and Applied Analysis 3 Definition 2.2. The fractional derivative of order q > 0 of a continuous function x : 0, ∞ → R is given by Dx t 1 Γ ( n − q) ( d dt )n∫ t 0 x s t − s q−n 1 ds, 2.2 where n q 1, provided that the right side is pointwise defined on 0,∞ . Lemma 2.3 see 7 . 1 If x ∈ L 0, 1 , ρ > σ > 0, then DIx t Iρ−σx t . 2 If ρ > 0, λ > 0, then Dρtλ−1 Γ λ /Γ λ − ρ tλ−ρ−1. Lemma 2.4 see 12 ....

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) ([5]), and so, via the Furstenberg correspondence principle introduced in [4], a new proof of the multi-dimensional Szemerédi Theorem. We bypass the detailed analysis of certain towers of factors of a probability-preserving...

ion by fresh free variables is the main tool to obtain a rigid system from one which is not, therefore explaining our notation RA. Example 8.5. Consider the system R = {f(λx.h(@(H, c(x)))) → g(H)}, in which H : α→ β is the only free variable. R can be abstracted by RA = {f(X)→ g(X), h(Y )→ Y, c(z)→ z} The rule in R is abstracted by the rule f(X) → g(X) added to RA with {X 7→ λz.h(@(H, c(z)))} a...

In this note, we prove that if a uniformly recurrent infinite word contains as a factor any finite permutation of words from an infinite family, then either this word is periodic, or its complexity (that is, the number of factors) grows faster than linearly. This result generalizes one of the lemmas of a recent paper by de Luca and Zamboni, where it was proved that such an infinite word cannot ...

Proof. Fix a pair of neighbouring datasets X ' X ′ and let E ⊆ R be any measurable set. Let ΘX,X′ be as in the statement and write ΘX,X′ = R \ΘX,X′ . Using the assumptions on ΘX,X′ we see that P[θX ∈ E] = P[θX ∈ E ∩ΘX,X′ ] + P[θX ∈ E ∩ΘX,X′ ] ≤ eP[θX′ ∈ E ∩ΘX,X′ ] + δ ≤ eP[θX′ ∈ E] + δ . Now we proceed with the proof of Lemma 1. Let X ' X ′ be two neighbouring datasets and let us write Z1 = ZX ...

ABSTRACT- Bromus danthoniae Trin. is an annual grass species which grows mainly on dry grassy rocky mountain slopes and grassy steppe, and is grazed by many herbivores and recognized as a useful pasture plant. The chromosome number, morphological and anatomical traits of 82 genotypes of B. danthoniae belonging to three sub-taxa were investigated. Twenty-seven quantitative and 20 qualitative mor...

mangrove sediments were collected during wet and dry seasons from nine stations in khamir,laft and natural reservoir mangrove-dense areas of hormozgan province in the south of iran. σ pcbs ranged from 5.33 to 15.5 ng/g dry weight and the dominant congener was no.153. average σ ddts for khamir and laft mangroves were 16.58 ± 1.51 and 18.8 ± 9.98 ng/g dry weight. ddt was more abundant than dde an...

and Applied Analysis 3 In the last section of this paper,we study the system 1.2 . Again, we introduce a notion of solution-tube of 1.2 which generalizes the notion of lower and upper solutions used by Atici et al. 8 . This notion generalizes also condition 1.5 and the notion of solution-tube of systems of second-order differential equations introduced in 10 . In addition, we assume that f sati...

Let K be a countable separably Hilbertian field. Denote the absolute Galois group of K by G(K). For each σ ∈ (σ1, . . . , σe) ∈ G(K) let Ks[σ] be the maximal Galois extension of K which is fixed by σ1, . . . , σe. We prove that for almost all σ ∈ G(K) (in the sense of the Haar measure) the field Ks[σ] is PAC and its absolute Galois group is isomorphic to F̂ω. Mathematische Zeitschrift 224 (1997)...

We prove an Alexander type theorem for the spectral unit ball Ωn showing that there are no non-trivial proper holomorphic mappings in Ωn, n ≥ 2. Let Mn denote the space of n× n complex matrices. In order to avoid some trivialities and ambiguities we assume in the whole paper that n ≥ 2. Let ρ(A) := max{|λ| : λ ∈ Spec(A)} be the spectral radius of A ∈ Mn. Denote also by Spec(A) := {λ ∈ C : det(A...