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28
result(s) for
"Cayley-Hamilton theorem."
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Blow-up theory for elliptic PDEs in Riemannian geometry
Elliptic equations of critical Sobolev growth have been the target of investigation for decades because they have proved to be of great importance in analysis, geometry, and physics. The equations studied here are of the well-known Yamabe type. They involve Schrödinger operators on the left hand side and a critical nonlinearity on the right hand side.
A significant development in the study of such equations occurred in the 1980s. It was discovered that the sequence splits into a solution of the limit equation--a finite sum of bubbles--and a rest that converges strongly to zero in the Sobolev space consisting of square integrable functions whose gradient is also square integrable. This splitting is known as the integral theory for blow-up. In this book, the authors develop the pointwise theory for blow-up. They introduce new ideas and methods that lead to sharp pointwise estimates. These estimates have important applications when dealing with sharp constant problems (a case where the energy is minimal) and compactness results (a case where the energy is arbitrarily large). The authors carefully and thoroughly describe pointwise behavior when the energy is arbitrary.
Intended to be as self-contained as possible, this accessible book will interest graduate students and researchers in a range of mathematical fields.
eBook
Non-adiabatic geometric phases in discrete time quantum walks
We present an analytical framework to investigate phase dynamics in discrete-time quantum walks, with a focus on geometric phases in both single- and two-particle systems. Leveraging the Cayley-Hamilton theorem, we construct the evolution operator in momentum space and derive exact closed-form solutions for the full time evolution. This allows us to explicitly extract total, dynamic, and most notably, non-adiabatic geometric phases throughout the walk. In the two-particle case, we further examine how initial entanglement influences the accumulation and interplay of these phases. Our results offer a unified and exact approach to understanding the geometric structure underlying quantum walk dynamics, with particular emphasis on non-adiabatic effects.
Journal Article
The MPWG inverse of third-order F-square tensors based on the T-product
We define the T-MPWG inverse of third-order F-square tensors by using the T-core EP decomposition of tensors via the T-product. Then, we present some characterizations and properties of the T-MPWG inverse. Moreover, the Cayley-Hamilton theorem of the third-order tensors is extended to T-MPWG inverses. Examples are also given to illustrate these results.
Journal Article
The superposition of Markovian arrival processes: moments and the minimal Laplace transform
The superposition of two independent Markovian arrival processes (MAPs) is also a Markovian arrival process of which the Markovian representation is given as the Kronecker sum of the transition rate matrices of the component processes. The moments of stationary intervals of the superposition can be obtained by differentiating the Laplace transform (LT) given in terms of the transition rate matrices. In this paper, we propose a streamlined procedure to determine the minimal LT of the merged process in terms of the minimal LT coefficients of the component processes. Combined with the closed-form transformation between moments and LT coefficients, our result enables us to determine the moments of the superposed process based on the moments of the component processes. The main contribution is that the whole procedure can be implemented without explicit Markovian representations. In order to transform the minimal LT coefficients of the component processes into the minimal LT representation of the merged process, we also introduce another minimal representation. A numerical example is provided to illustrate the procedure.
Journal Article
The m-WG Inverse in Minkowski Space
In this paper, we introduce the m-WG inverse in Minkowski space. Firstly, we show the existence and the uniqueness of the m-WG inverse. Secondly, we give representations of the m-WG inverse. Thirdly, we characterize the m-WG inverse by applying a bordered matrix. In addition, we extend the generalized Cayley-Hamilton theorem to the m-WG inverse matrix. Finally, we apply the m-WG inverse to solve linear equations in Minkowski space.
Journal Article
A new extension of the Cayley-Hamilton theorem to fractional different orders linear systems
The classical Cayley–Hamilton theorem is extended to fractional different order linear systems. The new theorems are applied to different orders fractional linear electrical circuits. The applications of new theorems are illustrated by numerical examples.
Journal Article
A Difference Equation Model of Infectious Disease
In the context of so much uncertainty with coronavirus variants and official mandate based on seemingly exaggerated predictions of gloom from epidemiologists, it is appropriate to consider a revised model of relative simplicity, because there can be dangers in developing models which endeavour to account for too many variables. Predictions and projections from any such models have to be in the context of relevant contingencies. The model presented here is based on relatively simple second order difference equations. The context here is as important as the content in that in many Western counties where the narrative currently seems more important than the truth, and the results of empirical science are valued more as a shield for politicians than a sword for protection of citizens.
Journal Article
Solving System of Mixed Variational Inclusions Involving Generalized Cayley Operator and Generalized Yosida Approximation Operator with Error Terms in Iq/I-Uniformly Smooth Space
In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by using the properties of q-uniformly smooth Banach space, and we discuss the convergence criteria for sequences generated by iterative algorithm. Two examples are constructed, and an application is provided.
Journal Article
Twisted Hypersurfaces in Euclidean 5-Space
The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.
Journal Article