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Yufei Zhao - One of the best experts on this subject based on the ideXlab platform.
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on the lower tail Variational Problem for random graphs
Combinatorics Probability & Computing, 2017Co-Authors: Yufei ZhaoAbstract:We study the lower tail large deviation Problem for subgraph counts in a random graph. Let XH denote the number of copies of H in an Erdős–Renyi random graph G(n, p). We are interested in estimating the lower tail probability P(XH ≤ (1− δ)EXH) for fixed 0 0, as p → 0 slowly, the main contribution to the lower tail probability comes from Erdős–Renyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite H and δ close to 1.
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On the Variational Problem for upper tails in sparse random graphs
Random Structures and Algorithms, 2016Co-Authors: Eyal Lubetzky, Yufei ZhaoAbstract:What is the probability that the number of triangles in Gn,p, the Erdi¾?s-Renyi random graph with edge density p, is at least twice its mean? Writing it as exp[-rn,p], already the order of the rate function rn, p was a longstanding open Problem when p=o1, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that rn,pi¾?n2p2log1/p for pi¾?lognn; the exact asymptotics of rn, p remained unknown. The following Variational Problem can be related to this large deviation question at pi¾?lognn: for i¾?>0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least 1+i¾?p3? A beautiful large deviation framework of Chatterjee and Varadhan 2011 reduces upper tails for triangles to a limiting version of this Problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n-αi¾?pi¾?1 for an explicit α>0, and plausibly it holds in all of the above sparse regime.
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On the Variational Problem for upper tails in sparse random graphs
Random Structures and Algorithms, 2016Co-Authors: Eyal Lubetzky, Yufei ZhaoAbstract:What is the probability that the number of triangles in Gn,p, the Erdi¾?s-Renyi random graph with edge density p, is at least twice its mean? Writing it as exp[-rn,p], already the order of the rate function rn, p was a longstanding open Problem when p=o1, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that rn,pi¾?n2p2log1/p for pi¾?lognn; the exact asymptotics of rn, p remained unknown. The following Variational Problem can be related to this large deviation question at pi¾?lognn: for i¾?>0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least 1+i¾?p3? A beautiful large deviation framework of Chatterjee and Varadhan 2011 reduces upper tails for triangles to a limiting version of this Problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n-αi¾?pi¾?1 for an explicit α>0, and plausibly it holds in all of the above sparse regime.
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on the lower tail Variational Problem for random graphs
arXiv: Combinatorics, 2015Co-Authors: Yufei ZhaoAbstract:We study the lower tail large deviation Problem for subgraph counts in a random graph. Let $X_H$ denote the number of copies of $H$ in an Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,p)$. We are interested in estimating the lower tail probability $\mathbb{P}(X_H \le (1-\delta) \mathbb{E} X_H)$ for fixed $0 < \delta < 1$. Thanks to the results of Chatterjee, Dembo, and Varadhan, this large deviation Problem has been reduced to a natural Variational Problem over graphons, at least for $p \ge n^{-\alpha_H}$ (and conjecturally for a larger range of $p$). We study this Variational Problem and provide a partial characterization of the so-called "replica symmetric" phase. Informally, our main result says that for every $H$, and $0 0$, as $p \to 0$ slowly, the main contribution to the lower tail probability comes from Erd\H{o}s-R\'enyi random graphs with a uniformly tilted edge density. On the other hand, this is false for non-bipartite $H$ and $\delta$ close to 1.
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on the Variational Problem for upper tails in sparse random graphs
arXiv: Combinatorics, 2014Co-Authors: Eyal Lubetzky, Yufei ZhaoAbstract:What is the probability that the number of triangles in $\mathcal{G}_{n,p}$, the Erd\H{o}s-R\'enyi random graph with edge density $p$, is at least twice its mean? Writing it as $\exp[- r(n,p)]$, already the order of the rate function $r(n,p)$ was a longstanding open Problem when $p=o(1)$, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that $r(n,p)\asymp n^2p^2 \log (1/p)$ for $p \gtrsim \frac{\log n}n$; the exact asymptotics of $r(n,p)$ remained unknown. The following Variational Problem can be related to this large deviation question at $p\gtrsim \frac{\log n}n$: for $\delta>0$ fixed, what is the minimum asymptotic $p$-relative entropy of a weighted graph on $n$ vertices with triangle density at least $(1+\delta)p^3$? A beautiful large deviation framework of Chatterjee and Varadhan (2011) reduces upper tails for triangles to a limiting version of this Problem for fixed $p$. A very recent breakthrough of Chatterjee and Dembo extended its validity to $n^{-\alpha}\ll p \ll 1$ for an explicit $\alpha>0$, and plausibly it holds in all of the above sparse regime. In this note we show that the solution to the Variational Problem is $\min\{\frac12 \delta^{2/3}\,,\, \frac13 \delta\}$ when $n^{-1/2}\ll p \ll 1$ vs. $\frac12 \delta^{2/3}$ when $n^{-1} \ll p\ll n^{-1/2}$ (the transition between these regimes is expressed in the count of triangles minus an edge in the minimizer). From the results of Chatterjee and Dembo, this shows for instance that the probability that $\mathcal{G}_{n,p}$ for $ n^{-\alpha} \leq p \ll 1$ has twice as many triangles as its expectation is $\exp[-r(n,p)]$ where $r(n,p)\sim \frac13 n^2 p^2\log(1/p)$. Our results further extend to $k$-cliques for any fixed $k$, as well as give the order of the upper tail rate function for an arbitrary fixed subgraph when $p\geq n^{-\alpha}$.
Faouzi Triki - One of the best experts on this subject based on the ideXlab platform.
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On the spectrum of the Poincaré Variational Problem for two close-to-touching inclusions in 2D
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Eric Bonnetier, Faouzi TrikiAbstract:We study the spectrum of the Poincaré Variational Problem for two close to touching inclusions in R^2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
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on the spectrum of the poincare Variational Problem for two close to touching inclusions in 2d
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Eric Bonnetier, Faouzi TrikiAbstract:We study the spectrum of the Poincare Variational Problem for two close to touching inclusions in R2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
Eric Bonnetier - One of the best experts on this subject based on the ideXlab platform.
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On the spectrum of the Poincaré Variational Problem for two close-to-touching inclusions in 2D
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Eric Bonnetier, Faouzi TrikiAbstract:We study the spectrum of the Poincaré Variational Problem for two close to touching inclusions in R^2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
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on the spectrum of the poincare Variational Problem for two close to touching inclusions in 2d
Archive for Rational Mechanics and Analysis, 2013Co-Authors: Eric Bonnetier, Faouzi TrikiAbstract:We study the spectrum of the Poincare Variational Problem for two close to touching inclusions in R2. We derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero.
D. M. Azimov - One of the best experts on this subject based on the ideXlab platform.
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Active Rocket Trajectory Arcs: A Review
Automation and Remote Control, 2005Co-Authors: D. M. AzimovAbstract:This paper reviews the development of analytical, approximate analytical, and numerical methods for solving the Variational Problem on the determination of optimal rocket trajectories in gravitational fields, and their application to study flight dynamics. Specifics of these methods as applied to solve modern and complex Problems are described. A Variational Problem is formulated and extremal thrust arcs are described. Papers containing results of analytical investigations on thrust arcs are reviewed in depth. Partially investigated Problems are described. Problems of great interest in the development of methods for solving the Variational Problem and Problems in the theory of optimal trajectories are mentioned.
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ANALYTIC SOLUTIONS FOR INTERMEDIATE-THRUST ARCS OF ROCKET TRAJECTORIES IN A NEWTONIAN FIELD
Journal of Applied Mathematics and Mechanics, 1996Co-Authors: D. M. AzimovAbstract:Abstract Mayer's Variational Problem of determining the optimum trajectories of a rocket moving with constant exhaust velocity and bounded mass flow rate. in a Newtonian field is considered. New analytic solutions are obtained for plane intermediate-thrust arcs, using the canonical system of equations of the Variational Problem and the properties of the switching function. These solutions represent certain spiral trajectories. In motion with a fixed time, at arbitrary angular distances, these solutions satisfy Robbins' necessary optimum condition. As an example the Problem of minimizing the characteristic velocity of flight between elliptic orbits is considered.
A A Deriglazov - One of the best experts on this subject based on the ideXlab platform.
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Variational Problem for hamiltonian system on so k m lie poisson manifold and dynamics of semiclassical spin
Modern Physics Letters A, 2014Co-Authors: A A DeriglazovAbstract:We describe the procedure for obtaining Hamiltonian equations on a manifold with so(k, m) Lie–Poisson bracket from a Variational Problem. This implies identification of the manifold with base of a properly constructed fiber bundle embedded as a surface into the phase space with canonical Poisson bracket. Our geometric construction underlies the formalism used for construction of spinning particles in [A. A. Deriglazov, Mod. Phys. Lett. A 28, 1250234 (2013); Ann. Phys. 327, 398 (2012); Phys. Lett. A 376, 309 (2012)], and gives precise mathematical formulation of the oldest idea about spin as the "inner angular momentum".
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kinematics of semiclassical spin and spin fiber bundle associated with so n lie poisson manifold
Journal of Physics: Conference Series, 2013Co-Authors: A A DeriglazovAbstract:We describe geometric construction underlying the Lagrangian actions for non-Grassmann spinning particles proposed in our recent works. If we discard the spatial variables (the case of frozen spin), the Problem reduces to formulation of a Variational Problem for Hamiltonian system on a manifold with so(n) Lie-Poisson bracket. To achieve this, we identify dynamical variables of the Problem with coordinates of the base of a properly constructed fiber bundle. In turn, the fiber bundle is embedded as a surface into the phase space equipped with canonical Poisson bracket. This allows us to formulate the Variational Problem using the standard methods of Dirac theory for constrained systems.
