Results 1 to 10 of about 682 (132)
On the Elementary Symmetric Polynomials and the Zeros of Legendre Polynomials
Journal of Mathematics, 2022 In this paper, we seek to present some new identities for the elementary symmetric polynomials and use these identities to construct new explicit formulas for the Legendre polynomials.
Maryam Salem Alatawi
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On multivariate orthogonal polynomials and elementary symmetric functions
Numerical Algorithms, 2022 AbstractWe study families of multivariate orthogonal polynomials with respect to the symmetric weight function indvariables$$\begin{aligned} B_{\gamma }(\mathtt {x}) = \prod \limits _{i=1}^{d} \omega (x_{i}) \prod \limits _{i<j} |x_{i}-x_{j}|^{2\gamma +1}, \quad \mathtt {x}\in (a,b)^{d}, \end{aligned}$$Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,for$
Miguel Pinar, Cleonice Bracciali
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Hyperbolicity cones of elementary symmetric polynomials are spectrahedral [PDF]
Optimization Letters, 2013 9 pages. Some typos corrected. Details added.
Petter Branden
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Computing Elementary Symmetric Polynomials with a Subpolynomial Numberof Multiplications
SIAM Journal on Computing, 2003 Summary: Elementary symmetric polynomials \(S_n^k\) are the building blocks of symmetric polynomials. In this work we prove that for constant \(k\)'s, \(S_n^k\) modulo composite numbers \(m=p_{1}p_{2}\) can be computed using only \(n^{o}(1)\) multiplications if the coefficients of monomials \(x_{i_1}x_{i_{2}}\dots x_{i_k}\) are allowed to be 1 either ...
Vince Grolmusz
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Observation of the topological aberrations of twisted light [PDF]
Nature CommunicationsWhen reflected from an interface, a laser beam generally drifts and tilts away from the path predicted by ray optics, an intriguing consequence of its finite transverse extent.
Rafael F. Barros +4 more
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On hyperbolicity cones associated with elementary symmetric polynomials [PDF]
Optimization Letters, 2007 zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Elementary symmetric polynomials in Shamir's scheme
Journal of Number Theory, 2010 Building on two papers of A. Schinzel, S. Spież, J. Urbanowicz [Finite Fields Appl. 16, No. 6, 449--462 (2010; Zbl 1209.94054)] and S. Spież, M. Srebrny, J. Urbanowicz (yet unpublished), the paper characterizes when the track \((1,2,\dots,n)\) is \(k\)-admissible over the field \(\mathbb F_p\).
Schinzel, Andrzej +2 more
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On the Weak Limits of Elementary Symmetric Polynomials
Annals of Probability, 1986 The paper extends recent results of the reviewer and others [see e.g. the reviewer, Z. Wahrscheinlichkeitstheor. Verw. Geb. 59, 355-359 (1982; Zbl 0468.60029)] on the weak limits of elementary symmetric polynomials of i.i.d. rv's. The most interesting result is the following: Let \(X,X_ 1,X_ 2,..\). be i.i.d. rv's, \(P(X=1)=P(X=-1)=(1/2)P(X\neq 0)=p/2\)
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On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials [PDF]
Statistica Neerlandica, 2000 We investigate the validity of the bootstrap method for the elementary symmetric polynomials S(k)n=(nk)−1Σ1≤i1< ... <ik≤nXi1 ... Xik of i.i.d. random variables X1, ..., Xn. For both fixed and increasing order k, as n→∞ the cases where μ=E X1[moe2]0, the nondegenerate case, and where μ=E X1=0, the degenerate case, are considered.
van Es, A.J., Helmers, R., Huskova, M.
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On partitions associated with elementary symmetric polynomials
Ramanujan JournalThe elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These elementary symmetric partition functions have been studied before, and are related to plethysm.
Cristina Ballantine +2 more
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