Small Littlewood–Richardson coefficients

We develop structural insights into the Littlewood–Richardson graph, whose number of vertices equals the Littlewood–Richardson coefficient \(c_{\lambda ,\mu }^{\nu }\) for given partitions \(\lambda \), \(\mu \), and \(\nu \). This graph was first introduced in Bürgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639–1681, 2013), where its connectedness was proved. Our insights are useful for the design of algorithms for computing the Littlewood–Richardson coefficient: We design an algorithm for the exact computation of \(c_{\lambda ,\mu }^{\nu }\) with running time \(\mathcal {O}\big ((c_{\lambda ,\mu }^{\nu })^2 \cdot {\textsf {poly}}(n)\big )\), where \(\lambda \), \(\mu \), and \(\nu \) are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge t\) whose running time is \(\mathcal {O}\big (t^2 \cdot {\textsf {poly}}(n)\big )\). Even the existence of a polynomial-time algorithm for deciding whether \(c_{\lambda ,\mu }^{\nu } \ge 2\) is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that \(c_{\lambda ,\mu }^{\nu }=2\) implies \(c_{M\lambda ,M\mu }^{M\nu } = M+1\) for all \(M \in \mathbb {N}\). Here, the stretching of partitions is defined componentwise.     

On hyperboundedness and spectrum of Markov operators

Consider an ergodic Markov operator \(M\) reversible with respect to a probability measure \(\mu \) on a general measurable space. It is shown that if \(M\) is bounded from \(\mathbb {L}^2(\mu )\) to \(\mathbb {L}^p(\mu )\), where \(p>2\), then it admits a spectral gap. This result answers positively a conjecture raised by Høegh-Krohn and Simon (J. Funct. Anal. 9:121–80, 1972) in the more restricted semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee et al. (Proceedings of the 2012 ACM Symposium on Theory of Computing, 1131–1140, ACM, New York, 2012). It provides a quantitative link between hyperboundedness and an eigenvalue different from the spectral gap in general. In addition, the usual Cheeger inequality is extended to the higher eigenvalues in the compact Riemannian setting and the exponential behaviors of the small eigenvalues of Witten Laplacians at small temperature are recovered.     

Chains,antichains, and complements in infinite partition lattices

We consider the partition lattice \(\Pi (\lambda )\) on any set of transfinite cardinality \(\lambda \) and properties of \(\Pi (\lambda )\) whose analogues do not hold for finite cardinalities. Assuming AC, we prove: (I) the cardinality of any maximal well-ordered chain is always exactly \(\lambda \); (II) there are maximal chains in \(\Pi (\lambda )\) of cardinality \(> \lambda \); (III) a regular cardinal \(\lambda \) is strongly inaccessible if and only if every maximal chain in \(\Pi (\lambda )\) has size at least \(\lambda \); if \(\lambda \) is a singular cardinal and \(\mu ^{< \kappa } < \lambda \le \mu ^\kappa \) for some cardinals \(\kappa \) and (possibly finite) \(\mu \), then there is a maximal chain of size \(< \lambda \) in \(\Pi (\lambda )\); (IV) every non-trivial maximal antichain in \(\Pi (\lambda )\) has cardinality between \(\lambda \) and \(2^{\lambda }\), and these bounds are realised. Moreover, there are maximal antichains of cardinality \(\max (\lambda , 2^{\kappa })\) for any \(\kappa \le \lambda \); (V) all cardinals of the form \(\lambda ^\kappa \) with \(0 \le \kappa \le \lambda \) occur as the cardinalities of sets of complements to some partition \(\mathcal {P} \in \Pi (\lambda )\), and only these cardinalities appear. Moreover, we give a direct formula for the number of complements to a given partition. Under the GCH, the cardinalities of maximal chains, maximal antichains, and numbers of complements are fully determined, and we provide a complete characterisation.     

Improved Moser–Trudinger inequality of Tintarev type in dimension <Emphasis Type="Italic">n</Emphasis> and the existence of its extremal functions

Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type

$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$

for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)

     

On the Rationality of the Spectrum

Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.     

Embedding of Möbius Invariant Function Spaces into Tent Spaces

On the unit disk we introduce a new class of tent spaces \(T^q_{s,t}(\mu )\) for any positive Borel measure \(\mu \), consider a class of Möbius invariant spaces \(Z_p\) of analytic functions, and show that \(Z_p\) is contained in \(T^1_{p,1}\) if and only if \(\mu \) is a p-Carleson measure. This could be considered a continuation or variation of the work initiated by Xiao (Adv Math 217:2075–2088, 2008).     

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of \(\mathbb R^m\) were studied in Aliprantis et al. (Macroecon Dyn 10(1):77–99, 2006), Aliprantis et al. (J Econometrics 136(2):431–456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr\(\ddot{\mathrm{a}}\)ge Algebra Geom 43:297–302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in \(C(\mathbb R^m)\), while piecewise affine functions are sequentially order dense in \(C(\mathbb R^m)\). This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)     

On Some Properties of a Class of Fractional Stochastic Heat Equations

We consider nonlinear parabolic stochastic equations of the form \(\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }\) on the ball \(B(0,\,R)\), where \(\dot{\xi }\) denotes some Gaussian noise and \(\sigma \) is Lipschitz continuous. Here \(\mathcal {L}\) corresponds to a symmetric \(\alpha \)-stable process killed upon exiting B(0, R). We will consider two types of noises: space-time white noise and spatially correlated noise. Under a linear growth condition on \(\sigma \), we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).     

The Fischer–Marsden conjecture and contact geometry

In this paper, we consider the Fischer–Marsden conjecture within the frame-work of K-contact manifolds and \((\kappa ,\mu )\)-contact manifolds. First, we prove that a complete K-contact metric satisfying \(\mathcal {L}^{*}_g(\lambda )=0\) is Einstein and is isometric to a unit sphere \(S^{2n+1}\). Next, we prove that if a non-Sasakian \((\kappa ,\mu )\)-contact metric satisfies \(\mathcal {L}^{*}_g(\lambda )=0\), then \( M^{3} \) is flat, and for \(n > 1\), \(M^{2n+1}\) is locally isometric to the product of a Euclidean space \(E^{n+1}\) and a sphere \(S^n(4)\) of constant curvature \(+\,4\).     

On a class of maximality principles

We study various classes of maximality principles, \(\mathrm {MP}(\kappa ,\Gamma )\), introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where \(\Gamma \) defines a class of forcing posets and \(\kappa \) is an infinite cardinal. We explore the consistency strength and the relationship of \(\textsf {MP}(\kappa ,\Gamma )\) with various forcing axioms when \(\kappa \in \{\omega ,\omega _1\}\). In particular, we give a characterization of bounded forcing axioms for a class of forcings \(\Gamma \) in terms of maximality principles MP\((\omega _1,\Gamma )\) for \(\Sigma _1\) formulas. A significant part of the paper is devoted to studying the principle MP\((\kappa ,\Gamma )\) where \(\kappa \in \{\omega ,\omega _1\}\) and \(\Gamma \) defines the class of stationary set preserving forcings. We show that MP\((\kappa ,\Gamma )\) has high consistency strength; on the other hand, if \(\Gamma \) defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP\((\kappa ,\Gamma )\) is consistent relative to \(V=L\).     

Zero varieties for the Nevanlinna class in weakly pseudoconvex domains of maximal type F in $$\mathbb {C}^2$$

Let \(\Omega \) be a bounded, uniformly totally pseudoconvex domain in \(\mathbb {C}^2\) with smooth boundary \(b\Omega \). Assume that \(\Omega \) is a domain admitting a maximal type F. Here, the condition maximal type F generalizes the condition of finite type in the sense of Range (Pac J Math 78(1):173–189, 1978; Scoula Norm Sup Pisa, pp 247–267, 1978) and includes many cases of infinite type. Let \(\alpha \) be a d-closed (1, 1)-form in \(\Omega \). We study the Poincaré–Lelong equation

$$\begin{aligned} i\partial \bar{\partial }u=\alpha \quad \text {on}\, \Omega \end{aligned}$$

in \(L^1(b\Omega )\) norm by applying the \(L^1(b\Omega )\) estimates for \(\bar{\partial }_b\)-equations in [11]. Then, we also obtain a prescribing zero set of Nevanlinna holomorphic functions in \(\Omega \).

     

Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem

In this paper, we will study the dependence of eigen-pairs \((\lambda _k(\rho ), \varphi _k(x,\rho ))\) of weighted Dirichlet eigenvalue problem on weights \(\rho \). It will be shown that \(\lambda _k(\rho )\) and \(\varphi _k(x,\rho )\) are completely continuous (CC) in \(\rho \). Precisely, when \(\rho _n\) is weakly convergent to \(\rho \) in some Lebesgue space, \(\lambda _k(\rho _n)\) is convergent to \(\lambda _k(\rho )\). As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from \(\varphi _k(x,\rho _n)\) to the eigen space \(V_k(\rho )\) of \(\lambda _k(\rho )\) is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given.     

Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Mar?enko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose \(\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}\) is a quaternion random matrix. For each \(n\), the entries \(\{x_{ij}^{(n)}\}\) are independent random quaternion variables with a common mean \(\mu \) and variance \(\sigma ^2>0\). It is shown that the empirical spectral distribution of the quaternion sample covariance matrix \(\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*\) converges to the Mar?enko–Pastur law as \(p\rightarrow \infty \), \(n\rightarrow \infty \) and \(p/n\rightarrow y\in (0,+\infty )\).     

A Defect Relation for Holomorphic Curves Intersecting General Divisors in Projective Varieties

In this paper, we introduce, for an effective Cartier divisor D on a normal projective variety X, the notion of its Nevanlinna constant, denoted by \(\text {Nev}(D)\). We then prove a defect relation \(\delta _f(D)\le \text {Nev}(D)\) for any Zariski-dense holomorphic mapping \(f: {\mathbb C}\rightarrow X\). It gives a unified proof (by simply computing \(\text {Nev}(D)\)) of the previous known results (see, for example, Ru Am J Math 126:215–226, 2004, Ann Math 2 169:255–267, 2009). More importantly, it also derives a new defect relation for holomorphic mappings \(f: {\mathbb C}\rightarrow X\) intersecting divisors \(D_1, \ldots , D_q\), where \(D_1, \ldots , D_q\) are not necessarily linearly equivalent.     

On a question of Gromov about the Wirtinger inequalities

We prove that Gromov’s \(\mathrm {Cycl}_4(0)\) condition implies \(\mathrm {Wir}_k\) inequalities for any \(k \ge 4\), which answers a question of Gromov (J Math Sci N Y 119(2):178–200, 2004).     

On the Stationary Navier–Stokes Problem in $$\mathbb {R}^3$$: An Approach in Weighted Sobolev Spaces

In this paper, we study the steady-state Navier–Stokes equations in \(\mathbb {R}^3\). First, we establish the existence of very weak solution in \(\varvec{L}^p(\mathbb {R}^3)\) with \(3/2< p < +\infty \) under smallness conditions on the data. A uniqueness result is also given in case the data belong to \(\mathbb {L}^r(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) with \(3/2<r<3\). We also discuss the case where the data are not necessarily small. In particular, these results enhance those obtained by Bjorland et al. (Commun Partial Differ Equ 26:216–246, 2011), and are in agreement with those obtained by Kim and Kozono (J Math Anal Appl 395(2):486–495, 2012). Second, we prove a result of existence and uniqueness of weak solution in the weighted Sobolev space \(\varvec{W}_0^{1,p}(\mathbb {R}^3)\cap \varvec{W}_0^{1,\,3/2}(\mathbb {R}^3)\) in case of small external forces given by \(\mathrm{div}\mathbb {F}\) with \(\mathbb {F} \in \mathbb {L}^p(\mathbb {R}^3)\cap \mathbb {L}^{3/2}(\mathbb {R}^3)\) and \(1<p<3\).     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

On <Emphasis Type="Italic">m</Emphasis>-ovoids of regular near polygons

We generalise the work of Segre (Ann Mat Pura Appl 4(70):1–201, 1965), Cameron et al. (J Algebra 55(2):257–280, 1978), and Vanhove (J Algebr Comb 34(3):357–373, 2011) by showing that nontrivial m-ovoids of the dual polar spaces \(\mathsf {DQ}(2d, q)\), \(\mathsf {DW}(2d-1,q)\) and \(\mathsf {DH}(2d-1,q^2)\) (\(d\geqslant 3\)) are hemisystems. We also provide a more general result that holds for regular near polygons.     

An $$O(\sqrt{n}L)$$ iteration Mehrotra-type predictor-corrector algorithm for monotone linear complementarity problem

In this paper we propose a new class of Mehrotra-type predictor-corrector algorithm for the monotone linear complementarity problems (LCPs). At each iteration, the method computes a corrector direction in addition to the Ai–Zhang direction (SIAM J Optim 16:400–417, 2005), in an attempt to improve performance. Starting with a feasible point \((x^0, s^0)\) in the wide neighborhood \(\mathcal {N}(\tau ,\beta )\), the algorithm enjoys the low iteration bound of \(O(\sqrt{n}L)\), where \(n\) is the dimension of the problem and \(L=\log \frac{(x^0)^T s^0}{\varepsilon }\) with \(\varepsilon \) the required precision. We also prove that the new algorithm can be specified into an easy implementable variant for solving the monotone LCPs, in such a way that the iteration bound is still \(O(\sqrt{n}L)\). Some preliminary numerical results are provided as well.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.