A generalization of Krasnosel’skii fixed point theorem for sums of compact and contractible maps with application

The existence of a fixed point for the sum of a generalized contraction and a compact map on a closed convex bounded set is proved. The result is applied to a kind of nonlinear integral equations.     

Bézoutians and injectivity of polynomial maps

We prove that an endomorphism f of affine space is injective on rational points if its Bézoutian is constant. Similarly, f is injective at a given rational point if its reduced Bézoutian is constant. We also show that if the Jacobian determinant of f is invertible, then f is injective at a given rational point if and only if its reduced Bézoutian is constant.     

Property (β) and uniform quotient maps

In 1999, Bates, Johnson, Lindenstrauss, Preiss and Schechtman asked whether a Banach space that is a uniform quotient of ? _p , 1 < p ≠ 2 < ∞, must be isomorphic to a linear quotient of ? _p . We apply the geometric property (β) of Rolewicz to the study of uniform and Lipschitz quotient maps, and answer the above question positively for the case 1 < p < 2. We also give a necessary condition for a Banach space to have c ₀ as a uniform quotient.     

Separation of algebraic sets and the Łojasiewicz exponent of polynomial mappings

We prove a sharp regular separation condition for arbitrary projective algebraic sets. As a corollary an extension of Kollár's results on the Łojasiewicz exponent of polynomial mappings is obtained. Oblatum 9-II-1998 & 3-VI-1998 / Published online: 14 January 1999     

Injectivity of real polynomial maps and Łojasiewicz exponents at infinity

In this paper we give a lower bound for the Łojasiewicz exponent at infinity of a special class of polynomial maps , s ≥ 1. As a consequence, we detect a class of polynomial maps that are global diffeomorphisms if their Jacobian determinant never vanishes. Work supported by DGICYT Grant BFM2003–02037/MATE.     

An algebraic construction of the ring of¶piecewise polynomial functions

In this note we will prove an algebraic characterization of the piecewiese polynomial functions of a real ℚ-algebra A. This characterization is related to the realm of investigations concerning the Pierce–Birkhoff conjecture. Received: 23 June 1997 / Revised version: 26 May 1998     

Forward Expansiveness and Entropies for Subsystems of ℤ + k -actions

Acta Mathematica Sinica, English Series - In this paper, forward expansiveness and entropies of “subsystems”2) of ℤ + -actions are investigated. Let α be a ℤ + -action...     

Virtual Poincaré polynomial of the link of a real algebraic variety

The link of an affine real algebraic variety at a point is defined to be the intersection of the variety with a small sphere centred at the point. The Euler characteristic of the link leads to local topological characterisation of real algebraic varieties in law dimensions. We prove in the paper that not only the Euler characteristic, but also the stronger virtual Poincaré polynomial is well-defined for the link at a point of an affine algebraic variety.     

Convergence theorems for rows of differential and algebraic Hermite-Padé approximations

The authors investigate the asymptotic behaviour of Hermite-Padé polynomials of Latin type, which are formed from differential or algebraic expressions involving a meromorphic function. The degrees of the ‘denominator’ polynomials are chosen to correspond to ‘rows’ of the Hermite-Padé table. For special choices of ‘denominator’ degrees, these asymptotic are used to prove de Montessus type theorems for Hermite-Padé approximants formed by solving differential or algebraic equations.     

Surjectivity of Generaized Expanding Maps and Some Characterizations of Monotone Operators

In [1] L. Nirenberg stated the following problem: Suppose T is a continuous map H→H (H is a Hilbert space), which is expanding,i.e. and T(0)=0.Suppose T map a neighborhood of the origin onto a neighborhood of the origin. Does T maps H onto H? In [2] Kung-Ching Chang and Li Shujie answered positively the problem when T is differentiable.     

Rigidity for rational maps with Cantor Julia sets

In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate.     

Étale endomorphisms of algebraic surfaces with $G_m$-actions

Let X be an algebraic surface defined over the complex field and endowed with an -fibration. As such a surface we have a Platonic -fiber space, a weighted hypersurface with its singular point deleted off and, more generally, an affine algebraic surface with an unmixed -action and its fixpoint deleted off. We consider an étale endomorphism and show that is an automorphism in most cases. Of particular interest is the case of a Platonic -fiber space, for which being an automorphism is closely related to the Jacobian Problem for the affine plane . We also investigate the automorphism group of such surfaces. Received January 7, 1999 / Published online October 30, 2000     

Regularity of harmonic maps with the potential

The aim of this work is to prove the partial regularity of the harmonic maps with potential. The main difficulty caused by the potential is how to find the equation satisfied by the scaling function. Under the assumption on the potential we can obtain the equation, however, for a general potential, even if it is smooth, the partial regularity is still open.     

Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper.a coniecture on trianguation is confirmed The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented.By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method,an upper     

Regev’s and Amitsur’s conjectures for codimensions of generalized polynomial identities

Let A be a finite-dimensional associative algebra over a field of characteristic 0. Then there exist C ∈ ℚ₊ and t ∈ ℤ₊ such that gc _n (A) ∼ Cn ^t d ⁿ as n → ∞, where d = PIexp(A). In particular, Amitsur’s and Regev’s conjectures hold for the codimensions gc _n (A) of generalized polynomial identities.     

P and Q polynomial schemes with q = −1

A theorem of D. Leonard, as it appears in Bannai and Ito (Benjamin-Cummings Lecture Note Series, No. 58, Menlo Park, Ca., 1984), says the intersection numbers of distance-regular graphs with the Q-polynomial property take seven possible forms, called 1, 1A, 2, 2A, 2B, 2C, and 3. In this paper we show the known list of examples of type 3 is complete.     

Liénard systems for quadratic systems with invariant algebraic curves

We study the character of the friction function f(x) and the restoring force g(x) in the Liénard system to which a quadratic system with an invariant second-order algebraic curve (an ellipse that is a limit cycle, a hyperbola defining two separatrix cycles, or a parabola) or fourth-order algebraic curve with an oval being a limit cycle can be reduced. Invariant curves are constructed for quadratic systems in a five-parameter canonical family, which can readily be reduced to Liénard systems.     

The Hodge–Poincaré polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group G and such that ${X^{ss} \neq X_{(0)}^{s}\neq \emptyset}$ . We give formulae for the Hodge–Poincaré series of the quotient ${X_{(0)}^{s}/G}$ . We use these computations to obtain the corresponding formulae for the Hodge–Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.