Medial groupoid varieties defined by finite cyclic groups

Let G be the group generated by δ of finite order n and let a and b be integers such that G is generated by δ ^a and δ ^b . We write ${\Sigma_{a,b}^n}$ for the set of groupoid identities that are satisfied in the group ring ${\mathbb{Z}[G]}$ when the binary operation is δ ^a x + δ ^b y. For every positive integer n, we show that ${\Sigma_{1,1}^n}$ and ${\Sigma_{0,1}^n}$ are finitely based. When n is not a multiple of 6, we give a finite basis for ${\Sigma_{n-1,1}^n}$ .     

{\mathbb{G}_a^ M} degeneration of flag varieties

Let ${\mathcal{F}_\lambda}$ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V _λ . We define a flat degeneration ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group G ^a acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group G ^a contains ${\mathbb{G}^M_a}$ as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

Sharing Set and Normal Families of Entire Functions

Let ${\mathcal{F}}$ be a family of holomorphic functions defined in a domain ${\mathcal{D}}$ , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each ${f \in \mathcal{F}}$ , all zeros of f(z) are of multiplicity at least k, and f and f ^(k) share the set S in ${\mathcal{D}}$ , then ${\mathcal{F}}$ is normal in ${\mathcal{D}}$ . As an application, we prove a uniqueness theorem.     

MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

Let G be a simple algebraic group defined over ?. Let e be a nilpotent element in $ \mathfrak{g} $ = Lie(G) and denote by U ( $ \mathfrak{g} $ , e) the finite W-algebra associated with the pair ( $ \mathfrak{g} $ , e). It is known that the component group Γ of the centraliser of e in G acts on the set ? of all one-dimensional representations of U ( $ \mathfrak{g} $ , e). In this paper we prove that the fixed point set ?^Γ is non-empty. As a corollary, all finite W-algebras associated with $ \mathfrak{g} $ admit one-dimensional representations. In the case of rigid nilpotent elements in exceptional Lie algebras we find irreducible highest weight $ \mathfrak{g} $ -modules whose annihilators in U ( $ \mathfrak{g} $ ) come from one-dimensional representations of U ( $ \mathfrak{g} $ , e) via Skryabin’s equivalence. As a consequence, we show that for any nilpotent orbit $ \mathcal{O} $ in $ \mathfrak{g} $ there exists a multiplicity-free (and hence completely prime) primitive ideal of U ( $ \mathfrak{g} $ ) whose associated variety equals the Zariski closure of $ \mathcal{O} $ in $ \mathfrak{g} $ .     

Irregular and Singular Loci of Commuting Varieties

Let $ {\user1{\mathcal{C}}} $ be the commuting variety of the Lie algebra $ \mathfrak{g} $ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ be the singular locus of $ {\user1{\mathcal{C}}} $ and let $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) $ {\user1{\mathcal{C}}}^{{{\text{sing}}}} $ is a nonempty subset of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ ; (b) $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ where the maximum is taken over all simple ideals $ \mathfrak{a} $ of $ \mathfrak{g} $ and $ l{\left( \mathfrak{a} \right)} $ is the “lacety” of $ \mathfrak{a} $ ; and (c) if $ \mathfrak{t} $ is a Cartan subalgebra of $ \mathfrak{g} $ and $ \alpha \in \mathfrak{t}^{*} $ root of $ \mathfrak{g} $ with respect to $ \mathfrak{t} $ , then $ \overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}} $ is an irreducible component of $ {\user1{\mathcal{C}}}^{{{\text{irr}}}} $ of codimension 4 in $ {\user1{\mathcal{C}}} $ . This yields the bound $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)} $ and, in particular, $ {\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2 $ . The latter may be regarded as an evidence in favor of the known longstanding conjecture that $ {\user1{\mathcal{C}}} $ is always normal. We also prove that the algebraic variety $ {\user1{\mathcal{C}}} $ is rational.     

On {\mathbb{F}_q} -Rational Structure of Nilpotent Orbits in the Witt Algebra

Let ${\mathfrak{g}=W_1}$ be the p-dimensional Witt algebra over an algebraically closed field ${k=\overline{\mathbb{F}}_q}$ , where p > 3 is a prime and q is a power of p. Let G be the automorphism group of ${\mathfrak{g}}$ . The Frobenius morphism F _G (resp. ${F_\mathfrak{g}}$ ) can be defined naturally on G (resp. ${\mathfrak{g}}$ ). In this paper, we determine the ${F_\mathfrak{g}}$ -stable G-orbits in ${\mathfrak{g}}$ . Furthermore, the number of ${\mathbb{F}_q}$ -rational points in each ${F_\mathfrak{g}}$ -stable orbit is precisely given. Consequently, we obtain the number of ${\mathbb{F}_q}$ -rational points in the nilpotent variety.     

On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C ^* (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C ^* with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T?=?(T, m _T , e _T ) is a monad and G?=?(G, δ _G , ε _G ) is a comonad on ${\mathbb{A}}$ , additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ^???, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

On the greatest prime factor of ab + 1

We prove that whenever $ \mathcal{A} $ and $ \mathcal{B} $ are dense enough subsets of {1, ..., N}, there exist a ∈ $ \mathcal{A} $ and b ∈ $ \mathcal{B} $ such that the greatest prime factor of ab + 1 is at least $ N^{1 + |\mathcal{A}|/(9N)} $ .     

Meromorphic Functions Sharing Two Values IM with Their Derivatives

Let f be a nonconstant meromorphic function such that ${\overline{N}(r, f) < \lambda T(r, f)}$ , where ${\lambda \in [0, \frac{1}{9})}$ , and let a, b be two distinct finite values. If f and f′ share a, b IM, then ${f \equiv f'}$ .     

Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,\mathfrak{G}, \equiv,\tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ \overline{K} := \overline{\infty,o}}$ there can be established an operation ??+?? such that ${(\overline{K},+)}$ becomes a commutative group and the map ${ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}$ is a motion on ${\overline{K}}$ . The separation ?? induces on ${\overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(\overline{K},+,\omega)}$ becomes a cyclic ordered group. For ${a,b \in K := \overline{K} {\setminus}\{\infty\}}$ we set ${a < b :\Longleftrightarrow [a,b,\infty] =1}$ and for all ${a\in K\,a < \infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$ with ?? ${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.     

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety $\mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $\mathcal L_{\mathcal H}$ of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $\mathcal L^{\rm Com}$ of commutative semi-Heyting algebras and the variety $\mathcal L^{\vee}$ generated by the chains in which a?<?b implies a → b?=?b. We also study the variety $\mathcal C$ generated within $\mathcal{SH}$ by $\mathcal L_{\mathcal H}$ , $\mathcal L_\vee$ and $\mathcal L_{\rm Com}$ . In particular we prove that $\mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $\mathcal C$ .     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

The geometric meaning of Zhelobenko operators

Let $ \mathfrak{g} $ be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, $ \mathfrak{b} $ a Borel subalgebra of $ \mathfrak{g} $ , $ \mathfrak{h}\subset \mathfrak{b} $ the Cartan sublagebra, and N ? G the unipotent subgroup corresponding to the nilradical $ \mathfrak{n}\subset \mathfrak{b} $ . We show that the explicit formula for the extremal projection operator for $ \mathfrak{g} $ obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence $ N\times \mathfrak{h}\to \mathfrak{b} $ given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^{p}(\mathbb{R}, w)}$ , where ${p \in (1, \infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{A}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a \in PSO^{\diamond}}$ ) and all convolution operators W ⁰(b) ( ${b \in PSO_{p,w}^{\diamond}}$ ), where ${PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}$ and ${PSO_{p,w}^{\diamond} \subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R} \cup \{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^{p}(\mathbb{R}, w)}$ . Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{A}_{p,w}}$ and establish a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ in terms of their Fredholm symbols. To study the Banach algebra ${\mathfrak{A}_{p,w}}$ we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of ${\mathfrak{A}_{p,w}}$ and necessary tools for studying local algebras.     

Diagram Groupoids and von Neumann Algebras

The main purpose of this paper is to study certain algebraic structures induced by directed graphs. We have studied graph groupoids, which are algebraic structures induced by given graphs. By defining a certain groupoid-homomorphism ?? on the graph groupoid ${\mathbb{G}}$ of a given graph G, we define the diagram of G by the image ${\delta(\mathbb{G})}$ of ??, equipped with the inherited binary operation on ${\mathbb{G}}$ . We study the fundamental properties of the diagram ${\delta(\mathbb{G})}$ , and compare them with those of ${\mathbb{G}}$ . Similar to Cho (Acta Appl Math 95:95?C134, 2007), we construct the groupoid von Neumann algebra ${\mathcal{M}_{G}=vN(\delta(\mathbb{G}))}$ , generated by ${\delta(\mathbb{G})}$ , and consider the operator algebraic properties of ${\mathcal{M}_{G}}$ . In particular, we show ${\mathcal{M}_{G}}$ is *-isomorphic to a von Neumann algebra generated by a family of idempotent operators and nilpotent operators, under suitable representations.     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$      

Flat orbits, minimal ideals and spectral synthesis

Let G = exp ${\mathfrak{g}}$ be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. In the weighted group algebra ${L^{1}_{\omega}(G)}$ we determine the minimal ideal of given hull ${\{\pi_{l'} \in \hat{G} | l' \in l + \mathfrak{n}^{\perp}\}}$ , where ${\mathfrak{n}}$ is an ideal contained in ${\mathfrak{g}(l)}$ , and we characterize all the L ^∞(G/N)-invariant ideals (where ${N = {\rm exp}\, \mathfrak{n}}$ ) of the same hull. They are parameterized by a set of G-invariant, translation invariant spaces of complex polynomials on N dominated by ω and are realized as kernels of specially built induced representations. The result is particularly simple if the co-adjoint orbit of l is flat.