The rational approximation of convex functions of the class Lip α

It is proved that if a function f(x) is convex on [a, b] and f ∈ Lip_K(f)α, 0<α<1, then the least uniform deviation of this function from rational functions of degree not higher than n does not exceed (v is a natural number; C(α, v) depends only onα andv; K(f) is a Lipschitz constant; and      

Lipschitz Stability of the Cauchy and Jensen Equations

Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ?: G → E be a given function. By C?: G × G → E we understand the Cauchy difference of the function /, i.e.: $$ {\cal C}f(x,y):=f(x+y)- f(x)- f(y)\ {\rm for}\ x,y\in G. $$ We prove that if the function C(f) is Lipschitz then there exists an additive function A: G → E such that f ? A is Lipschitz with the same constant. Analogous result for Jensen equation is also proved. As a corollary we obtain the stability of the Cauchy and Jensen equations in the Lipschitz norms.     

Weak Compactness and Variational Characterization of the Convexity

We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X ^*, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X ^* is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex.     

On functions of companion matrices

Presented in this paper are some new properties of a function f(C) of a companion matrix C, including (1) a representation of any entry of f(C) as a divided difference of f(λ) times a polynomial, (2) a similarity decomposition of f(C) generalizing that based on the Jordan form of C, and (3) characterization (construction) of all matrices that transform f(C) (by similarity) to companion form. The connection between functions of general and companion matrices is also dealt with, and a pair of dual relations is established.     

Random problems

A problem (a Boolean function f: {0, 1}^N → {0, 1}) is characterized by its randomness (à la Kolmogorov) R(f) and its entropy (à la Shannon) H(f). Random problems have large values of R(f) and are a good model for many natural pattern recognition problems. R(f) and H(f) are shown to be lower and upper bounds, respectively, for a minimum-size circuit that computes f False entropy, namely the hidden structure of a problem, is related to the difference between H(f) and R(f).     

Uniformity of a certain systems of functions of many-valued logic

For any finite system A of functions of many-valued logic taking values in the set {0,1} such that a projection of A generates the class of all monotone boolean functions, it is proved that there exists constants c and d such that for an arbitrary function f ε [A] the depth D(f) and the complexity L(f) of f in the class of formulas over A satisfy the relation D(f) ≤ clog₂ L(f) + d.     

Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators Studied Asymptotically

An Nth order asymptotic expansion is established for the error of weak approximation of a special class of functions by the well-known Cardaliaguet-Euvrard neural network operators. This class is made out of functions f that are N times continuously differentiable over R, so that all f,f′,…, f ^(N) have the same compact support and f ^(N) is of bounded variation. This asymptotic expansion involves products of integrals of the network activation bell-shaped function b and f. The rate of the above convergence depends only on the first derivative of involved functions.     

Courbure scalaire prescrite sur une variété C∞ compacte de dimension 2

Let (V₂, g) be a C^∞ compact Riemannian manifold of negative scalar curvature of dimension 2, and let f be a C^∞ function defined on V₂. We intend to find a condition on f in order that f be the scalar curvature of a metric conformal to the initial metric g.     

Applications of weighted estimates of Calderon’s commutators

The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition |V(z) ? V(??)| ?? w|z ? ??| and F is an entire function, then we put $$P_F^* f(z) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int\limits_{\left| {\zeta - z} \right| > \varepsilon } {F\left( {\frac{{V(\zeta ) - V(z)}} {{\zeta - z}}} \right)\frac{{f(\zeta )}} {{\left( {\zeta - z} \right)^2 }}d\sigma (\zeta )} } \right|.$$ It is shown that if the weight function ?? is a Muckenhoupt A _p weight for 1 < p < ??, then $$\left\| {P_F^* f} \right\|_{p,\omega } \leqslant C(F,w,p)\left\| f \right\|_{p,\omega } .$$ .     

Solvability of an integro-differential equation with partial derivatives on the half-axis

The paper considers the following integro-differential equation on the semi-axis: $ \frac{{\partial f(t,x)}} {{\partial t}} + \frac{{\partial f(t,x)}} {{\partial x}} + qf(t,x) = \int_0^\infty {k(x - x')f(t,x')dx'} , $ where 0 ≤ k(x) ? L ₁(?∞,∞), q = const and f(t, x) is the unknown function. This equation has significant applications in different fields of natural sciences, particularly in econometrics (see [1]), where the unknown function is considered as the density of the national income distribution function, q characterizes the mean savings etc., k(x) is the function of income rearrangement. A structural theorem of the solution is proved, whose asymptotic at infinity is found.     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

Roman Domination Dot-critical Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value ${f(V(G))=\sum_{u \in V(G)}f(u)}$ . The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. In this paper, we study graphs for which contracting any edge decreases the Roman domination number.     

Odd factors of a graph

LetG be a graph and letf be a function defined on V(G) such that f(x) is a positive odd integer for everyx ? V(G). A spanning subgraphF ofG is called a [l,f]-odd factor of G if d_F(x) ? {1,3,2026, f(x)} for every x ?V(G), whered _F (x) denotes the degree of x inF. We discuss several conditions for a graphG to have a [1,f]-odd factor.     

Optimal solution of a Monge-Kantorovitch transportation problem

Suppose that P and Q are probabilities on a separable Banach space $\text{E}$. It is known that if (P, Q) satisfies certain regularity conditions and a random variable X has law P, then there exists a function $\text{f :}\text{E}\text{→}\text{E}$, such that the function f(X) has the law Q and the random pair (X, f(X)) is an optimal coupling for the Monge-Kantorovitch problem. In this paper we provide an approximation of the function f when the law Q is discrete. Thenwe extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations.     

On arithmetical functions related to the Ramanujan sum

Let m and n be positive integers, and μ the M"bius function. And let S _f(m,n) be the function defined by , where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S _f(m,n) and taking f=μ, we obtain the Dirichlet series with the coefficients Sμ(m,n) and Sμ(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums. This revised version was published online in June 2006 with corrections to the Cover Date.     

The crossing function of a graph

The crossing function of a graphG with orientable genusn is defined as a mapping \(f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} \) for whichf(k)=cr _k (G) the crossing number ofG on the orientable surface of genusk. It is proved that any decreasing convex function \(f:\{ \not 0,1, \ldots ,n\} \to \{ \not 0,1,2, \ldots \} \) with \(f(n) = \not 0\) is the crossing function of some connected graph.     

On the growth of meromorphic functions of infinite order

Letf be a meromorphic function of infinite order,T(r, f) its Nevanlinna (or Ahlfors-Shimizu) characteristic, andM(r, f) its maximum modulus. It is proved that $$\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{rT'(r,f)}} \leqslant \pi and\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{T(r,f)\psi (log T(r,f))}} = 0$$ . if ? (x)/x is non-decreasing, ?′(x)<-√?(x) and ∝^∞ dx/?(x) < ∞.     

Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function

By using the variational calculus of fractional order, one derives a Hamilton-Jacobi equation and a Lagrangian variational approach to the optimal control of one-dimensional fractional dynamics with fractional cost function. It is shown that these two methods are equivalent, as a result of the Lagrange’s characteristics method (a new approach) for solving nonlinear fractional partial differential equations. The key of this results is the fractional Taylor’s seriesf(x + h) = E _α(h^αD^α)f(x) whereE _α(·) is the Mittag-Leffler function.     

Approximation of analytic functions and their real part

LetG?C be a finite quasidisk, andf(z) an analytic function inG whose real partu(z)?Ref(z) is continuous on \(\bar G\) . Connections between the approximation properties of the functionsu(z) andf(z) are obtained.