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This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

This paper inverstigates the automorphism groups of Cayley graphs of metracyclicp-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclicp-group for odd primep. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2pof a nonabelian metacyclicp-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.

The purpose of this paper is to describe the set of all Hilbert polynomials of stan-dard graded commutative algebras over a field and to prove some conjectures on Hilbertpolynomials.Throughout the paper Z, P, and N denote the sets of integers, positive integersand non-negative integers, respectively.A polynomial f(t) in one variable t with rational coefficients is called numerical iff(r) € Z for all sufficiently large r € N, that is, there exists s € N such that /(r) 6 Zfor all r € N,r ^ 5.It is clear that every polynomial with integer coefficients is numerical. As anexample of a numerical polynomial with non-integer coefficients one can consider apolynomial (£) = t(t - 1)... (t - k + \)/k\ where A; € P. In what follows, we use somerelationships between "binomial" numerical polynomials (£) that arise from well-knownidentities for binomial coefficients. In particular, the classica ("^j

In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.

Let C be a germ of curve singularity embedded in (kn, 0). It is well known that the blowing-up of C centred on its closed ring, Bl(C), is a finite union of curve singularities. If C is reduced we can iterate this process and, after a finite number of steps, we find only non-singular curves. This is the desingularization process. The main idea of this paper is to linearize the blowing-up of curve singularities Bl(C) → C. We perform this by studying the structure of [Oscr ]Bl(C)/[Oscr ]C as W-module, where W is a discrete valuation ring contained in [Oscr ]C. Since [Oscr ]Bl(C)/[Oscr ]C is a torsion W-module, its structure is determined by the invariant factors of [Oscr ]C in [Oscr ]Bl(C). The set of invariant factors is called in this paper as the set of micro-invariants of C (see Definition 1·2).In the first section we relate the micro-invariants of C to the Hilbert function of C (Proposition 1·3), and we show how to compute them from the Hilbert function of some quotient of [Oscr ]C (see Proposition 1·4).The main result of this paper is Theorem 3·3 where we give upper bounds of the micro-invariants in terms of the regularity, multiplicity and embedding dimension. As a corollary we improve and we recover some results of [6]. These bounds can be established as a consequence of the study of the Hilbert function of a filtration of ideals g = {g[r,i+1]}i [ges ] 0 of the tangent cone of [Oscr ]C (see Section 2). The main property of g is that the ideals g[r,i+1] have initial degree bigger than the Castelnuovo–Mumford regularity of the tangent cone of [Oscr ]C.Section 4 is devoted to computation the micro-invariants of branches; we show how to compute them from the semigroup of values of C and Bl(C) (Proposition 4·3). The case of monomial curve singularities is especially studied; we end Section 4 with some explicit computations.In the last section we study some geometric properties of C that can be deduced from special values of the micro-invariants, and we specially study the relationship of the micro-invariants with the Hilbert function of [Oscr ]Bl(C). We end the paper studying the natural equisingularity criteria that can be defined from the micro-invariants and its relationship with some of the known equisingularity criteria.

In this paper we construct a compact set K of zero Hausdorff dimension that satisfies certain ‘arithmetic-type’ thickness properties. The concept of ‘arithmetic thickness’ has its origins in applications to harmonic analysis, introduced in a paper by Lebedev and Olevskiĭ. For example, there are no spectral sets whose ‘essential boundary’ can contain the above set K.

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

Matrices provide essential tools in many branches of mathematics, and matrix semi-groups have applications in various areas. In this paper we give a complete descriptionof all infinite matrix semigroups satisfying a certain combinatorial property definedin terms of power graphs.Research on combinatorial properties of words in groups originates from the fol-lowing well-known theorem due to Bernhard Neumann [12], which was obtained as ananswer to a question of Paul Erdos: a group is centre-by-finite if and only if every infinitesequence contains a pair of elements that commute. Combinatorial properties of groupsand semigroups with all infinite subsets containing certain special elements have beenconsidered by Bell, Blyth, Curzio, de Luca, Gillam, Hall, Higgins, Justin, Longobardi,Maj, Okninski, Piochi, Pirillo, Restivo, Reutenauer, Rhemtulla, Robinson, Sapir, Shumy-atsky, Simon, Varricchio and other authors, and a survey of this direction was given bythe first author in [7] (see also [2, 3, 6, 11]).The following combinatorial property was introduced in [9] using power graphs. Thepower graph Pow(S) of a semigroup S has all elements of S as vertices, and it has edges(u, v) for all u,v € S such that u ^ v and v is a power of u. Let D be a directed graph.We say that an infinite semigroup 5 is power D-saturated if and only if, for every infinitesubset T of 5, the power graph of S has a subgraph isomorphic to D with all verticesin T. In this paper we describe all pairs (£>, S), where D is a directed graph and S is amatrix semigroup, such that S is power D-saturated.The reader is referred to [1, 4, 14] for standard graph, semigroup and group theoreticterminology, respectively. By the word 'graph' we mean a directed graph without loopsor multiple edges. A graph is said to be acyclic if it has no directed cycles.We refer to [10, 13] for preliminaries on fields and matrix semigroups, respectively.For a skew field K, the set of all n x n matrices with entries in K i

Let p be an odd prime. The results in this paper concern the units of the infinite extension of Q p generated by all p -power roots of unity. Let formula here where μ p n +1 denote the p n +1 th roots of 1. Let [pscr ] n be the maximal ideal of the ring of integers of Φ n and let U n be the units congruent to 1 modulo [pscr ] n . Let ζ n be a fixed primitive p n +1 th root of unity such that ζ p n = ζ n − 1 , ∀ n [ges ] 1. Put π n = ζ n − 1. Thus π n is a local parameter for Φ n . Let formula here Kummer already exploited the obvious fact that every u 0 ∈ U 0 can be written in the form formula here where f 0 ( T ) is some power series in Z p [[ T ]]. Here of course, the power series f 0 ( T ) is not uniquely determined. Let formula here the inverse limit being with respect to norm maps. Coates and Wiles (see [ 3 ]) discovered that any unit u = ( u n ) ∈ U ∞ has a unique power series f u ( T ) in Z p [[ T ]] with f u (π n ) = u n . The uniqueness of such a power series is obvious by Weierstrass Preparation Theorem, but the existence is in no way obvious. They worked with the formal group of height one attached to an elliptic curve with complex multiplication at an ordinary prime, but their ideas apply to any Lubin–Tate group defined over Z p . Almost immediately, Coleman [ 4 ] gave a totally different proof of the existence of the f u ( T ), which holds for all Lubin–Tate groups. We refer to such a power series as a Coleman power series. In this paper we adopt the same approach as [ 3 ]. We first prove the following result which is stronger than the original one in [ 3 ].

The aim of this paper is to prove the existence and uniqueness of mild and classical solutions of the non-local Cauchy problem for a semilinear integrodifferential equation with deviating argument. The results are established by using the method of semigroups and the contraction mapping principle. The paper generalizes certain results of Lin and Liu.AMS 2000 Mathematics subject classification: Primary 34K05; 34K30

The purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

There is an error in one of the major results in our original paper ‘Equivalent weights and standard homomorphisms for convolution algebras on ℝ+’. We describe the error and give a counterexample to the result as stated. We then give a substitute result which is in many ways stronger than the erroneous result. We will also indicate what changes need to be made in the original paper to accommodate the replacement of the erroneous result by the substitute.

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure. The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.

We construct Bratteli–Vershik models for Toeplitz flows and characterize a class of properly ordered Bratteli diagrams corresponding to these flows. We use this result to extend by a novel approach—using basic theory of dimension groups—an interesting and non-trivial result about Toeplitz flows, first shown by Downarowicz. (Williams had previously obtained preliminary results in this direction.) The result states that to any Choquet simplex $K$, there exists a $0$–$1$ Toeplitz flow $(Y,\psi)$, so that the set of invariant probability measures of $(Y,\psi)$ is affinely homeomorphic to $K$. Not only do we give a conceptually new proof of this result, we also show that we may choose $(Y,\psi)$ to have zero entropy and to have full rational spectrum.Furthermore, our Bratteli–Vershik model for a given Toeplitz flow explicitly exhibits the factor map onto the maximal equicontinuous (odometer) factor. We utilize this to give a simple proof of the existence of a uniquely ergodic 0–1 Toeplitz flow of zero entropy having a given odometer as its maximal equicontinuous factor and being strongly orbit equivalent to this factor. By the same token, we show the existence of 0–1 Toeplitz flows having the 2-odometer as their maximal equicontinuous factor, being strong orbit equivalent to the same, and assuming any entropy value in $[0,\ln 2)$.Finally, we show by an explicit example, using Bratteli diagrams, that Toeplitz flows are not preserved under Kakutani equivalence (in fact, under inducing)—contrasting what is the case for substitution minimal systems. In fact, the example we exhibit is an induced system of a 0–1 Toeplitz flow which is conjugate to the Chacon substitution system, thus it is prime, i.e. it has no non-trivial factors.The thrust of our paper is to demonstrate the relevance and usefulness of Bratteli–Vershik models and dimension group theory for the study of minimal symbolic systems. This is also exemplified in recent papers by Forrest and by Durand, Host and Skau, treating substitution minimal systems, and by papers by Boyle, Handelman and by Ormes.

In the present paper we give an improvement of a previous result of the paper [M. M. Skriganov. Ergodic theory on $SL(n)$, diophantine approximations and anomalies in the lattice point problem. Inv. Math.132(1), (1998), 1–72, Theorem 2.2] on logarithmically small errors in the lattice point problem for polyhedra. This improvement is based on an analysis of hidden symmetries of the problem generated by the Weyl group for $SL(n,\mathbb{B})$. Let $UP$ denote a rotation of a given compact polyhedron $P\subset\mathbb{B}^n$ by an orthogonal matrix $U\in SO(n)$, $tUP$ a dilation of $UP$ by a parameter $t>0$ and $N(tUP)$ the number of integer points $\gamma\in\mathbb{Z}^n$ which fall into the polyhedron $tUP$. We show that for almost all rotations $U$ (in the sense of the Haar measure on $SO(n)$) the following asymptotic formula \[ N(t\UP)=t^n{\rm vol} P+ O((\log t)^{n-1+\varepsilon}),\quad t\to\infty, \] holds with arbitrarily small $\varepsilon>0$.

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

This paper introduces a new concept of duality of generalised random fields using the geometric properties of Sobolev spaces of integer order. Under this duality condition, the covariance operators of a generalised random field and its dual can be factorised. The paper also defines a concept of generalised white noise relative to the geometries of the Sobolev spaces, and via the covariance factorisation, obtains a representation of the generalised random field as a stochastic equation driven by a generalised white noise. This representation is unique except for isometric isomorphisms on the parameter space.

This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

In a recent paper, Brown, Evans and Marletta extended the HardyEverittLittlewoodPolya inequality from 2nth-order formally self-adjoint ordinary differential equations to a wide class of linear Hamiltonian systems in 2n variables. The paper considered only problems on semi-infinite intervals [a, ∞) with a limit-point type singularity at infinity. In this paper we extend the theory to cover all types of endpoint ( lim-p for n ≤ p ≤ 2n ).

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each round (if any) retire from the rest of the game, and (3) the game ends when only one player (winner) is left. Let W n be the number of rounds played through the game. Among other things, it is proved that (2/3) n W n is asymptotically (as n → ∞) distributed according to the exponential distribution with mean ⅓, provided that each player chooses one of the three strategies (scissors, paper, rock) with equal probability and independently from other players in any round.

In the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements inU(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing. The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.

Throughout this paper, let M be a finitely generated module over a Noether local ring ( A , [mfr ]) with dim M = d . Let x = ( x 1 , …, x d ) be a system of parameters of M and n = ( n 1 , …, n d ) ∈ ℕ d a d-tuple of positive integers. This paper is concerned with the following two points of view. First, it is well-known that, the difference between lengths and multiplicities formula here considered as a function in n , gives a lot of information on the structure of M . This function in general is not a polynomial in n for all n 1 , …, n d large enough ( n [Gt ] 0 for short). But, it was shown in [ C3 ] that the least degree of all polynomials in n bounding above I M ( n , x ) is independent of the choice of x . This numerical invariant is denoted by p ( M ) and called the polynomial type of the module M . By [ C2 ] and [ C3 ] this polynomial type does not change by the [mfr ]-adic completion M ˆ of M and p ( M ) is just equal to the dimension of the non-Cohen–Macaulay locus of M ˆ (see [ C2 , C3 , C4 , CM ] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p ( M ) = − ∞ or p ( M ) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p ( M ) > 0. Second, following Sharp and Hamieh ([ SH ]), we consider the difference formula here as a function in n , where formula here is the cyclic submodule of the module of generalized fractions U ( M ) − d −1 d +1 M defined in [ SZ1 ].

In Part 1 of this paper we have investigated numerically the quasiperiodic and frequency locked solutions of mathematical model of a robot with one degree of freedom. In this paper we extend our investigations to the region of transient chaos. The zones of chaotic transients are very broad and lie beyond the parameter range of engineering significance. Transiently chaotic zones exhibit a complex structure, fractally intertwined with tongues of regular pattern and cover a broad range of control parameter L. The crisis point for the onset of sustained chaos lies extremely far from the point of onset of transient chaos.

In this paper we study the problem of the existence on non-innerautomorphisms for the class of torsion-free supersolvable groups, answering aquestion raised by Robinson.1991 Mathematics Subject Classification. 20F16, 20F28Introduction. The famous, seminal papers by Gaschu¨tz [2],[3] on the existenceof outer automorphisms in finite p–groups have been both the conclusion of a longprocess and the starting point of a number of extensions and generalisations in dif-ferent areas. The existence of non-inner automorphisms for infinite nilpotent p-groups was proved by Zalesskii [9]; moreover, apart from the obvious exceptions,infinite nilpotent p-groups admit outer automorphisms of p-power order [6]. Forgeneral nilpotent groups the situation is quite di•erent. In fact Zalesskii gives anexample of a torsion-free nilpotent group, all of whose automorphisms are inner (see[10]) while, on the other hand, the existence of non-inner automorphisms had beenproved by R. Ree [7] (but see also [1] and [4]) in the case of finitely generated, torsionfree nilpotent groups. It is worth mentioning that it is still unknown whether aninfinite finitely generated nilpotent group has an outer automorphism.The problem of the existence of outer automorphisms has been addressed forother classes of groups and in this short note we will consider this question for tor-sion-free supersoluble groups. Our interest is motivated by a result proved by D.J.S.Robinson in [8]. In this paper the author shows that a torsion-free supersolublegroup with trivial center always admits a non-inner automorphism. On this basis thefollowing question was also asked by Robinson:does every torsion-free supersoluble group have non-inner automorphisms?In this note we show that the answer to the above question is negative, by con-structing a class of torsion-free supersoluble groups, possessing only inner auto-morphisms.An invariant which seems to play a decisive role in this context is the torsion-freerank of the abelian factor group G=G

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.

In this paper we deduce the existence of analytic structure in a neighbourhood of a maximal ideal M in the spectrum of a commutative Banach algebra, A, from homological assumptions. We assume properties of certain of the cohomology groups H^n(A,A/M), rather than the stronger conditions on the homological dimension of the maximal ideal the first author has considered in previous papers. The conclusion is correspondingly weaker: in the previous work one deduces the existence of a Gel'fand neighbourhood with analytic structure, here we deduce only the existence of a metric neighbourhood with analytic structure. The main method is to consider products of certain co-cycles to deduce facts about the symmetric second cohomology, which is known to be related to the deformation theory of algebras.1991 Mathematics Subject Classification. 46J20, 46M20.

We study one-dimensional dynamical systems with indifferent fixed points $p$ and $q$. The dynamical systems we study in this paper have ergodic, infinite, invariant measures. We consider the limit of the ratio of the sojourn time of the trajectory in a small neighborhood of $p$ to that in a small neighborhood of $q$. We show that the limit does not exist under some conditions, which has been announced in a previous paper. In fact we prove that the limit supreme of the ratio is $\infty$ and that the limit infimum is 0.

In this paper we conduct a systematic computerized search for groups generated by small, but highly symmetric, sets of elements of order 3. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. Firstly, we introduce monomial modular representations as these will prove useful later in the paper. Then the techniques of symmetric generation developed elsewhere are described afresh. The results we obtain are presented in a convenient tabular form, together with relevant character tables.

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

This is the first of three papers in which we generalise the classical McAlister structure theory for E-unitary inverse semigroups to those 0-E-unitary inverse semigroups which admit a 0-restricted, idempotent pure prehomomorphism to a primitive inverse semigroup. In this paper, we concentrate on finding necessary and sufficient conditions for the existence of such prehomomorphisms in the case of 0-E-unitary inverse monoids. A class of inverse monoids which satisfy our conditions automatically are those which are unambiguous except at zero, such as the polycyclic monoids.

Throughout this paper all rings are commutative Noetherian of prime characteristic.Perhaps the most central notion in tight closure theory is that of test elements. Theexistence of uniform annihilation plays a key role in several important commutativealgebra theorems and so understanding test elements, which uniformly annihilate tightclosure, is of fundamental importance. Questions about test elements can be dividedinto two types, localization and existence, but ultimately these are facets of each other.At present, the best theorems that are available exist under a Gorenstein hypothesis.The goal of this paper is to demonstrate that the Gorenstein condition in thesetheorems can be weakened; in particular we show that it is sufficient that somesymbolic power of the canonical module be one-generated.What makes the theory difficult is the localization problem for tight closure: it isunknown whether I'R

In the mid 1970s Mark Mahowald constructed a new infinite family of elements in the 2-component of the stable homotopy groups of spheres, ηj∈πSj2 (S0)(2) [M]. Using standard Adams spectral sequence terminology (which will be recalled in Section 3 below), ηj is detected by h1hj∈Ext2,*[Ascr ] (Z/2, Z/2). Thus he had found an infinite family of elements all having the same Adams filtration (in this case, 2), thus dooming the so-called Doomsday Conjecture. His constructions were ingenious: his elements were constructed as composites of pairs of maps, with the intermediate spaces having, on one hand, a geometric origin coming from double loopspace theory and, on the other hand, mod2 cohomology making them amenable to Adams Spectral Sequence analysis and suggesting that they were related to the new discovered Brown–Gitler spectra [BG].In the years that followed, various other related 2-primary infinite families were constructed, perhaps most notably (and correctly) Bruner's family detected by h2h2j∈ Ext3,*[Ascr ](Z/2, Z/2) [B]. An odd prime version was studied by Cohen [C], leading to a family in πS∗(S0)(p) detected by h0bj∈ Ext3,*[Ascr ] (Z/p, Z/p) and a filtration 2 family in the stable homotopy groups of the odd prime Moore space. Cohen also initiated the development of odd primary Brown–Gitler spectra, completed in the mid 1980s, using a different approach, by Goerss [G], and given the ultimate ‘modern’ treatment by Goerss, Lannes and Morel in the 1993 paper [GLM]. Various papers in the late 1970s and early 1980s, e.g. [BP, C, BC], related some of these to loopspace constructions.Our project originated with two goals. One was to see if any of the later work on Brown–Gitler spectra led to clarification of the original constructions. The other was to see if taking advantage of post Segal Conjecture knowledge of the stable cohomotopy of the classifying space BZ/p would help in constructing new families at odd primes, in particular a conjectural family detected by h0hj∈ Ext2,*[Ascr ] (Z/p, Z/p). (This followed a paper [K1] by one of us on 2 primary families from this point of view.)

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

The paper is devoted to the following problem: \[ w'' (x) + c w'(x)+ F(w(x),x) = 0, \quad x\in{\mathbb R}^1,\quad w(\pm \infty) = w_{\pm}, \] where the non-linear term $F$ depends on the space variable $x$. A classification of non-linearities is given according to the behaviour of the function $F(w,x)$ in a neighbourhood of the points $w_+$ and $w_-$. The classical approach used in the Kolmogorov–Petrovsky–Piskunov paper [10] for an autonomous equation (where $F=F(u)$ does not explicitly depend on $x$), which is based on the geometric analysis on the $(w,w')$-plane, is extended and new methods are developed to analyse the existence and uniqueness of solutions in the non-autonomous case. In particular, we study the case where the function $F(w,x)$ does not have limits as $x \rightarrow \pm \infty$.

1. Introduction. It was proved by Harvey [8] that the order #…'† of an automorphism ' of a compact Riemann surface of genus g 2 is not bigger than 4g‡ 2. This bound is sharp for all values of g, and it follows from the proof that if ' attains this bound, it ®xes exactly one point. After that, many authors contributed to the study of the relationship between the order and the number of ®xed points of an automorphism, and we should mention here the papers of Macbeath [12], Moore [15] and Szemberg [18]. The latter, who also studied these questions for automorphisms of domains in the complex plane, proved that if ' has at least two ®xed points, then Harvey's bound can be strengthened to 4g, and again this bound is attained for all g 2. The starting point of this paper is the following result from [7, p. 245], which admits an elementary proof.

The purpose of this paper is to survey the key ideas which have played a role in the development of the theory of smooth unimodal maps in the 1990s so as to provide a starting point for more detailed study. Most papers underlying this survey are distinguished by great technical complexity; we have therefore attempted to extract their essence and present it in a way accessible to a non-specialist. For further study we compiled a long list of original references. Another goal of this paper is to present the multi-directional process in which ideas flowed and developed, placed in a historical order.

The purpose of this paper is to determine the optimal order of decrease of the L 2 norm for the following degenerate and singular oscillatory integral operators defined by formula here for f ∈ C ∞ 0 (ℝ), where ψ( x , y ) is a smooth compactly supported function on ℝ 2 , S ( x , y ) is a homogeneous polynomial of degree n formula here and K ( x , y ) is a distribution kernel which satisfies the following hypotheses: K is a C 2 function away from the diagonal and the estimates formula here hold for 0 i =1, or 2. This paper basically uses the methods of [ 6 ], which are very general and can be widely used. The main idea of these methods is that the sizes of operator norms after the decomposition are estimated in different ways and the decomposition is then summed back by balancing the estimates of two types. The first type estimate takes into account the support sizes of the integrand after the decomposition. The second type estimate is delicate, depending on how to keep track of the distance to the singular varieties of phase function. In our case, we have to include the singular variety arising from distribution kernel as well. The analytic tool of the second type estimate is the operator version of the Van der Corput lemma [ 7 ]. We notice that the crux of matters is to locate the singular varieties of phase function and kernel. For higher dimensional situations, the problem is very difficult. One expects that the powerful theory of algebraic geometry may play a significant role. Here we would like to remark that Mather's results [ 2 ] may be useful although the geometrical shapes of singular varieties are basically undecidable.

This paper is concerned with integral domains R, for which the factor group SL2(R)/U2(R) has a non-trivial, free quotient, where U2(R) is the subgroup of GL2(R) generated by the unipotent matrices. Recently, Krstić and McCool have proved that SL2(P[x])/U2(P[x]) has a free quotient of infinite rank, where P is a domain which is not a field. This extends earlier results of Grunewald, Mennicke and Vaserstein.Any ring of the type P[x] has Krull dimension at least 2. The purpose of this paper is to show that result of Krstić and McCool extends to some domains of Krull dimension 1, in particular to certain Dedekind domains. This result, which represents a two-dimensional anomaly is the best possible in the following sense. It is well known that SL2(R) = U2(R), when R is a domain of Krull dimension zero, i.e. when R is a field. It is already known that for some arithmetic Dedekind domains A, the factor group SL2(A)/U2(A) has a free quotient of finite (and not infinite) rank.

In this paper we will study the Brill–Noether theory of vector bundles on a smooth projective curve X. As usual in papers on this topic we are mainly interested in stable or at least semistable bundles. Let Wkr, d(X) be the scheme of all stable vector bundles E on X with rank (E)=r, deg (E)=d and h0(X, E)[ges ]k+1. For a survey of the main known results, see the introduction of [6]. The referee has pointed out that the results in [6] were improved by V. Mercat in [14]; he proved that Wkr, d(X) is non-empty for d

In this paper, the structure of $r$ equivalence, which was introduced by Vershik and which classifies group actions of the group $G=\sum_{n=1}^\infty{\Bbb Z}\slash r_n{\Bbb Z}$, $r_n\in{\Bbb N}\setminus\{1\}$, is examined. This is an equivalence relation that naturally arises from looking at certain sequences of $\sigma$-algebras. Vershik proved that if a sequence $r=(r_1,r_2,\ldots)$ does not satisfy a super-rapid growth rate, then entropy is an invariant for $r$ equivalence. In this paper, a strong converse of this is proven: for any $r$ which does satisfy this super-rapid growth rate, we can find a zero entropy action in every $r$ equivalence class.

Two given orbits of a minimal circle homeomorphism $f$ are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with $f$. By a well-known theorem due to Herman and Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C)., Comment: 38 pages, 5 figures. To appear in Ergodic Theory and Dynamical Systems

Let $\text{G}(n)$ be equal either to $\text{PO}(n,1),\text{PU}(n,1)$ or $\text{PSp}(n,1)$ and let $\Gamma \leq \text{G}(n)$ be a uniform lattice. Denote by $\mathbb{H}^n_K$ the hyperbolic space associated to $\text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=\dim_{\mathbb{R}} K$. Assume $d(n-1) \geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $\Gamma$-space $(X,\mu_X)$, we assume that a measurable cocycle $\sigma:\Gamma \times X \rightarrow \text{G}(m)$ admits an essentially unique boundary map $\phi:\partial_\infty \mathbb{H}^n_K \times X \rightarrow \partial_\infty \mathbb{H}^m_K$ whose slices $\phi_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are atomless for almost every $x \in X$. Then, there exists a $\sigma$-equivariant measurable map $F: \mathbb{H}^n_K \times X \rightarrow \mathbb{H}^m_K$ whose slices $F_x:\mathbb{H}^n_K \rightarrow \mathbb{H}^m_K$ are differentiable for almost every $x \in X$ and such that $\text{Jac}_a F_x \leq 1$ for every $a \in \mathbb{H}^n_K$ and almost every $x \in X$. The previous properties allow us to define the natural volume $\text{NV}(\sigma)$ of the cocycle $\sigma$. This number satisfies the inequality $\text{NV}(\sigma) \leq \text{Vol}(\Gamma \backslash \mathbb{H}^n_K)$. Additionally, the equality holds if and only if $\sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \text{G}(n) \leq \text{G}(m)$, modulo possibly a compact subgroup of $\text{G}(m)$ when $m>n$. Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context., Comment: 27 pages, to appear on J. Inst. Math. Jussieu

It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

A mathematical model is proposed for a revolute joint mechanism with an n-degree of freedom (DOF). The matrix approach is used for finding the relation between two consecutive links to determine desired link parameters such as position, velocity and acceleration using the forward kinematic approach. The matrix approach was confirmed for a proposed 10 DOF revolute type (R-type) human upper limb model with servo motors at each joint. Two DOFs are considered each at shoulder, elbow and wrist joint, followed by four DOF for the fingers. Two DOFs were considered for metacarpophalangeal (mcp) and one DOF each for proximal interphalangeal (pip) and distal interphalangeal (dip) joints. MATLAB script function was used to evaluate the mathematical model for determining kinematic parameters for all the proposed human upper limb model joints. The simplified method for kinematic analysis proposed in this paper will further simplify the dynamic modeling of any mechanism for determining joint torques and hence, easy to design control system for joint movements.

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].