In this paper we study the construction problem of spherical $t$-designs as a generalization of the Chebyshev problem of Gauss type quadrature formula. It is known that spherical $t$-designs can be constructed if we can construct interval $t$-designs for the weight function ${(1-x)}^{\frac{d-2}{2}}$. We shall show the exsitence and non-existence theorems of some interval $t$-desings.

The problem given by M. Matsumoto in his paper [10] is that whether there exist conformally equivalent Berwald, or locally Minkowski manifolds. In this paper we are interested in case of positive definite Berwald manifolds of dimension $n\ge 3$ solving the problem under a further condition: we shall suppose that one, and therefore all indicatrices have positive curvature. Then the conformal change must be homothetic unless the Berwald manifolds are Riemannian.

A group $G$ is said to be $(l,m,n)$-generated if it is a quotient group of the triangle group $T(l,m,n)=\langle x,y,z|x^l=y^m=z^n=xyz=1\rangle$. In 1993 J. Moori posed the question of finding all triples $(l,m,n)$ such that a given non-abelian finite simple group is $(l,m,n)$-generated. In this paper we partially answer this question for the Thompson group $Th$. In fact we study $(p,q,r)$-generation, where $p$, $q$ and $r$ are distinct primes, and $nX$-complementary generations of the Thompson group $Th$.

In this paper we continue the study of the $L^p$ mapping properties of singular integrals along surfaces of revolution which was initiated in [KWWZ] and obtain improvements over existing results by allowing kernels to be in certain block spaces.

Let $\{X^j=(X_t^j,{ℱ}_t),j\ge 1\}$ be a sequence of continuous local martingales and $\{\langle X^j\rangle \}$ the corresponding sequence of their quadratic variation processes and let $H_n(x,y),n=1,2,\dots$ be the Hermite polynomials with parametric variable $y$.

In this paper, we consider the series ${\displaystyle \sum _{j=1}^{\infty }{H}_n^2}(X^j,\langle X^j\rangle )$ of the continuous local martingales

$$H_n(X^j,\langle X^j\rangle )={(H_n(X_t^j,{\langle X^j\rangle }_t),{ℱ}_t)}_{t\ge 0},j=1,2,\dots ,$$

and its discrete analogue, and obtain some maximal inequalities.

In this paper, we consider the shifted barrier KKT conditions for nonlinear optimization. We propose a primal-dual interior point method based on these conditions. By choosing suitable parameters used in our method, we prove local and $q$-quadratic convergence of the Newton interior point method, and local and $q$-superlinear convergence of the quasi-Newton interior point method.

We will show that if symplectomorphisms on ${ℝ}^{2n}$ admit the generating function with the integrability condition, then these symplectomorphisms are Hamiltonian maps. This is an extension of results of J.Moser in [M].