çok sayıda esnek küme işlemi varyasyonu tanımlanmış ve kullanılmıştır. Bu çalışmada, esnek ikili parçalı simetrik fark işlemi adı verilen yeni bir esnek küme işlemi tanımlanıp, özellikleri klasik küme teorisinde var olan simetrik fark işleminin temel cebirsel özellikleri ile karşılaştırmalı olarak ele alınmış ve incelenmiştir. Ayrıca, esnek ikili parçalı simetrik fark işlemi ve kısıtlanmıs kesisim işlemleri ile birlikte sabit parametreye sahip tüm esnek kümelerin oluşturduğu cebirsel yapının, birimli ve değişmeli bir hemiring ve ayrıca Boole halkası olduğu gösterilmiştir.

Symmetric Difference Operators Derived from Overlap and Grouping Functions

This paper introduces the concept of symmetric difference operators in terms of overlap and grouping functions, for which the associativity property is not strongly required. These symmetric difference operators are weaker than symmetric difference operators in terms of positive and continuous t-norms and t-conorms. Therefore, in the sense of the characters of mathematics, these operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. We analyze several related important properties based on two models of symmetric differences.

Complex symmetry of differential operators on generalized Fock spaces

In this paper, we classify the family of entire functions [Formula: see text] such that the induced weighted composition map [Formula: see text], is a conjugation namely [Formula: see text], where [Formula: see text] is an arbitrary complex number and [Formula: see text] is a real number such that [Formula: see text]. In the last section, we characterize the [Formula: see text]-symmetric maximal differential operators on the Fock space [Formula: see text].

On principal value and standard extension of distributions

For a holomorphic function $f$ on a complex manifold $\mathscr {M}$ we explain in this article that the distribution associated to $\lvert f\rvert^{2\alpha } (\textrm{Log} \lvert f\rvert^2)^q f^{-N}$ by taking the corresponding limit on the sets $\{ \lvert f\rvert \geq \varepsilon \}$ when $\varepsilon $ goes to $0$, coincides for $\Re (\alpha ) $ non negative and $q, N \in \mathbb {N}$, with the value at $\lambda = \alpha $ of the meromorphic extension of the distribution $\lvert f\rvert^{2\lambda } (\textrm{Log} \lvert f\rvert^2)^qf^{-N}$. This implies that any distribution in the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution has the standard extension property. This implies a non $\mathcal {O}_\mathscr {M}$ torsion result for the $\mathcal {D}_{\mathscr {M}}$-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic $\mathcal {D}$-modules introduced and studied in [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] and [Barlet, D., On partial differential operators which annihilate the roots of the universal equation of degree $k$, arXiv:2101.01895] associated to the roots of universal equation of degree $k$, $z^k + \sum _{h=1}^k (-1)^h\sigma _hz^{k-h} = 0$.

Diffusion orthogonal polynomials in 3-dimensional domains bounded by developable surfaces

The following problem is studied: describe the triplets (Ω,g,μ), μ=ρdx, where g=(gij(x)) is the (co)metric associated with the symmetric second order differential operator L(f)=1ρ∑ij∂i(gijρ∂jf) defined on a domain Ω of Rn and such that there exists an orthonormal basis of L2(μ) made of polynomials which are eigenvectors of L, and the basis is compatible with the filtration of the space of polynomials with respect to some weighted degree.In a joint paper with D. Bakry and M. Zani this problem was solved in dimension 2 for the usual degree. In the author's subsequent paper this problem was solved in dimension 2 for any weighted degree. In the present paper this problem is solved in dimension 3 for the usual degree under the condition that ∂Ω contains a piece of a tangent developable surface. The proof is based on Plücker-like formulas in the form given by Ragni Piene. All the found solutions are generalized for any dimension.

A new symmetric differential operator of normalized functions with applications in image processing

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

Set algebra — based algebraic evolutionary algorithm for binary optimization problems

In order to design algebraic evolutionary algorithms by set algebra, the intersection, union, complement, difference and symmetric difference operations of 0-1 vectors on {0,1}n are firstly defined based on set operations. Then, the isomorphism between algebraic system defined on {0,1}n and set algebra defined on power set P(Ω) of set Ω is proved. Therefrom a simple and fast implement method of set algebra is proposed. Third, symmetric difference operator and asymmetric mutation operator are successively proposed based on set algebra, they have global exploration and local exploitation capabilities respectively. On this basis, a novel algebraic evolutionary algorithm, named set algebra-based heuristic algorithm (SAHA), is proposed based on the operations of 0-1 vectors on {0,1}n for solving binary optimization problems. For verifying the performance of SAHA, it is used to solve 0-1 knapsack problem (0-1KP) and knapsack problem with single continuous variable (KPC), respectively. The comparison with the state-of-the-art algorithms of solving 0-1KP and KPC shows that SAHA can not only obtain excellent calculation results, but also is faster speed, it is most competitive for solving binary optimization problems.

Integral representations of positive definite kernels

The paper proposes proof of the possibility of an integral representation of a positive definite kernel of two pairs of variables. Using this kernel, we use the technique of constructing a new Hilbert space in which symmetric differential operators formally commute. In this case, the kernel satisfies a system of differential equations with partial derivatives. It is known that a kernel given in a subdomain of the real plane, generally speaking, does not always imply an extension to the entire plane. This possibility is related to the problem of the existence of a commuting self-adjoint extension of symmetric operators. The author applies his own results related to a commuting self-adjoint extension in a wider Hilbert space. The resulting representation in the form of an integral of elementary positive-definite kernels with respect to the spectral measure generated by the resolution of the identity of the operators allows us to extend the positive-definite kernel to the entire plane.

Friedrichs extension of singular symmetric differential operators

For singular even order symmetric differential operators we find the matrices which determine all symmetric extensions of the minimal operator. And for each of these symmetric operators which is bounded below we find the boundary condition of its Friedrichs extension. The operators of regular problems are bounded below and thus each one has a symmetric extension and thus its symmetric extension has a Friedrichs extension. See also https://ejde.math.txstate.edu/special/02/b1/abstr.html

On New Symmetric Schur Functions Associated with Integral and Integro-Differential Functional Expressions in a Complex Domain

The symmetric Schur process has many different types of formals, such as the functional differential, functional integral, and special functional processes based on special functions. In this effort, the normalized symmetric Schur process (NSSP) is defined and then used to determine the geometric and symmetric interpretations of mathematical expressions in a complex symmetric domain (the open unit disk). To obtain more symmetric properties involving NSSP, we consider a symmetric differential operator. The outcome is a symmetric convoluted operator. Geometrically, studies are presented for the suggested operator. Our method is based on the theory of differential subordination.

Text encryption for lower text size: Design and implementation

For enterprises of all sizes and types in the IT sector, data security is essential. Data security is the process of giving databases and websites secure data privacy protections and preventing unauthorized data access. By encrypting digital data, software/hardware, and hard drives, unauthorized individuals or hackers can only access unreadable material, which is why encryption is a crucial data protection strategy. Data is produced by computing operations, which are then used to encrypt and send the data into the network. For low bandwidth channels, an encryption strategy has been put out in this work. The proposed method uses the properties of homomorphic transform to make it more unique and suitable for text encryption. Conversion of plain text to cipher text is done by using simple union and symmetric difference operators. Which makes it more reliable in terms of execution time.

Прямой метод вычисления циклов ячеек карты блока простого планарного графа

The proposed algorithm for calculating the cycles of the cells the simple planar graph block map is an extension of the classical depth-first search algorithm for cycles of the DFS-basis. The key idea of the modification of this algorithm is the strategy of right-hand traversal when passing the graph in depth. The vertex with the minimum coordinate on the OY axis is assigned as the starting vertex in the right-hand traversal. The exit from the initial vertex is performed along the edge with the minimum polar angle. The continuation of the traversal from each next vertex is carried out along an edge with a minimum polar angle relative to the edge along which arrived at the current vertex. A two-level structure of nested cycles is introduced. This is the main level and the zero level of nesting. All cycles of the basis belong to the main level. Each of the cycles can additionally have a zero level of nesting in another main cycle for it, if it is nested in the main cycle and not nested in any other cycle from the main cycle. With the right-hand traversal, zero nesting cycles are adjacent to the main cycle and do not have common vertices outside the main cycle. These two properties allowed in each basis cycle sequentially select and exclude from it all its zero nesting cycles, using the symmetric difference operation. It is shown that the rest of the basic cycle is the cycle of the block map cell. The complexity of each step of the proposed algorithm does not exceed the quadratic complexity with respect to the number of vertices of the simple planar graph.

Application of Comparison Algebras in Construction of Commutative Composites of Symmetric Higher Order Differential Operators

Application of Comparison Algebras in Construction of Commutative Composites of Symmetric Higher Order Differential Operators

A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus

In this paper, we develop theorems on finite and infinite summation formulas by utilizing the q and (q,h) anti-difference operators, and also we extend these core theorems to q(α) and (q,h)α difference operators. Several integer order theorems based on q and q(α) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q(α) difference operators. In order to develop the fractional order anti-difference equations for q and q(α) difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an infinite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for q and q(α) difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the (q,h) and (q,h)α difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the (q,h) and (q,h)α difference operators for verification.

On symmetric partial differential operators

Let \(\sigma _{1}, \dots , \sigma _{k}\) be the elementary symmetric functions of the complex variables \(x_{1}, \dots , x_{k}\). We say that \(F \in {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\) is a trace function if their exists \(f \in {\mathbb {C}}[z]\) such that \(F(\sigma _{1}, \dots , \sigma _{k}) = \sum _{j=1}^{k} f(x_{j})\) for all \(\sigma \in {\mathbb {C}}^{k}\). We give an explicit finite family of second order differential operators in the Weyl algebra \(W_{2}:= {\mathbb {C}}[\sigma _{1}, \dots , \sigma _{k}]\langle \frac{\partial }{\partial \sigma _{1}}, \dots , \frac{\partial }{\partial \sigma _{k}}\rangle \) which generates the left ideal in \(W_{2}\) of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As an application, we obtain for each integer k a holonomic system which is a quotient of \(W_{2}\) by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: \(z^{k} + \sum _{h=1}^{k} (-1)^{h}\sigma _{h}z^{k-h} = 0\).

A Class of Symmetric Fractional Differential Operator Formed by Special Functions

In light of a certain sort of fractional calculus, a generalized symmetric fractional differential operator based on Raina’s function is built. The generalized operator is then used to create a formula for analytic functions of type normalized. We use the ideas of subordination and superordination to show a collection of inequalities using the suggested differential operator. The new Raina’s operator is also used to the generalized kinematic solutions (GKS). Using the concepts of subordination and superordination, we provide analytic solutions for GKS. As a consequence, a certain hypergeometric function provides the answer. A fractional coefficient differential operator is also created. The geometric and analytic properties of the object are being addressed. The symmetric differential operator in a complex domain is shown to be a generalized fractional differential operator. Finally, we explore the characteristics of the Raina’s symmetric differential operator.

Symmetric Difference Operator in Quantum Calculus

The main focus of this paper is to develop certain types of fundamental theorems using q, q(α), and h difference operators. For several higher order difference equations, we get two forms of solutions: one is closed form and another is summation form. However, most authors concentrate only on the summation part. This motivates us to develop closed-form solutions, and we succeed. The key benefit of this research is finding the closed-form solutions for getting better results when compared to the summation form. The symmetric difference operator is the combination of forward and backward difference symmetric operators. Using this concept, we employ the closed and summation form for q, q(α), and h difference symmetric operators on polynomials, polynomial factorials, logarithmic functions, and products of two functions that act as a solution for symmetric difference equations. The higher order fundamental theorems of q and q(α) are difficult to find when the order becomes high. Hence, by inducing the h difference symmetric operator in q and q(α) symmetric operators, we find the solution easily and quickly. Suitable examples are given to validate our findings. In addition, we plot the figures to examine the value stability of q and q(α) difference equations.

Liouville-type results for spherical symmetric linear differential operators with constant coefficients

Liouville-type results for spherical symmetric linear differential operators with constant coefficients

Orthogonal polynomials and diffusion operators

We want to describe the triplets (\Omega, (g), \mu) where (g) is the (co)metric associated to some symmetric second order differential operator L defined on the domain \Omega of R^d and such that L is expandable on a basis of orthogonal polynomials of L_2(\mu), and \mu is some admissible measure. Up to affine transformation, we find 11 compact domains in dimension 2, and also give some non--compact cases in this dimension.