Vector-valued Functions Research Articles

Pricing of European Vulnerable Options under the Double Exponential Jump-diffusion Model with Stochastic Volatility and Stochastic Interest Rates

In this paper, the pricing formula for European vulnerable options is discussed under the framework of a double-exponential jump-diffusion model with stochastic interest rates and stochastic volatility. By utilizing the characteristic function of a multidimensional stochastic vector and the Fourier-Cosine method, among other techniques, we derive the pricing formula for European vulnerable options.

Vector Lattice Powers: Continuous and Measurable Vector Functions

In the study of order properties of homogeneous polynomials in vector lattices two constructions are of fundamental importance: the symmetric positive tensor product and the vector lattice power. Both associate a canonical $n$-homogeneous polynomial with each Archimedean vector lattice, such that any other homogeneous polynomial of an appropriate class defined on the same vector lattice is the composition of the canonical polynomial with a linear operator. With this so called ``linearization'' in hand, various tools of the theory of positive linear operators can be used to study homogeneous polynomials. Thus, the problem of description of the Fremlin symmetric tensor products and the vector lattice powers for special vector lattices arises. The former enables one to study a large class of order bounded homogeneous polynomials, but has a very complicated structure; the latter has a much more transparent structure, but handles a narrower class of homogeneous polynomials, namely orthogonally additive ones. The purpose of this note is to describe the power of the vector lattice of continuous or Bochner measurable vector functions with values in a Banach lattice and to apply this result to the representation of homogeneous orthogonally additive polynomials.

On Decompositions of Permutations in Quadratic Functions

The algebraic degree of a vectorial Boolean function is one of the main parameters driving the cost of its hardware implementation. Thus, finding decompositions of functions into sequences of functions of lower algebraic degrees has been explored to reduce the cost of implementations. In this paper, we consider such decompositions of permutations over F2n. We prove the existence of a decomposition of the inverse using quadratic and linear power permutations for all permutations when 2n-1 is a prime, and we prove the non-existence of such decompositions for power permutations of differential uniformity strictly lower than 16 when 4|n. We also prove that any permutation admits a decomposition into quadratic power permutations and affine permutations of the form ax+b if 4∤n. Furthermore, we prove that any permutation admits a decomposition into cubic power permutations and affine permutations. Finally, we present a decomposition of the PRESENT S-Box using the power permutation x7 and affine permutations.

A matrix-analytical sampling formula for time-homogeneous coalescent processes under the infinite sites mutation model.

A matrix-analytical sampling formula for time-homogeneous coalescent processes under the infinite sites mutation model.

Nonlinear approximation of vector-valued functions by Shepard operators based on max-product and max-min operations

Nonlinear approximation of vector-valued functions by Shepard operators based on max-product and max-min operations

Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions

In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. At first, we derive a necessary optimality condition for identifying weakly minimal solutions for the considered set optimization problem with the help of a family of vector optimization problems and the Gerstewitz scalarizing function. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function. Lastly, some numerical examples are given to exhibit the performance of the proposed method. Also, the performance of the proposed method with the existing steepest-descent method is given.

A new space of generalised vector-valued functions of bounded variation

In [18], the authors introduced the space of scalar-valued functions GBV⋆(A) to minimise a class of functionals whose study is motivated by fracture mechanics. In this paper, we extend the definition of GBV⋆(A) to the vectorial case, introducing the space GBV⋆(A;Rk). We study the main properties of GBV⋆(A;Rk) and prove a lower semicontinuity result useful for minimisation purposes. With the Direct Method in mind, we adapt the arguments of [18] to show that minimising sequences in GBV⋆(A;Rk) can be modified to obtain a minimising sequence converging Ld-a.e in A.

A system of interface cracks with different electric conditions at their faces in a magnetoelectroelastic bimaterial

A plane strain problem for a set of collinear cracks along the interface between dissimilar magnetoelectroelastic materials is considered. The number of cracks, their location, and lengths can be arbitrary. Remote mixed-mode mechanical loading, electric, and magnetic induction are prescribed at infinity. Two cases of magnetic and electric conditions at the crack faces are considered. First case assumes that the crack is electrically and magnetically permeable, and in the second case, the crack is assumed electrically permeable and magnetically impermeable. Using the presentations of mechanical, electric, and magnetic quantities via sectionally analytic vector functions, problems of linear relationship are formulated and solved analytically. The analytical expressions for mechanical, electric, and magnetic quantities at different parts of the material interface as well as their asymptotic expressions at the crack tips are obtained in the closed form. Using these expressions together with the crack closer integral, the energy release rates near right and left tips of the cracks are obtained in analytical form. The graphs of stresses and magnetic induction between cracks and also crack faces normal displacement jumps are presented for both crack models, different crack locations, their lengths, mechanical, and magnetic loading. The corresponding energy release rates are also found. These parameters are significant for assessing failure mechanisms and enhancing the reliability of magnetoelectroelastic composites during their operation.

Asymptotic analysis of a clamped thin multidomain allowing for fractures and discontinuities

We consider a thin multidomain of [Formula: see text] consisting of a vertical rod upon a horizontal disk. The equilibrium configurations of the thin hyperelastic multidomain, allowing for fracture and damage, are described by means of a bulk energy density of the kind [Formula: see text], where [Formula: see text] is a Borel function with linear growth and [Formula: see text] denotes the gradient of the displacement, i.e. a vector-valued function [Formula: see text]. By assuming that the two volumes tend to zero, under suitable boundary conditions and loads, and suitable assumptions of the rate of convergence of the two volumes, we prove that the limit model is well posed in the union of the limit domains, with dimensions, respectively, [Formula: see text] and [Formula: see text].

On the construction of solutions to a class of multidimensional generalized Cauchy-Riemann systems with singular coefficients

The article is devoted to the study of one class of multidimensional generalized Cauchy-Riemann systems with singular coefficients of the form -𝛼 -𝛽𝐴, here 𝛼,-| |𝛽 - real numbers such that + > 0 and 𝛽 - an integer, - a complex matrix of the order . By moving to the polar coordinates, the solutions of the system are searched for in the form of a trigonometric series of = 𝑔 with unknown coefficients ( ), depending on from = . For these unknown coefficients, the related Bessel equa-tions are obtained, in which the coefficient for the desired vector function has the form: (2𝛼 + ) + 4 2(1-𝛼-𝛽)𝐴𝐴, a unit matrix of the order of . Depending on 𝛼, 𝛽and the eigenvalues of of the matrix 𝐴𝐴, general solutions of the last equations are found,then the general solution of the original system is written out. It should be noted that in the case when the matrix has eigenvalues that are not semisimple, i.e. having unequal algebraic and geometric multiplicities, it is necessary to solve inhomogeneous related Bessel equations by the method of variation of arbitrary constants. From the found general solution of the system, it is possible to distinguish regular, i.e. having no singularities in the finite part of the plane, as well as solutions to boundary value problems.

Lattice Lipschitz operators on C(K)- spaces

Given a Banach lattice L, the space of lattice Lipschitz operators on L has been introduced as a natural Lipschitz generalization of the linear notions of diagonal operator and multiplication operator on Banach function lattices. It is a particular space of superposition operators on Banach lattices. Motivated by certain procedures in Reinforcement Learning based on McShane–Whitney extensions of Lipschitz maps, this class has proven to be useful also in the classical context of Mathematical Analysis. In this paper, we discuss the properties of such operators when defined on spaces of continuous functions, focusing attention on the functional bounds for the pointwise Lipschitz inequalities defining the lattice Lipschitz operators, the representation theorems for these operators as vector-valued functions, and the corresponding dual spaces. Finally, and with possible applications in Artificial Intelligence in mind, we provide a McShane–Whitney extension theorem for these operators.

Surrogate modeling of passive microwave circuits using recurrent neural networks and domain confinement

Electromagnetic (EM) simulation is widespread in microwave engineering. EM tools ensure evaluation reliability but incur significant expenses. These can be mitigated by employing surrogate modeling methods, especially to expedite design workflows like local/global optimization or uncertainty quantification. However, building accurate surrogates is a daunting task beyond simple cases (low dimensionality, narrow geometry parameter and frequency ranges). This research suggests a new technique for dependable modeling of microwave circuits. Its main ingredient is a recurrent neural network (RNN) with the main architectural components being bidirectional Long Short-Term Memory (LSTM) and Gated Recurrent Unit (GRU) layers. These are incorporated to accurately represent frequency relationship within circuit characteristics as well as dependencies between its dimensions and outputs considered as vector-valued functions parameterized by frequency. The network’s hyperparameters are adjusted through Bayesian Optimization (BO). Utilization of frequency as a sequential variable handled by RNN is a distinguishing feature of our approach, which leads to the enhancement of dependability and cost efficiency. Another critical factor is dimension- and volume-wise reduction of the model’s domain achieved through global sensitivity analysis. It allows for additional and dramatic accuracy improvements without diminishing the surrogate’s coverage regarding circuit’s operating parameters. Our methodology has been extensively validated using several microstrip structures. The results demonstrate its competitive performance over a range of kernel-based regression techniques and diverse neural networks. The proposed procedure ensures building models of outstanding predictive power while using small training datasets, which is beyond the capabilities of benchmark algorithms.

Correction: On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Correction: On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Vector Fourier analysis on compact groups and Assiamoua spaces

This paper shows how a family of function spaces, coined as Assiamoua spaces, plays a fundamental role in the Fourier analysis of vector-valued functions on compact groups. These spaces make it possible to transcribe the classical results of Fourier analysis in the framework of analysis of vector-valued functions and vector measures. The construction of Sobolev spaces of vector-valued functions on compact groups rests heavily on the members of the aforementioned family.

Increasing microplastics pollution: An emerging vector for potentially pathogenic bacteria in the environment.

Increasing microplastics pollution: An emerging vector for potentially pathogenic bacteria in the environment.

Dual time-delay Rényi entropy: a new wide-area information fusion and quantification method applied to intelligent fault diagnosis of rotating machinery

A new wide-area information fusion quantification method, dual time-delay Rényi entropy (DTDRE), is proposed to address the issue of the relatively high probability of faults in multi-stage planetary gear transmission systems under high-speed and heavy-load conditions. A state vector function with orthogonal relationships was proposed and a density matrix suitable for macroscopic non-linear systems was established, achieving data-level fusion of wide-area information. The application of DTDRE quantifies the information content, interaction relationship and correlation between different systems, solves the lifecycle monitoring problem of macroscopic non-linear systems and reveals the influence of time delay and fault characteristic information on the correlation between state vector functions. On this basis, an artificial intelligence diagnostic framework based on multivariate variational mode decomposition (MVMD), DTDRE and a deep residual network (DRN) was established. The experimental results show that the constructed intelligent learning framework can achieve accurate recognition of fault states and, compared with DenseNet and AlexNet, it has significant advantages in diagnostic accuracy and training stability.

Reference model of adaptive control system for unstable and deterministic chaotic processes in the catastrophe class hyperbolic umbilica

A method is proposed for solving the problem of synthesis of a self-organizing adaptive control system for unstable and deterministic chaotic processes in the catastrophe class hyperbolic umbilic for objects with m-inputs and n-outputs. The problem of synthesizing a self-organizing adaptive control system is solved by the velocity- gradient method of the Lyapunov vector function. From the boundary conditions of the aperiodic robust stability of the main contour and the generalized adjustable object, adjustable coefficients are calculated and the control goal is achieved. An urgent problem in the conditions of uncertainty of the parameters of the control object and external influences, for the control of unstable and deterministic chaotic processes, is the use of adaptation methods. At the same time, a self-organizing reference model with the desired dynamics is constructed in the class of catastrophes – a hyperbolic umbilic of a self-organizing system. But the aperiodic robust stability of a self-organizing reference model is studied using the Lyapunov vector function method. Collection of information about an object during operation, processing and its use to develop control impacts can improve the quality of management.A method is proposed for solving the problem of synthesizing the adaptation loop of a self-organizing control system for unstable and deterministic chaotic processes in the class of catastrophes “hyperbolic umbilic”. From the conditions of aperiodic robust stability of a generalized adjustable object, adjustable coefficients are calculated and the control goal is achieved.

METHOD OF ESTIMATING ELECTROMAGNETIC COMPATIBILITY OF RADIOE-LECTRONIC DEVICES WITH MULTIELEMENT ANTENNA SYSTEMS BASED ON LOW DIMENSION EQUATIONS

The required level of noise immunity of radio electronics is provided among other things, by ensuring electromagnetic compatibility of radio electronic equipment. However, its estimation becomes a rather time-consuming task when the number of radiating elements increase. The article presents methods for estimating electromagnetic compatibility of radio electronic equipment with multi-element antenna systems when exposed to mutual interference based on solving low-dimension equations. Methods for constructing a comparison system based on Lyapunov vector functions, as well as on a low-dimension Kalman filter, are presented. The results of mathematical modeling of the developed methods with the original 4x4 MIMO system are shown. The signal/noise ratio and interference power are considered as evaluation criteria. The results of the study show the possibility of using these methods when evaluating the electromagnetic compatibility of radio electronic equipment with multi-element antenna systems.

Secure Delivery Method for Preserving Data Integrity of a Video Frame with Sensitive Objects

Data integrity and authenticity verification are essential requirements to provide reliability for applications running in a video delivery system. We design a prototype system for handling data integrity as well as encoding sensitive objects in a video frame. This system is composed of two components, a logger and a verifier. The logger operates a cryptography hash function of a feature vector of a video data frame and an XOR encoding scheme used for hiding the privacy information of sensitive objects in the video data frame. The XOR encoding scheme is operated at the encoder as part of the logger. The hash value outputted from the hash function is signed by a private key owned by the logger. It is listed as one of the attributes in the metadata of the video frame. In the key distribution phase for the authenticity verification process, the logger maintains the private key, and the public key is stored on the verifier by handshaking. In order to reduce the processing load for data integrity verification, a hash tree configured for each stream of sensor data is used to minimize the verification time. This paper is mainly focused on the design of a logger for data integrity verification and encoding of sensitive objects. To evaluate performance of the proposed method, we build a prototype in a client–server manner. The designed method is operated on the logger, used to mitigate the risk of video data frame tampering and to significantly help in preserving sensitive objects securely.

Composition Operators on Vector-Valued Function Spaces

Composition Operators on Vector-Valued Function Spaces