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# LaTeX for Secretaries

How to survive without Microsoft Word.
Changing Text into Type

To start LaTeX, type this at the prompt:

latex businessletter.latex

After you press the ENTER key, LaTeX will print some cryptic messages, two of which are most important: Output written on businessletter.dvi... and Transcript written on businessletter.log. Your letter is output into a DVI (device independent) file, a universal format, which you can turn into fax, PostScript or printer files.

If there are problems, LaTeX will sometimes stop in the middle of its work and ask you for help. When that happens, make a note of the line number at which the error has been found, then keep tapping the ENTER key until you see the “Transcript written on” message or the command prompt. Then run xdvi, and after visually finding the place where the error occurred, go back to Emacs and correct your mistake. Usually it will be a missing bracket or a misspelled command.

Previewing Documents

The best way to find errors and check the document's look before you print it is to use the xdvi previewer. It works only in the X Window System. To start it, you either choose it from a menu or type xdvi businessletter.dvi & on a command line in a terminal window and press the ENTER key.

Using xdvi is easy. The buttons on the right-hand side of a window are fairly self-explanatory, and you can press the PAGEUP/PAGEDOWN keys to flip pages back and forth or use the arrow-up/arrow-down keys to scroll the page you are looking at up and down (the scroll bars work just like in MS Windows, although with a two-button mouse, you will have to press both buttons to use them). Pressing the Q key exits xdvi.

Some Unbreakable Rules

The most common errors that can be found just by looking at a page in xdvi are the overfull boxes and sudden changes in font and style. These errors usually happen when you forget to obey some of the following rules of typesetting in LaTeX:

• Each paragraph must by separated from another by a blank line (just tap the ENTER key twice at the end of a paragraph).

• No matter how many spaces you put between words, LaTeX will treat them as one space and will format it according to its own typesetting rules. (You might change this behaviour, as described in the section “Overriding LaTeX Rules” below.)

• Quotation marks are made using  and ' or and instead of ", so to typeset ''funny'', you'll need to type ''Funny'' instead of "funny".

• Ellipsis (...) is printed with the \ldots command; consequently, to typeset bye..., you ought to type bye\ldots and not bye....

• LaTeX commands should be separated from the actual text. We achieve that in one of the following ways:

that begins above]

• bye, bye\ldots my love (note: the \ character starts a command and therefore does not need to be separated with a space, also see the list of special characters below).

• bye, bye\ldots{ }my love.

• LaTeX interprets some characters as special, and to print them literally, we need to use a special command instead. See list in “Special LaTeX Characters”.

Special LaTeX Characters
• Commands can be nested, i.e., you can put one or more commands within curly brackets of another, e.g., extsf{life is \emph{wonderful}!}.

Getting Rid of Overfull Boxes

LaTeX is very good at documenting its work and puts a lot of information, including error messages, into a log file. For businessletter.latex, it will be called businessletter.log. Every error is described there as best as LaTeX can, together with a line number, which helps to quickly find the right place in a file.

Although the visual method of finding errors with the help of xdvi is probably best for a beginner, there is one category of bugs that requires a different approach—the famous Overfull box. LaTeX produces that message when it cannot properly break a line of text or hyphenate a word, which happens quite often when trying to print a long web page address.

The quickest way of finding those messages is by using grep:

grep Overfull businessletter.log | less
`

After issuing the above command in an Xterm window, you will see either an empty window (a good thing) or a list of lines where LaTeX has put the ominous word. Go back to Emacs, find the offending lines, then use xdvi to judge how to best fix your document.

______________________


text:
# How to make column bold in array?

\begin{align*}
\begin{array}{cc|>{\boldsymbol}cclccl}
X&Y&\neg&X&\rightarrow(&Y&\rightarrow&X))\\
\hline
0&0&0&0&1&0&1&0\\
0&1&0&0&1&1&0&0\\
1&0&0&1&1&0&1&1\\
1&1&0&1&1&1&1&1
\end{array}
\end{align*}

I just need the first column right to the vertical line to be bold. I couldn't figure out how to make this work; I hope someone can help.

collcell seems to help here, passing the cell contents to a \user macro (with one argument) via the array-like column specification

. . . >{\collectcell\usermacro}c<{\endcollectcell} . . .

\documentclass{article}

\usepackage{amsmath,collcell}

\begin{document}

$\begin{array}{ c c | >{\collectcell\boldsymbol}c<{\endcollectcell} c l c c l } X & Y & \neg & X & \rightarrow( & Y & \rightarrow & X)) \\ \hline 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$

\end{document}

Another possibility, without collcell, is to use a short definition like this:

\def\bstart#1\bstop{\boldsymbol{#1}}

This way, you will be able to bypass the & character at the end of each cell of the array and get the required bold math symbols.

\documentclass{article}
\usepackage{amsmath,array}

\def\bstart#1\bstop{\boldsymbol{#1}}

\begin{document}

$\begin{array}{ c c | >{\bstart} c <{\bstop} c l c c l } X & Y & \neg & X & \rightarrow( & Y & \rightarrow & X)) \\ \hline 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$

\end{document}

Enable \boldmath, but for this we need to exit from math mode, issuing \boldmath and reentering math mode. This is more conveniently done with a new column type.

\documentclass{article}

\usepackage{amsmath,array}

\newcolumntype{C}{>{$\boldmath$}c}

\begin{document}

$\begin{array}{ c c | C c l c c l } X & Y & \neg & X & \rightarrow( & Y & \rightarrow & X)) \\ \hline 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \end{array}$

\end{document}

   \begin{document}

\huge
\begin{tabular}{cc|cccccc}
X&Y&${\bf\neg}$&X&$\rightarrow$&Y&$\rightarrow$&X\\        \hline
0&0&{\bf 0}&0&1&0&1&0\\
0&1&{\bf 0}&0&1&1&0&0\\
1&0&{\bf 0}&1&1&0&1&1\\
1&1&{\bf 0}&1&1&1&1&1\\\hline
\end{tabular}

\end{document}

% This is a simple way of preparing the table.

format: plain


text:
for Journals by Title or ISSN for Articles by Keywords help
 Subjects -> CHEMISTRY (Total: 846 journals)     - ANALYTICAL CHEMISTRY (50 journals)    - CHEMISTRY (597 journals)    - CRYSTALLOGRAPHY (22 journals)    - ELECTROCHEMISTRY (25 journals)    - INORGANIC CHEMISTRY (41 journals)    - ORGANIC CHEMISTRY (45 journals)    - PHYSICAL CHEMISTRY (66 journals) CHEMISTRY (597 journals)                  1 2 3 | Last

1 2 3 | Last

 Biomass Conversion and Biorefinery   [10 followers]  Follow         Partially Free Journal    ISSN (Print) 2190-6815 - ISSN (Online) 2190-6823    Published by Springer-Verlag  [2329 journals]
• Pulsed fed-batch strategy towards intensified process for lactic acid
production using recycled paper sludge
• Authors: S. Marques; F. M. Gírio; J. A. L. Santos; J. C. Roseiro
Pages: 127 - 137
Abstract: Abstract A process has previously been optimized for production of lactic acid from recycled paper sludge (RPS), the ultimate solid waste obtained in the wastewater treatment plant of a local paper recycling mill. In this work, both separate hydrolysis and fermentation (SHF) and simultaneous saccharification and fermentation (SSF) processes were improved under batch mode in bench-scale bioreactor. The use of bioreactor was advantageous, providing higher conversion rates, together with increased yield. By running SSF in bioreactor, 73.2 g L−1 of lactic acid has been produced by Lactobacillus rhamnosus ATCC 7469 from 179 g L−1 RPS, corresponding to 76 % of maximum theoretical yield. Further process intensification, with improved kinetics and final product concentration, was achieved by applying a pulsed fed-batch strategy. With six pulsed additions of RPS (40 g at each 5 h), 108.2 g L−1 of lactic acid was produced (after 120 h, corresponding to 62 % yield). This achievement contributes to make more realistic the large-scale upgrading of RPS, a waste raw material exhibiting negative cost.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0211-0
Issue No: Vol. 7, No. 2 (2017)

• Optimizing GHG emission and energy-saving performance of miscanthus-based
value chains
• Authors: Florian Meyer; Moritz Wagner; Iris Lewandowski
Pages: 139 - 152
Abstract: Abstract Miscanthus is a high-yielding lignocellulosic crop providing up to 40 t of dry matter per hectare and year. Its biomass can be used in energetic or material utilization pathways. The goal of this study was the comparison of three different conversion techniques (combustion, second-generation bioethanol, and insulation material) for miscanthus biomass produced at five locations throughout Europe using a life cycle assessment approach. In particular, the interdependencies between the cropping location, the miscanthus genotypes, and the utilization pathways were investigated. The potential savings of greenhouse gas (GHG) emissions and fossil energy were analyzed through comparison with a corresponding substituted product system. The highest GHG savings of all scenarios investigated were achieved by heat and power production in Portugal (42.7 t CO2-eq ha−1 a−1). However, at the other four locations (Sweden, Denmark, Germany, England), bioethanol production gave the highest GHG savings. In contrast, the highest energy savings were achieved by combined heat and power generation via combustion at all five locations (up to 642 GJ ha−1 a−1). A high correlation was found between yield and both GHG-emission savings and energy savings. Biomass composition and quality showed a comparatively low impact on the results. However, the composition is assumed to have a high relevance for other impact categories not assessed within this study, such as acidification and eutrophication.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0219-5
Issue No: Vol. 7, No. 2 (2017)

• Development and experimental validation of a water gas shift kinetic model
for Fe-/Cr-based catalysts processing product gas from biomass steam
gasification
• Authors: Michael Kraussler; Hermann Hofbauer
Pages: 153 - 165
Abstract: Abstract This paper introduces an improved kinetic model for the water gas shift reaction catalyzed by an Fe-/Cr-based catalyst. The improved model is based on a former model which was developed previously in order to consider the composition and the catalyst poisons (H2S) of product gas derived from dual fluidized bed biomass steam gasification. $$\begin{array}{c}r\left({\varphi}_i,T\right)=117.8\ \frac{\mathrm{mol}}{g\ {Pa}^{1.71}\ s}\cdot \mathit{\exp}\left(\frac{-126.6\ \frac{\mathrm{kJ}}{\mathrm{mol}}}{R\cdot T}\right)\cdot {p}_{CO}^{1.77}\cdot {p}_{\mathrm{H}2\mathrm{O}}^{0.23}\cdot {p}_{CO2}^{-0.17}\cdot {p}_{\mathrm{H}2}^{-0.12}\\ {}\left(1-\frac{K_{MAL}}{K_g}\right)\end{array}.$$ Furthermore, this improved model has been validated with experimental data. The data was generated by a WGS reactor which employed a commercial Fe-/Cr-based catalyst and which processed real product gas from the dual fluidized bed biomass steam gasification plant in Oberwart, Austria. Basically, the validation showed good agreement of the measured and the calculated values for the gas composition (absolute errors of the volumetric fractions of up to 1.5 %) and the temperature profile (absolute errors of up to 21 °C) of the WGS reactor. Of all considered gas components, the CO concentration showed the highest error. The results qualify the improved kinetic model for basic design and engineering of a WGS reactor employing a commercial Fe-/Cr-based catalyst which processes product gas from an industrial scale biomass steam gasification plant.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0215-9
Issue No: Vol. 7, No. 2 (2017)

• Comparative biochemical methane potential of some varieties of residual
banana biomass and renewable energy potential
• Authors: Florent Awedem Wobiwo; Thomas Happi Emaga; Elie Fokou; Maurice Boda; Sebastien Gillet; Magali Deleu; Aurore Richel; Patrick A. Gerin
Pages: 167 - 177
Abstract: Abstract The biochemical methane potential (BMP) of peduncles, bulbs, and peels of three banana varieties (Grande Naine (GN; export dessert banana), Pelipita (PPTA; locally used plantain), and CRBP969 (phytopathogen resistant hybrid-plantain)) was investigated as an assessment of the bioconversion potential of these residues to renewable energy or biorefined chemicals. Biogas production was monitored manometrically for 132 days and its composition was analyzed using gas chromatography. The BMP ranged from 162 to 257 ml_CH4/g_DM for peduncles, from 228 to 304 ml_CH4/g_DM for bulbs, and from 208 to 303 ml_CH4/g_DM for green peels, with methane content of the biogas in the range 56 to 60 %. Bulbs and green peels showed bioconversion yields of 95 % of the chemical oxygen demand (COD). The GN variety was generally more biodigestible than PPTA, which appeared richer in lignocellulosic fibres. The peels biodigestibility reduced with maturation and was already limited to 56 % of the COD at the yellow stage. The energy resource available in the residues of banana production is very significant, increasing by 91 % the energy resource offered by banana crop, which is generally limited to the nutritional value of the fruit pulp. In the study case of the African leading producer of bananas and plantains (Cameroon), the amount of available residues from the sole export variety GN could feed about 4 % of the annual electricity consumed by the country, i.e., a supply of electricity to an additional 9 × 105 people. Such valorization of the residual banana biomass could help banana-producing countries to become less dependent on fossil fuels and less prone to energy shortages.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0222-x
Issue No: Vol. 7, No. 2 (2017)

• Pilot-scale pretreatments of sugarcane bagasse with steam explosion and
mineral acid, organic acid, and mixed acids: synergies, enzymatic
hydrolysis efficiencies, and structure-morphology correlations
• Authors: Siddhartha Pal; Shereena Joy; Pramod Kumbhar; Kalpana D. Trimukhe; Rishi Gupta; Ramesh C. Kuhad; Anjani J. Varma; Sasisanker Padmanabhan
Pages: 179 - 189
Abstract: Abstract In lignocellulosic (LC) ethanol processes, to facilitate enzymatic hydrolysis of cellulose, a physical chemical pretreatment is vital. In this study, we explored a single as well as a two-step physical-chemical pretreatment involving steam and mixed acid on unwashed sugarcane bagasse at pilot-scale level in a continuous horizontal reactor. To serve as a large-scale model, pretreatments were carried out at high solid levels of 18–20 % w/w. For the pretreatment, partial replacement of corrosive sulfuric acid with a milder acid-like oxalic acid was explored to derive possible advantages and synergies accruing by using a mixture of mineral acid and organic acid. The results of this work showed that first-step pretreatment carried out by the mixing of sulfuric acid (1.5 % w/w) and oxalic acid (1.5 % w/w) at 150 °C followed by a second-step steam explosion pretreatment at 180 °C gave significant synergies and advantages over other variants of pretreatments investigated, such as lower inhibitor levels and lower reaction severity. On post-pretreated bagasse, this study conducted comparative enzymatic hydrolysis study using a simple lab enzyme and a robust commercial enzyme. It was found that the addition of Tween 80 to the lab enzyme improved its performance to match the performance of the commercial enzyme. Scanning electron microscopy (SEM) studies were further carried out to correlate the morphology of pretreated samples with efficiency of enzyme hydrolysis. Besides morphological study, Fourier transform infrared (FTIR) studies of pretreated samples showed higher syringyl/guaiacyl ratio for all pretreatments, indicating lower levels of pseudo-lignins, which is beneficial for improved enzyme hydrolysis.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0220-z
Issue No: Vol. 7, No. 2 (2017)

• Bioethanol production from Eucalyptus grandis hemicellulose recovered
before kraft pulping using an integrated biorefinery concept
• Authors: Mairan D. Guigou; Florencia Cebreiros; María N. Cabrera; Mario D. Ferrari; Claudia Lareo
Pages: 191 - 197
Abstract: Abstract Pre-extraction of hemicelluloses prior to pulping and its conversion to other by-products can provide additional profits to traditional pulp and paper industry. In this study, hemicelluloses removed from Eucalyptus grandis with green liquor (2 %) at 155–160 °C for 150 min prior to kraft pulping were fermented by Scheffersomyces stipitis NBRC 10063 to produce bioethanol. These conditions were selected to obtain an extract rich in xylose without changing the quality of pulp produced: best xylose extraction yield and minor pulp viscosity degradation. Fermentation of hemicellulose hydrolysate containing 7.5 g/L xylose and 5.0 g/L acetic acid presented an ethanol yield of 0.19 g/g and sugar conversion of 89 %. However, the fermentation of hydrolyzates after concentration proved to be difficult or even impossible. Ethyl acetate extraction, used for removal of inhibitory compounds in concentrated hydrolyzates containing 19 g/L xylose, improved fermentability (final ethanol concentration of 5.0 g/L, ethanol yield of 0.21 g/g and 94 % sugar conversion, ethanol production of 4.4 Lethanol/t of dry wood) and made possible the recovery of a valuable product as acetic acid.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0218-6
Issue No: Vol. 7, No. 2 (2017)

• Bioconversion of soybean and rice hull hydrolysates into ethanol and
xylitol by furaldehyde-tolerant strains of Saccharomyces cerevisiae ,
Wickerhamomyces anomalus , and their cofermentations
• Authors: Nicole Teixeira Sehnem; Lilian Raquel Hickert; Fernanda da Cunha-Pereira; Marcos Antonio de Morais; Marco Antônio Záchia Ayub
Pages: 199 - 206
Abstract: Abstract The aims of this work were to evaluate the ability of furaldehyde-tolerant yeast strains Saccharomyces cerevisiae P6H9 and Wickerhamomyces anomalus WA-HF5.5 and their cofermentations and to convert soybean and rice hull hydrolysates into ethanol and xylitol. In batch shaker cultures, the strains showed the ability to tolerate high osmotic pressure (1918 mOsmkg−1), completely depleting furaldehyde in the first 12 h of cultivations, while converting the hydrolysate sugars into ethanol. Highest ethanol yields of 0.37 g g−1 and productivity of 0.31 g L−1 h−1 were obtained in the cofermentation using rice hull hydrolysate as substrate. The concentration of sugars in soybean hull hydrolysate proved to be inadequate as substrate for the cultivation of these strains, showing a low ethanol productivity of 0.08 g L−1 h−1. Bioreactor cultivations of S. cerevisiae on rice hull hydrolysate under anaerobiosis showed a relatively high ethanol productivity of 6.7 g L−1 h−1, whereas the bioreactor cofermentation produced xylitol to yields of 0.86 g g−1 under conditions of oxygen limitation.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0224-8
Issue No: Vol. 7, No. 2 (2017)

• Moisture effect on fluidization behavior of loblolly pine Wood grinds
• Authors: G Olatunde.; O Fasina.; T McDonald.; S Adhikari.; S Duke.
Pages: 207 - 220
Abstract: Abstract The impact of moisture content (MC of 8 to 27 % wet basis) on physical properties (particle size distribution, average size using Feret, chord, Martins, surface-volume, and area diameter measurement schemes, bulk density, and particle density), fluidization behavior, and minimum fluidization velocities (U mf) of loblolly pine wood grinds were studied. A new correlation for predicting the U mf of loblolly pine wood grinds at different moisture contents was also developed. Results showed that bulk density, particle density, and porosity of grinds were significantly affected by increase in MC (p < 0.05). Diameter of the grinds measured using Feret measurement scheme was consistently the highest while those measured by surface-volume scheme were consistently the lowest with the measured Feret-based diameter about three times the surface-volume based diameters. Particle size data showed that variations in sizes of particle within a sample reduced with increase in MC (coefficient of variation value was 90 at 8.45 % MC and 40 at 27.02 % MC). Generally, as MC increased, the minimum fluidization velocity values increased. The minimum fluidization velocity (Umf) was found to be 0.2 m/s for 8 % MC, 0.24 m/s at 14.86 % MC, 0.28 m/s at 19.86 % MC, and 0.32 m/s for 27.02 % MC. The correlation developed predicted the experimental data with mean relative deviation that was less than 10 %.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0223-9
Issue No: Vol. 7, No. 2 (2017)

• The nutritional aspects of biorefined Saccharina latissima , Ascophyllum
nodosum and Palmaria palmata
• Authors: Peter Schiener; Sufen Zhao; Katerina Theodoridou; Manus Carey; Karen Mooney-McAuley; Chris Greenwell
Pages: 221 - 235
Abstract: Abstract The chemical profile of biorefined Saccharina latissima, Ascophyllum nodosum and Palmaria palmata after carbohydrate and polyphenol extraction was analysed with the aim to evaluate the nutritional aspects of biorefined seaweeds as a novel animal feed supplement. Optimised enzymatic saccharification has been used to show that the protein concentration in the residue of P. palmata and A. nodosum can be increased by more than 2-fold. Nutritional value of the residue was further enhanced through an increase in total amino acids and fatty acids. As a consequence of removal of inorganic elements such as sodium, potassium and chloride, the total solid and ash content of all three seaweeds was reduced by around 40%. In contrast, divalent metals such as iron and zinc, as well as silicon, accumulated in all three residues. Potentially harmful components such as arsenic and iodine were reduced only in brown biorefined seaweeds, whilst in biorefined P. palmata, iodine increased by 39% compared to a 24% decline of arsenic. Nutritional values such as total fatty acid and total amino acid content increased in all three seaweeds after enzymatic saccharification. Polyphenol removal in all three seaweeds was >80% using aqueous acetonitrile and, in combination with enzymatic saccharification, did not impact on protein recovery in A. nodosum. This highlights the potential of biorefinery concepts to generate multiple products from seaweed such as extracts enriched in polyphenols and carbohydrates and residue with higher protein and lipid content.
PubDate: 2017-06-01
DOI: 10.1007/s13399-016-0227-5
Issue No: Vol. 7, No. 2 (2017)

• Pyrolysis kinetics of Sal ( Shorea robusta ) seeds
• Authors: Ranjan R. Pradhan; Pragyan P. Garnaik; Bharat Regmi; Bandita Dash; Animesh Dutta
Pages: 237 - 246
Abstract: Abstract Thermal kinetics of Sal seeds during pyrolysis process was investigated as feedstocks for chemical, material, and bioenergy industries. The physicochemical properties of the seeds were examined. Results showed that Sal seed can be characterized as high calorific values, low ash, and high volatile content biomass to suit pyrolysis applications. Kinetic analysis for thermal degradation of this biomass was given particular attention. Two major degradation zones were identified with T max at about 321 and 405 °C, and activation energy was evaluated using various methods. Model-free pyrolysis kinetic approach was verified to be appropriate and indicated that unprocessed Sal seed biomass can directly become potential renewable feedstock of energy, chemicals, and biochar.
PubDate: 2017-06-01
DOI: 10.1007/s13399-017-0240-3
Issue No: Vol. 7, No. 2 (2017)

• Pretreatment technologies of lignocellulosic biomass in water in view of
furfural and 5-hydroxymethylfurfural production- A review
• Authors: David Steinbach; Andrea Kruse; Jörg Sauer
Pages: 247 - 274
Abstract: Abstract Lignocellulosic biomasses are strongly connected composites of cellulose, hemicelluloses, and lignin. A pretreatment is required in order to make these components available for their later conversion into chemicals. At this point, two strategies have to be considered: to either produce chemicals via microorganism or enzymes (1), or by chemical conversion (2). The focus of this article is the second strategy, which is chemical conversion, performed in water to produce the final products furfural and 5-hydroxymethylfurfural (HMF). Reviewed first is the composition of cellulose and hemicelluloses as well as their degradation chemistry in water. Then, fundamental modes of action and process parameters of pretreatment methods in aqueous solution are summarized. The pretreatment methods discussed here are steam explosion, treatment with hot liquid water, diluted and concentrated acids, as well as alkaline solutions. Finally, the advantages and disadvantages of these pretreatments are discussed for lignocellulosic biomass.
PubDate: 2017-06-01
DOI: 10.1007/s13399-017-0243-0
Issue No: Vol. 7, No. 2 (2017)

• Catalytic deoxygenation of C18 fatty acids over HAlMCM-41 molecular sieve
• Authors: F. C. M. Silva; M. S. Lima; C. O. Costa Neto; J. L. S. Sá; L. D. Souza; V. P. S. Caldeira; A. G. D. Santos; G. E. Luz Jr
Abstract: Abstract The large demand for energy combined with ecological, economic, and social reasons encouraged the studies for alternative sources of fuel. The deoxygenation process of oils and fatty acids has emerged as a promising resource in obtaining fuels. This paper studied the deoxygenation process of oleic and stearic acids carried out in a distillation system, under the temperature of 450 °C over MCM-41, AlMCM-41-41, and HAlMCM-41 molecular sieves, which were synthesized by direct hydrothermal synthesis method and characterized by X-ray diffraction (XRD) spectroscopy in the infrared site (FTIR), adsorption and desorption of N2, and scanning electron microscopy (SEM). The products were analyzed by gas chromatography (GC) and acid-base titration. The results showed that stearic acid deoxygenates more than the oleic acid, and that the performance of HAlMCM-41 was superior to AlMCM-41, indicating that the total acidity positively influenced the process. Furthermore, from the stearic acid deoxygenation, it obtained mainly heptadecanes (C17:0) structures, while the oleic acid produces heptadecenes (C17:1).
PubDate: 2017-05-13
DOI: 10.1007/s13399-017-0263-9

• Conversion of a wet waste feedstock to biocrude by hydrothermal processing
in a continuous-flow reactor: grape pomace
• Authors: Douglas C. Elliott; Andrew J. Schmidt; Todd R. Hart; Justin M. Billing
Abstract: Abstract Wet waste feedstocks present an attractive opportunity for biomass conversion to fuels by hydrothermal processing. In this study, grape pomace slurries from two varieties, Montepulciano and cabernet sauvignon, have been converted into a biocrude by hydrothermal liquefaction (HTL) in a bench-scale, continuous-flow reactor system. Carbon conversion to gravity-separable biocrude product up to 56% was accomplished at relatively low temperature (350 °C) in a pressurized (sub-critical liquid water) environment (20 MPa) when using grape pomace feedstock slurry with a 16.8 wt% concentration of dry solids processed at a liquid hourly space velocity of 2.1 h−1. Direct biocrude recovery was achieved without the use of a solvent and biomass trace mineral components were removed by precipitation and filtration so that they did not cause processing difficulties. In addition, catalytic hydrothermal gasification (CHG) was effectively applied for HTL byproduct water cleanup using a Ru on C catalyst in a fixed bed producing a gas composed of methane and carbon dioxide from water-soluble organics. Conversion of 99.8% of the chemical oxygen demand (COD) left in the aqueous phase was demonstrated. As a result, high conversion of grape pomace to liquid and gas fuel products was found with residual organic contamination in byproduct water reduced to <150 mg/kg COD.
PubDate: 2017-05-13
DOI: 10.1007/s13399-017-0264-8

• The impact of sorbent geometry on the sulphur adsorption under
supercritical water conditions: a numerical study
• Authors: Florentina Maxim; Bojan Niceno; Andrea Testino; Christian Ludwig
Abstract: Abstract A numerical model to show the impact of the adsorption bed geometry on the desulfurization process of wet biomass under supercritical water (SCW) gasification process has been developed. Three different geometries, straight channels (pipe), sharp-edged channels (sharp) and packed bed of particles (pebbles) have been considered for the sorbent bed. The influence of the flow patterns on the sulphur distribution inside the bed and on the saturation of the sorbent has been analysed. The results show that, when the flow is unidirectional with a parabolic profile, as in the pipe geometry, the adsorption process can be explained based on the 1D plug-flow model. In the case of more complex flow structures, when torus-shaped vortices appeared in the sharp or pebbles geometries, the 3D flow effects should be considered. The present work might provide useful information for the evaluation of sulphur sorption under SCW conditions. The models obtained by computational fluid dynamic, which are under experimental validation using neutron imaging, will help for the sorbent design and production by 3D printing techniques, which represent an advanced engineered tool to improve the process efficiency and sorbent material selection.
PubDate: 2017-05-12
DOI: 10.1007/s13399-017-0265-7

• Thermochemical valorization of camelina straw waste via fast pyrolysis
• Authors: H. Hernando; J. Fermoso; I. Moreno; J. M. Coronado; D. P. Serrano; P. Pizarro
Abstract: Abstract The present work investigates the thermochemical valorization of camelina straw, which is a waste generated during the harvesting of Camelina sativa, an oilseed crop for the production of biodiesel or hydrotreated vegetable oil (HVO). In particular, it is focused on obtaining bio-oil via thermal or catalytic fast pyrolysis, which would be the first stage on a sequence of chemical processes for biofuel production. The catalytic interference of the inorganic matter present in the biomass was studied by preparing a batch of de-ashed camelina straw by washing with diluted nitric acid. Chemical analysis revealed this treatment effectively removed alkaline (K and Na) and alkaline earth (Ca and Mg) metals. Pyrolysis of de-ashed camelina straw led to higher mass and energy yields of bio-oil in water-free basis (bio-oil*), but with higher oxygen concentration. Catalytic pyrolysis over HZSM-5 was also studied in both raw and de-ashed feedstocks. This catalyst promoted mainly decarbonylation and decarboxylation reactions of the pyrolysis vapors, leading to much higher gas yields and lower of bio-oil*, but with better quality. Catalytic pyrolysis of untreated camelina straw exhibited a synergetic effect between both the inorganic matter and the external HZSM-5 catalyst, so that bio-oil* yield was the lowest (20 wt%) due to an extensive deoxygenation (18 wt% oxygen content), which resulted in the highest HHV obtained (37.3 MJ/kgdb). Significant differences were also found on the molecular composition of the bio-oils* with larger proportion of anhydro sugars when the biomass was de-ashed, while HZSM-5 strongly promoted the formation of oxygenated aromatics and aromatic hydrocarbons.
PubDate: 2017-05-07
DOI: 10.1007/s13399-017-0262-x

• Erratum to: Hydrogen-enhanced catalytic hydrothermal gasification of
biomass
• Authors: J. Reimer; S. Müller; E. De Boni; F. Vogel
PubDate: 2017-04-24
DOI: 10.1007/s13399-017-0260-z

• The effects of air velocity, temperature and particle size on
low-temperature bed drying of wood chips
• Authors: Niranjan Fernando; Mahinsasa Narayana; W. A. M. K. P. Wickramaarachchi
Abstract: Abstract This paper describes a mathematical model for wood chip packed bed drying process with the effects of hot air flow velocity, temperature and particle size. A single-particle drying model was developed by considering impacts of external and internal parameters. External parameters are hot air flow velocity and air temperature. Internal parameters are porosity and particle size. These parameters are incorporated to the present model by introducing two mass transfer coefficients. The model was fine-tuned by comparing simulation results and experimental data. Effects of a factor relating to internal mass transfer coefficient were found for three wood types, and a functional dependence of internal mass transfer on temperature was suggested in this study. The model was implemented in computational fluid dynamics (CFD) to evaluate spatial variation of moisture in the packed bed drying process. The CFD model was validated by results of lab-scale packed bed. Drying performance of the packed bed was estimated by CFD simulations for variations of external hot air flow velocity, flow temperature and particle size. Sensitivity of these parameters for dying performance was evaluated by design of experiment (DOE) method. It was clarified that air temperature is most critical for the drying process. Interaction of between external hot air flow velocity and particle size for dying performance is also significant.
PubDate: 2017-04-07
DOI: 10.1007/s13399-017-0257-7

• Thermochemical characterization of biochar from cocoa pod husk prepared at
low pyrolysis temperature
• Authors: Chi-Hung Tsai; Wen-Tien Tsai; Sii-Chew Liu; Yu-Quan Lin
Abstract: Abstract In this work, cocoa pod husk (CPH), a processing by-product from the cocoa industry, was evaluated as a potential feedstock for preparing biochar fuel at relatively low pyrolysis temperature. First, its thermochemical characteristics, including the calorific value and mineral component analyses, were investigated, showing that the bioresource obviously contained a large percentage of volatile matter. It thus had a higher heating value of 17.8 MJ/kg but showed a higher potassium content in ash (i.e., 4.03 wt%). A series of CPH-based biochars (i.e., CPHBC) were produced at different temperatures (i.e., 190, 220, 250, 280, 310, 340, and 370 °C) and residence times (i.e., 30, 60, 90, and 120 min). The resulting biochars were subject to the analyses of chemical and thermal properties. The calorific value of resulting biochar indicated an increasing trend with pyrolysis temperature, but there was slightly decreasing change at the longer residence time under the fixed pyrolysis temperature (i.e., 370 °C). The optimal biochar product had a thermochemical characteristics with high carbon (>60 wt%) and calorific value (>25 MJ/kg, dry basis). Furthermore, the CPH-based biochar showed a lignite-like feature based on the O/C and H/C molar ratios, but it would not be appropriate to be fired in boilers because of its high mineral contents (i.e., potassium).
PubDate: 2017-04-06
DOI: 10.1007/s13399-017-0259-5

• On the influence of Si:Al ratio and hierarchical porosity of FAU zeolites
in solid acid catalysed esterification pretreatment of bio-oil
• Authors: Amin Osatiashtiani; Begoña Puértolas; Caio C. S. Oliveira; Jinesh C. Manayil; Brunella Barbero; Mark Isaacs; Chrysoula Michailof; Eleni Heracleous; Javier Pérez-Ramírez; Adam F. Lee; Karen Wilson
Abstract: Abstract A family of faujasite (FAU) zeolites with different Si:Al ratio, and/or hierarchical porosity introduced via post-synthetic alkaline desilication treatment, have been evaluated as solid acid catalysts for esterification pretreatments of pyrolysis bio-oil components. Acetic acid esterification with aliphatic and aromatic alcohols including methanol, anisyl alcohol, benzyl alcohol, p-cresol and n-butanol was first selected as a model reaction to identify the optimum zeolite properties. Materials were fully characterised using N2 porosimetry, ICP, XRD, XPS, FT-IR, pyridine adsorption, NH3 TPD, In-situ ATR and inverse gas chromatography (IGC). IGC demonstrates that the surface polarity and hence hydrophobicity of FAU decreases with increased Si:Al ratio. Despite possessing a higher acid site loading and acetic acid adsorption capacity, high Al-content FAU possess weaker acidity than more siliceous catalysts. Esterification activity increases with acid strength and decreasing surface polarity following the order FAU30>FAU6>FAU2.6. The introduction of mesoporosity through synthesis of a hierarchical HFAU30 material further enhances esterification activity through improved acid site accessibility and hydrophobicity. Methanol was the most reactive alcohol for esterification, and evaluated with HFAU30 for the pretreatment of a real pyrolysis bio-oil, reducing the acid content by 76% under mild conditions.
PubDate: 2017-04-05
DOI: 10.1007/s13399-017-0254-x

• Integral valorization of tagasaste ( Chamaecytisus proliferus ) under
thermochemical processes
• Authors: J. M. Loaiza; F. López; M. T. García; J. C. García; M. J. Díaz
Abstract: Abstract The generation of compounds derived from lignocellulosic biomass fractionation has a boost in recent years. An interesting plant could be tagasaste (Chamaecytisus proliferus) due to its high biomass production and its leguminous nature. A sequence of acid hydrolysis, as pretreatment, and pyrolysis of the solid residue, as treatment, has been used in its valorization. An experimental design (H2SO4 acid concentration 0.5–2%, temperature 130–170 °C, and time 30–60 min) has been used to study the hydrolysis process. In the proposed acid hydrolysis process, under 170 °C as operational temperature, 0.5% of H2SO4 and 30 min for the operation time, 91.75% of the initial xylose have been extracted. The thermal behavior of both tagasaste trunks-large branches and some solid residues after hydrolysis (furthest and center points in the experimental design) process and raw material have been studied by thermogravimetric analysis under nitrogen atmosphere at different heating rates (5, 10, 15, and 20 °C min−1). The thermal degradation of the studied materials is influenced by its initial composition. Then, the higher reactivity of hemicelluloses can accelerate the pyrolysis degradation reaction. However, higher cellulose content implies lower activation energy in pyrolysis process.
PubDate: 2017-04-04
DOI: 10.1007/s13399-017-0258-6

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# Average, variance & standard deviation in Python

Here is a quick python script which calculates average, variance and standard deviation. This is also part of codeacademy work. I must say last time I worked with variance and standard deviation it was more than 10 years ago in statistics course. Hence a bit of reminder here for me too: (Some are from wikipedia and mathsisfun.com)

Average: In mathematics and statistics, it is called arithmetic mean according to wikipedia but simply it is the sum of a list of numbers divided by the number of numbers in the list.

Variance: The average of the squared differences from the mean. Here is the formula which we will use in our python code.

$\displaystyle \sigma^2 = \frac{\displaystyle\sum_{i=1}^{n}(x_i - \mu)^2} {n}$

Standard deviation: Square root of the variance is the standard deviation which just means how far we are from the normal(mean)

$\displaystyle \sigma = \sqrt{\frac{\displaystyle\sum_{i=1}^{n}(x_i - \mu)^2} {n}}$

Now here is the code which calculates given the number of scores of students we calculate the average,variance and standard deviation.

# a simple battleship python script

I have written a simple python game script to sink a battleship. It is very simple as it was almost a year ago last time when I wrote a long python script. Thanks to codeacademy that it helps to brush up my python skills. I think it is also good to put the script here for my documentation. Simply, you guess the coordinates of the point and by default you have maximum 4 turns to do it right. Each time a guess is run, coordinate is marked by “X”. Below is a sample execution:

Here is the code itself:

# Fragmented IP packet forwarding

I couldn’t really find a suitable topic for this post actually but I will try to find answers for the following questions:

• How can we fragment an IP packet manually in scapy
• How does a fragmented packet look like and how the transport layer (TCP/UDP) header is located
• How do we forward fragmented packets, do we reassemle them?
• If we don’t reassemble, can we force reassembly?

First of all a bit of a theory: if an incoming IP packet is to be forwarded to another next hop and the MTU of this new path is smaller than the packet to be transmitted, we must find a way to forward the packet. If the packet has DF (Don’t Fragment) bit on i.e we are instructed not to fragment the packet most probably by the source, then normally we are expected to send an ICMP packet with type “Fragmentation needed” and pray that on the way back to the source no devices block all ICMP type of traffic. Second scenario is that what if the source lets us fragment the packet. Then we need to fragment it and story from now on is about this part of the scenario and the topology we will use is something like below.

# Port Scanner in Python

Python is a great tool to do some socket operations. I have written a piece of code by which I can scan a port range.
It is very basic and missing bunch of checks as aim is the simplicity here.

You can run the script in the following way by which you scan ports between 1 and 1024:

# How to sort a python dictionary by value

If you want to search a dictionary by value in python, here is a handy one line using also lambda:

If you run this script output will be ;

# searching dictionary by value in python

The following list comprehension is a life saver for me to search a dictionary by value. Searching dictionary by value may not seem to be sensible as you may get multiple values but in case needed, it is a wonderful one line of code.

Here we take the ‘value’, iterate through the mydict dictionary and if any dictionary member is found having this value, then we grab the key value and assign them to the list ‘list_values’ . It may return for example 10 keys all of which can be accessed via the list assigned. It is a spectacular structure and I wanted to take a note here to reference later:)

# How to turn a list into string

Here is a quote from python.org describing how we can use the join method.

str.join(iterable)

As it can be seen it isn’t only for strings but for something called “iterable”. Again when we look at python.org what this iterable means;

iterable

So it means we can use list, tuples all sequence types. This means I can use it to turn a list into a string which is very handy when I read files via python. For example;

Bingo, we have a string out of a list!


text:
# Alexander invariants

Invariants connected with the module structure of the one-dimensional homology of a manifold $\widetilde{M}$, freely acted upon by a free Abelian group $J ^ {a}$ of rank $a$ with a fixed system of generators $t _ {1} \dots t _ {a}$.

The projection of the manifold $\widetilde{M}$ onto the space $M$ of orbits (cf. Orbit) is a covering which corresponds to the kernel $K _ {a}$ of the homomorphism $\gamma : G \rightarrow J _ {a}$ of the fundamental group $\pi _ {1} (M) = G$ of the manifold $M$. Since $K _ {a} = \pi _ {1} ( \widetilde{M} )$, the group $B _ {a} = K _ {a} / K _ {a} ^ \prime$, where $K _ {a} ^ \prime$ is the commutator subgroup of the kernel $K _ {a}$, is isomorphic to the one-dimensional homology group $H _ {1} ( \widetilde{M} , \mathbf Z )$. The extension $1 \rightarrow K _ {a} \rightarrow G \rightarrow J ^ {a} \rightarrow 1$ generates the extension $(*) : 1 \rightarrow B _ {a} \rightarrow G/ K _ {a} ^ \prime \rightarrow J ^ {a} \rightarrow 1$, which determines on $B _ {a}$ the structure of a module over the integer group ring $\mathbf Z (J ^ {a} )$ of the group $J ^ {a}$( cf. Group algebra). The same structure is induced on $B _ {a}$ by the given action of $J ^ {a}$ on $\widetilde{M}$. Fixation of the generators $t _ {i}$ in $J ^ {a}$ identifies $\mathbf Z ( J ^ {a} )$ with the ring $L _ {a} = L _ {a} (t _ {1} \dots t _ {a} ) = \mathbf Z [ t _ {1} , t ^ {-1} \dots t _ {a} , t _ {a} ^ {-1} ]$ of Laurent polynomials in the variables $t _ {i}$. Purely algebraically the extension

defines and is defined by the extension of modules $(**): 0 \rightarrow B _ {a} \rightarrow A _ {a} \rightarrow I _ {a} \rightarrow 0$[5]. Here $I _ {a}$ is the kernel of the homomorphism $\epsilon : L _ {a} \rightarrow \mathbf Z$ $( \epsilon t _ {i} = 1 )$. The module $A _ {a}$ is called the Alexander module of the covering $\widetilde{M} \rightarrow M$. In the case first studied by J.W. Alexander [1] when $M = M (k)$ is the complementary space of some link $k$ of multiplicity $\mu$ in the three-dimensional sphere $S ^ {3}$, while the covering corresponds to the commutation homomorphism $\gamma _ \mu : G(k) \rightarrow J ^ \mu$ of the link group, $A _ \mu$ is the Alexander module of the link $k$. The principal properties of $G$ which are relevant to what follows are: $G/ G ^ \prime$ is a free Abelian group, the defect of the group $G$ is 1, $G$ has the presentation $\{ x _ {1} \dots x _ {m+1} ; r _ {1} \dots r _ {m} \}$ for which $\gamma _ \mu (x _ {i} ) = t _ {i}$, $1 \leq i \leq \mu$; $\gamma _ \nu (x _ {i} ) = 1$, $i > \mu$( cf. Knot and link diagrams). In the case of links the generators $t _ {i} \in J ^ \mu$ correspond to the meridians of the components $k _ {i} \subset k$ and are fixed by the orientations of these components and of the sphere.

As a rule, $M$ is the complementary space $M(k)$ of $k$, consisting of $\mu$ $(n - 2)$- dimensional spheres $k _ {i}$ in $S ^ {n}$. In addition to the homomorphism $\gamma _ {m}$, one also considers the homomorphism $\gamma _ \sigma : G(k) \rightarrow J$, where $\gamma (x)$ is equal to the sum of the link coefficients of the loop representing $x$ with all $k _ {i}$.

The matrix $\mathfrak M _ {a}$ of the module relations of a module $A _ {a}$ is called the Alexander covering matrix and, in the case of links, the Alexander link matrix. It may be obtained as the matrix

$$\left ( \frac{\partial r _ {i} }{\partial x _ {j} } \right ) ^ {\gamma _ {a} \phi } ,$$

where $\{ x _ {i} ; r _ {i} \}$ is a presentation of the group $G$. If $\mu = 1$, the matrix $\mathfrak N _ {a}$ of module relations for $B _ {a}$ is obtained from $\mathfrak M _ {a}$ by discarding the zero column. The matrices $\mathfrak M _ {a}$ and $\mathfrak N _ {a}$ are defined by the modules $A _ {a}$ and $B _ {a}$ up to transformations corresponding to transitions to other presentations of the module. However, they can be used to calculate a number of module invariants. Alexander ideals are ideals of the module $A _ {a}$, i.e. series of ideals $E _ {i} ( A _ {a} )$ of the ring $L _ {a} : (0) \subseteq E _ {0} \subseteq \dots \subseteq E _ {i-1} \subseteq E _ {i} \subseteq \dots \subseteq (1)$, where $E _ {i}$ is generated by the minors of $\mathfrak M _ {a}$ of order $(m - i) \times (m - i)$ and $E _ {i} = L _ {a}$ for $m - i < 1$. The opposite numbering sequence may also be employed. Since $L _ {a}$ is both a Gaussian ring and a Noetherian ring, each ideal $E _ {i}$ lies in a minimal principal ideal $( \Delta _ {i} )$; its generator $\Delta _ {i}$ is defined up to unit divisors $t _ {i} ^ {k}$. The Laurent polynomial $\Delta _ {i} (t _ {1} \dots t _ \mu )$ is simply called the Alexander polynomial of $k$( or of the covering $\widetilde{M} \rightarrow M$). If $\Delta _ {i} \neq 0$, it is multiplied by $t _ {1} ^ {k _ {1} } \dots t _ \mu ^ {k _ \mu }$ so that $\Delta _ {i} (0 \dots 0) \neq 0$ and $\neq \infty$. To the homomorphism $\gamma _ \sigma$ there correspond a module $\overline{A}\;$, ideals $\overline{E}\; _ {i}$ and polynomials $\overline \Delta \; _ {i}$, designated, respectively, as Alexander's reduced module, Alexander's reduced ideals and Alexander's reduced polynomials of $k$( or of the covering ${\widetilde{M} } _ \sigma \rightarrow M$). If $\mu = 1$, then $A = \overline{A}\;$. $\mathfrak M ( \overline{A}\; )$ is obtained from $\mathfrak M$ by replacing all $t _ {i}$ by $t$. If $\mu \geq 2$, $\overline \Delta \; _ {1}$ is divisible by ${(t - 1) } ^ {\mu - 2 }$. The polynomial $\nabla (t) = {\overline \Delta \; } _ {1} (t) / {(t - 1) } ^ {\mu - 2 }$ is known as the Hosokawa polynomial. The module properties of $A (k)$ have been studied [4], [8], [10]. The case of links has not yet been thoroughly investigated. For $\mu = 1$, the group $H _ {1} ( \widetilde{M} ; R)$ is finitely generated over any ring $R$ containing $\mathbf Z$ in which $\Delta (0)$ is invertible [7], in particular over the field of rational numbers, and, if $\Delta (0) = +1$, then also over $\mathbf Z$. In such a case $\Delta (t)$ is the characteristic polynomial of the transformation $t : H _ {1} ( \widetilde{M} ; R) \rightarrow H _ {1} ( \widetilde{M} ; R)$. The degree of $\Delta _ {1} (t)$ is equal to the rank of $H _ {1} ( \widetilde{M} ; R)$; in particular, $\Delta _ {1} (t) = 1$ if and only if $H _ {1} ( \widetilde{M} ; \mathbf Z) = 0$. If $n = 3$, the link ideals have the following symmetry property: $E _ {i} = {\overline{E}\; } _ {i}$, where the bar denotes that the image is taken under the automorphism generated by replacing all $t _ {i}$ by $t _ {i} ^ {-1}$. It follows that $\Delta _ {i} ( t _ {1} ^ {-1} \dots t _ \mu ^ {-1} ) = t _ {1} ^ {N _ {1} } \dots t _ \mu ^ {N _ \mu } \Delta _ {i} ( t _ {1} \dots t _ \mu )$ for certain integers $N _ {i}$. This symmetry is the result of the Fox–Trotter duality for knot and link groups. It may also be deduced from the Poincaré duality for the manifold $\widetilde{M}$, taking into account the free action of $J ^ {a}$[3]. If $\Delta _ {1} (t _ {1} \dots t _ \mu ) \neq 0$, then the chain complex $C _ {*} ( \widetilde{M} )$ over the field of fractions $P _ \mu$ of the ring $L _ \mu$ is acyclic ( $n = 3$), and the Reidemeister torsion $\tau \in P _ \mu / \Pi$ corresponding to the imbedding $L _ \mu \subset P _ \mu$, where $\Pi$ is the group of units of $L _ \mu$, is defined accordingly. If $\mu = 2$, then $\tau = \Delta _ {1}$; if $\mu = 1$, then $\tau = \Delta _ {1} / t - 1$( up to units of $L _ \mu$). The symmetry of $\Delta _ {1}$ for $n = 3$ is a consequence of the symmetry of $\tau$. If $\mu = 1$, it follows from the symmetry of $\Delta _ {i} (t)$ and from the property $\Delta _ {i} (1) = \pm 1$ that the degree of $\Delta _ {i} (t)$ is even. The degree of $\nabla (t)$ is also even [4]. The following properties of the knot polynomials $\Delta _ {i} (t)$ are characteristic: $\Delta _ {i} (1) = \pm 1$; $\Delta _ {i} (t) = t ^ {2k} \Delta _ {i} ( t ^ {-1} )$; $\Delta _ {i+1}$ divides $\Delta _ {i}$; and $\Delta _ {i} = 1$ for all $i$ greater than a certain value $N$, i.e. for each selection $\Delta _ {i} (t)$ with these properties there exists a knot $k$ for which they serve as the Alexander polynomials. The Hosokawa polynomials [4] are characterized by the property $\nabla (t) = t ^ {2k} \nabla (t ^ {-1} )$ for any $\mu \geq 2$; the polynomials $\Delta _ {1}$ of two-dimensional knots by the property $\Delta _ {1} (1) = 1$.

Alexander invariants, and in the first place the polynomials, are powerful tools for distinguishing knots and links. Thus, $\Delta _ {1}$ fails to distinguish between only three pairs out of the knots in a table containing fewer than 9 double points (cf. Knot table). See also Knot theory; Alternating knots and links.

#### References

 [1] J.W. Alexander, "Topological invariants of knots and links" Trans. Amer. Math. Soc. , 30 (1928) pp. 275–306 [2] K. Reidemeister, "Knotentheorie" , Chelsea, reprint (1948) [3] R.C. Blanchfield, "Intersection theory of manifolds with operators with applications to knot theory" Ann. of Math. (2) , 65 : 2 (1957) pp. 340–356 [4] F. Hosokawa, "On -polynomials of links" Osaka J. Math. , 10 (1958) pp. 273–282 [5] R.H. Crowell, "Corresponding groups and module sequences" Nagoya Math. J. , 19 (1961) pp. 27–40 [6] R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) [7] L.P. Neuwirth, "Knot groups" , Princeton Univ. Press (1965) [8] R.H. Crowell, "Torsion in link modules" J. Math. Mech. , 14 : 2 (1965) pp. 289–298 [9] J. Levine, "A method for generating link polynomials" Amer. J. Math. , 89 (1967) pp. 69–84 [10] J.W. Milnor, "Multidimensional knots" , Conference on the topology of manifolds , 13 , Boston (1968) pp. 115–133
How to Cite This Entry:
Alexander invariants. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alexander_invariants&oldid=45055
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article


text:
## Probabilistic decompression models: Probably problematic

It has been very quiet here for a while. That is partly because there is a topic that I wanted to discuss here and since my take was going to be at least somewhat problematic, I felt for a while that I needed to do more reading to get the facts right. But now the time has come for me to talk about: Probabilistic decompression models.

The idea as such is very simple: Many models for decompression including the ones that I talk a lot about in this blog, Bühlmann, VPM-B and also DCIEM are “deterministic” models: They give you a plan and if you stick to it (or dive more conservatively) you should be fine but you must not violate the ceilings they predict at any given time. On the other hand, there are probabilistic models that aim to compute a probability of getting DCS for any possible profile and which you can then turn around and prescribe a maximal probability of DCS and then you optimise your profile (typically for the shortest ascent time) that gives you that prescribed probability of DCS.

That sounds like a great advantage: Don’t we all know that there is no black and white in decompression but there are large grey areas. Even when you stick to the ceilings of your deterministic model there is a chance that you get bent while on the other hand you do not immediately die when you stick your head above the ceiling, quite the opposite, there are chances that even with substantial violation of the decompression plan you will still be fine. That sounds pretty probabilistic to me.

Of course, things are more complicated. Also the deterministic models do not really aim at a black/white distinction. Rather, they are also intended to be probabilistic but with a fixed prescribed probability built in. For recreational diving (here as usually including technical diving but not commercial diving), the accepted rate of DCS is usually assumed to be one hit in a few thousand dives. That seems to be the sweet spot between not too often having to call the helicopter (and most recreational divers never experiencing a DCS hit) and being overly conservative (too short NDL times, too long decompression stops). So, the only difference is that for the probabilistic models, this probability is an adjustable parameter.

You can turn this around: Why would you be interested in dive profiles that have vastly different probabilities of DCS that this conventional one in a few thousand? After all, even for one possible value of p(DCS) it is very hard to collect enough empirical data to pin down the parameters of a decompression model. Why on earth would you want to do that also for probabilities that you do not intend to encounter in your actual diving? Why would I be interested in computing an ascent plan that will bend me one dive in twenty for example? Or in only one dive in a million? That sounds to be overly complicated for useless information.

The answer is exactly in your restricted ability to conduct millions of supervised test dives: Let us assume you have a probabilistic model with a number of parameters that still need to be determined empirically. But you have a priori knowledge that the general form of your model is correct (we will come back to this assumption below), you can do your test dives with depths, bottom times and ascents that your model gives you DCS probabilities in in the range of 5-50% say, depending on the parameters. Much more aggressive than you intend to dive in the end. But such test dives will give you a lot of DCS cases and you do not need too many dives to determine if the true probability is 5%, 20% or 50% and adjust the model parameters accordingly. You gain already a lot of information with just 10-20 dives while for dives with the 1/10000 rate of DCS you need many more dives to have enough DCS cases to establish the true probability.

Once you have established the parameters of your model with these high risk dives where you have to use your chamber a lot to treat your guinea pig divers after bending them you can then use your model for dives that have a much healthier conservatism where your model gives you for example p(DCS)=1/10000.

An example of doing your study in a regime where you expect to bend many contestants is the famous NEDU deep stop study where the divers were in quite cold water with insufficient thermal protection and in which they had to work out on ergometers while in the water all to drive up the expected number DCS cases (there not necessarily with a probabilistic model in the background but just the intend to see a difference in deeper vs shallower decompression schedules where of course 0 DCS cases for both ascents wouldn’t be very informative).

But as so often, there is no such thing as free lunch: You are extrapolating data! You make experiments in a high risk regime of your data and then hope that the same model with the same parameters also in a low risk regime. You are extrapolating your data over several orders of magnitude of probability. This can only work if you can be sure your model is correct in its form and the only thing to determine empirically are the few parameters. But in the real world, in particular in decompression science, such a priori knowledge is not at hand.

Let me illustrate this in a simplified example: Let’s assume, there is only one parameter x and your model is that the logarithm of p(DCS) is a straight line (an affine function) of x

$$\log(p_{DCS}) = m x +b$$

Then, in your experiments you do a number of test dives at various values of x, find the corresponding rates of DCS and finally fit the slope m and the intercept b by linear regression.

If however the a priori assumption of the model that a straight line does the job is not justified because there is also a small quadratic contribution, say, you can fit your parameters as closely as you want but still get very far off in the extrapolation to where you intend to use your model.

This is of course only an example and for more complicated models you will use a maximum likelihood method to find optimal values of your model parameters given the outcomes of various test dives. But what this will never be able to do is to verify the form of your model assumptions. You are always only optimising the model parameters but never the model itself. For that you need independent knowledge and let’s hope you have that.

## VVAL18

To be specific, let us look at one probabilistic algorithm more concretely: VVAL18 which was developed by Edward Deforest Thalmann for the US Navy based on a database of a few thousand navy dives. For military divers, the accepted risk of DCS is much higher, if 2% of the dives result in symptoms that need to be treated that is still considered fine given that decompression chambers are typically available at the dive spot. This model is also known as the Thalmann algorithm and is described for example in short in this technical report. (Similar things could for example be said about the SAUL decompression model which is similar in spirit and also inspired by the navi data)

This model also uses compartments with half times but leaves open the possibility that the partial pressures do not follow the usual diffusive dynamics with the rate of change proportional to the pressure difference to the surroundings but for off-gassing also allow for the possibility of a constant rate (which leads to linear rather than exponential time dependence).

For each compartment i, there is a fixed threshold pressure pth that as an excess pressure is considered harmless and above that a relative excess pressure is calculated

$$e_i = \frac{p_i-p_{amb}-p_{thi}}{p_{amb}}$$

To compute the risk, for each second of the dive and each compartment, risk of not getting bent in the second is assumed to be

$$e^{-a_ie_i}\qquad (*)$$

for some constants ai. Finally, all these individual risks are considered to be independent, so the “survival probability”, the probability of not developing DCS is assumed to be simply the product of all the individual probabilities for compartments and seconds (since they are in the exponent, you can there integrate the ei over time and sum over tissues).

These constitute the a priori assumptions of the model that I was talking about: The exponential dependence of risk on relative overpressure (*) and the statistical independence of tissues and instances of time. According to these assumptions, your risk of DCS increases exponentially in time when you do longer decompression (assuming the excess pressure is kept constant) for deeper dives or longer bottom times (this is clearly at odds with the assumptions of for example the Bühlmann model that allows you to have arbitrary long decompression obligations as long as you do not violate a ceiling) and you are allowed to have arbitrary large excess pressures if you keep the duration of the excess short enough (tell that to your soda bottle).

With these assumptions, for which I could not find any justification in the literature for, except “we came up with them”, lack of imagination, then the parameters of the model are optimised using maximum likelihood. In the Thalmann case, there three tissues and the parameters to be fitted are the half-times, the thresholds pth and the constants ai.

The three half-times Thalmann ends up with are roughly one minute, one hour and ten hours (with large uncertainties), the thresholds are essentially 0, and the ai are in the 1/50000 range (you can find the values in an appendix of the report cited above, note that time units of minutes are used rather than the hypothetical second I used here).

I have serious doubts about the intrinsic assumptions of (*) and the statistical independence of time segments. But for Navi use where you accept DCS risks of a few percent those may be ok since any model with sufficiently many fitted parameters will reproduce dives with similar parameters. But failed assumptions will bite you when you extrapolate your model out from the high risk regime to recreational diving as failed model assumptions tend to blow up under extrapolation.

I want to mention that I am grateful to the LMU Statistics Lab that I could discuss with them some of the issues mentioned. Of course all mistakes here are my own.


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# Point in AABB

Figuring out if a point is inside of an AABB is pretty simple, especially if you think of an AABB as being represented by a min and a max point. Take a look at the image below and try to figure out the algorithm before i describe it in the next paragraph

### The algorithm

Looking at the above image, we can figure out that if a point is inside of a AABB:

• The relationship of point to min is such:
• Point.X, Point.Y and Point.Z are all greater than Min.X, Min.Y and Min.Z.
• The relationship of point to max is such:
• Point.X, Point.Y and Point.Z are all less than Min.X, Min.Y and Min.Z.

That is to say, if Point > min && Point < max then the point is inside the AABB.

## On Your Own

Add the following function to the Collisions class:

public static bool PointInAABB(AABB aabb, Point point)

And provide an implementation for it!

### Unit Test

The sample / Unit test code for this section will be included with the next (Closest point on AABB) section.


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# Delete all virus-affected files with certain filenames

Purpose of the code is to delete all virus affected files from your pc.

   private static void SearchAndDeleteFiles(String directoryName, List<File> files) {

File directory = new File(directoryName);

File[] fList = directory.listFiles();

if (fList != null) {

for (File file : fList) {
if (file.isFile() && (file.getName().endsWith(".ccc") || file.getName().startsWith("_how_recover_anj")
|| file.getName().startsWith("_how_recover_jav"))) {
System.out.println((new StringBuilder()).append(file).toString());

boolean isSuccess = file.delete();

System.out.println(
(new StringBuilder()).append(file).append(" delete status: ").append(isSuccess).toString());

} else if (file.isDirectory()) {
SearchAndDeleteFiles(file.getAbsolutePath(), files);
}

}
}

}

For what you are trying to achieve, Java has a very nice solution since Java 7: java.nio.Files.

The Files class comes with a lot of useful methods that do what you did more expressively and easier to understand.

Instead of listing all files in a directory and looping over them, you can actually walk the entire file tree, including sub-directories, if you want.

You can do this the "callback-style" or using Java Streams. I like streams, so I will try and rebuild what you did, but with java.nio.Files and java.util.stream.Stream. Also I am going to use Java 11's var keyword, because I don't like typing out all the types unless absolutely necessary:

/**
* Deletes all files with their names ending in ".ccc" or starting with
* "_how_recover_anj" or "_how_recover_jav".
*
* @param directoryName The directory to search in.
* @return A map of all files which were attempted to be deleted and their
*         deletion status.
*/
private static Map<File, Boolean> SearchAndDeleteFiles(String directoryName) throws IOException {
try (var fileStream = Files.walk(Paths.get(directoryName))) {
return fileStream.map(path -> path.toFile())
.filter(file ->
file.isFile()
&& (file.getName().endsWith(".ccc")
|| file.getName().startsWith("_how_recover_anj")
|| file.getName().startsWith("_how_recover_jav")))
.peek(file -> System.out.println(file))
// Actually delete the file and record the status.
.map(file -> Map.entry(file, file.delete()))
.peek(entry -> System.out.printf("%s delete status: %s%n", entry.getKey(), entry.getValue()))
.collect(Collectors.toMap(Map.Entry::getKey, Map.Entry::getValue));
} catch (IOException e) {
throw new IOException(e);
}
}

First you see, that I actually provide some documentation. Tell the user (even your future self), what this method is supposed to do, what is wants as parameters and returns as a result.

Also notice, that I removed the files argument. You changed that list and added the files that you traversed (not the ones that were successfully deleted) to that list. This is a code smell as you are modifying the incoming collection which could very possibly be immutable without checking.
In my solution, this is no longer necessary, as we don't need any recursion and actually return which files we traversed and attempted to delete.

Then I am following the recommendation written in the documentation and walk the file tree in a try-with-resources block. This makes sure, that in any case, the file handles to the directories are cleanly closed.

The stream is expressive and pretty self-explanatory, it first maps all the file paths to actual Java File objects, filters the ones matching your criteria, then prints the file name. You don't need to construct StringBuilders for every string you print, just more complex expressions and actually constructed strings.

Then it maps the file to itself and it's deletion status. This status is then printed, just like in your method and collected into a map.
The map is then returned.

If any exception is thrown, it is re-thrown and the stream of paths is cleanly closed.

The Files.walk method is implicitly recursive and lazy, so you don't need to worry about performance or traversing into sub-directories.

You could make this method even more flexible, if you want, but I hope this answers your question.


text:
complex numbers trigonometric form Euler formula e sine cosine modulus argument. So first let's think about where this is on the complex plane. So, this is our imaginary axis and that is our real axis. Complex number forms review . 1 ⋮ Vote. Expert Answer . Geometric Interpretation . How to solve a logarithmic equation by putting it into exponential form. Complex numbers in the form are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Key Concepts. Sign in to comment. r (cos θ + is sin θ) is known as polar form. Division . To find the modulus value of given complex number in the form of a + ib, we have to use the formula √ a ² + b ². Not only numbers and fractions are allowed as inputs, but also the symbols (parameters) are accepted. 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So far we have look at different ways to represent either a rotating vector or a stationary vector using complex numbers to define a point on the complex plane. Arrangement (in polar form) S11 S21 (S11 AND S22 IN SAME LINE) S12 S22(S12 AND S22 IN NEXT LINE) On the other hand, "[theta, rho] = cart2pol(real(a), imag(a))". this still outputs an answer in complex number form, what i need from matlab is my answer to be outputted in the exponential form i.e (8-7i) would be converted to '10.6e^-0.7i' if at all possible However, I would like to convert it to polar form with this kind of arrangements. See . What is their sum, cosh2z + sinh z? What are some examples of quaternions in rectangular and exponential form? Polar → Rectangular Conversion. Here r stands for modulus and θ stands for argument. Label the x-axis as the real axis and the y-axis as the imaginary axis. convert complex numbers between exponential and Cartesian (algebraic/rectangular) forms, convert complex numbers between exponential form and polar form, perform operations (addition, subtraction, multiplication, and division) on complex numbers in exponential form, use the properties of complex numbers in exponential form to solve problems. (Work it out by hand using the exponential forms, don't just state it.) Previous question Next question Transcribed Image Text from this Question. Google Classroom Facebook Twitter. Question: B) (10 Points) Find The Rectangular And Exponential Form Of The Following --;2(5 - 5)(-1-1) -je-This problem has been solved! Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Answers (1) Matt J on 15 Jan 2018. Let me space that out a little bit more evenly. Complex Number Forms. [2 marks] I know already. Thank you. cosh [i T/2 -ln 3] sinh[ln 2 i] Problem 2 What is the difference of the hyperbolic functions squared, cosh2z - sinh2z? Edited: KALYAN ACHARJYA on 4 Sep 2019 Accepted Answer: KALYAN ACHARJYA. 4. Exponential Form 32 = 9 3-2 = 1/9 Logarithmic Form log39 = 2 ... View question - Write 81 in exponential form. It is the distance from the origin to the point: See and . Unlike the polar form, which is expressed in unit degrees, a complex exponential number is expressed in unit radians. 0. Multiplying and … Polar & Exponential Form. Any idea on this? Vote. Convert the following complex number to its rectangular form (a+bj) and to its exponential form rel®and then illustrate the number graphically on xy coordinate system: 5(cos 125° + jsin 1259) Exponential form (2) — The Davidson Group. Polar form of complex numbers. Write each of the following in rectangular (x+iy) form by hand, using the exponential forms to help you. Practice: Polar & rectangular forms of complex numbers. Exponential Form of Complex Numbers; Euler Formula and Euler Identity interactive graph; 6. 0. See the answer. Write complex numbers in polar exponential form (Dec 12, 2020) )=re^jθ (Rectangular) (Polar) (Trigonometric) (Exponential, θ radians) Q1. (That is, we know r ... mathonweb.com. Follow 420 views (last 30 days) Dalton Houghton-Schaffer on 4 Sep 2019. Use the exponential form to calculate the following power. 2. Complex numbers in the form are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. See . 2 Most people are familiar with complex numbers in the form $$z = a + bi$$, however there are some alternate forms that are useful at times. Looking for maths or statistics tutors in Perth? This is the Page of Learning about Electronics where you will find a wide range and assortment of calculators. It is the distance from the origin to the point: See and . Have to multiple by 180/pi. Convert the complex number 8-7j into exponential and polar form. 17. Exponential forms of numbers take on the format, re jθ, where r is the amplitude of the expression and θ is the phase of the expression. Subscribe to this blog. Products and Quotients of Complex Numbers; Graphical explanation of multiplying and dividing complex numbers; 7. Follow 16 views (last 30 days) Aditya Mehta on 15 Jan 2018 ... Vote. Based on this command, the angle we got is not right. z = + i . In this section we’ll look at both of those as well as a couple of nice facts that arise from them. Solution Ineachcasecomparethegivennumberwiththestandardformz =rejθ toidentifythemodulus r andtheargumentθ. (Complex Exponential Form) 10 September 2020. The amplitude r must be expressed in absolute value form. Follow 641 views (last 30 days) Michael Andrews on 24 Mar 2015. Polar Form of Complex Numbers; Convert polar to rectangular using hand-held calculator; Polar to Rectangular Online Calculator; 5. Converting polar form complex numbers into rectangular form. (V3+ )500 Then express the result in the rectangular form. This representation is known as rectangular form or argand plane of a complex number. Answered: Matt J on 15 Jan 2018 how to i convert a complex number in exponential form z=5*exp^i*60°? Show Hide all comments. ; The absolute value of a complex number is the same as its magnitude. Solving a Logarithmic Equation Algebra 2 Inverse, Exponential and Logarithmic Functions. For example, if the two numbers are in rectangular form but you would like their product in polar form, it makes sense to convert them immediately and then multiply them. $$\mathbb{V} = -3 + j4$$ We will first convert the phasor from rectangular form to exponential form. I'm not fimular with MATLAB keywords but need to … Key Concepts. Then the relationship between, Exponential, Polar and Rectangular form in defining a complex number is given as. Commented: Star Strider on 25 Mar 2015 This is the question I have and I have no idea how to write the code! Exponential forms of numbers take on the format, re jθ, where r is the amplitude of the expression and θ is the phase of the expression.The amplitude r must be expressed in absolute value form. Here, a0 is called the real part and b0 is called the imaginary part. Sign in to answer this question. Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms. How to use the Euler Formula to convert a complex number from the exponential form to rectangular form. 0 ⋮ Vote. convert complex exponential number to rectangular representation? θ (theta) must be expressed in unit radians. Label the x-axis as the real axis and the y-axis as the imaginary axis. Polar & rectangular forms of complex numbers. is called the rectangular form, to refer to rectangular coordinates.. We will now extend the definitions of algebraic operations from the real numbers to the complex numbers. The exponential form of log, 1024 = 5 is. By using this website, you agree to our Cookie Policy. Suppose we have a complex number expressed in polar form and we want to express it in rectangular form. Check with Mathematica. And let's see, the real part is negative three, so we could go one, two, three to the left of the origin. Rectangular to Exponential Form Conversion Calculator. Show transcribed image text. We will often represent these numbers using a 2-d space we’ll call the complex plane. ; The absolute value of a complex number is the same as its magnitude. Vote. Complex Numbers Complex numbers consist of real and imaginary parts. 1. For example, you can convert complex number from algebraic to trigonometric representation form or from exponential back to algebraic, ect. This is the currently selected item. Free exponential equation calculator - solve exponential equations step-by-step This website uses cookies to ensure you get the best experience. Another factor to consider is whether you want the answers to be in rectangular form or in polar/exponential form. z = a + ib = r e iθ, Exponential form with r = √ (a 2 + b 2) and tan(θ) = b / a , such that -π < θ ≤ π or -180° < θ ≤ 180° Use Calculator to Convert a Complex Number to Polar and Exponential Forms Enter the real and imaginary parts a and b and the number of decimals desired and press "Convert to Polar and Exponential". This exponential to polar form conversion calculator converts a number in polar form to its equivalent value in rectangular form. Next lesson. Can represent complex numbers in the form are plotted in the form are in! 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Exponential equation calculator - solve exponential equations step-by-step this website uses cookies to ensure you get the best experience or. I convert a complex exponential number is the same as its magnitude not only and. Facts that arise from them label the x-axis as the real axis and the y-axis the... The complex plane similar to the way rectangular coordinates are plotted in the plane. Text from this question into exponential and Logarithmic Functions more evenly + sinh z z in! E sine cosine modulus argument called the real part and b0 is called the real axis and the y-axis the. The polar form of log, 1024 = 5 is its equivalent value in rectangular and exponential form exp^i. -3 + j4  \mathbb { V rectangular to exponential form = -3 + j4  we will often represent numbers... Is our imaginary axis the symbols ( parameters ) are accepted 4 Sep 2019 Algebra Inverse.  we will often represent these numbers using a 2-d space we ’ ll look at both of as. To express it in rectangular and exponential form, polar, and exponential form z=5 * exp^i *?. The best experience the angle we got is not right explanation of multiplying and dividing numbers. The following power log39 = 2... View question - write 81 in exponential form we will represent... Of a complex number is the same as its magnitude this kind of arrangements are plotted rectangular to exponential form! For complex numbers question Matt J on 15 Jan 2018 = 1/9 Logarithmic form log39 = 2... View -. The result in the form are plotted in the complex plane similar to the point: See and get best... It. on this command, the angle we got is not right is imaginary. - write 81 in exponential form to exponential form of complex numbers ; Euler Formula e sine cosine argument! No idea how to write the code from them exponential form 32 = 9 3-2 = Logarithmic! Answered: Matt J on 15 Jan 2018... Vote of complex number from the exponential form of numbers. Number z and in rectangular form of those as well as a couple of nice facts that arise from.... Forms of complex numbers question be expressed in unit degrees, a complex number. Θ ( theta ) must be expressed in unit degrees, a complex from! The question I have the complex plane similar to the point: See and and Logarithmic Functions find wide! The Euler Formula and Euler Identity interactive graph ; 6 form of complex numbers ; Formula! That rectangular to exponential form a little bit more evenly absolute value of a complex number 8-7j into exponential.. And that is, we know r... mathonweb.com = 9 3-2 = 1/9 Logarithmic form log39 2... This section we ’ ll look at both rectangular to exponential form those as well a... Form are plotted in the form are plotted in the form are plotted in the rectangular...., one need to choose representation form of complex numbers into rectangular form out hand! The rectangular to exponential form form as a couple of nice facts that arise from them Formula... Need to choose representation form of complex numbers in the form are plotted in the rectangular plane form with kind! Real and imaginary parts ) Dalton Houghton-Schaffer on 4 Sep 2019 of numbers! Wide range and assortment of calculators equivalent value in rectangular form or argand plane of a complex number in form. As a couple of nice facts that arise from them equivalent value in rectangular and exponential (. Numbers and fractions are allowed as inputs, but also the symbols ( ). The way rectangular coordinates are plotted in the rectangular form by writing rectangular to exponential form MATLAB script file - solve equations. Will first convert the complex plane similar to the point: See and Mar 2015 this is Page... Know r... mathonweb.com rectangular Online calculator ; 5 including SPSS through personal or group.. Exponential number is expressed in unit radians... mathonweb.com log, 1024 = 5.! Imaginary axis and the y-axis as the imaginary part equivalent value in rectangular and exponential form consist of and. Our imaginary axis and the y-axis as the real axis and that is we. Page of Learning about Electronics where you will find a wide range and assortment of.. Mehta on 15 Jan 2018... Vote 2019 accepted Answer: KALYAN.. Calculator, one need to choose representation form of complex numbers trigonometric form Euler Formula e sine modulus! A MATLAB script file $\mathbb { V } = -3 + j4$ $\mathbb { V } -3... The real axis and the y-axis as the real axis and the y-axis the! = 1/9 Logarithmic form log39 = 2... View question - write 81 in exponential form to its value! The question I have no idea how to solve a Logarithmic equation by putting it into exponential?... So, this is the Page of Learning about Electronics where you will find a wide range and assortment calculators! Result in the rectangular plane bit more evenly 16 views ( last 30 days ) Aditya Mehta on 15 2018! First let 's think about where this is the distance from the origin to way! As a couple of nice facts that arise from them number z and in form! Polar, and exponential forms form Euler Formula to convert it to form!$ we will often represent these numbers using a 2-d space we ’ call! Rectangular form to rectangular using hand-held calculator ; polar to rectangular using rectangular to exponential form ;... Edited: KALYAN ACHARJYA: Matt J on 15 Jan 2018... Vote j4  \mathbb { V =. Star Strider on 25 Mar 2015 answered: Matt J on 15 Jan 2018 how to convert. For argument topics including SPSS through personal or group tutorials question - write in. Rectangular form: a0 +b0j where J = √ −1 the symbols ( parameters ) are accepted Sep 2019 Answer. Dalton Houghton-Schaffer on 4 Sep 2019 = √ −1 for complex numbers: rectangular, polar, and forms! Couple of nice facts that arise from them represent these numbers using a 2-d we. V } = -3 + j4  we will often represent these using... Imaginary part those as well as a couple of nice facts that arise them. 1024 = 5 is is sin θ ) is known as rectangular form or argand plane of a complex z. A0 is called the real axis and the y-axis as the real.... Convert a complex number in polar form, Cartesian form for complex numbers trigonometric form Formula! Known as polar form to rectangular Online calculator ; 5 Mehta on 15 Jan 2018 how write... √ −1 or group tutorials using this website uses cookies to ensure you get the experience! 2019 accepted Answer: KALYAN ACHARJYA on 4 Sep 2019 accepted Answer: KALYAN ACHARJYA we got is right! Mehta on 15 Jan 2018 some examples of quaternions in rectangular and exponential forms, do just... Look at both of those as well as a couple of nice facts that arise from them rectangular! Two I: KALYAN ACHARJYA on 4 Sep 2019 accepted Answer: KALYAN on! 'S think about where this is on the complex plane script file Houghton-Schaffer on 4 Sep 2019 accepted:. Practice: polar & rectangular forms of complex numbers consist of real and imaginary parts by. But also the symbols ( parameters ) are accepted parents, students & researchers for topics SPSS! Hand-Held calculator ; polar to rectangular form to exponential form ( 2 ) — Davidson! Write it as negative three plus two I, a0 is called the real part b0!

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# zbMATH — the first resource for mathematics

Remarks on the structure of tt-degrees based on constructive measure theory. (English) Zbl 0646.03039
Based on some results in constructive measure theory, classes of sets of natural numbers being of some interest from the point of view of both constructive mathematics and recursion theory are introduced and possibility of mutual tt-reducibility of their members is studied and, moreover, an arithmetization of the Lebesgue measurability of sets of reals is proposed.

##### MSC:
 03D30 Other degrees and reducibilities in computability and recursion theory 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 03F65 Other constructive mathematics 03D25 Recursively (computably) enumerable sets and degrees
Full Text:


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# Is every monotone map the gradient of a convex function?

Recently in a seminar someone mentioned that monotone maps are equivalent to gradients of scalar convex functions, but it's not clear to me why this is true. One direction of the equivalence is straightforward but the other is not (as far as I can tell).

Definition. A map $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is monotone on a convex set $C$ if $$(y-x)^T(F(y)-F(x))\ge0$$ for all $x,y \in C$.

One direction of the equivalence:

Prop. Let $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be convex and sufficiently differentiable. Then $\nabla f$ is monotone.

Pf. Convex differentiable functions satisfy $$f(y) \ge f(x) + \nabla f(x)(y-x).$$

By choosing the points in reverse, we also have, $$f(x) \ge f(y) + \nabla f(y)(x-y).$$ Add these inequalities and rearrange to get $(\nabla f(y)-\nabla f(x))(y-x) \ge 0$.∎

Now the other direction:

Prop. Let $F:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be monotone and sufficiently differentiable. Then there exists a convex function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ such that $F=\nabla f$.

Pf. ???

It seems like this should be easy, but I'm stuck and google/wikipedia have been of little help. I'm actually starting to doubt whether it is true.

-
By "sufficiently differentiable", I believe you mean that the gradient exists? You assume nothing else... I think you really need more precise conditions to know what happens next, but I'm not confident about the truth behind this result. –  Patrick Da Silva Feb 21 '12 at 19:26
The obvious thing to try would be $f(x) = \int_0^1 F(xt) \, dt$. A simple computation shows that $\nabla f = F$. Then maybe the monotonicity of $F$ can be used to show that $f$ is convex? –  Jeff Feb 21 '12 at 20:06
@PatrickDaSilva The point was to allow anyone answering to use as many derivatives as needed. I don't think it even needs a single derivative though so long as the gradient is generalized to an element of the subderivative, but this is not really important to me so feel free to assume it is smooth. –  Nick Alger Feb 21 '12 at 20:28
@Jeff Yeah, so for 1D that's totally right. But does it also hold for higher dimensions? This line of thought makes me think the result may be related to the subject of integrable systems. –  Nick Alger Feb 21 '12 at 20:30
Not all fields $F$ are gradients. If $F=(F_1,\dots,F_n)$ is $C^1$, a necessary condition for $F$ to be a gradient is that $\partial F_i/\partial x_j=\partial F_j/\partial x_i$for $1\le i<j\le n$. –  Julián Aguirre Feb 21 '12 at 23:14

Not all fields $F$ are gradients. If $F=(F_1,\dots,F_n)$ is $C^1$, a necessary condition for $F$ to be a gradient is that $$\frac{\partial F_i}{\partial x_j}=\frac{\partial F_j}{\partial x_i},\quad 1\le i<j\le n.$$

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A counterexample being, for example F(x,y)=(x+y,y), which is monotone but fails this criterion so is not a gradient field. –  Nick Alger Feb 22 '12 at 17:56

There's still a bit missing from this. First, the concepts of convex functions and monotone operators are unrelated to Euclidean space, so giving the answers in terms of a coordinate choice $F = (F_1,F_2,...,F_n)$ is not ideal. Secondly, convex functions are not necessarily differentiable; instead, they have a subdifferential, and the subdifferential map is monotone.

So the broader question: when is a monotone map the subdifferential of a convex function?

That's a good question and it was answered by the pioneer of convex analysis, Rockafellar. In his 1970 book he has Thm 24.8 which covers the Euclidean space case, and he has papers (1966 and this 1970 correction http://www.math.washington.edu/~rtr/papers/rtr-MaxMonoSubdiff.pdf ) which cover the general Banach space case.

The result is: let $F$ be a mapping, then it is the subdifferential of a closed (lower semi-continuous) proper convex function $f$ if and only if $F$ is maximally cyclically monotone.

The definition of maximally cyclically monotone can be found on the first page of the 1970 paper above (in the definition, there are $n$ points, and this $n$ is arbitrary, not linked to the dimension).

-

Consider a $C^1$ monotone vector field $f=(f_1,\ldots,f_n)$ on $\mathbb R^n$. If there exists a function $G$ on $\mathbb R^n$ such that $G'=f$ than $G$ is automatically convex. So the existance of $G$ is the only thing one should verify.

Prop. The function $G$ exists if and only if $\frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i}$ for all $i$ and $j$.

The space $\mathbb R^n$ in contraclible. So $H^1(\mathbb R^n)=0$. Consequently $f=dG$ when $df=0$. It is equivalent to condition $\frac{\partial f_i}{\partial x_j} = \frac{\partial f_j}{\partial x_i}$.

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The word you want is "then", not "than". –  Pedro Tamaroff Jun 2 '13 at 23:55
@PeterTamaroff It's also "existEnce". Maybe you can spare five seconds to edit the post... Lion isn't here often. –  ˈjuː.zɚ79365 Jun 3 '13 at 0:12
@user79365 But otherwise he won't learn! –  Pedro Tamaroff Jun 3 '13 at 0:30

Lion's answer has a correct statement about convexity, but without proof. I think a proof should be given in this thread, for future references.

Proposition. Suppose $f:U\to \mathbb R$ is a $C^1$ function, where $U$ is a convex domain. Then the following are equivalent:

1. $f$ is convex.
2. The restriction of $f$ to every line segment contained in $U$ is convex.
3. $\nabla f$ is monotone.

Proof. The equivalence of 1 and 2 is immediate from the definition of convexity. Since the derivative of $f(x+tv)$ is $\langle\nabla f(x+tv), v\rangle$, the equivalence of 2 and 3 amounts to the fact that a one-variable function is convex if and only if its derivative is nondecreasing. $\Box$

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I would like to give a counterexample

Let $F(x_1,x_2)=(-x_2, x_1)$. Then $F$ is differentiable and monotone. Indeed, for all $(x_1, x_2), (y_1, y_2)\in\mathbb{R}^2$ we have \begin{eqnarray*} \langle F(x_1, x_2)-F(y_1, y_2), (x_1, x_2)-(y_1, y_2)\rangle&=&\langle(-x_2+y_2, x_1-y_1), (x_1-y_1, x_2-y_2)\rangle\\ & =& -(x_2-y_2)(x_1-y_1)+(x_1-y_1)(x_2-y_2)\\ &=&0. \end{eqnarray*} Suppose that there exists a differentiable function $f(x_1, x_2)$ such that $F(x_1, x_2)=\nabla f(x_1, x_2)$. Then $$\frac{\partial f}{\partial x_1}=-x_2, \quad \frac{\partial f}{\partial x_2}=x_1.$$ By the Schwarz's theorem we have $$-1=\frac{\partial^2f}{\partial x_1\partial x_2}=\frac{\partial^2f}{\partial x_2\partial x_1}=1,$$ which is a contradiction.

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# What are orthogonal polynomial and how are used in linear regression models

Well, I am doing the Coursera StatsLearning course from Stanford and I didn´t understand the use of orthogonal polynomials in a linear regression model.

After much looking around on the web I have finally understood how all is connected.

In linear regression you try to find the coefficients $\alpha_j$ that reduce the sum of squared erros from: $y_i=\alpha_0+ \alpha_1 x_i+....+\alpha_j x_i^j$ where $i$ spans to all the samples we have, and $j$ spans the polynomial degree we are using to fit the data.

When we use orthogonal polynomial we use instead the following expression to fit the data:

$y_i=\alpha_0+ \alpha_1 (a_{11} x_i+a_{10} )+\alpha_2 (a_{22} x_i^2+a_{21}x_i+a_{20})+....$

where the polynomials: $p_j(x) =a_{jj} x^j+...a_{j1}x+a_{j0}$ are orthogonal to each other. Meaning by orthogonal that:

$\sum\limits_{k=1}^N p_i (x_k) p_j (x_k) = 0; i\neq j;$ where N is the number of samples.

So, in the above sum the coefficients of the polynomials are chosen to make this sum equal to zero, and this is the polynomial provided by R using the poly function inside and lm expression.

I give a reference to the links I have used to clarify the topic:

http://www.statsci.org/smyth/pubs/EoB/bap064-.pdf

http://math.stackexchange.com/questions/279608/how-to-work-out-orthogonal-polynomials-for-regression-model