Integral lattices in TQFT

We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3)-TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.     

A note on the Goormaghtigh equation

Periodica Mathematica Hungarica - Let p be an odd prime. By using a lower bound for linear forms in logarithms of two algebraic numbers, we prove that if $$p>10^{24}$$ , 2 is a primitive...     

Collineation groups irreducible on the components of a translation plane

We investigate finite translation planes of odd dimension over their kernels in which the translation complement induces on each component l a permutation group whose order is divisible by a p-primitive divisor. Using results of this investigation, we show that rank 3 affine planes of odd dimension over their kernels are either generalized André planes or semi-field planes. A similar result is given for translation planes having a collineation group which is doubly transitive on each affine line; besides the above two possibilities, there is a third possibility; the plane has order 27, the translation complement is doubly transitive on , and SL(2, 13) is contained in the translation complement.We also consider translation planes of odd dimension over their kernels which have a collineation group isomorphic to SL(2, w) with w prime to 5 and the characteristic, and having no affine perspectivity. We show that such planes have order 27, the prime power w=13, and the given group together with the translations forms a doubly transitive collineation group on {ie153-1}. This indicates quite strongly that the Hering translation plane of order 27 is unique with respect to the above properties.Both authors supported in part by NSF Grant No. MCS76-0661 A01.     

Gorenstein Isolated Quotient Singularities of Odd Prime Dimension are Cyclic

In this article, we shall prove that Gorenstein isolated quotient singularities of odd prime dimension are cyclic. In the case where the dimension is bigger than 1 and is not an odd prime number, then there exist Gorenstein isolated noncyclic quotient singularities.     

Some varieties and convexities generated by fractal lattices

We introduce definitions of semifractal, 0–1-fractal, quasifractal and fractal lattices. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for the other variants. We show that a semifractal generated nondistributive lattice variety cannot be of residually finite length. This easily implies that there are exactly continuously many lattice varieties which are not semifractal generated. On the other hand, for each prime field F, the variety generated by all subspace lattices of vector spaces over F is shown to be fractal generated. These countably many varieties and the class of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive fractal lattices are given each of which generates . After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals are given each of which has cardinality and generates the variety of all lattices. Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, thus answering a question of Jakubík. (A convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products, a notion due to Ervin Fried.) This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433 and K 60148.     

There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Self-dual codes over exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24, 12, 10] code over , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.      

n-distributivity,dimension and Carathéodory's theorem

A. Huhn proved that the dimension of Euclidean spaces can be characterized through algebraic properties of the lattices of convex sets. In fact, the lattice of convex sets of isn+1-distributive but notn-distributive. In this paper his result is generalized for a class of algebraic lattices generated by their completely join-irreducible elements. The lattice theoretic form of Carathéodory's theorem characterizesn-distributivity in such lattices. Several consequences of this result are studied. First, it is shown how infiniten-distributivity and Carathéodory's theorem are related. Then the main result is applied to prove that for a large class of lattices beingn-distributive means being in the variety generated by the finiten-distributive lattices. Finally,n-distributivity is studied for various classes of lattices, with particular attention being paid to convexity lattices of Birkhoff and Bennett for which a Helly type result is also proved.Presented by J. Sichler.     

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ₆-codes. Then extremal self-dual ₆-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.     

THE GOLDBACH-VINOGRDOV THEOREM WITH THREE PRIMES IN A THIN SUBSET

1.IntroductionandStatementofResultsIn1937,Vinogradovi7]provedthatJ(N),thenumberofrepresefltationsofanilltegerNassumsofthreeprimes,satisfiesthefollowingasymptoticformulawherea(N)isthesingularseries,andu(N)>>1foroddN.Itthereforefollowsthateverysufficientlylargeoddintegeristhesumofthreeprimes.ThissettledtheternaryGoldbachproblem,andtheresultisreferredtoastheGoldbach-Vinogradovtheorein.ManyauthorshaveconsideredthecorrespondingproblemswithrestrictedconditionsposedonthethreeprimesintheGoldbach…     

Some extremal self-dual codes and unimodular lattices in dimension 40

In this paper, binary extremal singly even self-dual codes of length 40 and extremal odd unimodular lattices in dimension 40 are studied. We give a classification of extremal singly even self-dual codes of length 40. We also give a classification of extremal odd unimodular lattices in dimension 40 with shadows having 80 vectors of norm 2 through their relationships with extremal doubly even self-dual codes of length 40.     

On the distribution of square-full and cube-full primitive roots

Periodica Mathematica Hungarica - A positive integer n is called an r-full integer if for all primes $$p\mid n$$ we have $$p^r\mid n.$$ Let p be an odd prime. For $$\gcd (n,p)=1$$, the smallest...     

Pro- $$\mathcal {C}$$ C congruence properties for groups of rooted tree automorphisms

We propose a generalisation of the congruence subgroup problem for groups acting on rooted trees. Instead of only comparing the profinite completion to that given by level stabilizers, we also compare pro- $$\mathcal {C}$$ completions of the group, where $$\mathcal {C}$$ is a pseudo-variety of finite groups. A group acting on a rooted, locally finite tree has the $$\mathcal {C}$$ -congruence subgroup property ( $$\mathcal {C}$$ -CSP) if its pro- $$\mathcal {C}$$ completion coincides with the completion with respect to level stabilizers. We give a sufficient condition for a weakly regular branch group to have the $$\mathcal {C}$$ -CSP. In the case where $$\mathcal {C}$$ is also closed under extensions (for instance the class of all finite p-groups for some prime p), our sufficient condition is also necessary. We apply the criterion to show that the Basilica group and the GGS-groups with constant defining vector (odd prime relatives of the Basilica group) have the p-CSP.     

On the Geometry of Cyclic Lattices

Cyclic lattices are sublattices of \(\mathbb Z^N\) that are preserved under the rotational shift operator. Cyclic lattices were introduced by Micciancio (FOCS, IEEE Computer Society, pp 356–365, 2002) and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen (Theory of Cryptography, Lecture Notes in Computer Science, vol 3876. Springer, Berlin, pp 145–166, 2006) showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices, proving that a positive proportion of them in every dimension is well-rounded. One implication of our main result is that SVP is equivalent to SIVP on a positive proportion of cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of cyclic lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of \(\mathbb Z^N\) closed under the action of subgroups of the permutation group \(S_N\) , which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any \(N\) -cycle.     

On Integral Well-rounded Lattices in the Plane

We investigate distribution of integral well-rounded lattices in the plane, parameterizing the set of their similarity classes by solutions of the family of Pell-type Diophantine equations of the form x ²+Dy ²=z ² where D>0 is squarefree. We apply this parameterization to the study of the greatest minimal norm and the highest signal-to-noise ratio on the set of such lattices with fixed determinant, also estimating cardinality of these sets (up to rotation and reflection) for each determinant value. This investigation extends previous work of the first author in the specific cases of integer and hexagonal lattices and is motivated by the importance of integral well-rounded lattices for discrete optimization problems. We briefly discuss an application of our results to planar lattice transmitter networks.     

Some permutations over $$\pmb {\mathbb {F}}_p$$ F p concerning primitive roots

The Ramanujan Journal - Let p be an odd prime and let $${\mathbb {F}}_p$$ denote the finite field with p elements. Suppose that g is a primitive root of $${\mathbb {F}}_p$$ . Define the permutation...     

Unimodular lattices with long shadow

Let L be an odd unimodular lattice of dimension n with shadow n−16. If min(L)?3 then dim(L)?46 and there is a unique such lattice in dimension 46 and no lattices in dimensions 44 and 45. To prove this, a shadow theory for theta series with spherical coefficients is developed.     

K-Theory of curves over number fields

We consider the algebraic K-groups with coefficients of smooth curves over number fields. We give a proof of the Quillen-Lichtenbaum conjecture at the prime 2 and prove explicit corank formulas for the algebraic K-groups with divisible coefficients. At odd primes these formulas assume the Bloch-Kato conjecture, at the prime 2 the formulas hold nonconjecturally.     

Hull mappings and dimension effect algebras

An effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.