Equivalence of weak and strong modes of measures on topological vector spaces

A strong mode of a probability measure on a normed space $X$ can be defined as a point $u$ such that the mass of the ball centred at $u$ uniformly dominates the mass of all other balls in the small-radius limit. Helin and Burger weakened this definition by considering only pairwise comparisons with balls whose centres differ by vectors in a dense, proper linear subspace $E$ of $X$, and posed the question of when these two types of modes coincide. We show that, in a more general setting of metrisable vector spaces equipped with measures that are finite on bounded sets, the density of $E$ and a uniformity condition suffice for the equivalence of these two types of modes. We accomplish this by introducing a new, intermediate type of mode. We also show that these modes can be inequivalent if the uniformity condition fails. Our results shed light on the relationships between among various notions of maximum a posteriori estimator in non-parametric Bayesian inference.Comment: 22 pages, 3 figure

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

Consider an infinite sequence (Un)n∈N of independent Cauchy random variables, defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scale parameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn≠0 for all n∈N. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn|−1)n∈N is square-summable

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.Comment: 13 page

Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.Comment: 25 page

Partonic effects on higher-order anisotropic flows in relativistic heavy-ion collisions

Higher-order anisotropic flows $v_{4}$ and $v_{6}$ in heavy ion collisions at the Relativistic Heavy Ion Collider are studied in a multiphase transport model that has previously been used successfully for describing the elliptic flow $v_2$ in these collisions. We find that the same parton scattering cross section of about 10 \textrm{mb} used in explaining the measured $v_2$ can also reproduce the recent data on $v_{4}$ and $v_{6}$ from Au + Au collisions at $\sqrt{s}=200$ \textrm{AGeV}. It is further found that the $$ is a more sensitive probe of the initial partonic dynamics in these collisions than $v_{2}$. Moreover, higher-order parton anisotropic flows are nonnegligible and satisfy the scaling relation $v_{n,q}(p_{T})\sim v_{2,q}^{n/2}(p_{T})$, which leads naturally to the observed similar scaling relation among hadron anisotropic flows when the coalescence model is used to describe hadron production from the partonic matter.Comment: 5 pages, 3 figures, version to appear in PRC as a Rapid Communicatio

Strategi Diferensiasi Produk dalam Menghadapi Persaingan Kuliner (Kasus Bisnis Sop Tunjang Tampan di Pekanbaru)

This study was conducted at the First Tunjang Sop located on Jl . Cinnamon Pekanbaru Riau , which is a company engaged in the trade . The purpose of this study was to determine the strategy of product differentiation in the face of the culinary competition in business tunjang perjama pekanbaru soup . The research methods include the location of the research conducted on the first attempt tunjang pekanbaru soup , types and sources of data used is primary data , where the data is obtained directly from the company include product differentiation strategy that made the first attempt tunjang pekanbaru culinary soup and secondary data is data annual obtained from the companys other sales data from the years 2008-2012 , and a general overview of the company , while the analysis of the data using the method of analyzing the data in this study would be conducted degan qualitative descriptive method .Based on the results of research on product differentiation strategy in the face of the culinary competition in the business case tunjang Handsome soup can be seen that most of the respondents to respond well enough that as many as 21 people , or 52.50 % , then the respondents who gave a good response ditangggapi many as 8 or 20 , 00 % and to respond less well addressed as many as 11 people or 27.50 %The obstacles in the face of the product differentiation strategy culinary competition among others : That the lack of parking, lack of ability of the company to maintain a corporate image , a lack of desire and willingness to further develop its business enterpriseFrom these results , the actual product being supported first Handsome culinary soup is considered to be able to compete , but business owners should still consider the quality of marketed products , as well as looking for new innovations . This is done so that businesses can compete soup first SOP tunjang To nex time .Key word : Product, differentiation, marketing strategy,business competitio

Recurrence of Preeclampsia in Northern Tanzania: A Registry-based Cohort Study.

Preeclampsia occurs in about 4 per cent of pregnancies worldwide, and may have particularly serious consequences for women in Africa. Studies in western countries have shown that women with preeclampsia in one pregnancy have a substantially increased risk of preeclampsia in subsequent pregnancies. We estimate the recurrence risks of preeclampsia in data from Northern Tanzania. A prospective cohort study was designed using 19,811 women who delivered singleton infants at a hospital in Northern Tanzania between 2000 and 2008. A total of 3,909 women were recorded with subsequent deliveries in the hospital with follow up through 2010. Adjusted recurrence risks of preeclampsia were computed using regression models. The absolute recurrence risk of preeclampsia was 25%, which was 9.2-fold (95% CI: 6.4 - 13.2) compared with the risk for women without prior preeclampsia. When there were signs that the preeclampsia in a previous pregnancy had been serious either because the baby was delivered preterm or had died in the perinatal period, the recurrence risk of preeclampsia was even higher. Women who had preeclampsia had increased risk of a series of adverse pregnancy outcomes in future pregnancies. These include perinatal death (RR= 4.3), a baby with low birth weight (RR= 3.5), or a preterm birth (RR= 2.5). These risks were only partly explained by recurrence of preeclampsia. Preeclampsia in one pregnancy is a strong predictor for preeclampsia and other adverse pregnancy outcomes in subsequent pregnancies in Tanzania. Women with previous preeclampsia may benefit from close follow-up during their pregnancies