Geometric Representation of Graphs in Low Dimension Using Axis Parallel Boxes

An axis-parallel k-dimensional box is a Cartesian product R-1 x R-2 x...x R-k where R-i (for 1 <= i <= k) is a closed interval of the form [a(i), b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension k, such that G is representable as the intersection graph of (axis-parallel) boxes in k-dimensional space. The concept of boxicity finds applications in various areas such as ecology, operations research etc. A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs. For example, the max-clique problem is polynomial time solvable on bounded boxicity graphs and the maximum independent set problem for boxicity d graphs, given a box representation, has a left perpendicular1 + 1/c log n right perpendicular(d-1) approximation ratio for any constant c >= 1 when d >= 2. In most cases, the first step usually is computing a low dimensional box representation of the given graph. Deciding whether the boxicity of a graph is at most 2 itself is NP-hard. We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in left perpendicular(Delta + 2) ln nright perpendicular dimensions, where Delta is the maximum degree of G. This algorithm implies that box(G) <= left perpendicular(Delta + 2) ln nright perpendicular for any graph G. Our bound is tight up to a factor of ln n. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree Delta, we show that for almost all graphs on n vertices, their boxicity is O(d(av) ln n) where d(av) is the average degree.

Geometric representations of graphs in low dimension

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.

Coherent propagation and energy transfer in low-dimension nonlinear arrays

We present a theory of coherent propagation and energy or power transfer in a low-dimension array of coupled nonlinear waveguides. It is demonstrated that in the array with nonequal cores (e.g., with the central core) stable steady-state coherent multicore propagation is possible only in the nonlinear regime, with a power-controlled phase matching. The developed theory of energy or power transfer in nonlinear discrete systems is rather generic and has a range of potential applications including both high-power fiber lasers and ultrahigh-capacity optical communication systems. © 2012 American Physical Society.

On homotopy spheres of low dimension

Let M be a smooth compact manifold homotopy equivalent to the 4-sphere S^4. Then M x R^1 is homeomorphism co S^4 x R^1.

ELECTRON TRANSPORT IN LOW-DIMENSIONAL NANOSTRUCTURES - THEORETICAL STUDY WITH APPLICATION

Graphene as a carbon monolayer has attracted extensive research interest in recent years. My research work within the frame of density functional theory has suggested that positioning graphene in proximity to h-BN may induce a finite energy gap in graphene, which is important for device applications. For an AB-stacked graphene/BN bilayer, a finite gap is induced at the equilibrium configuration. This induced gap shows a linear relationship with the applied strain. For a graphene/BN/graphene trilayer, a negligible gap is predicted in the ground state due to the overall symmetry of the system. When an electric field is applied, a tunable gap can be obtained for both AAA and ABA stackings. Enhanced tunneling current in the AA-stacked bilayer nanoribbons is predicted compared to either single-layer or AB-stacked bilayer nanoribbons. Interlayer separation between the nanoribbons is shown to have a profound impact on the conducting features. The effect of boron or nitrogen doping on the electronic transport properties of C60 fullerene is studied. The BC59 fullerene exhibits a considerably higher current than the pristine or nitrogen doped fullerenes beyond the applied bias of 1 V, suggesting it can be an effective semiconductor in p-type devices. The interaction between nucleic acid bases - adenine (A), guanine (G), cytosine (C), thymine (T) and uracil (U) - and a hydrogen-passivated silicon nanowire (SiNW) is investigated. The binding energy of the bases with the SiNW shows the order: G > A~C~T~U. This suggests that the interaction strength of a hydrogen passivated SiNW with the nucleic acid bases is nearly the same-G being an exception. The nature of the interaction is suggested to be electrostatic.

Modified Greedy Pursuits for Improving Sparse Recovery

Compressive Sensing (CS) theory combines the signal sampling and compression for sparse signals resulting in reduction in sampling rate. In recent years, many recovery algorithms have been proposed to reconstruct the signal efficiently. Subspace Pursuit and Compressive Sampling Matching Pursuit are some of the popular greedy methods. Also, Fusion of Algorithms for Compressed Sensing is a recently proposed method where several CS reconstruction algorithms participate and the final estimate of the underlying sparse signal is determined by fusing the estimates obtained from the participating algorithms. All these methods involve solving a least squares problem which may be ill-conditioned, especially in the low dimension measurement regime. In this paper, we propose a step prior to least squares to ensure the well-conditioning of the least squares problem. Using Monte Carlo simulations, we show that in low dimension measurement scenario, this modification improves the reconstruction capability of the algorithm in clean as well as noisy measurement cases.

Study of rapid carrier capture and relaxation in InAs/GaAs heterostructures

The rapid carrier capture and relaxation processes in InAs/GaAs quantum dots were studied at 77K by using a simple degenerate pump-probe technique. A rising process was observed in the transient reflectivity, following the initial fast relaxation associated with GaAs bulk matrix, and this rising process was assigned to be related to the carrier capture from the GaAs barriers to InAs layers. The assignment was modeled using Kramers-Kronig relation. By analyzing the rising process observed in the transient reflectivity, the carrier capture time constants were obtained. The measured capture times decrease with the increase of carrier concentration.

改善三维波动方程叠前深度偏移处理效果的方法研究

3D wave equation prestack depth migration is the effective tool for obtaining the exact imaging result of complex geology structures. It's a part of the 3D seismic data processing. 3D seismic data processing belongs to high dimension signal processing, and there are some difficult problems to do with. They are: How to process high dimension operators? How to improve the focusing? and how to construct the deconvolution operator? The realization of 3D wave equation prestack depth migration, not only realized the leap from poststack to prestack, but also provided the important means to solve the difficult problems in high dimension signal processing. In this thesis, I do a series research especially for the solve of the difficult problems around the 3D wave equation prestack depth migration and using it as a mean. So this thesis service for the realization of 3D wave equation prestack depth migration for one side and improve the migration effect for another side. This thesis expatiates in five departs. Summarizes the main contents as the follows: In the first part, I have completed the projection from 3D data point area to low dimension are using de big matrix transfer and trace rearrangement, and realized the liner processing of high dimension signal. Firstly, I present the mathematics expression of 3D seismic data and the mean according to physics, present the basic ideal of big matrix transfer and describe the realization of five transfer models for example. Secondly, I present the basic ideal and rules for the rearrange and parallel calculate of 3D traces, and give a example. In the conventional DMO focusing method, I recall the history of DM0 process firstly, give the fundamental of DMO process and derive the equation of DMO process and it's impulse response. I also prove the equivalence between DMO and prestack time migration, from the kinematic character of DMO. And derive the relationship between DMO base on wave equation and prestack time migration. Finally, I give the example of DMO process flow and synthetic data of theoretical models. In the wave equation prestak depth migration, I firstly recall the history of migration from time to depth, from poststack to prestack and from 2D to 3D. And conclude the main migration methods, point out their merit and shortcoming. Finally, I obtain the common image point sets using the decomposed migration program code.In the residual moveout, I firstly describe the Viterbi algorithm based on Markov process and compound decision theory and how to solve the shortest path problem using Viterbi algorithm. And based on this ideal, I realized the residual moveout of post 3D wave equation prestack depth migration. Finally, I give the example of residual moveout of real 3D seismic data. In the migration Green function, I firstly give the concept of migration Green function and the 2D Green function migration equation for the approximate of far field. Secondly, I prove the equivalence of wave equation depth extrapolation algorithms. And then I derive the equation of Green function migration. Finally, I present the response and migration result of Green function for point resource, analyze the effect of migration aperture to prestack migration result. This research is benefit for people to realize clearly the effect of migration aperture to migration result, and study on the Green function deconvolution to improve the focusing effect of migration.

A taxonomy for wavelet neural networks applied to nonlinear modelling

This article presents a novel classification of wavelet neural networks based on the orthogonality/non-orthogonality of neurons and the type of nonlinearity employed. On the basis of this classification different network types are studied and their characteristics illustrated by means of simple one-dimensional nonlinear examples. For multidimensional problems, which are affected by the curse of dimensionality, the idea of spherical wavelet functions is considered. The behaviour of these networks is also studied for modelling of a low-dimension map.

An improved TLBO with elite strategy for parameters identification of PEM fuel cell and solar cell models

Clean and renewable energy generation and supply has drawn much attention worldwide in recent years, the proton exchange membrane (PEM) fuel cells and solar cells are among the most popular technologies. Accurately modeling the PEM fuel cells as well as solar cells is critical in their applications, and this involves the identification and optimization of model parameters. This is however challenging due to the highly nonlinear and complex nature of the models. In particular for PEM fuel cells, the model has to be optimized under different operation conditions, thus making the solution space extremely complex. In this paper, an improved and simplified teaching-learning based optimization algorithm (STLBO) is proposed to identify and optimize parameters for these two types of cell models. This is achieved by introducing an elite strategy to improve the quality of population and a local search is employed to further enhance the performance of the global best solution. To improve the diversity of the local search a chaotic map is also introduced. Compared with the basic TLBO, the structure of the proposed algorithm is much simplified and the searching ability is significantly enhanced. The performance of the proposed STLBO is firstly tested and verified on two low dimension decomposable problems and twelve large scale benchmark functions, then on the parameter identification of PEM fuel cell as well as solar cell models. Intensive experimental simulations show that the proposed STLBO exhibits excellent performance in terms of the accuracy and speed, in comparison with those reported in the literature.

Factor models, VARMA processes and parameter instability with applications in macroeconomics

Avec les avancements de la technologie de l'information, les données temporelles économiques et financières sont de plus en plus disponibles. Par contre, si les techniques standard de l'analyse des séries temporelles sont utilisées, une grande quantité d'information est accompagnée du problème de dimensionnalité. Puisque la majorité des séries d'intérêt sont hautement corrélées, leur dimension peut être réduite en utilisant l'analyse factorielle. Cette technique est de plus en plus populaire en sciences économiques depuis les années 90. Étant donnée la disponibilité des données et des avancements computationnels, plusieurs nouvelles questions se posent. Quels sont les effets et la transmission des chocs structurels dans un environnement riche en données? Est-ce que l'information contenue dans un grand ensemble d'indicateurs économiques peut aider à mieux identifier les chocs de politique monétaire, à l'égard des problèmes rencontrés dans les applications utilisant des modèles standards? Peut-on identifier les chocs financiers et mesurer leurs effets sur l'économie réelle? Peut-on améliorer la méthode factorielle existante et y incorporer une autre technique de réduction de dimension comme l'analyse VARMA? Est-ce que cela produit de meilleures prévisions des grands agrégats macroéconomiques et aide au niveau de l'analyse par fonctions de réponse impulsionnelles? Finalement, est-ce qu'on peut appliquer l'analyse factorielle au niveau des paramètres aléatoires? Par exemple, est-ce qu'il existe seulement un petit nombre de sources de l'instabilité temporelle des coefficients dans les modèles macroéconomiques empiriques? Ma thèse, en utilisant l'analyse factorielle structurelle et la modélisation VARMA, répond à ces questions à travers cinq articles. Les deux premiers chapitres étudient les effets des chocs monétaire et financier dans un environnement riche en données. Le troisième article propose une nouvelle méthode en combinant les modèles à facteurs et VARMA. Cette approche est appliquée dans le quatrième article pour mesurer les effets des chocs de crédit au Canada. La contribution du dernier chapitre est d'imposer la structure à facteurs sur les paramètres variant dans le temps et de montrer qu'il existe un petit nombre de sources de cette instabilité. Le premier article analyse la transmission de la politique monétaire au Canada en utilisant le modèle vectoriel autorégressif augmenté par facteurs (FAVAR). Les études antérieures basées sur les modèles VAR ont trouvé plusieurs anomalies empiriques suite à un choc de la politique monétaire. Nous estimons le modèle FAVAR en utilisant un grand nombre de séries macroéconomiques mensuelles et trimestrielles. Nous trouvons que l'information contenue dans les facteurs est importante pour bien identifier la transmission de la politique monétaire et elle aide à corriger les anomalies empiriques standards. Finalement, le cadre d'analyse FAVAR permet d'obtenir les fonctions de réponse impulsionnelles pour tous les indicateurs dans l'ensemble de données, produisant ainsi l'analyse la plus complète à ce jour des effets de la politique monétaire au Canada. Motivée par la dernière crise économique, la recherche sur le rôle du secteur financier a repris de l'importance. Dans le deuxième article nous examinons les effets et la propagation des chocs de crédit sur l'économie réelle en utilisant un grand ensemble d'indicateurs économiques et financiers dans le cadre d'un modèle à facteurs structurel. Nous trouvons qu'un choc de crédit augmente immédiatement les diffusions de crédit (credit spreads), diminue la valeur des bons de Trésor et cause une récession. Ces chocs ont un effet important sur des mesures d'activité réelle, indices de prix, indicateurs avancés et financiers. Contrairement aux autres études, notre procédure d'identification du choc structurel ne requiert pas de restrictions temporelles entre facteurs financiers et macroéconomiques. De plus, elle donne une interprétation des facteurs sans restreindre l'estimation de ceux-ci. Dans le troisième article nous étudions la relation entre les représentations VARMA et factorielle des processus vectoriels stochastiques, et proposons une nouvelle classe de modèles VARMA augmentés par facteurs (FAVARMA). Notre point de départ est de constater qu'en général les séries multivariées et facteurs associés ne peuvent simultanément suivre un processus VAR d'ordre fini. Nous montrons que le processus dynamique des facteurs, extraits comme combinaison linéaire des variables observées, est en général un VARMA et non pas un VAR comme c'est supposé ailleurs dans la littérature. Deuxièmement, nous montrons que même si les facteurs suivent un VAR d'ordre fini, cela implique une représentation VARMA pour les séries observées. Alors, nous proposons le cadre d'analyse FAVARMA combinant ces deux méthodes de réduction du nombre de paramètres. Le modèle est appliqué dans deux exercices de prévision en utilisant des données américaines et canadiennes de Boivin, Giannoni et Stevanovic (2010, 2009) respectivement. Les résultats montrent que la partie VARMA aide à mieux prévoir les importants agrégats macroéconomiques relativement aux modèles standards. Finalement, nous estimons les effets de choc monétaire en utilisant les données et le schéma d'identification de Bernanke, Boivin et Eliasz (2005). Notre modèle FAVARMA(2,1) avec six facteurs donne les résultats cohérents et précis des effets et de la transmission monétaire aux États-Unis. Contrairement au modèle FAVAR employé dans l'étude ultérieure où 510 coefficients VAR devaient être estimés, nous produisons les résultats semblables avec seulement 84 paramètres du processus dynamique des facteurs. L'objectif du quatrième article est d'identifier et mesurer les effets des chocs de crédit au Canada dans un environnement riche en données et en utilisant le modèle FAVARMA structurel. Dans le cadre théorique de l'accélérateur financier développé par Bernanke, Gertler et Gilchrist (1999), nous approximons la prime de financement extérieur par les credit spreads. D'un côté, nous trouvons qu'une augmentation non-anticipée de la prime de financement extérieur aux États-Unis génère une récession significative et persistante au Canada, accompagnée d'une hausse immédiate des credit spreads et taux d'intérêt canadiens. La composante commune semble capturer les dimensions importantes des fluctuations cycliques de l'économie canadienne. L'analyse par décomposition de la variance révèle que ce choc de crédit a un effet important sur différents secteurs d'activité réelle, indices de prix, indicateurs avancés et credit spreads. De l'autre côté, une hausse inattendue de la prime canadienne de financement extérieur ne cause pas d'effet significatif au Canada. Nous montrons que les effets des chocs de crédit au Canada sont essentiellement causés par les conditions globales, approximées ici par le marché américain. Finalement, étant donnée la procédure d'identification des chocs structurels, nous trouvons des facteurs interprétables économiquement. Le comportement des agents et de l'environnement économiques peut varier à travers le temps (ex. changements de stratégies de la politique monétaire, volatilité de chocs) induisant de l'instabilité des paramètres dans les modèles en forme réduite. Les modèles à paramètres variant dans le temps (TVP) standards supposent traditionnellement les processus stochastiques indépendants pour tous les TVPs. Dans cet article nous montrons que le nombre de sources de variabilité temporelle des coefficients est probablement très petit, et nous produisons la première évidence empirique connue dans les modèles macroéconomiques empiriques. L'approche Factor-TVP, proposée dans Stevanovic (2010), est appliquée dans le cadre d'un modèle VAR standard avec coefficients aléatoires (TVP-VAR). Nous trouvons qu'un seul facteur explique la majorité de la variabilité des coefficients VAR, tandis que les paramètres de la volatilité des chocs varient d'une façon indépendante. Le facteur commun est positivement corrélé avec le taux de chômage. La même analyse est faite avec les données incluant la récente crise financière. La procédure suggère maintenant deux facteurs et le comportement des coefficients présente un changement important depuis 2007. Finalement, la méthode est appliquée à un modèle TVP-FAVAR. Nous trouvons que seulement 5 facteurs dynamiques gouvernent l'instabilité temporelle dans presque 700 coefficients.

A new set of orthogonal patterns in weather and climate: Optimally interpolated patterns.

A new spectral-based approach is presented to find orthogonal patterns from gridded weather/climate data. The method is based on optimizing the interpolation error variance. The optimally interpolated patterns (OIP) are then given by the eigenvectors of the interpolation error covariance matrix, obtained using the cross-spectral matrix. The formulation of the approach is presented, and the application to low-dimension stochastic toy models and to various reanalyses datasets is performed. In particular, it is found that the lowest-frequency patterns correspond to largest eigenvalues, that is, variances, of the interpolation error matrix. The approach has been applied to the Northern Hemispheric (NH) and tropical sea level pressure (SLP) and to the Indian Ocean sea surface temperature (SST). Two main OIP patterns are found for the NH SLP representing respectively the North Atlantic Oscillation and the North Pacific pattern. The leading tropical SLP OIP represents the Southern Oscillation. For the Indian Ocean SST, the leading OIP pattern shows a tripole-like structure having one sign over the eastern and north- and southwestern parts and an opposite sign in the remaining parts of the basin. The pattern is also found to have a high lagged correlation with the Niño-3 index with 6-months lag.

One-dimensional trapped atoms: critical coupling parameter and critical number of particles

In this paper we consider the case of a Bose gas in low dimension in order to illustrate the applicability of a method that allows us to construct analytical relations, valid for a broad range of coupling parameters, for a function which asymptotic expansions are known. The method is well suitable to investigate the problem of stability of a collection of Bose particles trapped in one- dimensional configuration for the case where the scattering length presents a negative value. The eigenvalues for this interacting quantum one-dimensional many particle system become negative when the interactions overcome the trapping energy and, in this case, the system becomes unstable. Here we calculate the critical coupling parameter and apply for the case of Lithium atoms obtaining the critical number of particles for the limit of stability.

Odor space and olfactory processing: Collective algorithms and neural implementation

Several basic olfactory tasks must be solved by highly olfactory animals, including background suppression, multiple object separation, mixture separation, and source identification. The large number N of classes of olfactory receptor cells—hundreds or thousands—permits the use of computational strategies and algorithms that would not be effective in a stimulus space of low dimension. A model of the patterns of olfactory receptor responses, based on the broad distribution of olfactory thresholds, is constructed. Representing one odor from the viewpoint of another then allows a common description of the most important basic problems and shows how to solve them when N is large. One possible biological implementation of these algorithms uses action potential timing and adaptation as the “hardware” features that are responsible for effective neural computation.

A Survey of Counterexamples to Hilbert's Fourteenth Problem

We survey counterexamples to Hilbert’s Fourteenth Problem, beginning with those of Nagata in the late 1950s, and including recent counterexamples in low dimension constructed with locally nilpotent derivations. Historical framework and pertinent references are provided. We also include 8 important open questions.