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b. N*2* = {(*x**1*, *x**2*,..., *xn*) : ∑ n

"... Proposition 1 ([DF], p342). Let R be a ring and let M be an R-module. A subset N of M is an R-submodule of M if and only if 1. N is nonempty and 2. x + ry ∈ N for all r ∈ R and all x, y ∈ N. Exercise 10.1.4. Let R be a ring with identity, let M be the R-module Rn with component-wise addition and mul ..."

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Proposition

*1*([DF], p342). Let R be a ring and let M be an R-module. A subset N of M is an R-submodule of M if and only if*1*. N is nonempty and*2*.*x*+ ry ∈ N for all r ∈ R and all*x*, y ∈ N. Exercise 10.*1*.4. Let R be a ring with identity, let M be the R-module Rn with component-wise addition###
Reisz s energy The Riesz s-energy of ωN = {*x**1*, *x**2*,..., *xN*} ⊂ Rp is, for s> 0, Es(ωN):= NX

, 2009

"... i=1 ..."

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∑ =*1* Example: Dice Probabilities Classification and Clustering in CL: Bayes / MLE 4

, 2004

"... • Sample space X with possible outcomes x1, x2,..., xn ..."

###
xixj (*1* − xi) (*1* − xj) ≥ n

"... Problem. Let x1, x2,..., xn be real numbers such that x1 +x2 +...+xn = 1 and such that xi < 1 for every i ∈ {1, 2,..., n}. Prove that ..."

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Problem. Let

*x**1*,*x**2*,...,*xn*be real numbers such that*x**1*+*x**2*+...+*xn*=*1*and such that xi <*1*for every i ∈ {*1*,*2*,..., n}. Prove that### Then

, 2006

"... • A sequence of random variables {X1, X2,.., XN} is said to be exchangeable if ..."

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• A sequence of random variables {

*X**1*,*X**2*,..,*XN*} is said to be exchangeable if### Tutorial Notes for the Workshop on Stein’s Method and Applications Stein’s Method and Normal Approximation

"... Let X1, X2, · · · , Xn be independent random variables with zero means and finite variances. Put ..."

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Let

*X**1*,*X**2*, · · · ,*Xn*be independent random variables with zero means and finite variances. Put###
A System of Equations Problem 07-006, by Apoloniusz Tyszka*1* (Technical Faculty, Hugo Koll ↪ataj University,

"... Prove or disprove the following conjecture: given n ≥ 1 and x1, x2,..., xn ∈ R (C) there ..."

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Prove or disprove the following conjecture: given n ≥

*1*and*x**1*,*x**2*,...,*xn*∈ R (C) there### J. Aust. Math. Soc. 75 (2003), 413–422 ON-DIRECT SUMS OF BANACH SPACES AND CONVEXITY

, 2003

"... Let X1; X2; : : : ; XN be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set inRN−1. Let.X1X2 XN / be the direct sum of X1; X2; : : : ; XN equipped with the norm associated with. We characterize the strict, uniform, and locally uniform convexit ..."

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Let

*X**1*;*X**2*; : : : ;*XN*be Banach spaces and a continuous convex function with some appropriate conditions on a certain convex set inRN−*1*. Let.*X**1**X**2**XN*/ be the direct sum of*X**1*;*X**2*; : : : ;*XN*equipped with the norm associated with. We characterize the strict, uniform, and locally uniform### On a generalized equation of Smarandache and its integer solutions

"... Abstract Let a 6 = 0 be any given real number. If the variables x1, x2, · · · , xn satisfy x1x2 · · ·xn = 1, the equation 1 x1 ax1 + 1 x2 ax2 + · · ·+ 1 xn axn = na has one and only one nonnegative real number solution x1 = x2 = · · · = xn = 1. This generalized the problem of Smarandache in ..."

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Abstract Let a 6 = 0 be any given real number. If the variables

*x**1*,*x**2*, · · · ,*xn*satisfy*x**1**x**2*· · ·*xn*=*1*, the equation*1**x**1*ax*1*+*1**x**2*ax*2*+ · · ·+*1**xn*axn = na has one and only one nonnegative real number solution*x**1*=*x**2*= · · · =*xn*=*1*. This generalized the problem of Smarandache### THE AGM INEQUALITY:

, 2009

"... x1 + x2 + · · ·+ xn n ≥ (x1x2 · · · xn)1/n ∀x ≥ 0. NEWTON’S INEQUALITIES: ek(x) ek(1) ek(x) ek(1) ≥ ek−1(x) ek−1(1) ek+1(x) ek+1(1) ∀x ≥ 0 ..."

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*x*

*1*+

*x*

*2*+ · · ·+

*xn*n ≥ (

*x*

*1*

*x*

*2*· · ·

*xn*)

*1*/n ∀

*x*≥ 0. NEWTON’S INEQUALITIES: ek(

*x*) ek(

*1*) ek(

*x*) ek(

*1*) ≥ ek−

*1*(

*x*) ek−

*1(1*) ek+

*1*(

*x*) ek+

*1(1*) ∀

*x*≥ 0