Let f (m) be an irreducible quadratic polynomial with integral coefficients and positive leading coefficient. Under the assumption of Extended Riemann Hypothesis, we obtain new remainder terms in the upper bounds on primes represented by f(m) or f(p) which greatly improve Bantle's recent results. As an application, we obtain, in the second part of the paper, a new result on the lower bound of the least primes in arithmetic progressions with some difference.
We first study the Grassmannian manifold Gn (Rn+p)as a submanifold in Euclidean space Λn (Rn+p). Then we give a local expression for each map from Riemannian manifold M toGn (Rn+p) ⊂Λn (Rn+p), and use the local expression to establish a formula which is satisfied by any harmonic map from M to Gn (Rn+p). As a consequence of this formula we get a rigidity theorem.
In this paper, we consider the following question. For some given n-bordism class α and (n-ki)-bordism classes βn-ki,(1<k1<...<ki),. under what condition can we find a representation M of α and an involution T on M, such that the bordism class of the fixed point set of T is . This paper also gives some examples not satisfying the above property.
In this paper we shall give a new proof of the well-known theorem of Faith-Utumi [1]. Using our method we can show that every right order of Kn is a prime right Goldie ring, where Kn is the n×n-matrix ring over division ring K. Specially,Dn is a prime right Goldie ring, if D is a right order of K.
In this paper, we continue the discussion of the conjecture which says that infinitesimal Ⅱ-isometry of surface is infinitesimal Ⅰ-isometry, i.e., infinitesimally rigid. We have some invariants by means of which some integral formulas are worked out. As an application to these integral formulas, we get some results on infinitesimal Ⅱ-isometry of closed surface. The theorems proved are just more or less obvious generalizations of known results.
This paper is to give the necessary and sufficient conditions for a square integrable martingale array to converge stably, which improves the results given by Jeganathan, Brown & Eagleson and others.
The complete convergence theorem implies that starting from any initial distribution the one dimensional contact process converges to a limit as t→∞. In this paper we give a necessary and sufficient condition on the initial distribution for the convergence to occur with exponential rapidity.
In this paper, we study the Hankel operators belonging to E which denotes all of the essential Toeplitz operators, and show that the class of symbols of these Hankel operators is a Douglas algebra. Meanwhile, using the concept of essential commutant and some knowledge of the theory of function algebras, we solve a problem raised by Jos? Barria and P. R. Halmos in[1].
In this paper, we study the free boundary problem for degenerate parabolic equations (1.1)-(1.4). The existence of generalized solutions in BV1, 1/2 is obtained by the means of parabolic regularization under certain restrictions. The uniqueness and regularity of generalized solutions are also discussed. In addition, a C1+α smoothness for the free boundary is obtained in the parabolic case.
