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In the mentioned paper of Hadwiger an upper bound is given for α (A)·N(A), where α (A) is the density of covering ℝn by lattice translates ofA⊂ℝn andN(A) is the number of neighbours ofA. Using “dual” descriptions of α (A) andN(A), in the paper a sharpening of Hadwiger's inequality as well as a lower bound for α (A)·N(A) are proved.

In a previous paper the authors introduced seven complete congruences on the lattice ℒev(ℛI of e-varieties of regular semigroups of the form ρP:UρPV⇔P∘U=P∘V, whereP is drawn from a small set of e-varieties: left zero, right zero, rectangular bands, groups, left groups, right groups and completely simple semigroups. Four new complete congruences are introduced here of the form αP:UαPV⇔P∩U=P∩V, whereP is one of the following classes of regular semigroups: left monoids, right monoids, monoids, idempotent generated semigroups. For each complete congruence ρ on ℒev(ℛI) and eachU∈ℒev(ℛI), the ρ-class ofU is an interval [Uρ,Uρ] so that there is associated with each such congruence an idempotent operatorU→Uρ on ℒev(ℛI). This paper establishes numerous results concerning the commutativity of operators of this form.

Starting from problem 4 ofK. Wagner [2],H. Fleischner andP. D. Vestergaard [1] introduce the notion of a value-true walk in a finite, connected graph, the edges of which are valuated with nonnegative integers. Their main theorem states that the existence of such a walk is equivalent to the existence of an orientation of the edges with the following property: For every vertex the sum of the valuations of the incoming edges equals the sum of the valuations of the outgoing edges. Let us call such an orientation a cyclic one. In the present paper we study finite, valuated graphs that admit a cyclic orientation. First, we give two necessary conditions for a valuated graphG to admit a cyclic orientation concerning the stars and the bonds ofG, respectively. (The starS (v) of a vertexv is the set of all edges ofG incident withv.) Then, as the main part of the paper we give a characterization of those graphs for which the star- and the bond-condition is sufficient, respectively (for any valuation of the graph). These characterizations are in terms of constructability from trees andK 3, respectively, as well as in terms of forbidden subgraphs.

The graphs considered are finite and undirected, loops do not occur. An induced subgraphI of a graphX is called animitation ofX, if In the first chapter some theorems concerning exclusive graphs and Euler graphs are stated. Chapters 2 deals withHG n′ s and bipartite graphs. In chapters 3 a useful concept—theX-graph of anHG n—is defined; in this paper it is the conceptual connection between exclusive graphs andHG n′ s, since a graphG is anHG n, if all itsX-graphs are exlusive. Furthermore, some theorems onX-graphs are proved. Chapter 4 contains the quintessence of the paper: If we want to construct a newHG n F from anotherHG n G, we can consider certain properties of theX-graphs ofG to decide whetherF is also anHG n.

For a latticeL in ℝn with determinantd(L), let η (L) denote the supremum of the values 2−2V(P)/d(L), taken over theL-admissible parallelepidesP, symmetric with respect to the origin and with faces parallel to the coordinate-axes. In 1936, Mordell asked for the constants ℵn= min ℵ(L) over alln-dimensional lattices. In this paper we investigate isolated minima of η (L) in all over alln-dimensional lattices. In this paper we (Satz 1) and some examples are given. In particular, forn

This paper is part of a program aiming at the classification of all higher-dimensional locally compact translation planes whose collineation groups have large dimension. In the present paper we determine all eight-dimensional locally compact translation planes which admit acompact collineation group Σ of dimension at least 5 acting almost effectively on the translation axis. In fact, Σ is isomorphic either to Spin4ℝ or toSO 4(ℝ). The case∑ ≅ Spin4(ℝ) has already been treated elsewhere ([6]). Here, the planes with∑ ≅SO 4(ℝ) are explicitly determined and studied in detail.

In this paper we prove three theorems, each of which gives a set of absolutely independent group axioms. Theorem 2 is an extension of Morgado's theorem obtained by using a weaker associativity axiom. The axioms of Theorem 3 are the first absolutely independent group axioms we have encountered for ,,~ich there is no k such that the axioms are absolutely independent (rood k) [2, p. 758]. The systems of Jacobson-Yoco~n and Morgado and the systems which appear in Theorems 1 and 2 of this paper are absolutely independent (rood ,~0). We conclude the paper with two axioms which define a group and which are very [l] (and in this case absolutely) independent (rood n) for every integer n ~ 1. We will write "xy" instead of " x , y" in all proofs. The following lemma is quite useful.

Let K be a commutative hypergroup with the Haar measure μ. In the present paper we investigate whether the maximal ideals in L1(K,μ) have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L∞(K,μ). Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.

In this paper we give a complete description of the set of blow up points of solutions of the problem $$\begin{gathered} u_t = \left( {u^m } \right)_{xx} {\text{ in }}D_T = [0,\infty ) \times [0,T) \hfill \\ - \left( {u^m } \right)_x \left( {0,t} \right) = u^m \left( {0,t} \right){\text{ on [0,}}T{\text{)}} \hfill \\ u\left( {x,0} \right) = u_0 \left( x \right) \geqslant 0{\text{ on[0, + }}\infty {\text{)}} \hfill \\ \end{gathered} $$ where m> I.

In this paper we introduce the notion of degree for C 1-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C 1-cocycle over an irrational rotation by α has nonzero degree, then the skew product is not ergodic and the group of essential values of ϕ is equal to the maximal Abelian subgroup of SU(2). Moreover, if ϕ is of class C 2 (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity. Possible values of degree are discussed, too.

The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the multidimensional discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums.

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators\(T_i : = \frac{\partial }{{\partial x_i }} + k\sum\nolimits_{j \ne i} {\frac{{1 - (ij)}}{{x_i - x_j }}} \), (i=1, ...,N, where (ij) denotes the transposition of the variablesxixj andk is a fixed parameter). We introduce a family of functions {pα}, indexed bym-tuples of non-negative integers α = (α1, ..., αm) form≤N, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which\(T_i V = V\frac{\partial }{{\partial x_i }}\),i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatTipα=0 fori>m, and $$V(x_1^{\alpha _1 } \cdots x_m^{\alpha _m } ) = \frac{{\lambda _1 !\lambda _2 ! \cdots \lambda _m !}}{{\left( {Nk + 1} \right)_{\lambda _1 } \left( {Nk - k + 1} \right)_{\lambda _2 } \cdots (Nk - (m - 1)k + 1)_{\lambda _m } }}p_\alpha + \sum\limits_\beta {A_{\beta \alpha } p_{\beta ,} } $$ where (λ1, λ2, ..., λm) is the partition whose parts are the entries of α (That is, λ1➮ λ2➮ ... λm➮0), β = (β1, ..., βm), ∑i=1m βi = ∑i=1m αm and the sorting of β is a partition strictly larger than λ in the dominance order. This triangular matrix representation ofV allows a detailed study. There is an inner product structure on span {pα} and a convenient set of self-adjoint operators, namelyTiρi, whereρipα ≔p(α1, ...., αi + 1, ..., αm). This structure has a bi-orthogonal relationship with the Jack polynomials inm variables. Values ofk for whichV fails to exist are called singular values and were studied byDe Jeu, Opdam, andDunkl in Trans. Amer. Math. Soc.346 (1994), 237–256. As a partial verification of a conjecture made in that paper, we construct, for anya=1,2,3,... such that gcd(N−m+1,a)

This paper deals with the inversive congruential method with power of two modulusm for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that there exist parameters in the inversive congruential method such that the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude at leastm−1/3. The method of proof relies on a detailed analysis of certain rational exponential sums.

On Roth's theorem. The celebrated theorem of Roth, together with its generalizations given by Mahler and Ridout, gives a lower bound for the degree of approximation of one or more algebraic numbers with respect to a fixed set of valuations by elements of a fixed number field. An analogous result holds for function fields in characteristic zero. In this paper we do the following: (1) generalize Roth's theorem to the case of fields with a product formula in characteristic zero, removing any technical hypothesis from a previous result of Lang: (2) give a unified proof of Roth's theorem in the number field and function field cases; (3) provide a quantitative version of the general Roth's theorem, extending, even in the number field case, previous results of Bombieri and Van Der Poorten.

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.

Letu(n) be a recurrent sequence of rational integers, i.e.,u(n+s)+a s−1 u(n+s−1)+...+a 0 u(n)=0,n≥0,a i∈ℤ,i=0,...,s−1. The polynomialP(x)=x s +a s−1xs +...+a 0 is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots ofP is a root of unity, then the setA={n,u(n)=0} is finite. A recent result of F. Beukers shows that ifs=3, then the setA has at most 6 elements and there exists, up to trivial transformations, only one recurrence of order 3 with 6 zeros, found by J. Berstel. In this paper, we construct for eachs, s≥2 a recurrent sequence of orders, with at leasts 2/2+s/2−1 zeroes, which generalize Berstel's sequence.

In this paper, we establish new upper bounds on the number ofS-integral solutions to hyper- and super-elliptic equations under the conditions ofLeVeque [3]. Our method is based upon that ofEvertse andSilverman [1].

Recently, the explicit inversive congruential method with power of two modulus for generating uniform pseudorandom numbers was introduced. Statistical independence properties of the generated sequences have been studied by estimating the discrepancy of all overlapping pairs of successive pseudorandom numbers. In the present paper a similar analysis is performed for the subsets of nonoverlapping pairs. The method of proof relies on a detailed discussion of the properties of certain exponential sums.

The classical Erdos-Turan-Koksma inequality gives us an upper bound for the discrepancy of a sequence in thes-dimensional unit cube in terms of exponential sums, more precisely, in terms of the trigonometric function system. In this paper, we shall prove the inequality of Erdos-Turan-Koksma for the extreme and the star discrepancy, for generalized Haar function systems. Further, we shall show the existence of the inequality of Erdos-Turan-Koksma for the isotropic discrepancy, for generalized Haar and Walsh function systems.

Let Ω(n) be the number of all prime divisors ofn and ω(n) the number of distinct prime divisors ofn. We definevq(x)∶=|{n≤x∶Ω(n)−ω(n)=q}|. In this paper, we give an asymptotic development ofvq(x); this improves on previous results.

ItG is a group written additively, the inner automorphisms and the endomorphisms additively generate near-ringsI(G) andE(G) respectively. IfI(G)=E(G), i.e., if every endomorphism is a sum of inner automorphisms, we callG anI-E group. In this paper we describe a class ofI-E groups which includes two of the four known classes ofI-E groups and which contains infinitely many other examples. The order ofI(G) is obtained and its radical determined.

Letn be a positive integer andSn a particular set of prime divisors ofn. We establish the average order off(n) wheref(n) stands for the cardinality ofSn. Thek-ary,k-free, semi-k-ary prime factors ofn are some of the classes of prime divisors studied in this paper.

Let Q(x) denote the number of 4-full numbers not exceeding x. It is well known that $$Q(x) = \sum\limits_{j = 4}^7 {r_j x^{1/j} + R(x)}$$ where $$r_j = \mathop {res}\limits_{s = 1/j} (F(s)/s), F(s) = \mathop \prod \limits_P \left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)$$ and R(x) is the remainder. This paper proves that $$R(x) \ll x^{3626/35461 + \varepsilon }$$ where e is any positive number.

Let Ω(n) be the number of prime divisors ofn, counted with multiplicity. We denote byS(x, k) the set of then≤x for which Ω(n)=k, and byVp(n) the exponent of the primep in the factorization ofn. In a previous paper we proved a result which implies that, ify=x/2k tends to infinity withk>2λloglogx where λ>1, then the distribution of the numbers\((V_2 (n) - k + 2\log \log y)/\sqrt {2 \log \log y} \) on the setS(x, k) converges to the normal distribution of Gauss. Here, besides a slight improvement of that result, we give, for the moment of orderq of the above mentioned distribution, a formula which holds uniformly for 2λloglogx≤k≤log (x/3)/log2 where 1

Leta 1,a 2 anda 3 be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp j. We obtain CardE(X)≤X 1-e. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx j with the above bound for CardE(X) better than that in [2].

The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs−2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort≥0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.

Let {Pn}n=0/∞ be a system of orthogonal polynomials.Lasser [5] observed that if the linearization coefficients of {Pn}n=0/∞ are nonnegative then each of thePn(x) is a linear combination of the Tchebyshev polynomials with nonnegative coefficients. The aim of this paper is to give a partial converse to this statement. We also consider the problem of determining when the polynomialsPncan be expressed in terms ofQnwith nonnegative coefficients, where {Qn}n=0/∞ is another system of orthogonal polynomials. New proofs of well known theorems are given as well as new results and examples are presented.

In this paper we study entire function of one complexe variable such thatf (k) (q k) belongs to the rational integers for allk, whereq is a given rational integer. We show that iff is of sufficiently slow growth, thenf is a polynomial.

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).

The purpose of this paper is to define a uniformity on a frame using entourages. A definition of uniformities in terms of covers has been given byA. Pultr ([8], [10]). We introduce two new symmetry conditions for a quasi-uniform space, open-set symmetry and small-set symmetry. We prove that a quasi-uniformityU is a uniformity if and only if it is both open-set symmetric and small-set symmetric. The category of (covering) uniform frames is isomorphic with the category of entourage uniform frames.

The object of this paper is to give a systematic treatment of permutation polynomials (over a finite fieldF q ) of the formx r f(x q−1)/d). In particular, a criterion is obtained for such a polynomial to be a permutation polynomial and it is proved that all such permutation polynomials form a group isomorphic to a generalized wreath product of certain abelian groups.

In this paper a characterization of the regular ω-semigroups whose congruence lattice is semimodular is given. The characterization obtained for such semigroups generalizes the one given by Scheiblich for bisimple ω-semigroups. Notice that we use the definition of semimodularity which other authors call double covering property.

In this paper we note some properties of fully invariant additive subgroups of near-rings and apply these results to d.g., medial, or subdirectly irreducible near-rings

Let ‖·‖ be the weightedL 2-norm with weightw(t). LetP n be the set of all complex polynomials whose degree does not exceedn and let $$\gamma _n^{(r)} : = \sup _{f \in P_n } $$ (‖f (r)‖/‖f‖). In this paper we given upper and lower bound for γ n (r) in the case of the Laguerre weight functionw(t)=exp (−t) and investigate its behaviour asn→∞. Moreover, we derive some identities concerningorthogonal polynomials.

In this paper we estimate the difference between the sum given in the title (whereg(t) is an arbitrary real-valued non-decreasing function,k is a positive integer and summation is extended over all positive integersn≤x) and the corresponding integral, obtaining the boundO(g(x)x 1/3logx). Furthermore, we show that these differences (for giveng and varyingk) are all “approximately equal”, with an error term ofO(g(x)x 3/10). Finally it is remarked without proof that these estimates can be refined toO(g(x)x θ) (with any θ>0,329...) resp.O(g(x)x 109/382).

In this paper the magnitude of the set of pointsx is studied for which the series\(\frac{1}{{V(N)}}\sum\limits_{n = - N}^N {c_n e^{2\pi inx} } \), with complex numberscn and an increasing sequenceV(N) of real numbers, is unbounded. An answer in terms of capacities is given. This result is then used to obtain results about exceptional sets in the theory of uniform distribution, e.g. it is shown that the spectrum of any sequence has dimension zero.

In this paper we give the connection between the zeros of the ζ-function and sequences(g(p)), p prime, mod 1 ifg(x)=αx σ for α≠0, σ>0 or ifg(X) is a polynomial in ℝ.

Bambah, Rogers, Woods, and Zassenhaus considered the general problem of covering planar convex bodiesC byk translates of a centrally-symmetric convex bodyK ofE 2 with the ramification that these translates cover the convex hullC k of their centres. They proved interesting inequalities for the volume ofC andC k . In the present paper some analogous results in euclideand-spaceE d are given. It turns out that on one hand extremal configurations ford≥5 are of quite different type than in the planar case. On the other hand inequalities similar to the planar ones seem to exist in general. Inequalities in both directions for the volume and other quermass-integrals are given.

Letd(n) denote the number of divisors ofn, then the asymptotic formula $$\sum\limits_{\mathop {n< x}\limits_{n = r(\bmod m)} } {d(n) = \xi _1 (r,m)} x\log x + \xi _2 (r,m)x + O(x^{1/2} )$$ is derived and, as the main result of the paper, the coefficients ξi(rm),i= 1,2, as functions of the powers of the prime numbers ofm and of g. c. d. (r, m) are determined.

In this paper we discuss (left) near-rings satisfying the identities:abcd=acbd,abc=bac, orabc=acb, called medial, left permutable, right permutable near-rings, respectively. The structure of these near-rings is investigated in terms of the additive and Lie commutators and the set of nilpotent elementsN (R). For right permutable and d.g. medial near-rings we obtain a “Binomial Theorem,” show thatN (R) is an ideal, and characterize the simple and subdirectly irreducible near-rings. “Natural” examples from analysis and geometry are produced via a general construction method.

Let 〈V; ∪, ∩〉 be a lattice, thenF(V), the set of all functions fromV toV, becomes a lattice by defining the operations ∪ and ∩ pointwise. If we also consider the composition of functions as an operation onF(V), we get the function algebra 〈F(V); ∪, ∩,·〉. In this paper we give a characterization of the lattices with nonsimple function algebras. Moreover, the congruence lattice of these function algebras turns out to be a three-element chain.

Forn a positive integer letp(n) denote the number of partitions ofn into positive integers and letp(n,k) denote the number of partitions ofn into exactlyk parts. Let $$P(n) = \sum\limits_{k = 1}^\infty {kp(n,k)} $$ , thenP(n) represents the total number of parts in all the partitions ofn. In this paper we obtain the following asymptotic formula for $${{P(n)} \mathord{\left/ {\vphantom {{P(n)} {p(n)}}} \right. \kern-\nulldelimiterspace} {p(n)}}:{{P(n)} \mathord{\left/ {\vphantom {{P(n)} {p(n)}}} \right. \kern-\nulldelimiterspace} {p(n)}} = \sqrt {{{3n} \mathord{\left/ {\vphantom {{3n} {2\pi }}} \right. \kern-\nulldelimiterspace} {2\pi }}} (\log n + 2\gamma - \log {\pi \mathord{\left/ {\vphantom {\pi 6}} \right. \kern-\nulldelimiterspace} 6}) + 0(\log ^3 n).$$ .

Letr *(x) denote the maximum number of pairwiserelatively prime integers which can exist in an interval (y,y+x] of lengthx, and let ϱ*(x) denote the maximum number ofprime integers in any interval (y,y+x] wherey≥x. Throughout this paper we assume the “primek-tuples hypothesis.” (This hypothesis could be avoided by using an alternative sievetheoretic definition of ϱ*(x); cf. the beginning of Section 1.) We investigate the differencer *(x)—ϱ*(x): that is we ask how many more relatively prime integers can exist on an interval of lengthx than the maximum possible number of prime integers. As a lower bound we obtainr *(x)—ϱ*(x)0 (whenx→∞). This improves the previous lower bound of logx. As an upper bound we getr *(x)—ϱ*(x)=o[x/(logx)2]. It is known that ϱ*(x)—π(x)>const.[x/(logx)2];.; thus the difference betweenr *(x) and ϱ*(x) is negligible compared to ϱ*(x)—π(x). The results mentioned so far involve the “upper bound” or “maximizing” sieve. In Section 2, similar comparisons are made between two types of “minimum” sieves. One of these is the “erasing” sieve, which completely eliminates an interval of lengthx; and the other, introduced by Erdos and Selfridge [1], involves a kind of “minimax” for sets of pairwise relatively prime numbers. Again these two sieving methods produce functions which are found to be closely related.

A well-known conjecture about the transformationT a :x ↦ax (1−x) on [0, 1], where 2≤a≤4, says that the mapa ↦h top (T a ) is monotone. In this paper we show that this is connected with a property of the polynomialsP k (t) (4≤t≤8) given byP 0 (t)=0 andP k+1 (t)=(t−P k t)2)/2, namely that they have in some sense a minimal number of zeros. Furthermore we show for a countable subset of [2, 4], whose limit points form a sequence converging to 4, to be in {a∈[2,4]:h top (T c ), ifc a}.

If one splits the nonwandering set of a piecewise monotonic map into maximal subsets, which are topologically transitive, one gets two kinds of subsets. The first kind of these subsets has periodic orbits dense, the second kind contains no periodic orbits. In this paper it is shown, that there are only finitely many subsets of the second kind, each of which is minimal and has only finitely many ergodic invariant Borel probability measures.

C. Hightower found two infinite sequences of gaps in the Markov spectrum, (α n , β n ) and (γ n , δ n ) with β n and γ n both Markov elements, converging to\(1 + \sqrt 5 \). This paper exhibits Markov elements α n * and δ n * such that, for alln ⩾ 1, (α n * , β n ) and (γ n δ n * ) are gaps in the Markov spectrum. Other results include showing that, for alln ⩾ 1, β n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.

In this paper we consider special subdivisions ofK4 in which some of the edges are left undivided. A best possible extremal-result for the case where the edges of a Hamiltonian path are left undivided is obtained. Moreover special subdivisions as subgraphs of 4-chromatic graphs are studied. Our main-result on 4-chromatic graphs says that any 4-critical graphG contains an odd cycleC without diagonals such thatG-V (C) is connected.

In this paper we will consider some properties of the maximum genus of those graphs which decompose into upper imbeddable subgraphs, any two of which have at most one vertex in common.

The object of this paper is to construct outer automorphisms for nilpotent torsion-free profinite groups, what reduces to nilpotent compact torsion-free pro-p-groups. If the class of such a group is greater than 2, a characteristic subgroupA can be found, such that the stabilizer of the series 1≤A≤C G (A)≤G in AutG contains a subgroup which modulo InnG is isomorphic to the center ofG. A similar result is obtained for class 2 groups with some exception for which AutG is given explicitely. In course of the proof for the class 2 case, the group Hom (A, B) of continuous homomorphisms is analysed, whereA is a locally direct product of a family {A i ;i∈I} and allA i 's andB are locally compact abelian groups.

The notion of dispersion, a measure of denseness of sequences, plays an important role in quasi-Monte Carlo optimization. In this paper, we obtain an explicit formula for the dispersion of an important low dispersion sequence, namely the Hammersley Sequence in the unit square. The dispersiondMof theM points of this sequence, whereM=2N withN a positive integer is given by $$d_M = \frac{{\sqrt {2M - 2\sqrt M + 1} }}{M},if N is even, d_M = \frac{{\sqrt {\left( {5/2} \right)M - \sqrt {8M} + 1} }}{M},if N is odd.$$ .