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).     

A Construction of Generalized Harish-Chandra Modules for Locally Reductive Lie Algebras

We study cohomological induction for a pair $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ , $ \mathfrak{g} $ being an infinitedimensional locally reductive Lie algebra and $ \mathfrak{k} \subset \mathfrak{g} $ being of the form $ \mathfrak{k}_{0} \subset C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ , where $ \mathfrak{k}_{0} \subset \mathfrak{g} $ is a finite-dimensional reductive in $ \mathfrak{g} $ subalgebra and $ C_{\mathfrak{g}} {\left( {\mathfrak{k}_{0} } \right)} $ is the centralizer of $ \mathfrak{k}_{0} $ in $ \mathfrak{g} $ . We prove a general nonvanishing and $ \mathfrak{k} $ -finiteness theorem for the output. This yields, in particular, simple $ {\left( {\mathfrak{g},\mathfrak{k}} \right)} $ -modules of finite type over k which are analogs of the fundamental series of generalized Harish-Chandra modules constructed in [PZ1] and [PZ2]. We study explicit versions of the construction when $ \mathfrak{g} $ is a root-reductive or diagonal locally simple Lie algebra.     

The Diederich–Fornaess index and the global regularity of the \bar{\partial }-Neumann problem

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$ , on a smooth $q$ -convex domain $D\subset \subset {\mathbb {C}}^{n}$ . This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$ . Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$ , the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$ . We are thus able to extend to general degree $k\ge 0$ of $B_k$ , the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$ : if for the Diederich–Fornaess index $\delta $ of $D$ , we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$ , then $B_k$ is $H^s$ -regular.     

Parabolic contractions of semisimple Lie algebras and their invariants

Let $G$ be a connected semisimple algebraic group with Lie algebra $\mathfrak{g }$ and $P$ a parabolic subgroup of $G$ with $\mathrm{Lie\, }P=\mathfrak{p }$ . The parabolic contraction $\mathfrak{q }$ of $\mathfrak{g }$ is the semi-direct product of $\mathfrak{p }$ and a $\mathfrak{p }$ -module $\mathfrak{g }/\mathfrak{p }$ regarded as an abelian ideal. We are interested in the polynomial invariants of the adjoint and coadjoint representations of $\mathfrak{q }$ . In the adjoint case, the algebra of invariants is easily described and it turns out to be a graded polynomial algebra. The coadjoint case is more complicated. Here we found a connection between symmetric invariants of $\mathfrak{q }$ and symmetric invariants of centralisers $\mathfrak{g }_e\subset \mathfrak{g }$ , where $e\in \mathfrak{g }$ is a Richardson element with polarisation $\mathfrak{p }$ . Using this connection and results of Panyushev et al. (J Algebra 313:343–391, 2007), we prove that the algebra of symmetric invariants of $\mathfrak{q }$ is free for all parabolic subalgebras in types $\mathbf A$ and $\mathbf C$ and some parabolics in type $\mathbf B$ . This technique also applies to the minimal parabolic subalgebras in all types. For $\mathfrak{p }=\mathfrak{b }$ , a Borel subalgebra of $\mathfrak{g }$ , one gets a contraction of $\mathfrak{g }$ recently introduced by Feigin (Selecta Math 18:513–537, 2012) and studied from invariant-theoretic point of view in our previous paper (Panyushev and Yakimova in Ann Inst Fourier 62(6):2053–2068, 2012).     

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.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

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.     

Complement of forms

The notions of the parallel sum, the parallel difference, and the complement of two nonnegative sesquilinear forms were introduced and studied by Hassi, Sebestyé and de Snoo in Hassi et al. (Oper Theory Adv Appl 198:211–227, 2010) and Hassi et al. (J Funct Anal 257(12):3858–3894, 2009). In this paper we continue these investigations. The Galois correspondence induced by the map ${\mathfrak{m} \mapsto \mathfrak{m}_\mathfrak{t}}$ (where ${\mathfrak{m}_\mathfrak{t}}$ denotes the ${\mathfrak{t}}$ -complement of ${\mathfrak{m}}$ ) is also studied. Inspired by the work of Eriksson and Leutwiler Eriksson and Leutwiler (Math Ann 274:301–317, 1986), we introduce the notion of quasi-unit for nonnegative sesquilinear forms. The quasi-units are characterized by means of the complement and the disjoint part. It is also shown that the ${{\mathfrak{t}}}$ -quasi-units coincide with the extreme points of the convex set ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

Infinite Families of Pellian Polynomials and Their Continued Fraction Expansions

We investigate families $ \lbrace D_k(X)\rbrace_{k\in{\rm N}} $ of quadratic integral polynomials and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is constant. Furthermore, we show that the period lengths of $ \sqrt {D_k(X)} $ go to infinity with k. For each member of the families involved, we show how to easily determine the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction expansion of $ \sqrt {D_k(X)} $ is related to that of $ \sqrt {C} $ . This continues work in [3]-[5].     

Geometric rigidity for sequences of W^{2,2} conformal immersions

We analyse sequences of discs conformally immersed in $ \mathbb{R }^ n$ with energy $ \int _{ D} |A_k |^ 2 \le \gamma _n$ , where $ \gamma _n = 8\pi $ if $ n=3$ and $ \gamma _n = 4 \pi $ when $n\ge 4$ . We show that if such sequences do not weakly converge to a conformal immersion, then by a sequence of dilations we obtain a complete minimal surface with bounded total curvature, either Enneper’s minimal surface if $ n=3$ or Chen’s minimal graph if $ n \ge 4$ . In the papers, (Kuwert and Li, Comm Anal Geom 20(2), 313–340, 2012; Rivière, Adv Calculus Variations 6(1), 1–31, 2013) it was shown that if a sequence of immersed tori diverges in moduli space then $\liminf _ {k\rightarrow \infty } \mathcal W ( f_k )\ge 8\pi $ . We apply the above analysis to show that in $ \mathbb{R }^3$ if the sequence diverges so that $ \lim _{ k \rightarrow \infty } \mathcal W (f_k) =8\pi $ then there exists a sequence of Möbius transforms $ \sigma _{k}$ such that $ \sigma _k\circ f _k$ converges weakly to a catenoid.     

Some applications of the double large sieve

We sharpen a procedure of Cao and Zhai (J Théorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with $|a_m|,\ |b_n| \le 1$ . We apply this to give bounds for the discrepancy (mod 1) of the sequence $\{p^c: p\le X\}$ where $p$ is a prime variable, in the range $\frac{130}{79}\le c \le \frac{11}{5}$ . An alternative strategy is used for the range $1.48 \le c \le \frac{130}{79}$ . We use further exponential sum estimates to show that for large $R>0$ , and a small constant $\eta >0$ , the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided $2<c\le 2.041$ . This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).     

Local behavior and hitting probabilities of the \text{ Airy}_1 process

We obtain a formula for the $n$ -dimensional distributions of the $\text{ Airy}_1$ process in terms of a Fredholm determinant on $L^2(\mathbb{R })$ , as opposed to the standard formula which involves extended kernels, on $L^2(\{1,\dots ,n\}\times \mathbb{R })$ . The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071–1106, 2002) for the $\text{ Airy}_2$ process. Using this formula we are able to prove that the $\text{ Airy}_1$ process is Hölder continuous with exponent $\frac{1}{2}$ —and that it fluctuates locally like a Brownian motion. We also explain how the same methods can be used to obtain the analogous results for the $\text{ Airy}_2$ process. As a consequence of these two results, we derive a formula for the continuum statistics of the $\text{ Airy}_1$ process, analogous to that obtained in Corwin et al. (Commun Math Phys 2011, to appear) for the $\text{ Airy}_2$ process.     

Toric partial density functions and stability of toric varieties

Let $(L, h)\rightarrow (X, \omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho }_{tk}:X\rightarrow \mathbb {R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$ , for fixed $t>0$ such that $tk\in \mathbb {N}$ . We prove the existence of a distributional expansion of $\hat{\rho }_{tk}$ as $k\rightarrow \infty $ , including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$ . This expansion is used to give a direct proof that if $\omega $ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$ (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.     

Conservative fragments of {{S}^{1}_{2}} and {{R}^{1}_{2}}

Conservative subtheories of ${{R}^{1}_{2}}$ and ${{S}^{1}_{2}}$ are presented. For ${{S}^{1}_{2}}$ , a slight tightening of Je?ábek??s result (Math Logic Q 52(6):613?C624, 2006) that ${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$ is presented: It is shown that ${T^{0}_{2}}$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this ${\forall\Sigma^{b}_{1}}$ -theory, we define a ${\forall\Sigma^{b}_{0}}$ -theory, ${T^{-1}_{2}}$ , for the ${\forall\Sigma^{b}_{0}}$ -consequences of ${S^{1}_{2}}$ . We show ${T^{-1}_{2}}$ is weak by showing it cannot ${\Sigma^{b}_{0}}$ -define division by 3. We then consider what would be the analogous ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ based on Pollett (Ann Pure Appl Logic 100:189?C245, 1999. It is shown that this theory, ${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$ , also cannot ${\Sigma^{b}_{0}}$ -define division by 3. On the other hand, we show that ${{S}^{0}_{2}+open_{\{||id||\}}}$ -COMP is a ${\forall\hat\Sigma^{b}_{1}}$ -conservative subtheory of ${R^{1}_{2}}$ . Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium?? 98, 262?C279, 2000) and show that ${\hat{C}^{0}_{2}}$ is ${\forall\hat\Sigma^{b}_{1}}$ -conservative over a theory based on open _cl-comprehension.     

Boundary Values of Harmonic Functions in Spaces of Triebel–Lizorkin Type

Triebel (J Approx Theory 35:275–297, 1982; 52:162–203, 1988) investigated the boundary values of the harmonic functions in spaces of the Triebel–Lizorkin type ${\mathcal F^{\alpha,q}_{p}}$ on ${\mathbb{R}^{n+1}_+}$ by finding an characterization of the homogeneous Triebel–Lizorkin space ${{\bf \dot{F}}^{\alpha,q}_p}$ via its harmonic extension, where ${0 < p < \infty, 0 < q \leq \infty}$ , and ${\alpha < {\rm min}\{-n/p, -n/q\}}$ . In this article, we extend Triebel’s result to α < 0 and ${0 < p, q \leq \infty}$ by using a discrete version of reproducing formula and discretizing the norms in both ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf{\dot{F}}}^{\alpha,q}_p}$ . Furthermore, for α < 0 and ${1 < p,q \leq \infty}$ , the mapping from harmonic functions in ${\mathcal{F}^{\alpha,q}_{p}}$ to their boundary values forms a topological isomorphism between ${\mathcal{F}^{\alpha,q}_{p}}$ and ${{\bf \dot{F}}^{\alpha,q}_p}$ .     

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.     

A construction for infinite families of semisymmetric graphs revealing their full automorphism group

We give a general construction leading to different non-isomorphic families $\varGamma_{n,q}(\mathcal{K})$ of connected q-regular semisymmetric graphs of order 2q ⁿ⁺¹ embedded in $\operatorname{PG}(n+1,q)$ , for a prime power q=p ^h , using the linear representation of a particular point set $\mathcal{K}$ of size q contained in a hyperplane of $\operatorname{PG}(n+1,q)$ . We show that, when $\mathcal{K}$ is a normal rational curve with one point removed, the graphs $\varGamma_{n,q}(\mathcal{K})$ are isomorphic to the graphs constructed for q=p ^h in Lazebnik and Viglione (J. Graph Theory 41, 249–258, 2002) and to the graphs constructed for q prime in Du et al. (Eur. J. Comb. 24, 897–902, 2003). These graphs were known to be semisymmetric but their full automorphism group was up to now unknown. For q≥n+3 or q=p=n+2, n≥2, we obtain their full automorphism group from our construction by showing that, for an arc $\mathcal{K}$ , every automorphism of $\varGamma_{n,q}(\mathcal{K})$ is induced by a collineation of the ambient space $\operatorname{PG}(n+1,q)$ . We also give some other examples of semisymmetric graphs $\varGamma _{n,q}(\mathcal{K})$ for which not every automorphism is induced by a collineation of their ambient space.     

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).     

Erratum to: Algèbres de lie quasi-réductives

In Corollary 12(ii) and Theorem 13(v) of [1] we omitted the hypothesis dim $ \mathfrak{z}\leq 1 $ . Moreover, in some places the symbol $ \mathbb{K} $ must be replaced by the symbol $ {{\mathbb{K}}^{\times }} $ .     

Sufficient conditions for Willmore immersions in \mathbb{R }^3 to be minimal surfaces

We provide two sharp sufficient conditions for immersed Willmore surfaces in $\mathbb{R }^3$ to be already minimal surfaces, i.e. to have vanishing mean curvature on their entire domains. These results turn out to be particularly suitable for applications to Willmore graphs. We can therefore show that Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with vanishing mean curvature on the boundary $\partial \varOmega $ must already be minimal graphs, which in particular yields some Bernstein-type result for Willmore graphs on $\mathbb{R }^2$ . Our methods also prove the non-existence of Willmore graphs on bounded $C^4$ -domains $\overline{\varOmega }$ with mean curvature $H$ satisfying $H \ge c_0>0 \,{\text{ on }}\, \partial \varOmega $ if $\varOmega $ contains some closed disc of radius $\frac{1}{c_0} \in (0,\infty )$ , and they yield that any closed Willmore surface in $\mathbb{R }^3$ which can be represented as a smooth graph over $\mathbb{S }^2$ has to be a round sphere. Finally, we demonstrate that our results are sharp by means of an examination of some certain part of the Clifford torus in $\mathbb{R }^3$ .