The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Tests of sunspot number sequences: 1. Using ionosonde data

More than 70 years ago it was recognised that ionospheric F2-layer critical frequencies [foF2] had a strong relationship to sunspot number. Using historic datasets from the Slough and Washington ionosondes, we evaluate the best statistical fits of foF2 to sunspot numbers (at each Universal Time [UT] separately) in order to search for drifts and abrupt changes in the fit residuals over Solar Cycles 17-21. This test is carried out for the original composite of the Wolf/Zürich/International sunspot number [R], the new “backbone” group sunspot number [RBB] and the proposed “corrected sunspot number” [RC]. Polynomial fits are made both with and without allowance for the white-light facular area, which has been reported as being associated with cycle-to-cycle changes in the sunspot number - foF2 relationship. Over the interval studied here, R, RBB, and RC largely differ in their allowance for the “Waldmeier discontinuity” around 1945 (the correction factor for which for R, RBB and RC is, respectively, zero, effectively over 20 %, and explicitly 11.6 %). It is shown that for Solar Cycles 18-21, all three sunspot data sequences perform well, but that the fit residuals are lowest and most uniform for RBB. We here use foF2 for those UTs for which R, RBB, and RC all give correlations exceeding 0.99 for intervals both before and after the Waldmeier discontinuity. The error introduced by the Waldmeier discontinuity causes R to underestimate the fitted values based on the foF2 data for 1932-1945 but RBB overestimates them by almost the same factor, implying that the correction for the Waldmeier discontinuity inherent in RBB is too large by a factor of two. Fit residuals are smallest and most uniform for RC and the ionospheric data support the optimum discontinuity multiplicative correction factor derived from the independent Royal Greenwich Observatory (RGO) sunspot group data for the same interval.

Characterisation of chicken riboflavin carrier protein gene structure and promoter regulation by estrogen

Estrogen is an important steroid hormone that mediates most of its effects on regulation of gene expression by binding to intracellular receptors. The consensus estrogen response element (ERE) is a 13 bp palindromic inverted repeat with a three nucleotide spacer. However, several reports suggest that many estrogen target genes are regulated by diverse elements, such as imperfect EREs and ERE half sites (ERE 1/2),which are either the proximal or the distal half of the palindrome. To gain more insight into ERE half site-mediated gene regulation, we used a region from the estrogen-regulated chicken riboflavin carrier protein (RCP) gene promoter that contains ERE half sites. Using moxestrol, an analogue of estrogen and transient transfection of deletion and mutation containing RCP promoter/reporter constructs in chicken hepatoma (LMH2A) cells, we identified an estrogen response unit (ERU) composed of two consensus ERE 1/2 sites and one non-consensus ERE 1/2 site. Mutation of any of these sites within this ERU abolishes moxestrol response. Further, the ERU is able to confer moxestrol responsiveness to a heterologous promoter. Interestingly, RCP promoter is regulated by moxestrol in estrogen responsive human MCF-7 cells, but not in other cell lines such as NIH3T3 and HepG2 despite estrogen receptor-alpha (ER-�) co transfection. Electrophoretic mobility shift assays (EMSAs) with promoter regions encompassing the half sites and nuclear extracts from LMH2A cells show the presence of a moxestrol-induced complex that is abolished by a polyclonal anti-ER� antibody. Surprisingly, estrogen receptor cannot bind to these promoter elements in isolation. Thus, there appears to be a definite requirement for some other factor(s) in addition to estrogen receptor, for the generation of a suitable response of this promoter to estrogen. Our studies therefore suggest a novel mechanism of gene regulation by estrogen, involving ERE half sites without direct binding of ER to the cognate elements.

Estrogen regulation of chicken riboflavin carrier protein gene is mediated by ERE half sites without direct binding of estrogen receptor

Estrogen is an important steroid hormone that mediates most of its effects on regulation of gene expression by binding to intracellular receptors. The consensus estrogen response element (ERE) is a 13 bp palindromic inverted repeat with a three nucleotide spacer. However, several reports suggest that many estrogen target genes are regulated by diverse elements, such as imperfect EREs and ERE half sites (ERE 1/2), which are either the proximal or the distal half of the palindrome. To gain more insight into ERE half site-mediated gene regulation, we used a region from the estrogen-regulated chicken riboflavin carrier protein (RCP) gene promoter that contains ERE half sites. Using moxestrol, an analogue of estrogen and transient transfection of deletion and mutation containing RCP promoter/reporter constructs in chicken hepatoma (LMH2A) cells, we identified an estrogen response unit (ERU) composed of two consensus ERE 1/2 sites and one non-consensus ERE 1/2 site. Mutation of any of these sites within this ERU abolishes moxestrol response. Further, the ERU is able to confer moxestrol responsiveness to a heterologous promoter. Interestingly, RCP promoter is regulated by moxestrol in estrogen responsive human MCF-7 cells, but not in other cell lines such as NIH3T3 and HepG2 despite estrogen receptor-alpha (ER-�) co transfection. Electrophoretic mobility shift assays (EMSAs) with promoter regions encompassing the half sites and nuclear extracts from LMH2A cells show the presence of a moxestrol-induced complex that is abolished by a polyclonal anti-ER� antibody. Surprisingly, estrogen receptor cannot bind to these promoter elements in isolation. Thus, there appears to be a definite requirement for some other factor(s) in addition to estrogen receptor, for the generation of a suitable response of this promoter to estrogen. Our studies therefore suggest a novel mechanism of gene regulation by estrogen, involving ERE half sites without direct binding of ER to the cognate elements.