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We consider a system of pseudomonotone equilibrium problem and split common fixed point problem in the framework of real Hilbert spaces. We propose a modified extragradient method with line searching technique for approximating a common element in the sets of solutions of the two nonlinear problems. The convergence result is proved without prior knowledge of the Lipschitz-like constants of the equilibrium bifunctions and the norm of the bounded linear operator of the split common fixed point problem. We further provide some application and numerical example to show the importance of the obtained results in the paper.

The purpose of the paper is to find an approximate solution of the two-dimensional nonlinear fuzzy Volterra integral equation, as homotopy analysis method (HAM) is applied. Studied equation is converted to a nonlinear system of Volterra integral equations in a crisp case. Using HAM we find approximate solution of this system and hence obtain an approximation for the fuzzy solution of the nonlinear fuzzy Volterra integral equation. The convergence of the proposed method is proved. An error estimate between the exact and the approximate solution is found. The validity and applicability of the HAM are illustrated by a numerical example.

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

The paper is a review of the latest results of I. R. Kayumov and S. Ponnusamy on the Bohr inequality. An exact estimate in the strong Bohr inequality is obtained and the Bohr–Rogosinski radius for a certain class of subordinations is examined. All results are exact.

In this paper, we examine the problem on the description in terms of the Laplace transform of functionals from the conjugate space for the Hilbert space of entire functions of n variables constructed by a convex function in ℂn, which depends on the modules of variables and grows at infinity faster than a‖z‖ for any a > 0.

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.

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages

In this paper, we consider the existence and exponential stability in mean square of mild solutions to second-order neutral stochastic functional differential equations with random impulses in Hilbert space. Firstly, the existence of mild solutions to the equations is proved by using the noncompact measurement strategy and the Monch fixed point theorem. Then, the mean square exponential stability for the mild solution of the considered equations is obtained by establishing an integral inequality. Finally, an example is given to illustrate our results.

In this paper, we propose a simple method to generate generalized Gaussian noises using the inverse transform of cumulative distribution. This inverse is expressible by means of the inverse incomplete Gamma function. Since the implementation of Newton's method is rather simple, for approximating inverse incomplete Gamma function, we propose a better and new initial value exploiting the close relationship between the incomplete Gamma function and its piecewise linear interpolant. The numerical results highlight that the proposed method simulates well the univariate and bivariate generalized Gaussian noises.

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure.

In the Kalman—Bucy filter problem, the observed process consists of the sum of a signal and a noise. The filtration begins at the same moment as the observation process and it is necessary to estimate the signal. As a rule, this problem is studied for the scalar and vector Markovian processes. In this paper, the scalar linear problem is considered for the system in which the signal and noise are not Markovian processes. The signal and noise are independent stationary autoregressive processes with orders of magnitude higher than 1. The recurrent equations for the filter process, its error, and its conditional cross correlations are derived. These recurrent equations use previously found estimates and some last observed data. The optimal definition of the initial data is proposed. The algebraic equations for the limit values of the filter error (the variance) and cross correlations are found. The roots of these equations make possible the conclusions concerning the criterion of the filter process convergence. Some examples in which the filter process converges and does not converge are given. The Monte Carlo method is used to control the theoretical formulas for the filter and its error.

In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.

This paper considers the nonlinear symmetric conic programming (NSCP) problems. Firstly, a type of strong sufficient optimality condition for NSCP problems in terms of a linear-quadratic term is introduced. Then, a sufficient condition of the nonsingularity of Clarke’s generalized Jacobian of the Karush–Kuhn–Tucker (KKT) system is demonstrated. At last, as an application, this property is used to obtain the local convergence properties of a sequential quadratic programming- (SQP-) type method.

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman’s matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

In this paper, small modification on Improved Euler’s method (Heun’s method) is proposed to improve the efficiency so as to solve ordinary differential equations with initial condition by assuming the tangent slope as an average of the arithmetic mean and contra-harmonic mean. In order to validate the conclusion, the stability, consistency, and accuracy of the system were evaluated and numerical results were presented, and it was recognized that the proposed method is more stable, consistent, and accurate with high performance.

In this paper, we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation $${{\cal S}^b}: = {\overline {\rm{\Delta }} ^{\alpha /2}} + b \cdot \nabla $$ where $${\overline {\rm{\Delta }} ^{\alpha /2}}$$ is the truncated fractional Laplacian, α ∈ (1, 2) and b ∈ K −1 . In the second part, for a more general finite range jump process, we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance ∣x − y∣ in short time.

Let ℌ be a Hilbert space, and let K(ℌ) be the C*-algebra of compact operators on ℌ. In this paper, we present some characterizations of the norm-parallelism for elements of a Hilbert K(ℌ)-module by employing the Birkhoff-James orthogonality. Among other things, we present a characterization of transitive relation of the norm-parallelism for elements in a certain Hilbert K(ℌ)-module. We also give some characterizations of the Schatten p-norms and the operator norm-parallelism for matrices.

In this paper we investigate the compact operators under Orlicz function, named noncommutative Orlicz sequence space (denoted by Sϕ(ℌ)), where ℌ is a complex, separable Hilbert space. We will show that the space generalizes the Schatten classes Sp(ℌ) and the classical Orlicz sequence space respectively. After getting some relations of trace and norm, we will give some operator inequalities, such as Holder inequality and some other classical operator inequalities. Also we will give the dual space and reflexivity of Sϕ(ℌ) which generalizes the results of Sϕ(ℌ). Finally, as an application, we will show that the Toeplitz operator $${T_{1 — {{\left| z \right|}^2}}}$$ on the Bergman space $$L_\alpha ^2\left(\mathbb{R} \right)$$ belongs to some Sϕ(ℌ), and the norm satisfies $$1 = \sum\limits_{n \ge 0} {\varphi \left({{1 \over {\left({n + 2} \right){{\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|}_\varphi}}}} \right)} $$ . Especially, if ϕ(T) = ∣T∣p, p > 1, the norm is $${\left\| {{T_{1 — {{\left| z \right|}^2}}}} \right\|_p} = {\left[{\sum\limits_{n \ge 0} {{1 \over {{{\left({n + 2} \right)}^p}}}}} \right]^{{1 \over p}}} = {\left({\zeta \left(p \right) — 1} \right)^{{1 \over p}}}$$ , where ζ(p) is the Riemann function.

In this paper, using the method of blow-up analysis, we obtained a Trudinger—Moser inequality involving Lp-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger—Moser functional.

A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet been found, but their nonexistence has also not been proved. The problem of a perfect cuboid is among unsolved mathematical problems. The problem has a natural S3-symmetry connected to permutations of edges of the cuboid and the corresponding permutations of face diagonals. In this paper, we give a survey of author’s results and results of J. R. Ramsden on using the S3 symmetry for the reduction and analysis of the Diophantine equations for a perfect cuboid.

Let $$x:M \to \mathbb{S}{^{n + p}}(1)$$ be an n-dimensional submanifold immersed in an (n + p)-dimensional unit sphere $$\mathbb{S}{^{n + p}}(1)$$ . In this paper, we study n-dimensional submanifolds immersed in $$\mathbb{S}{^{n + p}}(1)$$ which are critical points of the functional $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ , where S is the squared length of the second fundamental form of the immersion x. When $$x:M \to \mathbb{S}{^{2 + p}}(1)$$ is a surface in $$\mathbb{S}{^{2 + p}}(1)$$ , the functional $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ represents double volume of image of Gaussian map. For the critical surface of $${\cal S}(x)$$ , we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic. Furthermore, we establish a rigidity theorem for the critical surface of $${\cal S}(x)$$ .

In this paper, we present new properties of the space L1(M, τ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ-measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI −AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ ℝ. Under these conditions, [A,B] = AB − BA ∈ L1(M, τ) and τ ([A,B]) = 0. For τ -measurable operators A and B = B2, we find conditions that are sufficient for the validity of the relation τ ([A,B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1(M, τ) if and only if I − A, I − U ∈ L1(M, τ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A∗,A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X1/2, YX1/2] ∈ L1(M, τ). Then τ ([X1/2, YX1/2]) = it, where t ∈ ℝ and t = 0 for XY ∈ L1(M, τ).

In this paper, we prove a genuine analogue of the Wiener Tauberian theorem for L p , 1 ⁢ ( G ) {L^{p,1}(G)} ( 1 ≤ p < 2 {1\leq p ), with G = SL ⁢ ( 2 , ℝ ) {G=\mathrm{SL}(2,\mathbb{R})} .

This is the first of the two papers devoted to the study of global regularity of the 3 + 1 dimensional Einstein-Klein-Gordon system with a U(1) × ℝ isometry group. In this first part, the authors reduce the Cauchy problem of the Einstein-Klein-Gordon system to a 2 + 1 dimensional system. Then, the authors will give energy estimates and construct the null coordinate system, under which the authors finally show that the first possible singularity can only occur at the axis.

In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.

Solving a linear partial differential equation witness a noteworthy role of Wright function. Due to its usefulness and various applications, a variety of its extentions (and generalizations) have been investigated and presented. The purpose and design of the paper is intended to study and come up with a new extention of the genralized Wright function by using generalized beta function and obtain some integral representation of the freshly defined function. Also we present the Mellin transform of this function in the form of Fox Wright function. Furthermore, we obtain the recurrence relation, derivative formula for the said function and also by using an extended Riemann-Liouville fractional derivative, we present a fractional derivative formula for the extended Wright function.

In this paper we study the structure of positive vector solutions for nonlinear Schrodinger systems with 3 components when all interspecies interaction forces are positive and large while all intraspecies interaction forces are positive and fixed. We will show that the structure strongly depends on some relation of large interspecies interaction forces.

This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of $$L^p$$ -spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.

In this paper, we look at a probabilistic approach to a non-local quadratic form that has lately attracted some interest. This form is related to a recently introduced non-local normal derivative. The goal is to construct two Markov process: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neuman problem. We also study the Dirichlet-to-Neumann operator for non-local operators., Comment: 21 pages

We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Caratheodory domains. This operator is considered as one acting from a space of functions of Lipschitz–Holder type defined by a general modulus of a continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question. This criterion is also obtained in the paper. These results generalize and refine those obtained in the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019).

In this paper, we prove the existence and uniqueness results of entropy solutions to a class of nonlinear parabolic p(u)-Laplacian problem with Fourier-type boundary conditions and $$L^1$$ -data. The main tool used here is the Rothe method combined with the theory of variable exponent Sobolev spaces.

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

The paper deals with methods for calculating distributions of functionals of switching diffusions with jumps. The switching between two collections of diffusion coefficients occurs at the Poisson time moments which are independent of the initial diffusions. At the same moments, the diffusion can have jumps. The process controlling the switching is determined by a Markov chain.

In this paper, a stochastic multi-group AIDS model with saturated incidence rate is studied. We prove that the system is persistent in the mean under some parametric restrictions. We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function. Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system, which greatly improves upon previous results.

In this paper we obtain an Ito differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.

The goal of the paper is to study the convergence of finite volume approximations of the Navier–Stokes–Fourier system describing the motion of compressible, viscous and heat-conducting fluids. The numerical flux uses upwinding with an additional numerical diffusion of order $\mathcal O(h^{ \varepsilon +1})$, $0

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on the critical line are dense in the complex plane. In this paper, we give a result for the denseness of the values of the iterated integrals on the horizontal lines. By using this result, we obtain the denseness of the values of $\int_{0}^{t} \log \zeta(1/2 + it')dt'$ under the Riemann Hypothesis. Moreover, we show that, for any $m\geq 2$, the denseness of the values of an $m$-times iterated integral on the critical line is equivalent to the Riemann Hypothesis., Comment: 15 pages

National audience; This paper is devoted to the study of the internal stabilization of the Bernoulli-Euler plate equation in a square. The continuous and the space semi-discretizated problems are successively considered and analyzed using a frequency domain approach. For the infinite dimensional problem, we provide a new proof of the exponential stability result, based on a two dimensional Ingham's type result. In the second and main part of the paper, we propose a finite difference space semi-discretization scheme and we prove that this scheme yields a uniform exponential decay rate (with respect to the mesh size).

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal{F}$ in terms of the elements of $\mathcal{E}$ via a map $a \colon I \to G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of $\mathcal{F}$, the relative measure of $\mathcal{E}$ - both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements - and the density of the set $a(I)$ in $G$. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent paper, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this paper are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented., 37 pages, 1 figure

This work developes a quantitative framework for describing the overcompleteness of a large class of frames. A previous paper introduced notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal{F}$ in terms of the elements of $\mathcal{E}$ via a map $a \colon I \to G$. This paper shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this paper, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp., 34 pages, 1 figure

This paper addresses the existence of codimension one stable manifolds for the pseudo-conformal blow-up solution for critical one-dimensional NLS. By the work of Perelman and Merle, Raphael, the blow-up rate of these solutions is far from the generic one. However, Bourgain,Wang and Perelman formulated the possibility of codimension one stable manifolds for the non-generic blow-up solutions. This paper established the existence of such manifolds, albeit only in the measurable category.

In this paper, we study the transition threshold of the 3D Couette flow in Sobolev space at high Reynolds number $\text{Re}$. It was proved that if the initial velocity $v_0$ satisfies $\|v_0-(y,0,0)\|_{H^2}\le c_0\text{Re}^{-1}$, then the solution of the 3D Navier-Stokes equations is global in time and does not transition away from the Couette flow. This result confirms the transition threshold conjecture in physical literatures.

Let SF be the set of continuous bounded selections from the set-valued mapping F : T → 2H with nonempty convex closed values; here T is a paracompact Hausdorff topological space, and H is a Hilbert space. In this paper, we obtain a criterion for the existence of a 1-Lipschitz selection from the metric projection onto the set SF in C(T, H).

In this paper, we study a type of stochastic McKean–Vlasov equations with non-Lipschitz coefficients. Firstly, by an Euler–Maruyama approximation the existence of its weak solutions is proved. Then we observe the pathwise uniqueness of its weak solutions. Finally, it is shown that the Euler–Maruyama approximation has an optimal strong convergence rate.

In this paper, we present an oscillatory version of the celebrated Breuer-Major theorem that is motivated by the random corrector problem. As an application, we are able to prove new results concerning the Gaussian fluctuation of the random corrector. We also provide a variant of this theorem involving homogeneous measures., Comment: V2: Minor revision; to appear in Asymptotic analysis

We consider the following k-coupled nonlinear Schrodinger systems: { − Δ u j + λ j u j = μ j u j 3 + ∑ i = 1 , i ≠ j k β i , j u i 2 u j in R N , u j > 0 in R N , u j ( x ) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , k , where N ≤ 3 , k ≥ 3 , λ j , μ j > 0 are constants and β i , j = β j , i ≠ 0 are parameters. There have been intensive studies for the above systems when k = 2 or the systems are purely attractive ( β i , j > 0 , ∀ i ≠ j ) or purely repulsive ( β i , j 0 , ∀ i ≠ j ); however very few results are available for k ≥ 3 when the systems admit mixed couplings and the components are organized into groups, i.e., there exist ( i 1 , j 1 ) and ( i 2 , j 2 ) such that β i 1 , j 1 > 0 and β i 2 , j 2 0 . In this paper we give the first systematic and an (almost) complete study on the existence of ground states when the systems admit mixed couplings and the components are organized into groups. We first divide these systems into repulsive-mixed and total-mixed cases. In the first case we prove nonexistence of ground states. In the second case we give a necessary condition for the existence of ground states and also provide estimates for Morse index. The key idea is the block decomposition of the systems (optimal block decompositions, eventual block decompositions), and the measure of total interaction forces between different blocks. Finally the assumptions on the existence of ground states are shown to be optimal in some special cases.

A linear mapping ϕ from an algebra $${\cal A}$$ into its bimodule $${\cal M}$$ is called a centralizable mapping at G ∈ $${\cal A}$$ if ϕ(AB) = ϕ(A)B = Aϕ(B) for each A and B in $${\cal A}$$ with AB = G. In this paper, we prove that if $${\cal M}$$ is a von Neumann algebra without direct summands of type I1 and type II, $${\cal A}$$ is a *-subalgebra with $${\cal M}$$ ⊆ $${\cal A}$$ ⊆ LS ( $${\cal M}$$ ) and G is a fixed element in $${\cal A}$$ , then every continuous (with respect to the local measure topology t( $${\cal M}$$ )) centralizable mapping at G from $${\cal A}$$ into $${\cal M}$$ is a centralizer.

In the paper, the solvability problems of an nonhomogeneous boundary value problem in the first quadrant for a fractionally loaded heat equation are studied. Feature of this problem is that, firstly, the loaded term is presented in the form of the Caputo fractional derivative with respect to the spatial variable, secondly, the order of the derivative in the loaded term is less than the order of the differential part and, thirdly, the point of load is moving. The problem is reduced to the Volterra integral equation of the second kind, the kernel of which contains the generalized hypergeometric series. The kernel of the obtained integral equation is estimated and it is shown that the kernel of the equation has a weak singularity (under certain restrictions on the load), this is the basis for the statement that the loaded term in the equation is a weak perturbation of its differential part. In addition, the limiting cases of the order of the fractional derivative are considered. It is proved that there is continuity in the order of the fractional derivative.

Let Mob( $$\mathbb{S}^{{n + 1}}$$ ) denote the Mobius transformation group of $$\mathbb{S}^{{n + 1}}$$ . A hypersurface f: $${M^n} \to \mathbb{S}^{{n + 1}}$$ is called a Mobius homogeneous hypersurface, if there exists a subgroup $$G \triangleleft {\text{M}}\ddot o{\text{b}}{(^{n + 1}})$$ such that the orbit G(p) = {ϕ(p) ∣ ϕ ∈ G} = f (Mn),p ∈ f (Mn). In this paper, we classify the Mobius homogeneous hypersurfaces in $$\mathbb{S}^{{n + 1}}$$ with at most one simple principal curvature up to a Mobius transformation.

This paper develops asymptotics and approximations for ruin probabilities in a multivariate risk setting. We consider a model in which the individual reserve processes are driven by a common Markovian environmental process. We subsequently consider a regime in which the claim arrival intensity and transition rates of the environmental process are jointly sped up, and one in which there is (with overwhelming probability) maximally one transition of the environmental process in the time interval considered. The approximations are extensively tested in a series of numerical experiments.