The rigidity of Clifford torusS^1 \left( {\sqrt {\frac{1}{n}} } \right) \times S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)

In this paper, we prove that ifM is ann-dimensional closed minimal hypersurface with two distinct principal curvatures of a unit sphereS ⁿ⁺¹ (1), thenS=n andM is a Clifford torus ifn≤S≤n+[2n ²(n+4)/3(n(n+4)+4)], whereS is the squared norm of the second fundamental form ofM.     

Even unimodular 8-dimensional quadratic forms over\mathbb{Q}\left( {\sqrt 2 } \right)

Dedicated to Professor O. T. O'Meara on his 60th birthday     

On the diophantine equation F $$\left( {\left( {_n^x } \right)} \right) = b\left( {_m^y } \right)$$

As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions.     

On quadratic andO\left( {\sqrt {nL} } \right) convergence of a predictor—corrector algorithm for LCP

Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor—corrector algorithm for linear programming (LP) exhibits asymptotic quadratic convergence of the primal—dual gap to zero, without any assumptions concerning nondegeneracy, or the convergence of the iteration sequence. In this paper we prove a similar result for the monotone linear complementarity problem (LCP), assuming only that a strictly complementary solution exists. We also show by example that the existence of a strictly complementarity solution appears to be necessary to achieve superlinear convergence for the algorithm.Research supported in part by NSF Grants DDM-8922636 and DDM-9207347, and an Interdisciplinary Research Grant of the University of Iowa, Iowa Center for Advanced Studies.     

The Isometric Extension of an Into Mapping from the Unit Sphere S（t（2）^∞） to S（L^1（μ））

This is the first paper to consider the isometric extension problem of an into-mapping between the unit spheres of two different types of spaces. We prove that, under some conditions, an into-isometric mapping from the unit sphere S（t（2）^∞） to S（L^1（μ） can be （real） linearly isometrically extended.     

Approximations of some logarithms by numbers from the fields \mathbb{Q} and \mathbb{Q}\left( {\sqrt {d} } \right)

Estimates for approximations to logarithms of rational numbers by rational numbers and quadratic irrationalities are established.     

Some remarks on the moments of \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| in short intervals

Some new results on power moments of the integral $$ J_k (t,G) = \frac{1} {{\sqrt {\pi G} }}\int_{ - \infty }^\infty { \left| {\varsigma \left( {\tfrac{1} {2} + it + iu} \right)} \right|^{2k} } e^{ - (u/G)^2 } du $$ (t ? T, T ^? ≦ G ? T, κ ∈ N) are obtained when κ = 1. These results can be used to derive bounds for moments of $ \left| {\varsigma \left( {\tfrac{1} {2} + it} \right)} \right| $ .     

Effect of a random real on\kappa \to \left( {\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right)} \right)^2

It is consistent that $\kappa \to (\kappa ,{\text{ }}\left( {_{\omega _1 }^\alpha } \right))^2 $ holds in the random extension.     

On the Evaluation of the Integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$

The integral $$\int_0^{{\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-\nulldelimiterspace} 4}} {\ln \left( {\cos ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta \pm \sin ^{{m \mathord{\left/ {\vphantom {m n}} \right. \kern-\nulldelimiterspace} n}} \theta } \right)d\theta } $$ where m and n are relatively prime positive integers, is evaluated exactly in terms of elementary functions and the Catalan constant G.     

An interior point algorithm ofO(\sqrt m \left| {\ln \varepsilon } \right|) iterations forC 1-convex programming

We present a theoretical result on a path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial nonsmooth Newton step in each iteration, and converges to a point whose cost is within accuracy of the optimal cost in iterations, wherem is the number of constraints in the problem. Unlike other interior point methods, the analyzed algorithm only requires a first-order Lipschitzian condition and a generalized Hessian similarity condition on the objective and constraint functions. Therefore, our result indicates the theoretical feasibility of applying interior point methods to certainC ¹-optimization problems instead ofC ²-problems. Since the complexity bound is unchanged compared with similar algorithms forC ²-convex programming, the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods.This author's work is supported in part by the National Science Foundation of the USA under grant DDM-8721709.This author's work is supported in part by the Australian Research Council.     

On the sum of two integral squares in the imaginary quadratic field \mathbb{Q}\left( {\sqrt { - 2p} } \right)

We determine the sum of two integral squares over imaginary quadratic fields Q(√-2p),where p≡1 mod 8 is a prime satisfying 2p=r2+s2with r,s≡±3 mod 8.     

Factorization of the universalR-matrix forU_q \left( {\hat sl_2 } \right)^1

The factorization of the universal R-matrix corresponding to the so-called Drinfeld Hopf structure is described in the example of the quantum affine algebra . As a result of the factorization procedure, we deduce certain differential equations on the factors of the universal R-matrix that allow uniquely constructing these factors in the integral form. This article was written at the request of the Editorial Board Translated from Teoreticheskava i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 179–214, August, 2000     

A note on the congruence \left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right) (mod p r )

In the paper we discuss the following type congruences: $$\left( {_{mp^k }^{np^k } } \right) \equiv \left( {_m^n } \right)(\bmod p^r ),$$ where p is a prime, n, m, k and r are various positive integers with n ? m ? 1, k ? 1 and r ? 1. Given positive integers k and r, denote by W(k, r) the set of all primes p such that the above congruence holds for every pair of integers n ? m ? 1. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets W(k, r) and inclusion relations between them for various values k and r. In particular, we prove that W(k + i, r) = W(k ? 1, r) for all k ? 2, i ? 0 and 3 ? r ? 3k, and W(k, r) = W(1, r) for all 3 ? r ? 6 and k ? 2. We also noticed that some of these properties may be used for computational purposes related to congruences given above.     

Central limit theorem: Covergence in the norm\left\| u \right\|\left( {\int\limits_{ - \infty }^\infty {u^2 \left( x \right)e^{\frac{{x^2 }}{2}} dx} } \right)^{{\raise0.7ex\hbox{

Sufficient conditions for convergence in the central limit theorem (for identically distributed random variables) with respect to the topology specified in the title are given. These conditions hold for the uniform distribution although there exist distributions with smooth densities concentrated on a bounded interval for which the convergence result does not hold.     

An analogue of the Moutard transformation for the Goursat equation\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y }

We present a new Bäcklund-type transformation for the nonlinear equation $\vartheta _{xy} = 2\sqrt {\lambda \left( {x,y} \right)\vartheta _x \vartheta _y } $ studied by É. Goursat. Goursat found a linearization transformation and some properties of this equation, which make it similar to the Moutard equation u_xy=M(x, y)u. However, this Goursat transformation does not provide proper superposition formulas. We give the necessary extended superposition formulas.