Solutions Of Certain Type Research Articles

Drazin theta and theta Drazin matrices

<p style='text-indent:20px;'>In this paper, two new classes of matrices - Drazin - theta matrix and theta - Drazin matrix are introduced for a square matrix of index <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>. Whenever the index is equal to one, we get special case of matrices called Group - theta matrix and theta - Group matrix respectively. Several characterizations of these matrices, the integral representations, representation in limit form and the representation in terms of rank factorization are obtained. Also, the relationship of Drazin -theta and theta - Drazin matrices with other well known generalized inverses are investigated. By applying the concept of Drazin - theta matrix, general solutions of certain types of matrix equations are characterized here.</p>

MODELING OF SNOW LOAD ON ROOFS OF UNIQUE BUILDINGS

Introduction: The purpose of the work is to study the processes of snow transfer and snow deposition on the roofing of a unique building. Currently, the problem of assigning snow loads to the roofs of buildings and structures remains relevant, since cases of roof collapses in winter are still recorded annually, including those in the Russian Federation. A circus building with the roof in the form of a suspended reinforced concrete shell was chosen as the object of study. Methods: The design diagram for this type of roof is contained in SP 20.13330.2016; the snow load distribution diagrams for this roof were adopted according to Appendix B to SP 20.13330.2016. The author considered several options of snow load distribution and chose the most unfavorable one, when an increased value of the snow load is observed on one half of the roofing. During one winter period, field measurements of the thickness of the snow cover were carried out at the site. Results:It was established that in general, with the exception of local zones, the actual distribution of snow cover coincides with the adopted design solution, while the actual value of the weight of the snow cover for the current winter season was significantly lower than the calculated one. Discussion: The obtained result demonstrates that when developing design solutions for certain types of suspended roofing, it is permissible, without conducting specialized experimental studies, to use the data given in the scientific and technical literature, based on the results of monitoring the thickness of the snow cover on the roofing of the building.

Randomized Algorithms for Rounding in the Tensor-Train Format

The tensor-train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

On entire solutions of certain types of nonlinear differential equation

On entire solutions of certain types of nonlinear differential equation

On the Fly Ellipsometry Imaging for Process Deviation Detection

Spectroscopic ellipsometry is a very sensitive optical metrology technique commonly used in semiconductor manufacturing lines to accurately measure the thickness and refractive index of different layers present on specific dedicated metrology targets on the wafers. In parallel, optical defectivity techniques are widely implemented in production lines to inspect a significant amount of dies representative of the full wafer and detect physical and patterning defects. A new approach can then simply emerge which is to apply ellipsometry metrology techniques at a full or die wafer scale. This strategy, at the frontier between metrology and defectivity field is expected to bring solutions for certain types of process deviation. In our case, ellipsometry’s optical response was collected on large areas of product wafers to capture specific deviations such as film properties, thickness, and patterning variation. This is an innovative strategy that relies on a model-less approach to detect process drifts, using ellipsometry’s sensitivity to material properties and design architecture variations. In this paper, we will present this approach on three industrial cases.

Transcendental meromorphic solutions of certain types of differential equations

In this paper, we analyze the transcendental meromorphic solutions of nonlinear complex differential equation of the formfn+P(z,f)=p1eα1zk+p2eα2zk, where n≥2 and k≥1 are integers, f is a meromorphic function, P(z,f) is a differential polynomial in f of degree at most n−1 with rational functions of f as the coefficients, α1,α2,p1,p2 are nonzero constants. We also present a number of examples showing the accuracy of our results.

Some results on the transcendental entire solutions of certain type of non-linear shift-differential equations

Some results on the transcendental entire solutions of certain type of non-linear shift-differential equations

On meromorphic solutions of certain type of nonlinear differential-difference equations

On meromorphic solutions of certain type of nonlinear differential-difference equations

Characterization of exponential polynomial as solution of certain type of nonlinear delay-differential equation

In this paper, we have characterized the nature and form of solutions of the following nonlinear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ are a linear delay-differential polynomial of $f$; $n$ is positive integers; $q$, $Q$ and $P$ respectively are nonzero, nonconstant and any polynomials. Different special cases of our result will accommodate all the results of [J. Math. Anal. Appl., 452(2017), 1128-1144; Mediterr. J. Math., 13(2016), 3015-3027; Open Math., 18(2020), 1292-1301]. Thus our result can be considered as an improvement of all of them. We have also illustrated a handful number of examples to show that all the cases as demonstrated in our theorem actually occur and consequently the same are automatically applicable to the previous results.

Fixed Point Results for F-Contractions in Cone Metric Spaces over Topological Modules and Applications to Integral Equations

In this paper, the concept of F-contraction was generalized for cone metric spaces over topological left modules and some fixed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being a fractional integral equation.

Characterization on transcendental entire solutions of certain types of non-linear generalized delay-differential equations

Characterization on transcendental entire solutions of certain types of non-linear generalized delay-differential equations

Numerical Solution of Fuzzy Delay Differential Equations by Fifth Order Runge-Kutta Method

In this paper, we develop the numerical solutions of certain type called Fuzzy Delay Differential Equations(FDE) by using fifth order Runge-Kutta method for fuzzy differential equations. This method based on the seikkala derivative and finally we discuss the numerical examples to illustrate the theory.

The Exact Entire Solutions of Certain Type of Nonlinear Difference Equations

In this paper, we consider the entire solutions of nonlinear difference equation $$f^{3}+q(z)\Delta f=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ , where $$q$$ is a polynomial, and $$p_{1},p_{2},\alpha_{1},$$ and $$\alpha_{2}$$ are nonzero constants with $$\alpha_{1}\neq\alpha_{2}$$ . It is showed that if $$f$$ is a nonconstant entire solution of $$\rho_{2}(f)<1$$ to the above equation, then $$f(z)=e_{1}e^{\frac{\alpha_{1}z}{3}}+e_{2}e^{\frac{\alpha_{2}z}{3}}$$ , where $$e_{1}$$ and $$e_{2}$$ are two constants. Meanwhile, we give an affirmative answer to the conjecture posed by Zhang et al in [18].

Some results for initial value problem of nonlinear fractional equation in Sobolev space

This paper focuses on the existence and uniqueness of solutions for certain types of fractional differential equations involving Riemann–Liouville derivative. The main results are obtained by some fixed point theorems in Sobolev space.

Meromorphic solutions of certain types of complex functional equations

Meromorphic solutions of certain types of complex functional equations

一类微分–差分方程的整函数解

一类微分–差分方程的整函数解

Using the assumption G(q, r, s) = e ∫ y(q)dq + ∫x(r)dr + ∫ z(s)ds for solving some kinds of linear third order P.D.Es. with homogenous terms

The preset paper aims at finding the complete solution of certain types of linear partial differential equations of third order with constant coefficients that have three independent variables (q, r, s) of the general form:By using the assumption: G(q,r,s) = e ∫y(q)dq + ∫x(r)dr + z(s)ds , the above equation, according to this assumption, will be transformed to the non-linear second order ordinary differential equation with three independent functions. Thus, they have the general form: Note:- we used y instead of y(q) and χ instead of x(r) also ζ instead z(s).

Two dimensional second order Fibonacci summation formula with extorial functions

This paper aims to obtain the extorial type solutions for Fibonacci difference equations on real valued function defined in two dimensional space having shift value ℓ. Using two dimensional second order Fibonacci nabla difference operator and its inverse, we derived Second order fibonacci summation formula of product of real valued polynomial. A function obtained by replacing polynomial into polynomial factorials in the expansion of exponential function is called extorial function. Using extorial functions defined in ℝ2, we obtain solution of certain type of two dimensional second order difference equations. Suitable examples and numerical verifications are given to validate the findings.

Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

In this paper, we study the algebraic, rational and formal Puiseux series solutions of certain type of systems of autonomous ordinary differential equations. More precisely, we deal with systems which associated algebraic set is of dimension one. We establish a relationship between the solutions of the system and the solutions of an associated first order autonomous ordinary differential equation, that we call the reduced differential equation. Using results on such equations, we prove the convergence of the formal Puiseux series solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them. In addition, we bound the degree of the possible algebraic and rational solutions, and we provide an algorithm to decide their existence and to compute such solutions if they exist. Moreover, if the reduced differential equation is non trivial, for every given point (x_0,y_0) in mathbb {C}^2, we prove the existence of a convergent Puiseux series solution y(x) of the original system such that y(x_0)=y_0.

Meromorphic Solutions of Certain Types of Non-linear Differential Equations

Let $$p_1, p_2$$ and $$\alpha _1, \alpha _2$$ be non-zero constants, and $$P_d(z, f)$$ be a differential polynomial in f of degree d. Li obtained the forms of meromorphic solutions with few poles of the non-linear differential equations $$f^n+P_d(z, f)=p_1e^{\alpha _1 z}+p_2e^{\alpha _2 z}$$ provided $$\alpha _1\ne \alpha _2$$ and $$d\le n-2$$. In this paper, given $$d=n-1$$, we find the forms of meromorphic solutions with few poles of the above equations under some restrictions on $$\alpha _1, \alpha _2$$. Some examples are given to illustrate our results.