Covariance operator estimation of a functional autoregressive process with random coefficients

We deal with the covariance and cross covariance operators estimation of a Hilbert space valued autoregressive process with random coefficients. We establish bounds for empirical estimators in mean square error and almost sure convergence in Hilbert–Schmidt norm. Consistent estimators of the eigenvalues are also derived.     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

Frequency domain Walsh functions and sequences: An introduction

In this letter, a new set of orthogonal band-limited basis functions is introduced. This set of basis functions is derived from the inverse Fourier transform of the frequency domain Walsh functions. The Fourier transforms of the Walsh functions were calculated by Siemens and Kitai in 1973 but they have been overlooked in the literature. Some of the properties of these functions are studied in this paper. Moreover, the orthogonal discrete version of these functions is obtained by truncation, sampling and orthogonalization utilizing the orthogonal Procrustes problem.     

Short note on Hilbert’s inequality

Considering five different parameters, we obtain some new Hilbert-type integral inequalities for functions f(x), g(x) in L²[0, ∞). Then, we extract from our results some special cases which have been proved before.     

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert <Emphasis Type="Italic">C</Emphasis>*-module <Emphasis Type="Italic">l</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">A</Emphasis>)

We characterize A-linear symmetric and contraction module operator semigroup{Tt}t∈R+L(l2(A)),where A is a finite-dimensional C-algebra,and L(l2(A))is the C-algebra of all adjointable module maps on l2(A).Next,we introduce the concept of operator-valued quadratic forms,and give a one to one correspondence between the set of non-positive definite self-adjoint regular module operators on l2(A)and the set of non-negative densely defined A-valued quadratic forms.In the end,we obtain that a real and strongly continuous symmetric semigroup{Tt}t∈R+L(l2(A))being Markovian if and only if the associated closed densely defined A-valued quadratic form is a Dirichlet form.     

Hilbert function spaces and the Nevanlinna–Pick problem on the polydisc II

In [19], a geometric procedure for constructing a Nevanlinna–Pick problem on Dⁿ${\mathbb{D}}^n$ with a specified set of uniqueness was established. In this sequel we conjecture a necessary and a sufficient condition for a Nevanlinna–Pick problem on D²${\mathbb{D}}^2$ to have a unique solution. We use the results of [19] and Bezout?s theorem to establish three special cases of this conjecture.     

New lower bounds for the Hilbert number of polynomial systems of Liénard type

In this paper, we study the maximal number of limit cycles of some kinds of polynomial Liénard systems with arbitrary degree and obtain some new lower bounds for the Hilbert number of the systems, which improve truly the certain existing results.     

Hilbert空间中的不可微最优化问题的基路径增量目标水平算法

主要研究了Hilbert空间中的一类不可微最优化问题的基路径增量目标水平算法.证明了当约束集合为有界集时,问题的最优解集非空,且这时通过算法生成的迭代点列是弱收敛于最优解的..     

拟绝对-*-κ-仿正规算子的谱性质

引入了拟绝对-*-k-仿正规算子,获得了拟绝对-*-k-仿正规算子的一个充要条件.并证明了拟绝对-*-k-仿正规算子在0≤k≤1上是有限上升的,作为此性质的应用,证明了若T是拟绝对-*-k-仿正规算子,其中0≤k≤1,则Weyl谱和本质近似点谱的谱映射定理成立.最后证明了若T是拟绝对-*-k-仿正规算子,其中0≤k≤1,则σ_(ja)(T)\{0}=σ_a(T)\{0}.     

LENTICULAR NONCOMMUTATIVE TORI

All C*-algebras of sections of locally trivial C* -algebra bundles over ∏i=1sLki(ni) with fibres Aw Mc(C) are constructed, under the assumption that every completely irrational noncommutative torus Aw is realized as an inductive limit of circle algebras, where Lki (ni) are lens spaces. Let Lcd be a cd-homogeneous C*-algebra over whose cd-homogeneous C*-subalgebra restricted to the subspace Tr × T2 is realized as C(Tr) A1/d Mc(C), and of which no non-trivial matrix algebra can be factored out.The lenticular noncommutative torus Lpcd is defined by twisting in by a totally skew multiplier p on Tr+2 × Zm-2. It is shown that is isomorphic to if and only if the set of prime factors of cd is a subset of the set of prime factors of p, and that Lpcd is not stablyisomorphic to if the cd-homogeneous C*-subalgebra of Lpcd restricted to some subspace LkiLki (ni) is realized as the crossed product by the obvious non-trivial action of Zki on a cd/ki-homogeneous C*-algebra over S2ni+1 for ki an integer greater than