H2 Convergence of Solutions of a Biharmonic Problem on a Truncated Convex Sector Near the Angle π

We consider a biharmonic problem Δ²u_ω = f_ω with Navier type boundary conditions u_ω = Δu_ω = 0, on a family of truncated sectors Ω_ω in ?² of radius r, 0 < r < 1 and opening angle ω, ω ∈ (2π/3, π] when ω is close to π. The family of right-hand sides (f_ω)_{ω∈(2π/3, π]} is assumed to depend smoothly on ω in L²(Ω_ω). The main result is that u_ω converges to u_π when ω → π with respect to the H²-norm. We can also show that the H²-topology is optimal for such a convergence result.

     

Cut‐elimination Theorems for Some Infinitary Modal Logics

In this article, a cut‐free system TLM_ω1 for infinitary propositional modal logic is proposed which is complete with respect to the class of all Kripke frames.The system TLM_ω1 is a kind of Gentzen style sequent calculus, but a sequent of TLM_ω1 is defined as a finite tree of sequents in a standard sense. We prove the cut‐elimination theorem for TLM_ω1 via its Kripke completeness.     

The Qualitative Behaviour of Newton Flows for Weierstrass’ ℘-Functions

We study the continuous, desingularized Newton method for Weierstrass' ?-functions. This leads to a family of autonomous differential equations in the plane, which depends on two complex parameters ω ₁ and ω ₂. For the associated flows there are, up to conjugacy, precisely three possibilities. These are determined by the form of the parallelogram spanned by ω ₁ and ω ₂: square, rectangular but not square, and non-rectangular.     

Derivations,automorphisms, and representations of complex ω-Lie algebras

Let (𝔤,ω) be a finite-dimensional non-Lie complex ω-Lie algebra. We study the derivation algebra Der(𝔤) and the automorphism group Aut(𝔤) of (𝔤,ω). We introduce the notions of ω-derivations and ω-automorphisms of (𝔤,ω) which naturally preserve the bilinear form ω. We show that the set Der_ω(𝔤) of all ω-derivations is a Lie subalgebra of Der(𝔤) and the set Aut_ω(𝔤) of all ω-automorphisms is a subgroup of Aut(𝔤). For any three-dimensional and four-dimensional nontrivial ω-Lie algebra 𝔤, we compute Der(𝔤) and Aut(𝔤) explicitly, and study some Lie group properties of Aut(𝔤). We also study representation theory of ω-Lie algebras. We show that all three-dimensional nontrivial ω-Lie algebras are multiplicative, as well as we provide a four-dimensional example of ω-Lie algebra that is not multiplicative. Finally, we show that any irreducible representation of the simple ω-Lie algebra C_α(α≠0,?1) is one-dimensional.     

Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols

It is known that for a tridiagonal Toeplitz matrix, having on the main diagonal the constant a₀ and on the two first off‐diagonals the constants a₁(lower) and a₋₁(upper), which are all complex values, there exist closed form formulas, giving the eigenvalues of the matrix and a set of associated eigenvectors. For example, for the 1D discrete Laplacian, this triple is (a₀,a₁,a₋₁)=(2,−1,−1). In the first part of this article, we consider a tridiagonal Toeplitz matrix of the same form (a₀,a_ω,a_−ω), but where the two off‐diagonals are positioned ω steps from the main diagonal instead of only one. We show that its eigenvalues and eigenvectors can also be identified in closed form and that interesting connections with the standard Toeplitz symbol are identified. Furthermore, as numerical evidences clearly suggest, it turns out that the eigenvalue behavior of a general banded symmetric Toeplitz matrix with real entries can be described qualitatively in terms of the symmetrically sparse tridiagonal case with real a₀, a_ω=a_−ω, ω=2,3,…, and also quantitatively in terms of those having monotone symbols. A discussion on the use of such results and on possible extensions complements the paper.     

Fragments of Martin's Maximum in generic extensions

We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly (ω₁ + 1)‐game‐closed forcings. PFA can be destroyed by a strongly (ω₁ + 1)‐game‐closed forcing but not by an ω₂‐closed. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

Long Borel hierarchies

We show that there is a model of ZF in which the Borel hierarchy on the reals has length ω₂. This implies that ω₁ has countable cofinality, so the axiom of choice fails very badly in our model. A similar argument produces models of ZF in which the Borel hierarchy has exactly λ + 1 levels for any given limit ordinal λ less than ω₂. We also show that assuming a large cardinal hypothesis there are models of ZF in which the Borel hierarchy is arbitrarily long. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Arens multiplication on Banach algebras related to locally compact semigroups

Let S be a locally compact semigroup, let ω be a weight function on S, and let M_a (S, ω) be the weighted semigroup algebra of S. Let L^∞₀ (S;M_a (S, ω)) be the C^*‐algebra of allM_a (S, ω)‐measurable functions g on S such that g /ω vanishes at infinity. We introduce and study an Arens multiplication on L^∞₀ (S;M_a (S, ω))^* under which M_a (S, ω) is a closed ideal. We show that the weighted measure algebra M (S, ω) plays an important role in the structure of L^∞₀ (S;M_a (S, ω))^*. We then study Arens regularity of L^∞₀ (S;M_a (S, ω))^* and ist relation with Arens regularity of M_a (S, ω), M (S, ω) and the discrete convolution algebra ℓ¹(S, ω). As the main result, we prove that L^∞₀ (S;M_a (S, ω))^* is Arens regular if and only if S is finite, or S is discrete and Ω is zero cluster. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zolotarevω-Polynomials inWH[0, 1]

The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblemwhereω(t) is a concave modulus of continuity,r, m: 1?m?r,are integers, andB?B₀(r, m, ω). We show that theextremal functionsZ_Bhaver+1 points of alternance andthe full modulus of continuity ofZ^(r)_B: ω(Z^(r)_B; t)=ω(t) for allt∈[0, 1]. This generalizesthe Karlin's result on the extremality of classical Zolotarevpolynomials in the problem () forω(t)=tand allB?B_r.     

A Frequency Selection Criterion in Spatially Developing Flows

The possible existence of global modes or self-excited linear resonances in spatially developing systems is explored within the framework of the WKBJ approximation. It is shown that the existence and properties of the dominant global mode may be deduced from the variations of the local absolute frequency ω₀ with distance X. The main results are summarized in two theorems: (1) A system with no region of absolute instability does not sustain temporally growing global modes with an O(1) growth rate. (2) If the singularity X, closest to the real X-axis of the complex function ω₀(X) is a saddle point, the most unstable global mode has, to leading order in the WKBJ approximation, a complex frequency ω₀(X_s). Thus, it will be temporally growing only if ω₀(X_s) is positive.     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

Definitive Conditions on Maxwell's Tangential E or H for Solutions to His Equations Inside a Bounded Region

Let U be a connected, closed, bounded region in ℝ³ with smooth boundary 𝛛U. Consider Maxwell's equations on U for smooth fields and smooth sources, which are time harmonic, i.e., the fields E,B,H,D and the current density J and charge density ρ on U temporally varying as e^−iωt. We assume ω ≠ 0; when it is real, ω is the frequency of the pure oscillation; when ω is not real, one has exponentially decreasing or increasing oscillatory fields. We treat very general electric permittivity ɛ_ω and magnetic permeability μ_ω, which may both depend upon ω. Both can be arbitrary, time-independent, smoothly varying with position and described by complex Hermitian tensors subject only to weak positivity conditions specified below at each point of U. In this paper necessary and sufficient conditions are found for the tangential components, E_T or H_T on 𝛛U, to be realized by solutions of Maxwell's equations on all of U with these given sources J, ρ in the time-harmonic realm with ω ≠ 0. Also, all solutions E,H on U with specified E_T, H_T, respectively, satisfying the necessary finite-dimensional conditions are derived by an effective natural algorithm. In the electric E_T case, the positivity condition is this: the complex Hermitian matrix μ_ω(p) is to be positive definite while only the real part of ɛ_ω(p), i.e., Re(ɛ_ω(p)), necessarily real symmetric, need be positive definite. In the magnetic-type H_T case, the two roles are reversed. These positivity conditions are more general than those commonly considered. In the final section Maxwell's equations and all the present results are reformulated yet more generally in a context of compact, smooth, Riemannian manifolds with boundary. Sample examples include periodic lattices and wave guides. © 2019 Wiley Periodicals, Inc.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Double‐mode modeling of nonlinear flexural vibration analysis for a symmetric rectangular honeycomb sandwich thin panel by the homotopy analysis method

Nonlinear flexural vibration of a symmetric rectangular honeycomb sandwich thin panel with simply supported along all four edges is studied in this paper. The nonlinear governing equations of the symmetric rectangular honeycomb sandwich panel subjected to transverse excitations are simplified to a set of two ordinary differential equations by the Galerkin method. Based on the homotopy analysis method, the average equations of the primary resonance and harmonic resonance are obtained. The influence of structural parameters, the transverse exciting force amplitude, and transverse damping to the symmetric rectangular honeycomb sandwich panel are discussed by using the analytic approximation method. Compared with the results obtained by single‐mode modeling technique, the results obtained by double‐mode modeling technique change the softening and hardening nonlinear characteristics when Ω ≈ ω₁, ω₁/3, and ω₂/3.     

Reflection of Long Game Formulas

We study game formulas the truth of which is determined by a semantical game of uncountable length. The main theme is the study of principles stating reflection of these formulas in various admissible sets. This investigation leads to two weak forms of strict-II¹₁ reflection (or ∑₁-compactness). We show that admissible sets such as H(ω₂) and L_ω2 which fail to have strict-II¹₁ reflection, may or may not, depending on set-theoretic hypotheses satisfy one or both of these weaker forms. Mathematics Subject Classification : 03C70, 03C75.     

On Internal Resonance of Two Damped Oscillators

The damped motion of two oscillators with natural frequencies ω₁ and ω₂ is studied on the assumptions that: the oscillators are uncoupled for infinitesimal displacements and quadratically coupled for finite displacements; the frequencies are approximately in the ratio 2:1, such that ω₂?2ω₁=O(εω₁), where ε is a dimensionless measure of the displacements; the logarithmic decrements of the two modes, λ₁ and λ₁, are O(ε). The motion may be described by modulated sine waves with carrier frequencies ω_1,2 and slowly varying energies and phases that satisfy four first-order, nonlinear differential equations. These equations admit one invariant and may be reduced to two first-order equations if ω₂=2ω₁ and λ_1,2<0; they admit two invariants and can be completely integrated in terms of elliptic functions if ω₂=2ω₁ and 2λ₂=λ₁. Numerical results are presented for typical parametric combinations.