A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials

Let??? and ?? = (?? ₁, . . . , ?? _k ) be partitions such that??? is obtained from ?? by adding m parts of size r. Descouens and Morita proved algebraically that the modified Macdonald polynomials ${{\tilde{H}_\mu}(X; q, t)}$ satisfy the identity ${{\tilde{H}_\mu} = \tilde{H}_\nu \tilde{H}_{(r^m)}}$ when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ?? ?? _k and ${r \in \{1, 2\}}$ . This note gives a bijective proof of the formula for all r ?? ?? _k .     

On definability in some lattices of semigroup varieties

An identity of the form x ₁?x _n ??x _1?? x _2?? ?x _n?? where ?? is a non-trivial permutation on the set {1,??,n} is called a permutation identity. If u??v is a permutation identity, then ?(u??v) [respectively r(u??v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If $\mathcal{V}$ is a permutative variety, then $\ell=\ell(\mathcal{V})$ [respectively $r=r(\mathcal{V})$ ] is the least ? [respectively r] such that $\mathcal{V}$ satisfies a permutation identity ?? with ?(??)=? [respectively r(??)=r]. A?variety that consists of nil-semigroups is called a nil-variety. If ?? is a set of identities, then $\operatorname {var}\varSigma$ denotes the variety of semigroups defined by ??. If $\mathcal{V}$ is a variety, then $L (\mathcal{V})$ denotes the lattice of all subvarieties of $\mathcal{V}$ . For ?,r??0 and n>1 let $\mathfrak{B}_{\ell,r,n}$ denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where ?? is a permutation on the set {1,??,n}. We prove that for each permutative nil-variety $\mathcal{V}$ and each $\ell\ge\ell(\mathcal{V})$ and $r\ge r(\mathcal{V})$ there exists n>1 such that $\mathcal{V}$ is definable by a first-order formula in $L(\operatorname{var}{\mathfrak{B}}_{l,r,n})$ if ???r or $\mathcal{V}$ is definable up to duality in $L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})$ if ?=r.     

The best polynomial approximations of entire transcendental functions with generalized growth order in Banach spaces \mathcal{E}_p^{\prime}(G) and {\mathcal{E}_p}(G) , p ? 1

A theorem of the Hadamard type for entire transcendental functions f, which have a generalized ??-order of growth ?? _?? (f), has been obtained. This theorem connects the values $ \widetilde{M}\left( {f,r} \right)\;\left( {r > 1} \right) $ and the coefficients a _n (f) $ \left( {n \in {\mathbb{Z}_{+} }} \right) $ of the expansion of f in Faber series in a finite domain D whose boundary ?? belongs to the Al??per class. This result is the extension of a result obtained by M. N. Sheremeta onto a simply connected domain. The necessary and sufficient conditions for an analytic function $ f \in \mathcal{E}_p^{\prime}(G) $ or $ f \in {\mathcal{E}_p}(G)\;\left( {1 \leqslant p \leqslant \infty } \right) $ to be entire transcendental with a generalized ??-order of growth ?? _?? (f) are obtained. These conditions include the best polynomial approximations of the function f and determine the rate of their convergence to zero, as the degree of polynomials increases.     

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

Higher Order Coherent Pairs

In this paper, we study necessary and sufficient conditions for the relation $$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$ where {P _n (x)} _n??0 and {R _n (x)} _n??0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $\mathcal{U},\mathcal{V}$ , respectively, or associated with two positive Borel measures ?? ₀,?? ₁ supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.     

Existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval

In this paper we consider the following m-point fractional boundary value problem with p-Laplacian operator on infinite interval where 0<????1, 2<????3, $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional derivative, ?? _p (s)=|s| ^p?2 s,p>1, (?? _p )^?1=?? _q , $\frac{1}{p}+\frac{1}{q}=1$ . 0<?? ₁<?? ₂<?<?? _m?2<+??, ?? _i ??0, i=1,2,??,m?2 satisfies $0 <\sum_{i=1}^{m-2}\beta_{i}\xi_{i}^{\alpha-1} < \Gamma(\alpha)$ . We establish solvability of the above fractional boundary value problems by means of the properties of the Green function and some fixed-point theorems.     

On the Fourth Moment of Coefficients of Symmetric Square L-Function

Let f(z) be a holomorphic Hecke eigencuspform of weight k for the full modular group. Let ?? _f (n) be the nth normalized Fourier coefficient of f(z). Suppose that L(sym² f, s) is the symmetric square L-function associated with f(z), and $ \lambda _{sym^2 f} (n) $ (n) denotes the nth coefficient L(sym² f, s). In this paper, it is proved that $$ \sum\limits_{n \leqslant x} {\lambda _{sym^2 f}^4 (n)} = xP2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where P ₂(t) is a polynomial in t of degree 2. Similarly, it is obtained that $$ \sum\limits_{n \leqslant x} {\lambda _f^4 (n^2 )} = x\tilde P2(\log x) + O(x^{\frac{{79}} {{81}} + \varepsilon } ), $$ , where $ \tilde P_2 (t) $ is a polynomial in t of degree 2.     

Bounds for Zeros of Solutions of Second Order Differential Equations with Polynomial Coefficients

We consider the equation ${u''=P(z)u\;\;(z\in\mathbb{C})}$ where P(z) is a polynomial. Let z _k (u), k = 1, 2,... be the zeros of a solution u(z) to that equation. Bounds for the sums $$\sum_{k=1}^{j} \frac {1} {|z_k(u)|}\;(j=1, 2, \ldots)$$ are established. Some applications of these bounds are also considered.     

Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities

In this paper, we provide the Euler?CMaclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals $I[f]=\int^{b}_{a}f(x)\,dx$ , where f??C ^??(a,b) but can have general algebraic-logarithmic singularities at one or both endpoints. These integrals may exist either as ordinary integrals or as Hadamard finite part integrals. We assume that f(x) has asymptotic expansions of the general forms where $\widehat{P}(y),P_{s}(y)$ and $\widehat{Q}(y),Q_{s}(y)$ are polynomials in y. The ?? _s and ?? _s are distinct, complex in general, and different from ?1. They also satisfy The results we obtain in this work extend the results of a recent paper [A.?Sidi, Numer. Math. 98:371?C387, 2004], which pertain to the cases in which $\widehat{P}(y)\equiv0$ and $\widehat{Q}(y)\equiv0$ . They are expressed in very simple terms based only on the asymptotic expansions of f(x) as x??a+ and x??b?. The results we obtain in this work generalize, and include as special cases, all those that exist in the literature. Let $D_{\omega}=\frac{d}{d\omega}$ , h=(b?a)/n, where n is a positive integer, and define $\check{T}_{n}[f]=h\sum^{n-1}_{i=1}f(a+ih)$ . Then with $\widehat{P}(y)=\sum^{\hat{p}}_{i=0}{\hat{c}}_{i}y^{i}$ and $\widehat{Q}(y)=\sum^{\hat{q}}_{i=0}{\hat{d}}_{i}y^{i}$ , one of these results reads where ??(z) is the Riemann Zeta function and ?? _i are Stieltjes constants defined via $\sigma_{i}= \lim_{n\to\infty}[\sum^{n}_{k=1}\frac{(\log k)^{i}}{k}-\frac{(\log n)^{i+1}}{i+1}]$ , i=0,1,???.     

Generators and commutators in finite groups; abstract quotients of compact groups

The first part of the paper establishes results about products of commutators in a d-generator finite group G, for example: if H?G=??g ₁,??,g _r ?? then every element of the subgroup [H,G] is a product of f(r) factors of the form $[h_{1},g_{1}][h_{1}^{\prime},g_{1}^{-1}]\ldots\lbrack h_{r},g_{r}][h_{r}^{\prime },g_{r}^{-1}]$ with $h_{1},h_{1}^{\prime},\ldots,\allowbreak h_{r},h_{r}^{\prime }\in H$ . Under certain conditions on H, a similar conclusion holds with the significantly weaker hypothesis that G=H??g ₁,??,g _r ??, where f(r) is replaced by f ₁(d,r). The results are applied in the second part of the paper to the study of normal subgroups in finitely generated profinite groups, and in more general compact groups. Results include the characterization of (topologically) finitely generated compact groups which have a countably infinite image, and of those which have a virtually dense normal subgroup of infinite index. As a corollary it is deduced that a compact group cannot have a finitely generated infinite abstract quotient.     

The Zagier modification of Bernoulli numbers and a polynomial extension. Part I

The modified Bernoulli numbers $$ B_{n}^{*} = \sum_{r=0}^{n} \binom{n+r}{2r} \frac{B_{r}}{n+r}, \quad n > 0 $$ introduced by D. Zagier in 1998 are extended to the polynomial case by replacing B _r by the Bernoulli polynomials B _r (x). Properties of these new polynomials are established using the umbral method as well as classical techniques. The values of x that yield periodic subsequences $B_{2n+1}^{*}(x)$ are classified. The strange 6-periodicity of $B_{2n+1}^{*}$ , established by Zagier, is explained by exhibiting a decomposition of this sequence as the sum of two parts with periods 2 and 3, respectively. Similar results for modifications of Euler numbers are stated.     

Macdonald operators at infinity

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x ₁,x ₂,… and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N→∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x ₁,…,x _N . They are differential operators in terms of the power sum variables \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall–Littlewood symmetric functions of the variables x ₁,x ₂,…. Our result also yields elementary step operators for the Macdonald symmetric functions.     

Some properties of biorthogonal polynomials and their application to Padé approximations

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N?1/N],N ε ?, for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ wheref(z) is a function with known Padé approximants of the indicated orders,T _j [f;z] are Taylor polynomials of degreej for the functionf(z), and α _{m, M} = $\overline {1,M} $ are constants.     

Exponent of a finite group with a four-group of automorphisms

Let A be a group isomorphic with either S ₄, the symmetric group on four symbols, or D ₈, the dihedral group of order 8. Let V be a normal four-subgroup of A and ?? an involution in ${A\setminus V}$ . Suppose that A acts on a finite group G in such a manner that C _G (V)?=?1 and C _G (??) has exponent e. We show that if ${A\cong S_4}$ then the exponent of G is e-bounded and if ${A\cong D_8}$ then the exponent of the derived group G?? is e-bounded. This work was motivated by recent results on the exponent of a finite group admitting an action by a Frobenius group of automorphisms.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Commutators of BMO functions and degenerate Schr?dinger operators with certain nonnegative potentials

Let ${\mathcal{L}f(x)=-\frac{1}{\omega}\sum_{i,j} \partial_i(a_{i,j}(\cdot)\partial_jf)(x)+V(x)f(x)}$ with the non-negative potential V belonging to reverse H?lder class with respect to the measure ??(x)dx, where ??(x) satisfies the A ₂ condition of Muckenhoupt and a _i,j (x) is a real symmetric matrix satisfying ${\lambda^{-1}\omega(x)|\xi|^2\le \sum^n_{i,j=1}a_{i,j}(x)\xi_i\xi_j\le\lambda\omega(x)|\xi|^2. }$ We obtain some estimates for ${V^{\alpha}\mathcal{L}^{-\alpha}}$ on the weighted L ^p spaces and we study the weighted L ^p boundedness of the commutator ${[b, V^{\alpha} \mathcal{L}^{-\alpha}]}$ when ${b\in BMO_\omega}$ and 0?<??? ?? 1.     

Приближение алгебра ическими многочлена ми классов функций, являющихся д робными интегралами от суммируемых функц ий

Approximation in the mean (E _n(f)₁) by algebraic polynomials of order ≦n is studied in the paper, for classesW ₁ ^r of functionsf, which can be represented as $$f(x) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{^{r - 1} } } \varphi (t)dt,$$ where??L ₁-1, 1], ∥?∥₁≧1, (x-t) ₊ ^r1 =[max(0, x-t)]^r1, Г (r) stands for Euler's gamma-function. It is proved that for all realr≧1 and positive integersn≧[r]?1 the relation sup E_n(f)₁:f?W₁ ^r=∥(S_n)_r∥_t8, is valid, where $$(s_\Lambda )_{_r } (t) = \frac{1}{{\Gamma (r)}}\int\limits_{ - 1}^1 {(x - t)_ + ^{r - 1} } $$ sgn sin (n+2) arc cosx dx.     

Approximating the maximum size of a k-regular induced subgraph by an upper bound on the co-k-plex number

Let ?? _k and $ {\hat{\alpha }_k} $ denote respectively the maximum cardinality of a k-regular induced subgraph and the co-k-plex number of a given graph. In this paper, we introduce a convex quadratic programming upper bound on $ {\hat{\alpha }_k} $ , which is also an upper bound on ?? _k . The new bound denoted by $ {\hat{\upsilon }_k} $ improves the bound ?? _k given in [3]. For regular graphs, we prove a necessary and sufficient condition under which $ {\hat{\upsilon }_k} $ equals ?? _k . We also show that the graphs for which $ {\hat{\alpha }_k} $ equals $ {\hat{\upsilon }_k} $ coincide with those such that ?? _k equals ?? _k . Next, an improvement of $ {\hat{\upsilon }_k} $ denoted by $ {\hat{\vartheta }_k} $ is proposed, which is not worse than the upper bound ? _k for ?? _k introduced in [8]. Finally, some computational experiments performed to appraise the gains brought by $ {\hat{\vartheta }_k} $ are reported.     

Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem [10].     

On a problem due to Littlewood concerning polynomials with unimodular coefficients

Littlewood raised the question of how slowly $\lVert f_{n}\rVert_{4}^{4}-\lVert f_{n}\rVert_{2}^{4}$ (where $\lVert.\rVert _{r}$ denotes the L ^r norm on the unit circle) can grow for a sequence of polynomials f _n with unimodular coefficients and increasing degree. The results of this paper are the following. For $$g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k $$ the limit of $(\lVert g_{n}\rVert_{4}^{4}-\lVert g_{n}\rVert_{2}^{4})/\lVert g_{n}\rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=\sum_{j=0}^{n-1}\sum _{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} $$ the limit of $(\lVert h_{n}\rVert_{4}^{4}-\lVert h_{n}\rVert_{2}^{4})/\lVert h_{n}\rVert_{2}^{3}$ is shown to be 4/π ². No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists.