16,853 research outputs found
Theoretical and Experimental Analysis of a Randomized Algorithm for Sparse Fourier Transform Analysis
We analyze a sublinear RAlSFA (Randomized Algorithm for Sparse Fourier
Analysis) that finds a near-optimal B-term Sparse Representation R for a given
discrete signal S of length N, in time and space poly(B,log(N)), following the
approach given in \cite{GGIMS}. Its time cost poly(log(N)) should be compared
with the superlinear O(N log N) time requirement of the Fast Fourier Transform
(FFT). A straightforward implementation of the RAlSFA, as presented in the
theoretical paper \cite{GGIMS}, turns out to be very slow in practice. Our main
result is a greatly improved and practical RAlSFA. We introduce several new
ideas and techniques that speed up the algorithm. Both rigorous and heuristic
arguments for parameter choices are presented. Our RAlSFA constructs, with
probability at least 1-delta, a near-optimal B-term representation R in time
poly(B)log(N)log(1/delta)/ epsilon^{2} log(M) such that
||S-R||^{2}<=(1+epsilon)||S-R_{opt}||^{2}. Furthermore, this RAlSFA
implementation already beats the FFTW for not unreasonably large N. We extend
the algorithm to higher dimensional cases both theoretically and numerically.
The crossover point lies at N=70000 in one dimension, and at N=900 for data on
a N*N grid in two dimensions for small B signals where there is noise.Comment: 21 pages, 8 figures, submitted to Journal of Computational Physic
A local moment approach to the degenerate Anderson impurity model
The local moment approach is extended to the orbitally-degenerate [SU(2N)]
Anderson impurity model (AIM). Single-particle dynamics are obtained over the
full range of energy scales, focussing here on particle-hole symmetry in the
strongly correlated regime where the onsite Coulomb interaction leads to
many-body Kondo physics with entangled spin and orbital degrees of freedom. The
approach captures many-body broadening of the Hubbard satellites, recovers the
correct exponential vanishing of the Kondo scale for all N, and its universal
scaling spectra are found to be in very good agreement with numerical
renormalization group (NRG) results. In particular the high-frequency
logarithmic decays of the scaling spectra, obtained here in closed form for
arbitrary N, coincide essentially perfectly with available numerics from the
NRG. A particular case of an anisotropic Coulomb interaction, in which the
model represents a system of N `capacitively-coupled' SU(2) AIMs, is also
discussed. Here the model is generally characterised by two low-energy scales,
the crossover between which is seen directly in its dynamics.Comment: 23 pages, 7 figure
List decoding of noisy Reed-Muller-like codes
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are
two fundamental error-correcting codes which arise in communication as well as
in probabilistically-checkable proofs and learning. In this paper, we take the
first steps toward extending the quick randomized decoding tools of RM(1) into
the realm of quadratic binary and, equivalently, Z_4 codes. Our main
algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin
and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and
RM(2). That is, given signal s of length N, we find a list that is a superset
of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times
the norm of s, in time polynomial in k and log(N). We also give a new and
simple formulation of a known Kerdock code as a subcode of the Hankel code. As
a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm
for finding a sparse Kerdock approximation. That is, for k small compared with
1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k
log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at
most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such
approximation
Approximate Sparse Recovery: Optimizing Time and Measurements
An approximate sparse recovery system consists of parameters , an
-by- measurement matrix, , and a decoding algorithm, .
Given a vector, , the system approximates by , which must satisfy , where denotes the optimal -term approximation to . For
each vector , the system must succeed with probability at least 3/4. Among
the goals in designing such systems are minimizing the number of
measurements and the runtime of the decoding algorithm, .
In this paper, we give a system with
measurements--matching a lower bound, up to a constant factor--and decoding
time , matching a lower bound up to factors.
We also consider the encode time (i.e., the time to multiply by ),
the time to update measurements (i.e., the time to multiply by a
1-sparse ), and the robustness and stability of the algorithm (adding noise
before and after the measurements). Our encode and update times are optimal up
to factors
Finitely presented subgroups of automatic groups and their isoperimetric functions
We describe a general technique for embedding certain amalgamated products
into direct products. This technique provides us with a way of constructing a
host of finitely presented subgroups of automatic groups which are not even
asynchronously automatic. We can also arrange that such subgroups satisfy, at
best, an exponential isoperimetric inequality.Comment: DVI and Post-Script files only. To appear in J. London Math. So
Non-Symbolic Fragmentation
This paper reports on the use of non-symbolic fragmentation of data for securing communications. Non-symbolic fragmentation, or NSF, relies on breaking up data into non-symbolic fragments, which are (usually irregularly-sized) chunks whose boundaries do not necessarily coincide with the boundaries of the symbols making up the data. For example, ASCII data is broken up into fragments which may include 8-bit fragments but also include many other sized fragments. Fragments are then separated with a form of path diversity. The secrecy of the transmission relies on the secrecy of one or more of a number of things: the ordering of the fragments, the sizes of the fragments, and the use of path diversity. Once NSF is in place, it can help secure many forms of communication, and is useful for exchanging sensitive information, and for commercial transactions. A sample implementation is described with an evaluation of the technology
Berkeley's social philosophy
Thesis (M.A.)--Boston University, 1939. This item was digitized by the Internet Archive
Studies of Ionic Interactions in Solutions
The object of the present work was to examine the interactions which occur in aqueous solutions which contain alkaline earth cations and anions of dicarboxylic acids. The calcium-oxalate system was to be examined in detail with special reference to the protonated complexes formed by the species hydrogen oxalate. In order to examine the calcium hydrogen oxalate system, reliable values of the dissociation constants of oxalic acid were required so that the concentrations of anionic species in the solutions could be calculated. The thesis is divided into three parts. The first part describes the determination of the dissociation constants of oxalic acid in an ionic medium maintained at 0.1 molar by sodium perchlorate. A potentiometric titration method has been used and the results of the titrations are compared with earlier work on the dissociation constants of oxalic acid. Good agreement has been obtained. The second and principal part of the thesis involves the determination of the stability constants of a number of complexes formed between alkaline earth cations and anionic dicarboxylic acid ligands. The cation exchange method has been used and the theory of the method is described. The experimental difficulties which were encountered are introduced and the methods which were used to overcome these difficulties are described. The systems which were examined were the following: calcium oxalate, calcium malonate, calcium succinate, calcium tartrate, strontium oxalate, strontium tartrate, calcium hydrogen oxalate and strontium hydrogen oxalate. Where comparison with other published work was possible, good agreement was found. A critical comparison was made of the stabilities of corresponding acetate and hydrogen oxalate complexes and use was made of such other data in the literature as were relevant. The denticity of the hydrogen oxalate ligand was examined and it was deduced that bidenticity was possible. Dissociation of the hydrogen oxalate complexes to the unprotonated complex and a hydrogen ion was considered in the light of a recent paper by some Russian workers and it was found that the empirical relationship which they had suggested was obeyed. The relative stabilities of all the complexes with the exception of the protonated complexes were considered in terms mainly of entropy contributions to the total free energy change in the reactions. The effect of the cross linking of the cation exchanger on the stability constants obtained by the ion exchange method was examined in the third part of the thesis. A large number of experiments were performed on resins with cross linking of 8%, 10% and 12% DVB. The results of the experiments were not conclusive because excellent agreement was observed between the results obtained for the 8% and 10% cross linked resins, but the 12% resin appeared to be out of line, but to an inconclusive extent. As far as can be seen this is in agreement with some earlier unpublished work on the same topic
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