and Applied Analysis 3 The evolution system R t, s is said to be equicontinuous if for all bounded set Q ⊂ X, {s → R t, s x : x ∈ Q} is equicontinuous for t > 0. x ∈ C −q, b , X is said to be a mild solution of the nonlocal problem 1.1 , if x t φ t g x t for t ∈ −q, 0 , and, for t ∈ J , it satisfies the following integral equation: x t R t, 0 [ φ 0 g x 0 ] ∫ t 0 R t, s f ( s, xs, ∫ s 0 K s, r, ...

We present a short proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in details: Theorem. Let M be a connected oriented closed smooth 3-manifold. Let L1(M) be the set of framed links in M up to a framed cobordism. Let deg : L1(M) → H1(M ;Z) be the map taking a framed link to its homology class. Then for each α ∈ H1(M ;Z) there is a 1-1 ...

Let {X(t), 1 ≤ t ≤ T} be a sequence of Rp-valued independent random variables. Let Θ ⊆ Rd be an open, non-empty convex parameter space equipped with the Euclidean inner product 〈·, ·〉, and norm‖ · ‖2. We will also use the `1-norm ‖θ‖1 def = ∑d j=1 |θj |, and the `∞-norm ‖θ‖∞ def = max1≤j≤d |θj |. We assume that there exists a change point τ? ∈ {1, . . . , T − 1}, parameters θ ? , θ (2) ? ∈ Θ, s...

(Def. 2) lenpr2(h1) = lenh1 and for every n such that n ∈ dompr2(h1) holds pr2(h1)(n) = h1(n)2. Let us consider A, B, let f1 be a homogeneous quasi total non empty partial function from A∗ to A, and let f2 be a homogeneous quasi total non empty partial function from B∗ to B. Let us assume that arity f1 = arity f2. The functor ee f1, f2dd yields a homogeneous quasi total non empty partial functi...

Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum number of edges in an F -saturated graph of order n. For graphs G,H1, . . . , Hk, we write that G → (H1, . . . , Hk) if every k-coloring of E(G) contains a monochromatic copy of Hi in color i for some i. A...

Let A be a finite or countable alphabet and let θ be a literal (anti)morphism onto A∗ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under θ (θ-invariant for short). We establish a formula which allows to embed any non-complete θ-invariant code into a complete one: this brings a positive answer to the open ...

Let Θ be an arbitrary variety of algebras and let Θ0 be the category of all free finitely generated algebras from Θ. We study automorphisms of such categories for special Θ. The cases of the varieties of all groups, all semigroups, all modules over a noetherian ring, all associative and commutative algebras over a field are completely investigated. The cases of associative and Lie algebras are ...

Given graphs H1, . . . ,Hk, let f(H1, . . . ,Hk) be the minimum order of a graph G such that for each i, the induced copies of Hi in G cover V (G). We prove constructively that f(H1,H2) ≤ 2(n(H1) + n(H2) − 2); equality holds when H1 = H2 = Kn. We prove that f(H1,Kn) = n + 2 √ δ(H1)n + O(1) as n → ∞. We also determine f(K1,m−1,Kn) exactly.

Abstract. Let σ δ k denote the Cesàro means of order δ > −1 of the spherical harmonic expansions on the unit sphere Sd−1, and let Ej ( f, H1) denote the best approximation of f in the Hardy space H1(Sd−1) by spherical polynomials of degree at most j . It is known that λ := (d −2)/2 is the critical index for the summability of the Cesàro means on H1(Sd−1). The main result of this paper states th...

Let n ≥ 2, let K,K′ be fields such that K′ is a quadratic Galoisextension of K and let θ denote the unique nontrivial element in Gal(K′/K). Suppose the symplectic dual polar space DW (2n− 1,K) is fully and isometrically embedded into the Hermitian dual polar space DH(2n − 1,K′, θ). We prove that the projective embedding of DW (2n − 1,K) induced by the Grassmann-embedding of DH(2n − 1,K′, θ) is ...