Check Character Systems Using Quasigroups: II

In this part of the article we continue to research check character systems with one check character over quasigroups under check equations which have one permutation. These systems always detect all single errors (i.e. errors in only one component of a code word) and can detect some other errors arising during transmission of data. We study check character systems over T-quasigroups. These quasigroups are isotopic to abelian groups and generalize the well-known class of medial quasigroups. We establish some properties of a T-quasigroup so that the check character systems over it are able to detect transpositions, jump transpositions, twin errors and jump twin errors. We also give some models of T-quasigroups, which satisfy all of the required properties for detection of errors of each of the considered types. Communicated by: P. Wild     

Check Character Systems using Quasigroups: I

We consider check character systems with one check character over an arbitrary quasigroup under different check equations which have one permutation. The aim of using such a system is to discover transmission errors. These systems always detect all single errors (that is errors only one component of a code word). We establish necessary and sufficient conditions for a general quasigroup check character system over this quasigroup to be able to detect all transpositions. jump transpositions, twin errors and jump twin errors. We also study capabilities of check character systems over loops, in particular, over Moufang loops and groups when the permutation in the check equations is the identify permutation.Communicated by: P. Wild     

(2, 3, k)-generated groups of large rank

For any fixed k 3 7k \geq 7 there exist integers n_k and a_k such that if the ring R is generated by a set of m elements t₁,...,t_m, where 2t₁-t₁²2t_1-t_1^2 is a unit of finite multiplicative order, and n 3 n_k+ma_kn \geq n_k+ma_k, then the group E_n(R) generated by elementary transvections is an epimorphic image of the triangle group D(2,3,k).\Delta (2,3,k).     

Extension theorems for functional equations with bisymmetric operations

Summary. In this paper we deal with the extension of the following functional equation¶¶ f (x) = M (f (m₁(x, y)), ..., f (m_k(x, y)))        (x, y ? K) f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) , (*)¶ where M is a k-variable operation on the image space Y, m₁,..., m_k are binary operations on X, K ì X K \subset X is closed under the operations m₁,..., m_k, and f : K ? Y f : K \rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m₁,..., m_k, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m₁,..., m_k)-affine hull K^* of K, such that (*) holds over K^*. (The set K^* is defined as the smallest subset of X that contains K and is (m₁,..., m_k)-affine, i.e., if x ? X x \in X , and there exists y ? K^* y \in K^* such that m₁(x, y), ?, m_k(x, y) ? K^* m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* then x ? K^* x \in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.     

Permutations of finite fields for check digit systems

Let q be a prime power. For a divisor n of q ? 1 we prove an asymptotic formula for the number of polynomials of the form

$f(X)=\frac{a-b}{n}\left(\sum_{j=1}^{n-1}X^{j(q-1)/n}\right)X+\frac{a+b(n-1)}{n}X\in\mathbb{F}_q[X]$

such that the five (not necessarily different) polynomials f(X), f(X)±X and f(f(X))±X are all permutation polynomials over \({\mathbb{F}_q}\) . Such polynomials can be used to define check digit systems that detect the most frequent errors: single errors, adjacent transpositions, jump transpositions, twin errors and jump twin errors.

     

Eigenvalues of Random Power law Graphs

Many graphs arising in various information networks exhibit the "power law" behavior -the number of vertices of degree k is proportional to k^-# for some positive #. We show that if # > 2.5, the largest eigenvalue of a random power law graph is almost surely(1+ o(1))?m (1+ o(1))\sqrt{m} where m is the maximum degree. Moreover, the klargest eigenvalues of a random power law graph with exponent # have power law distribution with exponent 2# if the maximum degree is sufficiently large, where k is a function depending on #, mand d, the average degree. When 2<#< 2.5, the largest eigenvalue is heavily concentrated at cm^3-# for some constant c depending on # and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and [(d)\tilde] \tilde{d} , the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely (1+ o(1))?{m_k} (1+ o(1))\sqrt{m_k} where m_k is the k-th largest expected degree provided m_k is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.     

Auto-equivalences of derived categories acting on cohomology

Let A be a k-algebra which is projective as a k-module, let M be an A-module whose endomorphisms are given by multiplication by central elements of A, and let TrPic_k(A) be the group of standard self-equivalences of the derived category of bounded complexes of A-modules. Then we define an action of the stabilizer of M in TrPic_k(A) on the Ext-algebra of M. In case M is the trivial module for the group algebra kG = A, this defines an action on the cohomology ring of G which extends the well-known action of the automorphism group of G on the cohomology group.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

Non-horizontal submanifolds and coarea formula

Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.     

Finding Large p-Colored Diameter Two Subgraphs

Given a coloring of the edges of the complete graph K on n vertices in k colors, a p-colored subgraph of K_n is any subgraph whose edges only use colors from some p element set. We show for k̿ and k\2hphk that there is always a p-colored diameter two subgraph of K_n containing at least [((k+p)n)/(2k)]\displaystyle{(k+p)n \over 2k} vertices and that this is best possible up to an additive constant l satisfying 0hl<k\2.     

On the number of conjugacy classes of maximal subgroups in a finite soluble group

We show that for many formations \frak F\frak F, there exists an integer n = [`(m)](\frak F)n = \overline m(\frak F) such that every finite soluble group G not belonging to the class \frak F\frak F has at most n conjugacy classes of maximal subgroups belonging to the class \frak F\frak F. If \frak F\frak F is a local formation with formation function f, we bound [`(m)](\frak F)\overline m(\frak F) in terms of the [`(m)](f(p))(p ? \Bbb P )\overline m(f(p))(p \in \Bbb P ). In particular, we show that [`(m)](\frak N^k) = k+1\overline m(\frak N^k) = k+1 for every nonnegative integer k, where \frak N^k\frak N^k is the class of all finite groups of Fitting length ￡ k\le k.     

On formal quantum group laws

There exist natural generalizations of the concept of formal groups laws for noncommutative power series. This is a note on formal quantum group laws and quantum group law chunks. Formal quantum group laws correspond to noncommutative (topological) Hopf algebra structures on free associative power series algebras ká áx₁,...,x_m ? ?k\langle\! \langle x_1,\dots,x_m \rangle\! \rangle , k a field. Some formal quantum group laws occur as completions of noncommutative Hopf algebras (quantum groups). By truncating formal power series, one gets quantum group law chunks. ¶If the characteristic of k is 0, the category of (classical) formal group laws of given dimension m is equivalent to the category of m-dimensional Lie algebras. Given a formal group law or quantum group law (chunk), the corresponding Lie structure constants are determined by the coefficients of its chunk of degree 2. Among other results, a classification of all quantum group law chunks of degree 3 is given. There are many more classes of strictly isomorphic chunks of degree 3 than in the classical case.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

On Relative Difference Sets in Dihedral Groups

In this paper, we study extensions of trivial difference sets in dihedral groups. Such relative difference sets have parameters of the form (uλ,u,uλ, λ) or (uλ+2,u, uλ+1, λ) and are called semiregular or affine type, respectively. We show that there exists no nontrivial relative difference set of affine type in any dihedral group. We also show a connection between semiregular relative difference sets in dihedral groups and Menon–Hadamard difference sets. In the last section of the paper, we consider (m, u, k, λ) difference sets of general type in a dihedral group relative to a non-normal subgroup. In particular, we show that if a dihedral group contains such a difference set, then m is neither a prime power nor product of two distinct primes.     

Non-Commutative Ribbons and Quasi-Differential Operators

This paper is devoted to a non-commutative generalization of a classical result occurring in the context of the modular representation theory of the symmetric group. We prove that a non-commutative Schur ribbon function R_I is annihilated by the quasi-differential operator D_Pk if and only if the composition I is the external border of a k-core.     

The character values of the irreducible constituents of a transitive permutation representation

In this article we determine the irreducible ordinary characters c_r \chi_r of a finite group G occurring in a transitive permutation representation (1_M )^G of a given subgroup M of G, and their multiplicities m_r = ((1_M)^G, c_r) 1 0 m_r = ((1_{M})^G, \chi_r) \neq 0 by means of a new explicit formula calculating the coefficients a_rk of the central idempotents e_r = ?_k=1^d a_rk D_k e_r = \sum\limits_{k=1}^{d} a_{rk} D_k in the intersection algebra B \cal B of (1_M )^G generated by the intersection matrices D_k corresponding to the double coset decomposition G = è_k=1^d Mx_k M G = \bigcup\limits_{k=1}^{d} Mx_{k} M .¶Furthermore, an explicit formula is given for the calculation of the character values c_r(x) \chi_{r}(x) of each element x ? G x \in G . Using this character formula we obtain a new practical algorithm for the calculation of a substantial part of the character table of G.     

On the Gap in the Spectrum of the Translations

The spectrum of the translations in local quantum field theory will be analyzed in order to show that in a positive energy representation without vacuum vector and with lowest mass m₁ there is no gap in the spectrum which is larger than 2m₁. In particular in a zero mass representation there is no hole at all. These results are obtained with methods of analytic functions of several complex variables.     

Groups with anti-symmetric mappings

Conclusion The above theorems are sufficient to demonstrate the existence of antisymmetric mappings for all non-Abelian group of order less than 36, except for the group <a, b\a ³ =b ⁸ =e, ab =ba ²> of order 24, and this group can be shown to have one. On the basis of our results, and the fact that we have no conditions on a non-Abelian group that would eliminate any from having anti-symmetric mappings, we conjecture that allnon-Abelian groups have anti-symmetric mappings.     

Abelian Kernels of Some Monoids of Injective Partial Transformations and an Application

In this paper we compute the abelian kernels of the monoids POI_n and POPI_n of all injective order preserving and respectively, orientation preserving, partial transformations on a chain with n elements. As an application, we show that the pseudovariety POPI generated by the monoids POPI_n (n epsilon N) is not contained in the Mal'cev product of the pseudovariety POI generated by the monoids POI_n (n epsilon N) with the pseudovariety Ab of all finite abelian groups.     

Reducibility of polynomials a0(x) + a1 (x)y + a2 (x) y2 modulo p

Let f=a₀(x)+a₁(x)y+a₂(x)y² ? \Bbb Z[x,y]f=a_0(x)+a_1(x)y+a_2(x)y^2\in {\Bbb Z}[x,y] be an absolutely irreducible polynomial of degree m in x. We show that the reduction f mod p will also be absolutely irreducible if p 3 c_m·H(f)^e_mp\ge c_m\cdot H(f)^{e_m} where H (f) is the height of f and e₁ = 4,e₂ = 6, e₃ = 6 [2/3]{2}\over{3} and e_m = 2 m for m S 4. We also show that the exponents e_m are best possible for m 1 3m\ne 3 if a plausible number theoretic conjecture is true.