Rectangular group congruences on a semigroup

We study rectangular group congruences on an arbitrary semigroup. Some of our results are an extension of the results obtained by Masat (Proc. Am. Math. Soc. 50:107–114, 1975). We show that each rectangular group congruence on a semigroup S is the intersection of a group congruence and a matrix congruence and vice versa, and this expression is unique, when S is E-inversive. Finally, we prove that every rectangular group congruence on an E-inversive semigroup is uniquely determined by its kernel and trace.     

Amenability of semigroups and their algebras modulo a group congruence

We investigate the amenability of the semigroup algebras \({\ell^1(S/\rho)}\) , where \({\rho}\) is a group congruence (not necessarily minimal) on a semigroup S. We relate this to a new notion of amenability of Banach algebras modulo an ideal, to prove a version of Johnson’s theorem for a large class of semigroups, including inverse semigroups, E-inversive semigroup and E-inversive E-semigroups.     

Directional Lipschitzness of Minimal Time Functions in Hausdorff Topological Vector Spaces

In a general Hausdorff topological vector space E, we associate to a given nonempty closed set S???E and a bounded closed set Ω???E, the minimal time function T _S,Ω defined by $T_{S,\Omega}(x):= \inf \{ t> 0: S\cap (x+t\Omega)\not = \emptyset\}$ . The study of this function has been the subject of various recent works (see Bounkhel (2012, submitted, 2013, accepted); Colombo and Wolenski (J Global Optim 28:269–282, 2004, J Convex Anal 11:335–361, 2004); He and Ng (J Math Anal Appl 321:896–910, 2006); Jiang and He (J Math Anal Appl 358:410–418, 2009); Mordukhovich and Nam (J Global Optim 46(4):615–633, 2010) and the references therein). The main objective of this work is in this vein. We characterize, for a given Ω, the class of all closed sets S in E for which T _S,Ω is directionally Lipschitz in the sense of Rockafellar (Proc Lond Math Soc 39:331–355, 1979). Those sets S are called Ω-epi-Lipschitz. This class of sets covers three important classes of sets: epi-Lipschitz sets introduced in Rockafellar (Proc Lond Math Soc 39:331–355, 1979), compactly epi-Lipschitz sets introduced in Borwein and Strojwas (Part I: Theory, Canad J Math No. 2:431–452, 1986), and K-directional Lipschitz sets introduced recently in Correa et al. (SIAM J Optim 20(4):1766–1785, 2010). Various characterizations of this class have been established. In particular, we characterize the Ω-epi-Lipschitz sets by the nonemptiness of a new tangent cone, called Ω-hypertangent cone. As for epi-Lipschitz sets in Rockafellar (Canad J Math 39:257–280, 1980) we characterize the new class of Ω-epi-Lipschitz sets with the help of other cones. The spacial case of closed convex sets is also studied. Our main results extend various existing results proved in Borwein et al. (J Convex Anal 7:375–393, 2000), Correa et al. (SIAM J Optim 20(4):1766–1785, 2010) from Banach spaces and normed spaces to Hausdorff topological vector spaces.     

Left Reductive Congruences on Semigroups

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ? a congruence on S. Consider the sequence ? ⁽⁰⁾?? ⁽¹⁾???? ⁽ⁿ⁾?? of congruences on S, where ? ⁽⁰⁾=? and, for an arbitrary non-negative integer n, ? ⁽ⁿ⁺¹⁾ is defined by (a,b)∈? ⁽ⁿ⁺¹⁾ if and only if (xa,xb)∈? ⁽ⁿ⁾ for all x∈S. We show that $\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )$ for an arbitrary congruence ? on a semigroup S, where lrc(?) denotes the least left reductive congruence on S containing ?. We focuse our attention on congruences ? on semigroups S for which the congruence $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is left reductive. We prove that, for a congruence ? on a semigroup S, $\bigcup_{i=0}^{\infty}\varrho^{(i)}$ is a left reductive congruence of S if and only if $\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}$ is a left reductive congruence on the factor semigroup S/? (here ι _(S/?) denotes the identity relation on S/?). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S ⁿ =S ⁿ⁺¹ is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.     

Generating sets of finite singular transformation semigroups

J.M. Howie proved that $\operatorname {Sing}_{n}$ , the semigroup of all singular mappings of {1,…,n} into itself, is generated by its idempotents of defect 1 (in J. London Math. Soc. 41, 707–716, 1966). He also proved that if n≥3 then a minimal generating set for $\operatorname {Sing}_{n}$ contains n(n?1)/2 transformations of defect 1 (in Gomes and Howie, Math. Proc. Camb. Philos. Soc. 101. 395–403, 1987). In this paper we find necessary and sufficient conditions for any set for transformations of defect 1 in $\operatorname {Sing}_{n}$ to be a (minimal) generating set for $\operatorname {Sing}_{n}$ .     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

A new approach to the description of one-parameter groups of formal power series in one indeterminate

The aim of the paper is to describe one-parameter groups of formal power series, that is to find a general form of all homomorphisms \({\Theta_G : G \to \Gamma}\) , \({\Theta_G(t) = \sum_{k=1}^{\infty} c_k(t)X^k}\) , \({c_1 : G \to \mathbb{K} \setminus\{0\}}\) , \({c_k : G \to \mathbb{K}}\) for k ≥ 2, from a commutative group (G, + ) into the group \({(\Gamma, \circ)}\) of invertible formal power series with coefficients in \({\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}}\) . Considering one-parameter groups of formal power series and one-parameter groups of truncated formal power series, we give explicit formulas for the coefficient functions c _k with more details in the case where either c ₁ = 1 or c ₁ takes infinitely many values. Here we give the results much more simply than they were presented in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008). Also the case im c ₁ = E _m (here E _m stands for the group of all complex roots of order m of 1), not considered in Jab?oński and Reich (Abh. Math. Sem. Univ. Hamburg 75:179–201, 2005; Result Math 47:61–68, 2005; Publ Math Debrecen 73(1–2):25–47, 2008), will be discussed.     

A semigroup characterization of well-posed linear control systems

We study the well-posedness of a linear control system Σ(A,B,C,D) with unbounded control and observation operators. To this end we associate to our system an operator matrix $\mathcal{A}$ on a product space $\mathcal{X}^{p}$ and call it p-well-posed if $\mathcal{A}$ generates a strongly continuous semigroup on $\mathcal{X}^{p}$ . Our approach is based on the Laplace transform and Fourier multipliers. The results generalize and complement those of Curtain and Weiss (Int. Ser. Numer. Math. vol. 91. Birkhäuser, Basel, 1989), Staffans and Weiss (Trans. Am. Math. Soc. 354:3229–3262, 2002) and are illustrated by a heat equation with boundary control and point observation.     

An alternative Cauchy functional equation on a semigroup

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation ${f(xy) - f(x) - f(y) \in \{ 0, 1\}}$ where ${f : S \to \mathbb{R}, S}$ is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti in Abh Math Sem Univ Hamburg 57:215–226, 1987). It is well known that the Cauchy functional equation is not stable on the free semigroup generated by two elements. At the 44th ISFE in Louisville, USA, Professor G. L. Forti and R. Ger asked to solve this functional equation on a semigroup where the Cauchy functional equation is not stable. In this paper, we present the first result in this direction providing an answer to the problem of G. L. Forti and R. Ger. In particular, we determine the solutions ${f : H \to \mathbb{R}}$ of the alternative functional equation on a semigroup ${H = \langle a, b| a^2 = a, b^2 = b \rangle }$ where the Cauchy equation is not stable.     

The general iterative methods for nonexpansive semigroups in Banach spaces

Let E be a real reflexive strictly convex Banach space which has uniformly Gâteaux differentiable norm. Let ${\mathcal{S} = \{T(s): 0 \leq s < \infty\}}$ be a nonexpansive semigroup on E such that ${Fix(\mathcal{S}) := \cap_{t\geq 0}Fix( T(t) ) \not= \emptyset}$ , and f is a contraction on E with coefficient 0 <  α <  1. Let F be δ-strongly accretive and λ-strictly pseudo-contractive with δ + λ >  1 and ${0 < \gamma < \min\left\{\frac{\delta}{\alpha}, \frac{1-\sqrt{ \frac{1-\delta}{\lambda} }}{\alpha} \right\} }$ . When the sequences of real numbers {α _n } and {t _n } satisfy some appropriate conditions, the three iterative processes given as follows : $${\left.\begin{array}{ll}{x_{n+1} = \alpha_n \gamma f(x_n) + (I - \alpha_n F)T(t_n)x_n,\quad n\geq 0,}\\ {y_{n+1} = \alpha_n \gamma f(T(t_n)y_n) + (I - \alpha_n F)T(t_n)y_n,\quad n\geq 0,}\end{array}\right.}$$ and $$ z_{n+1} = T(t_n)( \alpha_n \gamma f(z_n) + (I - \alpha_n F)z_n),\quad n\geq 0 $$ converge strongly to ${\tilde{x}}$ , where ${\tilde{x}}$ is the unique solution in ${Fix(\mathcal{S})}$ of the variational inequality $${ \langle (F - \gamma f)\tilde {x}, j(x - \tilde{x}) \rangle \geq 0,\quad x\in Fix(\mathcal{S}).}$$ Our results extend and improve corresponding ones of Li et al. (Nonlinear Anal 70:3065–3071, 2009) and Chen and He (Appl Math Lett 20:751–757, 2007) and many others.     

3-Nets realizing a group in a projective plane

In a projective plane $\mathit{PG}(2,\mathbb{K})$ defined over an algebraically closed field $\mathbb{K}$ of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky (Compos. Math. 140:1614–1624, 2004), arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky (Adv. Math. 219:672–688, 2008), comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzúa’s 3-nets (Adv. Geom. 10:287–310, 2010) realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds in characteristic p>0 apart from three possible exceptions $\rm{Alt}_{4}$ , $\rm{Sym}_{4}$ , and $\rm{Alt}_{5}$ . Motivation for the study of finite 3-nets in the complex plane comes from the study of complex line arrangements and from resonance theory; see (Falk and Yuzvinsky in Compos. Math. 143:1069–1088, 2007; Miguel and Buzunáriz in Graphs Comb. 25:469–488, 2009; Pereira and Yuzvinsky in Adv. Math. 219:672–688, 2008; Yuzvinsky in Compos. Math. 140:1614–1624, 2004; Yuzvinsky in Proc. Am. Math. Soc. 137:1641–1648, 2009).     

Some Recent Developments on Hopf��s Holomorphic Form

Hopf??s theorem on surfaces in ${\mathbb{R}^3}$ with constant mean curvature (Hopf in Math Nach 4:232?C249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141?C174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in ${M_k^n \times \mathbb{R}, n \ge 2}$ , where ${M_k^n}$ is a complete, simply-connected Riemannian manifold with constant sectional curvature k ?? 0. The case n?=?2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.     

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ be the empirical process associated to an ? ^d -valued stationary process (X _i ) _i≥0. In the present paper, we introduce very general conditions for weak convergence of $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ , which only involve properties of processes (f(X _i )) _i≥0 for a restricted class of functions $f\in\mathcal{G}$ . Our results significantly improve those of Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011) and provide new applications. The central interest in our approach is that it does not need the indicator functions which define the empirical process $(U_{n}(t))_{t\in\mathbb{R}^{d}}$ to belong to the class  $\mathcal{G}$ . This is particularly useful when dealing with data arising from dynamical systems or functionals of Markov chains. In the proofs we make use of a new application of a chaining argument and generalize ideas first introduced in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011). Finally we will show how our general conditions apply in the case of multiple mixing processes of polynomial decrease and causal functions of independent and identically distributed processes, which could not be treated by the preceding results in Dehling et al. (Stoch. Proc. Appl. 119(10):3699–3718, 2009) and Dehling and Durieu (Stoch. Proc. Appl. 121(5):1076–1096, 2011).     

A tableau formula for eta polynomials

We use the Pieri and Giambelli formulas of Buch et al. (Invent Math 178:345–405, 2009; J Reine Angew, 2013) and the calculus of raising operators developed in Buch et al. (A Giambelli formula for isotropic Grassmannians, arXiv:0811.2781, 2008) and Tamvakis (J Reine Angew Math 652, 207–244, 2011) to prove a tableau formula for the eta polynomials of Buch et al. (J Reine Angew, 2013) and the Stanley symmetric functions which correspond to Grassmannian elements of the Weyl group $\widetilde{W}_n$ of type $\text {D}_n$ . We define the skew elements of $\widetilde{W}_n$ and exhibit a bijection between the set of reduced words for any skew $w\in \widetilde{W}_n$ and a set of certain standard typed tableaux on a skew shape $\lambda /\mu $ associated to $w$ .     

On the tolerance lattice of tolerance factors

Although the notion of a tolerance is a natural generalization of the notion of a congruence, many properties of factor lattices modulo congruences are not, in general, valid for factor lattices modulo tolerances. In this paper, for a lattice L of a finite length, we define a new partial order ? on $\operatorname{Tol}\, (L)$ such that for every ${S\in \operatorname{Tol}\, (L)}$ with T?S, a tolerance S/T is induced on the factor lattice L/T. This partial order is a particular restriction of ? and thus we can prove for tolerances some analogous results to the homomorphism theorem and the second isomorphism theorem for congruences. The poset $(\operatorname{Tol}\, (L), \sqsubseteq)$ is not always a lattice, but it can be converted into a specific commutative join-directoid. Then, for every ${T\in \operatorname{Tol}\, (L)}$ , $(\operatorname{Tol}\, (L/T),\sqsubseteq)$ constitutes a subdirectoid of the directoid based on the poset $(\operatorname{Tol}\, (L),\sqsubseteq)$ and this specific directoid structure is preserved by the direct product of lattices.     

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sⁿ and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces $ \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace $ in S^m+1 (which can be obtained by exchanging radii in the Clifford tori $ S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})$ and show that they are stable Willmore hypersurfaces. In case of surfaces in S³, the stability of the Clifford torus $ S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} $ was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sⁿ.     

On Simultaneous Digital Expansions of Polynomial Values

Let s _q denote the q-ary sum-of-digits function and let \({P_1(X), P_2(X) \in \mathbb{Z}[X]}\) with \({P_1(\mathbb{N}), P_2(\mathbb{N}) \subset \mathbb{N}}\) be polynomials of degree \({h, l \geqq 1, h \neq l}\) , respectively. In this note we show that ( \({s_q(P_1(n))/s_q(P_2(n)))_{n \geqq 1}}\) is dense in \({\mathbb{R}^+}\) . This extends work by Stolarsky [9] and Hare, Laishram and Stoll [6].     

Power Series of Bernstein Operators and Approximation of Resolvents

According to the theory developed by F. Altomare and his school, certain C ₀-semigroups can be approximated by iterates of positive linear operators. A. Albanese, M. Campiti and E. Mangino [J. Appl. Funct. Anal. 1 (2006), 343-358] proved that the resolvent (λ?A)^?1 of the infinitesimal generator of such a semigroup can be also approximated, for λ > 0, by suitable iterates. What happens when ${\lambda \to 0^{+}?}$ We give an answer in the case of the semigroup approximated by the classical Bernstein operators B _n on the canonical simplex S of ${\mathbb{R}^{d}}$ . Specifically, we show that $$-A^{-1}h = \lim\limits_{n \to \infty}\frac{1}{n}{\sum\limits^{\infty}_{k=0}}{B^{k}_{n}h}$$ for h in a certain subspace of C(S). This gives a new method to investigate the qualitative properties of the inverse of A.     

Wave Equation and Multiplier Estimates on Damek–Ricci Spaces

Let S be a Damek–Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators of the form ${\rm {e}}^{it\sqrt{ L}}\psi\big(\sqrt{ L}/{\lambda}\big)$ for arbitrary time t and arbitrary λ>0, where ψ is a smooth bump function supported in [?2,2] if λ<1 and supported in [1,2] if λ≥1. This generalizes previous results in Müller and Thiele (Studia Math. 179:117–148, 2007). We also prove pointwise estimates for the gradient of these convolution kernels. As a corollary, we reprove basic multiplier estimates from Hebish and Steger (Math. Z. 245:37–61, 2003) and Vallarino (J. Lie Theory 17:163–189, 2007) and derive Sobolev estimates for the solutions to the wave equation associated to L.