Weyl modules and q-Whittaker functions

Let $G$ be a semi-simple simply connected group over $\mathbb {C}$ . Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the $q$ -Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of $q$ -Whittaker functions $\varPsi _{\check{\lambda }}(q,z)$ . This is a family of invariant polynomials on the maximal torus $T\subset G$ (here $z\in T$ ) depending on a dominant weight $\check{\lambda }$ of $G$ whose coefficients are rational functions in a variable $q\in \mathbb {C}^*$ . For a conjecturally the same (but a priori different) definition of the $q$ -Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the $q$ -Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by $\varPsi '_{\check{\lambda }}(q,z)$ ]. For $G=SL(N)$ these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when $G$ is simply laced, the function $\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$ (here $I$ denotes the set of vertices of the Dynkin diagram of $G$ ) is equal to the character of a certain finite-dimensional $G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$ -module $D(\check{\lambda })$ (the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\varPsi _{\check{\lambda }}$ replaced by $\varPsi '_{\check{\lambda }}$ (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum $K$ -theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\check{\lambda })]$ .     

Global solutions to the Navier–Stokes-{\bar \omega} and related models with rough initial data

We establish the global well-posedness of the Navier–Stokes- ${\bar \omega}$ model with initial data ${u_0 \in H^{1-s}(\mathbb{R}^3)}$ with ${0 < s < \frac{1}{2}}$ which improves the existence results in Fan and Zhou (Appl Math Lett 24:1915–1918, 2011), Layton et al. (Commun Pure Appl Anal 10:1763–1777, 2011) where the initial data are required belonging to ${H^2(\mathbb{R}^3)}$ . We also obtain the similar results for a family of Navier–Stokes-α-like and magnetohydrodynamic-α models.     

Harmonic mappings in Bergman spaces

In this paper, we investigate the properties of mappings in harmonic Bergman spaces. First, we discuss the coefficient estimate, the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit disk $\mathbb D $ of $\mathbb C $ . Our results are generalizations of the corresponding ones in Chen et al. (Proc Am Math Soc 128:3231–3240, 2000), Chen et al. (J Math Anal Appl 373:102–110, 2011), Chen et al. (Ann Acad Sci Fenn Math 36:567–576, 2011). Then, we study the Schwarz-Pick Lemma and the Landau-Bloch theorem for mappings in harmonic Bergman spaces in the unit ball $\mathbb B ^{n}$ of $\mathbb C ^{n}$ . The obtained results are generalizations of the corresponding ones in Chen and Gauthier (Proc Am Math Soc 139:583–595 2011). At last, we get a characterization for mappings in harmonic Bergman spaces on $\mathbb B ^{n}$ in terms of their complex gradients.     

Porosity and products of Orlicz spaces

Let $(\Omega , \Sigma , \mu )$ be a measure space and let $\varphi _1, \ldots , \varphi _n$ and $\varphi $ be Young functions. In this paper, we, among other things, prove that the set $E=\{(f_1, \ldots ,f_n)\in M^{\varphi _1}\times \cdots \times M^{\varphi _n}:\, N_\varphi (f_1\cdots f_n)<\infty \}$ is a $\sigma $ - $c$ -lower porous set in $M^{\varphi _1}\times \cdots \times M^{\varphi _n}$ , under mild restrictions on the Young functions $\varphi _1, \ldots , \varphi _n$ and $\varphi $ . This generalizes a recent result due to G? a? b and Strobin (J Math Anal Appl 368:382–390, 2010) to more general setting of Orlicz spaces. As an application of our results, we recover a sufficient and necessary condition for Orlicz spaces to be closed under the pointwise multiplication due to Hudzik (Arch Math 44:535–538, 1985) and Arens et al. (J Math Anal Appl 177:386–411, 1993).     

Weak Fenchel and weak Fenchel-Lagrange conjugate duality for nonconvex scalar optimization problems

In this work, by using weak conjugate maps given in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), weak Fenchel conjugate dual problem, ${(D_F^w)}$ , and weak Fenchel Lagrange conjugate dual problem ${(D_{FL}^w)}$ are constructed. Necessary and sufficient conditions for strong duality for the ${(D_F^w)}$ , ${(D_{FL}^w)}$ and primal problem are given. Furthermore, relations among the optimal objective values of dual problem constructed by using Augmented Lagrangian in (Azimov and Gasimov, in Int J Appl Math 1:171–192, 1999), ${(D_F^w)}$ , ${(D_{FL}^w)}$ dual problems and primal problem are examined. Lastly, necessary and sufficient optimality conditions for the primal and the dual problems ${(D_F^w)}$ and ${(D_{FL}^w)}$ are established.     

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator

In a paper due to Jeong et al. (Kodai Math J 34(3):352–366, 2011) we have shown that there does not exist a hypersurface in $G_{2}({\mathbb{C }}^{m+2})$ with parallel shape operator in the generalized Tanaka–Webster connection (see Tanaka in Jpn J Math 20:131–190, 1976; Tanno in Trans Am Math Soc 314(1):349–379, 1989). In this paper, we introduce the notion of the Reeb parallel in the sense of generalized Tanaka–Webster connection for a hypersurface $M$ in $G_{2}({\mathbb{C }}^{m+2})$ and prove that $M$ is an open part of a tube around a totally geodesic $G_2(\mathbb{C }^{m+1})$ in $G_2(\mathbb{C }^{m+2})$ .     

An isoperimetric result for the fundamental frequency via domain derivative

The Faber–Krahn deficit $\delta \lambda $ of an open bounded set $\Omega $ is the normalized gap between the values that the first Dirichlet Laplacian eigenvalue achieves on $\Omega $ and on the ball having same measure as $\Omega $ . For any given family of open bounded sets of $\mathbb R ^N$ ( $N\ge 2$ ) smoothly converging to a ball, it is well known that both $\delta \lambda $ and the isoperimetric deficit $\delta P$ are vanishing quantities. It is known as well that, at least for convex sets, the ratio $\frac{\delta P}{\delta \lambda }$ is bounded by below by some positive constant (Brandolini et al., Arch Math (Basel) 94(4): 391–400, 2010; Payne and Weinberger, J Math Anal Appl 2:210–216, 1961), and in this note, using the technique of the shape derivative, we provide the explicit optimal lower bound of such a ratio as $\delta P$ goes to zero.     

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 pure critical exponent problem for the p-Laplacian

In this paper we prove existence and multiplicity of positive and sign-changing solutions to the pure critical exponent problem for the $p$ -Laplacian operator with Dirichlet boundary conditions on a bounded domain having nontrivial topology and discrete symmetry. Pioneering works related to the case $p=2$ are Brezis and Nirenberg (Comm Pure Appl Math 36, 437–477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209–212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253–294, 1988). A global compactness analysis is given for the Palais-Smale sequences in the presence of symmetries.     

The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited

In this paper we use the approach introduced in (Goerss et al., Ann Math 162(2):777–822, 2005) in order to analyze the homotopy groups of $L_{K(2)}V(0)$ , the mod- $3$ Moore spectrum $V(0)$ localized with respect to Morava $K$ -theory $K(2)$ . These homotopy groups have already been calculated by Shimomura (J Math Soc Japan 52(1): 65–90, 2000). The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of Shimomura (J Math Soc Japan 52(1): 65–90, 2000). An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the $K(2)$ -localization of the spectrum $TMF$ of topological modular forms and related spectra. Even more, the Adams–Novikov differentials for $L_{K(2)}V(0)$ can be read off from those for $TMF$ .     

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).     

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 ${\mathfrak{z}: 0 \leq \mathfrak{z} \leq \mathfrak{t}\}}$ .     

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).     

The effect of linear perturbations on the Yamabe problem

In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact, aspherical Riemannian manifold $\left( M,g\right) $ is compact. Established in the locally conformally flat case by Schoen (Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin 1989, Surveys Pure Application and Mathematics, 52 Longman Science, Technology, pp. 311–320. Harlow 1991) and for $n\le 24$ by Khuri–Marques–Schoen (J Differ Geom 81(1):143–196, 2009), it has revealed to be generally false for $n\ge 25$ as shown by Brendle (J Am Math Soc 21(4):951–979, 2008) and Brendle–Marques (J Differ Geom 81(2):225–250, 2009). A stronger version of it, the compactness under perturbations of the Yamabe equation, is addressed here with respect to the linear geometric potential $\frac{n-2}{4(n-1)} {{\mathrm{Scal}}}_g,\, {{\mathrm{Scal}}}_g$ being the Scalar curvature of $\left( M,g\right) $ . We show that a-priori $L^\infty $ –bounds fail for linear perturbations on all manifolds with $n\ge 4$ as well as a-priori gradient $L^2$ –bounds fail for non-locally conformally flat manifolds with $n\ge 6$ and for locally conformally flat manifolds with $n\ge 7$ . In several situations, the results are optimal. Our proof combines a finite dimensional reduction and the construction of a suitable ansatz for the solutions generated by a family of varying metrics in the conformal class of $g$ .     

A ternary additive problem

The problem of representing a large integer $n$ in the form $n=m^2+x^3+y^5$ has been studied by a number of authors in the past decades. In this paper, we restrict $m$ to square-free integers, and $x, y$ to primes, and show that there is such a representation for all $n\le N$ with at most $O(N^{1-\frac{1}{45}+\varepsilon })$ exceptions. We also improve the recent results of Liu (Acta Math Hungar 130(1–2):118–139, 2011) and Bauer (J Math 38(4):1073–1090, 2008) on related problems.     

Packing dimensions of the divergence points of self-similar measures with OSC

Let $\mu $ be the self-similar measure supported on the self-similar set $K$ with open set condition. In this article, we discuss the packing dimension of the set $\{x\in K: A(\frac{\log \mu (B(x,r))}{\log r})=I\}$ for $I\subseteq \mathbb R ,$ where $A(\frac{\log \mu (B(x,r))}{\log r})$ denotes the set of accumulation points of $\frac{\log \mu (B(x,r))}{\log r}$ as $r\searrow 0.$ Our main result solves the conjecture about packing dimension posed by Olsen and Winte (J London Math Soc, 67(2), pp 103–122, 2003) and generalizes the result in (Adv Math, 214, pp 267–287, (2007)).     

Destruction Of Lagrangian Torus For Positive Definite Hamiltonian Systems

For an integrable Hamiltonian ${H_0=\frac{1}{2} \sum_{i=1}^dy_i^2}$ ${(d \geq 2)}$ , we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily ${C^{2d-\delta}}$ -small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under ${C^{2d+\delta}}$ -small perturbations by Pöschel (Comm Pure Appl Math 35:653–696, 1982).     

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).