Using Nonlinear Diffusion Model to Identify Music Signals

In this paper, combined with the partial differential equation music signal smoothing model, a new music signal recognition model is proposed. Experimental results show that this model has the advantages of the above two models at the same time, which can remove noise and enhance music signals. This paper also studies the music signal recognition method based on the nonlinear diffusion model. By distinguishing the flat area and the boundary area of the music signal, a new diffusion coefficient equation is obtained by combining these two methods, and the corresponding partial differential equation is discretized by the finite difference method with numerical solution. The application of partial differential equations in music signal processing is a relatively new topic. Because it can accurately model the music signal, it solves many complicated problems in music signal processing. Then, we use the group shift Fourier transform (GSFT) to transform this partial differential equation into a linear homogeneous differential equation system, and then use the series to obtain the solution of the linear homogeneous differential equation system, and finally use the group shift inverse Fourier transform to obtain the noise frequency modulation time-dependent solution of the probability density function of the interference signal. This paper attempts to use the mathematical method of stochastic differentiation to solve the key problem of the time-dependent solution of the probability density function of noise interference signals and to study the application of random differentiation theory in radar interference signal processing and music signal processing. At the end of the thesis, the application of stochastic differentiation in the filtering processing of music signals is tried. According to the inherent self-similarity of the music signal system and the completeness and stability of the empirical mode decomposition (EMD) algorithm, a new kind of EMD music using stochastic differentiation is proposed for signal filtering algorithm. This improved anisotropic diffusion method can maintain and enhance the boundary while smoothing the music signal. The filtering results of the actual music signal show that the algorithm is effective.

Degenerate Canonical Forms of Ordinary Second-Order Linear Homogeneous Differential Equations

For each fundamental and widely used ordinary second-order linear homogeneous differential equation of mathematical physics, we derive a family of associated differential equations that share the same “degenerate” canonical form. These equations can be solved easily if the original equation is known to possess analytic solutions, otherwise their properties and the properties of their solutions are de facto known as they are comparable to those already deduced for the fundamental equation. We analyze several particular cases of new families related to some of the famous differential equations applied to physical problems, and the degenerate eigenstates of the radial Schrödinger equation for the hydrogen atom in N dimensions.

THE PROBLEM OF ADAPTING THE METHODOLOGY OF TEACHING DIFFERENTIAL EQUATIONS AT SCHOOL

The general properties of the solution of a homogeneous equation are given. Solutions of a homogeneous linear ordinary differential equation of the second order with constant coefficients in three cases related to the coefficients of the equation are investigated. The obtained results are justified in the form of a theorem. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the work was to develop methods for solving a linear homogeneous differential equation of the second-order at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material, and providing them with a simple teaching method.

ON THE CONSTRUCTION OF A FUNDAMENTAL SYSTEM OF SOLUTIONS OF A LINEAR HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS OF AN ARBITRARY ORDER

The paper proposes a method of presentation topics «On the construction of a fundamental system of solutions of a linear homogeneous differential equation with constant coefficients of an arbitrary order». In the traditional presentation of this topic in the case when the characteristic equation has complex roots, the particular solutions of the equation corresponding to them are constructed by applying the elements of complex analysis. In consequence of that, for students in the field, whose training programs included the theory of linear differential equations with constant coefficients and at the same time does not include the study of the theory of complex analysis, types of private solving the equation in this case is given without substantiation, or as a known fact, only for this case, previously issued elements complex analysis. Offered in the presentation technique differs from the traditional presentation of the topic in that it partial solutions scheme for constructing fundamental system of homogeneous linear equation with constant coefficients of arbitrary order is based only on the basis of the properties of the differential form corresponding to the left side of the equation, without using the elements of the theory of complex analysis.

About unbounded complex operators

The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.

About the general solution of a linear homogeneous differential equation in a Banach space in the case of complex characteristic operators

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

A Further Property of Functions in Class B(m): An Application of Bell Polynomials

We say that a function $\alpha(x)$ belongs to the set ${\bf A}^{(\gamma)}$ if it has an asymptotic expansion of the form $\alpha(x)\sim \sum^\infty_{i=0}\alpha_ix^{\gamma-i}$ as $x\to\infty$, which can be differentiated term by term infinitely many times. A function  $f(x)$ is in the class ${\bf B}^{(m)}$ if it satisfies a linear homogeneous differential equation of the form $f(x)=\sum^m_{k=1}p_k(x)f^{(k)}(x)$, with $p_k\in {\bf A}^{(i_k)}$, $i_k$ being integers satisfying $i_k\leq k$. These functions appear in many problems of applied mathematics and other scientific disciplines. They have been shown to have many interesting properties,  and their integrals $\int^\infty_0 f(x)\,dx$, whether convergent or divergent,  can be evaluated very efficiently via the Levin--Sidi $D^{(m)}$-transformation,  a most effective convergence acceleration method. (In case of divergence, these integrals  are defined in some summability sense, such as Abel summability or Hadamard finite part or a mixture of these two.) In this note, we show that if $f(x)$ is in ${\bf B}^{(m)}$, then so is $(f\circ g)(x)=f(g(x))$, where $g(x)>0$ for all large $x$ and $g\in {\bf A}^{(s)}$,  $s$ being a positive integer. This enlarges the scope of the $D^{(m)}$-transformation considerably to include functions of complicated arguments. We demonstrate  the validity of our result with an application of the $D^{(3)}$ transformation to two integrals $I[f]$ and $I[f\circ g]$, for some $f\in{\bf B}^{(3)}$ and $g\in{\bf A}^{(2)}$. The Fa\`{a} di Bruno formula and Bell polynomials play a central role in our study.

A generalisation of a theorem of O. Perron for semilinear evolution equations

We generalise a well-known result of O. Perron from the 30s that connects the asymptotic behavior of a linear homogeneous differential equation with the response of the inhomogeneous associated equation to a certain class of inhomogeneities (for this reason, Perron's result is also referred to as “input-output method”, “test function method” or “admissibility”).Our extension is twofold, on the one hand, through the means of a (non)linear evolution family, we deal with the mild solution of a nonautonomous semilinear evolution equation and on the other hand, we collect a very general class of inhomogeneities, eligible for a Perron-type approach in this case.From a technical point of view, the Perron input-output scenario is achieved here by using the Green operator.

On Functionally Commutative Quantum Systems

In this paper, a method to solve functionally commutative time-dependent linear homogeneous differential equation is discussed. We apply this technique to solve some dynamical quantum problems.