On a class of strongly contractive quadratic recurrent systems

We consider a class of nonlinear recurrent systems of the form \( {\Lambda_p} = \frac{1}{p}\sum\limits_{{p_1} = 1}^{p - 1} {f\left( {\frac{{{p_1}}}{p}} \right){\Lambda_{{p_1}}}{\Lambda_{p - {p_1}}}} \), p > 1, where f is a given function on the interval [0, 1] and Λ₁ = x is an adjustable real-valued parameter. Under some suitable assumptions on the function f, we show that there exists an initial value x ^* for which Λ _p = Λ _p (x ^* ) → const as p → ∞. More precise asymptotics of Λ _p is also derived.     

A note on K<Emphasis Type="Italic">ä</Emphasis>hler manifolds with almost nonnegative bisectional curvature

In this note, we prove the following result. There is a positive constant ε(n, Λ) such that if M ⁿ is a simply connected compact Kähler manifold with sectional curvature bounded from above by Λ, diameter bounded from above by 1, and with holomorphic bisectional curvature H ≥ ?ε(n, Λ), then M ⁿ is diffeomorphic to the product M ₁ × ? × M _k , where each M _i is either a complex projective space or an irreducible Kähler–Hermitian symmetric space of rank ≥ 2. This resolves a conjecture of Fang under the additional upper bound restrictions on sectional curvature and diameter.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

Let Λ={λ ₁,…,λ _p } be a given set of distinct real numbers. This work deals with the problem of constructing a real matrix A of order n such that each element of Λ is a Pareto eigenvalue of A, that is to say, for all k∈{1,…,p} the complementarity system

$x\geq \mathbf{0}_n,\quad Ax-\lambda_k x\geq \mathbf{0}_n,\quad \langle x, Ax-\lambda_k x\rangle = 0$

admits a nonzero solution x∈? ⁿ .

     

Endpoint asymptotics for Müntz–Legendre polynomials

Müntz–Legendre polynomials L _n (Λ;x) associated with a sequence Λ={λ _k } are obtained by orthogonalizing the system \((x^{\lambda_{0}},x^{\lambda_{1}},x^{\lambda_{2}},\dots)\) in L ₂[0,1] with respect to the Legendre weight. Under very mild conditions on Λ, we establish the endpoint asymptotics close to x=1. The main result is

$\lim_{n\to\infty}L_n\left(1-\frac{y^2}{4S_n}\right)=J_0\big(|y|\big)$

where \(S_{n}=\sum_{k=0}^{n-1}(2\lambda_{k}+1)+\frac{2\lambda_{n}+1}{2}\) and J ₀ is the Bessel function of order 0.

     

Results of Diophantine approximation by unlike powers of primes

Let k be an integer with k ≥ 6: Suppose that λ₁, λ₂,..., λ₅ be nonzero real numbers not all of the same sign, satisfying that λ₁/λ₂ is irrational, and suppose that η is a real number. In this paper, for any ε > 0; we consider the inequality |λ₁p₁ + λ₂p ₂ ² + λ₃p ₃ ³ + λ₄p ₄ ⁴ + λ₅p ₅ ^k + η | < (max p_j)^-σ(k)+ε has infinitely many solutions in prime variables p₁, p₂,...,p₅, where σ(k) depends on k. Our result gives an improvement of the recent result. Furthermore, using the similar method in this paper, we can refine some results on Diophantine approximation by unlike powers of primes, and get the related problem.     

Towering Phenomena for the Yamabe Equation on Symmetric Manifolds

Let (M, g) be a compact smooth connected Riemannian manifold (without boundary) of dimension N ≥ 7. Assume M is symmetric with respect to a point ξ ₀ with non-vanishing Weyl’s tensor. We consider the linear perturbation of the Yamabe problem

$$ (P_{\epsilon })\qquad -\mathcal {L}_{g} u+\epsilon u=u^{\frac {N+2}{N-2}}\ \text { in }\ (M,g) . $$

We prove that for any k ∈ ?, there exists ε _k > 0 such that for all ε ∈ (0, ε _k ) the problem (P _?? ) has a symmetric solution u _ε , which looks like the superposition of k positive bubbles centered at the point ξ ₀ as ε → 0. In particular, ξ ₀ is a towering blow-up point.

     

Asymptotic behavior on (−∞, 0) of the spectral measures of a family of singular Sturm-Liouville operators

We find the asymptotics as λ/? → ?∞ of the density of the spectral measure of the Sturm-Liouville operator in L ²(0,+∞) generated by the expression ?y″ + ?q(x)y, ? > 0, with the boundary condition y(0) cos α+y′(0) sinα = 0. The potential q(x) tends to ?∞ as x → +∞ and is assumed to satisfy the Sears condition and some additional regularity conditions.     

Singularly Perturbed Parabolic Problems with Multidimensional Boundary Layers

The first boundary value problem for a multidimensional parabolic differential equation with a small parameter ε multiplying all derivatives is studied. A complete (i.e., of any order with respect to the parameter) regularized asymptotics of the solution is constructed, which contains a multidimensional boundary layer function that is bounded for x = (x¹, x₂) = 0 and tends to zero as ε → +0 for x ≠ 0. In addition, it contains corner boundary layer functions described by the product of a boundary layer function of the exponential type by a multidimensional parabolic boundary layer function.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Integrable Crystals and Restriction to Levi Subgroups Via Generalized Slices in the Affine Grassmannian

Let G be a connected reductive algebraic group over ?, and let Λ _G ⁺ be the monoid of dominant weights of G. We construct integrable crystals B^G(λ), λ ∈ Λ _G ⁺ , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group of G. We also construct tensor product maps \(P{\lambda _1},{\lambda _2}:{B^G}({\lambda _2}) \to {B^G}({\lambda _1} + {\lambda _2}) \cup \{ 0\} \) in terms of multiplication in generalized transversal slices. Let L ? G be a Levi subgroup of G. We describe the functor Res _L ^G : Rep(G) → Rep(L) of restriction to L in terms of the hyperbolic localization functors for generalized transversal slices.     

Modified Kantorovich theorem and asymptotic approximations of solutions to singularly perturbed systems of ordinary differential equations

The functional equation f(x,ε) = 0 containing a small parameter ε and admitting regular and singular degeneracy as ε → 0 is considered. By the methods of small parameter, a function x _n ⁰(ε) satisfying this equation within a residual error of O(ε ⁿ⁺¹) is found. A modified Newton’s sequence starting from the element x _n ⁰(ε) is constructed. The existence of the limit of Newton’s sequence is based on the NK theorem proven in this work (a new variant of the proof of the Kantorovich theorem substantiating the convergence of Newton’s iterative sequence). The deviation of the limit of Newton’s sequence from the initial approximation x _n ⁰(ε) has the order of O(ε ⁿ⁺¹), which proves the asymptotic character of the approximation x _n ⁰(ε). The method proposed is implemented in constructing an asymptotic approximation of a system of ordinary differential equations on a finite or infinite time interval with a small parameter multiplying the derivatives, but it can be applied to a wider class of functional equations with a small parameters.     

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ¹ over a Hopf algebra A which are quotients of the augmentation ideal A ⁺ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation Open image in new window provides a bicovariant differential graded algebra on A. We introduce Λ _{m a x} providing the maximal prolongation, while the canonical braided-exterior algebra Λ _min = B _?(Λ¹) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ? by super-braided Fourier transform on B _?(Λ¹) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S ₃] with its 3D calculus and obeying the q-Hecke relation ?² = 1 + (q ? q ^?1)? in middle degree on k _q [S L ₂] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra \(\mathcal {L}\subseteq A\) defines a sub-braided Lie algebra and \({\Lambda }^{1}\subseteq \mathcal {L}^{*}\) provides the required data A ⁺ → Λ¹.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm–Liouville Operator

For semicontinuous summation methods generated by Λ = {λ_n(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λ_n(h) = exp(?u^α(n)h) are obtained. An application to a generalized heat equation is given.     

On Homogenization for Non-Self-Adjoint Periodic Elliptic Operators on an Infinite Cylinder

We consider an operator A^ε on L₂(\({\mathbb{R}^{{d_1}}} \times {T^{{d_2}}}\)) (d₁ is positive, while d₂ can be zero) given by A^ε = ?div A(ε^?1x₁,x₂)?, where A is periodic in the first variable and smooth in a sense in the second. We present approximations for (A^ε ? μ)^?1 and ?(A^ε ? μ)^?1 (with appropriate μ) in the operator norm when ε is small. We also provide estimates for the rates of approximation that are sharp with respect to the order.