On the differential properties of continuous functions

We introduce and investigate some new differential properties of continuous functions by means of the geometrical properties of their derivatives.

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Про диференціальні властивості неперервних функцій

Введено та досліджено деякі нові диференціальні властивості неперервних функцій за допомогою геометричних властивостей їх похідних.

     

Elastic waveguides: history and the state of the art. II

In this paper, the normal modes of an elastic rectangular waveguide are analyzed. We retrace the key aspects of the almost 150-year history of this problem. Using the superposition method, we have obtained an analytical solution of the problem for four types of symmetry of the wave field. In addition, we have established important differences of the dispersion characteristics of normal modes in a rectangle from the Rayleigh–Lamb modes for an infinite plate and the Pochhammer–Chree modes for a cylinder. We give also an estimate of a series of approximate theories for a rectangular waveguide.

The numerical interpretation of the results of analysis is however necessary, and it is a degree of perfection which it would be very important to give to every application of analysis to the natural sciences. So long as it is not obtained, the solutions may be said to remain incomplete and useless, and the truth which it is proposed to discover is no less hidden in the formulas of analysis than it was in the physical problem itself.

                             J. Fourier [28, Sec. 13]

     

The Wirtinger-Steklov inequality between the norm of a periodic function and the norm of the positive cutoff of its derivative

We study the sharp constant in the inequality between the L _p -mean (p ≥ 0) of a 2π-periodic function with zero mean value and the L _q -norm (q ≥ 1) of the positive cutoff of its derivative. We obtain estimates of the constant from below for 0 ≤ p ≤ ∞ and from above for 1 ≤ p ≤ ∞ for an arbitrary 1 ≤ q ≤ ∞. We write out the values of the sharp constant in the cases p = 2, 1 ≤ q ≤ ∞ and p = ∞, 1 ≤ q ≤ ∞.     

Weak generators of the algebral ∞ and the commutant of a model operator

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H^∞. It is known that ϕ+BH^∞ is a weak^* generator of the algebra H^∞/BH^∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak^* generator of the algebra l^∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B₁·…·B_N, where N ∈ℕ, and B₁, …, B_N are Blaschke products satisfying condition (C), there exists a function ϕ∈H^∞ such that ϕ(Λ) is a weak^* generator of the algebra l^∞, and ϕ+BH^∞ is not a weak^* generator of the algebra H^∞/BH^∞. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85. Translated by M. F. Gamal'.     

On the distribution of values of the derivative of the Riemann zeta function at its zeros. I

Let ζ′(s) be the derivative of the Riemann zeta function ζ(s). A study on the value distribution of ζ′(s) at the non-trivial zeros ρ of ζ(s) is presented. In particular, for a fixed positive number X, an asymptotic formula and a non-trivial upper bound for the sum Σ_{0<Im ρ≤T} ζ′(ρ)X ^ρ as T → ∞ are given. We clarify the dependence on the arithmetic nature of X.     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Indecomposability and the number of links

Let (ℋ, ℳ) be a linear matrix problem induced from a finite dimensional algebra ∧. Then anṉ ×ṉ matrix M in R(ℋ, ℳ) is indecomposable if and only if the number of links in the canonical formM ^(∞) of M is equal to. ℳ-dimṉ − 1. On the other hand, the dimension of the endomorphism ring of M is equal to ℋ-dimṉ − σ(M).     

Admissibility criteria for model subspaces with fast growth of the argument of the generating inner function

Let Θ be an inner function in the upper half-plane and let K_Θ = H² ⊖ ΘH² be the associated model subspace of the Hardy space H². We call a nonnegative function ω Θ-admissible if in the space K_Θ there exists a nonzero function f ∈ K_Θ such that |f| ≤ ω a.e. on ℝ. We give some sufficient conditions of Θ-admissibility for the case where Θ is a meromorphic function and arg Θ grows fast ((argΘ)′ tends to infinity). Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 55–84.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

An asymptotic model is found for the Neumann problem for the second-order differential equation with piecewise constant coefficients in a composite domain Ω∪ω, which are small, of order ε, in the subdomain ω. Namely, a domain Ω(ε) with a singular perturbed boundary is constructed, the solution for which provides a two-term asymptotic, that is, of increased accuracy O(ε²| log ε|^3/2), approximation to the restriction to Ω of the solution of the original problem. As opposed to other singularly perturbed problems, in the case of contrasting stiffness, the modeling requires the construction of a contour ∂Ω(ε) with ledges, i.e., with boundary fragments of curvature O(ε⁻¹). Bibliography: 33 titles.     

On the value set of the Ramanujan function

We give lower bounds on the number of distinct values of the Ramanujan function τ(n), n ≦ x, and on the number of distinct residues of τ(n), n ≦ x, modulo a prime ℓ. We also show that for any prime ℓ the values τ(n), n ≦ ℓ⁴, form a finite additive basis modulo ℓ. Received: 6 October 2004     

On the Modular Version of the Gluck-Wolf Theorem

Let N be a normal subgroup of a finite group G. Let ϕ be an irreducible Brauer character of N. Assume π is a set of primes and χ(1)/ϕ(1) is a π′-number of any χ∈IBr _p (G/ϕ). If p∤|G:N|, and N is p-solvable, then G/N has an abelian-by-metabelian Hall-π subgroup; If p∉π then G/N has a metabelian Hall-π subgroup. Received February 22, 2000, Accepted May 9, 2001     

The subgroups of the special linear group over a skew field that contain the group of diagonal matrices

For any (noncommutative) skew field T, the lattice of subgroups of the special linear group Λ=SL(n,T) that contain the subgroup Δ=SD(n,T) of diagonal matrices (with Dieudonné determinants equal to 1) is studied. It is established that for any subgroup H, Δ≤H≤Λ, there exists a uniquely determined unital net σ such that Λ(σ)≤H≤N(σ), where Λ(σ) is the net subgroup associated with the net σ and N(σ) is its normalizer in Λ. Bibliography: 11 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 91–103. Translated by Bui Xuan Hai.     

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

On the local convergence of a family of Euler-Halley type iterations with a parameter

This paper aims to study the local convergence of a family of Euler-Halley type methods with a parameter α for solving nonlinear operator equations under the second-order generalized Lipschitz assumption. The radius r _α of the optimal convergence ball and the error estimation of the method corresponding to α are estimated for each α ∈ ( − ∞ , + ∞ ). For each α > 0, we get r _α  ≥ r _− α and the upper bound of the error estimation of the method with α > 0 is not larger than the one with α < 0. For each α ≤ 0, we get the precise value of r _α , which is closely linked to the dynamical property of the method applied to a real or a complex function, and the optimal error estimation, which decreases when α→0⁻. Results show that the method corresponding to α is better than the one corresponding to − α for each α > 0 and the Chebyshev-Euler method is the best among all methods in the family with α ∈ ( − ∞ , 0] from the view of both safe choice of the initial point and error estimation.     

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem w_t=ℵ₀Δw+ℵ₁w-ℵ₂w|w|²,w|∂Ω₀=0 in the domain Ω₀={(x,y):0 ≤ x ≤ l₁,0 ≤ y ≤ l₂}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵ_j, j=0,1,2, are complex constants withRe ℵ_j > 0. We show that under a rather general choice of the parameters l₁ and l₂, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ₀ → 0 andRe ℵ₀ → 0. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 2, pp. 205–220, November, 2000.     

Selfadjoint extensions of the operator of the Dirichlet problem in a 3-dimensional region with an edge

Taking various viewpoints, we study the selfadjoint extensions $ \mathcal{A} $ \mathcal{A} of the operator A of the Dirichlet problem in a 3-dimensional region Ω with an edge Γ. We identify the infinite dimensional nullspace def A with the Sobolev space H ^−ϰ(Γ) on Γ with variable smoothness exponent −ϰ ∈ (−1, 0); while the selfadjoint extensions, with selfadjoint operators $ \mathcal{T} $ \mathcal{T} on the subspaces of H ^−ϰ(Γ). To the boundary value problem in the region with a “smoothed” edge we associate a concrete extension, which yields a more precise approximate solution to the singularly perturbed problem.     

Spectral stability of the Robin Laplacian

We consider the Robin Laplacian in two bounded regions Ω₁ and Ω₂ of ℝ ^N with Lipschitz boundaries and such that Ω₂ ⊂ Ω₁, and we obtain two-sided estimates for the eigenvalues λ _n,2 of the Robin Laplacian in Ω₂ via the eigenvalues λ _{n, 1} of the Robin Laplacian in Ω₁. Our estimates depend on the measure of the set difference Ω\Ω₂ and on suitably defined characteristics of vicinity of the boundaries ∂Ω₁ and ∂Ω₂, and of the functions defined on ∂Ω₁ and on ∂Ω₂ that enter the Robin boundary conditions.     

Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.