An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

A Categorical Approach to Groupoid Frobenius Algebras

In this paper, we show that $\mathcal{G}$ -Frobenius algebras (for $\mathcal{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\mathcal{G}])$ , where $D(k[\mathcal{G}])$ is the Drinfeld double of the quantum groupoid $k[\mathcal{G}]$ (Nikshych et al. 2000).     

Harmonic maps of foliated Riemannian manifolds

We study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps between foliated Riemannian manifolds ${(M, {\mathcal{F}}, g)}$ and ${(N, {\mathcal{G}}, h)}$ i.e. smooth critical points ? : M → N of the functional ${E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}$ with respect to variations through foliated maps. In particular we study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ _B u =  0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps. We study ${({\mathcal{F}}, {\mathcal{G}}_0 )}$ -harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.     

Inductive Moving Frames

An inductive implementation of the equivariant moving frame method is introduced for both finite-dimensional Lie group actions and infinite-dimensional Lie pseudo-groups. Given two Lie (pseudo-)groups ${\mathcal{G}}$ and ${\mathcal{H}}$ with ${\mathcal{G} \subset \mathcal{H}}$ , the inductive method streamlines the construction of a moving frame for ${\mathcal{H}}$ using the already constructed moving frame for ${\mathcal{G}}$ . As a by-product, a systematic procedure for expressing ${\mathcal{H}}$ -invariant quantities in terms of their ${\mathcal{G}}$ -invariant counterparts is obtained.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Modular Decomposition of the Orlik-Terao Algebra

Let ${\mathcal{A}}$ be a collection of n linear hyperplanes in ${\mathbb{k}^\ell}$ , where ${\mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${\mathcal{A}}$ is the subalgebra ${{\rm R}(\mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${\mathcal{A}}$ . It determines an irreducible subvariety ${Y (\mathcal{A})}$ of ${\mathbb{P}^{n-1}}$ . We show that a flat X of ${\mathcal{A}}$ is modular if and only if ${{\rm R}(\mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${\mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${\mathcal{A}}$ is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.     

On the Left and Right Brylinski-Kostant Filtrations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ a Borel subalgebra, and $\mathfrak{h}\subset\mathfrak{b}$ a Cartan subalgebra. Let V be a finite dimensional simple $U(\mathfrak{g})$ module. Based on a principal s-triple (e,h,f) and following work of Kostant, Brylinski (J Amer Math Soc 2(3):517–533, 1989) defined a filtration $\mathcal{F}_e$ for all weight subspaces V _μ of V and calculated the dimensions of the graded subspaces for μ dominant. In Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) these dimensions were calculated for all μ. Let δM(0) be the finite dual of the Verma module of highest weight 0. It identifies with the global functions on a Weyl group translate of the open Bruhat cell and so inherits a natural degree filtration. On the other hand, up to an appropriate shift of weights, there is a unique $U(\mathfrak{b})$ module embedding of V into δM(0) and so the degree filtration induces a further filtration $\mathcal{F}$ on each weight subspace V _μ . A casual reading of Joseph et al. (J Amer Math Soc 13(4):945–970, 2000) might lead one to believe that $\mathcal{F}$ and $\mathcal{F}_e$ coincide. However this is quite false. Rather one should view $\mathcal{F}_e$ as coming from a left action of $U(\mathfrak{n})$ and then there is a second (Brylinski-Kostant) filtration $\mathcal{F}'_e$ coming from a right action. It is $\mathcal{F}'_e$ which may coincide with $\mathcal{F}$ . In this paper the above claim is made precise. First it is noted that $\mathcal{F}$ is itself not canonical, but depends on a choice of variables. Then it is shown that a particular choice can be made to ensure that $\mathcal{F}=\mathcal{F}'_e$ . An explicit form for the unique left $U(\mathfrak{b})$ module embedding $V\hookrightarrow\delta M(0)$ is given using the right action of $U(\mathfrak{n})$ . This is used to give a purely algebraic proof of Brylinski’s main result in Brylinski (J Amer Math Soc 2(3):517–533, 1989) which is much simpler than Joseph et al. (J Amer Math Soc 13(4):945–970, 2000). It is noted that the dimensions of the graded subspaces of $\rm{gr}_{\mathcal{F}_e} V_{\!\mu}$ and $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ can be very different. Their interrelation may involve the Kashiwara involution. Indeed a combinatorial formula for multiplicities in tensor products involving crystal bases and the Kashiwara involution is recovered. Though the dimensions for the graded subspaces of $\rm{gr}_{\mathcal{F}'_e} V_{\!\mu}$ are determined by polynomial degree, their values remain unknown.     

Covering functions by countably many functions from some families

Let $ \mathcal{A} $ be a nonempty family of functions from $ \mathbb{R} $ to $ \mathbb{R} $ . A function $ f:\mathbb{R}\to \mathbb{R} $ is said to be strongly countably $ \mathcal{A} $ -function if there is a sequence (f _n ) of functions from $ \mathcal{A} $ such that $ \mathrm{Gr}(f)\subset {\cup_n}\mathrm{Gr}\left( {{f_n}} \right) $ (Gr(f) denotes the graph of f). If $ \mathcal{A} $ is the family of all continuous functions, the strongly countable $ \mathcal{A} $ -functions are called strongly countably continuous and were investigated in [Z. Grande and A. Fatz-Grupka, On countably continuous functions, Tatra Mt. Math. Publ., 28:57–63, 2004], [G. Horbaczewska, On strongly countably continuous functions, Tatra Mt. Math. Publ., 42:81–86, 2009], and [T.A. Natkaniec, On additive countably continuous functions, Publ. Math., 79(1–2):1–6, 2011]. In this article, we prove that the families $ \mathcal{A}\left( \mathbb{R} \right) $ of all strongly countably $ \mathcal{A} $ -functions are closed with respect to some operations in dependence of analogous properties of the families $ \mathcal{A} $ , and, in particular, we show some properties of strongly countably differentiable functions, strongly countably approximately continuous functions, and strongly countably quasi-continuous functions.     

Exact value of the resistance exponent for four dimensional random walk trace

Let S be a simple random walk starting at the origin in ${\mathbb{Z}^{4}}$ . We consider ${{\mathcal G}=S[0,\infty)}$ to be a random subgraph of the integer lattice and assume that a resistance of unit 1 is put on each edge of the graph ${{\mathcal G}}$ . Let ${R_{{\mathcal G}}(0,S_{n})}$ be the effective resistance between the origin and S _n . We derive the exact value of the resistance exponent; more precisely, we prove that ${n^{-1}E(R_{{\mathcal G}}(0,S_{n}))\approx (\log n)^{-\frac{1}{2}}}$ . As an application, we obtain sharp heat kernel estimates for random walk on ${\mathcal G}$ at the quenched level. These results give the answer to the problem raised by Burdzy and Lawler (J Phys A Math Gen 23:L23–L28, 1990) in four dimensions.     

Forbidden Induced Subgraphs for Perfect Matchings

Let ${\mathcal{F}}$ be a family of connected graphs. A graph G is said to be ${\mathcal{F}}$ -free if G is H-free for every graph H in ${\mathcal{F}}$ . We study the problem of characterizing the families of graphs ${\mathcal{F}}$ such that every large enough connected ${\mathcal{F}}$ -free graph of even order has a perfect matching. This problems was previously studied in Plummer and Saito (J Graph Theory 50(1):1–12, 2005), Fujita et al. (J Combin Theory Ser B 96(3):315–324, 2006) and Ota et al. (J Graph Theory, 67(3):250–259, 2011), where the authors were able to characterize such graph families ${\mathcal{F}}$ restricted to the cases ${|\mathcal{F}|\leq 1, |\mathcal{F}| \leq 2}$ and ${|\mathcal{F}| \leq 3}$ , respectively. In this paper, we complete the characterization of all the families that satisfy the above mentioned property. Additionally, we show the families that one gets when adding the condition ${|\mathcal{F}| \leq k}$ for some k ≥ 4.     

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.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

On additive decompositions of the set of primitive roots modulo p

It is conjectured that the set ${\mathcal {G}}$ of the primitive roots modulo p has no decomposition (modulo p) of the form ${\mathcal {G}= \mathcal {A} +\mathcal {B}}$ with ${|\mathcal {A}|\ge 2}$ , ${|\mathcal {B} |\ge 2}$ . This conjecture seems to be beyond reach but it is shown that if such a decomposition of ${\mathcal {G}}$ exists at all, then ${|\mathcal {A} |}$ , ${|\mathcal {B} |}$ must be around p ^1/2, and then this result is applied to show that ${\mathcal {G}}$ has no decomposition of the form ${\mathcal {G} =\mathcal {A} + \mathcal {B} + \mathcal {C}}$ with ${|\mathcal {A} |\ge 2}$ , ${|\mathcal {B} |\ge 2}$ , ${|\mathcal {C} |\ge 2}$ .     

Vector partition functions and generalized dahmen and micchelli spaces

This is the first of a series of papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory are in [4], while in [5] and [6] we shall investigate the cohomological formulas generated by this theory. Here we introduce a space of functions on a lattice which generalizes the space of quasipolynomials satisfying the difference equations associated to cocircuits of a sequence of vectors X, introduced by Dahmen and Micchelli [8]. This space $ \mathcal{F}(X) $ contains the partition function $ {\mathcal{P}_{(X)}} $ . We prove a “localization formula” for any f in $ \mathcal{F}(X) $ , inspired by Paradan's decomposition formula [12]. In particular, this implies a simple proof that the partition function $ {\mathcal{P}_{(X)}} $ is a quasi-polynomial on the Minkowski differences $ \mathfrak{c} - B(X) $ , where c is a big cell and B(X) is the zonotope generated by the vectors in X, a result due essentially to Dahmen and Micchelli.     

Newform theory for Hilbert Eisenstein series

In his thesis, Weisinger (Thesis, 1977) developed a newform theory for elliptic modular Eisenstein series. This newform theory for Eisenstein series was later extended to the Hilbert modular setting by Wiles (Ann. Math. 123(3):407–456, 1986). In this paper, we extend the theory of newforms for Hilbert modular Eisenstein series. In particular, we provide a strong multiplicity-one theorem in which we prove that Hilbert Eisenstein newforms are uniquely determined by their Hecke eigenvalues for any set of primes having Dirichlet density greater than $\frac{1}{2}$ . Additionally, we provide a number of applications of this newform theory. Let denote the space of Hilbert modular Eisenstein series of parallel weight k≥3, level $\mathcal{N}$ and Hecke character Ψ over a totally real field K. For any prime $\mathfrak{q}$ dividing $\mathcal{N}$ , we define an operator $C_{\mathfrak{q}}$ generalizing the Hecke operator $T_{\mathfrak{q}}$ and prove a multiplicity-one theorem for with respect to the algebra generated by the Hecke operators $T_{\mathfrak{p}}$ ( $\mathfrak{p}\nmid\mathcal{N}$ ) and the operators $C_{\mathfrak{q}}$ ( $\mathfrak{q}\mid\mathcal{N}$ ). We conclude by examining the behavior of Hilbert Eisenstein newforms under twists by Hecke characters, proving a number of results having a flavor similar to those of Atkin and Li (Invent. Math. 48(3):221–243, 1978).     

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors $\mathrm{Mod}(\mathcal{C})$ , from a skeletally small preadditive category $\mathcal{C}$ to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category $\mathcal{T}$ , and we concentrate here on extending Happel’s theorem to $\mathrm{Mod}(\mathcal{C})$ ; more specifically, we prove that there is an equivalence of triangulated categories $\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))$ . We then add some restrictions on our category $\mathcal{C}$ , in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors $\mathrm{mod}(\mathcal{C})$ , with $\mathcal{C}$ a dualizing variety.     

On the functional codes defined by quadrics and Hermitian varieties

In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q ²). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q ²) in the polynomials defining the Hermitian varieties of PG(N, q ²).     

Topological Aspects of Differential Chains

In this paper we investigate the topological properties of the space of differential chains $\,^{\prime}\mathcal{B}(U)$ defined on an open subset U of a Riemannian manifold M. We show that $\,^{\prime}\mathcal {B}(U)$ is not generally reflexive, identifying a fundamental difference between currents and differential chains. We also give several new brief (though non-constructive) definitions of the space $\,^{\prime}\mathcal{B}(U) $ , and prove that it is a separable ultrabornological (DF)-space. Differential chains are closed under dual versions of the fundamental operators of the Cartan calculus on differential forms (Harrison in Geometric Poincare lemma, Jan 2011, submitted; Operator calculus??the exterior differential complex, Jan 2011, submitted). The space has good properties, some of which are not exhibited by currents $\mathcal{B}'(U)$ or? $\mathcal{D}'(U)$ . For example, chains supported in finitely many points are dense in $\,^{\prime}\mathcal{B}(U)$ for all open U?M, but not generally in the strong dual topology of? $\mathcal{B}'(U)$ .     

Alternating groups and flag-transitive triplanes

Let ${\mathcal{D}}$ be a nontrivial triplane, and G be a subgroup of the full automorphism group of ${\mathcal{D}}$ . In this paper we prove that if ${\mathcal{D}}$ is a triplane, ${G\leq Aut(\mathcal{D})}$ is flag-transitive, point-primitive and Soc(G) is an alternating group, then ${\mathcal{D}}$ is the projective space PG ₂(3, 2), and ${G\cong A_7}$ with the point stabiliser ${G_x\cong PSL_3(2)}$ .