Structural properties of Riordan matrices and extending the matrices

We consider an infinite lower triangular matrix L=[?_n,k]_n,k∈N0 and a sequence Ω=(ω_n)_n∈N0 called the (a,b)-sequence such that every element ?_n+1,k+1 except lying in column 0 can be expressed as

     

Modified boundary element method for problems on oscillations of flat membranes

In the space L ² of real-valued measurable 2π-periodic functions that are square summable on the period [0, 2π], the Jackson-Stechkin inequality

$$E_n (f) \leqslant \mathcal{K}_n (\delta ,\omega )\omega (\delta ,f), f \in L^2 $$

, is considered, where E _n (f) is the value of the best approximation of the function f by trigonometric polynomials of order at most n and ω(δ, f) is the modulus of continuity of the function f in L ² of order 1 or 2. The value

$$\mathcal{K}_n (\delta ,\omega ) = \sup \left\{ {\frac{{E_n (f)}}{{\omega (\delta ,f)}}:f \in L^2 } \right\}$$

is found at the points δ = 2π/m (where m ∈ ?) for m ≥ 3n ² + 2 and ω = ω ₁ as well as for m ≥ 11n ⁴/3 ? 1 and ω = ω ₂.

     

Approximation orders of the unit in the <Emphasis Type="Italic">β</Emphasis>-dynamical systems

For any real number β > 1, let S _n (β) be the partial sum of the first n items of the β-expansion of 1. It was known that the approximation order of 1 by S _n (β) is β ^?n for Lebesgue almost all β > 1. We consider the size of the set of β > 1 for which 1 can be approximated with the other orders \({\beta^{-\varphi(n)}}\) , where \({\varphi}\) is a positive function defined on \({\mathbb N}\) . More precisely, the size of the sets

$$\left\{\beta\in \mathfrak{B}:\limsup_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

and

$$\left\{\beta\in \mathfrak{B}:\liminf_{n\rightarrow\infty}\frac{\log_{\beta}(1-S_n(\beta))}{\varphi(n)}=-1\right\}$$

are determined, where \({\mathfrak{B}=\{ \beta>1:\beta \text{ is not a simple Parry number}\}}\) .

     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and

     

Hamiltonian cycles in Dirac graphs

We prove that for any n-vertex Dirac graph (graph with minimum degree at least n/2) G=(V,E), the number, Ψ(G), of Hamiltonian cycles in G is at least

$exp_2 [2h(G) - n\log e - o(n)],$

where h(G)=maxΣ _e x _e log(1/x _e ), the maximum over x: E → ?⁺ satisfying Σ _e?υ x _e = 1 for each υ ∈ V, and log =log₂. (A second paper will show that this bound is tight up to the o(n).)

We also show that for any (Dirac) G of minimum degree at least d, h(G) ≥ (n/2) logd, so that Ψ(G) > (d/(e + o(1))) ⁿ . In particular, this says that for any Dirac G we have Ψ(G) > n!/(2 + o(1)) ⁿ , confirming a conjecture of G. Sárközy, Selkow, and Szemerédi which was the original motivation for this work.     

Mapping properties of certain oscillatory integrals on modulation spaces

Suppose β1 α1 ≥0,β2 α2 ≥ 0 and(k,j) ∈R2. In this paper, we mainly investigate the mapping properties of the operator T_αβf(x,y,z)=∫_Q~2f(x-t,y-s,z-t~ks~j)e~(-2πit-β1_s-β2)t~(-1-α1)s~(-1-α2)dtds on modulation spaces, where Q~2 = [0,1] x [0,1] is the unit square in two dimensions.     

Quasi-Periodic Solutions for Non-Autonomous mKdV Equation

In this paper, we consider the non-autonomous modified Korteweg-de Vries (mKdV) equation

$${u_t} = {u_{xxx}} - 6f\left( {\omega t} \right){u^2}{u_x},x \in \mathbb{R}/2\pi \mathbb{Z}$$

, where f(ωt) is real analytic and quasi-periodic in t with frequency vector ω = (ω₁,ω₂, · · ·; ω _m ). Basing on an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field, we obtain the existence of Cantor families of smooth quasi-periodic solutions.

     

Matrix refinement type equations

Taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the Fourier transform of an L¹-solution of the following random matrix refinement type equation

f(x)=_∫Ω|detL(ω)|C(ω)f(L(ω)x-M(ω))P(dω),     

Functional Integrals for the Bogoliubov Gaussian Measure: Exact Asymptotic Forms

We prove theorems on the exact asymptotic forms as u → ∞ of two functional integrals over the Bogoliubov measure μ_B of the forms

$$\int_{C[0,\beta ]} {[\int_0^\beta {|x(t){|^p}dt{]^u}d{\mu _B}(x)} } ,\;\int_{C(0,\beta )} {\exp \left\{ {\mu {{(\int_0^\beta {|x(t){|^p}dt} )}^{a/p}}} \right\}d{\mu _B}(x)} $$

for p = 4, 6, 8, 10 with p > p0, where p0 = 2+4π²/β²ω² is the threshold value, β is the inverse temperature, ω is the eigenfrequency of the harmonic oscillator, and 0 < α < 2. As the method of study, we use the Laplace method in Hilbert functional spaces for distributions of almost surely continuous Gaussian processes.

     

Two-weight inequality for operator-valued positive dyadic operators by parallel stopping cubes

We study the operator-valued positive dyadic operator

$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$

where the coefficients {λ _Q : C → D} _Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.

In the two-weight case, we prove that the L _C ^p (σ) → L _D ^q (ω) boundedness of the operator T _λ( · σ) is characterized by the direct and the dual L ^∞ testing conditions:

$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$

,

$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$

.

Here L _C ^p (σ) and L _D ^q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.

In the unweighted case, we show that the L _C ^p (μ) → L _D ^p (μ) boundedness of the operator T _λ( · μ) is equivalent to the end-point direct L ^∞ testing condition:

$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$

.

This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way.     

Phase dynamics in the complex Ginzburg-Landau equation

For αβ>−1, stable time periodic solutions A(X,T)=A_qe^iqX+iω_qT are the locally preferred planform for the complex Ginzburg-Landau equation

     

Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Let −Dω(·,z)D+q be a differential operator in L²(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form

     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

On random perturbation of some intersective sets

Given a subset S of Z and a sequence I = (I_n)_n=1^∞ of intervals of increasing length contained in Z, let

     

Stability and unobstructedness of syzygy bundles

It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle E_d1,…,dn on P^N defined as the kernel of a general epimorphism

     

Asymptotic Properties of Orthogonal Polynomials with Respect to a Non-discrete Jacobi-Sobolev Inner Product

Let {Q _n ^(α,β) (x)} _n=0 ^∞ denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product

$\langle f,g\rangle=\int_{-1}^{1}f(x)g(x)d\mu_{\alpha,\beta}(x)+\lambda\int_{-1}^{1}f'(x)g'(x)d\nu_{\alpha,\beta}(x)$

where λ>0 and d μ _α,β(x)=(x?a)(1?x)^α?1(1+x)^β?1 dx, d ν _α,β(x)=(1?x) ^α (1+x) ^β dx with aα,β>0. Their inner strong asymptotics on (?1,1), a Mehler-Heine type formula as well as some estimates of the Sobolev norms of Q _n ^(α,β) are obtained.

     

Forcing notions in inner models

There is a partial order \({\mathbb{P}}\) preserving stationary subsets of ω ₁ and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω ₁ over V also collapses ω ₁ over \({V^{\mathbb{P}}}\) . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B.     

Estimation of the algorithmic complexity of classes of computable models

We estimate the algorithmic complexity of the index set of some natural classes of computable models: finite computable models (Σ ₂ ⁰ -complete), computable models with ω-categorical theories (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), prime models (Δ _ω ⁰ -complex Π _ω+2 ⁰ -set), models with ω ₁-categorical theories (Δ _ω ⁰ -complex Σ _ω+1 ⁰ -set. We obtain a universal lower bound for the model-theoretic properties preserved by Marker’s extensions (Δ _ω ⁰ .