The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain.

Novel Stable Approach with Probability Distribution for Multi-Criteria Decision-Making Problems of Multi-Valued Neutrosophic Sets

In this paper, a novel approach and framework based on interval-dependent degree and probability distribution for multi-criteria decision-making problems with multi-valued neutrosophic sets (MVNSs) is proposed. First, a simplified dependent function and distribution function are given and integrated into a concise formula, which is called the interval-dependent function and contains interval computing and probability distribution information in an interval. Then a transformation operator is defined and it is shown how to convert MVNSs into an interval set. Subsequently, the interval-dependent function with the probability distribution of MVNSs is deduced. Finally, an example and comparative analysis are provided to verify the feasibility and effectiveness of the proposed method. In addition, uncertainty analysis, which reflects the dynamic change of the ranking result with decision-makers’ preferences, is performed by setting different distribution functions, which increases the reliability and accuracy of the proposed method.

The nonlocal boundary problem with perturbations of antiperiodicity conditions for the eliptic equation with constant coefficients

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

Simplified Neutrosophic Sets Based on Interval Dependent Degree for Multi-Criteria Group Decision-Making Problems

In this paper, a new approach and framework based on the interval dependent degree for multi-criteria group decision-making (MCGDM) problems with simplified neutrosophic sets (SNSs) is proposed. Firstly, the simplified dependent function and distribution function are defined. Then, they are integrated into the interval dependent function which contains interval computing and distribution information of the intervals. Subsequently, the interval transformation operator is defined to convert simplified neutrosophic numbers (SNNs) into intervals, and then the interval dependent function for SNNs is deduced. Finally, an example is provided to verify the feasibility and effectiveness of the proposed method, together with its comparative analysis. In addition, uncertainty analysis, which can reflect the dynamic change of the final result caused by changes in the decision makers’ preferences, is performed in different distribution function situations. That increases the reliability and accuracy of the result.

On the similarity transformations in second quantized fermion-boson interacting hamiltonian and the BCS theory

A similarity transformation is an equivalence relation between square matrices which preserves determinant, trace and eigenvalues, playing a key role in quantum mechanics in simplifying complex hamiltonian systems and improving analytical results attainable from the use of perturbation theory. As a prototypical example, the conventional BCS theory of superconductivity is usually derived from a similarity transformation of the original electron-phonon hamiltonian, written in second quantized version. Here we discuss the general method for writing the similarity transformation operator in second quantized form, allowing one to recast a hamiltonian describing an interacting fermion-boson system into an effective theory in which only the desired degrees of freedom are kept after the transformation.