Analytic Representations in the 3-dim Frobenius Problem

We consider the Diophantine problem of Frobenius for semigroup ${\sf S}({\bf d}^3)$ where ${\bf d}^3$ denotes the tuple $(d_1,d_2,d_3)$, $\gcd(d_1,d_2,d_3)=1$. Based on the Hadamard product of analytic functions we have found the analytic representation for the diagonal elements $a_{kk}({\bf d}^3)$ of the Johnson's matrix of minimal relations in terms of $d_1,d_2,d_3$. Bearing in mind the results of the recent paper this gives the analytic representation for the Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and the Hilbert series $H({\bf d}^3;z)$ for the semigroups ${\sf S}({\bf d}^3)$. This representation does complement the Curtis' theorem on the non-algebraic representation of the Frobenius number $F({\bf d}^3)$. We also give a procedure to calculate the diagonal and off-diagonal elements of the Johnson's matrix.Comment: 16 pages, 3 figure

Symmetric (not Complete Intersection) Semigroups Generated by Five Elements

We consider symmetric (not complete intersection) numerical semigroups S_5, generated by five elements, and derive inequalities for degrees of syzygies of S_5 and find the lower bound F_5 for their Frobenius numbers. We study a special case W_5 of such semigroups, which satisfy the Watanabe Lemma, and show that the lower bound F_{5w} for the Frobenius number of the semigroup W_5 is stronger than F_5.Comment: 7 pages, 2 Table

Duality Relation for the Hilbert Series of Almost Symmetric Numerical Semigroups

We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set d^m to generate such semigroups.Comment: 28 page

Piezoelectricity and Piezomagnetism : Duality in Two-Dimensional Checkerboards

The duality approach in 2-{\it dim} two-component regular checkerboards was extended onto piezoelectricity and piezomagnetism problems. There are found a relation for effective piezoelectric and piezomagnetic modules for the checkerboard with $p6^{\prime} mm^{\prime}$-plane symmetry group ({\em dichromatic triangle}). \pacs{Pacs: 73.50.Bk,Jt, 75.70.Ak, 77.65.-j, 77.84.Lf}Comment: 3 pages, two-columns, 1 figure. J.Phys.A - submitte

Frobenius Problem for Semigroups {\sl S}(d_1,d_2,d_3)

The matrix representation of the set $\Delta({\bf d}^3)$, ${\bf d}^3=(d_1,d_2, d_3)$, of the integers which are unrepresentable by $d_1,d_2,d_3$ is found. The diagrammatic procedure of calculation of the generating function $\Phi({\bf d}^3;z)$ for the set $\Delta({\bf d}^3)$ is developed. The Frobenius number $F({\bf d}^3)$, genus $G({\bf d}^3)$ and Hilbert series $H({\bf d}^3;z)$ of a graded subring for non--symmetric and symmetric semigroups ${\sf S}({\bf d}^3)$ are found. The upper bound for the number of non--zero coefficients in the polynomial numerators of Hilbert series $H({\bf d}^m;z)$ of graded subrings for non--symmetric semigroups ${\sf S} ({\bf d}^m)$ of dimension, $m\geq 4$, is established.Comment: 43 pages, 10 Figure

Self-Dual Symmetric Polynomials and Conformal Partitions

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$ and skew-reciprocal ${\sf S}^{\{m\}}_{S_n}$ algebraic polynomials based on the polynomial invariants of the symmetric group $S_n$. These polynomials form an infinite commutative semigroup. Real solutions $\lambda_n(x_i)$ of corresponding algebraic Eqns have many important properties: homogeneity of 1-st order, duality upon the action of the conformal group ${\sf W}$, inverting both function $\lambda_n$ and the variables $x_i$, compatibility with trivial solution, {\it etc}. Making use of the relationship between Gaussian generating function for conformal partitions and Molien generating function for usual restricted partitions we derived the analytic expressions for ${\cal P}_n^m(s)$. The unimodality indices for the reciprocal and skew-reciprocal equations were found. The existence of algebraic functions $\lambda_n(x_i)$ invariant upon the action of both the finite group $G\subset S_n$ and conformal group ${\sf W}$ is discussed.Comment: 30 page

Symmetric Numerical Semigroups Generated by Fibonacci and Lucas Triples

The symmetric numerical semigroups S(F_a,F_b,F_c) and S(L_k,L_m,L_n) generated by three Fibonacci (F_a,F_b,F_c) and Lucas (L_k,L_m,L_n) numbers are considered. Based on divisibility properties of the Fibonacci and Lucas numbers we establish necessary and sufficient conditions for both semigroups to be symmetric and calculate their Hilbert generating series, Frobenius numbers and genera.Comment: 10 page

New Identities for Degrees of Syzygies in Numerical Semigroups

We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series H(d^m;z) of nonsymmetric numerical semigroups S(d^m) of arbitrary generating set of positive integers d^m={d_1,...,d_m}, m\geq 3. These identities were obtained by studying together the rational representation of the Hilbert series H(d^m;z) and the quasipolynomial representation of the Sylvester waves in the restricted partition function W(s,d^m). In the cases of symmetric semigroups and complete intersections these identities become more compact.Comment: Primary -- 20M14, Secondary -- 11P8

Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number. Assuming an asymptotic behavior of summatory function, F{f;N,1}\stackrel{N\to \infty}{=}G_1(N) [1+ {\cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2\geq 0, -\infty < b_1, b_2< \infty, we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\infty}(f;p^m) when N\to \infty. We prove that the renormalization function is multiplicative, i.e., R_{\infty}(f;\prod_{i=1}^n p_i^{m_i})= \prod_{i=1}^n R_{\infty}(f;p_i^{m_i}) with n distinct primes p_i. We extend these results on the others summatory functions \sum_{k\leq N}f(p^m k^l), m,l,k\in \mathbb Z}_+ and \sum_{k\leq N}\prod_{i=1}^n f_i(k p^{m_i}), f_i\neq f_j, m_i\neq m_j. We apply the derived formulas to a large number of basic summatory functions including the Euler \phi(k) and Dedekind \psi(k) totient functions, divisor \sigma_n(k) and prime divisor \beta(k) functions, the Ramanujan sum C_q(n) and Ramanujan \tau(k) Dirichlet series, and others.Comment: 46 pages, 6 figure

Arnold's Conjectures on Weak Asymptotics and Statistics of Numerical Semigroups S(d_1,d_2,d_3)

Three conjectures #1999--8, #1999--9 and #1999--10 which were posed by V. Arnold [2] and devoted to the statistics of the numerical semigroups are refuted for the case of semigroups generated by three positive integers d_1,d_2,d_3 with gcd(d_1,d_2,d_3)=1. Weak asymptotics of conductor C(d_1,d_2,d_3) of numerical semigroup and fraction p(d_1,d_2,d_3) of a segment [0;C(d_1,d_2,d_3)-1] occupied by semigroup are found.Comment: 25 page