The arithmetic of distributions in free probability theory

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I ₀, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I ₀ consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.     

Arithmetical semigroups V: Multiplicative functions

Several classical results on multiplicative functions ℕ → ℂ are transposed to multiplicative functionsG → ℂ where (G, σ) denotes an additive arithmetical semigroup as introduced by John Knopfmacher.     

On Schwarz's domain decomposition methods for elliptic boundary value problems

Summary. We study the additive and multiplicative Schwarz domain decomposition methods for elliptic boundary value problem of order 2 r based on an appropriate spline space of smoothness . The finite element method reduces an elliptic boundary value problem to a linear system of equations. It is well known that as the number of triangles in the underlying triangulation is increased, which is indispensable for increasing the accuracy of the approximate solution, the size and condition number of the linear system increases. The Schwarz domain decomposition methods will enable us to break the linear system into several linear subsystems of smaller size. We shall show in this paper that the approximate solutions from the multiplicative Schwarz domain decomposition method converge to the exact solution of the linear system geometrically. We also show that the additive Schwarz domain decomposition method yields a preconditioner for the preconditioned conjugate gradient method. We tested these methods for the biharmonic equation with Dirichlet boundary condition over an arbitrary polygonal domain using cubic spline functions over a quadrangulation of the given domain. The computer experiments agree with our theoretical results. Received December 28, 1995 / Revised version received November 17, 1998 / Published online September 24, 1999     

The Ubiquity of Free Subsemigroups of Infinite Triangular Matrices

Abstract. In this note we show, that in a suitable sense, almost all k -generator subsemigroups of the multiplicative semigroup of infinite upper triangular matrices over a finite field are free of rank k .     

Averaging of random walks and shift-invariant measures on a Hilbert space

We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.     

Ring structures on groups of arithmetic functions

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q-algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin(R)-algebra, where Bin(R) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powersfα for α∈Bin(R) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z.     

Quotient space of LMC-compactification as a space of z-filters

The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup. In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters.     

Matrix semigroup homomorphisms into higher dimensions

We study non-degenerate irreducible homomorphisms from the multiplicative semigroup of all n-by-n matrices over an algebraically closed field of characteristic zero to the semigroup of m-by-m matrices over the same field. We prove that every non-degenerate homomorphism from the multiplicative semigroup of all n-by-n matrices to the semigroup of (n + 1)-by-(n + 1) matrices when n ? 3 is reducible and that every non-degenerate homomorphism from the multiplicative semigroup of all 3-by-3 matrices to the semigroup of 5-by-5 matrices is reducible.     

On mean values of multiplicative functions on the symmetric group

Mean values of nonnegative multiplicative functions defined on the symmetric group are explored in the paper. The result gives a sharp quantitative upper bound for their Cesàro mean. An approach that originated in number theory is adopted. It can be further applied for mappings defined on general decomposable structures, in particular, for estimating mean values with respect to multiplicative measures defined on additive partitions of a natural number.     

Arithmetic Processes in Semigroups

We consider the asymptotic behavior of the distributions of stochastic processes defined by multiplicative functions in an arithmetic semigroup.     

Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces

The purpose of this paper is to study iterative schemes of Browder and Halpern types for a semigroup of nonexpansive mappings on a compact convex subset of a smooth (and strictly convex) Banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented.     

Semigroups whose endomorphisms are power functions

For any commutative semigroup S and any positive integer m, the power function f:S→S defined by f(x)=x ^m is an endomorphism of S. In this paper we characterize finite cyclic semigroups as those finite commutative semigroups whose endomorphisms are power functions. We also prove that if S is a finite commutative semigroup with 1≠0, then every endomorphism of S preserving 1 and 0 is equal to a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. Immediate consequences of the results are, on the one hand, a characterization of commutative rings whose multiplicative endomorphisms are power functions given by Greg Oman in the paper (Semigroup Forum, 86 (2013), 272–278), and on the other hand, a partial solution of Problem 1 posed by Oman in the same paper.     

An axiomatic basis for multiattribute value functions defined on binary attributes

The insufficiency of solvability axioms for defining an additive conjoint structure in the case of binary attributes motivates an alternative approach to development of an axiomatic basis for multilinear, multiplicative and additive value functions defined on binary attributes. The alternative approach is to define four sufficiency conditions with respect to binary attributes with alternative combinations of these conditions allowing derivation of the multilinear, multiplicative and additive models.     

Ring semigroups whose subsemigroups intersect

A multiplicative semigroup S is said to be a ring semigroup provided there exists an addition + on S such that (S,+,⋅) is a ring. In this note, we characterize the ring semigroups S with the property that every two nonzero subsemigroups intersect.     

Near integral domains II

A direct product decomposition is given for the multiplicative semigroup of a finite near integral domain in terms of the subsemigroup of left identities and a group of automorphisms on the additive group of the domain. Conditions are given which insure that every element will have a uniquen-th root. If there existsx≠0 such that (?x)y=?(xy), for eachy, then the additive group of the near integral domain is abelian. Other conditions sufficient for the commutativity of the additive group are given. An example illustrates that non-isomorphic finite near integral domains can have a left ideal decomposition into Sylow subgroups which are isomorphic as near-rings. Another example shows that an infinite near integral domain need not have a nilpotent additive group, even in the d. g. case. It is conjectured that for each natural numbern there is a near integral domain whose additive group is of nilpotent classn.     

Extension of additive functionals and semicharacters on commutative semigroups

Let S be a commutative semigroup and S_o a subsemigroup. The present paper establishes a necessary and sufficient condition in order that any real additive functional ϕ₀ defined on S_o and dominated there by a real subadditive functional p defined on S, admit an additive extension to S such that ϕ≤p. This result is a strengthening of a result of R.Kaufman [3]. From this easily follow recent results of Kobayashi [5] and Putcha, Tamura [9] on extension of semigroup homomorphisms. Also a result of Ross on extension of semicharacters can be deduced from this result. As an application, the existence of continuous semicharacters on an open subsemigroup of an abelian topological group is derived. This too, generalizes previous results (Gleason in [10]). Supported in part by the Danish Natural Science Research Council     

Multiplicative Representation of a Hyperbolic non Distributive Algebra

The multiplicative or polar representation of hyperbolic scator algebra in 1 + n dimensions is introduced. The transformations between additive and multiplicative representations are presented. The addition and product operations are consistently defined in either representation using additive or multiplicative variables. The product is shown to produce a rotation and scaling for equal director components and solely a scaling in the orthogonal components.     

On semigroups admitting ring structure

A right-chain semigroup is a semigroup whose right ideals are totally ordered by set inclusion. The main result of this paper says that if S is a right-chain semigroup admitting a ring structure, then either S is a null semigroup with two elements or sS=S for some s∈S. Using this we give an elementary proof of Oman’s characterization of semigroups admitting a ring structure whose subsemigroups (containing zero) form a chain. We also apply this result, along with two other results proved in this paper, to show that no nontrivial multiplicative bounded interval semigroup on the real line ℝ admits a ring structure, obtaining the main results of Kemprasit et al. (ScienceAsia 36: 85–88, 2010).     

Optimal strategies for a game on amenable semigroups

The semigroup game is a two-person zero-sum game defined on a semigroup ${(S,\cdot)}$ as follows: Players 1 and 2 choose elements ${x \in S}$ and ${y \in S}$ , respectively, and player 1 receives a payoff f (x y) defined by a function f : S → [?1, 1]. If the semigroup is amenable in the sense of Day and von Neumann, one can extend the set of classical strategies, namely countably additive probability measures on S, to include some finitely additive measures in a natural way. This extended game has a value and the players have optimal strategies. This theorem extends previous results for the multiplication game on a compact group or on the positive integers with a specific payoff. We also prove that the procedure of extending the set of allowed strategies preserves classical solutions: if a semigroup game has a classical solution, this solution solves also the extended game.