Existence of the D0-D4 Bound State: a detailed Proof

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, we present a detailed proof of the existence of a normalizable ground state for this system

Recommendation Subgraphs for Web Discovery

Recommendations are central to the utility of many websites including YouTube, Quora as well as popular e-commerce stores. Such sites typically contain a set of recommendations on every product page that enables visitors to easily navigate the website. Choosing an appropriate set of recommendations at each page is one of the key features of backend engines that have been deployed at several e-commerce sites. Specifically at BloomReach, an engine consisting of several independent components analyzes and optimizes its clients' websites. This paper focuses on the structure optimizer component which improves the website navigation experience that enables the discovery of novel content. We begin by formalizing the concept of recommendations used for discovery. We formulate this as a natural graph optimization problem which in its simplest case, reduces to a bipartite matching problem. In practice, solving these matching problems requires superlinear time and is not scalable. Also, implementing simple algorithms is critical in practice because they are significantly easier to maintain in production. This motivated us to analyze three methods for solving the problem in increasing order of sophistication: a sampling algorithm, a greedy algorithm and a more involved partitioning based algorithm. We first theoretically analyze the performance of these three methods on random graph models characterizing when each method will yield a solution of sufficient quality and the parameter ranges when more sophistication is needed. We complement this by providing an empirical analysis of these algorithms on simulated and real-world production data. Our results confirm that it is not always necessary to implement complicated algorithms in the real-world and that very good practical results can be obtained by using heuristics that are backed by the confidence of concrete theoretical guarantees

Time-dependent transonic flow solutions for axial turbomachinery

Three-dimensional unsteady transonic flow through an axial turbomachine stage is described in terms of a pair of two-dimensional formulations pertaining to orthogonal surfaces, namely, a blade-to-blade surface and a hub-to-casing surface. The resulting systems of nonlinear, inviscid, compressible equations of motion are solved by an explicit finite-difference technique. The blade-to-blade program includes the periodic interaction between rotor and stator blade rows. Treatment of the boundary conditions and of the blade slipstream motion by a characteristic type procedure is discussed in detail. Harmonic analysis of the acoustic far field produced by the blade row interaction, including an arbitrary initial transient, is outlined. Results from the blade-to-blade program are compared with experimental measurements of the rotating pressure field at the tip of a high-speed fan. The hub-to-casing program determines circumferentially averaged flow properties on a meridional plane. Blade row interactions are neglected in this formulation, but the force distributions over the entire blade surface for both the rotor and stator are obtained. Results from the hub-to-casing program are compared with a relaxation method solution for a subsonic rotor. Results are also presented for a quiet fan stage which includes transonic flow in both the rotor and stator and a normal shock in the stator

Evolving Network With Different Edges

We proposed an evolving network model constituted by the same nodes but different edges. The competition between nodes and different links were introduced. Scale free properties have been found in this model by continuum theory. Different network topologies can be generated by some tunable parameters. Simulation results consolidate the prediction.Comment: 14 pages, 9 figures, some contents revised, fluctuation of x degree adde

Scaling of load in communications networks

We show that the load at each node in a preferential attachment network scales as a power of the degree of the node. For a network whose degree distribution is p(k) ~ k^(-gamma), we show that the load is l(k) ~ k^eta with eta = gamma - 1, implying that the probability distribution for the load is p(l) ~ 1/l^2 independent of gamma. The results are obtained through scaling arguments supported by finite size scaling studies. They contradict earlier claims, but are in agreement with the exact solution for the special case of tree graphs. Results are also presented for real communications networks at the IP layer, using the latest available data. Our analysis of the data shows relatively poor power-law degree distributions as compared to the scaling of the load versus degree. This emphasizes the importance of the load in network analysis.Comment: 4 pages, 5 figure

Correlation, Network and Multifractal Analysis of Global Financial Indices

We apply RMT, Network and MF-DFA methods to investigate correlation, network and multifractal properties of 20 global financial indices. We compare results before and during the financial crisis of 2008 respectively. We find that the network method gives more useful information about the formation of clusters as compared to results obtained from eigenvectors corresponding to second largest eigenvalue and these sectors are formed on the basis of geographical location of indices. At threshold 0.6, indices corresponding to Americas, Europe and Asia/Pacific disconnect and form different clusters before the crisis but during the crisis, indices corresponding to Americas and Europe are combined together to form a cluster while the Asia/Pacific indices forms another cluster. By further increasing the value of threshold to 0.9, European countries France, Germany and UK constitute the most tightly linked markets. We study multifractal properties of global financial indices and find that financial indices corresponding to Americas and Europe almost lie in the same range of degree of multifractality as compared to other indices. India, South Korea, Hong Kong are found to be near the degree of multifractality of indices corresponding to Americas and Europe. A large variation in the degree of multifractality in Egypt, Indonesia, Malaysia, Taiwan and Singapore may be a reason that when we increase the threshold in financial network these countries first start getting disconnected at low threshold from the correlation network of financial indices. We fit Binomial Multifractal Model (BMFM) to these financial markets.Comment: 32 pages, 25 figures, 1 tabl

Scaling Invariance in Spectra of Complex Networks: A Diffusion Factorial Moment Approach

A new method called diffusion factorial moment (DFM) is used to obtain scaling features embedded in spectra of complex networks. For an Erdos-Renyi network with connecting probability $p_{ER} < \frac{1}{N}$, the scaling parameter is $\delta = 0.51$, while for $p_{ER} \ge \frac{1}{N}$ the scaling parameter deviates from it significantly. For WS small-world networks, in the special region $p_r \in [0.05,0.2]$, typical scale invariance is found. For GRN networks, in the range of $\theta\in[0.33,049]$, we have $\delta=0.6\pm 0.1$. And the value of $\delta$ oscillates around $\delta=0.6$ abruptly. In the range of $\theta\in[0.54,1]$, we have basically $\delta>0.7$. Scale invariance is one of the common features of the three kinds of networks, which can be employed as a global measurement of complex networks in a unified way.Comment: 6 pages, 8 figures. to appear in Physical Review

Random Topologies and the emergence of cooperation: the role of short-cuts

We study in detail the role of short-cuts in promoting the emergence of cooperation in a network of agents playing the Prisoner's Dilemma Game (PDG). We introduce a model whose topology interpolates between the one-dimensional euclidean lattice (a ring) and the complete graph by changing the value of one parameter (the probability p to add a link between two nodes not already connected in the euclidean configuration). We show that there is a region of values of p in which cooperation is largely enhanced, whilst for smaller values of p only a few cooperators are present in the final state, and for p \rightarrow 1- cooperation is totally suppressed. We present analytical arguments that provide a very plausible interpretation of the simulation results, thus unveiling the mechanism by which short-cuts contribute to promote (or suppress) cooperation

A Quantum Lovasz Local Lemma

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of $\Omega(2^k/k^2)$. Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.Comment: 19 page

Opinion diversity and community formation in adaptive networks

It is interesting and of significant importance to investigate how network structures co-evolve with opinions. The existing models of such co-evolution typically lead to the final states where network nodes either reach a global consensus or break into separated communities, each of which holding its own community consensus. Such results, however, can hardly explain the richness of real-life observations that opinions are always diversified with no global or even community consensus, and people seldom, if not never, totally cut off themselves from dissenters. In this article, we show that, a simple model integrating consensus formation, link rewiring and opinion change allows complex system dynamics to emerge, driving the system into a dynamic equilibrium with co-existence of diversified opinions. Specifically, similar opinion holders may form into communities yet with no strict community consensus; and rather than being separated into disconnected communities, different communities remain to be interconnected by non-trivial proportion of inter-community links. More importantly, we show that the complex dynamics may lead to different numbers of communities at steady state with a given tolerance between different opinion holders. We construct a framework for theoretically analyzing the co-evolution process. Theoretical analysis and extensive simulation results reveal some useful insights into the complex co-evolution process, including the formation of dynamic equilibrium, the phase transition between different steady states with different numbers of communities, and the dynamics between opinion distribution and network modularity, etc.Comment: 12 pages, 8 figures, Journa