A class of tridiagonal matrices are examined and characterized. Their spectrum, the left and right eigenvectors as well as their scalar products will be determined. The determinants of the two ...matrices composed by the left and right eigenvectors are also evaluated in closed forms.
Privacy-preserving polynomial interpolation refers to a process which requires two parties to jointly finding out a polynomial over their private coordinate pairs. Unfortunately, the existing general ...approach remains impractical. To date, no practical solution to privacy-preserving polynomial interpolation exists. In this paper, we aim to fill this gap by presenting an efficient solution to enable this process. To this end, we first transform the privacy-preserving polynomial interpolation into privacy-preserving calculation of function values, and design a succinct privacy-preserving scalar product protocol. Then, we tackle the original problem by employing Lagrange interpolation in combination with our privacy-preserving scalar product protocol. Finally, we offer some application examples of how our protocol can be used to conduct privacy-preserving predictive analysis.
•Through mathematics, the paper aims to bring attention to the general problems of invariance in hydraulic engineering.•Section 3 presents a mathematical-like formalization of a water network and ...provides the change formulas for the fundamental energy paprameters.•As an explicit case, considered two widely used resilience indices, they turn out to be non-invariant in a PDA approach.•In addition to numerical calculations and simulations, the mathematical reasons for their non-invariance are investigated.•Two new invariant resilience indices, which modify one of those considered, are presented as an example of mathematical solutions in a new specific framework.
In recent decades many mathematical models, both theoretical and computational, have been applied to hydraulic networks with considerable success, and recently this trend appears to be growing exponentially. Yet there are important problems of a mathematical nature that have very often had little consideration in this field, such as that of the invariance of the models with respect to the reference adopted. In this paper, a mathematical framework new in the field is used and, starting from the discussion of the invariance problem of local indices, the behavior of some widespread global ones that evaluate the resilience of a network will be investigated from both a theoretical and computational point of view. The authors also give suitable changing formulas in the local and global case and describe the conditions that ensure invariance. Through a mathematical-like formalization of the hydraulic network concept, the new framework finally allows to find a series of mathematical solutions to problems of this kind, two of which will be provided in the text.
The first measurement of the ϒ(1S) elliptic flow coefficient (v2) is performed at forward rapidity (2.5 < y < 4) in Pb–Pb collisions at √sNN=5.02 TeV with the ALICE detector at the LHC. The results ...are obtained with the scalar product method and are reported as a function of transverse momentum (pT) up to 15 GeV/c in the 5%–60% centrality interval. The measured ϒ(1S)v2 is consistent with 0 and with the small positive values predicted by transport models within uncertainties. The v2 coefficient in 2 < pT < 15 GeV/c is lower than that of inclusive J/ψ mesons in the same pT interval by 2.6 standard deviations. These results, combined with earlier suppression measurements, are in agreement with a scenario in which the ϒ(1S) production in Pb–Pb collisions at LHC energies is dominated by dissociation limited to the early stage of the collision, whereas in the J/ψ case there is substantial experimental evidence of an additional regeneration component.
We present some applications of the Kudla–Millson and the Millson theta lift. The two lifts map weakly holomorphic modular functions to vector valued harmonic Maass forms of weight 3/2 and 1/2, ...respectively. We give finite algebraic formulas for the coefficients of Ramanujan's mock theta functions f(q) and ω(q) in terms of traces of CM-values of a weakly holomorphic modular function. Further, we construct vector valued harmonic Maass forms whose shadows are unary theta functions, and whose holomorphic parts have rational coefficients. This yields a rationality result for the coefficients of mock theta functions, i.e., harmonic Maass forms whose shadows lie in the space of unary theta functions. Moreover, the harmonic Maass forms we construct can be used to evaluate the Petersson inner products of unary theta functions with harmonic Maass forms, giving formulas and rationality results for the Weyl vectors of Borcherds products.
The aim of this work is to study a very special family of odd-quadratic Lie superalgebras g=g0¯⊕g1¯ such that g1¯ is a weak filiform g0¯-module (weak filiform type). We introduce this concept after ...having proved that the unique non-zero odd-quadratic Lie superalgebra (g,B) with g1¯ a filiform g0¯-module is the abelian 2-dimensional Lie superalgebra g=g0¯⊕g1¯ such that dimg0¯=dimg1¯=1. Let us note that in this context the role of the center of g is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight.
The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions ...leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.
PDF
