Quadratic Mappings Research Articles

Stability of Quadratic Mappings in 2-Banach Spaces and Related Topics

Functional analysis is an important branch of mathematics widely used to study the stability of different functional equations. This includes the stability of various quadratic functional equations, which typically involve a specific number of variables to reach results more easily in this field. One of the methods used to study the stability of functional equations is the direct method, which is known for its simplicity in proving the stability of this type of functional equation. In this research paper, we have successfully proven the Hyers-Ulam-Rassias stability of the quadratic functional equation in 2-Banach spaces. Specifically, we have shown that the equation: f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(x+z) holds true within this context. Our approach involved using either the usual or the direct method to establish this stability. Furthermore, we have also demonstrated the generalized Hyers-Ulam stability of the quadratic functional equation in 2-Banach spaces using the usual process by considering various conditions. This has led to many exciting results and revealed many related applications.

Criteria for the Existence of Internal Fixed Points of Lotka-Volterra Quadratic Stochastic Mappings with Homogeneous Tournaments Acting in an (m−1) - Dimensional Simplex

Criteria for the Existence of Internal Fixed Points of Lotka-Volterra Quadratic Stochastic Mappings with Homogeneous Tournaments Acting in an (m−1) - Dimensional Simplex

C1-smooth isogeometric spline functions of general degree over planar mixed meshes: The case of two quadratic mesh elements

Splines over triangulations and splines over quadrangulations (tensor product splines) are two common ways to extend bivariate polynomials to splines. However, combination of both approaches leads to splines defined over mixed triangle and quadrilateral meshes using the isogeometric approach. Mixed meshes are especially useful for representing complicated geometries obtained e.g. from trimming. As (bi-)linearly parameterized mesh elements are not flexible enough to cover smooth domains, we focus in this work on the case of planar mixed meshes parameterized by (bi-)quadratic geometry mappings. In particular we study in detail the space of C1-smooth isogeometric spline functions of general polynomial degree over two such mixed mesh elements. We present the theoretical framework to analyze the smoothness conditions over the common interface for all possible configurations of mesh elements. This comprises the investigation of the dimension as well as the construction of a basis of the corresponding C1-smooth isogeometric spline space over the domain described by two elements. Several examples of interest are presented in detail.

On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula.

The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.

Bernstein algebras that are algebraic and the Kurosh problem

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication algebra. Several equivalent conditions for a Bernstein algebra to be algebraic are given. In particular, known characterizations of Bernstein train algebras in terms of nilpotency are generalized to the case of locally train algebras. Along the way, we show that if a Banach Bernstein algebra is algebraic (respectively, locally train), then it is of bounded degree (respectively, train).Then we investigate the Kurosh problem for Bernstein algebras: whether a finitely generated Bernstein algebra which is algebraic of bounded degree is finite-dimensional. This problem turns out to have a closed link with a question about associative algebras. In particular, when the bar-ideal is nil, the Kurosh problem asks whether a finitely generated Bernstein train algebra is finite-dimensional. We prove that the answer is positive for some specific cases and for low degrees, and construct counter-examples in the general case.On the other hand, by results of Yagzhev the Jacobian conjecture is equivalent to a certain statement about Engel and nilpotence identities of multioperator algebras. We show that the generalized Jacobian conjecture for quadratic mappings holds for Bernstein algebras.

Characterization and Stability of Multi-Euler-Lagrange Quadratic Functional Equations

The aim of the current article is to characterize and to prove the stability of multi-Euler-Lagrange quadratic mappings. In other words, it reduces a system of equations defining the multi-Euler-Lagrange quadratic mappings to an equation, say, the multi-Euler-Lagrange quadratic functional equation. Moreover, some results corresponding to known stability (Hyers, Rassias, and Gӑvruta) outcomes regarding the multi-Euler-Lagrange quadratic functional equation are presented in quasi-β-normed and Banach spaces by using the fixed point methods. Lastly, an example for the nonstable multi-Euler-Lagrange quadratic functional equation is indicated.

Approximation of almost quadratic mappings via modular functional

In this paper, we present generalized stability results of refined quadratic functional equation \[ f(ax-by)=abf(x-y)+a(a-b)f(x)+b(b-a)f(y), \] for any fixed nonzero integer numbers \(a,b\in\mathbb{Z}\) with \(a\neq b\) in modular spaces. As results, we generalize stability results of a quadratic functional equation in [{K.-W. Jun, H.-M. Kim, J. Son, Functional Equations in Mathematical Analysis, \(\bf{2012}\) (2012), 153--164}].

A Novel Surrogate-Based Approach to Yield Estimation and Optimization of Microwave Structures Using Combined Quadratic Mappings and Matrix Transfer Functions

Yield estimation and yield-driven design optimization play important roles in microwave design. Existing surrogate-based yield estimation/optimization methods need to develop separate surrogate models or a large surrogate model with high complexity to deal with multiport problems, which is computationally inefficient. This article proposes a novel surrogate-based method to expedite yield estimation/optimization of multiport microwave structures using combined quadratic mappings and matrix-valued transfer functions. For multiport structures, the responses of different transfer functions corresponding to different pairs of input–output ports are computed with separate residues, while the responses of transfer functions for all pairs of input–output ports are computed with a common set of poles. Taking advantage of this, we propose to formulate a set of quadratic functions to map the relationship between the separate residues and statistical/geometrical parameters while employing merely one quadratic function to map the relationship between the common poles and statistical/geometrical parameters. The resultant mappings together with the matrix-format transfer function form an efficient surrogate to expedite the yield estimation and optimization processes for microwave structures. Compared with existing surrogate-based methods, the proposed method can achieve similar yield estimation accuracy in a shorter time due to fewer electromagnetic (EM) simulations. Three microwave structures are used to demonstrate the advantages of the proposed method. Based on accurate yield estimations, we further perform yield-driven design optimization incorporating the proposed surrogate for all the three examples.

On ranges of set-valued mappings

We derive conditions ensuring that the range of a set-valued mapping with a compact convex domain covers a prescribed set. In Fréchet spaces, we consider approximations by one single-valued mapping such that the inverse of it has convex fibers. Subsequently, in Banach and finite-dimensional spaces, we focus on approximations determined by a convex set of bounded linear mappings such as Páles-Zeidan Jacobian, Clarke's generalized Jacobian, shields by T.H. Sweetser, or Neumaier's interval extensions of the derivative of a smooth mapping. As easy corollaries in Euclidean spaces, we obtain perturbation stability of the property of metric semiregularity under the additive perturbation by a single-valued mapping having sufficiently small calmness modulus; as well as the non-smooth Lyusternik-Graves theorem and Robinson's theorem by A.F. Izmailov. Finally, given two quadratic mappings f and g, a polyhedral convex set Ω, and an ordered interval Θ, we provide conditions guaranteeing that an ordered interval Γ is such that for each y∈Γ there is an x∈Ω with y=f(x) and g(x)∈Θ. This theorem has direct applications in power network security management such as preventing the electricity blackout.

Dynamic behaviors of a symmetrically coupled period-doubling system

A system of two coupled mappings demonstrates a variety of nonlinear phenomena such as the inverse state, spatiotemporal intermittence, traveling wave and the synchronization. In this paper, we are concerned with a system of symmetrically coupled quadratic mappings. Bezruchko et al. employed numerical method to study the bifurcation problem of such a system, but did not give a full investigation in theory because of the complicated computation. In this paper, we adopt the complete discrimination system theory and the real root isolation algorithm to overcome the difficulty. We will give a completed description of the bifurcations in theory for such a system, including the transcritical bifurcation, pitchfork bifurcation, flip bifurcation and the Neimark-Sacker bifurcation.

On perturbations of two general equations in several variables

In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated.

Asymptotic behavior of generalized quadratic mappings

We show in this paper that a mapping $f$ satisfiesthe following functional equationbegin{eqnarray*}biguplus_{x_2,cdots,x_{d+1}}^{d}f(x_1) = 2^{d} sum_{i=1}^{d+1}f(x_i),end{eqnarray*}if and only if it is quadratic. In addition, we investigate generalized Hyers-Ulam stability problem for the equation, and thus obtain an asymptotic property of quadratic mappings as applications.

Topological Classifications for a Class of 2-dimensional Quadratic Mappings and an Application to Iterative Roots

Usually, it is difficult to find a topological relation among high dimensional mappings because of the complexity of existence conditions for conjugacies. In this paper, we investigate the conjugation for a class of 2-dimensional quadratic mappings via the theory of polynomial algebra. Two kinds of conjugate classifications are given. Finally, an application to the problem of iterative roots is also presented.

Ulam stability of a functional equation deriving from quadratic and additive mappings in random normed spaces

The aim of this work is to introduce a new mixed type quadratic-additive functional equation, to obtain its general solution and to investigate Ulam stability by using Hyers method in random normed spaces.

Ulam Type Stability of ?-Quadratic Mappings in Fuzzy Modular ∗-Algebras

In this paper, we find the solution of the following quadratic functional equation n∑1≤i<j≤nQxi−xj=∑i=1nQ∑j≠ixj−(n−1)xi, which is derived from the gravity of the n distinct vectors x1,⋯,xn in an inner product space, and prove that the stability results of the A-quadratic mappings in μ-complete convex fuzzy modular ∗-algebras without using lower semicontinuity and β-homogeneous property.

SYNTHESIS OF STRIP-BAND TELECOMMUNICATION SYSTEM WITH INCREASED CONFIDENTIALITY OF INFORMATION TRANSMISSION THROUGH USE OF PSSEVDOVIDOVYPOVY

Currently, special attention is paid to the creation of broadband telecommunications systems with a noise-like signal. Such systems have indisputable advantages in noise immunity, in the confidentiality of information transmission, electromagnetic compatibility with neighboring electronic devices. The formation of a broadband signal in most cases is carried out by the method of direct spreading of the spectrum by manipulating the carrier frequency signal by a pulse pseudo-random sequence (PVP) –m-sequence, Gold, Kasami, Walsh sequence. Sequences such as a well-known attacker can pick up in the receiver and they can not be considered reliable protection of information. The use of the phenomenon of dynamic chaos provides an opportunity to find a new class of PVP, the structure of which is almost impossible to reproduce, so their use provides increased confidentiality of information transmission. PVP generators are built on the basis of one-dimensional chaotic mappings - logistic, quadratic and cubic, which are chaotic systems. When creating a PVP based on chaos, three secret encryption keys are used: for example, for logistic separation, this is the initial value, the equation parameter, and the start of the sequence. To increase the cryptosecurity of telecommunication systems, it is proposed to create PVP based on 2 chaotic signals (by logistic and quadratic mappings), which provides an increase in secret keys from three to six. The paper proposes a scheme for building a telecommunications system with high confidentiality of information transmission through the use of such PVP.

Kato's chaos created by quadratic mappings associated with spherical orthotomic curves

Kato's chaos created by quadratic mappings associated with spherical orthotomic curves

Bifurcation of the Birth of a Closed Invariant Curve in a One-Parameter Family of Quadratic Mappings of the Plane

We give an example of a one-parameter family of quadratic endomorphisms of the plane, in which a closed invariant curve is born from an elliptic fixed point.

Some Fixed Point Theorems for Quadratic Quasicontractive Mappings

In this paper, we introduce the notion of quadratic quasicontractive mapping and prove two generalizations of some classical fixed point theorems. Furthermore, we present some examples to support our main results.

Remarks on Quadratic Mappings

We present several results concerning the geometry and topology of quadratic mappings. The main attention is given to certain basic properties, namely, properness, surjectivity, stability, and topology of fibers. The structure of singular sets, discriminants, bifurcation diagrams, and Pareto sets is also discussed. After describing the setting and algebraic methods for computing the topological degree and Euler characteristic that are crucial for our approach, we concentrate on the study of quadratic endomorphisms and quadratic mappings into the plane. We begin by considering homogeneous quadratic endomorphisms of the plane and give criteria of properness and possible values of topological degree, as well as some geometric information about the structure of singular sets and discriminants. Next, we deal with analogs of the above results for proper quadratic endomorphisms in arbitrary dimension. In particular, we obtain an explicit estimate for the topological degree of quadratic endomorphism in terms of dimension and present examples showing that this estimate is exact. After this we discuss homogeneous quadratic mappings from ℝn into the plane and obtain a number of results on the Euler characteristic and topology of fibers. Finally, we derive some corollaries in the case of numerical range mapping of a complex square matrix.