Decidability results in non-classical logic. III

A Kripke type semantics is given to a large class of tense logics with statability operators (including PriorsQK _t) in such a manner as to obtain their decidability using Rabin’s theorem. Dedicated to the Memory of A. N. Prior     

Algebraic logic and logical geometry. Two in one

The aim of the paper is to define the notion of isotypic algebras and to formulate a series of new problems related to this notion.     

Two inverse results

Let A be a finite subset of a group G ₀ with |A ^?1 A|≤2|A?2. We show that there are an element α∈A and a non-null proper subgroup H of G such that one of the following holds:

• x ^?1 Hy?A ^?1 A, for all x,y∈A not both in Hα
• x Hy ^?1?AA ^?1, for all x,y∈A not both in αH

where G is the subgroup generated by A ^?1 A. Assuming that A ^?1 A≠G and that $\left| {A^{ - 1} A} \right| < \tfrac{{5|A|}} {3} $ , we show that there are a normal subgroup K of G and a subgroup H with K?H?A ^?1 A and 2|K|≥|H| such that $A^{ - 1} AK = KA^{ - 1} A = A^{ - 1} Aand6|K| \geqslant |A^{ - 1} A| = 3|H|$ .     

Two results in metric fixed point theory

We establish two fixed point theorems for certain mappings of contractive type. The first result is concerned with the case where such mappings take a nonempty, closed subset of a complete metric space X into X, and the second with an application of the continuation method to the case where they satisfy the Leray–Schauder boundary condition in Banach spaces.     

Two results on arithmetical functions

Generalizing two results of Rieger [8] and Selberg [10] we give asymptotic formulas for sums of type $${\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n)\qquad {\rm and} {\matrix {\sum \limits_{n\leq x}\cr n\equiv l({\rm mod}k)\cr f_{\kappa}(n)\equiv s_{\kappa}({\rm mod}p_{\kappa})\cr (\kappa=1,\dots,r)\cr}}\qquad \chi(n),$$ where χ is a suitable multiplicative function, f₁,…, f _r are “small” additive, prime-independent arithmetical functions and k, l are coprime. The proofs are based on an analytic method which consists of considering the Dirichlet series generated by $ \chi(n)z_{1}^{f_{1}(n)}\cdot... \cdot z_{r}^{f_{r}(n)},z_{1}\dots z_{r} $ complex.     

Two results for monocompact measures

For every finite measure space (Ω,A, P) whereA is K₁-generated we prove the equivalence of compactness and monocompactness for P . Moreover, we prove the existence of a perfect, not monocompaot probability, thus answering an open question in [6]. Let P be a charge on the algebraA andK ?A be a monocompact class. We show that P is o-additive ifK ^S P-approximatesK ^S, the family of finite unions inK , needs not to be monocompact.     

Two decidable classes of formulas of the modal logic S5

With the use of S. Yu. Maslov's inverse method, we prove the decidability by derivability of two classes of formulas of the quantifier modal logic S5 containing only one-place predicate variables. Translated from Lietuvos Matematikos Rinkinys, Vol. 40, No. 3, pp. 350–360, July–September, 2000. Translated by Remigijus Lapinskas     

Two new associated Laguerre integral results

New associated Laguerre integral results are obtained by eliminating an unnecessary restriction in [1]. The expression used to obtain these new results is compared to that given in [2] and shown to be different and of greater generality. Finally, some physical moments are discussed that require such integrals.     

Two results on maximum nonlinear functions

Maximum nonlinear functions are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions , gcd(k, m) = 1, and the Kasami power functions , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.      

Two results on powers of 2 in Waring-Goldbach problem

In this paper, it is proved that every sufficiently large odd integer is a sum of a prime, four cubes of primes and 106 powers of 2. What is more, every sufficiently large even integer is a sum of two squares of primes, four cubes of primes and 211 powers of 2.     

Two Lil-type results for the Brownian bridge

The necessary and sufficient condition for a function to be upper class relative to a Brownian motion process was obtained by T. Sirao and T. Nisida (1952, Nagoya Math. J. 4, 97–101) and a related result on first passage distribution was obtained by P. K. Sen and M. J. Wichura (in press, Estimation of First Crossing Time Distribution for Brownian Motion Processes Relative to Upper Class Boundaries). Corresponding results for the Brownian bridge are discussed.     

Two results on homogeneous Hessian nilpotent polynomials

Let z=(z₁,…,z_n) and , the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix is nilpotent. In recent developments in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (8) (2005) 2201-2205. [MR2138860]; G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (6) (2006) 503-510. [MR2170971]. See also math-ph/0308035; W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249-274. [MR2247890]. See also math.CV/0409534], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture (VC) of HN polynomials: for any homogeneous HN polynomialP(z) (of degreed=4), we haveΔ^mP^m+1(z)=0for anym?0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties Z_P and Z_σ2 of CPⁿ⁻¹ determined by the principal ideals generated by P(z) and , respectively, intersect only at regular points of Z_P. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z−∇P with P(z) HN if F has no non-zero fixed point w∈Cⁿ with . Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), Δ^m(f(z)P(z)^m)=0 when m?0.     

Two results on centralisers of nilpotent elements

Let g be a semisimple Lie algebra. We provide a short proof of McNinch’s result on centralisers of the sum of commuting nilpotent elements (J. Pure Appl. Algebra 206 (2006), 123-140) and characterise nilpotent elements e∈g having the property that the orbit G⋅e is the largest nilpotent orbit meeting the centraliser of e.     

Two consistency results concerning thin-tall Boolean algebras

In this paper we show that the following is relatively consistent withZFC +CH: There is no superatomic Boolean algebra of height ₂+1 and width, and there is no superatomic Boolean algebraA with for 0<<₁ and Presented by J. Mycielski.     

Two results on the equivariant Ginzburg-Landau vortex in arbitrary dimension

We characterize the O(N)-equivariant vortex solution for Ginzburg-Landau type equations in the N-dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.