In the higher rank paper [17], we reduced Dwork’s conjecture from higher rank case over any smooth affine variety X to the rank one case over the simplest affine space A. In the present paper, we finish our proof by proving the rank one case of Dwork’s conjecture over the affine space A, which is called the key lemma in [17]. The key lemma had already been proved in [16] in the special case whe...

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T1, T2, . . . , Td : Z y (X,Σ, μ) ([6]), and so, via the Furstenberg correspondence principle introduced in [5], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the careful manipulation of certain towers of factors of a probability-preserv...

We point out that the theory of difference fields with algebraically closed fixed field has no model companion. By a difference field we mean a field K equipped with an automorphism σ. It is well-known ([?]) that the class of existentially closed difference fields is an elementary class (ACFA), and moreover all completions are unstable. The fixed field (set of a ∈ K such that σ(a) = a) is respo...

In [7] G.Navarro proposed a refinement of the McKay conjecture involving a special class of automorphisms. In [6] this new conjecture was verified for the alternating groups A(Π) when p = 2. In this paper, it is verified for the p-singular characters of A(Π) when |Π| = wp, w < p and p is odd. 1. McKay and Navarro conjectures Let G be a finite group, where |G| = n. Let p be a prime dividing n, D...

We assume the (i, j)th element and the (j, i)th element of Σ are functionally unrelated. The results can be extended to the case where symmetric matrix elements are considered functionally equal (see e.g., McCulloch [5]). In the following, we use ⊗ to denote the Kronecker product in matrix algebra and use vec to denote the operator that transforms a matrix into a column vector by stacking the c...

and Applied Analysis 3 and integrating over [ρ(a), T , we get p (r) θ Δ (r) = p (ρ (a)) θ Δ (ρ (a)) + ∫ r ρ(a) q (t) θ (t) ∇t. (22) Since p(ρ(a)) > 0, θΔ(ρ(a)) ≥ 0, q(t) > 0, and θ(t) > 0, we obtain p(r)θΔ(r) > 0. Thus, we determine θΔ(r) > 0. This contradiction shows that the solution θ(t) is strictly increasing and positive on [ρ(a), T as desired. Similar arguments can be applied for the proo...

and Applied Analysis 3 involving fractional differential equations the same applies to the boundary value problems of fractional differential equations . Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see 17 . Lemma 2.4 see 28 . For q > 0, the general solution of the fractional differential equ...

Proof. [Solution due to Claudiu Raicu] Consider two polynomials f, g ∈ C[X±] such that the spherical graphs they determine intersect transversely at the points where the vectors vi intersect the sphere. This implies that T (f) and T (g) intersect transversely ([1]) at each point and T (f)∩T (g) = Σ. Using the Transverse Intersection Lemma (lemma 3.2, corollary 3.3 of [1]) we get that Σ = T (f, ...

ion in BTA Abstraction [73, 20, 8] plays an important role in process algebras. It allows a simpler view of a thread, ignoring internal details. In [17] abstraction is used to emulate the interaction between clients and servers, assuming that clients and servers are threads in BTA. We assume the existence of a concrete internal action tau ∈ Σ that does not have any side effects and always repli...

We consider a partition of a space X consisting of a meager subset of X and obtain a sufficient condition for the existence of a subfamily of this partition which gives a non-Baire subset of X. The condition is formulated in terms of the theory of J. Morgan [1]. All notions concerning category bases come from Morgan’s monograph (see [1]). We establish the following theorem. Theorem A. Let (X,S)...