We propose generating functions, $\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.

Simulated tempering is a popular method of allowing Markov chain Monte Carlo algorithms to move between modes of a multimodal target density $\pi$ . Tawn, Moores and Roberts (2021) introduces the Annealed Leap-Point Sampler (ALPS) to allow for rapid movement between modes. In this paper we prove that, under appropriate assumptions, a suitably scaled version of the ALPS algorithm converges weakly to skew Brownian motion. Our results show that, under appropriate assumptions, the ALPS algorithm mixes in time $O(d [\log d]^2)$ or O(d), depending on which version is used.

We define positively expansive semigroups of linear operators on Banach spaces. We characterize these semigroups in terms of the point spectrum of the infinitesimal generator. In particular, we prove that a positively expansive semigroup is neither uniformly bounded nor equicontinuous. We apply our results to the Lasota equation.

We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

This paper presents a circuit architecture for a new integrated on chip test method for microwave circuits. The proposed built-in-self-test (BIST) cell targets a direct low-cost measurement technique of the gain and the 1 dB input compression point (CP1) of a K-band satellite receiver in the 18–22 GHz frequency bandwidth. A signal generator at the radiofrequency (RF) front end input of the device under test (DUT) has been integrated on the same chip. To inject this RF signal, a loopback technique has been used and the design has been accommodated for it. This paper focuses on the design of the most sensitive block of the BIST circuit, i.e. the RF signal generator. This circuit, fabricated in a SIGe:C BiCMOS process, consumes 10 mA. It presents a dynamic power range of 17 dB (−41; −24 dBm) and operates in a frequency range of 5.6 GHz (17.5; 23 GHz). This BIST circuit gives new perspectives in terms of test strategy, cost reduction, and measurement accuracy for microwave-integrated circuits and could be adapted for mm-wave circuits.

A simple model for a randomly oscillating variable is suggested, which is a variant of the two-state random velocity model. As in the latter model, the variable keeps rising or falling with constant velocity for some time before randomly reversing its direction. In contrast however, its propensity to reverse depends on its current value and it is for this desirable feature that the model is proposed here. This feature has two implications: (a) neither the changing variable nor its velocity is Markovian, although the joint process is, and (b) the linear differential equations arising in the case of our model do not have constant coefficients. The results given in this paper are meant to illustrate the straightforward nature of some of the calculations involved and to highlight the relationship with one-dimensional diffusions.

Most rhythmic behaviors are produced by a specialized ensemble of neurons found in the central nervous system. These central pattern generators (CPGs) have become a cornerstone of neuronal circuit analysis. Studying simple invertebrate nervous systems may reveal the interactions of the neurons involved in the production of rhythmic motor output. There has recently been progress in this area, but due to certain intrinsic features of CPGs it is unlikely that present techniques will ever yield a complete understanding of any but the simplest of them. The chief impediment seems to be our inability to identify and characterize the total interneuronal pool making up a CPG. In addition, our general analytic strategy relies on a descriptive, reductionist approach, with no analytical constructs beyond phenomenological modeling. Detailed descriptive data are usually not of sufficient depth for specific model testing, giving rise instead to ad hoc explanations of mechanisms which usually turn out to be incorrect. Because they make too many assumptions, modeling studies have not added much to our understanding of CPCs; this is due not so much to inadequate simulations as to the poor quality and incomplete nature of the data provided by experimentalists.

A basic strategy that would provide sufficient information for neural modeling would include: (1) identifying and characterizing each element in the CPG network; (2) specifying the synaptic connectivity between the elements; and (3) analyzing nonlinear synaptic properties and interactions by means of the connectivity matrix. Limitations based on our present technical capabilities are also discussed.

Haploid population models with non-overlapping generations and fixed population size N are considered. It is assumed that the family sizes ν1,…,νN within a generation are exchangeable random variables. Rates of convergence for the finite-dimensional distributions of a properly time-scaled ancestral coalescent process are established and expressed in terms of the transition probabilities of the ancestral process, i.e., in terms of the joint factorial moments of the offspring variables ν1,…,νN.

The Kingman coalescent appears in the limit as the population size N tends to infinity if and only if triple mergers are asymptotically negligible in comparison with binary mergers. In this case, a simple upper bound for the rate of convergence of the finite-dimensional distributions is derived. It depends (up to some constants) only on the three factorial moments E((ν1)2), E((ν1)3) and E((ν1)2(ν2)2), where (x)k := x(x-1)…(x-k+1). Examples are the Wright-Fisher model, where the rate of convergence is of order N-1, and the Moran model, with a convergence rate of order N-2.

Let τr denote the hitting time of B(0:r) for a multidimensional diffusion process. We give verifiable criteria for finiteness/infiniteness of As an application we exhibit classes of diffusion processes which are recurrent but is infinite for all p > 0, |x| > r > 0; this includes the two-dimensional Brownian motion and the reflecting Brownian motion in a wedge with a certain parameter α = 0.

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.

The long-run average cost per unit time of operating a finite dam controlled by a PlM policy (Faddy (1974), Zuckerman (1977)) is determined when the cumulative input process is (a) a Wiener process with drift and (b) the integral of a Markov chain. It is shown how the cost for (a) can be obtained as the limit of the costs associated with a sequence of input processes of the type (b).

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt} of an infinite dam whose cumulative input {At} is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = xα and {At} is stable with index β ∊ (0, 1). In general if EAt, < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt} as t → ∞. For {At} stable with index β and r(x) = xα, α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt} in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.