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Elementary proof of convergence to the mean-field model for the SIR process  [PDF]
Ekkehard Beck,Benjamin Armbruster
Computer Science , 2015,
Abstract: The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (2015) to the SIR process, we show convergence in mean-square using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.
Elementary proof of logarithmic Sobolev inequalities for Gaussian convolutions on $\mathbb{R}$  [PDF]
David Zimmermann
Mathematics , 2014,
Abstract: In a 2013 paper, the author showed that the convolution of a compactly supported measure on the real line with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). In a 2014 paper, the author gave bounds for the optimal constants in these LSIs. In this paper, we give a simpler, elementary proof of this result.
An elementary proof of convergence to the mean-field equations for an epidemic model  [PDF]
Benjamin Armbruster,Ekkehard Beck
Quantitative Biology , 2015,
Abstract: It is common to use a compartmental, fluid model described by a system of ordinary differential equations (ODEs) to model disease spread. In addition to their simplicity, these models are also the mean-field approximations of more accurate stochastic models of disease spread on contact networks. For the simplest case of a stochastic susceptible-infected-susceptible (SIS) process (infection with recovery) on a complete network, it has been shown that the fraction of infected nodes converges to the mean-field ODE as the number of nodes increases. However the proofs are not simple, requiring sophisticated probability, partial differential equations (PDE), or infinite systems of ODEs. We provide a short proof in this case for convergence in mean-square on finite time intervals using a system of two ODEs and a moment inequality.
A unified proof of inequalities and some new inequalities involving Neuman-Sándor mean  [PDF]
Wen-Hui Li,Feng Qi
Mathematics , 2013,
Abstract: In the paper, by finding linear relations of differences between some means, the authors supply a unified proof of some double inequalities for bounding Neuman-S\'andor means in terms of the arithmetic, harmonic, and contra-harmonic means and discover some new sharp inequalities involving Neuman-S\'andor, contra-harmonic, root-square, and other means of two positive real numbers.
An Elementary Proof of the Quantum Adiabatic Theorem  [PDF]
Andris Ambainis,Oded Regev
Physics , 2004,
Abstract: We provide an elementary proof of the quantum adiabatic theorem.
Elementary proofs of one weight norm inequalities for fractional integral operators and commutators  [PDF]
David Cruz-Uribe
Mathematics , 2015,
Abstract: We give new and elementary proofs of one weight norm inequalities for fractional integral operators and commutators. Our proofs are based on the machinery of dyadic grids and sparse operators used in the proof of the A2 conjecture.
An elementary proof that P =/= NP  [PDF]
Bhupinder Singh Anand
Mathematics , 2006,
Abstract: We show that, if PA has no non-standard models, then P=/=NP. We then give an elementary proof that PA has no non-standard models.
Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy  [PDF]
Mary Beth Ruskai
Physics , 2006, DOI: 10.1016/S0034-4877(07)00019-5
Abstract: A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz inequality in elementary courses. Several consequences are proved in a way which allow an elementary proof of strong subadditivity in a few more lines. Some expository material on Schwarz inequalities for operators and the Holevo bound for partial measurements is also included.
Elementary proof of the bound on the speed of quantum evolution  [PDF]
Piotr Kosinski,Magdalena Zych
Physics , 2005, DOI: 10.1103/PhysRevA.73.024303
Abstract: An elementary proof is given of the bound on "ortogonalization time".
An Elementary Proof of an Asymptotic Formula of Ramanujan  [PDF]
Adrian Dudek
Mathematics , 2014,
Abstract: We derive the mean square of the divisor function using only elementary techniques.
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