Let $\mathcal{L} = -\partial_x^2 - V(x) \partial_y^2$ be the Grushin operator on $\mathbb{R}^2$ with coefficient $V : \mathbb{R} \to [0,\infty)$. Under sole assumptions that $V(-x) \simeq xV'(x)$ and $x^2 |V''(x)| \lesssim V(x)$, we prove a spectral multiplier theorem of Mihlin--H\"ormander type for $\mathcal{L}$, whose smoothness requirement is optimal independent $V$. The assumption second de...

Most existing universal approximation results for fuzzy systems are based on the assumption that we use t-norms and t-conorms to represent \and" and \or". Yager has proposed to use, within the fuzzy systems modeling paradigm, more general operations based on uninorms. In this paper, we show that the universal approximation property holds for an arbitrary choice of a uninorm.

This paper will deal with the consequences of the Uniformization Theorem, which is a major result in complex analysis and differential geometry. We will proceed by stating the theorem, which is that for any simply connected Riemann surface, there exists a biholomorphic map to one (and only one) of the following three: the Riemann sphere, the open unit disk, and the complex plane. After the theo...

A theorem from control theory relating the Lie algebra generated by vector fields on a manifold to the controllability of the dynamical system is shown to apply to Holonomic Quantum Computation. Conditions for deriving the holonomy algebra are presented by taking covariant derivatives of the curvature associated to a non-Abelian gauge connection. When applied to the Optical Holonomic Computer, ...

We consider a homology theory h : T ! A on a triangulated category T with values in a graded abelian category A . If the functor h re ects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category, all idempotents in T split and the kernel of h is a square zero ideal which as a ...

Given a cardinal κ, a set A ⊂ ω is κ-universally Baire if there exist trees S, T which project to A and its complement respectively, and which continue to project to complements after the forcing Coll(ω, κ). Todorcevic [3] has shown that under Martin’s Maximum every set of reals of cardinality א1 is א1-universally Baire but not א2-universally Baire. Here we present alternate proofs of these two...

The aim of this paper is solving the predicate calculus formal system based on 1-level universal AND operator. Firstly, universal logic and propositional calculus formal deductive system (0 1] h UL∈ , are introduced. Secondly, a predicate calculus formal deductive system (0 1] h UL∈ , ∀ based on 1-level universal AND operator is built. Thirdly, the soundness theorem and deduction theorem of sys...

In this paper we study admissible consecutions (inference rules) in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic (as a set of theorems) is closed. We propose an approach based on the context effective finite model property. Theore...

∑ n≤x λ(n) is non-positive for all x ≥ 2, which also implies the Riemann Hypothesis). In [4] Haselgrove showed that both the Turán and Pólya conjectures are false; therefore we know that truncations to L(1, χ) may get negative. Let F denote the set of all completely multiplicative functions f(·) with −1 ≤ f(n) ≤ 1 for all positive integers n, let F1 be those for which each f(n) = ±1, and F0 be ...

For a graph G, a partiteness k ≥ 2 and a number of colours c, we define the multipartite Ramsey number r k (G) as the minimum value m such that, given any colouring using c colours of the edges of the complete balanced k-partite graph with m vertices in each partite set, there must exist a monochromatic copy of G. We show that the question of the existence of r k (G) is tied up with what monoch...