Real Numbers Research Articles

An SDP relaxation in the complex domain for the efficient coordination of BESS and DGs in single-phase distribution grids while considering reactive power capabilities

This research proposes an efficient energy management system design for medium-voltage distribution networks from a perspective of convex optimization. The exact nonlinear programming problem that represents the operation of distributed energy resources (DERs), such as batteries and photovoltaic sources, is approximated as an equivalent semi-definite programming (SDP) problem. This SDP approximation is achieved in the complex domain using Hermitian matrices, which are the equivalent of positive semi-definite matrices in the real number domain. Two test feeders, one with 27 nodes and the other with 33 nodes, are modified to match the characteristics of Colombian rural and urban networks. The 27-bus grid is adapted to accommodate the average demand and solar generation conditions of the municipality of Capurganá, while those of the city of Medellín are used for the 33-bus grid. One of the main contributions of this research is that it considers a DER operation with variable power factors, which allows for significant improvements compared to the traditional use of a unitary power factor scheme. Numerical comparisons are conducted using metaheuristic optimizers, including parallel versions of the particle swarm optimizer, the vortex search algorithm, and the continuous version of the genetic algorithm. These comparisons demonstrate the effectiveness of the proposed SDP approximation. All simulations are carried out in the MATLAB software (version 2022b), using the disciplined convex optimizer (CVX) and the SDPT3 and SEDUMI solvers.

The descriptive complexity of the set of Poisson generic numbers

Let [Formula: see text] be an integer. We show that the set of real numbers that are Poisson generic in base [Formula: see text] is [Formula: see text]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base [Formula: see text] and not Poisson generic in base [Formula: see text] is complete for the class given by the differences between [Formula: see text] sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.

On Semi-Vector Spaces and Semi-Algebras with Applications in Fuzzy Automata

In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R0+. More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences.

Typical Sequence of Real Numbers From the Unit Interval Has All Distribution Functions

This note is devoted to the study of typical properties (in Baire category sense) of sequences of real numbers in [0, 1]. We prove that the subset of sequences that have all distribution functions forms a residual set.

The Hyperreals enlargement sets and its application on compact topology

 This paper presents a study of hyperreal numbers and their relations to real numbers. The hyperreals are a number system extension of the real number system. With this system, simpler and more intuitively natural definition (topological notions of special points and compactness), proofs (Heine-Borel theorem), and new concepts (limited, unlimited numbers, enlargement sets, and halos) of mathematical interest are offered.

Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators A and B, both of which are positive and have spectra within the interval m,M, denoted as σA and σB. In addition, let us introduce two monotone continuous functions, namely, g and h, defined on the interval m,M. Let f be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers t and s, we have fts≤ftfs. Under these specified conditions, we establish the following inequality: for all 0≤ν≤1, this outcome highlights the intricate relationship between the numerical range of the expression gνAXh1−ν when transformed by the convex function f and the norm of X. Importantly, this inequality holds true for a broad range of values of ν. Furthermore, we provide supportive examples to validate these results.

The intrinsic connection between the fractal dimension of a real number sequence and convergence or divergence of the series formed by it

In this paper, we mainly make research on the relationship between convergence or divergence of a class of series and the fractal dimension of the corresponding real number sequence. A new discriminant method named the fractal dimension test to judge convergence or divergence of certain positive series has been proposed. Some examples have also been provided to illustrate this test.

Analysis and qualitative behaviour of a tenth-order rational difference equation

In this article, we examine the qualitative behavior of the solutionsof the following di¤erence equationzn+1 = aZn-4 +bZn-4/cZn-4-dzn4; n = 0,1,....where the initial conditions Z_9; Z_8; Z_7; Z_6; Z_5; Z_4; Z_3; Z_2; Z_1;Z0 are arbitrary non-zero real numbers and a, b, c, d are positive constants.

Design of Intelligent Political Test Paper Generation Method Based on Improved Intelligent Optimization Algorithm

With the development of artificial intelligence, computer intelligent grouping, as a research hotspot of political ideology examination paper proposition, can greatly shorten the time of generating examination papers, reduce the human cost, reduce the human factor, and improve the quality of political ideology teaching evaluation. Aiming at the problem that the current political ideology examination paper-grouping strategy method easily falls into the local optimum, a kind of intelligent paper-grouping method for political ideology examination based on the improved stock market trading optimisation algorithm is proposed. Firstly, by analyzing the traditional steps of political thought grouping, according to the index genus of the grouping problem and the condition constraints, we construct the grouping model of political thought test questions; then, combining the segmented real number coding method and the fitness function, we use the securities market trading optimization algorithm based on the Circle chaotic mapping initialization strategy and adaptive t-distribution variability strategy to solve the grouping problem of the political thought test. The experimental results show that the method can effectively find the optimal strategy of political thought exam grouping, and the test questions have higher knowledge point coverage, moderate difficulty, and more stable performance.

Computing Galois cohomology of a real linear algebraic group

AbstractLet be a linear algebraic group, not necessarily connected or reductive, over the field of real numbers . We describe a method, implemented on computer, to find the first Galois cohomology set . The output is a list of 1‐cocycles in . Moreover, we describe an implemented algorithm that, given a 1‐cocycle , finds the cocycle in the computed list to which is equivalent, together with an element of realizing the equivalence.

On the generalized interlacing property for the zeros of Bessel functions

This paper investigates a generalized interlacing property between Bessel functions, particularly Jν and Jμ, where the difference |ν−μ| exceeds 2. This interlacing phenomenon is marked by a compensatory interaction with the zeros of Lommel polynomials, extending our understanding beyond the traditional |ν−μ|≤2 regime. The paper extends the generalized interlacing property to cylinder functions and the derivative of Bessel functions, as an application. It is also discussed that Siegel's extension of Bourget hypothesis to rational numbers of ν cannot be further improved to arbitrary real numbers.

Uniqueness of the Eigenvalue Inverse Problem for a Class of X-type Matrices with Linear Relations

Constraining a matrix by adding additional conditions to determine the matrix if given eigenpairs is called the eigenvalue inverse problem for matrices. The eigenvalue inverse problem of a matrix can be based on a given combination of different eigenvectors and real numbers to inverse a matrix method, and the method of inversion varies for different types of matrices. In this paper, we investigate the eigenvalue inverse problem for a class of X-type matrices characterized by linear relations, Invert the matrix based on its characteristics, determine eigenvalues and eigenvectors, and finally, prove that the solution of the problem exists and is unique, derive a series of expressions and recursive formulas, and we also check the algorithm’s accuracy correctness by giving different instances.

An Algorithm for Computing the n^th Root of Real Numbers using a General Method

Abstract: Based on an old article about extracting the ࢔ ࢎ࢚ of real numbers using a general method, we introduce an algorithm for computing such roots, using a different approach, through introducing an operator which we define as the inverse binomial operator .We illustrate the procedure by computing ࢔ ࢎ࢚ roots of some ,randomly, selected real numbers. To show the beauty of using such a new procedure, we compare our calculations with the calculations obtained, for the a certain example, via well-known iteration methods such as the bisection and Newton-Raphson methods; moreover a comparison is made for same purpose using continued fractions method. Finally, a conclusion is given to clarify the advantages of the inverse binomial operator.

On eigenvalues of certain special matrices

Let g : R → [ 0 , ∞ ) be a conditionally negative definite function and f : [ 0 , ∞ ) → [ 0 , ∞ ) be a Bernstein function. We prove that the function h = f ∘ g is conditionally negative definite and that for distinct real numbers p 1 , p 2 , … , p n the inertia of the matrix [ h ( p i − p j ) ] is ( 1 , 0 , n − 1 ) if f is non-linear and g ( x ) = 0 only for x = 0. We also present a new and simple proof to show that the matrix [ log ⁡ ( 1 − p i p j ) ] is negative definite for n distinct positive real numbers p i < 1 , ∀ i . These results supplement and unify earlier results proved by a number of authors including Dyn, Goodman and Michelli, Bhatia and Jain, Garg and Aujla, Garg and Agarwal for operator monotone functions.

Square Root Expressions and Irrational Numbers in Middle School Mathematics Textbooks Throughout the History of the Republic

Irrational numbers, a fundamental set that extends rational numbers to real numbers, have been a subject of limited study in the context of textbook analyses. This study, therefore, examines how square root expressions and irrational numbers have been addressed in middle school mathematics textbooks throughout the history of the Republic of Türkiye. The research was conducted using a comprehensive approach, with qualitative research methods and content analysis was applied to 37 mathematics textbooks (seven 5th grade, twelve 6th grade, nine 7th grade, and nine 8th grade) from 1932 to 2022. Data were collected from mathematics textbooks available at the Ferit Ragıp Tuncor Archive and Documentation Library affiliated with the Ministry of National Education. The results of this study indicate that changes in mathematics curricula are reflected in textbooks, and this reflection is directly influenced by the learning theory adopted in the curriculum. The study also provides insights into how irrational numbers are introduced in textbooks based on the sequence of topics, and recommendations are made based on the findings.

Parametrized multiplicative integral inequalities

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers’ comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus.

Conditions When the Problems of Linear Programming Are Algorithmically Unsolvable

We study the properties of the constructive linear programming problems. The parameters of linear functions in such problems are constructive real numbers. Solving such a problem involves finding the optimal plan with the constructive real number components. We show that it is impossible to have an algorithm that solves an arbitrary constructive real programming problem.

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

Hyperholomorphicity by Proposing the Corresponding Cauchy–Riemann Equation in the Extended Quaternion Field

In algebra, the sedenions, an extension of the octonion system, form a 16-dimensional noncommutative and nonassociative algebra over the real numbers. It can be expressed as two octonions, and a function and differential operator can be defined to treat the sedenion, expressed as two octonions, as a variable. By configuring elements using the structure of complex numbers, the characteristics of octonions, the stage before expansion, can be utilized. The basis of a sedenion can be simplified and used for calculations. We propose a corresponding Cauchy–Riemann equation by defining a regular function for two octonions with a complex structure. Based on this, the integration theorem of regular functions with a sedenion of the complex structure is given. The relationship between regular functions and holomorphy is presented, presenting the basis of function theory for a sedenion of the complex structure.