A generalized Cuntz–Krieger uniqueness theorem for higher-rank graphs

We present a uniqueness theorem for k  -graph C^?$C^{?}$-algebras that requires neither an aperiodicity nor a gauge invariance assumption. Specifically, we prove that for the injectivity of a representation of a k  -graph C^?$C^{?}$-algebra, it is sufficient that the representation be injective on a distinguished abelian C^?$C^{?}$-subalgebra. A crucial part of the proof is the application of an abstract uniqueness theorem, which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient C^?$C^{?}$-algebra, each of which is the unique extension of a state on the distinguished abelian C^?$C^{?}$-subalgebra.     

Stable categories of higher preprojective algebras

We introduce (n+1)$(n+1)$-preprojective algebras of algebras of global dimension n$n$. We show that if an algebra is n$n$-representation-finite then its (n+1)$(n+1)$-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)$(n+1)$-preprojective algebra is (n+1)$(n+1)$-Calabi–Yau, and, more precisely, it is the (n+1)$(n+1)$-Amiot cluster category of the stable n$n$-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)$(n+1)$-cluster tilting object. We show that even if the (n+1)$(n+1)$-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}$n\in \{1,2\}$) the results above still hold for the stable category of Cohen–Macaulay modules.     

MF actions and K-theoretic dynamics

We study the interplay of C^?$C^{?}$-dynamics and K  -theory. Notions of chain recurrence for transformations groups (X,Γ)$(X,\Gamma )$ and MF actions for non-commutative C^?$C^{?}$-dynamical systems (A,Γ,α)$(A,\Gamma ,\alpha )$ are translated into K-theoretical language, where purely algebraic conditions are shown to be necessary and sufficient for a reduced crossed product to admit norm microstates. We are particularly interested in actions of free groups on AF algebras, in which case we prove that a K-theoretic coboundary condition determines whether or not the reduced crossed product is a matricial field (MF) algebra. One upshot is the equivalence of stable finiteness and being MF for these reduced crossed product algebras.     

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear (n,δ)$(n,\delta )$-width and p$p$-average linear n$n$-width of diagonal matrix M$M$ are determined.     

Generalized Zernike or disc polynomials: An application in quantum optics

An application of the “generalized Zernike or disc polynomials”, recently introduced in the literature, is shown, resorting to the Lie algebra based investigation of the dynamics of quantum systems driven by two-mode interaction Hamiltonians. Further properties of the associated “disc functions” are deduced. Also, a generalization of the disc polynomials towards the Hahn polynomials is suggested.     

Almost limited sets in Banach lattices

We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak^?${\mathrm{weak}}^{?}$ null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L  -weakly compact sets coincide. In particular, in terms of almost Dunford–Pettis operators into _c0$c_0$, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E   is almost limited if and only if every continuous linear operator T:E→_c0$T:E\to c_0$ is an almost Dunford–Pettis operator.     

p-embeddings

This is the sequel to Bhattacharjee et al. (in press) [3] where the notion of a p  -extension of commutative rings was investigated: a unital extension of commutative rings, say R?S$R?S$, is a p  -extension if for every s∈S$s\in S$ there is an r∈R$r\in R$ such that rS=sS$rS=sS$. In this article we apply the theory of p  -extensions to rings of continuous functions. We show that this concept lays between the concepts of C^?$C^{?}$-embeddings and z-embeddings.     

A new method for computing Moore–Penrose inverse matrices

The Moore–Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram–Schmidt process and the Moore–Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m×n$m\times n$ real matrix A$A$ with m≥n$m\ge n$ and rank r≤n$r\le n$. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.     

Second-order multiobjective symmetric duality with cone constraints

In this paper, we formulate Wolfe and Mond–Weir type second-order multiobjective symmetric dual problems over arbitrary cones. Weak, strong and converse duality theorems are established under η$\eta$-bonvexity/η$\eta$-pseudobonvexity assumptions. This work also removes several omissions in definitions, models and proofs for Wolfe type problems studied in Mishra [9]. Moreover, self-duality theorems for these pairs are obtained assuming the function involved to be skew symmetric.     

Bernstein–Nagumo conditions and solutions to nonlinear differential inequalities

For Ω$\Omega$, an open bounded subset of R^N${\mathbb{R}}^N$ with smooth boundary and 1<p<∞$1<p<\infty$, we establish W^1,p(Ω)$W^{1,p}\left(\Omega \right)$a priori bounds and prove the compactness of solution sets to differential inequalities of the form

|divA(x,∇u)|≤F(x,u,∇u),$\left|\text{div}A(x,\nabla u)\right|\le F(x,u,\nabla u)\text{,}$

which are bounded in L^∞(Ω)$L^{\infty }\left(\Omega \right)$. The main point in this work is that the nonlinear term F$F$ may depend on ∇u$\nabla u$ and may grow as fast as a power of order p$p$ in this variable. Such growth conditions have been used extensively in the study of boundary value problems for nonlinear ordinary differential equations and are known as Bernstein–Nagumo growth conditions. In addition, we use these results to establish a sub-supersolution theorem.     

A gap theorem on submanifolds with finite total curvature in spheres

We study a complete noncompact submanifold Mⁿ$M^n$ in a sphere S^n+p${\mathbb{S}}^{n+p}$. We prove that there admit no nontrivial L²$L^2$-harmonic 1-forms on M if the total curvature is bounded from above by a constant depending only on n. The gap theorem is a generalized version of Carron?s, Yun?s, Cavalcante?s and the first author?s results on submanifolds in Euclidean spaces and Seo?s result on submanifolds in hyperbolic space without the condition of minimality.     

An energy-based computational method in the analysis of the transmission of energy in a chain of coupled oscillators

In this paper we study the phenomenon of nonlinear supratransmission in a semi-infinite discrete chain of coupled oscillators described by modified sine-Gordon equations with constant external and internal damping, and subject to harmonic external driving at the end. We develop a consistent and conditionally stable finite-difference scheme in order to analyze the effect of damping in the amount of energy injected in the chain of oscillators; numerical bifurcation analyses to determine the dependence of the amplitude at which supratransmission first occurs with respect to the frequency of the driving oscillator are carried out in order to show the consequences of damping on harmonic phonon quenching and the delay of appearance of critical amplitude.     

Nonlinear maps preserving Jordan *-products

Let η$\eta$ be a non-zero scalar. In this paper, we investigate a bijective map ?$?$ between two von Neumann algebras, one of which has no central abelian projections, satisfying ?(AB+ηBA^∗)=?(A)?(B)+η?(B)?(A)^∗$?(AB+\eta B{A}^{\ast })=?\left(A\right)?\left(B\right)+\eta ?\left(B\right)?{\left(A\right)}^{\ast }$ for all A,B$A,B$ in the domain. It is showed that ?$?$ is a linear *-isomorphism if η$\eta$ is not real and ?$?$ is a sum of a linear *-isomorphism and a conjugate linear *-isomorphism if η$\eta$ is real.