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A Detailed Proofs and Derivations
"... For the QP edges, it holds that µij(xi, xj) = µi(xi)µj(xj). Instead of having a linear objective function (µ·θ), we can substitute µij(xi, xj) by µi(xi)µj(xj) in the objective. Thus we no longer need to store the parameter µij(xi, xj) nor the meanfield constraint explicitly for QP edges. Therefore, ..."
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For the QP edges, it holds that µij(xi, xj) = µi(xi)µj(xj). Instead of having a linear objective function (µ·θ), we can substitute µij(xi, xj) by µi(xi)µj(xj) in the objective. Thus we no longer need to store the parameter µij(xi, xj) nor the meanfield constraint explicitly for QP edges. Therefore, the total number of parameters is O(k2 L  + nk) and the total number of constraints is O(2Lk + n) where L=E\Q. As the size of Q increases by 1, the size of the set L decreases by 1. This proves the proposition. A.2 Proposition 4 The optimization problem involving the function g(µ, y; θ, Q) over Ω ′ is given by 1: min µ,y (i,j)∈Q xi,xj (i,j)∈L xi,xj θ(xi, xj)e y(xi)+y(xj) − θ(xi, xj)µ(xi, xj) (19) subject to: ∑ µ(xi, xj) = 1 ∀(i, j) ∈ L; xi,xj µij(xi, ˆxj) = e y(xi) ∀i ∈ V, ∀xi, ∀Nbl(i) (20)
Option Pricing in ARCHtype Models: with Detailed Proofs
, 1995
"... ARCHmodels have become popular for modelling financial time series. They seem, at first, however, to be incompatible with the option pricing approach of Black, Scholes, Merton et al., because they are discretetime models and posess too much variability. We show that completeness of the market hold ..."
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(1995). It includes additional comments and detailed proofs. It also includes a chapter concerning "the equality of filtrations" which deals with the following issue. Trading strategies should be based on information (filtration) that traders posess. In practice, however, one typically assumes
Detailed Proof of Two Dimensional Jacobian Conjecture 1
, 2006
"... Dedicated to my teacher Professor Zhexian Wan for his 80th birthday Abstract. We give a full proof of the two dimensional Jacobian conjecture. We also give an algorithm to compute the inverse map of a polynomial map. 1. ..."
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Dedicated to my teacher Professor Zhexian Wan for his 80th birthday Abstract. We give a full proof of the two dimensional Jacobian conjecture. We also give an algorithm to compute the inverse map of a polynomial map. 1.
Factorized Diffusion Map Approximation: Detailed Proofs
"... Proof. Let Tk = {T1, T2,..., Tk} be a partition of the variables in V and q(V) = ∏k i=1 pi(Ti). Now assume x = [x1, x2,..., xk] T and y = [y1, y2,..., yk] T are two random vectors partitioned according to Tk (note that xi’s and yi’s are subvectors). Then we will have: and therefore, aε(x, y) = k∏ ..."
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Proof. Let Tk = {T1, T2,..., Tk} be a partition of the variables in V and q(V) = ∏k i=1 pi(Ti). Now assume x = [x1, x2,..., xk] T and y = [y1, y2,..., yk] T are two random vectors partitioned according to Tk (note that xi’s and yi’s are subvectors). Then we will have: and therefore, aε(x, y) = k
Detailed proofs The most general results History Algorithms
, 1999
"... Overview of (some of) state of the art Where it fits in the grander scheme ..."
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Overview of (some of) state of the art Where it fits in the grander scheme
Appendix Formal Statement of Results and Detailed Proofs
"... Proposition 1. If a symmetric Nash equilibrium (SNE) exists, advertising levels and prices are given by the following system of equations: a / 0 p Xn 1 n F ð^xÞ ða Þ 1 F ð^xÞ n k 1 F ð^xÞ n k () Š 0; ð1Þ n k k0 1 F ðp =lÞ n nðp =lÞ f ð^xÞ 1 F ð^xÞ n n 1 F ð^xÞ þ ..."
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Proposition 1. If a symmetric Nash equilibrium (SNE) exists, advertising levels and prices are given by the following system of equations: a / 0 p Xn 1 n F ð^xÞ ða Þ 1 F ð^xÞ n k 1 F ð^xÞ n k () Š 0; ð1Þ n k k0 1 F ðp =lÞ n nðp =lÞ f ð^xÞ 1 F ð^xÞ n n 1 F ð^xÞ þ
Featherweight Java: A Minimal Core Calculus for Java and GJ
 ACM Transactions on Programming Languages and Systems
, 1999
"... Several recent studies have introduced lightweight versions of Java: reduced languages in which complex features like threads and reflection are dropped to enable rigorous arguments about key properties such as type safety. We carry this process a step further, omitting almost all features of the fu ..."
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Cited by 659 (23 self)
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, and Wadler) and give a detailed proof of type safety. The extended system formalizes for the first time some of the key features
Existence of the D0–D4 Bound State: a detailed Proof ∗
, 2004
"... We consider the supersymmetric quantum mechanical system which is obtained by dimensionally reducing d=6, N=1 supersymmetric gauge theory with gauge group U(1) and a single charged hypermultiplet. Using the deformation method and ideas introduced by Porrati and Rozenberg [1], we present a detailed p ..."
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We consider the supersymmetric quantum mechanical system which is obtained by dimensionally reducing d=6, N=1 supersymmetric gauge theory with gauge group U(1) and a single charged hypermultiplet. Using the deformation method and ideas introduced by Porrati and Rozenberg [1], we present a detailed
Text S1: Detailed proofs for “The time scale of evolutionary innovation”
"... We will present detailed proofs of all our results. In this section we present an overview of the proof structure and the organization of our results. 1. In Section 2 we present relevant lower and upper bounds on hitting time for Markov chains on an onedimensional grid. The results of this section ..."
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We will present detailed proofs of all our results. In this section we present an overview of the proof structure and the organization of our results. 1. In Section 2 we present relevant lower and upper bounds on hitting time for Markov chains on an onedimensional grid. The results of this section
Results 1  10
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274,448