Universally prestarlike functions as convolution multipliers

Universally prestarlike functions (of order α ≤ 1) in the slit domain L:=\mathbbC\[1,￥]{\Lambda:=\mathbb{C}{\setminus}[1,\infty]} have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk \mathbbD{\mathbb{D}} (and other circular domains in \mathbbC{\mathbb{C}}). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with α fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.     

Relative universality and universality obtained by adding constants

A variety ${\mathbb{V}}${\mathbb{V}} is var-relatively universal if it contains a subvariety \mathbbW{\mathbb{W}} such that the class of all homomorphisms that do not factorize through any algebra in \mathbbW{\mathbb{W}} is algebraically universal. And \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} if adding α nullary operations to all algebras in \mathbbV{\mathbb{V}} gives rise to a class a\mathbbV{\alpha\mathbb{V}} of algebras that is algebraically universal. The first two authors have conjectured that any varrelative universal variety \mathbbV{\mathbb{V}} has an algebraically universal α-expansion a\mathbbV{\alpha\mathbb{V}} . This note contains a more general result that proves this conjecture.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

All 4-Edge-Connected HHD-Free Graphs are {\mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called \mathbbZ₃{\mathbb{Z}_3}-connected if for all b: V ? \mathbbZ₃{b: V \rightarrow \mathbb{Z}_3} with ?_{v ? V}b(v)=0{\sum_{v \in V}b(v)=0}, an orientation D = (V, A) of G has a \mathbbZ₃{\mathbb{Z}_3}-valued nowhere-zero flow f: A? \mathbbZ₃-{0}{f: A\rightarrow \mathbb{Z}_3-\{0\}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)} for all v ? V{v \in V}. We show that all 4-edge-connected HHD-free graphs are \mathbbZ₃{\mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \mathbbZ₃{\mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.     

NTRU over rings beyond $${\mathbb{Z}}$$

The NTRU cryptosystem is constructed on the base ring \mathbbZ{\mathbb{Z}} . We give suitability conditions on rings to serve as alternate base rings. We present an example of an NTRU-like cryptosystem based on the Eisenstein integers \mathbbZ[z₃]{\mathbb{Z}[\zeta_3]} , which has a denser lattice structure than \mathbbZ{\mathbb{Z}} for the same dimension, and which furthermore presents a more difficult lattice problem for lattice attacks, for the same level of decryption failure security.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Metrizability of the space of {\mathbb{R}} -places of a real function field

For n = 1, the space of ${\mathbb{R}}For n = 1, the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x₁,?, x_n){\mathbb{R}(x_1,\ldots, x_n)} is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of \mathbbR{\mathbb{R}} -places of the rational function field \mathbbR(x, y){\mathbb{R}(x, y)} is not metrizable. We explain how the proof generalizes to show that the space of \mathbbR{\mathbb{R}} -places of any finitely generated formally real field extension of \mathbbR{\mathbb{R}} of transcendence degree ≥ 2 is not metrizable. We also consider the more general question of when the space of \mathbbR{\mathbb{R}} -places of a finitely generated formally real field extension of a real closed field is metrizable.     

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

Singularities of secant maps of immersed surfaces

The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion $X\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c}The secant map of an immersion sends a pair of points to the direction of the line joining the images of the points under the immersion. The germ of the secant map of a generic codimension-c immersion X:\mathbb Rⁿ ? \mathbb R^n+cX\!\!:{\mathbb R}^n \to {\mathbb R}^{n+c} at the diagonal in the source is a \mathbb Z₂{\mathbb Z}_2 stable map-germ \mathbb R²ⁿ ? \mathbb R^n+c-1{\mathbb R}^{2n} \to {\mathbb R}^{n+c-1} in the following cases: (i) c≥ 2 and (2n,n + c − 1) is a pair of dimensions for which the \mathbb Z₂{\mathbb Z}_2 stable germs of rank at least n are dense, and (ii) for generically immersed surfaces (i.e., n = 2 and any c≥ 1). In the latter surface case the A^{\mathbb Z₂}{\mathcal A}^{{\mathbb Z}_2}-classification of germs of secant maps at the diagonal is described and it is related to the A{\mathcal A}-classification of certain singular projections of the surfaces.     

On some inequalities for Doob decompositions in Banach function spaces

Let ${\Phi : \mathbb{R} \to [0, \infty)}Let F: \mathbbR ? [0, ￥){\Phi : \mathbb{R} \to [0, \infty)} be a Young function and let f = (f_n)_n ? \mathbbZ₊{f = (f_n)_n\in\mathbb{Z}_{+}} be a martingale such that F(f_n) ? L₁{\Phi(f_n) \in L_1} for all n ? \mathbbZ₊{n \in \mathbb{Z}_{+}} . Then the process F(f) = (F(f_n))_n ? \mathbbZ₊{\Phi(f) = (\Phi(f_n))_n\in\mathbb{Z}_{+}} can be uniquely decomposed as F(f_n)=g_n+h_n{\Phi(f_n)=g_n+h_n} , where g=(g_n)_n ? \mathbbZ₊{g=(g_n)_n\in\mathbb{Z}_{+}} is a martingale and h=(h_n)_n ? \mathbbZ₊{h=(h_n)_n\in\mathbb{Z}_{+}} is a predictable nondecreasing process such that h ₀ = 0 almost surely. The main results characterize those Banach function spaces X such that the inequality ||h_￥||_X ￡ C ||F(Mf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Mf)} \|_X} is valid, and those X such that the inequality ||h_￥||_X ￡ C ||F(Sf) ||_X{\|{h_{\infty}}\|_{X} \leq C \|{\Phi(Sf)} \|_X} is valid, where Mf and Sf denote the maximal function and the square function of f, respectively.     

Almost Periodicity of Parabolic Evolution Equations with Inhomogeneous Boundary Values

We show the existence and uniqueness of the (asymptotically) almost periodic solution to parabolic evolution equations with inhomogeneous boundary values on \mathbbR{\mathbb{R}} and \mathbbR_±\mathbb{R}_{\pm}, if the data are (asymptotically) almost periodic. We assume that the underlying homogeneous problem satisfies the ‘Acquistapace–Terreni’ conditions and has an exponential dichotomy. If there is an exponential dichotomy only on half intervals ( − ∞, − T] and [T, ∞), then we obtain a Fredholm alternative of the equation on \mathbbR{\mathbb{R}} in the space of functions being asymptotically almost periodic on \mathbbR₊{\mathbb{R}}_{+} and \mathbbR_-\mathbb{R}_{-}.     

Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}

We show that if A is a closed analytic subset of \mathbbPⁿ{\mathbb{P}^n} of pure codimension q then Hⁱ(\mathbbPⁿ\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that H^n-2(\mathbbPⁿ\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .     

An Algorithm for Computing a Gröbner Basis of a Polynomial Ideal over a Ring with Zero Divisors

An algorithm for computing a Gr?bner basis of an ideal of polynomials whose coefficients are taken from a ring with zero divisors, is presented; such rings include \mathbb Z_n\mathbb {Z}_n and \mathbb Z_n[i]\mathbb {Z}_n[i], where n is not a prime number. The algorithm is patterned after (1) Buchberger’s algorithm for computing a Gr?bner basis of a polynomial ideal whose coefficients are from a field and (2) its extension developed by Kandri-Rody and Kapur when the coefficients appearing in the polynomials are from a Euclidean domain. The algorithm works as Buchberger’s algorithm when a polynomial ideal is over a field and as Kandri-Rody–Kapur’s algorithm when a polynomial ideal is over a Euclidean domain. The proposed algorithm and the related technical development are quite different from a general framework of reduction rings proposed by Buchberger in 1984 and generalized later by Stifter to handle reduction rings with zero divisors. These different approaches are contrasted along with the obvious approach where for instance, in the case of \mathbb Z_n{\mathbb {Z}}_n, the algorithm for polynomial ideals over \mathbb Z{\mathbb {Z}} could be used by augmenting the original ideal presented by polynomials over \mathbb Z_n{\mathbb {Z}}_n with n (similarly, in the case of \mathbb Z_n[i]{\mathbb {Z}}_n[i], the original ideal is augmented with n and i² + 1).     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

Compact Operators that Commute with a Contraction

Let T be a C₀–contraction on a separable Hilbert space. We assume that I_H − T*T is compact. For a function f holomorphic in the unit disk \mathbbD{\mathbb{D}} and continuous on [`(\mathbbD)]\overline{{\mathbb{D}}}, we show that f(T) is compact if and only if f vanishes on s(T)?\mathbbT\sigma(T)\cap{\mathbb{T}}, where σ(T) is the spectrum of T and \mathbbT{\mathbb{T}} the unit circle. If f is just a bounded holomorphic function on \mathbbD{\mathbb{D}}, we prove that f(T) is compact if and only if lim_{n? ￥}||Tⁿf(T)|| = 0\lim\limits_{n\rightarrow \infty}\|T^{n}f(T)\| = 0.     

Boxicity of Circular Arc Graphs

A k-dimensional box is a Cartesian product R ₁ × · · · × R _k where each R _i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least p(\fraca-1a){\pi(\frac{\alpha-1}{\alpha})} for some a ? \mathbbN _{3 2}{\alpha\in\mathbb{N}_{\geq 2}}, then box(G) ≤ α (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree D < ?\fracn(a-1)2a?{\Delta < \lfloor{\frac{n(\alpha-1)}{2\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha \in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. We also demonstrate a graph having box(G) > α but with D = n\frac(a-1)2a+ \fracn2a(a+1)+(a+2){\Delta=n\frac{(\alpha-1)}{2\alpha}+ \frac{n}{2\alpha(\alpha+1)}+(\alpha+2)}. For a proper circular arc graph G, we show that if D < ?\fracn(a-1)a?{\Delta < \lfloor{\frac{n(\alpha-1)}{\alpha}}\rfloor} for some a ? \mathbbN _{3 2}{\alpha\in \mathbb{N}_{\geq 2}}, then box(G) ≤ α. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) ≤ r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k ≤ 3 arcs covers the circle, then box(G) ≤ 3 and if G admits a circular arc representation in which no family of k ≤ 4 arcs covers the circle, then box(G) ≤ 2. We also show that both these bounds are tight.     

Solutions to Polynomial Generalized Bers–Vekua Equations in Clifford Analysis

In this paper, we mainly study polynomial generalized Vekua-type equation _boxclose)w=0{p(\mathcal{D})w=0} and polynomial generalized Bers–Vekua equation p(D)w=0{p(\mathcal{\underline{D}})w=0} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} where D{\mathcal{D}} and D{\mathcal{\underline{D}}} mean generalized Vekua-type operator and generalized Bers–Vekua operator, respectively. Using Clifford algebra, we obtain the Fischer-type decomposition theorems for the solutions to these equations including (D-l)^kw=0,(D-l)^kw=0(k ? \mathbbN){\left(\mathcal{D}-\lambda\right)^{k}w=0,\left(\mathcal {\underline{D}}-\lambda\right)^{k}w=0\left(k\in\mathbb{N}\right)} with complex parameter λ as special cases, which derive the Almansi-type decomposition theorems for iterated generalized Bers–Vekua equation and polynomial generalized Cauchy–Riemann equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}. Making use of the decomposition theorems we give the solutions to polynomial generalized Bers–Vekua equation defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}} under some conditions. Furthermore we discuss inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}, and develop the structure of the solutions to inhomogeneous polynomial generalized Bers–Vekua equation p(D)w=v{p(\mathcal{\underline{D}})w=v} defined in W ì \mathbbRⁿ⁺¹{\Omega\subset\mathbb{R}^{n+1}}.