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by rector

Mathematical & Chemical API Integration

December 23, 2018 in Uncategorized

Math Equations on all webpages are now functional thanks to Prof. Horstmann's Math API integration

Beautiful Math Equations in all browsers

MathML Integral Formula

f ( a ) = 1 2 π i γ f ( z ) z a d z

or Tex base ... Integral Formula

\begin{align} f(a) = \frac{1}{2\pi i} \oint_{\gamma}\frac{f(z)}{z-a}dz \end{align}

Dynamic Equations - step by step

Expand the following:

\begin{align} (x+1)^2 &\cssId{Step1}{= (x+1)(x+1)}\\ &\cssId{Step2}{= x(x+1) + 1(x+1)}\\ &\cssId{Step3}{= (x^2+x) + (x+1)}\\ &\cssId{Step4}{= x^2 + (x + x) + 1}\\ &\cssId{Step5}{= x^2+2x+1}\\ \end{align}

The Quadratic Formula

\begin{align} When $a \ne 0$, there are two solutions to $ax^2 bx c = 0$ and they are \end{align}

\begin{align} x = {-b \pm \sqrt{b^2-4ac} \over 2a}. \end{align}

Double angle formula for Cosines

\begin{align} \cos(θ+φ)=\cos(θ)\cos(φ)−\sin(θ)\sin(φ) \end{align}

Gauss' Divergence Theorem

\begin{align} \int_D ({\nabla\cdot} F)dV=\int_{\partial D} F\cdot ndS \end{align}

Curl of a Vector Field

\begin{align} \vec{\nabla} \times \vec{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \end{align}

Standard Deviation

\begin{align} \sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2} \end{align}

Definition of Christoffel Symbols

\begin{align} (\nabla_X Y)^k = X^i (\nabla_i Y)^k = X^i \left( \frac{\partial Y^k}{\partial x^i} + \Gamma_{im}^k Y^m \right) \end{align}

The Lorenz Equations

\begin{align} \dot{x} & = \sigma(y-x) \\ \dot{y} & = \rho x - y - xz \\ \dot{z} & = -\beta z xy \end{align}

Cauchy's Integral Formula

\begin{align} f(a) = \frac{1}{2\pi i} \oint\frac{f(z)}{z-a}dz \end{align}

The Cauchy-Schwarz Inequality

\[ \left( \sum_{k=1}^n a_k b_k \right)^{\!\!2} \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right) \]

A Cross Product Formula

\[ \mathbf{V}_1 \times \mathbf{V}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ \frac{\partial X}{\partial u} & \frac{\partial Y}{\partial u} & 0 \\ \frac{\partial X}{\partial v} & \frac{\partial Y}{\partial v} & 0 \\ \end{vmatrix} \]

The probability of getting \(k\) heads when flipping \(n\) coins is:

\[P(E) = {n \choose k} p^k (1-p)^{ n-k} \]

An Identity of Ramanujan

\[ \frac{1}{(\sqrt{\phi \sqrt{5}}-\phi) e^{\frac25 \pi}} = 1 \frac{e^{-2\pi}} {1 \frac{e^{-4\pi}} {1 \frac{e^{-6\pi}} {1 \frac{e^{-8\pi}} {1 \ldots} } } } \]

A Rogers-Ramanujan Identity

\[ 1 \frac{q^2}{(1-q)} \frac{q^6}{(1-q)(1-q^2)} \cdots = \prod_{j=0}^{\infty}\frac{1}{(1-q^{5j 2})(1-q^{5j 3})}, \quad\quad \text{for $|q|<1$}. \]

Maxwell's Equations

\begin{align} \nabla \times \vec{\mathbf{B}} -\, \frac1c\, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \\ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \\ \nabla \times \vec{\mathbf{E}}\, \, \frac1c\, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \\ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align}

In-line Mathematics

Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression \(\sqrt{3x-1} (1 x)^2\) is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.

Trig Identity Formulas

Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form.

Basic Identities











Pythagorean Identities

\[\sin^2(x) \cos^2(x)=1\]

\[1 \tan^2(x)=\sec^2(x)\]

\[1 \cot^2(x)=\csc^2(x)\]

Sum and Difference Formulas

\[\sin(a b)=\sin(a)\cos(b) \cos(a)\sin(b)\]


\[\cos(a b)=\cos(a)\cos(b)-\sin(a)\sin(b)\]

\[\cos(a-b)=\cos(a)\cos(b) \sin(a)\sin(b)\]

\[\tan(a b)=\frac{\tan(a) \tan(b)}{1-\tan(a)\tan(b)}\]

\[\tan(a-b)=\frac{\tan(a)-\tan(b)}{1 \tan(a)\tan(b)}\]

\[\sin(x) \sin(y)=2\sin(\frac{x y}{2})\cos(\frac{x-y}{2})\]

\[\sin(x)-\sin(y)=2\cos(\frac{x y}{2})\sin(\frac{x-y}{2})\]

\[\cos(x) \cos(y)=2\cos(\frac{x y}{2})\cos(\frac{x-y}{2})\]

\[\cos(x)-\cos(y)=-2\sin(\frac{x y}{2})\sin(\frac{x-y}{2})\]

Double Angle Formulas


\[\cos(2x)=\cos^2(x)-\sin^2(x)=1-2\sin^2(x) = 2\cos^2(x)-1\]

Half Angle Formulas


\[\cos(\frac{x}{2})=\pm\sqrt{\frac{1 \cos(x)}{2}}\]

\[\tan(\frac{x}{2})=\pm\sqrt{\frac{1-\cos(x)}{1 \cos(x)}}=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1 \cos(x)}\]

Trigonometric Products

\[\sin(x)\cos(y)=\frac{\sin(x y) \sin(x-y)}{2}\]

\[\cos(x)\cos(y)=\frac{\cos(x y) \cos(x-y)}{2}\]

\[\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x y)}{2}\]

Chemical Equations on all webpages are now functional thanks to Prof. Horstmann's Chemical API integration

Beautiful Chemical Equations in all browsers

$$\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}$$

Organic chemistry molecule ...

$$\ce{CH3\bond{1}} {\stackrel{ \;\;\;\large\ce{CH3} }{\stackrel{|}{\underset{\underset{\huge\ce{Cl}}{|}}{\ce{C}}}}}\ce{-CH3}$$

Chemical Equations (ce) ...

$$\ce{CO2 C -> 2 CO}$$

$$\ce{Hg^2 ->[I-] HgI2 ->[I-] [Hg^{II}I4]^2-}$$

$$C_p[\ce{H2O(l)}] = \pu{75.3 J // mol K}$$

Chemical Formulae

$$\ce{H2O}$$ $$\ce{Sb2O3}$$


$$\ce{H }$$ $$\ce{CrO4^2-}$$ $$\ce{[AgCl2]-}$$

$$\ce{Y^99 }$$ $$\ce{Y^{99 }}$$

Stoichiometric Numbers

$$\ce{2 H2O}$$ $$\ce{2H2O}$$ $$\ce{0.5 H2O}$$

$$\ce{1/2 H2O}$$ $$\ce{(1/2) H2O}$$ $$\ce{$n$ H2O}$$


$$\ce{^{227}_{90}Th }$$ $$\ce{^227_90Th }$$ $$\ce{^{0}_{-1}n^{-}}$$ $$\ce{^0_-1n-}$$ $$\ce{H{}^3HO}$$ $$\ce{H^3HO}$$

Reaction Arrows

$$\ce{A -> B}$$     $$\ce{A <- B}$$     $$\ce{A <-> B}$$     $$\ce{A <--> B}$$     $$\ce{A <=> B}$$     $$\ce{A <=>> B}$$     $$\ce{A <<=> B}$$     $$\ce{A ->[H2O] B}$$     $$\ce{A ->[{text above}][{text below}] B}$$     $$\ce{A ->[$x$][$x_i$] B}$$

Parentheses, Brackets, Braces

$$\ce{(NH4)2S}$$ $$\ce{[\{(X2)3\}2]^3 }$$ $$\ce{CH4 2 $\left( \ce{O2 79/21 N2} \right)$}$$

States of Aggregation

$$\ce{H2(aq)}$$ $$\ce{CO3^2-_{(aq)}}$$ $$\ce{NaOH(aq,$\infty$)}$$

Crystal Systems

$$\ce{ZnS($c$)}$$ $$\ce{ZnS(\ca$c$)}$$

Variables like x, n, 2n 1

$$\ce{NO_x}$$ $$\ce{Fe^n }$$ $$\ce{x Na(NH4)HPO4 ->[\Delta] (NaPO3)_x x NH3 ^ x H2O}$$

Greek Characters

$$\ce{\mu-Cl}$$ $$\ce{[Pt(\eta^2-C2H4)Cl3]-}$$ $$\ce{\beta }$$ $$\ce{^40_18Ar \gamma{} \nu_e}$$

(Italic) Math

$$\ce{NaOH(aq,$\infty$)}$$ $$\ce{Fe(CN)_{$\frac{6}{2}$$}}$$ $$\ce{X_{$i$}^{$x$}}$$ $$\ce{X_$i$^$x$}$$

Italic Text

$$\ce{$cis${-}[PtCl2(NH3)2]}$$ $$\ce{CuS($hP12$)}$$

Upright Text, Escape Parsing

$$\ce{{Gluconic Acid} H2O2}$$ $$\ce{X_{{red}}}$$ $$\ce{{( )}_589{-}[Co(en)3]Cl3}$$


$$\ce{C6H5-CHO}$$ $$\ce{A-B=C#D}$$ $$\ce{A\bond{-}B\bond{=}C\bond{#}D}$$ $$\ce{A\bond{1}B\bond{2}C\bond{3}D}$$ $$\ce{A\bond{~}B\bond{~-}C}$$ $$\ce{A\bond{~--}B\bond{~=}C\bond{-~-}D}$$ $$\ce{A\bond{...}B\bond{....}C}$$ $$\ce{A\bond{->}B\bond{<-}C}$$

Addition Compounds

$$\ce{KCr(SO4)2*12H2O}$$ $$\ce{KCr(SO4)2.12H2O}$$ $$\ce{KCr(SO4)2 * 12 H2O}$$

Oxidation States


Unpaired Electrons, Radical Dots

$$\ce{OCO^{.-}}$$ $$\ce{NO^{(2.)-}}$$

Kröger-Vink Notation

$$\ce{Li^x_{Li,1-2x}Mg^._{Li,x}$$V'_{Li,x}Cl^x_{Cl}}$$ $$\ce{O''_{i,x}}$$ $$\ce{M^{..}_i}$$ $$\ce{$V$^{4'}_{Ti}}$$ $$\ce{V_{V,1}C_{C,0.8}$$V_{C,0.2}}$$

Equation Operators

$$\ce{A B}$$ $$\ce{A - B}$$ $$\ce{A = B}$$ $$\ce{A \pm B}$$

Precipitate and Gas

$$\ce{SO4^2- Ba^2 -> BaSO4 v}$$ $$\ce{A v B (v) -> B ^ B (^)}$$

Other Symbols and Shortcuts

$$\ce{NO^*}$$ $$\ce{1s^2-N}$$ $$\ce{n-Pr}$$ $$\ce{iPr}$$ $$\ce{\ca Fe}$$ $$\ce{A, B, C; F}$$ $$\ce{{and others}}$$

Complex Examples

$$\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}$$ $$K = \frac{[\ce{Hg^2 }][\ce{Hg}]}{[\ce{Hg2^2 }]}$$ $$K = \ce{\frac{[Hg^2 ][Hg]}{[Hg2^2 ]}}$$ $$\ce{Hg^2 ->[I-] $\underset{\mathrm{red}}{\ce{HgI2}}$$ ->[I-] $\underset{\mathrm{red}}{\ce{[Hg^{II}I4]^2-}}$$}$$

Physical Units (pu)

$$\pu{123 kJ}$$ $$\pu{123 mm2}$$ $$\pu{123 J s}$$ $$\pu{123 J*s}$$ $$\pu{123 kJ/mol}$$ $$\pu{123 kJ//mol}$$ $$\pu{123 kJ mol-1}$$ $$\pu{123 kJ*mol-1}$$ $$\pu{1.2e3 kJ}$$ $$\pu{1,2e3 kJ}$$ $$\pu{1.2E3 kJ}$$ $$\pu{1,2E3 kJ}$$

Crystal Systems

$$\ce{ZnS($c$)}$$ $$\ce{ZnS(\ca$c$)}$$

In-line Chemical Equations

Finally, while display equations look good for a page of samples, the ability to mix Chemical Equations and text in a paragraph is also important. This expression \(\ce{Zn^2 <=>[ 2OH-][ 2H ] $\underset{\text{amphoteres Hydroxid}}{\ce{Zn(OH)2 v}}$$ <=>[ 2OH-][ 2H ] $\underset{\text{Hydroxozikat}}{\ce{[Zn(OH)4]^2-}}$$}\) is an example of an inline equation. As you see, equations can be used this way as well, without unduly disturbing the spacing between lines.

by rector

Remove WordPress Logo & admin bar…

December 15, 2018 in Uncategorized


  • add_filter(‘show_admin_bar’, ‘__return_false’);
  • /

/* Disable WordPress Admin Bar for all users but admins. / / show_admin_bar(false); */

add_filter('show_admin_bar', '__return_false');


  • add_action( 'admin_bar_menu', 'remove_wp_logo', 999 );
  • function remove_wp_logo( $wp_admin_bar ) {
  • $wp_admin_bar->remove_node( 'wp-logo' );
  • }
    add_action( 'admin_bar_menu', 'remove_wp_logo', 999 ); function remove_wp_logo( $wp_admin_bar ) { $wp_admin_bar->remove_node( 'wp-logo' ); }

by rector


December 24, 2016 in Uncategorized

by rector

Biodiesel: Explore Its Untapped Potential

October 23, 2014 in Uncategorized

For too long, animal fat-based biodiesel has been hiding in the shadow of its plant oil-based sisters, wrongly perceived as a lesser-quality option. The truth is, that bad reputation isn’t justified.
According to scientific studies, both animal fat- and plant-based biodiesel contain the same chemical constituents-only the proportions vary. Both reduce unburned hydrocarbons, carbon monoxide and particulates. Both are biodegradable, non toxic and safe to use in diesel engines with no modifications. Both are completely renewable sources of energy.
Animal fat-based biodiesel gained a poor reputation in the early years of the industry because some producers ignored quality standards and did not adequately manage their chemistry or downstream refining. Today, there are many high-quality animal fat-based biodiesel producers that offer ASTM and/or EN specification product to their customers.a rather slow fading reaction and becomes completely colorless above 13.0 pH.

Pendant trop longtemps, le biodiesel à base de graisse animale a été caché dans l’ombre de ses sœurs à base d’huile végétale, à tort, perçue comme une option de moindre qualité. La vérité, c’est que la mauvaise réputation n’est pas justifiée.
Selon des études scientifiques, à la fois FAT animale et le biodiesel à base de plantes contiennent les mêmes constituants chimiques ne-les proportions varient. Les deux réduisent les hydrocarbures imbrûlés, monoxyde de carbone et de particules. Les deux sont biodégradable, non toxique et sans danger à utiliser dans les moteurs diesel sans aucune modification. Les deux sont complètement sources d’énergie renouvelables.
Biodiesel à base de graisse animale a acquis une mauvaise réputation dans les premières années de l’industrie parce que certains producteurs ignorés normes de qualité et n’ont pas réussi adéquatement leur chimie ou du raffinage en aval. Aujourd’hui, il ya beaucoup de producteurs de biodiesel à base de graisses animales de haute qualité qui offrent ASTM et / ou FR produit des spécifications pour leurs clients.

by rector

Bio Diesel from Animal Fat, consider the facts:

October 23, 2014 in Uncategorized

Consider the facts:
Cetane number and emissions: The cetane number measures ignition quality of the fuel. The higher the number, the better. Soybean oil-based biodiesel averages between 46 and 52, while that of conventional diesel falls between 40 and 52. Animal fat-based biodiesel is between 56 and 60. Inadequate cetane numbers result in poor ignition quality and delay, abnormal combustion, excessive engine knock and smoke on cold starts. In addition to improving fuel combustion, increasing cetane levels often reduces emissions of nitrogen oxides and particulate matter. Animal fat-based biodiesel also provides added lubricity, a measure of protective compounds in the fuel that reduce engine wear and tear.

Cold-weather performance: Winter performance is an issue for all biodiesel, regardless of feedstock. While its higher cloud point means animal fat-based biodiesel freezes at a higher temperature than its plant oil-based counterparts, extensive testing has shown virtually no impact in B5 blends.

Engine suitability: If it meets ASTM standards, animal fat-based biodiesel is safe to use in all diesel engines. Most original equipment manufacturer dealers and customer service departments recommend B5 blends, while some promote B20 blends. Like other biodiesel, it acts as a solvent, releasing the deposits that build up after using petroleum-based diesel. For that reason, based on how biodiesel use is implemented, filters should be checked regularly and may need to be changed more often.

Sustainability: While all biodiesel is greener than petroleum-based diesel, animal fat-based biodiesel offers the most sustainable choice because it uses byproducts as a feedstock instead of relying on virgin materials. In addition, scientific studies continue to show better greenhouse gas reductions from the use of animal fats to produce biodiesel.

Stability: Without modifications or treatment, biodiesel can form sediments during storage. However, the higher percentage of saturated fats in animal fat-based biodiesel provides greater oxidative stability than its plant oil-based counterparts, reducing the risk of sedimentation.

Supply: American meat production shows no signs of slowing as the USDA predicts continued growth of 1 percent per year. That guarantees a steady supply of tallow and fats. With only 3 percent to 8 percent of animal fats and tallow being used for biodiesel production, there’s no shortage of feedstock to supply North America’s booming biodiesel industry.

The bottom line is that if biodiesel meets the prevalent ASTM specification, the feedstock used to produce it should not be a question. Animal fat-based biodiesel continues to measure up. In order to meet our renewable fuel obligations, we need to start taking advantage of a promising resource.

Examinons les faits:
L’indice de cétane et les émissions: Le nombre de cétane mesure la qualité du carburant d’allumage. Plus le nombre est élevé, mieux c’est. Biodiesel à base de graisse entre 40 et 52 animaux l’huile de soja à base de moyennes de biodiesel entre 46 et 52, tandis que celui du diesel classique tombe entre 56 et 60 indices de cétane inadéquates entraînent une mauvaise qualité de l’allumage et de retard, une combustion anormale, frappe excessive du moteur et de la fumée sur les démarrages à froid. En plus d’améliorer la combustion du carburant, ce qui augmente les niveaux de cétane souvent réduit les émissions d’oxydes d’azote et les matières particulaires. Biodiesel à base de graisse animale fournit également un pouvoir lubrifiant supplémentaire, une mesure de protection des composés dans le carburant qui permettent de réduire l’usure du moteur et à la déchirure.

Temps froid performances: les performances de l’hiver est un problème pour tout le biodiesel, indépendamment de la matière première. Bien que son point de trouble élevé signifie animales à base de graisse biodiesel gèle à une température plus
élevée que ses homologues de la centrale à base d’huile, de nombreux tests ont montré pratiquement aucun impact dans les mélanges B5.

L’aptitude du moteur: Si il répond aux normes de l’ASTM, le biodiesel à base de graisse animale est sûr à utiliser dans tous les moteurs diesel. La plupart des concessionnaires du fabricant d’équipement d’origine et les départements de service à la clientèle recommandent mélanges B5, tandis que d’autres favorisent B20 mélanges. Comme autre biodiesel, il agit comme un solvant, libérant les dépôts qui se forment après l’utilisation de diesel à base de pétrole. Pour cette raison, basé sur la façon dont l’utilisation du biodiesel est mis en œuvre, les filtres doivent être vérifiés régulièrement et peuvent avoir besoin d’être changé plus souvent.

Durabilité: Alors que tout le biodiesel est plus verte que le diesel à base de pétrole, biodiesel à base de graisse animale offre le choix le plus durable, car il utilise des sous-produits comme matière première au lieu de compter sur les matériaux vierges. En outre, des études scientifiques continuent de montrer de meilleures réductions de gaz à effet de serre provenant de l’utilisation de graisses animales pour produire du biodiesel.

Stabilité: Sans modifications ou de traitement, le biodiesel peut former des sédiments pendant le stockage. Cependant, le pourcentage d’acides gras saturés dans le biodiesel à base de graisse animale offre une plus grande stabilité à l’oxydation par rapport à ses homologues de plantes à base d’huile, ce qui réduit le risque de sédimentation.

Alimentation: la production de viande américaine montre aucun signe de ralentissement que l’USDA prévoit une croissance continue de 1 pour cent par an. Cela garantit un approvisionnement régulier de suif et les graisses. Avec seulement 3 pour cent à 8 pour cent de graisses et le suif animal qui est utilisé pour la production de biodiesel, il n’y a pas de pénurie de matières premières à fournir à l’industrie du biodiesel en plein essor en Amérique du Nord.

La ligne de fond est que si le biodiesel est conforme aux spécifications de la norme ASTM répandue, la charge utilisée pour produire il ne devrait pas être une question. Biodiesel à base de graisse animale continue de se mesurer. Afin de répondre à nos obligations en matière de carburants renouvelables, nous devons commencer à prendre avantage d’une ressource prometteuse.