ENERGY OF ELECTRIC CHARGE RADIATION AND ITS EFFECTS

It is believed that an electric charge moving along a circular path, i.e. with centripetal acceleration, it is necessary to emit electromagnetic waves. This applies, inter alia, to cyclotron radiation. The purpose of the work is to establish the conditions for the radiation of an electric charge, based on the significant differences between its tangential and centripetal accelerations. The relevance of the work is determined by the widespread use of devices that generate electromagnetic radiation due to the acceleration of electric charges, including X-ray units and magnetrons. The starting point is a credible statement. A number of mathematically correct transformations are performed with it. Therefore, the result is necessarily reliable. Sad experience shows that this logic is not available for many specialists. In the event that such a necessary reliable result contradicts the existing paradigm, preference is almost always given to the paradigm, regardless of the persuasiveness of the evidence. This circumstance is an almost insurmountable obstacle to obtaining new knowledge. After all, if it does not contradict the paradigm, then it is not new and does not represent any value. Electromagnetic radiation carries away energy. It follows from this that the energy of the radiating system changes during radiation. Associated with this is the well-known rule: the change in energy is equal to the perfect work. Four theorems are proved. Theorem 1. A tangentially accelerated charge emits electromagnetic waves. Theorem 2. A normally accelerated charge does not emit electromagnetic waves. Theorem 2 formalizes a circumstance well-known in mechanics that the centripetal force does not perform work (since the scalar product of orthogonal vectors must be zero). Theorem 3. Electric charge satisfies Newton's second law. When a hydrogen-like atom passes from one stationary state to another, the orbital angular momentum changes. The difference is attributed to a photon and is called the photon's spin. Theorem 4. The spin of a photon is zero. The defect in the angular momentum of an atom during radiation can easily be attributed to the nucleus of an atom and even to an electron.

Interval Version of the Operational Calculation Method for Solving Ordinary Differential Equations

This article discusses the interval variant of solving ordinary differential equations with given initial conditions, i.e. the Cauchy problem, by the method of operational calculus. This is where the interval version of the operational calculus is motivated and built. As a result, on the basis of the proved theorem in this article, an analytic interval set of solutions is obtained that is guaranteed to contain a real solution to the problem.

Solution of a Linear Nondegenerate Matrix Equation Based on the Zero Divisor

New formulas were obtained to solve the linear non-degenerate matrix equations based on zero divisors of numerical matrices. Two theorems were formulated, and a proof to one of them is provided. It is noted that the proof of the second theorem is similar to the proof of the first one. The proved theorem substantiates new formula in solving the equation equivalent in the sense of the solution uniqueness to the known formulas. Its fundamental difference lies in the following: any explicit matrix inversion or determinant calculation is missing; solution is "based" not on the left, but on the right side of the matrix equation; zero divisor method is used (it was never used in classical formulas for solving a matrix equation); zero divisor calculation is reduced to simple operations of permutating the vector elements on the right-hand side of the matrix equation. Examples are provided of applying the proposed method for solving a nondegenerate matrix equation to the numerical matrix equations. High accuracy of the proposed formulas for solving the matrix equations is demonstrated in comparison with standard solvers used in the MATLAB environment. Similar problems arise in the synthesis of fast and ultrafast iterative solvers of linear matrix equations, as well as in nonparametric identification of abnormal (emergency) modes in complex technical systems, for example, in the power systems

On electromagnetic radiation of individual charges

The aim of the study is to establish the conditions for synchrotron radiation based on significant differences between tangential and centripetal accelerations of electric charges. From the fact that electromagnetic radiation carries away energy, it follows that the energy of the radiating system changes during radiation. Related to this is the following well-known rule: a change in energy is equal to work done. Three relevant theorems are proved. Theorem 1 states that a tangentially accelerated electric charge emits electromagnetic waves. Theorem 2 states that a normally accelerated electric charge does not emit electromagnetic waves. It is a well-known circumstance that the centripetal force does not perform work (since the scalar product of orthogonal vectors must be equal to zero). The proofs of Theorems 1 and 2 are performed in terms of forces. For electric charges, the transition to the terms of accelerations is carried out in accordance with Theorem 3which states that an electric charge satisfies Newton's second law. The tangential acceleration of an electric charge leads to the emission of electromagnetic waves. Generalization of the phenomenon of radiation to acceleration in general, including. normal charge acceleration, is false. The cause of synchrotron radiation should be sought in the tangential acceleration due to Coulomb interactions between the beam charges.

Indignation moving singularity and analytical approximate solution of a nonlinear third-order equations

В данной работе представлено исследование рассматриваемого класса нелинейных дифференциальных уравнений с подвижными особыми точками. Учитывая авторскую разработку теоремы существования и единственности решения построена структура аналитического приближенного решения, для которой, в данной работе, было установлено влияние возмущения подвижной особой точки. Представленные теоретические положения подтверждены с помощью численного эксперимента. Для оптимизации априорных оценок применялась апостериорная оценка. This paper presents a study of one class of nonlinear differential equations with movable singular points. On the basis of the previously proved theorem of existence and uniqueness of the solution, the structure of the analytical approximate solution was obtained, for which, in this work, the influence of the perturbation of a moving singular point was established. Results are tested using a numerical experiment. To optimize the prior estimates, the posterior estimate was used.

Block circulant matrices and the spectra of multivariate stationary sequences

Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.

Crosscap number of knots and volume bounds

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

Imaginary Points in Cartesian Coordinate System

Geometric models are considered that allow symbolic representation of imaginary points on a real Cartesian coordinate plane XY. The models are based on the fact that through every pair of imaginary conjugate points A~B with complex coordinates x = a ± jb, y = c ± jd one unique real line m passes. For the image of imaginary points, it is proposed to use the graphic symbol m{OL} consisting of the line m passing through the imaginary points, the center O of the elliptic involution σ with imaginary double points A~B on the line m, and the Laguerre point L, from which the corresponding points involutions σ are projected by an orthogonal pencil of lines. According to A.G. Hirsch, the symbol m{OL} is called the marker of imaginary conjugate points A~B. A theorem is proved that establishes a one-to-one correspondence between the real Cartesian coordinates of the points O, L of the marker, and the complex Cartesian coordinates of the pair of imaginary conjugate points represented by this marker. The proved theorem allows us to solve both the direct problem (the construction of a marker depicting these imaginary points) and the inverse problem (the determination of the Cartesian coordinates of imaginary points represented by the marker). A graphical algorithm for constructing a circle passing through a real point and through a pair of imaginary conjugate points is proposed. An example of the graph-analytical determination of the Cartesian coordinates of imaginary points of intersection of two conics that have no common real points is considered.

Deep Neural Network Quantization via Layer-Wise Optimization Using Limited Training Data

The advancement of deep models poses great challenges to real-world deployment because of the limited computational ability and storage space on edge devices. To solve this problem, existing works have made progress to prune or quantize deep models. However, most existing methods rely heavily on a supervised training process to achieve satisfactory performance, acquiring large amount of labeled training data, which may not be practical for real deployment. In this paper, we propose a novel layer-wise quantization method for deep neural networks, which only requires limited training data (1% of original dataset). Specifically, we formulate parameters quantization for each layer as a discrete optimization problem, and solve it using Alternative Direction Method of Multipliers (ADMM), which gives an efficient closed-form solution. We prove that the final performance drop after quantization is bounded by a linear combination of the reconstructed errors caused at each layer. Based on the proved theorem, we propose an algorithm to quantize a deep neural network layer by layer with an additional weights update step to minimize the final error. Extensive experiments on benchmark deep models are conducted to demonstrate the effectiveness of our proposed method using 1% of CIFAR10 and ImageNet datasets. Codes are available in: https://github.com/csyhhu/L-DNQ