Symmetric Differential Operators Research Articles

Difference operators on fuzzy sets

Based on the properties of the difference operator on crisp sets, a fuzzy difference operator in fuzzy logic is defined as a continuous binary operator on the closed unit interval with some boundary conditions. In this paper, the structures and properties of fuzzy difference operators are studied. The main results are: (1) Using the axiomatic approach, some generalizations of classical tautologies for fuzzy difference operators are obtained. (2) Based on the model theoretic approach, the fuzzy difference operator constructed by a nilpotent t-norm and a strong negation is characterized. (3) the paper discusses the relationship between the fuzzy difference operator and symmetric difference operator which was raised in [3].

Symmetric operator extensions of composites of higher order difference operators

In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.

Z2 classification of FTR symmetric differential operators and obstruction to Anderson localization

This paper concerns the Z2 classification of Fermionic Time-Reversal (FTR) symmetric partial differential Hamiltonians on the Euclidean plane. We consider the setting of two insulators separated by an interface. Hamiltonians that are invariant with respect to spatial translations along the interface are classified into two categories depending on whether they may or may not be gapped by continuous deformations. Introducing a related odd-symmetric Fredholm operator, we show that the classification is stable against FTR-symmetric perturbations. The property that non-trivial Hamiltonians cannot be gapped may be interpreted as a topological obstruction to Anderson localization: no matter how much (spatially compactly supported) perturbations are present in the system, a certain amount of transmission in both directions is guaranteed in the nontrivial phase. We present a scattering theory for such systems and show numerically that transmission is indeed guaranteed in the presence of FTR-symmetric perturbations while it no longer is for non-symmetric fluctuations.

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies.

A distribution network traveling wave localization method based on CEEMD-DEO3S

A traveling wave head calibration way according to Complementary Ensemble Empirical Mode Decomposition (CEEMD) and Three-point Symmetric Differential Energy Operator (DEO3S) is proposed to address the issues of noise interference and difficulty in precise positioning in local area network wave positioning. The way filters out low-frequency bandwidth in the fault data by performing CEEMD decomposition on the signal and then uses the DEO3S algorithm to define the time of arrival of the fault wave head. The emulation results indicate the proposed way can precisely extract wave heads even in the presence of a large amount of noise interference, and has good robustness and accuracy. Compared with traditional traveling wave positioning methods, it has obvious advantages and further improves the accuracy and reliability of traveling wave positioning.

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.