In this paper our aim is to establish new integral representations for the Fox–Wright function $${}_{p}\Psi_{q}[^{(\alpha_{p},A_{p})}_{(\beta_{q},B_{q})}|z]$$ when $$\mu=\sum_{j=1}^{q}\beta_{j}-\sum_{k=1}^{p}\alpha_{k}+\frac{p-q}{2}=-m,\;\;m\in\mathbb{N}_{0}.$$ particular, closed-form expressions are derived four parameter Wright under a special restriction on parameters. Exponential bounding i...

In this paper we investigate a property named GL(p, q) which is closely related to the Gordon-Lewis property. Our results on GL(p, q) are then used to estimate volume ratios relative to lp, 1 < p ≤ ∞, of unconditional direct sums of Banach spaces. Introduction In this paper we investigate a property named GL(p, q), 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞, closely related to the Gordon-Lewis property GL, and the b...

The Galois number Gn(q) is defined to be the number of subspaces of the n-dimensional vector space over the finite field GF (q). When q is prime, we prove that Gn(q) is equal to the number Ln(q) of n-dimensional mod q lattices, which are defined to be lattices (that is, discrete additive subgroups of n-space) contained in the integer lattice Z and having the property that given any point P in t...

We show that the q-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.

A set of points, S ⊆ PG(r, q), is said to be %-saturating if, for any point x ∈ PG(r, q), there exist %+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q, %). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction pres...

For a regular κ, a partial ordering P is said to have the κ Freese-Nation property (the κ-FN for short) if there is a mapping (κ-FN mapping) f : P → [P ] such that for any p, q ∈ P if p ≤ q then there is r ∈ f(p) ∩ f(q) such that p ≤ r ≤ q. Freese and Nation [5] used the א0-FN in a characterization of projective lattices and asked if this property alone already characterizes the projectiveness....

Razborov and Rudich have famously shown that so-called natural proofs are not useful for separating P from NP unless hard pseudorandom number generators do not exist. Their result is widely regarded as a serious barrier to proving strong lower bounds in circuit complexity theory. By definition, a natural combinatorial property satisfies two conditions, constructivity and largeness. We show unco...

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...

The present paper is concerned with the n–localization property and its preservation in countable support (CS) iterations. This property was first introduced in Newelski and Ros lanowski [10, p. 826]. Definition 0.1. Let n be an integer greater than 1. (1) A tree T is an n–ary tree provided that (∀s ∈ T)(|succ T (s)| ≤ n). (2) A forcing notion P has the n–localization property if P " ∀f ∈ ω ω ∃...