Abstract:
Injectivity of the continuous wavelet transform acting on a square integrable signal is proved under weak conditions on the Fourier transform of the wavelet, namely that it is nonzero somewhere in almost every direction. For a bounded signal (not necessarily square integrable), we show that if the continuous wavelet transform vanishes identically, then the signal must be constant.

Abstract:
We prove a sufficient condition for frame-type wavelet series in $L^p$, the Hardy space $H^1$, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.

Abstract:
We prove a technical estimate needed in our recent solution of the completeness question for the non-orthogonal Mexican hat wavelet system, in $L^p$ for $1

Abstract:
Inspired by the hierarchical hidden Markov models (HHMM), we present the hierarchical semi-Markov conditional random field (HSCRF), a generalisation of embedded undirectedMarkov chains tomodel complex hierarchical, nestedMarkov processes. It is parameterised in a discriminative framework and has polynomial time algorithms for learning and inference. Importantly, we consider partiallysupervised learning and propose algorithms for generalised partially-supervised learning and constrained inference. We demonstrate the HSCRF in two applications: (i) recognising human activities of daily living (ADLs) from indoor surveillance cameras, and (ii) noun-phrase chunking. We show that the HSCRF is capable of learning rich hierarchical models with reasonable accuracy in both fully and partially observed data cases.

Abstract:
objective: a cross-sectional survey was conducted in three districts of quang ninh province, viet nam, to find out what proportion of the people who lived there engaged in behaviour that put them at risk of becoming infected with hiv, and to measure their knowledge about hiv infection and aids. methods: the survey was conducted in a rural district, yen hung; a mountainous district inhabited primarily by ethnic minority groups, binh lieu; and an urban district, ha long. participants aged 15-45 years were randomly selected from the general population to be interviewed. findings: a total of 630 people from 707 households were interviewed; 8% were not home despite repeated visits and 3% refused to participate. the prevalence of premarital intercourse ranged from 9% to 16% among married men and 4% to 7% among married women. among single men the proportion who had ever had intercourse ranged from 6% to 16%. fewer than 3% reported having ever had sex with a sex worker. the median number of extramarital sex partners was 1. knowledge about hiv/aids was high in the urban and rural areas but low in the mountainous area. being male and being 20-29 years old were associated with having multiple sex partners. conclusion: the low prevalence of individuals reporting that they had had intercourse with sex workers and partners other than their spouse may explain the low rates of hiv infection among the heterosexual population; these rates are in contrast to the high rates of hiv infection found among injecting drug users. the association between having extramarital partners and being a younger man suggests that the tendency to have more sexual partners may increase in the future. if this happens, the potential for hiv to be spread through heterosexual sex will increase.

Abstract:
Let $\chi$ be a primitive Dirichlet character modulo $q$ and $L(s,\chi)$ be the Dirichlet L-function associated to $\chi$. Using a new two-piece mollifier we show that $L(\tfrac{1}{2},\chi)\ne0$ for at least 34% of the characters in the family.

Abstract:
Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.

Abstract:
Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of \xi'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive proportion of gaps are greater than 1.18 times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than 0.7203 (1.5, respectively).

Abstract:
We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet $L$-functions.

Abstract:
We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character \chi modulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.\chi,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.\chi,s), and from that derive an upper bound for the average analytic rank of the studied L-functions.