Let $A=U|A|$ be the polar decomposition of an operator $A$ on a Hilbert space $mathscr{H}$ and $lambdain(0,1)$. The $lambda$-Aluthge transform of $A$ is defined by $tilde{A}_lambda:=|A|^lambda U|A|^{1-lambda}$. In this paper we show that emph{i}) when $mathscr{N}(|A|)=0$, $A$ is self-adjoint if and only if so is $tilde{A}_lambda$ for some $lambdaneq{1over2}$. Also $A$ is self adjoint if and onl...

In this paper, we give the spectral theory for eigenvalues and eigenfunctions of a boundary value problem consisting of the linear fractional Bessel operator. Moreover, we show that this operator is self-adjoint, the eigenvalues of the problem are real, and the corresponding eigenfunctions are orthogonal. In this paper, we give the spectral theory for eigenvalues and eigenfunctions...

This tutorial describes the classic method of conjugate directions: the generalization of the conjugate-gradient method in iterative least-square inversion. I derive the algebraic equations of the conjugate-direction method from general optimization principles. The derivation explains the “magic” properties of conjugate gradients. It also justifies the use of conjugate directions in cases when ...

The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section the action semisimple Lie algebra on its dual using principal s-triple. Actually it is enough to have an “adapted pair” and indeed construction in Joseph Shafrir (2010) [25] works rather well co-adjoint algebraic, but not necessarily r...

Let V be a G-module where G is a complex reductive group. Let Z := V//G denote the categorical quotient and let π : V → Z be the morphism dual to the inclusion O(V ) ⊂ O(V ). Let φ : Z → Z be an algebraic automorphism. Then one can ask if there is an algebraic map Φ: V → V which lifts φ, i.e., π(Φ(v)) = φ(π(v)) for all v ∈ V . In Kuttler [Kut11] the case is treated where V = rg is a multiple of...

a mathematical model is presented in the present study to control‎ ‎the light propagation in an inhomogeneous media‎. ‎the method is ‎based on the identification of the optimal materials distribution‎ ‎in the media such that the trajectories of light rays follow the‎ ‎desired path‎. ‎the problem is formulated as a distributed parameter ‎identification problem and it is solved by a numerical met...

This work proposes a new adaptive remeshing strategy based on the sensitivity of the point wise error in stresses with respect to the nodal coordinates, which is computed using the adjoint-state method. This sensitivity provides information about the influence of the discretization of each zone of the computational domain in the estimation of the error in stresses at a specific point. This info...

We introduce a new algebraic framework based on the deformation of pre-Lie products. This allows us to provide construction objects at play in regularity structures works by Bruned, Hairer and Zambotti (2019) Bruned Schratz (2022) for deriving general scheme dispersive PDEs low regularity. also explains how structure et al. cited above can be viewed as Butcher–Connes–Kreimer extraction-contract...

following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

We propose a kinematic algebra for the Bern-Carrasco-Johansson (BCJ) numerators of tree-level amplitudes and form factors in Yang-Mills theory coupled with bi-adjoint scalars. The algebraic generators contain two parts: first part is simply flavour factor scalars, second that maps to non-trivial structures BCJ obeys extended quasi-shuffle fusion products. underlying allows us present closed for...