Linear Sparse Differential Resultant Formulas

Let P be a system of n linear nonhomogeneous ordinary differential polynomials in a set U of n-1 differential indeterminates. Differential resultant formulas are presented to eliminate the differential indeterminates in U from P. These formulas are determinants of coefficient matrices of appropriate sets of derivatives of the differential polynomials in P, or in a linear perturbation Pe of P. In particular, the formula dfres(P) is the determinant of a matrix M(P) having no zero columns if the system P is ``super essential". As an application, if the system PP is sparse generic, such formulas can be used to compute the differential resultant dres(PP) introduced by Li, Gao and Yuan.

Sparse differential resultant formulas: between the linear and the nonlinear case

A matrix representation of the sparse differential resultant is the basis for efficient computation algorithms, whose study promises a great contribution to the development and applicability of differential elimination techniques. It is shown how sparse linear differential resultant formulas provide bounds for the order of derivation, even in the nonlinear case, and they also provide (in many cases) the bridge with results in the nonlinear algebraic case.

Determinación de los periodos propios de presas bóveda simétricas mediante formulas empíricas

El artículo presenta una formulación sencilla que permite obtener los seis primeros períodos propios de vibración de una presa bóveda simétrica diseñada según las recomendaciones del U.S. Bureau of Reclamation. Se indican las expresiones polinómicas aproximadas de estos períodos, tanto para embalse vacío como para embalse lleno. El efecto del embalse se modeliza mediante la técnica de Westergaard modificada. Asimismo se indica una expresión que intenta tener en cuenta, de modo tentativo, el efecto de la flexibilidad del terreno

Checking existing formulas for estimating overtopping rates in harbour breakwaters

The purpose of the research work resulting from various studies undertaken in the CEDEX, as summarized in this article, is to make a comparative analysis of methods for calculating overtopping rates developed by different authors. To this effect, in the first place, existing formulas for estimating the overtopping rate on rubble mound and vertical breakwaters were summarised and analysed. Later, the above mentioned formulas were compared using the results obtained in a series of hydraulic model tests at the CEDEX. The results obtained in the Ferrol outer harbour breakwater and Melilla harbour breakwater tests are presented here. A calculation method based on the neural network theory, developed in the European CLASH Project, was applied to a series of sloping breakwater tests in order to complete this research and the results obtained in the Ferrol outer harbour breakwater test are presented in this article. A series of additional tests was also carried out in a physical model on the standard cross section of the Bilbao harbour sloping breakwater’s cross section, the results of which are under study using the empirical formulas applicable to the cross section, as well as the NN-OVERTOPPING neural network

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Democracy Promotion and the EU, US Formulas: Photo Opportunities or Potential. Jean Monnet/Robert Schuman Paper Series Vol. 14 No. 5, May 2014

Why do we think more of the United States (US) than the European Union (EU) in discussing Afghani or Iraqi democratization, and EU more than US when it is East European? Should not democratization be the same? A comparative study asks what democracy has historically meant in the two regions, how democratization has been spelled out, why instruments utilized differ, and democracy within global leadership contexts. Neither treats democracy as a vital interest, but differences abound: (a) While the US shifted from relative bottom-up to top-down democracy, the EU added bottom-up to its top-down approach; (b) the US interprets democracy as the ends of other policy interests, the EU treats it as the means to other goals; and (c) flexible US instruments contrast with rigid EU counterparts. Among the implications: (a) the 4-stage US approach reaches globally wider than EU’s multi-dimensional counterpart, but EU’s regional approach sinks deeper than the US’s; (b) human rights find better EU than US anchors; (c) whereas the US approach makes intergovernmental actions the sine qua non of democratization, EU’s intergovernmental, transnational, and supranational admixture promotes quid pro quo dynamics and incremental growth; and (d) competitive democratization patterns creates lock-ins for both recipient and supplier countries.

The coal and metal miners' pocketbook of principles : rules, formulas, and tables.

"Specially compiled and prepared for the convenient use of mine officials, mining engineers, and students preparing themselves for certificates of competency as mine inspectors or mine foremen."

Selected problems of minimization of variable-valued logic formulas /

"UIUCDCS-R-75-726"

Equivalent forms of multistep formulas /

"UIUL-ENG 78 1730."

Unified modified divided difference implementation of Adams and BDF formulas /

"March 1980."

Determination of symmetric VL1 formulas : algorithm and program SYM4 /

Mode of access: Internet.

Brewing formulas practically considered,

Advertising matter: p. 56-59.

A system of mineralogy, comprising the most recent discoveries: including full descriptions of species and their localities, chemical analyses and formulas, tables for the determination of minerals, with a treatise on mathematical crystallography and the drawing of figures of crystals.

Mode of access: Internet.

D. Johannis Michaelis ... Opera quotquot haberi potuerunt omnia : I. Praxis clinica ad Jo. Jonstoni Ideam comparata. II. Apparatus medicamentarius ad P. Morelli methodum praescribendi formulas remediorum. III. Animadversiones in Joh. Schroederi Pharmacopoeiam. IV. Notationes ad Guerneri Rolfincii Chymiam. V. Officinas pharmaceuticas visitandi ratio. VI. Medicamentorum familiar. Syllabus.

Mode of access: Internet.

Hieronymi Davidis Gaubii ... Libellus de methodo concinnandi formulas medicamentorum.

Mode of access: Internet.