Solvability Of Equations Research Articles

On the Properties of Two-Dimensional Normalized Boubaker Polynomials

The aim of the present work deals with newly defined two-variable polynomials for normalized Boubaker . The operational matrices of derivatives with respect to the two variables are presented at first with explicit expression. Then, a normalized Boubaker polynomial approximation for the numerical solution of a class of partial differential equations is proposed, depending on a truncated, normalized Boubaker function series in the equation together with the operational matrices in the proposed partial differential equation. The original partial differential equation is reduced under consideration of a system of simply solvable algebraic equations. Due to the interesting derived properties of normalized Boubaker polynomials in two variables, the suggested method can achieve good results with few complexities. Using operational matrices of derivatives, one can save computation and more memory. Two-dimensional examples are listed to show the satisfactory level of the suggested method.

Приложение теории Рисса-Шаудера к исследованию разрешимости уравнений типа романовского с частными интегралами

Приложение теории Рисса-Шаудера к исследованию разрешимости уравнений типа романовского с частными интегралами

A note on the surjectivity of operators on vector bundles over discrete spaces

In this note we give a short and self-contained proof for a criterion of Eidelheit on the solvability of linear equations in infinitely many variables. We use this criterion to study the surjectivity of magnetic Schrodinger operators on bundles over graphs.

The complexity of the equation solvability and equivalence problems over finite groups

We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a [Formula: see text]-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic presentation of these finite solvable groups.

Investigation of relativistic fermions with various energy–momentum distributions in q-deformed approach

In this paper, we investigate the [Formula: see text]-deformed Dirac equation. We obtain a quasi-exactly solvable equation. The energy–momentum distribution is considered using Landau–Lifshitz, Einstein and Papapetrou complexes and the results are discussed.

Robust output regulation for containment control of heterogeneous discrete-time nonlinear multi-agent systems

ABSTRACTIn existing researches on containment control of heterogeneous multi-agent systems (MASs), the solution is usually dependent on the solvability of regulator equations. However, the closed-form solution of many nonlinear regulator equations of systems is rarely obtained. Towards this end, in this paper the containment control problem of heterogeneous discrete-time nonlinear MASs subject to parameter uncertainties is considered, and the power series approach is adopted to solve complex regulator equations by decomposing them into a series of solvable linear equations. Then, a distributed robust control law based on internal model principle is presented by utilising the solution of the linear equations. Theoretical analysis shows that under certain assumptions asymptotic containment control is achieved for the heterogeneous discrete-time nonlinear MASs with sufficiently small parameter perturbations. Finally, a numerical simulation is implemented to verify the proposed control law.

On discrete groups and solvable equations of nonlinear oscillations

The autonomous ordinary differential equation of Lienard is considered. This equation and its generalizations arise in the theory of nonlinear oscillations, dynamical systems, etc. Currently for equations of the Lienard type, the problem of finding exact solutions is relevant. In view of this, the problem is to find connections in Lienard type equations with investigated nonlinear equations. These problems are solved in this article for the Lienard equation with three-parametrical coefficients of polynomial type. We found a transformation that reduces the equation under consideration of the well-studied generalized Emden-Fowler equation. We indicated the method of finding exact solutions of Lienard equations with various parameters. The transformations discrete group of dihedron for the class of generalized Emden-Fauler equations induces a discrete group of transformations for the corresponding class of Lienard equations. Likewise, known solutions of some particular generalized Emden-Fauler equations induce solutions of the Lienard equations. As examples, we found exact solutions in elementary (including polynomials) and special functions of several Lienard equations with certain parameters.

Linear and nonlinear statistical response theories with prototype applications to sensitivity analysis and statistical control of complex turbulent dynamical systems.

Statistical response theory provides an effective tool for the analysis and statistical prediction of high-dimensional complex turbulent systems involving a large number of unresolved unstable modes, for example, in climate change science. Recently, the linear and nonlinear response theories have shown promising developments in overcoming the curse-of-dimensionality in uncertain quantification and statistical control of turbulent systems by identifying the most sensitive response directions. We offer an extensive illustration of using the statistical response theory for a wide variety of challenging problems under a hierarchy of prototype models ranging from simple solvable equations to anisotropic geophysical turbulence. Directly applying the linear response operator for statistical responses is shown to only have limited skill for small perturbation ranges. For stronger nonlinearity and perturbations, a nonlinear reduced-order statistical model reduction strategy guaranteeing model fidelity and sensitivity provides a systematic framework to recover the multiscale variability in leading order statistics. The linear response operator is applied in the training phase for the optimal nonlinear model responses requiring only the unperturbed equilibrium statistics. The statistical response theory is further applied to the statistical control of inherently high-dimensional systems. The statistical response in the mean offers an efficient way to recover the control forcing from the statistical energy equation without the need to run the expensive model. Among all the testing examples, the statistical response strategy displays uniform robust skill in various dynamical regimes with distinct statistical features. Further applications of the statistical response theory include the prediction of extreme events and intermittency in turbulent passive transport and a rigorous saturation bound governing the total statistical growth from initial and external uncertainties.

Unlikely intersections in families of abelian varieties and the polynomial Pell equation

Let S be a smooth irreducible curve defined over a number field k and consider an abelian scheme A over S and a curve C inside A, both defined over k. In previous works, we proved that when A is a fibered product of elliptic schemes, if C is not contained in a proper subgroup scheme of A, then it contains at most finitely many points that belong to a flat subgroup scheme of codimension at least 2. In this article, we continue our investigation and settle the crucial case of powers of simple abelian schemes of relative dimension g bigger or equal than 2. This, combined with the above mentioned result and work by Habegger and Pila, gives the statement for general abelian schemes. These results have applications in the study of solvability of almost-Pell equations in polynomials.

On a global Lagrangian construction for ordinary variational equations on 2-manifolds

Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, how-ever, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of solvability of exactness equation for Lepage 2-forms, and the top-cohomology theorems. Examples from geometry and mechanics are discussed.

On a class of systems of rational second‐order difference equations solvable in closed form

We show that the following nonlinear system of difference equations urn:x-wiley:mma:media:mma5809:mma5809-math-0001 where parameters a,b,c,d and initial values x−1,x0,y−1,y0 are real numbers, is solvable in closed form, considerably generalizing some recent results. To do this, we use the method of transformation along with several tricks, transforming the system to some known solvable difference equations, by use of which we obtain some closed‐form formulas for general solution to the system. The following five cases are considered separately: (1) c=0; (2) d=0; (3) a=0; (4) b=0; and (5) abcd≠0.

New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass

Infinite families of quasi-exactly solvable position-dependent mass Schrodinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Poschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W+(x) (and its accompanying function W-(x) . The generalized trigonometric Poschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.

The ‘flea on the elephant’ effect

To localise a particle subject to a symmetric double-well potential a symmetry break is needed. If the wells are very distant from one another even a tiny difference between their depths causes a wavefunction concentration in the deeper well. This flea on the elephant (FOTE) effect can be studied in elementary terms by using a couple of unequal delta-like wells. After reviewing the case of equal delta functions, and introducing the Lambert W function, the case of different delta functions is studied. There, an easily solvable asymptotic equation is derived, which allows quantitative evaluations of the FOTE effect to be performed. Finally, the effect of the symmetry breaking on the time evolution of a wave packet is shown.

Adaptive Batch Size Image Merging Steganography and Quantized Gaussian Image Steganography

In digital image steganography, the statistical model of an image is essential for hiding data in less detectable regions and achieving better security. This has been addressed in the literature where different cost-based and statistical model-based approaches were proposed. However, due to the usage of heuristically defined distortions and non-constrained message models, resulting in numerically solvable equations, there is no closed-form expression for security as a function of payload. The closed-form expression is crucial for a better insight into image steganography problem and also improving performance of batch steganography algorithms. Here, we develop a statistical framework for image steganography in which the cover and the stego messages are modeled as multivariate Gaussian random variables. We propose a novel Gaussian embedding model by maximizing the detection error of the most common optimal detectors within the adopted statistical model. Furthermore, we extend the formulation to cost-based steganography, resulting in a universal embedding scheme that improves empirical results of current cost-based and statistical model-based approaches. This methodology and its presented solution, by reason of assuming a continuous hidden message, remains the same for any embedding scenario. Afterward, the closed-form detection error is derived within the adopted model for image steganography and it is extended to batch steganography. Thus, we introduce Adaptive Batch size Image Merging steganographer, AdaBIM , and mathematically prove it outperforms the state-of-the-art batch steganography method and further verify its superiority by experiments.

Insert a Root to Extract a Root of Quintic Quickly

Abstract The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation. We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u 2 − g 2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

Solvability and Controller Action on Local Saddle-Node Bifurcation of Reduced Order Two-Dimensional Differential-Algebraic System

This paper focuses on the solvability and local Saddle-Node bifurcation ensurance of two-dimensional differential-algebraic equations of index-1 with its feedback controller. The feedback controller action has been designed to satisfy the necessary requirements for Saddle-Node bifurcation ensurance. The mathematical assumptions and proofs have proposed with systematic illustrations to demonstrate the applicability of the proposed approach.

Tavis-Cummings models and their quasi-exactly solvable Schrödinger Hamiltonians

We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrodinger equations. The connection between them is established through the biconfluent Heun equation. We found that each invariant n-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to n potentials, each with one known solution, or to one potential with n known solutions. Among these Schrodinger potentials the quarkonium and the sextic oscillator appear.

Existence, uniqueness and Malliavin differentiability of Lévy-driven BSDEs with locally Lipschitz driver

ABSTRACTWe investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the Z and U variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value ξ and its Malliavin derivative . Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in U. BSDEs of the latter type find use in exponential utility maximization.

Dirac fermions in Som\u2013Raychaudhuri space-time with scalar and vector potential and the energy momentum distributions

The main object of the present paper is to investigate the Dirac equation (Dirac fermions) in presence of scalar and vector potential in a class of flat Gödel-type space-time called Som–Raychaudhuri space-time by using the methods quasi-exactly solvable (QES) differential equations and the Nikiforov Uvarov (NU) form. In addition, we evaluate the Einstein, and the Papapetrou.