Macaulay style formulas for sparse resultants

We present formulas for computing the resultant of sparse polyno- mials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.

A numerical characterization of hypersurface singularities

In this note we give a numerical characterization of hypersurface singularities in terms of the normalized Hilbert-Samuel coefficients, and we interpret this result from the point of view of rigid polynomials.

Optimal strategies for sending information through a quantum channel

Quantum states can be used to encode the information contained in a direction, i.e., in a unit vector. We present the best encoding procedure when the quantum state is made up of N spins (qubits). We find that the quality of this optimal procedure, which we quantify in terms of the fidelity, depends solely on the dimension of the encoding space. We also investigate the use of spatial rotations on a quantum state, which provide a natural and less demanding encoding. In this case we prove that the fidelity is directly related to the largest zeros of the Legendre and Jacobi polynomials. We also discuss our results in terms of the information gain.

Chaotic Kabanov formula for the Azéma martingales

We derive the chaotic expansion of the product of nth- and first-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes.

Oeuvres de La Bruyère (Nouvelle édition précédée d'un avant-propos)

Collection : Bibliothèque des chefs-d'oeuvre

Exact results for Wilson loops in arbitrary representations

We compute the exact vacuum expectation value of 1/2 BPS circular Wilson loops of TeX = 4 U(N) super Yang-Mills in arbitrary irreducible representations. By localization arguments, the computation reduces to evaluating certain integrals in a Gaussian matrix model, which we do using the method of orthogonal polynomials. Our results are particularly simple for Wilson loops in antisymmetric representations; in this case, we observe that the final answers admit an expansion where the coefficients are positive integers, and can be written in terms of sums over skew Young diagrams. As an application of our results, we use them to discuss the exact Bremsstrahlung functions associated to the corresponding heavy probes.

Pediatric reference intervals for plasma free and total metanephrines established with a parametric approach: Relevance to the diagnosis of neuroblastoma.

BACKGROUND: Urine catecholamines, vanillylmandelic, and homovanillic acid are recognized biomarkers for the diagnosis and follow-up of neuroblastoma. Plasma free (f) and total (t) normetanephrine (NMN), metanephrine (MN) and methoxytyramine (MT) could represent a convenient alternative to those urine markers. The primary objective of this study was to establish pediatric centile charts for plasma metanephrines. Secondarily, we explored their diagnostic performance in 10 patients with neuroblastoma. PROCEDURE: We recruited 191 children (69 females) free of neuroendocrine disease to establish reference intervals for plasma metanephrines, reported as centile curves for a given age and sex based on a parametric method using fractional polynomials models. Urine markers and plasma metanephrines were measured in 10 children with neuroblastoma at diagnosis. Plasma total metanephrines were measured by HPLC with coulometric detection and plasma free metanephrines by tandem LC-MS. RESULTS: We observed a significant age-dependence for tNMN, fNMN, and fMN, and a gender and age-dependence for tMN, fNMN, and fMN. Free MT was below the lower limit of quantification in 94% of the children. All patients with neuroblastoma at diagnosis were above the 97.5th percentile for tMT, tNMN, fNMN, and fMT, whereas their fMN and tMN were mostly within the normal range. As expected, urine assays were inconstantly predictive of the disease. CONCLUSIONS: A continuous model incorporating all data for a given analyte represents an appealing alternative to arbitrary partitioning of reference intervals across age categories. Plasma metanephrines are promising biomarkers for neuroblastoma, and their performances need to be confirmed in a prospective study on a large cohort of patients. Pediatr Blood Cancer 2015;62:587-593. © 2015 Wiley Periodicals, Inc.

A computationally efficient shear deformable beam element for large deformation multibody applications

In this paper, a new two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation is proposed. The nonlinear elastic forces of the beam element are obtained using a continuum mechanics approach without employing a local element coordinate system. In this study, linear polynomials are used to interpolate both the transverse and longitudinal components of the displacement. This is different from other absolute nodal-coordinate-based beam elements where cubic polynomials are used in the longitudinal direction. The accompanying defects of the phenomenon known as shear locking are avoided through the adoption of selective integration within the numerical integration method. The proposed element is verified using several numerical examples, and the results are compared to analytical solutions and the results for an existing shear deformable beam element. It is shown that by using the proposed element, accurate linear and nonlinear static deformations, as well as realistic dynamic behavior, can be achieved with a smaller computational effort than by using existing shear deformable two-dimensional beam elements.

Generalization of Vélu’s formulae for isogenies between elliptic curves

Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P + G as the difference between the abscissa of IG (P ) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We generalize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.