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In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.

The current study deals with exact soliton solutions for Schrodinger-Hirota (SH) equation via two modified integration methods. Those methods are known as the improved ( G ′ / G ) -expansion method and the Kudryashov method. This model is a generalized version of the nonlinear Schrodinger (NLS) equation with higher order dispersion and cubic nonlinearity. It can be considered as a more accurate approximation than the NLS equation in explaining wave propagation in the ocean and optical fibers. A novel derivative operator named as the conformable truncated M-fractional is used to study the above mentioned model. The obtained results can be used in describing the Schrodinger-Hirota equation in some better way. Moreover the obtained results are verified through symbolic computational software. Also, the obtained results show that the suggested approaches have broaden capacity to secure some new soliton type solutions for the fractional differential equations in an effective way. In the end, the results are also explained through their graphical representations.

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for thep-Laplace equation and system, with the right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder,$$\text {BMO}$$BMOand$${{\,\mathrm{VMO}\,}}$$VMOspaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when$$p=2$$p=2, and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces. A distinctive trait of our results is their sharpness, which is demonstrated by a family of apropos examples.

We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial proof of an identity of Hilbert, which leads to a relation of Cayley on semi-invariants. This identity plays a crucial role in the original proof of Sylvester’s theorem on semi-invariants in connection with the Gaussian coefficients. Moreover, we show that the additivity lemma of Pak and Panova which yields the strict unimodality of the Gaussian coefficients for $$n,k \ge 8$$ can be deduced from the ring property of semi-invariants.

In this paper, we investigate topological transitivity of operators on nonseparable Hilbert spaces which are similar to backward weighted shifts. In particular, we show that abstract differential operators and dual operators to operators of multiplication in graded Hilbert spaces are similar to backward weighted shift operators.

Let £ be a second order positive, elliptic differential operator that is self-adjoint with respect to some C∞ density dx on a compact connected manifold $$\mathbb{M}$$ . We proved that if 0 < α < 1, α/2 < s < α and $$f \in {H^s}(\mathbb{M})$$ then the fractional Schrodinger propagator $${{\rm{e}}^{{\rm{i}}t{{\cal L}^{\alpha /2}}}}$$ on $$\mathbb{M}$$ satisfies $${{\rm{e}}^{{\rm{i}}t{{\cal L}^{\alpha /2}}}}f(x) - f(x) = o({t^{s/\alpha - \varepsilon }})$$ almost everywhere as t → 0+, for any e > 0.

We generalize the unifying relations for tree amplitudes to the $1$-loop Feynman integrands. By employing the $1$-loop CHY formula, we construct differential operators which transmute the $1$-loop gravitational Feynman integrand to Feynman integrands for a wide range of theories, include Einstein-Yang-Mills theory, Einstein-Maxwell theory, pure Yang-Mills theory, Yang-Mills-scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory, bi-adjoint scalar theory, non-linear sigma model, as well as special Galileon theory. The unified web at $1$-loop level is established. Under the well known unitarity cut, the $1$-loop level operators will factorize into two tree level operators. Such factorization is also discussed., Comment: 42 pages, 3 figures, 4 tables

Aún no está publicado oficialmente. No se conoce volumen, número ni páginas, sólo el año. We construct new examples of exceptional Hahn and Jacobi polynomials. Exceptional polynomials are orthogonal polynomials with respect to a measure which are also eigenfunctions of a second-order difference or differential operator. In mathematical physics, they allow the explicit computation of bound states of rational extensions of classical quantum-mechanical potentials. The most apparent difference between classical or classical discrete orthogonal polynomials and their exceptional counterparts is that the exceptional families have gaps in their degrees, in the sense that not all degrees are present in the sequence of polynomials. The new examples have the novelty that they depend on an arbitrary number of continuous parameters.

In recent years, two types of nonlocal differential operators and their theories and formulations have been proposed and used in numerical modeling and computations, particularly simulations of material and structural failures, such as fracture and crack propagations in solids. Since the differences of these nonlocal operators are subtle, and they often cause confusion and misunderstandings. The first type of nonlocal differential operators is derived from the Taylor series expansion of nonlocal interpolation, e.g., Bergel and Li (Comput Mech 58(2):351–370, 2016), Madenci et al. (Comput Methods Appl Mech Eng 304:408–451, 2016) and Ren et al. (Comput Methods Appl Mech Eng 358:112621, 2020). The second type of nonlocal operators is based on the nonlocal operator theory in peridynamic theory, which is a class of antisymmetric nonlocal operators stemming from the nonlocal balance laws, e.g., Gunzburger and Lehoucq (Multiscale Model Simul 8(5):1581–1598, 2010) and Du et al. (SIAM Rev 54(4):667–696, 2012; Math Models Methods Appl Sci 23(03):493–540, 2013a). In this work, a comparative study is conducted for evaluating the computational performances of these two types of nonlocal differential operators. It is found that the first type of nonlocal differential operators can yield convergent results in both uniform and non-uniform particle distributions. In contrast, the second type of nonlocal differential operators can only converge in uniform particle distributions. Specifically, we have evaluated the performance of the two types of nonlocal differential operators in three crack propagation simulation examples. The results show that the second type of nonlocal differential operators is more suitable to deal with complex crack branching patterns than the first type of nonlocal differential operators, and the simulation results obtained by using the second type of nonlocal differential operators have better agreement with experimental observations. For modelings of simple crack growth and conventional elastic deformation problems, both nonlocal operators can provide good results in simulations.

Let a reductive group $G$ act on a smooth affine complex algebraic variety $X.$ Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mu:T^*(X)\to \mathfrak{g}$ be the moment map. If the moment map is flat, and for a generic character $\chi:\mathfrak{g}\to\mathbb{C}$, the action of $G$ on $\mu^{-1}(\chi)$ is free, then we show that for very generic characters $\chi$ the corresponding quantum Hamiltonian reduction of the ring of differential operators $D(X)$ is simple., Comment: 5 pages, to appear in Comptes Rendus Math

In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hormander classes.

The topic of this paper are non-self-adjoint second-order differential operators with constant delay generated by $$-y''+q(x)y(x-\tau )$$ where potential q is complex-valued function, $$q\in L^{2}[0,\pi ]$$ . We study inverse problems of these operators for $$\tau \in \left[ \frac{2\pi }{5},\pi \right) $$ . We investigate the inverse spectral problems of recovering operators from their two spectra, firstly under Dirichlet–Dirichlet and second under Dirichlet/Polynomial boundary conditions. We will prove theorem of uniqueness, and we will give procedure for constructing potential. In the first case, for $$\tau \in \left[ \frac{\pi }{2},\pi \right) :$$ we will show that Fourier coefficients of a potential are uniquely0 determined by spectra. In the second case for $$\tau \in \left[ \frac{2\pi }{5},\frac{\pi }{2}\right) ,$$ we will construct integral equation under potential and we will prove that this integral equation has a unique solution. Also, we will show that other parameters are uniquely determined by spectra.

UDC 519.114; 512.622We introduce the notion of Chebyshev derivations of the first and second kinds based on the polynomial algebra andcorresponding specific differential operators, derive the elements of their kernels, and prove that any element of the kernelof the derivations defines a polynomial identity for Chebyshev polynomials of both kinds. We obtain several polynomialidentities involving the Chebyshev polynomials of both kinds, a partial case of the Jacobi polynomials, and the generalizedhypergeometric function.

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.

The aim of this paper is to introduce a new wavelet method for presenting approximate solutions of multitype variable-order (VO) fractional partial differential equations arising from the modeling of phenomena. In specific, this paper focuses on the numerical solution of the VO-fractional mobile-immobile advection-dispersion equation, Klein Gordon equation and Burgers equation. These equations are converted into a system of algebraic equations with the assistance of the bivariate Genocchi wavelet functions, their operational matrices, and the variable-order fractional Caputo derivative operator. Also, we present a new technique to get the operational matrix of integration and VO-fractional derivative. The modified operational matrices for solving the proposed equations are powerful and effective. So that, the accuracy of these matrices directly affects the implementation process. Finally, we consider numerical examples to confirm the superiority of the scheme, and for each example, exhibit the results through graphs and tables.

This work is concerned with the development of a scheme to obtain boundary conditions adapted representations of derivatives (differential operators) in the orthonormal wavelet bases of Daubechies family in $$\Omega = [0, \ 1] \subset {\mathbb{R}}$$ . The scheme is employed to find approximate solution of the (1+1)-D advection-diffusion-transport equation arising in the flow of contaminant fluid (water) through a porous medium (ground). Representation of derivatives adopting Dirichlet’s boundary conditions have been derived first. These representations are then used to reduce the aforesaid advection-diffusion-transport equation to a system of nonhomogeneous first order coupled ordinary differential equations. It is observed that representations of the derivatives derived here shift the non-homogeneous terms in the boundary conditions to non-homogeneous terms of the reduced ordinary differential equations. The resulting ordinary differential equations can be solved globally in time, in general, which helps to prevent the necessary analysis of time discretization such as estimate of step lengths, error etc. In case of convection dominated problems, the system of ordinary differential equations can be solved numerically with the aid of any efficient ODE solver. An estimate of $$L^2$$ -error in wavelet-Galerkin approximation of the unknown solution has been presented. A number of examples is given for numerical illustration. It is found that the scheme is efficient and user friendly.

The quantum averaging method is applied to the Hamiltonian of the multifrequency resonance harmonic oscillator perturbed by a differential operator with polynomial coefficients. The twisted product is used to transfer the averaging procedure to the space of graded algebra of symbols. As a result, the averaged Hamiltonian is expressed in terms of generators of the quantum algebra of symmetries of the harmonic part of the Hamiltonian. The proposed approach to the operator averaging is used to solve the spectral problem for the Hamiltonian of the cylindrical Penning trap.

In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator. The reaction has a parametric concave term and negative sublinear perturbation. In contrast to the case of a positive perturbation, we show that now for all big values of the parameter $$\lambda >0$$ , we have at least two positive solutions which do not vanish in the domain. In the process we prove a nonlinear maximum principle which is of independent interest.

We study classes of functions satisfying the generalized Cauchy–Riemann system and construct solutions by successively applying quaternionic differential operators. We extract some classes of solutions to this system.

In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

In this paper, we investigate a problem on small motions of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. The initial boundary value problem is reduced to the Cauchy problem for a first-order differential operator equation on a Hilbert space. Properties of the resulting operator matrices, which are coefficients of the equation, are studied. Theorems on solvability of the Cauchy problem and the initial boundary value problem are proven.

In this study, a nonlocal model is presented for the steady-state thermoelastic analysis of functionally graded materials (FGM) by using peridynamic differential operator. The displacement–temperature equations of coupled thermo-elasticity and boundary conditions for a two-dimensional FGM square plate under mechanical and thermal loads are converted from classical local differential forms into nonlocal integral forms with peridynamic differentical operator. The temperature, displacement and stress fields are solved by introducing Lagrange multiplier and employing variational analysis. A comparison study is conducted to validate the accuracy and convergence of this nonlocal model by comparing the nonlocal analysis results with finite element results as well as analytical solutions in literature. The effects of different material gradients and loads on the physical fields of FGM plates are investigated further by introducing the Mori-Tanaka method to estimate the effective properties, and the influences of the degree of nonlocality on the stress singularity at the crack tip in FGM plates are analysed finally.

Let $I_0$ and $K_0$ be modified Bessel functions of the zeroth order. We use Vanhove's differential operators for Feynman integrals to derive upper bounds for dimensions of the $\mathbb Q$-vector space spanned by certain sequences of Bessel moments \[ \left\{\left.\int_0^\infty [I_0(t)]^a[K_0(t)]^b t^{2k+1}\mathrm{d}\, t\right|k\in\mathbb Z_{\geq0}\right\},\]where $a$ and $b$ are fixed non-negative integers. For $ a\in\mathbb Z\cap[1,b)$, our upper bound for the $ \mathbb Q$-linear dimension is $\lfloor (a+b-1)/2\rfloor$, which improves the Borwein-Salvy bound $\lfloor (a+b+1)/2\rfloor$. Our new upper bound $\lfloor (a+b-1)/2\rfloor$ is not sharp for $ a=2,b=6$, due to an exceptional $ \mathbb Q$-linear relation $\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t\mathrm{d}\, t=72\int_0^\infty [I_0(t)]^2[K_0(t)]^6 t^{3}\mathrm{d}\, t$, which is provable by integrating modular forms., Comment: (v1) i+21 pages. Simplification and extension of some results in Section 5 of arXiv:1706.08308; (v2) i+29 pages; (v3) 20 pages, accepted version

This paper concerns the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $$p_n$$ -Laplacian when $$p_n\rightarrow \infty $$ as a particular case, tough it could be extended to a function of the Holder quotient of order s, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Holder infinity Laplacian.

In this paper we present a general extrapolated elliptic regularity result for second order differential operators in divergence form on fractional Sobolev-type spaces of negative order Xs-1,qD(Ω) for s > 0 small, including mixed boundary conditions and with a fully nonsmooth geometry of Ω and the Dirichlet boundary part D. We expect the result to find applications in the analysis of nonlinear parabolic equations, in particular for quasilinear problems or when treating coupled systems of equations. To demonstrate the usefulness of our result, we give a new proof of local-in-time existence and uniqueness for the van Roosbroeck system for semiconductor devices which is much simpler than already established proofs.

In L2(Rd;Cn), we consider a self-adjoint matrix strongly elliptic second-order differential operator Aϵ with periodic coefficients depending on x/ϵ. We find approximations of the exponential e−iτAϵ...

Consider the eigenvalue problem generated by a fixed differential operator with a sign-changing weight on the eigenvalue term. We prove that as the negative part of the weight is rescaled towards negative infinity on some subregion, the spectrum converges to that of the original problem restricted to the complementary region. On the interface between the regions the limiting problem acquires Dirichlet-type boundary conditions. Our main theorem concerns eigenvalue problems for sign-changing bilinear forms on Hilbert spaces. We apply our results to a wide range of PDEs: second and fourth order equations with both Dirichlet and Neumann-type boundary conditions, and a problem where the eigenvalue appears in both the equation and the boundary condition., 34 pages, 3 figures, 2 tables

In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has been described. Then, the deficiency indices of the minimal operator have been calculated. Moreover, the space of boundary values of the minimal operator have been constructed. Afterwards, by using the method of Calkin–Gorbachuk, the general form of all selfadjoint extensions of the minimal operator in terms of boundary values has been found. Later on, the structure of spectrum of these extensions has been investigated.

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

An inverse problem for determining the order of time-fractional derivative in a nonhomogeneous subdiffusion equation with an arbitrary elliptic differential operator with constant coefficients in $$N$$ -dimensional torus is considered. Using the classical Fourier method it is proved, that the value of the solution at a fixed time instant as the observation data recovers uniquely the order of fractional derivative. Generalization to an arbitrary $$N$$ -dimensional domain and to elliptic operators with variable coefficients is considered.

In this paper, certain spectral properties related with the first order linear differential expression in the weighted Hilbert space at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential expression in the weighted Hilbert space have been determined. Then, the deficiency indices of the minimal operator have been calculated. Moreover, a space of boundary values of the minimal operator has been constructed. Afterwards, by using the Calkin–Gorbachuk’s method, the general form of all maximally dissipative extensions of the minimal operator in terms of boundary values has been found. Later on, the structure of spectrum of these extensions has been investigated.

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n-Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n. In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

In this paper, we study the coefficient estimates for a class of analytic functions defined by using the q-Ruscheweyh derivative operator. In particular, we investigate the first four coefficient bounds, the Fekete–Szegő problem and some other coefficient inequalities for the functions in this class. Several known and new consequences of the results are also pointed out. Further, the results in this work improve or generalize some recent works in the literature.

An asymptotic method based on the calculus of pseudo-differential operators is used to derive the well-known relations between one-particle supersymmetric partner Hamiltonians. This alternative approach will be applied to one of the standard examples of supersymmetric quantum mechanics, where all the known analytical results will be reproduced. We also note a fundamental uncertainty relation between an operator and its formal left and right inverses.

In this paper, we establish the existence of two positive constants $$\lambda _0$$ and $$\lambda _1$$ with $$\lambda _0\leqslant \lambda _1$$ , such that any $$\lambda \in [\lambda _1, \infty )$$ is an eigenvalue, while any $$\lambda \in (0,\lambda _0)$$ is not an eigenvalue, for a Kirchhoff type problem driven by nonlocal operators of elliptic type in a fractional Orlicz-Sobolev space, with Dirichlet boundary conditions.

We study solvability of some linear nonhomogeneous elliptic problems and establish that under reasonable technical conditions the convergence in L 2 ( R d ) of their right sides implies the existence and the convergence in H 4 ( R d ) of the solutions. The problems contain the squares of the sums of second order non-Fredholm differential operators and we use the methods of the spectral and scattering theory for Schrodinger type operators. We especially emphasize that here we deal with the fourth order operators in contrast to the second order operators in [29] and investigate the dependence of the solvability conditions on the dimension of our problem when the constant a = 0 . We also consider the case of solvability with a single potential in an arbitrary dimension.

When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ when the ground manifold has dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. If this is possible, that is when the differential module defined by ${\cal{D}}_1$ is torsion-free, one shall say that the operator ${\cal{D}}_1$ is parametrized by ${\cal{D}}$ and there is no relation in general between ${\cal{D}}$ and ${\cal{D}}_2$. The parametrization is said to be " minimum " if the differential module defined by ${\cal{D}}$ has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator which is the formal adjoint of the Killing operator ${\cal{D}}$ parametrizing the Riemann operator ${\cal{D}}_1$ for an arbitrary $n$. Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment., Comment: The purpose of this paper is to prove that Einstein equations had already been exhibited by E. Beltrami 20 years before A. Einstein and to solve the minimum parametrization problem in arbitrary dimension

In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular linear differential operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse operator in this case. Going beyond linear order, we argue that the operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem., Comment: 51 pages, 7 figures and 2 appendices. v2: Matches published version

One of the methods of mathematical analysis in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator d/dt and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the Laplace transform; J. Mikusinski (1953) put operational calculus into algebraic form, using the concept of a function ring [1]. Thereupon I’m suggesting here the use of the integration operator dt to make an extension for the systematic use of operational calculus. Throughout designing and analyzing a control system, we need to treat the functionality of the system with respect to time. The reaction of the system instructs us how to stable it by amplifiers and feedbacks [2], thereafter the Differential Transform is a good tool for doing this, and it’s a technique to frustrate difficulties we may bump into, also it has the methods to find the immediate reaction of the system with respect to infinitesimal (tiny) time which spares us from the hard work in finding the time dependent function, this could be done by producing finite series.

A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule., Comment: 10 pages, no figure

We extend the Poincare group to the complex Minkowski space-time. Special attention is paid to the corresponding algebra that we achieve through matrices as well as differential operators. We also point out the generalizations of the two Casimir operators.

In this paper, we study spectral decompositions with respect to a system of generalized eigenvectors of second-order differential operators on an interval whose resolvents possess convolution representations. We obtain the convolution representation of resolvents of second-order differential operators on an interval with integral boundary conditions. Then, using the convolution generated by the initial differential operator, we construct the Fourier transform. A connection between the convolution operation in the original functional space and the multiplication operation in the space of Fourier transforms is established. Finally, the problem on the convergence of spectral expansions generated by the original differential operator is studied. Examples of convolutions generated by operators are also presented.

A nonlinear integro-differential equation with a zero operator of the differential part and several rapidly changing kernels is considered. The work is a continuation of the research conducted earlier for a single rapidly changing core. The main ideas of this generalization and the subtleties that arise in the development of the corresponding algorithm for the regularization method are fully visible in the case of two rapidly changing kernels, so for the sake of reducing the calculations, this particular case is taken. A similar problem with a single spectral value of the kernel of an integral operator is analyzed in one of the authors ' papers. In this case, the singularities in the solution of the problem are described only by the spectral value of the kernel. However, the influence of the zero operator of the differential part affects the fact that in the first approximation, the asymptotics of the solution of the problem under consideration will not contain the functions of the boundary layer, and the limit operator itself will become degenerate (but not zero). The conditions for the solvability of the corresponding iterative problems, as in the linear case, will not be in the form of differential equations (as was the case in problems with a non-zero operator of the differential part), but integro-differential equations, and the formation of these equations is significantly influenced by nonlinearity. Note that, in contrast to the linear case, there is no inhomogeneity of the corresponding linear problem in the right part of the problem under study. As it was shown earlier, its presence in the problem would lead to the appearance in the asymptotic solution of terms with negative powers of a small parameter, and in the nonlinear case there would be innumerable such powers, and the corresponding formal asymptotic solution would have the form of a Laurent series. This would make the creation of an algorithm for asymptotic solutions problematic, so in this paper, wanting to remain within the framework of asymptotic solutions of the type of Taylor series, inhomogeneity is excluded. In addition, in the nonlinear case, so-called resonances may occur, which significantly complicate the development of the corresponding algorithm for the regularization method. This publication deals with the non-resonant case. It is assumed that the study of an alternative variant (a more complex resonant problem) will be carried out in the future.

The quasilinear differential system x ′ = a x + b | y | p ⁎ − 2 y + k ( t , x , y ) , y ′ = c | x | p − 2 x + d y + l ( t , x , y ) is considered, where a, b, c and d are real constants with b 2 + c 2 > 0 , p and p ⁎ are positive numbers with ( 1 / p ) + ( 1 / p ⁎ ) = 1 , and k and l are continuous for t ≥ t 0 and small x 2 + y 2 . When p = 2 , this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin ( 0 , 0 ) is very similar to the behavior of solutions to the unperturbed system, that is, the system with k ≡ l ≡ 0 , near ( 0 , 0 ) , provided k and l are small in some sense. It is emphasized that this system can not be linearized at ( 0 , 0 ) when p ≠ 2 , because the Jacobian matrix can not be defined at ( 0 , 0 ) . Our result will be applicable to study radial solutions of the quasilinear elliptic equation with the differential operator r − ( γ − 1 ) ( r α | u ′ | β − a u ′ ) ′ , which includes p-Laplacian and k-Hessian.

The problem of separability of differential operators is considered for the first time in the works of V. N. Everitt and M. Hirz. Further development of this theory belongs to K. H. Boymatov, M. Otelbaev and their students. The main part of the published papers on this theory relates to linear operators. The nonlinear case was considered mainly when studied operator was a weak perturbation of the linear one. The case when the operator under study is not a weak perturbation of the linear operator is considered only in some works. The results obtained in this paper also relate to this little-studied case. The paper studies the coercive properties of the nonlinear Laplace-Beltrami operator in the space L2(R^n) $$L[u]=-\frac{1}{\sqrt{det g(x)}}\sum_{i,j=1}^n\frac{\partial}{\partial x_i}\left[\sqrt{det g(x)}g^{-1}(x)\frac{\partial u}{\partial x_j}\right]+V(x,u)u(x)$$, and proves its separability in this space by coercivity estimates. The operator under study is not a weak perturbation of the linear operator, i.e. it is strongly nonlinear. Based on the obtained coercive estimates and separability, the solvability of the nonlinear Laplace-Beltrami equation in the space L2(R^n) is studied