相关论文: Equidistribution of primitive lattices in $\mathbb…
A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error…
We study the statistical distribution of primitive sublattices in the space of lattices $\mathrm{SL}(n,\mathbb Z)\backslash\mathrm{SL}(n,\mathbb R)$. A central difficulty in this area is that the second moment of the counting function for…
Lattices and Z-modules in Euclidean space possess an infinitude of subsets that are images of the original set under similarity transformation. We classify such self-similar images according to their indices for certain 4D examples that are…
We outline the theory of sets with distributive operations: multishelves and multispindles, with examples provided by semi-lattices, lattices and skew lattices. For every such a structure we define multi-term distributive homology and show…
In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In…
We investigate the representation of lattices as sublattices of the lattice of all convex subsets (intervals) of a linearly ordered set $(X,\le)$. We introduce the purely lattice-theoretic notion of a \textit{loc-lattice} and prove that…
Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since…
We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions.…
Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets…
D.Happel and L.Unger defined a partial order on the set of basic tilting modules. We study the poset of basic pre-projective tilting modules over path algebra of infinite type. We give an equivalent condition for that this poset is a…
The distribution of the deformations of elementary cells is studied in an abstract lattice constructed from the existence of the empty set. One combination rule determining oriented sequences with continuity of set-distance function in such…
We initiate a systematic study of lattices of thick subcategories for arbitrary essentially small triangulated categories. To this end we give several examples illustrating the various properties these lattices may, or may not, have and…
Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable…
We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of…
We associate a certain tensor product lattice to any primitive integer lattice and ask about its typical shape. These lattices are related to the tangent bundle of Grassmannians and their study is motivated by Peyre's programme on…
We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: The discriminant - an integer - and the type - an integer vector. We then…
Modular lattices, introduced by R. Dedekind, are an important subvariety of lattices that includes all distributive lattices. Heitzig and Reinhold developed an algorithm to enumerate, up to isomorphism, all finite lattices up to size 18.…
The evaluation of the interaction between objects arranged on a lattice requires the computation of lattice sums. A scenario frequently encountered are systems governed by the Helmholtz equation in the context of electromagnetic scattering…
It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list: uniform stability with respect to the family of unitary operators on…
The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ belongs to a special class of algebraic cross-polytopes and simplices. It is shown that…