Sur une décomposition topologique des fonctions de Sobolev et sur la localisation de l'énergie dans les plasmas idéaux

Motivated by properties of “frozen topological invariants” in ideal plasmas theory, we consider the problem of an intrinsic characterization of the weak closure in H₀¹ (Ω) of all functions ψ₀ ∘ ϕ, with ϕ : Ω → Ω a diffeomorphism. We show that one may associate with ψ₀ a class W^weak (ψ₀) of functions which captures robust topological properties, of the level sets of {ψ₀} and, moreover, is closed under weak limits.     

Mixed problem for the wave equation with a summable potential and nonzero initial velocity

The resolvent approach in the Fourier method, combined with Krylov’s ideas concerning convergence acceleration for Fourier series, is used to obtain a classical solution of a mixed problem for the wave equation with a summable potential, fixed ends, a zero initial position, and an initial velocity ψ(x), where ψ(x) is absolutely continuous, ψ'(x) ∈ L ₂[0,1], and ψ(0) = ψ(1) = 0. In the case ψ(x) ∈ L[0,1], it is shown that the series of the formal solution converges uniformly and is a weak solution of the mixed problem.     

Oscillation criteria for sublinear differential equations with damping

We present some criteria for the oscillation of the second order nonlinear differential equation [a(t)ψ(x(t))x'(t)]' + p(t)x'(t) + q(t)f (x(t)) =0, t≧t ₀> 0 with damping where a∈C ¹ ([t ₀,∞)) is a nonnegative function, p, q∈ C([t ₀,∞)) are allowed to change sign on [t ₀,∞), ψ, f∈C(R) with ψ(x) ≠ 0, xf(x)/ψ(x) > 0 for x≠ 0, and ψ, f have continuous derivatives on R{0} with [f(x) / ψ(x)]' ≧ 0 for x≠ 0. This criteria are obtained by using a general class of the parameter functions H(t,s) in the averaging techniques. An essential feature of the proved results is that the assumption of positivity of the function ψ(x) is not required. Consequently, the obtained criteria cover new classes of equations to which known results do not apply.     

Laws of large numbers in self-correcting point processes

For a point process N(·) with the conditional intensity function ψ(t> − N(t)), the process t − N(t) has the Markov property, where ψ(·) is a function satisfying suitable conditions. It follows from the properties of ψ(·) that the Markov process t − N(t) is ergodic (i.e. positive recurrent). This result leads to the law of large numbers for functionals of the Markov process t − N(t).     

A Transference Result of the L p Continuity from Jacobi Riesz Transform to the Gaussian and Laguerre Riesz Transforms

In this paper we develop a transference method to obtain the Gaussian-Riesz transform’s L ^p -continuity and the Laguerre-Riesz transform’s L ^p -continuity from the L ^p -continuity of the Jacobi-Riesz transform, in dimension one, using the well known asymptotic relations between Jacobi polynomials and Hermite and Laguerre polynomials.     

On weak neighborhood systems and spaces

We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s. This work was supported by a grant from Research Institute for Basic Science at Kangwon National University.     

Graphoidal bipartite graphs

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths inG such that every path in ψ has at least two vertices, every vertex ofG is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. Let Ω (ψ) denote the intersection graph of ψ. A graph G is said to be graphoidal if there exists a graphH and a graphoidal cover ψof H such that G is isomorphic to Ω(ψ). In this paper we study the properties of graphoidal graphs and obtain a forbidden subgraph characterisation of bipartite graphoidal graphs.     

A note on θ(g, g′)-continuity in generalized topological spaces

The concept of θ(g, g′)-continuity was introduced by Császár [1]. In this paper, we investigate characterizations for θ(g, g′)-continuous functions and introduce the concept of weak θ(g, g′)-continuity, and study characterizations for weak θ(g, g′)-continuity and the relationships among θ(g, g′)-continuity, weak (g, g′)-continuity and weak θ(g, g′)-continuity.     

A characterization of reflexive spaces of operators

We show that for a linear space of operators M ? B(H₁, H₂) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ₁, ψ₂) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ₁(P,Q) and Q ≤ ψ₂(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator T ∈ B(H₁, H₂) lies in M if and only if ψ₂(P,Q)Tψ₁(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.     

Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ? 2 ^X with τ ? C we have f ^?1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ? C* ? 2 ^X , C*-continuity is a dual to C-continuity if C?C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.     

A completely monotonic function involving the tri- and tetra-gamma functions

The di-gamma function ψ(x) is defined on (0,∞) by $\psi (x) = \frac{{\Gamma '(x)}} {{\Gamma (x)}} $ and ψ ⁽ⁱ⁾(x) for i ∈ ? denote the polygamma functions, where Γ(x) is the classical Euler’s gamma function. In this paper we prove that a function involving the difference between [ψ′(x)]² + ψ″(x) and a proper fraction of x is completely monotonic on (0,∞).     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

Sums of Element Orders in Finite Groups

Given a finite group G, write ψ(G) to denote the sum of the orders of the elements of G. Our main result is that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ψ(C).     

On a Constructive Proof of Kolmogorov’s Superposition Theorem

Kolmogorov (Dokl. Akad. Nauk USSR, 14(5):953–956, 1957) showed that any multivariate continuous function can be represented as a superposition of one-dimensional functions, i.e., $$f(x_{1},\ldots,x_{n})=\sum_{q=0}^{2n}\varPhi _{q}\Biggl(\sum_{p=1}^{n}\psi_{q,p}(x_{p})\Biggr).$$ The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Φ _q and ψ _q,p , respectively. Sprecher (Neural Netw. 9(5):765–772, 1996; Neural Netw. 10(3):447–457, 1997) gave a constructive proof of Kolmogorov’s superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ by ψ _p,q :=λ _p ψ(x _p +qa) with appropriate values λ _p ,a∈?. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Köppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474–479, 2002) suggested a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen’s ψ, and we present a correct constructive proof of Kolmogorov’s superposition theorem for continuous inner functions ψ similar to Sprecher’s approach.     

On weakly (τ, m)-continuous functions

In this paper we introduce a new notion of weakly (τ, m)-continuous functions as functions from a topological space into a set satisfying some minimal conditions. We obtain some characterizations and several properties of such functions. This function leads to the formulation of a unified theory of weak continuity [20], almosts-continuity [33],p(θ)-continuity [10] andp-continuity [41].     

Two linear equations in formal power series

The equations [gradφ(x)]^TF(x)=h(x) and F(ψ(x))–ψ(x) are considered. They arise in the stability theory of differential and difference equations. The scalar function h(x) is a given, and the function ψ(x) an unknown, formal power series in the n indeterminates x=(x₁,…,x_n)^T, and h(0)=ψ=0; the elements of the n×n matrix F(x) are also formal power series in x, F(0)=0. It is shown that the solvability of both equations depends on the eigenvalues of the Jacobian F_x(0).     

On the rank of odd hyper-quasi-polynomials

Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+???+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.     

Factorization of mappings of topological spaces and homomorphisms of topological groups in accordance with weight and dimension ind

Symbols w(X), nw(X), and hl(X) denote the weight, the network weight, and the hereditary Lindelöf number of a space X, respectively. We prove the following factorization theorems.

1. Let X and Y be Tychonoff spaces, φ: X→Y a continuous mapping, hl(X)≤τ, and w(Y)≤τ. Then there exist a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤τ andind Z≤ind X. Moreover, if nw(X)≤τ, then mapping ψ is one-to-one.
2. Let π: G→H be a continuous homomorphism of a Hausdorff topological group G to a Hausdorff topological group H, hl(G)≤τ and w(H)≤τ. Then there are a Hausdorff topological group G^* and continuous homomorphisms g: G→G^*, h: G^*→H so that π=h o g, G^*=g(G), w(G^*)≤τ andind G^*≤ind G. If nw(G)≤τ, then g is one-to-one.
3. For every continuous mapping φ: X→Y of a regular Lindelöf space X to a Tychonoff space Y one can find a Tychonoff space Z and continuous mappings ψ: X→Z, χ: Z→Y such that φ=χ o ψ, Z=ψ(X), w(Z)≤w(Y),dim Z≤dim X, andind ₀ Z≤ind ₀ X, whereind ₀ is the dimension function defined by V.V.Filippov with the help of G_δ-partitions. If we additionally suppose that X has a countable network, then ψ can be chosen to be one-to-one. The analogous result also holds for topological groups.
4. For each continuous homomorphism π: G→H of a Hausdorff Lindelöf Σ-group G (in particular, of a σ-compact group G) to a Hausdorff group H there exist a Hausdorff group G^* and continuous homomorphisms g: G→G^*, h:G^*→H so that π=h o g, G^*=g(G), w(G^*)≤w(H),dimG^*≤dimG, andind G^*≤ind G. Bibliography: 25 titles.

     

Verdier specialization via weak factorization

Let X?V be a closed embedding, with V?X nonsingular. We define a constructible function ψ _X,V on X, agreeing with Verdier’s specialization of the constant function 1 _V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–W?odarczyk. The main property of ψ _X,V is a compatibility with the specialization of the Chern class of the complement V?X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when X is the zero-locus of a function on V. Our definition has a straightforward counterpart Ψ _X,V in a motivic group. The function ψ _X,V and the corresponding Chern class c _SM(ψ _X,V ) and motivic aspect Ψ _X,V all have natural ‘monodromy’ decompositions, for any X?V as above. The definition also yields an expression for Kai Behrend’s constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.     

The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for ∝ λ (dx)6Pⁿ(x, ·) - π6 to be of order o(ψ(n)^-1), where Pⁿ (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, π is the invariant probability measure, λ an initial distribution and ψ belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order ψ, which in a countable state space is equivalent to $\text{Σ ψ(n)}\text{P}{}_{\mathrm{i}}\text{{τ}{}_{\mathrm{i}}\text{?n} <∞}$ for some i, where τ_i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order ψ it suffices that a corresponding condition is satisfied by a small set.We apply these results to non-singular renewal measures on $\text{R}$ providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies ∝ tF(dt) 0; > 0 and ∝ ψ(t)(1- F(t))dt< ∞.