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Let H2(ⅅ2) be the Hardy space over the bidisk ⅅ2, and let Mψ,ϕ = [(ψ(z) − ϕ(w))2] be the submodule generated by (ψ(z) − ϕ(w))2, where ψ(z) and ϕ(w) are nonconstant inner functions. The related quotient module is denoted by Nψ,ϕ = H2(ⅅ2) ⊖ Mψ,ϕ. In this paper, we give a complete characterization for the essential normality of Nψ,ϕ. In particular, if ψ(z)= z, we simply write Mψ,ϕ and Nψ,ϕ as Mϕ and Nϕ respectively. This paper also studies compactness of evaluation operators L(0)∣nϕ and R(0)ϕnϕ, essential spectrum of compression operator Sz on Nϕ, essential normality of compression operators Sz and Sw on Nϕ.

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number $\sqrt{q}$. This establishes a key step that one may extend our previous explicit calculations of superspecial abelian surfaces to those of supersingular abelian surfaces.The second part is to introduce the notion of genera and ideal complexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the results of Yu on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigation., Comment: 23 pages. Section 5.4 corrected

This paper is concerned with one-dimensional derivative quintic nonlinear Schrodinger equation, $${\rm{i}}u_t-u_{xx}+{\rm{i}}(|u|^4u)_x=0, \;\; x\in\mathbb{T}.$$ The existence of a large amount of quasi-periodic solutions with two frequencies for this equation is established. The proof is based on partial Birkhoff normal form technique and an unbounded KAM theorem. We mention that in the present paper the mean value of u does not need to be zero, but small enough, which is different from the assumption (1.7) in Geng-Wu [J. Math. Phys., 53, 102702 (2012)].

Let G be a graph with vertex set V(G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …, |E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to $${{1 + \left| {E(G)} \right|} \over 2} \cdot d(v)$$ , where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V(G), |{e : e ∈ Ev, f(e) ≤ Δ/2}| = |{e : e ∈ Ev, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.

Zhao and Ho asked in a recent paper that for each T0 space X, whether KB(X) (the set of all irreducible closed sets of X whose suprema exist) is the canonical k-bounded sobrification of X in the sense of Keimel and Lawson. In this paper, we construct a counterexample to give a negative answer. We also consider the subcategory Topκ of the category Top0 of T0 spaces, and prove that the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category KBSob of k-bounded sober spaces is a full reflective subcategory of the category Topκ.

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical semigroup solutions to the fragmentation–coagulation equations.

In 2002, in the paper entitled “A subspace theorem approach to integral points on curves”, Corvaja and Zannier started the program of studying integral points on algebraic varieties by using Schmidt’s subspace theorem in Diophantine approximation. Since then, the program has led a great progress in the study of Diophantine approximation. It is known that the counterpart of Schmidt’s subspace in Nevanlinna theory is H. Cartan’s Second Main Theorem. In recent years, the method of Corvaja and Zannier has been adapted by a number of authors and a big progress has been made in extending the Second Main Theorem to holomorphic mappings from C into arbitrary projective variety intersecting general divisors by using H. Cartan’s original theorem. We call such method “a Cartan’s Second Main Theorem approach”. In this survey paper, we give a systematic study of such approach, as well as survey some recent important results in this direction including the recent work of the author with Paul Voja.

This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kahler Calabi–Yau surfaces and Hopf surfaces.

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy’s EVP in [On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

The aim of this paper is to introduce and study Hom-Gel’fand–Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing Hom-Gel’fand–Dorfman super-bialgebras. Also, we obtain some infinite-dimensional Hom-Lie superalgebras from affinization of Hom-Gel’fand–Dorfman super-bialgebras. Finally, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish the equivalence between quadratic Hom-Lie conformal superalgebras and Hom-Gel’fand–Dorfman super-bialgebras.

In this paper, distributed estimation of high-dimensional sparse precision matrix is proposed based on the debiased D-trace loss penalized lasso and the hard threshold method when samples are distributed into different machines for transelliptical graphical models. At a certain level of sparseness, this method not only achieves the correct selection of non-zero elements of sparse precision matrix, but the error rate can be comparable to the estimator in a non-distributed setting. The numerical results further prove that the proposed distributed method is more effective than the usual average method.

The well-known Hartman-Grobman Theorem says that a C1 hyperbolic diffeomorphism F can be locally linearized by a homeomorphism Φ. For parameterized systems Fθ, known results show that the corresponding homeomorphisms Φθ exist uniquely in a functional space equipped with the supremum norm and depend continuously on the parameter θ. In this paper, we further extend the results to Holder dependence of Φθ on θ by Pugh’s strategy, but introducing a kind of special Holder norm instead of the usual supremum norm in the proof to control the linear parts of Fθ. This requires a new Holder linearization result for every Fθ.

In this paper, we study the distribution function of the time of explosion of a stochastic differential equation modeling the length of the dominant crack due to fatigue. The main novelty is that initial condition is regarded as an anticipating random variable and the stochastic integral is in the forward sense. Under suitable conditions, we use the substitution formula from Russo and Vallois to find the local solution of this equation. Then, we find the law of blow up time by proving some results on barrier crossing probabilities of Brownian bridge.

In this paper, the mixed Polya-Szego principle is established. By the mixed Polya-Szego principle, the mixed Morrey-Sobolev inequality and some new analytic inequalities are obtained.

In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent and Poisson stable) solutions for semilinear stochastic differential equations driven by infinite dimensional Levy noise with large jumps. Under suitable conditions on drift, diffusion and jump coefficients, we prove that there exist solutions which inherit the Poisson stability of coefficients. Further we show that these solutions are globally asymptotically stable in square-mean sense. Finally, we illustrate our theoretical results by several examples.

In this paper, we study the regularity of the complex Hessian equation when the right hand has pole singularity. We show the Holder continuity of the solution to the Dirichlet problem. In particular, for the complex Monge-Ampere equation, we improve a result of [7].

Recently, Geher and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L2-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.

This paper is mainly about holomorphic mappings associated with conic regions which are closely connected with k − ST(α). We introduce new subclasses of starlike (spirallike) functions, namely, S (k,α)(S (k,α,β)), and discuss their coefficient estimates and the Fekete-Szego-Goluzin’s problem. Then we generalize S (k,α,β) on the unit ball Bn in ℂn, that is, k-conic spirallike mappings of type β and order α. We obtain the growth, covering and distortion theorems of the generalized mappings. Besides that, we construct k-conic spirallike mappings of type β and order α on Bn through Sc(k,α,β) by the generalized Roper-Suffridge extension operators.

In this paper we introduce two metrics: the max metric dn,q and the mean metric $${\bar d_{n,q}}$$ . We give an equivalent characterization of rigid measure preserving systems by the two metrics. It turns out that an invariant measure μ on a topological dynamical system (X, T) has bounded complexity with respect to dn,q if and only if μ has bounded complexity with respect to $${\bar d_{n,q}}$$ if and only if (X, $${\cal B}x$$ , μ, T) is rigid. We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system (resp. the topological entropy of a topological dynamical system) by the two metrics dn,q and $${\bar d_{n,q}}$$ .

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of |G| is a maximal subgroup.

In this paper we introduce the notions of (Banach) density-equicontinuity and density-sensitivity. On the equicontinuity side, it is shown that a topological dynamical system is density-equicontinuous if and only if it is Banach density-equicontinuous. On the sensitivity side, we introduce the notion of density-sensitive tuple to characterize the multi-variant version of density-sensitivity. We further look into the relation of sequence entropy tuple and density-sensitive tuple both in measure-theoretical and topological setting, and it turns out that every sequence entropy tuple for some ergodic measure on an invertible dynamical system is density-sensitive for this measure; and every topological sequence entropy tuple in a dynamical system having an ergodic measure with full support is density-sensitive for this measure.

This paper is mainly concerned with the following nonlinear p-Laplacian equation $$ - {\Delta _p}u(x) + (\lambda a(x) + 1){\left| u \right|^{p - 2}}(x)u(x) = f(x,u(x)),\;\;\;{\rm{in}}\;V$$ on a locally finite graph G =(V, E) with more general nonlinear term, where Δp is the discrete p-Laplacian on graphs, p ≥ 2. Under some suitable conditions on f and a(x), we can prove that the equation admits a positive solution by the Mountain Pass theorem and a ground state solution uλ via the method of Nehari manifold, for any λ > 1. In addition, as λ → + ∞, we prove that the solution uλ converge to a solution of the following Dirichlet problem $$\left\{ {\matrix{ { - {\Delta _p}u(x) + {{\left| u \right|}^{p - 2}}(x)u(x) = f(x,u(x)),} \;\;\;\;\;\;\;\; {{\rm{in}}\;\Omega ,} \hfill \cr {u(x) = 0,} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\; \;\;\;\;\; {{\rm{on}}\;\partial \Omega ,} \hfill \cr } } \right.$$ where Ω = {x ∈ V: a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.

This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ) space. The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane $$S_\kappa^2$$ , including the limit shape of each step. As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on $$S_1^2$$ . At last we make a contrast of geometric properties of CBA with CBB spaces, including a comparison between their defining properties and a discussion about Alexandrov’s Lemma.

In this paper, we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract, and that it also admits the uniform compact approximation property. These can be regarded as extensions of Lindenstrauss and Kalton’s corresponding results.

The goal of this paper is threefold. First, we survey the existing results on Hunt’s hypothesis (H) for Markov processes and Getoor’s conjecture for Levy processes. Second, we investigate (H) for multidimensional Levy processes from the viewpoint of projections. Third, we present a few open questions for further study.

In this paper we study on gradient quasi-Einstein solitons with a fourth-order vanishing condition on the Weyl tensor. More precisely, we show that for n ≥ 4, the Cotton tensor of any n-dimensional gradient quasi-Einstein soliton with fourth order f-divergence free Weyl tensor is flat, if the manifold is compact, or noncompact but the potential function satisfies some growth condition. As corollaries, some local characterization results for the quasi-Einstein metrics are derived.

In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is -1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions., Comment: 13 Pages. Comments welcome

Pilsniak and Woźniak put forward the concept of neighbor sum distinguishing (NSD) total coloring and conjectured that any graph with maximum degree Δ admits an NSD total (Δ + 3)-coloring in 2015. In 2016, Qu et al. showed that the list version of the conjecture holds for any planar graph with Δ ≥ 13. In this paper, we prove that any planar graph with Δ Δ 7 but without 6-cycles satisfies the list version of the conjecture.

In this paper, we investigate the isolated closed orbits of two types of cubic vector fields in ℝ3 by using the idea of central projection transformation, which sets up a bridge connecting the vector field X (x) in ℝ3 with the planar vector fields. We have proved that the cubic vector field in ℝ3 can have two isolated closed orbits or one closed orbit on the invariant cone. As an application of this result, we have shown that a class of 3-dimensional cubic system has at least 10 isolated closed orbits located on 5 invariant cones, and another type of 3-dimensional cubic system has at least 26 isolated closed orbits located on 13 invariant cones or 26 invariant cones.

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.

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lame—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting.

In this paper, let m ≥ 1 be an integer, M be an m-dimensional compact Riemannian manifold. Firstly the linearized Poincare map of the Lagrangian system at critical point x $${d \over {dt}}{L_q}\left( {t,x,\dot x} \right) - {L_p}\left( {t,x,\dot x} \right) = 0$$ is explicitly given, then we prove that Morse index and Maslov-type index of x are well defined whether the manifold M is orientable or not via the parallel transport method which makes no appeal to unitary trivialization and establish the relation of Morse index and Maslov-type index, finally derive a criterion for the instability of critical point and orientation of M and obtain the formula for two Maslov-type indices.

Let G be a nontrivial connected and vertex-colored graph. A subset X of the vertex set of G is called rainbow if any two vertices in X have distinct colors. The graph G is called rainbow vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is rainbow and x and y belong to different components of G − S; whereas when x and y are adjacent, S + x or S + y is rainbow and x and y belong to different components of (G − xy) − S. For a connected graph G, the rainbow vertex-disconnection number of G, denoted by rvd(G), is the minimum number of colors that are needed to make G rainbow vertex-disconnected. In this paper, we characterize all graphs of order n with rainbow vertex-disconnection number k for k ∈ {1, 2, n}, and determine the rainbow vertex-disconnection numbers of some special graphs. Moreover, we study the extremal problems on the number of edges of a connected graph G with order n and rvd(G)= k for given integers k and n with 1 ≤ k ≤ n.

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.

This paper is devoted to studying Bergman spaces $$A_{\omega_{1,2}}^{p}(M)(1

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 $${\cal S}(x)$$ .

Doubly warped product of Finsler manifolds is useful in theoretical physics, particularly in general relativity. In this paper, we study doubly warped product of Finsler manifolds with isotropic mean Berwald curvature or weak isotropic S-curvature.

In this paper we extend the Fibonacci-like maps to a wider class with the so-called “bounded combinatorics”. The Fibonacci-like renormalization operator $${\cal R}$$ is defined and we show that the orbit of each map from this class converges to a universal limit under iterates of $${\cal R}$$ .

Let f and g be functions in Fock—Sobolev space F2,m. In this paper, we completely characterize the boundedness and compactness of $${H_{\mathop f\limits^ - }}{T_{\mathop g\limits^ - }}$$ .

Let (M,g) be a compact Kahler manifold and (E,F) be a holomorphic Finsler vector bundle of rank r ≥ 2over M. In this paper, we prove that there exists a Kahler metric Φ defined on the projective bundle P(E)of E, which comes naturally from g and F. Moreover, a necessary and sufficient condition for Φ having positive scalar curvature is obtained, and a sufficient condition for Φ having positive Ricci curvature is established.

In this paper, assuming there is a fiberwise Morse function, we extend Bismut-Lott index theorem to the complex valued case.

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers (ADMM) algorithm. Furthermore, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided.

In this paper, we consider the hypersurfaces of Randers space with constant flag curvature. (1) Let $$({\overline M ^{n + 1}},\overline F )$$ be a Randers-Minkowski space. If (Mn, F)isa hypersurface of $$({\overline M ^{n + 1}},\overline F )$$ with constant flag curvature K = 1, then we can prove that M is Riemannian. (2) Let $$({\overline M ^{n + 1}},\overline F )$$ be a Randers space with constant flag curvature. Assume (M, F) is a compact hypersurface of $$({\overline M ^{n + 1}},\overline F )$$ with constant mean curvature ∣H∣. Then a pinching theorem is established, which generalizes the result of [Proc. Amer. Math. Soc., 120, 1223–1229 (1994)] from the Riemannian case to the Randers space.

In this paper, we study complex symmetric C0-semigroups on the Bergman space A2(ℂ+) of the right half-plane ℂ+. In contrast to the classical case, we prove that the only involutive composition operator on A2(ℂ+) is the identity operator, and the class of J-symmetric composition operators does not coincide with the class of normal composition operators. In addition, we divide semigroups {ϕt} of linear fractional self-maps of ℂ+ into two classes. We show that the associated composition operator semigroup {Tt} is strongly continuous and identify its infinitesimal generator. As an application, we characterize Jσ-symmetric C0-semigroups of composition operators on A2(ℂ+).

In this paper, we give the complete classifications of isoparametric hypersurfaces in Randers space forms. By studying the principal curvatures of anisotropic submanifolds in a Randers space (N, F) with the navigation data (h, W), we find that a Randers space form (N, F, dµBH) and the corresponding Riemannian space (N, h) have the same isoparametric hypersurfaces, but in general, their isoparametric functions are different. We give a necessary and sufficient condition for an isoparametric function of (N, h) to be isoparametric on (N, F, dµBH), from which we get some examples of isoparametric functions.

Among recent measures for risk management, value at risk (VaR) has been criticized because it is not coherent and expected shortfall (ES) has been criticized because it is not robust to outliers. Recently, [Math. Oper. Res., 38, 393–417 (2013)] proposed a risk measure called median shortfall (MS) which is distributional robust and easy to implement. In this paper, we propose a more generalized risk measure called quantile shortfall (QS) which includes MS as a special case. QS measures the conditional quantile loss of the tail risk and inherits the merits of MS. We construct an estimator of the QS and establish the asymptotic normality behavior of the estimator. Our simulation shows that the newly proposed measures compare favorably in robustness with other widely used measures such as ES and VaR.

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.