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Following a result of Hatori et al. (J Math Anal Appl 326:281–296, 2007), we give here a spectral characterization of an isomorphism from a $$C^\star $$ -algebra onto a Banach algebra. We then use this result to show that a $$C^\star $$ -algebra A is isomorphic to a Banach algebra B if and only if there exists a surjective function $$\phi :A\rightarrow B$$ satisfying (i) $$\sigma \left( \phi (x)\phi (y)\phi (z)\right) =\sigma \left( xyz\right) $$ for all $$x,y,z\in A$$ (where $$\sigma $$ denotes the spectrum), and (ii) $$\phi $$ is continuous at $$\mathbf 1$$ . In particular, if (in addition to (i) and (ii)) $$\phi (\mathbf 1)=\mathbf 1$$ , then $$\phi $$ is an isomorphism. An example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. The results presented here also elaborate on a paper of Bresar and Spenko (J Math Anal Appl 393:144–150, 2012), and a paper of Bourhim et al. (Arch Math 107:609–621, 2016).

This paper studies a discrete-time batch arrival GI/Geo/1 queue where the server may take multiple vacations depending on the state of the queue/system. However, during the vacation period, the server does not remain idle and serves the customers with a rate lower than the usual service rate. The vacation time and the service time during working vacations are geometrically distributed. Keeping note of the specific nature of the arrivals and departures in a discrete-time queue, we study the model under late arrival system with delayed access and early arrival system independently. We formulate the system using supplementary variable technique and apply the theory of difference equation to obtain closed-form expressions of steady-state system content distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the associated characteristic equations. We discuss the stability conditions of the system and develop few performance measures as well. Through some numerical examples, we illustrate the feasibility of our theoretical work and highlight the asymptotic behavior of the probability distributions at pre-arrival epochs. We further discuss the impact of various parameters on the performance of the system. The model considered in this paper covers a wide class of vacation and non-vacation queueing models which have been studied in the literature.

In this paper, we prove normality criteria for families of meromorphic functions involving sharing of a holomorphic function by a certain class of differential polynomials. Results in this paper extend the works of different authors carried out in recent years, for example Charak and Sharma (Turk J Math 41: 404–411, 2017), Fang and Yuan (J Aust Math Soc Ser A 68(3): 321–333, 2000) Jiang and Gao (Acta Math Sci 32(4): 1503–1512, 2012), Meng and Hu (Bull Malays Math Sci Soc 38:1331–1347, 2015), Sun (Kragujev J Math 38(2):273–282, 2014), Xiao et al. (Abstr Appl Anal Article ID 373910, 1–8, 2011), Yuan (Arch Math 96(5): 435–444 2011).

In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.

In this paper, a general five-step discrete-time Zhang neural network is proposed for time-varying nonlinear optimization (TVNO). First, based on the bilinear transform and Routh–Hurwitz stability criterion, we propose a general five-step third-order Zhang dynamic (ZeaD) formula, which is shown to be convergent with the truncation error $$O(\tau ^3)$$ with $$\tau >0$$ the sampling gap. Second, based on the designed ZeaD formula and the continuous-time Zhang dynamic design formula, a general five-step discrete-time Zhang neural network (DTZNN) model with quartic steady-state error pattern is presented for TVNO, which includes many multi-step DTZNN models as special cases. Third, based on the Jury criterion, we derive the effective domain of step size in the DTZNN model, which determines its convergence and convergence speed. Fourth, a nonlinear programming is established to determine the optimal step size. Finally, simulation results and discussions are given to validate the theoretical results obtained in this paper.

Let P be a Borel probability measure on ℝ generated by an infinite system of similarity mappings {Sj : j ∈ ℕ} such that $$P=\Sigma_{j=1}^{\infty}\frac{1}{2^{j}}P\circ{S}_j^{-1}$$ , where for each j ∈ ℕ and x ∈ ℝ, $$S_j(x)=\frac{1}{3^j}x+1-\frac{1}{3^{j-1}}$$ . Then, the support of P is the dyadic Cantor set C generated by the similarity mappings f1, f2 : ℝ → ℝ such that f1(x) = 1/3x and f2(x) = 1/3x+ 2/3 for all x ∈ ℝ. In this paper, using the infinite system of similarity mappings {Sj : j ∈ ℕ} associated with the probability vector $$(\frac{1}{2},\frac{1}{{{2^2}}},...)$$ , for all n ∈ ℕ, we determine the optimal sets of n-means and the nth quantization errors for the infinite self-similar measure P. The technique obtained in this paper can be utilized to determine the optimal sets of n-means and the nth quantization errors for more general infinite self-similar measures.

In this paper, the authors give the local L2 estimate of the maximal operator $$S_{\phi ,\gamma }^ * $$ of the operator family {St,ϕ, γ} defined initially by $${S_{t,\phi ,\gamma }}f(x): = {{\rm{e}}^{{\rm{i}}\,t\phi (\sqrt { - \Delta } )}}f(\gamma (x,t)) = {(2\pi )^{ - 1}}\int_\mathbb{R} {{{\rm{e}}^{{\rm{i}}\gamma (x,t) \cdot \xi + {\rm{i}}\,t\phi ({\rm{|}}\xi {\rm{|}})}}} \hat f(\xi ){\rm{d}}\xi ,\;\;\;\;\;\;\;\;f \in {\cal S}(\mathbb{R}),$$ which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}: $$\left\{ {\matrix{ {{\rm{i}}{\partial _t}u + \phi (\sqrt { - {\rm{\Delta }}} )u = 0,} \hfill \;\;\;\;\; {(x,t) \in \mathbb{R}{^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$$ where {itϕ}: ℝ+ → ℝ satisfies some suitable conditions and $$\phi (\sqrt { - {\rm{\Delta }}} )$$ is a pseudo-differential operator with symbol {itϕ}(∣{itξ}∣). As a consequence of the above result, the authors give the pointwise convergence of the solution (when {itn} = 1) of the equation (*) along curve {itγ}. Moreover, a global {itL}2 estimate of the maximal operator $$S_{\phi ,\gamma }^ * $$ is also given in this paper.

In [BGLM] and [GLNP] it was conjectured that if $H$ is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in $H$ of covolume at most $x$ is $x^{(\gamma(H)+o(1))\log x/\log\log x}$ where $\gamma(H)$ is an explicit constant computable from the (absolute) root system of $H$. In [BLu] we disproved this conjecture. In this paper we prove that for most groups $H$ the conjecture is actually true if we restrict to counting only non-uniform lattices., Comment: 23 pages, revised following referee's comments. Dedicated to Aner Shalev on his 60th birthday. This paper is related to our previous work arXiv:0905.1841 with which it shares some preliminaries

We begin with the reference measure P0 induced by simple, symmetric nearest neighbor continuous time random walk on Zd starting at 0 with jump rate 2d and then define, for β ⩾ 0, t > 0, the Gibbs probability measure Pβ,t by specifying its density with respect to P0 as $${{d{P_{\beta, t}}} \over {d{P^0}}} = {Z_{\beta, t}}{(0)^{-1}}{{\rm{e}}^{\beta \int_0^t {{\delta _0}({x_s})ds}}},$$ (0.1) where $${Z_{\beta, t}}(0) \equiv {E^0}{\rm{[}}{{\rm{e}}^{\beta \;\int_0^t {{\delta _0}({x_s})ds}}}].$$. This Gibbs probability measure provides a simple model for a homopolymer with an attractive potential at the origin. In a previous paper (Cranston and Molchanov, 2007), we showed that for dimensions d ⩾ 3 there is a phase transition in the behavior of these paths from the diffusive behavior for β below a critical parameter to the positive recurrent behavior for β above this critical value. The critical value was determined by means of the spectral properties of the operator Δ + βδ0, where Δ is the discrete Laplacian on Zd. This corresponds to a transition from a diffusive or stretched-out phase to a globular phase for the polymer. In this paper we give a description of the polymer at the critical value where the phase transition takes place. The behavior at the critical parameter is dimension-dependent.

In this paper, we consider the following logarithmic Choquard equation $$\begin{aligned} - \,\Delta u + a(x)u + \lambda (\log |\cdot |*|u|{^2})u = b|u|^{p-2}u, ~~\ \text{ in }~\ {\mathbb {R}}^n, \end{aligned}$$where $$n\ge 3, \lambda >0, 2 2$$, this equation has been studied extensively. In this paper, we prove the existence of a mountain-pass solution and a ground state solution for $$n\ge 3$$ by using the variational method.

In this paper, we are concerned with a bounded strongly continuous semigroup of continuous module homomorphisms on a complex complete random normed module. The main results of this section are a theorem, which establishes a differential equation associated with the bounded s.c. semigroup generalizing a central result in the paper [X. Zhang and M. Liu, On almost surely bounded semigroups of random linear operators, J. Math. Phys., 54:053517, 2013], and a theorem, which is a version of the Hille–Yosida theorem for the contraction semigroup of continuous module homomorphisms on complex complete RN module.

We consider the Siegel upper half space $$H_{2m}$$ of degree 2m and a subset $$H_m\times H_m$$ of $$H_{2m}$$ consisting of two $$m\times m$$ diagonal block matrices. We consider two actions of $$Sp(m,{\mathbb R})\times Sp(m,{\mathbb R}) \subset Sp(2m,{\mathbb R})$$ , one is the action on holomorphic functions on $$H_{2m}$$ defined by the automorphy factor of weight k on $$H_{2m}$$ and the other is the action on vector valued holomorphic functions on $$H_m\times H_m$$ defined on each component by automorphy factors obtained by $$det^k \otimes \rho $$ , where $$\rho $$ is a polynomial representation of $$GL(n,{\mathbb C})$$ . We consider vector valued linear holomorphic differential operators with constant coefficients on holomorphic functions on $$H_{2m}$$ which give an equivariant map with respect to the above two actions under the restriction to $$H_m\times H_m$$ . In a previous paper, we have already shown that all such operators can be obtained either by a projection of the universal automorphic differential operator or alternatively by a vector of monomial basis corresponding to the partition $$2m=m+m$$ . Here in this paper, based on a completely different idea, we give much simpler looking one-line formula for such operators. This is obtained independently from our previous results. The proofs also provide more algorithmic approach to our operators.

The waveguide considered coincides with a strip having two narrows of width e. An electron wave function satisfies the Dirichlet boundary value problem for the Helmholtz equation. The part of the waveguide between the narrows serves as a resonator, and conditions for the electron resonant tunneling may occur. In the paper, asymptotic formulas as e → 0 for characteristics of the resonant tunneling are used. The asymptotic results are compared with the numerical ones obtained by approximate calculation of the scattering matrix for energies in the interval between the second and third thresholds. The comparison allows us to state an interval of e, where the asymptotic and numerical approaches agree. The suggested methods can be applied to more complicated models than that considered in the paper. In particular, the same approach can be used for asymptotic and numerical analysis of the tunneling in three-dimensional quantum waveguides of variable cross-sections. Bibliography: 3 titles.

The present paper offers a new approach to the construction of the coordinate asymptotics of the kernel of the resolvent of the Schrodinger operator in the scattering problem of three onedimensional quantum particles with short-range pair potentials. Within the framework of this approach, the asymptotics of eigenfunctions of the absolutely continuous spectrum of the Schrodinger operator can be constructed. In the paper, the possibility of a generalization of the suggested approach to the case of the scattering problem of N particles with arbitrary masses is discussed.

Nonconforming grids with hanging nodes are frequently used in adaptive finite element computations. In all earlier works on such methods, proper constraints should be enforced on degrees of freedom on edges/faces with hanging nodes to keep continuity, which yield numerical computations much complicated. In 2014, Zhao et al. (2014) presented quadrilateral constraint-free finite element methods on quadrilateral grids with hanging nodes. This paper further develops a hexahedral constraint-free finite element method on hexahedral grids with hanging nodes, which is of greater challenge than the two-dimensional case. Residual-based a posteriori error reliability and efficiency are also established in this paper.

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications., Comment: 20 pp.23 Ref

Let $${\widetilde{k}}$$ be a fixed cubic field, F a quadratic field and $$L=\widetilde{k}\cdot F$$. In this paper and its companion paper, we determine the density of more or less the ratio of the residues of the Dedekind zeta functions of L, F where F runs through quadratic fields.

Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples., Comment: 23 pages, 3 figures

In the present paper, we apply Hopf’s lemma to the contact CR-warped product submanifolds of generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. The indicated inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iranian Math. Soc., 39, No. 3, 415–429 (2013), M. Atceken, Collect. Math., 62, No. 1, 17–26 (2011), and S. Sular and C. Ozgur, Turkish J. Math., 36, 485–497 (2012)]. Moreover, we also deduce another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities obtained in [I. Mihai, Geom. Dedicata, 109, 165–173 (2004) and K. Arslan, R. Ezentas, I. Mihai, and C. Murathan, J. Korean Math. Soc., 42, No. 5, 1101–1110 (2005)]. The inequalities deduced in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for the contact CR-warped product submanifolds of any almost contact metric manifold.

In this paper we determine the largest sets of points of finite thick buildings of type F4 such that no two points of the set are at maximal distance. The motivation for studying these sets comes from [9], where a general Erdős–Ko–Rado problem was formulated for finite thick buildings. The result in this paper solves this problem for points (and dually for symplecta) in finite thick buildings of type F4.

This paper is the first of a series of papers culminating in the result that measure preserving diffeomorphisms of the disc or 2-torus are unclassifiable. It addresses another classical problem: which abstract measure preserving systems are realizable as smooth diffeomorphisms of a compact manifold? The main result gives symbolic representations of Anosov–Katok diffeomorphisms.

The paper is devoted to the investigations of holomorphic functions on complete n-circular domains G of ℂn which are solutions of some partial differential equations in G. Our considerations concern a collection M G , k ≥ 2, of holomorphic solutions of equations corresponding to planar Sakaguchi’s conditions for starlikeness with respect to k-symmetric points. In an earlier paper of the first author some embedding theorems for M G k were given. In this paper we solve the problem of finding some sharp estimates of m-homogeneous polynomials in a power series expansion of f from M G . We obtain a formula of the extremal function which includes some special functions. Moreover, its construction is based on properties of hypergeometric functions and (j, k)-symmetric functions. The (j, k)-symmetric functions were considered in several papers of the second author and his co-author, J. Polubinski.

This paper mainly focuses on the strong convergence of the Euler-Maruyama method for nonlinear stochastic convolution Ito-Volterra integral equations with constant delay. It is well known that the strong approximation of the Ito integral usually leads to 0.5-order approximation for stochastic problems. However, in this paper, we will show that 1-order strong superconvergence can be obtained for nonlinear stochastic convolution Ito-Volterra integral equations with constant delay under some mild conditions on the kernel of the diffusion term. Finally, some numerical experiments are given to illustrate our theoretical results.

The present paper deals with the performance evaluation of a cognitive radio network with the help of a queueing model. The queueing system contains two interconnected, not independent sub-systems. The first part is for the requests of the Primary Units (PU). The number of sources is finite, and each source generates high priority requests after a exponentially distributed time. The requests are sent to a single server unit or Primary Channel Service (PCS) with a preemptive priority queue. The service times are assumed to be exponentially distributed. The second sub-system is for the requests of the Secondary Units (SU), which is finite sources system too; the inter-request times and service times of the single server unit or Secondary system Channel Service (SCS) are assumed to be exponentially distributed, respectively. A generated high priority packet goes to the primary service unit. If the unit is idle, the service of the packet begins immediately. If the server is busy with a high priority request, the packet joins the preemptive priority queue. When the unit is engaged with a request from SUs, the service is interrupted and the interrupted low priority task is sent back to the SCS. Depending on the state of the secondary channel, the interrupted job is directed to either the server or the orbit. In case the requests from SUs find the SCS idle, the service starts, and if the SCS is busy, the packet looks for the PCS. In the case of an idle PCS, the service of the low-priority packet begins at the high-priority channel (PCS). If the PCS is busy, the packet goes to the orbit. From the orbit it retries to be served after an exponentially distributed time. The novelty of our investigation is that each server is subject to random breakdowns, in which case the interrupted request is sent to the queue or orbit, respectively. The operating and repair times of the servers are assumed to be generally distributed. Finally, all the random times included in the model construction are assumed to be independent of each other. The main aim of the paper is to analyze the effect of the nonreliability of the servers on the mean and variance of the response time for the SUs by using simulation.

Let $A$ be a (left and right) Noetherian ring that is semiperfect. Let $e$ be an idempotent of $A$ and consider the ring $\Gamma:=(1-e)A(1-e)$ and the semi-simple right $A$-module $S_e : = eA/e{\rm rad}A$. In this paper, we investigate the relationship between the global dimensions of $A$ and $\Gamma$, by using the homological properties of $S_e$. More precisely, we consider the Yoneda ring $Y(e):={\rm Ext}^*_A(S_e,S_e)$ of $e$. We prove that if $Y(e)$ is artinian of finite global dimension, then $A$ has finite global dimension if and only if so is $\Gamma$. We also investigate the situation where both $A,\Gamma$ have finite global dimension. When $A$ is Koszul and finite dimensional, this implies that $Y(e)$ has finite global dimension. We end the paper with a reduction technique to compute the Cartan determiant of artin algebras. We prove that if $Y(e)$ has finite global dimension, then the Cartan determinants of $A$ and $\Gamma$ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture., Comment: 14 pages

In this paper, we find all positive squarefree integers d such that the Pell equation X2-dY2 = +-1 has at least two positive integer solutions (X,Y) and (X',Y') such that both X and X' have Zeckendorf representations with at most two terms. This paper has been accepted for publication in SCIENCE CHINA Mathematics.

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl-Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.

International audience; In this paper, we first introduce the concept and properties of ω-periodic limit process. Then we apply specific criteria obtained to investigate asymptotically ω-periodic mild solutions of a Stochastic Differential Equation driven by a Brownian motion. Finally, we give an example to show usefulness of the theoritical results that we obtain in the paper. MSC : 34C25, 34C27, 60H30, 34 F05.

In this paper, we aim to investigate the mean-variance portfolio selection in an economy with inflation risk. In the financial market, the inflation index can only be partially observed by a signal process. We transform the initial problem into an equivalent completely observed problem. The effect of the partially observed price index on the optimization problem is twofold. Firstly, the equivalent completely observed problem involves more estimation error. Secondly, the mean-variance criterion is distorted. Higher moment is assigned with a bigger weight. The optimization goal does not satisfy the assumption in the Bellman’s optimality condition and we derive the equilibrium strategy based on the extended HJB equation. Besides, we also show the results of the efficient frontier and strategy in the precommitment case. In the end of this paper, we present a sensitivity analysis to show the economic behaviors of the investor and compare the efficient strategies and frontiers in precommitment case and equilibrium case.

Glider representations can be defined for a finite algebra filtration FKG determined by a chain of subgroups 1 subset of G(1) subset of horizontal ellipsis subset of G(d) = G. In this paper we develop the generalized character theory for such glider representations. We give the generalization of Artin's theorem and define a generalized inproduct. For finite abelian groups G with chain 1 subset of G, we explicitly calculate the generalized character ring and compute its semisimple quotient. The papers ends with a discussion of the quaternion group as a first non-abelian example.

In this paper, we present a new mathematical model generalizing the model of a single unit repairable device system, with preventive maintenance under warranty period, and the model of a repairable multi-state device system. To study the time dependance solution of the resulting system, we present in this paper a new analytic approach. So, by using the $$C_0$$ -semigroup theory of linear operators and some spectral properties, we proof that the dynamic solution of the system converges to steady state solution when the time go to infinity.

Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences $$\mathfrak {c}$$ and $$\mathfrak {d}$$ on a vector lattice, we study $$\mathfrak {d}$$ -asymptotic properties of operator nets formed by $$\mathfrak {c}$$ -continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on $$\mathfrak {d}$$ -martingale and $$\mathfrak {d}$$ -Lotz–Rabiger nets.

In the present paper, our boundary value problem is Sturm–Liouville problem with retarded argument and transmission conditions at the finitely many points of discontinuity. This is the first work containing several points of discontinuity in the theory of differential equations with retarded argument. The main purpose of this paper is to obtain asymptotic formulas for the eigenvalues and corresponding eigenfunctions of this boundary value problem. The simplicity of the eigenvalues is proved. In special case when our problem with retarded argument and transmission conditions at one point of discontinuity, the obtained results coincide with the corresponding results in Bayramov et al. (Appl Math Comput 191, 592–600, 2007).

In the paper, we consider a nonlinear integrodifferential equation of the third order with delay. We establish sufficient conditions guaranteeing the global existence and boundedness of the solutions of the analyzed equation. We use the Lyapunov second method to prove the main result. An example is also given to illustrate the applicability of our result. The result of this paper is new and improves previously known results.

In this paper, we study a dynamic coloring of the vertices of a graph G that starts with an initial subset S of colored vertices, with all remaining vertices being uncolored. At each discrete time interval, a colored vertex with exactly one uncolored neighbor forces this uncolored neighbor to be colored. The initial set S is a zero forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set of G. In this paper, we prove that if G is a connected, cubic, claw-free graph of order $$n \ge 6$$, then $$Z(G) \le \alpha (G) + 1$$ where $$\alpha (G)$$ denotes the independence number of G. Further we prove that if $$n \ge 10$$, then $$Z(G) \le \frac{1}{3}n + 1$$. Both bounds are shown to be best possible.

Let k be a field of characteristic $$ p\not =0 $$. For a (purely) inseparable extension K / k the notion of modularity, defined by M.E. Sweedler in the 60s, is a very important property, very much like being Galois for a separable extension. We have defined, together with M. Chellali, a generalization of the notion of modularity, called lower quasi-modularity: K / k is lower quasi-modular (lq-modular) if for some finite extension $$k'$$ over k we have that $$K/k'$$ is modular. In subsequent papers M. Chellali and the author have studied various properties of lq-modular field extensions, including the existence of lq-modular closures in case $$[k{:}k^p]$$ is finite. In this paper we prove a similar result, without the hypothesis on k but with extra assumptions on K / k: the extension needs to be q-finite, that is, there must exist an integer M such that for every positive integer n the field $$K\cap k^{p^{-n}}$$ is generated by at most M elements on k. A number of properties of lq-modular closures are determined and examples are presented illustrating the results.

The Unitary Dual Problem is one of mathematics’ most important open problems: classify the irreducible unitary representations of a group. The general approach has been to classify all representations admitting non-degenerate invariant Hermitian forms, compute the signatures of those forms, and then determine which forms are positive definite. Signature character algorithms and formulas arising from deforming representations and analysing changes at reducibility points, as in Adams et al. (Unitary representations of real reductive groups (ArXiv e-prints), 2012) and Yee (Represent Theory 9:638–677, 2005), produce very complicated formulas or algorithms from the resulting recursion. This paper shows that in the case of irreducible Verma modules all of the complexity can be encapsulated by the affine Hecke algebra: for compact real forms and for alcoves corresponding to translations of the fundamental alcove by a regular weight, signature characters of irreducible Verma modules are in fact “negatives” of Hall–Littlewood polynomial summands evaluated at $$q=-\,1$$ times a version of the Weyl denominator, establishing a simple signature character formula and drawing an important connection between signature characters and the affine Hecke algebra. Signature characters of irreducible highest weight modules are shown to be related to Kazhdan-Lusztig basis elements. This paper also handles noncompact real forms. The current state of the art for the unitary dual is a computer algorithm for determining if a given representation is unitary. These results suggest the potential to move the state of the art to a closed form classification for the entire unitary dual.

We have, in this paper, constructed a new non-geometric embedding of some braid group into the mapping class group of a surface which is induced by the 3-fold branched covering over a disk with some branch points. There is a lift $$\tilde{\beta }_i$$ of the half-Dehn twist $$\beta _i$$ on the disk with some marked points to some surface via the 3-fold covering. We show how this lift $$\tilde{\beta }_i$$ acts on the fundamental group of the surface, and also show that $$\tilde{\beta }_i$$ equals the product of two (inverse) Dehn twists. Two adjacent lifts satisfy the braid relation, hence such lifts induce a homomorphism $$\phi : B_k \rightarrow \Gamma _{g,b}$$ . In this paper we give a concrete description of this homomorphism and show that it is injective by the Birman–Hilden theory. Furthermore, we show that the map on the level of classifying spaces of groups is compatible with the action of little 2-cube operad so that it induces a trivial homomorphism on the stable homology.

The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on $${\mathbb {R}}_{+}$$ and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Kothe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.

The variance-gamma (VG) distributions form a four-parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein’s method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper (Gaunt 2018), which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

This paper is concerned with a parabolic–parabolic Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{ll} u_t = \nabla \cdot ( \nabla u - u \nabla v ) &{} \quad \text{ in } \, \Omega \times (0,T), \\ v_t = \Delta v - \alpha v + u &{} \quad \text{ in } \, \Omega \times (0,T) \end{array} \right. \end{aligned}$$with a constant $$ \alpha \ge 0 $$ and nonnegative initial data in a smoothly bounded domain $$ \Omega \subset \mathbb {R}^2 $$ under the Neumann boundary condition or in $$ \Omega = \mathbb {R}^2 $$. It was introduced as a model of aggregation of bacteria, which is mathematically translated as finite-time blowup. A solution (u, v) is said to blow up at $$ t = T 0 $$, and type II otherwise. It was shown in Mizoguchi (J Funct Anal 271:3323–3347, 2016) that each blowup is type II in radial case. In this paper, we obtain the conclusion in general case.

In this paper, we mainly study a family of unbounded non-hyperbolic domains in $$\mathbb {C}^{n+m}$$, called Fock–Bargmann–Hartogs domains $$D_{n,m}(\mu )$$ ($$\mu >0$$) which are defined as a Hartogs type domains with the fiber over each $$z\in \mathbb {C}^{n}$$ being a ball of radius $$e^{-\frac{\mu }{2} {\Vert z\Vert }^{2}}$$. The purpose of this paper is twofold. Firstly, we obtain necessary and sufficient conditions for Rawnsley’s $$\varepsilon $$-function $$\varepsilon _{(\alpha ,g)}(\widetilde{w})$$ of $$\big (D_{n,m}(\mu ), g(\mu ;\nu )\big )$$ to be a polynomial in $$\Vert \widetilde{w}\Vert ^2$$, where $$g(\mu ;\nu )$$ is a Kahler metric associated with the Kahler potential $$\nu \mu {\Vert z\Vert }^{2} -\ln (e^{-\mu {\Vert z\Vert }^{2}}-\Vert w\Vert ^2)$$. Secondly, using above results, we study the Berezin quantization on $$D_{n,m}(\mu )$$ with the metric $$\beta g(\mu ;\nu )$$$$(\beta >0)$$.

The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized.

In the present paper, we consider a class of second-order singular differential operators which generalize the well-known Bessel differential operator. The associated eigenfunctions are the Bessel–Wright functions. These functions can be obtained by the action of the Riemann–Liouville operator on the normalized Bessel functions. We introduce a Bessel–Wright transform with Bessel–Wright functions as kernel which is connected to the classical Bessel–Fourier transform via the dual of the Riemann–Liouville operator. The Bessel–Wright transform leaves invariant the Schwartz space and sends the set of functions indefinitely differentiable with compact support into the Paley–Wiener space. We conclude the paper by proving two variants of the inversion formulas.

For any symplectic manifold $${(M,\omega )}$$ , the set of Hamiltonian diffeomorphisms $${{\text {Ham}}^c(M,\omega )}$$ forms a group and $${{\text {Ham}}^c(M,\omega )}$$ contains an important subset $${{\text {Aut}}(M,\omega )}$$ which consists of time one flows of autonomous(time-independent) Hamiltonian vector fields on M. One might expect that $${{\text {Aut}}(M,\omega )}$$ is a very small subset of $${{\text {Ham}}^c(M,\omega )}$$ . In this paper, we estimate the size of the subset $${{\text {Aut}}(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric which was introduced by Hofer. Polterovich and Shelukhin proved that the complement $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed symplectically aspherical manifold where Conley conjecture is established (Polterovich and Schelukhin in Sel Math 22(1):227–296, 2016). In this paper, we generalize above theorem to general closed symplectic manifolds and general conv! ex symplectic manifolds. So, we prove that the set of all non-autonomous Hamiltonian diffeomorphisms $${{\text {Ham}}^c\backslash {\text {Aut}}(M,\omega )}$$ is a dense subset of $${{\text {Ham}}^c(M,\omega )}$$ in $${C^{\infty }}$$ -topology and Hofer’s metric if $${(M,\omega )}$$ is a closed or convex symplectic manifold without relying on the solution of Conley conjecture.

This paper investigates the transitivity of the relation defined by degeneration of finitely generated modules over an associative algebra. It is proved in this paper that if L degenerates to M and M degenerates to N, then $$L^{\oplus e}$$ degenerates to $$N^{\oplus e}$$ for some (but explicitly given) integer $$e>0$$ .

In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the $${H^\infty}$$ functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form $$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$ where $${e_{\ell}, {\ell} = 1, 2, 3}$$ are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables $${x = (x_{1}, x_{2}, x_{3})}$$ and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version $${T^{\alpha}, {\rm for} \alpha \in (0, 1)}$$ , of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.