Narrow Results

At the 2011 Durham Conference "Geometry and Arithmetic of Lattices" M. Kapovich formulated the following

Question. Does there exist an embedding ℤ² * ℤ ↪ SL(3, ℤ)?

The goal of the paper is to prove the following

Main Theorem.If p and m are arbitrary positive integers then there exists an embeddingℤ² * F_m ↪ SL(3, ℤ[1/p]).

Every isometry of a finite-dimensional Euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this paper we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimum length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let $(X,\le ,c)$ be a finite poset, where $c:X\to X$ is an order-reversing and involutive map such that $c(x)\ne x$ for each $x\in X$. Let $B_2=\{N<P\}$ be the Boolean lattice with two elements and ${{𝒲}}_{+}(X,B_2)$ the family of all the order-preserving 2-valued maps $A:X\to B_2$ such that $A(c(x))=P$ if $A(x)=N$ for all $x\in X$. In this paper, we build a family ${{ℬ}}_{w+}(X)$ of particular subsets of $X$, that we call $w_{+}$-bases on $X$, and we determine a bijection between the family ${{ℬ}}_{w+}(X)$ and the family ${{𝒲}}_{+}(X,B_2)$. In such a bijection, a $w_{+}$-basis $\mathrm{\Omega }$ on $X$ corresponds to a map $A\in {{𝒲}}_{+}(X,B_2)$ whose restriction of $A$ to $\mathrm{\Omega }$ is the smallest 2-valued partial map on $X$ which has $A$ as its unique extension in ${{𝒲}}_{+}(X,B_2)$. Next we show how each $w_{+}$-basis on $X$ becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Several structural properties of an algebraic structure can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In this paper, we exhibit finite representations of this sequence for finite algebras from congruence modular varieties.

Common meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. The inverse of zero is an error term $a$ which is absorbent for addition. We study the problem of enumerating all finite common meadows of order n (that is, common meadows with n elements). This problem turns out to be deeply connected with both the number of finite rings of order n and with the number of a certain kind of partition of positive integers.

We derive the equivalent energy of a square lattice that either deforms into the three- dimensional Euclidean space or remains planar. Interactions are not restricted to pairs of points and take into account changes of angles. Under some relationships between the local energies associated with the four vertices of an elementary square, we show that the limit energy can be obtained by mere quasiconvexification of the elementary cell energy and that the limit process does not involve any relaxation at the atomic scale. In this case, it can be said that the Cauchy–Born rule holds true. Our results apply to classical models of mechanical trusses that include torques between adjacent bars and to atomistic models.

This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.

Let be a Kac–Moody Lie algebra. We give an interpretation of Tits' associated group functor using representation theory of and we construct a locally compact "Kac–Moody group" G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B^-,N) for G we show that if k is "sufficiently large", then the subgroup B^- is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and non-uniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.

This paper presents three main ideas. They are the Metatheorem, the lattice embedding for sets, and the lattice embedding for algebras.

The Metatheorem allows you to convert existing theorems about classical subsets into corresponding theorem about fuzzy subsets. The concept of a fuzzyfiable operation on a powerset is defined. The main result states that any implication or identity which can be stated using fuzzyfiable operations is true about fuzzy subsets if and only if it is true about classical subsets.

The lattice embedding theorem for sets shows that for any set X, there is a set Y such that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of Y. In fact it is further proved that if X is infinite, then we can choose Y = X and get the surprising result that the lattice of fuzzy subsets of X is isomorphic to a sublattice of the classical subsets of X itself. The idea is illustrated with an example explicitly showing how the lattice of fuzzy subsets of the closed unit interval 𝕀 = [0,1] embeds into the lattice of classical subsets of 𝕀.

The lattice embedding theorem for algebras shows that under certain circumstances the lattice of fuzzy subalgebras of an algebra A embeds into the lattice of classical subalgebras of a closely related algebra A′. The following sample use of this embeding theorem is given. It is a well known fact that the lattice of normal subgroups of a group is a modular lattice. The embeding theorem is used here to conclude that lattice of fuzzy normal subgroups of a group is a modular lattice too.

A new constructive family of asymptotically good lattices with respect to sphere packing density is presented. The family has a lattice in every dimension n ≥ 1. Each lattice is obtained from a conveniently chosen integral ideal in a subfield of the cyclotomic field ℚ(ζ_q) where q is the smallest prime congruent to 1 modulo n.

Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl’s bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.

In a recent paper by Curran and Goldmakher, the cardinality of $h$-fold sumsets $hA$ was given when $A\subset {\mathbb{Z}}^d$ has $d+2$ elements. In this paper we provide a different method for doing this and obtain a more general result.

We also obtain an upper bound for the value of $\left|hA\right|$ when $A\subset {\mathbb{Z}}^d$ is a set of $d+3$ elements with simplicial hull.

Let K be a totally real number field of degree n over $\mathbb{Q}$, with discriminant and regulator ${\mathrm{\Delta }}_K,R_K$, respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by $|{\mathrm{\Delta }}_K{|}^{-1∕2}{\gamma }_K^{n}exp(2n{\rho }_{\infty }({\mathrm{\Lambda }}_K))$, where ${\mathrm{\Delta }}_K$ is the discriminant of the field K, ${\gamma }_K$ is the additive Hermite–Humbert constant over positive-definite unary forms for K and ${\rho }_{\infty }({\mathrm{\Lambda }}_K)$ is the covering radius of the log-unit lattice. In particular, when K is Galois over $\mathbb{Q}$ and n is a prime number, the number of homothety classes of unary forms is upper bounded by ${(\frac{2}{3}n)}^{n}exp(O(n^{\frac{3}{2}}\sqrt{log(n)}{R}_K^{\frac{1}{n-1}}))|{\mathrm{\Delta }}_K{|}^{1∕2}$, where $R_K$ is the regulator of K. Moreover, if K is a maximal totally real subfield of a cyclotomic field, the number of homothety classes of perfect unary forms is upper bounded by ${(\frac{2}{3}n)}^{n}exp(O(n^{\frac{3}{2}}\sqrt{log(n)}))|{\mathrm{\Delta }}_K{|}^{1∕2}$.

We show that for every n-dimensional lattice the torus can be embedded with distortion into a Hilbert space. This improves the exponential upper bound of O(n^3n/2) due to Khot and Naor (FOCS 2005, Math. Ann. 2006) and gets close to their lower bound of . We also obtain tight bounds for certain families of lattices.

Our main new ingredient is an embedding that maps any point to a Gaussian function centered at u in the Hilbert space . The proofs involve Gaussian measures on lattices, the smoothing parameter of lattices and Korkine–Zolotarev bases.

We take a Galois ring $GR(p^m,g)$ and discuss about the self-dual codes and its properties over the ring. We will also describe the relationship between Clifford-Weil group and Jacobi forms by constructing the invariant polynomial ring with the complete weight enumerator.

In this paper, we showed that any clone of operations preserving a nontrivial $n$-equivalence on a finite set is categorically equivalent to the clone of operations preserving a nontrivial $n$-equivalence on a set having the cardinality of $3$.

In this paper, we consider an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-m})$ with $-m\equiv 3$ (mod $4$). In particular, we study the ring of integers corresponding to the field $K$ and visualize the form of ${𝜖}_K/p{𝜖}_K$. We also consider lattices over the ring of integers ${𝜖}_K$ and discuss the theta series to see its relation with the weight enumerator. As a consequence, we will see how the theta series differs for different $m$ and $m^{\prime }$.