Bifurcation theory of nonlinear boundary value problems

The theory of bifurcation of solutions to two-point boundary value problems is developed for a system of nonlinear first order ordinary differential equations in which the bifurcation parameter is allowed to appear nonlinearly. An iteration method is used to establish necessary and sufficient conditions for bifurcation and to construct a unique bifurcated branch in a neighborhood of a bifurcation point which is a simple eigenvalue of the linearized problem. The problem of bifurcation at a degenerate eigenvalue of the linearized problem is reduced to that of solving a system of algebraic equations. Cases with no bifurcation and with multiple bifurcation at a degenerate eigenvalue are considered.

The iteration method employed is shown to generate approximate solutions which contain those obtained by formal perturbation theory. Thus the formal perturbation solutions are rigorously justified. A theory of continuation of a solution branch out of the neighborhood of its bifurcation point is presented. Several generalizations and extensions of the theory to other types of problems, such as systems of partial differential equations, are described.

The theory is applied to the problem of the axisymmetric buckling of thin spherical shells. Results are obtained which confirm recent numerical computations.

Distinct intron DNA structures in simian virus 40 T-antigen and adenovirus 2 E1A genes

Distinct structures delineating the introns of Simian Virus 40 T-antigen and Adenovirus 2 E1A genes have been discovered. The structures, which are centered around the branch points of the genes inserted in supercoiled double-stranded plasmids, are specifically targeted through photoactivated strand cleavage by the metal complex tris(4,7-diphenyl-1,10-phenanthroline)rhodium(III). The DNA sites that are recognized lack sequence homology but are similar in demarcating functionally important sites on the RNA level. The single-stranded DNA fragments corresponding to the coding strands of the genes were also found to fold into a structure apparently identical to that in the supercoiled genes based on the recognition by the metal complex. Further investigation of different single-stranded DNA fragments with other structural probes, such as another metal complex bis(1,10-phenanthroline)(phenanthrenequinone diimine)rhodium(III), AMT (4'aminomethyl-4,5',8 trimethylpsoralen), restriction enzyme Mse I, and mung bean nuclease, showed that the structures require the sequ ences at both ends of the intron plus the flanking sequences but not the middle of the intron. The two ends form independent helices which interact with each other to form the global tertiary structures. Both of the intron structures share similarities to the structure of the Holliday junction, which is also known to be specifically targeted by the former metal complex. These structures may have arisen from early RNA intron structures and may have been used to facilitate the evolution of genes through exon shuffling by acting as target sites for recombinase enzymes.

Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse

The effect of the mixing of pulsed two color fields on the generation of an isolated attosecond pulse has been systematically investigated. One main color is 800 nm and the other color (or secondary color) is varied from 1.2 to 2.4 mu m. This work shows that the continuum length behaves in a similar way to the behavior of the difference in the square of the amplitude of the strongest and next strongest cycle. As the mixing ratio is increased, the optimal wavelength for the extended continuum shifts toward shorter wavelength side. There is a certain mixing ratio of intensities at which the continuum length bifurcates, i.e., the existence of two optimal wavelengths. As the mixing ratio is further increased, each branch bifurcates again into two sub-branches. This 2D map analysis of the mixing ratio and the wavelength of the secondary field easily allows one to select a proper wavelength and the mixing ratio for a given pulse duration of the primary field. The study shows that an isolated sub-100 attosecond pulse can be generated mixing an 11 fs full-width-half-maximum (FWHM), 800 laser pulse with an 1840 nm FWHM pulse. Furthermore the result reveals that a 33 fs FWHM, 800 nm pulse can produce an isolated pulse below 200 as, when properly mixed. (c) 2008 Optical Society of America.

Towards conditional RNAi : shape and sequence transduction with small conditional RNAs

RNA interference (RNAi) is a powerful biological pathway allowing for sequence-specific knockdown of any gene of interest. While RNAi is a proven tool for probing gene function in biological circuits, it is limited by being constitutively ON and executes the logical operation: silence gene Y. To provide greater control over post-transcriptional gene silencing, we propose engineering a biological logic gate to implement “conditional RNAi.” Such a logic gate would silence gene Y only upon the expression of gene X, a completely unrelated gene, executing the logic: if gene X is transcribed, silence independent gene Y. Silencing of gene Y could be confined to a specific time and/or tissue by appropriately selecting gene X.

To implement the logic of conditional RNAi, we present the design and experimental validation of three nucleic acid self-assembly mechanisms which detect a sub-sequence of mRNA X and produce a Dicer substrate specific to gene Y. We introduce small conditional RNAs (scRNAs) to execute the signal transduction under isothermal conditions. scRNAs are small RNAs which change conformation, leading to both shape and sequence signal transduction, in response to hybridization to an input nucleic acid target. While all three conditional RNAi mechanisms execute the same logical operation, they explore various design alternatives for nucleic acid self-assembly pathways, including the use of duplex and monomer scRNAs, stable versus metastable reactants, multiple methods of nucleation, and 3-way and 4-way branch migration.

We demonstrate the isothermal execution of the conditional RNAi mechanisms in a test tube with recombinant Dicer. These mechanisms execute the logic: if mRNA X is detected, produce a Dicer substrate targeting independent mRNA Y. Only the final Dicer substrate, not the scRNA reactants or intermediates, is efficiently processed by Dicer. Additional work in human whole-cell extracts and a model tissue-culture system delves into both the promise and challenge of implementing conditional RNAi in vivo.

Branes and supersymmetric quantum field theories

Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.

One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.

First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.

Part I. A study of the velocity structure of the earth by the use of core phases. Part II. The 1971 San Fernando earthquake series focal mechanisms and tectonics

The initial objective of Part I was to determine the nature of upper mantle discontinuities, the average velocities through the mantle, and differences between mantle structure under continents and oceans by the use of P'dP', the seismic core phase P'P' (PKPPKP) that reflects at depth d in the mantle. In order to accomplish this, it was found necessary to also investigate core phases themselves and their inferences on core structure. P'dP' at both single stations and at the LASA array in Montana indicates that the following zones are candidates for discontinuities with varying degrees of confidence: 800-950 km, weak; 630-670 km, strongest; 500-600 km, strong but interpretation in doubt; 350-415 km, fair; 280-300 km, strong, varying in depth; 100-200 km, strong, varying in depth, may be the bottom of the low-velocity zone. It is estimated that a single station cannot easily discriminate between asymmetric P'P' and P'dP' for lead times of about 30 sec from the main P'P' phase, but the LASA array reduces this uncertainty range to less than 10 sec. The problems of scatter of P'P' main-phase times, mainly due to asymmetric P'P', incorrect identification of the branch, and lack of the proper velocity structure at the velocity point, are avoided and the analysis shows that one-way travel of P waves through oceanic mantle is delayed by 0.65 to 0.95 sec relative to United States mid-continental mantle.

A new P-wave velocity core model is constructed from observed times, dt/dΔ's, and relative amplitudes of P'; the observed times of SKS, SKKS, and PKiKP; and a new mantle-velocity determination by Jordan and Anderson. The new core model is smooth except for a discontinuity at the inner-core boundary determined to be at a radius of 1215 km. Short-period amplitude data do not require the inner core Q to be significantly lower than that of the outer core. Several lines of evidence show that most, if not all, of the arrivals preceding the DF branch of P' at distances shorter than 143° are due to scattering as proposed by Haddon and not due to spherically symmetric discontinuities just above the inner core as previously believed. Calculation of the travel-time distribution of scattered phases and comparison with published data show that the strongest scattering takes place at or near the core-mantle boundary close to the seismic station.

In Part II, the largest events in the San Fernando earthquake series, initiated by the main shock at 14 00 41.8 GMT on February 9, 1971, were chosen for analysis from the first three months of activity, 87 events in all. The initial rupture location coincides with the lower, northernmost edge of the main north-dipping thrust fault and the aftershock distribution. The best focal mechanism fit to the main shock P-wave first motions constrains the fault plane parameters to: strike, N 67° (± 6°) W; dip, 52° (± 3°) NE; rake, 72° (67°-95°) left lateral. Focal mechanisms of the aftershocks clearly outline a downstep of the western edge of the main thrust fault surface along a northeast-trending flexure. Faulting on this downstep is left-lateral strike-slip and dominates the strain release of the aftershock series, which indicates that the downstep limited the main event rupture on the west. The main thrust fault surface dips at about 35° to the northeast at shallow depths and probably steepens to 50° below a depth of 8 km. This steep dip at depth is a characteristic of other thrust faults in the Transverse Ranges and indicates the presence at depth of laterally-varying vertical forces that are probably due to buckling or overriding that causes some upward redirection of a dominant north-south horizontal compression. Two sets of events exhibit normal dip-slip motion with shallow hypocenters and correlate with areas of ground subsidence deduced from gravity data. Several lines of evidence indicate that a horizontal compressional stress in a north or north-northwest direction was added to the stresses in the aftershock area 12 days after the main shock. After this change, events were contained in bursts along the downstep and sequencing within the bursts provides evidence for an earthquake-triggering phenomenon that propagates with speeds of 5 to 15 km/day. Seismicity before the San Fernando series and the mapped structure of the area suggest that the downstep of the main fault surface is not a localized discontinuity but is part of a zone of weakness extending from Point Dume, near Malibu, to Palmdale on the San Andreas fault. This zone is interpreted as a decoupling boundary between crustal blocks that permits them to deform separately in the prevalent crustal-shortening mode of the Transverse Ranges region.

Arbeitsgebiete Ichthyologie und Fischsammlung am Institut für Seefischerei werden Ende 1992 aufgegeben

On 31 Oecember 1992, the Institut für Seefischerei Hamburg (ISH) will dissolve its branch of systematic ichthyology at the Zoologieal Museum Hamburg University. Remaining ISH ichthyology staff and facilities will move back in January 1993 to Institut für Seefischerei headquarters in Palmaille 9, D-·2000 Hamburg 50 (Tel. 040·389050, Fax 040·38905 129). The research field systematic ichthyology will not be continued at ISH. The ISH fish collection will be handed over to the Zoological Museum of Hamburg University (ZMH) as future responsible holder from 1 January 1993 onward. Loan requests for former ISH material should be addressed from January 1993 on only to: Prof. Dr. H. Wilkens, Kustos Ichthyologie, Zoological Museum der Universität, Martin-Luther-King-Platz 3, D·2000 Hamburg 13 (Tel. 040·4123 3870, Fax 040·4123 3937). A couple of months will be needed before requests for loan of former ISH holdings can be handled again. It should, however, be noted that even for a longer period there may be problems in serving loan requests promptly due to logistic difficulties and shortage of staff at the ZMH ichthyology department.

A intervenção judicial no âmbito das políticas públicas orientadas à concretização dos direitos fundamentais

O presente estudo se propõe a desvelar o espaço legítimo de controle de políticas públicas destinadas à concretização de direitos fundamentais pelo Poder Judiciário. Para tanto, inicialmente é apresentada uma teoria das políticas públicas, que compreende a busca de um conceito para a categoria e a apresentação de suas características e elementos mais relevantes. O estudo não prescinde da análise da teoria dos direitos fundamentais, em especial das questões atinentes à eficácia dos direitos ditos prestacionais, e também da chamada análise institucional, um campo de estudos recentemente reavivado nos Estados Unidos. Na segunda parte do trabalho, de natureza marcadamente propositiva, as políticas públicas são divididas segundo a sua natureza, e em seguida sugeridos diferentes níveis de controle jurídico. Para as políticas ligadas ao mínimo existencial, sustenta-se o controle por meio dos princípios da proibição da proteção insuficiente e vedação do retrocesso. Para as demais políticas públicas, o controle é analisado sob o prisma dos princípios da isonomia, eficiência e transparência. Após o estudo de questões incidentais, o trabalho segue para as modalidades de controle de políticas públicas, distinguindo-se entre o controle forte, em que a discricionariedade dos órgãos políticos é reduzida a zero, e o controle fraco, onde o Poder Judiciário apenas comprime o espaço de liberdade decisória.

O argumento pragmático ou consequencialista de cunho econômico e a modulação temporal dos efeitos das decisões do Supremo Tribunal Federal em matéria tributária

A tese analisa a relação íntima que há entre o pragmatismo ou o conseqüencialismo e a modulação temporal dos efeitos das decisões judiciais. Nesta relação, interessa ressaltar o ponto de interseção que certamente sobressai em várias ocasiões: o argumento de cunho econômico. Tal tipo de argumento pode assumir especial relevo quando do exame da oportunidade e conveniência na tomada das decisões eminentemente políticas. No âmbito jurisdicional, no entanto, o argumento pragmático ou consequencialista de cunho econômico não deve prevalecer como fundamento das decisões judiciais, especialmente cuidando-se de matéria tributária. Os problemas que centralizam o estudo podem ser colocados através das seguintes indagações: é possível que o Supremo Tribunal Federal compute, no julgamento de certa matéria tributária, argumento como o eventual rombo de X bilhões de reais que a decisão contrária ao Fisco possa acarretar para os cofres públicos? A fundamentação de eventual decisão judicial calcada exclusiva ou predominantemente em tal argumento é legítima ou ilegítima? Que importância pode ter na tomada de decisão judicial? Quando aplicada, há parâmetros a serem seguidos? Quais? Demonstramos que a prevalência de tal argumento é inadequada na seara judicial, ou seja, deve ter peso reduzido ou periférico, servindo para corroborar ou reforçar os argumentos jurídicos que centralizam o debate submetido ao exame do Poder Judiciário de modo geral, e do Supremo Tribunal Federal, de maneira particular. Em busca de esclarecer quais os principais limites e possibilidades de tal argumento, especialmente relacionando-o à modulação temporal dos efeitos da decisão judicial, explicitamos algumas regras necessárias para a sua adequada utilização, sob pena de inconcebível subversão de variados princípios e direitos fundamentais assegurados em sede constitucional. No exame das questões submetidas à apreciação da Corte Suprema em matéria tributária, o seu parâmetro consiste na maior efetividade e concretude ao texto constitucional. A modulação temporal dos efeitos se aplica a uma decisão que, declarando a inconstitucionalidade do ato normativo, se afastaria ainda mais da vontade constitucional, caso fosse aplicado o tradicional efeito ex tunc (retroativo até o nascimento da lei). Nestas situações específicas e excepcionais se justifica aplicar a modulação, com vistas a dar maior concretude e emprestar maior eficácia à Constituição. A tese proposta, ao final, consiste na reunião das regras explicitadas no trabalho e em proposta legislativa.

Para uma teoria do direito internacional do consumidor: a proteção do consumidor no livre comércio internacional

O consumidor é o agente vulnerável na relação de consumo internacional. O processo de globalização se apresenta, para o consumidor, como uma globalização do consumo. A globalização do consumo se caracteriza pelo comércio e fornecimento internacional de produtos e serviços por empresários/fornecedores transnacionais/globais, utilizando marcas de renome mundial, acessíveis a todos os consumidores do planeta, e agrava a vulnerabilidade do consumidor no mercado. A proteção jurídica do consumidor internacional é uma necessidade que os sistemas jurídicos nacionais não se mostram aptos a prover adequadamente, assim como o Direito Internacional também não. A presente tese demonstra a deficiência da Ciência do Direito na proteção do consumidor no contexto da globalização; demonstra como o próprio comércio internacional é prejudicado ao não priorizar de maneira absoluta e efetiva a proteção do consumidor na OMC, bem como ao mostrar-se apático diante dos diferentes níveis de proteção proporcionada aos consumidores em cada diferente sistema jurídico nacional; demonstra, também, como a proteção do consumidor de maneira uniforme e global por um direito comum aos Estados é possível e será capaz de tornar mais eficiente economicamente o processo de globalização do consumo, ao encorajar a participação mais intensa do consumidor no mercado internacional; e propõe a construção de um novo ramo do Direito dedicado ao problema, o Direito Internacional do Consumidor (DIC), por meio da elaboração de uma Teoria do Direito Internacional do Consumidor. O Direito Internacional do Consumidor pretende ser um direito comum e universal de proteção ao consumidor, fundado em métodos, conceitos, institutos, normas e princípios jurídicos universais. O DIC dialogará com outros ramos do Direito Público e Privado, especialmente o Direito Internacional Econômico, o Direito Internacional do Comércio, o Direito Internacional Privado, o Direito Processual Civil Internacional, e o Direito do Consumidor. Pretende-se com isto atender ao ideal de promover o livre comércio internacional com respeito aos Direitos Humanos.

Conflito de interesses na sociedade de economia mista: interesse público versus interesse privado

A presente tese tem por objetivo tratar do conflito de interesses na sociedade de economia mista. Referida sociedade tem na sua base constitutiva o conflito como elemento inerente. Por possuir capital público e privado, não é fácil solucionar facilmente os problemas que se apresentam no decorrer de sua existência frente à possibilidade de que o detentor do poder de controle pode decidir em prol do interesse público. E é visando limitar a má utilização do interesse público como resposta à tomada de decisões por parte do acionista controlador que se propõe uma mudança de paradigma. Para tanto, propõe-se analisar o papel do Estado empresário na atual conjuntura de limitação de intervenção do Estado na economia. Também é abordada a forma com que o poder Executivo vem intervindo no mercado, de forma a limitar a livre iniciativa e, por vezes, até mesmo eivada de certa inconstitucionalidade. No entanto, para evitar que haja afronta à Constituição no que diz respeito à exploração de atividade econômica por parte do Estado sem que sejam observados os limites constitucionais impostos, apresenta-se o meta-interesse como meio de solução. Sendo o meta-interesse o interesse da própria companhia, e considerando que o interesse público que fundamenta a autorização para a criação da sociedade de economia mista se extingue com a criação da referida companhia, tem-se que as normas que devem reger as sociedades de economia mista são as normas de direito privado. Com o meta-interesse o Estado passa a intervir na seara privada em igualdade de condições com as demais companhias, não podendo mais se valer de sua posição de acionista majoritário para tomar decisões que conflitem com o interesse da companhia e que privilegiem o interesse público secundário ou até mesmo o interesse político do Estado em detrimento do interesse social e dos acionistas minoritários. Dessa forma, o meta-interesse tem por finalidade colocar fim aos conflitos em relação à aplicação das normas jurídicas e as indefinições da própria natureza da sociedade de economia mista.

基于统计光学的无透镜鬼成像数值模拟与实验验证

作为量子信息领域分支的鬼成像,由于物体的像将出现在不包含物体的光路上的特点,使得这一领域的研究引人入胜。一度认为,只有基于纠缠态双光子的纠缠光源,才能实现鬼成像;但近年来的研究表明,经典热光场也能实现这一过程。从经典统计光学入手,建立了热光场的数值模型,模拟符合热光特性的光场变化、光场传播、以及物体透射函数对热光场的调制,进而从光强度起伏的关联函数中,分别重现振幅型物体和纯相位型物体的傅里叶变换图像;通过与真实实验结果的对比,表明基于统计光学原理的该数值模型所预测的实验结果,与真实的实验结果完全一致。

Feira de São Cristóvão: uma incursão sobre os estudos na área e a tentativa de aproximação à relação global-local

O presente trabalho se propõe fazer uma aproximação aos estudos sobre a tradicionalmente chamada Feira de São Cristóvão, levando em consideração a relação entre os agentes partícipes e a dinâmica no espaço. O objetivo é compreender melhor a força que move os atores no sentido da mudança da forma da feira para a forma do Centro Luiz Gonzaga de Tradições Nordestinas (CLGTN), e assim me aproximar da seguinte questão: Por que uma forma de comércio como a modelada na Feira de São Cristóvão, popularmente conhecida como feira dos paraíbas, sobrevive à força das mudanças operadas no ramo, dando lugar aos sofistificados shopping-centers? Para isso, faço uma aproximação ao estado de arte da pesquisa na área com base nas seguintes categorias de análise: a Feira como forma de produção do espaço e a relação Forma-Conteúdo, a relação espaço-identidade na feira e a relação entre agentes internos e externos no Espaço. Vendo a migração como movimento responsável pela constituição de um espaço social e territorialmente diferenciado do nordestino na cidade do Rio de Janeiro, é por ela que inicio a exposição. Esta se continua com a análise dos trabalhos escolhidos para estudo, via as categorias indicadas e, com um capítulo no qual procuro repensar a feira a partir da relação global-local que vem referenciando as reflexões acumuladas na área do urbano a partir da segunda metade do século XX. Por fim, concluo observando que a permanência da feira se deve a um duplo movimento de resistência-absorção que, se redefinindo no tempo e no espaço, não consegue, entretanto, apagar os traços da tradição que lhe deu origem.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.