Enumerating Range Modes

Given a sequence of elements, we consider the problem of indexing the sequence to support range mode queries - given a query range, find the element with maximum frequency in the range. We give indexing data structures for this problem; given a sequence, we construct a data structure that can be used later to process arbitrary queries. Our algorithms are efficient for small maximum frequency cases. We also consider a natural generalization of the problem: the range mode enumeration problem, for which there has been no known efficient algorithms. Our algorithms have query time complexities which are linear in the output size plus small terms

SU(n)SU(n) symmetry breaking by rank three and rank two antisymmetric tensor scalars

We study $SU(n)$ symmetry breaking by rank three and rank two antisymmetric tensor fields. Using tensor analysis, we derive branching rules for the adjoint and antisymmetric tensor representations, and explain why for general $SU(n)$ one finds the same $U(1)$ generator mismatch that we noted earlier in special cases. We then compute the masses of the various scalar fields in the branching expansion, in terms of parameters of the general renormalizable potential for the antisymmetric tensor fields.Comment: Latex, 11 pages; v2 has a minor revision above Eq. (30

Mapping local Hamiltonians of fermions to local Hamiltonians of spins

We show how to map local fermionic problems onto local spin problems on a lattice in any dimension. The main idea is to introduce auxiliary degrees of freedom, represented by Majorana fermions, which allow us to extend the Jordan-Wigner transformation to dimensions higher than one. We also discuss the implications of our results in the numerical investigation of fermionic systems.Comment: Added explicit mappin

Analysis of Nonlinear Synchronization Dynamics of Oscillator Networks by Laplacian Spectral Methods

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronizationat low coupling noted elsewhere is associated with high-eigenvalue modes.Comment: Revised version: References added, introduction rewritten, additional minor changes for clarit

Linear-Space Data Structures for Range Mode Query in Arrays

A mode of a multiset $S$ is an element $a \in S$ of maximum multiplicity; that is, $a$ occurs at least as frequently as any other element in $S$. Given a list $A[1:n]$ of $n$ items, we consider the problem of constructing a data structure that efficiently answers range mode queries on $A$. Each query consists of an input pair of indices $(i, j)$ for which a mode of $A[i:j]$ must be returned. We present an $O(n^{2-2\epsilon})$-space static data structure that supports range mode queries in $O(n^\epsilon)$ time in the worst case, for any fixed $\epsilon \in [0,1/2]$. When $\epsilon = 1/2$, this corresponds to the first linear-space data structure to guarantee $O(\sqrt{n})$ query time. We then describe three additional linear-space data structures that provide $O(k)$, $O(m)$, and $O(|j-i|)$ query time, respectively, where $k$ denotes the number of distinct elements in $A$ and $m$ denotes the frequency of the mode of $A$. Finally, we examine generalizing our data structures to higher dimensions.Comment: 13 pages, 2 figure