A Categorification of $\displaystyle {\mathfrak q} (2)$-Crystals

We provide a categorification of \(\mathfrak {q}(2)\)-crystals on the singular \(\mathfrak {gl}_{n}\)-category \({\mathcal O}_{n}\). Our result extends the \(\mathfrak {gl}_{2}\)-crystal structure on \(\text {Irr} ({\mathcal O}_{n})\) induced from the work of Bernstein-Frenkel-Khovanov. Further properties of the \({\mathfrak q}(2)\)-crystal \(\text {Irr} ({\mathcal O}_{n})\) are also discussed.     

Examples of Ricci-Mean Curvature Flows

Let \(\pi :{\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\rightarrow {\mathbb {P}}^{n-1}\) be a projective bundle over \({\mathbb {P}}^{n-1}\) with \(1\le k \le n-1\). We denote \({\mathbb {P}}({\mathcal {O}}(0)\oplus {\mathcal {O}}(k))\) by \(N_{k}^{n}\) and endow it with the U(n)-invariant gradient shrinking Kähler Ricci soliton structure constructed by Cao (Elliptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, 1996) and Koiso (Recent topics in differential and analytic geometry. Advanced studies in pure mathematics, Boston, 1990). In this paper, we show that lens space \(L(k\, ;1)(r)\) with radius r embedded in \(N_{k}^{n}\) is a self-similar solution. We also prove that there exists a pair of critical radii \(r_{1}<r_{2}\), which satisfies the following. The lens space \(L(k\, ;1)(r)\) is a self-shrinker if \(r<r_{2}\) and self-expander if \(r_{2}<r\), and the Ricci-mean curvature flow emanating from \(L(k\, ;1)(r)\) collapses to the 0-section of \(\pi \) if \(r<r_{1}\) and to the \(\infty \)-section of \(\pi \) if \(r_{1}<r\). This paper gives explicit examples of Ricci-mean curvature flows.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Spherically Balanced Hilbert Spaces of Formal Power Series in Several Variables-II

We continue the study of spherically balanced Hilbert spaces initiated in the first part of this paper. Recall that the complex Hilbert space \(H^2(\beta )\) of formal power series in the variables \(z_1, \ldots , z_m\) is spherically balanced if and only if there exist a Reinhardt measure \(\mu \) supported on the unit sphere \(\partial {\mathbb {B}}\) and a Hilbert space \(H^2(\gamma )\) of formal power series in the variable \(t\) such that

$$\begin{aligned} \Vert f\Vert ^2_{H^2(\beta )} = \int _{\partial {\mathbb {B}}}\Vert {f_z}\Vert ^2_{H^2(\gamma )}~d\mu (z)~(f \in H^2(\beta )), \end{aligned}$$

where \(f_z(t)=f(t z)\) is a formal power series in the variable \(t\). In the first half of this paper, we discuss operator theory in spherically balanced Hilbert spaces. The first main result in this part describes quasi-similarity orbit of multiplication tuple \(M_z\) on a spherically balanced space \(H^2(\beta ).\) We also observe that all spherical contractive multi-shifts on spherically balanced spaces admit the classical von Neumann’s inequality. In the second half, we introduce and study a class of Hilbert spaces, to be referred to as \({\mathcal {G}}\)-balanced Hilbert spaces, where \({\mathcal {G}}={\mathcal {U}}(r_1) \times {\mathcal {U}}(r_2) \times \cdots \times {\mathcal {U}}(r_k)\) is a subgroup of \({\mathcal {U}}(m)\) with \(r_1 + \cdots + r_k=m.\) In the case in which \({\mathcal {G}}={\mathcal {U}}(m),\) \({\mathcal {G}}\)-balanced spaces are precisely spherically balanced Hilbert spaces.

     

Aspects of the Segre variety $${\mathcal{S}_{1,1,1}(2)}$$

We consider various aspects of the Segre variety \({\mathcal{S}:=\mathcal{S} _{1,1,1}(2)}\) in PG(7, 2), whose stabilizer group \({\mathcal{G}_{\mathcal{S}}<{\rm GL}(8,2)}\) has the structure \({\mathcal{N}\rtimes{\rm Sym}(3),}\) where \({\mathcal{N} :={\rm GL}(2,2)\times{\rm GL}(2,2)\times{\rm GL} (2,2).}\) In particular we prove that \({\mathcal{S}}\) determines a distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2)}\) such that \({A\mathcal{Z}A^{-1}=\mathcal{Z},}\) for all \({A\in\mathcal{G}_{\mathcal{S}},}\) and in consequence \({\mathcal{S}}\) determines a \({\mathcal{G}_{\mathcal{S}}}\)-invariant spread of 85 lines in PG(7, 2). Furthermore we see that Segre varieties \({\mathcal{S}_{1,1,1}(2)}\) in PG(7, 2) come along in triplets \({\{\mathcal{S},\mathcal{S}^{\prime},\mathcal{S}^{\prime\prime}\}}\) which share the same distinguished Z ₃-subgroup \({\mathcal{Z}<{\rm GL}(8,2).}\) We conclude by determining all fifteen \({\mathcal{G}_{\mathcal{S}}}\)-invariant polynomial functions on PG(7, 2) which have degree < 8, and their relation to the five \({\mathcal{G}_{\mathcal{S}}}\)-orbits of points in PG(7, 2).     

Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance

Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a \(C^*\)-algebra \({\mathcal {B}}\subset \mathcal B(H)\) has the form \( L(b) = \Psi (b) + k^*b+bk\), \(b\in {\mathcal {B}}\) for some completely positive map \(\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)\) and \(k\in {\mathcal {B}}(H)\).     

Centralizing traces with automorphisms on triangular algebras

Let \({\mathcal{T}}\) be a triangular algebra over a commutative ring \({\mathcal{R}}\), \({\xi}\) be an automorphism of \({\mathcal{T}}\) and \({\mathcal{Z}_{\xi}(\mathcal{T})}\) be the \({\xi}\)-center of \({\mathcal{T}}\). Suppose that \({\mathfrak{q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}}\) is an \({\mathcal{R}}\)-bilinear mapping and that \({\mathfrak{T}_{\mathfrak{q}}\colon \mathcal{T}\longrightarrow \mathcal{T}}\) is a trace of \({\mathfrak{q}}\). The aim of this article is to describe the form of \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the commuting condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}\) (resp. the centralizing condition \({[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}\in \mathcal{Z}_\xi(\mathcal{T})}\)) for all \({x\in \mathcal{T}}\). More precisely, we will consider the question of when \({\mathfrak{T}_{\mathfrak{q}}}\) satisfying the previous condition has the so-called proper form.     

Homomorphisms of $$\ell ^1$$-Munn algebras and Rees matrix semigroup algebras

Let \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) be an \(\ell ^1\)-Munn algebra over an arbitrary unital Banach algebra \({\mathcal {A}}\). We characterize homomorphisms from \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \) into an arbitrary Banach algebra \({\mathcal {B}}\) in terms of homomorphisms from \({\mathcal {A}}\) into \({\mathcal {B}}\). Then we discuss homomorphisms from arbitrary Banach algebras into \({\mathcal {LM}}\left( {\mathcal {A}}, P\right) \). Existence and uniqueness of homomorphisms under certain conditions are also discussed. We apply these results to the concrete case of \(\ell ^1(S)\) where S is a Rees matrix semigroup, to identify characters of \(\ell ^1(S)\) in both cases where S is with or without zero. As a consequence if the sandwich matrix of S has a zero entry, then \(\ell ^1(S)\) is character amenable.     

Bernstein components via the Bernstein center

Let G be a reductive p-adic group. Let \(\Phi \) be an invariant distribution on G lying in the Bernstein center \({\mathcal {Z}}(G)\). We prove that \(\Phi \) is supported on compact elements in G if and only if it defines a constant function on every component of the set \({\text {Irr}}(G)\); in particular, we show that the space of all elements of \({\mathcal {Z}}(G)\) supported on compact elements is a subalgebra of \({\mathcal {Z}}(G)\). Our proof is a slight modification of the argument from Section 2 of Dat (J Reine Angew Math 554:69–103, 2003), where our result is proved in one direction.     

A decomposition theorem for Lie ideals in nest algebras

Let \({\mathcal {N}}\) be a nest and let \({\mathcal {L}}\) be a weakly closed Lie ideal of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We explicitly construct the greatest weakly closed associative ideal \({\mathcal {J} (\mathcal {L})}\) contained in \({\mathcal {L}}\) and show that \({\mathcal {J} (\mathcal {L}) \subseteq \mathcal {L} \subseteq \mathcal {J} (\mathcal {L})\oplus {\breve{\mathcal{D}}} (\mathcal {L})}\) , where \({{\breve{\mathcal{D}}}} (\mathcal {L})\) is an appropriate subalgebra of the diagonal \({\mathcal {D} (\mathcal {N})}\) of the nest algebra \({\mathcal {T} (\mathcal {N})}\) . We show that norm-preserving linear extensions of elements of the dual of \({\mathcal {L}}\) , satisfying a certain condition, are uniquely determined on the diagonal of the nest algebra by the ideal \({\mathcal {J} (\mathcal {L})}\) .     

Completely regular clique graphs,II

Let \(\varGamma = (X,R)\) be a connected graph. Then \(\varGamma \) is said to be a completely regular clique graph of parameters (s, c) with \(s\ge 1\) and \(c\ge 1\), if there is a collection \({\mathcal {C}}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \({\mathcal {C}}\). In the previous paper (Suzuki in J Algebr Combin 40:233–244, 2014), we showed, among other things, that a completely regular clique graph is distance-regular if and only if it is a bipartite half of a certain distance-semiregular graph. In this paper, we show that a completely regular clique graph with respect to \({\mathcal {C}}\) is distance-regular if and only if every \({\mathcal {T}}(C)\)-module of endpoint zero is thin for all \(C\in {\mathcal {C}}\). We also discuss the relation between a \({\mathcal {T}}(C)\)-module of endpoint 0 and a \({\mathcal {T}}(x)\)-module of endpoint 1 and study examples of completely regular clique graphs.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

Finite elements in some vector lattices of nonlinear operators

We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) in the vector lattice \({\mathcal {U}}(E,F)\) of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices E and F, where F is Dedekind complete. In particular, for an atomic vector lattice E it is proved that for a finite element in \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of \(\sigma \)-laterally continuous abstract Uryson functionals. We also describe the ideal \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Littlewood–Paley Theory for Triangle Buildings

For the natural two-parameter filtration \(\left( {\mathcal {F}_\lambda }: {\lambda \in P}\right) \) on the boundary of a triangle building, we define a maximal function and a square function and show their boundedness on \(L^p(\Omega _0)\) for \(p \in (1, \infty )\). At the end, we consider \(L^p(\Omega _0)\) boundedness of martingale transforms. If the building is of \({\text {GL}}(3, \mathbb {Q}_p)\), then \(\Omega _0\) can be identified with p-adic Heisenberg group.     

On the global convergence rate of the gradient descent method for functions with Hölder continuous gradients

The gradient descent method minimizes an unconstrained nonlinear optimization problem with \({\mathcal {O}}(1/\sqrt{K})\), where K is the number of iterations performed by the gradient method. Traditionally, this analysis is obtained for smooth objective functions having Lipschitz continuous gradients. This paper aims to consider a more general class of nonlinear programming problems in which functions have Hölder continuous gradients. More precisely, for any function f in this class, denoted by \({{\mathcal {C}}}^{1,\nu }_L\), there is a \(\nu \in (0,1]\) and \(L>0\) such that for all \(\mathbf{x,y}\in {{\mathbb {R}}}^n\) the relation \(\Vert \nabla f(\mathbf{x})-\nabla f(\mathbf{y})\Vert \le L \Vert \mathbf{x}-\mathbf{y}\Vert ^{\nu }\) holds. We prove that the gradient descent method converges globally to a stationary point and exhibits a convergence rate of \({\mathcal {O}}(1/K^{\frac{\nu }{\nu +1}})\) when the step-size is chosen properly, i.e., less than \([\frac{\nu +1}{L}]^{\frac{1}{\nu }}\Vert \nabla f(\mathbf{x}_k)\Vert ^{\frac{1}{\nu }-1}\). Moreover, the algorithm employs \({\mathcal {O}}(1/\epsilon ^{\frac{1}{\nu }+1})\) number of calls to an oracle to find \({\bar{\mathbf{x}}}\) such that \(\Vert \nabla f({{\bar{\mathbf{x}}}})\Vert <\epsilon \).     

Some Characterizations of Integral Operators Associated with Certain Classes of <Emphasis Type="Italic">p</Emphasis>-Valent Functions Defined by the Srivastava–Saigo–Owa Fractional Differintegral Operator

The purpose of this paper is to introduce new integral operators associated with Srivastava–Saigo–Owa fractional differintegral operator. We investigate some properties for the integral operators \({\mathcal {F}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) and \({\mathcal {G}}_{p,\eta ,\mu }^{\lambda ,\delta }(z)\) to be in the classes \({\mathcal {R}}_{k}^{\zeta }\left( p,\rho \right) \) and \({\mathcal {V}}_{k}^{\zeta }\left( p,\rho \right) \).