A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C ₁, C ₂, C ₃ be three symmetric matrices in ℜ ^n×n , if max{x ^T C ₁ x,x ^T C ₂ x,x ^T C ₃ x}≥0 for all x∈ℜ ⁿ , it is proved that there exist t _i ≥0 (i=1,2,3) such that ∑ _i=1 ³ t _i =1 and ∑ _i=1 ³ t _i C _i has at most one negative eigenvalue. Received February 18, 1997 / Revised version received October 1, 1997? Published online June 11, 1999     

A note on Pontrjagin forms

Let be a principal bundle over a manifold of dimension , and let be its -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class can be realised as the Pontrjagin form of some connection on ?

     

A note on biquadratic forms

It is shown that semidefinite quadratic forms in two by n variables are sums of squares of bilinear forms.     

A note on moment inequality for quadratic forms

Moment inequality for quadratic forms of random vectors is of particular interest in covariance matrix testing and estimation problems. In this paper, we prove a Rosenthal-type inequality, which exhibits new features and certain improvement beyond the unstructured Rosenthal inequality of quadratic forms when dimension of the vectors increases without bound. Applications to test the block diagonal structures and detect the sparsity in the high-dimensional covariance matrix are presented.     

A note on Siegel modular forms

This paper gives a new identification for Siegel modular forms with respect to any congruence subgroup by investigating the properties of their Fourier-Jacobi expansions, and verifies a comparison theorem for the dimensions of the spaces Skn (Γn) and J0k, 1 (Γn) with small weight k. These results can be used to estimate the dimension of the space of modular forms.     

A note on the torsion of differential forms

Archiv der Mathematik -     

A note on fan's identity for symmetric bilinear forms

The identity recently discovered by Ky Fan is shown to be a combination of shorter identities expressing determinants that vanish.     

A note on tight extensive game forms

Associated with each extensive game form are the α- and β-effectivity functions,E _α andE _β. These functions characterize the effective capabilities which coalitions command under pessimistic and optimistic cooperative behavior respectively. A game form is said to be tight whenE _α ≡E _β, in which case pessimistic, optimistic, and intermediate types of behavior all support the same effective power. Dalkey's condition for Nash consistency, essentially perfect information, is found to be a necessary and sufficient condition for a game form to be tight.     

A note on the Drazin inverses with Banachiewicz-Schur forms

Let H be a Hilbert space, M the closed subspace of H with orthocomplement M^⊥. According to the orthogonal decomposition H=M⊕M^⊥, every operator M∈B(H) can be written in a block-form . In this note, we give necessary and sufficient conditions for a partitioned operator matrix M to have the Drazin inverse with Banachiewicz-Schur form. In addition, this paper investigates the relations among the Drazin inverse, the Moore-Penrose inverse and the group inverse when they can be expressed in the Banachiewicz-Schur forms.     

A note on stronger forms of sensitivity for dynamical systems

Let (X,d) be a compact metric space and (κ(X),d_H) be the space of all non-empty compact subsets of X equipped with the Hausdorff metric d_H. The dynamical system (X,f) induces another dynamical system $(\kappa (X)\text{,}\overline{f})$, where f:X → X is a continuous map and $\overline{f}:\kappa (X)\to \kappa (X)$ is defined by $\overline{f}(A)=\{f(a):a\in A\}$ for any A ∈ κ(X). In this paper, we introduce the notion of ergodic sensitivity which is a stronger form of sensitivity, and present some sufficient conditions for a dynamical system (X,f) to be ergodically sensitive. Also, it is shown that $\overline{f}$ is syndetically sensitive (resp. multi-sensitive) if and only if f is syndetically sensitive (resp. multi-sensitive). As applications of our results, several examples are given. In particular, it is shown that if a continuous map of a compact metric space is chaotic in the sense of Devaney, then it is ergodically sensitive. Our results improve and extend some existing ones.     

A note on harmonic forms on complete manifolds

In this note, we will prove that under certain conditions, the space of polynomial growth harmonic functions and harmonic forms with a fixed growth rate on manifolds which are asymptotically nonnegatively curved is finite dimensional. This is a partial generalization of the works of Li and Colding-Minicozzi. We will also give an explicit estimate for the dimension in case the manifold is a complete surface of finite total curvature. This is a generalization to harmonic forms of the work of Li and the author.

     

A note on the norm-continuity for evolution families arising from non-autonomous forms

We consider evolution equations of the form $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T],\ \ u(0)=u_0, \end{aligned}$$where $${\mathcal {A}}(t),\ t\in [0,T],$$ are associated with a non-autonomous sesquilinear form $${\mathfrak {a}}(t,\cdot ,\cdot )$$ on a Hilbert space H with constant domain $$V\subset H.$$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces V, H and on the dual space $$V'$$ of V. The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.