Suppose $f\in L^1(\Bbb{R}^d)$, $\Lambda\subset\Bbb{R}^d$ is a finite union of translated lattices such that $f+\Lambda$ tiles with weight. We prove there exists lattice $L\subset\Bbb{R}^d$ $f+L$ also tiles, possibly different As corollary, together result Kolountzakis, it implies any convex polygon multi-tiles the plane by translations admits multi-tiling, multiplicity.

Let X be a set. Note that there exists a subset of X which is trivial. Let X be a trivial set. Observe that every subset of X is trivial. Let L be a 1-sorted structure. Note that there exists a subset of L which is trivial. Let L be a relational structure. Note that there exists a subset of L which is trivial. Let L be a non empty 1-sorted structure. Note that there exists a subset of L which i...

A frame $L$ is called {it coz-dense} if $Sigma_{coz(alpha)}=emptyset$ implies $alpha=mathbf 0$. Let $mathcal RL$ be the ring of real-valued continuous functions on a coz-dense and completely regular frame $L$. We present a description of the socle of the ring $mathcal RL$ based on minimal ideals of $mathcal RL$ and zero sets in pointfree topology. We show that socle of $mathcal RL$ is an essent...

asiatic citrus canker is a devastating disease resulting in drastic economic losses in citriculture worldwide. amongst three different types of the disease, i.e. a, a* and aw, the a* type is genetically less known. in order to comprehend the behavior of the asiatic citrus canker a*-type strain (xanthomonas citri subsp. citri) in the vicinity of the host cells, a targeted semi-quantitative trans...

The following proposition is true (1) For every non empty poset L and for every element x of L holds compactbelow(x) = ↓x ∩ the carrier of CompactSublatt(L). Let L be a non empty reflexive transitive relational structure and let X be a subset of 〈Ids(L),⊆〉. Then ⋃ X is a subset of L. The following propositions are true: (2) For every non empty relational structure L and for all subsets X, Y of ...

This paper uses the weighted L$_1-$norm to propose an algorithm for finding a well-dispersed subset of non-dominated solutions of multiple objective mixed integer linear programming problem. When all variables are integer it finds the whole set of efficient solutions. In each iteration of the proposed method only a mixed integer linear programming problem is solved and its optimal solutions gen...

is also a fuzzy lattice. Throughout this paper, if {λj : j ∈ J} is a collection of L-sets in X , then (∨λj)(x) = ∨{λj(x) : j ∈ J}, x ∈ X ; and (∧λj)(x)=∧{λj(x) : j ∈ J}, x ∈ X . If r ∈ L, then rX denotes the fuzzy set given by rX(x) = r for all x ∈ X ; that is, rX denotes the “constant” L-set of level r, that is, the smallest and the largest elements of LX are denoted, respectively, by 0X and 1...