Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

Rate of convergence of the mean curvature flow

We study the flow M_t of a smooth, strictly convex hypersurface by its mean curvature in ?^{n + 1}. The surface remains smooth and convex, shrinking monotonically until it disappears at a critical time T and point x^* (which is due to Huisken). This is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere Sⁿ of radius √n. In this paper we will study the rate of exponential convergence of a rescaled flow. We will present here a method that tells us that the rate of the exponential decay is at least 2/n. We can define the “arrival time” u of a smooth, strictly convex, n‐dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u(x) = t if x ∈ M_t for x ∈ Int(M₀). Huisken proved that, for n ≥ 2, u(x) is C² near x^*. The case n = 1 has been treated by Kohn and Serfaty [11]; they proved C³‐regularity of u. As a consequence of the obtained rate of convergence of the mean curvature flow, we prove that u is not necessarily C³ near x^* for n ≥ 2. We also show that the obtained rate of convergence 2/n, which arises from linearizing a mean curvature flow, is the optimal one, at least for n ≥ 2. © 2007 Wiley Periodicals, Inc.     

3-(v, 4, 1) covering designs with chromatic numbers 2 and 3

A t-(v, k, λ) covering design is a pair (X, B) where X is a v-set and B is a collection of k-sets in X, called blocks, such that every t element subset of X is contained in at least λ blocks of B. The covering number, C_λ(t, k, v), is the minimum number of blocks a t-(v, k, λ) covering design may have. The chromatic number of (X, B) is the smallest m for which there exists a map φ: X → Z_m such that ∣φ((β)∣ ≥2 for all β ∈ B, where φ(β) = {φ(x): x ∈ β}. The system (X, B) is equitably m-chromatic if there is a proper coloring φ with minimal m for which the numbers ∣φ^?1(c)∣ c ∈ Z_m differ from each other by at most 1. In this article we show that minimum, (i.e., ∣B∣ = C λ (t, k, v)) equitably 3-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0 (mod 6), v ≥ 18 for v ≥ 1, 13 (mod 36), v ≡ 13 and for all numbers v = n, n + 1, where n ≡ 4, 8, 10 (mod 12), n ≥ 16; and n = 6.5^a 13^b 17^c ?4, a + b + c > 0, and n = 14, 62. We also show that minimum, equitably 2-chromatic 3-(v, 4, 1) covering designs exist for v ≡ 0, 5, 9 (mod 12), v ≥ 0, v = 2.5^a 13^b 17^c + 1, a + b + c > 0, and v = 23. © 1993 John Wiley & Sons, Inc.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

Complexity of Finding Irreducible Components of a Semialgebraic Set

Let W ⊂ Rⁿ be a semialgebraic set defined by a quantifier-free formula with k atomic polynomials of the kind f ∈ Z[X₁, . . . , X_n] such that deg_{X1, . . . , Xn}(f) < d and the absolute values of coefficients of f are less than 2^M for some positive integers d, M. An algorithm is proposed for producing the complexification, Zariski closure, and also for finding all irreducible components of W. The running time of the algorithm is bounded from above by M^O(1)(kd)^nO(1). The procedure is applied to computing a Whitney system for a semialgebraic set and the real radical of a polynomial ideal.     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

Calculus of variations and descriptive set theory

If X is a locally compact Polish space, then LSC(X, ?) denotes the compact Polish space of lower semi‐continuous real‐valued functions on X equipped with the topology of epi‐convergence. Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ? \ {0}, then the set CV of all f ∈ LSC([α, β ] × [a, b ] × ?, ?) for which there is u ∈ C^r ([α, β ], [a, b ]) such that for any v ∈ C^r ([α, β ], [a, b ]) we have that ∫_α^β f (x, u (x), v ′(x))dx ≥ ∫_α^β f (x, v (x), v (x))dx is not Borel (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Modification of the Lyons-Sullivan Discretization of Positive Harmonic Functions

A modification of the Lyons-Sullivan discretization of positive harmonic functions on a Riemannian manifold M is proposed. This modification, depending on a choice of constants C = {C _n :n = 1,2,..}, allows for constructing measures n_x^C, x ? M\nu_x^\mathbf{C},\ x\in M, supported on a discrete subset Γ of M such that for every positive harmonic function f on M

q-Bernstein polynomials and their iterates

Let B_n( f,q;x), n=1,2,… be q-Bernstein polynomials of a function f : [0,1]→C. The polynomials B_n( f,1;x) are classical Bernstein polynomials. For q≠1 the properties of q-Bernstein polynomials differ essentially from those in the classical case. This paper deals with approximating properties of q-Bernstein polynomials in the case q>1 with respect to both n and q. Some estimates on the rate of convergence are given. In particular, it is proved that for a function f analytic in {z: |z|<q+} the rate of convergence of {B_n( f,q;x)} to f(x) in the norm of C[0,1] has the order q⁻ⁿ (versus 1/n for the classical Bernstein polynomials). Also iterates of q-Bernstein polynomials {B_n^j_n( f,q;x)}, where both n→∞ and j_n→∞, are studied. It is shown that for q(0,1) the asymptotic behavior of such iterates is quite different from the classical case. In particular, the limit does not depend on the rate of j_n→∞.     

Approximations by convolutions and antiderivatives

Let g be a given function in L ¹ = L ¹(0, 1), and let B be one of the spaces L ^p (0, 1), 1 ≤ p < ∞, or C ₀[0, 1]. We prove that the set of all convolutions f * g, f ∈ B, is dense in B if and only if g is nontrivial in an arbitrary right neighborhood of zero. Under an additional restriction on g, we prove the equivalence in B of the systems f _n * g and I f _n , where f _n ∈ L ¹, n ∈ ?, and I f = f * 1 is the antiderivative of f. As a consequence, we obtain criteria for the completeness and basis property in B of subsystems of antiderivatives of g.     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decomposition tree of a lexicographic product of binary structures

Given a set S and a positive integer k, a binary structure is a function . The set S is denoted by V(B) and the integer k is denoted by . With each subset X of V(B) associate the binary substructure B[X] of B induced by X defined by B[X](x,y)=B(x,y) for any x≠y∈X. A subset X of V(B) is a clan of B if for any x,y∈X and v∈V(B)?X, B(x,v)=B(y,v) and B(v,x)=B(v,y). A subset X of V(B) is a hyperclan of B if X is a clan of B satisfying: for every clan Y of B, if X∩Y≠0?, then X⊆Y or Y⊆X. With each binary structure B associate the family Π(B) of the maximal proper and nonempty hyperclans under inclusion of B. The decomposition tree of a binary structure B is constituted by the hyperclans X of B such that Π(B[X])≠0? and by the elements of Π(B[X]). Given binary structures B and C such that , the lexicographic product B⌊C⌋ of C by B is defined on V(B)×V(C) as follows. For any (x,y)≠(x^′,y^′)∈V(B)×V(C), B⌊C⌋((x,x^′),(y,y^′))=B(x,y) if x≠y and B⌊C⌋((x,x^′),(y,y^′))=C(x^′,y^′) if x=y. The decomposition tree of the lexicographic product B⌊C⌋ is described from the decomposition trees of B and C.     

Existence, uniqueness and stability ofC m solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f ⁿ (x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ?,G∈C ^m (J ⁿ⁺¹, ?) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.     

Centralizers of generalized derivations on multilinear polynomials in prime rings

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x ₁, …, x _n ) a noncentral multilinear polynomial over C. If δ(G(f(r ₁, …, r _n ))f(r ₁, …, r _n )) = 0 for all r ₁, …, r _n ∈ R, then f(x ₁, …, x _n )² is central-valued on R. Moreover there exists a ∈ U such that G(x) = ax for all x ∈ R and δ is an inner derivation of R such that δ(a) = 0.     

Construction of spherical 4- and 5-designs

A finite subsetX of thed-dimensional unit sphereS ^d-1 is called a sphericalt-design, if and only if $$\frac{1}{{\left| {S^{d - 1} } \right|}}\int_{S^{d - 1} } {f(x)d\omega (x)} = \frac{1}{{\left| x \right|}}\sum\limits_{x \in X} {f(x)} $$ holds for all polynomialsf(x) =f(x ₁,x ₂,...,x _d ) of degree at mostt. In 1984 Seymour and Zaslavsky proved the existence of sphericalt-designs for anyt andd, but for sufficiently large |X|. Since spherical designs can be used for numerical integration, it is of interest to give explicit constructions. Mimura gave a construction fort = 2,d ∈ ? and |X| ≥n ₂ for somen ₂ ∈ ? (n ₂ is sharp). Here we will give an explicit construction fort = 4 and 5,d ∈ ? and |X| ≥n ₄ for somen ₄ ∈ ?.     

Adjacency preserving functions on symmetric elements and linear preservers of non-zero decomposable symmetric tensors

Let A be a non-empty set and m be a positive integer. Let ≡ be the equivalence relation defined on A ^m such that (x ₁, …, x _m ) ≡ (y ₁, …, y _m ) if there exists a permutation σ on {1, …, m} such that y _σ(i) = x _i for all i. Let A ^(m) denote the set of all equivalence classes determined by ≡. Two elements X and Y in A ^(m) are said to be adjacent if (x ₁, …, x _m?1, a) ∈ X and (x ₁, …, x _m?1, b) ∈ Y for some x ₁, …, x _m?1 ∈ A and some distinct elements a, b ∈ A. We study the structure of functions from A ^(m) to B ⁽ⁿ⁾ that send adjacent elements to adjacent elements when A has at least n + 2 elements and its application to linear preservers of non-zero decomposable symmetric tensors.