The background for the introduction of Kadison-Singer algebras (KSalgebras, for short) is reviewed. Definitions and basics properties are explained. Studies on hyperfinite KS-algebras, non-hyperfinite case, KS-lattices and strong KS-algebras are described in details, as well as their connections with classical open problems such as the invariant subspace problem, Kadison's transitive algebra problem, von Neumann algebra generator problem. Operations on non-selfadjoint operator algebras are also discussed and two new operations are included in the discussion. Sixteen open problems are listed with some explanations in the end.
We are concerned with a class of Kirchhoff problem
where a, b > 0 are constants. The existence of ground state solutions, i.e., nontrivial solutions with least possible energy of this Kirchhoff problem is obtained. Moreover, when Q ≡ 1, under suitable conditions on h(x), we prove the existence of ground state solutions for the the following Kirchhoff problem
.
We give a geometric definition of sub-orbifold groupoid, and a criterion for determining a sub-orbifold groupoid. Then we prove the existence of orbifold tubular neighborhoods of compact sub-orbifold groupoids, the existence of symplectic neighborhoods of compact symplectic sub-orbifold groupoids in symplectic orbifold groupoids, and, the existence of Lagrangian neighborhoods of compact Lagrangian sub-orbifold groupoids.
We are interested in considering the following nonlocal critical problem
where Ω ⊂ R4 is a smooth bounded domain with 0 ∈ Ω, a ≥ 0, b, λ, μ > 0, 1 < q < 2, 0 < β < 2. By using the variational method, some existence and multiplicity results are obtained.
Computing height of piecewise monotonic functions is difficult because the value of functions may interact each other under iteration. In this paper we consider the set of continuous functions with a single non-monotonic point. We first present a sufficient and necessary condition for heights which gives a classification of those functions. Then we provide an equivalent condition and a new construction method for topological conjugacy for a nonempty subset of the mentioned continuous functions. Furthermore, we prove that topological conjugacy is a sufficient but not necessary condition for equal heights of piecewise monotonic functions. Finally, some examples are given to illustrate our results.
Convergence for identically distributed negatively associated (NA) random variables and independent and identically distributed (i.i.d.) random variables are studied under general moment conditions, the extension of the Baum-Katz Theorem and strong law of large numbers are obtained. The theorems extend the related known works in the literature.
Ye etc introduced the Lp-mixed geominimal surface areas of multiple convex bodies for any real p(p≠-n). In this paper, we define the Lp-dual mixed geominimal surface areas of multiple star bodies for any real p(p≠n), and establish some inequalities related to this concept.
Sampling in shift-invariant subspaces of Lp is commonly studied under the requirement that the generator is in a Wiener amalgam space independent of p, but such condition is too strong because it does not allow us to control p. In this paper, we mainly discuss the nonuniform average sampling and reconstruction of signals in a non-decaying shift-invariant space under the assumption that the generator is in a hybrid-norm space. The new condition is weaker than the Wiener amalgam space and depends on the parameter p. Based on some lemmas in hybrid-norm spaces, we first give the sampling stability for two kinds of average sampling functionals, and then the corresponding iterative reconstruction algorithms with exponential convergence are established.
We present the canonical forms of coupled self-adjoint boundary conditions for regular differential operators of order four by using new methods. In this new canonical forms, the four small block matrices are symmetric, and the determinant has absolute value equal to 1. This fourth order form is quite similar to the new second order one, and it provides a new way to give the canonical forms of the self-adjoint boundary condition for general high-order differential operators.
Assume TΩ is the Calderón-Zygmund singular integral operator with rough kernel and I is the closed arc of unit circle that I ? S1. In this paper, we prove that if Ω is supported in I and monotonous on I, then TΩ is bounded from Hardy space H1(R2) to L1(R2) if and only if||Ω||L log L < ∞.
The limiting behaviors for controlled branching processes in random environments are discussed. Under the normalization factor {Sn:n ∈ N}, the normalization processes {?n:n ∈ N} of are studied, and the sufficient conditions of {?n:n ∈ N} a.s., L1 and L2 convergence are given; A sufficient condition and a necessary condition for convergence to a non-degenerate at 0 random variable are got; Under the normalization factor {In:n ∈ N}, the normalization processes {Wn:n ∈ N} are discussed, and the sufficient conditions of {Wn:n ∈ N} a.s., and L1 convergence are obtained.
We study the stochastic analysis of the measure determined by fractional diffusion process (fractional diffusion measure). We first give the pull back formula by Bismut method, then establish the integration by parts formula for fractional diffusion measure. Finally, we generalize the classic Clark-Ocone theorem to martingale representation theorem under fractional diffusion measure.
We apply the coupling method in inhomogeneous Markov processes, and generalize the fundamental coupling theorem of homogeneous Markov processes, which provides a theoretical basis for the further study of the coupling methord for inhomogeneous Markov processes.
We define the abscissa of convergence σc, the uniform abscissa of convergence σu and the absolute abscissa of convergence σa on Dirichlet series Σn=0∞ aneλns,use the relationship between the index λn and the coefficients an to estimate the three convergence abscissas, and obtain that the convergence conditions of two Dirichlet series Σn=0∞ aneλns and Σn=0∞ ane-λns are the same.
