Generalizations of methods of best approximation to convex optimization in locally convex spaces. II: Hyperplane theorems

We show that for a continuous convex functional f on a locally convex space E and a convex subset G of E such that inf f(E) < inf f(G), the problems of computing inf f(G) and of characterizing the elements g₀?G with f(g₀) = inf f(G) can be reduced to the same problems for a suitable hyperplane H₀ in E.     

Value distribution and shared sets of differences of meromorphic functions

We investigate value distribution and uniqueness problems of difference polynomials of meromorphic functions. In particular, we show that for a finite order transcendental meromorphic function f with λ(1/f)<ρ(f) and a non-zero complex constant c, if n?2, then fn(z)f(z+c) assumes every non-zero value a∈C infinitely often. This research also shows that there exist two sets S₁ with 9 (resp. 5) elements and S₂ with 1 element, such that for a finite order nonconstant meromorphic (resp. entire) function f and a non-zero complex constant c, Ef(z)(Sj)=Ef(z+c)(Sj)(j=1,2) imply f(z)≡f(z+c). This gives an answer to a question of Gross concerning a finite order meromorphic function f and its shift.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Small Box-Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box-Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2_III³⁻¹ designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.     

Lower negative decision number in a graph

Let G=(V,E) be a graph. A function f:V(G)→{?1,1} is called bad if ∑ _v∈N(v) f(v)≤1 for every v∈V(G). A bad function f of a graph G is maximal if there exists no bad function g such that g≠f and g(v)≥f(v) for every v∈V. The minimum of the values of ∑ _v∈V f(v), taken over all maximal bad functions f, is called the lower negative decision number and is denoted by β _D ^* (G). In this paper, we present sharp lower bounds on this number for regular graphs and nearly regular graphs, and we also characterize the graphs attaining those bounds.     

Wavelet estimations for density derivatives

Donoho et al. in 1996 have made almost perfect achievements in wavelet estimation for a density function f in Besov spaces Bsr,q(R). Motivated by their work, we define new linear and nonlinear wavelet estimators flin,nm, fnonn,m for density derivatives f(m). It turns out that the linear estimation E(‖flinn,m-f(m)‖p) for f(m) ∈ Bsr,q(R) attains the optimal when r≥ p, and the nonlinear one E(‖fnonn,m-f(m)‖p) does the same if r≤p/2(s+m)+1 . In addition, our method is applied to Sobolev spaces with non-negative integer exponents as well.     

On the distinguished character of the function spaces of holomorphic mappings of bounded type

Given two complex normed spaces E and F, F complete, and a balanced open subset U of E, we prove that the space $\text{H}$_(b(U, F) of the holomorphic mappings f: U → F of bounded type, endowed with its natural topology τ_b, is a distinguished quasi-normable Fréchet space, which is not a Schwartz space unless dim E < ∞ and dim F < ∞.     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

Graph-theoretical methods in general function theoryThéorie des graphes dans la théorie generale des fonctions

Consider two maps f and g from a set E into a set F such that f(x)≠g(x) for every x in E. What is the maximal cardinal of a subset A of E such that the images of the restriction of f and g to A are disjoint? Mekler, Pelletier and Taylor have shown that it is card(E) when the set E is infinite; in the finite case, we have proved that it is greater than or equal to card(E)/4. In this paper, using graph theoretical technics, we find these results as a direct application of a lemma of Erdös. Moreover, we show that if $\text{E=F=}\text{R}$, then there exists a countable partition {E_n}_n?1 of $\text{R}$ such that f(E_n)∩g(E_n)=φ, for every n?1. To cite this article: A. El Sahili, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 859–861.     

Optimum designs when the observations are second-order processes

Let the process {Y(x,t) : t?T} be observable for each x in some compact set X. Assume that Y(x, t) = θ₀f₀(x)(t) + … + θ_kf_k(x)(t) + N(t) where f_i are continuous functions from X into the reproducing kernel Hilbert space H of the mean zero random process N. The optimum designs are characterized by an Elfving's theorem with $\text{R}$ the closed convex hull of the set {(φ, f(x))_H : 6φ 6_H ≤ 1, x?X}, where (·, ·)_H is the inner product on H. It is shown that if X is convex and f_i are linear the design points may be chosen from the extreme points of X. In some problems each linear functional c′θ can be optimally estimated by a design on one point x(c). These problems are completely characterized. An example is worked and some partial results on minimax designs are obtained.     

Divergence of the greedy algorithm in the Faber-Schauder system on a continuum cardinality set

The paper establishes that there exist a continuum cardinality set E ₀ ? [0, 1] and a function f ₀(x) ∈ C _[0,1], such that the greedy algorithm of f ₀(x) with respect to the Faber-Schauder system converges to +∞ at all points of E ₀.     

Polynomial approximation errors for functions of low-order continuity

Given a functionf defined on [-1, 1] we obtain, in terms of (n+1)st divided differences, expressions for the minimax errorE _n(f) and the errorS _n(f) obtained by truncating the Chebyshev series off aftern+1 terms. The advantage of using divided differences is thatf is required to have no more than a continuous second derivative on [-1, 1].     

Injective hulls with distinct ring structures

It is well known from Osofsky’s work that the injective hull E(R_R) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(R_R) has a ring structure and its multiplication extends its R-module scalar multiplication, must the ring structure be unique? In this paper, we utilize the properties of Morita duality to explicitly describe an injective hull of a ring R with R=Q(R) (where Q(R) is the maximal right ring of quotients of R) such that every injective hull of R_R has (possibly infinitely many) distinct compatible ring structures which are mutually ring isomorphic and quasi-Frobenius. Further, these rings have the property that the ring structures for E(R_R) also are ring structures on E(_RR).     

The extended gamma class and applications

The gamma class Γ _α (g) consists of positive and measurable functions that satisfy f(x+yg(x))/f(x)→exp(αy). In most cases, the auxiliary function g is Beurling varying, i.e. g(x)/x→0 and g∈Γ₀(g). Taking h=logf, we find that h∈EΓ _α (g,1), where EΓ _α (g,a) is the class of ultimately positive and measurable functions that satisfy (f(x+yg(x))?f(x))/a(x)→αy. In this paper, we discuss local uniform convergence for functions in the classes Γ _α (g) and EΓ _α (g,a). From this we obtain several representation theorems. We also prove some higher order relations for functions in the classes Γ _α (g) and EΓ _α (g,a). Some applications conclude the paper.     

Some properties onf-edge covered critical graphs

LetG(V, E) be a simple graph, and letf be an integer function onV with 1 ≤f(v) ≤d(v) to each vertexv ∈V. An f-edge cover-coloring of a graphG is a coloring of edge setE such that each color appears at each vertexv ∈V at leastf(v) times. Thef-edge cover chromatic index ofG, denoted by χ′ _fc (G), is the maximum number of colors such that anf-edge cover-coloring ofG exists. Any simple graphG has anf-edge cover chromatic index equal to δ_f or δ _f - 1, where $\delta _f = \mathop {\min }\limits_{\upsilon \in V} \{ \left\lfloor {\frac{{d(v)}}{{f(v)}}} \right\rfloor \} $ . LetG be a connected and not complete graph with χ′ _fc (G)=δ _f-1, if for eachu, v ∈V and e =uv ?E, we have ÷ _fc (G + e) > ÷ _fc (G), thenG is called anf-edge covered critical graph. In this paper, some properties onf-edge covered critical graph are discussed. It is proved that ifG is anf-edge covered critical graph, then for eachu, v ∈V and e =uv ?E there existsw ∈ {u, v } withd(w) ≤ δ _f (f(w) + 1) - 2 such thatw is adjacent to at leastd(w) - δ _f + 1 vertices which are all δ _f -vertex inG.     

Adding machines, kneading maps, and endpoints

Given a unimodal map f, let I=[c₂,c₁] denote the core and set E={(x₀,x₁,…)∈(I,f)|xi∈ω(c,f) for all i∈N}. It is known that there exist strange adding machines embedded in symmetric tent maps f such that the collection of endpoints of (I,f) is a proper subset of E and such that limk→∞Q(k)≠∞, where Q(k) is the kneading map.We use the partition structure of an adding machine to provide a sufficient condition for x to be an endpoint of (I,f) in the case of an embedded adding machine. We then show there exist strange adding machines embedded in symmetric tent maps for which the collection of endpoints of (I,f) is precisely E. Examples of this behavior are provided where limk→∞Q(k) does and does not equal infinity, and in the case where limk→∞Q(k)=∞, the collection of endpoints of (I,f) is always E.     

On a family of solutions of a linear second-order differential equation with constant unbounded operator coefficients in a Banach space

In a Banach space E, we study the equation 1 $$ u''(t) + Bu'(t) + Cu(t) = f(t), 0 \leqslant t < \infty $$ , where f(t) ∈ C([0,∞);E), B,C ∈ N(E), and N(E) is the set of closed unbounded linear operators from E to E with dense domain in E. We find a two-parameter family of solutions of Eq. (1) in two cases: (a) the operator discriminant D = B ² ? 4C of Eq. 1 is zero; (b) D = F ², where F is some operator in N(E). We suggest a method for increasing the smoothness of such solutions by imposing more restrictive conditions on the input data W = (B,C,f(t)) and the parameters x ₁, x ₂ ∈ E.     

Stably average shadowable homoclinic classes

Let p be a hyperbolic periodic saddle of a diffeomorphism of f on a closed smooth manifold M, and let H_f(p) be the homoclinic class of f containing p. In this paper, we show that if H_f(p) is locally maximal and every hyperbolic periodic point in H_f(p) is uniformly far away from being nonhyperbolic, and H_f(p) has the average shadowing property, then H_f(p) is hyperbolic.