On the relation between K-means and PLSA

Non-negative matrix factorization [5](NMF) is a well known tool for unsupervised machine learning. It can be viewed as a generalization of the K-means clustering, Expectation Maximization based clustering and aspect modeling by Probabilistic Latent Semantic Analysis (PLSA). Specifically PLSA is related to NMF with KL-divergence objective function. Further it is shown that K-means clustering is a special case of NMF with matrix L2 norm based error function. In this paper our objective is to analyze the relation between K-means clustering and PLSA by examining the KL-divergence function and matrix L2 norm based error function.

On the SIG-dimension of trees under the L (a)-metric

Let where be a set of points in d-dimensional space with a given metric rho. For a point let r (p) be the distance of p with respect to rho from its nearest neighbor in Let B(p,r (p) ) be the open ball with respect to rho centered at p and having the radius r (p) . We define the sphere-of-influence graph (SIG) of as the intersection graph of the family of sets Given a graph G, a set of points in d-dimensional space with the metric rho is called a d-dimensional SIG-representation of G, if G is isomorphic to the SIG of It is known that the absence of isolated vertices is a necessary and sufficient condition for a graph to have a SIG-representation under the L (a)-metric in some space of finite dimension. The SIG-dimension under the L (a)-metric of a graph G without isolated vertices is defined to be the minimum positive integer d such that G has a d-dimensional SIG-representation under the L (a)-metric. It is denoted by SIG (a)(G). We study the SIG-dimension of trees under the L (a)-metric and almost completely answer an open problem posed by Michael and Quint (Discrete Appl Math 127:447-460, 2003). Let T be a tree with at least two vertices. For each let leaf-degree(v) denote the number of neighbors of v that are leaves. We define the maximum leaf-degree as leaf-degree(x). Let leaf-degree{(v) = alpha}. If |S| = 1, we define beta(T) = alpha(T) - 1. Otherwise define beta(T) = alpha(T). We show that for a tree where beta = beta (T), provided beta is not of the form 2 (k) - 1, for some positive integer k a parts per thousand yen 1. If beta = 2 (k) - 1, then We show that both values are possible.

The projective plane, J-holomorphic curves and Desargues' theorem

By a theorem of Gromov, for an almost complex structure J on CP2 tamed by the standard symplectic structure, the J-holomorphic curves representing the positive generator of homology form a projective plane. We show that this satisfies the Theorem of Desargues if and only if J is isomorphic to the standard complex structure. This answers a question of Ghys. (C) 2013 Published by Elsevier Masson SAS on behalf of Academie des sciences.

Dynamical symmetry of screened Coulomb potential and isotropic harmonic oscillator

It is shown that for the screened Coulomb potential and isotropic harmonic oscillator, there exists an infinite number of closed orbits for suitable angular momentum values. At the aphelion (perihelion) points of classical orbits, an extended Runge-Lenz vector for the screened Coulomb potential and an extended quadrupole tensor for the screened isotropic harmonic oscillator are still conserved. For the screened two-dimensional (2D) Coulomb potential and isotropic harmonic oscillator, the dynamical symmetries SO3 and SU(2) are still preserved at the aphelion (perihelion) points of classical orbits, respectively. For the screened 3D Coulomb potential, the dynamical symmetry SO4 is also preserved at the aphelion (perihelion) points of classical orbits. But for the screened 3D isotropic harmonic oscillator, the dynamical symmetry SU(2) is only preserved at the aphelion (perihelion) points of classical orbits in the eigencoordinate system. For the screened Coulomb potential and isotropic harmonic oscillator, only the energy (but not angular momentum) raising and lowering operators can be constructed from a factorization of the radial Schrodinger equation.

Transport Phenomena in Ammonothermal Growth of Gallium Nitride

GaN can be used to fabricate blue/green/UV LEDs and high temperature, high power electronic devices. Ideal substrates are needed for high quality III-nitride epitaxy, which is an essential step for the manufacture of LEDs. GaN substrates are ideal to be lattice matched and isomorphic to nitride-based films. Bulk single crystals of GaN can be grown from supercritical fluids using the ammonothermal method, which utilizes ammonia as fluid rather than water as in the hydrothermal process. In this process, a mineralizer such as amide, imide or azide is used to attack a bulk nitride feedstock at temperatures from 200 - 500癈 and pressures from 1 - 4 kbar. Baffle design is essential for successful growth of GaN crystals. Baffle is used to separate the dissolving zone from the growth zone, and to maintain a temperature difference between the two zones. For solubility curve with a positive coefficient with respect to temperature, the growth zone is maintained at a lower temperature than that in the dissolving zone, thus the nutrient becomes supersaturated in the growth zone. The baffle opening is used to control the mixing of nutrients in the two zones, thus the transfer of nutrient from the lower part to the upper part. Ammonothermal systems have been modeled here using fluid dynamics, thermodynamics and heat transfer models. The nutrient is considered as a porous media bed and the flow is simulated using the Darcy-Brinkman-Forchheimer model. The resulting governing equations are solved using the finite volume method. We investigated the effects of baffle opening and position on the transport phenomena of nutrient from dissolving zone to the growth zone. Simulation data have been compared qualitatively with experimental data.

Studies of ambient and chamber aerosol composition using the aerosol mass spectrometer

This thesis presents composition measurements for atmospherically relevant inorganic and organic aerosol from laboratory and ambient measurements using the Aerodyne aerosol mass spectrometer. Studies include the oxidation of dodecane in the Caltech environmental chambers, and several aircraft- and ground-based field studies, which include the quantification of wildfire emissions off the coast of California, and Los Angeles urban emissions.

The oxidation of dodecane by OH under low NO conditions and the formation of secondary organic aerosol (SOA) was explored using a gas-phase chemical model, gas-phase CIMS measurements, and high molecular weight ion traces from particle- phase HR-TOF-AMS mass spectra. The combination of these measurements support the hypothesis that particle-phase chemistry leading to peroxyhemiacetal formation is important. Positive matrix factorization (PMF) was applied to the AMS mass spectra which revealed three factors representing a combination of gas-particle partitioning, chemical conversion in the aerosol, and wall deposition.

Airborne measurements of biomass burning emissions from a chaparral fire on the central Californian coast were carried out in November 2009. Physical and chemical changes were reported for smoke ages 0 – 4 h old. CO₂ normalized ammonium, nitrate, and sulfate increased, whereas the normalized OA decreased sharply in the first 1.5 - 2 h, and then slowly increased for the remaining 2 h (net decrease in normalized OA). Comparison to wildfire samples from the Yucatan revealed that factors such as relative humidity, incident UV radiation, age of smoke, and concentration of emissions are important for wildfire evolution.

Ground-based aerosol composition is reported for Pasadena, CA during the summer of 2009. The OA component, which dominated the submicron aerosol mass, was deconvolved into hydrocarbon-like organic aerosol (HOA), semi-volatile oxidized organic aerosol (SVOOA), and low-volatility oxidized organic aerosol (LVOOA). The HOA/OA was only 0.08–0.23, indicating that most of Pasadena OA in the summer months is dominated by oxidized OA resulting from transported emissions that have undergone photochemistry and/or moisture-influenced processing, as apposed to only primary organic aerosol emissions. Airborne measurements and model predictions of aerosol composition are reported for the 2010 CalNex field campaign.

Diagonal forms, linear algebraic methods and Ramsey-type problems

This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.

As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.

One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.

Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.

Ordinary mod p representations of the metaplectic cover of p-adic SL_2

We classify the genuine ordinary mod p representations of the metaplectic group SL(2,F)-tilde, where F is a p-adic field, and compute its genuine mod p spherical and Iwahori Hecke algebras. The motivation is an interest in a possible correspondence between genuine mod p representations of SL(2,F)-tilde and mod p representations of the dual group PGL(2,F), so we also compare the two Hecke algebras to the mod p spherical and Iwahori Hecke algebras of PGL(2,F). We show that the genuine mod p spherical Hecke algebra of SL(2,F)-tilde is isomorphic to the mod p spherical Hecke algebra of PGL(2,F), and that one can choose an isomorphism which is compatible with a natural, though partial, correspondence of unramified ordinary representations via the Hecke action on their spherical vectors. We then show that the genuine mod p Iwahori Hecke algebra of SL(2,F)-tilde is a subquotient of the mod p Iwahori Hecke algebra of PGL(2,F), but that the two algebras are not isomorphic. This is in contrast to the situation in characteristic 0, where by work of Savin one can recover the local Shimura correspondence for representations generated by their Iwahori fixed vectors from an isomorphism of Iwahori Hecke algebras.

Geometric quantization and foliation reduction

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kähler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kähler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.

In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kähler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or “admissible”, values of momentum.

We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.

We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.

Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.

A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.

Rings faithfully represented on their left socle

In 1964 A. W. Goldie [1] posed the problem of determining all rings with identity and minimal condition on left ideals which are faithfully represented on the right side of their left socle. Goldie showed that such a ring which is indecomposable and in which the left and right principal indecomposable ideals have, respectively, unique left and unique right composition series is a complete blocked triangular matrix ring over a skewfield. The general problem suggested above is very difficult. We obtain results under certain natural restrictions which are much weaker than the restrictive assumptions made by Goldie.

We characterize those rings in which the principal indecomposable left ideals each contain a unique minimal left ideal (Theorem (4.2)). It is sufficient to handle indecomposable rings (Lemma (1.4)). Such a ring is also a blocked triangular matrix ring. There exist r positive integers K₁,..., K_r such that the i,j^th block of a typical matrix is a K_i x K_j matrix with arbitrary entries in a subgroup D_ij of the additive group of a fixed skewfield D. Each D_ii is a sub-skewfield of D and D_ri = D for all i. Conversely, every matrix ring which has this form is indecomposable, faithfully represented on the right side of its left socle, and possesses the property that every principal indecomposable left ideal contains a unique minimal left ideal.

The principal indecomposable left ideals may have unique composition series even though the ring does not have minimal condition on right ideals. We characterize this situation by defining a partial ordering ρ on {i, 2,...,r} where we set iρj if D_ij ≠ 0. Every principal indecomposable left ideal has a unique composition series if and only if the diagram of ρ is an inverted tree and every D_ij is a one-dimensional left vector space over D_ii (Theorem (5.4)).

We show (Theorem (2.2)) that every ring A of the type we are studying is a unique subdirect sum of less complex rings A₁,...,A_s of the same type. Namely, each A_i has only one isomorphism class of minimal left ideals and the minimal left ideals of different A_i are non-isomorphic as left A-modules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings A_i having these properties to be faithfully represented on the right side of its left socle. We show ((4.F), p. 42) that up to technical trivia the rings A_i are matrix rings of the form

[...]. Each Q_j comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D. The bottom row is filled in by arbitrary elements of D.

In Part V we construct an interesting class of rings faithfully represented on their left socle from a given partial ordering on a finite set, given skewfields, and given additive groups. This class of rings contains the ones in which every principal indecomposable left ideal has a unique minimal left ideal. We identify the uniquely determined subdirect summands mentioned above in terms of the given partial ordering (Proposition (5.2)). We conjecture that this technique serves to construct all the rings which are a unique subdirect sum of rings each having the property that every principal-indecomposable left ideal contains a unique minimal left ideal.

Class two p groups as fixed point free automorphism groups

Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p^c ≠ r^d + 1 for any c = 1, 2 and any prime r where r^2d+1 divides |G| and if C_G(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.

The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A₁, a subgroup of A, where A₁ centralizes D(R), then all irreducible characters of A₁R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.

Decoding cosets of first-order Reed-Muller code

Proper encoding of transmitted information can improve the performance of a communication system. To recover the information at the receiver it is necessary to decode the received signal. For many codes the complexity and slowness of the decoder is so severe that the code is not feasible for practical use. This thesis considers the decoding problem for one such class of codes, the comma-free codes related to the first-order Reed-Muller codes.

A factorization of the code matrix is found which leads to a simple, fast, minimum memory, decoder. The decoder is modular and only n modules are needed to decode a code of length 2ⁿ. The relevant factorization is extended to any code defined by a sequence of Kronecker products.

The problem of monitoring the correct synchronization position is also considered. A general answer seems to depend upon more detailed knowledge of the structure of comma-free codes. However, a technique is presented which gives useful results in many specific cases.

Implementação paralela de um código de elementos finitos em 2D para as Equações de Navier-Stokes para fluidos incompressíveis com transporte de escalares.

O estudo do fluxo de água e do transporte escalar em reservatórios hidrelétricos é importante para a determinação da qualidade da água durante as fases iniciais do enchimento e durante a vida útil do reservatório. Neste contexto, um código de elementos finitos paralelo 2D foi implementado para resolver as equações de Navier-Stokes para fluido incompressível acopladas a transporte escalar, utilizando o modelo de programação de troca de mensagens, a fim de realizar simulações em um ambiente de cluster de computadores. A discretização espacial é baseada no elemento MINI, que satisfaz as condições de Babuska-Brezzi (BB), que permite uma formulação mista estável. Todas as estruturas de dados distribuídos necessárias nas diferentes fases do código, como pré-processamento, solução e pós-processamento, foram implementadas usando a biblioteca PETSc. Os sistemas lineares resultantes foram resolvidos usando o método da projeção discreto com fatoração LU por blocos. Para aumentar o desempenho paralelo na solução dos sistemas lineares, foi empregado o método de condensação estática para resolver a velocidade intermediária nos vértices e no centróide do elemento MINI separadamente. Os resultados de desempenho do método de condensação estática com a abordagem da solução do sistema completo foram comparados. Os testes mostraram que o método de condensação estática apresenta melhor desempenho para grandes problemas, às custas de maior uso de memória. O desempenho de outras partes do código também são apresentados.

Tutorial on loop shaping using H-infinity robust stabilization

A new method for multivariable control was proposed in McFarlane and Glover (1988 CDC). This method involved shaping the open-loop singular values with pre and/or post compensators and then designing a controller to robustly stabilize a normalized coprime factorization of this weighted plant. The method has many attractive features including guaranteed loop shape and robust stability and performance. This talk will outline the rationale of this method and illustrate its use on a number of applications.

Robust control of plants with saturation nonlinearity based on coprime factor representations

We consider the robust control of plants with saturation nonlinearities from an input/output viewpoint. First, we present a parameterization for anti-windup control based on coprime factorizations of the controller. Second, we propose a synthesis method which exploits the freedom to choose a particular coprime factorization.