Numerical semigroups whose fractions are of maximal embedding dimension

Each saturated (resp., Arf) numerical semigroup S has the property that each of its fractions \(\frac{S}{k}\) is saturated (resp., Arf), but the property of being of maximal embedding dimension (MED) is not stable under formation of fractions. If S is a numerical semigroup, then S is MED (resp., Arf; resp., saturated) if and only if, for each 2≤k∈?, \(S = \frac{T}{k}\) for infinitely many MED (resp., Arf; resp., saturated) numerical semigroups T. Let \(\mathcal{A}\) (resp., \(\mathcal{F}\)) be the class of Arf numerical semigroups (resp., of numerical semigroups each of whose fractions is of maximal embedding dimension). Then there exists an infinite strictly ascending chain \(\mathcal{A} =\mathcal{C}_{1} \subset\mathcal{C}_{2} \subset\mathcal{C}_{3}\subset \,\cdots\, \subset\mathcal{F}\), where, like \(\mathcal{A}\) and \(\mathcal{F}\), each \(\mathcal{C}_{n}\) is stable under the formation of fractions.     

Regularity for free interface variational problems in a general class of gradients

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form

$$\begin{aligned} (u,A) \quad \mapsto \quad \int _\Omega 2fu \,\mathrm {d}x \; - \int _{\Omega \cap A} \sigma _1\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; - \int _{\Omega {\setminus } A} \sigma _2\mathscr {A}u\cdot \mathscr {A}u \, \,\mathrm {d}x \; + \; \text {Per }(A;\overline{\Omega }), \end{aligned}$$

where \(\Omega \) is a bounded Lipschitz domain, \(A\subset \mathbb {R}^N\) is a Borel set, \(u:\Omega \subset \mathbb {R}^N \rightarrow \mathbb {R}^d\), \(\mathscr {A}\) is an operator of gradient form, and \(\sigma _1, \sigma _2\) are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on f, we show for a solution (w, A), that the topological boundary of \(A \cap \Omega \) is locally a \(\mathrm {C}^1\)-hypersurface up to a closed set of zero \(\mathscr {H}^{N-1}\)-measure.

     

Inflation of Finite Lattices Along All-or-nothing Sets

We introduce a new generalization of Alan Day’s doubling construction. For ordered sets \(\mathcal {L}\) and \(\mathcal {K}\) and a subset \(E \subseteq \ \leq _{\mathcal {L}}\) we define the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) arising from inflation of \(\mathcal {L}\) along E by \(\mathcal {K}\). Under the restriction that \(\mathcal {L}\) and \(\mathcal {K}\) are finite lattices, we find those subsets \(E \subseteq \ \leq _{\mathcal {L}}\) such that the ordered set \(\mathcal {L} \star _{E} \mathcal {K}\) is a lattice. Finite lattices that can be constructed in this way are classified in terms of their congruence lattices.A finite lattice is binary cut-through codable if and only if there exists a 0?1 spanning chain \(\left \{\theta _{i}\colon 0 \leq i \leq n \right \}\) in \(Con(\mathcal {L})\) such that the cardinality of the largest block of ?? _i /?? _i?1 is 2 for every i with 1≤i≤n. These are exactly the lattices that can be constructed by inflation from the 1-element lattice using only the 2-element lattice. We investigate the structure of binary cut-through codable lattices and describe an infinite class of lattices that generate binary cut-through codable varieties.     

Largest initial segments pointwise fixed by automorphisms of models of set theory

Given a model \(\mathcal {M}\) of set theory, and a nontrivial automorphism j of \(\mathcal {M}\), let \(\mathcal {I}_{\mathrm {fix}}(j)\) be the submodel of \(\mathcal {M}\) whose universe consists of elements m of \(\mathcal {M}\) such that \(j(x)=x\) for every x in the transitive closure of m (where the transitive closure of m is computed within \(\mathcal {M}\)). Here we study the class \(\mathcal {C}\) of structures of the form \(\mathcal {I}_{\mathrm {fix}}(j)\), where the ambient model \(\mathcal {M}\) satisfies a frugal yet robust fragment of \(\mathrm {ZFC}\) known as \(\mathrm {MOST}\), and \(j(m)=m\) whenever m is a finite ordinal in the sense of \(\mathcal {M}.\) Our main achievement is the calculation of the theory of \(\mathcal {C}\) as precisely \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\). The following theorems encapsulate our principal results: Theorem A. Every structure in \(\mathcal {C}\) satisfies \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm { Collection}\). Theorem B. Each of the following three conditions is sufficient for a countable structure \(\mathcal {N}\) to be in \(\mathcal {C}\):(a) \(\mathcal {N}\) is a transitive model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(b) \(\mathcal {N}\) is a recursively saturated model of \(\mathrm {MOST+\Delta }_{0}^{\mathcal {P}}\)-\(\mathrm {Collection}\).(c) \(\mathcal {N}\) is a model of \(\mathrm {ZFC}\). Theorem C. Suppose \(\mathcal {M}\) is a countable recursively saturated model of \(\mathrm {ZFC}\) and I is a proper initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is closed under exponentiation and contains \(\omega ^\mathcal {M}\) . There is a group embedding \(j\longmapsto \check{j}\) from \(\mathrm {Aut}(\mathbb {Q})\) into \(\mathrm {Aut}(\mathcal {M})\) such that I is the longest initial segment of \(\mathrm {Ord}^{\mathcal {M}}\) that is pointwise fixed by \(\check{j}\) for every nontrivial \(j\in \mathrm {Aut}(\mathbb {Q}).\) In Theorem C, \(\mathrm {Aut}(X)\) is the group of automorphisms of the structure X, and \(\mathbb {Q}\) is the ordered set of rationals.     

Malcev products of unipotent monoids and varieties of bands

Let \(\mathcal{U}\) be the class of all unipotent monoids and \(\mathcal{B}\) the variety of all bands. We characterize the Malcev product \(\mathcal{U} \circ \mathcal{V}\) where \(\mathcal{V}\) is a subvariety of \(\mathcal{B}\) low in its lattice of subvarieties, \(\mathcal{B}\) itself and the subquasivariety \(\mathcal{S} \circ \mathcal{RB}\), where \(\mathcal{S}\) stands for semilattices and \(\mathcal{RB}\) for rectangular bands, in several ways including by a set of axioms. For members of some of them we describe the structure as well. This succeeds by using the relation \(\widetilde{\mathcal{H}}= \widetilde{\mathcal{L}} \cap \widetilde{\mathcal{R}}\), where \(a\;\,\widetilde{\mathcal{L}}\;\,b\) if and only if a and b have the same idempotent right identities, and \(\widetilde{\mathcal{R}}\) is its dual.We also consider \((\mathcal{U} \circ \mathcal{RB}) \circ \mathcal{S}\) which provides the motivation for this study since \((\mathcal{G} \circ \mathcal{RB}) \circ \mathcal{S}\) coincides with completely regular semigroups, where \(\mathcal{G}\) is the variety of all groups. All this amounts to a generalization of the latter: \(\mathcal{U}\) instead of \(\mathcal{G}\).     

Sign-changing solutions to a partially periodic nonlinear Schrödinger equation in domains with unbounded boundary

We consider the problem

$$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$

where \(\Omega \) is either \(\mathbb {R}^{N}\) or a smooth domain in \(\mathbb {R} ^{N}\) with unbounded boundary, \(N\ge 3,\) \(V_{\infty }>0,\) \(V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),\) \(\inf _{\mathbb {R}^{N}}V>-V_{\infty }\) and \(2<p<\frac{2N}{N-2}\). We assume V is periodic in the first m variables, and decays exponentially to zero in the remaining ones. We also assume that \(\Omega \) is periodic in the first m variables and has bounded complement in the other ones. Then, assuming that \(\Omega \) and V are invariant under some suitable group of symmetries on the last \(N-m\) coordinates of \(\mathbb {R}^{N}\), we establish existence and multiplicity of sign-changing solutions to this problem. We show that, under suitable assumptions, there is a combined effect of the number of periodic variables and the symmetries of the domain on the number of sign-changing solutions to this problem. This number is at least \(m+1\)

     

Morrey Spaces on Domains: Different Approaches and Growth Envelopes

We deal with Morrey spaces on bounded domains \(\Omega \) obtained by different approaches. In particular, we consider three settings \(\mathcal {M}_{u,p}(\Omega )\), \(\mathbb {M}_{u,p}(\Omega )\) and \(\mathfrak {M}_{u,p}(\Omega )\), where \(0<p\le u<\infty \), commonly used in the literature, and study their connections and diversities. Moreover, we determine the growth envelopes \(\mathfrak {E}_{\mathsf {G}}(\mathcal {M}_{u,p}(\Omega ))\) as well as \(\mathfrak {E}_{\mathsf {G}}(\mathfrak {M}_{u,p}(\Omega ))\), and obtain some applications in terms of optimal embeddings. Surprisingly, it turns out that the interplay between p and u in the sense of whether \(\frac{n}{u}\ge \frac{1}{p}\) or \(\frac{n}{u} < \frac{1}{p}\) plays a decisive role when it comes to the behaviour of these spaces.     

On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem

$$\begin{aligned} \epsilon ^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad \text{ in }\quad \Omega , \quad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\quad \partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary, the exponent p is greater than 1, \(\epsilon >0\) is a small parameter, V is a uniformly positive, smooth potential on \(\bar{\Omega }\), and \(\nu \) denotes the outward unit normal of \(\partial \Omega \). Let \(\Gamma \) be a curve intersecting orthogonally \(\partial \Omega \) at exactly two points and dividing \(\Omega \) into two parts. Moreover, \(\Gamma \) satisfies stationary and non-degeneracy conditions with respect to the functional \(\int _{\Gamma }V^{\sigma }\), where \(\sigma =\frac{p+1}{p-1}-\frac{1}{2}\). We prove the existence of a solution \(u_\epsilon \) concentrating along the whole of \(\Gamma \), exponentially small in \(\epsilon \) at any fixed distance from it, provided that \(\epsilon \) is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004).

     

Recurrent Dirichlet Forms and Markov Property of Associated Gaussian Fields

For the extended Dirichlet space \(\mathcal {F}_{e}\) of a general irreducible recurrent regular Dirichlet form \((\mathcal {E},\mathcal {F})\) on L ²(E;m), we consider the family \(\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}\) of centered Gaussian random variables defined on a probability space \(({\Omega }, \mathcal {B}, \mathbb {P})\) indexed by the elements of \(\mathcal {F}_{e}\) and possessing the Dirichlet form \(\mathcal {E}\) as its covariance. We formulate the Markov property of the Gaussian field \(\mathbb {G}(\mathcal {E})\) by associating with each set A ? E the sub-σ-field σ(A) of \(\mathcal {B}\) generated by X _u for every \(u\in \mathcal {F}_{e}\) whose spectrum s(u) is contained in A. Under a mild absolute continuity condition on the transition function of the Hunt process associated with \((\mathcal {E}, \mathcal {F})\), we prove the equivalence of the Markov property of \(\mathbb {G}(\mathcal {E})\) and the local property of \((\mathcal {E},\mathcal {F})\). One of the key ingredients in the proof is in that we construct potentials of finite signed measures of zero total mass and show that, for any Borel set B with m(B) >?0, any function \(u\in \mathcal {F}_{e}\) with s(u) ? B can be approximated by a sequence of potentials of measures supported by B.     

Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

Given a smooth, symmetric and homogeneous of degree one function \(f\left( \lambda _{1},\ldots ,\lambda _{n}\right) \) satisfying \(\partial _{i}f>0\quad \forall \,i=1,\ldots , n\), and a properly embedded smooth cone \({\mathcal {C}}\) in \({\mathbb {R}}^{n+1}\), we show that under suitable conditions on f, there is at most one f self-shrinker (i.e. a hypersurface \(\Sigma \) in \({\mathbb {R}}^{n+1}\) satisfying \(f\left( \kappa _{1},\ldots ,\kappa _{n}\right) +\frac{1}{2}X\cdot N=0\), where \(\kappa _{1},\ldots ,\kappa _{n}\) are principal curvatures of \(\Sigma \)) that is asymptotic to the given cone \({\mathcal {C}}\) at infinity.     

Geometry of Coadjoint Orbits and Multiplicity-one Branching Laws for Symmetric Pairs

Consider the restriction of an irreducible unitary representation π of a Lie group G to its subgroup H. Kirillov’s revolutionary idea on the orbit method suggests that the multiplicity of an irreducible H-module ν occurring in the restriction π|_H could be read from the coadjoint action of H on \(\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H})\), provided π and ν are ‘geometric quantizations’ of a G-coadjoint orbit \(\mathcal {O}^{G}\) and an H-coadjoint orbit \(\mathcal {O}^{H}\), respectively, where \(\text {pr} \colon \sqrt {-1}\mathfrak {g}^{\ast } \to \sqrt {-1}\mathfrak {h}^{\ast }\) is the projection dual to the inclusion \(\mathfrak {h} \subset \mathfrak {g}\) of Lie algebras. Such results were previously established by Kirillov, Corwin and Greenleaf for nilpotent Lie groups. In this article, we highlight specific elliptic orbits \(\mathcal {O}^{G}\) of a semisimple Lie group G corresponding to highest weight modules of scalar type. We prove that the Corwin–Greenleaf number \(\sharp (\mathcal {O}^{G} \cap \text {pr}^{-1}({\mathcal {O}}^{H}))/H\) is either zero or one for any H-coadjoint orbit \(\mathcal {O}^{H}\), whenever (G,H) is a symmetric pair of holomorphic type. Furthermore, we determine the coadjoint orbits \(\mathcal {O}^{H}\) with nonzero Corwin–Greenleaf number. Our results coincide with the prediction of the orbit philosophy, and can be seen as ‘classical limits’ of the multiplicity-free branching laws of holomorphic discrete series representations (Kobayashi [Progr. Math. 2007]).     

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Frame completions with prescribed norms: local minimizers and applications

Let \(\mathcal {F}_{0}=\{f_{i}\}_{i\in \mathbb {I}_{n_{0}}}\) be a finite sequence of vectors in \(\mathbb {C}^{d}\) and let \(\mathbf {a}=(a_{i})_{i\in \mathbb {I}_{k}}\) be a finite sequence of positive numbers, where \(\mathbb {I}_{n}=\{1,\ldots , n\}\) for \(n\in \mathbb {N}\). We consider the completions of \(\mathcal {F}_{0}\) of the form \(\mathcal {F}=(\mathcal {F}_{0},\mathcal {G})\) obtained by appending a sequence \(\mathcal {G}=\{g_{i}\}_{i\in \mathbb {I}_{k}}\) of vectors in \(\mathbb {C}^{d}\) such that ∥g _i ∥² = a _i for \(i\in \mathbb {I}_{k}\), and endow the set of completions with the metric \(d(\mathcal {F},\tilde {\mathcal {F}}) =\max \{ \,\|g_{i}-\tilde {g}_{i}\|: \ i\in \mathbb {I}_{k}\}\) where \(\tilde {\mathcal {F}}=(\mathcal {F}_{0},\,\tilde {\mathcal {G}})\). In this context we show that local minimizers on the set of completions of a convex potential P _φ , induced by a strictly convex function φ, are also global minimizers. In case that φ(x) = x ² then P _φ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn’s conjecture on the FOD.     

Sandwich semigroups in locally small categories II: transformations

Fix sets X and Y, and write \(\mathcal P\mathcal T_{XY}\) for the set of all partial functions \(X\rightarrow Y\). Fix a partial function \({a:Y\rightarrow X}\), and define the operation \(\star _a\) on \(\mathcal P\mathcal T_{XY}\) by \(f\star _ag=fag\) for \(f,g\in \mathcal P\mathcal T_{XY}\). The sandwich semigroup \((\mathcal P\mathcal T_{XY},\star _a)\) is denoted \(\mathcal P\mathcal T_{XY}^a\). We apply general results from Part I to thoroughly describe the structural and combinatorial properties of \(\mathcal P\mathcal T_{XY}^a\), as well as its regular and idempotent-generated subsemigroups, \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). After describing regularity, stability and Green’s relations and preorders, we exhibit \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) as a pullback product of certain regular subsemigroups of the (non-sandwich) partial transformation semigroups \(\mathcal P\mathcal T_X\) and \(\mathcal P\mathcal T_Y\), and as a kind of “inflation” of \(\mathcal P\mathcal T_A\), where A is the image of the sandwich element a. We also calculate the rank (minimal size of a generating set) and, where appropriate, the idempotent rank (minimal size of an idempotent generating set) of \(\mathcal P\mathcal T_{XY}^a\), \({\text {Reg}}(\mathcal P\mathcal T_{XY}^a)\) and \(\mathbb E(\mathcal P\mathcal T_{XY}^a)\). The same program is also carried out for sandwich semigroups of totally defined functions and for injective partial functions. Several corollaries are obtained for various (non-sandwich) semigroups of (partial) transformations with restricted image, domain and/or kernel.     

Symmetric Pairs and Self-Adjoint Extensions of Operators,with Applications to Energy Networks

We provide a streamlined construction of the Friedrichs extension of a densely-defined self-adjoint and semibounded operator A on a Hilbert space \(\mathcal {H}\), by means of a symmetric pair of operators. A symmetric pair is comprised of densely defined operators \(J: \mathcal {H}_1 \rightarrow \mathcal {H}_2\) and \(K: \mathcal {H}_2 \rightarrow \mathcal {H}_1\) which are compatible in a certain sense. With the appropriate definitions of \(\mathcal {H}_1\) and J in terms of A and \(\mathcal {H}\), we show that \((\textit{JJ}^\star )^{-1}\) is the Friedrichs extension of A. Furthermore, we use related ideas (including the notion of unbounded containment) to construct a generalization of the construction of the Krein extension of A as laid out in a previous paper of the authors. These results are applied to the study of the graph Laplacian on infinite networks, in relation to the Hilbert spaces \(\ell ^2(G)\) and \(\mathcal {H}_{\mathcal {E}}\) (the energy space).     

Resolutions of Hilbert Modules and Similarity

Let \(H^{2}_{m}\) be the Drury–Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function \((z, w) \in\mathbb{B}^{m} \times\mathbb{B}^{m} \rightarrow (1 - \sum_{i=1}^{m}z_{i} \bar{w}_{i})^{-1}\). We investigate for which multipliers \(\theta: \mathbb{B}^{m} \rightarrow \mathcal{L}(\mathcal{E}, \mathcal {E}_{*})\) with ran?M _θ closed, the quotient module \(\mathcal{H}_{\theta}\), given by

$\cdots\longrightarrow H^2_m \otimes\mathcal{E} \stackrel{M_{\theta }}{\longrightarrow}H^2_m \otimes\mathcal{E}_* \stackrel{\pi_{\theta}}{\longrightarrow}\mathcal{H}_{\theta}\longrightarrow0,$

is similar to \(H^{2}_{m} \otimes \mathcal {F}\) for some Hilbert space \(\mathcal{F}\). Here M _θ is the corresponding multiplication operator in \(\mathcal{L}(H^{2}_{m} \otimes\mathcal{E}, H^{2}_{m} \otimes\mathcal{E}_{*})\) for Hilbert spaces \(\mathcal{E}\) and \(\mathcal{E}_{*}\) and \(\mathcal {H}_{\theta}\) is the quotient module \((H^{2}_{m} \otimes\mathcal{E}_{*})/ M_{\theta}(H^{2}_{m} \otimes\mathcal{E})\), and π _θ is the quotient map. We show that a necessary condition is the existence of a multiplier ψ in \(\mathcal{M}(\mathcal{E}_{*}, \mathcal{E})\) such that

$\theta\psi\theta= \theta.$

Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of \(H^{2}_{m} \otimes\mathcal{E}\) for a Hilbert space \(\mathcal {E}\), which is valid for the case of m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator m-tuple (or algebra) is not necessarily commuting (or commutative). In this case the converse to the similarity result is always valid.

     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of $$\mathbb {C}^{n}$$

The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f :  \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).     

On Shalika germs

Let G be (the group of F-points of) a reductive group over a local field F satisfying the assumptions of Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002), sections 2.2, 3.2, 4.3. Let \(G_{{\text {reg}}}\subset G\) be the subset of regular elements. Let \(T\subset G\) be a maximal torus. We write \(T_{{\text {reg}}}=T\cap G_{{\text {reg}}}\). Let dg, dt be Haar measures on G and T. They define an invariant measure Open image in new window on Open image in new window . Let \(\mathcal {H}\) be the space of complex valued locally constant functions on G with compact support. For any \(f\in \mathcal {H}\), \(t\in T_{{\text {reg}}}\), we put \(I_t(f)=\int _{G/T}f(\dot{g}t\dot{g}^{-1})dg/dt\). Let \(\mathcal U\) be the set of conjugacy classes of unipotent elements in G. For any \(\Omega \in \mathcal U\) we fix an invariant measure \(\omega \) on \(\Omega \). It is well known—see, e.g., Rao (Ann Math 96:505-510, 1972)—that for any \(f\in \mathcal {H}\) the integral

$$\begin{aligned} I_\Omega (f)=\int _\Omega f\omega \end{aligned}$$

is absolutely convergent. Shalika (Ann Math 95:226–242, 1972) showed that there exist functions \(j_\Omega (t)\), \(\Omega \in \mathcal U\), on \(T\cap G_{{\text {reg}}}\), such that

$$\begin{aligned} I_t(f)=\sum _{\Omega \in \mathcal U}j_\Omega (t)I_\Omega (f) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad ({\star }) \end{aligned}$$

for any \(f\in \mathcal {H}\), \(t\in T\) near to e, where the notion of near depends on f. For any \(r\ge 0\) we define an open \({\text {Ad}}(G)\)-invariant subset \(G_r\) of G, and a subspace \(\mathcal {H}_r\) of \(\mathcal {H}\), as in Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002). Here I show that for any \(f\in \mathcal {H}_r\) the equality \((\star )\) holds for all \(t\in T_{{\text {reg}}}\cap G_r\).

     

Irreducible Morphisms and Locally Finite Dimensional Representations

Let \(\mathcal {A}\) be a Hom-finite additive Krull-Schmidt k-category where k is an algebraically closed field. Let \(\text {mod}\mathcal {A}\) denote the category of locally finite dimensional \(\mathcal {A}\)-modules, that is, the category of covariant functors \(\mathcal {A} \to \text {mod}k\). We prove that an irreducible monomorphism in \(\text {mod}\mathcal {A}\) has a finitely generated cokernel, and that an irreducible epimorphism in \(\text {mod}\mathcal {A}\) has a finitely co-generated kernel. Using this, we get that an almost split sequence in \(\text {mod}\mathcal {A}\) has to start with a finitely co-presented module and end with a finitely presented one. Finally, we apply our results to the study of rep(Q), the category of locally finite dimensional representations of a strongly locally finite quiver. We describe all possible shapes of the Auslander-Reiten quiver of rep(Q).