The supermassive binary black hole system OJ287

Abstract This doctoral thesis concerns the active galactic nucleus (AGN) most often referred to with the catalogue number OJ287. The publications in the thesis present new discoveries of the system in the context of a supermassive binary black hole model. In addition, the introduction discusses general characteristics of the OJ287 system and the physical fundamentals behind these characteristics. The place of OJ287 in the hierarchy of known types of AGN is also discussed. The introduction presents a large selection of fundamental physics required to have a basic understanding of active galactic nuclei, binary black holes, relativistic jets and accretion disks. Particularly the general relativistic nature of the orbits of close binaries of supermassive black holes is explored with some detail. Analytic estimates of some of the general relativistic effects in such a binary are presented, as well as numerical methods to calculate the effects more precisely. It is also shown how these results can be applied to the OJ287 system. The binary orbit model forms the basis for models of the recurring optical outbursts in the OJ287 system. In the introduction, two physical outburst models are presented in some detail and compared. The radiation hydrodynamics of the outbursts are discussed and optical light curve predictions are derived. The precursor outbursts studied in Paper III are also presented, and tied into the model of OJ287. To complete the discussion of the observable features of OJ287, the nature of the relativistic jets in the system, and in active galactic nuclei in general, is discussed. Basic physics of relativistic jets are presented, with additional detail added in the form of helical jet models. The results of Papers II, IV and V concerning the jet of OJ287 are presented, and their relation to other facets of the binary black hole model is discussed. As a whole, the introduction serves as a guide, though terse, for the physics and numerical methods required to successfully understand and simulate a close binary of supermassive black holes. For this purpose, the introduction necessarily combines a large number of both fundamental and specific results from broad disciplines like general relativity and radiation hydrodynamics. With the material included in the introduction, the publications of the thesis, which present new results with a much narrower focus, can be readily understood. Of the publications, Paper I presents newly discovered optical data points for OJ287, detected on archival astronomical plates from the Harvard College Observatory. These data points show the 1900 outburst of OJ287 for the first time. In addition, new data points covering the 1913 outburst allowed the determination of the start of the outburst with more precision than was possible before. These outbursts were then successfully numerically modelled with an N-body simulation of the OJ287 binary and accretion disc. In Paper II, mechanisms for the spin-up of the secondary black hole in OJ287 via interaction with the primary accretion disc and the magnetic fields in the system are discussed. Timescales for spin-up and alignment via both processes are estimated. It is found that the secondary black hole likely has a high spin. Paper III reports a new outburst of OJ287 in March 2013. The outburst was found to be rather similar to the ones reported in 1993 and 2004. All these outbursts happened just before the main outburst season, and are called precursor outbursts. In this paper, a mechanism was proposed for the precursor outbursts, where the secondary black hole collides with a gas cloud in the primary accretion disc corona. From this, estimates of brightness and timescales for the precursor were derived, as well as a prediction of the timing of the next precursor outburst. In Paper IV, observations from the 2004–2006 OJ287 observing program are used to investigate the existence of short periodicities in OJ287. The existence of a _50 day quasiperiodic component is confirmed. In addition, statistically significant 250 day and 3.5 day periods are found. Primary black hole accretion of a spiral density wave in the accretion disc is proposed as the source of the 50 day period, with numerical simulations supporting these results. Lorentz contracted jet re-emission is then proposed as the reason for the 3.5 day timescale. Paper V fits optical observations and mm and cm radio observations of OJ287 with a helical jet model. The jet is found to have a spine–sheath structure, with the sheath having a much lower Lorentz gamma factor than the spine. The sheath opening angle and Lorentz factor, as well as the helical wavelength of the jet are reported for the first time. Tiivistelmä Tässä väitöskirjatutkimuksessa on keskitytty tutkimaan aktiivista galaksiydintä OJ287. Väitöskirjan osana olevat tieteelliset julkaisut esittelevät OJ287-systeemistä saatuja uusia tuloksia kaksoismusta-aukkomallin kontekstissa. Väitöskirjan johdannossa käsitellään OJ287:n yleisiä ominaisuuksia ja niitä fysikaalisia perusilmiöitä, jotka näiden ominaisuuksien taustalla vaikuttavat. Johdanto selvittää myös OJ287-järjestelmän sijoittumisen aktiivisten galaksiytimien hierarkiassa. Johdannossa käydään läpi joitakin perusfysiikan tuloksia, jotka ovat tarpeen aktiivisten galaksiydinten, mustien aukkojen binäärien, relativististen suihkujen ja kertymäkiekkojen ymmärtämiseksi. Kahden toisiaan kiertävän mustan aukon keskinäisen radan suhteellisuusteoreettiset perusteet käydään läpi yksityiskohtaisemmin. Johdannossa esitetään joitakin analyyttisiä tuloksia tällaisessa binäärissä havaittavista suhteellisuusteoreettisista ilmiöistä. Myös numeerisia menetelmiä näiden ilmiöiden tarkempaan laskemiseen esitellään. Tuloksia sovelletaan OJ287-systeemiin, ja verrataan havaintoihin. OJ287:n mustien aukkojen ratamalli muodostaa pohjan systeemin toistuvien optisten purkausten malleille. Johdannossa esitellään yksityiskohtaisemmin kaksi fysikaalista purkausmallia, ja vertaillaan niitä. Purkausten säteilyhydrodynamiikka käydään läpi, ja myös ennusteet purkausten valokäyrille johdetaan. Johdannossa esitellään myös Julkaisussa III johdettu prekursoripurkausten malli, ja osoitetaan sen sopivan yhteen OJ287:n binäärimallin kanssa. Johdanto esittelee myös relativististen suihkujen fysiikkaa sekä OJ287- systeemiin liittyen että aktiivisten galaksiydinten kontekstissa yleisesti. Relativististen suihkujen perusfysiikka esitellään, kuten myös malleja kierteisistä suihkuista. Julkaisujen II, IV ja V OJ287-systeemin suihkuja koskevat tulokset esitellään binäärimallin kontekstissa. Kokonaisuutena johdanto palvelee suppeana oppaana, joka esittelee tarvittavan fysiikan ja tarpeelliset numeeriset menetelmät mustien aukkojen binäärijärjestelmän ymmärtämiseen ja simulointiin. Tätä tarkoitusta varten johdanto yhdistää sekä perustuloksia että joitakin syvällisempiä tuloksia laajoilta fysiikan osa-alueilta kuten suhteellisuusteoriasta ja säteilyhydrodynamiikasta. Johdannon sisältämän materiaalin avulla väitöskirjan julkaisut, ja niiden esittämät tulokset, ovat hyvin ymmärrettävissä. Väitöskirjan julkaisuista ensimmäinen esittelee uusia OJ287-systeemistä saatuja havaintopisteitä, jotka on paikallistettu Harvardin yliopiston observatorion arkiston valokuvauslevyiltä. OJ287:n vuonna 1900 tapahtunut purkaus nähdään ensimmäistä kertaa näissä havaintopisteissä. Uudet havaintopisteet mahdollistivat myös vuoden 1913 purkauksen alun ajoittamisen tarkemmin kuin aiemmin oli mahdollista. Havaitut purkaukset mallinnettiin onnistuneesti simuloimalla OJ287-järjestelmän mustien aukkojen paria ja kertymäkiekkoa. Julkaisussa II käsitellään mekanismeja OJ287:n sekundäärisen mustan aukon spinin kasvamiseen vuorovaikutuksessa primäärin kertymäkiekon ja systeemin magneettikenttien kanssa. Julkaisussa arvioidaan maksimispinin saavuttamisen ja spinin suunnan vakiintumisen aikaskaalat kummallakin mekanismilla. Tutkimuksessa havaitaan sekundäärin spinin olevan todennäköisesti suuri. Julkaisu III esittelee OJ287-systeemissä maaliskuussa 2013 tapahtuneen purkauksen. Purkauksen havaittiin muistuttavan vuosina 1993 ja 2004 tapahtuneita purkauksia, joita kutsutaan yhteisnimityksellä prekursoripurkaus (precursor outburst). Julkaisussa esitellään purkauksen synnylle mekanismi, jossa OJ287-systeemin sekundäärinen musta aukko osuu primäärisen mustan aukon kertymäkiekon koronassa olevaan kaasupilveen. Mekanismin avulla johdetaan arviot prekursoripurkausten kirkkaudelle ja aikaskaalalle. Julkaisussa johdetaan myös ennuste seuraavan prekursoripurkauksen ajankohdalle. Julkaisussa IV käytetään vuosina 2004–2006 kerättyjä havaintoja OJ287- systeemistä lyhyiden jaksollisuuksien etsintään. Julkaisussa varmennetaan systeemissä esiintyvä n. 50 päivän kvasiperiodisuus. Lisäksi tilastollisesti merkittävät 250 päivän ja 3,5 päivän jaksollisuudet havaitaan. Julkaisussa esitetään malli, jossa primäärisen mustan aukon kertymäkiekossa oleva spiraalitiheysaalto aiheuttaa 50 päivän jaksollisuuden. Mallista tehty numeerinen simulaatio tukee tulosta. Systeemin relativistisen suihkun emittoima aikadilatoitunut säteily esitetään aiheuttajaksi 3,5 päivän jaksollisuusaikaskaalalle. Julkaisussa V sovitetaan kierresuihkumalli OJ287-systeemistä tehtyihin optisiin havaintoihin ja millimetri- sekä senttimetriaallonpituuden radiohavaintoihin. Suihkun rakenteen havaitaan olevan kaksijakoinen ja koostuvan ytimestä ja kuoresta. Suihkun kuorella on merkittävästi pienempi Lorentzin gamma-tekijä kuin suihkun ytimellä. Kuoren avautumiskulma ja Lorentztekijä sekä suihkun kierteen aallonpituus raportoidaan julkaisussa ensimmäistä kertaa.

A 37-kb restriction map of the human immunoglobulin lambda variable locus, VB cluster, harboring four functional genes and two non-coding Vl sequences

The human immunoglobulin lambda variable locus (IGLV) is mapped at chromosome 22 band q11.1-q11.2. The 30 functional germline v-lambda genes sequenced untill now have been subgrouped into 10 families (Vl1 to Vl10). The number of Vl genes has been estimated at approximately 70. This locus is formed by three gene clusters (VA, VB and VC) that encompass the variable coding genes (V) responsible for the synthesis of lambda-type Ig light chains, and the Jl-Cl cluster with the joining segments and the constant genes. Recently the entire variable lambda gene locus was mapped by contig methodology and its one- megabase DNA totally sequenced. All the known functional V-lambda genes and pseudogenes were located. We screened a human genomic DNA cosmid library and isolated a clone with an insert of 37 kb (cosmid 8.3) encompassing four functional genes (IGLV7S1, IGLV1S1, IGLV1S2 and IGLV5a), a pseudogene (VlA) and a vestigial sequence (vg1) to study in detail the positions of the restriction sites surrounding the Vl genes. We generated a high resolution restriction map, locating 31 restriction sites in 37 kb of the VB cluster, a region rich in functional Vl genes. This mapping information opens the perspective for further RFLP studies and sequencing

Effects of prenatal exposure to a mild chronic variable stress on body weight, preweaning mortality and rat behavior

Early stimulation has been shown to produce long-lasting effects in many species. Prenatal exposure to some strong stressors may affect development of the nervous system leading to behavioral impairment in adult life. The purpose of the present work was to study the postnatal harmful effects of exposure to variable mild stresses in rats during pregnancy. Female Holtzman rats were submitted daily to one session of a chronic variable stress (CVS) during pregnancy (prenatal stress; PS group). Control pregnant rats (C group) were undisturbed. The pups of PS and C dams were weighed and separated into two groups 48 h after delivery. One group was maintained with their own dams (PS group, N = 70; C group, N = 36) while the other PS pups were cross-fostered with C dams (PSF group, N = 47) and the other C pups were cross-fostered with PS dams (CF group, N = 58). Pups were undisturbed until weaning (postnatal day 28). The male offspring underwent motor activity tests (day 28), enriched environment tests (day 37) and social interaction tests (day 42) in an animal activity monitor. Body weight was recorded on days 2, 28 and 60. The PS pups showed lower birth weight than C pups (Duncan's test, P<0.05). The PS pups suckling with their stressed mothers displayed greater preweaning mortality (C: 23%, PS: 60%; c2 test, P<0.05) and lower body weight than controls at days 28 and 60 (Duncan's test, P<0.05 and P<0.01, respectively). The PS, PSF and CF groups showed lower motor activity scores than controls when tested at day 28 (Duncan's test, P<0.01 for PS group and P<0.05 for CF and PSF groups). In the enriched environment test performed on day 37, between-group differences in total motor activity were not detected; however, the PS, CF and PSF groups displayed less exploration time than controls (Duncan's test, P<0.05). Only the PS group showed impaired motor activity and impaired social behavior at day 42 (Duncan's test, P<0.05). In fact, CVS treatment during gestation plus suckling with a previously stressed mother caused long-lasting physical and behavioral changes in rats. Cross-fostering PS-exposed pups to a dam which was not submitted to stress counteracted most of the harmful effects of the treatment. It is probable that prenatal stress plus suckling from a previously stressed mother can induce long-lasting changes in the neurotransmitter systems involved in emotional regulation. Further experiments using neurochemical and pharmacological approaches would be interesting in this model.

Expression of anti-Z-DNA single chain antibody variable fragment on the filamentous phage surface

We describe the expression of an anti-Z-DNA single chain variable region antibody fragment (scFv) on a filamentous phage surface. Four vectors for phage display were constructed. Two of them are able to display multiple copies of the antibody fragment, and the others can be used to make monovalent libraries. The vectors use different promoter/leader sequences to direct the expression of the fused proteins. All were able to promote the assembly of fusion virion particles. In this paper we also show the affinity selection (biopanning) of those phage-antibodies based on the capacity of their products to recognize the antigen. We used biotinylated Z-DNA and the selection was performed in a solution phase fashion. The data presented here indicate that these vectors can be further used to construct anti-nucleic acid antibody fragment libraries that can be used to study the basis of nucleic acid-protein interaction and its role in autoimmunity mechanisms.

Association between EcoRI fragment-length polymorphism of the immunoglobulin lambda variable 8 (IGLV8) gene family with rheumatoid arthritis and systemic lupus erythematosus

The human immunoglobulin lambda variable 8 (IGLV8) subgroup is a gene family containing three members, one of them included in a monomorphic 3.7-kb EcoRI genomic fragment located at the major lambda variable locus on chromosome 22q11.1 (gene IGLV8a, EMBL accession No. Z73650) at 100% frequency in the normal urban population. The second is a polymorphic RFLP allele included in a 6.0-kb EcoRI fragment at 10% frequency, and the third is located in a monomorphic 8.0-kb EcoRI fragment at 100% frequency, the last being translocated to chromosome 8q11.2 and considered to be an orphan gene. Our Southern blot-EcoRI-RFLP studies in normal individuals and in patients with rheumatoid arthritis (RA) or with systemic lupus erythematosus (SLE), using a specific probe for the IGLV8 gene family (probe pVL8, EMBL accession No. X75424), have revealed the two monomorphic genomic fragments containing the IGLV8 genes, i.e., the 3.7-kb fragment from chromosome 22q11.1 and the 8.0-kb fragment from 8q11.2, both occurring at 100% frequency (103 normal individuals, 48 RA and 28 SLE patients analyzed), but absence of the 6.0-kb IGLV8 polymorphic RFLP allele in all RA or SLE patients. As expected, the frequency of the 6.0-kb allele among the normal individuals was 10%. These findings suggest an association between the absence of the 6.0-kb EcoRI fragment and rheumatoid arthritis and systemic lupus erythematosus.

Lebesgue and Sobolev spaces with variable exponents

Kirjallisuusarvostelu

Local Binary Patterns in Focal-Plane Processing. Analysis and Applications

Feature extraction is the part of pattern recognition, where the sensor data is transformed into a more suitable form for the machine to interpret. The purpose of this step is also to reduce the amount of information passed to the next stages of the system, and to preserve the essential information in the view of discriminating the data into different classes. For instance, in the case of image analysis the actual image intensities are vulnerable to various environmental effects, such as lighting changes and the feature extraction can be used as means for detecting features, which are invariant to certain types of illumination changes. Finally, classification tries to make decisions based on the previously transformed data. The main focus of this thesis is on developing new methods for the embedded feature extraction based on local non-parametric image descriptors. Also, feature analysis is carried out for the selected image features. Low-level Local Binary Pattern (LBP) based features are in a main role in the analysis. In the embedded domain, the pattern recognition system must usually meet strict performance constraints, such as high speed, compact size and low power consumption. The characteristics of the final system can be seen as a trade-off between these metrics, which is largely affected by the decisions made during the implementation phase. The implementation alternatives of the LBP based feature extraction are explored in the embedded domain in the context of focal-plane vision processors. In particular, the thesis demonstrates the LBP extraction with MIPA4k massively parallel focal-plane processor IC. Also higher level processing is incorporated to this framework, by means of a framework for implementing a single chip face recognition system. Furthermore, a new method for determining optical flow based on LBPs, designed in particular to the embedded domain is presented. Inspired by some of the principles observed through the feature analysis of the Local Binary Patterns, an extension to the well known non-parametric rank transform is proposed, and its performance is evaluated in face recognition experiments with a standard dataset. Finally, an a priori model where the LBPs are seen as combinations of n-tuples is also presented

Variable positive end-expiratory pressure can maintain oxygenation in experimental acute respiratory distress syndrome induced by oleic acid in dogs

The use of positive end-expiratory pressure (PEEP) or lung recruitment maneuvers (RM) to improve oxygenation in acute respiratory distress syndrome (ARDS) is used but it may reduce cardiac output (CO). Intermittent PEEP may avoid these complications. Our objective was to determine if variable PEEP compared with constant PEEP is capable of maintaining arterial oxygenation and minimizing hemodynamic alterations with or without RM. Eighteen dogs with ARDS induced by oleic acid were randomized into three equal groups: group 1, low variable PEEP; group 2, high variable PEEP, and group 3, RM + high variable PEEP. All groups were submitted to constant PEEP, followed by variable PEEP (PEEP was increased from 5 to 10 cmH2O in group 1, and from 5 to 18 cmH2O in the other two groups). PaO2 was higher in group 3 (356.2 ± 65.4 mmHg) than in group 1 (92.7 ± 29.7 mmHg) and group 2 (228.5 ± 72.4 mmHg), P < 0.05. PaO2 was maintained during variable PEEP except in group 2 (318.5 ± 82.9 at constant PEEP to 228.5 ± 72.4 at variable PEEP). There was a reduction in CO in group 3 after RM (3.9 ± 1.1 before to 2.7 ± 0.5 L·min-1·(m2)-1 after; P < 0.05), but there was not any difference between constant and variable PEEP periods (2.7 ± 0.5 and 2.4 ± 0.7 L·min-1·(m2)-1; P > 0.05. Variable PEEP is able to maintain PaO2 when performed in combination with RM in dogs with ARDS. After RM, CO was reduced and there was no relevant difference between the variable and constant PEEP periods.

A conjugate of an anti-midkine single-chain variable fragment to doxorubicin inhibits tumor growth

Doxorubicin (DOX) was conjugated to a single-chain variable fragment (scFv) against human midkine (MK), and the conjugate (scFv-DOX) was used to target the chemotherapeutic agent to a mouse solid tumor model in which the tumor cells expressed high levels of human MK. The His-tagged recombinant scFv was expressed in bacteria, purified by metal affinity chromatography, and then conjugated to DOX using oxidative dextran (Dex) as a linker. The molecular formula of this immunoconjugate was scFv(Dex)1.3(DOX)20. In vitro apoptosis assays showed that the scFv-DOX conjugate was more cytotoxic against MK-transfected human adenocarcinoma cells (BGC823-MK) than untransfected cells (55.3 ± 2.4 vs 22.4 ± 3.8%) for three independent experiments. Nude mice bearing BGC823-MK solid tumors received scFv-DOX or equivalent doses of scFv + DOX for 2 weeks and tumor growth was more effectively inhibited by the scFv-DOX conjugate than by scFv + DOX (51.83% inhibition vs 40.81%). Histological analysis of the tumor tissues revealed that the highest levels of DOX accumulated in tumors from mice treated with scFv-DOX and this resulted in more extensive tumor cell death than in animals treated with the equivalent dose of scFv + DOX. These results show that the scFv-DOX conjugate effectively inhibited tumor growth in vivo and suggest that antigen-specific scFv may be competent drug-carriers.

Expression, production, and renaturation of a functional single-chain variable antibody fragment (scFv) against human ICAM-1

Intercellular adhesion molecule-1 (ICAM-1) is an important factor in the progression of inflammatory responses in vivo. To develop a new anti-inflammatory drug to block the biological activity of ICAM-1, we produced a monoclonal antibody (Ka=4.19×10−8 M) against human ICAM-1. The anti-ICAM-1 single-chain variable antibody fragment (scFv) was expressed at a high level as inclusion bodies in Escherichia coli. We refolded the scFv (Ka=2.35×10−7 M) by ion-exchange chromatography, dialysis, and dilution. The results showed that column chromatography refolding by high-performance Q Sepharose had remarkable advantages over conventional dilution and dialysis methods. Furthermore, the anti-ICAM-1 scFv yield of about 60 mg/L was higher with this method. The purity of the final product was greater than 90%, as shown by denaturing gel electrophoresis. Enzyme-linked immunosorbent assay, cell culture, and animal experiments were used to assess the immunological properties and biological activities of the renatured scFv.

Doubly fed wind turbine performance in variable grid conditions

Wind turbines based on doubly fed induction generators (DFIG) become the most popular solution in high power wind generation industry. While this topology provides great performance with the reduced power rating of power converter, it has more complicated structure in comparison with full-rated topologies, and therefore leads to complexity of control algorithms and electromechanical processes in the system. The purpose of presented study is to present a proper vector control scheme for the DFIG and overall control for the WT to investigate its behavior at different wind speeds and in different grid voltage conditions: voltage sags, magnitude and frequency variations. The key principles of variable-speed wind turbine were implemented in simulation model and demonstrated during the study. Then, based on developed control scheme and mathematical model, the set of simulation is made to analyze reactive power capabilities of the DFIG wind turbine. Further, the rating of rotor-side converter is modified to not only generate active rated active power, but also to fulfill Grid Codes. Results of modelling and analyzing of the DFIG WT behavior under different speeds and different voltage conditions are presented in the work.

Variable-speed-drive-based detection methods for problems in fluid handling systems

Fluid handling systems account for a significant share of the global consumption of electrical energy. They also suffer from problems, which reduce their energy efficiency and increase life-cycle costs. Detecting or predicting these problems in time can make fluid handling systems more environmentally and economically sustainable to operate. In this Master’s Thesis, significant problems in fluid systems were studied and possibilities to develop variable-speed-drive-based detection methods for them was discussed. A literature review was conducted to find significant problems occurring in fluid handling systems containing pumps, fans and compressors. To find case examples for evaluating the feasibility of variable-speed-drive-based methods, queries were sent to industrial companies. As a result of this, the possibility to detect heat exchanger fouling with a variable-speed drive was analysed with data from three industrial cases. It was found that a mass flow rate estimate, which can be generated with a variable speed drive, can be used together with temperature measurements to monitor a heat exchanger’s thermal performance. Secondly, it was found that the fouling-related increase in the pressure drop of a heat exchanger can be monitored with a variable speed drive. Lastly, for systems where the flow device is speed controlled with by a pressure measurement, it was concluded that increasing rotational speed can be interpreted as progressing fouling in the heat exchanger.

"Man of science, man of faith" : Lost, consumer agency and the fate/free will binary in the post-9/11 context

In 2004, Lost debuted on ABC and quickly became a cultural phenomenon. Its postmodem take on the classic Robinson Crusoe desert island scenario gestures to a variety of different issues circulating within the post-9II1 cultural consciousness, such as terrorism, leadership, anxieties involving air travel, torture, and globalization. Lost's complex interwoven flashback and flash-forward narrative structure encourages spectators to creatively hypothesize solutions to the central mysteries of the narrative, while also thematically addressing archetypal questions of freedom of choice versus fate. Through an examination of the narrative structure, the significance of technological shifts in television, and fan cultures in Lost, this thesis discusses the tenuous notion of consumer agency within the current cultural context. Furthermore, I also explore these issues in relation to the wider historical post-9/II context.

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map.

La vivienda como variable dependiente de un eje industrial en formación hacia nuevas ciudades [por] Cecilia Andrade Cedeño

Tesis (Maestría en Ciencias para la Planificación de Asentamientos Humanos) U.A.N.L.