Khovanskii–Rolle Continuation for Real Solutions

We present a new continuation algorithm to find all real solutions to a nondegenerate system of polynomial equations. Unlike homotopy methods, the algorithm is not based on a deformation of the system; instead, it traces real curves connecting the solutions to one system of equations to those of another, eventually leading to the desired real solutions. It also differs from homotopy methods in that it follows only real paths and computes no complex solutions to the original equations. The number of curves traced is essentially bounded above by the fewnomial bound for real solutions, and the method takes advantage of any slack in that bound.     

Which Real Lie Groups are Reductive?

The present author introduced in [4] the following version of reductivity: a real Lie group G with reductive Lie algebra g should be called reductive if its each irreducible continuous unitary representation is admissible. Furthermore, he showed there that his really reductive groups are reductive. Here we show the converse: each reductive real Lie group is really reductive, i.e., contains an open normal subgroup G1 of finite index such that G1 = Z(G1)G0, where G0 is the connected component of G containing 1, and Z(G1) is the center of G1.     

Tangent and Normal Cones to Real Surfaces

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Complete Totally Real Submanifolds with Parallel Mean Curvature

i. Introduction A submanifold M in a Kaehler manifold M is said to be totally real if every tangent space of M is mapped into its normal space by the complex structure of M. Some fundamental properties of totally real submanifolds can be found in [1], [2]. Let σ be the second fundamental form of M. The mean curvature η of M is defined by η=tr σ , and M is called a submanifold with     

Real normalized differentials and Arbarello’s conjecture

Using meromorphic differentials with real periods, we prove Arbarello’s conjecture that any compact complex cycle of dimension g - n in the moduli space M _g of smooth algebraic curves of genus g must intersect the locus of curves having a Weierstrass point of order at most n.     

Totally Real Minimal Submanifolds in a Quaternion Projective Space

51.IntroductionAquaternionKaehlermanifoldisdefinedasa4m-dimensionalRiemannianmanifoldwhoseholonomygroupiscontainedinSp(m).Sp(1)withtheadditionalconditionform=lthatitisaself-dualEinsteinspace.AquaternionprojectivespaceQP:)isaquaternionKeahlermanifoldwithconstantquaternionsectionalcurvaturec>O.AcomplexprojectivespaceCP2)withconstantholomorphicsectionalcurvatureccanbeisometricallyimbeddedinQP:)asatthtallygeodesicsubmanifold.LetMbeannsidimensionalRiemannianmanifoldander:M-Qer).-.'an1sometr1…     

Construction of Real Band Anti-Symmetric Matrices from Spectral Data

In this paper,we describe how to construct a real anti-symmetric(2p-1)-band matrix with prescribed eigenvalues in its ρ leading principal submatrices.This is done in two steps.First,an anti-symmetric matrix B is constructed with the specified spectral data but not necessary a band matrix.Then B is transformed by Householder transformations to a (2ρ-1)-band matrix with the prescribed eigenvalues.An algorithm is presented.Numerical results are presented to demonstrate that the proposed method is effective.     

The Heat Kernels of the Real Second Classical Domains

L3t[1]istherealsecondclassicaldomains.ThemapdefinedbyW~(I T)--'(I--T)carriesRillontotheunboundeddomainWhichisinvariantfinderthelineartransformationW=ATA',whereAisannxnupertrianglarmatrixwithpositivediagonalelements.SuchtransformationsformaLiegroupRill(n),whichisequivalenttoRillunderthemapdefinedbyW=AA'.TheAiscalledthehoro-hypercirclecoordinateofRRII.Itiswell-knownthatinRillthereisanillvariantmetricdsZ=tr[(I--KW)--'dW]'.InRHI,,whichisequaltothemetricdsZ=fir[(W--'dW)'].Inthehoro-h…     

Real Lightlike Hypersurfaces of Paraquaternionic Kähler Manifolds

The main purpose of this paper is to give basic properties of real lightlike hypersurfaces of paraquaternionic manifold and to prove the nonexistence of real lightlike hypersurfaces in paraquaternionic space forms under some conditions. Dedicated to the memory of Professor Aldo Cossu     

On the Convergence of ω1 Sequences of Real Functions

We study sets of points at which ω₁ sequences of real functions from a given class F converge. As F we consider continuous functions, first class of Baire, Borel measurable functions, functions with Baire property and Lebesgue measurable functions. Connections of those problem with additional set-theoretic axioms are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Remark on Concircularly Flat Totally Real Minimal Submanifolds in CP^n

Let M be a concircularly fiat totally real minimal submanifold in CP4. The infimum Vm of the volume V （M） of M is obtained, also the necessary and sufficient conditions of ＂V（M）=Vm＂ is given.     

Fréchet-Valued Real Analytic Functions on Fréchet Spaces

The aim of this paper is to establish the equivalence between the real and topologically real analyticity of Fréchet-valued functions on Fréchet spaces. This is an extension of recent results of Bonet and Domanski [1, 2] to infinite dimension.Received February 19, 2002; in revised form June 25, 2002 Published online May 16, 2003     

Real analytic curves in Fréchet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT u=f which depends holomorphically on the parameterz wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.     

2—Harmonic Totally Real Submanifolds in a Complex Projective Space

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Real polynomial iterative roots in the case of nonmonotonicity height ≥ 2

It is known that a strictly piecewise monotone function with nonmonotonicity height ≥ 2 on a compact interval has no iterative roots of order greater than the number of forts. An open question is: Does it have iterative roots of order less than or equal to the number of forts? An answer was given recently in the case of "equal to". Since many theories of resultant and algebraic varieties can be applied to computation of polynomials, a special class of strictly piecewise monotone functions, in this paper we investigate the question in the case of "less than" for polynomials. For this purpose we extend the question from a compact interval to the whole real line and give a procedure of computation for real polynomial iterative roots. Applying the procedure together with the theory of discriminants, we find all real quartic polynomials of non-monotonicity height 2 which have quadratic polynomial iterative roots of order 2 and answer the question.     

Indivisibility of Class Numbers and Iwasawa λ-Invariants of Real Quadratic Fields

Let D>0 be the fundamental discriminant of a real quadratic field, and h(D) its class number. In this paper, by refining Ono's idea, we show that for any prime p>3, {0<D<X|h(D)0(mod p)}>> _p (X)/logX.     

Real solutions of the equation ϕ(X)=1/nJ n

For ann x n real matrixX, let ?(X)=X ο (X ^?1) ^T , where ο stands for the Hadamard (entrywise) product. SupposeA, B, C andD aren x n real nonsingular matrices, and among them there are at least one solutions to the equation ?(X)=1/nJ _n . An equivalent condition which enable $M = \left( {\begin{array}{*{20}c} A & B \\ C & D \\ \end{array} } \right)$ become a real solution to the equation ?(X)=1/2nJ _2n , is given. As applications, we get new real solutions to the matrix equation ?(X)-1/2nJ _2n by applying the results of Zhang, Yang and Cao [SIAM. J. Matrix Anal. Appl, 21 (1999), pp: 642–645] and Chen [SIAM. J. Matrix Anal. Appl, 22 (2001), pp:965–970]. At the same time, all solutions of the matrix equation ?(X)=1/4J ₄ are also given.