A New Class of Harmonic Measure Distribution Functions

Let Ω be a planar domain containing 0. Let h _Ω (r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h _Ω is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X _n such that \(h_{X_{n}}\) converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h _Ω , where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X _n . Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h _Ω for a unique symmetric bounded simply connected domain Ω.     

A Class of Bipartite 2-Path-Transitive Graphs

A graph Γ is called half-arc-transitive if it’s automorphism group Aut Γ is transitive on the vertex set and edge set, but not on the arc set of the graph Γ, and it is called 2-path-transitive if Aut Γ is transitive on the set of the 2-paths. In this paper we construct a class of 2-path-transitive graphs from some symmetric groups, based on which a new class of half-arc-transitive graphs is given.     

Plane sections of centrally symmetric convex bodies

The note contains an example of three plane convex centrally symmetric figuresP ₁,P ₂,P ₃ such that no centrally symmetric 3-dimensional body has three coaxial central affinely equivalent toP ₁,P ₂,P ₃ respectively.     

Further Pieri-type formulas for the nonsymmetric Macdonald polynomial

The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial P _?? (z) are known explicitly. These formulas generalise the known r=1 case of the Pieri-type formulas for the nonsymmetric Macdonald polynomials E _?? (z). In this paper, we extend beyond the case r=1 for the nonsymmetric Macdonald polynomials, giving the full generalisation of the Pieri-type formulas for symmetric Macdonald polynomials. The decomposition also allows the evaluation of the generalised binomial coefficients $\tbinom{\eta }{\nu }_{q,t}$ associated with the nonsymmetric Macdonald polynomials.     

On the symmetric Laplace derivative

Let f:?R??R be integrable in a neighbourhood of x??R. If there are real numbers ?? ₀,?? ₂,??,?? _2n?2 such that $$\lim_{s\to\infty}s^{2n+1} \int_0^\delta e^{-st}\left[\frac{f(x+t)+f(x-t)}{2}-\sum_{i=0}^{n-1}\frac{t^{2i}}{(2i)!}\alpha_{2i}\right]\, dt$$ exists for some ??>0 then the limit is called the 2n-th symmetric Laplace derivative at x. There is a corresponding definition of (2n+1)-th symmetric Laplace derivative. It is shown that this derivative is a generalization of the symmetric d.l.V.P. derivative. Some properties of this derivative are studied.     

A canonical expansion of the product of two Stanley symmetric functions

We study the problem of expanding the product of two Stanley symmetric functions F _w ?F _u into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_{w}=\lim_{n\to\infty}\mathfrak{S}_{1^{n}\times w}$ , and study the behavior of the expansion of $\mathfrak {S}_{1^{n}\times w}\cdot \mathfrak {S}_{1^{n}\times u}$ into Schubert polynomials as n increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. We then study some other related stability properties, providing a second proof of the main result.     

Central Units,Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S _n in characteristic zero, and we show that they are units in very special instances.     

On complementability of subspaces in symmetric spaces with the Kruglov property

We show that, for a broad class of symmetric spaces on [0, 1], the complementability of the subspace generated by independent functions f _k (k = 1, 2,…) is equivalent to the complementability of the subspace generated by the disjoint translates $\bar f_k (t) = f_k (t - k + 1)\chi _{[k - 1,k]} (t)$ of these functions in some symmetric space Z _X ² on the semiaxis [0,∞). Moreover, if Σ _k=1 ^∞ m(supp f _k ) ? 1, then Z _X ² can be replaced by X itself. This result is new even in the case of L _p -spaces. A series of consequences is obtained; in particular, for the class of symmetric spaces, a result similar to a well-known theorem of Dor and Starbird on the complementability in L _p [0, 1] (1 ? p < ∞) of the subspace [f _k ] generated by independent functions provided that it is isomorphic to the space l _p is obtained.     

Symmetric quasi-norms of sums of independent random variables in symmetric function spaces with the Kruglov property

Let X be a symmetric Banach function space on [0, 1] and let E be a symmetric (quasi)-Banach sequence space. Let f = {f _k } _k=1 ⁿ , n ≥ 1 be an arbitrary sequence of independent random variables in X and let {e _k } _k=1 ^∞ ? E be the standard unit vector sequence in E. This paper presents a deterministic characterization of the quantity

$||||\sum\limits_{k = 1}^n {{f_k}{e_k}|{|_E}|{|_X}} $

in terms of the sum of disjoint copies of individual terms of f. We acknowledge key contributions by previous authors in detail in the introduction, however our approach is based on the important recent advances in the study of the Kruglov property of symmetric spaces made earlier by the authors. Authors acknowledge support from the ARC.

     

Symmetries on almost symmetric numerical semigroups

The notion of an almost symmetric numerical semigroup was given by V. Barucci and R. Fröberg in J. Algebra, 188, 418–442 (1997). We characterize almost symmetric numerical semigroups by symmetry of pseudo-Frobenius numbers. We give a criterion for H ^? (the dual of M) to be an almost symmetric numerical semigroup. Using these results we give a formula for the multiplicity of an opened modular numerical semigroup. Finally, we show that if H ₁ or H ₂ is not symmetric, then the gluing of H ₁ and H ₂ is not almost symmetric.     

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.     

Asymptotics of the Smallest Singular Value of a Class of Toeplitz-generated Matrices II

Square matrices of the form ${X_n = T_n + f_n(T_n^{-1})^*}$ , where T _n is a ${n \times n}$ invertible banded Toeplitz matrix and f _n some positive sequence are considered. Convergence via an order estimate is proven for the difference of ${\|X_n^{-1}\|}$ and a function depending only on f _n . Fredholmness of the infinite counterpart of T _n is shown to greatly affect this result. A correction of a proof in the paper on which the current research is based, is appended as well.     

Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

Asymptotics of Solutions of One Class of Linear Differential Equations of Arbitrary Order

In this paper, we study the problem on the asymptotic behavior of solutions of the equation l _n y = λy of arbitrary order (even or odd) with complex-valued coefficients in a neighborhood of the point +∞. The result obtained allows one to find defect numbers of the corresponding closed minimal symmetric differential operator.     

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.

     

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ₃-metric measuring the difference between expectations of sufficiently smooth functions, like |·|³, of a sum of independent random variables X ₁,..., X _n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X _i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X ₁,..., X _n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).     

Symmetric spaces with maximal projection constants

In this paper we introduce a special class of finite-dimensional symmetric subspaces of L₁, so-called regular symmetric subspaces. Using this notion, we show that for any k?2, there exist k-dimensional symmetric subspaces of L₁ which have maximal projection constant among all k-dimensional symmetric spaces. Moreover, L₁ is a maximal overspace for these spaces (see Theorems 4.4 and 4.5.) Also a new asymptotic lower bound for projection constants of symmetric spaces is obtained (see Theorem 5.3). This result answers the question posed in [12, p. 36] (see also [15, p. 38]) by H. Koenig and co-authors. The above results are presented both in real and complex cases.     

Functional Identities and Rings of Quotients

The fundamental theorem on functional identities states that a prime ring R with \(\deg (R)\ge d\) is a d-free subset of its maximal left ring of quotients Q _{m l} (R). We consider the question whether the same conclusion holds for symmetric rings of quotients. This indeed turns out to be the case for the maximal symmetric ring of quotients Q _{m s} (R), but not for the symmetric Martindale ring of quotients Q _s (R). We show, however, that if the maps from the basic functional identities have their ranges in R, then the maps from their standard solutions have their ranges in Q _s (R). We actually prove a more general theorem which implies both aforementioned results. Its proof is somewhat shorter and more compact than the standard proof used for establishing d-freeness in various situations.