Topological invariants from quantum group $$\mathcal {U}_{\xi }\mathfrak {sl}(2|1)$$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2|1)\). The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \)-modular category of previous authors.     

Tangents to Subsolutions Existence and Uniqueness,II

This second part of the paper (see Ann Math de Toulouse, arXiv:1408.5797 for part I) is concerned with questions of existence and uniqueness of tangents in the special case of \(\varvec{\mathbb {G}}\)-plurisubharmonic functions, where \(\varvec{\mathbb {G}}\subset G(p,\mathbf{R}^n)\) is a compact subset of the Grassmannian of p-planes in \(\mathbf{R}^n\). An u.s.c. function u on an open set \(\Omega \subset \mathbf{R}^n\) is \(\varvec{\mathbb {G}}\)-plurisubharmonic if its restriction to \(\Omega \cap W\) is subharmonic for every affine \(\varvec{\mathbb {G}}\)-plane W. Here \(\varvec{\mathbb {G}}\) is assumed to be invariant under a subgroup \(K\subset \mathrm{O}(n)\) which acts transitively on \(S^{n-1}\). Tangents to u at a point \(x\) are the cluster points of u under a natural flow (or blow-up) at \(x\). They always exist and are \(\varvec{\mathbb {G}}\)-harmonic at all points of continuity. A homogeneity property is established for all tangents in these geometric cases. This leads to principal results concerning the Strong Uniqueness of Tangents, which means that all tangents are unique and of the form \(\Theta K_p\) where \(K_p\) is the Riesz kernel and \(\Theta \) is the density of u at the point. Strong uniqueness is a form of regularity which implies that the sets \(\{\Theta (u,x)\ge c\}\) for \(c>0\) are discrete. When the invariance group \(K= \mathrm{O}(n), \mathrm{U}(n)\) or Sp(n) strong uniqueness holds for all but a small handful of cases. It also holds for essentially all interesting \(\varvec{\mathbb {G}}\) which arise in calibrated geometry. When strong uniqueness fails, homogeneity implies that tangents are characterized by a subequation on the sphere, which is worked out in detail. In the cases corresponding to the real, complex, and quaternionic Monge–Ampère equations (convex functions, and complex and quaternionic plurisubharmonic functions), tangents, which are far from unique, are then systematically studied and classified.     

A note on the weight spectrum of the Schubert code $$C_{\alpha }(2, m)$$

We consider the Schubert code \(C_{\alpha }(2, m)\) associated to the \(\mathbb {F}_q\)-rational points of the Schubert variety \(\Omega _{\alpha }(2,m)\) in the Grassmannian \(G_{2,m}\). A correspondence between codewords of \(C_{\alpha }(2, m)\) and skew-symmetric matrices of certain special form is given. Using this correspondence, we give a formula for all possible weights of codewords in \(C_{\alpha }(2, m)\). It is shown that the weight of each codeword is divisible by certain power of q. Further, a formula for the weight spectrum of the Schubert code \(C_{\alpha }(2, m)\) is given.     

Vizing Bound for the Chromatic Number on Some Graph Classes

We are interested in hereditary classes of graphs \({\mathcal {G}}\) such that every graph \(G \in {\mathcal {G}}\) satisfies \(\varvec{\chi }(G) \le \omega (G) + 1\), where \(\chi (G)\) (\(\omega (G)\)) denote the chromatic (clique) number of G. This upper bound is called the Vizing bound for the chromatic number. Apart from perfect graphs few classes are known to satisfy the Vizing bound in the literature. We show that if G is (\(P_6, S_{1, 2, 2}\), diamond)-free, then \(\chi (G) \le \omega (G)+1\), and we give examples to show that the bound is sharp.     

Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

Groups with a ternary equivalence relation

We consider in a group \((G,\cdot )\) the ternary relation

$$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ | \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$

and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure

$$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ | \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$

We study the automorphism group of \((G,\kappa )\), i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \). We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\), \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\). This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps

$$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$

of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ | \ \gamma \in G \}\)) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\), \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\), the pair \((G,\odot )\) is a K-loop (cf. Theorem 5).

     

Congruences for $$\ell $$-regular overpartitions and Andrews’ singular overpartitions

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and 9. Recently, Andrews introduced the partition function \(\overline{C}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be over-lined. He also proved that \(\overline{C}_{3, 1}(9n+3)\) and \(\overline{C}_{3, 1}(9n+6)\) are divisible by 3. In this paper, we prove that \(\overline{C}_{3, 1}(12n+11)\) is divisible by 144 which was conjectured to be true by Naika and Gireesh.     

Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

The Finite Hankel Transform Operator: Some Explicit and Local Estimates of the Eigenfunctions and Eigenvalues Decay Rates

For fixed real numbers \(c>0,\)\(\alpha >-\frac{1}{2},\) the finite Hankel transform operator, denoted by \(\mathcal {H}_c^{\alpha }\) is given by the integral operator defined on \(L^2(0,1)\) with kernel \(K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).\) To the operator \(\mathcal {H}_c^{\alpha },\) we associate a positive, self-adjoint compact integral operator \(\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.\) Note that the integral operators \(\mathcal {H}_c^{\alpha }\) and \(\mathcal Q_c^{\alpha }\) commute with a Sturm-Liouville differential operator \(\mathcal D_c^{\alpha }.\) In this paper, we first give some useful estimates and bounds of the eigenfunctions \(\varphi ^{(\alpha )}_{n,c}\) of \(\mathcal H_c^{\alpha }\) or \(\mathcal Q_c^{\alpha }.\) These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator \(\mathcal D_c^{\alpha }.\) If \((\mu _{n,\alpha }(c))_n\) and \(\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2\) denote the infinite and countable sequence of the eigenvalues of the operators \(\mathcal {H}_c^{(\alpha )}\) and \(\mathcal Q_c^{\alpha },\) arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer \(n\ge 0,\)\(\lambda _{n,\alpha }(c)\) is decreasing with respect to the parameter \(\alpha .\) As a consequence, we show that for \(\alpha \ge \frac{1}{2},\) the \(\lambda _{n,\alpha }(c)\) and the \(\mu _{n,\alpha }(c)\) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.     

On color-preserving automorphisms of Cayley graphs of odd square-free order

An automorphism \(\alpha \) of a Cayley graph \(\mathrm{Cay}(G,S)\) of a group G with connection set S is color-preserving if \(\alpha (g,gs) = (h,hs)\) or \((h,hs^{-1})\) for every edge \((g,gs)\in E(\mathrm{Cay}(G,S))\). If every color-preserving automorphism of \(\mathrm{Cay}(G,S)\) is also affine, then \(\mathrm{Cay}(G,S)\) is a Cayley color automorphism (CCA) graph. If every Cayley graph \(\mathrm{Cay}(G,S)\) is a CCA graph, then G is a CCA group. Hujdurovi? et al. have shown that every non-CCA group G contains a section isomorphic to the non-abelian group \(F_{21}\) of order 21. We first show that there is a unique non-CCA Cayley graph \(\Gamma \) of \(F_{21}\). We then show that if \(\mathrm{Cay}(G,S)\) is a non-CCA graph of a group G of odd square-free order, then \(G = H\times F_{21}\) for some CCA group H, and \(\mathrm{Cay}(G,S) = \mathrm{Cay}(H,T)\mathbin {\square }\Gamma \).     

On spt-crank-type functions

In a recent paper, Andrews, Dixit, and Yee introduced a new spt-type function \({\mathrm {spt}}_{\omega }(n)\), which is closely related to Ramanujan’s third-order mock theta function \(\omega (q)\). Garvan and Jennings-Shaffer introduced a crank function which explains congruences for \({\mathrm {spt}}_{\omega }(n)\). In this article, we study the asymptotic behavior of this crank function and confirm a positivity conjecture of the crank asymptotically. We also study a sign pattern of the crank and congruences for \({\mathrm {spt}}_{\omega }(n)\).     

Some new Kirkman signal sets

A decomposition of the blocks of an \(\textsf {STS}(v)\) into partial parallel classes of size m is equivalent to a Kirkman signal set \(\textsf {KSS}(v,m)\). We give decompositions of \(\textsf {STS}(4v-3)\) into classes of size \(v-1\) when \(v \equiv 3 \pmod {6}\), \(v \not = 3\). We also give decompositions of \(\textsf {STS}(v)\) into classes of various sizes when v is a product of two arbitrary integers that are both congruent to \(3 \pmod {6}\). These results produce new families of \(\textsf {KSS}(v,m)\).     

On the principal components of sample covariance matrices

We introduce a class of \(M \times M\) sample covariance matrices \({\mathcal {Q}}\) which subsumes and generalizes several previous models. The associated population covariance matrix \(\Sigma = \mathbb {E}{\mathcal {Q}}\) is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of \(\Sigma - I_M\) may depend on \(M\) in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of \({\mathcal {Q}}\). We derive precise large deviation estimates on the generalized components \(\langle {\mathbf{{w}}} , {\varvec{\xi }_i}\rangle \) of the outlier and non-outlier eigenvectors \(\varvec{\xi }_i\). Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of \(\Sigma \). The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix \({\mathcal {Q}}\), satisfying \(\mathbb {E}{\mathcal {Q}} = I_M\), as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.     

Representations of groups with CAT(0) fixed point property

We show that certain representations over fields with positive characteristic of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image. In particular, we obtain rigidity results for representations of the following groups: the special linear group over \({\mathbb {Z}}\), \({\mathrm{SL}}_k({\mathbb {Z}})\), the special automorphism group of a free group, \(\mathrm{SAut}(F_k)\), the mapping class group of a closed orientable surface, \(\mathrm{Mod}(\Sigma _g)\), and many other groups. In the case of characteristic zero, we show that low dimensional complex representations of groups having CAT\((0)\) fixed point property \(\mathrm{F}\mathcal {B}_{\widetilde{A}_n}\) have finite image if they always have compact closure.     

Stochastic Ordering of Infinite Geometric Galton–Watson Trees

We consider Galton–Watson trees with Geom\((p)\) offspring distribution. We let \(T_{\infty }(p)\) denote such a tree conditioned on being infinite. We prove that for any \(1/2\le p_1 <p_2 \le 1\), there exists a coupling between \(T_{\infty }(p_1)\) and \(T_{\infty }(p_2)\) such that \({\mathbb {P}}(T_{\infty }(p_1) \subseteq T_{\infty }(p_2))=1\).     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).