Non-isomorphic Abelian Groups Research Articles

On some sums involving the counting function of nonisomorphic Abelian groups*

On some sums involving the counting function of nonisomorphic Abelian groups*

An Invariant of Topologically Ordered States Under Local Unitary Transformations

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups, must have depth that is at least linear in the system's diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness.

The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals

The result is: The distribution of values of the enumerating function of finite, non-isomorphic abelian groups in short intervals is similar to the distribution of square-free numbers in short intervals. Moreover, a new estimate of the error term in the corresponding asymptotic formula is given, which improves former estimates.

On the enumeration of finite Abelian and solvable groups

Let a( n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a( n). More generally, we get short interval version of results of Ivić on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a( n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.

On the number of finite non-isomorphic Abelian groups in short intervals

Let a(n) denote the number of non-isomorphic Abelian groups of order n. It is well-known thatfor a natural number k we defineand

Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Let a(n) denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA_k(x, h):=sum from n=x<n≤x+h,a(n)=k to (1)If h≥x~(581/1744)logx=x~(0.33314…)logx as x→∞,it was proved by A,Ivic thatA_k(x, h)=(d_k+o(1))h, (1)whered_k=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e~(ikt g_t(n)dt≥0))),g_t(n)=sum from n=d/n to (μ(n/d)e~(ita)(d)).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x~(877/2653)(logx)~c=x~(0.3305…)(logx)~c,then Eq.(1)is true, where C is a computable constant. Based on the estimate for △(1, 2, 2;x) in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.

The number of finite non-isomorphic abelian groups in mean square.

Let $\Delta(x)=\sum_{n\leq x}a(n)-\sum_{j=1}^6 c_jx^{1/j}$ denote the error term in the abelian group problem. Using zeta-function methods it is proved that $$\int_1^X\Delta^2(x)\,dx~&lt;\!\!&lt;~ X^{39/29} \log^2X$$ where the exponent $39/29=1.344827\ldots$ is close to the best possible exponent $4/3$ in this problem.

On the number of Abelian groups of a given order and on certain related multiplicative functions

Let a( n) denote the number of nonisomorphic Abelian groups with n elements. Several problems concerning a( n) and a class of related multiplicative functions are discussed. These include precise characterizations of local densities, distribution of values (where the Tauberian theorem of Hardy-Ramanujan is used), sharp formulas for the sums Σ n≤ x F( a( n)) for a wide class of functions F, and a comparison of values taken on the average by a( n) and some other common arithmetic functions.

An Asymptotic Formula for Reciprocals of Logarithms of Certain Multiplicative Functions

Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)&gt;0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

Symplectic groups, symmetric designs, and line ovals

symmetric design P(2m) having r as its full automorphism group, we will prove these results. The symmetric designs 9+(2m) will be studied and characterized in terms of an interesting property concerning the symmetric difference of distinct blocks. Other properties and characterizations of these desings will also be given, and an application made to rank 3 linear groups. There are several ways to construct Yf(2m). One way is in terms of difference sets [3, p. 1081; from this point of view, Ye(2m) has the unusual property of arising from difference sets in m + 1 nonisomorphic abelian groups. Using a description in terms of the incidence matrix, Block [l] observed that the automorphism group of F(2m) is 2-transitive. The present work was motivated by Block’s result. In the course of studying the full automorphism group Yf(2m), a description was found in terms of GOs(2m, 2) and Sp(2m, 2) (see Section 4). More recently, the designs P(2m) were discovered in terms of the latter description by A. Rudvalis (unpublished) and Cameron and Seidel [2]. Yet another description of the designs P(2m) arises from using suitable dual ovals in desarguesian and Liineburg-Tits planes. The construction in terms of desarguesian planes was obtained by the author [3, p. 951 at the same time that the incidence matrix and symplectic group descriptions were first considered; however, it was not originally known that the descriptions produced the same designs. The construction from the Liineburg-Tits planes requires the use of the designs to prove a new property of the planes. We note

Estimates of convolutions of certain number-theoretic error terms

Several estimates for the convolution functionC [f(x)]:=∫1xf(y) f(x/y)(dy/y)and its iterates are obtained whenf(x)is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for∫0T|ζ(1/2+it)|2kdt(k=1,2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution.