LetX be a Banach space with dualX∗, and letK be a nonempty subset ofX. A gauge function is a continuous strictly increasing function φ : R → R such that φ 0 0 and limt→∞φ t ∞. The duality mapping Jφ : X → X∗ associated with a gauge function φ is defined by Jφ x : {f ∈ X∗ : 〈x, f〉 ‖x‖‖f‖, ‖f‖ φ ‖x‖ }, x ∈ X, where 〈·, ·〉 denotes the generalized duality pairing. In the particular case φ t t, the ...

Let N0 be the set of all non-negative integers, let X N0 and P(X) be the the power set of X. An integer additive set-labeling (IASL) of a graph G is an injective function f : V (G) ! P(N0) such that the induced function f+ : E(G) ! P(N0) is defined by f+(uv) = f(u) + f(v), where f(u) + f(v) is the sum set of f(u) and f(v). An IASL f is said to be an integer additive set-indexer (IASI) of a grap...

We prove the weak convergence of Mann iteration for a hybrid pair of maps to a common fixed point of a selfmap f and a multivalued f nonexpansive mapping T in Banach space E. Keywords—Common fixed point, Mann iteration, Multivalued mapping, weak convergence.

Let V be a vector space over a field F . If G≤GL(V, F ), the central dimension of G is the F -dimension of the vector space V/CV (G). In [DEK] and [KS], soluble linear groups in which the set Licd(G) of all proper infinite central dimensional subgroups of G satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally...

The dynamical systems of the form¨r = F(r, ˙ r) in R n accepting the normal shift are considered. The concept of weak normality for them is introduced. The partial differential equations for the force field F(r, ˙ r) of the dynamical systems with weak and complete normality are derived.

Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f : weak f -congruence, f -congruence, and strong f -congruence. If f is odd, weak f -congruence preserves the ring structure on cohomology with Zf -coefficients. We show that strong f -congruence coincides with a relation previously stu...

We consider the task of transforming a weak one-way function (which may be easily inverted on all but a polynomial fraction of the range) into a strong one-way function (which can be easily inverted only on a negligible fraction of the range). The previous known transformation [Yao 82] does not preserve the security (i.e., the running-time of the inverting algorithm) within any polynomial. Its ...

We extend the boosting paradigm to the realistic setting of agnostic learning, that is, to a setting where the training sample is generated by an arbitrary (unknown) probability distribution over examples and labels. We deene a-weak agnostic learner with respect to a hypothesis class F as follows: given a distribution P it outputs some hypothesis h 2 F whose error is at most erP(F) + , where er...

Let P be a polynomial over the ring of mod m integers. P weakly Abstract-1 represents Boolean function f : {0, 1}n → {0, 1} if there is a subset S ⊆ {0, 1, . . . , m − 1} such that f(x) = 0 if and only if P (x) ∈ S. The smallest degree of polynomials P weakly representing f is called the weak mod m degree of f . We give here an Ω(logn) lower bound for the weak degree of the generalized inner pr...

0.1 Definition. Weak Diamond (Φ) is the statement that for every function F : 12 → 2 there exists an h : ω1 → 2 such that for all η : ω1 → 2 the set {α < ω1 | F (η α) = h(α)} is stationary. Note that by replacing h with 1 − h, Φ is equivalent to statement that for every F : 12 → 2 there exists an h : ω1 → 2 such that for all η : ω1 → 2 the set {α < ω1 | F (η α) ̸= h(α)} is stationary. Weak Diamo...