scholarly journals Symmetric Polynomials and Symmetric Mean Inequalities

10.37236/2915 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Karl Mahlburg ◽  
Clifford Smyth
Keyword(s):  
Geometric Mean ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Combinatorial Result ◽  
A Cell ◽  
Generalized Arithmetic

We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of non-homogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids.Suppose that the cells of a $n \times n$ checkerboard are each independently filled or empty, where the probability that a cell is filled depends only on its column. We prove that for any $0 \leq \ell \leq n$, the probability that each column has at most $\ell$ filled sites is less than or equal to the probability that each row has at most $\ell$ filled sites.

scholarly journals GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

2011 ◽  
Vol 07 (01) ◽  
pp. 173-202
Author(s):  
ROBERT CARLS
Keyword(s):  
Galois Theory ◽  
Theoretical Foundation ◽  
Abelian Varieties ◽  
Geometric Mean ◽  
Null Point ◽  
Point Counting ◽  
Characteristic 2 ◽  
Theoretic Characterization ◽  
Generalized Arithmetic

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

scholarly journals ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

2010 ◽  
Vol 21 (07) ◽  
pp. 939-949 ◽  
Author(s):  
KEIJI MATSUMOTO ◽  
TOMOHIDE TERASOMA
Keyword(s):  
Projective Space ◽  
Hyperelliptic Curve ◽  
Geometric Mean ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Geometric Means ◽  
Dimensional Projective Space ◽  
Period Integral ◽  
Generalized Arithmetic

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomae's formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

A Generalized Arithmetic-Geometric Mean

SIAM Review ◽  
1983 ◽  
Vol 25 (3) ◽  
pp. 401-401 ◽  
Author(s):  
D. Borwein ◽  
P. B. Borwein
Keyword(s):  
Geometric Mean ◽  
Generalized Arithmetic

A Generalized Arithmetic-Geometric Mean

SIAM Review ◽  
1984 ◽  
Vol 26 (3) ◽  
pp. 433-433
Author(s):  
D. Borwein
Keyword(s):  
Geometric Mean ◽  
Generalized Arithmetic

The hit problem for symmetric polynomials over the Steenrod algebra

2002 ◽  
Vol 133 (2) ◽  
pp. 295-303 ◽  
Author(s):  
A. S. JANFADA ◽  
R. M. W. WOOD
Keyword(s):  
Positive Integer ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Left Module ◽  
Symmetric Version ◽  
The Right ◽  
Hit Problem ◽  
Prime 2

We cite [18] for references to work on the hit problem for the polynomial algebra P(n) = [ ]2[x1, ;…, xn] = [oplus ]d[ges ]0Pd(n), viewed as a graded left module over the Steenrod algebra [Ascr ] at the prime 2. The grading is by the homogeneous polynomials Pd(n) of degree d in the n variables x1, …, xn of grading 1. The present article investigates the hit problem for the [Ascr ]-submodule of symmetric polynomials B(n) = P(n)[sum ]n , where [sum ]n denotes the symmetric group on n letters acting on the right of P(n). Among the main results is the symmetric version of the well-known Peterson conjecture. For a positive integer d, let μ(d) denote the smallest value of k for which d = [sum ]ki=1(2λi−1), where λi [ges ] 0.

scholarly journals Hyperelliptic integrals and generalized arithmetic–geometric mean

2012 ◽  
Vol 28 (1) ◽  
pp. 61-78 ◽  
Author(s):  
Jeroen Spandaw ◽  
Duco van Straten
Keyword(s):  
Geometric Mean ◽  
Generalized Arithmetic

scholarly journals Note on bases in algebras of analytic functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 42-47 ◽  
Author(s):  
I.V. Chernega ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Symmetric Polynomials ◽  
Homogeneous Polynomials ◽  
Absolute Basis

Let $\{P_n\}_{n=0}^\infty$ be a sequenceof continuous algebraically independent  homogeneous polynomials on a complex Banach space $X.$ We consider the following question: Under which conditions polynomials $\{P_1^{k_1}\cdots P_n^{k_n}\}$ form a Schauder (perhaps absolute) basis in the minimal subalgebra of entire functions of bounded type on $X$ which contains the sequence $\{P_n\}_{n=0}^\infty$? In the paper we study the following examples: when $P_n$ are coordinate functionals on $c_0,$ and when $P_n$ are symmetric polynomials on $\ell_1$ and on $L_\infty[0,1].$ We can see that for some cases $\{P_1^{k_1}\cdots P_n^{k_n}\}$ is a Schauder basis which is not absolute but for some cases it is absolute.

scholarly journals Interval Oscillation Criteria for Second-Order Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integrals

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Yuangong Sun
Keyword(s):  
Continuous Function ◽  
Forced Oscillation ◽  
Geometric Mean ◽  
Continuous Functions ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Interval Oscillation ◽  
Special Case ◽  
Generalized Arithmetic

By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form[p(t)ϕα(xΔ(t))]Δ+q(t)ϕα(x(τ(t)))+∫aσ(b)r(t,s)ϕγ(s)(x(g(t,s)))Δξ(s)=e(t), wheret∈[t0,∞)T=[t0,∞)  ⋂  T,Tis a time scale which is unbounded from above;ϕ*(u)=|u|*sgn u;γ:[a,b]T1→ℝis a strictly increasing right-dense continuous function;p,q,e:[t0,∞)T→ℝ,r:[t0,∞)T×[a,b]T1→ℝ,τ:[t0,∞)T→[t0,∞)T, andg:[t0,∞)T×[a,b]T1→[t0,∞)Tare right-dense continuous functions;ξ:[a,b]T1→ℝis strictly increasing. Some interval oscillation criteria are established in both the cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

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