Transitive quasi-uniform structures depending on a parameter

Aequationes mathematicae - In a category $${\mathcal {C}}$$ with an ( $${\mathcal {E}}$$ , $${\mathcal {M}}$$ )-factorization structure for morphisms, we prove that any subclass $${\mathcal {N}}$$...     

Generation in singularity categories of hypersurfaces of countable representation type

The Orlov spectrum and Rouquier dimension are invariants of a triangulated category to measure how big the category is, and they have been studied actively. In this paper, we investigate the singularity category $$\textsf {D} _{\textsf {sg} }(R)$$ of a hypersurface R of countable representation type. For a thick subcategory $${\mathcal {T}}$$ of $$\textsf {D} _{\textsf {sg} }(R)$$ and a full subcategory $$\mathcal {X}$$ of $${\mathcal {T}}$$, we calculate the Rouquier dimension of $${\mathcal {T}}$$ with respect to $$\mathcal {X}$$. Furthermore, we prove that the level in $$\textsf {D} _{\textsf {sg} }(R)$$ of the residue field of R with respect to each nonzero object is at most one.     

Congruences with intervals and arbitrary sets

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).     

Indefinite Stein fillings and $$\text {PIN}(2)$$PIN(2)-monopole Floer homology

Let $$W(z_1, \ldots , z_n): ({\mathbb {C}}^*)^n \rightarrow {\mathbb {C}}$$ be a Laurent polynomial in n variables, and let $${\mathcal {H}}$$ be a generic smooth fiber of W. Ruddat et al. (Geom Topol 18:1343–1395, 2014) give a combinatorial recipe for a skeleton for $${\mathcal {H}}$$. In this paper, we show that for a suitable exact symplectic structure on $${\mathcal {H}}$$, the RSTZ-skeleton can be realized as the Liouville Lagrangian skeleton.     

Characterizing linear mappings through zero products or zero Jordan products

Periodica Mathematica Hungarica - Let $${\mathcal {A}}$$ be a $$*$$ -algebra and $${{\mathcal {M}}}$$ be a $$*$$ - $${\mathcal {A}}$$ -bimodule. We study the local properties of $$*$$ -derivations...     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

Approximate Representation of Bergman Submodules

In the present paper, the author shows that if a homogeneous submodule M of the Bergman module L_a~2(B_d) satisfies P_M-sum from i to ( M_(zi)P_MM*_(zi))≤c/(N + 1)P_M for some number c 0, then there is a sequence {f_j } of multipliers and a positive number c such that c'P_M ≤sum from j to ( M_(fj)M*_(fj))≤ P_M, i.e., M is approximately representable. The author also proves that approximately representable homogeneous submodules are p-essentially normal for p d.     

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

Given an inclusion of (graded) local nets, we analyse the structure of the corresponding inclusion of scaling limit nets , giving conditions, fulfilled in free field theory, under which the unicity of the scaling limit of implies that of the scaling limit of . As a byproduct, we compute explicitly the (unique) scaling limit of the fixpoint nets of scalar free field theories. In the particular case of an inclusion of local nets with the same canonical field net , we find sufficient conditions which entail the equality of the canonical field nets of . Work supported by MIUR, GNAMPA-INDAM, the EU and SNS. Submitted: August 29, 2008. Accepted: March 23, 2009.     

Algebraic entropy of sign-stable mutation loops

Geometriae Dedicata - In the theory of cluster algebras, a mutation loop induces discrete dynamical systems via its actions on the cluster $${\mathcal {A}}$$ - and $${\mathcal {X}}$$ -varieties. In...     

Z∗$$ {\mathcal{Z}}^{\ast } $$-Semilocal Modules and the Proper Class ℛS$$ \mathrm{\mathcal{R}}\mathcal{S} $$

Ukrainian Mathematical Journal - Over an arbitrary ring, a module M is said to be $$ {\mathcal{Z}}^{\ast } $$-semilocal if every submodule U of M has a $$ {\mathcal{Z}}^{\ast } $$ -supplement V in...     

The $${\mathcal {F}}$$ F -transitivity and recurrence of translation semigroups on complex sectors

Semigroup Forum - In this paper, we study the $${\mathcal {F}}$$ -transitive behaviour of the translation semigroups on complex sectors, where $${\mathcal {F}}$$ is a Furstenberg family of the...     

Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes and graph classes , an ()-system is a Boolean dynamical system such that all local transition functions lie in and the underlying graph lies in . Let be a class of Boolean functions which is closed under composition and let be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If contains the self-dual functions and contains the planar graphs, then the fixed-point existence problem for ()-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If contains the self-dual functions and contains the graphs having vertex covers of size one, then the fixed-point existence problem for ()-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

Choquet representability of submodular functions

Mathematical Programming - Let $$\Omega $$ be an arbitrary set, equipped with an algebra $${\mathcal {A}}\subseteq 2^{\Omega }$$ and let $$f: B({\mathcal {A}}) \rightarrow {\mathbb {R}}$$ be a...     

Schlichting’s theorem for approximate subgroups

We prove Schlichting’s theorem for approximate subgroups: if $${\mathcal {X}}$$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with $${\mathcal {X}}$$.     

A note on strong conciseness in virtually nilpotent profinite groups

Archiv der Mathematik - A group word w is said to be strongly concise in a class $${\mathcal {C}}$$ of profinite groups if, for every group G in $${\mathcal {C}}$$ such that w takes less than...     

Beck-type identities: new combinatorial proofs and a modular refinement

The Ramanujan Journal - Let $${\mathcal {O}}_r(n)$$ be the set of r-regular partitions of n, $${\mathcal {D}}_r(n)$$ the set of partitions of n with parts repeated at most $$r-1$$ times,...     

Weak law of large numbers for iterates of random-valued functions

Aequationes mathematicae - Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$ , a complete and separable metric space X with the $$ \sigma $$ -algebra $$ {\mathcal {B}} $$ of all its...