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.     

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.     

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 $${\mathcal {N}}$$...     

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.      

Prime ideals in the quantum grassmannian

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a natural torus action of on the quantum grassmannian and the cell decomposition of the set of -primes leads to a parameterisation of the -spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the nonnegative cells in recent studies concerning the totally nonnegative grassmannian. Finally, we use the cell decomposition to establish that the quantum grassmannian satisfies normal separation and catenarity.     

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.     

Quantum observable generalized orthoalgebras

Positivity - Let $${{{\mathcal {S}}}}({\mathcal {H}})$$ denote the set of all self-adjoint operators (not necessarily bounded) on a Hilbert space $${\mathcal {H}}$$, which is the set of all...     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

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...     

Chains and antichains in partial orderings

We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is or , and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the difference of two sets. Our main result is that there is a computably axiomatizable theory K of partial orderings such that K has a computable model with arbitrarily long finite chains but no computable model with an infinite chain. We also prove the corresponding result for antichains. Finally, we prove that if a computable partial ordering has the feature that for every , there is an infinite chain or antichain that is relative to , then we have uniform dichotomy: either for all copies of , there is an infinite chain that is relative to , or for all copies of , there is an infinite antichain that is relative to .     

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.     

A conic bundle degenerating on the Kummer surface

Let C be a genus 2 curve and the moduli space of semi-stable rank 2 vector bundles on C with trivial determinant. In Bolognesi (Adv Geom 7(1):113–144, 2007) we described the parameter space of non stable extension classes of the canonical sheaf ω of C by ω⁻¹. In this paper, we study the classifying rational map that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up along a certain cubic surface S and at the point p corresponding to the bundle , then the induced morphism defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in . Furthermore we construct the -bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.     

Solutions of affine stochastic functional differential equations in the state space

We consider solutions of affine stochastic functional differential equations on . The drift of these equations is specified by a functional defined on a general function space which is only described axiomatically. The solutions are reformulated as stochastic processes in the space . By representing such a process in the bidual space of we establish that the transition functions of this process form a generalized Gaussian Mehler semigroup on . This way the process is characterized completely on since it is Markovian. Moreover we derive a sufficient and necessary condition on the underlying space such that the transition functions are even an Ornstein-Uhlenbeck semigroup. We exploit this result to associate a Cauchy problem in the function space to the stochastic functional differential equation.      

Neighbour-transitive codes and partial spreads in generalised quadrangles

Designs, Codes and Cryptography - A code C in a generalised quadrangle $${\mathcal {Q}}$$ is defined to be a subset of the vertex set of the point-line incidence graph $${\Gamma }$$ of $${\mathcal...     

On the Stability of Deformations of Cuntz–Toeplitz Algebras

Ukrainian Mathematical Journal - We study C*-algebras $$ {\mathcal{O}}_{n+m}^{\hat{q}} $$ generated by isometries $$ {\left\{{S}_i\right\}}_{i=1}^n\bigcup {\left\{{t}_j\right\}}_{j=1}^m, $$ where...     

Weighted Triangle-free 2-matching Problem with Edge-disjoint Forbidden Triangles

Mathematical Programming - The weighted $${\mathcal {T}}$$ -free 2-matching problem is the following problem: given an undirected graph G, a weight function on its edge set, and a set $${\mathcal...     

Negatively Correlated Random Variables and Mason’s Conjecture for Independent Sets in Matroids

Mason’s Conjecture asserts that for an m-element rank r matroid the sequence is logarithmically concave, in which I _k is the number of independent k-sets of . A related conjecture in probability theory implies these inequalities provided that the set of independent sets of satisfies a strong negative correlation property we call the Rayleigh condition. This condition is known to hold for the set of bases of a regular matroid. We show that if ω is a weight function on a set system that satisfies the Rayleigh condition then is a convex delta-matroid and ω is logarithmically submodular. Thus, the hypothesis of the probabilistic conjecture leads inevitably to matroid theory. We also show that two-sums of matroids preserve the Rayleigh condition in four distinct senses, and hence that the Potts model of an iterated two-sums of uniform matroids satisfies the Rayleigh condition. Numerous conjectures and auxiliary results are included. Research supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392.