The reduced expressions in a Coxeter system with a strictly complete Coxeter graph

Let (W,S)$(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red(z)$\mathrm{Red}(z)$ of all reduced expressions for any z∈W$z\in W$. By associating each bc-expression to a certain symbol, we describe the set Red(z)$\mathrm{Red}(z)$ and compute its cardinal |Red(z)|$|\mathrm{Red}(z)|$ in terms of symbols. An explicit formula for |Red(z)|$|\mathrm{Red}(z)|$ is deduced, where the Fibonacci numbers play a crucial role.     

A verified method for bounding clusters of zeros of analytic functions

In this paper, we propose a verified method for bounding clusters of zeros of analytic functions. Our method gives a disk that contains a cluster of m   zeros of an analytic function f(z)$f(z)$. Complex circular arithmetic is used to perform a validated computation of n  -degree Taylor polynomial p(z)$p(z)$ of f(z)$f(z)$. Some well known formulae for bounding zeros of a polynomial are used to compute a disk containing a cluster of zeros of p(z)$p(z)$. A validated computation of an upper bound for Taylor remainder series of f(z)$f(z)$ and a lower bound of p(z)$p(z)$ on a circle are performed. Based on these results, Rouché's theorem is used to verify that the disk contains the cluster of zeros of f(z)$f(z)$. This method is efficient in computation of the initial disk of a method for finding validated polynomial factor of an analytic function. Numerical examples are presented to illustrate the efficiency of the proposed method.     

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if −A   generates a _C0$C_0$-semigroup on a Hilbert space, then for each τ>0$\tau >0$ the operator A   has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy f(z)=O(e^−τRe(z))$f(z)=O(e^{-\tau \mathrm{Re}(z)})$ as |z|→∞$|z|\to \infty$. The bound of this calculus grows at most logarithmically as τ↘0$\tau \searrow 0$. As a consequence, f(A)$f(A)$ is a bounded operator for each holomorphic function f (on a right half-plane) with polynomial decay at ∞. Then we show that each semigroup generator has a so-called (strong) m  -bounded calculus for all m∈N$m\in \mathbb{N}$, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called γ-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.     

Characterizing sequences for precompact group topologies

Motivated from [31], call a precompact group topology τ on an abelian group G ss-precompact (abbreviated from single sequence precompact  ) if there is a sequence u=(_un)$\mathbf{u}=(u_n)$ in G such that τ is the finest precompact group topology on G   making u=(_un)$\mathbf{u}=(u_n)$ converge to zero. It is proved that a metrizable precompact abelian group (G,τ)$(G,\tau )$ is ss-precompact iff it is countable. For every metrizable precompact group topology τ on a countably infinite abelian group G there exists a group topology η such that η is strictly finer than τ   and the groups (G,τ)$(G,\tau )$ and (G,η)$(G,\eta )$ have the same Pontryagin dual groups (in other words, (G,τ)$(G,\tau )$ is not a Mackey group in the class of maximally almost periodic groups).     

On spaces of modular forms spanned by eta-quotients

An eta-quotient of level N   is a modular form of the shape f(z)=_∏δ|Nη(δz)^_rδ$f(z)={\prod }_{\delta |N}\eta {(\delta z)}^{r_{\delta }}$. We study the problem of determining levels N   for which the graded ring of holomorphic modular forms for _Γ0(N)${\Gamma }_0(N)$ is generated by (holomorphic, respectively weakly holomorphic) eta-quotients of level N  . In addition, we prove that if f(z)$f(z)$ is a holomorphic modular form that is non-vanishing on the upper half plane and has integer Fourier coefficients at infinity, then f(z)$f(z)$ is an integer multiple of an eta-quotient. Finally, we use our results to determine the structure of the cuspidal subgroup of _J0(2^k)(Q)$J_0(2^k)(\mathbb{Q})$.     

Test ideals of non-principal ideals: Computations,jumping numbers,alterations and division theorems

Given an ideal a⊆R$\mathfrak{a}\subseteq R$ in a (log) Q$\mathbb{Q}$-Gorenstein F  -finite ring of characteristic p>0$p>0$, we study and provide a new perspective on the test ideal τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ for a real number t>0$t>0$. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F  -jumping numbers of τ(R,a^t)$\tau (R,{\mathfrak{a}}^t)$ as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.     

Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.     

Selections and their absolutely continuous invariant measures

Let I=[0,1]$I=[0,1]$ and let P be a partition of I   into a finite number of intervals. Let _τ1${\tau }_1$, _τ2${\tau }_2$; I→I$I\to I$ be two piecewise expanding maps on P  . Let G⊂I×I$G\subset I\times I$ be the region between the boundaries of the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$. Any map τ:I→I$\tau :I\to I$ that takes values in G is called a selection of the multivalued map defined by G  . There are many results devoted to the study of the existence of selections with specified topological properties. However, there are no results concerning the existence of selection with measure-theoretic properties. In this paper we prove the existence of selections which have absolutely continuous invariant measures (acim). By our assumptions we know that _τ1${\tau }_1$ and _τ2${\tau }_2$ possess acims preserving the distribution functions F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$. The main result shows that for any convex combination F   of F⁽¹⁾$F^{(1)}$ and F⁽²⁾$F^{(2)}$ we can find a map η   with values between the graphs of _τ1${\tau }_1$ and _τ2${\tau }_2$ (that is, a selection) such that F is the η-invariant distribution function. Examples are presented. We also study the relationship of the dynamics of our multivalued maps to random maps.     

A sharp upper bound on the signless Laplacian spectral radius of graphs

Let G be a simple connected graph of order n   with degree sequence _d1,_d2,…,_dn$d_1,d_2,\dots ,d_n$ in non-increasing order. The signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ of G   is the largest eigenvalue of its signless Laplacian matrix Q(G)$Q(G)$. In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G))$\rho (Q(G))$ in terms of _di$d_i$, which improves and generalizes some known results.     

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are _A∞$A_{\infty }$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg?(_∂φ(f(z))/z)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.