Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right] for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(\setCount\cdot(\log \setCount)^{O(1)}) running time and $O(\log m)$ rounds, where $m$ is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O\left(\setCount^{1+\xi}\right) and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$

Local-set-based Graph Signal Reconstruction

Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios.Comment: 28 pages, 9 figures, 6 tables, journal manuscrip

Approximate Set Union via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem |A1 U A2 U...UAm|, which given a list of sets A1,...,Am with approximate set size m i for Ai with mi ∈ ((1–βL)|A i|,(1+βR)|Ai|), and biased random generators with Prob(x = RandomElement(Ai)) ∈ [1–a L/Ai, 1 +aR/Ai] for each input set Ai and element x ∈ Ai, where i = 1,2,...,m. The approximation |Ai | |Ai | ratio for |A1 U A2 U...UAm| is in the range [(1–ϵ)(1–aL)(1–βL),(1+ϵ)(1+β R)(1+βR)] for any ϵ ∈ (0,1), where α L,αR,βL,βR ∈ (0,1). The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(m˙(logm) O(1)) running time and O(logm) rounds, where m is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O m1 and round complexity O 1 for any (0, 1). We prove that our algorithm runs sublinear in time under certain condition that each element in A 1 U A2 U ... U Am belong to ma for any fixed a \u3e 0. A O r(r + l|)3l3d4 running time dynamic programming algorithm is proposed to deal with an interesting problem in number theory area that is to count the number of lattice points in a d—dimensional ball Bd( r,p,d) of radius r with center at p ∈ D(λ,d,l), where D(λ, d,l) = {(x1,˙˙˙ , xd) : (x1,˙˙˙ ,xd) with xk = ik + jkλ for an integer jk ∈ [–l, l], and another arbitrary integer ik for k = 1,2,...,d.} We prove that it is #P-hard to count the number of lattice points in a set of balls, and we also show that there is no polynomial time algorithm to approximate the number of lattice points in the intersection of n-dimenisonal k-degree balls unless P=NP

Lie Symmetry to second-order nonlinear differential equations and its first integrals

There are many well-known techniques for obtaining exact solutions of differential equations, but most of them are merely special cases of a few powerful symmetry methods. In this paper, we focus our attention on a second-order nonlinear ordinary differential equation of special forms with arbitrary parameters, which is a combination of Liénard-type equation and equation with quadratic friction. With the help of Lie Symmetry methods, we identify several integrable cases of this equation. And for each case, we use the Lie Symmetry method to derive the associated determining system, and apply it further to find infinitesimal generators under the given parametric conditions. After reducing them to canonical variables, we obtain a autonomous equation. Further, through the inverse transformations we identify the explicit first integrals form for each case

Follow Whom? Chinese Users Have Different Choice

Sina Weibo, which was launched in 2009, is the most popular Chinese micro-blogging service. It has been reported that Sina Weibo has more than 400 million registered users by the end of the third quarter in 2012. Sina Weibo and Twitter have a lot in common, however, in terms of the following preference, Sina Weibo users, most of whom are Chinese, behave differently compared with those of Twitter. This work is based on a data set of Sina Weibo which contains 80.8 million users' profiles and 7.2 billion relations and a large data set of Twitter. Firstly some basic features of Sina Weibo and Twitter are analyzed such as degree and activeness distribution, correlation between degree and activeness, and the degree of separation. Then the following preference is investigated by studying the assortative mixing, friend similarities, following distribution, edge balance ratio, and ranking correlation, where edge balance ratio is newly proposed to measure balance property of graphs. It is found that Sina Weibo has a lower reciprocity rate, more positive balanced relations and is more disassortative. Coinciding with Asian traditional culture, the following preference of Sina Weibo users is more concentrated and hierarchical: they are more likely to follow people at higher or the same social levels and less likely to follow people lower than themselves. In contrast, the same kind of following preference is weaker in Twitter. Twitter users are open as they follow people from levels, which accords with its global characteristic and the prevalence of western civilization. The message forwarding behavior is studied by displaying the propagation levels, delays, and critical users. The following preference derives from not only the usage habits but also underlying reasons such as personalities and social moralities that is worthy of future research.Comment: 9 pages, 13 figure

Operator Size Distribution in Large NN Quantum Mechanics of Majorana Fermions

Under the Heisenberg evolution in chaotic quantum systems, initially simple operators evolve into complicated ones and ultimately cover the whole operator space. We study the growth of the operator ``size'' in this process, which is related to the out-of-time-order correlator (OTOC). We derive the full time evolution of the size distribution in large $N$ quantum mechanics of Majorana fermions. As examples, we apply the formalism to the Brownian SYK model (infinite temperature) and the large $q$ SYK model (finite temperature).Comment: 14 pages, 4 figure

Water-Energy Nexus Management for Power Systems

The water system management problem has been widely investigated. However, the interdependencies between water and energy systems are significant and the effective co-optimization is required considering strong interconnections. This paper proposes a two-stage distributionally robust operation model for integrated water-energy nexus systems including power, gas and water systems networked with energy hub systems at a distribution level considering wind uncertainty. The presence of wind power uncertainty inevitably leads to risks in the optimization model. Accordingly, a coherent risk measure, i.e., conditional value-at-risk, is combined with the optimization objective to determine risk-averse operation schemes. This two-stage mean-risk distributionally robust optimization is solved by Bender's decomposition method. Both the day-ahead and real-time operation cost are minimized with an optimal set of scheduling the multi-energy infrastructures. Case studies focus on investigating the strong interdependencies among the four interconnected energy systems. Numerical results validate the economic effectiveness of IES through optimally coordinating the multi-energy infrastructures. The proposed model can provide system operators a powerful two-stage operation scheme to minimise operation cost under water-energy nexus considering risk caused by renewable uncertainties, thus benefiting customers with lower utility bills

Coordinated Risk Mitigation Strategy for Integrated Energy Systems under Cyber-Attacks

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