Measuring and quantifying dependencies between random variables (RVs) can give critical insights into a dataset. Typical questions are: ‘Do underlying relationships exist?’, ‘Are some variables redundant?’, and ‘Is some target variable Y highly or weakly dependent on variable X?’ Interestingly, despite the evident need for a general-purpose measure of dependency between RVs, common practice is that most data analysts use the Pearson correlation coefficient to quantify dependence between RVs, while it is recognized that the correlation coefficient is essentially a measure for linear dependency only. Although many attempts have been made to define more generic dependency measures, there is no consensus yet on a standard, general-purpose dependency function. In fact, several ideal properties of a dependency function have been proposed, but without much argumentation. Motivated by this, we discuss and revise the list of desired properties and propose a new dependency function that meets all these requirements. This general-purpose dependency function provides data analysts with a powerful means to quantify the level of dependence between variables. To this end, we also provide Python code to determine the dependency function for use in practice.

A substantial database of published excavation and other reports has been used to map the character and distribution in Roman Britain of whetstones, those unprepossessing implements essential in the home, farmstead, workshop and barracks for the maintenance of edge-tools and weapons. The quality of the geological identifications in the reports varies considerably, but a wide range of lithologies are reported as put to use: granite, basalts-dolerites, lava, tuff, mica-schist, slates/phyllites, Brownstones, Pennant sandstone, micaceous sandstones, grey sandstones/siltstones, Millstone Grit, Coal Measures, red sandstones, ferruginous sandstones, sarsen, Weald Clay Formation sandstones, sandy limestones, shelly limestones, cementstones, and (Lower) Carboniferous Limestone. On distributional evidence, some of these categories are aliases for alternatively and more familiarly named lithologies. Bringing ‘high-end’ products to the market, the long-running industry based on sandstones from the Weald Clay Formation (Lower Cretaceous) emerges as a British economic feature, evidenced from the Channel coast to the Scottish Borders, and with a recently demonstrated, substantial representation on the Roman near-continent. The distribution maps point to another and more complete British industry, based on the Brownstones (Old Red Sandstone, Devonian) and Pennant sandstone (Upper Carboniferous), outcropping close together in the West Country. A more systematic and geology-based treatment of excavated whetstones in the future is likely to yield yet more insights into the role these artefacts played in the economy of Roman Britain.

Ventenata [Ventenata dubia (Leers) Coss.] is a nonindigenous, invasive grass in the inland Pacific Northwest (PNW) of the United States. It appears to be present in the PNW without any evidence of disease expression. Surveys of V. dubia in the PNW (Idaho, Montana, Oregon, and Washington) were entirely negative for fungi, including types of pathogens that might be expected in grasses (e.g., rust, powdery mildew, choke). In Europe, where V. dubia is native, fungi were documented (i.e., Septoria ventenatae Sandu, Tilletia fusca Ellis & Everh., and Tilletia elisabethae T. Denchev & Denchev) on V. dubia. In its native range there likely are natural enemies that may limit V. dubia abundance, and these may include fungal pathogens. Pathogens of V. dubia from its native range may hold potential for use as classical biological control agents in North America, and if deemed safe, could be introduced. To ascertain V. dubia’s native range, we compiled data from herbarium specimens, consulted with herbarium curators in the region, and searched relevant literature. We found that V. dubia primarily is reported in southern Europe and western Asia. Ventenata dubia has been reported only occasionally from North Africa in Algeria and Morocco. The common name “North Africa grass” likely originated from references to V. dubia in the 19th-century botanical explorations in Algeria of the French botanist, Ernest Cosson, who published the current scientific name based on a preexisting name in 1854. Another finding of interest is that the latitudinal range of collections from Europe and North Africa of V. dubia spans Tunisia to Finland. The plant may thus be adapted to a range of environments, indicating it could become more widely distributed in North America. Efforts to search its native range for pathogens should also consider the range of environmental conditions found within its native and introduced ranges.

Despite the large and growing returns to deferring Social Security benefits, most individuals claim Social Security before the full retirement age. In this paper, we use a panel of administrative tax data on individuals likely to financially benefit from delaying Social Security claiming to explore the relationship between Social Security claiming and distributions from tax-advantaged retirement savings accounts. We find that the majority of our sample claim Social Security prior to taking distributions from Individual Retirement Accounts (IRAs). We also find that a third of our sample have IRA balances equivalent to at least two additional years of Social Security benefits, and a quarter have IRA balances equivalent to at least 4 years of Social Security benefits. We complement our analysis with data from the Health and Retirement Study and find that these percentages are considerably higher when other financial assets are taken into account.

In this article we prove an embedding theorem for inhomogeneous Besov and Triebel–Lizorkin spaces on $\text{RD}$-spaces. The crucial idea is to use the geometric density condition on the measure.

Data on intra-specific variability for seed oil content, physical characteristics and fatty acid composition in Cucurbita moschata and Cucurbita argyrosperma are lacking in the scientific literature. We examined 528 genebank accessions of C. moschata and 166 accessions of C. argyrosperma – which included members of both subsp. argyrosperma and subsp. sororia – for seed oil content, oil physical characteristics and fatty acid composition. The oil of both species had near-identical viscosities, viscosity indices, colour and oxidative stabilities while the oil of C. argyrosperma had a slightly higher pour point, cloud point, percentage of free fatty acids and acid value when compared with C. moschata. Mean oil content values of the two species were similar at 28.7 ± /2.7 and 29.8 ± /2.6% for C. moschata and C. argyrosperma, respectively. The mean seed oil content of C. argyrosperma subsp. argyrosperma var. palmeri (32.1%) was significantly higher than that of the other taxa examined. The average (mean) percentage of total seed weight attributable to the kernel was 77.2% in C. moschata (n= 34) and 74.5% in C. argyrosperma (n= 46). The percentage of total seed weight attributable to the hull was correlated with seed oil content, in both species. Linoleic was the predominant fatty acid in all the samples analysed. Means for individual fatty acids in C. moschata were linoleic 48.5%, oleic 22.6%, palmitic 20.7% and stearic 7.5%. Means for individual fatty acids in C. argyrosperma were linoleic 47.3%, oleic 27.5%, palmitic 16.5% and stearic 8.0%.

Littlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

Decapoda taken in Continuous Plankton Recorder (CPR) samples from the Pacific in 1997 and 2000–2003 have been identified and measured. Some previously un-described larval stages were referred to species and characteristics of these are described. Distributions and seasonal occurrence of decapod taxa in the samples are described and discussed with particular emphasis on the dendrobranchiate shrimp Sergestes similis and the brachyurans Cancer spp. and Chionoecetes spp. There is a prolonged larval season at low levels of abundance off the Californian coast but in the more northern waters there is a shorter productive period but numbers of larvae per sample are high, particularly in June. Larvae of Chionoecetes and other Oregoninae were found only from May to July.

This article discusses the proposed changes to the capital maintenance regime applicable to the Dutch BV. The draft legislation introduces a liability of directors for unlawful distributions and an obligation for shareholders to repay distributions made to them in the suspect period of one year prior to the opening of bankruptcy proceedings. The current enhanced balance sheet test is replaced with a simple balance sheet test and a liquidity test. With the proposed new rules on creditor protection, the Dutch legislature aims to balance welfare and fairness: the law should not place unnecessary restrictions on company's financing decisions, but at the same time it should offer effective protection to creditors. The draft legislation has been subjected to a consultation process. The reactions to the consultation document show that it is not easy to strike the right balance between the interests of the three parties involved: the shareholders, the directors and the creditors.

One of the standard requirements of company law is the restriction of distributions to shareholders in order to protect the legitimate interests of the company's creditors. As lawful dividends don't have to be paid back when the company runs into losses at a later stage, we need a measuring rod in order to decide on the availability of funds for distribution. The traditional balance sheet test is running into criticism due to the rigidity of the old rules and the conflicts between the philosophy of IAS/IFRS and the concept of creditor protection. Newly offered devices like the solvency test aim at giving a better view of the business prospects of the company but they suffer from a limited time horizon and a wide range of discretion for directors. This makes them particularly problematic when long-term obligations have to be addressed. In the end, a combination of balance sheet test and solvency test seems to be a reasonable solution.

Let $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.

AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

Let F be a non-Archimedean local field with ring of integers R and prime ideal $\frak p$. Suppose T is a GLn(F)-invariant distribution on ${\frak g}$=Mn(F), the Lie algebra of GLn(F). If T has support in the set of topologically nilpotent elements, then the restriction of T to the set of functions which are compactly supported and invariant under Mn($\frak p$) may be expressed as a linear combination of nilpotent orbital integrals restricted to the same set of functions.

The classical limit of the scaled elliptic algebra${\mathcal A}$ħ,η ($\widehat{\mathfrak sl}$2) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra ${\mathcal A}$q,p ($\widehat{\mathfrak sl}$2) is also briefly presented.

This paper is concerned with the distibutions and the moments of the area and the local time of a random walk, called the Bernoulli meander. The limit behavior of the distributions and the moments is determined in the case where the number of steps in the random walk tends to infinity. The results of this paper yield explicit formulas for the distributions and the moments of the area and the local time for the Brownian meander.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].