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AAS 204th Meeting, June 2004
Session 5 Eclipsing Binaries and Friends
Poster, Monday, May 31, 2004, 9:20am-6:30pm, Ballroom

## [5.12] Six Years of HST/STIS Observations of the Eclipsing Binary VV Cephei

P.D. Bennett, A. Brown (University of Colorado), W.H. Bauer (Wellesley College)

VV Cephei (M2~Iab + B0.5~V) is the brightest M~supergiant in an eclipsing binary system in the sky (V=4.90). Its orbital period of 20.3 years is one of the longest known for an eclipsing binary. This system is of great interest because of the possible use of the eclipse-mapping technique developed for the \zeta~Aurigae binaries to construct empirical models of the M~supergiant's extended atmosphere and wind. The method uses circumstellar absorption and scattering features seen superimposed on the early-type companion's continuum near eclipse (when the hot companion passes behind the cool supergiant) to map absorber column densities along the line of sight. During ingress, the line of sight to the hot companion sweeps through increasingly thick layers of the M~supergiant's chromosphere prior to second contact; this sequence proceeds again in reverse during egress. Observing in the ultraviolet is particularly advantageous since there the cool supergiant contributes negligibly to the flux, thereby avoiding the problem of disentangling composite spectra. To this end, we have obtained HST/STIS echelle observations of the ultraviolet spectrum of VV~Cep at 21 epochs over a 6-year period from mid-eclipse in late 1997 through quadrature in 2003. The application of the eclipse-mapping technique to VV~Cep is not straightforward. The high mass loss rate and corresponding massive wind column densities produce a complex, heavily blended forest of circumstellar absorption lines, mainly from singly-ionized iron group elements. Accretion near the hot companion produces additional absorption and a highly-variable accretion continuum. Nevertheless, the science objectives of the project have been realized: there are sufficient, unblended lines of appropriate strengths (near optical depth unity) to permit the reconstruction of column-density maps along the various sightlines, and thus the construction of detailed models of the supergiant's chromosphere and wind. We show some sample spectra and present preliminary modeling results in this poster.


text: Last 7 days     Results 1101-1120 of 42586.   51 52 53 54 55 56 57 58 59 60 61   The development of Structural Fire Engineering over the past 25 years and issues for the futureFranssen, Jean-Marc Conference (2015, October 15)This talk presents a series of thoughts of the author from his observation of the evolution of structural fire engineering since the beginning of his career Experimental research works are discussed ... [more ▼]This talk presents a series of thoughts of the author from his observation of the evolution of structural fire engineering since the beginning of his career Experimental research works are discussed, either on the behavior of materials or on the behavior of structures. The rest of the presentation is dedicated to calculation methods: tabulated data, simple calculation models and general calculation models. For numerical calculation models, the presentation contains a brief history of appearance of different software, of their objectives, of their capabilities… Some examples are presented of structures recently modelled in real applications. Possible abusive utilizations of numerical modelling are also shown. The presentation ends with a presentation of some challenges that the discipline is facing for the future. [less ▲]Detailed reference viewed: 13 (0 ULg) Mise au point sur les neuropathies héréditairesWANG, François-Charles ; BOUQUIAUX, Olivier Scientific conference (2015, October 15)Detailed reference viewed: 12 (0 ULg) Belgian International Lawyers and WWI. Networks, Experience and MemoryGenin, Vincent Poster (2015, October 15)Detailed reference viewed: 21 (3 ULg) La médecine de famille, une connaissance complexe et multi facettesJamoulle, Marc Conference (2015, October 15)Dans les conditions difficiles que doit affronter la Tunisie, soit faire face a l'augmentation de la demande en soins de santé, à l'amélioration de la qualité et à la transition démographique et ... [more ▼]Dans les conditions difficiles que doit affronter la Tunisie, soit faire face a l'augmentation de la demande en soins de santé, à l'amélioration de la qualité et à la transition démographique et épidémiologique, la Médecine de Famille représente un atout certain. L'image internationale peu flatteuse de la médecine de famille Tunisienne est présentée au travers de sa présence dans les publications scientifiques. Une série de concepts spécifiques de la Médecine de Famille sont exposés ; la complexité, l'extension de la connaissance, l'incertitude, les différents niveaux de l'action. Leur prise en compte permet d'appeler à un autre regard et à un appel à écrire le métier d'une façon différente. Pour en sortir définitivement avec l'image délétère de ce métier exigeant. [less ▲]Detailed reference viewed: 77 (11 ULg) The impact of factors related to retirement on cognitive functioningGrotz, Catherine ; Adam, Stéphane ; Matharan, Fanny et alConference (2015, October 15)Detailed reference viewed: 38 (0 ULg) Do Burundians have a more positive perception of older people than immigrants or Belgians?Marquet, Manon ; Missotten, Pierre ; Nindaba, Desiderate et alPoster (2015, October 15)Detailed reference viewed: 26 (8 ULg) Efficient standard error formulas of ability estimators in item response theoryMagis, David Conference (2015, October 15)This talk focuses on the computation of asymptotic standard errors (ASE) of ability estimators with dichotomous item response models. A general framework is considered, and ability estimators are defined ... [more ▼]This talk focuses on the computation of asymptotic standard errors (ASE) of ability estimators with dichotomous item response models. A general framework is considered, and ability estimators are defined from a very restricted set of assumptions and formulas. This approach encompasses most standard methods such as maximum likelihood, weighted likelihood, maximum a posteriori, and robust estimators. A general formula for the ASE is derived from the theory of M-estimation. Well-known results are found back as particular cases for the maximum and robust estimators, while new ASE proposals for the weighted likelihood and maximum a posteriori estimators are presented. These new formulas are compared to traditional ones by means of a simulation study under Rasch modeling. [less ▲]Detailed reference viewed: 18 (1 ULg) EFFECTS OF AGRICULTURAL PRACTICES AND HEAVY METAL CONTAMINATION ON THE COMMUNITY DYNAMICS OF EARTHWORMS IN RELATION TO SOIL PHYSICAL AND CHEMICAL FACTORS IN AGRICULTURAL FIELDS (BELGIUM)Lemtiri, Aboulkacem Doctoral thesis (2015)We investigated the effect of different agricultural practices on the abundances, biomass, and species diversity of earthworms. Specifically, we aimed to identify the relationship between certain soil ... [more ▼]We investigated the effect of different agricultural practices on the abundances, biomass, and species diversity of earthworms. Specifically, we aimed to identify the relationship between certain soil physico-chemical properties and earthworm communities in agricultural soils. Two tillage systems and crop residue management were investigated. After conducting the study over four years, we found that the abundance, biomass, and diversity of earthworms were negatively affected by tillage application and the removal of crop residues. All ecological groups were negatively affected by conventional tillage system and crop residues exportation. However, crop residues removal had a greater impact than the conventional tillage system. In this study area, the earthworm community was dominated by the endogeic species A. c. caliginosa (64%), while few epigeic and anecic species were observed (5%). Endogeic and epi-anecic (L. terrestris) species appeared to be highly influenced by tillage and of crop residues exportation. When crop residues were exported from the field, the concentrations of chemical elements were low, particularly P and K nutrients. Earthworm activity contributed to nutrient dynamics and soil structure after four years of incorporating of crop residues to the fields and reduced tillage application. No consistent relationship was detected between soil and earthworm variables, even though different soil properties responded differently with respect to the tillage systems, crop residues removal and the presence of certain earthworm species. The number of years that our field was managed might have also affected our results.On the basis of the primarily research focused on understanding how earthworms participate and contribute towards improving soil quality (structure, nutrient dynamics and fertility), we subsequently focused on investigating how two plants (Vicia faba and Zea mays) and the Eisenia fetida earthworm contribute to uptake of different metals: Cd, Zn, Pb and Cu from the land surrounding of a former Zn-Pb ore treatment plant. Specifically, we tested whether the earthworm Eisenia fetida could act as a catalyzor to enhance phytoremediation efficiency. After 42 days of exposure, our results showed that certain earthworm life-cycle traits are affected by metal contamination and by the addition of plants. Specifically, the concentrations of metals in earthworm tissues decreased in the presence of plants. Our findings demonstrate that earthworm activities modify the availability of metals in soils, enhancing metal uptake by plants. This innovative system offers new investigation possibilities by considering earthworm-plant-soil interaction. In conclusion, this work confirmed that earthworms are important catalyzor optimizing the phytoremediation processes of polluted soils. [less ▲]Detailed reference viewed: 44 (10 ULg) L’insécurité de l’exclusion sociale : comment sortir de l’impuissance ?Glowacz, Fabienne Scientific conference (2015, October 15)Detailed reference viewed: 16 (0 ULg) Figuras de traductor (no) profesionalWillson, Patricia Scientific conference (2015, October 15)Detailed reference viewed: 14 (3 ULg) Public Finance Tools Reshape the Belgian Polity: A Sociological Perspective on Public FinancePiron, Damien Poster (2015, October 15)Detailed reference viewed: 12 (0 ULg) Shear banding modelling in cross-anisotropic rocksPardoen, Benoît ; Seyedi, D.M.; Collin, Frédéric in International Journal of Solids and Structures (2015), 72Sedimentary geomaterials such as rocks frequently exhibit cross-anisotropic properties and their behaviour depends on the direction of loading with respect to their microstructure. As far as material ... [more ▼]Sedimentary geomaterials such as rocks frequently exhibit cross-anisotropic properties and their behaviour depends on the direction of loading with respect to their microstructure. As far as material rupture is concerned, localised deformation in shear band mode appears generally before cracks and material failure. The influence of cross-anisotropy on the shear strain localisation remains an important issue and is investigated in the present study. To do so, a constitutive elastoplastic cross-anisotropic model that includes anisotropy both on the elastic and plastic characteristics is defined. For the plastic part of the model, the anisotropy of a strength parameter is introduced with a microstructure fabric tensor. Then, the fractures are modelled with finite element methods by considering the development of shear strain localisation bands and an enriched model is used to properly reproduce the shear banding. The cross-anisotropy influence on shear banding is studied through numerical applications of small and large-scale geotechnical problems that engender fractures. The two considered applications are a plane-strain biaxial compression test and an underground gallery excavation. The numerical results provide information about the influence of cross-anisotropy on the appearance and development of shear bands. It has been noticed, among other observations, that the material strength vary with the loading direction and that the development and the shape of the excavation fractured zone that develops around a gallery is strongly influenced by the material anisotropy. [less ▲]Detailed reference viewed: 36 (11 ULg) Environmental impact assessment of bio-based binders: from production to industrial applicationsGerbinet, Saïcha ; Belboom, Sandra ; Briard, Vincent et alPoster (2015, October 15)A binder is used to hold together the fibers forming the mineral wool products (see figure 1). These fibers can be produced from sand and recycled glass for glass wool products (see figure 2) or from rock ... [more ▼]A binder is used to hold together the fibers forming the mineral wool products (see figure 1). These fibers can be produced from sand and recycled glass for glass wool products (see figure 2) or from rock (volcanic rock, typically basalt or dolomite) for stone wool products. Traditionally, the binders used in mineral wool products are based on phenol-formaldehyde. Due to sanitary and environmental considerations and increased focus on indoor air quality, the producers developed new alternative binders. Especially, Knauf Insulation, a worldwide building insulation manufacturer, developed a binder based on plant starch and called ECOSE. In addition of not using added phenol-formaldehyde , this new binder is also supposed to reduce the environmental impacts of Knauf Insulation mineral wools. Moreover, due to its properties, others applications are now considered for ECOSE such as composite wood panels [1]. The aim of this study is to determine the environmental impact of ECOSE and to compare it with more traditional binders using Life Cycle Assessment (LCA) methodology. LCA analyzes the environmental aspects and potential impacts associated with all the stages of a product's life. In this type of environmental assessment the energy and material flows for the entire life-cycle are surveyed and analyzed with special attention to possible environmental hazards or human health problems. The ISO 14040 and 14044 norms [2, 3] provide the general guidance for performing an LCA. The LCA methodology is first applied to ECOSE main component: glucose. The presentation will present results for glucose production from cereals starches After that, first ECOSE application, glass mineral wool products will be presented in details, including production process (see figure 3). The modelling of the glass mineral wool production process in LCA software GaBi 6 [4] is then described. One of the model specificity is that it allows to perform LCA of any glass wool products produced in Knauf Insulation plants in Europe. The adaptations to the model to allow studying former glass wool product when using phenol-formaldehyde binders will also be presented such as the advantages of this model. Moreover, preliminary results about ECOSE and phenol-formaldehyde glass wool products are explained. Références [1] Knauf Insulation. [cited 2014; Available from: http://www.knaufinsulation.ua/en. [2] ISO 14044, Environmental management - Life cycle assessment - Requirements and guidelines. 2006. [3] ISO 14040, Environmental management - Life cycle assessment - Principles and framework. 2006. [4] LBP, University of Stuttgart, and PE INTERNATIONAL AG, GaBI 6. 2012: Leinfelden-Echterdingen. p. GaBi 6: Software and database for life cycle engineering. [less ▲]Detailed reference viewed: 15 (0 ULg) Belgian children’s and adolescents’ views on the elderlyFlamion, Allison ; Missotten, Pierre ; Schroyen, Sarah et alPoster (2015, October 15)Detailed reference viewed: 16 (3 ULg) "Lirécrire pour apprendre" : d'un outil d'enseignement à sa mise en oeuvre au 1er degré de l'enseignement secondaireDe Croix, Séverine ; Wyns, MarielleDiverse speeche and writing (2015)Formation à la mise en oeuvre, dans les classes de 1re secondaire, de l'outil "Lirécrire pour apprendre" (Chaire Lirécrire, Fondation Louvain, UCL). Ce dispositif didactique et pédagogique est centré sur ... [more ▼]Formation à la mise en oeuvre, dans les classes de 1re secondaire, de l'outil "Lirécrire pour apprendre" (Chaire Lirécrire, Fondation Louvain, UCL). Ce dispositif didactique et pédagogique est centré sur l'enseignement explicite, dans l'ordinaire de la classe, des stratégies de lecture nécessaires à la compréhension des textes et des opérations d'écriture requises pour réussir à l'école. [less ▲]Detailed reference viewed: 10 (2 ULg) L’apport de la revalidation multidisciplinaire pour les lombalgies - Ecole du DosKaux, Jean-François ; Tomasella, Marco ; Vanderthommen, Marc et alConference (2015, October 15)Detailed reference viewed: 34 (7 ULg) Télémétrie RFID au niveau d'un barrage hydroélectrique : efficacité d'un exutoire de dévalaison et essais d'identification de voies de passagesOvidio, Michaël Scientific conference (2015, October 15)Detailed reference viewed: 24 (5 ULg) HPLC-DAD IDENTIFICATION OF SOME FLAVONOIDS FROM THE LEAVES AND AERIAL PARTS OF BRYONIA ALBA L. SPECIES SPONTANEOUS IN THE ROMANIAN FLORAIelciu, Irina-Ioana ; Păltinean, Ramona; Cieckiewicz, Ewa et alPoster (2015, October 14)Bryonia alba L. is a climbing species, spontaneous in the Romanian flora, which can be found throughout the whole country [1]. It is known for its cytotoxic, analgesic, antipyretic, anti-inflammatory ... [more ▼]Bryonia alba L. is a climbing species, spontaneous in the Romanian flora, which can be found throughout the whole country [1]. It is known for its cytotoxic, analgesic, antipyretic, anti-inflammatory, antibacterial, anti-rheumatic, laxative-purgative and smooth muscle relaxant proprieties, being used both in traditional medicine and in homeopathy [2]. The main objective of this study consists in the evaluation of the flavonoid profile of this species. The vegetal material was harvested from the spontaneous flora of Cluj county (Romania). The vegetal extracts were obtained by ultrasonication, in methanol. Analysis of flavonoids was performed by a HPLC-DAD method and revealed mainly the presence of C-glycosides, of which saponarine was found as the main compound. Quantification of saponarin was also performed, using the HPLC method, on samples collected at different periods of time. Variation of the quantity of saponarine according to harvested samples was determined. Further analysis are under process in order to investigate the structure of these flavonoids and the pharmacological effects of the Bryonia alba L. plant extracts. References: 1. *** Flora Europea, vol. 2, Cambridge, Univ. Press. Cambridge London-New York Melbourne, 1979, p. 298-299 2. Demarque D, Jouanny J, Poitevin B, Saint Jean Y. Pharmacologie et matière médicale homéopathique, 3ième edition, France, CEDH, 2007. [less ▲]Detailed reference viewed: 22 (1 ULg) Guided group purchases of energy renovation services and works in deprived urban neighbourhoodsRuelle, Christine ; Teller, Jacques in Energy Efficiency (2015)The energy retrofitting of housings is progressing very slowly in Wallonia (Belgium) despite individual fiscal measures and subsidies offered by public authorities to support households engaging in this ... [more ▼]The energy retrofitting of housings is progressing very slowly in Wallonia (Belgium) despite individual fiscal measures and subsidies offered by public authorities to support households engaging in this process. This paper explores the use of a method called ‘Guided Group Purchases’ (GPPs), i.e. an approach combining support, guidance and group purchasing of energy renovation services and works, as a possible method to accelerate the retrofitting of the existing housing stock. It reports on an action research experiment conducted in the context of the Interreg project ‘SUN’ (Sustainable Urban Neighbourhoods). The results of this experiment are quite promising. Within a few months, some 80 energy retrofitting interventions (energy audits or insulation works) were implemented in one neighbourhood of the city of Liège. Our evaluation of the experiment showed that the prospect of achieving cost reductions was one of the main initial expectations of participants to the GGPs. Still the participants rather insisted on other benefits after taking part to the initiative. These benefits were mostly related to the guidance they received, to the facilitation with contractors and to the social dynamics generated by the initiative. Several weaknesses were also identified by participants, like the lack of follow-up during and after the works. Our evaluation also highlighted that the main landlords did not participate in the initiative and that the influence of GGPs on costs was limited to specific types of interventions. Despite observed limitations and possible improvements, such a neighbourhood-based approach certainly constitutes a promising avenue for public authorities to support and accelerate the energy retrofitting of private residential buildings in urban neighbourhoods. [less ▲]Detailed reference viewed: 35 (11 ULg) Direct exoplanet detection and characterization using the ANDROMEDA method: Performance on VLT/NaCo dataCantalloube, F.; Mouillet, D.; Mugnier, L. M. et alin Astronomy and Astrophysics (2015), 582Context. The direct detection of exoplanets with high-contrast imaging requires advanced data processing methods to disentangle potential planetary signals from bright quasi-static speckles. Among them ... [more ▼]Context. The direct detection of exoplanets with high-contrast imaging requires advanced data processing methods to disentangle potential planetary signals from bright quasi-static speckles. Among them, angular differential imaging (ADI) permits potential planetary signals with a known rotation rate to be separated from instrumental speckles that are either statics or slowly variable. The method presented in this paper, called ANDROMEDA for ANgular Differential OptiMal Exoplanet Detection Algorithm is based on a maximum likelihood approach to ADI and is used to estimate the position and the flux of any point source present in the field of view. Aims. In order to optimize and experimentally validate this previously proposed method, we applied ANDROMEDA to real VLT/NaCo data. In addition to its pure detection capability, we investigated the possibility of defining simple and efficient criteria for automatic point source extraction able to support the processing of large surveys. Methods. To assess the performance of the method, we applied ANDROMEDA on VLT/NaCo data of TYC-8979-1683-1 which is surrounded by numerous bright stars and on which we added synthetic planets of known position and flux in the field. In order to accommodate the real data properties, it was necessary to develop additional pre-processing and post-processing steps to the initially proposed algorithm. We then investigated its skill in the challenging case of a well-known target, $\beta$ Pictoris, whose companion is close to the detection limit and we compared our results to those obtained by another method based on principal component analysis (PCA). Results. Application on VLT/NaCo data demonstrates the ability of ANDROMEDA to automatically detect and characterize point sources present in the image field. We end up with a robust method bringing consistent results with a sensitivity similar to the recently published algorithms, with only two parameters to be fine tuned. Moreover, the companion flux estimates are not biased by the algorithm parameters and do not require a posteriori corrections. Conclusions. ANDROMEDA is an attractive alternative to current standard image processing methods that can be readily applied to on-sky data. [less ▲]Detailed reference viewed: 21 (0 ULg)

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## CryptoDB

### Alessandro Chiesa

#### Publications

Year
Venue
Title
2019
EUROCRYPT
We design, implement, and evaluate a zero knowledge succinct non-interactive argument (SNARG) for Rank-1 Constraint Satisfaction (R1CS), a widely-deployed NP language undergoing standardization. Our SNARG has a transparent setup, is plausibly post-quantum secure, and uses lightweight cryptography. A proof attesting to the satisfiability of n constraints has size $O(\log ^2 n)$O(log2n); it can be produced with $O(n \log n)$O(nlogn) field operations and verified with O(n). At 128 bits of security, proofs are less than ${250}\,\mathrm{kB}$250kB even for several million constraints, more than $10{\times }$10× shorter than prior SNARGs with similar features.A key ingredient of our construction is a new Interactive Oracle Proof (IOP) for solving a univariate analogue of the classical sumcheck problem [LFKN92], originally studied for multivariate polynomials. Our protocol verifies the sum of entries of a Reed–Solomon codeword over any subgroup of a field.We also provide $\texttt {libiop}$libiop, a library for writing IOP-based arguments, in which a toolchain of transformations enables programmers to write new arguments by writing simple IOP sub-components. We have used this library to specify our construction and prior ones, and plan to open-source it.
2019
TCC
Succinct non-interactive arguments (SNARGs) are highly efficient certificates of membership in non-deterministic languages. Constructions of SNARGs in the random oracle model are widely believed to be post-quantum secure, provided the oracle is instantiated with a suitable post-quantum hash function. No formal evidence, however, supports this belief.In this work we provide the first such evidence by proving that the SNARG construction of Micali is unconditionally secure in the quantum random oracle model. We also prove that, analogously to the classical case, the SNARG inherits the zero knowledge and proof of knowledge properties of the PCP underlying the Micali construction. We thus obtain the first zero knowledge SNARG of knowledge (zkSNARK) that is secure in the quantum random oracle model.Our main tool is a new lifting lemma that shows how, for a rich class of oracle games, we can generically deduce security against quantum attackers by bounding a natural classical property of these games. This means that in order to prove our theorem we only need to establish classical properties about the Micali construction. This approach not only lets us prove post-quantum security but also enables us to prove explicit bounds that are tight up to small factors.We additionally use our techniques to prove that SNARGs based on interactive oracle proofs (IOPs) with round-by-round soundness are unconditionally secure in the quantum random oracle model. This result establishes the post-quantum security of many SNARGs of practical interest.
2019
TCC
We study the problem of delegating computations via interactive proofs that can be probabilistically checked. Known as interactive oracle proofs (IOPs), these proofs extend probabilistically checkable proofs (PCPs) to multi-round protocols, and have received much attention due to their application to constructing cryptographic proofs (such as succinct non-interactive arguments). The relevant complexity measures for IOPs in this context are prover and verifier time, and query complexity.We construct highly efficient IOPs for a rich class of nondeterministic algebraic computations, which includes succinct versions of arithmetic circuit satisfiability and rank-one constraint system (R1CS) satisfiability. For a time-T computation, we obtain prover arithmetic complexity $O(T \log T)$ and verifier complexity polylog(T). These IOPs are the first to simultaneously achieve the state of the art in prover complexity, due to [14], and in verifier complexity, due to [7]. We also improve upon the query complexity of both schemes.The efficiency of our prover is a result of our highly optimized proof length; in particular, ours is the first construction that simultaneously achieves linear-size proofs and polylogarithmic-time verification, regardless of query complexity.
2017
EUROCRYPT
2017
EUROCRYPT
2017
TCC
2017
JOFC
2016
TCC
2016
TCC
2015
EPRINT
2015
EUROCRYPT
2014
CRYPTO
2014
EPRINT
2014
EPRINT
2014
EPRINT
2013
TCC
2013
CRYPTO
2012
CRYPTO

Crypto 2019
Eurocrypt 2018
TCC 2017


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As of May 4 2007 the scripts will autodetect your timezone settings. Nothing here has to be changed, but there are a few things

## Wednesday, August 31, 2011

### Fermat on arithmeticians

Thanks to Sol Robeson ( in PI ) we call mathematicians who lost it "numerologists".

In 1657, Fermat challenged William Brouncker, of Castle Lynn in Ireland, and John Wallis to find integral solutions to the equations $$x^2 − 151y^2 = 1$$ and $$x^2 − 313y^2 = −1.$$ He ( Fermat ) cautioned them not to submit rational solutions because even the lowest type of arithmetician could devise such answers.

"An Introduction to Diophantine Equations, A Problem-Based Approach, Andreescu, Andrica & Cucurezeanu, Springer 2010"

Considering that Fermat used the qualification the lowest type of arithmetician there must have been a ranking in the computational branch those days. Until at least WW2, a computer, was the job description of someone who did "computational work" in banking, insurance, trading, logistics and what have you. Jobs like that exist even now, think of the actuarial sciences, but most of them if not all require a degree in mathematics. I am not sure but I suppose that in Fermat's days there must have been people responsible for the basic addition and multiplication type of calculations. Fermat called them "arithmeticians, of the lowest kind".

I am speculating of course. Fermat could have been a terrible arrogant man looking down on the working class. Considering that he was not a mathematician himself but that he wrote, on his own initiative, letters to the great minds of his time says at least something of his self-image.

Link: My previous post on Fermat

## Monday, August 29, 2011

### Continued fractions (3)

Each rational number can be represented as a finite continued fraction (FCF) and each FCF represents a rational number. We have seen how to calculate the rational number from a given FCF, in this post we show how to calculate the FCF for any rational number.

For example, the FCF representation of $\frac{17}{13}$ can be calculated as follows:

 $a$ $q$ $b$ $r$ $17$ $1$ $13$ $4$ $13$ $3$ $4$ $1$ $4$ $4$ $1$ $0$

The value of the FCF is contained in the second column from top to bottom: $\frac{17}{13}$ is $\left[ 1,3,4 \right]$. This is clearly an application of Euclid's algorithm for calculating the GCD of two integers. The algorithm for calculating the GCD stops at row $3$ but by adding one more row containing $4 = 4 \times 1 + 0$ the column containing the FCF is complete.

- Continued fractions (1)
- Continued fractions (2)
- Continued fractions (2a)

## Sunday, August 28, 2011

### Contined fractions (2a)

I found a better way to present the table which shows the algorithm for calculating continued fractions:

 $k$ $a_k$ $p_k$ $q_k$ $C_k$ $-1$ $0$ 1 $0$ $1$ $0$ $k$ $a_k$ $a_k \cdot p_{k-1} + p_{k-2}$ $a_k \cdot q_{k-1} + q_{k-2}$ $\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

- Continued fractions (1)
- Continued fractions (2)

## Saturday, August 27, 2011

### Continued fractions (2)

A finite continued fraction (FCF) is a map $$f: \mathbf{N}^m \rightarrow \mathbf{Q}$$ $$\left( a_1, a_2, \cdots a_m \right) \mapsto a_1 + \frac{1}{a_2 + \frac{1}{\ddots + \frac{1}{a_m}}}$$

Continued fractions are calculated by creating a table of convergents, as follows:

 $k$ $a_k$ $p_k$ $q_k$ $C_k$ $-1$ $0$ $0$ $1$ $0$ $1$ $a_1$ $a_1 \cdot p_0 + p_{-1}$ $1$ $\frac{p_1}{q_1}$ $k$ $a_k$ $a_k \cdot p_{k-1} + p_{k-2}$ $a_k \cdot q_{k-1} + q_{k-2}$ $\frac{p_k}{q_k}$

The table consists of $m+2$ rows. The value of the FCF is $\frac{p_m}{q_m}$.

To be continued.

## Friday, August 26, 2011

### Study Tip - 2

Do you know any professional musicians, dancers perhaps? History shows that art and mathematics thrive in the same places. I, sadly, don't. Although yesterday I witnessed a pianist's daily practice routine. It started with loosening up the muscles. Then, slowly, player and instrument become one sound generating machine. To me this explained why musicians can have such deep relationships with their instruments. The musicians we see on stage ( all of them, not just the 'stars' ) have practiced at least ten-thousand hours to reach that level. I don't think it's such a bad guess to say that, any mathematician who is at the forefront and is creating new mathematics, carries at least the same weight of practice hours on his belt as a professional in the arts or an athlete.

#### Tip 2: Exercise daily

Work on a difficult exercise every day. Find a booklet with Olympiad level exercises or any book with exercises that are a challenge for -you-. Exercises you can do won't make you better. Hard exercises do.

## Thursday, August 25, 2011

### Continued fractions (1)

Generating functions are "mathematical data structures" that can store an infinite amount of data.

For example $$\frac{1}{1-x} = \left\{ 1,1,1, \cdots \right\}$$ and $$\frac {1}{1-x-x^2} = \left\{ 1,1,2,3,5,8,13, \cdots \right\}$$ nicely represents the Fibonacci series. ( The existence of tools like the GF's made me sort of addicted on mathematics. ) If you think this is the most compact way to describe the Fibonacci series, then let mathematics surprise you. The most compact way to describe the Fibonacci series is $$\left[ <1> \right]$$ which means $$1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \cdots }}}.$$ Objects like this are called continued fractions, more on these and why $\left[ <1> \right]$ is related to the Fibonacci series in the next post.

## Wednesday, August 24, 2011

### Creating new mathematics ( ... )

Suppose your assignment was to teach a group of friendly aliens, just arrived from the Pleiadians, the rules of the game of chess. A student has completed the course with success if he is able to play a game according to the rules. That's not too difficult you think considering the number of six year olds able to play a descent game of chess. Your study materials are: lots of chalk and a blackboard. No chess pieces, no boards are available in class. Your teaching assistant will type your lecture as you speak, therefore it is not allowed to use drawings or symbols that can't be typed instantly.

This may seem difficult ( it is ), but compare it with the creation of new mathematics (...)

Mathematics is a parallel universe which we can enter with our mind only. Although bodiless, exterior, we are free to travel in this spectacular universe. When we come back however we lack the words to describe our observations, to communicate what we have seen with our mental ( mathematical ) eyes. Each and every observation must be recorded and analyzed before we can attempt to describe it. Definition by definition we try to create a consistent picture of what we have 'seen'. In this notion of mathematics, for example the number e always existed, it just took an Euler to describe it properly. Obviously the mathematical world is not some parallel library where we can go to and lookup the answers to the current unsolved problems. - In a BBC documentary about Andrew Wiles and Fermat's last theorem, Wiles describes his research as entering some space, then by touching things by hand in the dark he had to form a mental picture of what's in that space, and so on.

 Describe what you observe (...)

## Tuesday, August 23, 2011

### Study tip - 1

( For a while I have been thinking about writing -the- great (...) post listing zillions of study tips. Since it is not hard to imagine such a post will never be written I just start with tip 1 ( in random order ), then see how far I will get and maybe, one day, compile them into one post or page. )

#### Tip-1: The next item on the (study-)list

If you want to start studying immediately in the time you have allocated for study make sure you know -exactly- what you are going to do when you start. Don't lose time on deciding if you are going to read, revise, do exercises, work on assignments or whatever it is that you do for studying. The best time to plan a session is at the end of each study session. This already structures a session into study / plan next session. This plan can be as short as 'Do TMA questions 2 and 3'. Or 'Read pages 12-28'. Very quickly go through it and write your plan down in your agenda ( whatever system you use ). Visualize yourself starting the next session and starting with these tasks. - The trick is that your subconscious already starts working on it. Programs, prepares you for the task. Next session, starting the task will be easy and enjoyable. It works. Make a habit of it.

## Saturday, August 20, 2011

### Mandelbrot fractals in 3D

The laptops we use today are extremely powerful machines if you measure them against the standards of a decade ago. In these days producing ( rendering ) Mandelbrot fractals was hard work for any computer. Also, Mandelbrot fractals were 2D, by definition, case closed. Experimenting with '3d-type-of-Mandelbrots' was impossible due to the limitations of the hardware. A lot has happened since then. - Daniel White created a website about the topic, called 'The unravelling of the Real 3D Mandelbulb" where he explains the interesting ( and surprising ! ) history of 3D Mandelbrots.

### Exercise

Exercise:
Find $x, y$ such that $$\frac{1}{x} + \frac{1}{y} = \frac{1}{pq}$$ where $x,y \in \mathbf{Z}$ and $p,q$ are prime.

Hint: there are nine different solutions. I'll publish the method and solution on request ( comment ).

## Friday, August 19, 2011

### About mathematics at the Open University

There are of course many differences between studying ( mathematics ) at the Open University and studying math at a brick university. The main difference is of course the main method of delivering knowledge: course booklets versus lectures with accompanying lecture notes. A brick university course is often based upon some textbook. Homework includes reading assignments and exercises. An Open University booklet is a mix of theory, worked examples and exercises. If the method of presentation matches the way you like to learn math following a course is easy. - The way mathematics is presented in textbooks (at the advanced undergraduate or graduate level ) is however completely different. If you learn all your math from Open University booklets this may come as a shock, since the skill to read mathematics books hasn't been developed. Compare a graduate math book in your field of interest with one of the level 3 booklets to see what I mean. Or to put it differently: you have not been initiated in the ( secret ) protocols of how mathematicians communicate.

The challenge can best be met by attempting to solve the exercises without recourse to the hints. The density of information in the text is rather high; a newcomer may need one hour for one page. Make sure to have paper and pencil at hand when reading the text.

Wolfgang Rautenberg in "A Concise Introduction to Mathematical Logic 3rd edition, Springer 2010, preface"

### When nerds fall in love...

Most shapes can be described by one or more equations, the human imagination does the rest. The equation (x^2 + 9/4 y^2 + z^2 - 1)^3 - x^2 z^3 - 9/80 y^2 z^3 == 0 describes a surface representing the form of a heart.

 Click to enlarge

( From an idea in the Mathematica docs.)

## Tuesday, August 16, 2011

Just found out that my personality resembles the fictional characters George Smiley, John le Carre's master spy but alas also Professor Moriarty, Sherlock Holmes' nemesis or the worst of all Hannibal Lecter (Silence of the Lambs). Real characters in my personality group aren't much better: Arnold Schwarzenegger, Donald Rumsfeld, Hillary Clinton and Michelle Obama. - I wouldn't have said that I look like any of them, not counting George Smiley. The description of the personality however is correct.
I am an INTJ type. The predicted job matched too: computer programming. Lots of scientists have this profile as well. Famous ones are Stephen Hawking and Isaac Newton.

Do the test yourself here. The test is based on 16 types originally described by Jung.

## Sunday, August 14, 2011

### Mathematics: how hard is it really?

Let's be honest: mathematics is hard.
I've been postponing a full entrance into the life of a mathematician since the beginning of my excursions into mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. I admire the men and women who have come before me and were able to put in the work necessary to stand in their tower and look upon the landscape they greatly desire to see farther and farther into. That landscape is, of course, the mathematical landscape and it is a beautiful and terrifying scene to gaze upon. It is terrifying, do to its clean, sterile, and powerful nature. I am ready, however, to fully embark on this journey, which I am afraid will consume my life. But it is a necessary sacrifice to make. I am not the most talented mathematician. And because of the extent of the work that is out there and the pace at which mathematics is moving today, to truly make a mark in the mathematical world I must devote more time and energy to this calling than I have to any other task thus far. - David Andrews ( Read his blog if you like his thoughts. )
But: is it, really?

Isn't mathematics easier than say particle physics? Look at the thousands of people working with billion dollar equipment at CERN. What about the rules and regulations in biotechnology? All sciences use mathematics to the limit, mathematicians have only math to worry about. - Or what about this one: economy must be the most difficult science because economists fail again and again in their forecasts and there is no consensus among economists which way leads us out of the depression.

What do you think? Is mathematics hard? If so, what in particular makes mathematics hard?

## Friday, August 12, 2011

I found an interesting book.

It's hard to describe the book or assign it to a category. Let me give you two quotes from the book.
Mathematicians always strive to confuse their audiences; where there is no confusion there is no prestige.
All numbers are interesting, since the first uninteresting number would be interesting.
If you like mathematics and you are ready for some light reading while giving the impression you are reading really hard stuff then this book might be for you.

A handbook for the perplexed
Carl E. Linderholm
( Unbelievable it is out of print in the age of Kindle, iPad and what have you. )

### The Code - Episode 3 - Prediction ( with Marcus du Sautoy )

Just like the orbits of the planets life follows a pattern. It can be reduced to cause and effect. Everything can be represented by numbers, and thus has mathematics at his heart. Strip everything away and mathematics remains.

The Code

Beautiful TV, the BBC at it's best. Series verdict 8.5/10. Not a 10 because given previous series with Marcus du Sautoy, and the promising title "The Code", my expectations were too high. I had the idea it was mostly about physics, ( of which some say is just another branch of mathematics ).

Episode 3 starts off with a tale about Columbus, lunar tables and the moon eclipse. Given the regular movement of the planets it is possible to forecast a moon eclipse.

"The Code is such a powerful thing that I entrust my life to it" says du Sautoy. He calculates the arc of a ball which is rolled off some ramp and takes a seat close by where the ball should land. Classical mechanics is predictable.

Denmark. Flocks of starlings. A single flock can contain a million birds. The Black Sun they call them. Suprisingly flocks can be modelled mathematically.

On to America. We meet a detective with a Ph.D in mathematics hunting for serial killers. ( I would rather consult Charlie Eppes though. Charlie is an expert in -all- branches of mathematics. ) This detective said that he studied the Jack the Ripper case and worked out the address of Jack the Ripper. Not bad for a case as old as 888. My prediction is that Jack the Ripper is dead by now.

Knowing the series is part of an Open University course I am not surprised that the logistic map made an appearance. If you intend to study math at the Open University be prepared. The logistic map is hard, very hard. The logistic map is what makes mathematicians modest, humble almost. - Suppose there is a God, a creator, who used mathematics as a language in the creation of the universe. Wouldn't he/ she/ it be so clever to -secure- the creation from having it cracked from creatures like us? If you assume that then the logistic map, chaos theory, the complexity of the primes, Goedel's theorems and all that could be just firewalls.

Anyway, du Sautoy shows that we can use the logistics map to understand the dynamics of ( lemming ) populations but we won't be able to predict populations with it. Weather systems ( and I suppose stress systems related to earthquakes ) are bound in a similar manner.

New York. Patterns everywhere, of course. ( Echoes of Max Cohen? ) There seems to be a 15% rule for cities. It says that when a city doubles in size everything gets better by 15%. Then I lost it:

You have 15% more restaurants to choose from, 15% more art galleries ( .... ).

Here the producers had to show off their class, I suppose. A bit insensitive in such harsh economic times. Not good in my opinion. - The evening The Code part 3 was broadcast a friendly between England and The Netherlands was canceled. Due to class related riots in London.

## Thursday, August 11, 2011

### High IQ

I have seen the new ape movie 'Rise of the Planet of the Apes'. ( Average user rating on IMDB is 8.0, today. ) It's a fun movie really, for every member of the family, not that it's a comedy, it's an action thriller with a touch of Splice and Avatar.

To the point, this is a mathematics blog.

When the movie started to work on me I thought about the great minds of tomorrow, but still children today. Some of which may be struggling with the fact that they seem somehow 'different'. Some are lucky and are recognized as children with high IQ. But others may be entirely surrounded with average or low IQ teachers, parents and friends. The most gifted child is probably ( as in statistics ) born in India, China or Africa, which makes the chance we'll ever benefit from his or her gifst quite remote. - Again, I thought of Ramanujan. - Part of being gifted with a high IQ is waking up to it. Life is miserable when you have to live with monkeys. Although officially diagnosed with some disorder in the DSM there is nothing wrong with most middle-aged, highly sensitive, but depressed people, except that they did not wake up to their IQ.

The main character in the movie is a research scientist working for a pharmaceutical company. His father has Alzheimer's disease so that's the area he is researching. He is working on a promising drug which reached the stage for testing on apes. Something goes wrong and all test apes must be killed. One of the apes was carrying a baby and is saved and raised by people. The movie is the story of that ape, Caesar who inherited the gene modifications from his mother. Caesar physically develops as an ape but his intelligence is higher than that of any human. He is of course completely aware of his situation. Society demands Caesar is locked up with other apes. Despite his intelligence he is powerless among his peers. The first thing the apes do is humiliating him by ripping of his clothes and tearing them apart. - The most intelligent person on the planet is also the most lonely person on the planet.

I know of a group that has all sorts of programs to develop the mental capabilities of its members but the message is that you can't use them when you are on your own. Only when you are in a group of equals you can flourish and prosper. - Low IQ people bring you down, if they must by force.

## Monday, August 8, 2011

### Proof that there are only five regular polyhedra.

In my previous post I wrote that Euler's formula implies that there can only be five regular polyhedra and that this can be shown by simply solving a Diophantine equation. In this post I will demonstrate ( read: prove ) this and I will show how to solve Diophantine equations in Mathematica.

We start with Euler's polyhedron formula $$F - E + V = 2,$$ where $F=$ number of faces, $E=$ number of edges and $V=$ number of vertices. What exactly makes a polyhedron regular? Clearly, all the faces are equal ( i.e. all triangles ) of a regular polyhedron. This property alone is not enough though. Also, on a regular polyedron the same number of faces meet at each vertex. Clearly the number of edges remains an unknown for now. These requirements can be related to E, the number of edges, if we introduce two ( new ) variables:
- $A$ : the number of edges surrounding a face. ( i.e. $A = 4$ for a cube ).
- $B$ : the number of edges meeting at a vertex. ( i.e.$B = 3$ for a cube ).

We know the following about a regular polyhedron: $$FA= 2E$$ $$VB = 2E$$ $$F-E+V=2E$$ $$A,B \geq 3$$ $$F,E,V \geq 1$$ ( Note that I have used $FA=2E$ ( instead of $FA=E$ ) because each edge is part of two faces, as well as $VB = 2E$ because each edge connects two vertices. )

By simply algebraically reworking the equations above we get $$\frac{2E}{A}-E+\frac{2E}{B}=2.$$ In this particular case there are two approaches to attack this Diophantine equation. ( Named after Diophantus, 3rd century. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations.) We can use trial and error or we can use a tool like Mathematica. I have used Mathematica as follows. The tool to solve Diophantine equations in Mathematica is Reduce. In Mathematica Reduce[expr, vars, dom] reduces the statement expr by solving equations or inequalities for vars and eliminating quantifiers and does this over the domain dom. We solve $\frac{2E}{A}-E+\frac{2E}{B}=2$ in Mathematica as follows using Reduce:
Reduce[{(2 e)/a - e + (2 e)/b == 2, a >= 3, b>= 3, e >= 0}, {a, b, e}, Integers]

Out[1]=
(a == 3 && b == 3 && e == 6) ||
(a == 3 && b == 4 && e == 12) ||
(a == 3 && b == 5 && e == 30) ||
(a == 4 && b == 3 && e == 12) ||
(a == 5 &$b == 3 && e == 30)  So there are indeed five solutions! Let's look at them more closely.  F V E Type$\frac{2 \cdot 6}{3} = 4$triangle$\frac{2 \cdot 6}{3}= 4$6 Tetrahedron$\frac{2 \cdot 12}{3} = 8$triangle$\frac{2 \cdot 12}{4}= 6$12 Octahedron$\frac{2 \cdot 30}{3} = 20$triangle$\frac{2 \cdot 30}{5}= 12$30 Icosahedron$\frac{2 \cdot 12}{4} = 6$square$\frac{2 \cdot 12}{3}= 8$12 Cube$\frac{2 \cdot 30}{5} = 12$pentagon$\frac{2 \cdot 30}{3}= 20$30 Dodecahedron  A=5 (edges per face), B=3 (edges to a vertice ) = Dodecahedron. This proof clearly shows that the various disciplines ( in this case topology, number theory and geometry ) in mathematics are related and thus pointing to mathematics at a deeper layer . It has been said that graph theory, geometry, algebra and number theory are just different manifestations of the same mathematical concepts. I recently read that the Riemann Hypothesis ( analytical number theory ) can be proved by proving an equivalent theorem in graph theory. Deep stuff, surely. ## Thursday, August 4, 2011 ### The Code - Episode 2 - with Marcus du Sautoy The Code part 2 is about shapes. Descartes ( rediscovered by Euler ) stated that for any polyhedron the following identy holds $$F-E+V=2.$$ Where$F$is the number of faces,$E$is the number of edges and$V$is the number of vertices. A tetrahedron, for example has$4$vertices,$4$faces and$6$edges:$4 - 6 + 4 = 2$. One of the most beautiful and fascinating mathematical theorems I know states that this formula implies that there are only five regular polyhedra. A theorem that doesn't involve any deep topology or geometry, given$F-E+V=2\$ all it takes is some number theory, i.e. solving some Diophantine equations and modular congruences.

Du Sautoy demonstrates the 2D version of the formula above in this episode. He shows that in 2D there can only be three regular lattices created from a regular polyhedron. He visits a beehive and shows how bees create lattices based on perfect pentagons. He then calculates the amount of required wax for the three possibilities and concludes that the pentagon is the best solution. "Nature is lazy", he says and is obviously part of The Code. The fascinating fact here is that nature, in the form of the bee, "knows" this, and for thousands of years. The knowledge about the pentagon and the skill to create pentagons seems encoded in the bee lifeform. Then, what is nature? And what exactly is the role of mathematics? That seem to be the questions he is trying to answer.

From there on Du Sautoy shows us more regular polyhedra in nature. Like a virus in the shape of an icosahedron.

He visits a salt mine with perfectly cube shaped crystals and using a model of the molecule explains the creation of the cube.

If you look closer to the shapes however the creations are not mathematically perfect. Are we using mathematics merely to understand nature? Or is nature driven by mathematics? Fractals are introduced to explain tree like shapes. But although close mathematically perfect fractal-type-of trees don't exist either.

Next week episode 3.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)


text:
# I Problem with Planck units

1. Jul 9, 2016

### facenian

Since $c=1$ we have $[l]=[t]$,
$\hbar=1$ and from Einstein law $E=\hbar\omega\rightarrow [E]=[t^{-1}]$,
$G=1\rightarrow [E]=[m]^2[t^{-1}]$
then we must have $[m]=1$ but since $E=mc^2\rightarrow [E]=[t]^2$
Can someone please point out my mistake

2. Jul 9, 2016

### Staff: Mentor

To my knowledge Planck units are not geometrized, so for example c still has dimensions of L/T. So Energy has the usual dimensions of L^2 M/T^2

3. Jul 10, 2016

### facenian

I should have said "but since $E=mc^2\rightarrow [E]=[1]\neq [t^{-1}]$"

4. Jul 10, 2016

### vanhees71

I think you've got everything right. What Planck intrigued so much about his units was that then physics is in a sense "absolute", i.e., when measuring all quantities in Planck units they are all dimensionless "absolute" numbers.

5. Jul 10, 2016

### Staff: Mentor

They are if you are referring to these Planck units:

https://en.wikipedia.org/wiki/Planck_units

Saying that the Planck units referred to above set $G = 1$ is a little misleading. Actually, what they do is replace $G$ with $1 / m_p^2$, where $m_p$ is the Planck mass (possibly with a factor of $16 \pi$ stuck in there somewhere, depending on the equation). For example, consider the Newtonian equation for gravitational potential energy:

$$U = - \frac{G m_1 m_2}{r}$$

Here $U$ has conventional units of energy. Now suppose we write this same equation in Planck units, in which $[E] = [l^-1]$ (which, as you note, is implied by $c = \hbar = 1$); we will have (if we keep the units of mass the same as the units of energy, which is implied by $c = 1$):

$$U = - \frac{m_1 m_2}{m_p^2 r}$$

The Wikipedia page on Planck units describes this as measuring masses in units of the Planck mass--i.e., it would write $U = - m_1 m_2 / r$, making it look like the units have $G = 1$, but only because $m_1$ and $m_2$ do not have units of mass but are dimensionless numbers. I would prefer to describe it as setting $G = 1 / m_p^2$; this is what many QFT textbooks do, for example, when discussing quantum gravity.

6. Jul 10, 2016

### Staff: Mentor

Hmm, to me tables 1 and 2 both indicate that the Planck units are considered to have SI-like dimensionality.

However, I should be candid that I don't have an authoritative source, just that Wiki page.

7. Jul 10, 2016

### facenian

I think table two only gives the equivalents of plank units in SI units, but Planck units are dimensionless

I read the Wiki page and I believe things have cleared up for me now. I will tell you how I understand it now and ask for you(all) opinion about it.
In Planck units rather than putting physical constants like c,G,etc. equal to one(dimensionless) we rather get rid of them by division making them disappear from the equations so that the physical equations become relations between pure numbers with no dimensions at all.

Last edited: Jul 10, 2016
8. Jul 10, 2016

### facenian

I think I haven't got it right because I tried to assign dimensionality to the magnitudes which must be, as you pointed it out, dimensionless

9. Jul 10, 2016

### facenian

I believe that $[G]=\frac{1}{[m]^2}$ comes from the requirement $c=\hbar=1$ the problem arises when we put $c=\hbar=G=1$ because this leads to a contradiction in dimensionality that caused me the problem that made me post this thread. The Wiki page puts $c=\hbar=G=etc.=1$ and think this is misleading as you said(it migtht be plain wrong)

10. Jul 10, 2016

### Staff: Mentor

Yes, that is my understanding. Basically, any time a "length" appears in an equation, we put in a number representing the ratio of that length to the Planck length; for a "time", we use the ratio of that time to the Planck time; for a "mass", the ratio of that mass to the Planck mass; etc. This makes all numbers appearing in all equations dimensionless.

11. Jul 11, 2016

### Staff: Mentor

That is a good approach. It completely sidesteps the question of dimensionality. It doesn't matter if energy is considered to have dimensions of M or L^2 M/T^2 or even just E. It always shows up as the dimensionless ratio E/Ep

12. Jul 11, 2016

### vanhees71

Well, I think it's pretty simple, at least from the point of view of a theorist. You can define a complete unit system by fixing some purely conventional constants which appear in the SI units, because you have to convert between man-made units that don't fit to the fundamental laws of nature. This is understandable since in Newtonian mechanics there is no fundamental constant. The first time a fundamental constant occured in the history of physics was Maxwell electromagnetism, where the "speed of light" entered the game, which is a universal unit in dimensions of a speed. Nowadays we know it's much more fundamental than that. It's a fundamental property of spacetime as described in the general (and thus also special) theory of relativity. In the SI it's defined as an exact value which was chosen such as to make the relation between the base unit of time (second) compatible with the base unit of length (metre). However, these are conventional units, and thus you can as well set $c=1$, which is very natural, because space and time occur in the theory of relativity in a very "symmetric" way, although there is a distinction, because you need a causality structure to do physics. That's implemented in GR by using a pseudometric with signature (1,3) (or (3,1) which is equivalent) in the description of spacetime as a pseudo-Riemannian continuum.

The next fundamental constant entering the history of physics was Planck's "quantum of action". It's also purely conventional and will probably be fixed to a certain value very soon by the international committee defining the SI. Anyway, for a theoretical physicist it's as inconvenient to keep it as it is inconvenient to keep $c$. So we set it to 1. Now we have only one unit left. That you can choose as some length, time, or energy (momentum or mass). In high-energy particle and nuclear physics we are used to such units and usually choose GeV as units for energies, momenta, masses and fm as units for lengths and times (as well as barns for cross sections. To convert between these choicses you only need one number, namely $\hbar c \simeq 0.197 \mathrm{GeV} \, \mathrm{fm}$ and $10\mathrm{mb}=10^{-30} \mathrm{m}^2=1 \mathrm{fm}^2$.

The SI units are even more inconvenient, because they introduce some (from the point of view of the now established laws of nature superfluous) additional base units. One is the Ampere for electric current. In natural units it's totally superfluous, and you need to remember one more conversion factor between components of the electromagnetic field defined by $\mu_0$ and $\epsilon_0$ with the constraint that $1/(\mu_0 \epsilon_0)=c^2$ (i.e., you measure parts of one and the same quantity in different units; that's a bit like in the US where you measure distances in miles and heights in feet and inches). Another unnecessary base unit is Kelvin for temperature, which needs another conversion factor (the Boltzmann constant $k_{\text{B}}$) between units of energy and temperatures. It's pretty arbitrary using the triple point of water and thus also pretty likely to be substituted by using simply a definition of $k_{\text{B}}$ as with the speed of light $c$.

Anyway, in Planck units all quantities are expressed in terms of dimensionless numbers and in this sense it's the most natural system of units, defining all quantities in terms of fundamental universal constants (as far as we know today in terms of our most fundamental theories, which however can turn out to be wrong in future observations!).

Now in General Relativity there's one more fundamental unit, namely the universal coupling between the gravitational field (represented by the curvature tensor of spacetime) and the energy-momentum-stress tensor of matter and radiation, which conventionally is given by Newton's gravitational constant $G$. Setting also this constant to 0, we are done in the sense that this also fixes the remaining freedom of the choice in the system of natural units described before. Now all quantities are given by dimensionless numbers. To convert to conventional SI units, you only need in addition the corresponding length or energy scale which is Planck length or Planck mass.


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# Rotational Kinetic Energy

Written by Jerry Ratzlaff on . Posted in Kinematics

Rotational kinetic energy, abbreviated as $$KE_r$$, is the energy an object has because it is rotating and does not matter if its moving from one place to another.

## Rotational Kinetic Energy formula

$$\large{ KE_r = \frac {1}{2}\; I \; \omega^2 }$$
Symbol English Metric
$$\large{ KE_r }$$ = rotational kinetic energy $$\large{ lbf-ft }$$ $$\large{ J }$$
$$\large{ \omega }$$  (Greek symbol omega) = angular velocity $$\large{\frac{deg}{sec}}$$ $$\large{\frac{rad}{s}}$$
$$\large{ I }$$ = moment of inertia $$\large{ in^4 }$$ $$\large{ mm^4 }$$


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# Carpool ***

Some friends have organized carpool them-self.
At the beginning of each accounting period, they write down the numbers of the car's odometer.
Every time somebody drives the car, she/he writes down the numbers when the car is returned and hers/his name.

carpool.dat

17000  start
17100  Daniel
17220  Sara
17310  David
17410  Daniel
17550  Sara
17800  David

Try to write an awk script that calculates what distance has everyone traveled with the car. Hints: Differences between numbers in two consecutive lines might be a good start. Associative arrays might come handy.

           Sara: 260
Daniel: 200
David: 340

The problem is rather easy to solve with awk (it is my humble opinion). Can it be solved easier with some of your favorite tools? (I am interested of alternative approaches)

Possible solution:
awk '{dist[$2]=dist[$2]+($1-prev); prev=$1} END{ for (i in dist) if (i !="start") printf("%15s: %g\n",i,dist[i]) }' carpool.dat

solution in Python (credits to Mihai Croicu)

import pandas as pd


text:
My PhD field involves a lot of statistic, so I usually need to write math equations both by hands and, more convenient, on the internet. This is my solution.

### Use MathJax for Jekyll

• insert the code in the _layouts/default.html
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
tex2jax: {
inlineMath: [['$','$'], ['\$','\$']],
processEscapes: true
}
});
</script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.0/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>

The first part is used to enable inline equation.

### Use LaTex to write equations

I found LaTex/Mathematics is wildly used to write equations on the website, such as rmarkdown.

For example, use $$mean = \frac{\displaystyle\sum_{i=1}^{n} x_{i}}{n}$$ to write

Use k_{n+1} = n^2 + k_n^2 - k_{n-1} to write


text:
# How does QFT interpret the Negative probability problem of the real scalar fields' Klein-Gordon equation?

I am totally a beginner in QFT, here's the problem that I got: for the real scalar fields, are there any elementary particles descriped by them. If so, how to understand the negative probability problem?

• What is the negative probability problem? (And yes, the Higgs particle is the excitation of a scalar field) – ACuriousMind Jan 19 '15 at 15:05
• QFT does not have a negative probability problem. On the other hand relativistic quantum mechanics (RQM), meaning taking wave function solutions rather than operator solutions to the KG equation, does have that problem. This is one of the reasons QFT was invented. – BMS Jan 19 '15 at 17:45

Quantum field theory solves the problem by giving a different interpretation to the "probability". In the case of complex fields, quantum field theory also introduces antiparticles.

In the first-quantized Klein-Gordon equation, the time component $j^0$ of the probability current vector $j^\mu$ may indeed be both positive and negative and negative probabilities are bad, as you point out.

However, $j^\mu$ is a bilinear expression constructed from the field $\phi$ and its derivatives, roughly $\phi\cdot \partial^\mu\phi$, and when $\phi$ becomes a quantum field (instead of a wave function), which is an operator (or operator-distribution), $j^\mu$ becomes an operator, too. There is nothing wrong about $j^0$ being positive or negative (indefinite) because it defines the charge density for a complex field $\phi$.

In quantum field theory, the wave functions that could have had both positive and negative probabilities are used as prefactors in formulae for quantum fields and the positive-energy (and positive-probability) and negative-energy (and negative-probability) solutions for the wave function are treated asymetrically. The latter must be multiplied by the creation operator and the former by the annihilation operators.

In effect, it means that in quantum field theory, we may "create" an arbitrary number of particles in wave functions that are allowed by the first-quantized (one-particle) quantum mechanical theory but we are only allowed to use the positive-energy (positive-probability) wave function to excite the vacuum. The negative-energy ones are multiplied by annihilation operators which annihilate the vacuum so we get no state.

Those things may only be properly understood along with the full mathematical apparatus of QFT and it is covered in every course or textbook on quantum field theory.

• so that's how it is.It sounds like I need more work on QFT. – Shane Jan 20 '15 at 14:05


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# $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists? [duplicate]

The question is the following: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, and for all $x \neq 0$, $f'(x)$ exists. If $\lim_{x \rightarrow 0} f'(x) = L$ exists, does it follow that $f'(0)$ exists?

My intuition is that this does not have to hold in general. However, I keep finding counter examples such that $f'(0)$ exists where $\lim_{x \rightarrow 0} f'(x) = L$ does not exist, which is obviously not the original question.

## marked as duplicate by Hans Lundmark, José Carlos Santos, Sahiba Arora, Yujie Zha, Antonios-Alexandros RobotisAug 6 '17 at 13:38

Note

$$\lim_{h \to 0^{+}} \left(\frac{f(h) - f(0)}{h}\right) = \lim_{h \to 0^{+}} f'(\alpha) = L$$

where $\alpha \in (0,h)$. To be precise, the hypotheses of the mean value theorem are satisfied since $f$ is assumed continuous on $[0,h]$ and differentiable on $(0,h)$.

The above limit is necessarily $L$, since the $\alpha$ is forced to $0$ as $h \to 0^{+}$, and we know $\lim_{h \to 0} f'(h) = L$. This can easily be formalized with $\epsilon-\delta$ if so desired.

Similarly, $$\lim_{h \to 0^{-}} \left(\frac{f(h) - f(0)}{h}\right) = \lim_{h \to 0^{-}} f'(\alpha) = L$$

where $\alpha \in (h, 0)$.

This implies

$$\lim_{h \to 0} \left(\frac{f(h) - f(0)}{h}\right) = L$$

and so $f'(0)$ exists.


text:
# Re: How to pass marginparwidth to includegraphics

Great. Thanks a lot for the quick reply.

------- Original Message -------
On Thursday, February 24th, 2022 at 11:25 PM, Paul A. Rubin via lyx-users
<lyx-users@lists.lyx.org> wrote:

> On 2/24/22 16:18, tush via lyx-users wrote:
>
>> I have some figures which I want to include in the marginal part of the
>> document.
>> The graphics to be included should be of width \marginparwidth.
>> How can I pass this width to the include graphics tool (Insert->Graphics)
>> within the editor?
>>
>> In plain LaTeX I would have typed
>>
>> \begin{marginfigure}
>> \includegraphics[width=\marginparwidth]{file.png}
>> \end{marginfigure}
>>
>> (I created a customized marginfigure inset with a module file)
>>
>> So after inserting the graphics file, I don't see any way to specify the
>> above width.
>>
>> The only options I have are those in the attached picture, non of which is
>> marginparwidth.
>
> Click the "LaTeX and LyX options" tab and enter "width=\marginparwidth" in
> the "LaTeX options" box.
>
> Paul
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text:
Showing:
Handle Score
@Ibrahim Nash 6424
@mb1973 5704
@Quandray 5245
@akhayrutdinov 5111
@saiujwal13083 5046
@sanjay05 3762
@kirtidee18 3673
@mantu_singh 3532
@marius_valentin_dragoi 3523
@sushant_a 3459
Sum of Lengths of Non-Overlapping SubArrays
Easy Accuracy: 44.42% Submissions: 862 Points: 2

Given an array arr[] of N elements, the task is to find the maximum sum of lengths of all non-overlapping subarrays with K as the maximum element in the subarray.

Example 1:

Input: N = 9, K = 4
arr[] = {2, 1, 4, 9, 2, 3, 8, 3, 4}
Output: 5
Explanation: {2, 1, 4} => Length = 3
{3, 4} => Length = 2
So, 3 + 2 = 5 is the answer.

Example 2:

Input: N = 7, K = 4
arr[] = {1, 2, 3, 2, 3, 4, 1}
Output:  7
Explanation: {1, 2, 3, 2, 3, 4, 1}
=> Length = 7.

This is a function problem. You don't need to take any input, as it is already accomplished by the driver code. You just need to complete the function calculateMaxSumLength() that takes array arr, integer N, and integeras parameters and returns the sum of the length of all such subarrays.

Expected Time Complexity: O(N).
Expected Auxiliary Space: O(1).

Constraints:
1 ≤ N ≤ 106


text:
# Cheaha Quick Start (Deprecated)

Cheaha is a shared cluster computing environment for UAB researchers. A shared cluster environment uses a job scheduler to run tasks on the cluster to provide optimal resource sharing among users. Cheaha uses a job scheduling system call SGE to schedule and manage jobs. A user needs to tell SGE about resource requirements (e.g. CPU, memory) so that it can schedule jobs effectively. These resource requirements along with actual application code can be specified in a single file commonly referred as 'Job Script/File'. Following is a simple job script that prints job number and hostname.

#!/bin/bash
#
# Define the shell used by your compute job
#
#$-S /bin/bash # # Tell the cluster to run in the current directory from where you submit the job # #$ -cwd
#
# Tell the scheduler only need 10 minutes
#
#$-l h_rt=00:01:00,s_rt=0:00:55,vf=512M # # Set your email address and request notification when you job is complete or if it fails # #$ -M YOUR_EMAIL_ADDRESS
#$-m eas # # Tell the scheduler to use the environment from your current shell # #$ -V

echo "The job $JOB_ID is running on$HOSTNAME"

Lines starting with '#$' have a special meaning in the SGE world. SGE specific configuration options are specified after the '#$' characters. SGE offers many configuration options but above mentioned options should be useful for basic job scripts. A job script is submitted to the cluster using SGE specific commands. There are many commands available, but following three commands are the most common:

• qsub - to submit job
• qdel - to delete job
• qstat - to view job status

We can submit above job script using qsub command:

$qsub HelloCheaha.sh Your job 9043385 (HelloCheaha.sh) has been submitted  When the job script is submitted, SGE queues it up and assigns it a job number (e.g. 9043385 in above example). The job number is available inside job script using environment variable$JOB_ID. This variable can be used inside job script to create job related directory structure or file names.


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