Vector Argument Research Articles

Threshold ring signature: generic construction and logarithmic size instantiation

A ring signature is a variant of normal digital signature and protects the privacy of a specific signer in the sense that a ring signature can be verified, but the signer’s identity can only be traced to a limited set. The concept was further enhanced to threshold setting to distribute signing ability among several signers. Since threshold ring signature was introduced, it was a hard problem whether one can have efficient constructions for it. In this paper, we introduce a new generic construction of threshold ring signature, named GTRS, based on canonical identification of a specific form. Our signature consists of a polynomial (represented by n-t+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n - t + 1$$\\end{document} coefficients) and a single response, resulting in significantly shorter threshold ring signatures. Instantiating the generic construction with specific DL-based components, e.g. Schnorr identification and a novel vector argument of knowledge developed in this paper, we obtain GTRS-EC, which is shorter than all existing threshold ring signatures without any trusted setup.

Fourier series for the multidimensional-matrix functions of the vector variable

In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix (mdm) polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series theory are extended to the case of the mdm functions, what allows us to solve more general approximation problems. The general case of the approximation of the mdm function of the vector argument by the Fourier series on the orthogonal mdm polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for possible analytical studies are obtained.

On the functional basis of isotropic vector and tensor functions by Shariff (2023)

In the paper Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) a functional basis of a system of vectors and symmetric tensors is proposed. The functional basis is expressed in terms of eigenvalues and eigenvectors of the first tensor and includes a smaller number of terms in comparison to the classical irreducible representation (see, e.g., Boehler, J. Appl. Math. Mech. 57:323–327, 1977; Pennisi and Trovato, Int. J. Eng. Sci. 25:1059–1065, 1987). In the present contribution, we show that elements of the functional basis by Shariff (Q. J. Mech. Appl. Math. 76:143–161, 2023) do not represent isotropic invariants of the vector and tensor arguments and cannot thus be referred to as the functional basis. To this end, a counterexample with two symmetric tensors is considered. Under an arbitrary orthogonal transformation the functional basis (Shariff, Q. J. Mech. Appl. Math. 76:143–161, 2023) of these two tensors should remain constant but it does change in contrast to the classical representation.

Orthogonal Polynomials and Fourier Series for Functions of Vector Variable: Multidimensional-Matrix Approach

In the article, the theory of the Fourier series on the orthogonal multidimensional-matrix polynomials is developed. The known results from the theory of the orthogonal polynomials of the vector variable and the Fourier series are given and the new results are presented. In particular, the known results of the Fourier series are extended to the case of the multidimensional-matrix functions, what allows us to solve more general approximation problems. The general case of the approximation of the multidimensional-matrix function of the vector argument by the Fourier series on the orthogonal multidimensional-matrix polynomials is realized programmatically as the program function and its efficiency is confirmed. The analytical expressions for the coefficients of the second degree orthogonal polynomials and Fourier series for the potential studies are obtained.

Stochastic Time Complexity Surfaces of Computing Node

The paper is devoted to the formal description of the running time of the user task on some virtual nodes in the computing network. Based on the probability theory framework, this time represents a random value with a finite mean and variance. For any class of user task, these moments are the functions of the node resources, task numerical characteristics, and the parameters of the current node state. These functions of the vector arguments can be treated as some surfaces in the multidimensional Euclidean spaces, so the proposed models are called the stochastic time complexity surfaces. The paper also presents a class of functions suitable for the description of both the mean and variance. They contain unknown parameters which should be estimated. The article includes the statement of the parameter identification problem given the statistical results of the node stress testing, recommendations concerning the test planning, and preprocessing of the raw experiment data. To illustrate the performance of the proposed model, the authors design it for an actual database application—the prototype of the passengers’ personal data anonymization system. Its application functions are classified into two user task classes: the data anonymization procedures and fulfillment of the statistical queries. The authors identify the stochastic time complexity surfaces for both task types. The additional testing experiments confirm the high performance of the suggested model and its applicability to the solution of the practical providers’ problems.

MULTI-CRITERIA EVALUATION OF THE MULTIFACTOR STOCHASTIC SYSTEMS EFFECTIVENESS

Subject of the research in the article is the evaluation of complex multifactor stochastic systems functioning effectiveness according to a variety of correlated criteria. The problem actuality is determined by the fact that an independent evaluation of system effectiveness for each of the mutually correlated criteria for the system under consideration is not informative. In well-known works in the direction of multiple correlation, a relatively simple problem of estimating the correlation between one resulting parameter and a set of influencing factors is considered, which is not enough for the analysis and management of multicriteria systems. In addition, the known results do not take into account possible significant differences in influencing factors mutual correlation values. Purpose of Work is to develop a methodology for a comprehensive assessment of system effectiveness according to a variety of interrelated criteria. Tasks to be Solved: splitting the set of system parameters into two subsets (parameters determining the effectiveness of system and parameters affecting it), forming additive convolutions of parameters included in subsets, developing a methodology for calculating the multiple correlation coefficient between the components of the selected subsets, developing a method for differentiating a scalar function from a vector argument by this argument. Applied Methods: nonlinear programming, multidimensional correlation analysis, method of differentiation of scalar functions by vector argument. These methods are used for forming and calculating a multiple correlation coefficient between the set of system effectiveness complex criterion components values and its control parameters set values. Results Obtained: proposed methodology provides the possibility of solving the problems of system management, taking into account the revealed relationship between the multi-criteria evaluation of system effectiveness and values of its controlled parameters. At the same time, an important advantage of the obtained result lies in the possibility that arises when using it to take into account the joint (group) influence of control variables on the complex criterion of system efficiency. The developed technology of scalar functions differentiation by vector argument has great practical importance which expands the arsenal of computational mathematics.

Analytic functions of a vector argument and partially conformal mappings in continuum complex spaces

Analytic functions of a vector argument and partially conformal mappings in continuum complex spaces

Analytic functions of a vector argument and partially conformal mappings in continuum complex spaces

A vector generalization of the main concepts in the theory of functions of a complex variable - the concepts of the modulus and the argument of the complex number - is proposed. The authors introduce a certain generalization of the concept of holomorphic functions and mappings in the case of continuum complex spaces.

Second‐order design sensitivity analysis using diagonal hyper‐dual numbers

AbstractAlthough sensitivity analysis provides valuable information for structural optimization, it is often difficult to use the Hessian in large models since many methods still suffer from inaccuracy, inefficiency, or limitation issues. In this context, we report the theoretical description of a general sensitivity procedure that calculates the diagonal terms of the Hessian matrix by using a new variant of hyper‐dual numbers as derivative tool. We develop a diagonal variant of hyper‐dual numbers and their arithmetic to obtain the exact derivatives of tensor‐valued functions of a vector argument, which comprise the main contributions of this work. As this differentiation scheme represents a general black‐box tool, we supply the computer implementation of the hyper‐dual formulation in Fortran. By focusing on the diagonal terms, the proposed sensitivity scheme is significantly lighter in terms of computational costs, facilitating the application in engineering problems. As an additional strategy to improve efficiency, we highlight that we perform the derivative calculation at the element‐level. This work can contribute to many studies since the sensitivity scheme can adapt itself to numerous finite element formulations or problem settings. The proposed method promotes the usage of second‐order optimization algorithms, which may allow better convergence rates to solve intricate problems in engineering applications.

MATHEMATICAL MODEL TO CALCULATE THE TRAJECTORIES OF ELECTROMAGNETIC MILL OPERATING ELEMENTS

The purpose of the research under consideration is to develop a mathematical model to calculate the trajectories of the ferromagnetic operating elements (millstones) of an electromagnetic mill, moving in a rotating magnetic field under electrodynamic and hydrodynamic resistance forces being limited by the space of the mill’s working chamber. The millstone motion is described through the equations of plane motion of arbitrary-shaped two-dimensional body. The driving forces of this motion are determined on the basis of the approximation of the tabulated functions connecting the module and the orientation of the equivalent force applied to the millstone, with its position in the working chamber and composite MMF phase of mill inductor winding. These tabulated functions are derived from the estimation of the magnetic field inside a working chamber with millstones, in two-dimensional quasi-stationary approximation, using FEM analysis. The publication contains the approximation algorithm for these tabulated vector functions of a vector argument, mathematical statement of millstones trajectories calculating, and analysis of mathematical experiments results that make it possible to evaluate the adequacy of the model. The developed tool enables conducting quantitative analysis of grinding/mixing process and will help to establish relationships between the electromagnetic mill design parameters and its performance. References 21, figures 6.

Sliding Space-Frequency Processing of Finite Two-Dimensional Real Discrete Signals Based on the Fourier Transform with Variable Parameters

In the field of Fourier processing of finite signals, three main directions of scientific research have been identified: Fourier processing of one-dimensional finite signals - processing of scalar functions of a scalar argument, Fourier processing of two-dimensional finite signals - processing of scalar functions of a vector argument, multichannel Fourier processing - processing of vector functions of a scalar argument. As part of the creation of a generalized theory of Fourier processing of finite signals, the authors proposed: the theory of spectral analysis of discrete signals at finite intervals in the bases of parametric exponential functions and the theory of two-dimensional digital signal processing in Fourier bases with variable parameters. The developed theories, generalizing the theory of Fourier processing of one-dimensional and two-dimensional signals, are based: on the introduction of new concepts of the shift of finite discrete signals in one-dimensional and two-dimensional cases and the introduction of new basic Fourier processing systems of discrete signals, which have the properties of multiplicativity, functions in the system. The mathematical apparatus of two-dimensional discrete Fourier transform with variable parameters in matrix and algebraic form is considered. A new method for processing finite two-dimensional real discrete signals in the spatial-frequency domain based on the discrete Fourier transform with variable parameters, the method of sliding spatial-frequency processing, has been introduced. An efficient method and algorithm for fast diagonal sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters has been developed. The estimation of the efficiency and effectiveness of the algorithm of the diagonal sliding two-dimensional discrete Fourier transform with variable parameters from the point of view of computational costs is carried out. As a result of experimental studies on model two-dimensional discrete finite signals, the validity, efficiency and reliability of the proposed method of sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters have been proved. A comparison (from the point of view of computational costs) of the developed method of sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters with the standard method of sliding processing of this type of signals is carried out.

Estimating the Needed Volume of Investment in Developing a Chain of Charging Stations for Electric Vehicles along a Highway

ABSTRACTThe problem of developing a chain of charging stations for electric vehicles along a highway crossing a geographic region is considered, and a tool for determining an optimal structure of such a chain is proposed. The tool allows, particularly, the regional administration and interested private investors to estimate the needed volume of investment in developing the chain under uncertainty in a) a demand for electricity in the chain, b) the market cost of the equipment to be acquired and installed in the chain, and c) the maintenance cost of all the equipment and the cost of operating the chain. The problem under consideration is formulated as that of finding the maximin of a sum of several linear and two bilinear functions of vector arguments some coordinates of which are integer. Finding this maximin is proven to be reducible to solving a mixed programming problem having a linear structure.

Caristi’s Inequality and α-Contraction Mappings

A new Caristi-type inequality is considered and Caristi’s fixed point theorem for mappings of complete metric spaces is developed (in both the single- and set-valued cases). On the basis of this development mappings of complete metric spaces which are contractions with respect to a function of two vector arguments are studied. This function is not required to be a metric or even a continuous function. The proved theorems are generalizations of the Banach contraction principle and Nadler’s theorem.

On the differentiation of a composite function with a generalized vector argument on homogeneous time scales

On the differentiation of a composite function with a generalized vector argument on homogeneous time scales

An equilibrium analysis of investment opportunities in a regional electrical grid

A tool for quantitatively analyzing perspectives of investing in a regional electrical grid that allows regional authorities to estimate (a) the level of potential investments in developing this system, and (b) the volumes and the prices of electricity to be provided for the population and for the industry in the region (in the framework of this system) in a certain period of time in the future (as a result of these investments) is presented. The problem of estimating such perspectives is modeled as a static, two-person, non-cooperative game on disjoint polyhedra with the payoff functions being a sum of a bilinear and a linear function of vector arguments (for the first player), and a bilinear function of the same vector arguments (for the second player). It is proved that equilibria in this game, which determine (a) optimal investment strategies proceeding from financial capabilities of the electricity producers in the region and their ability to borrow money from external investors, and (b) electricity prices expected to be mutually acceptable to both the consumers and the producers of electricity in the developed electrical grid, can be found by solving linear programming problems forming a dual pair. An illustrative numerical example of developing the proposed game model and calculating both optimal investment strategies and equilibrium electricity prices is considered.

A discrete Faà di Bruno's formula

We derive a discrete Faa di Bruno's formula that rules the behaviour of finite differences under composition of functions with vector values and arguments.

Physical-statistical model of a vector field for single scattering by random inhomogeneities in the propagation channel

A simple Gaussian model for engineering calculations of radar signals passed through the propagation channel with random inhomogeneities is considered. The set of parameters introduced to the model allows spatiotemporal and polarization properties of the signal to be described by a random vector function of the vector argument.

Visualizing Poynting vector energy flow in electric circuits

According to standard Poynting vector arguments, energy in a circuit flows from the batteries to the resistors, not through the connecting wires, but through empty space between the wires. The computer simulation CircuitSurveyor helps to visualize this counterintuitive fact. The simulation also demonstrates the electric fields present near a circuit.

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.

Analyzing the composition-property diagrams of multicomponent liquid mixtures

We consider the local regularities of the composition-intensive/specific property diagrams of multicomponent mixtures in which composition is a scalar function of a vector argument. A topographical method used in differential geometry is used to produce one-dimensional, two-dimensional, and three-dimensional scalar fields. The scalar field gradient is used to study the composition-property diagrams of five-component mixtures (producing four-dimensional scalar fields).