Recursion Rules for the Hypergeometric Zeta Functions

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of Beta distributions

Ancient Roman Munificence: The Development of the Practice and Law of Charity

This article traces Roman charity from its incipient meager beginnings during Rome’s infancy to the mature legal formula it assumed after intersecting with the Roman emperors and Christianity. During this evolution, charity went from being a haphazard and often accidental private event, to a broad undertaking of public, religious, and legal commitment. Charitable giving within ancient Rome was quite extensive and longstanding, with some obvious differences from the modern definition and practice of the activity. The main differences can be broken into four key aspects. First, as regards the republican period, Roman charity was invariably given with either political or ego-driven motives, connected to ambitions for friendship, political power or lasting reputation. Second, charity was almost never earmarked for the most needy. Third, Roman largesse was not religiously derived, but rather drawn from personal, or civic impetus. Last, Roman charity tended to avoid any set doctrine, but was hit and miss in application. It was not till the imperium’s grain dole, or cura annonae, and the support of select Italian children, or alimenta were established in the later Empire that the approach became more or less fixed in some basic areas. It was also in the later Empire that Christianity made an enormous impact, helping motivate Constantine – who made Christianity the state religion – and Justinian to develop legal doctrines of charity. This study of Roman charitable activities will concern itself with several streams of enquiry, one side being the historical, societal, and religious, versus the legal. From another angle, it will follow the pagan versus Christian developments. The first part is a reckoning of Roman largesse in its many expressions, with explanations of what appeared to motivate Roman benefactors. This will be buttressed by a description of the Roman view of society and how charity fit within it. The second part will deal with the specific legal expressions of euegertism (or ‘private munificence for public benefit’ ) that typify and reveal the particular genius that Romans had for casting their activities in a legal framework. This is important because Rome is the starting point of much of charity as we understand the term, both legally and institutionally in the modern world. So studying Roman giving brings into highlight and contrast the beginnings of Charity itself – arguably one of the most important developments of the civilized world, and the linchpin of the Liberal ethos

Ancient Roman Munificence: The Development of the Practice and Law of Charity

This article traces Roman charity from its incipient meager beginnings during Rome’s infancy to the mature legal formula it assumed after intersecting with the Roman emperors and Christianity. During this evolution, charity went from being a haphazard and often accidental private event, to a broad undertaking of public, religious, and legal commitment. Charitable giving within ancient Rome was quite extensive and longstanding, with some obvious differences from the modern definition and practice of the activity. The main differences can be broken into four key aspects. First, as regards the republican period, Roman charity was invariably given with either political or ego-driven motives, connected to ambitions for friendship, political power or lasting reputation. Second, charity was almost never earmarked for the most needy. Third, Roman largesse was not religiously derived, but rather drawn from personal, or civic impetus. Last, Roman charity tended to avoid any set doctrine, but was hit and miss in application. It was not till the imperium’s grain dole, or cura annonae, and the support of select Italian children, or alimenta were established in the later Empire that the approach became more or less fixed in some basic areas. It was also in the later Empire that Christianity made an enormous impact, helping motivate Constantine – who made Christianity the state religion – and Justinian to develop legal doctrines of charity. This study of Roman charitable activities will concern itself with several streams of enquiry, one side being the historical, societal, and religious, versus the legal. From another angle, it will follow the pagan versus Christian developments. The first part is a reckoning of Roman largesse in its many expressions, with explanations of what appeared to motivate Roman benefactors. This will be buttressed by a description of the Roman view of society and how charity fit within it. The second part will deal with the specific legal expressions of euegertism (or ‘private munificence for public benefit’ ) that typify and reveal the particular genius that Romans had for casting their activities in a legal framework. This is important because Rome is the starting point of much of charity as we understand the term, both legally and institutionally in the modern world. So studying Roman giving brings into highlight and contrast the beginnings of Charity itself – arguably one of the most important developments of the civilized world, and the linchpin of the Liberal ethos

The Private Foundation\u27s Topsy Turvy Road in the American Political Process

This article studies this American political debate on the charitable tax exemption from 1864 to 1969, in particular, the debate regarding philanthropic, private foundations. The article\u27s premise is that the debate\u27s core has little evolved since that between the 1850s and 1870s. To create perspective, a short brief of the modern economic significance of the foundation sector follows. Thereafter, the article begins with a review of the preand post-colonial attitudes toward charitable institutions leading up to the 1800s debates, illustrating the incongruity of American policy regarding whether and to what extent to grant charities tax exemption. The 1800s state debates are referenced and correlated to parts of the 1900s federal debate to show the similarity if not sameness of the arguments against and justifications for exemption. The twentieth century legislative examination primarily focuses upon the regulatory evolution for foundations. Finally, the article concludes with a brief discussion of the 1969 tax reform\u27s changes to the foundation rules and the significant twentieth century legislation regulating both public and private foundations

The Private Foundation\u27s Topsy Turvy Road in the American Political Process

This article studies this American political debate on the charitable tax exemption from 1864 to 1969, in particular, the debate regarding philanthropic, private foundations. The article\u27s premise is that the debate\u27s core has little evolved since that between the 1850s and 1870s. To create perspective, a short brief of the modern economic significance of the foundation sector follows. Thereafter, the article begins with a review of the preand post-colonial attitudes toward charitable institutions leading up to the 1800s debates, illustrating the incongruity of American policy regarding whether and to what extent to grant charities tax exemption. The 1800s state debates are referenced and correlated to parts of the 1900s federal debate to show the similarity if not sameness of the arguments against and justifications for exemption. The twentieth century legislative examination primarily focuses upon the regulatory evolution for foundations. Finally, the article concludes with a brief discussion of the 1969 tax reform\u27s changes to the foundation rules and the significant twentieth century legislation regulating both public and private foundations

Interactive digital signal processor

The Interactive Digital Signal Processor (IDSP) is examined. It consists of a set of time series analysis Operators each of which operates on an input file to produce an output file. The operators can be executed in any order that makes sense and recursively, if desired. The operators are the various algorithms used in digital time series analysis work. User written operators can be easily interfaced to the sysatem. The system can be operated both interactively and in batch mode. In IDSP a file can consist of up to n (currently n=8) simultaneous time series. IDSP currently includes over thirty standard operators that range from Fourier transform operations, design and application of digital filters, eigenvalue analysis, to operators that provide graphical output, allow batch operation, editing and display information

The negative Bogoliubov dispersion in exciton-polariton condensates

Bogoliubov's theory states that self-interaction effects in Bose-Einstein condensates produce a characteristic linear dispersion at low momenta. One of the curious features of Bogoliubov's theory is that the new quasiparticles in the system are linear combinations of creation and destruction operators of the bosons. In exciton-polariton condensates, this gives the possibility of directly observing the negative branch of the Bogoliubov dispersion in the photoluminescence (PL) emission. Here we theoretically examine the PL spectra of exciton-polariton condensates taking into account of reservoir effects. At sufficiently high excitation densities, the negative dispersion becomes visible. We also discuss the possibility for relaxation oscillations to occur under conditions of strong reservoir coupling. This is found to give a secondary mechanism for making the negative branch visible

Primordial black holes as a tool for constraining non-Gaussianity

Primordial Black Holes (PBH's) can form in the early Universe from the collapse of large density fluctuations. Tight observational limits on their abundance constrain the amplitude of the primordial fluctuations on very small scales which can not otherwise be constrained, with PBH's only forming from the extremely rare large fluctuations. The number of PBH's formed is therefore sensitive to small changes in the shape of the tail of the fluctuation distribution, which itself depends on the amount of non-Gaussianity present. We study, for the first time, how quadratic and cubic local non-Gaussianity of arbitrary size (parameterised by f_nl and g_nl respectively) affects the PBH abundance and the resulting constraints on the amplitude of the fluctuations on very small scales. Intriguingly we find that even non-linearity parameters of order unity have a significant impact on the PBH abundance. The sign of the non-Gaussianity is particularly important, with the constraint on the allowed fluctuation amplitude tightening by an order of magnitude as f_nl changes from just -0.5 to 0.5. We find that if PBH's are observed in the future, then regardless of the amplitude of the fluctuations, non-negligible negative f_nl would be ruled out. Finally we show that g_nl can have an even larger effect on the number of PBH's formed than f_nl.Comment: 9 pages, 5 figures, v2: version to appear in Phys. Rev. D with minor changes, v3: typos corrected (including factor of 1/2 in erfc prefactor), no changes to result