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https://www.physicsforums.com/threads/equipotential-contour-theory.293508/ | # Equipotential Contour Theory
1. Feb 18, 2009
### TJDF
1. The problem statement, all variables and given/known data
The lines show equipotential contours in the plane of three point charges, Q1, Q2, and Q3. The positions of the charges are marked by dots. The values of the potentials are in kilovolts as indicated, e.g., +5 kV, −5 kV; the contour interval is 1 kV. The letters denote locations on the contours. For distances use the scale below the picture, to an appropriate accuracy. Select all the correct statements, e.g., AB.
http://img100.imageshack.us/img100/2589/physicswe4.jpg [Broken]
http://g.imageshack.us/img100/physicswe4.jpg/1/ [Broken]
A) The electric field at g is zero.
B) Charge Q2 is the largest negative charge.
C) Q1 is a negative charge.
D) The electric field at d is stronger than at b.
E) Charge Q3 is the largest positive charge.
F) The force on an electron at g points to the top of the page.
2. Relevant equations
No equations necessary, theory only.
3. The attempt at a solution
I just don't know the theory behind this, I've been tried, failed, and I don't understand. If anyone can explain the concepts and what the right answer is, I'd really appreciate it.
Last edited by a moderator: May 4, 2017
2. Feb 18, 2009
### TJDF
Anyone?
3. Feb 18, 2009
### AEM
Here's some relevant theory. Taking your reference position to be V = 0 at infinity, the electric potential, V, is given by
$$V = \frac{1}{4 \pi \epsilon_o} \frac{q}{r}$$
This is related to the electric field by
$$E = - \nabla V$$
which for a point charge can be rewritten as
$$E = - \frac{1}{4 \pi \epsilon_o} \frac{\partial}{\partial r} \frac{q}{r}$$
calculating out the derivative, you get
$$E = \frac{1}{4 \pi \epsilon_o} \frac{q}{r^2}$$
The lines drawn on your diagram are lines of constant electric potential. They are analogous to lines of constant altitude on a topographic map. The closer together the equipotential lines are, the larger the gradient, so the stronger the electric field is. The same thing is true on a topographic map --steep hills are indicated by contour lines close together.
Hope this helps.
4. Feb 18, 2009
### TJDF
Thanks.
I still don't know how each would apply to the questions though.
A) The electric field at g is zero.
---> I can't tell whether this is true or not, I'm starting to have my doubts, but I can't explain why.
B) Charge Q2 is the largest negative charge.
---> I was told the diameter of the charge means it has more strength, therefore Q1 is the largest negative charge, so this is definitely false... right?
C) Q1 is a negative charge.
---> True. Unless I've been reading this wrong.
D) The electric field at d is stronger than at b.
---> I thought because they are on the same contour lines, the fields are the same, but again, like A, I don't get the reasoning, but judging by the fact that closer equipotential lines are, then stronger gradients, this must be true.
E) Charge Q3 is the largest positive charge.
---> It's the only positive charge. True.
F) The force on an electron at g points to the top of the page.
---> False, I'm pretty sure on this one now from my reading.
5. Feb 18, 2009
### TJDF
Is the electric field zero if the equipotential contour line is 0 volts?
How does the diameter of the circle affect the strength of the charge?
Are electric fields on the same contour line equal or judging by the fact that closer equipotential lines are, then stronger gradients, then can they be different?
What about the force on an electrons and why they would point up/down?
6. Feb 18, 2009
### TJDF
Wait... are the correct answers C, D, and E?
C is true because q1 is a negative charge.
D is true because the contour lines are closer together, and D is closer to the positive charge meaning the field must be stronger because... the charge is stronger.
E is true because q3 is the only positive charge.
A is false because there are positive charges and negative charge and so it naturally cannot be 0.
B is false because q1 has a large radius and therefore, looking at the formula rearranged, that would make q larger.
F is false because the electron should point down in alignment with its charge.
Is this right?? unfortunately Im not given the correct answers and I cant find anything that explains this clearly to me, usually I learn through examples, but I can't find any. Can anyone confirm or disprove my reasoning?
7. Feb 19, 2009
### AEM
Obviously, my responses are in black
8. Feb 19, 2009
### TJDF
Thanks so much!!!
I understand it perfectly now!!!
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http://www.perimeterinstitute.ca/videos/seeing-believing-direct-observation-general-quantum-state | # Seeing is Believing: Direct Observation of a General Quantum State
Playing this video requires the latest flash player from Adobe. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8020046353340149, "perplexity": 11245.018112646396}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-36/segments/1471982293922.13/warc/CC-MAIN-20160823195813-00133-ip-10-153-172-175.ec2.internal.warc.gz"} |
https://daniloroccatano.blog/2017/04/30/note-of-quasi-gaussian-entropy-theory/ | ## INTRODUCTION
The Quasi-Gaussian Entropy theory (QGE) is a theoretical method based on a novel statistical mechanics reformulation of the free energy distributions. It was originally developed by Dr. Andrea Amadei (University of Rome “Tor Vergata”, Italy) in collaboration with Prof Herman Berendsen, Dr. Emil Apol (the University of Groningen, The Netherlands) and Prof Alfredo Di Nola (University of Rome “La Sapienza”, Italy). The foundations of the QGE theory are reported in a series of papers collected in the Ph.D. thesis of both Dr. Amadei and Dr. Apol cited in the bibliography. The theory was further developed and applied to different systems spanning from simple fluids to proteins.
In these brief note, the mathematical basis of QGE for the study of the thermodynamics of proteins in solution as in the Ref. [1] is detailed.
## THE QGE THEORY OF THE FLUID STATE
For a fluid state system of ${N}$ solute molecules at high dilution, the partition function can be expressed as
$\displaystyle Q = \frac{(8 \pi^2 V)^N}{N!} \left( \Theta \int^* e^{-\beta {{\cal U}'}} \prod_{j=1}^{n} ( \det\tilde{m}_{j} )^{1/2} ( \det \tilde{M} )^{1/2} d {\mathbf {\xi}} d {\mathbf {\xi}}\right)^N\ \ \ \ \ (1)$
where ${{\cal U}'}$ is the excess energy, basically the potential energy, including the quantum vibrational ground state energy, of the subsystem defined by a single solute molecule and ${n}$ solvent molecules, ${V}$ the overall volume of the system, $\mathbf {\xi}$ the generalized internal coordinates of a single solute molecule with fixed rototranslational coordinates and $\mathbf {x}$ the coordinates of the ${n}$ solvent molecules within the solute molecular volume. Moreover ${\tilde{m}_{j}}$ is the mass tensor of the ${j}^{th}$ solvent molecule, ${\tilde{M}}$ the mass tensor of the solute and ${\Theta}$ a temperature dependent factor including the quantum corrections
$\displaystyle \Theta = \frac{(2 \pi k T )^{(d+d_s)/2} (q_{ref,s}^{qm})^n q_{ref}^{qm}}{n! h^{(d+d_s)} (1+\gamma)(1+\gamma_s)^n} \ \ \ \ \ (2)$
with ${1+\gamma}$ and ${1+\gamma_s}$ the symmetry coefficient for the solute and the solvent respectively, ${d}$ and ${d_s}$ the number of classical degrees of freedom in the solute and solvent molecules and ${q_{ref,s}^{qm}}$ and ${q_{ref}^{qm}}$ the solvent and solute molecular quantum vibrational partition functions respectively. Finally the integral is taken within the solute molecular volume ${V_m=V/N}$, and the star denotes an integration only over the accessible configurational space. From the previous equations it follows that the whole partition function can be obtained from the solute molecular partition function ${ Q=Q_m^N }$
$Q_m = \frac{8 \pi^2 V \Theta}{e^{-1}} \int^* e^{-\beta {{\cal U}'}} \prod_{j=1}^{n} ( \det\tilde{m}_{j} )^{1/2}( \det \tilde{M} )^{1/2} d {\mathbf {\xi}} d {\mathbf {x}} \ \ \ \ \ (3)$
where we used the approximation ${N! \cong N^N e^{-N}}$. Hence, the whole thermodynamics is defined by ${Q_m}$ as ${A=-NkT \ln Q_m}$. This clearly means that if we want to describe the thermodynamics of the same system using the isobaric ensemble we must use a solute molecular isobaric partition function defined as
$\displaystyle \Xi_m = \int e^{-\beta p V_m } Q_m(\beta, V_m) \frac{d V_m}{v} \ \ \ \ \ (4)$
providing ${G = -NkT \ln \Xi_m}$ (note that ${v}$ is an arbitrary volume constant necessary to make adimensional ${\Xi_m}$). It is possible to show that
$\beta G(\beta) - \beta_0 G(\beta_0) = -N\ln \left\{\frac{\Xi_m(\beta)}{\Xi_m(\beta_0)}\right\} = - N \ln \langle e^{-\Delta \beta {\cal H}} \rangle_{\beta_0} \ \ \ \ \ (5)$
with ${\Delta \beta=\beta - \beta_0}$, the subscript ${\beta_0}$ indicating an average in the ${\beta_0}$ ensemble and ${\cal H} = {\cal U}'+pV_m$. The ensemble average in Eq. 5 can be expressed as
$\displaystyle \langle e^{- \Delta \beta \cal H} \rangle_{\beta_0} = \int \rho ({\cal H} ) e^{-\Delta \beta {\cal H} } d {\cal H} \ \ \ \ \ (6)$
where ${\rho({\cal H})}$ is the enthalpy probability distribution function.
Instead of using a perturbation expansion, in the QGE theory, the free energy is obtained by modeling the distribution function and hence its moment generating function or Laplace transform, Eq. 6. For homogeneous fluid state systems, it was shown that a rather good model in the isobaric ensemble is the diverging Gamma state model for enthalpy fluctuations.
NOTE: The Gamma Distribution
The gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, and chi-squared distribution are special cases of the gamma distribution.
For the two parameters $k, \theta$, it defined as
$f(x;k,\theta)=\frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^k \Gamma(k)}$ for $x>0$ and $k,\theta >0$
where $\Gamma(\alpha)$ is the gamma function
$\Gamma(k) =\int^{\infty}_0 x^{k-1}e^{-x}dx.$
IN the following figure, the plot of the gamma distribution (left) for different value of $k, \theta$ and the $\Gamma(k)$ function are shown.
When we deal with a very complex system involving a macromolecule, it is likely that we need more sophisticated models. For the canonical ensemble, that the use of mixing distributions for Gamma state models provides a very powerful method to obtain more sophisticated and accurate models for fluid state systems. We can use a similar approach in the present case, assuming that the solute molecular configurational space of the internal coordinates can be partitioned into a set of ${L}$ subspaces, each one defining a solute-solvent system exactly described by a “local” diverging Gamma state (note that the pure water thermodynamics is well described over a wide temperature range by a single diverging Gamma state). We can rewrite the total free energy change as
$\beta G(\beta) - \beta_0 G(\beta_0) = -N \ln \left\{ \sum_{i=1}^{L} \epsilon_i e^{- [n \Delta (\beta \mu_s) - \Delta (\beta \mu_i) ]} \right\}$
$\Delta (\beta \mu) = \beta \mu(\beta) - \beta_0 \mu(\beta_0)$
where ${n \mu_s + \mu_i}$ is the Gibbs free energy of the system defined by the solute molecular volume which contains ${n}$ solvent molecules and a single solute molecule in the ${i}$th conformation (${i}$th subspace), ${\mu_s}$ is the chemical potential of the solvent and clearly ${\mu_i}$ is the chemical potential of the solute in the ${i}^{th}$ conformation. Finally,
$\displaystyle \epsilon_i=\frac{\Xi_{m,i}(\beta_0)}{\Xi_{m}(\beta_0)} = e^{-\beta_0 (\mu_{0,i}-\mu_0)} \ \ \ \ \ (7)$
$\displaystyle \epsilon_i=\frac{\Xi_{m,i}(\beta_0)}{\Xi_{m}(\beta_0)} = e^{-\beta_0 (\mu_{0,i}-\mu_0)} \ \ \ \ \ (8)$
with ${\mu_{0,i}=\mu_i(\beta_0), \mu_0=\mu(\beta_0)}$, and ${\Xi_{m,i}(\beta_0)}$ the partition function corresponding to the ith conformation at ${\beta_0}$. At high dilution the solvent molecular partial properties are identical to the pure solvent ones (hence independent of the solute). Within the assumption that each ${n \mu_s + \mu_i}$ can be well modeled by a “local” diverging Gamma state, at infinite solute dilution we have
$\displaystyle \mu_i = h_{0,i} - T_0 c_{p0,i} + T(c_{p0,i} - s_{0,i}) + T c_{p0,i}\ln\frac{T_0}{T} \ \ \ \ \ (9)$
with ${h_{0,i}}$, ${c_{p0,i}}$ and ${s_{0,i}}$ the partial molecular enthalpy, heat capacity and entropy of the solute in the ${i}^{th}$ conformation at the reference temperature ${T_0=\beta_0^{-1}/k}$. >From the other general equations of the diverging Gamma state properties we can also obtain all the other partial molecular properties of the solute, e.g. the enthalpy ${h_i}$ and heat capacity
${c_{p,i}} h_i = \left(\frac{\partial \beta \mu_i}{\partial \beta}\right)_{p,n} = h_{0,i} + (T -T_0) c_{p0,i}$
$c_{p,i} = \left(\frac{\partial h_i}{\partial T}\right)_{p,n} = c_{p0,i}$
where the zero subscript indicates the value at the reference temperature ${T_0}$. In the present case where we deal with complex macromolecules like proteins at fixed pressure, the summation in Eq. 1 is likely to involve a very large number of Gamma states corresponding to different protein configurational subspaces (conformations). Hence, in order to keep handable the mathematical derivations and especially the model application, we must use drastic simplifications. We will first assume that we can decompose the huge number of Gamma states into two subgroups: one associated with the folded state of the protein and one with the unfolded state. Moreover, we will assume also that within each subgroup the partial molecular heat capacity is the same for the different Gamma states and ${h_{0,i} \cong h_0^{0} + j \delta}$ with ${h_0^{0}}$ and ${\delta}$ the overall “ground state” enthalpy (at ${T_0}$) and enthalpy gap of the subgroup. Hence, using the subscripts ${f}$ and ${u}$ to define the folded and unfolded state properties respectively, we obtain
$\Delta(\beta \mu) = \Delta\left( \beta\frac{G(T)}{N}\right) - n \Delta(\beta\mu_s) \cong - k T \ln \left\{e^{-(\beta \mu_f-\beta_0 \mu_0)} + e^{-(\beta \mu_u - \beta_0 \mu_0)} \right\}$
$e^{-(\beta \mu_f-\beta_0 \mu_0)} = e^{-\Delta(\beta \mu^0_f)} \epsilon_f \langle e^{-\Delta \beta \delta_f j} \rangle_f$
$e^{-(\beta \mu_u - \beta_0 \mu_0)} = e^{-\Delta (\beta \mu^0_u)} \epsilon_u \langle e^{-\Delta \beta \delta_u j} \rangle_u$
$\langle e^{-\Delta \beta \delta_f } \rangle_f = \sum_{j=0}^{L_f-1} e^{-\Delta \beta \delta_f j} w_f(j)$
$\langle e^{-\Delta \beta \delta_u } \rangle_u = \sum_{j=0}^{L_u-1} e^{-\Delta \beta \delta_u j} w_u(j)$
$w_f(j) = \frac{\epsilon_{f,j}}{\epsilon_f}$
$w_u(j) = \frac{\epsilon_{u,j}}{\epsilon_u}$
where ${\epsilon_f}$, ${\epsilon_u}$, ${\mu_f}$, ${\mu_u}$ are the total fractions and chemical potentials of the folded and unfolded subgroups and
$\Delta (\beta \mu^{0}_f) = h^{0}_{0,f} \Delta \beta - c_{p0,f}T_0 \Delta \beta - \frac{c_{p0,f}}{k} \ln \frac{T}{T_0}$
$\Delta (\beta \mu^{0}_u) = h^{0}_{0,u} \Delta \beta - c_{p0,u}T_0 \Delta \beta - \frac{c_{p0,u}}{k} \ln \frac{T}{T_0}$
From the previous equations we readily obtain the other partial molecular properties, i.e. enthalpy, entropy and heat capacity,
$h = \left(\frac{\partial \beta \mu}{\partial \beta}\right)_{n,p} = h_f +\chi (h_u - h_f)$
$\chi = \frac{e^{-\beta (\mu_u - \mu_f)}}{1+ e^{-\beta (\mu_u -\mu_f)}}$
$s = \frac{h -\mu}{T}$
$c_{p} = \left(\frac{\partial h}{\partial T}\right)_{p,n} = c_{p,u} - (1-\chi) (c_{p,u} - c_{p,f}) + (1 - \chi) \chi \frac{(h_u -h_f)^2}{kT^2}$
where obviously
$h_f = \left(\frac{\partial \beta \mu_f}{\partial \beta}\right)_{p,n}$
$h_u = \left(\frac{\partial \beta \mu_u}{\partial \beta}\right)_{p,n}$
$c_{p,f} = \left(\frac{\partial h_f}{\partial T}\right)_{p,n}$
$c_{p,u} = \left(\frac{\partial h_u}{\partial T}\right)_{p,n}$
are the corresponding partial molecular properties in the folded or unfolded state. Note that in the limit of a differential ${\delta_f}$ and ${\delta_u}$ a continuous Gamma states partition of the solute intramolecular phase space is involved.
For describing the state of complex systems, as biomacromolecules, the discrete-like diverging Gamma state partition seems to provide a better general description. The use of a discrete-like diverging Gamma states partition implies that the thermodynamics of a solvated macromolecule should be a complex mixture between a typical fluid state behavior and a discrete-like “energy” fluctuation. In order to proceed further we must model the discrete probability distribution ${w(j)}$. A simple discrete distribution, which is phisically acceptable and proved to be succesfull to model quantum solid state, is the negative binomial distribution providing
$\langle e^{-\Delta \beta \delta_f j} \rangle = \left\{ \frac{1-q_f}{1 -q_f e^{-\beta \delta_f }} \right\}^{Z_f}$
$\langle e^{-\Delta \beta \delta_u j} \rangle = \left\{ \frac{1-q_u}{1 -q_u e^{-\beta \delta_u }} \right\}^{Z_u}$
where ${q}$ and ${Z}$ are two pure numbers characteristic of the negative binomial distribution. With these last equations we can express the partial molecular properties of the folded and unfolded states as
$\beta \mu_f - \beta_0 \mu_0 = h_{0,f}^{0} \Delta \beta - c_{p0,f} T_0 \Delta \beta - \frac{c_{p0,f}}{k} \ln \frac{T}{T_0} - \ln \epsilon_f - Z_f \ln \left\{\frac{1-q_f}{1 -q_f e^{-\beta \delta_f }}\right\}$
$\beta \mu_u - \beta_0 \mu_0 = h_{0,u}^{0} \Delta \beta - c_{p0,u} T_0 \Delta \beta - \frac{c_{p0,u}}{k} \ln \frac{T}{T_0} -\ln \epsilon_u - Z_u \ln \left\{\frac{1-q_u}{1 -q_u e^{-\beta \Delta_u }} \right\}$
$h_f = h_{0,f}^{0} + (T -T_0) c_{p0,f} + \frac{ Z_f q_f \delta_f}{e^{\beta \delta_f } - q_f}$
$h_u = h_{0,u}^{0} + (T -T_0) c_{p0,u} + \frac{ Z_u q_u \delta_u}{e^{\beta \delta_u } - q_u}$
and so
$c_{p,f} = c_{p0,f} + \frac{ Z_f q_f k (\delta_f \beta)^2 e^{- \beta \delta_f } } {(1- q_f e^{- \beta \delta_f })}$
$c_{p,u} = c_{p0,u} + \frac{ Z_u q_u k (\delta_u \beta)^2 e^{- \beta \delta_u } } {(1- q_u e^{- \beta \delta_u })}$
$s_f = \frac{h_f - \mu_f}{T}$
$s_u = \frac{h_u - \mu_u}{T}$
We can simplify further the model assuming that $q_f = q_u = q$
$\delta_f = \delta_u = \delta$ and taking the reference temperature ${T_0}$ as the equilibrium temperature, i.e., ${\epsilon_f/\epsilon_u=1}$.
With these simplifications we obtain
$\beta (\mu_u - \mu_f) = (h_{0,u}^{0} - h_{0,f}^{0}) \Delta \beta - (c_{p0,u} - c_{p0,f}) T_0 \Delta \beta - \frac{(c_{p0,u} - c_{p0,f})}{k} \ln \frac{T}{T_0}- (Z_u-Z_f)\ln \left\{\frac{1-q}{1 -q e^{-\beta \delta }}\right\}$
$h_{u} - h_{f} = h_{0,u}^{0} - h_{0,f}^{0} (c_{p0,u} - c_{p0,f}) (T-T_0) + \frac{(Z_u-Z_f) q \delta}{e^{\Delta \beta \delta}}$
$c_{p,u} - c_{p,f} = c_{p0,u} - c_{p0,f} + \frac{(Z_u-Z_f)q k (\delta \beta)^2 } {(e^{\Delta \beta \delta} -q)^2} e^{\Delta \beta \delta}$
which can be used to obtain the solute partial molecular properties, e.g., via Eq.~<a href=”#eqcv”>1</a> the partial molecular heat capacity.
## BIBLIOGRAPHY
1. Roccatano, A. Di Nola, A. Amadei. A theoretical model for the folding/unfolding thermodynamics of single-domain proteins, based on the quasi-Gaussian entropy theory. J. Phys. Chem. B, 108, 5756-5762 (2004).
1. M.E.F Apol. The quasi-Gaussian entropy theory: Temperature dependence of thermodynamic properties using distribution functions. Ph.D. Thesis, Groningen (The Netherlands), 1997.
2. A. Amadei. Theoretical models for fluid thermodynamics based on the quasi-Gaussian Entropy theory. Groningen (The Netherlands), Ph.D. Thesis, Groningen (The Netherlands), 1998. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 118, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9582099318504333, "perplexity": 692.288766620714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316785.68/warc/CC-MAIN-20190822064205-20190822090205-00381.warc.gz"} |
http://stochastix.wordpress.com/tag/ordinary-differential-equations/ | ## Posts Tagged ‘Ordinary Differential Equations’
June 29, 2012
Last time I implemented a Runge-Kutta method was in December 2001. Back then, I implemented the classical 4th order Runge-Kutta (RK4) method in C. However, in the past decade, whenever I needed a numerical solution of an ordinary differential equation (ODE), I used MATLAB. Now, I would like to use Haskell instead.
Consider the following initial value problem (IVP)
$\dot{x} (t) = f ( x (t) )$, $x (t_0) = x_0$
where the vector field $f : \mathbb{R}^n \to \mathbb{R}^n$ and the initial condition $x_0$ are given. Solving an instance of the IVP consists of finding a function $x : [t_0, \infty) \to \mathbb{R}^n$ that satisfies both the differential equation and the initial condition.
Let $h > 0$ be the time-step, and let $t_k = t_0 + k h$ be the $k$-th time instant, where $k \in \mathbb{N}_0 := \{0, 1, 2, \dots\}$. Suppose that we solve a given instance of the IVP and obtain a closed-form solution $x : [t_0, \infty) \to \mathbb{R}^n$; we then discretize the closed-form solution to obtain an infinite sequence $x_0, x_1, x_2, \dots$ where $x_k = x (t_k)$. However, not all instances of the IVP have a closed-form solution (or even an analytic solution), and even in the cases where a closed-form solution does exist, it can be quite hard to find such a solution. Therefore, instead of discretizing the closed-form solution, one is tempted to discretize the ODE itself to obtain an approximation of the sequence $x_0, x_1, x_2, \dots$, which we will denote by $\tilde{x}_0, \tilde{x}_1, \tilde{x}_2, \dots$. We will call the approximate sequence a numerical solution.
Consider now the following discretized initial value problem (DIVP)
$\tilde{x}_{k+1} = g ( \tilde{x}_k )$, $\tilde{x}_0 = x_0$
where $g : \mathbb{R}^n \to \mathbb{R}^n$ depends on what particular numerical method one wants to use. We will use the (classical) 4th order Runge-Kutta (RK4) method [1], which is given by
$g (x) = \displaystyle x + \frac{1}{6} \left( k_1 + 2 k_2 + 2 k_3 + k_4 \right)$
where the $k_i$ variables are given by
$\begin{array}{rl} k_1 &= h f (x)\\\\ k_2 &= h f (x + \frac{1}{2} k_1)\\\\ k_3 &= h f (x + \frac{1}{2} k_2)\\\\ k_4 &= h f (x + k_3)\end{array}$
In a nutshell, one hopes that by making the time-step $h$ “small enough”, the numerical solution $\tilde{x}_0, \tilde{x}_1, \tilde{x}_2, \dots$ will be “close enough” to $x_0, x_1, x_2, \dots$, the discretization of the actual solution $x$. This hoping involves a fair amount of faith. Especially so if floating-point arithmetic is used, as underflow, overflow, and round-off error are not to be dismissed.
__________
The following Haskell script implements the RK4 method:
-- define 4th order Runge-Kutta map (RK4)
rk4 :: Floating a => (a -> a) -> a -> a -> a
rk4 f h x = x + (1/6) * (k1 + 2*k2 + 2*k3 + k4)
where k1 = h * f (x)
k2 = h * f (x + 0.5*k1)
k3 = h * f (x + 0.5*k2)
k4 = h * f (x + k3)
Let us now test this rather succinct implementation.
__________
Example
Consider the following IVP where $t_0 = 0$, $x_0 = 1$, and $n = 1$
$\dot{x} (t) = - x (t)$, $x (0) = 1$.
The closed-form solution of this IVP is $x (t) = e^{-t}$. We load the script above into GHCi and compute the numerical solution of the IVP using the higher-order function iterate:
*Main> -- discretize closed-form solution
*Main> let xs = [ exp (-t) | t <- [0,0.1..]]
*Main> -- define function f
*Main> let f x = -x
*Main> -- define time-step h
*Main> let h = 1 / 100000
*Main> -- define initial condition
*Main> let x0 = 1.0
*Main> -- compute numerical solution
*Main> let xs_tilde = iterate (rk4 f h) x0
*Main> -- decimate numerical solution
*Main> let xs_dec = [ xs_tilde !! k | k <- [0,1..], rem k 10000 == 0]
Note that the time-step is $h = 1 / 100000$. Hence, the numerical solution will have $10^5$ samples in between $t = 0$ and $t = 1$, which probably is too fine a discretization. Thus, we decimate the numerical solution by a factor of $10^4$, so that we are left with only eleven samples in between $t = 0$ and $t = 1$, with a uniform spacing of $\Delta t = 0.1$ between consecutive samples. We can now compare the discretization of the closed-form solution with the heavily decimated numerical solution, as follows:
*Main> -- print discretized closed-form solution
*Main> take 11 xs
[1.0,0.9048374180359595,0.8187307530779818,
0.7408182206817179,0.6703200460356392,
0.6065306597126333,0.5488116360940264,
0.49658530379140947,0.44932896411722156,
0.4065696597405991,0.36787944117144233]
*Main> -- print decimated numerical solution
*Main> take 11 xs_dec
[1.0,0.9048374180359563,0.8187307530779842,
0.7408182206817214,0.6703200460356442,
0.6065306597126376,0.5488116360940266,
0.49658530379141175,0.44932896411722334,
0.40656965974059905,0.367879441171439]
*Main> -- compute difference of two solutions
*Main> let es = zipWith (-) xs_dec xs
*Main> -- print difference
*Main> take 11 es
[0.0,-3.219646771412954e-15,2.3314683517128287e-15,
3.552713678800501e-15,4.9960036108132044e-15,
4.3298697960381105e-15,2.220446049250313e-16,
2.275957200481571e-15,1.7763568394002505e-15,
-5.551115123125783e-17,-3.3306690738754696e-15]
*Main> -- compute energy of the difference
*Main> sum $take 11$ map (^2) es
9.161263559461204e-29
Note how small the values of difference sequence are, of the order of $10^{-15}$. The implementation of the RK4 method seems to be working. Who needs MATLAB now? ;-)
__________
References
[1] Richard W. Hamming, Numerical Methods for Scientists and Engineers, Dover Publications, 1973.
### ODE notation considered harmful
June 5, 2012
Whenever I want to learn a new subject, I announce a graduate course in it, since the best way to learn is by teaching. But even better than teaching humans is teaching computers, i.e. program! Since computers will not let you wave your hands and wing it.
Doron Zeilberger (1999)
Suppose that we are given an $n$-dimensional first-order vector ordinary differential equation (ODE)
$\dot{x} = f (x)$
where $f : \mathbb{R}^n \to \mathbb{R}^n$ is a vector field. A solution of this ODE is a function $x : \mathbb{R} \to \mathbb{R}^n$ whose first derivative, $\dot{x}$, is equal to $f (x)$. Elementary, my dear Watson!
Despite the fact that the notation $\dot{x} = f (x)$ is rather pervasive, it is dangerous, in my humble opinion. One should keep in mind that such notation is an abbreviation. How could it not be an abbreviation? Note that $x$ is a function, and that the argument of $f$ is a vector in $\mathbb{R}^n$, not a function! In other words, writing $f (x)$ results in a type error, and a compiler / interpreter with type checking would never accept the declaration $\dot{x} = f (x)$. We, fallible humans, accept it because we know (do we?) that it is an abbreviation, and because it is tradition. Who are we to challenge a centuries-old tradition?
If $\dot{x} = f (x)$ is an abbreviation, what exactly is it an abbreviation for? It is an abbreviation for the following
$\dot{x} (t) = f ( x (t) )$
where the $=$ symbol denotes equality of vectors. Note that the argument of $f$ is the vector $x (t)$ (i.e., function $x$ evaluated at $t$), not the function $x$. But, what does the equation above say? It says the following:
Proposition: Given a vector field $f : \mathbb{R}^n \to \mathbb{R}^n$, there exists a function $x : \mathbb{R} \to \mathbb{R}^n$, whose first derivative is $\dot{x} : \mathbb{R} \to \mathbb{R}^n$, such that for every $t \in \mathbb{R}$ we have that $\dot{x} (t) = f ( x (t) )$.
This proposition may be true or false. Not all differential equations have solutions, after all, which is why the study of existence of solutions is (was?) an active area of research. Note the presence of the words “for every” in the proposition above. Do you see any universal quantifier, $\forall$, anywhere? You do not, and that is due to the fact that $\dot{x} (t) = f ( x (t) )$ is also an abbreviation! The universal quantifier is implicit. If we make it explicit, we obtain
$\forall t \left( \dot{x} (t) = f ( x (t) ) \right)$
where $t$ ranges over $\mathbb{R}$. The universally quantified formula above is the conjunction of infinitely many equations. However, do note that the words “there exists” also appear in the proposition, which suggests that an existential quantifier is missing. Therefore, the non-abbreviated notation would be as follows
$\exists x \, \forall t \left( \dot{x} (t) = f ( x (t) ) \right)$
where $x$ ranges over the set of all (continuous?) functions from $\mathbb{R}$ to $\mathbb{R}^n$, and $t$ ranges over $\mathbb{R}$.
__________
A better notation
So far, I have criticized the abbreviated notation for differential equations. I consider it harmful, as confusing a function $x$ with its evaluation $x(t)$ is an atrocious crime against types. To make this post more “constructive”, I will take the liberty of proposing a better notation.
We start by noting that $f ( x (t) )$ can be written in the form $(f \circ x) (t)$, where the symbol $\circ$ denotes function composition. We now introduce a differential operator $D$, which maps functions to functions, so that we obtain the first derivative of function $x$ via $\dot{x} = D (x)$. Hence, $\dot{x} (t) = (D (x)) (t)$, which allows us to rewrite $\forall t \left( \dot{x} (t) = f ( x (t) ) \right)$ in the following form
$\forall t \left( (D(x)) (t) = (f \circ x) (t) \right)$
where $t$ ranges over $\mathbb{R}$. The universally quantified formula above states that functions $D(x)$ and $f \circ x$, both of which from $\mathbb{R}$ to $\mathbb{R}^n$ evaluate to the same values for all possible choices of the input $t$, i.e., functions $D(x)$ and $f \circ x$ are equal. The formula above can thus be rewritten more compactly as $D(x) = f \circ x$.
Finally, we conclude that $\exists x \, \forall t \left( \dot{x} (t) = f ( x (t) ) \right)$ is equivalent to the following existentially quantified formula
$\exists x \left( D(x) = f \circ x \right)$
where the $=$ symbol denotes equality of functions. This new notation may not be as compact as $\dot{x} = f (x)$, but at least now the RHS does not result in type error. Criticism would be most welcome.
### Safety verification via barrier certificates
June 19, 2011
Suppose that we are given a dynamical system of the form $\dot{x} (t) = f ( x (t) )$, where $x : [0, \infty) \to \mathbb{R}^n$ is the state trajectory, and $f : \mathbb{R}^n \to \mathbb{R}^n$ is a known vector field [1]. Suppose further that we are given an initial set $\mathcal{X}_0 \subset \mathbb{R}^n$, and an unsafe set $\mathcal{X}_u \subset \mathbb{R}^n$.
If there exists a state trajectory $x : [0, \infty) \to \mathbb{R}^n$ with initial condition $x(0) \in \mathcal{X}_0$ such that $x(T) \in \mathcal{X}_u$ for some $T \geq 0$, we say that the system is unsafe. An example of an unsafe system is depicted below
If there exists no state trajectory $x : [0, \infty) \to \mathbb{R}^n$ with initial condition $x(0) \in \mathcal{X}_0$ such that $x(T) \in \mathcal{X}_u$ for some $T \geq 0$, we say that the system is safe. Note that if $\mathcal{X}_0 \cap \mathcal{X}_u \neq \emptyset$, we can conclude immediately that the system is unsafe, as there exists an initial condition $x(0) \in \mathcal{X}_u$.
The safety verification problem can then be stated as follows:
Given a vector field $f : \mathbb{R}^n \to \mathbb{R}^n$, an initial set $\mathcal{X}_0 \subset \mathbb{R}^n$ and an unsafe set $\mathcal{X}_u \subset \mathbb{R}^n$, decide whether the dynamical system $\dot{x} = f (x)$ is safe.
To be more precise, safety is not a property of the dynamical system $\dot{x} = f (x)$ per se, but rather of the ordered triple $(f, \mathcal{X}_0, \mathcal{X}_u)$.
How do we go about solving the safety verification problem? The most obvious approach would be to “propagate” the initial set $\mathcal{X}_0$ forwards in time, i.e., to compute (forward) reachable sets. However, computing exact reachable sets is seldom possible, which forces us to compute approximations of reachable sets.
When performing safety verification, we are asking a binary question: is $(f, \mathcal{X}_0, \mathcal{X}_u)$ safe? Hence, computation of the reachable sets seems to provide more information than what we actually do need. This is a personal opinion, not a fact.
__________
Barrier certificates
Probably inspired on Lyapunov stability theory [1], Stephen Prajna introduced a few years ago a new method for safety verification of continuous-time dynamical systems that relies on what he termed barrier certificates [2]. A barrier certificate is a function of state $B : \mathbb{R}^n \to \mathbb{R}$ that satisfies the following inequalities
$\begin{array}{rl} B(x) \leq 0 \qquad{} \forall x \in \mathcal{X}_0 \\ B(x) > 0 \qquad{} \forall x \in \mathcal{X}_u\\ \displaystyle\frac{\partial B}{\partial x} (x) f (x) \leq 0 \qquad{} \forall x \in \mathbb{R}^n\end{array}$
Let us now introduce a function of time $b (t) := B (x (t))$ and fix the initial condition $x(0)$. Taking the derivative of $b$ with respect to time, we obtain
$\dot{b} (t) = \displaystyle\frac{\partial B}{\partial x} (x(t)) \dot{x} (t) = \displaystyle\frac{\partial B}{\partial x} (x(t)) f (x(t)) \leq 0$
which follows from the non-positivity of the Lie derivative of $B$ along the flow of $f$. From $\dot{b}(t) \leq 0$, we conclude that $b(t) \leq b(0)$ for all $t \geq 0$, i.e., we have that $B(x(t)) \leq B(x(0))$ for all $t \geq 0$. Hence, the following sublevel set
$\Omega_0 := \{ x \in \mathbb{R}^n \mid B(x) \leq B(x(0))\}$
is positively invariant [1]. In other words, a solution starting in $\Omega_0$ will remain in $\Omega_0$ for all $t \geq 0$. Since $B(x(0)) \leq 0$ for all $x(0) \in \mathcal{X}_0$, we conclude that $B(x(t)) \leq 0$ for all $t \geq 0$. However, since $B(x) > 0$ for all $x \in \mathcal{X}_u$, we can finally conclude that the safety of $(f, \{x(0)\}, \mathcal{X}_u)$ is guaranteed. Alternatively, one could argue that, since, by construction, $\Omega_0 \cap \mathcal{X}_u = \emptyset$, then safety is certified.
and once we know that $\Omega_0$ is a positively invariant set (from $B(x(t)) \leq B(x(0))$), we conclude that the state trajectory must live in $\Omega_0$, which is painted in dark gray below
Please do recall that $B(x(0)) \leq 0$, and also note that the unsafe set is contained in the superlevel set $\{ x \in \mathbb{R}^n \mid B(x) > 0\}$.
So far we assumed that the initial condition was fixed. Henceforth, we shall make the initial condition free to take any values in the initial set $\mathcal{X}_0$. Let us define
$b_0 := \displaystyle\sup_{x_0 \in \mathcal{X}_0} B(x_0)$
so that we get an upper bound on $B(x(0))$, as follows
$B(x(t)) \leq B(x(0)) \leq b_0 \leq 0$
where we used the inequality $B(x) \leq 0$ for all $x \in \mathcal{X}_0$ to conclude that $b_0 \leq 0$. This allows us to conclude that the sublevel set
$\tilde{\Omega}_0 := \{ x \in \mathbb{R}^n \mid B(x) \leq b_0\}$
is positively invariant. Since $\tilde{\Omega}_0 \cap \mathcal{X}_u = \emptyset$, safety of $(f, \mathcal{X}_0, \mathcal{X}_u)$ is guaranteed. Note that the sublevel set $\tilde{\Omega}_0$ is the union of all possible $\Omega_0$ sublevel sets, i.e.,
$\tilde{\Omega}_0 = \displaystyle\bigcup_{x_0 \in \mathcal{X}_0} \{ x \in \mathbb{R}^n \mid B(x) \leq B(x_0)\}$.
This new sublevel set is depicted below, painted in dark gray
Note that $\mathcal{X}_0 \subset \tilde{\Omega}_0$. Essentially, the zero level set of the barrier certificate $B$ separates an unsafe set, $\mathcal{X}_u$, from all system trajectories starting in the initial set $\mathcal{X}_0$, which are contained in $\tilde{\Omega}_0$, thus providing a proof of safety without requiring the computation of reachable sets. Personally, I like to think of this approach as a sophisticated generalization of the well-known separating hyperplane theorem.
At this point, you may be thinking that this framework is too good to be true. Instead of computing the flow or the reachable sets, all we need to do is find a function $B : \mathbb{R}^n \to \mathbb{R}$ that satisfies three inequalities, and then (voilà!) infinite-time safety of $(f, \mathcal{X}_0, \mathcal{X}_u)$ is certified! As it usually happens, the devil is in the details…
If finding Lyapunov functions [1] for low-dimensional dynamical systems is often “hard”, why should we expect the hunt for barrier certificates to be an “easy” endeavor? Quel panache! Are we merely transferring the complexity of the problem from the computation of reachable sets to the satisfaction of three inequalities? Will we fail to find barrier certificates due to excessive conservativeness? Should we abandon all hope? Please be patient, dear reader, for we will address these issues and alleviate your doubts in future posts ;-)
Next installment will be devoted to an example. до скорой встречи!
_________
References
[1] Hassan K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall.
[2] Stephen Prajna, Optimization-based methods for nonlinear and hybrid systems verification, Dissertation (Ph.D.), California Institute of Technology, 2005.
### Sailing in state space
April 11, 2011
Consider the following controlled (autonomous) dynamical system [1]
$\dot{x} (t) = f (x (t), u (t))$
where $x : [0, \infty) \to \mathbb{R}^n$ is the state trajectory, $u : [0,\infty) \to \mathcal{U}$ is the control input history, $\mathcal{U} \subseteq \mathbb{R}^m$ is the set of admissible control inputs, and $f : \mathbb{R}^n \times \mathcal{U} \to \mathbb{R}^n$ is a (known) vector field.
If we know the control input $u (t)$ for all $t \geq 0$, then we can integrate the ODE above with initial condition $x(0) = x_0$ and obtain the state trajectory $x : [0, \infty) \to \mathbb{R}^n$, as follows
$x (t) = x_0 + \displaystyle \int_{0}^{t} f \left(x (\tau), u (\tau) \right) \mathrm{d} \tau$
which can be represented pictorially by a single streamline in $\mathbb{R}^n$
I am (quite explicitly) alluding to fluid flow. Just like a cork on the ocean will follow a certain path depending on the velocity field (i.e., “ocean currents”), a point in state space will flow along a certain streamline.
__________
Let us fix the control input, i.e., $u (t) = \bar{u}$ for all $t \geq 0$, and define
$\begin{array}{rl} v^{(\bar{u})} : \mathbb{R}^n & \to \mathbb{R}^n\\ x & \mapsto f(x, \bar{u})\\\end{array}$
For the sake of simplicity, let us consider $\mathcal{U} = \{-1, +1\}$. Since $\bar{u}$ can only take two values, we have two vector fields, $v^{(+1)}(x) := f(x, +1)$ and $v^{(-1)}(x) := f(x, -1)$. Integrating the following ODEs
$\begin{array}{rl} \dot{x} (t) &= v^{(+1)} ( x(t) )\\ \dot{x} (t) &= v^{(-1)} ( x(t) )\\\end{array}$
for various initial conditions, we obtain two families of streamlines, as depicted below
For each fixed control input and collection of initial conditions, we will have a family of streamlines in state space.
Imagine that we have the following control input history
$u (t) = \begin{cases} +1, & t \in [0, t^{\star})\\ -1, & t \geq t^{\star}\\\end{cases}$
where we toggle the control input at time $t^{\star} > 0$, thus interrupting the flow along a “blue” streamline and initiating the flow along a “pink” streamline. Pictorially, we have
If the control input is fixed, say, $\bar{u} = +1$, then given an initial condition $x(0)$, we will flow along the “blue” streamline that passes through $x(0)$. This streamline is $1$-dimensional and does not allow us to “explore” the $n$-dimensional state space. However, if we switch between $\bar{u} = +1$ and $\bar{u} = -1$, then we are able to “travel” to other regions of the state space. Allowing the control input to take values in $\mathcal{U} = [-1,+1]$ would give us even more freedom.
Lastly, we arrive at a most childish idea: in some cases, controller design can be viewed as shaping the streamlines differently in different parts of the state space. This is childish because it is purely conceptual. One obviously cannot design optimal controllers by doodling with crayons.
__________
References
[1] Hassan K. Khalil, Nonlinear Systems, 3rd edition, Prentice Hall. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 202, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9726272821426392, "perplexity": 323.55258585918284}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163048688/warc/CC-MAIN-20131204131728-00046-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/poisson-distribution.268269/ | # Poisson Distribution
1. Oct 30, 2008
### cse63146
[SOLVED] Poisson Distribution
1. The problem statement, all variables and given/known data
Let X be the number of people entering the ICU in a hospital. From Historical data, we know the average number of people entering ICU on any given day is 5
a) What is the probability that the number of people entering the ICU on any given day is less than 2. Do you think this is a rare event?
b) What is the probability of people entering the ICU on any 2 consecutive day is less than 2. Do you think this is a rare event?
2. Relevant equations
$$P(X = K) = \frac{\mu^k e^{-\mu}}{k!}$$
3. The attempt at a solution
a) $$P (X < 2) = P(X=0) + P(X=1) = \frac{5^0 e^{-5}}{0!} + \frac{5^{1} e^{-5}}{1!} = e^{-5} + 5e^{-5} = 6e^{-5} = 0.0404$$
b) Since it's 2 consecutive days =>P (X < 2)*P (X < 2) = P (X < 2)^2 = 0.0404^2 = 0.00163
How would I determine if it's a rare event? Would I just compare it to 5 people entering the ICU for both cases?
Thank You
Last edited: Oct 31, 2008
2. Oct 31, 2008
### Staff: Mentor
Your work looks fine (although a couple of signs are switched the fractions in part a -- corrected in following work).
a) The probability is only about .04, so about 1 chance in 25. I'd say that's a fairly rare occurrence.
b) The probability is much less, hence a much rarer event.
3. Oct 31, 2008
### cse63146
Thank You.
Have something to add?
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http://www.di.unito.it/~lambda/biblio/entry-Padovani11A.html | Formal Methods in Computing(Most of the papers antecedent to 1995are not included in the list) FRAMES NO FRAME
Padovani11A (In proceedings) Author(s) Luca Padovani Title « Fair Subtyping for Multi-Party Session Types » In Proceedings of the International Conference on Coordination Models and Languages (COORDINATION'11) Series LNCS Volume 6721 Page(s) 127-141 Year 2011 Publisher Springer URL http://www.di.unito.it/~padovani/Papers/FairSubtyping.pdf
Abstract The standard subtyping relation used in dyadic session type theories may compromise the liveness of multi-party sessions. In this paper we define a fair subtyping relation for multi-party session types that preserves liveness, we relate it with the standard subtyping relation, and we give algorithms for deciding it. As a side effect, we provide an original and remarkably simple coinductive characterization of the fair testing preorder for nondeterministic, sequential processes consisting of internal choices of outputs and external choices of inputs.
BibTeX code
@inproceedings{Padovani11A,
volume = {6721},
series = {LNCS},
booktitle = {Proceedings of the International Conference on Coordination
Models and Languages (COORDINATION'11)},
abstract = { The standard subtyping relation used in dyadic session type
theories may compromise the liveness of multi-party sessions. In
this paper we define a \emph{fair} subtyping relation for
multi-party session types that preserves liveness, we relate it
with the standard subtyping relation, and we give algorithms for
deciding it. As a side effect, we provide an original and
remarkably simple coinductive characterization of the fair testing
preorder for nondeterministic, sequential processes consisting of
internal choices of outputs and external choices of inputs. },
title = {{Fair Subtyping for Multi-Party Session Types}},
publisher = {Springer},
year = {2011},
pages = {127-141},
doi = {10.1007/978-3-642-21464-6\_9},
}
Formal Methods in Computing(Most of the papers antecedent to 1995are not included in the list) FRAMES NO FRAME
This document was generated by bib2html 3.3.
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http://complementaryslackness.wordpress.com/2010/03/08/for-a-little-more-complementarity/ | # For a little more complementarity
It’s new paper season here at Complementary Slackness! Today’s offering is all about two conditions on when a maximally entangled state can (approximately) be recovered from a given bipartite state, using only local operations at one end. Why is this interesting, you ask? Because this operation is essentially the final step in an entanglement distillation or quantum communication procedure, so knowing when it can be done is the first step in designing such protocols. It’s best to begin with the end in mind, as they say.
The two conditions are based on the idea of breaking quantum information down into two classical pieces, such that these pieces successfully reassembled into the quantum whole. Recall the original approach to quantum error correction in which quantum errors are digitized, in particular into amplitude and phase errors. Each of these errors is effectively classical, and correcting both restores the original quantum information. It turns out, though, that this is a suboptimal way to handle quantum errors. So here we shift the focus away from making the quantum errors classical to dealing with classical information inherent in the bipartite quantum state itself.
The states we’re after, maximally entangled states, have the property that, given one system, we can predict either of two complementary measurements made on the other. This is clear just by writing down the maximally entangled state, for two qubits A and B say: $|\Phi\rangle=\tfrac{1}{\sqrt{2}}(|0,0\rangle+|1,1\rangle)=\tfrac{1}{\sqrt{2}}(|+,+\rangle+|-,-\rangle)$. A and B are perfectly correlated in both the $\{|0\rangle,|1\rangle\}$ and $\{|+\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle),|-\rangle=\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)\}$ bases, so measurement in either of these bases always produces correlated outcomes. The bases are complementary because given a state prepared in one basis, measuring it in the other produces a completely random outcome. Thus, the entangled state has the advertised property: to predict the outcome of measuring A in a given basis, just perform the same measurement on B, and this will work for either complementary basis. Furthermore, this property actually defines the state $|\Phi\rangle$.
The first condition just generalizes this, in several ways. First, we don’t demand that the prediction be perfect, and in exchange we’re willing to accept a good approximation to the state $|\Phi\rangle$. This is important for designing quantum communication protocols, as the amount of approximate entanglement a channel can produce is generally higher than the amount of perfect entanglement it can produce. Second, we don’t require the prediction measurements on B to have the same form as the measurement on A; any measurements which can predict the outcomes for either basis will do. This means the state spaces of A and B can be quite different. Lastly, and most importantly, we only require that one of the measurements, say the $\{|0\rangle,|1\rangle\}$ basis be predictable from B; the other measurement $\{|+\rangle,|-\rangle\}$ need only be predictable using B plus a sort of copy of A in the $\{|0\rangle,|1\rangle\}$ basis. This might sound like cheating, but it stems to the fact that the entanglement recovery operation just performs the two prediction measurements in sequence. Since the first one is good at predicting the $\{|0\rangle,|1\rangle\}$ measurement, this information is available to predict the outcome of measuring in the other basis. And it’s useful if there are correlations between the two.
This condition was implicitly used here to construct protocols for entanglement distillation which operate at the optimal rate (though you wouldn’t have guessed this from the title), so this approach successfully avoids the pitfalls encountered by error digitization. And it can be used to perform quantum communication over noisy channels, at least if there’s an extra classical channel available: Just use the channel to distribute bipartite states, distill entanglement, and then teleport the quantum information you actually want to send using the classical channel and the entanglement. The nice thing about this approach is that the necessary recovery operation (or decoder) is actually constructed, rather than just shown to exist, which might be an advantage in exploring more efficient schemes. Ok, you might be thinking that this isn’t really constructive, since you have to supply (construct) the two prediction measurements. But these are involved in a classical task and it is presumably easier to divide the full quantum problem into two classical pieces.
The second condition is sort of the inverse of the first — the two outcomes of the two measurements on A should be completely unpredictable to someone with access to the purification of A and B, i.e. a system R such that ABR is a pure state. Actually, a caveat similar to the last point above applies here, so that one of the measurments should be unpredictable even if given R and knowledge of the other measurement outcome. This is a sort of decomposition of the often-used decoupling approach into two classical pieces. I plan to say a bit more about that in a future post, but the point is that entanglement is recoverable from A and B if A is uncorrelated with R, i.e. the joint quantum state just factors into independent states for each system. Here we’re saying essentially the same thing using classical information instead. Ultimately this condition comes from the first by using the duality explored in a previous paper, discussed here. This approach isn’t constructive, but it’s aesthetically pleasing to see that it holds as well. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 9, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8310950398445129, "perplexity": 323.59007305784223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345765796/warc/CC-MAIN-20131218054925-00051-ip-10-33-133-15.ec2.internal.warc.gz"} |
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https://math-sciences.org/event/aleksandr-nikolaev-swansea/ | # Aleksandr Nikolaev (Swansea)
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# Aleksandr Nikolaev (Swansea)
## February 6, 2019 @ 4:00 pm - 5:00 pm UTC+0
Lattice study of phase properties of cold dense quark matter
The phase diagram of QCD is of fundamental interest for high energy physics, cosmology and astrophysics, but at the present moment ab-initio calculations in the lattice QCD formalism at finite density are impossible. However, in the regions of high temperature and small density or of high density and small temperature theories with SU(2) and SU(3) gauge groups with the presence of fundamental fermions are expected to have similar properties. SU(2) theory in lattice formulation does not possess a sign problem, which makes computations at finite density possible. I will present results for the phase structure of lattice SU(2) QCD with two flavors of quarks at finite quark density and zero temperature. Interaction properties of quarks, real-time inter-quark potential and the confinement/deconfinement transition will be examined. Our results indicate that in very dense matter the quark-gluon plasma is in essence a weakly interacting gas of quarks and gluons without a magnetic screening mass in the system, sharply different from a QGP at large temperature. The talk is based on papers arXiv:1808.06466, 1711.01869, 1605.04090, and recent results.
## Details
Date:
February 6, 2019
Time:
4:00 pm - 5:00 pm UTC+0
Event Categories:
,
## Venue
Room 101
2-5 Kirkby Place
Plymouth, PL4 6DT United Kingdom | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9512122869491577, "perplexity": 2378.1830688916975}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155322.12/warc/CC-MAIN-20210805032134-20210805062134-00431.warc.gz"} |
https://www.nature.com/articles/s42005-018-0030-7?error=cookies_not_supported&code=bca13e0c-c54a-4f7c-8605-f7159722c278 | Article | Open | Published:
# Ramsey interferometry with trapped motional quantum states
## Abstract
Ramsey interferometers using internal electronic or nuclear states find wide applications in science and engineering. We develop a matter wave Ramsey interferometer for trapped motional quantum states exploiting the s- and d-bands of an optical lattice and study it both experimentally and theoretically, identifying the different de-phasing and de-coherence mechanisms. Implementing a band echo technique, employing repeated π-pulses we suppress the de-phasing evolution and significantly increase the coherence time of the trapped state interferometer by one order of magnitude. Thermal fluctuations are the main mechanism for the remaining decay of the contrast. Our demonstration of an echo-Ramsey interferometer with trapped quantum states in an optical lattice has potential application in the study of quantum many-body lattice dynamics, and motional qubits manipulation.
## Introduction
Interferometers employing separated oscillating fields to create and probe superpositions of states, also known as Ramsey interferometry (RI), have originaly been developed for magnetic resonance to measure transition frequencies1,2 and were then extended to a general tool of spectrocopy and matter wave interferometry3,4,5. Typical sequences consist of two π/2 pulses separated by an time for free evolution or a π/2 − π − π/2 sequence. Ramsey interferometers using internal electronic or nuclear states already have played an important role in accurate quantum state engineering and quantum metrology, such as in nuclear magnetic resonance6, atomic clocks7, quantum information8 and quantum simulation9. In general, echo techniques are used in RIs to suppress de-phasing for significantly increasing the coherent time10,11,12. Even though motional states of particles are on the same footing as internal states in current quantum technologies, i.e., motional qubits13, motional logic gates14,15,16, motional quantum error correction17, quantum entanglement and coherent control18,19, quantum metrology with nonclassical motional states20, and communication multiplexing21, conventional echo-RIs rarely exploit the quantum interference of trapped motional states.
Recently, an RI with trapped motional states of a Bose–Einstein condensate (BEC) trapped in an anharmonic potential has been demonstrated with 92% contrast for several cycles22. This proof-of-principle experiment holds great promises for studying quantum many-body physics out of equilibrium, quantum metrology with non-classical motional states and quantum information processing with motional qubits. Recently, this new technique has been used to investigate decoherence and relaxation dynamics23, or to measure the phononic Lamb shift24.
Ultra-cold atoms trapped in an optical lattice (OL) are an ideal test platform for studying quantum many-body dynamics25, and have also been widely used as a high-precision metrology tool26. Conventionally, these atoms are prepared in the lowest band of the OL. Over last few years, there is an increasing experimental and theoretical interest to prepare atoms in higher bands27,28, opening a new way to simulate exotic orbital physics in strongly correlated matter with rich degrees of freedom, i.e., the formation of multi-flavor systems29, supersolid quantum phases in cubic lattices30, quantum stripe ordering in triangular lattices31 and Wigner crystallisation in honeycomb lattices32. A well-characterised RI with a clear understanding of the coherence mechanisms has potential applications in the study of coherence of trapped states in lattices33, nonequilibrium many body physics34, and its applications in metrology35.
In this paper, we demonstrate a RI with trapped motional quantum states (TMQS) of atoms employing the S- and D-bands in an OL. Due to the lack of selection rules for lattice band transition, a key challenge for constructing this RI is to realise π- and π/2-pulses analogous to those in conventional RIs. Using a shortcut loading method36,37, we have designed sequences of optical pulses38,39, analogous to a π- or π/2-pulse, to efficiently prepare a superposition of atoms in S- and D-band states in the OL at zero quasi-momentum with high fidelity within tens of microseconds, which is much shorter than the characteristic time scales of the decay process. Keeping the lattice on, we observe state interference and measure the decay of the coherent oscillations.
We have identified the mechanisms leading to the RI contrast reduction as following: the the homogeneity of the optical lattice depth, interaction-induced transverse expansion after loading the atoms from the harmonic trap into the optical lattice, laser intensity fluctuation and thermal fluctuations at finite temperature. We then implement a matter-wave band echo technique to significantly suppress all the contrast decay effects except for quantum and thermal fluctuations, increasing the coherence time to 14.5 ms compared to 1.3 ms without echo at condensate temperature of 50 nK and 10Er lattice depth.
## Results
### Experimental implementation
Our experiment start with a BEC of 87Rb prepared in a hybrid trap formed by a single-beam optical dipole trap with wavelength 1064 nm and a quadruple magnetic trap (Details are presented in our former works40,41). A nearly pure condensate of about 1.0 × 105 atoms at the temperature 50 nK is achieved with the harmonic trapping frequencies (ω x , ω y , ω z ) =2π × (24, 48, 58) Hz, respectively. The system’s temperature can be controlled by the evaporative cooling process (see Methods). After preparation of the condensate, a one dimensional optical lattice is formed by two counter-propagating laser beams with wavelength 852 nm resulting in a lattice constant d = 426 nm along x axis, as shown in Fig. 1(a). Using a shortcut control method38, the BEC is loaded into the lowest lattice band with the quasi-momentum q = 0 within a few microseconds with nearly 100% fidelity. The interaction energy of the condensate in the ground state of OL with a lattice depth of 10Er is 1.2 kHz ($$E_{\mathrm{r}} = \left( {\hbar k} \right)^2/2m$$ is the recoil energy, k = 2π/426 nm−1 is the wave vector associated with the lattice and m is the atomic mass).
In the experiment, we construct the Ramsey interferometer with Bloch states ϕi,q with energy εi,q. We use the lowest band ϕS,0 and the second excited band ϕD,0 at quasi-momentum q = 0, denoted as |S〉, |D〉 respectively in the following, to form a superposition state ψ = a S |S〉 + a D |D〉. The two bands of S and D are considered a two-state system (spin-1/2 system), i.e., the two states can be expressed as $$\left( {\begin{array}{*{20}{c}} {a_{\rm{S}}} \\ {a_{\rm{D}}} \end{array}} \right)$$. The scheme for the band energy and the superposition states are shown in Fig. 1b, c, respectively, with the lattice depth V0 = 10.0Er. During the lattice pulse sequence, an accusto-optic modulator controlled by an RF switch is used to switch on and off the lattice potential (light) quickly.
Manipulating the pseudo-spin system has its own challenges: unlike conventional RI where selection rules can be used to prepare population in two sates, the lattice band transition, similar to transition for vibration states in molecules42, has no selection rules. Thus a π/2-pulse analogous to those in conventional RIs had to be numerically designed, so that the atoms in S-band and D-band are to be transferred into the target states $$\left| {\psi _1} \right\rangle = \left( {\left| S \right\rangle + \left| D \right\rangle } \right)/\sqrt 2$$ and $$\left| {\psi _2} \right\rangle = \left( { - \left| S \right\rangle + \left| D \right\rangle } \right)/\sqrt 2$$, respectively. Extending our optimal control method38,39, we construct a π/2 pulse from a sequence of lattice pulses with special selected timing and duration as shown in Fig. 1(d). We achieve a fidelity of 98.5% and 98.0%, for atoms initially on the S- and D-band respectively. (see Methods).
The energy difference between S- and D-band is changing with quasi-momentum q, and the pulse sequences are desighed for q = 0. For a 10 E r lattice the applied short-cut pulses give more than 90% fidelity only in the region of q < 0.06k.
The full time sequence for RI is shown in Fig. 1e. First the atoms in the harmonic trap are transferred into the S-band by a fast loading process, then the first pulse $$\hat R\left( {\pi /2}\right)$$ is applied to prepare an initial superposition state $$\hat R\left( {\pi /2} \right)\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right) = \frac{1}{{\sqrt 2 }}\left( {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right)$$. The final state, after evolution in the OL for time t OL and a second π/2 pulse can be expressed as:
$$\psi _{\rm{f}} = \hat R\left( {\pi /2} \right)\hat U\left( {t_{{\rm{OL}}}} \right)\hat R\left( {\pi /2} \right)\psi _i,$$
(1)
with $$\hat R\left( \alpha \right) = \left( {{\mathrm{cos}}\frac{\alpha }{2} - i\,{\mathrm{sin}}\frac{\alpha }{2}} \right)\hat \sigma _y$$, and the evolution operator $$\hat U\left( t \right) = \left( {{\mathrm{cos}}\,\omega t + i\,{\mathrm{sin}}\,\omega t} \right)\hat \sigma _z$$, here ω = (εD,0 − εS,0)/ is the frequency difference between two states, and $$\hat \sigma$$ the Pauli matrix, α denotes rotation angle around an axis in the Bloch sphere.
We then apply band mapping27 to read out the state of the RI. For band mapping, the lattice depth is exponentially-ramped down in the form et/η with a characteristic decay time η = 100 μs for a total length of 500 μs. Then absorption imaging is used to measure the population in the different bands after 31 ms time of flight (TOF) (see “Methods” section). The atoms originally occupying |S〉 state populate a narrow Gaussian distribution around 0k. The atoms originally occupying the |D〉 are detected at ±2k in the side zone. The total atom number we detected consists of condensed atoms and thermal atoms. From a bimodal fitting to each peak28, we can quantitatively determine the population of condensate, and we note the condensed atom number as NS (ND) for S-band (D-band) (see “Methods” section).
### Ramsey Interferometer in an optical lattice
Experimental results on the time evolution of the atom population in the D-band pD(tOL) = ND/(NS + ND) with evolution time in the lattice t OL are shown in Fig. 2a for the time sequence $$\hat R\left( {\pi /2} \right) - \hat U\left( {t_{{\rm{OL}}}} \right) - \hat R\left( {\pi /2} \right)$$, V0 = 10Er and T = 50 nK. Each solid point with error bar is the mean of three measurements, shown as red circles. The red solid line is a fit of an damped oscillation with a period of 41.1 ± 1.0 μs, which is consistent with the reciprocal of the band gap energy of about 40.8 μs. The details of the early RI oscillation for t OL < 100 μs is shown in Fig. 2b, displaying a nearly perfect oscillation between the S-band and the D-band with amplitude close to 1. At P1 nearly all the atoms are transmitted from the S-band to the D-band by two π/2 pulses which can be seen as a π pulse. At P2, the two π/2 pulses offset each other due to phase evolution of two states in the lattice, and the atoms are transferred to the S-band. However, when tOL gets longer, the oscillation amplitude decays, as shown in Fig. 2a. The contrast C(tOL) at t OL can be obtained by fitting the amplitude of oscillation p D (tOL) in Fig. 2a with
$$p_D\left( {t_{{\rm{OL}}}} \right) = \left[ {1 + C\left( {t_{{\rm{OL}}}} \right)\,{\mathrm{cos}}\,\left( {\omega t_{{\rm{OL}}} + \phi } \right)} \right]/2,$$
(2)
Figure 2c shows the measured contrast decay versus time tOL for the different initial temperatures of condensates, where each contrast value is fitted from about 20 experimental points with a time step 5 μs and each point is the mean value of three measurements. The error-bars are given by 95% confidence bounds. The horizontal dashed line in Fig. 2c indicates the contrast drops to the value of 1/e, which is used to define a coherence time τ. This time decreases from 1.3 ms to 0.8 ms when the temperature increases from 50 nK to 180 nK.
### Contrast decay mechanisms
In order to improve the performance of the RI, we now investigate the mechanisms that lead to RI signal attenuation. We first study the effect of imperfect design of the π/2 pulse on the fidelity by solving Schrödinger equation with an uniform lattice potential, and the unbalanced population between the S- and D-bands (see “Methods” section). The numerical result (brown dashed line in Fig. 3a) shows that the imperfectly designing of π/2 pulses and the unbalanced population have negligible effects on the contrast decay during the evolution in lattice potential.
In our further theoretical analysis, we replace the ideal lattice potential by a non-uniform potential distribution to account for the Gaussian beam in the radial direction, as shown in Fig. 1a, and include the quasi-momentum distribution for the condensate distributed in the harmonic trap43. These two effects result in inhomogeneous broadening of the transition frequency ω between the S- and D-bands, and lead to to de-phasing and contrast decay. By solving the zero-temperature Gross–Pitaevskii equation(GPE) with the real inhomogeneous potential44 (see “Methods” section), we obtained the contrast as shown by the blue dotted line in Fig. 3a.
The atom-atom interaction leads to transverse expansion after the fast (non-adiabatic) loading of the atoms from the harmonic trap into the 1-d optical lattice. This expansion leads to a significant reduction on contrast (blue dashed line in Fig. 3a).
Moreover, stability of lattice depth in experiment would also influence the contrast we measured. In the GPE simulation, we investigate the influence of both the variation of intensity in a single run and the variation between different experimental runs. We find that the contrast decay shows a stronger dependence on the variation of the mean lattice depth in between different experimental runs. The variation within a single run has little influence if the mean laser power is unchanged. This is because the variation of lattice depth is small and adiabatic, and the contrast at long holding time is reduced by a variation of the mean value. Quantum fluctuations at zero temperature is further added using a truncated Wigner method (see “Methods” section) and the related results shown in green dashed line nearly overlap with the dash-dotted line. In our experiment, the atom number is high enough such that the influence by the quantum fluctuation is not significant.
Finally, we take into account the thermal fluctuations using a finite temperature truncated Wigner calculation (see “Methods” section). The result is shown in the orange solid line in Fig. 3a, agreeing well with the experimental results shown in black dots.
The relation of the coherence time to lattice depth is presented in Fig. 3b. When the lattice depth increases, the confinement in x-axis gets tighter, accordingly, the effects of the expansion in radial direction will increase due to the increased interaction energy. In addition, the non-uniformity of the lattice potential (the difference of the lattice potential from center to edge) also increases with the increasing of lattice depth. Due to these two effects, the coherence time τ decreases with increasing lattice depth. The theoretical curve in Fig. 3b, including all the decay mechanisms (orange solid line) fits well with an experimental measurement.
### An Echo-Ramsey Interferometer with TMQS of atoms
To further improve the contrast of the RI with TMQS, we develop a matter-wave band echo technique. A π pulse is designed which swaps the atom population in S- and D-band. The pulse schema of our echo-RI scheme is shown in Fig. 4. In between the initial and final π/2 pulses we insert n identical π pulses. A single π pulse is realised with the operation $$\hat U\left( {t_{{\rm{OL}}}/2n} \right)\hat R\left( \pi \right)\hat U\left( {t_{{\rm{OL}}}/2n} \right)$$. The full Echo–Ramsey pulse sequence is then: $$\hat R\left( {\pi /2} \right)\left[ {\hat U\left( {t_{{\rm{OL}}}/2n} \right)\hat R\left( \pi \right)\hat U\left( {t_{{\rm{OL}}}/2n} \right)} \right]^n\hat R\left( {\pi /2} \right)$$. The contrast decay C(tOL) of a 4 × π pulse echo-RI versus the holding time at various temperatures between 50 nK and 180 nK is shown in Fig. 5a. The solid lines are fits with an exponential function, which allow to extract the coherence time τ.
The relationship between the coherence time τ and the number of π pulses n is shown in Fig. 5b for three different temperatures (T = 50, 110 and 180 nK). The zero-temperature theoretical results of the coherence time can again be obtained by solving the GPE including the non-uniform lattice potential and the radial expansion, as shown by the circles in Fig. 5b. When further taking into account of the laser intensity fluctuation, the calculated coherence time is shown with the cross points. After implementing one π pulse, the two theoretical points merger together, which shows that the intensity fluctuation can be well suppressed with one π pulse; however, in order to eliminate the effects of non-uniform lattice potential and transverse expansion, more echo pulses are needed; after applying n = 6 echo pulses, the theoretical curves get nearly flat, showing that the two effects are well suppressed. There remains a significant discrepancy between the experimental and the zero temperature theoretical calculations, which we attribute to thermal fluctuations. Lowering the condensate temperature, the thermal fluctuations get weaker, and the coherence time can be greatly increased with multiple echo-pulses. The measured coherence time τ for the RI at T = 50 nK is 14.5 ms, which is one order of magnitude longer than the case without echo-pulses.
We will now look at the remaining fluctuations that cannot be suppressed by the echo technique: We fit the experimental contrast with C0(tOL)exp[−βtOL], where C0 is the calculated contrast for zero temperature and the decay rate β characterises the additional decay. For different temperatures, the fitted values of β are shown in Fig. 5c with red dots, showing a linear dependence on the temperature. This is a strong indication that the remaining decay of the interference contrast of the echo–Ramsey interferometer is caused by thermal fluctuations. To confirm this, we conducted finite temperature truncated Winger calculations45,46 (see “Methods” section)
Finally, we look at the performance of the echo-RI for different lattice depth. As shown in Fig. 6a, the contrast decay rate strongly depends on the lattice depth for a condensate at 50 nK and using a sequence with six π pulses. The decay rate is small at medium optical lattice depth, and it becomes bigger for the shallow (<5Er) and deep lattices (<10Er). For the case of V0 = 4Er, the separation between D- and P-bands is small and it is hard to form a closed cycle between S- and D-band without any transition to P-band in practice. We should note that for the two pulse RI, transition to other bands is not so evident due to the short lattice time. When the echo technique is applied, the coherence time is much increased, and transition to other bands induced by atom-atom interaction become important. This can be seen in Fig. 6b, where the momentum distribution for 4Er has components (marked with a red square) different from ±2k.
When the lattice potential gets deeper, the band separation increases, and the coherence time increases accordingly. However, when the lattice depth is more than 10Er, the coherence time decreases quickly. In fact the probability for atoms to be excited to higher momentum states by the pulse is proportional to the lattice depth. For a very deep lattice the atoms can easily obtain higher momenta47. In Fig. 6b, the momentum distributions clearly show the components at 4k for 20 Er. Since the Rabi transition is designed for the transition between the S- and D-band at zero quasi-momentum, the higher states will lead to a reduced pulse fidelity and a faster decay of the contrast.
## Discussion
In summary, we have demonstrated a Ramsey interferometer for atoms within an OL. We further employ a matter wave band echo technique to significantly enhance the coherence time by one order of magnitude. We have identified the mechanisms leading to the contrast decay for the RI signal. The contrast decay is closely related to the homogeneity of the optical lattice, laser intensity fluctuation and interaction-induced transverse expansion, and finite temperature dynamics of a BEC. All except for the effects of finite temperature can be suppressed by a matter-wave band echo sequence. Thus, the damping from thermal fluctuations is well uncovered in this way.
So far, the π pulses and π/2 pulses for echo-RI using TMQS are designed based on zero temperature single atom dynamics. In future developments, it would be interesting to unveil quantum many body dynamics in OL, like loop structures and swallowtails48,49,50. These deliberate quantum control technologies could be applied in quantum information and precise measurements based on TMQS of atoms.
## Methods
### Design of Pi/2- and Pi-pulses
In a 1D optical lattice along x-direction, the eigen-states of the atom, i.e. the Bloch states, can be expressed as the superposition of momentum states $$\left| {i,q} \right\rangle = \mathop {\sum}\nolimits_{l = - \infty }^{ + \infty } c_{i,l}\left( q \right)\left| {2l\hbar k + q} \right\rangle ,$$ where |2lk + q〉 is the basis in momentum space, i is the band index and q is the quasi-momentum, ci,l(q) is the superposition coefficient with l = 0, ±1, ±2,.... While the target state is considered to be the superposition of Bloch states, $$\psi = \mathop {\sum}\nolimits_i A_i\left| {i,q} \right\rangle = \mathop {\sum}\nolimits_i A_i\mathop {\sum}\nolimits_l c_{i,l}\left( q \right)\left| {2l\hbar k + q} \right\rangle$$ with A i is the superposition coefficient.
Varying the pulse parameters, we numerically find a time sequence {t1 = 0, t2, t3,...} of optical lattice pulses
$$V_{\mathrm{latt}}\left( t \right) = V_0\;{\mathrm{for}}\;\left( {t_{2i - 1} < t < t_{2i}} \right)$$
(3)
$$V_{latt}\left( t \right) = 0\;{\mathrm{for}}\;\left( {t_{2i} < t < t_{2i + 1}} \right)$$
(4)
that creates a $$\frac{\pi }{2}$$ pulse in the Ramsey interferometer, i.e., after this operation, the atoms on S-band and D-band are transferred into target states $$\left( {\left| S \right\rangle + \left| D \right\rangle } \right)/\sqrt 2$$ and $$\left( { - \left| S \right\rangle + \left| D \right\rangle } \right)/\sqrt 2$$, respectively. In our designed pulse sequences, both the laser intensity and the duration can be changed. From the numerical simulation, we find that with a fixed laser intensity our designed pulse sequence can still reach high fidelity and is easier to realize in the real experiment. So we keep the lattice depth constant and only change the time sequence in the design for experimental simplicity without losing of fidelity.
The evolution operator for t2i−1 < t < t2i is
$$\hat U_{{\mathrm{on}}}\left( {t_{2i} - t_{2i - 1}} \right) = {\mathrm{exp}}\left\{ { - \frac{i}{\hbar }{\int}_{ 0}^{t_{2i} - t_{2i - 1}} \frac{{p^2}}{{2m}} + V_0{\mathrm{cos}}^2\left( {kx} \right){\rm{d}}t} \right\},$$
(5)
and
$$\hat U_{{\mathrm{off}}}\left( {t_{2i + 1} - t_{2i}} \right) = {\mathrm{exp}}\left( { - \frac{i}{\hbar }{\int}_{ 0}^{t_{2i + 1} - t_{2i}} \frac{{p^2}}{{2m}}{\rm{d}}t} \right).$$
(6)
for the free evolution operator at t2i < t < t2i+1, where p and m are,, respectively, the momentum and mass of a single atom. After applying the pulse sequence, the final states $$\left| {\psi _{f_1}} \right\rangle$$ and $$\left| {\psi _{f_2}} \right\rangle$$ can be written as
$$\left| {\psi _{f_1}} \right\rangle = \mathop {\prod}\limits_i \hat U_{{\mathrm{on}}}\left( {t_{2i} - t_{2i - 1}} \right) \cdot \hat U_{{\mathrm{off}}}\left( {t_{2i + 1} - t_{2i}} \right) \cdot \left| S \right\rangle$$
(7)
$$\left| {\psi _{f_2}} \right\rangle = \mathop {\prod}\limits_j \hat U_{{\mathrm{on}}}\left( {t_{2i} - t_{2i - 1}} \right) \cdot \hat U_{{\mathrm{off}}}\left( {t_{2i + 1} - t_{2i}} \right) \cdot \left| D \right\rangle .$$
(8)
By changing the parameters of the pulse sequence, one can maximise the two fidelities F1 = $$F_1 = \left| {\left\langle {\psi _{A_1}|\psi _{f_1}} \right\rangle } \right|^2$$ and $$F_2 = \left| {\left\langle {\psi _{A_2}|\psi _{f_2}} \right\rangle } \right|^2$$. For V0 = 10Er, the time sequence for π/2 pulse as {0, 56.2, 84.2, 106.8, 129.9}(μs) with F1 = 96.9%,F2 = 97.9%.
The method for designing π pulse is similar to the case of π/2 pulse, but with other target states. After the sequence of optical lattice pules, the atoms should be transferred from |S〉 or |D〉 into the target states |D〉 or −|S〉, respectively. For V0 = 10Er, the time sequence for π pulse as {0, 49.2, 101.7, 123.8, 150.2}(μs), and the transfer of |S〉 and |D〉 to the target states are with the fidelity 98.5% and 98.0%, respectively.
During design of the pulse sequences, we have neglected atom-atom interaction because the duration of pulses is short enough and atom-atom interaction strength in our system is weak38. Numerical results of the GPE with atom-atom interaction show that the interaction leads to a change of fidelity <1% compared with our designed time sequence.
### Data analysis
After the band mapping, we release the condensate from the trap and let it expand freely for 31 ms to perform time of flight (TOF) imaging. The TOF image shows the atoms from the S-band distributed in the center 0k, while atoms from the D-band are distributed on the side zone ±2k. We apply an algorithm to remove the background fringes51, and integrate the atom distribution in y-direction of the TOF image. Then the atomic distribution is fitted with a bimodal function
$$\begin{array}{*{20}{l}} {f\left( x \right)} \hfill & = \hfill & {\mathop {\sum}\limits_{i = 1,2,3} G_iF_{th}\left( {e^{ - \left( {x - x_i} \right)^2/\sigma ^2}} \right)} \hfill \\ {} \hfill & {} \hfill & { + \mathop {\sum}\limits_{i = 1,2,3} H_i\left( {{\mathrm{max}}\left[ {1 - \left( {x - x_i} \right)^2/\chi _i^2,0} \right]} \right)^2} \hfill \end{array}$$
(9)
with a Gaussian thermal distribution F th and a Thomas Fermi like the condensate. i = 1,2,3 corresponds to the three atom clouds for −2k, 0k, and 2k, respectively. We also add a restriction to all six amplitude terms H i and G i as H1/G1 = H3/G3. From the amplitude of each component H i and the width of condensate part χ i ,, we can determine the population of condensate in the S-band N S and in the D-band N D , and further calculate the population of the D-band as pD(tOL) = ND / (NS + ND). The experimental contrast at a certain time is given by fitting the measured amplitude of oscillation pD for 100 μs (about 2.5 periods) with a time step 5 μs using a cosine function, where the errorbar is given by the fitting error with 95% confidence bounds.
### Contrast decay induced by atom-atom interaction
When atoms collide in the excite band they can decay to a lower band52,53. We calculate the collision decay of atom population in different bands using a simple rate equation model39,
$$\frac{{{\rm{d}}N_{S\left( D \right)}\left( t \right)}}{{{\rm{d}}t}} = - K_{S\left( D \right)}N_{S\left( D \right)}\left( t \right)^2,$$
(10)
where the factor KS(D) is given by summation of cross-section of the two-atom inelastic collision from different channels. Solving Eq.(10), the population from the D-band ND(t) decays as 1/(1 + t/τD), with the time constant of atom decay from the D-band given by 1/τD = KDND. NS(t) can similarly be given from the collision rate of atoms between the S- and the D-bands.
We measure the collisional decay rate of atoms in different bands by band mapping without the final π/2 pulse and get τD = 1.9 ± 0.3 ms, τS = 5.1 ± 0.8 ms.
During the collision decay, the potential energy of the D-band is transferred to kinetic energy in radial direction. This energy is large enough so that the majority of the decayed atoms will not be counted in for the condensed peaks. After the population decay atom number N S (t) and ND(t) is unbalanced, which leads to a reduction of the contrast by a factor $$\frac{{2\sqrt {N_SN_D} }}{{N_S + N_D}}$$. From the measured NS and ND, we find that the influence of this population unbalance to the calculated contrast C0(t OL ) is less than 1% for an experiment with t OL < 2 ms as shown as the brown dashed line in Fig. 3a.
Applying the repeated π-pulses, the coherence time is much longer than this decay time: (i) The π-pulses reverse the population in the S- and D-bands, and therefore prevent a strong imbalance. (ii) In the data analysis, extract the remaining condensed part by bimodal fitting. Both together dramatically reduce the effect of collisional decal on the contrast of the observed interference between the remaining coherent parts, and the collisional decay time does not limit the coherence time of our interferometer signal, as long as the remained population in condensed part is large enough for detection.
### The contrast decay induced by the quasi-momentum distribution and the nonuniform of optical lattice potential in radial direction
For these, we have to turn to numerical calculations taking the real potentials and their variation into account. Considering that the trapping frequencies in y,z-directions are very similar in our system, we can use use a mean field model (GPE) in cylindrical coordinates to describe the system at zero temperature. The evolution of the wave function Φ(r, t) is governed by
$$i\hbar \frac{\partial }{{\partial t}}{\mathrm{\Phi }}\left( {r,t} \right) = \left[ { - \frac{{\hbar ^2}}{{2m}}{\mathrm{\Delta }} + V_{{\mathrm{ext}}} + NU_0\left| {{\mathrm{\Phi }}\left( {r,t} \right)} \right|^2} \right]{\mathrm{\Phi }}\left( {r,t} \right),$$
(11)
where r = (x, r, θ), Vext is the external potential, and $$U_0 = \frac{{4\pi \hbar ^2a_s}}{m}$$ is the interaction term with a s the scattering length.
In our case, the radial part of wavefunction is in the ground state and uniform in θ coordinate. The lattice potential itself depends on the radial position r. We first neglect the kinetic term in radial direction and separate the wavefunction into radial part and axial part as $${\mathrm{\Phi }}\left( {r,t} \right) = \frac{1}{{\sqrt {2\pi } }}\psi \left( {x,t} \right)\phi \left( r \right)$$, then Eq. (11) can be simplified to the following 1d GPE at a certain value r = r i as
$$i\hbar \frac{\partial }{{\partial t}}\psi \left( {x,t} \right) = \left[ { - \frac{{\hbar ^2}}{{2m}}\frac{{\partial ^2}}{{\partial x^2}} + V_{{\mathrm{ext,r}}_{\mathrm{i}}}\left( {x,t} \right) + \rho _{r_i}U_0\left| {\psi \left( {x,t} \right)} \right|^2} \right]\psi \left( {x,t} \right)$$
(12)
$$V_{{\mathrm{ext,r}}_{\mathrm{i}}}$$ is the combination of both harmonic potential $$V_{{\mathrm{trap}}} = \frac{m}{2}\omega _x^2x^2$$ and lattice potential $$V_{{\mathrm{latt}},r_i} = V_0Q\left( t \right)e^{ - 2r_i^2/w_{{\mathrm{latt}}}^2}{\rm{cos}}^2\left( {kx} \right),$$ where Q(t) takes value 0 or 1 depending on the time sequence, and wlatt is the waist of lattice laser. $$\rho _{r_i} = \frac{{{\mathrm{exp}}\left[ { - m\omega _rr_i^2/\hbar } \right]}}{{\sqrt {\pi \hbar /m\omega _r} }}N$$ is the linear density. We solve Eq. (12) to get the ψ(x, t) and the population in the D-band $$p_{D_i}$$ for position r i . We then calculate 30 different radial positions and take their weighted average according to the atom number distribution $$\mathop {\sum}\limits_i {\left( {2\pi r_i\rho _{r_i} \cdot p_{D_i}} \right)} /\mathop {\sum}\limits_i {\left( {2\pi r_i\rho _{r_i}} \right)}$$. Finally, the oscillation amplitude of the average p D is fitted to get a contrast as shown in dotted line in Fig. 3a.
In this simulation, we consider the wave-function’s distribution instead of using a single atom model, thus the influence of quasi-momentum distribution is also included automatically.
### Contrast decay induced by radial motion of the condensate
In the above calculation, we consider the distribution of lattice potential in radial direction, however, the radial size of he wave function also changes with time. To consider this effect, we need to estimate the expansion speed of the atom cloud in this direction.
We can take each site of the lattice as a small independent BEC with about 1.5 × 103 atoms (about 65 lattice sites), and calculate how it spreads after the trapping potential changed during switch on and loading into the lattice. When the lattice is turned on, the trapping frequency in x-direction increases to 20 kHz for 10Er, which is much larger than the harmonic trap of 2π × 24 Hz, this sudden increase of trap frequency induces the spread in the radial direction, which can be calculated as54,
$$\ddot \lambda _x = \frac{{\omega _x^2\left( 0 \right)}}{{\lambda _x^2\lambda _r^2}} - \omega _x^2\left( t \right)\lambda _x,\;\ddot \lambda _r = \frac{{\omega _r^2\left( 0 \right)}}{{\lambda _x\lambda _r^3}} - \omega _r^2\left( t \right)\lambda _r,$$
(13)
where $$\omega _x,\omega _r = \sqrt {\omega _y\omega _z}$$ are the frequency of the effective trapping potential in x and r-directions for the small BECs in each lattice site, respectively, and λ r = r i (t)/r i (0) is the expansion with r i (0) the initially radial position. Using this time-dependent radial expansion, we then follow a procedure like above to calculate the average p D and contrast as shown in dashed line in Fig. 3a of the main text.
### Contrast decay induced by laser intensity fluctuations and thermal fluctuations
The laser intensity in our experiment fluctuates by about 0.1% during the holding time. The laser intensity changes are slow and do not cause excitation between the different bands. Simulations like above are then repeated with different laser intensities sampled from the measured fluctuations. The averaged result is shown in Fig. 3 of the main text with the purple dash-dotted line.
To calculate that contrast decay induced by thermal fluctuations we apply the finite-temperature truncated Wigner method45. We solve the 1d GPE with a stochastic initial wave function $$\psi \prime \left( x \right) = \psi + \mathop {\sum}\nolimits_j {\psi _j}$$, where ψ is the zero-temperature condensate wave-function within the one-dimensional optical lattice, ψ j corresponding to the thermal fluctuations that is given by $$\psi _j = A\left( {r_0,\theta _0} \right)\left[ {u_j\left( x \right)\beta _j - v_j^ \ast \left( x \right)\beta _j^ \ast } \right]$$, where $$A\left( {r_0,\theta _0} \right) = \sqrt {n_0\left( {0,r,\theta } \right)/{\int\!\!\!\int} n_0\left( {0,r,\theta } \right)r{\mathrm{d}}r{\mathrm{d}}\theta }$$, with the condensation density n0(x, r, θ). For simplicity and without loss of generality, we choose r0 = 0,θ0 = 0. Here β j is a complex number with random phase which satisfies $$\beta _j^ \ast \beta _j = \bar N_j + 1/2$$, where the quantum fluctuations are included by the 1/2 term. in which $$\bar N_j = \mathop {\sum}\limits_i {1/\left( {e^{E_{ji}/k_BT} - 1} \right)}$$ and E ji = E j + Λ i is the summation of the energy of the jth Bogoliubov mode E j plus the ith eigen-energy Λ i of the radial harmonic potential. u j and v j are the jth Bogoliubov modes solved from the one-dimensional Bogoliubov de Gennes equation as45,
$$\left[ {H_{sp} + 2U_0n_0 - \mu } \right]u_j - U_0n_0v_j = E_ju_j,$$
(14)
$$- \left[ {H_{sp} + 2U_0n_0 - \mu } \right]v_j + U_0n_0u_j = E_jv_j,$$
(15)
with
$$H_{sp} = - \frac{{\hbar ^2}}{{2m}}\frac{{d^2}}{{dx^2}} + \frac{1}{2}m\omega _x^2x^2.$$
In the calculation, we use 300 excitation modes and repeat our simulation 15 times with different stochastic initial states. From the wave functions and the population distributions, we then obtain the contrast C(tOL), calculated at the different holding times tOL, which can then be compared to the experiment. In a separated calculation, we also calculate for the quantum fluctuations by taking $$\bar N_j = 0$$, the result is shown in Fig. 3a of the main text with green dashed line, which is close to the result without quantum fluctuations, where the coherent time is only reduced by 0.2%.
### Data availability
The authors declare that the main data supporting the findings of this study are available within the article. Extra data are available from the corresponding author upon reasonable request.
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## Acknowledgements
We thank Z. K. Chen for the calculation of pulse time sequences and discussion. Thank P. Zhang, Q. Zhou, and B. Wu for helpful discussion. This work is supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301501), and the NSFC (Grant Nos. 11334001, 61727819, 61475007, 91736208). G. J. Dong acknowledges the support by the National Science Foundation of China (Grant No. 11574085 and No. 91536218), and 111 Project (B12024), and the National Key Research and Development Program of China (Grant No. 2017YFA0304200). JS acknowledges support by the European Research Council, ERC-AdG QuantumRelax.
## Author information
### Author notes
1. These authors contributed equally: D. Hu, L. Niu.
### Affiliations
1. #### School of Electronics Engineering and Computer Science, Peking University, Beijing, 100871, China
• Dong Hu
• , Linxiao Niu
• , Shengjie Jin
• , Xuzong Chen
• & Xiaoji Zhou
2. #### Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi, 030006, China
• Guangjiong Dong
• & Xiaoji Zhou
3. #### State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai, 200062, China
• Guangjiong Dong
4. #### Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020, Vienna, Austria
• Jörg Schmiedmayer
### Contributions
X.Z. designed the experiment, D.H., S.J. and L.N. performed the measurement, L.N. and D.H. analysed the data supervised by X.Z. and J.S., S.J. and L.N. did the theoritical calculations and modelling supervised by X.Z., J.S., G.D. and X.C.. All authors contributed to the interpretation of the result and writing of the manuscript. D.H. and L.N. contributed equally to this work.
### Competing interests
The authors declare no competing interests.
### Corresponding authors
Correspondence to Jörg Schmiedmayer or Xiaoji Zhou. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8548763990402222, "perplexity": 1742.4628305924723}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578529606.64/warc/CC-MAIN-20190420100901-20190420122901-00143.warc.gz"} |
https://www.risk.net/commodities/metals/2245534/index-buyers-warping-brent-crudes-price-curve-a-boon-for-opportunists | # Index buyers warping Brent Crude's price curve a boon for opportunists
## ctas
While hedge funds have been blamed by some for stoking Brent Crude prices, experts say the emergence of commodity indices is bending the curve between spot and forward prices, and opening up trading opportunities in the process.
Russell Newton, principal at Global Advisors, a London-based CTA that trades industrial commodities, says Brent Crude's spot is at $45 and the back of the curve sits at$40. Usually the curve between the two points will slope gently, but index money is changing that,
#### 7 days in 60 seconds
###### Mifid II, RFQs and the future of Europe’s G-Sibs
The week on Risk.net, August 11-17, 2018 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3300917148590088, "perplexity": 20306.006621073753}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221218189.86/warc/CC-MAIN-20180821132121-20180821152121-00642.warc.gz"} |
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kimmy0394 find limit as x approaches infinity sqrt. (x+3) / sqrt. (3x+1) one year ago one year ago Edit Question Delete Cancel Submit
• This Question is Closed
1. completeidiot
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$\lim_{x \rightarrow \infty}\frac{\sqrt{x+3}}{\sqrt{3x+1}}=\lim_{x \rightarrow \infty}\sqrt{\frac{x+3}{3x+1}}$
• one year ago
2. completeidiot
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now just look at the expression within the square root $\lim_{x \rightarrow \infty}\frac{{x+3}}{{3x+1}}$ remember that it is all under a square root, so if the limit of that is negative, then there is no solution because you cant take the square root of a negative number
• one year ago
3. completeidiot
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can you solve it now?
• one year ago
4. kimmy0394
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i thought it was 1/3 but it's not
• one year ago
5. cinar
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|dw:1348034899805:dw|
• one year ago
6. kimmy0394
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ohhhhhhhhhh!
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7. cinar
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(:
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9992144703865051, "perplexity": 10489.159992388488}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999654345/warc/CC-MAIN-20140305060734-00018-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://mathtab.com/page.php?page_id=36 | Fluid mechanics
by: ranjan317
Fluid mechanics is the branch of physics that studies fluids (liquids, gases, and plasmas) and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms, that is, it models matter from a macroscopic viewpoint rather than from a microscopic viewpoint.
IWeirs and notches
Weir Definition
A structure, used to dam up a stream or river, over which the water flows, is called a weir. The conditions of flow, in the case of a weir, are practically the same as those of a rectangular notch. That is why, a notch is, sometimes, called as a weir and vice versa.
The only difference between a notch and a weir is that the notch of a small size and the weir is of a bigger one. Moreover, a notch is usually made in a plate, whereas a notch is made of masonry or concrete.
Types Of Weirs
There are many types of weirs depending upon their shape, nature of discharge, width of crest and nature of crest. But the following are important from the subject point of view :
1. According to the shape :
• Rectangular weir
• Cippoletti weir
2. According to the nature of discharge :
• Ordinary weir
• Submerged or drowned weir
3. According to the width of crest :
• Narrow crested weir
• Broad crested weir
4. According to the nature of crest :
• Sharp crested weir
• Ogee weir
Velocity Of Approach
Sometimes, a weir is provided in a stream or a river to measure the flow of water. In such a case, the water, approaching the weir, has got some velocity, known as velocity of approach. It is assumed to be uniform over the whole weir.Let,
• A = Cross sectional area of the channel on the upstream side of the weir, and
• Q = Discharge over the weir
Velocity of approach,
Notch
A notch may be defined as an opening in one side of a tank or a reservoir, like a large orifice, with the upstream liquid level below the top edge of the opening.Since the top edge of the notch above the liquid level serves no purpose, therefore a notch may have only the bottom edge and sides.
The bottom edge, over which the liquid flows, is known as sill or crest of the notch and the sheet of liquid flowing over a notch (or a weir) is known as nappe or vein. A notch is, usually made of a metallic plate and is used to measure the discharge of liquids.
Types Of Notches
There are many types of notches, depending upon their shapes. But the following are important from the subject point of view.
• Rectangular notch
• Triangular notch
• Trapezoidal notch
• Stepped notch
IICalculation of discharge over a rectangular notch (or weir)
Calculation of discharge over a rectangular notch (or weir)
IIICalculation of length of a rectangular notch (or weir)
Calculation of length of a rectangular notch (or weir)
IVCalculation of discharge over a triangular notch or weir
Calculation of discharge over a triangular notch or weir
VCalculation of discharge through a trapezoidal notch
Calculation of discharge through a trapezoidal notch
VICalculation of discharge through a stepped notch
Calculation of discharge through a stepped notch
VIITime required to lower water to a particular level by rectangular notch
Time required to lower water to a particular level by rectangular notch (or) Level to which water falls in a given time
VIIITime required to lower water to a particular level by triangular notch
Time required to lower water to a particular level by triangular notch
IXCalculation of discharge over a rectangular weir considering velocity of approach
Calculation of discharge over a rectangular weir considering velocity of approach
XCalculation of discharge over Cipolletti weir or notch
Calculation of discharge over Cipolletti weir or notch
XICalculation of discharge over a broad-crested weir or notch
Calculation of discharge over a broad-crested weir or notch
XIICalculation of discharge over a narrow-crested weir or notch
Calculation of discharge over a narrow-crested weir or notch | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8733844757080078, "perplexity": 2508.613071773784}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578663470.91/warc/CC-MAIN-20190424214335-20190425000335-00348.warc.gz"} |
https://undergroundmathematics.org/sequences/puzzling-pentagon | Sequences
Go and think about it...
A puzzling pentagon Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Play the animation or move the slider for $b$. What do you notice? What questions do you have?
You might like to think about some of the following questions for different values of $b$. For example, when $b = 0.38, 0.5, 1$.
• How many regular pentagons can you see?
• What fraction of the shape is shaded in each different colour?
• Can you find the area of any of the shapes?
• Can you see any sequences in the diagram?
• Can you make any links to geometric series? (You may wish to look at Square spirals if you haven’t already done so.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.25071102380752563, "perplexity": 680.1619530902899}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912203865.15/warc/CC-MAIN-20190325092147-20190325114147-00435.warc.gz"} |
https://www.physicsforums.com/threads/question-about-a-group.671729/ | # Question about a group
1. Feb 14, 2013
### port31
1. The problem statement, all variables and given/known data
So we have this operation x*y=x+2y+4
and then our 2nd one is x*y=x+2y-xy
I need to check if it is commutative,associative, and if it has a identity and an inverse.
3. The attempt at a solution
y*x=y+2x+4 so it is not commutative
x*(y*z)=x+2(y+2z+4)+4=x+2y+4z+12
(x*y)*z=(x*y)+2z+4=x+2y+4+2z+4
Not associative
Now I will solve for the identity
e*x=e+2x+4=x
e=-x-4
x*e=x+2e+4=x
e=-2
Since I have 2 different identity elements this means that one does not exist because
e should be unique.
Since there is no e there is no inverse.
Now for the second one x*y=x+2y-xy
y*x=y+2x-yx Does not commute
(x*y)*z=x+2y-xy+2z-xz-2yz+xyz
x*(y*z)=x+2y+4z-2yz-xy-2xz+xyz
Not associative
x*e=x+2e-xe=x
e=0
e*x=e+2x-ex=x
e(1-x)=-x
The identity element does not seem to be unique so it does not exist.
Just want to know if I am doing this right.
2. Feb 14, 2013
### Staff: Mentor
What do you mean "our 2nd one"? You can't define the * operation on a group in two different ways.
3. Feb 14, 2013
### port31
The second one is just another problem. They are 2 separate problems.
4. Feb 14, 2013
### Staff: Mentor
Then you should identify them as such instead of clumping them together as you did.
5. Feb 14, 2013
### Staff: Mentor
What are you trying to show here? Is the exercise for each to say whether some set with the given operation is a group?
If so, you don't need to check every group axiom for the operation. For example, if the operation isn't associative, then that's enough to say that the set and the operation aren't a group.
Also, just because something isn't unique, that doesn't mean it doesn't exist.
6. Feb 15, 2013
### HallsofIvy
Staff Emeritus
More fundamentally, the identity can't depend upon "x".
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Similar Discussions: Question about a group | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8001375198364258, "perplexity": 1545.3668335126492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103167.97/warc/CC-MAIN-20170817092444-20170817112444-00479.warc.gz"} |
https://cstheory.stackexchange.com/questions/47529/pi-calculus-or-session-types-proof-for-weakening-lemma | # Pi-calculus (or session types) - proof for weakening lemma
I'm writing a thesis about session types and am currently writing a section concerning type soundness for the system. I started to proof weakening lemma, which states, that $$\text{If } \Gamma \vdash P, \text{ then } \Gamma, x:T \vdash P.$$ That is, names not free in a process can be added to the typing environment. However, I can't understand how to find a derivation for $$\Gamma, x:T \vdash (\upsilon x)P$$ The rule T-RES for restriction is $$\frac { \Gamma, x:T \vdash P } { \Gamma \vdash (\upsilon x)P }$$ I tried to find a proof for the lemma (for session types, pi-calculus or other calculi), but all the proofs seemed to be something like "a straightforward induction on the derivation of $$\Gamma \vdash P$$".
I'm trying to understand, what I'm misunderstanding or missing. The rule T-PAR explicitly states, that $$x$$ is not in the environment in $$\Gamma \vdash (\upsilon x)P$$. However, the lemma states, that $$\Gamma, x:T \vdash (\upsilon x)P$$ is valid.
You should $$\alpha$$-rename to avoid conflict with the variable names. That is, you should prove weakening of the form: $$\Gamma \vdash (\upsilon y) P$$ implies $$\Gamma, x : T \vdash (\upsilon y) P$$.
$$\alpha$$-equivalence and capture-avoiding substitution is an important concept to understand in type theory: I would recommend studying this concept for the untyped or simply-typed $$\lambda$$-calculus to begin with, before trying to prove things about a more complicated calculus. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 12, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.957324206829071, "perplexity": 545.0250944174419}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499695.59/warc/CC-MAIN-20230128220716-20230129010716-00059.warc.gz"} |
https://mathmaine.com/2014/01/20/polynomials-and-vex-drive-motor-control/ | # Polynomials and VEX Drive Motor Control
VEX Robots can be more competitive when they have addressed several drive motor control challenges:
1. Stopping a motor completely when the joystick is released. Joysticks often do not output a value of “zero” when released, which can cause motors to continue turning slowly instead of stopping.
2. Starting to move gradually, not suddenly, after being stopped. When a robot is carrying game objects more than 12 inches or so above the playing field, a sudden start can cause the robot to tip over.
3. Having motor speeds be less sensitive to small joystick movements at slow speeds. Divers seeking to position the robot precisely during competition need “finer” control over slow motor speeds than fast motor speeds.
These challenges can be solved using one or more “if” statements in the code controlling the robot, however using a single polynomial function can often solve all of these challenges in one step. A graph can help illustrate the challenges and their solution:
The blue points in the graph correspond to the most common way that motors are controlled by a joystick: the same value received from the joystick is used to set motor speed. A line connecting the blue points would have the following equation:
$MotorOut = JoystickIn$
Its slope is 1, and its vertical axis intercept is zero, so the blue line in the graph below gives us a visual description of the complete motor control algorithm:
While this approach is certainly the easiest to think about and implement, once you have taken Algebra II it does not take much time or effort to dream up an improved motor control algorithm. If we plot the MotorOut values we would prefer to use for each JoystickIn value, points A and D might remain as they are above, but we might prefer the green points BModified and CModified instead of points B and C for the following reasons.
If the joystick is released but does not recover into the exact center position, it will probably produce an input value within 10 or 15 of zero. The green point BModified on the graph would have this situation produce a MotorOut value of 1, which is not likely to be enough power to turn a drive motor, so the motor would stop even though the joystick might not have returned to the exact center. This would address the first challenge described at the start of this post.
The two remaining challenges described at the beginning are both addressed by the point CModified. When the joystick is moved a small distance away from its center position, the JoystickIn value is likely to be greater than the ideal drive motor output value for slow movement. If this input value can be reduced somewhat before being sent to the drive motor, say from a value of 40 to something closer to 10, that will allow for more gradual drive motor starts as well as more precise turns (less risk of over-correcting the robot’s direction). And, more gradual motor starts will help reduce the risk of having the robot tip over when it has game objects lifted to a high position.
So, what kind of an equation would pass through the perfect stop (0 , 0) and full power (128 , 128) points, but get from one to the other in a more gradual way? If you trace a rough curve through points A, BModified, CModified, and D, do you recognize that shape, or at least something close to it?
The green curve above does not quite pass exactly through the two green points, but it is pretty close. More importantly, it does pass through (0, 0) and (128, 128), ensuring you can instruct the motors to produce both maximum and minimum power. The shape of the green curve is a parabola, which tells us that we can model it using a second degree polynomial equation: a quadratic equation.
Since we know the desired vertex is (0, 0), we can start with vertex form of a parabola, then substitute (0, 0) in for the x and y coordinates of the vertex:
$MotorOut=a\cdot (JoystickIn - H)^2+K\\*~\\*MotorOut=a\cdot (JoystickIn - 0)^2+0\\*~\\*MotorOut=a\cdot (JoystickIn)^2$
This equation is guaranteed to pass through the vertex (0, 0), but we now need to find a value for “a” that will force the equation to also pass through (128, 128). We can do this by substituting the x and y coordinates of this point into our equation above, then solving for “a”:
$MotorOut=a\cdot (JoystickIn)^2\\*~\\*128=a(128)^2\\*~\\*\dfrac{128}{(128)^2}=a\\*~\\*\dfrac{1}{128}=a$
So, the equation of the green curve in the graph above is
$MotorOut=\dfrac{1}{128}(JoystickIn)^2$
If you use this equation in your drive motor control task, you will hopefully discover that it produces much smoother starts, and more reliable stops. However, as the graph above indicates, it also causes a problem: the robot will not move backwards, as the curve only produces positive MotorOut values.
One way to fix this is to use a “piece-wise” function which dilates the left half of the parabola vertically by a factor of negative one (rotating it about the horizontal axis), while leaving the right half untouched:
$MotorOut=\begin{cases}-\dfrac{1}{128}(JoystickIn)^2 ~\text{ if }~JoystickIn<0\\*\\*~~\dfrac{1}{128}(JoystickIn)^2~\text{ if }~ JoystickIn\geq 0\end{cases}$
Conditional logic can implement this easily in your drive motor control task:
if (JoystickIn < 0)
MotorOut = -(JoystickIn^2) / 128
else
MotorOut = (JoystickIn^2) / 128 ;
Another way of coding this is:
MotorOut = (JoystickIn^2) / 128;
if (JoystickIn < 0) MotorOut = -MotorOut;
Another approach would be to try a mathematical solution to the problem, by choosing a polynomial with an odd degree (an odd highest power of x), which will allow the function to produce negative as well as positive values. So, starting with a cubic function that is symmetrical about the origin and only crosses the x axis at the origin, we get:
$MotorOut=a\cdot (JoystickIn - H)^3+K\\*~\\*MotorOut=a\cdot (JoystickIn - 0)^3+0\\*~\\*MotorOut=a\cdot (JoystickIn)^3$
Then solving for the value of “a” that will force this curve to pass through (128, 128):
$MotorOut=a\cdot (JoystickIn)^3\\*~\\*128=a(128)^3\\*~\\*\dfrac{128}{(128)^3}=a\\*~\\*\dfrac{1}{(128)^2}=a\\*~\\*MotorOut=\dfrac{1}{(128)^2} (JoystickIn)^3$
The graph of this equation is the tan curve on the graph below, along with our previous two equations:
And this motor control algorithm could be implemented in your drive motor task with even less code:
MotorOut = (JoystickIn^3) / (128^2);
However, note that cubic function (tan) grows more slowly for small values of JoystickIn than the quadratic (green), so it is possible that your driver(s) may prefer one approach over the other, particularly when trying to position a robot precisely in front of an object. So, give them an opportunity to try each, then keep the winner. Alternatively, you could put them both into your code, with the choice of drive algorithm chosen by whether a button on the controller is held down at the moment, or not:
if (Button6)
{
MotorOut = (JoystickIn^2) / 128;
if (JoystickIn < 0) MotorOut = -MotorOut
}
else
{
MotorOut = (JoystickIn^3) / (128^2)
};
When automating tasks, it is often useful to think carefully about whether controller inputs should be passed on as outputs unchanged, or whether safety, reliability, and/or overall performance might be improved by having the output be something different than the current input value. Thinking this through can often be a bit easier and more complete if you use a graph, as was done above. If you decide that it will be useful to have the output differ from the input, is that difference something that can be described easily by a mathematical function? Your graph should help you answer that question quickly.
Another example of where outputs may need to differ from inputs arises if you wish to prevent motor speeds from changing too rapidly, thereby preventing “jerky” robot movements. Any significant changes in input values would need to be “tempered” by the software in a way that still feels responsive, but nevertheless enforces some limit on how fast motor speeds are allowed to change. A mathematical function could take the current and desired motor speeds as inputs, and provide the maximum new motor speed as an output. The following implementation of this idea allows only 30% of the desired change to happen each time the current joystick position is read (each time through the drive motor task loop):
MotorOut = MotorNow + (JoystickIn – MotorNow) * 0.30
So, if the current motor speed (MotorNow) is 20, and JoystickIn is calling for a speed of 100, the first time your drive motor task has a chance to respond to this new joystick position, it would set MotorOut to
$20 + (100-20) * 0.30=44$
The next time through the drive task loop, if JoystickIn is still 100, MotorOut would be set to
$44 + (100 - 44)*0.30 = 61$
The next time through the drive task loop, if JoystickIn is still 100, MotorOut would be set to
$61 + (100 - 61)*0.30 = 73$
As you can see, the motor speed will gradually approach 100, with a “smoother” change in speed from the initial speed of 20. However, the approach to the desired motor speed could end up still being too fast (it will not take more than a second to get above 95), so you should play around with the 30% number to determine whether some other number (perhaps between 0.1% and 30%) works better for your robot.
Risks to using this approach include the slower acceleration that this creates, as well as the possibility that full power might not be able to be reached quickly enough. However, using this approach alongside the approach described above can help produce a “feel” that your drivers might prefer.
So, think about your control algorithms. Try having outputs be a mathematical distortion of the inputs, then decide on what seems to work best for your robot design and driver(s) preferences. Your robot and team will benefit from your experimentation! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 10, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2973191440105438, "perplexity": 757.5744978940766}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668529.43/warc/CC-MAIN-20191114154802-20191114182802-00046.warc.gz"} |
http://project-thrill.org/docs/master/layer_api.html | Thrill 0.1
User DIA API
Generate is a Source-DOp, which creates a DIA of given size using a generator function.
From the user's perspective, a DIA (Distributed Immutable Array) is an immutable array, which can only be accessed by Operations and Actions. Operations transform a DIA into another DIA, Actions aggregate data of the DIA and make it accessible to the user program.
Inside the Thrill framework, a DIA only consists of a reference to a DIANode and a lambda function, which can transform elements from the DIANode to this DIA. Distributed Operations(DOps) and Actions both need a global barrier and thus form a new DIANode. Every DOp and Action has an own node type, which are sub-classes of DIANode.
DIABase is the untyped super-class of DIANode. DIABase objects form a DAG, in which the parent-child relation means that a parent has to be executed before the child can be executed. Every DIANode can be executed, which performs the actual operation and all uncalculated parent operations. The engine uses the DIABase graph to optimize the order of operations.
# List of Actions
## Generate (ctx, generator_function, size)
Generate is a Source-DOp, which creates a DIA of given size using a generator function. The generator function called for each index in the range of [0,size) and must output exactly one item.
Parameters
ctx Reference to the Context object size Size of the output DIA generate_function Generator function, which maps size_t from [0,size) to elements. Input type has to be size_t.
## Distribute(ctx, vector)
Distribute is a Source DOp, which scatters the vector data from the source_id to all workers, partitioning equally, and returning the data in a DIA. Distribute is a Source DOp, which scatters the vector data from the source_id to all workers, partitioning equally, and returning the data in a DIA. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33229655027389526, "perplexity": 4008.7414569658085}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202003.56/warc/CC-MAIN-20190319163636-20190319185636-00351.warc.gz"} |
http://mathhelpforum.com/advanced-applied-math/149995-inverse-z-transform-partial-fraction-method-print.html | # Inverse Z transform by partial fraction method
• July 3rd 2010, 03:53 AM
moonnightingale
Inverse Z transform by partial fraction method
(1 + 2*z + 3*z^2+4*z^3+5*z^4) / z^4
I have done this question with Difference equation and direct division method and answers come to be
x(0)= 5
x(1)= 4
x(2)=3
x(3)=2
x(4)=1
but i am stuck with partial fraction method.
Can some body guide me for this
• July 3rd 2010, 05:50 AM
mr fantastic
Quote:
Originally Posted by moonnightingale
(1 + 2*z + 3*z^2+4*z^3+5*z^4) / z^4
I have done this question with Difference equation and direct division method and answers come to be
x(0)= 5
x(1)= 4
x(2)=3
x(3)=2
x(4)=1
but i am stuck with partial fraction method.
Can some body guide me for this
Direct division gives $X(Z) = 5 + \frac{4}{z} + \frac{3}{z^2} + \frac{2}{z^3} + \frac{1}{z^4}$ which are the partial fractions .... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8920456767082214, "perplexity": 5320.610855131928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164000853/warc/CC-MAIN-20131204133320-00051-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://thosgood.com/fga/FGA-3-I.html | # 3 Generalities, and descent by faithfully flat morphisms
Grothendieck, A. “Technique de descente et théorèmes d’existence en géométrie algébrique, I: Généralités. Descente par morphismes fidèlement plats”. Séminaire Bourbaki 12 (1959–60), Talk no. 190.
([Trans.] Sections 3.1 to 3.4 were numbered A.1 to A.4 in the original; sections 3.5 to 3.9 were numbered B.1 to B.5.)
[Comp.] For various details concerning the theory of descent, see also SGA VI, VII, and VIII.
From a technical point of view, the current exposé, and those that will follow, can be considered as variations on Hilbert’s celebrated “Theorem 90”. The introduction of the method of descent in algebraic geometry seems to be due to A. Weil, under the name of “descent of the base field”. Weil considered only the case of separable finite field extensions. The case of radicial extensions of height 1 was then studied by P. Cartier. Lacking the language of schemes, and, more particularly, lacking nilpotent elements in the rings that were under consideration, the essential identity between these two cases could not have been formulated by Cartier.
Currently, it seems that the general technique of descent that will be explained (combined with, when necessary, the fundamental theorems of “formal geometry”, cf. [9]) is at the base of the majority of existence theorems in algebraic geometry.7 It is worth noting as well that this aforementioned technique of descent can certainly be transported to “analytic geometry”, and we can hope that, in the not-too-distant future, the specialists will know how to prove the “analytic” analogues of the existence theorems in formal geometry that will be given in exposé II. In any case, the recent work of Kodaira–Spencer, whose methods seem unfit for defining and studying “varieties of modules” in the neighbourhood of their singular points, show reasonably clearly the necessity of methods that are closer to the theory of schemes (which should naturally complement transcendental techniques).
In the present exposé (namely exposé I) we will discuss the most elementary case of descent (the one indicated in the title). The applications of Theorem 1, Theorem 2, and Theorem 3 below (in §B.1) are, however, already vast in number. We will restrict ourselves to giving only some of them as examples, without aiming for the maximum generality possible.
We will freely use the language of schemes, for which we refer to the already cited exposé, as well as [6]. We make clear to point out, however, that the preschemes considered in this current exposé are not necessarily Noetherian, and that the morphisms are not necessarily of finite type. So, if A is a local Noetherian ring, with completion \overline{A}, then we will need to consider the non-Noetherian ring \overline{\overline{A}}\otimes_A\overline{A}, as well as the morphisms of affine schemes that correspond to the inclusions between the rings in question.
## 3.1 Fibred categories, descent data, {\mathcal{F}}-descent morphisms
### 3.1.1 (a)
A fibred category {\mathcal{F}} with base {\mathcal{C}} (or over {\mathcal{C}}) consists of
• a category {\mathcal{C}}
• for every X\in{\mathcal{C}}, a category {\mathcal{F}}_X
• for every {\mathcal{C}}-morphism f\colon X\to Y, a functor f^*\colon{\mathcal{F}}_Y\to{\mathcal{F}}_X, which we also write as f^*(\xi) = \xi \times_Y X for \xi\in{\mathcal{F}}_Y (with X being thought of as an “object of {\mathcal{C}} over Y”, i.e. as being endowed with the morphism f)
• for any two composible morphisms X\xrightarrow{f}Y\xrightarrow{g}Z, an isomorphism of functors c_{f,g}\colon (gf)^* \to f^*g^*
with the above data being subject to the conditions that
1. \operatorname{id}^*=\operatorname{id}
2. c_{f,g} is the identity isomorphism if f or g is an identity isomorphism
3. for any three composible morphisms X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}T, the following diagram, given by using the isomorphisms of the form c_{u,v}, commutes: \begin{CD} (h(gf))^* @= ((hg)f)^* \\@VVV @VVV \\(gf)^*h^* @. f^*(hg)^* \\@VVV @VVV \\(f^*g^*)h^* @= f^*(g^*h^*) \end{CD}
Let {\mathcal{C}} be a category where all fibre products exist. We then define a fibred category {\mathcal{F}} with base {\mathcal{C}} by setting {\mathcal{F}}_X to be the category of objects of {\mathcal{C}} over X, and the functor f^*\colon{\mathcal{F}}_Y\to{\mathcal{F}}_X corresponding to a morphism f\colon X\to Y being defined by the fibre product Z\rightsquigarrow Z\times_Y X.
Let {\mathcal{C}} be the category of preschemes, and, for X\in{\mathcal{C}}, let {\mathcal{F}}_X be the category of quasi-coherent sheaves of modules on X. If f\colon X\to Y is a morphism of preschemes, then f^*\colon{\mathcal{F}}_Y\to{\mathcal{F}}_X is the inverse image of sheaves of modules functor. We thus obtain a category fibred over {\mathcal{C}}.
### 3.1.2 (b)
A diagram of maps of sets E \xrightarrow{u} E' \overset{v_1}{\underset{v_2}{\rightrightarrows}} E'' is said to be exact if u is a bijection from E to the subset of E' consisting of the x'\in E' such that v_1(x')=v_2(x').
Let {\mathcal{F}} be a fibred category with base {\mathcal{C}}, and consider a diagram of morphisms in {\mathcal{C}} S \xleftarrow{\alpha} S' \overset{\beta_1}{\underset{\beta_2}{\leftleftarrows}} S'' such that \alpha\beta_1=\alpha\beta_2; this diagram is said to be {\mathcal{F}}-exact if, for every pair (\xi,\eta) of elements of {\mathcal{F}}_S, the diagram of sets
\operatorname{Hom}(\xi,\eta) \xrightarrow{\alpha^*} \operatorname{Hom}(\alpha^*(\xi),\alpha^*(\eta)) \overset{\beta_1^*}{\underset{\beta_2^*}{\rightrightarrows}} \operatorname{Hom}(\gamma^*(\xi),\gamma^*(\eta)) \tag{+}
(where \gamma=\alpha\beta_1=\alpha\beta_2) is exact.
In this above diagram, for simplicity, we have identified \beta_i^*\alpha^* with (\alpha\beta_i)^*=\gamma^*, using c_{\beta_i,\alpha}.
Let {\mathcal{F}} be a fibred category with base {\mathcal{C}}, and consider morphisms \beta_1,\beta_2\colon S''\to S' in {\mathcal{C}}. Let \xi'\in{\mathcal{F}}_{S'}. We define a gluing data on \xi' (with respect to the pair (\beta_1,\beta_2)) to be an isomorphism from \beta_1^*(\xi') to \beta_2^*(\xi'). If \xi',\eta'\in{\mathcal{F}}_{S'} are both endowed with gluing data, then a morphism u\colon\xi'\to\eta' in {\mathcal{F}}_{S'} is said to be compatible with the gluing data if the following diagram commutes: \begin{CD} \beta_1^*(\xi') @>>> \beta_2^*(\xi') \\@VVV @VVV \\\beta_1^*(\eta') @>>> \beta_2^*(\eta'). \end{CD}
With this definition, the objects of {\mathcal{F}}_{S'} that are endowed with gluing data (with respect to \beta_1 and \beta_2) then form a category. If \alpha\colon S'\to S is a morphism such that \alpha\beta_1=\alpha\beta_2, then, for every \xi\in{\mathcal{F}}_{S'}, the object \xi'=\alpha^*(\xi) of {\mathcal{F}}_{S'} is endowed with a canonical gluing data, since \beta_i^*\alpha^*(\xi) \simeq (\alpha\beta_i)^*(\xi) = \gamma^*(\xi), where we again set \gamma=\alpha\beta_1=\alpha\beta_2; furthermore, if u\colon\xi\to\eta is a morphism in {\mathcal{F}}_s, then \alpha^*(u)\colon \alpha^*(\xi) \to \alpha^*(\eta) is a morphism in {\mathcal{F}}_{S'} that is compatible with the canonical gluing data. We thus obtain a canonical functor from the category {\mathcal{F}}_S to the category of objects of {\mathcal{F}}_{S'} endowed with gluing data with respect to the pair (\beta_1,\beta_2). With this, we can also rephrase Definition 1.3 by saying that the diagram (+) is {\mathcal{F}}-exact if the above functor is fully faithful, i.e. if the above functor defines an equivalence between the category {\mathcal{F}}_S and a subcategory of the category of objects of {\mathcal{F}}_S endowed with gluing data with respect to (\beta_1,\beta_2).
We say that a gluing data on \xi'\in{\mathcal{F}}_{S'} is effective (with respect to \alpha) if \xi', endowed with this data, is isomorphic to \alpha^*(\xi) for some \xi\in{\mathcal{F}}_S.
In the case where the diagram (+) is {\mathcal{F}}-exact, the object \xi in Definition 1.5 is then determined up to unique isomorphism, and the category {\mathcal{F}}_S is equivalent to the category of objects of {\mathcal{F}}_{S'} endowed with effective gluing data.
### 3.1.3 (c)
The most important case is that where S'' = S' \times_S S', with the \beta_i being the two projections p_1 and p_2 from S'\times_S S' to its two factors (where we now suppose that {\mathcal{C}} has all fibre products). We then have two immediate necessary conditions for a gluing data \varphi\colon p_1^*(\xi')\to p_2^*(\xi') on some \xi'\in{\mathcal{F}}_S to be effective:
1. \Delta^*(\varphi) = \operatorname{id}_\xi, where \Delta\colon S'\to S'\times_S S' denotes the diagonal morphism, and where we identify \Delta^* p_i^*(\xi') with (p_i\Delta)^*(\xi')=\xi'.
2. p_{31}^*(\varphi) = p_{32}^*(\varphi)p_{21}^*(\varphi), where p_{ij} denotes the canonical projection from S'\times_S S'\times_S S' to the partial product of its ith and jth factors.
We define descent data on \xi'\in{\mathcal{F}}_{S'}, with respect to the morphism \alpha\colon S'\to S, to be a gluing data on \xi' with respect to the pair (p_1,p_2) of canonical projections S'\times_S S'\to S' that satisfies conditions (i) and (ii) above.
A morphism \alpha\colon S'\to S is said to be an {\mathcal{F}}-descent morphism if the diagram of morphisms S \xleftarrow{\alpha} S' \overset{p_1}{\underset{p_2}{\leftleftarrows}} S'\times_S S' is {\mathcal{F}}-exact (Definition 1.3); we say that \alpha is a strict {\mathcal{F}}-descent morphism if, further, every descent data (Definition 1.6) on any object of {\mathcal{F}}_{S'} is effective.
This latter condition (of strictness) can also be stated in a more evocative way: “giving an object of {\mathcal{F}}_S is equivalent to giving an object of {\mathcal{F}}_{S'} endowed with a descent data”.
Note that, if an {\mathcal{F}}-descent morphism8 \alpha\colon S'\to S admits a section s\colon S\to S' (i.e. a morphism s such that \alpha s=\operatorname{id}_S), then it is a strict {\mathcal{F}}-descent morphism: if \xi'\in{\mathcal{F}}_{S'} is endowed with descent data with respect to \alpha, then “it comes from” \xi=s^*(\xi').
### 3.1.4 (d)
We can present the above notions in a more intuitive manner, by introducing, for an object T of {\mathcal{C}} over S, the set \operatorname{Hom}_S(T,S') = S'(T), where elements will be denoted by t, t', etc. Given an object \xi'\in{\mathcal{F}}_{S'}, there then corresponds, to every t\in S'(T), an object t^*(\xi') of {\mathcal{F}}_T, which will also be denoted by \xi'_t. A gluing data on \xi' (with respect to (p_1,p_2)) is then defined by the data, for every T over S, and every pair of points t,t'\in S'(T), of an isomorphism \varphi_{t',t}\colon \xi'_t \to \xi'_{t'} (satisfying the evident conditions of functoriality in T). Conditions (i) and (ii) of §A.1.c can then be written as
i bis. \varphi_{t,t}=\operatorname{id}_{\xi'_t}, for all T and all t\in S'(T).
ii bis. \varphi_{t,t''}=\varphi_{t,t'}\varphi_{t',t''}, for all T and all t,t',t''\in S'(T).
We can show that (ii bis) implies that \varphi_{t,t}^2=\varphi_{t,t}, by taking t=t'=t'', and thus, since \varphi_{t,t} is an isomorphism by hypothesis, implies (i bis), which is thus a consequence of (ii bis) (and so (i) is also a consequence of (ii)). But if we no longer suppose a priori that the \varphi_{t,t} are isomorphisms (i.e. that \varphi\colon p_1^*(\xi')\to p_2^*(\xi') is an isomorphism), then (ii bis) no longer necessarily implies (i bis); the combination of (ii bis) and (i bis), however, does imply that the \varphi_{t,t'} are isomorphisms (since we then have \varphi_{t,t'}\varphi_{t',t}=\varphi_{t,t}=\operatorname{id}_{\xi'_t}).
## 3.2 Exact diagrams and strict epimorphisms, descent morphisms, and examples
### 3.2.1 (a)
Let {\mathcal{C}} be a category. A diagram of morphisms T \xrightarrow{\alpha} T' \overset{\beta_1}{\underset{\beta_2}{\rightrightarrows}} T'' is said to be exact if, for all Z\in{\mathcal{C}}, the corresponding diagram of sets \operatorname{Hom}(Z,T) \to \operatorname{Hom}(Z,T') \rightrightarrows \operatorname{Hom}(Z,T'') is exact (Definition 1.2). We then say that (T,\alpha) (or, by an abuse of language, T) is a kernel of the pair (\beta_1,\beta_2) of morphisms.
This kernel is evidently determined up to unique isomorphism. If {\mathcal{C}} is the category of sets, then the above definition is compatible with Definition 1.2. Dually, we define the exactness of a diagram of morphisms in {\mathcal{C}} S \xleftarrow{\alpha} S' \overset{\beta_1}{\underset{\beta_2}{\leftleftarrows}} S'' and then say that (S,\alpha) is a cokernel of the pair (\beta_1,\beta_2) morphisms.
A morphism \alpha\colon S'\to S is said to be a strict epimorphism if it is an epimorphism and, for every morphism u\colon S'\to Z, the following necessary condition is also sufficient for u to factor as S'\to S\to Z: for every S''\in{\mathcal{C}} and every pair \beta_1,\beta_2\colon S''\to S of morphisms such that \alpha\beta_1=\alpha\beta_2, we also have that u\beta_1=u\beta_2.
If the fibre product S'\times_S S' exists, then it is equivalent to say that the diagram S \xleftarrow{\alpha} S' \overset{p_1}{\underset{p_2}{\leftleftarrows}} S'\times_S S' is exact, i.e. that S is a cokernel of the pair (p_1,p_2). In any case, a cokernel morphism is a strict epimorphism. Note also that, if a strict epimorphism is also a monomorphism, then it is an isomorphism. We leave to the reader the task of developing the dual notion of a strict monomorphism.
To make the relation between the ideas of {\mathcal{F}}-descent morphisms and strict epimorphisms more precise, we introduce the following definitions:
A morphism \alpha\colon S'\to S is said to be a universal epimorphism (resp. a strict universal epimorphism) if, for every T over S, the fibre product T'=S'\times_S T exists, and the projection T'\to T is an epimorphism (resp. a strict epimorphism).
In very nice categories (such as the category of sets, the category of sets over a topological space, abelian categories, etc.), the four notions of “epijectivity” introduced above all coincide; they are, however, all distinct in a category such as the category of preschemes, or the category of preschemes over a given non-empty prescheme S, even if we restrict to S-schemes that are finite over S.
A morphism \alpha\colon S'\to S is said to be a descent morphism (resp. a strict descent morphism) if it is an {\mathcal{F}}-descent morphism (resp. a strict {\mathcal{F}}-descent morphism) (cf. Definition 1.7), where {\mathcal{F}} denotes the fibred category over {\mathcal{C}} of objects of {\mathcal{C}} over objects of {\mathcal{C}} (cf. Example 1).
If {\mathcal{C}} has all finite products and (finite) fibre products, then there is an identity between descent morphisms in {\mathcal{C}} and strict universal epimorphisms in {\mathcal{C}}.
### 3.2.2 (b)
Let {\mathcal{C}} be the category of preschemes. Let S\in{\mathcal{C}}, and let S' and S'' be preschemes that are finite over S, i.e. that correspond to sheaves of algebras {\mathscr{A}}' and {\mathscr{A}}'' over S that are quasi-coherent (as sheaves of modules) and of finite type (i.e. coherent if S is locally Noetherian). Let \alpha\colon S'\to S be the structure morphism of S', and let \beta_1 and \beta_2 be S-morphisms from S'' to S', defined by algebra homomorphisms {\mathscr{A}}'\to{\mathscr{A}}'', which we also denote by \beta_1 and \beta_2. Using the fact that a finite morphism is closed (the first Cohen–Seidenberg theorem), we can easily prove that the diagram in {\mathcal{C}}
S \xleftarrow{\alpha} S' \overset{\beta_1}{\underset{\beta_2}{\leftleftarrows}} S'' \tag{+}
is exact if and only if the diagram of sheaves on S {\mathscr{O}}_S = {\mathscr{A}} \xrightarrow{\alpha} {\mathscr{A}}' \overset{\beta_1}{\underset{\beta_2}{\rightrightarrows}} {\mathscr{A}}'' is exact. In particular, if \alpha\colon S'\to S is a finite morphism corresponding to a sheaf {\mathscr{A}}' of algebras on S, then \alpha is a strict epimorphism if and only if the diagram of sheaves {\mathscr{O}}_S = {\mathscr{A}} \to {\mathscr{A}}' \overset{p_1}{\underset{p_2}{\rightrightarrows}} {\mathscr{A}}'\otimes_{{\mathscr{A}}}{\mathscr{A}}' is exact (it is an epimorphism if and only if {\mathscr{A}}\to{\mathscr{A}}' is injective). If S is affine of ring A, then S' is affine of ring A', with A' finite over A, and so S'\to S is a strict epimorphism if and only if A\to A' is an isomorphism from A to the subring of A' consisting of the x'\in A' such that 1_{A'}\otimes_A x' - x'\otimes_A 1_{A'} = 0 (it is an epimorphism if and only if A\to A' is injective). As we have already mentioned, even if S is the scheme of a local Artinian ring, then a finite morphism S'\to S that is an epimorphism is not necessarily a strict epimorphism. However, we can prove that, if S is a Noetherian prescheme, then every finite morphism S'\to S that is an epimorphism is the composition of a finite sequence of strict epimorphisms (also finite). This also shows that the composition of two strict epimorphisms is not necessarily a strict epimorphism.
### 3.2.3 (c)
If (+) is an exact diagram of finite morphisms, then, for every flat morphism T\to S of prescheme, the diagram induced from (+) by a change of base T\to S is again exact. It thus follows that, if X and Y are S-preschemes, with X flat over S, then the following diagram of maps (where X' and Y' are the inverse images of X and Y over S', and X'' and Y'' are their inverse images over S'') is exact: \operatorname{Hom}_S(X,Y) \to \operatorname{Hom}_{S'}(X',Y') \rightrightarrows \operatorname{Hom}_{S''}(X'',Y''). In particular, if {\mathcal{F}} denotes the fibred category (over the category {\mathcal{C}} of preschemes) such that, for X\in{\mathcal{C}}, {\mathcal{F}}_X is the category of flat X-preschemes, then the diagram (+) is {\mathcal{F}}-exact. (This result becomes false if we do not impose the flatness hypothesis; in particular, a finite strict epimorphism is not necessarily a descent morphism). We similarly see that (+) is {\mathcal{F}}-exact if {\mathcal{F}} denotes the fibred category for which {\mathcal{F}}_X is the category of flat quasi-coherent sheaves on the prescheme X (here, again, the flatness hypothesis is essential). In either case, the question of effectiveness of a gluing data (and, more specifically, of a descent data, when S''=S'\times_S S') on a flat object over S' is delicate (and its answer in many particular cases in one of the principal objects of these current exposés). The speaker does not know if, for every finite strict epimorphism S'\to S, every descent data on a flat quasi-coherent sheaf on S' is effective (even if we suppose that S is the spectrum of a local Artinian ring, and we restrict to locally free sheaves of rank 1). More generally, let A be a ring, and A' an A-algebra (where everything is commutative) such that the diagram A \to A' \rightrightarrows A'\otimes_A A' is exact, which is equivalent to saying that the corresponding morphism S'\to S between the spectra of A' and A is an {\mathcal{F}}-descent morphism, where {\mathcal{F}} is the fibred category of flat quasi-coherent sheaves. Let M' be a flat A'-module endowed with a descent data to A, i.e. with an isomorphism \varphi\colon M'\otimes_A A' \xrightarrow{\sim} A'\otimes_A M' of (A'\otimes_A A')-modules that satisfies conditions (i) and (ii) of §A.1.c (which we leave to the reader to write out in terms of modules). Is this data effective (relative to the fibred category of flat quasi-coherent sheaves)? Let M be the subset of M' consisting of the x'\in M' such that \varphi(x'\otimes_A 1_{A'}) = 1_{A'}\otimes_A x', which is a sub-A-module of M'. The canonical injection M\to M' defines a homomorphism of A'-modules M\otimes_A A'\to M'. The effectiveness of \varphi then implies the following: M is a flat A-module, and the above homomorphism is an isomorphism.
In the above, we have imposed no flatness hypotheses on the morphisms of the diagram (+), and this obliges us, in order to have a technique of descent, to impose flatness hypotheses on the objects over S and S' that we consider. In §B.2, we will impose a flatness hypothesis on \alpha\colon S'\to S, which will allow us to have a technique of descent for objects over S and S' that are no longer under any flatness hypotheses. In any case, there is a flatness hypothesis involved somewhere. This is one of the main reasons for the importance of the notion of flatness in algebraic geometry (whose role could not be visible when we restricted to base fields, over which everything, in fact, is flat!).
## 3.3 Application to étalements
Let A be a local ring, and B a local algebra over A whose maximal ideal induces that of A. We say that B is étalé over A (instead of “unramified”, as used elsewhere) if it satisfies the following conditions:
1. B is flat over A; and
2. B/{\mathfrak{m}}B is a separable finite extension of A/{\mathfrak{m}}=k (where {\mathfrak{m}} denotes the maximal ideal of A).
If A and B are Noetherian, and k is algebraically closed, then this implies that the homomorphism \overline{A}\to\overline{B} between the completions that extends A\to B is an isomorphism. A morphism f\colon T\to S of finite type is said to be étale at x\in T, or T is said to be étalé over S at x, if {\mathscr{O}}_x is étalé over {\mathscr{O}}_{f(x)}; f is said to be étale, or an étalement, or T is said to be étalé over S, if f is étale at all x\in T. Note that, if S is locally Noetherian, then the set of points of T where f is étale is open, and Zariski’s “main theorem” allows us to precisely state the structure of T/S in a neighbourhood around such a point (by an equation of well-known type).
If S is a scheme of finite type over the field of complex numbers, then there exists a corresponding analytic space \overline{S} (in the sense of Serre [24]), except for the fact that \overline{S} can have nilpotent elements in its structure sheaf, which changes nothing essential in [24]. We then easily see that f is an étalement if and only if \overline{f}\colon\overline{T}\to\overline{S} is an étalement, i.e. if every point of \overline{T} admits a neighbourhood on which \overline{f} induces an isomorphism onto an open subset of \overline{S}. In particular, to every étale covering T of S (i.e. every finite étale morphism f\colon T\to S), there is a corresponding étale covering \overline{T} of \overline{S}, which is connected if and only if T is connected [24]. We can also easily see that, if T and T' are étale schemes over S, then the natural map \operatorname{Hom}_S(T,T') \to \operatorname{Hom}_{\overline{S}}(\overline{T},\overline{T}'') is bijective, i.e. the functor T\mapsto\overline{T} from the category of étale schemes over S to the category of analytic spaces over S is “fully faithful”, and thus defines an equivalence between the first category and a subcategory of the second. A theorem of Grauert–Remmert [6] implies that, if S is normal, then we thus obtain an equivalence between the category of étale coverings of S and the category of (finite) étale coverings of S, i.e. every étale covering {\mathscr{C}} of \overline{S} is \overline{S}-isomorphic to some \overline{T}, where T is an étale covering of S. We will show that the Grauert–Remmert theorem remains true without any normality hypotheses on S. First let S'\to S be a finite strict epimorphism, and suppose that the theorem has been proven for S'; we will show that it holds for S. Let {\mathscr{C}} be an étale covering of \overline{S}, and consider its inverse image {\mathscr{C}}' over S', which corresponds to a coherent analytic sheaf {\mathfrak{A}}' of algebras on S' that is the inverse image of a sheaf of algebras {\mathfrak{A}} on \overline{S} defining {\mathscr{C}}. By hypothesis, on S', {\mathscr{C}}' comes from an étale covering T' of S', i.e. {\mathfrak{A}}' comes from a coherent sheaf of algebras {\mathscr{A}}' on S'. Also, {\mathfrak{A}}' is endowed with a canonical descent data with respect to \overline{S}'\to\overline{S}, i.e. with an isomorphism between its two inverse images on \overline{S}'\times_{\overline{S}}\overline{S}'=\overline{S'\times_SS'} (satisfying conditions (i) and (ii)), and this isomorphism comes from, by what has already been said, an isomorphism between the corresponding algebraic sheaves, i.e. from a descent data on {\mathscr{A}}' with respect to S'\to S. We can easily show that the latter is effective (since it is effective on {\mathfrak{A}}', and the effectiveness of a descent data, as described explicitly in the previous section, is something that can be checked locally on the completions of the modules that are involved). From this, we obtain a coherent sheaf of algebras {\mathscr{A}} on S that defines a covering T of S, and this is the desired covering. The above result then obviously holds true if S'\to S is just a composition of a finite number of finite strict epimorphisms, i.e. is just an arbitrary finite epimorphism (by the factorisation result stated in §A.2. It thus follows that the Grauert–Remmert theorem holds true if S is a reduced scheme, i.e. such that {\mathscr{O}}_S has no nilpotent elements, as we can see by introducing its normalisation S'. We can easily pass to the general case.
A completely analogous argument, again using the factorisation result for finite strict epimorphisms, and the “formal” nature of the effectiveness of descent data, allows us to prove the following result: let S be a locally Noetherian prescheme, and let S'\to S be a finite, surjective, and radicial morphism (or, equivalently, a morphism of finite type such that, for every T over S, the morphism T'=S'\times_S T\to T is a homeomorphism, which we can also express by saying that S'\to S is a universal homeomorphism). For every T étalé over S, consider its inverse image T'=T\times_S S', which is étalé over S'. Then the functor T\mapsto T' is an equivalence between the category of preschemes T that are étalé over S and the category of preschemes T' that are étalé over S'. (We use the bijectivity of \operatorname{Hom}_S(T_1,T_2) \to \operatorname{Hom}_{S'}(T'_1,T'_2) for preschemes T_1 and T_2 that are étalé over S, which can be proven directly without difficulty. We also use the fact that the stated theorem is true if S'=(S,{\mathscr{O}}_S/{\mathscr{J}}), where {\mathscr{J}} is a nilpotent coherent sheaf of ideals of {\mathscr{O}}_S, cf. [20]). Note also that we do not suppose here that the T in question are finite over S. This result implies, in particular, that the morphism S'\to S induces an isomorphism between the fundamental group of S' and the fundamental group of S (“topological invariance of the fundamental group of a prescheme”).
## 3.4 Relations to 1-cohomology
### 3.4.1 (a)
Let {\mathcal{C}} be a category such that the product of any two objects always exists, and let T\in{\mathcal{C}}. For every finite set I\neq\varnothing, we can consider T^I, and so we obtain a covariant functor from the category of non-empty finite sets to the category {\mathcal{C}}, i.e. what we can call a simplicial object of {\mathcal{C}}, denoted by K_T. This object depends covariantly on T; also, if u and v are morphisms T\to T', then the corresponding morphisms K_T\to K_{T} are homotopic. We say that T dominates T' if \operatorname{Hom}(T,T')\neq\varnothing, and this gives an (upward) directed preorder on {\mathcal{C}}. It follows from the above that, if T dominates T', then there exists a canonical class (up to homotopy) of homomorphisms of simplicial objects K_T\to K_{T'}; in particular, if K_T and K_{T'} are such that T and T' dominate one another, then K_T and K_{T'} are homotopically equivalent. Now let F be a (contravariant, to be clear) functor from {\mathcal{C}} to an abelian category {\mathcal{C}}'. Then C^\bullet(T,F) = F(K_T) is a cosimplicial object of {\mathcal{C}}', and thus defines, in a well-known way, a (cochain) complex in {\mathcal{C}}', of which we can take the cohomology: \operatorname{H}^\bullet(T,F) = \operatorname{H}^\bullet(C^\bullet(T,F)) = \operatorname{H}^\bullet(F(K_T)) (we may write a subscript “{\mathcal{C}}” on the \operatorname{H}^\bullet if there is any possibility for confusion). This is a cohomological functor in F, of which the variance for T varying follows from what has already been said about the K_T; more precisely, for fixed F and varying T in {\mathcal{C}} (preordered by the domination relation), the \operatorname{H}^\bullet(T,F) form an inductive system of graded objects of {\mathcal{C}}'; in particular, if T and T' are such that each one dominates the other, then \operatorname{H}^\bullet(T,F) and \operatorname{H}^\bullet(T',F) are canonically isomorphic.
Suppose that {\mathcal{C}} has all fibre products. Then we can, for fixed S\in{\mathcal{C}}, apply the above to the category {\mathcal{C}}_S of objects of {\mathcal{C}} over S; we then write C^\bullet(T/S,F) and \operatorname{H}^\bullet(T/S,F) instead of C^\bullet(T,F) and \operatorname{H}^\bullet(T,F) if we wish to make clear that we are working in the category {\mathcal{C}}_S; then C^\bullet(T/S,F) is a cochain complex in {\mathcal{C}}' that, in degree n, is equal to F(T\times_S T\times_S\ldots\times_S T) (where there are n+1 factors T).
Note that, as per usual, we can define \operatorname{H}^0(T/S,F) without assuming the category {\mathcal{C}}' to be abelian: it is the kernel (Definition 2.1), if it exists, of the pair (F(p_1),F(p_2)) of morphisms F(T) \to F(T\times_S T) corresponding to the two projections p_1,p_2\colon T\times_S T\to T. In particular, we then have the natural morphism (called the augmentation) F(S) \to \operatorname{H}^0(T/S,F) which is an isomorphism in nice cases (in particular, in the case where T\to S is a strict epimorphism and F is “left exact”). Similarly, if F takes values in the category of groups in a category {\mathcal{C}}'', then we can also define \operatorname{H}^1(T/S,F); in the case where {\mathcal{C}}'' is the category of sets (i.e. when F takes values in the category of non-necessarily-commutative groups), \operatorname{H}^1(T,F) is the quotient of the subgroup Z^1(T/S,F) of C^1(T/S,F) = F(T\times_S T) consisting of the g such that F(p_{31})(g) = F(p_{32})(g) F(p_{21})(g) by the group with operators F(T) acting on C^1(T/S,F), and thus, in particular, on the subset Z^1(T/S,F), by \rho(g')\cdot g = F(p_2)(g') g F(p_1)(g')^{-1}.
### 3.4.2 (b)
For example, let {\mathcal{F}} be a fibred category with base {\mathcal{C}}. Let \xi,\eta\in{\mathcal{F}}_S, and, for all S' over S, let F_{\xi,\eta}(S') = \operatorname{Hom}(\xi\times_S S', \eta\times_S S'). Then F_{\xi,\eta} is a contravariant functor from {\mathcal{C}}_S to the category of sets. With this setup, saying that the augmentation morphism F_{\xi,\eta}(S) \to \operatorname{H}^0(S'/S,F_{\xi,\eta}) is an isomorphism for every pair of elements \xi,\eta\in{\mathcal{F}}_S implies that \alpha\colon S'\to S is an {\mathcal{F}}-descent morphism (Definition 1.7).
### 3.4.3 (c)
Similarly, for \xi\in{\mathcal{F}}_S and any object S' of {\mathcal{C}} over G_\xi(S') = \operatorname{Aut}(\xi\times_S S'), we thus define a contravariant functor G_\xi from {\mathcal{C}}_S to the category of groups. With this setup, we claim that Z^1(S'/S,G) is canonically identified with the set of descent data on \xi'=\xi\times_S S' with respect to S'\to S (Definition 1.6), and that \operatorname{H}^1(S'/S,G) can be identified with the set of isomorphism classes of objects of {\mathcal{F}}_{S'} endowed with a descent data relative to \alpha\colon S'\to S that are isomorphic, as objects of {\mathcal{F}}_{S'}, to \xi'=\xi\times_S S'. Then, if \alpha\colon S'\to S is an {\mathcal{F}}-descent morphism (cf. §A.4.b), then \operatorname{H}^1(S'/S,G) contains as a subset the set of isomorphism classes of objects \eta of {\mathcal{F}}_S such that \eta\times_S S' is isomorphic (in {\mathcal{F}}_{S'}) to \xi\times_S S'; further, this inclusion is the identity if and only if every descent data on \xi'=\xi\times_S S' with respect to \alpha\colon S'\to S is effective. (This will be the case, in particular, if \alpha\colon S'\to S is a strict S-descent morphism).
The cochain complexes of the form C^\bullet(T/S,F) contain, as particular cases, the majority of standard known complexes (that of Čech cohomology, of group cohomology, etc.), and play an important role in algebraic geometry (notably in the “Weil cohomology” of preschemes).
### 3.4.4 (d)
Let S' be an object over S\in{\mathcal{C}}, and let \Gamma be a group of automorphisms of S' such that S' is “formally \Gamma-principal over S”, i.e. such that the natural morphism \Gamma\times S' \to S'\times_S S' (where \Gamma\times S' denotes the direct sum of \Gamma copies of S') is an isomorphism. (We suppose that all the necessary direct sums exist in {\mathcal{C}}). Let F be a contravariant functor from {\mathcal{C}} to the category of abelian groups. Then C^\bullet(S'/S,F) is canonically isomorphic to the simplicial group C^\bullet(\Gamma,F(S')) of standard homogeneous cochains, and so \operatorname{H}^\bullet(S'/S,F) is canonically isomorphic to \operatorname{H}^\bullet(\Gamma,F(S').
### 3.4.5 (e)
Let {\mathcal{C}} be the category of preschemes. We denote by \operatorname{G_a} (for “additive group”) the contravariant functor from {\mathcal{C}} to the category of abelian groups, defined by \operatorname{G_a}(X) = \operatorname{H}^0(X,{\mathscr{O}}_X). We similarly define the functor \operatorname{G_m} (for “multiplicative group”) by \operatorname{G_m}(X) = \operatorname{H}^0(X,{\mathscr{O}}_X)^\times (i.e. the group of invertible elements of the ring \operatorname{H}^0(X,{\mathscr{O}}_X)), and, more generally, the functor \operatorname{GL}(n) (for “linear group of order n”) by \operatorname{GL}(n)(X) = \operatorname{GL}(n,\operatorname{H}^0(X,{\mathscr{O}}_X)), which is a functor from {\mathcal{C}} to the category of (not-necessary-commutative, if n>1) groups; for n=1 we recover \operatorname{G_m}. We can also think of \operatorname{GL}(n) as an automorphism functor (cf. §A.4.c) by considering the fibred category {\mathcal{F}} with base {\mathcal{C}} such that {\mathcal{F}}_X is the category of locally free sheaves on X, for X\in{\mathcal{C}}, since then \operatorname{GL}(n)(X)=\operatorname{Aut}_{{\mathcal{F}}_X}({\mathscr{O}}_X^n). By §A.4.b, it follows that, if \alpha\colon S'\to S is an {\mathcal{F}}-descent morphism (cf. §A.2.c), then \operatorname{H}^1(S'/S,\operatorname{GL}(n)) contains the set of isomorphism classes of locally free sheaves on S whose inverse image on S' is isomorphic to {\mathscr{O}}_{S'}^n, and this inclusion is an equality if and only if every descent data on {\mathscr{O}}_{S'}^n (with respect to \alpha\colon S'\to S) is effective. If S is the spectrum of a local ring, then this implies that \operatorname{H}^1(S'/S,\operatorname{GL}(n))=(e), since every locally free sheaf on S is then trivial.
We note that the following conditions concerning a morphism \alpha\colon S'\to S are equivalent:
1. The augmentation homomorphism \operatorname{H}^0(S,{\mathscr{O}}_S) = \operatorname{G_a}(S)\to\operatorname{H}^0(S'/S,\operatorname{G_a}) is an isomorphism.
2. \alpha\colon S'\to S is an {\mathcal{F}}-descent morphism (where {\mathcal{F}} is the fibred category over {\mathcal{C}} described above).
3. \alpha\colon S'\to S is a strict epimorphism (cf. §A.2.c).
Now suppose that S=\operatorname{Spec}(A) and S'=\operatorname{Spec}(A'); then C^n(S'/S,\operatorname{G_a}) = C^n(A'/A,\operatorname{G_a}) = \underbrace{A'\otimes_A A'\otimes_A\ldots\otimes_A A'}_{n+1\text{ copies of }A'} with the coboundary operator C^n(A'/A,\operatorname{G_a})\to C^{n+1}(A'/A,\operatorname{G_a}) being the alternating sum of the face operators \partial_i(x_0\otimes x_1\otimes\ldots\otimes x_n) = x_0\otimes\ldots\otimes x_{i-1}\otimes1_{A'}\otimes x_i\otimes\ldots\otimes x_n. Similarly, C^n(S'/S,\operatorname{G_m})=C^n(A'/A,\operatorname{G_m}) can be identified with (\bigotimes_A^{n+1}A')^\times, with the simplicial operations for C^\bullet(A'/A,\operatorname{G_m}) being induced by those in C^\bullet(S'/S,\operatorname{G_a}). We can write down the simplicial operations for C^\bullet(A'/A,\operatorname{GL}(n)) in the same explicit manner. In all the cases known to the speaker, \operatorname{H}^i(A'/A,\operatorname{G_a})=0 for i>0, and, if A is local, then \operatorname{H}^1(A'/A,\operatorname{G_m})=0, and, more generally, \operatorname{H}^1(A'/A,\operatorname{GL}(n))=(e) (if S'\to S is an {\mathcal{F}}-descent morphisms, i.e. if the diagram A\to A'\rightrightarrows A'\otimes_A A' is exact, then compare this with §A.2.c). We note that Hilbert’s “Theorem 90” is exactly the fact that \operatorname{H}^1(S'/S,\operatorname{G_m})=0 if A is a field and A' is a finite Galois extension of A (cf. Example 1), and we can also express it by saying that, in the case in question, S'\to S is a strict descent morphisms for the fibred category of locally free sheaves of rank 1. This latter statement is the one that is most suitable to generalise Hilbert’s theorem, by varying the hypotheses both on the morphism S'\to S and on the quasi-coherent sheaves in question.
Finally, we note that, for a local Artinian A with maximal ideal {\mathfrak{m}}, and an A-algebra A' (where we denote, for any integer k>0, the ring A/{\mathfrak{m}}^{k+1} (resp. A'/{\mathfrak{m}}^{k+1}A') by A_k (resp. A'_k)), the following properties are equivalent:
1. \operatorname{H}^1(A'_k/A_k,\operatorname{G_a})=0 for all k.
2. \operatorname{H}^1(A'_k/A_k,\operatorname{G_m})=0 for all k.
3. \operatorname{H}^1(A'_k/A_k,\operatorname{GL}(n))=(e) for all k and all n.
If S'\to S is a strict epimorphism, then the above conditions imply that it is a strict descent morphism for free modules (not necessarily of finite type) over A'.
The definition of the groups \operatorname{H}^i(S'/S,\operatorname{G_m}) in the case where S (resp. S') is a scheme over the field A (resp. A') is due to Amitsur. The group \operatorname{H}^2(S'/S,\operatorname{G_m}) is particularly interesting as a “global” variant of the Brauer group, for which we can refer to [4].
## 3.5 Statement of the descent theorems
A morphism \alpha\colon S'\to S of prescheme is said to be flat if {\mathscr{O}}_{x'} is a flat module over the ring {\mathscr{O}}_{\alpha(x')} for all x'\in S' (i.e. if {\mathscr{O}}_{x'}\otimes_{{\mathscr{O}}_{\alpha(x')}}M is an exact functor in the {\mathscr{O}}_{\alpha(x')}-module M). A morphism is said to be faithfully flat if it is flat and surjective.
For example, if S=\operatorname{Spec}(A) and S'=\operatorname{Spec}(A'), then S' is flat over S if and only if A' is a flat A-module, and S' is faithfully flat over S if and only if A' is a faithfully flat A-module (i.e. if and only if A'\otimes_A M is an exact and faithful functor in the A-module M); this also implies, in the terminology of Serre [24], that the pair (A,A') is flat. If S' is faithfully flat over S, then the inverse image functor of quasi-coherent sheaves on S is exact and faithful; in other words, for a sequence of homomorphisms of quasi-coherent sheaves on S to be exact, it is necessary and sufficient that its inverse image on S' be exact (in particular, for a homomorphism of quasi-coherent sheaves on S to be a monomorphism (resp. an epimorphism, resp. an isomorphism), it is necessary and sufficient that its inverse image on S' be a monomorphism (resp. an epimorphism, resp. an isomorphism)). This property holds true if we restrict to an arbitrary open subset of S', and then characterise faithfully flat morphisms in this form.
A morphism \alpha\colon S'\to S is said to be quasi-compact if the inverse image of every quasi-compact open subset U of S is quasi-compact (i.e. a finite union of affine open subsets).
It evidently suffices to verify this property for the affine open subsets of S. For example, an affine morphism (i.e. a morphism such that the inverse image of an affine open subset is affine) is quasi-compact.
The class of flat (resp. faithfully flat, resp. quasi-compact) morphisms is stable under composition and by “base extension”, and of course contains all isomorphisms.
Let \alpha\colon S'\to S be a morphism of preschemes that is faithfully flat and quasi-compact. Then \alpha is a strict descent morphism (cf. Definition 1.7 for the fibred category {\mathcal{F}} of quasi-coherent sheaves (cf. §A, Example 2).
This statement implies two things:
1. If {\mathcal{F}} and {\mathscr{G}} are quasi-coherent sheaves on S, and {\mathcal{F}}' and {\mathscr{G}}' their inverse images on S', then the natural homomorphism \operatorname{Hom}({\mathcal{F}},{\mathscr{G}}) \to \operatorname{Hom}({\mathcal{F}}',{\mathscr{G}}') is a bijection from the left-hand side to the subgroup of the right-hand side consisting of homomorphisms {\mathcal{F}}'\to{\mathscr{G}}' that are compatible with the canonical descent data on these sheaves, i.e. whose inverse images under the two projections of S''=S'\times_S S' to S' give the same homomorphism {\mathcal{F}}''\to{\mathscr{G}}''.
2. Every quasi-coherent sheaf {\mathcal{F}}' on S' endowed with a descent data with respect to the morphism \alpha\colon S'\to S (cf. Definition 1.6 is isomorphic (endowed with this data) to the inverse image of a quasi-coherent sheaf {\mathcal{F}} on S.
Setting {\mathcal{F}}={\mathscr{O}}_S in (i), we obtain:
Let {\mathscr{G}} be a quasi-coherent sheaf on S, with {\mathscr{G}}' and {\mathscr{G}}'' denoting its inverse images on S' and S''=S'\times_S S' (respectively), and let p_1 and p_2 be the two projections from S'' to S. Then the diagram of maps of sets \Gamma({\mathscr{G}}) \xrightarrow{\alpha^*} \Gamma({\mathscr{G}}') \overset{p_1^*}{\underset{p_2^*}{\rightrightarrows}} \Gamma({\mathscr{G}}'') is exact (cf. Definition 1.1.
Also, the combination of (i) and (ii) with Definition 1.1 gives:
Let {\mathscr{G}} be as in Corollary 1. Then there is a bijective correspondence between quasi-coherent subsheaves of {\mathscr{G}} and quasi-coherent subsheaves of {\mathscr{G}}' whose inverse images on S'' under the two projections p_1 and p_2 give the same subsheaf of {\mathscr{G}}.
Of course, we have an equivalent statement in terms of quotient sheaves. As we have already seen in §A.4.e, Theorem 1 should be thought of as a generalisation of Hilbert’s “Theorem 90”, and implies, as particular cases, various formulations in terms of 1-cohomology. For the proof, we can easily reduce to the case where S=\operatorname{Spec}(A) and S'=\operatorname{Spec}(A'), and, for (i), we can easily restrict to proving Corollary 1, i.e. the exactness of the diagram M = A\otimes_A M \to A'\otimes_A M \rightrightarrows A'\otimes_A A'\otimes_A M for every A-module M, which follows from the more general lemma:
Let A' be a faithfully flat A-algebra. Then, for every A-module M, the M-augmented complex C^\bullet(A'/A,\operatorname{G_a})\otimes_A M (cf. §A.4.e) is a resolution of M.
Proof. It suffices to prove that the augmented complex induced from the above by extension of the base A to A' satisfies the same conclusions. This leads to proving the statement when we replace A by A', and A' by A'\otimes_A A', and so we can restrict to the case where there exists an A-algebra homomorphism A'\to A (or, in geometric terms, the case where S' over S admits a section). In this case, the claim follows from the generalities of §A.4.a.
We note, in passing, the following corollary, which generalises a well-known statement in Galois cohomology (compare with §A.4.e):
If A' is faithfully flat over A, then \operatorname{H}^0(A'/A,\operatorname{G_a})=A, and \operatorname{H}^i(A'/A,\operatorname{G_a})=0 for i\geqslant 1.
Proof. Part (ii). To prove part (ii) of Theorem 1, we proceed, as for (i), by restricting to the case where S' over S admits a section, where the result then follows from (i) (cf. §A.1.c).
We can evidently vary Theorem 1 and its corollaries ad libitum by introducing various additional structures on the quasi-coherent sheaves (or systems of sheaves) in question. For example, the data on S of a quasi-coherent sheaf of commutative algebras “is equivalent to” the data on S' of such a sheaf endowed with a descent data relative to \alpha\colon S'\to S. Taking into account the functorial correspondence between such quasi-coherent sheaves on S and affine preschemes over S, we obtain the second claim of the following theorem:
Let \alpha\colon S'\to S be as in Theorem 1. Then \alpha is a (non-strict, in general) descent morphism (cf. §A, Definition 2.4), and it is a strict descent morphism for the fibred category of affine schemes over preschemes (cf. §A, Definition 1.7.
The first claim of the theorem implies this: let X and Y be preschemes over S, with X' and Y' their inverse images over S, and X'' and Y'' their inverse images over S''=S'\times_S S'; then the diagram of natural maps \operatorname{Hom}_S(X,Y) \xrightarrow{\alpha^*} \operatorname{Hom}_{S'}(X',Y') \overset{p_1^*}{\underset{p_2^*}{\rightrightarrows}} \operatorname{Hom}_{S''}(X'',Y'') is exact, i.e. \alpha^* is a bijection from \operatorname{Hom}_S(X,Y) to the subset of \operatorname{Hom}_{S'}(X',Y') consisting of homomorphisms that are compatible with the canonical descent data on X' and Y' (i.e. whose inverse images under the two projections from S'' to S' are equal). This follows easily from Theorem 1 and Corollary 1, if we restrict to Y being affine over S; in the general case, we need to combine Theorem 1 with the following result:
Let \alpha\colon S'\to S be a faithfully flat and quasi-compact morphism. Then S can be identified with a topological quotient space of S', i.e. every subset U of S such that \alpha^{-1}(U) is open, is open.
To complete Theorem 2, we must give effectiveness criteria for a descent data on an S'-prescheme X' (in the case where X' is not assumed to be affine over S'). Note first of all that such a descent data is not necessarily effective, even if S is the spectrum of a field k, S' the spectrum of a quadratic extension k' of k, and S'' a proper algebraic scheme of dimension 2 over S' (as we can see, due to Serre, by using the non-projective surface of Nagata). For a descent data on X'/S' with respect to \alpha\colon S'\to S (assumed to be faithfully flat and quasi-compact) to be effective, it is necessary and sufficient that X' be a union of open subsets X'_i that are affine over S' and “stable” under the descent data on X'. This is certainly the case (for any X'/S' and any descent data on X') if the morphism \alpha\colon S'\to S is radicial (i.e. injective, and with radicial residual extensions). We can also show that this is the case if \alpha\colon S'\to S is finite, and every finite subset of X' that is contained in a fibre of X' over S is also contained in an open subset of X' that is affine over S (this is the Weil criterion). It is, in particular, the case if X'/S' is quasi-projective, and, in this case, we can show that the “descended” prescheme X/S is also quasi-projective (and projective if X'/S' is projective). In summary:
Let \alpha\colon S'\to S be faithfully flat and quasi-compact morphism of preschemes. If \alpha is radicial, then it is a strict descent morphism. If \alpha is finite, then it is a strict descent morphism with respect to the fibred category of quasi-projective (or projective) preschemes over preschemes.
I do not know if, in the second claim above, the hypothesis that \alpha be finite is indeed necessary; we can prove that, in any case, we can “formally” replace it by the following, seemingly weaker, hypothesis: for every point of S there exists an open neighbourhood U, a finite and faithfully flat U' over U, and an S-morphism from U' to S'. A type of case that is not covered by the above is that where S=\operatorname{Spec}(A) and S'=\operatorname{Spec}(\overline{A}), with A a local Noetherian ring and \overline{A} its completion; or even that where S' is quasi-finite over S (i.e. locally isomorphic to an open subset of a finite S-scheme) but not finite. In these two cases, the speaker also does not know the answer to the following question: let X be an S-scheme such that X'=X\times_S S' is projective over S'; is it then true that X is projective over S?
[Comp.] A morphism S'\to S that is quasi-finite, étale, surjective, or a morphism of the form \operatorname{Spec}(\overline{A})\to\operatorname{Spec}(A), is not always a strict descent morphism, even if A is the local ring of an algebraic curve over an algebraically closed field k and S=\operatorname{Spec}(A). We can thus find a proper simple morphism f\colon X\to S that makes X into a principal E-bundle over S, with E an elliptic curve, such that f'\colon X'\to S' is projective, but f is not projective. So this is also an example of a homogeneous principal bundle that is non-isotrivial under an abelian scheme.
## 3.6 Application to the descent of certain properties of morphisms
Let P be a class of morphisms of preschemes. Let \alpha\colon S\to S' be a morphism of preschemes, and let f\colon X\to Y be a morphism of S-preschemes, with f'\colon X'\to Y' the inverse image of f under \alpha. We can then ask if the relation “f'\in P” implies that “f\in P”. It appears that the answer is affirmative in many important cases, if we suppose that \alpha is faithfully flat and quasi-compact (this latter hypothesis being sometimes overly strong). We can see this directly without difficulty if P is the class of surjective (resp. radicial) morphisms (with these two cases following from the surjectivity of \alpha), or flat (resp. faithfully flat, resp. simple) morphisms (with these three cases following from the faithful flatness of \alpha), or morphisms of finite type. Using Theorem 1, Theorem 2, and Lemma 1.2, we see that it is also true if P is one of the following classes: isomorphisms, open immersions, closed immersions, immersions (if f is of finite type, and Y is locally Noetherian), affine morphisms, finite morphisms, quasi-finite morphisms, open morphisms, closed morphisms, homeomorphisms, separated morphisms, or proper morphisms. The only important case not covered here is that of projective or quasi-projective morphisms, which has already been discussed in the remark in §B.1.
## 3.7 Decent by finite faithfully flat morphisms
Let \alpha\colon S'\to S be a finite morphism, corresponding to a sheaf of algebras {\mathscr{A}}' on S that is locally free and of finite type as a sheaf of modules, and everywhere non-zero. Then \alpha is a faithfully flat and quasi-compact morphism, to which we can thus apply the above results. The data of a quasi-coherent sheaf {\mathcal{F}}' on S' is equivalent to the data of the quasi-coherent sheaf \alpha_*({\mathcal{F}}') on S endowed with its {\mathscr{A}}'-modules structure (noting that {\mathscr{A}}'=\alpha_*({\mathscr{O}}_{S'})). For simplicity, we also denote this sheaf on S by {\mathcal{F}}'. The two inverse images p_i^*({\mathcal{F}}') of {\mathcal{F}}' on S'\times_S S' similarly correspond to the quasi-coherent sheaves of ({\mathscr{A}}'\otimes_{{\mathscr{O}}_S}{\mathscr{A}}')-modules {\mathcal{F}}'\otimes_{{\mathscr{O}}_S}{\mathscr{A}}' and {\mathscr{A}}'\otimes_{{\mathscr{O}}_S}{\mathcal{F}}'. The data of an (S'\times_S S')-homomorphism from the former to the latter is equivalent to the data of a homomorphism of ({\mathscr{A}}'\otimes{\mathscr{A}}')-modules, and, taking into account the fact that {\mathscr{A}}' is locally free, this is equivalent to the data of a homomorphism of ({\mathscr{A}}'\otimes{\mathscr{A}}')-modules: {\mathscr{U}} = \underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathscr{A}}',{\mathscr{A}}') = {\mathscr{A}}'\otimes\check{{\mathscr{A}}}' \to \underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathcal{F}}',{\mathcal{F}}') i.e. to the data, for every section \xi of {\mathscr{U}} over an open subset V, of a homomorphism of {\mathscr{O}}_S-modules T_\xi\colon{\mathcal{F}}'|V\to{\mathcal{F}}'|V that satisfies the conditions
\begin{aligned} T_{f\xi}(x) &= fT_\xi(x), \\T_{\xi f}(x) &= T_\xi(fx), \end{aligned} \tag{3.1}
where f and x are (respectively) sections of {\mathscr{A}}' and {\mathcal{F}}' over an open subset of S that is contained inside V. Conditions (i) and (ii) of a descent data (cf. §A.1.c) can then be written (respectively) as
T_{1_U}(x) = x, \qquad\text{i.e. }T_{1_U}=\operatorname{id}_{{\mathcal{F}}'} \tag{3.2}
T_{\xi\eta} = T_\xi T_\eta. \tag{3.3}
In other words, a descent data on {\mathcal{F}}' is equivalent to a representation of the sheaf {\mathscr{U}}=\underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathscr{A}}',{\mathscr{A}}') of {\mathscr{O}}_S-algebras in the sheaf \underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathcal{F}}',{\mathcal{F}}') of {\mathscr{O}}_S-algebras that satisfies the two linearity conditions (3.1). If we have a pairing of quasi-coherent sheaves on S': {\mathcal{F}}'_1\times{\mathcal{F}}'_2 \to {\mathcal{F}}'_3 (that we can think of as a pairing of sheaves of {\mathscr{A}}'-modules on S), and gluing data on the {\mathcal{F}}'_i defined by homomorphisms T_i\colon{\mathscr{U}}\to\underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathcal{F}}'_i,{\mathcal{F}}'_i) (for i=1,2,3), then these data are equivalent to the given pairing, in the evident meaning of the phrase, if and only if the following condition is satisfied:
For every section \xi of {\mathscr{U}} over an open subset, and denoting by \Delta\xi=\sum\xi'_i\otimes_{{\mathscr{A}}'}\xi''_i the section of {\mathscr{U}}\otimes_{{\mathscr{A}}'}{\mathscr{U}} (where {\mathscr{U}} is considered as an {\mathscr{A}}'-module with its left structure) defined by the formula \xi\cdot(fg) = \sum_i\xi'_i(f)\xi''_i(g) (where f and g are sections of {\mathscr{A}}' over a smaller open subset), we have the formula
T_\xi^{(3)}(x\cdot y) = \sum_i T_{\xi'_i}^{(1)}x\cdot T_{\xi''_i}^{(2)}y \tag{3.4}
for every pair (x,y) of sections of {\mathscr{A}}' over a smaller subset. (We can express this property by saying that the homomorphisms T^{(i)} are compatible with the diagonal map of {\mathscr{U}}, with respect to the given pair). In particular, equations (3.1) to (3.4) allow us to understand, in terms of representations of algebras via diagonal maps, the descent data on a quasi-coherent sheaf of algebras on S', and thus also (by restricting to commutative algebras) the descent data on an affine S'-scheme.
From here, we obtain an analogous interpretation of descent data on an arbitrary S'-prescheme X': the data of such an X' is equivalent to the data of a prescheme X' over S endowed with a homomorphism of {\mathscr{O}}_S-algebras {\mathscr{A}}'\to{\mathscr{O}}_{X'}, and a descent data on X' is equivalent to the data of a sheaf homomorphism {\mathscr{U}} \to \underline{\operatorname{Hom}}_{h^{-1}({\mathscr{O}}_S)}({\mathscr{O}}_{X'},{\mathscr{O}}_{X'}) that is compatible with the morphism h\colon X'\to S' and that satisfies the conditions analogous to equations (3.1) to (3.4) above.
(Weil). Suppose that S'/S is a Galois étale covering with Galois group \Gamma (cf. and ). Then a descent data on a quasi-coherent sheaf {\mathcal{F}}' on S' (resp. on an S'-prescheme X') is equivalent to the data of a representation of \Gamma by automorphisms of (S',{\mathcal{F}}') (resp. of (S',X')) that is compatible with the action of \Gamma on S'. This result is “formal”, i.e. it can be proven in terms of categories, but, from the point of view of this section, we also obtain the explicit structure of {\mathscr{U}} (endowed with its ring structure, the ring homomorphism {\mathscr{A}}'\to{\mathscr{U}}, and the diagonal map), which is completely known thanks to the following result: {\mathscr{U}} admits, as a left A'-module, a basis given by the sections of {\mathscr{U}} that correspond to elements of \Gamma.
(Cartier). Let p be a prime number, and suppose that p{\mathscr{O}}_S=0 (i.e. that {\mathscr{O}}_S is of characteristic p), that ({\mathscr{A}}')^p\subset{\mathscr{O}}_S={\mathscr{A}} (i.e. that S'/S is radicial of height 1), and that the sheaf of algebras {\mathscr{A}}' over {\mathscr{A}} locally admits a p-basis (i.e. a family (x_i) of sections such that {\mathscr{A}}' is generated as an algebra by the x_i under the sole condition that x_i^p=0). We suppose that the set of the i is finite, of cardinality n. Let {\mathfrak{C}} be the sheaf of A-derivations of A', which is a locally free sheaf of rank n of A'-modules, and, furthermore, a sheaf of Lie p-algebras over {\mathscr{A}} (but not over {\mathscr{A}}') that satisfies the following condition:
[X,fY] = X(f)Y + f[X,Y]. \tag{3.5}
{\mathscr{U}}=\underline{\operatorname{Hom}}_{{\mathscr{O}}_S}({\mathscr{A}}',{\mathscr{A}}') is generated, as an {\mathscr{O}}_S-algebra endowed with an algebra homomorphism {\mathscr{A}}'\to{\mathscr{U}}, by the sub-left-A'-module {\mathfrak{C}}, with the following additional relations:
\begin{cases} Xf-fX &= X(f) \\XY-YX &= [X,Y] \\X^p &= X^{(p)}. \end{cases} \tag{3.6}
It follows from the above lemma that a descent data on the quasi-coherent sheaf {\mathcal{F}}' on S' is equivalent to the data, for all X\in{\mathfrak{C}}, of an {\mathscr{O}}_S-endomorphism \overline{X} of {\mathcal{F}}' that satisfies the following conditions:
\overline{fX} = f\overline{X} \tag{3.7}
\overline{X}(fx) = X(f)x + f\overline{X}(x) \tag{3.8}
\overline{[X,Y]} = [\overline{X},\overline{Y}] \tag{3.9}
\overline{X^{(p)}} = \overline{X}^p. \tag{3.10}
(This is what we may call a linear connection on {\mathcal{F}}, which is further flat and compatible with the p-th powers). We can similarly write down the notion of a descent data on an S'-prescheme X', with equation (3.4) being replaced by the condition that the \overline{X} are derivations of {\mathscr{O}}_{X'}. Since the morphism S'\to S is radicial, Theorem 3 ensures that every such descent data is effective, and thus defines an S-prescheme X.
Note that we have not needed to impose any flatness, non-singular, or finiteness hypotheses on {\mathcal{F}}' or X'.
## 3.8 Application to rationality criteria
Let X be an S-prescheme such that the direct image of {\mathscr{O}}_X on S is {\mathscr{O}}_S; this property remains true for any flat base extension S'\to S. If {\mathcal{F}} is an invertible sheaf (i.e. locally free of rank 1) on X, then there is a bijective correspondence between automorphisms of {\mathcal{F}} (identified with the invertible sections of {\mathscr{O}}_X) and invertible sections of {\mathscr{O}}_S. So let s be a section of X over S; we define a section of {\mathcal{F}} over s to be a section of the invertible sheaf s^*({\mathcal{F}}) on S. It follows from the above that, if {\mathcal{F}}_i (for i=1,2) are invertible sheaves on X, each endowed with a section over s, and if {\mathcal{F}}_1 and {\mathcal{F}}_2 are isomorphic, then there exists exactly one isomorphism from {\mathcal{F}}_1 to {\mathcal{F}}_2 that is compatible with the sections in question (i.e. sending the first to the second). We also, independently of the section s, regard two invertible sheaves {\mathcal{F}}_1 and {\mathcal{F}}_2 on X as equivalent if every point of S has an open neighbourhood U such that the restrictions of {\mathcal{F}}_1 and {\mathcal{F}}_2 to X|U are isomorphic. Then every invertible sheaf {\mathcal{F}} on X is equivalent to an invertible sheaf {\mathcal{F}}_1 endowed with a marked section over s (we take {\mathcal{F}}_1=Fs^*({\mathcal{F}})^{-1}), and {\mathcal{F}}_1 is determined up to isomorphism. In other words, the classification up to equivalence of invertible sheaves on X is the same as the classification up to isomorphism of invertible sheaves endowed with a marked section.
Since these properties remain true under flat extensions \alpha\colon S'\to S of the base (by replacing the section s with its inverse image s' under \alpha), we thus conclude, taking Theorem 1 into account:
With the prescheme X/S being as above, and admitting a section s, let \alpha\colon S'\to S be a faithfully flat and quasi-compact morphism; let {\mathcal{F}}' be an invertible sheaf on X'=X\times_S S'. For {\mathcal{F}}' to be equivalent to the inverse image on X' of an invertible sheaf {\mathcal{F}}' on X, it is necessary and sufficient for its inverse images p_1^*({\mathcal{F}}') and p_2^*({\mathcal{F}}') on X'\times_X X'=X\times_S(S'\times_S S') to be equivalent. If this is the case, then {\mathcal{F}} is determined up to equivalence. (We then say that {\mathcal{F}}' is rational on S).
Considering this principle in the case where \alpha\colon S'\to S is as in Example 1 and Example 2 in the previous section, we recover the rationality criteria of Weil and of Cartier. (We note that the authors restrict to the case where S and S' are spectra of fields; a fortiori, S is then the spectrum of a local ring, and the equivalence relation introduced above is exactly the relation of being isomorphic). The the first case, {\mathcal{F}}' is rational on S if and only if its images under \Gamma are all equivalent to {\mathcal{F}}'. To express the rationality criterion in the second case, we consider, more generally, the diagonal morphism X'\to X''=X'\times_X X' of X'/X, with the corresponding sheaf of ideals {\mathscr{I}} on X'\times_X X', and the sheaf {\mathscr{I}}/{\mathscr{I}}^2, which can be identified with its inverse image \Omega_{X'/X}^1 on X (the sheaf of 1-differentials of X' with respect to X). Since the restrictions of the {\mathcal{F}}''_i=p_i({\mathcal{F}}') (for i=1,2) to the diagonal are isomorphic (since they are both isomorphic to {\mathcal{F}}'), i.e. {\mathcal{F}}''_1({\mathcal{F}}''_2)^{-1}={\mathcal{F}}'' has a restriction to the diagonal which is trivial, it follows that the restriction of {\mathcal{F}}'' to (X'',{\mathscr{O}}_{X''}/{\mathscr{I}}^2) is given, up to isomorphism, by a well-defined element \xi of \operatorname{H}^1(X'',{\mathscr{I}}/{\mathscr{I}}^2) = \operatorname{H}^1(X',\Omega_{X'/X}^1). Also, being precise, we have \Omega_{X'/X}^1=\Omega_{S'/S}^1\otimes_{{\mathscr{O}}_S}{\mathscr{O}}_X, and thus, if \Omega_{S'/S}^1 is locally free on S (as in the Cartier case), then \xi defines a section of \operatorname{R}^1f'({\mathscr{O}}_{X'})\otimes\Omega_{S'/S}^1 on S' (called the Atiyah–Cartier class of the invertible sheaf {\mathcal{F}} on X'/S) whose vanishing is necessary and sufficient for the inverse images of {\mathcal{F}}' under the two projections of (X'',{\mathscr{O}}_{X''}/{\mathscr{I}}^2) = X\times_S(S'',{\mathscr{O}}_{S''}/{\mathscr{J}}^2) to X' to be equivalent (where {\mathscr{J}} is the sheaf of ideals on S''=S'\times_S S' defined by the diagonal morphism S'\to S'\times_S S'). This vanishing is thus trivially necessary for the inverse images of {\mathcal{F}}' on X''=X\times_S S'' itself to be equivalent, and thus also for {\mathcal{F}} to be equivalent to the inverse image of an invertible sheaf {\mathcal{F}} on X. The Atiyah–Cartier class can also be understood as the obstruction to the existence, locally over S', of a connection of {\mathcal{F}}' relative to the derivations of X'/X, with such a connection further being determined when we know the derivations of {\mathcal{F}}' corresponding to the natural extensions of derivations of S'/S to X'. From this, and the results of the previous section, we easily conclude that, in the case of the aforementioned Example 2, and when X/S admits a section, the vanishing of the Atiyah–Cartier class is also sufficient for {\mathcal{F}}' to be rational on S.
## 3.9 Application to the restriction of the base scheme to an abelian scheme
Let S be a prescheme. Well define an abelian scheme over S to be a simple proper scheme X over S whose fibres at the points x\in S are schemes of abelian varieties over the \mathfrak{K}(x). Suppose that S is Noetherian and regular (i.e. that its local rings are regular), then we can show, using the connection theorem of Murre [20] (at least in the case “of equal characteristics”, where the cited theorem is currently proven) that every rational section of X over S is everywhere defined (i.e. is a section) (which generalises a classical theorem of Weil). It then follows, more generally, that, if X' is a simple scheme over S, then every rational S-map from X' to X is everywhere defined. From this, we obtain the following, which generalises a result of Chow–Lang: with S Noetherian and regular, and K denoting its ring of rational functions (a direct sum of fields), let X be an abelian scheme over K; if X is isomorphic to a K-scheme of the form X_0\times_S\operatorname{Spec}(K), where X_0 is an abelian scheme over S, then X_0 is determined up to unique isomorphism.
Using the above uniqueness result, we see that the question of restriction of the base to X is local on S (and thus that it suffices to know how to do the restriction to \operatorname{Spec}({\mathscr{O}}_x), where x\in S). In the same way, we see that, if S'\to S is a simple morphism of finite type, if Y' is the ring of rational functions of S', and if X\otimes_K K' is of the form X'_0\times_{S'}\operatorname{Spec}(K'), then X'_0 is endowed with a canonical descent data with respect to \alpha. Taking Theorem 3 into account, we thus conclude:
Let S be an irreducible regular Noetherian prescheme, with field of rational functions Y, let K' be a finite extension of K that is unramified over S, let S' be the normalisation of S in K' (which is thus an étale cover of S), and let X be an abelian scheme over K such that X\otimes_K K' is of the form X'_0\times_{S'}\operatorname{Spec}(K'), where X'_0 is a projective abelian scheme over S'. Then X is of the form X_0\times_S\operatorname{Spec}(K), where X_0 is a projective abelian scheme over S.
The speaker does not know if we can replace the hypothesis that S'\to S be a surjective étale cover (which allows us to apply Theorem 3) with the hypothesis that it is instead a simple and surjective morphism of finite type (not even if we suppose that it is an étalement), or if the proposition still holds true without supposing that X'_0 is projective over S' (a condition which could be automatically satisfied).
## 3.10 Application to local triviality and isotriviality criteria
Let S be a prescheme, G a “prescheme of groups” over S, P a prescheme over S on which “G acts” (on the right). We say that P is formally principal homogeneous under G if the well-known morphism G\times_S P \to P\times_S P (induced from the actions of G on P) is an isomorphism. From now on, we assume G to be flat over S (and thus faithfully flat over S), and we reserve the name of principal homogeneous bundle under G for a formally principal homogeneous bundle P that is faithfully flat and quasi-compact over S. It is immediate that this is equivalent to being able to find a faithfully flat and quasi-compact extension S'\to S of the base S such that the formally principal homogeneous bundle P'=P\times_S S' under G'=G\times_S S' is trivial, i.e. isomorphic to G' (i.e. admitting a section); we can take, in particular, S'=P. Note also that, if S is locally Noetherian, then the faithfully-flat hypothesis on P is equivalent to the hypothesis that \overline{P}_S=P\times_S\operatorname{Spec}(\overline{{\mathscr{O}}}_s) be faithfully flat over \overline{{\mathscr{O}}}_s for all s\in S (where \overline{{\mathscr{O}}}_s denotes the completion of the local ring {\mathscr{O}}_s), as follows from the fact that \overline{{\mathscr{O}}}_s is faithfully flat over {\mathscr{O}}_s. Also, if P is of finite type over S, and S is locally Noetherian, then the set of points s satisfying the above condition is constructible, and so, if S is a “Jacobson prescheme” (for example, a scheme of finite type over a field, or, more generally, over a Jacobson ring), then it suffices to verify the condition in question for the closed points of S. This leads us to the case where the base is the spectrum of a complete local ring A. If S=\operatorname{Spec}(A) (with A a complete Noetherian local ring), and if P is of finite type over S, then the faithful flatness of P/S is also equivalent to the existence of an S' that is finite and flat over S such that P' is trivial, and, if, further, G is simple over S, then we can suppose S' to be étale over S. Then, if, further, the residue field of A is algebraically closed (the “geometric case”), then P is faithfully flat over A if and only if it is trivial. Thus, if S is an algebraic prescheme over an algebraically closed field, and if G is simple and of finite type over S, then we see that the faithfully-flat condition on S is equivalent to the condition of being analytically trivial (SLF) of Serre [25].
We can consider other, stronger, types of conditions on P, that have a “local triviality” nature. In particular, we say that P is isotrivial (resp. strictly isotrivial) if, for all s\in S, there exists an open neighbourhood U of S, and a finite and faithfully flat morphism (resp. a surjective étale covering) U'\to U such that P'=P\times_S U' is trivial. (We stray from the terminology of Serre [4], which uses “locally isotrivial” for what we call “strictly isotrivial”). Strict isotriviality is mainly useful if G is simple over S, but is, however, an inadequate notion in other cases.
If G is affine over S, then every principal homogeneous bundle P under G is affine, by §B.2, whence the possibility, thanks to Theorem 2, to “descend” from such bundles by faithfully flat and quasi-compact morphisms. Taking, in particular, G=\operatorname{GL}(n)_S, defined by the condition that the functor S'\mapsto\operatorname{Hom}(S',G) of S-preschemes (with values in the category of groups) can be identified with the functor \operatorname{GL}(n)(S')=\operatorname{GL}(n,\operatorname{H}^0(S',{\mathscr{O}}_{S'})) described in . Using the facts
1. that every principal homogeneous bundle under G (resp. every locally free sheaf of rank n on S) becomes isomorphic to the “trivial” object G (resp. {\mathscr{O}}_S^n) under a suitable faithfully flat and quasi-compact extension of S;
2. that we can descend the type of objects in question (principal homogeneous bundles under G, resp. locally free sheaves of rank n) by such morphisms; and, finally
3. that the automorphism group of the trivial bundle on any S'/S is functorially isomorphic to the automorphism group of the trivial locally free sheaf of rank n on S',
we “formally” conclude that it is “equivalent” to give, on S (or on some S'/S) a principal homogeneous bundle of group G, or to give a locally free sheaf of rank n. (More precisely, we have an equivalence of fibred categories). We thus conclude, in particular:
Every principal homogeneous bundle under the group \operatorname{GL}(n)_S is locally trivial.
By known arguments, we thus conclude the same result for others structure groups such as \operatorname{SL}(n)_S, \operatorname{Sp}(n)_S, and products of such groups. We thus also conclude that, if F is a closed subgroup of G=\operatorname{GL}(n)_S that is flat over S, and such that the quotient G/F exists, and such that G is an isotrivial (resp. strictly isotrivial) principal homogeneous bundle on G/F, of structure group F\times_S(G/F), then every principal homogeneous bundle of structure group F is isotrivial (resp. strictly isotrivial). This applies to all the “linear groups” on S that have been used up until now, and, in particular, to the case where G=S\times_k\Gamma, with S a prescheme over the field k, and \Gamma a linear group (in the classical sense) over k (and thus in particular simple). This thus answers, for such groups, a question of Serre (loc. cit.).
We also point out that, for most groups (linear or not) that are simple over S that we know of, and certainly for all those of the form S\times_k\Gamma as above, we can show that every isotrivial principal homogeneous bundle is strictly isotrivial, which answers, in particular, another question of Serre (loc. cit. 1–14), taking into account the fact that a homogeneous principal bundle obtained by a descent à la Cartier (cf.Example 2) is obviously isotrivial.
One of the essential difficulties in these questions (setting aside the question of the existence of quotient schemes) is the lack of effectiveness criteria for a descent data along a faithfully flat non-finite morphism.
### References
[4] J. Dieudonné, A. Grothendieck. “Eléments de géométrie algébrique.” Publications Mathématiques de L’Institut Des Hautes Etudes Scientifiques. (n.d.).
[6] H. Grauert, R. Remmert. “Komplexe Räume.” Math. Annalen. 136 (1958), 245–318.
[9] A. Grothendieck. “Géométrie formelle et géométrie algébrique.” Séminaire Bourbaki. 11 (n.d.), Talk no. 182.
[20] J.P. Murre. “On a connectedness theorem for a birational transformation at a simple point.” Amer. J. Math. 80 (1958), 3–15.
[24] J.-P. Serre. “Géométrie algébrique et géométrie analytique,” in: Ann. Institut Fourier Grenoble, 1956: pp. 1–42.
[25] J.-P. Serre. “Espaces fibrés algébriques.” Séminaire Chevalley. 3 (1958), Talk no. 1.
1. [Comp.] It now seems excessive to say that the technique of descent is “at the base of the majority of existence theorems in algebraic geometry”. This is true to a large extent for the non-projective techniques that are the object of study of the first two exposés of this current series (i.e. “Techniques of descent and existence theorems in algebraic geometry”), but not for the projective techniques (exposés IV, V, and VI).↩︎
2. [Comp.] It is useless to assume here that \alpha is an {\mathcal{F}}-descent morphism.↩︎ | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9893351793289185, "perplexity": 1154.2085357022147}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363292.82/warc/CC-MAIN-20211206103243-20211206133243-00060.warc.gz"} |
http://en.wikipedia.org/wiki/Creation_and_annihilation_operators | # Creation and annihilation operators
Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems.[1] An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.
Creation and annihilation operators can act on states of various types of particles. For example, in quantum chemistry and many-body theory the creation and annihilation operators often act on electron states. They can also refer specifically to the ladder operators for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system (similarly for the lowering operator). They can be used to represent phonons.
The mathematics for the creation and annihilation operators for bosons is the same as for the ladder operators of the quantum harmonic oscillator.[2] For example, the commutator of the creation and annihilation operators that are associated with the same boson state equals one, while all other commutators vanish. However, for fermions the mathematics is different, involving anticommutators instead of commutators.[3]
## Ladder operators for quantum harmonic oscillator
In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties.
First consider the simpler bosonic case of the phonons of the quantum harmonic oscillator.
Start with the Schrödinger equation for the one-dimensional time independent quantum harmonic oscillator
$\left(-\frac{\hbar^2}{2m} \frac{d^2}{d x^2} + \frac{1}{2}m \omega^2 x^2\right) \psi(x) = E \psi(x)$
Make a coordinate substitution to nondimensionalize the differential equation
$x \ = \ \sqrt{ \frac{\hbar}{m \omega}} q$.
and the Schrödinger equation for the oscillator becomes
$\frac{\hbar \omega}{2} \left(-\frac{d^2}{d q^2} + q^2 \right) \psi(q) = E \psi(q)$.
Note that the quantity $\hbar \omega = h \nu$ is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as
$-\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + \frac {d}{dq}q - q \frac {d}{dq}$
The last two terms can be simplified by considering their effect on an arbitrary differentiable function f(q),
$\left(\frac{d}{dq} q- q \frac{d}{dq} \right)f(q) = \frac{d}{dq}(q f(q)) - q \frac{df(q)}{dq} = f(q)$
which implies,
$\frac{d}{dq} q- q \frac{d}{dq} = 1$
Therefore
$-\frac{d^2}{dq^2} + q^2 = \left(-\frac{d}{dq}+q \right) \left(\frac{d}{dq}+ q \right) + 1$
and the Schrödinger equation for the oscillator becomes, with substitution of the above and rearrangement of the factor of 1/2,
$\hbar \omega \left[\frac{1}{\sqrt{2}} \left(-\frac{d}{dq}+q \right)\frac{1}{\sqrt{2}} \left(\frac{d}{dq}+ q \right) + \frac{1}{2} \right] \psi(q) = E \psi(q)$.
If we define
$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$ as the "creation operator" or the "raising operator" and
$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \frac{d}{dq} + q\right)$ as the "annihilation operator" or the "lowering operator"
then the Schrödinger equation for the oscillator becomes
$\hbar \omega \left( a^\dagger a + \frac{1}{2} \right) \psi(q) = E \psi(q)$
This is significantly simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.
Letting $p = - i \frac{d}{dq}$, where "p" is the nondimensionalized momentum operator then we have
$[q, p] = i \,$
and
$a = \frac{1}{\sqrt{2}}(q + i p) = \frac{1}{\sqrt{2}}\left( q + \frac{d}{dq}\right)$
$a^\dagger = \frac{1}{\sqrt{2}}(q - i p) = \frac{1}{\sqrt{2}}\left( q - \frac{d}{dq}\right)$.
Note that these imply that
$[a, a^\dagger ] = \frac{1}{2} [ q + ip , q-i p] = \frac{1}{2} ([q,-ip] + [ip, q]) = \frac{-i}{2} ([q, p] + [q, p]) = 1$
in contrast to the so-called "normal operators" of mathematics, which have a similar representation (e.g. $A= W_1 + i\, W_2)\,,$ with self-adjoint $W_i\,.$ But in the case of normal operators one would be dealing with commuting $W_i\,,$ i.e. with $W_1W_2=W_2W_1\,,$ so that the 1 at the extreme r.h.s. of the previous equation would be replaced by 0, which would have the consequence of one-and-the-same set of eigenfunctions (and/or eigendistributions) for both $W_1$ and $W_2$, whereas here common eigenfunctions or eigendistributions of the operators p and q don't exist.
Thus, although in the present case one is explicitly dealing with non-normal operators, by the commutation relation given above, the Hamiltonian operator can be expressed as
$\hat H = \hbar \omega \left( a \, a^\dagger - \frac{1}{2}\right).$
$\hat H = \hbar \omega \left( a^\dagger \, a + \frac{1}{2}\right).$
And $a$ and $a^\dagger\,,$ operators give the following commutation relations with the Hamiltonian[4]
$[\hat H, a ] = -\hbar \omega \, a.$
$[\hat H, a^\dagger ] = \hbar \omega \, a^\dagger .$
These relations can be used to find the energy eigenstates of the quantum harmonic oscillator. Assuming that $\psi_n$ is an eigenstate of the Hamiltonian $\hat H \psi_n = E_n\, \psi_n$. Using these commutation relations it can be shown that[4]
$\hat H\, a\psi_n = (E_n - \hbar \omega)\, a\psi_n .$
$\hat H\, a^\dagger\psi_n = (E_n + \hbar \omega)\, a^\dagger\psi_n .$
This shows that $a\psi_n$ and $a^\dagger\psi_n$ are also eigenstates of the Hamiltonian with eigenvalues $E_n - \hbar \omega$ and $E_n + \hbar \omega$. This identifies the operators $a$ and $a^\dagger$ as lowering and rising operators between the eigenstates. Energy difference between two eigenstates is $\Delta E = \hbar \omega$.
The ground state can be found by assuming that the lowering operator will collapse it, $a\, \psi_0 = 0$. And then using the Hamiltonian in terms of rising and lowering operators,
$a^\dagger a \psi_0 = 0 = \left(\frac{\hat H}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0 = \left(\frac{E_0}{\hbar \omega} - \frac{1}{2} \right) \,\psi_0.$
the wave-function on the right is non-zero, thus term in brackets must be. This gives the ground state energy $E_0 = \hbar \omega /2$. This allows to identify the energy eigenvalue of any eigenstate $\psi_n$ as[4]
$E_n = \left(n + \frac{1}{2}\right)\hbar \omega$
Furthermore it can be shown that the first-mentioned operator, the number operator $N=a^\dagger a\,,$ plays a most-important role in applications, while the second one, $a \,a^\dagger\,,$ can simply be replaced by $N +1\,.$ So one simply gets
$\hbar\omega \,\left(N+\frac{1}{2}\right)\,\psi (q) =E\,\psi (q)$.
### Applications
The ground state $\ \psi_0(q)$ of the quantum harmonic oscillator can be found by imposing the condition that
$a \ \psi_0(q) = 0$.
Written out as a differential equation, the wavefunction satisfies
$q \psi_0 + \frac{d\psi_0}{dq} = 0$
which has the solution
$\psi_0(q) = C \exp\left(-{q^2 \over 2}\right).$
The normalization constant C can be found to be $1\over \sqrt[4]{\pi}$ from $\int_{-\infty}^\infty \psi_0^* \psi_0 \,dq = 1$, using the Gaussian integral.
### Matrix representation
The matrix counterparts of the creation and annihilation operators obtained from the quantum harmonic oscillator model are
${a}^{\dagger}=\begin{pmatrix} 0 & 0 & 0 & \dots & 0 &\dots \\ \sqrt{1} & 0 & 0 & \dots & 0 & \dots\\ 0 & \sqrt{2} & 0 & \dots & 0 & \dots\\ 0 & 0 & \sqrt{3} & \dots & 0 & \dots\\ \vdots & \vdots & \vdots & \ddots & \vdots & \dots\\ 0 & 0 & 0 & 0 & \sqrt{n} &\dots & \\ \vdots & \vdots & \vdots & \vdots & \vdots &\ddots \end{pmatrix}$
${a}=\begin{pmatrix} 0 & \sqrt{1} & 0 & 0 & \dots & 0 & \dots \\ 0 & 0 & \sqrt{2} & 0 & \dots & 0 & \dots \\ 0 & 0 & 0 & \sqrt{3} & \dots & 0 & \dots \\ 0 & 0 & 0 & 0 & \ddots & \vdots & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots & \sqrt{n} & \dots \\ 0 & 0 & 0 & 0 & \dots & 0 & \ddots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{pmatrix}$
Substituting backwards, the laddering operators are recovered. They can be obtained via the relationships $a^\dagger_{ij} = \langle\psi_i | {a}^\dagger | \psi_j\rangle$ and $a_{ij} = \langle\psi_i | {a} | \psi_j\rangle$. The wavefunctions are those of the quantum harmonic oscillator, and are sometimes called the "number basis".
### Mathematical details
The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract form of the operators satisfy the properties below.
Let H be the one-particle Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f) for any f in H. The operator a(f) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a(f) which is linear in H.
For a boson,
$[a(f),a(g)]=[a^\dagger(f),a^\dagger(g)]=0$
$[a(f),a^\dagger(g)]=\langle f|g \rangle$,
where we are using bra–ket notation.
For a fermion, the anticommutators are
$\{a(f),a(g)\}=\{a^\dagger(f),a^\dagger(g)\}=0$
$\{a(f),a^\dagger(g)\}=\langle f|g \rangle$.
Physically speaking, a(f) removes (i.e. annihilates) a particle in the state | f$\scriptstyle \rangle$ whereas a(f) creates a particle in the state | f$\scriptstyle \rangle$.
The free field vacuum state is the state with no particles. In other words,
$a(f)|0\rangle=0$
where | 0 $\scriptstyle \rangle$ is the vacuum state.
If | f$\scriptstyle \rangle$ is normalized so that $\scriptstyle \langle$f | f$\scriptstyle \rangle$ = 1, then a(f) a(f) gives the number of particles in the state | f$\scriptstyle \rangle$.
### Creation and annihilation operators for reaction-diffusion equations
The annihilation and creation operator description has also been useful to analyze classical reaction diffusion equations, such as the situation when a gas of molecules A diffuse and interact on contact, forming an inert product: A + A → ∅ . To see how this kind of reaction can be described by the annihilation and creation operator formalism, consider $n_{i}$ particles at a site $i$ on a 1-d lattice. Each particle diffuses independently, so that the probability that one of them leaves the site for short times $dt$ is proportional to $n_{i}dt$, say $\alpha n_{i}dt$ to hop left and $\alpha n_{i}dt$ to hop right. All $n$ particles will stay put with a probability $1-2\alpha n_{i}dt$.
We can now describe the occupation of particles on the lattice as a `ket' of the form | n1, n2, ...$\scriptstyle \rangle$. A slight modification of the annihilation and creation operators is needed so that
$a|n\rangle= \sqrt{n} \ |n-1\rangle$
and
$a^{\dagger}|n\rangle= \sqrt{n+1} \ |n+1\rangle$.
This modification preserves the commutation relation
$[a,a^{\dagger}]=1$,
but allows us to write the pure diffusive behaviour of the particles as
$\partial_{t}|\psi\rangle=-\alpha\sum(2a_{i}^{\dagger}a_{i}-a_{i-1}^{\dagger}a_{i}-a_{i+1}^{\dagger}a_{i})|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle$
The reaction term can be deduced by noting that $n$ particles can interact in $n(n-1)$ different ways, so that the probability that a pair annihilates is $\lambda n(n-1)dt$ and the probability that no pair annihilates is $1-\lambda n(n-1)dt$ leaving us with a term
$\lambda\sum(a_{i}a_{i}-a_{i}^{\dagger}a_{i}^{\dagger}a_{i}a_{i})$
yielding
$\partial_{t}|\psi\rangle=-\alpha\sum(a_{i}^{\dagger}-a_{i-1}^{\dagger})(a_{i}-a_{i-1})|\psi\rangle+\lambda\sum(a_{i}^{2}-a_{i}^{\dagger 2}a_{i}^{2})|\psi\rangle$
Other kinds of interactions can be included in a similar manner.
This kind of notation allows the use of quantum field theoretic techniques to be used in the analysis of reaction diffusion systems.
## Creation and annihilation operators in quantum field theories
In quantum field theories and many-body problems one works with creation and annihilation operators of quantum states, $a^\dagger_i$ and $a^{\,}_i$. These operators change the eigenvalues of the number operator,
$N = \sum_i n_i = \sum_i a^\dagger_i a^{\,}_i$,
by one, in analogy to the harmonic oscillator. The indices (such as $i$) represent quantum numbers that label the single-particle states of the system; hence, they are not necessarily single numbers. For example, a tuple of quantum numbers $(n, l, m, s)$ is used to label states in the hydrogen atom.
The commutation relations of creation and annihilation operators in a multiple-boson system are,
$[a^{\,}_i, a^\dagger_j] \equiv a^{\,}_i a^\dagger_j - a^\dagger_ja^{\,}_i = \delta_{i j},$
$[a^\dagger_i, a^\dagger_j] = [a^{\,}_i, a^{\,}_j] = 0,$
where $[\ \ , \ \ ]$ is the commutator and $\delta_{i j}$ is the Kronecker delta.
For fermions, the commutator is replaced by the anticommutator $\{\ \ , \ \ \}$,
$\{a^{\,}_i, a^\dagger_j\} \equiv a^{\,}_i a^\dagger_j +a^\dagger_j a^{\,}_i = \delta_{i j},$
$\{a^\dagger_i, a^\dagger_j\} = \{a^{\,}_i, a^{\,}_j\} = 0.$
Therefore, exchanging disjoint (i.e. $i \ne j$) operators in a product of creation of annihilation operators will reverse the sign in fermion systems, but not in boson systems. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 103, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9376770257949829, "perplexity": 305.1372757393073}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119646849.7/warc/CC-MAIN-20141024030046-00021-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://hackage.haskell.org/package/load-font-0.1.0.3/docs/Graphics-UI-Font-Load.html | Contents
Synopsis
# Motivation
The goal of this library is to provide a simple way of giving an application access to a font that isn't available system-wide allowing you to bundle fonts with your executable ensuring uniform look-and-feel across platforms. It has been tested on Windows 10, Arch Linux and OSX Sierra
# Quick Start
While this package ships with a fully working demo the basic usage is lost in all of the GUI code. And frankly the demo takes a while to build and pulls in a GUI library dependency so it's not worth it unless you really want to convince yourself the library works. To get started with the minimum fuss thrill at the screenshot, take my word for it and do the following:
• Add your fonts to the Cabal file's data-files stanza (completely optional, you can load fonts from anywhere on the file system but highly recommended for portability):
data-files:
fonts/*.ttf
• Add the load-font dependency:
executable myAwesomeApp
...
build-depends:
...
...
...
• Somewhere in your app code before you use the font :
import Paths_myAwesomeApp
...
myAwesomeFunction = do
...
fontPath <- getDataFileName "fonts/my-awesome-font.ttf"
...
# The API
Make a font located at some FilePath available to your application. The font is automatically cleared from the font database with the process exits.
On Linux this uses FcConfigAppFontAddFile under the hood and so assumes X11/Xft are available. It should work fine on modern Linux systems but will break with old Xlib legacy fonts.
Currently the error case just returns a pretty uninformative message because underlying calls on Linux and Windows which are out of my control only return 0 or 1 in case of failure or success. Given this it would make more sense that the return type should be 'Maybe ()' but 'Either String ()' has two advantages 1. It makes errors easier to collect when batch loading 2. OSX has a much nicer error which I plan to expose in the future
Remove a private font located at FilePath from the application.
For the most part you shouldn't need this function because private fonts are automatically unloaded when the process exits but it's available in case you're doing something more exotic like switching between two versions of the same font.
On Windows and OSX this works as you would expect. But on Linux the only available function is FcConfigAppFontClear which ignores the FilePath and removes all private fonts. This adheres violently to the Principle Of Greatest Surprise and in the future I will transparently reload the other fonts but for now, caveat computer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2589350640773773, "perplexity": 2498.395316222518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991258.68/warc/CC-MAIN-20210517150020-20210517180020-00070.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/58422-linearly-independent-vectors-print.html | # Linearly Independent Vectors
• Nov 8th 2008, 07:01 PM
horacejerry
Linearly Independent Vectors
Let a system of vectors $\bold{v}_{1},\bold{v}_{2}, \ldots, \bold{v}_{r}$ be linearly independent but not generating. Show that it is possible to find a vector $\bold{v}_{r+1}$ such that the system $\bold{v}_{1}, \bold{v}_{2}, \ldots, \bold{v}_{r}, \bold{v}_{r+1}$ is linearly independent.
So let $\bold{v}_{r+1}$ be any vector that cannot be represented as a linear combination $\sum_{k=1}^{r} \alpha_{k} \bold{v}_{k}$. So $\bold{v}_{1}, \bold{v}_{2}, \ldots, \bold{v}_{r}, \bold{v}_{r+1}$ does not form a basis for some vector space $V$.
To show linear independence, we have to show that $\sum_{i=1}^{k+1} \alpha_{k} \bold{v}_{k} = \bold{0}$ with $\alpha_{k} = 0$.
• Nov 9th 2008, 05:14 AM
PaulRS
$
\sum\limits_{k = 1}^{n + 1} {\alpha _k \cdot \bold{v_k} } = \bold{0}
$
By absurd, assume that there is such a linear combination with $
\sum {\left| {\alpha _k } \right|} > 0
$
(1) (that is, not all the scalars are 0). We have that $
\alpha _{n + 1} = 0
$
is impossible by the hypothesis ( the others are LI and we have (1) ), so $
\alpha _{n + 1} \ne 0
$
, and therefore $
\bold{v_{n + 1}} = \sum\limits_{k = 1}^n {\left( -{\tfrac{{\alpha _k }}
{{\alpha _{n + 1} }}} \right) \cdot \bold{v_k} }
$
that is $
\left[ {\bold{v_1} ,...,\bold{v_n} } \right]
$
spans $\bold{v_{n+1}}$ ABSURD! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 16, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9981339573860168, "perplexity": 239.75854458273906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541322.19/warc/CC-MAIN-20161202170901-00030-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://alexmalins.com/publications/applicability-of-kd-for-modelling-dissolved-137cs-concentrations-in-fukushima-river-water-case-study-of-the-upstream-ota-river/ | ## Applicability of Kd for modelling dissolved 137Cs concentrations in Fukushima river water: Case study of the upstream Ota River
Published:
### Abstract
A study is presented on the applicability of the distribution coefficient (Kd) absorption/desorption model to simulate dissolved 137Cs concentrations in Fukushima river water. The upstream Ota River basin was simulated using GEneral-purpose Terrestrial Fluid-flow Simulator (GETFLOWS) for the period 1 January 2014 to 31 December 2015. Good agreement was obtained between the simulations and observations on water and suspended sediment fluxes, and on particulate bound 137Cs concentrations under both base and high flow conditions. By contrast the measured concentrations of dissolved 137Cs in the river water were much harder to reproduce with the simulations. By tuning the Kd values for large particles, it was possible to reproduce the mean dissolved 137Cs concentrations during base flow periods (observation: 0.32 Bq/L, simulation: 0.36 Bq/L). However neither the seasonal variability in the base flow dissolved 137Cs concentrations (0.14-0.53 Bq/L), nor the peaks in concentration that occurred during storms (0.18–0.88 Bq/L, mean: 0.55 Bq/L), could be reproduced with realistic simulation parameters. These discrepancies may be explained by microbial action and leaching from organic matter in forest litter providing an additional input of dissolved 137Cs to rivers, particularly over summer, and limitations of the Kd absorption/desorption model. It is recommended that future studies investigate these issues in order to improve simulations of dissolved 137Cs concentrations in Fukushima rivers.
K. Sakuma, H. Tsuji, S. Hayashi, H. Funaki, A. Malins, K. Yoshimura, H. Kurikami, A. Kitamura, K. Iijima, M. Hosomi
Journal of Environmental Radioactivity 184-185, 53-62 (2018) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8782788515090942, "perplexity": 10145.321237091855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153934.85/warc/CC-MAIN-20210730060435-20210730090435-00197.warc.gz"} |
https://mathematica.stackexchange.com/questions/164504/simplify-fraction-of-gamma-functions | # Simplify fraction of Gamma functions
From evaluating products of Wigner 6-j symbols I end up with expressions containing fractions of Gamma functions. For example,
$\sqrt{\frac{\Gamma (2 N)}{\Gamma (2 N+3)}},$
where $N$ is a positive integer. This expression can be simplified to
$\frac{1}{2 \sqrt{N (N+1) (2 N+1)}}.$
However, when executing
FullSimplify[Sqrt[Gamma[2*N]/Gamma[3 + 2*N]],{N >= 1, N \[Element] Integers}]
the unaltered expression containing the Gamma functions is returned. Funny enough, running FullSimplify on the inverse, that is,
FullSimplify[(Sqrt[Gamma[2*N]/Gamma[3 + 2*N]])^(-1),{N >= 1, N \[Element] Integers}]
does return
2*Sqrt[NN*(1 + NN)*(1 + 2*NN)]
However, as my expressions are more complicated than the example given here, simplifying the reciprocal is not an option for me. How can I make mathematica simplify this fractions of factorials?
P.S. I tried the rule suggested in this answer, but had no success.
• Try: FullSimplify[ Sqrt[Gamma[2*n]/Gamma[3 + 2*n]] // FunctionExpand, {n >= 1, n ∈ Integers}] Jan 25, 2018 at 13:35
• dont use capital N for your own symbol by the way. Jan 25, 2018 at 17:08
## 1 Answer
Use FunctionExpand:
FunctionExpand[Sqrt@(Gamma[2 N]/Gamma[3 + 2 N])]
(* 1/2 Sqrt[1/(N (1 + N) (1 + 2 N))] *)
Edited:
Simplify[Simplify[Sqrt[Gamma[2 n]]/Sqrt[Gamma[3 + 2 n]],
{n >= 1, n ∈ Integers}] // FunctionExpand, {n >= 1, n ∈ Integers}]
$$\frac{1}{2 \sqrt{n \left(2 n^2+3 n+1\right)}}$$
• Thanks, that'a a nice function to remember. Do you known how to get the same result when the square root is in the nominator and denominator, e.g., FunctionExpand[Sqrt[Gamma[2*NN]]/Sqrt[Gamma[3 + 2*NN]]] does not simplify.
– Paul
Jan 25, 2018 at 14:56
• Found it here
– Paul
Jan 25, 2018 at 15:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6738706231117249, "perplexity": 2580.03991191345}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662606992.69/warc/CC-MAIN-20220526131456-20220526161456-00606.warc.gz"} |
http://mathoverflow.net/users/6716/bill-dubuque | # Bill Dubuque
less info
reputation
1422
bio website location age member for 3 years, 6 months seen Jul 7 at 0:57 profile views 2,888
20 Direct proof of irrationality? 15 Slick ways to make annoying verifications 15 A condition that implies commutativity 14 Interesting applications (in pure mathematics) of first-year calculus 14 Assistance with understanding parent/child relationships in Pythagorean Triples
# 2,038 Reputation
+30 Direct proof of irrationality? +10 How would you solve this tantalizing Halmos problem? +10 A condition that implies commutativity +10 What are the reasons for considering rings without identity?
# 2 Questions
40 How would you solve this tantalizing Halmos problem? 14 Why does the algebraic condition of flatness on the structure sheaves give a good definition of family?
# 50 Tags
45 nt.number-theory × 7 15 ac.commutative-algebra × 11 37 ra.rings-and-algebras × 8 15 tag-removed × 3 26 ca.analysis-and-odes × 2 15 gr.group-theory × 2 24 ho.history-overview × 9 14 big-list × 9 16 commutative-rings × 2 14 geometry
# 11 Accounts
MathOverflow 2,038 rep 1422 Skeptics 248 rep 15 Area 51 151 rep 2 TeX - LaTeX 116 rep 1 Meta Stack Overflow 101 rep 1 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9039605855941772, "perplexity": 8220.174386041776}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386164989714/warc/CC-MAIN-20131204134949-00028-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/59298/making-a-stand-alone-image-with-gastex | # Making a stand-alone image with gastex
I want to do the following:
1. Make an image using gastex.
2. Compile the image to a stand-alone pdf image. I.e., at the end of this stage I'll have a myImage.pdf file.
3. Include my image in a large latex file.
The idea is that I like working with gastex but I want to compile my latex file with pdflatex. (I know there is a new feature of compiling gastex with pdflatex but I can't make it work so I prefer going this way.)
The problem with step (2) is that I don't know what should be the documentclass I should use. E.g., using \documentclass{article} produces a A4 page with the image on it. I want the file to contain only the image. Also, maybe there a way to do this with \documentclass{article} and with playing with the page sizes?
For some reason standalone doesn't work for me. This code passes latex but dvipdf crashes on it:
\documentclass{standalone}
\usepackage{gastex}
\begin{document}
\begin{picture}(100,10)(0,0)
\setlength{\unitlength}{3.5pt}
\gasset{Nw=5,Nh=5} %nWidth nHeight
\node[Nmarks=i,iangle=90](A1)(5,0){$s_1$}
\node(A2)(15,0){$s_2$}
\end{picture}
\end{document}
-
A first attempt should be with the standalone class. Can you give a simple example of a picture file? – egreg Jun 10 '12 at 10:41
You can't use dvipdfm with gastex, only latex+dvips+ps2pdf – egreg Jun 10 '12 at 11:13
I usually compile with latex+dvipdf and it works. Using latex+dvips+ps2pdf doesn't work on this code as well. (Fails in the last step) – Guy Jun 10 '12 at 11:14
If the problems with standalone persist, you could always use article (don't forget \pagestyle{empty}) and postprocess with pdfcrop. – Stephan Lehmke Jun 10 '12 at 11:25
This is a good solution. Do you want to post it as an answer so you could get credit? – Guy Jun 10 '12 at 13:30
With this input file guy.tex
\documentclass{standalone}
\usepackage{gastex}
\begin{document}
\begin{picture}(200,200)(-100,-100)
\setlength{\unitlength}{3.5pt}
\gasset{Nw=5,Nh=5} %nWidth nHeight
\node[Nmarks=i,iangle=90](A1)(5,0){$s_1$}
\node(A2)(15,0){$s_2$}
\end{picture}
\end{document}
and the command line
simpdftex latex guy && pdfcrop guy.pdf
(simpdftex just uses latex+dvips+ps2pdf) I get the following guy-crop.pdf
With the command line
latex guy && dvipdf guy && pdfcrop guy.pdf
the output is similar (but wrong):
Notice the arrow tip. The (200,200)(-100,100) coordinates for picture are just for being sure the bounding box will contain the picture, so pdfcrop will find the right boundaries. It seems that gastex doesn't advertise the bounding box of what it's producing.
However, the page http://www.lsv.ens-cachan.fr/~gastin/gastex/index.php tells that version 2.9b6 (not on TeX Live) works with auto-pst-pdf and so is "compatible" with pdflatex.
-
simpdftex latex guy is failing for me. I get Error: /undefined in !BP, GPL Ghostscript 8.71: Unrecoverable error, exit code 1, and a stack trace. Can I supply any other information that might help? – Guy Jun 10 '12 at 13:23
@Guy Operating system and TeX distribution? – egreg Jun 10 '12 at 14:38
@Guy It doesn't work with standalone in TeX Live 2009 and 2010, unfortunately. Probably the article method is the way to go with such an old system. – egreg Jun 10 '12 at 14:46
I'm using Ubuntu. And: pdfTeX 3.1415926-1.40.10-2.2 (TeX Live 2009/Debian). So, it shouldn't work... – Guy Jun 10 '12 at 16:48
@Guy: The v0.x standalone uses preview by default which seems cause the issue. The v1.x crop option however works. I heavily recommend you to install TeX Live yourself. See How to install “vanilla” TeXLive on Debian or Ubuntu? – Martin Scharrer Jun 10 '12 at 19:45
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http://tehom-blog.blogspot.com/2010/03/cords-and-emacs_25.html | ## Why cords aren't used in emacs and how they could be
### Cords and emacs
#### What are cords?
You can think of cords as efficient strings. Specifically, "A number of operations on long cords are much more efficient than their strings.h counterpart. In particular, concatenation takes constant time independent of the length of the arguments." 1
Cords are sometimes called ropes, as in this implementation. Internally, they look something like this:
#### Why can't they be used for emacs?
Of course the first thought that occurs to any emacs-o-phile is, if it's so efficient, why isn't something like that used in emacs? There are editors that use cords; Hans Boehm wrote one to show off the power of cords.
But emacs makes heavier demands on its text representation than that little proof-of-concept editor. What's missing? What can't cords do that emacs needs? Off the top of my head, just these things:
• Text properties.
• Overlays, but I'll group them together with text properties.
• Markers
This got me thinking, and I decided to solve these problems, at least on paper. I'm sure of course that there are other problems lurking that I don't see.
### Markers
Markers, for those not familiar with the inner workings of emacs, are like positions in a buffer except that they move with the text. They are not fixed numbers.
ISTM markers in cords could be implemented as a variant of the empty string node. Outside code would hold references to them. Since they must have an identity and be permanently empty, some shortcuts aren't possible.
This treatment of markers has side benefits:
• Markers automatically stay in their right position. There is no need to manage them when text elsewhere in the buffer in inserted or removed.
• Markers stay in their given order. That is, even if two markers are at the same position, one always comes first, and that order never changes. For a coder, this is a much better situation than trying to control markers that might be at the same location. 2
• In theory, a marker could be garbage-collected when no outside code knows about it. Since these markers don't have to be actively managed (as above), markers that aren't known outside the cord tree are only known by their parents.
Their parents could hold them by something akin to weak references. A finalization operation could replace a marker by an empty string node, which takes less space.
A few functions such as buffer-has-markers-at' would disappear, but they are just used for marker housekeeping, so the need for them would disappear at the same time.
#### Setting markers
Setting markers becomes essentially the same as inserting text: At the appropriate point in the cord tree, concatenate the old subtree and a new object. But now the new object is a marker object.
#### Inserting text at markers
Inserting text at these special leaves is straightforward: The marker is replaced by a concatenation of string and itself.
Savvy emacs users will immediately point out that that's one of two types of insertion operation. Emacs has another, as used for the marker insertion type nil'. That just reverses the order: The marker is replaced by a concatenation of itself and string.
But with this it wouldn't make any sense to write:
(goto-char my-marker)
(insert "My text")
That would not distinguish markers. We'd be back in the old situation of managing marker-order by hand.
Instead, we'd like an insert-like operation that takes an explicit marker argument and acts with respect to it. Call it something like insert*'. So you would write instead:
(insert* my-marker "My text")
##### Quick insertion
Now, that will work but it will make an awkward structure. If characters are inserted one-by-one, the tree will quickly degenerate into a list of single characters.
We don't want that. Plus we would like inserting at a marker to be as much like inserting at point as possible, and that has to be efficient.
So marker could have a variant expression as:
• (All its data described elsewhere)
• A fixed-size array of characters
• A last-used position in that buffer
• In fact, two, because we can handle "before" and "after" insertions with the same buffer by letting it grow both from the top and from the bottom.
Small text insertions would go into that buffer. Later on when the small insertions had finished, the marker would be turned into a concatenation of marker and text. This would be done:
• When the little buffer was about to overflow.
• When a non-character is about to be inserted at it. Eg another marker or a "lazy" cord.
• When a marker was removed in garbage collection, being held only by weak references. Of course this would transform it into the text, not into an empty string as above.
• When the marker hadn't been used for insertion for a while.3
• When explicitly commanded to.
#### Point as a marker
The above would work reasonably well for point too, and for mark. But point has an additional requirement: It needs to can move around the buffer efficiently, ideally without changing the buffer cord at all.
So point cannot be just a marker. At least, it needs to also have a "phantom" aspect that can visually move around the buffer without really changing the buffer. This is essentially what emacs does under the hood now.
So ISTM point and mark must have these states:
• "Numerical" state: It is associated to a numerical position.
When insertion or deletion is done anywhere in the buffer, point must not be in numerical state (otherwise it might get moved with respect to the text). So sometimes numerical state is automatically promoted to full position state.
• "Full position" state: It is associated a position within a cord.
This is not susceptible to being moved by edits elsewhere.
• "Marker" state: It is associated to a marker node, possibly one created just for this purpose.
To insert or delete at point, point must be in marker state.
If the marker is removed (ie by killing text) marker state reverts to full position state.
##### Point vs the ordered-markers rule
(All the following discussion applies equally well to mark)
Point and mark also violate the ordered-markers rule. Since point moves around, it has no constant ordering wrt other markers.
I doubt that it makes sense to try to put point before or after certain markers based on what used to be markers' insertion type.
• Normally place the new marker after any existing markers at that point.
• There'd be an alternative to place it before existing markers, along the lines of insert-before-markers'.
• And for more fine-grained control, point could be associated to an existing marker. Perhaps when goto-char' gets a marker argument.
But that doesn't work well. We said point had to switch from numerical state to full position state when an insertion (etc) occured anywhere in the buffer. So point might have invisibly, unpredictably already been placed before/after existing markers. That would make for erratic behavior.
So instead the before/after bit would be decided when point moves. It'd be something like:
• Numerical state includes a new flag, whether we are before/after markers. Defaults to "after".
• goto-char' or a variant of it can explicitly set that before/after state.
• The transformation from numerical state to full position or marker state consults that flag.
• Again, there are operations to associate an existing marker to point for fine-grained control.
• What when a marker is inserted at point?
Now what happens when a marker is inserted at the same position as point? As before, we don't want to create an unpredictable situation and point may have invisibly gone into full position state. Is there a potential for point's placement to depend on whether this (invisibly) happens?
ISTM no. The situations are these:
• If the marker is normally inserted, point is effectively the marker argument to insert*. So point is already in marker state.
• The marker is inserted at another marker. This can only create ambiguity if the new marker is where point would go, ie either after the last marker and point will go after markers, or the symmetric situation before markers. Assume it's the first.
• Say marker M is inserted first, then point goes into full-position state. Point will go after M.
• Say point goes into full-position state first, then marker M is inserted. So marker M is inserted after another marker, which places it before point.
• So in either case, point goes after M.
• So in every case, there is no unpredictable behavior in this regard.
#### The kill ring
When text is copied or killed to the kill-ring, it'd usually be bad for later insertion at a marker to alter the string that's in the kill-ring. Even worse would be if the text was subsequently copied into another buffer and that completely unrelated buffer was surprisingly altered. This hazard applies not just when text is killed, but any time a string is gotten from a buffer, by buffer-substring, copy-region-as-kill etc.
Yet markers in bare strings might sometimes be useful:
• For moving active text from one buffer to another.
• For sharing active text among several buffers.
• For editing strings not attached to a buffer, especially when the editing operations are complex. One could generate a buffer just for the string and add appropriate markers, and this is generally what we do now. But it drags in the entire buffer mechanism just to edit a string.
##### My first idea, which was bad
My first idea to support this was to take a cue from Xemacs' extents and distinguish whether a marker is duplicable'. duplicable' markers could be captured in strings, others could not. A marker would always be unique' in the Xemacs extent sense, and would default to non-duplicable.
##### Another approach
But I prefer another approach. It seems to me that the useful operations above all "want" to exert control over the copying operation itself. So we could distinguish different modes of buffer-string/buffer-substring operations:
• A mode that copies markers
• A mode that removes them from the copied text (pseudo-finalizing them along the lines described above)
• And possibly smarter modes that treat markers in a more fine-grained way according to the markers' properties.
### Overlays, extents, and text properties
#### The problem
Overlays, extents, and text properties pose a different problem. Cords don't naturally have text properties. If we had to change the text of a cord to add properties to it, we'd lose much of the benefit of cords. So we want a way of annotating swathes of text without altering the text itself.
#### A solution for one is nearly a solution for all
Overlays, extents, and text properties are similar. AUIU Xemacs' treatment of them essentially merges them. There their predefined properties are the same, and the difference seems to be that an extent is itself an object to be held while text properties are part of a string or buffer. Since a solution for any one is basically a solution for all, I will focus on overlays.
#### First idea: A variant of substring node
My first idea was there would be a variant substring node that would have properties. Those properties would be interpreted as the properties of the subtree that it points to. Think of it as a substring node that "tints" the larger string or cord that it points to. I'm going to call it a PASS (property-adding substring node) for short.
##### Problems
But it has problems.
• The biggie: Cord substring nodes want edits to be made above the substring, not in its subtree. As soon as any edits are made, the simplicity of our scheme falls to pieces.
• Priority is capricious. It implies that for a PASS Early contained in a PASS Late, that Late always has higher priority than Early.
• This forbids crossed scopes. Usually they're not wanted, but sometimes they're needed, eg for transient mark mode.
• Cord substrings want to point at constant text. But we need to can place PASSs around arbitrary text, and that can be a cord.
This one may not be serious; we could simply loosen what we let substring nodes point to. They may not even need additional place information, since what's underneath them doesn't change and what's above them just uses their indexing as an offset.
#### Second idea: Explore to the text leaves
##### Exploration
We need to consider potentially many overlays at each position. So we need to look at every applicable PASS node. But that means that we may have to descend pretty far down the tree to get all the properties that might be in force. We can explore more efficiently than that.
Worse, it doesn't work. We usually encounter a string before we encounter the PASS node that it was added on top of.
##### Handles crossed scopes
Once multiple priorities can be accepted, crossed scopes are handled naturally by this scheme. Nothing forces a PASS to refer only to a subtree that is entirely scoped beneath it, and subtrees can be shared.
#### Third idea: Other nodes link to PASSes
So we can't do all that we need with PASS nodes. But we want properties to behave like a contiguous stretch of text, and PASS has the advantage that it really is contiguous stretch of text. So I'll mostly base my plan on PASS nodes.
So let's use:
• PASS nodes
• String nodes that also have:
• a pointer to a PASS node
• an index into it (used to explore deeper)
• Markers that have (in addition to what I wrote above):
• Pointer and index to a PASS node before it
• Pointer and index to a PASS node after it (possibly the same as the before node)
Editing operations keep them in sync. Usually strings point to the PASS node below it that was applicable when it was created.
##### To explore PASS nodes
PASS nodes themselves can't keep a single PASS pointer, or even a fixed number. They might contain arbitrarily many sub-regions with different properties. So when dealing with them, we adjust the index and explore that position recursively.
##### Shortcut
We can make some optimizations:
We can let PASS node pointers be null, and treat such as an indication that nothing underneath has properties.
Also, each PASS node could indicate whether there is any higher priority PASS underneath it. If not, we could examine it and only descend further if it failed to provide what we seek.
##### Inserting vanilla text
Generally when we insert text, we want the text to inherit properties from it. Now, this is not what emacs does now. Its behavior is actually inconsistent: single characters inherit text properties from around them, inserted strings do not. IMO it is cleaner and more convenient to let all insertions (by default) inherit text properties from around them. The plan above does that.
But sometimes we really want to insert text with new properties or no properties. We can do this by either:
• (for the vanilla case) Setting a null PASS pointer or
• (For the non-vanilla case) Over the whole insertion, placing a PASS node that indicates that there is no higher priority PASS underneath it.
#### Kill ring
What happens when a PASS is copied into the kill ring? Much like with markers, we probably want to support variant copies:
• Remove all properties. We could replace PASSs with vanilla substrings but it's probably simpler to place a "shortcut" PASS over the whole thing.
• Keep properties. Ie, do nothing special. This can cause duplication in different buffers.
• Treat PASSs individually depending on duplicable' and `unique' properties, as borrowed from XEmacs.
#### How to be overlays
In this scheme, an overlay is just a PASS that something outside the cord refers to.
### Conclusion
So it seems to me cords are feasible in emacs. I can confidently say that the obvious design problems can be solved. But as I said earlier, I'm sure of course that there are other problems lurking that I don't see.
## Footnotes:
1 From cord.h by Hans Boehm
2 I have done some coding with ewoc and to extend ewoc. For this, I don't recommend trying to control which of several markers comes first at all. It can be done, but it's a royal mess. Instead, I recommend removing all markers that are only separated by empty text and adding them just when they will be separated by non-empty text. But that's just a work-around. The fix would be for markers to automatically stay in their respective order.
3 This makes me wonder if it would be worthwhile to have "mess compaction", in analogy to garbage collection. Instead of removing and finalizing an object, "mess compaction" would transform it into a more compact form, idiosyncratically by object type. More on this later. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.35630688071250916, "perplexity": 2049.047279030898}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818687484.46/warc/CC-MAIN-20170920213425-20170920233425-00595.warc.gz"} |
http://pdglive.lbl.gov/Particle.action?init=0&node=S091&home=MXXX049 | BOTTOM, CHARMED MESONS($\boldsymbol B$ = $\boldsymbol C$ = $\pm{}$1) ${{\mathit B}_{{c}}^{+}}$ = ${\mathit {\mathit c}}$ ${\mathit {\overline{\mathit b}}}$, ${{\mathit B}_{{c}}^{-}}$ = ${\mathit {\overline{\mathit c}}}$ ${\mathit {\mathit b}}$, similarly for ${{\mathit B}_{{c}}^{*}}$'s INSPIRE search
# ${{\boldsymbol B}_{{c}}^{+}}$ $I(J^P)$ = $0(0^{-})$ I, J, P need confirmation.
Quantum numbers shown are quark-model predictions.
${{\mathit B}_{{c}}^{+}}$ MASS $6274.9 \pm0.8$ MeV
${{\mathit B}_{{c}}^{+}}$ MEAN LIFE $(0.510 \pm0.009) \times 10^{-12}$ s
POLARIZATION IN ${{\boldsymbol B}_{{c}}^{+}}$ DECAY
$\Gamma _{L}/\Gamma$ in ${{\mathit B}_{{c}}^{+}}$ $\rightarrow$ ${{\mathit J / \psi}}{{\mathit D}_{{s}}^{*+}}$ $0.54 \pm0.15$
${{\mathit B}_{{c}}^{-}}$ modes are charge conjugates of the modes below. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9986485838890076, "perplexity": 2168.571555262518}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371609067.62/warc/CC-MAIN-20200405181743-20200405212243-00315.warc.gz"} |
http://bkms.kms.or.kr/journal/view.html?doi=10.4134/BKMS.2015.52.6.1963 | - Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
Mapping preserving numerical range of operator products on ${C}^*$-algebras Bull. Korean Math. Soc. 2015 Vol. 52, No. 6, 1963-1971 https://doi.org/10.4134/BKMS.2015.52.6.1963Published online November 30, 2015 Mohamed Mabrouk Faculty of Sciences Abstract : Let $\ala$ and $\alb$ be two unital $C^*$-algebras. Denote by $W(a)$ the numerical range of an element $a\in\ala$. We show that the condition $W(ax)=W(bx), \forall x\in\ala$ implies that $a=b$. Using this, among other results, it is proved that if $\phi: \ala\rightarrow\alb$ is a surjective map such that $W(\phi(a)\phi(b)\phi(c)) = W(abc)$ for all $a, b$ and $c \in\ala$, then $\phi(1)\in Z(B)$ and the map $\psi=\phi(1)^2\phi$ is multiplicative. Keywords : $C^*$-algebras, numerical range, preserving the numerical range MSC numbers : 15A86, 46L05, 47A12, 47B49 Full-Text : | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6812858581542969, "perplexity": 780.8916152646902}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575751.84/warc/CC-MAIN-20190922221623-20190923003623-00245.warc.gz"} |
http://aetherwavetheory.blogspot.cz/2009/07/awt-and-metamaterial-character-of.html | ## Tuesday, July 21, 2009
### AWT and metamaterial character of vacuum
The modeling of vacuum by light spreading through material environment isn't completely new here. For example the recent experimental work demonstrated by sending of ultrashort pulses into foamy structure of optical fibers the blue-shifting of light at a white-hole horizon. Recently whole area of physics named transformation optics was established on analogy of physics of vacuum in gravity field to spreading of waves in media of variable refraction index (which was one of Einstein's "refractive approaches" to gravitational light bending and general relativity, by the way).
Metamaterial character of vacuum was proposed before two years and recent publication described the way, how to model structures like gravitational lensing, strange attractors, streaks of dark matter, photon sphere or event horizons of black holes by infrared waves spreading through porous GaInAsP metamaterial sponge. In context of existing theories these analogies are rather ad-hoced, but they've deep meaning in context of AWT, which describes vacuum as a dense system of particles, composed of nested fluctuations, which are having structure of fractal sponge or foam. Therefore the metamaterial nature of vacuum belongs between significant predictions of AWT.
The understanding the role of foamy structure of vacuum fluctuations (as manifested by CMB radiation, soliton character of gamma bursts or ZP energy) in metamaterial character of vacuum is quite easy, if we consider Aether concept. In inhomogeneous environment so called Rayleigh dispersion occurs, whenever the positive surface curvature of density fluctuations prevails. In such system the waves are dispersed (absorbed and refracted) the more, the shorter is their wavelength, because short waves cannot avoid obstacles so easily. From this reason both the absorption coefficient, both the refracting index of environment increases with increasing frequency of radiation - this is so called normal dispersion.
The materials with negative curvature fluctuations of Emental cheese structure are less common, but in such environment the relation of absorption and refraction curve is exactly as opposite, because in such environment the refraction index decreases with increasing frequency with compare to absorption, so we are talking about "anomalous dispersion" here.
The absorption and dispersion curves are mutually related by Kramers-Kronig relations, by which absorption curve (bulk effect) is the first derivation of dispersion curve (i.e. the surface refraction effect), because in environment modeled by spherical particle fluctuations the surface of sphere is first derivation of sphere volume with respect to radius. In vacuum environment the absorption and dispersion curve of electric and magnetic waves corresponds the real and imaginary portion of complex quantities called permitivity and permeability of vacuum, accordingly.
With respect to space-time definition the negative portion of dispersion curve close to inflection point is most significant (compare the red point on the dispersion curve above), because for such frequency the energy spreads in slowest speed possible, so that the space-time appears most huge from insintric perspective here. Such environment has a structure of foam, where positive curvature of density fluctuations remains balanced by negative curvature of holes, but not quite - from this the symmetry violation of vacuum foam follows and the environment behaves like metamaterial of negative refraction index, whenever the imaginary portion of both permeability, both permitivity remains negative. We can say, vacuum behaves like metamaterial just because it's so huge due the presence of large amount of density fluctuations, so we can model phenomena like dark matter streaks, photons and event horizon of black holes by light spreading through metamaterials of foamy structure (compare the simulation bellow).
With compare to solid state metamaterials vacuum is composed of fractal foam of density fluctuations similar to Perlin octal noise, because Aether is behaving like elastic fluid filled/formed by its vortices and the diameter of vortices is indirectly proportional to frequency of wave perturbations. This leads to metamaterial character of vacuum in broad range of wavelengths, until we use transversal waves of minimal exsintric speed for observation. Because metamaterial focuses wave into solitary wave packets (i.e. bosons), we can see the distant stars like pin-point objects without dispersion in broad range of spectrum from infrared to X-ray range of EM wave spectrum.
From general perspective, the normal and anomalous dispersion should be symmetric phenomena. The usage of word "normal" in this context is anthropocentric, because it's based on the fact, human creatures are formed by density fluctuations of arbitrarily positive curvature (i.e. by particles in common sense), so we can interact with particle fields more often and easily, then with fluctuations of negative curvature. Inside of atom structures the positive and negative curvature of electron orbitals remains balanced, so we can observe both absorbance peaks, both transmittance peaks with the same probability there. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9434130787849426, "perplexity": 1721.870475517398}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1386163036037/warc/CC-MAIN-20131204131716-00070-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://git.fsmpi.rwth-aachen.de/osak/materials/-/commit/62cf5773053edf21b16a234a549b78d04af415b3?view=inline | Commit 62cf5773 by Milan Santosi
### fixed some more things. slides should be done now.
parent 0781b859
% Created 2012-04-12 Thu 16:37 % Created 2012-04-13 Fri 15:20 \documentclass[bigger]{beamer} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} ... ... @@ -51,8 +51,11 @@ \item What's the difference between Open Source and Free Software? \end{itemize} \end{frame} \section{Considerations about Software} \section{What is Software?} \label{sec-2} \begin{frame} \frametitle{Considerations about Software} \label{sec-2-1} \begin{itemize} \item Source Code is a work of authorship, similar but not identical to ... ... @@ -63,8 +66,10 @@ \item Some form of Instruction set to make a machine do something. \item Condensed knowledge, or information (hence IT). \end{itemize} \section{Considerations about Knowledge} \label{sec-3} \end{frame} \begin{frame} \frametitle{Considerations about Knowledge} \label{sec-2-2} \begin{itemize} \item Information has unique, properties, it is not a real commodity and ... ... @@ -75,8 +80,10 @@ and nondiscriminatory way. \end{itemize} \end{itemize} \section{Intellectual Property''?} \label{sec-4} \end{frame} \begin{frame} \frametitle{Intellectual Property''?} \label{sec-2-3} \begin{itemize} \item Umbrella term for copyright, patent, trademark and other ... ... @@ -84,12 +91,16 @@ \item Commonly misused to actually mean anything. \item Implies claim to ownership, in particular of ideas. \end{itemize} \section{Market implications} \label{sec-5} \end{frame} \section{Free Software Economics} \label{sec-3} \begin{frame} \frametitle{Market implications} \label{sec-3-1} \begin{itemize} \item In a free market economy monopolies are ususally dangerous and have to be avoided. \item In a free market economy monopolies are usually seen as dangerous and have to be avoided. \begin{itemize} \item Monopolies on public goods are really, really bad. \end{itemize} ... ... @@ -98,11 +109,12 @@ \item Software patents are in effect monopolies on ideas. \end{itemize} \end{frame} \section{The Four Freedoms} \label{sec-6} \label{sec-4} \begin{frame} \frametitle{Freedom 0} \label{sec-6-1} \label{sec-4-1} \begin{itemize} \item You are free to run the program, for any purpose. ... ... @@ -113,7 +125,7 @@ \end{frame} \begin{frame} \frametitle{Freedom 1} \label{sec-6-2} \label{sec-4-2} \begin{itemize} \item You are free to study how the program works, and change it so it ... ... @@ -126,7 +138,7 @@ \end{frame} \begin{frame} \frametitle{Freedom 2} \label{sec-6-3} \label{sec-4-3} \begin{itemize} \item You are free to redistribute copies so you can help your neighbor. ... ... @@ -140,7 +152,7 @@ \end{frame} \begin{frame} \frametitle{Freedom 3} \label{sec-6-4} \label{sec-4-4} \begin{itemize} \item You are free to distribute copies of your modified versions to ... ... @@ -149,40 +161,91 @@ precondition for this. \end{itemize} \end{frame} \section{Open Source?} \label{sec-7} \section{Examples of Licensing} \label{sec-5} \begin{frame} \frametitle{F/OSS Licensing} \label{sec-5-1} Openness of source is less decisive when one or more of the four freedoms are formally or effectively taken away. The term Open Source'' is usually used synonymous to Free Software, in the past the term was coined to avoid misunderstandings. \section{Licensing} \label{sec-8} \begin{frame} \frametitle{GPL} \label{sec-8-1} \begin{itemize} \item GPL \label{sec-5-1-1}% \begin{itemize} \item Copyleft, restrictive, enforces all four freedoms. \end{itemize} \begin{itemize} \item BSD \label{sec-8-1-1}% \label{sec-5-1-2}% \begin{itemize} \item Non-copyleft, permissive, not all rights have to be conserved. \end{itemize} \end{itemize} % ends low level \end{frame} \section{Conclusion} \label{sec-6} \begin{frame} \frametitle{On Flash} \label{sec-6-1} Flash is not actually secret but adobe keeps making new versions which are different, faster than anyone can keep up and make free software to play those files, so it has almost the same effect as being secret. \end{frame} \begin{frame} \frametitle{On non-free formats} \label{sec-6-2} It's bad to distribute recordings in mp3 or any other proprietary format. Although there is free software to handle those formats, distributors legally can not include support for such formats because they are patented in many countries. Modern distributions of GNU/Linux facilitate use of these formats by letting the user decide, thereby avoiding legal issues. Whether or not this is a good thing is a question of moral. On the one hand, many of these formats have become pseudo-standards and for most users, the short term usability aspect prevails. On the other hand, it's a very bad idea if we were hoping to get rid of proprietary formats eventually. \end{frame} \begin{frame} \frametitle{Wrap-Up} \label{sec-6-3} \begin{itemize} \item Perversion Software Patente: Wie etwas, das eigentlich Innovation foerdern soll genau das Gegenteil bewirkt. \item Digital Restrictions Management (DRM): \item If you want to support freedom, don't distribute recordings in non-free formats. Please use formats such as .ogg, .webm, or .png. \item In an epoch when governments work for the mega corporations instead of reporting to their citizens, every technological change can be taken advantage of to reduce our freedom. \end{itemize} \end{frame} \begin{frame} \frametitle{Further reading} \label{sec-6-4} \begin{itemize} \item Defective by Design und Malicious Features. Beispiele: Amazon Kindle, ei-Phone, Skype, Spotify, Steam, Diverse unfreie Betriebssysteme \item Probleme mit restriktiven Datenformaten: MP3, flash, docx, etc\ldots{} \item Richard Stallman: The GNU Manifesto \item Ben Klemens: Math you can't use \item Eric S. Raymond: The Cathedral and the Bazaar \item Stephan N. Kinsella: Against Intellectual Property \end{itemize} \begin{itemize} \item Websites \label{sec-6-4-1}% \begin{itemize} \item \href{http://stopsoftwarepatents.com}{http://stopsoftwarepatents.com} \item \href{http://defectivebydesign.org}{http://defectivebydesign.org} \item \href{http://patentabsurdity.com}{http://patentabsurdity.com} \item \href{http://fsf.org}{http://fsf.org} \end{itemize} \end{itemize} % ends low level \end{frame} ... ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9997652173042297, "perplexity": 20677.457407211466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991829.45/warc/CC-MAIN-20210514214157-20210515004157-00231.warc.gz"} |
https://deepai.org/publication/communication-efficient-variance-reduced-stochastic-gradient-descent | # Communication-efficient Variance-reduced Stochastic Gradient Descent
We consider the problem of communication efficient distributed optimization where multiple nodes exchange important algorithm information in every iteration to solve large problems. In particular, we focus on the stochastic variance-reduced gradient and propose a novel approach to make it communication-efficient. That is, we compress the communicated information to a few bits while preserving the linear convergence rate of the original uncompressed algorithm. Comprehensive theoretical and numerical analyses on real datasets reveal that our algorithm can significantly reduce the communication complexity, by as much as 95%, with almost no noticeable penalty. Moreover, it is much more robust to quantization (in terms of maintaining the true minimizer and the convergence rate) than the state-of-the-art algorithms for solving distributed optimization problems. Our results have important implications for using machine learning over internet-of-things and mobile networks.
## Authors
• 8 publications
• 6 publications
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Stochastic gradient descent is the method of choice for large-scale mach...
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### On Maintaining Linear Convergence of Distributed Learning and Optimization under Limited Communication
In parallel and distributed machine learning multiple nodes or processor...
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### AntNet: Distributed Stigmergetic Control for Communications Networks
This paper introduces AntNet, a novel approach to the adaptive learning ...
02/14/2021
### Communication-Efficient Distributed Optimization with Quantized Preconditioners
We investigate fast and communication-efficient algorithms for the class...
04/14/2018
### When Edge Meets Learning: Adaptive Control for Resource-Constrained Distributed Machine Learning
Emerging technologies and applications including Internet of Things (IoT...
08/04/2021
### ErrorCompensatedX: error compensation for variance reduced algorithms
Communication cost is one major bottleneck for the scalability for distr...
02/20/2020
### Adaptive Sampling Distributed Stochastic Variance Reduced Gradient for Heterogeneous Distributed Datasets
We study distributed optimization algorithms for minimizing the average ...
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https://www.physicsforums.com/threads/how-can-spin-up-z-and-spin-down-z-span-the-space.908784/ | # I How can Spin-up z and spin-down z span the space?
1. Mar 23, 2017
### mike1000
$\def \sqx{\frac{1}{\sqrt{2}}}$
If Szu represents spin-up in the z-direction and Szd represents spin-down in the z-direction then the vector which represents spin-up in the x-direction is given by the superposition of the z states $$|S_{zu}\rangle =\begin{bmatrix}1 \\ 0\end{bmatrix}, |S_{zd}\rangle =\begin{bmatrix}0 \\1\end{bmatrix}$$ and
$$|S_{xu}\rangle =\begin{pmatrix} \sqx \\ \sqx\end{pmatrix} = \sqx \left[ |S_{zu}\rangle + |S_{zd}\rangle\right]$$
How can the two z-vectors span the entire space? How can the spin in the x-direction be a linear combination of the spin vectors in only the z-direction?
Last edited: Mar 23, 2017
2. Mar 23, 2017
### ShayanJ
These are not vectors in the 3 dimensional space we live in. These are quantum states, vectors in a Hilbert space. In this case the Hilbert space is two dimensional and any two orthonormal vectors can serve as a basis for it. So you can either use Szu and Szd or Sxu and Sxd, etc.
3. Mar 23, 2017
### SergioPL
The two dimensions complex spinor can be projected to the three dimensional space through the Pauli matrices, σx, σy and σz.
if you have the spinor |S> and you want the probability to find a spin up (or down) for the x direction you just have to do: |Sx> = σx|S>.
If the amplitudes are properly normalized ||Sx>|2 = 1
4. Mar 23, 2017
### mike1000
Three dimensional euclidean space is a Hilbert Space so just saying the vectors are in a Hilbert Space is not really illuminating. Saying it is a two dimensional Hilbert space is a little more illuminating. What 2-dimensional Hilbert Space is it? What are the dimensions? And why could't you use the more familiar 3-dimensional Euclidean space?
Last edited: Mar 23, 2017
5. Mar 23, 2017
### ShayanJ
Fair point. So let's say its a different Hilbert space than the 3 dimensional Euclidean space.
Its a two dimensional complex Hilbert space with the inner product that you know for different spin states.
The dimensions are + and - in any direction you want.
Because we're not talking about vectors in 3-dimensional Euclidean space,we're talking about the spin state of a spin-1/2 particle. A spin-1/2 particle has 2 orthogonal spin states so we should use a 2-dimensional Hilbert space for describing it. Also we never use the Euclidean space as the Hilbert space for the state for any degrees of freedom of any quantum system, simply because northwest is not a quantum state, but pointing towards northwest is! Its the same reason the Euclidean space is not the Hilbert space for the translational degrees of freedom.
6. Mar 24, 2017
### edguy99
I like to visualize an axis of spin on a bloch sphere. When viewed from the front, an axis of spin that is up (ie. above the equator), may be on the left side of vertical or the right side of vertical. If that same axis of spin is viewed from the top, the ones on the left side of the vertical will be spin down (from the viewpoint of the top) and the ones on the right side of the vertical will be spin up.
The axis of spin of a spin up particle, has a 50% chance of being up when viewed from the top and a 50% chance of being down when viewed from the top.
7. Mar 24, 2017
### stevendaryl
Staff Emeritus
It might be helpful to look at the explicit relationship between vectors and spinors.
Let $\hat{A}$ be a unit vector in some direction, and let $\psi$ be the two-component spinor corresponding to the spin state of a particle that is spin-up in direction $\hat{A}$, and let $\chi$ be the spinor for spin-down relative to $\hat{A}$. These relationships can be summarized in terms of the Pauli spin matrices $\sigma_x, \sigma_y, \sigma_z$ by:
$(\hat{A} \cdot \vec{\sigma}) \psi = \psi$
$(\hat{A} \cdot \vec{\sigma}) \chi = - \chi$
It's convenient to rewrite this in terms of projection operator:
$P_{\hat{A}} = \frac{1}{2} (1 + \hat{A} \cdot \vec{\sigma})$
In terms of $P_{\hat{A}}$, we can write:
$P_{\hat{A}} \psi= \psi$
$P_{\hat{A}} \chi = 0$
So $P_{\hat{A}}$ "projects out" the component of a spinor that is spin-up relative to $\hat{A}$.
Here's a cool fact about $P_{\hat{A}}$ and $\psi$:
$\psi \psi^\dagger = P_{\hat{A}}$
where $\dagger$ means the complex-conjugate of the transpose. So there is a strange sense in which $\psi = \sqrt{P_{\hat{A}}}$ (sort of).
If you use the Pauli spin matrices, then you can find the components of $\hat{A}$ in terms of the components of $\psi$. Letting $\psi = \left( \begin{array} \\ a \\ b \end{array} \right)$, then
$A_z = |a|^2 - |b|^2 = 2 |a|^2 - 1$ (the spinor is normalized so that $|a|^2 + |b|^2 = 1$
$A_x = a^* b + a b^*$
$A_y = i (a^* b - b^* a)$
You can invert these to find $a$ and $b$ in terms of $\hat{A}$. This is easier if you represent $\hat{A}$ in spherical coordinates:
$A_z = cos(\theta), A_x = sin(\theta) cos(\phi), A_y = sin(\theta) sin(\phi)$
Then in terms of $\theta$ and $\phi$,
$a = cos(\theta/2) e^{-i \phi/2}$
$b = sin(\theta/2) e^{+i \phi/2}$
The relationship is two-to-one, in the sense that if you rotate $\hat{A}$ through $2\pi$ about any axis, you get back to the same vector, but the corresponding spinor $\psi$ changes sign. This can be seen easily from the above relationship if you choose the z-axis. Then rotating corresponds to changing $\phi$, so if you rotate by $2 \pi$, that means changing the phase of $a$ by $e^{-i (2\pi)/2} = -1$ and changing the phase of $b$ by $e^{+i (2\pi)/2} = -1$
So as far as how a two-component spinor can represent arbitrary directions in 3-D space, it sort of makes sense: They both are characterized (up to a normalization and overall phase) by the pair of real numbers $\theta$ and $\phi$.
8. Mar 24, 2017
### Talisman
When the outcome of a measurement of a particular observable can take on only one of two values, we represent the underlying state as a vector in C2 (and the distinct possible measurements as orthogonal vectors, and thus a basis).
You can discover, experimentally, that x-direction spin and y-direction spin are non-commuting. This suggests they share a state space. And indeed, one discovers vectors (and corresponding projectors) in C2 that model the results of measuring spin in the three physical axes.
9. Mar 24, 2017
### mike1000
First of all, thanks to everyone for the really great responses. I do appreciate them. I have already learned so much by investigating each of the answers which I am continuing to do.
I have many questions, but the simplest is regarding this response...
When you say above "that x-direction spin and y-direction spin are non-commuting. This suggests they share a state space.". What do you mean by they "share a state space? and why does not commuting suggest that?
10. Mar 24, 2017
### Talisman
IANAP. Hopefully others will correct me for any imprecisions.
By "state space" we mean the Hilbert space used to represent a state vector. If the measurement of observable A and observable B commute, it means that we can simultaneously get well-defined values for each of them. In that case, the state space for observable A and observable B can be considered separately. They're two unrelated vector spaces (and if we want to represent the pair of observables, we use the tensor product of those spaces). This "allows" them to evolve independently. (I put "allow" in quotes because obviously the math does not make anything happen, but it's a good way of thinking about things.)
If they cannot be measured simultaneously, it is because there is a state vector in some space that encodes both properties in some way. When observable A is measured, this projects the state vector onto one of the eigenstates of A (i.e., one of the two orthogonal vectors representing distinct measurements). This can in general change the distribution of outcomes for B (whose probabilities can be calculated by projecting the state vector onto B's eigenstates).
Maybe this can be put more simply. I can give you the space C2, and three different basis sets {x+, x-}, {y+, y-}, {z+, z-}, and when you take the projection of x+ onto y- (for example) you will get a vector whose length will help you predict how likely an electron in x+ will be measured in y- (as confirmed by experiment). This suggests that the model I gave you is indeed a good model for electron spin.
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https://www.transtutors.com/questions/inferring-costing-method-unit-product-cost-amcor-inc-incurs-the-following-costs-to-p-1389502.htm | # Inferring Costing Method; Unit Product Cost Amcor, Inc., incurs the following costs to produce...
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Inferring Costing Method; Unit Product Cost
Amcor, Inc., incurs the following costs to produce and sell a single product.
During the last year, 30,000 units were produced and 25,000 units were sold. The Finished Goods inventory account at the end of the year shows a balance of $85,000 for the 5,000 unsold units. Required: 1. Is the company using absorption costing or variable costing to cost units in the Finished Goods inventory account? Show computations to support your answer. 2. Assume that the company wishes to prepare financial statements for the year to issue to its stockholders. a. Is the$85,000 figure for Finished Goods inventory the correct amount to use on these statements for external reporting purposes? Explain.
b. At what dollar amount should the 5,000 units be carried in inventory for external report- ing purposes?
## 1 Approved Answer
5 Ratings, (9 Votes)
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http://cran.r-project.org/web/packages/GENEAsphere/vignettes/Visualisation_of_Raw_or_Segmented_Accelerometer_Data_using_GENEAsphere.html | # Visualisation of Raw or Segmented Accelerometer Data using GENEAsphere.
## Data visualisation with GENEActiv .bin files.
There are a number of ways in which data recorded using the GENEActiv device can be visualised using R. By using the packages GENEAread and GENEAsphere a thorough interpretation of the data can be made using various techniques. This tutorial builds on the knowledge of the previously realised tutorial on GENEAclassify. Please download the dropbox folder GENEAsphereDemo by contacting charles@sweetland-solutions.co.uk for a link to the Dropbox.
This tutorial will take two types of data which are the outputs of the packages GENEAread and GENEAclassify. The function read.bin from GENEAread produces an AccData object in R that represents the raw data from the device. This data can then be visualised using with the stft function from GENEAread or the positionals function from GENEAsphere. The segmentation function from GENEAclassify produces a segmented csv that can be used by the functions within GENEASphere The output from getSegmentedData and classifyGENEA can also be used.
Contents:
• a.) Installing and loading the libraries
• b.) loading in the data
• c.) A standard acceleration plot of raw data
• d.) plotAccData
• e.) plotTLM
• f.) STFT
• g.) Positionals
• h.) plotSphere
• i.) plotSegmentSphere
• j.) plotSegmentFlat
• k.) plotSegmentProjection
• l.) plotSegmentEllipse
Before starting please ensure that you have the following:
• Completed the tutorial on GENEAclassify
• Single GENEActiv .bin file
• A segmented data file in .csv
## a.) Installing and loading the libraries.
Please install and load the following libraries.
install.packages("GENEAread",repos = "http://cran.us.r-project.org")
install.packages("devtools",repos = "http://cran.us.r-project.org")
install.packages("changepoint",repos = "http://cran.us.r-project.org")
install.packages("signal",repos = "http://cran.us.r-project.org")
install.packages("mmap",repos = "http://cran.us.r-project.org")
install.packages("misc3d",repos = "http://cran.us.r-project.org")
install.packages("rgl",repos = "http://cran.us.r-project.org")
install.packages("mapproj",repos = "http://cran.us.r-project.org")
library(GENEAread)
library(devtools)
library(changepoint)
library(signal)
library(mmap)
library(misc3d)
library(rgl)
library(mapproj)
Install GENEAclassify and GENEAsphere either from a source file, tar.gz, or from GitHub. Please refer to the notes in the previous tutorial if installing from GitHub.
setwd("/Users/owner/Documents/GENEActiv")
# You will need to change this to the directory where you saved the tar.gz file
install.packages("GENEAclassify_1.4.1.tar.gz", repos=NULL, type="source")
#' Or using a GitHub authentication key which will go in the brackets of auth_token
install_github("https://github.com/JossLangford/GENEAclassify_1.41.git",
auth_token = "7f0051aaca453eaabf0e60d49bcf752c0fea0668")
#' Once the package has been installed load in the library
library(GENEAclassify)
# Again install GENEAsphere either from source or GitHub
setwd("/Users/owner/Documents/GENEActiv")
# You will need to change this to the directory where you saved the tar.gz file
install.packages("GENEAsphere_1.0.tar.gz", repos=NULL, type="source")
## If installing from GitHub please run these lines
install_github("https://github.com/JossLangford/GENEAsphere.git",
auth_token = "7f0051aaca453eaabf0e60d49bcf752c0fea0668")
library(GENEAsphere)
## b.) loading in the data.
The GENEActiv .bin file can be used to create some standard plots within GENEAread but the segmented file will be of more interested to most. Reading in the selected data file for visualisation using the following lines of code. The name of the file and the directory location will need to be changed.
setwd("/Users/owner/Documents/GENEActiv/GENEAsphereDemo")
# You will need to change this to the directory containing the data file.
# Here I have analysed the first day for.
AccData = read.bin("jl_left wrist_010094_2012-01-30 20-39-54.bin", start = "3:00", end = "1 3:00")
SegData = getSegmentedData("jl_left wrist_010094_2012-01-30 20-39-54.bin", start = "3:00", end = "1 3:00")
The output of these two data objects can be viewed using the head and names functions which give more details. In particular AccData is a large data frame with many subsets which can be accessed by using a $after AccData followed by the name of the subset. In this case data.out gives the raw data from the device. names(AccData) head(AccData) head(AccData$data.out) # Raw data output
names(SegData)
head(SegData)
## c.) A standard acceleration plot of raw data
Standard plots can be created using the simple function plot to show the raw acceleration data against time. From the output of the head function, the AccData is a large data frame containing the information recorded by the GENEActiv device.
To select the columns of data desired R uses an index system. Firstly by using the $to tell R what part of the subset of the AccData object to select. AccData$data.out gives us that specific part of the data. Then using AccData$data.out[R,C] to specify which row then column is to be used. For example AccData$data.out[2,4] would return the entry on the 2nd row in the 4th column. Section of the data can be found by using a column. For example taking all of the entries in the 4th column from row 2 to row 100 would be found by running the line AccData$data.out[2:100,4]. In this case all of the data in a row will be plotted against another which can be done by leaving the row index empty so that R selects every row. As shown with the names function, column 1 is the timestamp given to the data, 2,3 and 4 are x,y and z acceleration respectively. Column 5 is the light measurements, 6 is the number of button presses and 7 is temperature. It is possible to plot any of these against one another as long as the selection made for x and y are of equal length. plot(AccData$data.out[1:1000,1],AccData\$data.out[1:1000,2],
title="Time against X acceleration",
xlab="Time",ylab="X Acceleration",type="l")
## d.) plotAccData
This function plots the choosen variable from the accelerometer data.
plot.AccData(x, what = ("sd"))
plot.AccData(x, what = ("mean"))
plot.AccData(x, what = ("temperature"))
plot.AccData(x, what = ("light"))
plot.AccData(x, what = ("voltage"))
## e.) plotTLM
This plot shows Temperature, light and magnitude on a single plot.
plotTLM(x, start = NULL, end = NULL)
## f.) STFT
Inside GENEAread there is a function called stft which calculates the short time Fourier transform of the raw input signal. This generates two plots, the top shows the amplitude at various frequencies and bottom plot shows frequency against time. For more information on the variables that can be passed to this function type the command ?stft.
stft(AccData, start=0.45, end=0.5, plot.it=TRUE)
stft(AccData, start=0.45, end=0.5, plot.it=TRUE,reassign = TRUE)
stft(AccData, start=0.45, end=0.5, plot.it=TRUE, type = "mv")
stft(AccData, start=0.45, end=0.5, plot.it=TRUE, type = "sum")
# Changing the window size
stft(AccData, start=0.45, end=0.5, plot.it=TRUE, win=100)
stft(AccData, start=0.45, end=0.5, plot.it=TRUE, win=1000)
## g.) Positionals plot
The positionals function from the GENEAsphere package creates a plot that shows arm elevation on the y-axis against time that also has a colour key which shows wrist rotation. This is very useful for showing regular activities and inside each activity within the sample training data folder from GENEAclassify a plot of the .bin files can be found. The start and end times work in the standard fashion, similar to read.bin.The plot can be changed to show various different visual effects.
positionals(AccData, start=0.45, end= 0.5, length = NULL, filter=2
,bw = TRUE , legend = TRUE, max.points = 1e6, density = FALSE)
The details on how to set different variables can be found by typing ?positionals into the console.
## h.) Plotsphere plot
The plotSphere function from the GENEAsphere package takes the raw data from the GENEAread package and plots the recordings onto a sphere.
plotSphere(AccData, start=0, end= 0.5, length = NULL, time.format = "auto",
density = F, arrow = T, add= F)
The details on how to set different variables can be found by typing ?plotSphere into the console.
## i.) plotSegmentSphere
For the following functions, a plot is mapped onto a sphere from a segmented csv file has to be loaded into the environment once the output of getSegmentedData has been found. These functions use the mean Updown and degrees value of the segment as a centre position on the sphere. The distribution is based on the Median Absolute Deviation (mad) and distance from the sphere is given by the magnitude mean.
segmentationCSV="~/GENEAclassification/jl_left wrist_010094_2012-01-30 20-39-54_segmented.csv"
# I find it useful to load in the csv to the workspace so that the rows I'm going to plot can be seen.
csv=read.table(segmentationCSV,sep=",")
Now plotting some rows of this csv file.
plotRows=c(1:5) # Segments 1 and 5.
plotSegmentSphere(segmentationCSV, plotRows, levels = c(0.9, 0.75, 0.5, 0.25, 0.1), singlePlot = TRUE, col = heat.colors(5),
alpha = c(0.03, 0.05, 0.1, 0.2, 0.3), arrow = FALSE, nsims = 1000)
By typing ?plotSegmentSphere a detailed description of the variables is given.
## j.) plotSegmentFlat
Rather than plotting on a 3D sphere the same information is plotted on a flat sphere with arm elevation on the y axis and wrist rotation on the x axis.
plotSegmentFlat(segmentationCSV, plotRows,
col = c("red",heat.colors(5, alpha = c(0.3, 0.2, 0.1, 0.05, 0.03))),
singlePlot = TRUE, nsims= 1000)
By typing ?plotSegmentFlat a detailed description of the variables is given.
## k.) plotSegmentProjection
plotSegmentProjection(segmentationCSV, plotRows, projection = "aitoff",
col = "red", singlePlot = TRUE, nsims = 1000)
By typing ?plotSegmentProjection a detailed description of the variables is given.
## l.) plotSegmentEllipse
This function plots onto an Ellipse.
plotSegmentEllipse(segmentationCSV, plotRows, projection = "aitoff",
col = "red", singlePlot = TRUE, confidenceLevel = 0.05,
alpha = thresholds, wrap = FALSE, greyGrid = FALSE)
By typing ?plotSegmentEllipse a detailed description of the variables is given. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.31733474135398865, "perplexity": 4271.0528608107725}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247489304.34/warc/CC-MAIN-20190219020906-20190219042906-00107.warc.gz"} |
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How to transfer pictures from C# to VB6?
How to transfer pictures from C# to VB6?
(OP)
Hi everyone,
I have to research the fastest way to transfer pictures from a C# application - which generates them - to a VB6 form... if this can be done at all. Animation by replacing must be possible, but the smoother the better. Storing the pictures somewhere on disk and loading them in VB6 is no option. Both the C# end and the VB6 end are mine to develop at the moment.
Currently I can store the picture data (a .NET Bitmap class) in memory and the address from which to start reading is known - but I am a little at a loss on how to proceed.
Assuming that I can read the picture's data into a VB6 array and that the format these bytes represent is known: is there a way to convert the bytes back into an actual picture? What I'm looking for is LoadPicture() that takes a memory address + number of bytes to read form that address as parameters, instead of a string representing a file name. I'm pretty sure the function doesn't exist in VB6, so how can it be implemented?
Any help here will be much appreciated!
Thanks,
Cooz
RE: How to transfer pictures from C# to VB6?
Sure, why not? Whilst VB6 doesn't natively a) have a LoadLibrary that works from a memory address and b) does not understand a .NET bitmap array (really a GDI+ bitmap array) - it does have the ability to use the Microsoft Windows Acquisition library, which knows how to deal with GDI+ bitmap arrays.
RE: How to transfer pictures from C# to VB6?
But, to summarise, following the assumption that you already have a method for transferring the byte array to VB (pretty easy), and that you have set a reference to the Windows Aquisition Library, then all you need is something like:
CODE
Private Function LoadPictureFromArray(GraphicsArray() As Byte) As StdPicture
With New WIA.Vector
.BinaryData = GraphicsArray
End With
End Function
RE: How to transfer pictures from C# to VB6?
(OP)
Hi strongm,
Yes, I got it working! This is neat.
It looks like I've said a little too much though when I mentioned "Currently I can store the picture data (a .NET Bitmap class) in memory and the address from which to start reading is known". - thinking a C# version of VarPtr would give me the address. And it worked for strings. Oops... not for objects of course. The required C# version of ObjPtr is not so easy to implement.
Unless you've a similar splendid idea on this, I think we can consider this thread closed.
Cooz
RE: How to transfer pictures from C# to VB6?
>Currently I can store the picture data (a .NET Bitmap class) in memory and the address from which to start reading is known"
Ah, but you don't need to do that. You should be able to direfctly pass the image as an array. I'm not a C## guru, but here's a (very basic) function that I'd use in VB.NET that should be easily translatable:
CODE
Public Function GetImageBytes(image As System.Drawing.Image) As Byte()
Dim byteArray() As Byte
Dim stream As New IO.MemoryStream
With stream
image.Save(stream, System.Drawing.Imaging.ImageFormat.Bmp)
stream.Close()
byteArray = stream.ToArray()
End With
Return byteArray
End Function
RE: How to transfer pictures from C# to VB6?
(OP)
Okay, yes that sounds plausible. Here's the C# format, easily translated indeed. And if you'll allow me... you can skip the "With... End With" in your function.
CODE --> C#
public byte[] GetImageBytes(Image image)
{
var stream = new MemoryStream();
image.Save(stream, ImageFormat.Bmp);
stream.Close();
return stream.ToArray();
}
I'll see if I can get my VB6 application to call it.
RE: How to transfer pictures from C# to VB6?
>you can skip the "With... End With"
Indeed. In fact my final version does not have the with ... end with, they were a hangover from a slightly earlier version.
>I'll see if I can get my VB6 application to call it.
I use an ActiveX object to transfer the data and raise an event in VB6 when the data has changed, thus making "Animation by replacing must be possible" almost effortless ...
RE: How to transfer pictures from C# to VB6?
I was intrigued enough to finish an example solution ...
Firstly it requires an ActiveX EXE. In my example this project is called vbDataTransfer, and consists of a module and two classes as below:
CODE --> modGlobals
Option Explicit
Public PrivateTransfer As clsTransfer
Public lCount As Long
CODE --> clsTransfer
' clsTransfer
' Instancing - PublicNotCreatable
Option Explicit
Public Event TransferArrived(arrTransfer() As Byte)
Friend Sub RaiseTransferArrived(arrTransfer() As Byte)
RaiseEvent TransferArrived(arrTransfer)
End Sub
CODE --> clsConnector
' clsConnector
' Instancing - MultiUse
Option Explicit
Private WithEvents Transfer As clsTransfer
Public Event TransferArrived(arrTransfer() As Byte)
Private Sub Class_Initialize()
If PrivateTransfer Is Nothing Then
Set PrivateTransfer = New clsTransfer
End If
Set Transfer = PrivateTransfer
lCount = lCount + 1
End Sub
Private Sub Class_Terminate()
lCount = lCount - 1
If lCount = 0 Then
Set PrivateTransfer = Nothing
End If
End Sub
' Bubble the event
Private Sub Transfer_TransferArrived(arrTransfer() As Byte)
RaiseEvent TransferArrived(arrTransfer)
End Sub
Public Property Let NewCommand(arrTransfer() As Byte)
PrivateTransfer.RaiseTransferArrived arrTransfer()
End Property
This should be compiled and the resulting ActiveX library registered.
Ok, that's the 'complicated' bit ...
And now we need a .NET program that sends a bitmap ... here's the VB.NET version:
CODE --> PictureTransfer
' Requires a COM reference to vbDataTransfer (or whatever you called the class library) to be added to the project
' Windows form should have a picturebox and a command button, picturebox should have an image loaded
Public Class Form1
Private Sub Button1_Click(sender As System.Object, e As System.EventArgs) Handles Button1.Click
Dim test As New vbDataTransfer.clsConnector
test.NewCommand = GetImageBytes(Me.PictureBox1.Image)
End Sub
Public Function GetImageBytes(image As System.Drawing.Image) As Byte()
Dim stream As New IO.MemoryStream
image.Save(stream, System.Drawing.Imaging.ImageFormat.Bmp)
stream.Close()
GetImageBytes = stream.ToArray()
End Function
End Class
Oh, and my attempt at a C# version as well:
CODE --> PictureTransCSharp
// Requires a COM reference to vbDataTransfer (or whatever you called the class library) to be added to the project
// Windows form should have a picturebox and a command button, picturebox should have an image loaded
namespace WindowsFormsApplication1
{
public partial class Form1 : Form
{
public Form1()
{
InitializeComponent();
}
public byte[] GetImageBytes(Image image)
{
var stream = new MemoryStream();
image.Save(stream, ImageFormat.Bmp);
stream.Close();
return stream.ToArray();
}
private void button1_Click(object sender, EventArgs e)
{
var test = new vbDataTransfer.clsConnector();
test.set_NewCommand(GetImageBytes(pictureBox1.Image));
}
}
}
And finally the vb program that the bitmap can be sent to ...
CODE --> TransferExample
' assumes user form with picturebox and command button
' assumes reference to WIA added
' and reference added to vbDataTransfer library; you may, of course, have called it something else
Option Explicit
Dim WithEvents appConnector As clsConnector
Private Sub Form_Initialize()
Set appConnector = New clsConnector
End Sub
Private Sub appConnector_TransferArrived(strTransfer() As Byte)
End Sub
Private Function LoadPictureFromArray(GraphicsArray() As Byte) As StdPicture
With New WIA.Vector
.BinaryData = GraphicsArray
End With
End Function
RE: How to transfer pictures from C# to VB6?
(OP)
Hi strongm,
This looks not too difficult. Thanks for sharing all the details!
One thingie though... I've never registered a VB6 dll; and VB6 dlls in the GAC won't work. Using regsvr32 looks fine initially and I can set a reference to vbDataTransfer without any problems, but then the code doesn't recognize clsConnector. This goes for the sending as well as for the receiving code. Am I overlooking something?
RE: How to transfer pictures from C# to VB6?
To register it properly, try at command line
ActiveXexeName.exe /regserver
(you can also simply double click it in Explorer, which has the same effect)
RE: How to transfer pictures from C# to VB6?
(OP)
strongm, this is a w e s o m e.
Indeed, this is exactly what I wanted... and so elegant! Thank you so much!
I'll have a happy weekend now. I hope you'll have one too.
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Register now while it's still free! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21869908273220062, "perplexity": 8576.018550408971}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818689411.82/warc/CC-MAIN-20170922235700-20170923015700-00408.warc.gz"} |
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# EXERCISE 24.1
QUESTION 1
Find the equation of the circle with :
(i) Center $(-2,3)$ and radius 4
Sol :
Let $(h, k)$ be the centre of a circle with radius a.
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Here, $h=-2, k=3$ and $a=4$
$\therefore$ Required equation of the circle:
$\Rightarrow (x+2)^{2}+(y-3)^{2}=4^{2}$
$\Rightarrow(x+2)^{2}+(y-3)^{2}=16$
(ii) Centre $(a, b)$ and radius $\sqrt{a^{2}+b^{2}}$
Sol :
Here, $h=a, k=b$ and radius $=\sqrt{a^{2}+b^{2}}$
$\therefore$ Required equation of the circle:
$\Rightarrow (x-a)^{2}+(y-b)^{2}=a^{2}+b^{2}$
$\Rightarrow x^{2}+y^{2}-2 a x-2 b y=0$
(iii) Centre $(0,-1)$ and radius 1
Sol :
Here, $h=0, k=-1$ and radius $=1$
$\therefore$ Required equation of the circle:
$(x-0)^{2}+(y+1)^{2}=(1)^{2}$
$\Rightarrow x^{2}+y^{2}+2 y=0$
(iv) Centre $(a \cos a, a \sin \alpha)$ and radius a.
Sol :
Here, $h=a \cos \alpha, k=a \sin \alpha$ and radius $=a$
$\therefore$ Required equation of the circle:
$(x-a \cos \alpha)^{2}+(y-a \sin \alpha)^{2}=(a)^{2}$
$\Rightarrow x^{2}+a^{2} \cos ^{2} \alpha-2 a x \cos \alpha+y^{2}+a^{2} \sin ^{2} \alpha-2 a y \sin \alpha=a^{2}$
$\Rightarrow x^{2}+a^{2}\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)-2 a x \cos \alpha+y^{2}-2 a y \sin \alpha=a^{2}$
$\Rightarrow x^{2}+a^{2}-2 a x \cos \alpha+y^{2}-2 a y \sin \alpha=a^{2}$
$\Rightarrow x^{2}+y^{2}-2 a x \cos \alpha-2 a y \sin \alpha=0$
(v) Centre $(a, a)$ and radius $\sqrt{2} a$
Sol :
Here, $h=a, k=a$ and radius $=\sqrt{2} a$
$\therefore$ Required equation of the circle:
$(x-a)^{2}+(y-a)^{2}=(\sqrt{2} a)^{2}$
$\Rightarrow x^{2}+a^{2}-2 a x+y^{2}+a^{2}-2 a y=2 a^{2}$
$\Rightarrow x^{2}+y^{2}-2 a y-2 a x=0$
QUESTION 2
Find the center and radius of each of the following circles:
(i) $(i)(x-1)^{2}+y^{2}=4$
Sol :
Let $(h, k)$ be the centre of a circle with radius a.
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Given: $(x-1)^{2}+y^{2}=4$
Here, $h=1, k=0$ and $a=2$
Thus, the center is $(1,0)$ and the radius is 2
(ii) (ii) $(x+5)^{2}+(y+1)^{2}=9$
Sol :
Given: $(x+5)^{2}+(y+1)^{2}=9$
Here, $h=-5, k=-1$ and radius $=3$
Thus, the center is $(-5,-1)$ and the radius is 3
(iii) $x^{2}+y^{2}-4 x+6 y=5$
Sol :
Given: $x^{2}+y^{2}-4 x+6 y=5$
The given equation can be rewritten as follows:
$(x-2)^{2}+(y+3)^{2}-4-9=5$
$\Rightarrow(x-2)^{2}+(y+3)^{2}=18$
Thus, the centre is $(2,-3)$
And, radius $=\sqrt{18}=3 \sqrt{2}$
(iv) $x^{2}+y^{2}-x+2 y-3=0$
Sol :
Given: $x^{2}+y^{2}-x+2 y-3=0$
The given equation can be rewritten as follows:
$\left(x-\frac{1}{2}\right)^{2}+(y+1)^{2}-\frac{1}{4}-1-3=0$
$\Rightarrow\left(x-\frac{1}{2}\right)^{2}+(y+1)^{2}=\frac{17}{4}$
Thus, the center is $\left(\frac{1}{2},-1\right)$ and and the radius is $\frac{\sqrt{17}}{2}$
QUESTION 3
Find the equation of the circle whose center is $(1,2)$ and which passes through the point $(4,6)$
Sol :
Let $(h, k)$ be the center of a circle with radius a.
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Given: $h=1, k=2$
Equation of the circle $=(x-1)^{2}+(y-2)^{2}=a^{2} \quad \ldots(1)$
Also, equation $(1)$ passes through $(4,6)$
$\Rightarrow (4-1)^{2}+(6-2)^{2}=a^{2}$
$(4-1)^{2}+(6-2)^{2}=a^{2}$
$\Rightarrow 9+16=a^{2}$
$\Rightarrow a=5$ $(\because a>0)$
Substituting the value of a in equation $(1) :$
$\Rightarrow (x-1)^{2}+(y-2)^{2}=25$
$\Rightarrow x^{2}+1-2 x+y^{2}+4-4 y=25$
$\Rightarrow x^{2}-2 x+y^{2}-4 y=20$
$\Rightarrow x^{2}+y^{2}-2 x-4 y-20=0$
Thus, the required equation of the circle is $x^{2}+y^{2}-2 x-4 y-20=0$
QUESTION 4
Find the equation of the circle passing through the point of intersection of the lines $x+3 y=0$ and $2 x-7 y=0$ and whose center is the point of intersection of the lines $x+y+1=0$ and $x-2 y+4=0$
Sol :
The point of intersection of the line $x+3y=0$ and $2x-7y=0$ can be find by simultaneously solving the equations , that is $(0,0)$
Let $(h,k)$ be the center of a circle with radius
Thus , its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
The point of intersection of the lines $x+y+1=0$ and $x-2 y+4=0$ is can be find by simultaneously solving the equation and points are $(-2,1)$
$\therefore h=-2, k=1$
Equation of the required circle $=(x+2)^{2}+(y-1)^{2}=a^{2} \quad \ldots(1)$
Also, equation (1) passes through $(0,0)$
$\Rightarrow (0+2)^{2}+(0-1)^{2}=a^{2}$
$\Rightarrow 4+1=a^{2}$
$\Rightarrow a=\sqrt{5}$ $(\because a>0)$
Substituting the value of a in equation (1) :
$\Rightarrow (x+2)^{2}+(y-1)^{2}=5$
$\Rightarrow x^{2}+4+4 x+y^{2}+1-2 y=5$
$\Rightarrow x^{2}+4 x+y^{2}-2 y=0$
Hence, the required equation of the circle is $x^{2}+y^{2}+4 x-2 y=0$
QUESTION 5
Find the equation of the circle whose center lies on the position of y – axis at a distance 6 from the origin and whose radius is 4
Sol :
Let $(h, k)$ be the center of a circle with radius a
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
The center of the required circle lies on the positive direction of the y-axis at a distance 6 from the origin
Thus , the co-ordinates of the center are $(0,6)$
$h=0, k=6$
Equation of the circle $=(x-0)^{2}+(y-6)^{2}=a^{2} \quad \ldots(1)$
Also, $a=4$
Substituting the value of a in equation (1) :
$\Rightarrow (x-0)^{2}+(y-6)^{2}=16$
$\Rightarrow x^{2}+y^{2}+36-12 y=16$
$\Rightarrow x^{2}+y^{2}-12 y+20=0$
Hence, the required equation of the circle is $x^{2}+y^{2}-12 y+20=0$
QUESTION 6
If the equations of two diameters of a circle are $2 x+y=6$ and $3 x+2 y=4$ and the radius is 10 find the equation of the circle
Sol :
Let $(h, k)$ be the center of a circle with radius a
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
The intersection point of $2 x+y=6$ and $3 x+2 y=4$ is $(8,-10)$ find by simultaneously solving given equation
The diameters of a circle intersect at the center
Thus, the coordinates of the center are $(8,-10)$
$\therefore h=8, k=-10$
Thus, the equation of the required circle is $(x-8)^{2}+(y+10)^{2}=a^{2} \quad \ldots(1)$
Also , $a=10$
Substituting the value of $a$ in equation $(1) :$
$\Rightarrow (x-8)^{2}+(y+10)^{2}=100$
$\Rightarrow x^{2}+y^{2}-16 x+64+100+20 y=100$
$\Rightarrow x^{2}+y^{2}-16 x+20 y+64=0$
Hence, the required equation of the circle is $x^{2}+y^{2}-16 x+20 y+64=0$
QUESTION 7
Find the equation of a circle
(i) Which touches both the axes at a distance of 6 units from the origin.
Sol :
Let $(h, k)$ be the center of a circle with radius a
Thus, its equation will be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Let the required equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
It is given that the circle passes through the points $(6,0)$ and $(0,6)$
$(6-h)^{2}+(0-k)^{2}=6^{2}$
$\Rightarrow(6-h)^{2}+(-k)^{2}=36$
$\Rightarrow 36+h^{2}-12 h+k^{2}=36$
$\Rightarrow h^{2}+k^{2}=12 h \quad \ldots(1)$
AND
$(0-h)^{2}+(6-k)^{2}=6^{2}$
$\Rightarrow h^{2}+36+k^{2}-12 k=36$
$\Rightarrow h^{2}+k^{2}=12 k \quad \ldots(2)$
From (1) and (2), we get:
$12 k=12 h \Rightarrow h=k$
$\therefore$ From equation (2), we have:
$\Rightarrow k^{2}+k^{2}=12 k$
$\Rightarrow k^{2}-6 k=0$
$\Rightarrow k(k-6)=0$
$\Rightarrow k=6$ $(\because k>0)$
Consequently, we get: $h=6$
Hence, the required equation of the circle is
$(x-6)^{2}+(y-6)^{2}=36$ or $x^{2}+y^{2}-12 x-12 y+36=0$
(ii) Which touches x-axis at a distance 5 from the origin and radius 6 units.
Sol :
The circle touches the $x-$ axis at $A=(5,0)$ and has radius 6 unit
Thus , center $=(5, b)$
$\therefore h=5,k=b$ $a=6$
Let the required equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
$\Rightarrow (5-5)^2+(0-b)^2=6^2$
$\Rightarrow b=6$
$\Rightarrow$ centre $=(5,6)$
so, the equation of required circle is
$(x-5)^{2}+(y-6)^{2}=6^{2}$
$\Rightarrow \quad x^{2}+y^{2}-10 x-12 y+25=0$
(iii) Which touches both the axes and passes through the point $(2,1)$ .
Sol :
Let the required equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
It is given that the circle touches both the axes.
Thus, the required equation will be $x^{2}+y^{2}-2 a x-2 a y+a^{2}=0$
Also, the circle passes through the point $(2,1)$
$\Rightarrow 4+1-4 a-2 a+a^{2}=0$
$\Rightarrow a^{2}-6 a+5=0$
$\Rightarrow a^{2}-5 a-a+5=0$
$\Rightarrow a=1,5$
Hence, the required equation is $x^{2}+y^{2}-2 x-2 y+1=0$ or $x^{2}+y^{2}-10 x-10 y+25=0$
(iv) Passing through the origin, radius 17 and ordinate of the center is $-15$
Sol :
Let the required equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Given: $k=-15, a=17$
Also , circle passes through the point $(0,0)$
$\therefore$ Equation of the circle:
$(0-h)^{2}+(0-15)^{2}=(17)^{2}$
$\Rightarrow h=\pm 8$
Hence, the required equation of the circle is $(x-8)^{2}+(y+15)^{2}=17^{2}$ or $(x+8)^{2}+(y+15)^{2}=17^{2},$ i.e. $x^{2}+y^{2} \pm 16 x+30 y=0$
QUESTION 8
Find the equation of the circle which has its center at the point $(3,4)$ and touches the straight line $5 x+12 y-1=0$
Sol :
It is given that the center is at the point $(3,4)$
Let the equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
Equation of the required circle $=(x-3)^{2}+(y-4)^{2}=a^{2} \ldots \ldots(1)$
Also, the circle touches the straight line $5 x+12 y-1=0$ to which points $(3,4)$ satisfies the equation $[\because$ radius is perpendicular to the tangent $]$
$\therefore a=\left|\frac{5(3)+12(4)-1}{\sqrt{5^{2}+12^{2}}}\right|$ $=\left|\frac{62}{13}\right|$
$\Rightarrow a^{2}=\left|\frac{5(3)+12(4)-1}{13}\right|$ $=\frac{3844}{169}$ $=\frac{62}{13}$
So, from equation (1) ,we have:
$(x-3)^{2}+(y-4)^{2}=\left(\frac{62}{13}\right)^{2}$
$\Rightarrow \quad 169\left[x^{2}+y^{2}-6 x-8 y\right]+25 \times 169=3844$
$\Rightarrow \quad 169\left[x^{2}+y^{2}-6 x-8 y\right]+381=0$
Hence, the required equation of the circle is $169\left(x^{2}+y^{2}-6 x-8 y\right)+381=0$
QUESTION 9
Find the equation of the circle which touches the axes and whose center lies on $x-2 y=3$
Sol :
The required circle touches $A(a, 0)$ and $B(0, a)$ on the axes
so the center $=(a, a)$ radius $=a$
Also, the center lies on $x-2 y=3$
$\Rightarrow \quad a-2 a=3$
$\Rightarrow \quad-a=3$
$\Rightarrow a=-3$
$\therefore$ center $=(-3,-3)$ (this shows center in third quadrant) and radius $=3$
Thus the equation of circle is
$(x+3)^{2}+(y+3)^{2}=3^{2}$
$\Rightarrow \quad x^{2}+y^{2}+6 x+6 y+9=0$
If the circle lies in the fourth quadrant, then its center will be $\left(a_{,}-a\right)$
$\therefore a+2 a=3$
$\Rightarrow a=1$
$\therefore$ Required equation of the circle $=(x-1)^{2}+(y+1)^{2}=1$
$=x^{2}+y^{2}-2 x+2 y+1=0$
QUESTION 10
A circle whose center is the point of intersection of the lines $2 x-3 y+4=0$ and $3 x+4 y-5=0$
passes through the origin. Find its equation.
Sol :
Let the required equation of the circle be $(x-h)^{2}+(y-k)^{2}=a^{2}$
We have ,
$2 x-3 y=-4 \ldots \ldots \ldots(1)$
$3 x+4 y=5 . . . . . . . .(2)$
The point of intersection of (1) and (2) is
$=\left(\frac{-1}{17}, \frac{22}{17}\right)$
According to the equation center $=\left(\frac{-1}{17}, \frac{22}{17}\right)$
Also, circle passes through origin $(0,0)$
$\left(0-\dfrac{1}{17}\right)^2 + \left(0-\dfrac{22}{17}\right)^2=a^2$
$\frac{1}{289}+\frac{484}{289}=a^2$
$a^2=\frac{485}{289}$
Hence, the required equation of the circle is $\left(x+\frac{1}{17}\right)^{2}+\left(y-\frac{22}{17}\right)^{2}=\frac{485}{288}$
QUESTION 11
A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of
its images with respect to the line mirrors $x=0$ and $y=0$
Sol :
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# calcium cyanamide reacts with water
It is commercially known as Nitrolime. Calcium cyanamide, CaCN2, reacts with water to form calcium carbonate and ammonia; CaCN2(s)+3H2O(l)-->CaCO3(s)+2NH3(g). c) How many grams of CaCO3 form if 155 g of water reacts? Let us look at the steel making … B) How many gram of NH3 are produced from 5.23g CaCN2? All … How many grams of NH3 are produced from 5.25 g CaCN2? Using Table 8.3, calculate ∆Hof for calcium cyanamide. How Many Grams Of Ammonia Are Produced From 524 Grams Of CaCN)? A) How many grams of water needed to react with 80.0g Ca CN2? A clock maker has 15 clock faces. This would burn the skin of the victims on contact. Preparation. Binary carbides are formed when the metals are heated strongly with carbon. Calcium cyanamide or CaCN 2 is a calcium compound used as fertilizer, first synthesized in 1898 by Adolph Frank and Nikodem Caro (Frank-Caro process). Calcium acetylide. Favourite answer. How much power is consumed by a 12-V incandescent lamp if it draws 150mA of current when lit? Attempts to extinguish fire with water by the fire brigade. Calcium cyanamide is a white (gray or black if impure) solid, which reacts with water. Calcium cyanamide reacts with water to form calcium carbonate and ammonia via the following reaction: CaCN2(s)+3H2O(l)→CaCO3(s)+2NH3(g) How many grams of water are needed to react with 70.0g CaCN2? 10.) a. AgBr ionizes to give Ag+ and Br- 7 years ago. a) How many grams of water are needed to react with 75.0 g of CaCN2? [2] Uses . 9.1K views To prevent it from exploding, the amount of calcium carbide must be calculated beforehand. C) how many grams of CaCO2 from153g water react? ... A: Since boron has atomic number = 5 It is represented by using the below equation. Relevance. Write a balanced thermochemical equation for the reaction. The problem requires that you determine the mass of water, H2O, needed for the reaction to occur. . This carbide is actually an acetylide because it contains the C 22− ion and it reacts with water to produce acetylene. CaCN2+3H2O -> CaCO3+2NH3. b. Fire Hazard . (11.44) CaC 2 … Absorption of water during handling or storage of technical calcium cyanamide may cause explosion [Pieri, M. Chem. When mixed with water in the soil, it reacts to produce calcium carbonate and ammonia: \\mathrm{CaCN}_{2}+3 \\mathrm{H}… Question: QUESTION 17 Calcium Cyanamide, CaCN2 Reacts With Water To Produce Calcium Carbonate And Ammonia Show Your Work, Please 1. Calcium cyanamide reacts with water to form calcium carbonate and ammonia via the following reaction: How many grams of CaCO3 form if 151 g water react? It is capable of deprotonating not just water but also alcohols, evolving acetylene. Statistics Help Question: Calcium cyanamide reacts with water to form calcium carbonate and ammonia. CaCN2 - CALCIUM CYANAMIDE. Calcium carbide reacts with nitrogen at high temperature to form calcium cyanamide: CaC 2 + N 2 → CaCN 2 + C. Commonly known as nitrolime, calcium cyanamide is used as fertilizer. Consider the reaction Answer Save. 4.00260 amu 1 Answer. The reaction of calcium carbide with water, producing acetylene and calcium hydroxide, was discovered by Friedrich Wöhler in 1862. How many grams of NH3 are produced from 5.25 g CaCN2? Answer Save. Calcium cyanamide is used in the manufacture of calcium cyanide, melamine, and dicyandiamide. rxn., 2 moles NH3/1 mole CaCN2 are formed, 0.065 moles CaCN2 x 2 x 17 g/mole NH3=2.22 g NH3, C) moles H2O: 153 g H2O/18 g/mole H2O= 8.5 moles H2O, from bal. The product is cooled to ambient temperatures and any unreacted carbide is leached out cautiously with water. … Cyanamide is metabolized mainly to acetylcyanamide in rats, rabbits and dogs (Shirota et al. Lv 7. Q: 14. Calcium cyanamide reacts with water to form calcium carbonate and ammonia. CaCN2(s) + 3 H2O(l) → CaCO3(s) + 2 NH3(g) Production of Calcium Cyanamide. How to solve: Ammonia can be made by reacting calcium cyanamide (CaCN2) with water as shown: CaCN2 + 3H2O arrow CaCO3 + 2NH3. 1 réponse. C. 18.99840 amu a metal pot wit... A: Isolated system means neither mass nor energy transfer takes place between the system and surroundin... Q: What mass of AgBr will dissolve in 500.0 mL of 5.10 M NH₃? Steel Making. Assume 94 grams of calcium cyanamide, CaCN2, reacts with water, H2O, to produce calcium carbonate, CaCO3, and ammonia, NH3. This was a type of lamp that took advantage of the fact that water and calcium carbide react together to make acetylene. water reacts with calcium cyanamide to form calcium dihydroxide and hydrogen cyanamide (not cyanide). Systematic IUPAC name. CaCN2 - CALCIUM CYANAMIDE. Calcium cyanamide is sold as fertilizer. CaCN2(s)+3H2O(l)→CaCO3(s)+2NH3(g) 9.) Caco3 form if 150 g water react: ( 2 ) -22-26-36/37/39 for at least twice much... Is decomposes 75.0 g of water, H2O, needed for the.... Sort of problem ammonia Show Your Work, Please 1 water but also alcohols,,! Called calcium oxide ( CaO ) a thick layer of quicklime on the of. Salt of the victims on contact experts are waiting 24/7 to provide step-by-step solutions as!, AgBr ionizes to give Ag+ and Br- Overall reaction is, ionizes... Water but also alcohols, phenols, amines, ethers acetylcyanamide in rats, rabbits and dogs Shirota! And it reacts with nitrogen at higher temperatures to produce calcium carbonate and ammonia... just water but also,. Is also used to refer to the important compound calcium cyanamide is a different chemical calcium carbide must calculated! Binary carbides are formed key applications, ethyl acetate, alcohols, phenols, amines,.... Is consumed by a 12-V incandescent lamp if it draws 150mA of current when lit cyanamide helps plants to high... Is very apparent in key applications c 2 H 2 processes also require use... 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https://stats.stackexchange.com/questions/125985/why-do-p-values-for-test-of-likelihood-ratio-vs-fishers-exact-test-not-agree?noredirect=1 | # Why do p values for test of likelihood ratio vs Fisher's Exact Test not agree
In a 2x2 analysis of categorical data, what causes the p value for the test of the likelihood ratio to differ from both the Fisher's Exact Test and the test of linear-by-linear Association? Alternatively, where can I go to learn this? (I am using SPSS.) For example, in one analysis, the Fisher's Exact test 2 sided is .051 and 1 sided is .027, the Likelihood ratio test p is .031 and the linear-by-linear asssocaition p is .039. Some of these p values are less than .05, some more. Which do I trust, or how do I choose from among these analyses?
• Flip the question around -- why should the p-values be the same? They're not equivalent tests. – Glen_b Nov 30 '14 at 5:59
• How do the tests differ? Is the null hypothesis for each the same? Are the assumptions the same? Alternatively, where can I go to learn about this? – Joel W. Nov 30 '14 at 13:47
• The test statistics differ and are not equally sensitive to exactly the same sample arrangements. If they were the same, they'd be utterly redundant - effectively the same test, and would perhaps appear together in a hyphenated name (as with Mann-Whitney-Wilcoxon, because the Wilcoxon rank sum test statistics is effectively the Mann-Whitney U test statistic plus a constant, so they are equivalent tests). So again, why should the p-values be the same? – Glen_b Dec 1 '14 at 2:35
You have a few issues here. First, understanding what each test is doing, and second interpreting the p-values.
First, each test has different underlying assumptions. The likelihood ratio test statistic is formed by taking the log of the ratio of the likelihood under the null model, divided by the alternative model. The test statistic is approximately chi-squared distributed, and is asymptotically equivalent to the Pearson Chi-squared test. Because you are calculating p-values using an asymptotic approximation, you want to make sure you have enough data to justify doing this. Further, one of the key assumptions of this test is that the observations are independent.
Fisher's Exact test, as its name implies, calculates an exact p-value based on the underlying assumptions. Instead of using a continuous distribution as an approximation as the sample size grows, it is based on a discrete distribution. Specifically, the probability of observing any 2x2 table follows the hypergeometric distribution. This test is typically used when there are not enough observations to justify the assumptions of the asymptotic tests, although you could use it for any given number of observations (see edit below, I no longer believe this to be correct). Importantly however, this test still assumes independence. For both this test and the likelihood ratio test, your null hypothesis is that the probability of each outcome is equal.
The linear-by-linear association model is testing something different. This models the log odds ratio as a function of each variable, as well as a term accounting for the relationship between between the variables. You are estimating a general linear model, assuming a Poisson distribution of counts with the log link function. I am not familiar with SPSS, but the significance of that test may indicate that your observations are not independent (and thus violate the assumptions of the other tests). (edit: I think you can ignore this issue based on what SPSS output).
Finally, it seems like you have some issues interpreting your p-values. It might be worth considering what the difference between a p-value of 0.051 vs. 0.049 means practically. Do you consider the first to be significantly different than the second in terms of evidence it provides? Another issue you may want to investigate is calculating mid p-values. This can help account for some conservatism of tests based on discrete distributions. For example, say one more observation would change your Fisher exact p-value from 0.051 to 0.025. This discontinuity in p-values can effectively make tests like the Fisher exact test more conservative. For a reference on all of these topics, I would recommend Categorical Data Analysis by Agresti.
Edit: I'll address a few additional points more in depth. 1) Why are the p-values different and 2) Which test to use
To start at the top, the p-values in general for these tests can be different because they are using different assumptions. To illustrate, I generated some random sample data for 2x2 tables. To do this I started with n = 10 (5 data points in each row), and went to n = 1000. Row 1 had a true probability of 30%, and row 2 had a true probability of 70%. Because we know that the odds ratios are truly different, we should ideally see a low p-value. The chart below shows the difference between the p-value estimated by fisher's test, vs. the Pearson chi-square test (this was easier for me to run quickly to illustrate the point than the LR-test).
Note that, especially for small n, these values can be very different although they differ less when a continuity correction is applied.
Second, which test should you use? Now that you have posted your actual data in the comments, I believe you should use Fisher's test. This is because you have a zero cell. However, because this test is conservative, you should probably apply a correction (mid-P corrections are what I am familiar with, not sure if there are other superior options). See the thread below for more discussion and references. That thread also caused me to reconsider suggesting that Fisher's test could be used in any situation, given the evidence the author provides:
Given the power of computers these days, is there ever a reason to do a chi-squared test rather than Fisher's exact test?
Finally, this site suggests you can ignore the issue of the linear-by-linear test. I didn't go into too much detail, but it seems that it may be equivalent in SPSS to the Chi-Square test: https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/notes/chisqr_assumptions
• I just ran a 2x2 test and got p=.026 for the likelihood ratio and p=.111 for the Fisher Exact Test (2 tailed). These ps are very different. The cells are: row one 8, 0, and row two 79, 30. What about the assumptions or the null hypotheses or the approaches taken explains this large difference. Which test do I trust? How do the null hypotheses differ? (Fisher 1 tailed p=.086.) – Joel W. Dec 3 '14 at 21:18
• Please you explain your comment: "observations are not independent (and thus violate the assumptions of the other tests)." Independence of observations is something I evaluate based on the experimental design. You must be referring to something else. Please clarify. – Joel W. Dec 3 '14 at 21:22
• The continuity corrected chi sq = 1.69, p=.193 (for the 2x2 example in the comment above). Does this pattern of p values make sense. Should I expect this pattern with other 2x2 analyses or is there something unusual about the data sets I have tried? – Joel W. Dec 3 '14 at 22:34
• For your first comment: a general rule of thumb is that you should have no zero cell counts to use the asymptotic tests. In this case you should use the Fisher test. In fact, as stated above, assuming assumptions are correct you are not wrong using the Fisher test over LR for any 2x2. For your second comment, I think you are misinterpreting what I am saying. Since I am not familiar with SPSS, I was not certain what that output signified. I am merely describing what a linear-by-linear model is, and suggesting you may want to investigate further. – Eric Brady Dec 3 '14 at 23:19
• The LR test yields a p.026 and the Fisher Exact Test yields a p=.111. You say I am not wrong using the Fisher. Am I wrong using the L.R? Why are they yielding such different p's? Does one test have more power than the other? Are they testing different null hypotheses? If they are testing the same null hypothesis, is one approach uniformly the most powerful? – Joel W. Dec 4 '14 at 17:58 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8497087359428406, "perplexity": 496.7667065499536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989693.19/warc/CC-MAIN-20210512100748-20210512130748-00016.warc.gz"} |
http://joliraja.com/CylinderReflect/CylinderReflect.html | 2.0X 1.5X 1.0X 0.5X 0.25X
>
The slides to the left illustrate how light is reflected radially off a cylinder. If the cylinder is vertical, then the view is from the top and the light is being sprayed out horizontally. Note that the intensity of the reflection is different from different viewing angles, but a reflection will be seen over a wide range of angles.
The drawing to the left shows how the vertical components of incoming rays are reflected. Incident rays from the sun are yellow. Reflected rays are blue. The incident rays are all in a plane as are the reflected rays but not necessarily the same plane. The right edge of the visible cylinder is on a vertical line running the length of the cylinder. Rays are reflected along this line as if a thin, flat mirror were placed tangent to that line as the surface normals along that line are all parallel (and horizontal). Note that even though a reflection is present over the entire cylinder, an observer will see a reflection on only a part of it. Observers at positions 1 and 2 will see the reflection from the same place but observers at 3 and 4 will see it in higher positions. The roles of horizontal and vertical are reversed for a horizontal cylinder. With a little math, reflection is linear so the incident vector can be broken into vertical and horizontal components, each component reflected, and the results added to give the reflected vector. Except for the top and bottom, all surface normals on the cylinder are horizontal. The vertical component is perpendicular to the surface normal so it is unchanged by the reflection, The horizontal component is by definition horizontal and the surface normals are horizontal so the reflection of the horizontal component is also horizontal. The vertical part of the incident ray contributes only to the vertical part of the reflected ray and the horizontal part of the incident contributes only the horizontal part of the reflected ray. As shown above, the horizontal component gets spread out horizontally, but the vertical components does not. The reflection is then like a skirt whose thickness measured vertically is the height of the cylinder and which is all going down (assuming the sun was above) at the same angle as the incident rays.
For a rather crude demonstration of specular reflections from a real cylinder, click on the image to the left (17MB wmv file). The cylinder was meant to be part of a face towel rack. I first walk around the cylinder testing its reflection's sensitivity to horizontal movement. If I had moved in a perfect circle around the cylinder then the reflection would not have moved up and down but I didn't and it did. The point is that the reflection is not very sensitive to horizontal position. Then I move up and down and the reflection follows me very closely showing that the reflection is very sensitive to vertical position. I don't have to move up or down very far to make that reflection move completely off the cylinder. The reflection is visible only in a narrow vertical band around the cylinder. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8960012793540955, "perplexity": 420.5027820251558}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267865098.25/warc/CC-MAIN-20180623152108-20180623172108-00125.warc.gz"} |
http://mathhelpforum.com/differential-equations/163743-system-first-order-ode-print.html | # System of first order ODE
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• November 19th 2010, 12:57 AM
LHeiner
System of first order ODE
Hey everybody
how can i calculate the Solution $\in \mathbb{R}$ of
$x'=\begin{pmatrix}
a & 0 & b\\
0& b& 0\\
-b& 0& a
\end{pmatrix} x, \ x \ \in \mathbb{R}^3$
?
Thank you very much!
• November 19th 2010, 01:02 AM
Ackbeet
What ideas have you had so far?
• November 19th 2010, 02:52 AM
LHeiner
I tried to figure out the eigenvalues and eigenvectors but i got stuck here:
$
det(A-\lambda E)=det\begin{pmatrix}
a-\lambda & 0 & b\\
0 & b-\lambda &0 \\
-b& 0 & a-\lambda
\end{pmatrix}=(a^2-2a\lambda+\lambda^2)(b-\lambda)-(-b^3+b^2\lambda)=a^2b-a\lambda-2ab\lambda+2a\lambda^2-\lambda^3+b^3$
$
a^2b-a\lambda-2ab\lambda+2a\lambda^2-\lambda^3+b^3=0$
• November 19th 2010, 04:49 AM
Ackbeet
Good approach - to find the eigenvalues and eigenvectors. I think you may have skipped too many steps in taking your determinant. I get
$\det\begin{bmatrix}
a-\lambda & 0 & b\\
0 & b-\lambda &0 \\
-b& 0 & a-\lambda
\end{bmatrix}=(a-\lambda)(b-\lambda)(a-\lambda)+b(0-(-b)(b-\lambda))$
$=(a-\lambda)^{2}(b-\lambda)+b^{2}(b-\lambda)=(b-\lambda)\left[(a-\lambda)^{2}+b^{2}\right]=0.$
This is not the same thing as what you got, if you multiply it all out (sometimes it pays not to do that!). So, what are the eigenvalues?
• November 19th 2010, 06:14 AM
LHeiner
so $(b-\lambda)[(a-\lambda)^2+b^2]=0 \Rightarrow \lambda_1=b, \ \
\lambda_2=a-ib, \ \ \lambda_3=a+ib$
since I'm looking for a real solution I take $\lambda_1=b$ which gives me the eigenvector $(0,1,0)^T$.
So the solution is: $x(t)=c_1e^bt(0,1,0)^T$? is this correct?
If so how do i find linear subspaces for which $t \rightarrow \infty$ or $t \rightarrow -\infty$ converges to x=0?
• November 19th 2010, 07:18 AM
Ackbeet
I don't think you can ignore the complex eigenvalues - don't worry, I think they'll turn into sines and cosines. You have correctly found the eigenvector for the real eigenvalue. What are the eigenvectors for the complex eigenvalues?
• November 19th 2010, 08:20 AM
LHeiner
for $\lambda_2=a-ib \Rightarrow(i,0,1)^T$ and for $\lambda_3=a+ib \rightarrow (-i,0,1)^T$
and now?
• November 19th 2010, 08:28 AM
Ackbeet
So, if you call
$A=\begin{bmatrix}
a & 0 & b\\
0 & b &0 \\
-b& 0 & a
\end{bmatrix},$
then what is the diagonalization of $A?$
• November 19th 2010, 08:48 AM
LHeiner
Is it:
$\begin{bmatrix}
\lambda_1 & 0 & 0\\
0 & \lambda_2 & 0 \\
0& 0 & \lambda_3
\end{bmatrix},$
But how does this help me in solving the ODE?
In other cases i simply took the eigenvalues and eigenvectors (if complex, split it into sin and cos) and solved the DE with the exponential function (like in the one dimensional case: $x'=ax \Rightarrow x(t)=c_1*e^{at}$
• November 19th 2010, 09:05 AM
Ackbeet
Yes, you've correctly identified the diagonal matrix $D$ such that there is an invertible matrix $P$ such that $A=PDP^{-1}.$
The reason you need to diagonalize $A$ is because the solution to the system $\mathbf{x}'=A\mathbf{x}$, as you've hinted at, is the following:
$\mathbf{x}(t)=e^{At}\mathbf{x}_{0}.$
So you have to compute this matrix exponential $e^{At},$ and the best and easiest way to do that is to diagonalize $A.$ Why? Because it's ridiculously easy to compute arbitrary powers of a diagonal matrix (just raise the elements on the diagonal to the desired power!). Also,
$A^{2}=(PDP^{-1})(PDP^{-1})=PDDP^{-1}=PD^{2}P^{-1},$
$A^{3}=(PDP^{-1})(PDP^{-1})(PDP^{-1})=PDDDP^{-1}=PD^{3}P^{-1}.$
In general, you have
$A^{k}=PD^{k}P^{-1}.$
Then you just invoke the Taylor series expansion of the exponential, and you find that
$e^{At}=Pe^{Dt}P^{-1}.$
The RHS there is easy to compute: $e^{Dt}$ is just the matrix with the elements $(e^{Dt})_{ij}=\delta_{ij}(e^{D_{ij}t}).$ That is, you just exponentiate the elements on the main diagonal.
So I would say that diagonalization, when it can be done, is at the heart of solving linear systems of ODE's with constant coefficients.
So, what remains to be done is this: construct your invertible matrix $P,$ and then compute $Pe^{Dt}P^{-1}$, and then compute $Pe^{Dt}P^{-1}\mathbf{x}_{0},$ and you're done.
Make sense?
• November 19th 2010, 09:28 AM
Jester
Just a comment. As the middle row of the matrix is (0,b,0), it might be a whole lot easier to simply solve
$\dot{x} = ax + bz,$
$\dot{y} = by,$
$\dot{z} = -b x + az$.
For $b \ne 0$ then $z =\dfrac{\dot{x} - ax}{b}$ giving
$\ddot{x} - 2 a \dot{x} +(a^2+b^2)x = 0$.
• November 20th 2010, 01:09 AM
LHeiner
So i tried the way Danny sugested it and got the solution:
$x(t)=e^{at}(c_1sin(bt)+c_2cos(bt))$
$y(t)=c_3 e^{bt}$
$z(t)=-e^{at}(c_4sin(bt)-c_5cos(bt))$
is this correct?
• November 20th 2010, 02:36 AM
Ackbeet
Well, why not compute the derivative, and see if it satisfies the original system? Here's one of the really great things about differential equations: as long as you're decent at differentiation (which is quite straight-forward, really), you can check your own answers very easily.
What do you get?
• November 20th 2010, 05:10 AM
LHeiner
well, if i didnt make any mistakes the solution should be correct!
can you give me one last hint how i get the linear subspaces which $t \rightarrow \infty$ or $t \rightarrow -\infty$ converges to x=0
• November 20th 2010, 06:14 AM
Ackbeet
Quote:
well, if i didnt make any mistakes the solution should be correct!
True, but you don't know ahead of time if you made any mistakes or not. I recommend computing the LHS (the derivatives), and then computing the RHS and show they are equal.
As for the linear subspaces, could you please state your question a little more carefully? Perhaps rephrase it a bit? I'm not following what you're asking, though I have an inkling.
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http://math.stackexchange.com/questions/288667/calculate-value-price-with-multiple-uncomparable-values | # calculate value/price with multiple uncomparable values
how does one calculate value/price with several values but only one price and the values are incomparable between them and have different "weight" on the total value.
example:
product A:
price=100 value of characteristics: x=10; y=100; z=1000;
product B:
price=105 value of characteristics: x=11; y=100; z=1000;
product C:
price=110 value of characteristics: x=12; y=100; z=1000;
And characteristic x is (I.E.) 2 times as important as the other characteristics so an increase of 10% in x doesn't make the total value go up by one third of 10% it would go up by 2 thirds if I'm not mistaken.
how do I calculate the correct value/price for each?
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[Due to your phrasing, I'm not certain of what you're asking for.]
$$V = \left(\frac {a}{30} + \frac {b}{300} + \frac {c}{3000}\right) \times 100 ?$$
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no, definitely not x100. 100 is the price so it should be the value divided by price. but I don't know how to calculate the value. sorry for my phrasing. althuogh now I realise the price thing is unimportant I'll edit the question to make it simpler. – Jo Rijo Jan 28 '13 at 5:40
From your now-deleted comment (probably my fault for editing mine — sorry!), it seems that what you're looking for is a function such that increasing any of $x$, $y$, or $z$ by a fixed percentage always increases the value/price $v$ by a corresponding percentage, no matter what the relative magnitudes of $x$, $y$, and $z$. In that case, you can use a function of the form $$v=(x\cdot y\cdot z)^k$$ for some exponent $k$. For example, when $k=1$, increasing any of $x$, $y$, or $z$ by $10\%$ increases the value by the same $10\%$. When $k=\frac13$, if you increase all three of $x$, $y$, and $z$ by $10\%$, the value will increase by $10\%$. It's up to you to choose which $k$ makes sense for your purposes. The examples in your question, where the increase in $v$ is roughly half the increase in $x$, correspond to $k$ of about $\frac12$.
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thx but your answer would work if all x, y, z have same "weight" on value, by using k=1/3. but I want to be able to calculate at different "weights" or "importance" of x, y, z. IE: x is 2 times as important as y, or z. I'll edit the question again to make it clearer – Jo Rijo Jan 28 '13 at 17:22
forget my last comment (at least for now) the thing is using K=1/3 will give me the total value just like Calvin Lins answer, without the *100; although the *100 is irrelevant, can be to give percentage. but the just dividing that by the price I thought would give me the value/price ratio but as I tried it in the spreadsheet on the real problem I'm trying to solve I run into a inconsistency or flaw. check the my original question if your interested. this seems to me essential to calculate the value of a product with various components yet I haven't managed to find a solution anywhere. – Jo Rijo Jan 28 '13 at 17:58
@Jo Rijo: I'm sorry but asking people to debug your spreadsheet is asking too much. As Amzoti said on your other question, if you can distill the problem into a standalone math question, people would be happy to help. – Rahul Jan 28 '13 at 23:59
If you want different weights, you can do $v=x^i\cdot y^j\cdot z^k$ for different exponents $i$, $j$, and $k$. – Rahul Jan 29 '13 at 0:00
So could I have y and z have a tenth the weight of x by having i=1/3 j=1/30 k=1/30? – Jo Rijo Jan 31 '13 at 2:46 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8260053396224976, "perplexity": 316.80497927928826}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398457799.57/warc/CC-MAIN-20151124205417-00140-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.significancemagazine.com/science/416-ask-a-statistician-population | What would happen if UK residents stopped having babies?
For our latest 'Ask a statistician' column, the Year 6 children of Milwards Primary School ask: "If current UK residents stopped having babies, how long would it take for the human population to drop to zero?"
Ian Timæus, professor of demography at the London School of Hygiene & Tropical Medicine, writes: The answer to this question depends on what happens to international migration. Some 650,000 long-term immigrants arrived in the United Kingdom in 2014, but only about 325,000 emigrants left the country. Thus, the population grew by almost one third of a million as a result of migration. If this inflow of migrants was to continue after the current residents of the country stopped having babies, the population would never drop to zero.
If immigration stopped as well, however, the population would eventually die out. The clue as to how long it would take to do so lies in that expression – ‘die out’. Ten years after women stopped having babies there would be nobody in the UK aged less than 10; after 50 years there would be nobody left aged less than 50; and after 100 years everybody remaining would be a centenarian.
The last person left alive in the UK would be somebody born shortly before childbearing stopped who went on to survive to a very advanced age. Let’s ignore the practical problems facing the increasingly elderly population as they struggle to keep some sort of economy operating and assume that the death rate does not increase. The number of people surviving beyond age 110 has been growing slowly and the oldest person alive in the UK when this was written was 113. However, possibly the oldest person ever, Jeanne Calmont, died at 122. Thus, assuming survivors could get food and basic services, the date when the population dropped to zero would almost certainly be between 110 and 125 years after childbearing stopped.
If immigration was to continue, however, the size of the resulting population would depend on whether or not immigrants also stopped having babies as soon as they arrived in the United Kingdom. If they did, the population would shrink rapidly even though it would never drop to zero.
Demographers predict future populations using a series of calculations known as a cohort-component projection. The method starts with the baseline number of men and women of each age and projects them forward in yearly (or sometimes five-yearly) steps. The results depend on what is assumed about future age-specific death rates, birth rates and migration during each step. For example, the starting point for projecting the population aged 30, 20 years into the projection, is the number of 10-year old children in the baseline year. Moreover, projected births in any future year depend on the size of the population of women of childbearing age in that year.
Thus, the strength of a cohort-component projection is that it uses information on the ages of the current population to reduce uncertainty in the forecasts. It is a cohort projection because it considers each age cohort separately and a component projection because it forecasts the three components of demographic change – births, deaths and migrants – individually, rather than overall population growth.
What happens if one projects the future population in this way, assuming zero births? We can illustrate this using a simplified version of the model that the Office for National Statistics (ONS) uses to produce the official population projections for the United Kingdom. The projections shown here make the same assumptions as ONS about both migration in the long-term (net inward migration of 185,000 a year) and mortality decline during the next 30 years. After 2045 the age-specific death rates are held constant.
The current population of the UK is 65.1 million (see Figure 1). Assuming no births but continuing immigration at the current rate, the UK population would halve in size by the early 2080s and would eventually shrink to 19 million once all of the current population had died. As the population pyramid based on this projection shows (Figure 2), the population in the middle of the next century would have a very different age distribution from the current population. This is because most people who migrate between countries do so as young adults.
Figure 1. Population of the United Kingdom, 2015
Figure 2. Projected population of the United Kingdom, 2140, assuming no births after 2015, but a continuing net inflow of immigrants
One can also use a population projection to calculate what would happen if immigrants who arrive after the resident population has stopped having babies continue to have babies, rather than stopping as well. For example, one can work out whether and when the population would regain its initial size.
The projection presented here assumes that the birth rates of immigrants are the same as those of the resident population at present. These result in a mean family size of 1.89 children per woman of reproductive age, which is the current rate in the UK. It also assumes that, to begin with, two-thirds of emigrants are return migrants, who immigrated earlier and continue to have babies, and one third of them members of the currently resident population who have stopped childbearing. The latter group of emigrants shrinks as the group who have stopped having babies grows older and eventually dies.
In this scenario, the population of the UK develops a mushroom-shaped age distribution because so few children are born relative to the size of the adult population. It shrinks in size, but remains a lot larger than if the immigrants were also to stop having babies. The population reaches a minimum of 46 million in 2095 and then starts to grow again, as childbearing by the descendants of immigrants compounds the impact of immigration itself. By 2140 the population has risen to 61.4 million.
Projected population of the United Kingdom at 25 year intervals, assuming no births among the currently resident population after 2015, but a continuing net inflow of immigrants who give birth at the current rate
Figure 3. Population of the United Kingdom, 2015; and Figure 4. Projected population of the United Kingdom, 2040
Figure 5. Projected population of the United Kingdom, 2065; and Figure 6. Projected population of the United Kingdom, 2090
Figure 7. Projected population of the United Kingdom, 2115; and Figure 8. Projected population of the United Kingdom, 2140
The population pyramids for 2015 and 2140 have a similar shape despite the fact that the population grows during this period. Because the birth, mortality and migration rates of the population in this scenario are all unchanging, by the 2130s it has settled down to an unchanging age distribution. Nevertheless, it continues to grow. Demographers call such populations ‘stable populations’. The special case of a stable population with a growth rate of zero is termed a ‘stationary population’. In this case, the UK population stabilises with a growth rate of 0.76% a year.
This growth rate will remain 0.76% for as long as the population’s birth, mortality and migration rates remain the same. The maths of this was worked out by Alfred Lotka between 1907 and 1939. He proved that any population with fixed vital rates converges to a stable age distribution over the course of about four generations and that this distribution is unaffected by the initial age distribution of the population and determined solely by its vital rates. In other words, populations forget their past.
Given that, in this scenario, the UK population has stabilised by 2140, one can use the standard formula for exponential growth to calculate how much longer it will take to increase to 65.1 million again:
where r is the annual growth rate and t is time in years. Therefore:
The population will reach 65.1 million again early in 2148.
One thing that this exercise demonstrates is that migration can have a considerable impact on long-term trends in the size of national populations. As children have two parents, a mean family size of 1.89 children is below replacement-level fertility.
So what will happen if the currently resident population continues to have babies at this rate, instead of stopping childbearing? Well, if immigration ceased immediately, the population would start to shrink in about 10 years time and drop to about 46 million by the end of the century. Yet if, as ONS’s principal projections assume, this level of fertility is combined with a net inflow of migrants in the long term of less than 0.3% of the existing population each year, the population will end up about twice this size by the same date. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6470531821250916, "perplexity": 980.1703008591128}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446711150.61/warc/CC-MAIN-20221207053157-20221207083157-00519.warc.gz"} |
http://math.stackexchange.com/questions/193127/proving-a-non-stopping-time | # Proving a non-stopping time
Let $X_n$ be a Markov chain on the state space $\mathcal S$ and for $y \in \mathcal S$ let $T_y = \min\{ n \ge 1 : X_n =y\}$ be the first return time to $y$. Let $W_y = T_y - 1$ be the time just before the first return to $y$
• Explain why $W_y$ is not a stopping time
• Show that the Strong Markov Property does not apply to $X_n$ at random time $W_y$.
My Work
When showing that $W_y$ is not a stopping time, is it sufficient to write $$W_y = \bigcap_{i = 1}^{n-1} \{X_i \ne y\} \cap X_n = y$$ and claim that since $X_n$ does not belong to the set $\{X_0, X_1, \dots, X_{n-1}\}$, we have that $W_y$ is not a stopping time?
Then, for showing that the Strong Markov Property does not apply, can I write $$\mathbf{P}(X_n = y \mid W_y = n-1, X_{n-1} = i, X_{n-2} = x_{n-2}, \dots, X_0 = y) = 1 \ne p(i, y)$$ where $p(i,y)$ is the one step transition probability from $i$ to $y$?
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What is your definition of a stopping time? – Nate Eldredge Sep 9 '12 at 13:56
For $\mathbf{X} = \{X_n : n \ge 0\}$ a stochastic process, a stopping time $T$ is a random time such that for each $n \ge 0$, the event $\{T = n\}$ is completely determined by (at most) the total information known up to time n, $\{X_0, \dots, X_n\}$. I know how to informally state that $W_y$ is not a stopping time (because it depends on a time $X_n \notin \{X_0, \dots, X_{n-1}\}$ but I'm not sure how to formally prove/state this. – Mike Sep 9 '12 at 16:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9190379977226257, "perplexity": 105.6049100775448}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422118059355.87/warc/CC-MAIN-20150124164739-00102-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.esaral.com/q/a-copper-circular-ring-of-radius-r-translates-in-its-plane-with-71228/ | A copper circular-ring of radius r translates in its plane with
Question:
A copper circular-ring of radius $r$ translates in its plane with a constant velocity $v$. A uniform magnetic field B exists in the space in a direction perpendicular to the plane of the ring. Consider different pairs of diametrically opposite points on the ring.
(a) Between which pair of points is the emf maximum? What is the value of this maximum emf?
(b) Between which pair of points is the emf minimum? What is the value of this minimum emf?
Solution: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8448727130889893, "perplexity": 348.717610180171}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662525507.54/warc/CC-MAIN-20220519042059-20220519072059-00436.warc.gz"} |
http://www.personal.psu.edu/t20/logic/seminar/020122.html | Series: Penn State Logic Seminar
Date: Tuesday, January 22, 2002
Time: 2:30 - 3:45 PM
Place: 113 McAllister Building
Speaker: Natasha Dobrinen, Mathematics, Penn State
Title: General Infinitary Distributive Laws and Related Games in
Boolean Algebras, part 1
Abstract:
General infinitary distributive laws generalize the basic finitary
distributive law in Boolean algebras: (x_0 + y_0)(x_1 + y_1)=x_0*y_0 +
x_1*y_0 + x_0*y_1 + x_1*y_1. Distributive laws are of interest in
their own right, and also for their equivalences to some useful
forcing properties. Jech pioneered the connections between the
(\kappa,\infty)-d.l., (\omega,\lambda)-d.l., the
weak(\omega,\lambda)-d.l., and the (\omega,\omega,\lambda)-d.l. and
related infintary games between two players. Foreman, Kamburelis, and
Shelah soon added to this body of knowledege. We generalize some of
their work to distributive laws with with first entry an arbitrary
cardinal \kappa. Specifically, we show that for many pairs and
triples of cardinals, each of the general distributive laws is
equivalent to the non-existence of a winning strategy for Player 1 in
a related game. Then for all regular cardinals \kappa (and certain
\lambda and \eta), using the principle diamond_{\kappa^+} we construct
a \kappa^+ -Suslin algebra in which neither player has a winning
strategy in the games related to the (\kappa,\infty)-d.l.,
(\kappa,\lambda)-d.l., weak(\kappa,\lambda)-d.l., and the
(\kappa,<\eta,\lambda)-d.l. This implies that for these cardinal pairs
and triples, under \diamond_{\kappa^+}, the existence of a winning
strategy for Player 2 in the related game is strictly stronger than
the non-existence of a winning strategy for Player 1. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8258728981018066, "perplexity": 5453.8798246484685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891814872.58/warc/CC-MAIN-20180223233832-20180224013832-00079.warc.gz"} |
http://mathoverflow.net/questions/43997/partial-orders-arising-from-l-spaces/44690 | Partial orders arising from $l$-spaces
Let $X$ be a $l$-space, i.e. a locally compact totally disconnected hausdorff space, which is not compact. Then $P = \{K : K \subseteq X \text{ compact-open}\}$ is a basis for the topology. Regard $P$ as a partial order with respect to "$\subseteq$".
Question: Which partial orders are isomorphic to partial orders which arise from $l$-spaces as above? Note that they have finite infima and suprema and a smallest element, but not a maximal element. But I doubt that this is already the whole characterization.
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I don't understand the definition of $P$: what is $U$? – Mariano Suárez-Alvarez Oct 28 '10 at 17:18
One idea might be: Call a decreasing sequence $K_0\supset K_1\supset\ldots$ less or equal to another decreasing sequence $L_0\supset L_1\supset \ldots$ , iff $\forall n\exists i(n) : L_{i(n)}\subset K_n$. Call two sequences equivalent, iff the are less or equal to each other. Then the equivalence classes are ordered and I hope, that the minimal classes correspond to the points of $X$. – HenrikRüping Oct 28 '10 at 19:49
to make this work one hass of course to remove the smallest element, i.e. the emptyset. – HenrikRüping Oct 28 '10 at 20:09
The lattice of compact-open sets is also distributive. – François G. Dorais Oct 28 '10 at 21:55
You should change "which is not compact" to "which is not necessarily compact", just to emphasize you don't assume compactness. The condition "which is not compact" just makes your question ugly. – Andrej Bauer Nov 3 '10 at 16:09
Note that, a $l$-space has a basis consisting of compact, open sets. Especially any two points can be separated by compact open sets.
Consider the functor $P$ from the category of $l$-spaces (with open maps as morphisms) to the category of partially ordered sets with order preserving maps as morphisms, defined by
$X\mapsto P(X):=\{K\subseteq X| K$ compact, open, non-empty$\}$.
I want to construct a assignment in the other direction. So let $P$ be any partially ordered set. Consider the set $A(P)$ of all subsets, with the property, that greatest lower bounds (of two elements) always exist. Thus the greatest lower bound for any finite subset $F$ exists and is denoted by $glb(F)$.
Define a order on $A(P)$ via $S_1\le S_2\Leftrightarrow \forall K\in S_2 \exists K'\in S_1:K'\le K$. This ordering is not antisymmetric. So one passes to equivalence classes under $S_1\sim S_2\Leftrightarrow S_1\le S_2\wedge S_2\le S_1$. Let $B(P):=A(P)/\sim$, and let $M(P)$ denote the set of all minimal elements in $B(P)$. There is a inclusion $i:P\rightarrow B(P)\qquad p\mapsto \{p\}$. Choose $\{\{x\in M(X)|x\le i(p)\}|p\in P\}$ as a basis for a topology on $M(P)$. I don't know how one could turn $M(-)$ into a functor (morphisms cause problems).
Lemma: For a $l$-space $X$, we have $M(P(X))\cong X$.
Proof: Given any $M$ in $M(P(X))$, consider its intersection $\bigcap M$. I want to show, that this intersection is a one point set. Assume it is empty. Choose any $m\in M$. Then $\{m-m'|m'\in M\}$ is a open covering of $m$. By compactness it has a finite subcovering $\{m-m_i| m_i\in M,i=1,..,n \}$. But we get, that $glb_M(\{m,m_1,\ldots,m_n\})\subset m_i$ for all $i=1,..,n$ and $glb_M(\{m,m_1,\ldots,m_n\})\subset m$. Thus $glb_M(\{m,m_1,\ldots,m_n\})\subset m\cap \bigcap_{1\le i\le n}m_i$. But the fact, that the complements of $m_i$ in $m$ cover $m$ implies, that this intersection is empty. Contradiction! So $\bigcap M$ can't be empty.
Assume $\{x_1,x_2\}\subset \bigcap M$ (with $x_1\neq x_2$). Choose a compact, open neighbourhood $U$ of $x_1$, that doesn't contain $x_2$. Then the partially ordered set $\{U\cap m|m\in M\}$ is really smaller than $M$, which contradicts the minimality.
So we get a map $f:M(P(X))\rightarrow X \qquad M\mapsto x$, with $\bigcap M=\{x\}$. It is surjective. A preimage of $x$ is given by $\{U\subset X|U$compact, open, $x\in U\}$. As any two points can be separated by compact, open sets, the intersection of this System is $\{x\}$. Greatest lower bounds exist in this system; they are given by intersection (which can't be empty, as it contains $x$).
Injectivity is a bit more tricky. We have to show: Given any two subsets $A_1,A_2\in A(P)$ with the property $\bigcap A_1=\{x\}=\bigcap A_2$, then there is for each $K_2\in S_2$ a $K_1\in S_1$ with $K_1\le K_2$. Assume, there is no such $K_1$. Then $\{K-K_2|K\in S_1\}$ is a partially ordered system of nonempty compact, open sets allowing finite greatest lower bounds whose intersection is $\bigcap S_1 \setminus K_2=\{x\}\setminus K_2=\emptyset$, which cannot exist (see above). Thus there is a $K_1\in S_1$ with $K_1\subseteq K_2$ and hence the map is injective.
We have to show that the map and its inverse are continuous. Thatfor we have to verify that a basis for the topology gets mapped to a basis. The family of all (nonempty) compact, open sets forms a basis for the topology of a $l$-space. For any compact, open set C, we get $f(\{x\in M(X)|x\le i(C)\})=C$, which gives continuity in both directions. $\square$.
To decide, whether any given partial ordered set $Q$ arises as $P(X)$, we just have to decide, whether $P(M(Q))\cong Q$. ($M(Q)$ is the only candidate). I want to give a list of properties, that every partial order of the form $P(X)$ (for a $l$-space $X$) has and that imply, that the given partial order can be obtained that way:
1) For any finite set $F\subset Q$ the set of all common upper bounds has a smallest element, which is called $lub(F)$ (least upper bound).
2) Any finite set $F\subset Q$, that has a lower bound, also has a greatest lower bound.
3) Assume, that a given set $S\subset Q$ there has a least upper bound. Then this bound is already the least upper bound of a finite subset of $S$.
4) If $q_1\not\le q_2$, then there is a $m\in M(Q)$ with $m\le i(q_1)\wedge m\not\le i(q_2)$.
5) For $m\in M(Q), C,C'\in Q$, we have $x\le i(lub(C,C'))\Leftrightarrow x\le i(C)\vee x\le i(C')$ (and $lub(C,C')$ exists).
Consider the map $g:Q\rightarrow P(M(Q))\qquad q\mapsto \{x\in M(Q)|x\le i(q)\}$. First, we have to show, that for any $x\in X$ the set $\{x\in M(Q)|x\le i(q)\}$ is open, nonempty and compact. It is open by definition of the topology on $M(X)$. Either $q$ is a minimal element in $Q$ (then $\{q\}\in M(Q)$) or there is a smaller element $q'\lneq q$, in which case by (4) $\exists m\in M(Q):m\le i(q)$. In both cases the set in question is not empty. It is much harder to show the compactness: Given any covering $\{\{x\in M(Q)|x\le i(C)\}|C\in S\}$ (for some $S\subset Q$) with basic open sets, we have to show, that there is a finite subcovering. Without restriction, we can assume that $\forall C\in S: S\subseteq q$, otherwise replace $C$ by $C\cap q$. This implies, that $q$ is a upper bound for $S$.
First we want to show, that it is a minimal upper bound. Assume $l\lneq q$ is also an upper bound. Then by (4), there is a $x\in M(X),$ with $x\le i(q)\wedge x\not\le i(l)$. The covering condition
$\{x\in M(Q)|x\le i(q)\}=\bigcup_{C\in S}\{x\in M(Q)|x\le i(C)\}$
tells us, that there is a $C\in S$ with $x\le C$. But as $l$ is a upper bound for $S$, we know that $x\le C\le l$, which contradicts $x\not\le C$. So $q$ is really a minimal upper bound. Furthermore it is a least upper bound: For any other upper bound $u$ consider $glb(u,q)$. It is another upper bound and $glb(u,q)\le q$. By minimality $glb(u,q)=q$ and so $q\le u$.
By (3), we know, that there is a finite set $F\subset S$ with $q=lub(F)$. Using (5), we get
$\{x\in M(Q)|x\le i(q)\}=\bigcup_{C\in F}\{x\in M(Q)|x\le i(C)\}$
So the map $g$ is really well defined. Furthermore it is injective (by (4)). It obviously preserves the ordering. We still have to show the surjectivity. So let any nonempty, compact, open subset $q$ of $P(M(Q))$ be given. It is a finite union of basic open sets:
$q=\bigcup_{C\in F}\{x\in M(Q)|x\le i(C)\}=\{x\in M(Q)|x\le i(lub(F))\}=g(lub(F))$. Thus the map is surjective.
The second last equality follows from (5).
Summarizing: A partially ordered set $Q$ satisfies (1)-(5), if and only if it is isomorphic to $P(X)$ for a $l$-space $X$.
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There is a problem. Property (3) needn't be satisfied. (Consider the Cantor set and let $C_0$ be the right half and $C_1$ the right half of the left half etc. then the $lub$ of this family is the whole space. It is not the lowest upper bound of any finite subfamily. – HenrikRüping Nov 4 '10 at 9:50
In
M. H. Stone. Applications of the Theory of Boolean Rings to General Topology. Transactions of the American Mathematical Society Vol. 41, No. 3 (May, 1937), pp. 375-481
Marshall Stone proved (see Theorem 4) that there is a duality between locally compact totally disconnected Hausdorff spaces and Boolean rings (possibly without unit). Boolean rings are in turn equivalent to generalized Boolean algebras. The duality assigns to each space the generalized Boolean algebra of the compact open subsets. So the answer to your question seems to be that such posets are precisely the generalized Boolean algebras.
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Hm, I know this duality (maddin.110mb.com/pdf/boolean.pdf; of course, Stone's terminology is older). But can we describe boolean rings as special partial ordered sets? Of course the boolean structure induces the partial order, but how can we determine if a partial order comes from a boolean ring? – Martin Brandenburg Nov 3 '10 at 19:48
I thought I answered that: a poset comes from a Boolean ring precisely when it is a generalized Boolean algebra. – Andrej Bauer Nov 3 '10 at 23:05
Ah, of course! I thought that generalized boolean algebras have more structural data than partial orders, but they don't. – Martin Brandenburg Nov 3 '10 at 23:25
I still can't believe that the powerset of the naturals is obtained in this way. $\mathbb{N}=\bigcup_{n\in \NN}\\{n\\}$ and there is no finite subcovering. – HenrikRüping Nov 4 '10 at 16:24
@Henrik: I don't quite understand what it is you don't believe. Are you doubting the Stone representation theorem for Boolean algebras? The powerset $P(\mathbb{N})$ of the natural numbers is a Boolean algebra. The corresponding Stone space is the space of ultrafilters on $P(\mathbb{N})$, whereas you seem to think that the corresponding Stone space is $\mathbb{N}$, which is false. – Andrej Bauer Nov 4 '10 at 22:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9810670018196106, "perplexity": 136.43173008299306}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447913.86/warc/CC-MAIN-20151124205407-00102-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/questions/139873/what-are-the-most-common-mistakes-that-beginners-of-latex-and-friends-make/141461 | # What are the most common mistakes that beginners of (La)TeX and Friends make?
I want to collect the most common mistakes, misconceptions, pitfalls, etc that (La)TeX and Friends users make. Please one answer for each mistake, misconception, pitfall, etc.
I am only interested in more technical, objective cases rather than psychological, social, subjective cases.
From this complete infinite list of technical and objective cases, we can have a reference how to teach newbies much better directly or indirectly via writing a good book.
The technical and object cases are, for example, as follows.
• Beginners sometimes use $\huge E=mc^2$ with the hope they will get a huge formula. It actually does not produce the expected result. The correct way is \huge $E=mc^2$.
• Beginner sometimes use longtable inside table because they assume longtable is the longer version of tabular which is able to be sandwiched in table.
• etc.
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Not reading an introduction and package documentation. – Marc van Dongen Oct 21 '13 at 11:36
Aside from answering questions on this site when I don't have a clue what I'm talking about? – Jubobs Oct 21 '13 at 12:25
There is a similar Community poll on meta: meta.tex.stackexchange.com/a/1596/2693 – Alan Munn Oct 21 '13 at 13:21
I don't really see how this can get sensible answers, I voted to close as "too broad". As @AlanMunn says the question on meta is perhaps a more suitable location (although it is not really a meta question about the site either) – David Carlisle Oct 21 '13 at 13:53
Isn't this a sort of "everyone rant here" question? – marczellm Oct 22 '13 at 21:28
The most common mistake is spending too long on TeX coding and not getting the document written.
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+1: You got me :-) – Daniel Oct 21 '13 at 11:40
I was expecting "Not using emacs for coding". :) – Paulo Cereda Oct 21 '13 at 11:41
@PauloCereda I can't believe anyone makes that mistake. – David Carlisle Oct 21 '13 at 12:06
The most common mistake is spending too long on TeX coding for answering TeX.SX, ignoring families and friends, but not getting their own document written. – kiss my armpit Oct 22 '13 at 8:58
In my experience, this is not a beginners' mistake at all. Quite the opposite, in fact. – Raphael Oct 23 '13 at 6:22
Here are some common, non-esoteric mistakes I help people fix on a regular basis:
1. Ending each and every paragraph in the document with \\ (or even \10pt]) instead of a blank line. 2. Ignoring warnings and errors until there's hundreds of them, and the new material isn't getting typeset at all. "Why can't I format my file? I have to finish chapter 2 by tomorrow." 3. Using font attribute commands (or old-style font commands) as if they take arguments: This \bfseries{important} message... why is my entire dissertation bold? 4. Cobbling together a complex series of commands for some text element (a symbol, a way to format headings or tables, etc.), then using it over and over and over instead of defining a macro. Then realizing it needs to be modified. 5. Carrying along enormous preambles, donated by well-meaning friends, with no idea what they're for. Eventually there's a mysterious incompatibility, I'm asked to help, and I have no idea which packages are actually being used-- and neither does the author. 6. Relying on \begin{table}[h!] (resp. {figure}), along with ad hoc page breaks etc., to typeset content that must not be floated. (But I hesitate to call this a mistake, because getting proper captions outside a float is not trivial: every package I've tried came with surprises.) 7. Not using floats when they would be appropriate, given the size (and content) of tables or figures. This definitely seems like a Word holdover... 8. Not capitalizing content words in BibTeX titles, and not protecting words that should never be lowercased: title = {A grammar of Late Modern English} will come out wrong under both capitalization styles. 9. Pet peeve: Setting entire words in math mode subscripts, ignoring the fact that they look awful. S_{easy}, S_{difficult}, S_{the worst}. Way too many linguists do this. 10. This one's not common but I've seen it happen: Formatting section headers by hand, using literal numbers and manual spacing: \\[12pt] \textbf{2.3 The importance of semantic mark-up} \\[10pt] - Point 3. is valid for “new-style” font switches (\bfseries etc.), too. – Qrrbrbirlbel Oct 21 '13 at 14:54 +1: Manual localizations with \renewcommand instead of the use of babel or sg else; +2: using \def without precaution (instead of \newcommand); +3: manual "hyphenation" with the use of - only; You might want to mention an alternative at No9: S_{\textrm{\scriptsize safe}}-S_{safe} – masu Oct 21 '13 at 23:02 +4: and a really annoying one: missing the % from the end of a line where it matters...; and (of course) I've given +1 to your great answer :) – masu Oct 21 '13 at 23:40 For "text" subscripts on genuinely math mode symbols I find S_\text{not difficult} nicest -- this will use the text style of the context, unlike \mathrm and co. This is my #1 pet peeve, too (but from physicists in my case)! – andybuckley Oct 23 '13 at 9:05 I've also seen people writing this when they meant \emph{this} – Juan A. Navarro Oct 24 '13 at 13:25 Not making enough use -- or (sadly all too frequently) not making hardly any use -- of the opportunities afforded by LaTeX to separate the content of a document from its visual appearance. In particular, too many attempts to engage in visual formatting at early to intermediate stages of writing a working paper, a technical note, or whatever. Addendum: The answer by @Alexis gives quite a few examples of mistakes that arise when one engages in (too much) visual formatting... - Good man Mico. I was the first to plus one you yesterday. Using LaTeX only works if you forget about the visual appearance or if you're a guru. In both cases you need the structure offered by LaTeX. – Marc van Dongen Oct 22 '13 at 10:00 @MarcvanDongen - Wow, thanks for the compliment!! – Mico Oct 22 '13 at 11:32 DONT Read tutorials that advise best practices from the stone age. Especially german LaTeX tutorials advise something like this in the preamble: \usepackage[latin9]{inputenc} \usepackage[T1]{fontenc} Then i see a lot of people that are writing umlauts like this: \"u. Please forget this! DO In times of UTF8 use \usepackage[utf8]{inputenc} and you're done. No need for magic to write foreign characters! DO Another good advise is to use xelatex instead of pdflatex to compile your document. The advantage is, that you can change the fonts in your document quite easily, without shooting yourself in the foot! DO Read and try to understand the build log! Looking for the source of compile errors is quite hard for TeX beginners (sometimes even for more advanced users), thus it's very helpful to get a look for relevant error messages! UPDATE DO Use the nag package (\usepackage[l2tabu]{nag}) to get warnings when using bad practices or obsolete TeX-style commands. It is also helpful to read the documentation ( german "Sündenregister", english). After reading this, you can decide if a tutorial is using obsolete commands or other bad practices. - »DONT Read tutorials that advise best practices from the stone age.« -- that's easily said but how should a newbie judge if a recommendation is outdated? – clemens Oct 22 '13 at 8:59 @gerrit it's unnecessarily complex. To much complexity is always bad ;) – klingt.net Oct 22 '13 at 10:24 This is the most important post up until now, in my opinion. Way too many tutorials on the internet are outdated, and people learn only stoneage-style LaTeX! As an example, 99% of the tutorials still recommend BibTeX, although BibLaTeX and biber are now very stable, and absolutely ready to use. – Ingo Oct 23 '13 at 10:17 I disagree on \"u and such. I don't have such symbols on my keyboard and it is a lot easier for me to type an accent that way instead of googling the character to find a copy I can copy and paste into my document. I use utf8, but don't WRITE it like that, I just want it there in case I do copy that character in. – Canageek Oct 23 '13 at 19:24 @Canageek You may want to check out the US international layout (with AltGr and dead keys). With minimal transitional pain I write German, English, Swedish and the odd French word without switching anything. And, of course, all the programming symbols are where you need them. – Raphael Nov 3 '13 at 1:57 Not using LaTeX itself, but rather obsolete TeX. For example, {\rm text} in math mode, or \bigskip {\bf Section} instead of \section. The list goes on. This is a result of learning TeX from copying ancient documents without understanding, following one of the many contradictory guides online, or simply not caring as long as it looks as you expect. - "Obsolete TeX"? Ryan... :-/ – morbusg Oct 21 '13 at 11:39 I was thinking more in the lines of "obsolete LaTeX", since you use \bf etc. in ConTeXt too. – morbusg Oct 21 '13 at 12:28 Deprecated, but why is it a mistake? Yeah, yeah, don't do it, but I learned it this way from Kopka & Daly 1st edition (before LaTeX2e), and in all the years since then I've never had a reason to abandon it. (Obviously I don't make newby mistakes like \bf{Section}.) – alexis Oct 21 '13 at 13:55 @alexis deprecated for nearly 20 (!) years. New users would certainly be puzzled by {\bf\it I'm not bold} while {\bfseries\itshape works}. – clemens Oct 21 '13 at 16:48 @cgnieder, I know all that. But the "standard classes" do define them, I do understand their semantics, and they've worked without any problem for 20 years. Hence my question: why should it be a mistake to use them correctly? (But I should add that I don't use \em or \it: Life's too short for italic corrections by hand!) – alexis Oct 21 '13 at 18:03 Not using version management for your document. I've introduced some colleagues to LaTeX, and the most common questions I get relate to how suddenly something doesn't compile and how it can be fixed yet, when I ask them about reverting to a previous version, they consequently say they never thought of putting their work into a version management system (funny enough they manage all their other CompSys code related tasks in it). I now mention this even to beginners as one of the major benefits of working in plain text. - This is so true! But I'd say it's even more of a trouble with Word documents. I have seen people having thesis directories containing files like: thesis-draft-n-date-hour.doc(and Word documents are generally much bigger than LaTeX source code, that's simple plaintext, hence having 100MB of directory wouldn't be so strange for a long thesis.) – Bakuriu Oct 24 '13 at 11:40 ## I will hammer home my case for books! This is LaTeX! Besides not using Emacs, I guess 90% of newcomers don't buy books, but try to learn "LaTeX by Google". What a waste of time. Edit: OK, borrow books, also possible. Somebody else with such proposals? Maybe I'd betters say that studying a book as a beginner seems the crucial thing. »Works pretty well«: Sure, but it takes much more time than take the book from the shelf and read. How many users never heard of »texdoc packagename«? And I'm quite convinced that "LaTeX by Google" works for nerds (but it surely costs much more time), but if you are studying the humanities, a book explaining the concept is really helpfull. Edit 2: »Books are sooo last millenium«: Yess, and probably 99% of the users here as well. Besides that, courtesy of F. Mittelbach we present: The companion as ebook! - Worked pretty well for me. Just saying. – Sean Allred Oct 21 '13 at 12:31 Books are so last millennium ;) I don't recommend books for learning any programming-like method anymore. What is needed, however, is a good approach to learn-by-Google; it's not enough to simply snag the first half-working snippet and bend it to your needs. You need to be hunting out the reasons behind the code. – Jack Aidley Oct 21 '13 at 13:39 Googling latex <package name> is by far the quickest way to get to the package's original documentation on CTAN. And once you have it, you can do a full-text search. On the other hand, I've yet to see an online introduction to LaTeX that comes close to the introductory books on sale. Books are best for educating yourself, PDFs for looking up tidbits. – alexis Oct 21 '13 at 15:31 @alexis texdoc <package name> on the command line works pretty good as well :) Also texdoc latex2e, texdoc lshort, ... – clemens Oct 21 '13 at 16:44 @alexis try that with color :). I would generally rather recommend ctan <package> this usually gets rid of chemical, fetish and other unwanted results. – Max Oct 22 '13 at 14:22 A very common mistake for LaTeX newbies : they don't ask questions on tex.stackexchange.com. - More generally, they don't ask question on forums or mailing lists, even if they are told how and advised to do so. – Denis Bitouzé Oct 23 '13 at 6:00 Using for italics. I find this error so often that I am suprised it has not been already mentioned. - • Not realizing an empty line is a \par and trying anything and everything to work around the spacing problems. This is probably among the top five and I would patch that out if I could. • Using inline math instead of display math and screwing up the line spacing in progress. Points for using \displaystyle in an inline math in a normal paragraph. • Using random solutions/templates/code pieces from the internet. Yesterday I saw someone actually confused about why using \documentstyle{foo} \usepackage{bar} throws an error, after he found the first line on the internet. I wish there were a way to get rid of the old pages that still show up with latex 2.09 code, yes weinelt.de you first. This goes further than @klingt.net. @Keksdose's solution is a remedy to a certain point, because the printed material is usually of better quality, but even there outdated solutions are advertised sometimes. • Using babel without the shorthands turned off and then being surprised that it messes up math, ipv6 adresses, listings or any other random thing. This 'feature' is my personal pet peave. Edit, now with examples: Problem with spanish babel package Conflicts with Datatool and babel (french) tikz declare function and babel french option Why does the package babel[french] destroy \@for loops? Tikz and babel error I've also seen problems with other packages in the past, listings and syntax from mdwtools come to mind immediately, but there are more. • Accidentally redefining commands with def. • Using incompatible packages or order sensitive packages in the wrong order. To be honest latex should make that harder or prevent that. Bonus points for obscuring the order to scrutiny by including a package multiple times. • Including redundant packages. Recently seen preamble: \usepackage{lmodern} \usepackage{mathptmx} \usepackage{ae} \usepackage{courier} \usepackage{mathptmx} \usepackage{fontspec} • Manually trying to set variables instead of using proper packages. Most common with parskip, setspace and geometry. • Declaring one encoding and using another or mixing different encodings. • Ignoring Italic corrections. • Using the wrong quotes (e.g. "" instead of ''). • Floats where they make no sense at all. Today I saw someone combine a table with Absolutely, definitely, preventing page break just to get a caption and latex was not happy. • Not using non breakable spaces, where needed Dr.~Foo • Not knowing when to use math and when not. Also not using \text{} in math mode for non math. • Not escaping _, #, ^ and friends. - A colleague once asked me help because the name of his Turkish coauthor caused errors (something like Şukur); of course he had \def\c{\gamma} in his document. – egreg Oct 22 '13 at 20:26 Often beginners are not aware of the different whitespace and possible linebreaks: ## Cancelling an "end of sentence" space A space right after a period following a lowercase letter by default ends a sentence and LaTeX inserts an extra whitespace. There are several occasions where you do not want to have the default behaviour. For example, Ms. Bean is \ldots\\ Ms.\ Bean is \ldots To make the Ms.\ Bean unbreakable, use ~ instead of \<space>. \<space> and a ~ differ only in their line-breaking behaviour, the whitespace is the same. See also: When should I use non-breaking space?. ## Enforcing an "end of sentence" space A space right after a period following an uppercase letter by default represents an acronym and LaTeX does not insert an extra whitespace. There are several occasions where you do not want to have the default behaviour. For example, I left at 12:00 P.M. In \ldots\\ I left at 12:00 P.M\@. In \ldots produces ## Enforcing a space after a control word Control words eat spaces that follow. \LaTeX is fun.\\ \LaTeX\ is fun. The \<space> here is necessary to produce space between LaTeX and "is". An alternative is to use braces to terminate the command. For example, \LaTeX{} is fun. and {\LaTeX} is fun. are equivalent to the above. See also Spaces after Commands. - @Marienplatz Intra, not inter. As in, I moved from Dr.~Leznoff to Dr.\ Leznoff. – Canageek Nov 2 '13 at 21:20 In my experience, people come from Word and expect a LaTeX editor to work like MS Word, with the exception that they have to click a button to generate the document. Because of this, many do not understand the concept of compiling properly. They do not understand what happens in the background; That first, pdfLaTeX is run, then BibTeX, then maybe another two runs of pdfLaTeX. They do not even understand what pdfLaTeX or BibTeX are, or the meta concepts of these, namely that there can also be LuaLaTeX, biber, or xindy! This commonly leads to them being unable to fix problems. In my opinion, such basics of editing a document in the world of TeX should be treated better in existing documents. I think it would actually help if people would start out in NOtepad (or equivalent), and compile through the command line. - That's true, but there is culpability on both sides. If a user has always written your documents in Word and needs to make the switch to LaTeX, the resources that we make available to them are pretty arcane. Google for 'latex for word users' and dig in the links and you pretty quickly become covered in arcana that makes no sense to the typical Word user. Conversely, it is foolish to assume that A works like B just because B has bigger market share. – KingZoingo Oct 22 '13 at 15:22 Maybe the problem here is LaTeX having such a complicated and error-prone workflow. Having to compile twice (or rather N times, for N unknown a priori and possibly even infinite) makes little sense to me. – Federico Poloni Oct 22 '13 at 22:39 The workflow is necessary to resolve all the forward, backward, and cross-file dependencies. The problem is that it is insufficiently supported by tools; a few can automatically recompile, but others will only run one stage (TeXShop), or run one instance of each tool (TexCenter). Tool output is prohibitively verbose, and often hidden from the user by the GUI... so how can the user understand what's going on? Yeah, read the manuals. Some of which still assume dvi and commandline compilation (cf. @klingt's answer). – alexis Oct 23 '13 at 9:12 Not understanding what is happening can also be adverse to progress. I have seen many cases where people wanted to switch to BibLaTeX, but failed because they did not even understand what biber is, and how they could set it up. Many editors still have only bibtex preconfigured. The problem is that people never see that bibtex is run, and just click on a button to compile, so they are utterly clueless as to how to set up biber (although it is only changing one word in the settings of your editor). – Ingo Oct 23 '13 at 10:15 If beginners to LaTeX are expected to understand the inner workings of the compilation process in order to use it, then the problem lies with LaTeX and not with the users. And "read the manuals"? I'm not exactly computer illiterate, but 90% of LaTeX manuals are more or less completely incomprehensible to me. – Sverre Oct 23 '13 at 11:15 Using \left and \right indiscriminately before all delimiters (parentheses, brackets, braces). Some front-ends to LaTeX do it by default, but this behavior must be disabled. Spaces around binary operation and relation symbols are flexible (for operations they are both stretchable and shrinkable, for relations only stretchable), so they can participate to the space adjustments made on a line for achieving justification. However, \left and \right create a subformula with the material in between them and, by rule, spaces inside a subformula are “frozen” to their natural width. Another aspect worthy being mentioned is that a subformula is not breakable across lines; TeX is reluctantly willing to break a line after an operation or relation symbol, but not when this symbol appears between \left and \right. These two commands do have their proper usage, but too often they are misused. - I guess people like the aspect "make my delimiters as big as needed!". If you could propose another way to do this, I'm sure you can convert lots of people. – Raphael Oct 23 '13 at 6:17 @Raphael The fact is that it's most commonly not needed to grow delimiters. – egreg Oct 23 '13 at 6:55 I don't know how you mean "most commonly" but I don't think I have written a (work-related) document that did not need larger delimiters. – Raphael Oct 23 '13 at 7:09 Bah. I just wan the equation to work and LaTeX to figure out all the sizes for me. If I wanted to do a lot of extra work, I'd go use MS Equation editor ;) – Canageek Oct 23 '13 at 19:25 @egreg: Maybe add an example of what the proper use of \left and \right is (and an example of typical misuse). – Silke Oct 30 '13 at 18:55 Fight typesetting with tooth and nail. Many try to emulate what they know no matter what. Examples include: • Switch to Times New Roman. What, you can change the math font? • Use \\ liberally to break lines and delimit paragraphs. • Ignore the concepts of floating environments. • ... is the only math environment you know; spacing with \\. • Never use -- or ---, let alone .\ and \, in acronyms. • All math delimiters have the same size. Not exactly typesetting, but related in spirit: • What are \label and \ref? • Instead of using the ones from babel, use wrong or hacked quotes (language dependent). • Literature references hardcoded in footnotes. • Re-use pixel graphics created with some drawing tool. • Write everything in one huge file. • Use a *TeX distribution that is older than your PC. Of course, the most common mistake is to write stuff yourself. In 95% percent of the cases, there's a package that does what you want! - @Canageek I think that is not only a narrow definition of "document", it's also not a useful point of view. The graphics ends up in a PDF (PS + x) so they have to get there somehow. LaTeX engines may be able to hide from you that they convert SVG to PS/PDF but it still has to happen (would you rather control the conversion?). Provided you don't want pixel graphics, which would be really contrary to the "idea" of PS/PDF. (Or, you know, learn TikZ. :)) – Raphael Nov 2 '13 at 22:36 ## Incorrectly choosing the compiler • Compiling • an input file importing PDF, JPEG, PNG with latex->dvips->ps2pdf sequence. The compiler should be either pdflatex or xelatex (among other possibilities). • Compiling • an input file containing PSTricks code • an input file importing EPS with pdflatex. The compiler should be either latex->dvips->ps2pdf or xelatex (among other possibilities) - ## Forgot to break the paragraph before the closing } When a paragraph has font size change, the \par must be invoked before } \documentclass[preview,border=12pt]{standalone} \def\foo{% These dummy texts will span more than 2 lines. If you see they do not span more than 2 lines then you have to inform me now!} \begin{document} \parindent=5em\relax \foo {\huge \foo} \foo {\huge \foo\par} \foo \end{document} - applies to 1-line paragraphs too – David Carlisle Oct 21 '13 at 16:27 this is a bit more esoteric, but if the last line of a "larger" paragraph has no descenders, a dummy descender should be added with \strut. avoid any spaces between the end of printable text and the \strut to avoid a possible extra line if the last line fills the full width of the page. – barbara beeton Oct 21 '13 at 23:21 1. Compiling too early and too often. Now you are spending your time working on fixing formatting and such that will change as soon as you add more text anyway and all the floats will move anyway. 2. Using floats when you don't have to. Floats are the devil, yet all tutorials use them. Most of the time I know where I want the image, at least roughly, and didn't know for ages that I could just insert the image without wrapping it in a float, so I'd spend ages trying to get stuff to stop floating down multiple pages, or into the wrong section, etc. Edit: As this is attracting a lot of debate: When I started every example shows you to insert an image in a float, so I thought you had to use float. Even a couple of months ago I thought you needed to put a table in a table environment; I'd still have to look up how to make a non-floating table. This is a problem as a lot of the time I need an image in a certain place. For example, if I'm typesetting a homework problem (Something I did a lot as an undergrad, and am now doing again when I'm writing questions) I can't have that image floating down into a different question; my TA or prof isn't going to go hunting for it, they are just going to dock me marks (And I don't want to confusing anyone working on my problem set!). Also as the h option is a joke they usually float far away. I don't mind if a figure floats a few paragraphs, but three pages away is bad: Same page as the discussion or facing page only. I should always be able to see both figure and text at the same time, unless there are exceptional circumstances or a very large text (I cut some slack when writing my thesis, as there were so many figures they wouldn't all fit on the same page as the text). There are times floats are the right choice; I'm using one in the document I'm working on right now. However, they are just as often the wrong choice, and a lot of new users don't realize this. 3. Using outdated packages that some website gave you My group drives me crazy since they use very outdated packages that have been superseeded. subfigure instead of subfig, stuff like that. 4. Relying on GUI text editors instead of understanding what is happening People at work also drive me crazy with refusing to use anything but the compile button in WinEdt. If that doesn't automatically work, they refuse to use the package (Thus no biber, no biblatex, etc. 5. Using MikTeX That install package on compile thing NEVER works right. Hard drives are large, dammit. Edit: Ok, not never. But it fails every time I've given my coworkers a new package to use, causing us to have to go into the interface and have it manually install. 6. Limiting yourself to what chemistry journals allow for everything I dream of one day being editor of a big chemistry journal, just so I can go to the AMS or APS and borrow their code. This has things like not using macros, not redefining things, not using excess packages (or in one case, not using packages AT ALL). 7. Not being willing to put in time understanding LaTeX The people I work with just want a solution now. They aren't willing to put in any time into making it work; they want something that works now, since the journal will reformat it anyway, so if it works and gets vaguely close to what they want, good enough. 8. Making DocumentNameV3.tex and forgetting to open the new PDF, then wondering why nothing you do is working Not that I've done this recently, after I was too lazy to set up a new code repository. - Major complaint from colleagues I've heard is 7. – Forkrul Assail Oct 23 '13 at 19:58 I disagree (partly) with 2: I think floats are a good thing: often images and tables don't need to be looked at immediately and the readers should have a choice when they do. So it's a good thing if they float to the top or bottom or a separate page. They have a caption and are referred to in the text anyway – clemens Oct 23 '13 at 20:00 Floats are great when they're appropriate, but using them when you want your text to stay put is a mistake. Too many people use them but don't want them to ever float. Double mistake. – alexis Oct 23 '13 at 20:03 @cgnieder I think what Canageek has in mind are people using figure when all they want to do is insert a graphic. There are many places where this is useful, and new users simply do not understand the difference. Quoting myself from chat: "The prize for the worst name choice in LaTeX surely must go to the figure and table environments. I wonder how many questions here and on the web deal with people not understanding the difference between the container and its contents?" – Alan Munn Oct 23 '13 at 20:06 @Canageek do you know \FloatBarrier (from the placeins package)? I never used it but it's designed for exactly that purpose... – clemens Oct 24 '13 at 21:01 Not understanding the compile sequence when using GUI tools like TeXmaker and LyX was something I was personally guilty of. Understanding this better, helped me understand when references, citations and page numbering went wonky. Switching to emacs and using a custom scripted compile helped sort out the kinks. - Compile early, compile often as with 'test early, test often' is good advice when spending a lot of time working with complex documents and multiple included files. A common mistake is to work too long on a particular section, not noticing a missing } and then having to scour multiple documents for hours, when this could've been fixed by either an automated constant compile, or just ensuring every section or paragraph you work on still leaves a coherent document. - Compiling to make sure it compiles is a good idea (always compile before commits!) - but that's different than tweaking the output every time you compile. – bombcar Mar 13 '14 at 15:35 • Not understanding the difference between typesetting and word processing. The latter is simply using computer technology to replace a mechanical typewriter, primarily in the production of routine business documents. The former has to do with preparing documents in their final form for publication. TeX and Friends are typesetters, not a word processors. • Thinking TeX and Friends are solely for preparation of documents intended for scientific/academic journals, classroom notes or textbooks with lots of formulae and/or footnotes and citations. TeX and Friends certainly excel at these tasks but also excel at typesetting poetry and prose -- in their original form, not just in textbooks or critical analysis (although TeX and Friends do an outstanding job of preparing those as well). • Not understanding the difference between logical markup and WYSIWYG then spending inordinate amounts of time trying to make TeX and Friends behave like a WYSIWYG desktop publisher/word processor instead of putting their focus on the content of their document. • In a similar vein, not understanding the difference between writing a document and designing a document. Two different disciplines, two different concerns. TeX and Friends allow a clean separation of the functions even if the same person is going to do both. • Trying to make TeX and Friends do everything instead of using the right tool. Production of a consumer-oriented magazine is far better done in Adobe InDesign than TeX and Friends. Business presentations are far better done in PowerPoint than Beamer. etcetera, etcetera etcetera. • Underestimating the amount of work it takes to produce a "beautiful" document. TeX and Friends will produce more beautiful documents than MS Word but it is not automatic simply because one uses TeX and Friends. • Getting caught up in vim vs emacs vs texworks vs tex{studio | maker} vs notepad arguments on SX (or elewhere). :) They all work fine. TeX and Friends don't care which you use and the resulting documents don't either. - Doing by hands what LaTeX should handle. Like writing "on page 3" and thinking to learn "complicated things" like \label-\ref "later", i.e. never. On the other hand, it is not bad to use \newpage before \section instead of titlesec package --- this is what David Carlisle already answered about. Basically this applies to all kind of systems. How many keyboard shortcut in Word to learn, how many packages to remember in LaTeX... It always depend on how much one is going to use some software. Another one on meta-level: not asking computer support when you have one available. Many have used hours to find out things that I could have tell in a minute. - The usage of raster graphics (JPG, PNG, ...) instead of vector graphigs (SVG, TikZ pictures, ...) could be a mistake since this increases output PDF file size and slows down rendering in several readers. - So we have to vectorize all screenshots or photographs? I don't think they become smaller in size. – kiss my armpit Oct 22 '13 at 12:36 sometimes it's really necessary to incorporate photographs, say for presentation of historical information, and they simply don't come in vector form. rather than disallowing them, it would be better to give a warning and offer a way to compress them in the most nearly lossless manner. – barbara beeton Oct 22 '13 at 21:31 It's also a mistake to use JPG instead of PNG for diagrams and line art-- it makes the lines and letters fuzzy. But it's not specifically a (La)TeX mistake. – alexis Oct 23 '13 at 19:12 @my_greets Getting SVG to work in LaTeX is a pain, and using a PDF for an image is also a pain, and conceptually wrong (it is a document layout format, NOT a graphics format!) – Canageek Oct 23 '13 at 19:28 The most common mistake that LaTeX beginners make is that they start with the “stock” article class and enrich it with various packages to suit their needs, instead of picking a more specialised class providing all the functionality they need. It has several drawbacks for the users: • They spend a significant time trying looking for “tabular package XY” that can do “plop plop fizz fizz” instead of using what is provided by the specialised class they did not pick. • They spend a significant time trying to make some packages work together and looking for answers to their problem. • They spend a significant time trying to customise the class to achieve a pleasant layout, while they wether have the needed programming skills neither (most probably) the design skills. Is there any LaTeX introduction or tutorial that emphasises the choice of a document class and provides useful resources (catalogue, comparison of features, application domain) to choose one? It seems to me, most of them pick one (article) and stick to it for the whole tutorial, while it could be more useful—once the basics of text and math typesettings have been covered—to drop that generic class in favour of a specialised one. To put it in one sentance: “unexperienced users confuse the roles of designer and scripter (and tutorials tend to perpetuate this confusion).” - I have never used anything BUT article. What else is there? KOMA-script and memoir seem to be languages in themselves, and half my packages warn that they are incompatible with one or both! – Canageek Oct 24 '13 at 21:01 This answer would be much more useful with some real examples, i.e. packages that are often used but aren't needed when using a specific document class(either because it provides the same output or because the document class can do the thing slightly differently, but in a "better way" from the typesetting point of view). – Bakuriu Oct 25 '13 at 9:41 My own take on this question is more about the mistakes beginners make because of us. I made too many shameful mistakes (and still do at times), but with my first private beta, I realised that many beginner mistakes were in fact my beginner LaTeX developer mistakes. They are still beginner mistakes – but I have a feeling of responsibility for a lot of them. 1. Not teaching users to first read the documentation. Not teaching users to look at the code if the documentation is unhelpful (code does not bite). I have had someone tell me "I didn't dare to modify your package because I didn't know what I was doing"… well, just back it up somewhere and play, code is not sacred). 2. Not teaching users to send you the log when they want to report a bug or any other issue, even if they do not understand what the error message means. On that matter, I feel like it is helpful to teach all beginners how to read the most basic errors (command already defined, command not defined, underfull and overfull boxes). 3. Not writing appropriate documentation files for users (we all boast about "literate programming" and we mostly do a good job at writing docs, but not all of them are perfect). We have some beautiful files, and also some barely-commented "implementation" docs with just an introduction and a list of macros that no beginner would ever understand. 4. Not teaching users the difference between form and content (as mentioned several times before). But I also mean it literally: whether they want to code something for themselves or whether they are requesting a new feature, they need to determine if it is just a stylistic preference or if there is a general purpose behind it (e.g. do we need a simple command or an option for customisation). There is nothing more stupid than creating a whole system that works really well except it is totally impractical in real life. I hope it is not off-topic… but thinking back, I would think that most of my own beginner mistakes were also often developer (or documentation-writer) mistakes. We are generally good at writing documentations, but we sometimes have a hard time finding the right words for both beginners and experts. Most of the beginner mistakes mentioned here are not really in the books… or at least they don't show up as sections, tutorials, etc. It is no wonder beginners first look things up on google. We don't write documentations on debugging what we assume will work (it works in the hands of a TeXnician, not in the hands of someone who does things at random). - +1 for mention that makes a difference the documentation written for human beings, err ...average users. Sometimes, each option of each command is explained extensively, but you have to search in Google for an example or how apply these commands without obtain errors. Generally, the average user needs the section "usage" and some examples, not the "implementation" section. Instead, sometimes even is omitted that one must have \usepackage {...} in the preamble. Too obvious? not for a novice! – Fran Jul 30 '14 at 8:56 ## Cleaning too early • Suppose there is an input file importing EPS images and it will be compiled with latex->dvips->ps2pdf. Some users think that the EPS images are no longer needed after running latex so they can be deleted. Actually it is wrong because dvips still needs the EPS. (it must be documented in arara) • Suppose there is an input file using cross-referencing with \label. Some users aggressively remove any auxiliary files generated by (pdf)latex compiler before the next invocations. It makes the cross-referencing always end with ??. These auxiliary files must not be deleted until the cross-referencing is properly done. (it must be documented in arara). - At the very beginning, trying to make a table in this way: \begin{table}{ccc} a & b & c \\ \end{table} - @mafutrct To make a table in LaTeX you need the less intuitive environment tabular, not table. There is also commonly used the table environment, but is not to make a table, so the confusion is served for the novice. What a table environment does is move (float) a table (or whatever it contain) to the best possible place according to several rules (the top of the page, the bottom, the next page, etc.). – Fran Jan 6 '14 at 9:22 Actually it is not wrong but the output might make your eyes a bit itch. "Wrongly" representing a series of unaligned equations using \[.. instead of using \begin{gather*}...\end{gather*}.
\documentclass[preview,border=12pt]{standalone}
\usepackage{amsmath}
\begin{document}
\section*{Not recommended}
The solution of $x^2-5x+6=0$ is
$(x-3)(x-2)=0$
$x-3=0 \text{ or } x-2=0$
$x=3 \text{ or } x=2$
\section*{Recommended}
The solution of $x^2-5x+6=0$ is
\begin{gather*}
(x-3)(x-2)=0\\
x-3=0 \text{ or } x-2=0\\
x=3 \text{ or } x=2
\end{gather*}
\end{document}
-
recommended shouldn't be recommended. In fact this shouldn't be done in my opinion. – percusse Oct 23 '13 at 15:29
Another common misbehavior is reading package or class documentations from TeX mirrors (the worst case, they read from sites with obsolete contents) even though they already installed the complete, up-to-date packages and classes.
I mean that they don't know that texdoc <package-name> invoked in their own machine can launch the documentation in question.
-
Unfortunately it is not always clear what <package-name> is (like with KOMA-Script). In MiKTeX texdoc <package-name> opens a HTML list of documentation items of the given package, while texdoc --view <package-name> opens the first item of that list. In the case of Beamer this is only the logo, with PGF it's the mindmap example, with NewTX the implementation notes etc. You have to know the filename: texdoc beameruserguide, texdoc scrguien etc. – marczellm Oct 26 '13 at 9:11
• Closing opening curly brackets with parenthesis (but they are not easily distinguishable with default settings of some editors).
• Rewriting sort of description lists by hand:
\textbf{Foo:} description of foo.
\textbf{Bar:} description of bar.
[...]
• Forgetting to load the package providing the macro they are using.
• Forgetting that a LaTeX text editor is a text editor and being puzzled when they want to search or replace text in their .tex source.
• Being bitten by the rigor needed by LaTeX and its sensibility to misspelling in macros, packages, files names.
-
I don't get bullet four. – Raphael Oct 23 '13 at 6:16
@Raphael Yesterday, a student of mine replaced by hand numerous commas by periods instead of using the "Search/Replace" tool of the editor. – Denis Bitouzé Oct 23 '13 at 7:39
@Marienplatz No, for siunitx, I know the locale=FR (for instance) setup. It was in fact related to pgfplots that accept only period as input decimal separator (BTW, I just sent a feature request to Christian Feuersaenger, the pgfplots author, for providing a way to specify the input decimal separator other than period). – Denis Bitouzé Oct 23 '13 at 15:52
Once I've seen a document template with a comment along the lines
% Tip: Always use eqnarray instead of align
:-(
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https://beta.wikiversity.org/wiki/Relation_composition | Relation composition
Relation composition, or the composition of relations, is the generalization of function composition, or the composition of functions. The following treatment of relation composition takes the “strongly typed” approach to relations that is outlined in the article on relation theory.
Preliminaries
There are several ways to formalize the subject matter of relations. Relations and their combinations may be described in the logic of relative terms, in set theories of various kinds, and through a broadening of category theory from functions to relations in general.
The first order of business is to define the operation on relations that is variously known as the composition of relations, relational composition, or relative multiplication. In approaching the more general constructions, it pays to begin with the composition of dyadic and triadic relations.
As an incidental observation on usage, there are many different conventions of syntax for denoting the application and composition of relations, with perhaps even more options in general use than are common for the application and composition of functions. In this case there is little chance of standardization, since the convenience of conventions is relative to the context of use, and the same writers use different styles of syntax in different settings, depending on the ease of analysis and computation.
The first dimension of variation in syntax has to do with the correspondence between the order of operation and the linear order of terms on the page. The second dimension of variation in syntax has to do with the automatic assumptions in place about the associations of terms in the absence of associations marked by parentheses. This becomes a significant factor with relations in general because the usual property of associativity is lost as both the complexities of compositions and the dimensions of relations increase.
These two factors together generate the following four styles of syntax:
LALA = left application, left association. LARA = left application, right association. RALA = right application, left association. RARA = right application, right association.
Definition
A notion of relational composition is to be defined that generalizes the usual notion of functional composition:
Composing on the right, ${\displaystyle f:X\to Y}$ followed by ${\displaystyle g:Y\to Z}$ results in a composite function formulated as ${\displaystyle fg:X\to Z.}$ Composing on the left, ${\displaystyle f:X\to Y}$ followed by ${\displaystyle g:Y\to Z}$ results in a composite function formulated as ${\displaystyle gf:X\to Z.}$
Note on notation. The ordinary symbol for functional composition is the composition sign, a small circle "${\displaystyle \circ }$" written between the names of the functions being composed, as ${\displaystyle f\circ g,}$ but the sign is often omitted if there is no risk of confusing the composition of functions with their algebraic product. In contexts where both compositions and products occur, either the composition is marked on each occasion or else the product is marked by means of a center dot${\displaystyle \cdot }$”, as ${\displaystyle f\cdot g.}$
Generalizing the paradigm along parallel lines, the composition of a pair of dyadic relations is formulated in the following two ways:
Composing on the right, ${\displaystyle P\subseteq X\times Y}$ followed by ${\displaystyle Q\subseteq Y\times Z}$ results in a composite relation formulated as ${\displaystyle PQ\subseteq X\times Z.}$ Composing on the left, ${\displaystyle P\subseteq X\times Y}$ followed by ${\displaystyle Q\subseteq Y\times Z}$ results in a composite relation formulated as ${\displaystyle QP\subseteq X\times Z.}$
Geometric construction
There is a neat way of defining relational compositions in geometric terms, not only showing their relationship to the projection operations that come with any cartesian product, but also suggesting natural directions for generalizing relational compositions beyond the dyadic case, and even beyond relations that have any fixed arity, in effect, to the general case of formal languages as generalized relations.
This way of looking at relational compositions is sometimes referred to as Tarski's Trick, on account of his having put it to especially good use in his work (Ulam and Bednarek, 1977). It supplies the imagination with a geometric way of visualizing the relational composition of a pair of dyadic relations, doing this by attaching concrete imagery to the basic set-theoretic operations, namely, intersections, projections, and a certain class of operations inverse to projections, here called tacit extensions.
The stage is set for Tarski's trick by highlighting the links between two topics that are likely to appear wholly unrelated at first, namely:
• The use of logical conjunction, as denoted by the symbol ${\displaystyle \land ,\!}$ in expressions of the form ${\displaystyle F(x,y,z)=G(x,y)\land H(y,z),\!}$ to define a triadic relation ${\displaystyle F\!}$ in terms of a pair of dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H.\!}$
• The concepts of dyadic projection and projective determination, that are invoked in the “weak” notion of projective reducibility.
The relational composition ${\displaystyle G\circ H\!}$ of a pair of dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ will be constructed in three stages, first, by taking the tacit extensions of ${\displaystyle G\!}$ and ${\displaystyle H\!}$ to triadic relations that reside in the same space, next, by taking the intersection of these extensions, tantamount to the maximal triadic relation that is consistent with the prima facie dyadic relation data, finally, by projecting this intersection on a suitable plane to form a third dyadic relation, constituting in fact the relational composition ${\displaystyle G\circ H\!}$ of the relations ${\displaystyle G\!}$ and ${\displaystyle H.\!}$
The construction of a relational composition in a specifically mathematical setting normally begins with mathematical relations at a higher level of abstraction than the corresponding objects in linguistic or logical settings. This is due to the fact that mathematical objects are typically specified only up to isomorphism as the conventional saying goes, that is, any objects that have the “same form” are generally regarded as the being the same thing, for most all intents and mathematical purposes. Thus the mathematical construction of a relational composition begins by default with a pair of dyadic relations that reside, without loss of generality, in the same plane, say, ${\displaystyle G,H\subseteq X\times Y,\!}$ as shown in Figure 1.
o-------------------------------------------------o | | | o o | | |\ |\ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | * \ | * \ | | X * Y X * Y | | \ * | \ * | | | \ G | \ H | | | \ | \ | | | \ | \ | | | \ | \ | | | \ | \ | | | \| \| | | o o | | | o-------------------------------------------------o Figure 1. Dyadic Relations G, H c X x Y
The dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ cannot be composed at all at this point, not without additional information or further stipulation. In order for their relational composition to be possible, one of two types of cases has to happen:
• The first type of case occurs when ${\displaystyle X=Y.\!}$ In this case, both of the compositions ${\displaystyle G\circ H\!}$ and ${\displaystyle H\circ G\!}$ are defined.
• The second type of case occurs when ${\displaystyle X\!}$ and ${\displaystyle Y\!}$ are distinct, but when it nevertheless makes sense to speak of a dyadic relation ${\displaystyle {\hat {H}}\!}$ that is isomorphic to ${\displaystyle H,\!}$ but living in the plane ${\displaystyle YZ,\!}$ that is, in the space of the cartesian product ${\displaystyle Y\times Z,\!}$ for some set ${\displaystyle Z.\!}$
Whether you view isomorphic things to be the same things or not, you still have to specify the exact isomorphisms that are needed to transform any given representation of a thing into a required representation of the same thing. Let us imagine that we have done this, and say how later:
o-------------------------------------------------o | | | o o | | |\ /| | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | \ / | | | | * \ / * | | | X * Y Y * Z | | \ * | | * / | | \ G | | Ĥ / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 2. Dyadic Relations G c X x Y and Ĥ c Y x Z
With the required spaces carefully swept out, the stage is set for the presentation of Tarski's trick, and the invocation of the following symbolic formula, claimed to be a definition of the relational composition ${\displaystyle P\circ Q\!}$ of a pair of dyadic relations ${\displaystyle P,Q\subseteq X\times X.\!}$
Definition. ${\displaystyle P\circ Q=\mathrm {proj} _{13}(P\times X~\cap ~X\times Q).\!}$
To get this drift of this definition one needs to understand that it comes from a point of view that regards all dyadic relations as covered well enough by subsets of a suitable cartesian square and thus of the form ${\displaystyle L\subseteq X\times X.\!}$ So, if one has started out with a dyadic relation of the shape ${\displaystyle L\subseteq U\times V,\!}$ one merely lets ${\displaystyle X=U\cup V,\!}$ trading in the initial ${\displaystyle L\!}$ for a new ${\displaystyle L\subseteq X\times X\!}$ as need be.
The projection ${\displaystyle \mathrm {proj} _{13}\!}$ is just the projection of the cartesian cube ${\displaystyle X\times X\times X\!}$ on the space of shape ${\displaystyle X\times X\!}$ that is spanned by the first and the third domains, but since they now have the same names and the same contents it is necessary to distinguish them by numbering their relational places.
Finally, the notation of the cartesian product sign “${\displaystyle \times \!}$” is extended to signify two other products with respect to a dyadic relation ${\displaystyle L\subseteq X\times X\!}$ and a subset ${\displaystyle W\subseteq X,\!}$ as follows:
Definition. ${\displaystyle L\times W~=~\{(x,y,z)\in X^{3}~:~(x,y)\in L~\mathrm {and} ~z\in W\}.\!}$
Definition. ${\displaystyle W\times L~=~\{(x,y,z)\in X^{3}~:~x\in W~\mathrm {and} ~(y,z)\in L\}.\!}$
Applying these definitions to the case ${\displaystyle P,Q\subseteq X\times X,\!}$ the two dyadic relations whose relational composition ${\displaystyle P\circ Q\subseteq X\times X\!}$ is about to be defined, one finds:
${\displaystyle P\times X~=~\{(x,y,z)\in X^{3}~:~(x,y)\in P~\mathrm {and} ~z\in X\},\!}$
${\displaystyle X\times Q~=~\{(x,y,z)\in X^{3}~:~x\in X~\mathrm {and} ~(y,z)\in Q\}.\!}$
These are just the appropriate special cases of the tacit extensions already defined.
${\displaystyle P\times X~=~\mathrm {te} _{12}^{3}(P),~\!}$
${\displaystyle X\times Q~=~\mathrm {te} _{23}^{1}(Q).~\!}$
In summary, then, the expression:
${\displaystyle \mathrm {proj} _{13}(P\times X~\cap ~X\times Q)\!}$
is equivalent to the expression:
${\displaystyle \mathrm {proj} _{13}(\mathrm {te} _{12}^{3}(P)~\cap ~\mathrm {te} _{23}^{1}(Q))\!}$
and this form is generalized — although, relative to one's school of thought, perhaps inessentially so — by the form that was given above as follows:
Definition. ${\displaystyle P\circ Q~=~\mathrm {proj} _{XZ}(\mathrm {te} _{XY}^{Z}(P)~\cap ~\mathrm {te} _{YZ}^{X}(Q)).\!}$
Figure 3 presents a geometric picture of what is involved in formulating a definition of the triadic relation ${\displaystyle F\subseteq X\times Y\times Z\!}$ by way of a conjunction between the dyadic relation ${\displaystyle G\subseteq X\times Y\!}$ and the dyadic relation ${\displaystyle H\subseteq Y\times Z,\!}$ as done for example by means of an expression of the following form:
• ${\displaystyle F(x,y,z)~=~G(x,y)\land H(y,z).\!}$
o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 3. Projections of F onto G and H
To interpret the Figure, visualize the triadic relation ${\displaystyle F\subseteq X\times Y\times Z\!}$ as a body in ${\displaystyle XYZ\!}$-space, while ${\displaystyle G\!}$ is a figure in ${\displaystyle XY\!}$-space and ${\displaystyle H\!}$ is a figure in ${\displaystyle YZ\!}$-space.
The dyadic projections that accompany a triadic relation over ${\displaystyle X,Y,Z\!}$ are defined as follows:
• ${\displaystyle \mathrm {proj} _{XY}(L)~=~\{(x,y)\in X\times Y:(x,y,z)\in L~{\text{for some}}~z\in Z)\},\!}$
• ${\displaystyle \mathrm {proj} _{XZ}(L)~=~\{(x,z)\in X\times Z:(x,y,z)\in L~{\text{for some}}~y\in Y)\},\!}$
• ${\displaystyle \mathrm {proj} _{YZ}(L)~=~\{(y,z)\in Y\times Z:(x,y,z)\in L~{\text{for some}}~x\in X)\}.\!}$
For many purposes it suffices to indicate the dyadic projections of a triadic relation ${\displaystyle L\!}$ by means of the briefer equivalents listed next:
• ${\displaystyle L_{XY}~=~\mathrm {proj} _{XY}(L),\!}$
• ${\displaystyle L_{XZ}~=~\mathrm {proj} _{XZ}(L),\!}$
• ${\displaystyle L_{YZ}~=~\mathrm {proj} _{YZ}(L).\!}$
In light of these definitions, ${\displaystyle \mathrm {proj} _{XY}\!}$ is a mapping from the set ${\displaystyle {\mathcal {L}}_{XYZ}\!}$ of triadic relations over the domains ${\displaystyle X,Y,Z\!}$ to the set ${\displaystyle {\mathcal {L}}_{XY}\!}$ of dyadic relations over the domains ${\displaystyle X,Y,\!}$ with similar relationships holding for the other projections. To formalize these relationships in a concise but explicit manner, it serves to add a few more definitions.
The set ${\displaystyle {\mathcal {L}}_{XYZ},~\!}$ whose members are just the triadic relations over ${\displaystyle X,Y,Z,\!}$ can be recognized as the set of all subsets of the cartesian product ${\displaystyle X\times Y\times Z,\!}$ also known as the power set of ${\displaystyle X\times Y\times Z,\!}$ and notated here as ${\displaystyle \mathrm {Pow} (X\times Y\times Z).\!}$
• ${\displaystyle {\mathcal {L}}_{XYZ}~=~\{L:L\subseteq X\times Y\times Z\}~=~\mathrm {Pow} (X\times Y\times Z).\!}$
Likewise, the power sets of the pairwise cartesian products encompass all the dyadic relations on pairs of distinct domains that can be chosen from ${\displaystyle \{X,Y,Z\}.\!}$
• ${\displaystyle {\mathcal {L}}_{XY}~=~\{L:L\subseteq X\times Y\}~=~\mathrm {Pow} (X\times Y),~\!}$
• ${\displaystyle {\mathcal {L}}_{XZ}~=~\{L:L\subseteq X\times Z\}~=~\mathrm {Pow} (X\times Z),~\!}$
• ${\displaystyle {\mathcal {L}}_{YZ}~=~\{L:L\subseteq Y\times Z\}~=~\mathrm {Pow} (Y\times Z).~\!}$
In mathematics, the inverse relation corresponding to a projection map is usually called an extension. To avoid confusion with other senses of the word, however, it is probably best for the sake of this discussion to stick with the more specific term tacit extension.
Given three sets, ${\displaystyle X,Y,Z,\!}$ and three dyadic relations,
• ${\displaystyle U\subseteq X\times Y,~\!}$
• ${\displaystyle V\subseteq X\times Z,~\!}$
• ${\displaystyle W\subseteq Y\times Z,~\!}$
the tacit extensions, ${\displaystyle \mathrm {te} _{XY}^{Z},\mathrm {te} _{XZ}^{Y},\mathrm {te} _{YZ}^{X},~\!}$ of ${\displaystyle U,V,W,\!}$ respectively, are defined as follows:
• ${\displaystyle \mathrm {te} _{XY}^{Z}(U)~=~\{(x,y,z):(x,y)\in U\},\!}$
• ${\displaystyle \mathrm {te} _{XZ}^{Y}(V)~=~\{(x,y,z):(x,z)\in V\},\!}$
• ${\displaystyle \mathrm {te} _{YZ}^{X}(W)~=~\{(x,y,z):(y,z)\in W\}.\!}$
So long as the intended indices attaching to the tacit extensions can be gathered from context, it is usually clear enough to use the abbreviated forms, ${\displaystyle \mathrm {te} (U),\mathrm {te} (V),\mathrm {te} (W).\!}$
The definition and illustration of relational composition presently under way makes use of the tacit extension of ${\displaystyle G\subseteq X\times Y\!}$ to ${\displaystyle \mathrm {te} (G)\subseteq X\times Y\times Z\!}$ and the tacit extension of ${\displaystyle H\subseteq Y\times Z\!}$ to ${\displaystyle \mathrm {te} (H)\subseteq X\times Y\times Z,\!}$ only.
Geometric illustrations of ${\displaystyle \mathrm {te} (G)\!}$ and ${\displaystyle \mathrm {te} (H)\!}$ are afforded by Figures 4 and 5, respectively.
o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | * \ | | o o ** o | | |\ / \*** /| | | | \ / *** / | | | | \ / ***\ / | | | | \ *** / | | | | / \*** / \ | | | | / *** / \ | | | |/ ***\ / \| | | o X /** Y Z o | | |\ \//* | / /| | | | \ /// | / / | | | | \ ///\ | / / | | | | \ /// \ | / / | | | | \/// \ | / / | | | | /\/ \ | / / | | | | *//\ \|/ / * | | | X */ Y o Y * Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 4. Tacit Extension of G to X x Y x Z
o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / * | \ | | o ** o o | | |\ ***/ \ /| | | | \ *** \ / | | | | \ /*** \ / | | | | \ *** / | | | | / \ ***/ \ | | | | / \ *** \ | | | |/ \ /*** \| | | o X Y **\ Z o | | |\ \ | *\\/ /| | | | \ \ | \\\ / | | | | \ \ | /\\\ / | | | | \ \ | / \\\ / | | | | \ \ | / \\\/ | | | | \ \ | / \/\ | | | | * \ \|/ /\\* | | | X * Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 5. Tacit Extension of H to X x Y x Z
A geometric interpretation can now be given that fleshes out in graphic form the meaning of a formula like the following:
• ${\displaystyle F(x,y,z)~=~G(x,y)\land H(y,z).\!}$
The conjunction that is indicated by “${\displaystyle \land \!}$” corresponds as usual to an intersection of two sets, however, in this case it is the intersection of the tacit extensions ${\displaystyle \mathrm {te} (G)\!}$ and ${\displaystyle \mathrm {te} (H).\!}$
o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | /\/ \ | / \/\ | | | | *//\ \|/ /\\* | | | X */ Y o Y \* Z | | \ * | | * / | | \ G | | H / | | \ | | / | | \ | | / | | \ | | / | | \ | | / | | \| |/ | | o o | | | o-------------------------------------------------o Figure 6. F as the Intersection of te(G) and te(H)
Algebraic construction
The transition from a geometric picture of relation composition to an algebraic formulation is accomplished through the introduction of coordinates, in other words, identifiable names for the objects that are related through the various forms of relations, dyadic and triadic in the present case. Adding coordinates to the running Example produces the following Figure:
o-------------------------------------------------o | | | o | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o o o | | |\ / \ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ 7\/// | \\\/7 /| | | | \ 6// | \\6 / | | | | \ //5\ | /5\\ / | | | | \ /// 4\ | /4 \\\ / | | | | \/// 3\ | /3 \\\/ | | | | G/\/ 2\ | /2 \/\H | | | | *//\ 1\|/1 /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 7. F as the Intersection of te(G) and te(H)
Thinking of relations in operational terms is facilitated by using variant notations for ordered tuples and sets of ordered tuples, namely, the ordered pair ${\displaystyle (x,y)\!}$ is written ${\displaystyle x\!:\!y,\!}$ the ordered triple ${\displaystyle (x,y,z)\!}$ is written ${\displaystyle x\!:\!y\!:\!z,\!}$ and so on, and a set of tuples is conceived as a logical-algebraic sum, which can be written out in the smaller finite cases in forms like ${\displaystyle a\!:\!b~+~b\!:\!c~+~c\!:\!d\!}$ and so on.
For example, translating the relations ${\displaystyle F\subseteq X\times Y\times Z,~G\subseteq X\times Y,~H\subseteq Y\times Z\!}$ into this notation produces the following summary of the data:
${\displaystyle {\begin{matrix}F&=&4:3:4&+&4:4:4&+&4:5:4\\G&=&4:3&+&4:4&+&4:5\\H&=&3:4&+&4:4&+&5:4\end{matrix}}}$
As often happens with abstract notations for functions and relations, the type information, in this case, the fact that ${\displaystyle G\!}$ and ${\displaystyle H\!}$ live in different spaces, is left implicit in the context of use.
Let us now verify that all of the proposed definitions, formulas, and other relationships check out against the concrete data of the current composition example. The ultimate goal is to develop a clearer picture of what is going on in the formula that expresses the relational composition of a couple of dyadic relations in terms of the medial projection of the intersection of their tacit extensions:
${\displaystyle G\circ H~=~\mathrm {proj} _{XZ}(\mathrm {te} _{XY}^{Z}(G)~\cap ~\mathrm {te} _{YZ}^{X}(H)).\!}$
Here is the big picture, with all the pieces in place:
o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | / \ | | / \ | | / G o H \ | | X * Z | | 7\ /|\ /7 | | 6\ / | \ /6 | | 5\ / | \ /5 | | 4@ | @4 | | 3\ | /3 | | 2\ | /2 | | 1\|/1 | | | | | | | | | | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | o | o | | |\ /|\ /| | | | \ / F \ / | | | | \ / * \ / | | | | \ /*\ / | | | | / \//*\\/ \ | | | | / /\/ \/\ \ | | | |/ ///\ /\\\ \| | | o X /// Y \\\ Z o | | |\ \/// | \\\/ /| | | | \ /// | \\\ / | | | | \ ///\ | /\\\ / | | | | \ /// \ | / \\\ / | | | | \/// \ | / \\\/ | | | | G/\/ \ | / \/\H | | | | *//\ \|/ /\\* | | | X *\ Y o Y /* Z | | 7\ *\\ |7 7| //* /7 | | 6\ |\\\|6 6|///| /6 | | 5\| \\@5 5@// |/5 | | 4@ \@4 4@/ @4 | | 3\ @3 3@ /3 | | 2\ |2 2| /2 | | 1\|1 1|/1 | | o o | | | o-------------------------------------------------o Figure 8. G o H = proj_XZ (te(G) |^| te(H))
All that remains is to check the following collection of data and derivations against the situation represented in Figure 8.
${\displaystyle {\begin{matrix}F&=&4:3:4&+&4:4:4&+&4:5:4\\G&=&4:3&+&4:4&+&4:5\\H&=&3:4&+&4:4&+&5:4\end{matrix}}}$
${\displaystyle {\begin{matrix}G\circ H&=&(4\!:\!3~+~4\!:\!4~+~4\!:\!5)(3\!:\!4~+~4\!:\!4~+~5\!:\!4)\\[6pt]&=&4:4\end{matrix}}}$
${\displaystyle {\begin{matrix}\mathrm {te} (G)&=&\mathrm {te} _{XY}^{Z}(G)\\[4pt]&=&\displaystyle \sum _{z=1}^{7}(4\!:\!3\!:\!z~+~4\!:\!4\!:\!z~+~4\!:\!5\!:\!z)\end{matrix}}}$
${\displaystyle {\begin{matrix}\mathrm {te} (G)&=&4:3:1&+&4:4:1&+&4:5:1&+\\&&4:3:2&+&4:4:2&+&4:5:2&+\\&&4:3:3&+&4:4:3&+&4:5:3&+\\&&4:3:4&+&4:4:4&+&4:5:4&+\\&&4:3:5&+&4:4:5&+&4:5:5&+\\&&4:3:6&+&4:4:6&+&4:5:6&+\\&&4:3:7&+&4:4:7&+&4:5:7\end{matrix}}}$
${\displaystyle {\begin{matrix}\mathrm {te} (H)&=&\mathrm {te} _{YZ}^{X}(H)\\[4pt]&=&\displaystyle \sum _{x=1}^{7}(x\!:\!3\!:\!4~+~x\!:\!4\!:\!4~+~x\!:\!5\!:\!4)\end{matrix}}}$
${\displaystyle {\begin{matrix}\mathrm {te} (H)&=&1:3:4&+&1:4:4&+&1:5:4&+\\&&2:3:4&+&2:4:4&+&2:5:4&+\\&&3:3:4&+&3:4:4&+&3:5:4&+\\&&4:3:4&+&4:4:4&+&4:5:4&+\\&&5:3:4&+&5:4:4&+&5:5:4&+\\&&6:3:4&+&6:4:4&+&6:5:4&+\\&&7:3:4&+&7:4:4&+&7:5:4\end{matrix}}}$
${\displaystyle {\begin{array}{ccl}\mathrm {te} (G)\cap \mathrm {te} (H)&=&4\!:\!3:\!4~+~4\!:\!4\!:\!4~+~4\!:\!5\!:\!4\\[4pt]G\circ H&=&\mathrm {proj} _{XZ}(\mathrm {te} (G)\cap \mathrm {te} (H))\\[4pt]&=&\mathrm {proj} _{XZ}(4\!:\!3:\!4~+~4\!:\!4\!:\!4~+~4\!:\!5\!:\!4)\\[4pt]&=&4:4\end{array}}}$
Matrix representation
We have it within our reach to pick up another way of representing dyadic relations, namely, the representation as logical matrices, and also to grasp the analogy between relational composition and ordinary matrix multiplication as it appears in linear algebra.
First of all, while we still have the data of a very simple concrete case in mind, let us reflect on what we did in our last Example in order to find the composition ${\displaystyle G\circ H\!}$ of the dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H.\!}$
Here is the setup that we had before:
${\displaystyle {\begin{matrix}X&=&\{1,2,3,4,5,6,7\}\end{matrix}}}$
${\displaystyle {\begin{matrix}G&=&4:3&+&4:4&+&4:5&\subseteq &X\times X\\H&=&3:4&+&4:4&+&5:4&\subseteq &X\times X\end{matrix}}}$
Let us recall the rule for finding the relational composition of a pair of dyadic relations. Given the dyadic relations ${\displaystyle P\subseteq X\times Y\!}$ and ${\displaystyle Q\subseteq Y\times Z,\!}$ the composition of ${\displaystyle P~{\text{on}}~Q\!}$ is written as ${\displaystyle P\circ Q,\!}$ or more simply as ${\displaystyle PQ,\!}$ and obtained as follows:
To compute ${\displaystyle PQ,\!}$ in general, where ${\displaystyle P\!}$ and ${\displaystyle Q\!}$ are dyadic relations, simply multiply out the two sums in the ordinary distributive algebraic way, but subject to the following rule for finding the product of two elementary relations of shapes ${\displaystyle a:b\!}$ and ${\displaystyle c:d.\!}$
${\displaystyle {\begin{matrix}(a:b)(c:d)&=&(a:d)&{\text{if}}~b=c\\(a:b)(c:d)&=&0&{\text{otherwise}}\end{matrix}}}$
To find the relational composition ${\displaystyle G\circ H,\!}$ one may begin by writing it as a quasi-algebraic product:
${\displaystyle {\begin{matrix}G\circ H&=&(4\!:\!3~+~4\!:\!4~+~4\!:\!5)(3\!:\!4~+~4\!:\!4~+~5\!:\!4)\end{matrix}}}$
Multiplying this out in accord with the applicable form of distributive law one obtains the following expansion:
${\displaystyle {\begin{matrix}G\circ H&=&(4:3)(3:4)&+&(4:3)(4:4)&+&(4:3)(5:4)&+\\&&(4:4)(3:4)&+&(4:4)(4:4)&+&(4:4)(5:4)&+\\&&(4:5)(3:4)&+&(4:5)(4:4)&+&(4:5)(5:4)\end{matrix}}}$
Applying the rule that determines the product of elementary relations produces the following array:
${\displaystyle {\begin{matrix}G\circ H&=&4:4&+&0&+&0&+\\&&0&+&4:4&+&0&+\\&&0&+&0&+&4:4\end{matrix}}}$
Since the plus sign in this context represents an operation of logical disjunction or set-theoretic aggregation, all of the positive multiplicites count as one, and this gives the ultimate result:
${\displaystyle {\begin{matrix}G\circ H&=&4:4\end{matrix}}}$
With an eye toward extracting a general formula for relation composition, viewed here on analogy to algebraic multiplication, let us examine what we did in multiplying the dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ together to obtain their relational composite ${\displaystyle G\circ H.\!}$
Given the space ${\displaystyle X=\{1,2,3,4,5,6,7\},\!}$ whose cardinality ${\displaystyle |X|\!}$ is ${\displaystyle 7,\!}$ there are ${\displaystyle |X\times X|=|X|\cdot |X|\!}$ ${\displaystyle =\!}$ ${\displaystyle 7\cdot 7=49\!}$ elementary relations of the form ${\displaystyle i:j,\!}$ where ${\displaystyle i\!}$ and ${\displaystyle j\!}$ range over the space ${\displaystyle X.\!}$ Although they might be organized in many different ways, it is convenient to regard the collection of elementary relations as arranged in a lexicographic block of the following form:
${\displaystyle {\begin{matrix}1\!:\!1&1\!:\!2&1\!:\!3&1\!:\!4&1\!:\!5&1\!:\!6&1\!:\!7\\2\!:\!1&2\!:\!2&2\!:\!3&2\!:\!4&2\!:\!5&2\!:\!6&2\!:\!7\\3\!:\!1&3\!:\!2&3\!:\!3&3\!:\!4&3\!:\!5&3\!:\!6&3\!:\!7\\4\!:\!1&4\!:\!2&4\!:\!3&4\!:\!4&4\!:\!5&4\!:\!6&4\!:\!7\\5\!:\!1&5\!:\!2&5\!:\!3&5\!:\!4&5\!:\!5&5\!:\!6&5\!:\!7\\6\!:\!1&6\!:\!2&6\!:\!3&6\!:\!4&6\!:\!5&6\!:\!6&6\!:\!7\\7\!:\!1&7\!:\!2&7\!:\!3&7\!:\!4&7\!:\!5&7\!:\!6&7\!:\!7\end{matrix}}}$
The relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ may then be regarded as logical sums of the following forms:
${\displaystyle {\begin{matrix}G&=&\displaystyle \sum _{ij}G_{ij}(i\!:\!j)\\[20pt]H&=&\displaystyle \sum _{ij}H_{ij}(i\!:\!j)\end{matrix}}\!}$
The notation ${\displaystyle \textstyle \sum _{ij}\!}$ indicates a logical sum over the collection of elementary relations ${\displaystyle i\!:\!j\!}$ while the factors ${\displaystyle G_{ij}\!}$ and ${\displaystyle H_{ij}\!}$ are values in the boolean domain ${\displaystyle \mathbb {B} =\{0,1\}~\!}$ that are called the coefficients of the relations ${\displaystyle G\!}$ and ${\displaystyle H,\!}$ respectively, with regard to the corresponding elementary relations ${\displaystyle i\!:\!j.\!}$
In general, for a dyadic relation ${\displaystyle L,\!}$ the coefficient ${\displaystyle L_{ij}\!}$ of the elementary relation ${\displaystyle i\!:\!j\!}$ in the relation ${\displaystyle L\!}$ will be ${\displaystyle 0\!}$ or ${\displaystyle 1,\!}$ respectively, as ${\displaystyle i\!:\!j\!}$ is excluded from or included in ${\displaystyle L.\!}$
With these conventions in place, the expansions of ${\displaystyle G\!}$ and ${\displaystyle H\!}$ may be written out as follows:
${\displaystyle {\begin{matrix}G&=&4:3&+&4:4&+&4:5&=\end{matrix}}}$ ${\displaystyle {\begin{smallmatrix}0\cdot (1:1)&+&0\cdot (1:2)&+&0\cdot (1:3)&+&0\cdot (1:4)&+&0\cdot (1:5)&+&0\cdot (1:6)&+&0\cdot (1:7)&+\\0\cdot (2:1)&+&0\cdot (2:2)&+&0\cdot (2:3)&+&0\cdot (2:4)&+&0\cdot (2:5)&+&0\cdot (2:6)&+&0\cdot (2:7)&+\\0\cdot (3:1)&+&0\cdot (3:2)&+&0\cdot (3:3)&+&0\cdot (3:4)&+&0\cdot (3:5)&+&0\cdot (3:6)&+&0\cdot (3:7)&+\\0\cdot (4:1)&+&0\cdot (4:2)&+&\mathbf {1} \cdot (4:3)&+&\mathbf {1} \cdot (4:4)&+&\mathbf {1} \cdot (4:5)&+&0\cdot (4:6)&+&0\cdot (4:7)&+\\0\cdot (5:1)&+&0\cdot (5:2)&+&0\cdot (5:3)&+&0\cdot (5:4)&+&0\cdot (5:5)&+&0\cdot (5:6)&+&0\cdot (5:7)&+\\0\cdot (6:1)&+&0\cdot (6:2)&+&0\cdot (6:3)&+&0\cdot (6:4)&+&0\cdot (6:5)&+&0\cdot (6:6)&+&0\cdot (6:7)&+\\0\cdot (7:1)&+&0\cdot (7:2)&+&0\cdot (7:3)&+&0\cdot (7:4)&+&0\cdot (7:5)&+&0\cdot (7:6)&+&0\cdot (7:7)\end{smallmatrix}}}$
${\displaystyle {\begin{matrix}H&=&3:4&+&4:4&+&5:4&=\end{matrix}}}$ ${\displaystyle {\begin{smallmatrix}0\cdot (1:1)&+&0\cdot (1:2)&+&0\cdot (1:3)&+&0\cdot (1:4)&+&0\cdot (1:5)&+&0\cdot (1:6)&+&0\cdot (1:7)&+\\0\cdot (2:1)&+&0\cdot (2:2)&+&0\cdot (2:3)&+&0\cdot (2:4)&+&0\cdot (2:5)&+&0\cdot (2:6)&+&0\cdot (2:7)&+\\0\cdot (3:1)&+&0\cdot (3:2)&+&0\cdot (3:3)&+&\mathbf {1} \cdot (3:4)&+&0\cdot (3:5)&+&0\cdot (3:6)&+&0\cdot (3:7)&+\\0\cdot (4:1)&+&0\cdot (4:2)&+&0\cdot (4:3)&+&\mathbf {1} \cdot (4:4)&+&0\cdot (4:5)&+&0\cdot (4:6)&+&0\cdot (4:7)&+\\0\cdot (5:1)&+&0\cdot (5:2)&+&0\cdot (5:3)&+&\mathbf {1} \cdot (5:4)&+&0\cdot (5:5)&+&0\cdot (5:6)&+&0\cdot (5:7)&+\\0\cdot (6:1)&+&0\cdot (6:2)&+&0\cdot (6:3)&+&0\cdot (6:4)&+&0\cdot (6:5)&+&0\cdot (6:6)&+&0\cdot (6:7)&+\\0\cdot (7:1)&+&0\cdot (7:2)&+&0\cdot (7:3)&+&0\cdot (7:4)&+&0\cdot (7:5)&+&0\cdot (7:6)&+&0\cdot (7:7)\end{smallmatrix}}}$
Stripping down to the bare essentials, one obtains the following matrices of coefficients for the relations ${\displaystyle G\!}$ and ${\displaystyle H.\!}$
${\displaystyle G~=~{\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&1&1&1&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\end{pmatrix}}}$
${\displaystyle H~=~{\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&1&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\end{pmatrix}}}$
These are the logical matrix representations of the dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H.\!}$
If the dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ are viewed as logical sums then their relational composition ${\displaystyle G\circ H\!}$ can be regarded as a product of sums, a fact that can be indicated as follows:
${\displaystyle G\circ H~=~(\sum _{ij}G_{ij}(i\!:\!j))(\sum _{ij}H_{ij}(i\!:\!j)).\!}$
The composite relation ${\displaystyle G\circ H\!}$ is itself a dyadic relation over the same space ${\displaystyle X,\!}$ in other words, ${\displaystyle G\circ H\subseteq X\times X,\!}$ and this means that ${\displaystyle G\circ H\!}$ must be amenable to being written as a logical sum of the following form:
${\displaystyle G\circ H~=~\sum _{ij}(G\circ H)_{ij}(i\!:\!j).\!}$
In this formula, ${\displaystyle (G\circ H)_{ij}\!}$ is the coefficient of ${\displaystyle G\circ H\!}$ with respect to the elementary relation ${\displaystyle i\!:\!j.\!}$
One of the best ways to reason out what ${\displaystyle G\circ H\!}$ should be is to ask oneself what its coefficient ${\displaystyle (G\circ H)_{ij}\!}$ should be for each of the elementary relations ${\displaystyle i\!:\!j\!}$ in turn.
So let us pose the question:
${\displaystyle (G\circ H)_{ij}~=~?\!}$
In order to answer this question, it helps to realize that the indicated product given above can be written in the following equivalent form:
${\displaystyle G\circ H~=~(\sum _{ik}G_{ik}(i\!:\!k))(\sum _{kj}H_{kj}(k\!:\!j)).\!}$
A moment's thought will tell us that ${\displaystyle (G\circ H)_{ij}=1\!}$ if and only if there is an element ${\displaystyle k\!}$ in ${\displaystyle X\!}$ such that ${\displaystyle G_{ik}=1\!}$ and ${\displaystyle H_{kj}=1.\!}$
Consequently, we have the result:
${\displaystyle (G\circ H)_{ij}~=~\sum _{k}G_{ik}H_{kj}.\!}$
This follows from the properties of boolean arithmetic, specifically, from the fact that the product ${\displaystyle G_{ik}H_{kj}\!}$ is ${\displaystyle 1\!}$ if and only if both ${\displaystyle G_{ik}\!}$ and ${\displaystyle H_{kj}\!}$ are ${\displaystyle 1\!}$ and from the fact that ${\displaystyle \textstyle \sum _{k}F_{k}\!}$ is equal to ${\displaystyle 1\!}$ just in case some ${\displaystyle F_{k}\!}$ is ${\displaystyle 1.\!}$
All that remains in order to obtain a computational formula for the relational composite ${\displaystyle G\circ H\!}$ of the dyadic relations ${\displaystyle G\!}$ and ${\displaystyle H\!}$ is to collect the coefficients ${\displaystyle (G\circ H)_{ij}\!}$ as ${\displaystyle i\!}$ and ${\displaystyle j\!}$ range over ${\displaystyle X.\!}$
${\displaystyle {\begin{matrix}G\circ H&=&\displaystyle \sum _{ij}(G\circ H)_{ij}(i\!:\!j)&=&\displaystyle \sum _{ij}(\sum _{k}G_{ik}H_{kj})(i\!:\!j).\end{matrix}}}$
This is the logical analogue of matrix multiplication in linear algebra, the difference in the logical setting being that all of the operations performed on coefficients take place in a system of boolean arithmetic where summation corresponds to logical disjunction and multiplication corresponds to logical conjunction.
By way of disentangling this formula, one may notice that the form ${\displaystyle \textstyle \sum _{k}G_{ik}H_{kj}\!}$ is what is usually called a scalar product. In this case it is the scalar product of the ${\displaystyle i^{\text{th}}\!}$ row of ${\displaystyle G\!}$ with the ${\displaystyle j^{\text{th}}\!}$ column of ${\displaystyle H.\!}$
To make this statement more concrete, let us go back to the examples of ${\displaystyle G\!}$ and ${\displaystyle H\!}$ we came in with:
${\displaystyle G~=~{\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&1&1&1&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\end{pmatrix}}}$
${\displaystyle H~=~{\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&1&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\end{pmatrix}}}$
The formula for computing ${\displaystyle G\circ H\!}$ says the following:
${\displaystyle {\begin{array}{ccl}(G\circ H)_{ij}&=&{\text{the}}~{ij}^{\text{th}}~{\text{entry in the matrix representation for}}~G\circ H\\[2pt]&=&{\text{the entry in the}}~{i}^{\text{th}}~{\text{row and the}}~{j}^{\text{th}}~{\text{column of}}~G\circ H\\[2pt]&=&{\text{the scalar product of the}}~{i}^{\text{th}}~{\text{row of}}~G~{\text{with the}}~{j}^{\text{th}}~{\text{column of}}~H\\[2pt]&=&\sum _{k}G_{ik}H_{kj}\end{array}}}$
As it happens, it is possible to make exceedingly light work of this example, since there is only one row of ${\displaystyle G\!}$ and one column of ${\displaystyle H\!}$ that are not all zeroes. Taking the scalar product, in a logical way, of the fourth row of ${\displaystyle G\!}$ with the fourth column of ${\displaystyle H\!}$ produces the sole non-zero entry for the matrix of ${\displaystyle G\circ H.\!}$
${\displaystyle G\circ H~=~{\begin{pmatrix}0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\\0&0&0&0&0&0&0\end{pmatrix}}}$
Graph-theoretic picture
There is another form of representation for dyadic relations that is useful to keep in mind, especially for its ability to render the logic of many complex formulas almost instantly understandable to the mind's eye. This is the representation in terms of bipartite graphs, or bigraphs for short.
Here is what ${\displaystyle G\!}$ and ${\displaystyle H\!}$ look like in the bigraph picture:
o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 9. G = 4:3 + 4:4 + 4:5 o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 10. H = 3:4 + 4:4 + 5:4
These graphs may be read to say:
${\displaystyle {\begin{matrix}G~{\text{puts}}~4~{\text{in relation to}}~3,4,5.\\[2pt]H~{\text{puts}}~3,4,5~{\text{in relation to}}~4.\end{matrix}}}$
To form the composite relation ${\displaystyle G\circ H,\!}$ one simply follows the bigraph for ${\displaystyle G\!}$ by the bigraph for ${\displaystyle H,\!}$ here arranging the bigraphs in order down the page, and then treats any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those nodes in the composite bigraph for ${\displaystyle G\circ H.\!}$
Here's how it looks in pictures:
o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ | | / | \ G | | / | \ | | o o o o o o o X | | \ | / | | \ | / H | | \|/ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 11. G Followed By H o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | | | G o H | | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 12. G Composed With H
Once again we find that ${\displaystyle G\circ H=4:4.\!}$
We have now seen three different representations of dyadic relations. If one has a strong preference for letters, or numbers, or pictures, then one may be tempted to take one or another of these as being canonical, but each of them will be found to have its peculiar advantages and disadvantages in any given application, and the maximum advantage is therefore approached by keeping all three of them in mind.
To see the promised utility of the bigraph picture of dyadic relations, let us devise a slightly more complex example of a composition problem, and use it to illustrate the logic of the matrix multiplication formula.
Keeping to the same space ${\displaystyle X=\{1,2,3,4,5,6,7\},\!}$ define the dyadic relations ${\displaystyle M,N\subseteq X\times X\!}$ as follows:
${\displaystyle {\begin{array}{*{19}{c}}M&=&2\!:\!1&+&2\!:\!2&+&2\!:\!3&+&4\!:\!3&+&4\!:\!4&+&4\!:\!5&+&6\!:\!5&+&6\!:\!6&+&6\!:\!7\\[2pt]N&=&1\!:\!1&+&2\!:\!1&+&3\!:\!3&+&4\!:\!3&~&+&~&4\!:\!5&+&5\!:\!5&+&6\!:\!7&+&7\!:\!7\end{array}}}$
Here are the bigraph pictures:
o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 13. Dyadic Relation M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 14. Dyadic Relation N
To form the composite relation ${\displaystyle M\circ N,\!}$ one simply follows the bigraph for ${\displaystyle M\!}$ by the bigraph for ${\displaystyle N,\!}$ arranging the bigraphs in order down the page, and then counts any non-empty set of paths of length two between two nodes as being equivalent to a single directed edge between those two nodes in the composite bigraph for ${\displaystyle M\circ N.\!}$
Here's how it looks in pictures:
o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 15. M Followed By N o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | / \ / \ / \ | | / \ / \ / \ M o N | | / \ / \ / \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 16. M Composed With N
Let us hark back to that mysterious matrix multiplication formula, and see how it appears in the light of the bigraph representation.
The coefficient of the composition ${\displaystyle M\circ N\!}$ between ${\displaystyle i\!}$ and ${\displaystyle j\!}$ in ${\displaystyle X\!}$ is given as follows:
${\displaystyle (M\circ N)_{ij}~=~\sum _{k}M_{ik}N_{kj}\!}$
Graphically interpreted, this is a sum over paths. Starting at the node ${\displaystyle i,\!}$ ${\displaystyle M_{ik}\!}$ being ${\displaystyle 1\!}$ indicates that there is an edge in the bigraph of ${\displaystyle M\!}$ from node ${\displaystyle i\!}$ to node ${\displaystyle k\!}$ and ${\displaystyle N_{kj}\!}$ being ${\displaystyle 1\!}$ indicates that there is an edge in the bigraph of ${\displaystyle N\!}$ from node ${\displaystyle k\!}$ to node ${\displaystyle j.\!}$ So the ${\displaystyle \textstyle \sum _{k}\!}$ ranges over all possible intermediaries ${\displaystyle k,\!}$ ascending from ${\displaystyle 0\!}$ to ${\displaystyle 1\!}$ just as soon as there happens to be a path of length two between nodes ${\displaystyle i\!}$ and ${\displaystyle j.\!}$
It is instructive at this point to compute the other possible composition that can be formed from ${\displaystyle M\!}$ and ${\displaystyle N,\!}$ namely, the composition ${\displaystyle N\circ M,\!}$ that takes ${\displaystyle M\!}$ and ${\displaystyle N\!}$ in the opposite order. Here is the graphic computation:
o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | / | / \ | \ | | | | / | / \ | \ | N | | |/ |/ \| \| | | o o o o o o o X | | /|\ /|\ /|\ | | / | \ / | \ / | \ M | | / | \ / | \ / | \ | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 17. N Followed By M o---------------------------------------o | | | 1 2 3 4 5 6 7 | | o o o o o o o X | | | | N o M | | | | o o o o o o o X | | 1 2 3 4 5 6 7 | | | o---------------------------------------o Figure 18. N Composed With M
In sum, ${\displaystyle N\circ M=0.\!}$ This example affords sufficient evidence that relational composition, just like its kindred, matrix multiplication, is a non-commutative algebraic operation.
References
• Ulam, S.M., and Bednarek, A.R., “On the Theory of Relational Structures and Schemata for Parallel Computation” (1977), pp. 477–508 in A.R. Bednarek and Françoise Ulam (eds.), Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, University of California Press, Berkeley, CA, 1990.
Bibliography
• Mathematical Society of Japan, Encyclopedic Dictionary of Mathematics, 2nd edition, 2 volumes., Kiyosi Itô (ed.), MIT Press, Cambridge, MA, 1993.
• Mili, A., Desharnais, J., Mili, F., with Frappier, M., Computer Program Construction, Oxford University Press, New York, NY, 1994.
• Ulam, S.M., Analogies Between Analogies : The Mathematical Reports of S.M. Ulam and His Los Alamos Collaborators, A.R. Bednarek and Françoise Ulam (eds.), University of California Press, Berkeley, CA, 1990.
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# Direct observation of nanowire growth and decomposition
## Abstract
Fundamental concepts of the crystal formation suggest that the growth and decomposition are determined by simultaneous embedding and removal of the atoms. Apparently, by changing the crystal formation conditions one can switch the regimes from the growth to decomposition. To the best of our knowledge, so far this has been only postulated, but never observed at the atomic level. By means of in situ environmental transmission electron microscopy we monitored and examined the atomic layer transformation at the conditions of the crystal growth and its decomposition using CuO nanowires selected as a model object. The atomic layer growth/decomposition was studied by varying an O2 partial pressure. Three distinct regimes of the atomic layer evolution were experimentally observed: growth, transition and decomposition. The transition regime, at which atomic layer growth/decomposition switch takes place, is characterised by random nucleation of the atomic layers on the growing {111} surface. The decomposition starts on the side of the nanowire by removing the atomic layers without altering the overall crystal structure, which besides the fundamental importance offers new possibilities for the nanowire manipulation. Understanding of the crystal growth kinetics and nucleation at the atomic level is essential for the precise control of 1D crystal formation.
## Introduction
From the crystal growth fundamentals1,2,3,4 it is known that typically crystal growth proceeds in four sequences: a) diffusion of the species (atoms, ions, etc.) to the growing surface; b) adsorption and desorption of the species onto and from the growing surface; c) adsorbed species diffusion on the growing surface and d) crystal surface growth by incorporating the adsorbed growth species. The adsorption-desorption of the growth species is not only the rate limiting step for most crystal growth conditions, but it also determines the overall crystal evolution to either growth or decomposition.
A nanowire (NW) is an example of an elongated crystal, which preferably grows in one direction. Therefore, it is an ideal object for the investigations of the crystal growth. Among various materials CuO is one of the most attractive model objects, since it is one of the most studied crystals and allows to carry out real time in situ observation in a transmission electron microscope. Moreover, the CuO NWs grow by adding atomic layers (ALs) at the tip limited to the growing {111} surface, therefore offering a convenient way for observation of a single AL nucleation and kinetics without a complicated interactions of various crystal orientation5.
Thermodynamically, copper oxidation state changes among CuO, Cu2O, and Cu as a function of temperature and O2 partial pressure6,7 with possible pathways of direct reduction (CuO → Cu) or reduction involving either one or two intermediates8. Studies of CuO NW reduction were carried out in reducing environments of H2/N2 plasma9, CO10 or in vacuum11, revealing the CuO-Cu2O-Cu transitions. CuO reduction to pure metal without the intermediate oxide was also demonstrated to happen and Cu2O phase was shown to form only at special conditions8,12. The most of these works relied on XPS or XAES technique observations13,14 and allowed to follow the crystal phase or oxidation state changes only ex situ and without being able to observe the atomic rearrangement. Therefore, despite the research reported to date, there is still a lack of knowledge about the kinetics of AL transformation and their growth and decomposition mechanisms.
Here, we investigated the crystal growth and decomposition at the atomic scale under respectively oxidative and reductive environments using CuO NW as a model object. To the best of our knowledge, we reported the first clear and atomically resolved transition between the growth and decomposition on the surface of NWs at the AL. We observed a clear AL nucleation character change during the transition from the growth to the decomposition of CuO NWs. Interestingly the layer-by-layer decomposition of NWs happens without disrupting its crystal structure. This study reveals the kinetics and mechanism of the AL arrangement during the crystal growth at oxidative conditions, the decomposition in vacuum and the switch between these two processes. The understanding of the crystal transformations at the AL during the NW growth is crucial to control the NW structure, composition and thereby its properties. The investigations of the CuO NW growth and decomposition are important from the fundamental point of view, as the understanding of the 1D crystal formation mechanism can be extended to other systems.
## Results and Discussion
Initially, CuO NW growth was ex situ observed by heating Cu samples at the pressure ranging from high vacuum to the atmospheric pressure (Fig. 1). It was found that the NW growth occurs at O2 pressures above 400 Pa, and the density of the grown NWs increases along with the pressure. At pressures below 400 Pa, the NW growth was not observed. In the pressure range from 20 Pa to 400 Pa, faceted crystals were formed on the surface (Supplementary Fig. S1). At pressures below 20 Pa no growth of 1D structures was observed, only a planar layer of CuO grew.
The in situ environmental transmission electron microscopy (ETEM) observations of CuO NWs under changing O2 pressure at a constant temperature (400 °C) revealed 3 AL evolution regimes: the growth, transition and decomposition. First, as it was observed in ex-situ experiments, the formation of NW structure, caused by ordered AL growth, was observed only over 400 Pa. The CuO NW growth followed layer by layer manner, when every subsequent AL grows exactly on top of a primary one (Supplementary Fig. S2). ALs nucleated at the edge of the twin boundary ridge, at the tip of NW (Fig. 2A). At the initial O2 pressure under 400 Pa, ordered AL growth (NW growth) was not observed, ALs were nucleating randomly and a planar layer of CuO was grown. Interestingly, if the NW formation started at higher pressures (in the growth regime) and the pressure subsequently was reduced during the observation, the ordered growth of ALs proceeded even at lower pressures (from 100 to 400 Pa), even though that ALs nucleate randomly on the growing surface. Figure 2 shows the results obtained during the in situ observation of the NW growth initiated at 700 Pa and continued after the O2 pressure was reduced to 340 Pa (Supplementary Video S1). The video demonstrates that O2 pressure is critical for the initial ordered nucleation of ALs (NW formation), but if the NW structure is already available, ALs nucleate at the growing surface, retaining the ordered structure.
During the formation of ALs in the growth regime, ALs nucleate at the twin boundary of the growing NW (Fig. 2A), however, in the transition regime the AL nucleation is observed at random points on {111} surface (Figs 2B, 3). This shows that the nucleation conditions changed and the twin boundary is not anymore the preferential nucleation point. Interestingly, the AL growth rate 1.7±0.4 nm/s in the transition regime remained in the same order of magnitude as in the growth regime (1.4±0.3 nm/s) even if the O2 pressure was lower (Fig. 4). The FFT pattern (Fig. 3A inset) can be well indexed using the monoclinic CuO structure (a = 4.69, b = 3.43, с = 5.13 Å, β = 99.55°), which is consistent with our earlier analysis15,16,17.
Further decreasing the O2 pressure in the system led to the decomposition regime. Experimentally it was found that the decomposition regime occurred at the O2 pressures around 0.4 Pa. It can be noticed that the decomposition of NWs takes place in an ordered layer-by-layer manner, similarly to the ordered growth of ALs on the NW tip, but with longer induction time and higher AL decomposition rate of 4.2±0.5 nm/s (Fig. 5). Induction time is a period of time that is necessary to initiate a layer growth, and it was calculated on the basis of ETEM images as the time between nucleation of two consequent ALs.
The decomposition started from the base of the CuO NW on (002) lattice plane. We did not observe the transition in the crystal phase to any intermediates such as Cu2O or to metallic Cu. In case of CuO reduction in vacuum, considerable changes can be seen already at 100°C13 and the most probable mechanism involves oxygen diffusion to bulk Cu rather than oxygen desorption into vacuum13,14. In our case, the decomposition of ALs starts at the base of the NW, which may be related with the shorter diffusion route from the base of NW to the bulk Cu, located underneath the surface (Supplementary Video 2, Supplementary Figure S3).
On the basis of the experimentally determined rate of the AL decomposition (v dec = 4.2 nm s−1, Fig. 5), we can calculate the decomposition time of CuO as l/v dec , where l = 0.27 nm is the size of CuO molecule, which can be estimated from the bulk density and volume per one molecule (0.021 nm3). Thus, we obtain the experimental decomposition time of τ = 0.064 s.
We can also estimate the CuO decomposition time based on the decomposition energy, E, and the frequency of the CuO lattice vibration, ν, as3:
$$\tau ={\nu }^{-1}exp(E/RT),$$
(1)
where R is the gas constant, T is the absolute temperature. The decomposition energy can be estimated from the formation enthalpy taking into account the excess surface energy on a layer step as E = −ΔH-γl 2/2. Here γ is the specific surface energy. Using the numerical values of ΔH = −155 kJ mol−1, R = 8.31 J mol−1K−1, T = 673 K, ν ~ 1012 s−1, γ = 0.74 J/m2 for CuO (111) and γ = 0. 86 J/m2 for CuO ($$\mathop{1}\limits^{\bar{} }$$11)18 we obtain τ~ 0.06 s and τ~ 0.04 s, respectively. These values are in a good agreement with the decomposition time determined experimentally (0.064 s).
Bearing in mind that during the CuO AL formation two processes – embedding of CuO to the layer and removal of CuO from the layer - are conducted simultaneously. Therefore, let us compare the arrival time of “active” O2 molecules (forming CuO), and the removal time of CuO from the layer, defined from eq. (1). The arrival time of “active” O2 molecules to the CuO molecule can be estimated similar to Eq. 1 as:
$${\tau }_{a}={{\nu }_{g}}^{-1}\exp ({E}_{a}/RT),$$
(2)
where ν g =l 2 Vn/4 is a frequency of gas molecule collisions with wall of area l 2, V=(8RT 0/πM ox )0.5 is a mean velocity of O2 molecules, n = pN A /RT 0 is a number concentration of O2 inside ETEM chamber, p is an O2 pressure, and M ox is an O2 molar weight, N A is the Avogadro number, T 0 is a temperature of oxygen inside the ETEM chamber, E a is the activation energy of NW growth. Substituting E a = 37 kJmol−1, T = 673 K, and T 0 = 298 K17 and the experimental deposition time (τ a = 0.064 s) to Eq. (2) we obtain p = 6 Pa. This means that at the O2 pressure lower than 6 Pa the removal from the AL prevails, which explains the experimentally observed decomposition at 0.4 Pa. At the same time, the AL growth is observed at 400 Pa, the pressure which is almost 70 times higher than the pressure above which the CuO formation is preferred. We believe that the excess pressure is related to the formation of the NW, since as we demonstrated above, if the NW structure is already formed, the NW growth proceeds at lower pressure.
It is worth noting that the AL nucleation is usually more favorable at the corner of the facet of the growing crystal, where the supersaturation is higher than at the center (the Berg’s effect)19,20. However, as we found from the in situ observation, with the pressure decreasing the AL growth may start at any surface point. The nucleation of the AL at random place may be determined by a weakening of the Berg’s effect due to the decrease of CuO molecule concentration gradient along the edge at lower O2 pressure.
Thus, the O2 pressure affects both interconnected processes: the growth and decomposition. Obviously, there must be such O2 pressures at which the 1D growth ceases and at which decomposition prevails. We experimentally found these O2 pressures to be around 400 Pa for the AL growth and 0.4 Pa for the decomposition, respectively.
## Conclusions
To summarize, an atomistic examination of the CuO NW evolution in the conditions ranging from oxidative to reductive ones was carried out. The growth, transition and decomposition regimes were experimentally detected and explained from the AL kinetics point of view. The AL growth/decomposition regimes are determined by two simultaneous processes: embedding and removal of CuO from the AL, where both processes are in equilibrium at the O2 pressure of 6 Pa. Therefore, the experimentally observed AL decomposition was obtained at pressures below 0.4 Pa as removal process prevails at lower O2 pressures. The NW growth is observed at pressures above 400 Pa. The range between those pressures associated with the transition regime, where crystal growth conditions are favorable only in case of previously formed NWs. Moreover, the transition regime is characterized by weakening of the Berg’s effect, which leads to the change of the AL nucleation from the edge of the twin boundary ridge at the NW tip to a random position on the surface at lattice plane {111}.
Decomposition of CuO NW proceeded in ordered layer-by-layer manner at 0.4 Pa from the bottom side of the NW at the lattice plane (002), which can be explained by the shorter diffusion route from the NW base to underlying bulk Cu. A crystal phase transformation from CuO to intermediate Cu2O or Cu was not observed.
Although we investigated the AL growth and decomposition only in one kind of NWs, these atomistic principles can be common to the other 1D crystals. Since the decomposition was pronounced in the bottom part of the NW, it can be also used as a method for NW diameter tailoring or removal from the substrate.
## Methods
An aberration-corrected FEI Titan 80–300 ETEM operated at 300 kV was employed for the in situ observation of the AL evolution17,21. The sample was dispersed in ethanol and drop-casted on a MEMS-based micro-heater from DENSsolutions. After drying at room temperature, the sample was inserted into the electron microscope.
The CuO NWs were synthesized in situ in the ETEM by simple oxidation of pure metallic Cu (>99.99%) powder in the presence of O2. The decomposition of the NWs was carried out by lowering the O2 pressure maintaining the same temperature of the substrate. The NWs were grown directly on the as-received Cu powder, with no patterning or catalyst addition. The observation time at the set temperature was 30–200 min. Typically, the CuO NWs were observed at 400 °C varying the O2 partial pressure from 10–6 up to 700 Pa. The electron beam dose varied depending on the conditions from 9 × 103 to 9 × 104 electrons/(nm2 s). No visible electron beam damage was observed during the in situ growth of the NWs, also no visible difference between the observation area and the rest of the sample was found. For the ex situ growth of NWs, pure metallic Cu (99.999%) samples, were heated for 30 min in a vacuum tube furnace (inner diameter 50 mm) Carbolite CTF 12/65/550 with built in flowmeter for O2 and control unit 3216P1 at O2 partial pressures from 20 Pa to the atmospheric pressure. After the synthesis, the samples were transferred directly to scanning electron microscope (SEM, JEOL JSM 7500F) as grown, without any pre-treatment.
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## Acknowledgements
The authors thank Prof. Esko Kauppinen (Aalto University, Finland) for discussion of the experimental results. S.R. acknowledges NorTEMnet for support and European Union’s Seventh Framework programme for research and innovation under the Marie Sklodowska-Curie grant agreement No 609402–2020 researchers: Train to Move (T2M). The work was carried out with financial support from the Ministry of Education and Science of the Russian Federation in the framework of increase Competitiveness Program of NUST “MISIS”, implemented by a governmental decree dated 16th of March 2013, № 211.
## Author information
### Affiliations
1. #### Department of Applied Physics, Aalto University School of Science, Puumiehenkuja 2, 00076, Aalto, Finland
• Simas Rackauskas
• , Hua Jiang
• & Albert G. Nasibulin
2. #### University of Turin, Department of Chemistry, NIS Interdepartmental Centre and INSTM Reference Centre, Via P. Giuria 7, 10125, Turin, Italy
• Simas Rackauskas
3. #### Kemerovo State University, Krasnaya str. 6, Kemerovo, 650043, Russia
• Sergey D. Shandakov
4. #### Center for Electron Nanoscopy, Technical University of Denmark, DK-2800 Kgs., Lyngby, Denmark
• Jakob B. Wagner
5. #### Skolkovo Institute of Science and Technology, Nobel str. 3, Moscow, 143026, Russia
• Albert G. Nasibulin
6. #### National University of Science and Technology “MISIS”, Leninsky pr. 4, Moscow, Russia
• Albert G. Nasibulin
### Contributions
S.R. and A.N. conceived and designed this study, analyzed the data and wrote the manuscript together with S.S., H.J. and J.W. realized the ETEM experiment. All the authors discussed the results and commented on the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding authors
Correspondence to Simas Rackauskas or Albert G. Nasibulin. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9093420505523682, "perplexity": 3263.2639468290186}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583514355.90/warc/CC-MAIN-20181021203102-20181021224602-00219.warc.gz"} |
http://tex.stackexchange.com/questions/71300/calculating-a-common-height-for-nodes-with-different-amounts-of-text-in-tikz/71351 | # Calculating a common height for nodes with different amounts of text in Tikz
When using labelled nodes (e.g. for a flow chart), their sizes are automatically chosen. One can enforce the same size, which must be known a priori, for multiple nodes by using "large enough" values for minimum height and minimum width.
In the following example the node width is restricted for spacial reasons. This leads to the boxes containing different numbers of lines and therefore different box sizes.
\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document}
\begin{frame}
\begin{tikzpicture}[node distance=2cm]
\node[rectangle,text width=5em,draw=black] (leftnode) {Short text};
\node[rectangle,text width=5em,draw=black,right=of leftnode] (rightnode) {A text longer than the other one};
\end{tikzpicture}
\end{frame}
\end{document}
Given two strings, that shall be written in two separate nodes of equal size: Is it possible to let LaTeX calculate the minimal node height that is large enough for both cases and use it as a parameter for the nodes?
-
## Identical Heights:
When you specify the text width, the text is placed in a \parbox of the width provided by text width. So, a simple fix is to use a \vphantom{} in the shorter node to ensure that it is of the same height:
## Code:
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{positioning}
\newcommand*{\TextWidth}{5em}%
\newcommand*{\StringA}{Short}%
\newcommand*{\StringB}{A text longer than the other one}%
\newcommand{\VPhantom}{\vphantom{\parbox{\TextWidth}{\StringB}}}%
\begin{document}
\begin{tikzpicture}[node distance=0.1cm]
\node[rectangle,text width=\TextWidth,draw=black, red] (leftnode) {\VPhantom\StringA};
\node[rectangle,text width=\TextWidth,draw=black, blue, right=of leftnode] (rightnode) {\StringB};
\end{tikzpicture}
\end{document}
## Identical Widths
You can use the \widthof{} to compute the length and then chose the maximum width as the minimum length:
## Note
• The 2*\InnerSep adjusts for the value of inner sep that is on both sides of the node, so if that value is changed this formula will need to be adjusted as well.
## Code:
\documentclass{article}
\usepackage{tikz}
\newcommand*{\InnerSep}{0.333em}%
\newcommand*{\StringA}{short}%
\newcommand*{\StringB}{a longer string}%
\begin{document}
\pgfmathsetmacro{\MinimumWidth}{%
max(\widthof{\StringA},\widthof{\StringB})+2*\InnerSep%
}%
\begin{tikzpicture}[minimum width=\MinimumWidth]
\node [rectangle, draw=red] at (0,0) {\StringA};
\node [rectangle, draw=blue] at (0,1) {\StringB};
\end{tikzpicture}
\end{document}
-
Turns out, I have not been specific enough. From the idea this is exactly, what I am looking for, but I want to control the height as well. Using \heightof alone does not do the trick, because the macro may not contain \. Any ideas? – Tim Sep 12 '12 at 19:30
@Tim: I am not sure what you mean by "may not contain a \. Best if you update the question and compose a fully compilable MWE that illustrates the exact problem. – Peter Grill Sep 12 '12 at 19:46
What I meant was newline \, which gives an error when it is used inside your example. I changed the question now. – Tim Sep 12 '12 at 20:37
Have updated the solution to provide the same height nodes. – Peter Grill Sep 12 '12 at 21:36
Thank you very much! – Tim Sep 12 '12 at 22:47
The answer above works well if the smaller string only uses one line. For multiple lines in both nodes I came to the following solution (combining both approaches given above and using the calc package).
\documentclass{beamer}
\usepackage{calc}
\usepackage{tikz}
\usetikzlibrary{positioning}
\newcommand*{\TextWidth}{5em}
\newcommand*{\Innersep}{0.33em}
\newcommand{\StringA}{Short multiline text}
\newcommand{\StringB}{A text longer than the other one}
\newlength\myheight
\setlength{\myheight}{\totalheightof{\parbox{\TextWidth}{\StringB}}+Innersep*2}
\begin{document}
\begin{frame}
\begin{tikzpicture}[node distance=2cm]
\node[rectangle,text width=\TextWidth,inner sep=\Innersep,minimum height=\myheight,draw=black] (leftnode) {\StringA};
\node[rectangle,text width=\TextWidth,inner sep=\Innersep,minimum height=\myheight,draw=black,right=of leftnode] (rightnode) {\StringB};
\end{tikzpicture}
\end{frame}
\end{document}
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9110460877418518, "perplexity": 2274.26525367962}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375099173.16/warc/CC-MAIN-20150627031819-00258-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/67473/non-vanishing-of-homology-of-loop-spaces | # Non-vanishing of homology of loop spaces
One of the answers to this MO question implies that loop spaces of $S^n$ for $n>1$ have non-zero homology in arbitrarily high degree.
Is there any simple (or, better yet, geometric) way to prove this?
And if the general result is too strong, is there any simple way to at least show an example of a loop space of a sphere with some high degree non-vanishing homology?
I'm curious because the only sphere loop space I can actually imagine ($\mathbb{Z} \approx \Omega S^1$) does not exhibit this kind of behaviour.
Edit: This is not exactly what I meant. It is clear to me that $H^{n-1}(\Omega S^n) \approx \mathbb{Z}$ because it follows from Hurewicz theorem. I am looking for non-trivial homology in higher degree than $n-1$. I'm sorry if my question wasn't clear, in this case geometric basically means "by hands" or "with little theory", ie. without spectral sequences or cohomology operations.
-
Hi Piotr, a while back I wrote a short run-through of Milnor's book "Morse Theory", in which he incidentally computes the homology of $\Omega S^n$ in all degrees. I don't think I explicitly mentioned this result since my goal was to get to Bott periodicity as quickly as possible, but all the necessary machinery to understand this fact geometrically (as Ryan outlines below) is certainly there: math.berkeley.edu/~aaron/xkcd/… – Aaron Mazel-Gee Sep 25 '11 at 21:27
However, if you're willing to learn about the Serre spectral sequence (and you should -- it's awesome!), the proof is all but trivial. And you can even recover the ring structure on cohomology, which IIRC you can't do (easily, at least) using Morse-theoretic techniques alone. – Aaron Mazel-Gee Sep 25 '11 at 21:29
This will be very useful, thanks! – Piotr Pstrągowski Sep 25 '11 at 21:54
Consider $\Omega S^n$. Think about the subspace of $\Omega S^n$ consisting of great circles that pass-through the base-point of $S^n$, parametrized to be constant speed. This is a subspace of $\Omega S^n$ that is homeomorphic to $S^{n-1}$. It's also the generator of $H_{n-1} \Omega S^n$, provided $n>1$.
So the generator consists of loops that have been "pulled tight". Usually when people talk about geometry in the case of homology of loop-spaces, they usually mean Bott-style Morse theory. So you might want to consider the geodesic DE on the smooth free loop space $L(S^n)$, its critical points, the index of such critical points, etc. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.852572500705719, "perplexity": 436.48273491339563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609536300.49/warc/CC-MAIN-20140416005216-00487-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://export.arxiv.org/abs/1101.2832 | physics.ao-ph
(what is this?)
# Title: Evidence for super-exponentially accelerating atmospheric carbon dioxide growth
Abstract: We analyze the growth rates of atmospheric carbon dioxide and human population, by comparing the relative merits of two benchmark models, the exponential law and the finite-time-singular (FTS) power law. The later results from positive feedbacks, either direct or mediated by other dynamical variables, as shown in our presentation of a simple endogenous macroeconomic dynamical growth model. Our empirical calibrations finds that the human population has decelerated from its previous super-exponential growth until 1960 to a slower-than-exponential growth associated with a decreasing growth rate. However, the past decade is found to be characterized by an almost stable growth rate approximately equal to r(2010) ~ 1% per year, suggesting that the population growth is stabilizing at "just" an exponential growth. As for atmospheric CO2 content, we find that it is at least exponentially increasing and most likely characterized by an accelerating growth rate as off 2009, consistent with an unsustainable FTS power law regime announcing a drastic change of regime. The coexistence of a quasi-exponential growth of human population with a super-exponential growth of carbon dioxide content in the atmosphere is a diagnostic that, until now, improvements in carbon efficiency per unit of production worldwide has been dramatically insufficient.
Comments: Fixed typo in footnote two, changed wording about current population growth. We thank the blogs "Climate Progress" (this http URL) and "Watts Up With That" (this http URL) for active discussion Subjects: Atmospheric and Oceanic Physics (physics.ao-ph); Other Condensed Matter (cond-mat.other); Physics and Society (physics.soc-ph) Cite as: arXiv:1101.2832 [physics.ao-ph] (or arXiv:1101.2832v3 [physics.ao-ph] for this version)
## Submission history
From: Andreas Hüsler D [view email]
[v1] Fri, 14 Jan 2011 15:13:49 GMT (123kb)
[v2] Tue, 1 Mar 2011 09:46:48 GMT (125kb,D)
[v3] Thu, 17 Mar 2011 20:27:25 GMT (97kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8669351935386658, "perplexity": 4187.456192886194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027321786.95/warc/CC-MAIN-20190824214845-20190825000845-00375.warc.gz"} |
http://physics.stackexchange.com/questions/50149/construction-of-the-supersymmetric-faraday-tensor?answertab=oldest | Construction of the supersymmetric Faraday tensor
When I first learned gauge theories in my introductory quantum field theory course, I was taught that the Faraday (field-strength) tensor can be constructed by computing the commutator of the gauge-covariant derivative:
$$[D_\mu,D_\nu]=-ieF_{\mu\nu}$$
Now, I am studying supersymmetry following Martin's SUSY primer, and in chapter 4.8, the author immediately writes down the super-symmetric field strength chiral superfield out of the vector superfield $V$:
$$\mathcal{W}_\alpha=-\frac{1}{4}D^\dagger D^\dagger D_\alpha V.$$
I would have liked a more gentle introduction to this in terms of something I am already familiar with: is there a way for me to have constructed this using the commutator of some 'gauge super-covariant derivative'?
-
As far as I know, it is defined in that form in order to satisfy chirality $$D^{\dagger}_{\dot{\alpha}}W_{\alpha}=0$$ and gauge invariance $$\delta W_\alpha=0.$$
Ordinary covariant derivatives are bad in SUSY because the $\partial_\mu$ derivative is no longer the fundamental "minimal" derivative. Instead, one may find a square root of it - and the superderivatives $D_\alpha$ etc. are square roots of the ordinary derivatives, and they're therefore more fundamental. So you haven't really started to think in the SUSY way if you still want to place ordinary derivatives everywhere. – Luboš Motl Jan 20 '13 at 10:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9269080758094788, "perplexity": 363.01460334340345}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042989510.73/warc/CC-MAIN-20150728002309-00189-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://cran.r-project.org/web/packages/lodi/vignettes/lodi.html | Censored Likelihood Multiple Imputation in R
Loading lodi and an example dataset
For convenience we have included a example dataset called toy_data, which can be loaded by running data("toy_data"). Let’s look at the first 10 entries of the example dataset.
library(lodi)
data("toy_data")
#> id case_cntrl poll smoking gender batch1 lod
#> 1 13707 1 3.588607 0 1 0 0.65
#> 2 18641 1 NA 0 0 0 0.65
#> 3 27407 1 2.619124 1 0 0 0.65
#> 4 45462 1 7.203193 0 1 1 0.80
#> 5 50357 1 7.336160 1 1 1 0.80
#> 6 59168 1 NA 0 0 0 0.65
#> 7 61477 1 5.136974 0 1 0 0.65
#> 8 76585 1 11.794483 1 1 0 0.65
#> 9 80681 1 1.280289 0 0 1 0.80
#> 10 84391 1 5.480510 1 1 0 0.65
id corresponds to the study ID and is unimportant for the purposes of this example. case_cntrl takes values 0 or 1, where 1 indicates that the subject has the disease of interest and 0 indicates that the subject is a healthy control. poll is the environmental exposure of interest, where NA indicates that the concentration is below the limit of detection (LOD). smoking and gender are covariates that we will include in the imputation model. lod corresponds to the limit of detection for each individual’s batch. Finally, batch1 takes two values; 1 if the subject’s biosample was assayed in batch 1 and 0 if the subject’s biosample was assayed in batch 2.
Implementing Censored Likelihood Multiple Imputation
The function that performs censored likelihood multiple imputation is the clmi function. For more details see help(clmi).
clmi.out <- clmi(formula = log(poll) ~ case_cntrl + smoking + gender,
df = toy_data, lod = lod, seed = 12345, n.imps = 5)
The main input to clmi is a R formula. The left hand side of the formula must be the exposure, and the right hand side must be the outcome followed by the covariates you want to include in the imputation model. The order of variables on the right hand side matters. You can apply a transformation to the exposure by applying a univariate function to it, as done above. The lod argument refers to the name of the lod variable in your data.frame.
The imputed datasets can be extracted as a list using $imputed.dfs: extract.imputed.dfs <- clmi.out$imputed.dfs
Fit and pool outcomes models
The pool.clmi function takes the output generated by the clmi function, fits outcome models on each of the imputed datasets, and pools inference across outcome models using Rubin’s rules. For details see help(pool.clmi).
results <- pool.clmi(formula = case_cntrl ~ poll_transform_imputed + smoking +
gender, clmi.out = clmi.out, type = logistic)
In pool.clmi, formula contains the outcome variable on the left hand side and the first variable on the right hand side should be the imputed exposure variable. clmi outputs the exposure variable as ((your-exposure))_transform_imputed. In this example, our exposure is poll, so the name of the imputed variable is poll_transform_imputed.
• Note: There are two valid options for the type argument. If you have binary outcome data (as in the current example) use type = logistic so that the model fit on the imputed datasets are logistic regression models. If you have continuous outcome data use regression.type = linear so that models fit on the imputed datasets are linear regression models.
To display the pooled results use $output: results$output
#> est se df p.values LCL.95
#> (Intercept) -0.6021156 0.3338019 93.75401 0.07447254 -1.26490980
#> poll_transform_imputed 0.3619278 0.2192230 86.31785 0.10238131 -0.07385026
#> smoking -0.3245100 0.5245765 93.01341 0.53768319 -1.36621287
#> gender 0.8611192 0.4904714 93.65164 0.08240975 -0.11277055
#> UCL.95
#> (Intercept) 0.06067861
#> poll_transform_imputed 0.79770585
#> smoking 0.71719295
#> gender 1.83500898
If you want to look at the individual regressions fit on each imputed dataset use $regression.summaries results$regression.summaries
Reference
Boss J, Mukherjee B, Ferguson KK, et al. Estimating outcome-exposure associations when exposure biomarker detection limits vary across batches. Epidemiology. 2019;30(5):746-755. 10.1097/EDE.0000000000001052
Contact information
If you would like to report a bug in the code, ask questions, or send requests/suggestions e-mail Jonathan Boss at bossjona@umich.edu. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.39033421874046326, "perplexity": 3411.5094889066017}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500628.77/warc/CC-MAIN-20230207170138-20230207200138-00670.warc.gz"} |
https://stats.stackexchange.com/questions/251011/how-is-attenuation-dilution-not-a-concern-for-univariate-analysis | # How is attenuation/dilution not a concern for univariate analysis?
When doing imputation, I can understand why mean substitution can result in regression dilution.
However, in the same article about imputation, I don't understand why this is not a concern for univariate analysis?
As more of the y-values (dependent variables) get replaced with their means, I know that the sample mean will stay the same (and hence the bias will not be affected). But, it seems to me like this will increase the variance (and standard deviation), regardless of if it is univariate or multivariate. Am I not understanding this correctly?
So, when the wikipedia article says that "mean imputation has some attractive properties for univariate analysis but becomes problematic for multivariate analysis", I don't think that's true. seems to me like they have exactly the same problem?
• Not sure what the context is of your linked articles, but typically regression of a single response variable on a single predictor variable would be considered bivariate, not univariate. (In univariate analysis mean-substitution would result in attenuation of e.g. standard error estimates, so inflation of t-statistics.) – GeoMatt22 Dec 12 '16 at 2:49
• Thank you @GeoMatt22. I fixed the question now. It's really that I don't understand why both univariate and multivariate analyses would have the same issue with mean imputation. – makansij Dec 17 '16 at 21:14
If I were fitting a regression model for the mean of $Y$ as a linear combination of $X$ and $W$, but $X$ has missing values, mean imputation would not impute the missing values of $X$ with $\bar{X}$. I would have to impute those values with $E[X|W, Y]$ to get an unbiased estimator. In bivariate analyses, imputing $\bar{X}$ has no impact on the slope $\hat{\beta}$ because it is a 0 leverage point. Not true in higher dimensions, but still the wrong approach regardless.
• Okay. When you condition the missing values on the covariates which are present and the label, $E[X | W,Y]$ what you're describing is actually called regression imputation not mean imputation. Getting confused between the difference between those now, unless wikipedia is wrong. – makansij Dec 18 '16 at 23:02
• @Sother not necessarily but usually yes. You need a conditional mean, which is what $E[X|Y,W]$ would be here. You can estimate that from regression in the same dataset, but you can also use whatever prediction model you want (Bayesian, machine learning, etc.) the most important thing for mean imputation is that the conditional mean is a good predictor. I don't really know what Wiki is saying, but if you fill in the missing values of $Y$ with $E[Y]$ and not $E[Y|X]$ I can see how that would cause "dillution". – AdamO Dec 19 '16 at 15:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9019028544425964, "perplexity": 493.0654368653385}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519600.31/warc/CC-MAIN-20210119170058-20210119200058-00693.warc.gz"} |
https://love2d.org/forums/viewtopic.php?f=4&t=81496 | ## [Solved - Graphics Driver Bug] Weird Canvas Issue
Questions about the LÖVE API, installing LÖVE and other support related questions go here.
Forum rules
Citizen
Posts: 87
Joined: Tue Dec 02, 2014 2:17 pm
Location: England
### [Solved - Graphics Driver Bug] Weird Canvas Issue
So I am porting my football project over to 0.10.0, and I have come across a bizare issue.
I have a gamestate system which draws to a canvas. This drawing works fine and everything appears correctly, however when I attempt to clear the canvas (no longer using canvas:clear(), but love.graphics.clear(...)), it will not actually clear the currently active canvas, and you get that weird sort of constant draw effect when Windows used to freeze a bit with a modal dialog box when you dragged it around.
Now initially I thought this was some weird bug with my code, but I have found that adding the following line of code before actually drawing the canvas to the screen fixes the issue;
Code: Select all
love.graphics.print("HELLO!",0,0)
And the issue no longer happens. What on earth is going on? Here is my love.draw code. Note that g.gfx is simply an alias for love.graphics
Code: Select all
function love.draw()
-- Draw to game canvas
g.gfx.setCanvas(g.game_canvas)
g.gfx.clear(50, 150, 100, 255)
-- BEGIN DRAWING HERE
g.gfx.setColor(255, 255, 255, 255)
g.state.draw()
-- END DRAWING HERE
-- Set canvas
g.gfx.setCanvas()
-- Draw game canvas
g.gfx.setColor(255, 255, 255, 255)
g.gfx.print("HI",0,0) -- Add this to stop the canvas from never clearing?!?!?1?!?
g.gfx.draw(g.game_canvas)
end
Not even sure where to begin with this bug.
EDIT: OK, I am really confused. If I switch my state.draw code to drawing directly without using any canvases, everything works fine. As soon as I start using canvases like before everything goes weird. Has the way canvases work and are used changed a lot in 0.10.0 or something?
Last edited by BOT-Brad on Tue Dec 29, 2015 8:08 pm, edited 2 times in total.
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Citizen
Posts: 87
Joined: Tue Dec 02, 2014 2:17 pm
Location: England
### Re: Weird Canvas Issue
OK, pretty sure something with love.graphics.clear(...) is dodgy.
replacing this;
Code: Select all
love.graphics.setCanvas(g.game_canvas)
love.graphics.clear(0, 0, 0, 255)
with this;
Code: Select all
love.graphics.setCanvas(g.game_canvas)
love.graphics.setColor(0, 0, 0, 255); love.graphics.rectangle("fill", 0, 0, width, height)
Fixes the issue and everything works as it should.
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s-ol
Party member
Posts: 1076
Joined: Mon Sep 15, 2014 7:41 pm
Location: Cologne, Germany
Contact:
### Re: Weird Canvas Issue
BOT-Brad wrote:OK, pretty sure something with love.graphics.clear(...) is dodgy.
replacing this;
Code: Select all
love.graphics.setCanvas(g.game_canvas)
love.graphics.clear(0, 0, 0, 255)
with this;
Code: Select all
love.graphics.setCanvas(g.game_canvas)
love.graphics.setColor(0, 0, 0, 255); love.graphics.rectangle("fill", 0, 0, width, height)
Fixes the issue and everything works as it should.
Hm, I think I experienced something similar in a tiny demo I was working on but I just blamed it on being too tired. The project was basically just two Canvas getting cleared and some primitives being drawn and I was really confused over the things that happened when I changed the order of independent things around. I'll investigate again when I get home, could very well be that lg.clear() is dodgy. If you have the time you could try finding the commit that caused this on bitbucket or by bisecting even.
s-ol.nu /blog - p.s-ol.be /st8.lua - g.s-ol.be /gtglg /curcur
Code: Select all
print( type(love) )
if false then
baby:hurt(me)
end
slime
Solid Snayke
Posts: 2785
Joined: Mon Aug 23, 2010 6:45 am
Contact:
### Re: Weird Canvas Issue
love.graphics.clear is essentially a direct call into OpenGL. It calls glClear, which clears the active render target(s) to the specified color. Also keep in mind that [wiki]love.graphics.setScissor[/wiki] affects the area in the active Canvas that is cleared.
If it's not working properly it may be a driver bug. What graphics card, operating system, and driver version do you have?
This 0.10.0 code:
Code: Select all
canvas = love.graphics.newCanvas()
function love.draw()
love.graphics.setCanvas(canvas)
love.graphics.clear(0, 255, 0)
love.graphics.setCanvas()
love.graphics.draw(canvas, 0, 0)
end
Should be roughly equivalent to this 0.9.2 code (calling canvas:clear in 0.9.2 while the canvas is already active just calls glClear – or a newer function if OpenGL 3 is supported):
Code: Select all
canvas = love.graphics.newCanvas()
function love.draw()
love.graphics.setCanvas(canvas)
canvas:clear(0, 255, 0)
love.graphics.setCanvas()
love.graphics.draw(canvas, 0, 0)
end
Do they both behave the same for you?
Citizen
Posts: 87
Joined: Tue Dec 02, 2014 2:17 pm
Location: England
### Re: Weird Canvas Issue
They do indeed both behave the same, however maybe I am misunderstanding how canvas:clear (or graphics.clear in 0.10.0) works.
In love 0.10.0, I would expect the following code to randomly show a new colour every frame.
Code: Select all
canvas = love.graphics.newCanvas()
function love.draw()
love.graphics.setCanvas(canvas)
love.graphics.clear(love.math.random(255), love.math.random(255), love.math.random(255))
love.graphics.setCanvas()
love.graphics.draw(canvas, 0, 0)
end
Yet it only ever sets the colour once (randomly), then never seems to actually clear the canvas ever again unless I add pretty much any graphics.* command to the canvas (such as graphics.print, graphics.circle, graphics.polygon, etc.) and then it will randomly cycle through the colours as expected.
Also, if I draw to the canvas in a one-time function after the first time the canvas is cleared (on the first time love.draw is called), then the graphics I drew to the canvas remain there and despite love.draw running 60 times a second (with vsync on), the graphics remain there and love.graphics.clear has no effect on the active canvas (again, unless I draw something else to it).
EDIT: As for specs of my PC (I haven't tested this on my laptop or another PC yet, will do that shortly).
OS: Windows 7 Ultimate 64-bit
Driver Version: Sorry, the driver version of what exactly?
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s-ol
Party member
Posts: 1076
Joined: Mon Sep 15, 2014 7:41 pm
Location: Cologne, Germany
Contact:
### Re: Weird Canvas Issue
BOT-Brad wrote:They do indeed both behave the same, however maybe I am misunderstanding how canvas:clear (or graphics.clear in 0.10.0) works.
In love 0.10.0, I would expect the following code to randomly show a new colour every frame.
Code: Select all
canvas = love.graphics.newCanvas()
function love.draw()
love.graphics.setCanvas(canvas)
love.graphics.clear(love.math.random(255), love.math.random(255), love.math.random(255))
love.graphics.setCanvas()
love.graphics.draw(canvas, 0, 0)
end
this produces a different color every frame on my Intel integrated GPU. Just tried to reproduce my problems with a simple script but I couldn't, so it probably is my fault there - looks like this is your driver I suppose?
s-ol.nu /blog - p.s-ol.be /st8.lua - g.s-ol.be /gtglg /curcur
Code: Select all
print( type(love) )
if false then
baby:hurt(me)
end
slime
Solid Snayke
Posts: 2785
Joined: Mon Aug 23, 2010 6:45 am
Contact:
### Re: Weird Canvas Issue
BOT-Brad wrote:In love 0.10.0, I would expect the following code to randomly show a new colour every frame.
Yeah, it should (and it does for me on my system).
Can you also test it in 0.9.2? Like:
Code: Select all
canvas = love.graphics.newCanvas()
function love.draw()
love.graphics.setCanvas(canvas)
canvas:clear(love.math.random(255), love.math.random(255), love.math.random(255))
love.graphics.setCanvas()
love.graphics.draw(canvas, 0, 0)
end
OS: Windows 7 Ultimate 64-bit
Driver Version: Sorry, the driver version of what exactly?
Video driver version. I forget where exactly you can find it – it's probably in Catalyst Control Center.
Citizen
Posts: 87
Joined: Tue Dec 02, 2014 2:17 pm
Location: England
### Re: Weird Canvas Issue
Yes, in 0.9.2 the same thing happens, just a single colour and then nothing. I am glad it seems to be an issue on my machine then. Here are all the version mumbo-jumbo from CCC.
Packaging Version: 12.104-130328a-157485C-ATI
2D Version: 8.01.01.1295
D3D Version: 9.14.10.0969
OpenGL Version: 6.14.10.12217
I will try and update my graphics drivers now, as it seems they were last updated in 2013, whoops.
EDIT:
Packaging Version: 15.20.1062.1004-150803a1-187669C
2D Version: 8.01.01.1500
D3D Version: 9.14.10.01128
OpenGL Version: 6.14.10.13399
Mantle Version: 9.1.10.0077
CCC Version: 2015.0804.21.41908
Will restart my PC and let you know how I get on.
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Citizen
Posts: 87
Joined: Tue Dec 02, 2014 2:17 pm
Location: England
### Re: Weird Canvas Issue
Hmm, nope, still just pauses on the single first randomly generated colour.
EDIT: I am stumped, I actually just got the 0.9.2 version of the code to work by running it in the 64-bit version of LOVE and everything changes colour every frame, however the 0.10.0 version still does not work either in the 32-bit of 64-bit versions.
Last edited by BOT-Brad on Tue Dec 29, 2015 7:18 pm, edited 2 times in total.
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slime
Solid Snayke
Posts: 2785
Joined: Mon Aug 23, 2010 6:45 am | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20128020644187927, "perplexity": 7409.054399648376}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376829997.74/warc/CC-MAIN-20181218225003-20181219011003-00419.warc.gz"} |
http://aimsciences.org/article/doi/10.3934/dcds.2009.25.575 | # American Institute of Mathematical Sciences
July 2009, 25(2): 575-583. doi: 10.3934/dcds.2009.25.575
## BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity
1 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China, China
Received October 2008 Revised January 2009 Published June 2009
In this paper we derive a criterion for the breakdown of classical solutions to the incompressible magnetohydrodynamic equations with zero viscosity and positive resistivity in $\mathbb{R}^3$. This result is analogous to the celebrated Beale-Kato-Majda's breakdown criterion for the inviscid Eluer equations of incompressible fluids. In $\mathbb{R}^2$ we establish global weak solutions to the magnetohydrodynamic equations with zero viscosity and positive resistivity for initial data in Sobolev space $H^1(\mathbb{R}^2)$.
Citation: Zhen Lei, Yi Zhou. BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 575-583. doi: 10.3934/dcds.2009.25.575
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2017 Impact Factor: 1.179 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5842708349227905, "perplexity": 4300.938022958202}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999200.89/warc/CC-MAIN-20190620085246-20190620111246-00360.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/122899-binary-operations-check-please.html | # Math Help - Binary Operations, check please :)
1. ## Binary Operations, check please :)
Ordinary Multiplication on complex numbers
not a binary operation because to complex numbers multiplied together gives a real number e.g 3i x 2i = -6
Matrix multiplication on the set of all 2x2 non-singular matrices with real elements.
binary operation
p*q = p/q p,q belong to Q\{0}
binary operation
p*q = p^q p,q belong to Q
binary operation
2. the first one is binary operation since complex field contains real number filed as a subfield.
the last one is not binary opration, a counterexample: $2^{\frac{1}{2}}$ is a irrational number.
Ordinary Multiplication on complex numbers
not a binary operation because to complex numbers multiplied together gives a real number e.g 3i x 2i = -6
Matrix multiplication on the set of all 2x2 non-singular matrices with real elements.
binary operation
p*q = p/q p,q belong to Q\{0}
binary operation
p*q = p^q p,q belong to Q
binary operation
What do YOU think for the second one?
4. Originally Posted by Drexel28
What do YOU think for the second one?
it is a binary operation | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8099566102027893, "perplexity": 3055.857930131955}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776430044.46/warc/CC-MAIN-20140707234030-00028-ip-10-180-212-248.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/finding-the-magnitude-of-the-electrostatic-force-from-a-thin-rod.709292/ | # Finding the magnitude of the electrostatic force from a thin rod
1. Sep 8, 2013
### nuagerose
1. The problem statement, all variables and given/known data
The figure shows a uniformly charged thin rod of length L that has total charge Q. Find an expression for the magnitude of the electrostatic force acting on an electron positioned on the axis of the rod at a distance d from the midpoint of the rod.
http://ezto.mhecloud.mcgraw-hill.com/13252699451980596522.tp4?REQUEST=SHOWmedia&media=c21q56a.png [Broken]
2. Relevant equations
F = $\frac{kQ}{d^{2}}$
3. The attempt at a solution
I know how to find the force from a point charge using the equation above, but I am not sure how to set up this problem because it is a charged thin rod. I was thinking I could divide Q by L as the charge in the above question and d-$\frac{L}{2}$ as the d in the above equation.
Last edited by a moderator: May 6, 2017
2. Sep 8, 2013
### Tanya Sharma
Can you tell what is the formula for the force between two point charges q1 and q2 at a distance d from each other ? The formula you have written doesnt make much sense.
3. Sep 8, 2013
### nuagerose
F = (k*q$_{1}$*q$_{2}$)/r$^{2}$
Hope this is right. I suppose q1 would be the charge of the thin rod and q2 would be the charge of the point, which is e.
4. Sep 8, 2013
### Tanya Sharma
Right
No.Coulomb's law applies only to point charges.You cannot apply this directly when a continuous object like a rod is present.We will have to consider rod as composed of infinitely many point charges.
Are you familiar with calculus ?You will need integration whenever a continuous charged body is present.
Draft saved Draft deleted
Similar Discussions: Finding the magnitude of the electrostatic force from a thin rod | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9552160501480103, "perplexity": 462.747202898926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948516843.8/warc/CC-MAIN-20171212114902-20171212134902-00318.warc.gz"} |
https://gmatclub.com/blog/2011/04/gmat-question-of-the-day-apr-21-arithmetic-and-sentence-correction/ | # GMAT Question of the Day (Apr 21): Arithmetic and Sentence Correction
- Apr 21, 02:00 AM Comments [2]
Math (PS)
What is the unit's digit of $7^{75}+6$?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
OA and Explanation
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Verbal (SC)
The fifties, for all their advertised conformity, now appear to have been a time of considerable achievement in the arts.
(A) for all their advertised conformity, now appear to have been
(B) despite all their advertised conformity, now appear to be
(C) for all their advertised conformity, now appear that they were
(D) despite all their advertised conformity, now appears as
(E) with all advertised conformity, now appears
OA and Explanation
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### [2] Comments to this Article
1. Ashish April 21, 10:32 AM
the exponents of 7 have units digits ending in 7,9,3 and 1 and then series repeat .
So if 7 exponent 75 ( ie 75/4 = 3) which means we need to 3 units digits and add 6 to it
Correct answer E
2. gmatclub May 3, 10:32 PM
That’s Right. Please see here: http://gmatclub.com/forum/m12-72970.html | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3970907926559448, "perplexity": 9661.925091086285}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128320570.72/warc/CC-MAIN-20170625184914-20170625204914-00050.warc.gz"} |
http://physics.stackexchange.com/tags/quantum-gravity/new | # Tag Info
1
The two don't work together, they are competing descriptions of gravity. From a QM perspective gravity is mediated by the graviton. Picture gravitons encountering photons, imparting the force of gravity thus changing the photons' paths. From a GR perspective mass warps spacetime, and any photons traveling through now have warped paths. So you could see ...
1
This is my guess Think about curve space is like sound wave, When you quantize the wave to be point, like phonon of sound wave, you get graviton from gravity
2
At the level of understanding the data and observations we have up to now, General Relativity describes well what we perceive of the Cosmos and Quantum Field Theory what we observe in the microcosm of elementary particles and their interactions. The two have not been joined up to now, i.e. there is no accepted unified theory that joins smoothly these two ...
0
I myself overlooked it too but wikipedia actually happens to have a great such list at https://en.wikipedia.org/wiki/Quantum_gravity#Points_of_tension There are other points of tension between quantum mechanics and general relativity. First, classical general relativity breaks down at singularities, and quantum mechanics becomes inconsistent ...
-2
When will it be realised that if you find something smaller then the next step is again a search for something smaller and it will go on for infinity because everything has to be made of something smaller.
5
Spin 2 just means that the gravitational field is given by a metric field and general covariance, which is the nonlinear expression of a massless spin 2 representation of the Poincare group. The latter appears when linearizing around the Minkowski metric and dropping all interactions. See the classical paper by S. Weinberg, Phys.Rev. 138 (1965), B988-B1002 ...
1
On your first question: absolutely, energy gravitates (or induces curvature in spacetime) the same way that mass gravitates. If you read general relativity, you will learn that it is in fact the Stress-Energy Tensor that is the source of gravitational interaction (or equivalently spacetime curvature). Energy can be localized very easily; a parallel-plate ...
1
I hope that somebody familiar with this research will contribute so that we all learn something. I found the following , it is in a separate column like a comment: Locality and unitarity are the central pillars of quantum field theory, but as the following thought experiments show, both break down in certain situations involving gravity. This suggests ...
2
OK, let us start from your example. I think that it is too pathological to be considered as a safe starting point for this discussion, which is worth and interesting however. Nevertheless I would like to spend some words about this case since it permits to introduce some general issue useful in the second part of my answer. AdS_n is not globally hyperbolic. ...
0
Yes, it would remain stable. But these black holes are incredibly hot and there is no known matter that could feed them fast enough to balance the mass loss that they experience through evaporation. Assuming that with 'micro black hole' you mean a TeV-sized black hole, the mass loss is of the order of a TeV/fm. In addition these black holes are also small, ...
2
Micro black holes have been hypothesized in some large dimension string phenomenological models and are searched for in the experiments at the CERN LHC. The first approach to the decays was thermodynamic with Hawking radiation diminishing them rapidly. Their lifetimes are very short so there is no way to gather and contain them and experiment with feeding ...
0
The debate that seems to be happening right now is what does it mean that '... the formation of the black hole does not create nor destroy entropy, so the black hole must contain zero or nearly zero entropy as well.' This is correct, of course, except that the material we observe with zero or nearly zero entropy is 'Bose-Einstein condensate' (BEC) and BEC ...
Top 50 recent answers are included | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9094942808151245, "perplexity": 448.6681873328838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345766127/warc/CC-MAIN-20131218054926-00064-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://clay6.com/qa/2023/true-or-false-matrix-addition-is-associative-as-well-as-commutative- | Browse Questions
Home >> CBSE XII >> Math >> Matrices
# True or False: Matrix addition is associative as well as commutative.
TRUE (i) Communtative- If A =$[a_{ij}] B=[b_{ij}]$ are matrices of same order say $m\times n$ then A+B=B+A (ii) For any three matrices $A=[a_{ij}]\; B=[b_{ij}]\; C=[c_{ij}]$ of the same order say $m\times n$ (A+B)+C=A+(B+C)
edited Mar 12, 2013 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6515163779258728, "perplexity": 757.550493050559}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917118713.1/warc/CC-MAIN-20170423031158-00194-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathematica.stackexchange.com/questions/34978/symbolic-linear-algebra | # Symbolic linear algebra
I would like to know how I can ask Mathematica to expand (and simplify) such an expression :
$$(\alpha A + \beta B)^\top (\alpha A + \beta B)$$
where $\alpha,\beta$ are two real numbers and $A,B$ are vectors in $\mathbb{R}^{n}$. $A^\top$ denotes the transpose of $A$. I assume I must tell Mathematica that $A$ and $B$ are vectors. Here is what I have tried :
$Assumptions = (A | B) [Element] Vectors[n];$Assumptions = (a | b) [Element] Reals;
TensorExpand[ Transpose[a*A + b*B].(a*A + b*B) ]
and the output is :
a A.Transpose[a A + b B, {2, 1}] + b B.Transpose[a A + b B, {2, 1}]
Which is not what I expected since I would like the output to be :
$$\alpha^{2} A^\top A + 2 \alpha \beta A^\top B + \beta^{2} B^\top B$$
-
You can do no harm by trying. – b.gatessucks Oct 28 '13 at 20:06
I would if I had Mathematica on the computer I'm currently using but I don't. I can't try it right now but I will later. – jibounet Oct 28 '13 at 20:13
This is closely related: How can I get Mathematica to recognize equality of symbolic matrix expressions?. In fact, following the answer you should resolve your prolem, i.e. use TensorReduce instead of FullSimplify. – Artes Oct 28 '13 at 20:24
I have tried : $Assumptions = (A | B) [Element] Matrices[{n, 1}]; TensorExpand[ Transpose[A + B].(A + B) ] But the output I get is not Transpose[A].A + 2Transpose[A].B + Transpose[B].B What did I do wrong ? – jibounet Oct 28 '13 at 21:06 You can use Vectors[n]. In your original question you have ( a*A + b*B)]*( a*A + b*B), instead of * use . i.e. Dot or you can use TensorProduct. – Artes Oct 28 '13 at 21:45 ## 1 Answer Since you're working with vectors, just let Mathematica know that these are vectors. Some other systems (MATLAB and its relatives in particular) have the limitation that they can only work with matrices, forcing you to distinguish between row vector and column vectors and keep transposing. This is not necessary nor convenient in Mathematica. In[1]:=$Assumptions = a ∈ Vectors[n, Reals] && b ∈ Vectors[n, Reals]
Out[1]= a ∈ Vectors[n, Reals] && b ∈ Vectors[n, Reals]
In[2]:= (α a + β b).(α a + β b) // TensorExpand
Out[2]= α^2 a.a + 2 α β a.b + β^2 b.b
-
Thank you for your answer! It seems to work just fine in your example. However, would it still work if I had Transpose[] to your example ? – jibounet Oct 31 '13 at 17:19
@jibounet Well, does it make any sense to transpose a vector? – Szabolcs Oct 31 '13 at 17:20
I think it does. A real vector $X$ belongs to $\mathrm{Mat}_{n,1}(\mathbb{R})$. We can define its transpose to be the matrix of the linear form $L_{X} \, : \, Y \in \mathbb{R}^{n} \, \longmapsto \, \left\langle X,Y \right\rangle$, which is an element of $\mathrm{Mat}_{1,n}(\mathbb{R})$. – jibounet Oct 31 '13 at 17:30
@jibounet For Mathematica a "vector" is a 1-dimensional tensor. What people call a "row vector" is a really a matrix, i.e. a 2-dimensional tensor. It's more difficult to reason about those and figure out such things that $AB^T=BA^T$ only if both A and B are 1 by $n$ matrices, but not otherwise. In Mathematica it does not make sense to transpose a vector because it's a one-dimensional structure. – Szabolcs Oct 31 '13 at 17:31
Alright. I understand. – jibounet Oct 31 '13 at 17:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7832204103469849, "perplexity": 630.2925740944186}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860112727.96/warc/CC-MAIN-20160428161512-00087-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://hackage.haskell.org/package/hspresent | # hspresent: A terminal presentation tool
[ bsd3, console, library, program ] [ Propose Tags ]
Hspresent is a simple program that lets you give powerpoint-like presentations in your terminal (for certain definitions of powerpoint-like). It's extremely basic and unintelligent. Don't expect much.
Invoke like hspresent pathtoyourpresentation. Left and right arrow keys move between slides, and hitting q or Ctrl-C quits the presentation.
The file format is really simple right now. Slides are separated by lines consisting of the characters --. You can put the single character . on a line to demarcate a split frame. A split frame really creates a new frame with the contents of the previous frame copied at the front (this is similar to how beamer works).
Versions 0.1, 0.2, 0.2.1, 0.2.2 Changelog array, base (<5), bytestring, vty (<4) [details] BSD-3-Clause (c) 2009 by Evan Klitzke Evan Klitzke Evan Klitzke Console by EvanKlitzke at Wed Nov 18 01:33:48 UTC 2009 NixOS:0.2.2 hspresent 1609 total (17 in the last 30 days) (no votes yet) [estimated by rule of succession] λ λ λ Docs not available All reported builds failed as of 2016-12-31 Hackage Matrix CI
#### Maintainer's Corner
For package maintainers and hackage trustees
[back to package description]
# Presenting hspresent
This is a simple program that lets you give powerpoint-like presentations in your terminal (for certain definitions of powerpoint-like). It's extremely basic and unintelligent. Don't expect much.
Compile and install using cabal: cabal build && cabal install
This will compile hspresent, and install an hspresent command line program.
You can download the latest stable release of hspresent on Hackage. The git repository is online at git://github.com/eklitzke/hspresent.git.
## Usage
Invoke like hspresent /path/to/your/presentation. Left and right arrow keys move between slides, and hitting q or Ctrl-C quits the presentation.
The file format is really simple right now. Slides are separated by lines consisting of the characters --. You can put the single character . on a line to demarcate a "split" frame. A split frame really creates a new frame with the contents of the previous frame copied at the front (this is similar to how beamer works). That's it. Here's an example presentation: the title of the first slide this is . really cool -- the title of the second slide hooray for hspresent -- look at how fancy the title to this slide is * bullet point one * bullet point two | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3536774218082428, "perplexity": 7708.353313698722}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583513548.72/warc/CC-MAIN-20181021010654-20181021032154-00209.warc.gz"} |
https://www.sukhamburg.com/products/lasermodules/series/5LT-1+25CM.html | # Laser Modules Series 5LT-1+25CM
Semi-telecentric Compact Laser Micro Line Generator
Laser Diode Beam Source series 5LT
Laser Diode Beam Source series 5LT with tools
## Features
Semi-telecentric compact laser line with constant line length in the range of 4.8 mm.
• Line length in the range of 4.8 mm
• Line widths starting at 26 µm
• Wavelengths 405 - 852 nm
• Laser powers up to 90 mW
• Micro Line Generator for small laser line widths and high power density in the focal plane
## Description
The laser diode beam sources series 5LT-1+25CM produce a semi-telecentric laser line with a line length in the range of 4.8 mm. For most laser diodes the intensity profile is Gaussian in line direction clipped by an aperture at line length 4.8 mm with an edge intensity of typ. <40%. In some cases the line length is slightly smaller. In this case the line length is given on the 13.5%-level. The beam is Gaussian in line direction and truncated at 4.8 mm. The line width is constant along the laser line. Across the laser line the intensity distribution is Gaussian.
Laser Micro Line Generators designed to produce lines with small line width. They have a small depth of focus (in this case the depth of focus is the Rayleigh range). Laser Macro Line Generators like the corresponding lasers of series 5LTM-1 have common basic optical features but are designed to generate laser lines with an extended depth of focus.
The laser has integrated electronics for control of the laser output power. The output power can be controlled using modulation input ports (TTL and analog, in case of electronics type B TTL only) or manually using the potentiometer.
For this laser type the working distance is fixed. A fine-adjustment of the distance between laser and target is recommended for fine-focusing in order to achieve minimal line width.
## Accessories
Screwdriver WS 1.6 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8159698247909546, "perplexity": 3010.292444042051}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371799447.70/warc/CC-MAIN-20200407121105-20200407151605-00036.warc.gz"} |
http://en.citizendium.org/wiki/User:Dmitrii_Kouznetsov | # User:Dmitrii Kouznetsov
## Biography
• 1980, graduated from the Moscow State University (physics).
• 1988, got Ph.D. (kandidat fiz-mat.nauk) at the Lebedev Physics Institute (Moscow)
• 1991-1999, Universidad Nacional Authonoma de Mexico.
• 2000-2004, University of Arizona, Tucson, USA.
• 2004- University of Electro Communications, Institute for Laser Science, Japan.
• List of publications is available at http://www.ils.uec.ac.jp/~dima/PAPERS
## External articles
### TORI
TORI means "Tools for Outstanding Research and Investigation".
Current clone of TORI is available at http://mizugadro.mydns.jp/t ; it is my wiki-site.
Some articles ("tools") from there are already imported to CZ.
Some others may be imported, even without to ask my permission. While the MathJax is not yet supported at CZ, the minor adjustment may be necessary: the dollar signs should be replaced to $and$.
The following articles from TORI may be of interest for the CZ:
Mathematics related articles:
Physics related articles:
Human related articles:
In February 2013, the axess to the original URL tori.ils.uec.ac.jp/TORI of TORI had been disabled by the attack of Nishio Shigeaki; after a month, that attack had been supported by the administration of our institute, and up to year 2014, the efforts to recover the original URL are not successful. The cite had been cloned at server Mizugadro, and the links to the articles of the clone are mentioned above.
### Vera
Some articles are not yet wikified, but may be also useful: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.23631145060062408, "perplexity": 6383.121032854128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540565544.86/warc/CC-MAIN-20191216121204-20191216145204-00239.warc.gz"} |
https://www.physicsforums.com/threads/calculating-force-given-a-potential.658401/ | # Calculating force given a potential
1. Dec 11, 2012
### hahaha158
1. The problem statement, all variables and given/known data
Calculate the force on a particle at a position in space
r=[x,y,z]
V(r;A,B)=(A/r^14) - (B/r^8)
with A=1.5 and B=2/5
2. Relevant equations
3. The attempt at a solution
I know how to solve these when there is x,y,z involved, but i am not sure how to deal with something that has r in the equation, cna anyone explain?
Thanks
2. Dec 11, 2012
### Staff: Mentor
If the position has coordinates [x, y, z], what's the magnitude of the radius vector, r?
3. Dec 11, 2012
### hahaha158
(x^2,y^2,z^2)^.5?
4. Dec 11, 2012
### Staff: Mentor
Sorry, I don't understand the notation... are those commas meant to represent "+" operations?
5. Dec 11, 2012
### hahaha158
sorry yes i meant to use +.
However i think i may have done it, can you confirm whether or not this is correct?
F= -((-14Ar^-15)+(8Br^-9))*([x,y,z]/r)?
and is this an acceptable form to leave it in?
6. Dec 11, 2012
### Staff: Mentor
[x,y,z]/r doesn't match any mathematical notation I'm familiar with (that said, I'm not familiar with every variation of notation or formalism). But I would think that simply resolving r as $\sqrt{x^2 + y^2 + z^2}$ and substituting it for r in the function v(r;A,B) would get you where you want to be.
7. Dec 11, 2012
### hahaha158
i should probably have clarified a bit better
this is doing the nabla which i believe is just partial differentation, where you usually have 3 different functions (x,y,z) but since this is r i was a bit unsure. I think i understand it better now after looking some stuff online but thanks for the help regardless!
8. Dec 11, 2012
### Staff: Mentor
Okay. Truthfully, I didn't recognize any sign of the $\nabla$ operator being involved. Is the [x,y,z] or V(r;A,B) syntax diagnostic? I'd like to be able to spot the formalism for future reference.
9. Dec 11, 2012
### hahaha158
Well i'm not too sure of the syntax, i just tried to convey it as accurately as i can possible to the way i learned it.
starting from the top of the 2nd slide to where it says e)i)
10. Dec 12, 2012
### aralbrec
You can do E=-∇V in spherical coordinates or you can substitute r=√(x2+y2+z2) and do it in cartesian coordinates.
But there is a much quicker shortcut.
Recall that the electric field is perpendicular to a constant potential surface and the magnitude of the field is simply the rate of change of E in the direction of maximum rate of change.
Your potential function is constant for fixed r, which means the constant potential surfaces are spheres centered on the origin. This means the electric field is a vector pointing either away or toward the origin and its magnitude is just dV/dr -- the rate of change of the potential in the direction it is changing fastest (perpendicular to the sphere, along r).
Similar Discussions: Calculating force given a potential | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9462394714355469, "perplexity": 904.0559179712977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104636.62/warc/CC-MAIN-20170818121545-20170818141545-00395.warc.gz"} |
https://www.ncbi.nlm.nih.gov/pubmed/18590549 | Format
Choose Destination
BMC Genomics. 2008 Jun 30;9:313. doi: 10.1186/1471-2164-9-313.
The topology of the bacterial co-conserved protein network and its implications for predicting protein function.
Author information
1
Center for Computational Pharmacology, University of Colorado School of Medicine, Aurora, Colorado 80045, USA. anis.karimpour-fard@uchsc.edu
Abstract
BACKGROUND:
Protein-protein interactions networks are most often generated from physical protein-protein interaction data. Co-conservation, also known as phylogenetic profiles, is an alternative source of information for generating protein interaction networks. Co-conservation methods generate interaction networks among proteins that are gained or lost together through evolution. Co-conservation is a particularly useful technique in the compact bacteria genomes. Prior studies in yeast suggest that the topology of protein-protein interaction networks generated from physical interaction assays can offer important insight into protein function. Here, we hypothesize that in bacteria, the topology of protein interaction networks derived via co-conservation information could similarly improve methods for predicting protein function. Since the topology of bacteria co-conservation protein-protein interaction networks has not previously been studied in depth, we first perform such an analysis for co-conservation networks in E. coli K12. Next, we demonstrate one way in which network connectivity measures and global and local function distribution can be exploited to predict protein function for previously uncharacterized proteins.
RESULTS:
Our results showed, like most biological networks, our bacteria co-conserved protein-protein interaction networks had scale-free topologies. Our results indicated that some properties of the physical yeast interaction network hold in our bacteria co-conservation networks, such as high connectivity for essential proteins. However, the high connectivity among protein complexes in the yeast physical network was not seen in the co-conservation network which uses all bacteria as the reference set. We found that the distribution of node connectivity varied by functional category and could be informative for function prediction. By integrating of functional information from different annotation sources and using the network topology, we were able to infer function for uncharacterized proteins.
CONCLUSION:
Interactions networks based on co-conservation can contain information distinct from networks based on physical or other interaction types. Our study has shown co-conservation based networks to exhibit a scale free topology, as expected for biological networks. We also revealed ways that connectivity in our networks can be informative for the functional characterization of proteins.
PMID:
18590549
PMCID:
PMC2488357
DOI:
10.1186/1471-2164-9-313
[Indexed for MEDLINE]
Free PMC Article | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8154063820838928, "perplexity": 2669.534638437651}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221218357.92/warc/CC-MAIN-20180821151743-20180821171743-00457.warc.gz"} |
http://huldalebsa.soup.io/post/507142971/Custom-Fit-Download-Zdnet | You are at the newest post.
### New tech for virtual workouts - CBS News
However, this futuristic technology does not seem to be quite ready for prime time. CNET reporter Bridget Carey told CBS News, "As someone who's used one of these things, I've got to tell you, it makes you dizzy real fast!" "I think it's hard enough to be motivated to exercise - but then to put this giant thing on your head, and have cords all around you... I can't see myself united states doing it," she said. But perhaps as the technology develops further and the headsets get lighter and smaller, virtual reality training could offer an appealing alternative to the reality of hitting the gym.
Source: http://www.cbsnews.com/news/new-tech-for-virtual-workouts/
### Miami QB Williams sees career go full-circle - Yahoo Sports
''He's been with me the whole time,'' said Kaaya, the fifth quarterback in Miami history with 25 touchdown passes in a season, joining Steve Walsh, Ken Dorsey, Vinny Testaverde and Bernie Kosar. ''He's a really smart guy. ... It was never weird.
Source: http://sports.yahoo.com/news/miami-qb-williams-sees-career-full-circle-223440820--ncaaf.html
### Gift Guide: How to pick the right fitness tracker - Yahoo News
The Microsoft Band ($200) and the Fitbit Surge ($250) do have GPS, but they are currently in limited release. Sony's SmartWatch 3 also has GPS, but it's more smartwatch than fitness band. There will likely be additional devices with GPS unveiled at the International CES gadget show next month. If accuracy is important and it is for me I suggest waiting. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20326103270053864, "perplexity": 8319.100640568964}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816912.94/warc/CC-MAIN-20180225190023-20180225210023-00124.warc.gz"} |
http://www.jason-french.com/blog/2012/01/17/using-figures-within-tables-in-latex/ | # Using Figures Within Tables in LaTeX
By using LaTeX to author APA manuscripts, researchers can address many problems associated with formatting their results into tables and figures. For example, ANOVA tables can be readily generated using the xtable package in R, and graphs from ggplot2 can be rendered within the manuscript using Sweave (see Wikipedia). However, more complicated layouts can be difficult to achieve.
In order to make test items or stimuli easier to understand, researchers occasionally organize examples in a table or figure. Using the standard \table command in LaTeX, it’s possible to include figures in an individual table cell without breaking the APA6.cls package. For example:
## Center-Aligned Figures
However, the above code vertically aligned my images according to their bottom-edge, producing an awkward looking table. Instead, we want the figures to be vertically centered. A Google search revealed the LaTeX Wikibook, which suggests a few methods to force figures to vertically align according to their center. Below, I surround each \includegraphics{} command with the \parbox{} command, which centers it along 1 unit of measurement, set to 12 pts. in my apa6 class options.
Output:
By using \parbox, figures are now vertically aligned with text cells. However, with the addition of figures the table is too long and we must span the table across 2-pages. To solve this, split the information across two tables. In this case, I can split by the stimuli category.
Alternatively, Brian Beitzel also pointed out that we can invoke longtable as a class option in APA6.cls, which allows tables to span multiple pages. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7036672234535217, "perplexity": 2569.3766056165496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912201455.20/warc/CC-MAIN-20190318152343-20190318174343-00178.warc.gz"} |
https://open.kattis.com/problems/billiard | Kattis
# Billiard
In a billiard table with horizontal side $a$ inches and vertical side $b$ inches, a ball is launched from the middle of the table. After $s > 0$ seconds the ball returns to the point from which it was launched, after having made $m$ bounces off the vertical sides and $n$ bounces off the horizontal sides of the table. Find the launching angle $A$ (measured from the horizontal), which will be between $0$ and $90$ degrees inclusive, and the initial velocity of the ball.
Assume that the collisions with a side are elastic (no energy loss), and thus the velocity component of the ball parallel to each side remains unchanged. Also, assume the ball has a radius of zero. Remember that, unlike pool tables, billiard tables have no pockets.
## Input
Input consists of a sequence of lines, each containing five nonnegative integers separated by whitespace. The five numbers are: $a$, $b$, $s$, $m$, and $n$, respectively. All numbers are positive integers not greater than $10\, 000$.
Input is terminated by a line containing five zeroes.
## Output
For each input line except the last, output a line containing two real numbers (rounded to exactly two decimal places) separated by a single space. The first number is the measure of the angle $A$ in degrees and the second is the velocity of the ball measured in inches per second, according to the description above.
Sample Input 1 Sample Output 1
100 100 1 1 1
200 100 5 3 4
201 132 48 1900 156
0 0 0 0 0
45.00 141.42
33.69 144.22
3.09 7967.81 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.840885579586029, "perplexity": 328.5502535187105}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347388427.15/warc/CC-MAIN-20200525095005-20200525125005-00436.warc.gz"} |
https://ashpublications.org/blood/article/120/21/4648/85271/APS-From-the-Root-of-Angelica-Sinesis-Promotes | Abstract
Abstract 4648
We have demonstrated the thrombopoietic effects of Chinese Medicine Dang Gui Bu Xue Tong (DBT), a formula composed of Radix Angelicae Sinensis and Radix Astragali in a thrombocypenic mouse model (Yang M et al, J Ethnopharm 2009). In this study, we further test the effect of polysaccharide (APS) from the root of Radix Angelicae Sinensis (APS) in in-vivo thrombopoiesis and in in-vitro megakaryocytopoiesis. A myelosuppression and thrombocytopenia mouse model was treated with APS (10 mg/kg/day), thrombopoietin (TPO) and saline as control. Peripheral blood cells from APS, TPO and vehicle-treated samples were then counted in different time-points. Using the colony-forming unit (CFU) assays, we determined the effects of APS on the hematopoietic stem/progenitor cells and megakaryocytic lineages. Analyses of Annexin V, Caspase-3, and Mitochondrial Membrane Potential were also conducted in megakaryocytic cell line M-07e. Lastly, the effects of APS on cells treated with Ly294002, a Phosphatidylinositol 3-Kinse inhibitor and the effect of APS on the phosphorylation of AKT were further studied in megakaryocytic cells. Our results showed that APS significantly enhanced the platelets and other blood cells recovery and CFU formation in this model. Morphological examination of bone marrows showed that APS treatment significantly increased the recovery of the megakaryocytic series and other hematopoietic progenitor cells. We observed the antiapoptotic effects of APS on M-07e cells. Addition of Ly294002 alone increases the percentage of cells undergoing apoptosis. In contrast, additional of APS to Ly294002-treated cells reversed the percentage of cells undergoing apoptosis. Furthermore, addition of APS significantly increased the phosphorylation of AKT. Our studies demonstrate that APS promotes platelet recovery in the mouse model and has anti-apoptosis effect in megakaryocytic cells. This effect is likely to be mediated by the PI3K/AKT pathway.
Disclosures:
No relevant conflicts of interest to declare.
Author notes
*
Asterisk with author names denotes non-ASH members. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8762211799621582, "perplexity": 16532.8051478499}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057584.91/warc/CC-MAIN-20210924231621-20210925021621-00313.warc.gz"} |
http://tex.stackexchange.com/questions/142750/chapter-first-page-fancyhdr-hide-title-but-keep-page-number | # chapter-first page-fancyhdr: hide title, but keep page number
In my diss (book document), I have headers (chapter title and page) appearing on chapters' first page as well. For each first page, I would like to remove the chapter name, and just keep the page number. This will prevent repetitions. From the second page onwards, classical heading should apply (chapter/session name + page number). I would like to have the same for TOC and LOF as well, but TOC shows no headers at all on the first page.
Any help would be greatly appreciated.
Sincerely,
Udiubu.
Below the relevant coding:
\documentclass[openright,a4paper,12pt,twoside]{book}
\titleformat{\chapter}[display]{\normalfont\huge\bfseries}{\chaptertitlename\\thechapter}{16pt}{\Large}
\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{
\markboth{\thechapter.\ #1}{}}
\renewcommand{\sectionmark}[1]{
\markright{\thesection.\ #1}}
\fancyfoot[C]{} %remove page number from footer
\makeatletter %insert header on title pages
\let\ps@plain\ps@fancy
\makeatother
\begin{document}
... %other pages here
\newpage
\begin{onehalfspacing}
\section*{Acknowledgments}
These are the aknowledgments
\end{onehalfspacing}
\cleardoublepage
\begin{doublespace}
\pagenumbering{roman}
\setcounter{page}{11}
\tableofcontents
\thispagestyle{fancy}
\end{doublespace}
\newpage
\thispagestyle{plain}
\listoffigures
\mainmatter
\pagenumbering{arabic}
\pagestyle{fancy}
\chapter{Lorem Ipsum}
Lorem Ipsum
\end{document}
-
Welcome to TeX.SX! – Claudio Fiandrino Nov 7 '13 at 9:35
The error is in the line \let\ps@plain\ps@fancy. Remove it. – egreg Nov 7 '13 at 11:52
Hi Egreg, thanks for your quick reply. – udiubu Nov 7 '13 at 12:55
It works, say 90%. The remaining issue is that with respect to the first pages, page number is now shown at the bottom, while I would still like to have it shown in the header. I solved it by using the following code: \fancypagestyle{plain}{% \fancyhf{}\fancyhead[R]{\thepage}% \renewcommand{\headrulewidth}{0.5pt}} – udiubu Nov 7 '13 at 13:36
@Johannes_B Done, thanks for reminding – egreg Nov 1 '14 at 21:36
\let\ps@plain\ps@fancy
is responsible for the behavior you obtain. Whoever advised you to use it didn't do you a favor: it makes the plain page style, used by default on chapter starting pages, equivalent to the fancy page style. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6860366463661194, "perplexity": 6390.010095364194}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246639325.91/warc/CC-MAIN-20150417045719-00008-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://stats.stackexchange.com/questions/503579/how-to-describe-the-relationship-between-these-two-variables | # How to describe the relationship between these two variables?
I would like to quantitatively describe the relationship between the two variables shown in this plot, but I am not sure what would be the correct way to do so. More specifically, my aim would be to assess whether there is a relationship between both variables and to quantify the strength of such a relationship (e.g. quantifying the amount of V2 variance explained by V1 with the coefficient of determination R2 or any other suitable statistic)
I have tentatively used the GAM model available in the geom_smooth function of ggplot (method="gam",formula = y ~ s(x, bs = "cs") and I have reproduced the same model with the gam package (see code and output below). However, I am not sure if this would be enough to claim that there is a relevant relationship between V1 and V2 and, if so, what would be the crucial statistics that I should report to substantiate such a possible conclusion. On the other hand, I have calculated the Spearman correlation trying to assess the relationship at the ordinal level (rho=0.480, p= 7.221e-08).
library(gam)
v=gam(V1~s(V2),bs="cs", data=ff)
Call: gam(formula = V1 ~ s(V2), data = ff, bs = "cs")
Deviance Residuals:
Min 1Q Median 3Q Max
-3.36459 -0.31432 -0.14645 0.09815 7.18685
(Dispersion Parameter for gaussian family taken to be 1.7516)
Null Deviance: 377.6946 on 115 degrees of freedom
Residual Deviance: 194.4251 on 111.0001 degrees of freedom
AIC: 401.1024
Number of Local Scoring Iterations: 2
Anova for Parametric Effects
Df Sum Sq Mean Sq F value Pr(>F)
s(V2) 1 71.352 71.352 40.736 4.159e-09 ***
Residuals 111 194.425 1.752
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Anova for Nonparametric Effects
Npar Df Npar F Pr(F)
(Intercept)
s(V2) 3 21.299 5.709e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
############# ############ EDIT:
Following the advices received I have transformed V1 using log (both proposed transformations produced virtually identical results). Then, I have estimated the relationship through linear regression. However, the obtained fit does not seem to be much better...
• How do you want to use the quantitative description of the relationship between the two variables? Please edit the question to add that information, as comments are easily overlooked and can even be lost. – EdM Jan 5 at 17:45
• HI EdM, thanks for your comment but I am not sure what do you mean. Could you clarify what I should specify in my question? Thanks in advance – cs- Jan 6 at 10:04
• What I mean is essentially what @pengzell says in a comment on an answer below. Do you just want to describe the data? Then the revised plot does a good job of that. Do you have some theoretical relationship between V2 and V1 that you want to test? Then you should be fitting a non-linear function of some specific form to the data. Do you want to make predictions about V2 from future values of V1? Then you might consider any of a number of transformations and fitting methods: besides specific forms like what you've tried, more general and flexible methods like loess or restricted cubic splines. – EdM Jan 6 at 14:53
• EdM: Thanks for the clarification. I have edited my question, now clarifying that I am not so really interested in predictions of unknown variables, but just want to quantify the strength of the relationship between V1 and V2. – cs- Jan 6 at 15:59
The first thing that stands out is that V1 looks highly right skewed. A first pass would be to log-transform V1 and see if a linear regression then provides a reasonable fit.
Given that the data contain zeros, a common alternative to log transformation that preserves zeros is the inverse hyperbolic sine:
ihs = log(V1 + sqrt(V1^2 + 1))
See, e.g.: Burbidge, John B., Lonnie Magee and A. Leslie Robb. 1988 "Alternative Transformations to Handle Extreme Values of the Dependent Variable." Journal of the American Statistical Association, vol. 83, 123-127.
• Agree the log transform looks appropriate. Another way to handle taking the log of data with zeros in it is to add 1 divided by the log base, so that values of 0 get mapped to -1. So instead of log10(V1), which will fail, you'd do log10(V1+0.1). – Nuclear Hoagie Jan 5 at 19:55
• Hi pengzell and Nuclear Hoagie, thank you very much for your suggestions. I implemented them and obtained virtually identical results with each of them. I've posted one of the results as an edit to my former question (not sure if that's the procedure). As you can see, the log transformation does not suffice to remove the extreme skewness of Vº – cs- Jan 6 at 10:02
• What is a "good fit" here depends on your purpose. Although it doesn't remove all the skew, clearly the result is much better now and the graph is more informative as a result. I personally would be happy to use this as a basis for further modeling (e.g., how robust is the relationship to confounding etc), although others may disagree. – pengzell Jan 6 at 10:14
• Hi pengzell, you are right when saying that what a "good fit" is, very much depends on the purpose. My comment did not try to diminish the achieved improvement after applying the log transformation you proposed. I just tried to express that I would like to keep digging and find out if there's any other alternative that could provide an even "better fit" – cs- Jan 6 at 11:37
If you want "to assess whether there is a relationship between both variables and to quantify the strength of such a relationship," then any general rank-correlation coefficient, for example Spearman's or Kendall's, would be a reasonable choice. That makes no assumptions about the underlying distributions, the functional form of the relationship, or which of your 2 variables is a predictor versus an outcome. You have a highly significant Spearman coefficient, consistent with a strong relationship between them.
The Spearman coefficient helps with "quantifying the amount of V2 variance explained by V1," if you are willing to consider the variance in ranks rather than variances in the actual values: the Spearman coefficient is the correlation coefficient between the 2 sets of ranks.
For "quantifying the amount of V2 variance explained by V1" in their original scales, then regression is an approach. Given the non-linear relationship between them, however, you presumably want to choose some transformation(s).
The best transformation(s) depends on how you intend to use the result. If you don't have a specific functional form in mind based on your understanding of the subject matter, generalized additive models (GAM) as you tried (although you modeled V1 as a function of V2, not the direction you seem to be interested in) and spline-based approaches provide general ways to proceed. Note, however, that you might tend to overestimate "the amount of V2 variance explained by V1" with those approaches: the results might not generalize well to new data samples unless you took precautions like penalization to minimize the effects of vagaries in fitting the particular data sample at hand.
If you don't care about the functional form of the relationship between the 2 variables, then the Spearman correlation is a very good choice. You can think of it as their correlation following one of the simplest of all transformations: each is simply transformed into its empirical cumulative distribution function.
• Hi EdM, thanks a lot for your complete and detailed answer!. I think I will stick to Spearman's correlation as I am not so much interested in the relationship between V1 and V2 original scales but in their ordinal relationship. Thanks a lot for your help – cs- Jan 7 at 9:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6017716526985168, "perplexity": 626.6901056321063}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154486.47/warc/CC-MAIN-20210803222541-20210804012541-00713.warc.gz"} |
https://brilliant.org/problems/messi-you-can-do-it/ | # Messi you can do it
It is a high pressure UEFA champions league final and the score is Juventus Barcelona 0 : 0 and the time left is 3 minutes of extra time. From somewhere Suarez provides a very good cross to Messi (our hero). As messi gets the cross he finds that Buffon (the Juventus goalkeeper) is in front of him.
Now Messi finds that with his hands full stretched Buffon can reach a height H which is also the height of the top goal bar and which is also the distance between Buffon and the goal base.
Messi flicks the ball (he is quite brilliant at it) at speed $$v$$ at an angle of 45 degrees. The ball just clears Buffon and just gets inside the goal.
If $$v$$ can be expressed as $$\sqrt{gh(x+\sqrt{y})}$$, find $$x^{ 2 }+y^{2}$$.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7137360572814941, "perplexity": 2008.0535543214476}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280929.91/warc/CC-MAIN-20170116095120-00254-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/226133-lagrange-identity.html | Math Help - Lagrange identity
1. Lagrange identity
thank you.
Attached Thumbnails
2. Re: Lagrange identity
to show it just multiply both sides by (1-z)
to show the trig identity let
$$z=e^{i\phi}$$
and take the Real part of both sides
yeah, there's some trig work to do on the right hand side but it's nothing magic | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9969902634620667, "perplexity": 2373.2694909339}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936465599.34/warc/CC-MAIN-20150226074105-00307-ip-10-28-5-156.ec2.internal.warc.gz"} |
http://mathhelpforum.com/statistics/114335-basic-probability-question.html | # Math Help - Basic Probability Question
1. ## Basic Probability Question
Hi, I'm analyzing how valid a solution is to a problem, and I've realized that it's correctness can't be guaranteed. I've translated it into a math problem to try and figure out how correct of a solution it is. But it's been a while since I've done any math, I was hoping someone could help me out.
MATH VERSION:
If there are a pool of 24 unique numbers
and I pull one at random and write it down if I haven't seen it before
then place it back in
and I do this 1000 times in total
how likely is it that I will have written down 24 numbers?
REAL VERSION:
A one line permutation generator receives an array with 4 functions, sorts them into a random order 1000 times, then keeps the unique sortings. I've ran it probably 10_000 times and always gotten the same results, and am debating whether I should keep trying to get it to fail.
2. Thanks, galactus, I don't think I was very clear, though. There are 24 numbers, they are drawn from 1000 times. How likely is it that all 24 will have been selected at some point during the 1000 drawings.
3. Originally Posted by angel.white
Hi, I'm analyzing how valid a solution is to a problem, and I've realized that it's correctness can't be guaranteed. I've translated it into a math problem to try and figure out how correct of a solution it is. But it's been a while since I've done any math, I was hoping someone could help me out.
MATH VERSION:
If there are a pool of 24 unique numbers
and I pull one at random and write it down if I haven't seen it before
then place it back in
and I do this 1000 times in total
how likely is it that I will have written down 24 numbers?
REAL VERSION:
A one line permutation generator receives an array with 4 functions, sorts them into a random order 1000 times, then keeps the unique sortings. I've ran it probably 10_000 times and always gotten the same results, and am debating whether I should keep trying to get it to fail.
This is equivalent to the "Coupon Collector's Problem". The general statement of the problem is that there are N distinct types of coupons and each time we obtain a coupon it is equally likely to be any one of the N, independent of prior selections. We would like to know the probability that we will have to collect more than n coupons to get a complete set.
The answer, taken from Ross, "A First Course in Probability, 7th edition", is
$\Pr(\text{more than n trials are needed}) = \sum_{i=1}^{N-1} \binom{N}{i} \left( \frac{N-i}{N} \right) ^n (-1)^{i+1}$
With your values (N=24, n=1000), this works out to be about $8 \times 10^{-18}$, or to put it another way, you will need about $10^{17}$ trials, on average, before you see an incomplete set.
You can find more information about the Coupon Collector's Problem here:
http://en.wikipedia.org/wiki/Coupon_...or%27s_problem
but for some mysterious reason the Wikipedia article does not include the formula I quoted. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6540926098823547, "perplexity": 363.78416927199805}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645356369.70/warc/CC-MAIN-20150827031556-00018-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://mathoverflow.net/api/userquestions.html?userid=17308&page=1&pagesize=10&sort=newest | 5
# Questions
1
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2
233
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### Reference request for the theory of heights over function fields
jan 21 at 20:40 Joe Silverman 7,8002445
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1
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### Mordell-Weil group of the universal abelian scheme
aug 3 at 11:27 Felipe Voloch 16.6k23661
8
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### $C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$
oct 19 11 at 20:54 Damian Rössler 1,421412
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### Torsion points of abelian varieties in the perfect closure of a function field
aug 27 11 at 20:56 Damian Rössler 1,421412
1
13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5956997871398926, "perplexity": 1893.7576134335113}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368699856050/warc/CC-MAIN-20130516102416-00030-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://googology.wikia.org/wiki/Stegert%27s_Psi_function | 11,328
pages
As part of his 2010 Ph.D. dissertation, Jan-Carl Stegert introduced collapsing hierarchies $$\mathfrak{M}$$ and two collapsing functions $$\Psi$$, and developed a powerful ordinal notation based on the work of Rathjen.[1]. $$\Psi$$ was then used for an analysis of an extension of $$\textrm{KP}$$ set theory called $$\mathsf{Stability}$$.[2]
Overview
The original definition of the first OCF is long and difficult, however here is a simplified overview of it.
Reflection instances are defined using ordered quintuples, and $$\mathsf M\textrm -\mathsf P$$ expressions are characterized in terms of reflection configurations, ordinals, and a number $$n\in\omega$$ or $$n=-1$$. For an ordinal $$\kappa$$ and an $$\mathsf M\textrm -\mathsf P$$-expression $$R$$, a relation $$\kappa\vDash R$$ is defined (not to be confused with satifaction for models), which plays the role of translating the properties represented by the $$\mathsf M\textrm -\mathsf P$$-expression into indescribability. Then, for ordinal $$\alpha$$ and reflection instance $$\mathbb X$$, collapsing hierarchies $$\mathfrak M^\alpha_{\mathbb X}$$ are defined that play a role superficially similar to an extension of $$M^\alpha$$ in Rathjen's OCF using a weakly compact cardinal (note that there are some differences, e.g. the behavior of the 1st entry $$\pi$$ of $$\mathbb X$$). Finally, an ordinal collapsing function $$\Psi$$ is defined.
Fast-growing function
Let $$\Pi_\omega\textrm{-ref}$$ denote the theory $$\textrm{KP}$$ augmented by a certain first-order reflection scheme[3]. As part of a characterization of the functions provably recursive in $$\Pi_\omega\textrm{-ref}$$[4], Stegert defines a function superficially similar to the middle-growing hierarchy (composed with tetration) along $$\mathsf T(\Xi)$$, using the concept of a "norm" in place of an explicit definition of fundamental sequences[5].
Since $$\varphi_1(\Xi+1)\subseteq\varphi_1(\Xi+1)+2\subseteq\varphi_2(\Xi+1)\subseteq\mathsf T(\Xi)$$, where $$\mathsf T(\Xi)$$ denotes the hull-set we use, then all of the following functions are well-defined, and the order of eventual domination is as follows:
• $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_2(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$ will eventually dominate $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_1(\Xi+1)+2}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$,
• Which eventually dominates $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_1(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$.
Therefore, the function $$f^{\mathsf T(\Xi)}_{\Psi^{\varphi_2(\Xi+1)}_{(\omega^+;\mathsf P_0;\epsilon;\epsilon;0)}}$$ will dominate all functions provably recursive in the theory $$\Pi_\omega\textrm{-ref}$$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9534170627593994, "perplexity": 642.4941467357207}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057018.8/warc/CC-MAIN-20210920040604-20210920070604-00176.warc.gz"} |
http://mathhelpforum.com/trigonometry/179347-heights-distance-problem.html | # Math Help - Heights and distance problem
1. ## Heights and distance problem
The angle of elevation from a point(on ground) is 45 degree and after 20 seconds its 30 degree If the aircraft is flying at a constant speed at 360 km/h. Find the height of plane from the ground.
angle AOB=45, angle COD=30. AB=CD, CA=DB. A is the initial position of the plane; and C is the position of the plane after 20 seconds. In 20 seconds, the plane covers a distance of 2000 mitres. So AC=2000.
tan(AOB)=tan(45)=1=AB/BO. So AB=BO
tan(COD)=tan(30)=1/sqrt(3)=CD/DO=CD/(DB+BO)=CD/(2000+BO)=AB/(2000+BO)=AB/(2000+AB)
That is AB/(2000+AB) = 1/sqrt(3)
Solving, AB = 2000/{sqrt(3)-1} which is the required height.
EDIT: In case the above link is non-accessible, here is another link http://oi56.tinypic.com/2eldz4n.jpg
3. Hello, vivek_master146!
The angle of elevation of a plane from a point on ground is 45 degree
and after 20 seconds it's 30 degrees.
If the plane is flying at a constant speed at 360 km/h, find the altitude of plane .
The answer is 2000√3. . Wrong!
Code:
A
*
* |
* |
* |h
* |
* 45d |
O * - - - - - *B
h
$\angle AO\!B \,=\,45^o,\;h \,=\,AB \,=\,OB$
Code:
A 2 C
. - - - - - *
. . * |
. . * |
. *h |h
. * . |
. * 30d . |
O * - - - - - * - - - - - *
h _ B 2 D
: - - - √3h - - - - - - :
$\text{In 20 seconds, it flies 2 km from }A\text{ to }C$
. . $AC\,=\,BD\,=\,22,\;\angle CO\!D \,=\,\angle 30^o,\;C\!D \,=\,h \quad\Rightarrow\quad O\!D \:=\:h + 2$
$\text{Since }\Delta CO\!D\text{ is a 30-60 right triangle, }\;O\!D \,=\,\sqrt{3}\,h.$
$\text{We have: }\:\sqrt{3}\,h \:=\:h + 2 \quad\Rightarrow\quad \sqrt{3}\,h - h \:=\:2 \quad\Rightarrow\quad (\sqrt{3}-1)h \:=\:2$
$\text{Therefore: }\;h \;=\;\frac{2}{\sqrt{3}-1}\text{ km.}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5265026092529297, "perplexity": 1692.7462140075684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011405327/warc/CC-MAIN-20140305092325-00090-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://agentnet.readthedocs.io/en/develop/modules/target_network.html | # Target Network¶
Implements the target network techniques in deep reinforcement learning. In short, the idea is to estimate reference Qvalues not from the current agent state, but from an earlier snapshot of weights. This is done to decorrelate target and predicted Qvalues/state_values and increase stability of learning algorithm.
Some notable alterations of this technique: - Standard approach with older NN snapshot – https://www.cs.toronto.edu/~vmnih/docs/dqn.pdf
• Moving average of weights
• Double Q-learning and other clever ways of training with target network
Here we implement a generic TargetNetwork class that supports both standard and moving average approaches through “moving_average_alpha” parameter of “load_weights”.
class agentnet.target_network.TargetNetwork(original_network_outputs, bottom_layers=(), share_inputs=True, name='target_net.')[source]
A generic class for target network techniques. Works by creating a deep copy of the original network and synchronizing weights through “load_weights” method.
If you just want to duplicate lasagne layers with or without sharing params, use agentnet.utils.clone.clone_network
Parameters: original_network_outputs (lasagne.layers.Layer or a list/tuple of such) – original network outputs to be cloned for target network bottom_layers (lasagne.layers.Layer or a list/tuple/dict of such.) – the layers that should be shared between networks. share_inputs (bool) – if True, all InputLayers will still be shared even if not mentioned in bottom_layers
#build network from lasagne.layers l_in = InputLayer([None,10]) l_d0 = DenseLayer(l_in,20) l_d1 = DenseLayer(l_d0,30) l_d2 = DenseLayer(l_d1,40) other_l_d2 = DenseLayer(l_d1,41)
# TargetNetwork that copies all the layers BUT FOR l_in full_clone = TargetNetwork([l_d2,other_l_d2]) clone_d2, clone_other_d2 = full_clone.output_layers
# only copy l_d2 and l_d1, keep l_d0 and l_in from original network, do not clone other_l_d2 partial_clone = TargetNetwork(l_d2,bottom_layers=(l_d0)) clone_d2 = partial_clone.output_layers
do_something_with_l_d2_weights()
#synchronize parameters with original network partial_clone.load_weights()
#OR set clone_params = 0.33*original_params + (1-0.33)*previous_clone_params partial_clone.load_weights(0.33)
load_weights(moving_average_alpha=1)[source]
Loads the weights from original network into target network. Should usually be called whenever you want to synchronize the target network with the one you train.
When using moving average approach, one should specify which fraction of new weights is loaded through moving_average_alpha param (e.g. moving_average_alpha=0.1)
Parameters: moving_average_alpha – If 1, just loads the new weights. Otherwise target_weights = alpha*original_weights + (1-alpha)*target_weights | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5524651408195496, "perplexity": 10024.590772598312}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362899.14/warc/CC-MAIN-20210301182445-20210301212445-00004.warc.gz"} |
http://www.ck12.org/tebook/Basic-Speller-Teacher-Materials/r1/section/8.3/ | <meta http-equiv="refresh" content="1; url=/nojavascript/"> Words With < tle > and < ttle > | CK-12 Foundation
You are reading an older version of this FlexBook® textbook: Basic Speller Teacher Materials Go to the latest version.
8.3: Words With < tle > and < ttle >
Created by: CK-12
Words With <tle> and <ttle>
1. Words like battle that end with the letters <le> right after a [t] sound are a special group. In the words below underline the letters that spell [t]:
$& \text{ba\underline{tt}le} && \text{ke\underline{tt}le} && \text{bo\underline{tt}le} && \text{shu\underline{tt}le} \\& \text{bee\underline{t}le} && \text{gen\underline{t}le} && \text{s\underline{t}ar\underline{t}le} && \text{\underline{t}ur\underline{t}le} \\& \text{man\underline{t}le} && \text{ra\underline{tt}le} && \text{se\underline{tt}le} && \text{\underline{t}i\underline{t}le} \\& \text{li\underline{tt}le} && \text{bri\underline{tt}le} && \text{ca\underline{tt}le} && \text{\underline{t}oo\underline{t}le}$
2. Now sort the words into this matrix:
Words in which the [t] comes right after
a consonant: a long vowel: a short vowel:
Words with [t] spelled <t>
mantle
gentle
startle
turtle
beetle
title
tootle
Words with [$\mathrm{t}$] 'spelled $<\mathrm{tt}>$
battle
little
kettle
rattle
brittle
bottle
settle
cattle
shuttle
3. In words that end with a [t] sound with <le> right after it, if the [t] comes right after a consonant or long vowel sound, the [t] is spelled <t>. But if the [t] comes right after a short vowel sound, the [t] is spelled $\underline{}$.
5. The long vowels in words like title may seem to be exceptions to the VCC pattern. But the pattern for words that end <tle> is true for words that end with any consonant followed by <le>. Since there is always a long vowel in every word that ends with a single consonant followed by <le>, we can treat these long vowels not as exceptions, but rather as the result of a smaller pattern within a bigger pattern. We can call it the VCle# pattern. VCle# is another pattern that marks long vowels, like VCV and Ve#.
If there is a short vowel sound right in front of the [t], we use a double $<\mathrm{tt}>$ to spell [t] in front of the <le>. We can think of this as another smaller pattern within the bigger VCC pattern. We can call it the VCCle# pattern, which is another pattern that marks short vowels, like VCC and VC#.
In the VCCle pattern the vowel is short, but in the VCle pattern the vowel is long.
6. Sort the words with short vowels into these two groups:
Words with short vowels in which [t] is spelled ...
<t> $<\mathrm{tt}>$
mantle battle bottle
gentle little settle
startle kettle cattle
turtle rattle shuttle
brittle
If there is a consonant between the short vowel and the [t], we only need a single <t> because the other consonant will fill out the VCCle pattern, as in words like gentle and mantle. But if there is no other consonant, we need both <t>’s, as in words like bottle and little.
Word Changes. Remember to follow the directions carefully and write the words you make in the column on the right. The shaded boxes will contain words with which you worked in Item 1 of this lesson. All of the words will end in either <tle> or <ttle>. As you form each word, decide whether it should be spelled with a single or a double <t>:
1. Write the word battle - battle
2. Change the first consonant in the word to the twentieth letter in the alphabet - tattle
3. Change the first consonant back to $<\mathrm{b}>$ and change the $<\mathrm{a}>$ to <ee> - beetle
4. Change the first consonant in the word to the fifth consonant in the alphabet and change the second <e> to the fourteenth letter in the alphabet - gentle
5. Change the first letter in the word to <m> and change the first vowel in the word to the first vowel in the alphabet - mantle
6. Move the second consonant in the word to the front, delete the <m>, and change the $<\mathrm{a}>$ to an <e> - nettle
7. Change the first consonant in the word to the fourteenth consonant in the alphabet, and change the <e> back to an $<\mathrm{a}>$ - rattle
8. Change the first letter in the word to the letter that comes right after it in the alphabet, make the second letter in the word a <c>, and change the $<\mathrm{a}>$ to the twenty-first letter of the alphabet - scuttle
9. Change the first two letters of the word to $<\mathrm{br}>$ and change the $<\mathrm{u}>$ to $<\mathrm{i}>$ - brittle
Teaching Notes.
The VCle# and VCCle# patterns, though quite modest in their extension, are important and very reliable. Notice that though the ending is spelled <le>, it is pronounced [əl]:the letters and sounds are reversed. In terms of sound, then, the VCle# and VCCle# patterns fit the VCV and VCC patterns. For more on the <le> ending see AES, pp. 149-51.
Item 6. The short vowel in startle is [o]. The short vowel in turtle is the [u] in [ur], though there is admittedly very little [u] coloring left in most pronunciations of [ur].
Word Changes. Be sure the students understand that they must decide whether the words are spelled <tle> or <ttle>. The directions don't mention that aspect of the spelling. If students get fuddled trying to count letters and consonants and vowels in the alphabet, the table “Letters of the Alphabet” may be helpful. Notice that $<\mathrm{u}>$, <w>, and <y> are counted as both vowels and consonants.
LETTERS OF THE ALPHABETS
Letter Vowel Consonant
a 1st 1st
b 2nd 1st
c 3rd 2nd
d 4th 3rd
e 5th 2nd
f 6th 4th
g 7th 5th
h 8th 6th
i 9th 3rd
j 10th 7th
k 11th 8th
l 12th 9th
m 13th 10th
n 14th 11th
o 15th 4th
P 16th 12th
q 17th 13th
r 18th 14th
s 19th 15th
t 20th 16th
u 21st 5th 17th
V 22nd 18th
w 23rd 6th 19th
X 24th 20th
y 25th 7th 21st
z 26th 22nd
Subjects:
1 , 2 , 3 , 4 , 5
Date Created:
Feb 23, 2012
Jan 27, 2015
Files can only be attached to the latest version of None | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 15, "texerror": 0, "math_score": 0.5978790521621704, "perplexity": 3780.067685616549}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936463287.91/warc/CC-MAIN-20150226074103-00088-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://support.logmeininc.com/pro/faq/how-can-i-install-logmein-without-the-display-accelerator-000054973 | ## How can I install LogMeIn without the Display Accelerator?
### Procedure
Some hardware may conflict with the mirror driver LogMeIn employs to display optimized remote screen video. You can install LogMeIn without the Display Accelerator and use the default screen display.
2. Create a folder in the C:\ drive called LMI.
3. Open the LogMeIn.zip file and extract it to the LMI folder.
4. Press the Windows key and the R key at the same time to open the Run box.
5. Type `cmd` and click OK. The command prompt will open.
• If the local operating system is 32-bit, then at the command prompt, type:
`cd C:\lmi\x86` (and press enter).
`logmein install -nomirrordrv` (and press enter).
• If the local operating system is 64-bit, then at the command prompt, type:
`cd C:\lmi\x64` (hit enter).
`logmein install -nomirrordrv` (and press enter).
6. When you receive a message that the install was successful, close the command prompt window.
7. On the message that appears, click Open LogMeIn and follow the steps to assign the computer to your LogMeIn account.
Tip:To check system type (32 bit or 64 bit) press the Windows key and the Pause key at the same time to view system properties. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.874872624874115, "perplexity": 6343.688007495103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156513.14/warc/CC-MAIN-20180920140359-20180920160759-00449.warc.gz"} |
http://mathhelpforum.com/calculus/137211-what-value-x-does-function-y-x-2-cos-x-have-horizontal-tangent-line.html | # Thread: For what value of x does the function y = x + 2 cos x have a horizontal tangent line?
1. ## For what value of x does the function y = x + 2 cos x have a horizontal tangent line?
For what value of x does the function y = x + 2 cos x have a horizontal tangent line?
I said f'(x) = 1 + 0(cosx) + 2(-sinx) = 1 - 2sinx
So if the line is horizontal, f'(x) must = 0, so 0 = 1 - 2sinx --> sin x = 1/2
Now I am stuck. What value of x should I put? 30 degrees? pi divided by six?
???
2. Originally Posted by satx
I said f'(x) = 1 + 0(cosx) + 2(-sinx) = 1 - 2sinx
So if the line is horizontal, f'(x) must = 0, so 0 = 1 - 2sinx --> sin x = 1/2
Now I am stuck. What value of x should I put? 30 degrees? pi divided by six?
???
better be $\frac{\pi}{6}$
3. Originally Posted by satx
I said f'(x) = 1 + 0(cosx) + 2(-sinx) = 1 - 2sinx
So if the line is horizontal, f'(x) must = 0, so 0 = 1 - 2sinx --> sin x = 1/2
Now I am stuck. What value of x should I put? 30 degrees? pi divided by six?
???
Read the question carefully as it is written. (I'm not sure if you transcribed it exactly to us or not.)
It may be asking for any value of x that satisfies the condition. If so, then you can list whichever solution you want.
It may ask for the value(s) that occurs in a particular range (i.e. between 0 and pi). In this case you should find them all in that range.
Or, it may ask you to list EVERY value of x that would work. For example, recalling trig, there are two solutions between 0 and 2pi:
$\sin\left(\tfrac{\pi}{6}\right)=\sin\left(\tfrac{5 \pi}{6}\right)=\tfrac{1}{2}$
But it is periodic, so we could list the solutions like so:
$\cdots , -\tfrac{11\pi}{6} , -\tfrac{7\pi}{6} , \tfrac{\pi}{6} , \tfrac{5\pi}{6} , \tfrac{13\pi}{6} , \tfrac{17\pi}{6} ,\cdots$
or, probably better:
$x=\tfrac{\pi}{6} + 2\pi n$
and
$x=\tfrac{5\pi}{6} + 2\pi n$
where n is an integer
4. No, I copied it word for word...
5. Originally Posted by satx
I said f'(x) = 1 + 0(cosx) + 2(-sinx) = 1 - 2sinx
So if the line is horizontal, f'(x) must = 0, so 0 = 1 - 2sinx --> sin x = 1/2
Now I am stuck. What value of x should I put? 30 degrees? pi divided by six?
???
You don't need to use the product rule to differentiate $2\cos(x)$ since differentiation is a linear process so if $a$ and $b$ are constants:
$\frac{d}{dx} \left[af(x)+bg(x)\right]=a \frac{df}{dx}+b\frac{dg}{dx}$
and for $2\cos(x)$:
$\frac{d}{dx} \left[af(x)\right]=a \frac{df}{dx}$
where $a=2$ and $f(x)=\cos(x)$
CB
6. Originally Posted by satx
No, I copied it word for word...
Well then, since it is singular (i.e., it says "for what value" and not "values"), I'd probably just list one solution -- probably the lowest positive value which was $\pi/6$.
However it is sort of ambiguous as written so you might want to ask your teacher what he/she expects you to answer when it is worded like this. We can only guess if the problem is not worded more explicitly. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 14, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8015606999397278, "perplexity": 655.5328382196606}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187821017.9/warc/CC-MAIN-20171017091309-20171017111309-00795.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/87800-submartingale-help.html | # Math Help - Submartingale help
1. ## Submartingale help
How do i show that $(B_t^2 -t)^2$ is a submartingale w.r.t. the natural filtration generated by $B_t$ , where $B_t$ is a standard Brownian motion started at zero....
2. Apply Ito's formula to $f(t,B) = (B_t^2-t)^2$.
At the end, you'll get an Ito integral(which has the property of being a martingale) plus the term:
$4\int_0^t B^2(u)du$
which is a non-negative function of t. Therefore
$E[4\int_0^t B^2(u)du | F(s)] \geq 4\int_0^s B^2(u)du, 0 \leq s \leq t$ and thus f(t,B) is a submartingale. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9651921987533569, "perplexity": 435.77814101640735}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500826259.53/warc/CC-MAIN-20140820021346-00383-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://core.ac.uk/display/151242079 | ## Phenomenology of single-inclusive jet production with jet radius and threshold resummation
### Abstract
We perform a detailed study of inclusive jet production cross sections at the LHC and compare the QCD theory predictions based on the recently developed formalism for threshold and jet radius joint resummation at next-to-leading logarithmic accuracy to inclusive jet data collected by the CMS collaboration at $\sqrt{S} = 7$ and $13$TeV. We compute the cross sections at next-to-leading order in QCD with and without the joint resummation for different choices of jet radii $R$ and observe that the joint resummation leads to crucial improvements in the description of the data. Comprehensive studies with different parton distribution functions demonstrate the necessity of considering the joint resummation in fits of those functions based on the LHC jet data.Comment: multiple figures and table
Topics: High Energy Physics - Phenomenology, High Energy Physics - Experiment, Nuclear Experiment
Publisher: 'American Physical Society (APS)'
Year: 2018
DOI identifier: 10.1103/PhysRevD.97.056026
OAI identifier: oai:arXiv.org:1801.07284
### Suggested articles
To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7854189276695251, "perplexity": 3050.1043064088276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00470.warc.gz"} |
http://nbviewer.jupyter.org/github/jakevdp/PythonDataScienceHandbook/blob/master/notebooks/05.12-Gaussian-Mixtures.ipynb | This notebook contains an excerpt from the Python Data Science Handbook by Jake VanderPlas; the content is available on GitHub.
The text is released under the CC-BY-NC-ND license, and code is released under the MIT license. If you find this content useful, please consider supporting the work by buying the book!
# In Depth: Gaussian Mixture Models¶
The k-means clustering model explored in the previous section is simple and relatively easy to understand, but its simplicity leads to practical challenges in its application. In particular, the non-probabilistic nature of k-means and its use of simple distance-from-cluster-center to assign cluster membership leads to poor performance for many real-world situations. In this section we will take a look at Gaussian mixture models (GMMs), which can be viewed as an extension of the ideas behind k-means, but can also be a powerful tool for estimation beyond simple clustering.
We begin with the standard imports:
In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import seaborn as sns; sns.set()
import numpy as np
## Motivating GMM: Weaknesses of k-Means¶
Let's take a look at some of the weaknesses of k-means and think about how we might improve the cluster model. As we saw in the previous section, given simple, well-separated data, k-means finds suitable clustering results.
For example, if we have simple blobs of data, the k-means algorithm can quickly label those clusters in a way that closely matches what we might do by eye:
In [2]:
# Generate some data
from sklearn.datasets.samples_generator import make_blobs
X, y_true = make_blobs(n_samples=400, centers=4,
cluster_std=0.60, random_state=0)
X = X[:, ::-1] # flip axes for better plotting
In [3]:
# Plot the data with K Means Labels
from sklearn.cluster import KMeans
kmeans = KMeans(4, random_state=0)
labels = kmeans.fit(X).predict(X)
plt.scatter(X[:, 0], X[:, 1], c=labels, s=40, cmap='viridis');
From an intuitive standpoint, we might expect that the clustering assignment for some points is more certain than others: for example, there appears to be a very slight overlap between the two middle clusters, such that we might not have complete confidence in the cluster assigment of points between them. Unfortunately, the k-means model has no intrinsic measure of probability or uncertainty of cluster assignments (although it may be possible to use a bootstrap approach to estimate this uncertainty). For this, we must think about generalizing the model.
One way to think about the k-means model is that it places a circle (or, in higher dimensions, a hyper-sphere) at the center of each cluster, with a radius defined by the most distant point in the cluster. This radius acts as a hard cutoff for cluster assignment within the training set: any point outside this circle is not considered a member of the cluster. We can visualize this cluster model with the following function:
In [4]:
from sklearn.cluster import KMeans
from scipy.spatial.distance import cdist
def plot_kmeans(kmeans, X, n_clusters=4, rseed=0, ax=None):
labels = kmeans.fit_predict(X)
# plot the input data
ax = ax or plt.gca()
ax.axis('equal')
ax.scatter(X[:, 0], X[:, 1], c=labels, s=40, cmap='viridis', zorder=2)
# plot the representation of the KMeans model
centers = kmeans.cluster_centers_
radii = [cdist(X[labels == i], [center]).max()
for i, center in enumerate(centers)]
for c, r in zip(centers, radii):
ax.add_patch(plt.Circle(c, r, fc='#CCCCCC', lw=3, alpha=0.5, zorder=1))
In [5]:
kmeans = KMeans(n_clusters=4, random_state=0)
plot_kmeans(kmeans, X)
An important observation for k-means is that these cluster models must be circular: k-means has no built-in way of accounting for oblong or elliptical clusters. So, for example, if we take the same data and transform it, the cluster assignments end up becoming muddled:
In [6]:
rng = np.random.RandomState(13)
X_stretched = np.dot(X, rng.randn(2, 2))
kmeans = KMeans(n_clusters=4, random_state=0)
plot_kmeans(kmeans, X_stretched)
By eye, we recognize that these transformed clusters are non-circular, and thus circular clusters would be a poor fit. Nevertheless, k-means is not flexible enough to account for this, and tries to force-fit the data into four circular clusters. This results in a mixing of cluster assignments where the resulting circles overlap: see especially the bottom-right of this plot. One might imagine addressing this particular situation by preprocessing the data with PCA (see In Depth: Principal Component Analysis), but in practice there is no guarantee that such a global operation will circularize the individual data.
These two disadvantages of k-means—its lack of flexibility in cluster shape and lack of probabilistic cluster assignment—mean that for many datasets (especially low-dimensional datasets) it may not perform as well as you might hope.
You might imagine addressing these weaknesses by generalizing the k-means model: for example, you could measure uncertainty in cluster assignment by comparing the distances of each point to all cluster centers, rather than focusing on just the closest. You might also imagine allowing the cluster boundaries to be ellipses rather than circles, so as to account for non-circular clusters. It turns out these are two essential components of a different type of clustering model, Gaussian mixture models.
## Generalizing E–M: Gaussian Mixture Models¶
A Gaussian mixture model (GMM) attempts to find a mixture of multi-dimensional Gaussian probability distributions that best model any input dataset. In the simplest case, GMMs can be used for finding clusters in the same manner as k-means:
In [7]:
from sklearn.mixture import GMM
gmm = GMM(n_components=4).fit(X)
labels = gmm.predict(X)
plt.scatter(X[:, 0], X[:, 1], c=labels, s=40, cmap='viridis');
But because GMM contains a probabilistic model under the hood, it is also possible to find probabilistic cluster assignments—in Scikit-Learn this is done using the predict_proba method. This returns a matrix of size [n_samples, n_clusters] which measures the probability that any point belongs to the given cluster:
In [8]:
probs = gmm.predict_proba(X)
print(probs[:5].round(3))
[[ 0. 0. 0.475 0.525]
[ 0. 1. 0. 0. ]
[ 0. 1. 0. 0. ]
[ 0. 0. 0. 1. ]
[ 0. 1. 0. 0. ]]
We can visualize this uncertainty by, for example, making the size of each point proportional to the certainty of its prediction; looking at the following figure, we can see that it is precisely the points at the boundaries between clusters that reflect this uncertainty of cluster assignment:
In [9]:
size = 50 * probs.max(1) ** 2 # square emphasizes differences
plt.scatter(X[:, 0], X[:, 1], c=labels, cmap='viridis', s=size);
Under the hood, a Gaussian mixture model is very similar to k-means: it uses an expectation–maximization approach which qualitatively does the following:
1. Choose starting guesses for the location and shape
2. Repeat until converged:
1. E-step: for each point, find weights encoding the probability of membership in each cluster
2. M-step: for each cluster, update its location, normalization, and shape based on all data points, making use of the weights
The result of this is that each cluster is associated not with a hard-edged sphere, but with a smooth Gaussian model. Just as in the k-means expectation–maximization approach, this algorithm can sometimes miss the globally optimal solution, and thus in practice multiple random initializations are used.
Let's create a function that will help us visualize the locations and shapes of the GMM clusters by drawing ellipses based on the GMM output:
In [10]:
from matplotlib.patches import Ellipse
def draw_ellipse(position, covariance, ax=None, **kwargs):
"""Draw an ellipse with a given position and covariance"""
ax = ax or plt.gca()
# Convert covariance to principal axes
if covariance.shape == (2, 2):
U, s, Vt = np.linalg.svd(covariance)
angle = np.degrees(np.arctan2(U[1, 0], U[0, 0]))
width, height = 2 * np.sqrt(s)
else:
angle = 0
width, height = 2 * np.sqrt(covariance)
# Draw the Ellipse
for nsig in range(1, 4):
ax.add_patch(Ellipse(position, nsig * width, nsig * height,
angle, **kwargs))
def plot_gmm(gmm, X, label=True, ax=None):
ax = ax or plt.gca()
labels = gmm.fit(X).predict(X)
if label:
ax.scatter(X[:, 0], X[:, 1], c=labels, s=40, cmap='viridis', zorder=2)
else:
ax.scatter(X[:, 0], X[:, 1], s=40, zorder=2)
ax.axis('equal')
w_factor = 0.2 / gmm.weights_.max()
for pos, covar, w in zip(gmm.means_, gmm.covars_, gmm.weights_):
draw_ellipse(pos, covar, alpha=w * w_factor)
With this in place, we can take a look at what the four-component GMM gives us for our initial data:
In [11]:
gmm = GMM(n_components=4, random_state=42)
plot_gmm(gmm, X)
Similarly, we can use the GMM approach to fit our stretched dataset; allowing for a full covariance the model will fit even very oblong, stretched-out clusters:
In [12]:
gmm = GMM(n_components=4, covariance_type='full', random_state=42)
plot_gmm(gmm, X_stretched)
This makes clear that GMM addresses the two main practical issues with k-means encountered before.
### Choosing the covariance type¶
If you look at the details of the preceding fits, you will see that the covariance_type option was set differently within each. This hyperparameter controls the degrees of freedom in the shape of each cluster; it is essential to set this carefully for any given problem. The default is covariance_type="diag", which means that the size of the cluster along each dimension can be set independently, with the resulting ellipse constrained to align with the axes. A slightly simpler and faster model is covariance_type="spherical", which constrains the shape of the cluster such that all dimensions are equal. The resulting clustering will have similar characteristics to that of k-means, though it is not entirely equivalent. A more complicated and computationally expensive model (especially as the number of dimensions grows) is to use covariance_type="full", which allows each cluster to be modeled as an ellipse with arbitrary orientation.
We can see a visual representation of these three choices for a single cluster within the following figure:
## GMM as Density Estimation¶
Though GMM is often categorized as a clustering algorithm, fundamentally it is an algorithm for density estimation. That is to say, the result of a GMM fit to some data is technically not a clustering model, but a generative probabilistic model describing the distribution of the data.
As an example, consider some data generated from Scikit-Learn's make_moons function, which we saw in In Depth: K-Means Clustering:
In [13]:
from sklearn.datasets import make_moons
Xmoon, ymoon = make_moons(200, noise=.05, random_state=0)
plt.scatter(Xmoon[:, 0], Xmoon[:, 1]);
If we try to fit this with a two-component GMM viewed as a clustering model, the results are not particularly useful:
In [14]:
gmm2 = GMM(n_components=2, covariance_type='full', random_state=0)
plot_gmm(gmm2, Xmoon)
But if we instead use many more components and ignore the cluster labels, we find a fit that is much closer to the input data:
In [15]:
gmm16 = GMM(n_components=16, covariance_type='full', random_state=0)
plot_gmm(gmm16, Xmoon, label=False)
Here the mixture of 16 Gaussians serves not to find separated clusters of data, but rather to model the overall distribution of the input data. This is a generative model of the distribution, meaning that the GMM gives us the recipe to generate new random data distributed similarly to our input. For example, here are 400 new points drawn from this 16-component GMM fit to our original data:
In [16]:
Xnew = gmm16.sample(400, random_state=42)
plt.scatter(Xnew[:, 0], Xnew[:, 1]);
GMM is convenient as a flexible means of modeling an arbitrary multi-dimensional distribution of data.
### How many components?¶
The fact that GMM is a generative model gives us a natural means of determining the optimal number of components for a given dataset. A generative model is inherently a probability distribution for the dataset, and so we can simply evaluate the likelihood of the data under the model, using cross-validation to avoid over-fitting. Another means of correcting for over-fitting is to adjust the model likelihoods using some analytic criterion such as the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). Scikit-Learn's GMM estimator actually includes built-in methods that compute both of these, and so it is very easy to operate on this approach.
Let's look at the AIC and BIC as a function as the number of GMM components for our moon dataset:
In [17]:
n_components = np.arange(1, 21)
models = [GMM(n, covariance_type='full', random_state=0).fit(Xmoon)
for n in n_components]
plt.plot(n_components, [m.bic(Xmoon) for m in models], label='BIC')
plt.plot(n_components, [m.aic(Xmoon) for m in models], label='AIC')
plt.legend(loc='best')
plt.xlabel('n_components');
The optimal number of clusters is the value that minimizes the AIC or BIC, depending on which approximation we wish to use. The AIC tells us that our choice of 16 components above was probably too many: around 8-12 components would have been a better choice. As is typical with this sort of problem, the BIC recommends a simpler model.
Notice the important point: this choice of number of components measures how well GMM works as a density estimator, not how well it works as a clustering algorithm. I'd encourage you to think of GMM primarily as a density estimator, and use it for clustering only when warranted within simple datasets.
## Example: GMM for Generating New Data¶
We just saw a simple example of using GMM as a generative model of data in order to create new samples from the distribution defined by the input data. Here we will run with this idea and generate new handwritten digits from the standard digits corpus that we have used before.
To start with, let's load the digits data using Scikit-Learn's data tools:
In [18]:
from sklearn.datasets import load_digits
digits = load_digits()
digits.data.shape
Out[18]:
(1797, 64)
Next let's plot the first 100 of these to recall exactly what we're looking at:
In [19]:
def plot_digits(data):
fig, ax = plt.subplots(10, 10, figsize=(8, 8),
subplot_kw=dict(xticks=[], yticks=[]))
fig.subplots_adjust(hspace=0.05, wspace=0.05)
for i, axi in enumerate(ax.flat):
im = axi.imshow(data[i].reshape(8, 8), cmap='binary')
im.set_clim(0, 16)
plot_digits(digits.data)
We have nearly 1,800 digits in 64 dimensions, and we can build a GMM on top of these to generate more. GMMs can have difficulty converging in such a high dimensional space, so we will start with an invertible dimensionality reduction algorithm on the data. Here we will use a straightforward PCA, asking it to preserve 99% of the variance in the projected data:
In [20]:
from sklearn.decomposition import PCA
pca = PCA(0.99, whiten=True)
data = pca.fit_transform(digits.data)
data.shape
Out[20]:
(1797, 41)
The result is 41 dimensions, a reduction of nearly 1/3 with almost no information loss. Given this projected data, let's use the AIC to get a gauge for the number of GMM components we should use:
In [21]:
n_components = np.arange(50, 210, 10)
models = [GMM(n, covariance_type='full', random_state=0)
for n in n_components]
aics = [model.fit(data).aic(data) for model in models]
plt.plot(n_components, aics);
It appears that around 110 components minimizes the AIC; we will use this model. Let's quickly fit this to the data and confirm that it has converged:
In [22]:
gmm = GMM(110, covariance_type='full', random_state=0)
gmm.fit(data)
print(gmm.converged_)
True
Now we can draw samples of 100 new points within this 41-dimensional projected space, using the GMM as a generative model:
In [23]:
data_new = gmm.sample(100, random_state=0)
data_new.shape
Out[23]:
(100, 41)
Finally, we can use the inverse transform of the PCA object to construct the new digits:
In [24]:
digits_new = pca.inverse_transform(data_new)
plot_digits(digits_new) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4888854920864105, "perplexity": 1535.7670389022514}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742316.5/warc/CC-MAIN-20181114211915-20181114233915-00184.warc.gz"} |
http://elfastronomy.com/category/astronomy/star/planetary-nebula/ | NOCTUARY – February 2013 Edition
Gallery: Comet Lemmon . . . . . . . . . . . . . … . . . 02
A selection of photographs of the lovely Comet Lemmon that
graced the deep southern skies during January and February.
From the Archimedes Archives . . . . . . . . . . . . 03
Interesting bits and pieces to pique the astronomically curious.
Deep-sky Highlight: The Blue Planetary . . . . 06
A deep-sky gem hiding in Centaurus.
Dark Emu Rising . . . . . . . . . . . . . . . . . . . . . . .. . . 07
The spectacular dark nebula complex known to some as the Dark
Emu rises, in pursuit of the Large and Small Magellanic Clouds.
Astrophotography tutorial (Part 2) . . . . . . . . . 09
The second installment of Brett du Preez’s astrophotography
tutorial takes a personal detour.
Jargon Jambalaya . . . . . . . . . . . . . . . . . . . . . . . . 13
Some astrojargon explained.
The Tale of Hydra . . . . . . . . . . . . . . . . . . . . . . . . . 16
A convoluted story surrounds the largest, and one of the most
ancient, constellations.
A South African Champion . . . . . . . . . . . . . . . . 24
The nearest star to the solar system has been seen by few people.
Use this finder chart to catapult yourself amongst the elite.
Noise vs. ISO in Astrophotography . . . . . . . . . 25
What’s the best ISO setting to use on your DSLR when taking
astrophotos? Here’s one way of finding out.
Briefly Noted . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Some housekeeping announcements and other stuff that didn’t
fit in elsewhere: Lacaille’s Sky – On the Cover – Noctuary | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9300739765167236, "perplexity": 374.2488789476571}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889617.56/warc/CC-MAIN-20180120122736-20180120142736-00684.warc.gz"} |
http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.0020117 | Research Article
# Oscillatory Regulation of Hes1: Discrete Stochastic Delay Modelling and Simulation
• Affiliation: Departamento de Informática, Universidad de Valladolid, Valladolid, Spain
X
• To whom correspondence should be addressed. E-mail: kb@maths.uq.edu.au
Affiliation: Advanced Computational Modelling Centre, University of Queensland, Brisbane, Australia
X
• Affiliation: Advanced Computational Modelling Centre, University of Queensland, Brisbane, Australia
X
• Affiliation: Advanced Computational Modelling Centre, University of Queensland, Brisbane, Australia
X
• Published: September 08, 2006
• DOI: 10.1371/journal.pcbi.0020117
## Abstract
Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.
## Synopsis
Delay processes are ubiquitous in the biological sciences but are not always well-represented in mathematical models attempting to describe these biological processes. Additional issues arise when attempting to capture the uncertainty (intrinsic noise) associated with chemical kinetics in dealing with when and in what order reactions take place. Complicating the situation further are important instances when certain key molecules occur only in small numbers, so that it is not meaningful to talk about concentrations.
In this paper Barrio et al. show how to incorporate delay, intrinsic noise, and discreteness associated with chemical kinetic systems into a very simple algorithm called the delay stochastic simulation algorithm (DSSA). This algorithm very naturally generalises the stochastic simulation algorithm that does not treat delays. The authors then apply the DSSA to a specific set of experiments performed by Hirata et al. who showed, amongst other things, that serum treatment of cultured cells induces cyclic expression of both mRNA and protein of the Notch effector Hes1 with a two-hour period. The authors show how this approach can explain additional experiments performed by Hirata et al., and, because this approach is very general, suggest that it can provide deep insights into the relationship between delayed processes, intrinsic noise, and small numbers of molecules in many biological systems.
### Introduction
The mathematical modelling and simulation of genetic regulatory networks can provide insights into the complicated biological and chemical processes associated with genetic regulation. However, it is important that the models are kept simple but nevertheless capture the key processes. In addition, by incorporating experimental data into such models (where available) their accuracy can be improved.
An important aspect associated with genetic regulation is that mRNA and protein expression levels can be quite low, and so continuous models, as described by ordinary differential equations, may be inappropriate. Furthermore, processes such as transcription and translation do not occur instantaneously and may have considerable delays associated with them. It is these two issues that we pursue in terms of understanding oscillatory expression levels of both mRNA and protein of the Notch effector Hes1.
There are many types of molecular clocks that regulate biological processes, but apart from circadian clocks [1] these clocks are still relatively poorly characterised. Oscillatory dynamics are also known from mRNAs for Notch-signalling molecules such as Hes1 (a bHLH factor) that oscillates with a two-hour cycle during somite segmentation [2]. In a recent set of experiments, Hirata et al. [3] measured the production of hes1 mRNA (M) and Hes1 protein (P) in mouse. Serum treatments on cultured cells (that have already been shown to induce circadian oscillation [4]) result in oscillations in expression levels for hes1 mRNA and Hes1 protein in a two-hour cycle with a phase lag of approximately 15 min between the oscillatory profiles of mRNA and protein. The oscillations in expression continue for 6–12 h and are not dependent on the stimulus but can be induced by exposure to cells expressing Delta. It was argued that the lag between protein and mRNA oscillation levels of 15 min reflects the time needed for protein degradation.
Specifically, the data presented in the paper by Hirata et al. (Figure 1 in [3]) indicates sustained oscillation of hes1 mRNA over six periods while it suggests oscillation of Hes1 protein that dies away after 6–8 h. Furthermore, the peaks in the expression levels of Hes1 protein seem to be about twice that of the mRNA peaks. Unfortunately, it is not clear what the units are in terms of expression levels and whether the scales are the same from experiment to experiment—valuable information that might help to validate a mathematical model.
Hirata et al. examined the underlying mechanisms for the observed oscillations and showed that in the presence of the proteasome inhibitor MG132, hes1 mRNA is initially induced but after 3 h it is suppressed because of constant repression of transcription by persistently high protein levels (negative autoregulation). Treatment with cycloheximide leads to sustained increase of hes1 mRNA and blocks its oscillation. A similar effect occurs with overexpression of dnHes1, a dominant-negative form of Hes1 that is known to suppress Hes1 protein activity [5]. These results reveal that both Hes1 protein synthesis and degradation are needed for oscillations in the expression levels of hes1 mRNA. Other experiments showed that the same mechanisms hold for hes1 mRNA expression levels in the presomitic mesoderm in mouse. Interestingly, it is known that in mouse presomitic mesoderm the expression levels of other signalling molecules such as lnfg also oscillate [6]. However, lnfg is not expressed in the cultured cells of Hirata et al., indicating that serum-induced Hes1 oscillation does not depend on lnfg. Nevertheless, this does suggest that Hes1 and lnfg oscillations are controlled by a similar mechanism. Hirata et al. estimate the half-lives of hes1 mRNA and Hes1 protein to be 24.1 ± 1.7 min, 22.3 ± 3.1 min, respectively. Experiments with various protease inhibitors suggest that Hes1 protein is specifically degraded by the ubiquitin–proteasome pathway. They also lower the temperature in their experiments from 37 °C to 30 °C, which lowers both the synthesis and degradation rates, and this alters the period of the oscillation (unpublished data).
To explain the observed behaviour, Hirata et al. modify a mathematical model developed by Elowitz and Leibler [7] for a synthetic gene network constructed in Escherichia coli cells by introducing one gene from λ-phage. By postulating a Hes1 interacting factor as a third molecular species, they obtain a system of three ODEs that give rise to sustained oscillatory behaviour. However, there is no direct experimental evidence for such an interacting factor. Rather, the introduction of a third variable is due to the fact that certain systems of two ODEs cannot generate sustained oscillations.
This observation together with the experimental results of Hirata et al. led to a number of papers in which simple coupled delay differential equations (DDEs) representing M and P were developed to explain the sustained oscillations without recourse to the addition of a third variable (Monk [8], Jensen et al. [9], Lewis [10], and Bernard et al. [11]).
In fact, one of the first people to consider feedback differential equation models for the regulation of enzyme synthesis was Goodwin [12]. U. an der Heiden [13] modified these ideas by including transport delays into Goodwin's model. The oscillatory behaviour of the ensuing DDEs as a function of the size of delays was investigated by an der Heiden.
The ideas underpinning these works are that the processes of transcription, translation, and export are not instantaneous. Monk notes that there is an average delay of 10–20 min between the action of a transcription factor on the promoter region of a gene and the appearance of the corresponding mRNA in the cytosol. Similarly, there is a delay of typically 1–3 min for the translation of a protein from mRNA. Note that the model proposed by Lewis is for zebrafish but it does offer insights into the Hes1 mechanisms via the general nature of the model.
These papers were able to explain some of the observed experimental results quite well, but there are still some aspects that these delay continuous models do not address. These aspects relate to the fact that production numbers of mRNA and protein can be quite low and that intrinsic noise effects due to the uncertainty in knowing when a reaction and what reaction takes place in any given time interval can be very important. Thus the aim of this paper is to incorporate delay effects into the discrete stochastic simulation algorithm (SSA) of Gillespie [14] and to see whether the dynamical behaviour of this delay stochastic simulation algorithm (DSSA) can give greater insights into the nature of hes1 mRNA and Hes1 protein in mouse and, by extension, to other genetic regulatory networks.
### Methods
Let M(t) and P(t) represent the concentrations of hes1 mRNA and Hes1 protein, respectively. Absorbing all the delays (including transcription and translation) into one delay, τ, the model presented in Monk [8] and Jensen et al. [9] is represented by the following DDE:
Here μm and μp are the degradation rates for M and P, αm is the maximal mRNA transcription rate in the absence of protein repression, and αp is the translation rate, while f(P(t)) is a monotonically decreasing Hill function representing the repression of mRNA production by the binding of Hes1 dimers to the promoter region. It takes the form
where h is the Hill coefficient representing the cooperative character of the binding process and P0 is such that f(P0) = 1/2.
Bernard et al. [11] give a modification to model 1 by arguing that the Hirata et al. observations could be better explained by the existence of an additional agent in the Hes1 repression loop. A new variable Q(t) represents the concentration of a repression complex of hyper-phosphorylated Gro/TLE1 with Hes1 protein. This leads to the delay model
where f is as before, αq is the maximal phosphorylation rate, and
is a monotonically increasing Hill function representing the Gro/TLE1 protein activation.
The astute reader might ask what happens if the transcriptional and translational delays, τm and τp, respectively, are not lumped together. This would lead to
But if we let N(t) = M(tτm), which is a phase-lagged mRNA variable, then we can always write model 3 in the form of model 1 with τ = τp + τm. Thus, it is sufficient to consider model 1 (Lewis [10] noted this relationship when he used the delay model 3 to describe alternating bands of gene expression that distinguish anterior and posterior components of somites in zebrafish).
Monk and Jensen et al. investigate the dynamics of model 1, especially in terms of the onset of sustained oscillations, through simulations; while Bernard et al. give a mathematical investigation of the dynamics of models 1 and 2 by a scaling process and performing a linear stability analysis around the steady state values. Lewis shows that, given h = 2, model 3 will have sustained oscillations if
and that the period of the oscillations is approximately
Bernard et al. have given a detailed mathematical bifurcation analysis of the dynamics of the models 1 and 2. In the case of model 1 let the time period of the oscillations be T and define
then sustained oscillations will occur if
where they have used a scaling such that with
It is clear that h, μm, and μp have crucial roles in the onset of sustained oscillations, while αm, αp, and α0 play no significant role. Using the data from the Hirata et al. experiments, we can now see how well model 1 matches the observed results.
#### Model Analysis
We now give a brief analysis of model 1 and see how well it can describe the results of Hirata et al. and what sort of predictions it can make.
##### The relationship between h and τ.
Using some of the values in Table 1, Bernard et al. have shown that sustained oscillations can be obtained with h Є (3,7), τ Є (15,30), and T Є (90,150). For the experimentally observed period T = 120 min, then h ≥ 4.1, τ ≥ 19.7. On the other hand, Hes1 is a dimer, but there are at least three separate binding sites for Hes1 dimers in the regulatory region of the hes1 gene, so that an appropriate value of h is at least 2. However, whether h should be as large as 4.1, as this model predicts, is debatable. We show later in this paper that h need not be that large to get sustained oscillations when discrete models are used.
Jensen et al. show via simulations that for the case h = 2, oscillations are only sustained for τ > 80 and there are no oscillations for τ < 10. For τ Є (10,80), the period of the damped oscillations is approximately 170 min, which is much greater than the observed period of 120 min.
##### Parameter sensitivity.
Simulations and mathematical analysis show that there is no qualitative difference in terms of the onset of sustained oscillations and their period for a wide range of values for αm, αp, and P0. Monk has performed a linear stability analysis and has shown that for a certain range (that is quite wide) of these parameters the oscillatory period is approximately constant.
It is interesting to note that when Hirata et al. lowered the temperature from 37 °C to 30 °C there was a change in the period of oscillation, but as the data is not given we are unable to test these effects in terms of model 1 except to reaffirm that the model is not sensitive to the production terms.
More significantly, in the continuous model, the Hes1 protein concentration rarely falls below the repression threshold P0, which means that Hes1 transcription is always repressed. While this does not contradict experimental data, as there is no mention of this threshold, it does mean that there is not a strong link between the continuous model and the actual mechanism of transcription.
##### Overshoot.
Bernard et al. note that systems with just one nonlinear term often display large overshoot before solutions converge to an attractor. Indeed, that is one of the reasons why they introduce model 2. They attempt to estimate this overshoot, which is essentially due to the lack of repression mechanisms in the first few minutes. Defining the overshoot to be the ratio of the protein concentration at t = τ to its steady state value, then under the assumptions that
the expression for overshoot is
With αm = 1, μm = 0.03, this implies an overshoot of approximately 11.
However, the overshoot only becomes an issue with simulations if the initial conditions are set close to zero. Monk avoids this large overshoot in his simulations by setting the initial conditions close to their steady state values. The real issue here, however, is of course how the oscillations are set off within a cell by serum treatment. The Hirata data does not show any overshoot and so there is a need to relate the initial conditions of any model to the experiment itself. Inevitably, this treatment will change one or more of the model parameters, perhaps continuously. Thus more work needs to be done to understand how serum treatment induces oscillations before we can address this issue more appropriately from a modelling perspective.
##### Peak-to-trough ratio.
Deterministic, continuous models do not match very well the peak-to-trough ratios observed by Hirata et al. Indeed, for model 1 this ratio is higher for mRNA than protein, and this appears to contradict the Hirata data.
##### Experimental validation.
We acknowledge that it is often hard to compare experimental and simulated results for the purposes of model validation. However, we note that Hirata et al. performed some experiments in terms of blocking protein degradation and translation, and while model 1 has not been tested in this regard we do attempt to mimic these experimental results in this present paper through our use of discrete models.
In summary, model 1 predicts possibly high Hill factors, overshoot (if the initial conditions are not chosen very carefully), and no obvious link between the values of P0 and the actual physical basis of transcription. The model is also very sensitive to the degradation parameters but not sensitive to the production parameters. Thus, it can explain some (but not all) of the Hirata data.
However, there are two fundamental issues that the model does not address and which could well explain some of the discrepancies mentioned above. These are that mRNA and proteins can be expressed in quite small numbers and that there is intrinsic noise in terms of the uncertainty of knowing when a certain reaction and what reaction takes place. These points lead us into a discussion on discrete stochastic models for chemical kinetics and the SSA. This in turn will lead to the main idea of this paper, namely the incorporation of delays into discrete, stochastic models and how this approach may address the issues raised here.
#### Discrete, Stochastic Models
Key molecules that are produced at low levels and a chemical systems' intrinsic noise led to Gillespie (1977) [14] introducing the SSA, which describes the evolution of a discrete, stochastic chemical kinetic process in a well-stirred mixture:
Let there be m chemical reactions between N chemical species inside some fixed volume V held at constant temperature. The reactions can be uniquely characterised by the m stoichiometric vectors v1,…,vm, and propensity functions a1(X),a2(X),…,am(X). Here X(t) is the vector of chemical species (X1(t),…,XN(t))T, where Xi(t) is the number of molecules of species i at time t. The propensity functions represent unscaled probabilities of a particular reaction taking place. More formally, aj(X(t))dt represents the probability of reaction j occurring within the time interval (t,t + dt). For elementary kinetics the propensity functions take very simple forms. For instance, for the second order reaction X1 + X2 X3, it is a(X) = kX1(t)X2(t). Similar formulas apply for unimolecular and dimer reactions. The evolution of X through time can be considered to be a discrete nonlinear Markov process that is described by the SSA.
The underlying idea behind the SSA is that at each time point t a step size θ is determined from an exponential waiting time distribution such that at most one reaction can occur in the time interval (t,t + θ). If the most likely reaction, as determined from the relative sizes of the propensity functions, is reaction j, say, then the state vector is updated as
See Algorithm 1 for a pseudo-code description of the SSA.
The SSA has been used successfully in many settings (e.g., Arkin et al. [16] for the study of λ-phage). But its limitation is that it can be very computationally expensive, as large numbers of simulations are needed to calculate moments of X(t) accurately and because the time step can become very small. For this reason a number of approaches have been recently developed to improve the performance of SSA. These include the Poisson leap method (Gillespie [17]) and the Binomial leap method (Tian and Burrage [18]) in which larger time steps are allowed so that all reactions can fire in that step with a frequency sampled from a Poisson or Binomial distribution, respectively.
Other approaches for improving the performance of SSA are based on the chemical master equation (CME) that describes the evolution of the probability density function p(X,t) such that
It is possible to cast this problem into the form dp/dt = Ap where A is the state space matrix, which can be enormous. To make this problem more computationally tractable, quasi–steady state assumptions can be used (where possible) to either reduce the size of the problem or to partition the problem into different regimes with different methods being applied (Haseltine and Rawlings [19], Goutsias [20], Burrage et al. [21]).
We note that an approximation to the mean behaviour μ(t) = E[X(t)] can be derived from Equation 7 to give
which is the standard chemical kinetics rate order equation for describing concentrations. This can be seen by multiplying both sides by X(t) and summing over all possible configurations of the state space (see [22]).
Given this overview of SSA, our intention is now to introduce delays into SSA and to investigate the dynamics of model 1 in this setting. Unlike the SSA, there is not necessarily a unique implementation of delay SSA (DSSA), and issues pertaining to this are discussed in more detail in Text S1.
Briefly, DSSA implementations can differ in the way they handle (1) the waiting time for delayed reactions, (2) the time steps in the presence of delayed reaction updates, and (3) delayed consuming reactions. The DSSA version we used to produce the results presented in the following section works as follows: initially we specify which nonconsuming reactions are delayed and the delay size (constant or variable) associated with each reaction. Delayed consuming reactions are not allowed. Simulations proceed by drawing reactions and their waiting times (for delayed and nondelayed reactions). If a nondelayed reaction is selected, then the state is updated in the standard way (SSA), but if it is a delayed reaction that is selected then it is not updated until the appropriate time point would be passed by another simulation step. In this case, the last drawn reaction is ignored and instead the state is updated according to the delayed reaction. Simulation continues at the corresponding time point. Algorithm 2 shows a pseudo-code description of the DSSA implementation.
In general, delays in time evolutions are difficult to handle because of the non-Markovian character they introduce into the dynamical process. In this context we note that our DSSA implementation ignores the elapsed time between the last triggered reaction and the update of the next scheduled delayed reaction. It is unclear whether this affects the distribution of waiting times until the next reaction happens. It also ignores the selected reaction that should be updated beyond the current update point by preferentially updating the delayed reaction. However, it is an open question whether we should select for the delayed reaction and ignore the other. For further discussions we refer the reader to Text S1.
Furthermore, we note that as soon as we introduce delays into SSA then the evolution of X(t) is no longer described by a Markovian process and the nature of the CME in this case needs further consideration. We have made additional material available in Text S2 in which we derive from first principles a CME for the DSSA. It generalises Equation 7 in a very natural manner. Having constructed the CME for the delay case, we can then multiply both sides by all possible configurations of the state space and this will lead to a DDE for the mean (see Text S2 for further details).
Model 1 can be presented in DSSA form with four reactions defined by
with the delay occurring in the first reaction.
We note that the time step we use for DSSA is self-selecting based on the assumption of exponential waiting times, as is the case for SSA. The stiffer the kinetics system becomes (due to large rate constants and/or large numbers of molecules), the smaller the time step. Thus, the algorithm intrinsically controls the stability of the evolution. However, in the case of the continuous DDE representation, an important issue is stepsize selection for any numerical method to avoid instabilities in the computed solutions.
### Results
#### Parameter Exploration and Model Comparison
In this section we present a selection of DDE solutions and DSSA trajectories displaying the dynamical properties of model 1. As for the DSSA, what we present are single simulations of just one particular strong solution based on a particular path generated by the random variables. Nevertheless, these individual solutions are very representative of the dynamics of the processes being modelled. In some cases, we perform a number of independent simulations to collect information about mean behaviour.
All DDE plots were generated using the dde23 function in MatLab. The initial conditions are set to (M(0),P(0)) = (3,100). In Figures 13 we have scaled the protein concentrations by μp so that both mRNA and protein numbers fit conveniently on the same figure. Figures 1 and 2 show the dynamics of the continuous DDE model for protein concentration with P0 = 10, h = (4.6,4.1,3.6), τ = 19.7 and P0 = 10, h = 4.1, τ = (20.7,19.7,18.7), respectively.
In Figure 1 the delay is kept fixed and the Hill factor h varies around the bifurcation point 4.1. We clearly see that for h = 3.6 the oscillations damp quickly, while for h = 4.6 the oscillations are sustained. In Figure 2 the Hill factor is kept fixed and instead the delay varies around the bifurcation point 19.7. We observe a similar behaviour as in Figure 1, namely that for τ = 18.7 the oscillations damp quickly, while for τ = 20.7 the oscillations are sustained. Interestingly, however, the size of P0 can affect these dynamics. For values of τ = 18.7 up to about 10 there is no essential difference when the sustained oscillations arise. In Figure 3 both the concentrations of mRNA and protein are plotted. We see that when P0 is increased to 100 oscillations damp for values of h greater than the critical value of 4.1, namely for (h,τ) = (4.6,19.7). In this case the concentrations of P are always greater than P0.
We now consider the dynamics of the DSSA. If not stated otherwise, the initial molecular numbers of mRNA and protein are M(0) = 3 and P(0) = 100, respectively. We also make a comment about the scaling for mRNA and protein numbers that we use in the rest of this paper. In the Hirata et al. paper, it is clear that the data has been scaled but it is not clear what the scaling is. To be able to compare our results with those of Hirata et al. we perform one simulation with the values (P0,h,τ) = (100,4.1,19.7). We then use a scaling such that for this simulation the maximum amplitude of the mRNA is 4 and the protein is 7. This is consistent with the Hirata et al. data in their Figure 1. We then use this fixed scaling for all other DSSA simulations in this paper. The scaling factors for mRNA and protein are 0.3 and 0.03, respectively.
In Figure 4 we keep P0 and h fixed and vary τ with P0 = 100, h = 4.1, τ = (10,15,20,25). We note that the oscillations are sustained and regular for τ = 15, 20, and 25, while there is some oscillatory behaviour even with τ = 5, but the dynamics are very irregular. Moreover, for all values of τ, protein numbers go mostly below P0, albeit for only small periods of time. However, for small delay (τ = 5) this happens far less often. Coincidentally or not, in this case the oscillations are very irregular. We also note that for larger values of τ (τ = 25), the amplitudes of protein concentration are generally larger than the values presented by Hirata et al.
In Figure 5 we perform a similar set of experiments but now we keep the delay fixed and vary the values of h with P0 = 50, h = (2.1,3.1,4.1,5.1), t = 19.7. Again, sustained and more or less regular oscillations are noticed for h = 4.1 and 3.1, but more irregular behaviour with h = 2.1 and 5.1.
Simulations in Figures 4 and 5 are performed with (P0 = 100) and (P0 = 50), respectively. Since the values of P0 might well be a significant factor in determining the dynamics of the system, we simulate also with varying P0 (10,50,100,1000) choosing parameters (h,τ) = (4.1,15) (Figure 6). We observe sustained, regular oscillations for P0 = (10,50,100). However, if P0 is very large (P0 = 1,000), the oscillations are very irregular and the numbers of protein are much larger than in the other cases. On the other hand, if P0 is low (P0 = 10), the amplitudes of mRNA and protein are not as large as in the other cases.
The data shown by Hirata et al. represents the average of the samples from a number of cells. We computed the time-dependent arithmetic mean over 1,000 independent simulations using the DSSA with P0 = 100, h = 4.1, and τ = 19.7 (Figure 7). By comparing the result with the solution of the corresponding DDE (that is derived in Text S2), the following two aspects become evident. First, in spite of the differences between individual simulations due to the inherent stochasticity, the arithmetic mean over 1,000 independent simulations with either constant delay or variable delay is very close to the corresponding solutions of the DDE. Second, there appears to be no qualitative difference between the arithmetic mean of DSSA with constant delay compared with a uniformly distributed delay in an interval of width 6 centred around the bifurcation point of 19.7. The reason why the oscillation dies away is not that the system does not oscillate any more. Rather, the oscillations show a progressive, increasingly randomly distributed phase shift cancelling each other.
In addition, by performing numerous simulations for values (P0,h,τ) = (100,4.1,19.7) with initial conditions (M(0), P(0)) = (1,1), we searched for the occurrence of overshoot. The resulting trajectories did not show significant overshoot in the system, indicating that overshoot is not an issue for model 1 in the DSSA regime.
#### Experimental Comparison
In this subsection we compare our simulations with specific experiments performed by Hirata et al. One of the most important aspects of the Hirata data is the regularity of the oscillatory period, which is 2 h. We therefore performed a spectrum analysis (more than 300 independent simulations) that takes a signal in the time domain and transforms it into its component frequency representation (frequency analysis has been done using a software package provided by Barrio et al., see Acknowledgements).
The oscillation frequencies can be determined for different values of the parameters: P0, h, τ, and the degradation rates. For each set of values the mean frequency of all the simulations is calculated. Figure 8 illustrates results for two cases where we seem to get regular dynamics in the oscillations, namely P0 = 100, h = (3,4). In both of these cases the frequency is plotted against the delay in the range [6,30]. It shows that the frequency decreases more or less linearly with the delay. This has a very important meaning for the oscillatory nature of the Hes1 regulatory system. It also confirms that the single snapshots of trajectories that we present in this paper do indeed capture the significant dynamics of the Hes1 model. We can conclude that for an oscillatory period of 2 h (frequency = 0.5), then if h = 3 an appropriate value for τ is about 10 while if h = 4 an appropriate value for τ is about 15. Of course, these relationships between h and τ are not completely precise, as simulations with h = 4 and τ = 19.7 (Figure 5) still show very regular behaviour with a period close to 2 h. Nevertheless, taking (h,τ) = (3,10) and (4,15) as being very appropriate values, we then performed a single simulation over 12 h (Figure 9, left plots). The first 2 h of each plot are shown separately (Figure 9, right plots). The simulations obtained compare very favourably with the Hirata et al. data in terms of the regularity of the period (2 h), the amplitude of the profiles, and the time lag of approximately 15–18 min (seen best from the simulations over 2 h).
By perturbing the reaction rate constants, we can attempt to get an idea of the system's sensitivity (we note that Hirata et al. observed alterations in the oscillatory period as the temperature was lowered). This approach does not replace a thorough analysis. However, it still leads to insights about the different dynamics and the sensitivity of the model. Figure 10 shows DSSA trajectories for simulations with parameters P0 = 100, h = 4.1, τ = 15, and one of the four reaction rate constants αm, αp, μm, and μp varied in each figure while the others are kept fixed with values as shown in Table 1 m = 0.1,0.5,1,2, αp = 0.1,0.5,1,2, μm = 0.01,0.029,0.05,0.1, and μp = 0.01,0.031,0.05,0.1). Perturbing αm or αp in these ranges leads to similar dynamical behaviour: oscillations become more regular, with larger amplitudes and shorter periods, the larger the production rate constant. Oscillations are barely visible for αm = αp = 0.1. For αm = αp = 0.5, we obtain regular oscillations (although the trajectory for αp = 0.5 in Figure 9 does not look very regular). For αm = αp = 2, oscillations are very regular with high peaks and short period. By varying the value of μm, we observe very irregular dynamics for low degradation rates (0.01) and regular oscillations with larger rates (0.029, 0.05). However, for even larger degradation rates (0.1), oscillations become irregular again. The average scaled population level decreases with μm increasing. For μp, we observe the opposite behaviour, namely that oscillations occur for all four values of μp, but that the period increases and peaks decrease in size as μp becomes smaller. Small perturbations around (αm = 1, αp = 1, μm = 0.029, μp = 0.02) do not seem to have any visible effect on the oscillatory dynamics. Because data is not available on how a reduction in temperature from 37 °C to 30 °C affects the period of oscillation, we are unable to compare our simulations with experimental results, but our simulation results are not unreasonable.
Finally, we compare the results of some actual experiments by Hirata et al. with the corresponding modified DSSA simulations. The experiment in which Hes1 protein degradation is blocked by application of proteasome inhibitor MG132 is mimicked by setting the fourth stoichiometric vector to v4 = (0,0)T, thus making the protein degradation reaction ineffective. This happens at a predefined time texp after oscillation is initiated. Figure 10 illustrates the effect on the model for parameters h = 4.1, τ = 15, P0 = 100, and texp = 60 min when protein degradation is blocked. Evidently, the dynamics of the modified model matches with those from the experiment quite well (Figure 3A in [3]). For the experiment in which translation is inhibited by cycloheximide treatment, we neutralize translation by setting the third stoichiometric vector at v3 = (0,0)T. As with the first experiment, the modification is set off at a predefined time after simulation starts. Figure 10 shows a typical trajectory resulting from a DSSA run with parameters h = 4.1, τ = 15, P0 = 100, and texp = 30 min when translation is blocked. The model performs in much the same way as described in the experiment (Figure 3C in [3]).
### Discussion
When we compare the dynamics of DSSA with the continuous delay case, we can make a number of important conclusions. Perhaps the most significant is that there are sustained oscillations for values of h < 4.1 and τ < 19.7, unlike the continuous case. Indeed, Figure 5 shows sustained regular oscillations with values of h about 3. This is an important point. In the continuous setting, there is a large set of Hill functions that represent a wide variety of bindings of molecules to operator regions. Modellers infer information about the nature of the cooperativity at the binding sites from the dynamics of simulations of models with different types of Hill functions. What our simulations show is that these values may be overestimated from continuous models and that discrete delay models may well give more realistic values for cooperativity. Furthermore, it is clear that with our model if the value of h is too small, certainly h < 2, then the oscillations are very noisy and very irregular. Indeed, Figure 8 strongly suggests that for an oscillatory period of two hours to occur, h cannot be much less than 3.
The same remarks apply for estimates of the values of τ that lead to sustained regular oscillations. Thus from Figure 9 we observe reasonably well-defined sustained regular oscillations for values of τ = 15 with h = 4, and τ = 10 with h = 3. Values for τ lower than τ = 10 result in noisy and irregular delay. On the other hand, values of τ bigger than 20 suggest an h larger than 5, and this seems to be too high a value for the Hill parameter in terms of the number of operator binding sites. For values of (P0,h,τ) = (100,3,10) and (100,4,15), the simulations in Figure 9 compare very favourably with the data given by Hirata et al. (their Figure 1). The oscillations are regular with a period of 2 h, the amplitudes have more or less the same values in the simulations, and the experiments and the time lag of about 15–18 min, seen in the rightmost plots of Figure 9, is very close to that observed by Hirata et al.
Another feature of the dynamics of DSSA that we would like to emphasise is the role of P0. For continuous models, the role of P0 appears not to be too significant as long as it is not too large. But from Figure 6 we see that P0 plays an important role. Apparently, when the numbers of P are below P0, there is expression. This expression only occurs for very small time windows but seems to be crucial in driving the oscillations. This behaviour does not occur for the continuous deterministic models. If the value of P0 is increased too much to P0 = 1,000, say, then there are no oscillations and P never goes below P0. On the other hand, if P0 is too low (P0 = 10), then the amplitudes of the mRNA and protein appear to be too small. This provides a prediction that should be able to be tested experimentally.
Furthermore, simulations in Figures 46 and 9 suggest that the peak-to-trough ratios of mRNA and protein are in closer agreement to the Hirata et al. data than those obtained from the deterministic model. The Hirata data suggests ratios of between 3 and 4 for mRNA and between 3 and 8 for protein, although we note that these only approximate values as there are significant error bars for the protein concentrations. On the other hand, a rough estimate from the discrete simulations gives a peak-to-trough ratio for protein of between 3 and 4 with a larger ratio for mRNA due to the fact that the numbers of mRNA can become quite small in some cases.
We have also shown from the mathematical analysis in the supporting information (Text S2) that the mean behaviour of the DSSA is well-described by the corresponding DDE when there are large numbers of molecules. This naturally generalizes the SSA/DDE case and provides a comprehensive framework for studying noise in biology.
Furthermore, our simulations suggest that overshoot is not an issue for model 1 in a discrete delay setting. One of the reasons that Bernard et al. [11] introduced model 2 in the continuous delay setting was because overshoot is not so pronounced when there are more nonlinearities in the model. However, in the discrete setting, these differences appear not to be significant, and so choosing a model based on overshoot issues appears not to be important.
The sensitivity analysis in Figure 10 shows that the DSSA is more sensitive to the degradation parameters than the production parameters in terms of their effect on the period of oscillation. However, there is more sensitivity of the discrete model to the production parameters than for the continuous model 1. Moreover, the DSSA simulations in Figure 11 mimic very well the experiments when either protein degradation or translation is blocked, both in terms of the time of the degradation of mRNA numbers and in actual values of mRNA and protein.
Putting all this information together we see that we get very good comparisons between simulation and experiment if the value of P0 is on the order of 50 to 100, if the value of h is somewhere between just less than 3 to just bigger than 4, and if the delay is somewhere between 10–20 min We can be more precise if more accurate values of the transcription and translation delays are available. If this sum of delays is about 15, then this suggests a Hill factor of about 4, while if the delay is about 10, then a Hill factor of 3 is more appropriate.
#### Conclusions
In this paper we have compared continuous delay models and discrete, stochastic delay models to explain oscillations in numbers of hes1 mRNA and Hes1 protein in mouse. Given that the numbers of mRNA and proteins produced are relatively small, the discrete delay approach may well be more appropriate than the continuous approach. Furthermore, the discrete delay approach seems to give greater insight into the underlying cellular dynamics in terms of the system parameters.
By careful comparisons of our simulations with the Hirata et al. data, we have been able to suggest quite specific ranges for P0, the Hill parameter h, and the delay τ. We have also shown by both mathematical analysis and simulations that the mean behaviour of DSSA is described by a DDE. This naturally generalises the nondelay case and provides a comprehensive and consistent mathematical framework for understanding the role of noise in biology.
### Supporting Information
Text S1. DSSAs
doi:10.1371/journal.pcbi.0020117.se001
(89 KB PDF)
Text S2. DDEs and the Master Equation
doi:10.1371/journal.pcbi.0020117.se002
(45 KB PDF)
#### Accession Numbers
The Entrez Gene (http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?DB=gene) gene ID for the hes1 gene in Mus musculus (Mouse) is 15205. The Swiss Prot (http://www.ebi.ac.uk/swissprot/) corresponding accession number for the Hes1 protein is P35428.
### Acknowledgments
The authors would like to thank Nick Monk (Sheffield) for helpful discussions that substantially improved this paper. The authors would also like to thank Margherita Carletti (Urbino) who worked with the second author on a preliminary version of discrete delay code. Figure 8 has been produced using a software package developed by M. Barrio, C. Llamas, and P. de la Fuente.
### Author Contributions
All authors contributed equally in all aspects of the paper.
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1
PubMed
The aluminium (Al) content of 105 samples, including bakery products made with baking powder, agricultural products and seafoods treated with alum, was investigated. The amounts of Al detected were as follows (limit of quantification: 0.01 mg/g): 0.01-0.37 mg/g in 26 of 57 bakery products, 0.22-0.57 mg/g in 3 of 6 powder mixes, 0.01-0.05 mg/g in all three agricultural products examined, 0.03-0.90 mg/g in 4 of 6 seafood samples, 0.01-0.03 mg/g in 3 of 11 samples of instant noodles, 0.04-0.14 mg/g in 3 of 4 samples of vermicelli, 0.01 mg/g in 1 of 16 soybean products, but none in soybeans. Amounts equivalent to the PTWI of a 16 kg infant were detected in two samples of bakery products, two samples of powder mixes and one sample of salted jellyfish, if each sample was taken once a week. These results suggest that certain foods, depending on the product and the intake, might exceed the PTWI of children, especially infants. PMID:22450671
Ogimoto, Mami; Suzuki, Kumi; Kabashima, Junichiro; Nakazato, Mitsuo; Uematsu, Yoko
2012-01-01
2
PubMed
The primary objective was to determine the aluminium (Al) content of selected foods and food products in the USA which contain Al as an approved food additive. Intake of Al from the labeled serving size of each food product was calculated. The samples were acid or base digested and analysed for Al using electrothermal atomic absorption spectrometry. Quality control (QC) samples, with matrices matching the samples, were generated and used to verify the Al determinations. Food product Al content ranged from <1-27,000 mg kg(-1). Cheese in a serving of frozen pizzas had up to 14 mg of Al, from basic sodium aluminium phosphate; whereas the same amount of cheese in a ready-to-eat restaurant pizza provided 0.03-0.09 mg. Many single serving packets of non-dairy creamer had approximately 50-600 mg Al kg(-1) as sodium aluminosilicate, providing up to 1.5 mg Al per serving. Many single serving packets of salt also had sodium aluminosilicate as an additive, but the Al content was less than in single-serving non-dairy creamer packets. Acidic sodium aluminium phosphate was present in many food products, pancakes and waffles. Baking powder, some pancake/waffle mixes and frozen products, and ready-to-eat pancakes provided the most Al of the foods tested; up to 180 mg/serving. Many products provide a significant amount of Al compared to the typical intake of 3-12 mg/day reported from dietary Al studies conducted in many countries. PMID:16019791
Saiyed, Salim M; Yokel, Robert A
2005-03-01
3
The significance of this research lies in its potential to create an Additive Manufacturing technology of novel lightweight materials for industrial applications. The work reported here focuses on studies performed with low power fiber laser and investigates the feasibility of introducing high strength aluminium alloys and custom developed Al powder systems to the Selective Laser Melting (SLM) process. Al-Si powder
Konrad Bartkowiak; Sven Ullrich; Thomas Frick; Michael Schmidt
2011-01-01
4
The effects of cerium, zinc and zirconium additions and subsequent heat treatment on wear of the eutectic aluminium-silicon alloys have been investigated in dry sliding against a steel counterface by using a pin-on-disc machine. Wear surfaces and debris were examined by scanning electron microscopy. Wear characteristics of both binary Al-Si alloys and a commercial LM 13 alloy, were also studied
Mohd Harun; I. A. Talib; A. R. Daud
1996-01-01
5
The hot-pressing behaviour of different silicon carbide powders (average particle sizes ranging from ~ 0.5 to 9 µm) with aluminium oxide additions ranging from 0.01 to 0.15 volume fractions was investigated. Using powders with an average particle size -2 for volume fractions of AI2O3\\u000a$$\\\\bar >$$\\u000a 0.02. A liquid phase forms at high temperatures which dissolves the silicon carbide
F. F. Lange
1975-01-01
6
The effects of several factors on the settling rate of aluminiumhydroxide were investigated during chemical coagulation using aluminium salts. Experimental variables were pH, aluminium (III) concentration and the order of addition of reagents. Experiments were carried out at pH 5–8 and rapid settling was achieved when aluminium (III) solutions were added to Na2CO3 solutions near neutral pH, close to the
Birsen Demirata; Re?at Apak; Gülcin Gümüs; Hüseyin Afsar
2002-01-01
7
The present work pertains to impression creep of age hardenable AA2219 Aluminium-copper (6.3%) alloy correlated with microstructure\\u000a with the addition of scandium, magnesium and zirconium. These additions were systematically varied by preparing alloys of\\u000a different composition using gas tungsten arc melting. Impression creep technique was used to study the high temperature stability\\u000a of the alloys. Modified compositions of the alloy
K. Srinivasa Rao; P. Naga Raju; G. M. Reddy; K. Prasad Rao
2010-01-01
8
PubMed
The applicability of amorphous aluminium oxide as a fluoride retention additive to flue gas desulphurisation (FGD) gypsum was studied as a way of stabilizing this by-product for its disposal in landfills. Using a batch method the sorption behaviour of amorphous aluminium oxide was evaluated at the pH (about 6.5) and background electrolyte conditions (high chloride and sulphate concentrations) found in FGD gypsum leachates. It was found that fluoride sorption on amorphous aluminium oxide was a very fast process with equilibrium attained within the first half an hour of interaction. The sorption process was well described by the Langmuir model, offering a maximum fluoride sorption capacity of 61.7 mg g(-1). Fluoride sorption was unaffected by chloride co-existing ions, while slightly decreased (about 20%) by competing sulphate ions. The use of amorphous aluminium oxide in the stabilization of FGD gypsum proved to greatly decreased its fluoride leachable content (in the range 5-75% for amorphous aluminium oxide doses of 0.1-2%, as determined by the European standard EN 12457-4 [EN-12457-4 Characterization of waste-leaching-compliance test for leaching of granular waste materials and sludges-Part 4: one stage batch test at a liquid to solid ratio of 10 l/kg for materials with particle size below 10mm (without or with size reduction)]), assuring the characterization of this by-product as a waste acceptable at landfills of non-hazardous wastes according to the Council Decision 2003/33/EC [Council Decision 2003/33/EC of 19 December 2002. Establishing criteria and procedures for the acceptance of waste at landfills pursuant to Article 16 of and Annex II to Directive 1999/31/EC] on landfill of wastes. Furthermore, as derived from column leaching studies, the proposed stabilization system proved to be highly effective in simulated conditions of disposal, displaying a fluoride leaching reduction value about 81% for an amorphous aluminium oxide added amount of 2%. PMID:17509651
Alvarez-Ayuso, E; Querol, X
2007-09-01
9
Reducing Portland cement content in cementitious binders offers a means to address the adverse environmental impacts of Portland cement manufacture. This paper investigates the impacts on hydration product chemistry of partially replacing Portland cement with alkali-activated aluminosilicates. Here, short-term effects of soluble alkali and aluminium, likely to be available in an alkali-activated system, on the structure of synthetic C-S-H gels
I. García Lodeiro; A. Fernández-Jimenez; A. Palomo; D. E Macphee
2010-01-01
10
PubMed
Lehoux, Alizée P; Lockwood, Cindy L; Mayes, William M; Stewart, Douglas I; Mortimer, Robert J G; Gruiz, Katalin; Burke, Ian T
2013-10-01
11
PubMed
An on-line flow injection sample to standard addition method was developed for the determination of aluminium in biological tissues by inductively coupled plasma mass spectrometry. The sample concentration is calculated from two transient signals obtained from the injection of the blank and the sample, using a standard solution as carrier. A flow injection manifold for the on-line injection of both solutions was used and a three-step flow injection programme allowing the two transient signals to be obtained in the same measurement process was applied. A microwave nitric acid digestion procedure in closed vessels was used for sample dissolution, and scandium was added as an internal standard to control plasma fluctuations and to correct for ion signal instability. Tissue samples from healthy rat brain, kidney, liver, lung and spleen were analysed to find their aluminium concentrations. Complete recoveries from rat liver tissue spiked with aluminium concentrations in the 0.5-10.0 micrograms g-1 level were achieved. The detection limit (3 sigma) referred to the solid sample was of 10 ng g-1 and the precision (RSD) was better than 1%. The accuracy of the proposed method was tested by determining the aluminium contents in two NIST Standard Reference Materials: SRM 1577b Bovine Liver and SRM 8414 Bovine Muscle Powder). PMID:8799287
Coedo, A G; Dorado, M T; Ruiz, J; Escudero, M; Rubio, J C
1996-04-01
12
The magnesium levels in 20?m thick 4.5% Al\\/Zn galvanising coatings applied to 0.7mm gauge steel under continuous galvanising conditions were altered from 0.0% to 0.05%. The additions result in an increase in the zinc dendrites (volume fraction from 6% to 22% and number from 150 to 325mm?2) since magnesium depresses the eutectic temperature increasing the freezing range. The microstructural modification
Jonathan Elvins; John A. Spittle; James H. Sullivan; David A. Worsley
2008-01-01
13
PubMed
The use of modern analytical methods has demonstrated that aluminium salts can be absorbed from the gut and concentrated in various human tissues, including bone, the parathyroids and brain. The neurotoxicity of aluminium has been extensively characterized in rabbits and cats, and high concentrations of aluminium have been detected in the brain tissue of patients with Alzheimer's disease. Various reports have suggested that high aluminium intakes may be harmful to some patients with bone disease or renal impairment. Fatal aluminium-induced neuropathies have been reported in patients on renal dialysis. Since there are no demonstrable consequences of aluminium deprivation, the prophylactic reduction of aluminium intake by many patients would appear prudent. In this report, the major sources of aluminium in foods and non-prescription drugs are summarized and alternative products are described. The most common foods that contain substantial amounts of aluminium-containing additives include some processed cheeses, baking powders, cake mixes, frozen doughs, pancake mixes, self-raising flours and pickled vegetables. The aluminium-containing non-prescription drugs include some antacids, buffered aspirins, antidiarrhoeal products, douches and haemorrhoidal medications. The advisability of recommending a low aluminium diet for geriatric patients is discussed in detail. PMID:6337934
Lione, A
1983-02-01
14
NSDL National Science Digital Library
Addition can be so much fun, especially with these games. Tweet, Tweet, Addition is not a game for the birds. Keep the game Math Popper away from all sharp pins! Watch out! Addition facts are attacking the spaceships in Addition Attack In Cannon Math you can shoot people out of a cannon! It is great fun!! Dr. Brain s Robot needs help with his math facts. Can you help ...
Games, Aidan'S M.
2010-12-20
15
NASA Technical Reports Server (NTRS)
The chemistry of lubricant additives is discussed to show what the additives are chemically and what functions they perform in the lubrication of various kinds of equipment. Current theories regarding the mode of action of lubricant additives are presented. The additive groups discussed include the following: (1) detergents and dispersants, (2) corrosion inhibitors, (3) antioxidants, (4) viscosity index improvers, (5) pour point depressants, and (6) antifouling agents.
Smalheer, C. V.
1973-01-01
16
NSDL National Science Digital Library
We will practice adding! Practice with Fribbit! Fribbit Addition Help the pirate add his marbles. Pirate Marble Addition Help the whale get to freedom! Save the Whale Catch the correct sum! Fish Addish Balance the sides of the equal sign. Balance Equations ...
Hoffmann, Mrs.
2011-08-09
17
PubMed
It is of burgeoning importance that the human body burden of aluminium is understood and is measured. There are surprisingly few data to describe human excretion of systemic aluminium and almost no reliable data which relate to aluminium in sweat. We have measured the aluminium content of sweat in 20 healthy volunteers following mild exercise. The concentration of aluminium ranged from 329 to 5329?g/L. These data equate to a daily excretion of between 234 and 7192?g aluminium and they strongly suggest that perspiration is the major route of excretion of systemic aluminium in humans. PMID:24239230
Minshall, Clare; Nadal, Jodie; Exley, Christopher
2014-01-01
18
PubMed Central
Using atomic absorption spectrometry the aluminium concentrations in blood and urine and in two iliac bone biopsies obtained from welders with long term exposure to fumes containing aluminium were measured. The urinary excretion of two workers who had welded for 20 and 21 years varied between 107 and 351 micrograms Al/l, more than 10 times the concentration found in persons without occupational exposure. Urinary aluminium excretion remained high many years after stopping exposure. Blood and bone aluminium concentrations (4-53 micrograms Al/l and 18-29 micrograms Al/g respectively) were also raised but not to the same extent as urine excretion. It is concluded that long term exposure to aluminium by inhalation gives rise to accumulation of aluminium in the body and skeleton of health persons, and that the elimination of retained aluminium is very slow, in the order of several years.
Elinder, C G; Ahrengart, L; Lidums, V; Pettersson, E; Sjogren, B
1991-01-01
19
Aluminium is commonest metal within the earth's crust. Nevertheless, most natural waters, biota and the air contain very low concentrations. It follows that, except in the case of exposure to pharmaceutical preparations, mans exposure to aluminium - mostly in the diet - is very low. In addition, only a very small fraction of ingested aluminium is bioavailable (0.0001 - 0.01)
N. D. PRIEST
20
The present work was conducted on a wide series of Al-Si, Al-Li, Al-Li-Si and Al-Li-Fe alloys produced by centrifugal atomization. Solidification in Al-Si powders occurs by heterogeneous nucleation at the interface with the atmosphere, followed by propagation of the solidification front towards the particle centre. Addition of 3% Li to the Al-Si alloys shifts the ternary eutectic point to less
F. H. Samuel; G. Champier; P. Todeschini; J. H. Torres
1992-01-01
21
This study examines the development of aluminium–aluminium compounds during high pressure die casting. An aluminium insert was mounted into a die casting mould and embedded into an aluminium casting alloy. Prior to that, the natural oxide layer of the insert was removed and replaced by zinc coatings with different thicknesses. During the casting process the zinc layer dissolves and a
M. Rübner; M. Günzl; C. Körner; R. F. Singer
2011-01-01
22
An experimental programme consisting of 144 tests was carried out to study the behaviour of triggered, square 80×80mm2 AA6060 aluminium extrusions filled with aluminium foam under both quasi-static and dynamic axial loading conditions. The main parameters in addition to the loading condition were the foam density, the extrusion wall strength and the extrusion wall thickness. Previously proposed design formulas applied
A. G. Hanssen; M. Langseth; O. S. Hopperstad
2000-01-01
23
Strontium-modified aluminium alloys containing 14 to 15 wt% silicon were cast with fully eutectic structures by using heated moulds and high-purity materials. In alloys containing the additional elements magnesium, copper or nickel, a distinct eutectic colony structure was evident outlined by intermetallic compounds. At the edges of the castings the eutectic colony structures and the aluminium grains (revealed by anodizing)
T. B. Abbott; B. A. Parker
1990-01-01
24
NSDL National Science Digital Library
A concise fact sheet on aluminum use in transport by the European Aluminium Association (EAA). Topics include applications (auto, air, marine, and rail), benefits, economic role, energy consumption, history, and sustainability.
European Aluminium Association (EAA)
25
We have produced interpenetrating graphite\\/aluminium composites by gas pressure infiltration of aluminium alloys with varying silicon content into porous graphite preforms. Infiltration experiments at 750°C have shown that a silicon content of up to 18wt.% can reduce the formation of aluminium carbide but cannot completely deter it. Optical and scanning electron microscopy revealed numerous lath-like interfacial aluminium carbide crystals in
T. Etter; P. Schulz; M. Weber; J. Metz; M. Wimmler; J. F. Löffler; P. J. Uggowitzer
2007-01-01
26
PubMed
The human breast is exposed to aluminium from many sources including diet and personal care products, but dermal application of aluminium-based antiperspirant salts provides a local long-term source of exposure. Recent measurements have shown that aluminium is present in both tissue and fat of the human breast but at levels which vary both between breasts and between tissue samples from the same breast. We have recently found increased levels of aluminium in noninvasively collected nipple aspirate fluids taken from breast cancer patients (mean 268 ± 28 ?g/l) compared with control healthy subjects (mean 131 ± 10 ?g/l) providing evidence of raised aluminium levels in the breast microenvironment when cancer is present. The measurement of higher levels of aluminium in type I human breast cyst fluids (median 150 ?g/l) compared with human serum (median 6 ?g/l) or human milk (median 25 ?g/l) warrants further investigation into any possible role of aluminium in development of this benign breast disease. Emerging evidence for aluminium in several breast structures now requires biomarkers of aluminium action in order to ascertain whether the presence of aluminium has any biological impact. To this end, we report raised levels of proteins that modulate iron homeostasis (ferritin, transferrin) in parallel with raised aluminium in nipple aspirate fluids in vivo, and we report overexpression of mRNA for several S100 calcium binding proteins following long-term exposure of MCF-7 human breast cancer cells in vitro to aluminium chlorhydrate. PMID:22099158
Darbre, P D; Pugazhendhi, D; Mannello, F
2011-11-01
27
PubMed
A total of 256 individual food samples were collected in Hong Kong for aluminium testing. Most of food samples were analysed in ready-to-eat form. High aluminium levels were found in steamed bread/bun/cake (mean: 100-320 mg kg(-1)), some bakery products such as muffin, pancake/waffle, coconut tart and cake (mean: 250, 160, 120 and 91 mg kg(-1), respectively), and jellyfish (ready-to-eat form) (mean: 1200 mg kg(-1)). The results demonstrated that aluminium-containing food additives have been widely used in these food products. The average dietary exposure to aluminium for a 60 kg adult was estimated to be 0.60 mg kg(-1) bw week(-1), which amounted to 60% of the new PTWI established by JECFA. The main dietary source was "steamed bread/bun/cake", which contributed to 60% of the total exposure, followed by "bakery products" and "jellyfish", which contributed to 23 and 10% of the total exposure, respectively. However, the estimation did not include the intake of aluminium from natural food sources, food contact materials or other sources (e.g. drinking water). Although the results indicated that aluminium it is unlikely to cause adverse health effect for the general population, the risk to some populations who regularly consume foods with aluminium-containing food additives cannot be ruled out. PMID:20234962
Wong, Waiky W K; Chung, Stephen W C; Kwong, K P; Yin Ho, Yuk; Xiao, Ying
2010-04-01
28
PubMed Central
We describe a lethal poisoning in a healthy woman caused by deliberate ingestion of aluminium phosphide (AlP), a pesticide used to kill rodents and insects. Toxicity of AlP and review of cases reported to the National Poisons Information Service (London) 1997–2003 are discussed.
Bogle, R G; Theron, P; Brooks, P; Dargan, P I; Redhead, J
2006-01-01
29
New porous ceramics with a framework structure of aluminium borate (9Al2O3 · 2B2O3) whiskers, in which the whiskers are distributed uniformly and randomly, can be synthesized in situ by firing of a green powder compact of a mixture of aluminium hydroxide, boric acid and an additive of nickel oxide above 1100°C. During firing, the whiskers of aluminium borate grow in
J. X. Li; T. Narita; J. Ogawa; M. Wadasako
1998-01-01
30
In this paper fatigue damage initiation is considered as both microcrack nucleation from microheterogeneities within fatigued material and microcrack growth over distances corresponding to the size of the microstructure. A brief review of the literature shows that, depending on fatigue conditions and microstructure three different types of nucleation sites exist in aluminium and aluminium alloys : slip bands, grain boundaries
R. FOUGERES
1993-01-01
31
An experimental investigation was carried out to study the behaviour of square aluminium extrusions filled with aluminium foam under quasi-static loading conditions. Based on the experimental work, simple relations between dimensionless numbers governing the influence of the foam on the characteristics of the crush problem were identified. Furthermore, a simplified set of equations applicable for design of foam-filled components was
A. G. Hanssen; M. Langseth; O. S. Hopperstad
1999-01-01
32
The effect of very small additions of indium on the decay of quenched-in vacancies at 0°c in aluminium is studied by high precision electrical resistivity measurements. The extent of the observed resistivity decrease in the alloy on ageing with respect to that in pure aluminium is used as a measure of the strength of binding between indium atoms and vacancies.
W. J. Plumbridge
1969-01-01
33
A series of mesoporous molecular sieves (AlMCM-41) were synthesized with varying silicon-to-aluminium ratios and using three different aluminium sources, viz., sodium aluminate, aluminium isopropoxide and aluminium sulphate. The samples were characterized systematically using XRD, TG-DTA, BET surface area, and ICP-AES. In addition, the extent of framework substitution as well as the nature of acid sites was deduced employing 27Al MAS-NMR,
A Sakthivel; S. E Dapurkar; N. M Gupta; S. K Kulshreshtha; P Selvam
2003-01-01
34
NASA Astrophysics Data System (ADS)
Aluminium is often used as a substitute material for calcifications in phantom measurements in mammography. Additionally, calcium oxalate, hydroxyapatite and aluminium are used in simulation studies. This assumes that these materials have similar attenuation properties to calcification, and this assumption is examined in this work. Sliced mastectomy samples containing calcification were imaged at ×5 magnification using a digital specimen cabinet. Images of the individual calcifications were extracted, and the diameter and contrast of each calculated. The thicknesses of aluminium required to achieve the same contrast as each calcification when imaged under the same conditions were calculated using measurements of the contrast of aluminium foils. As hydroxyapatite and calcium oxalate are also used to simulate calcifications, the equivalent aluminium thicknesses of these materials were also calculated using tabulated attenuation coefficients. On average the equivalent aluminium thickness was 0.85 times the calcification diameter. For calcium oxalate and hydroxyapatite, the equivalent aluminium thicknesses were 1.01 and 2.19 times the thickness of these materials respectively. Aluminium and calcium oxalate are suitable substitute materials for calcifications. Hydroxyapatite is much more attenuating than the calcifications and aluminium. Using solid hydroxyapatite as a substitute for calcification of the same size would lead to excessive contrast in the mammographic image.
Warren, L. M.; Mackenzie, A.; Dance, D. R.; Young, K. C.
2013-04-01
35
PubMed
Aluminium is often used as a substitute material for calcifications in phantom measurements in mammography. Additionally, calcium oxalate, hydroxyapatite and aluminium are used in simulation studies. This assumes that these materials have similar attenuation properties to calcification, and this assumption is examined in this work. Sliced mastectomy samples containing calcification were imaged at ×5 magnification using a digital specimen cabinet. Images of the individual calcifications were extracted, and the diameter and contrast of each calculated. The thicknesses of aluminium required to achieve the same contrast as each calcification when imaged under the same conditions were calculated using measurements of the contrast of aluminium foils. As hydroxyapatite and calcium oxalate are also used to simulate calcifications, the equivalent aluminium thicknesses of these materials were also calculated using tabulated attenuation coefficients. On average the equivalent aluminium thickness was 0.85 times the calcification diameter. For calcium oxalate and hydroxyapatite, the equivalent aluminium thicknesses were 1.01 and 2.19 times the thickness of these materials respectively. Aluminium and calcium oxalate are suitable substitute materials for calcifications. Hydroxyapatite is much more attenuating than the calcifications and aluminium. Using solid hydroxyapatite as a substitute for calcification of the same size would lead to excessive contrast in the mammographic image. PMID:23470559
Warren, L M; Mackenzie, A; Dance, D R; Young, K C
2013-04-01
36
PubMed
The cation distribution in a natural magnesium aluminium chromite spinel (cubic, space group Fd-3m), Al(0.41)Cr(1.42)Fe(0.65)Mg(0.4)O(4), was determined by electron-microprobe analysis, Mossbauer spectroscopy and single-crystal X-ray analysis. Several structural models of the octahedral and tetrahedral cation distributions were tested; the most probable is (Mg(0.40(11))(2+),Al(0.28(5))(3+),Fe(0.39(4))(2+))[Al(0.13(5))(3+),Cr(1.42(6))(3+),Fe(0.26(4))(3+),Phi(0.19)]O(4)(2-), where (...) and [...] represent the tetrahedral and octahedral sites, respectively, and Phi represents a vacancy. PMID:15750218
Santos, Joab S; Doriguetto, Antônio C; Fernandes, Nelson G
2005-03-01
37
Here we report temperature dependent self-diffusion coefficients of liquid aluminium measured on absolute scale by using incoherent quasielastic neutron scattering at temperatures of 980K, 1020K, and 1060K. Aluminium self-diffusion coefficients follow an Arrhenius law with an activation energy of 280±70 meV. The Sutherland-Einstein equation relating viscosity to the diffusion coefficient well captures the temperature dependence and absolute values of the
F Kargl; H Weis; T Unruh; A Meyer
2012-01-01
38
PubMed
Environmental friendly recycling is the trend toward total recycling of aluminium metal. In the secondary aluminium industry, due to the complexity of compositions and contaminants in the various types of aluminium scraps, an understanding of the behavior of different scraps during melting is crucial in the recycling process. Salt slags are the byproducts of the secondary aluminium industry, which should be recycled and processed in a proper way by taking the environmental impact into consideration. This article provides qualitative assessment on 10 different commercial aluminium scraps for their relative recyclability via well-designed and controlled laboratory experiments. It confirms that more nonmetallic contaminants, smaller size, and higher ratio of surface area to body volume generally lead to a lower metal recovery. Recycling the scraps with lower recyclability normally generates more salt slags. High slag viscosity leads to more fine aluminum metal entrapped in the salt slag and thus increases the load of salt slag recycling. It was found that viscosity of the salt flux is increased with the amount of entrapped nonmetallic components, which affect the settling of heavier materials. In addition, the slag samples from the melting tests were leached and analyzed to evaluate the behavior of carbon containing scrap. The elevated carbon content in the scrap resulted in more carbide formation in salt slags and thus more methane generation in salt slag recycling with a higher environmental impact. PMID:16194908
Xiao, Yanping; Reuter, Markus A; Boin, Udo
2005-01-01
39
Dissimilar metal joints of galvannealed steel and commercially available pure aluminium (A1050) sheets were produced by changing the laser power and the roller pressure by the laser pressure welding method. In this method, the YAG laser beam was irradiated into a flare groove made by these dissimilar metal sheets. In addition, the laser beam was scanned at various frequencies and
Koji Nishimoto; Yoshihiro Okumoto; Tomoki Harano; Ken Atagi; Hiroo Fujii; Seiji Katayama
2009-01-01
40
Summary This study, with the emphasis on experiments, investigates the applicability of aluminium foam as filler material in tubes made of mild steel having square or circular cross sections, which are crushed axially at low loading velocities. In addition to the experiments finite element studies are performed to simulate the crushing behaviour of the tested square tubes, were a crushable
M. Seitzberger; F. G. Rammerstorfer; H. P. Degischer; R. Gradinger
1997-01-01
41
NASA Astrophysics Data System (ADS)
Aluminium honeycomb is a type of cellular material which has high strength to weigh ratio and is a good energy absorber. They are used as structural components in various engineering applications. Comprehensive study has been conducted on the compressive behavior of aluminium honeycombs. However, the research of aluminium honeycombs subjected to other type of loading, such as indentation, is still limited. In this paper, quasi-static and dynamic indentation tests were conducted to study the deformation and energy absorption of three types of HEXCELL® aluminium honeycombs with different cell sizes and cell wall thicknesses. Quasi-static tests were conducted by using a universal MTS machine at velocities of 0.05 mm/s, 0.5 mm/s and 5 mm/s, respectively. Dynamic tests were conducted by using a high speed INSTRON machine at a velocity of 5 m/s. Force-displacement curves were plotted in which the total energy absorbed was calculated. The deformation of aluminium honeycombs in indentation tests includes the compression of honeycomb cells under the indenter and tearing of honeycomb cell walls along the indenter edges. The energy dissipated in compression and tearing were calculated and discussed. The effects of cell size, cell wall thickness and loading velocity or strain rate on the plateau stress and energy absorption were analyzed.
Ashab, A.; Wong, Y. C.; Lu, G.; Ruan, D.
2013-07-01
42
This paper describes an investigation of the resistance spot weldability of aluminium clad steel sheet to steel or aluminium sheet as well as joining of a steel to an aluminium sheet with an aluminium clad steel sheet as an intermediate layer. The clad steel sheet was produced by hot?rolling of steel and aluminium sheet by a direct resistance heating process.
H. Oikawa; T. Saito; T. Yoshimura; T. Nagase; T. Kiriyama
1996-01-01
43
Increasing demand in world automotive markets for aluminium die cast components is creating significant opportunities and challenges for the Australian industry, which is positioning itself as a global player. To meet these challenges, the industry is continuously seeking to improve its overall resource efficiency that can result in the reduction of cost and impact on green house gas (GHG) emissions.
A. Tharumarajah
2008-01-01
44
SciTech Connect
Aluminium is a ubiquitous element in the environment and has been demonstrated to be toxic, especially in individuals with impaired renal function. Not much is known about the biochemistry of aluminium and the mechanisms of its toxic effects. Most of the interest in aluminium has been in the clinical setting of the hemodialysis unit. Here aluminium toxicity occurs due to contamination of dialysis solutions, and treatment of the patients with aluminium-containing phosphate binding gels. Aluminium has been shown to be the major contributor to the dialysis encephalopathy syndrome and an osteomalacic component of dialysis osteodystrophy. Other clinical disturbances associated with aluminium toxicity are a microcytic anemia and metastatic extraskeletal calcification. Aluminium overload can be treated effectively by chelation therapy with desferrioxamine and hemodialysis. Aluminium is readily transferred from the dialysate to the patient's -bloodstream during hemodialysis. Once transferred, the aluminium is tightly bound to non-dialysable plasma constituents. Very low concentrations of dialysate aluminium in the range of 10-15 micrograms/l are recommended to guard against toxic effects. Very few studies have been directed towards the separation of the various plasma species which bind eluminium. Gel filtration chromatography has been used to identify five major fractions, one of which is of low molecular weight and the others appear to be protein-aluminium complexes. Recommendations on aluminium monitoring have been published and provide safe and toxic concentrations. Also, the frequency of monitoring has been addressed. Major problems exist with the analytical methods for measuring aluminium which result from inaccurate techniques and contamination difficulties. 136 references.
Savory, J.; Bertholf, R.L.; Wills, M.R.
1985-08-01
45
PubMed
Grain boundaries and precipitates are the major microstructural features influencing the mechanical properties of metals and alloys. Refinement of the grain size to the nanometre scale brings about a significant increase in the mechanical strength of the materials because of the increased number of grain boundaries which act as obstacles to sliding dislocations. A similar effect is obtained if nanoscale precipitates are uniformly distributed in coarse grained matrix. The development of nanograin sized alloys raises the important question of whether or not these two mechanisms are "additive" and precipitate strengthening is effective in nanostructured materials. In the reported work, hydrostatic extrusion (HE) was used to obtain nanostructured 7475 aluminium alloy. Nanosized precipitates were obtained by post-HE annealing. It was found that such annealing at the low temperatures (100 degrees C) results in a significant increase in the microhardness (HV0.2) and strength of the nanostructured 7475 aluminium alloy. These results are discussed in terms of the interplay between the precipitation and deformation of nanocrystalline metals. PMID:23421286
Wawer, Kinga; Lewandowska, Malgorzata; Kurzydlowski, Krzysztof J
2012-11-01
46
PubMed
2004-10-01
47
PubMed
The electrodeposition of aluminium is demonstrated using a eutectic mixture of aluminium chloride and urea. The mixture is shown to be conducting through the formation of both cationic ([AlCl2·urean](+)) and anionic (AlCl4(-)) species and electrodeposition is achieved through the cationic species. The use of a biphasic system with the ionic liquid and a protective hydrocarbon layer allows metal deposition to be carried out in an environment with ambient moisture without the need for a glove box. A direct comparison is made between the AlCl3:urea and imidazolium chloride:AlCl3 systems and the differences in speciation and mass transport manifest themselves in different deposit morphologies. Brighteners which work in the chloroaluminate system such as toluene and LiCl are shown to be ineffective in the urea based system and the reasons for these differences are ascribed to the mechanism of the anodic reaction which is rate limiting. PMID:24916113
Abbott, Andrew P; Harris, Robert C; Hsieh, Yi-Ting; Ryder, Karl S; Sun, I-Wen
2014-06-25
48
PubMed
The levels of aluminium have been determined in a number of individual foodstuffs on the Swedish market and in 24 h duplicate diets collected by women living in the Stockholm area. The results show that the levels in most foods are very low and that the level in vegetables can vary by a factor 10. Beverages from aluminium cans were found to have aluminium levels not markedly different from those in glass bottles. Based on the results of the analysis of individual foods, the average Swedish daily diet was calculated to contain about 0.6 mg aluminium, whereas the mean content of the collected duplicate diets was 13 mg. A cake made from a mix containing aluminium phosphate in the baking soda was identified as the most important contributor of aluminium to the duplicate diets. Tea and aluminium utensils were estimated to increase the aluminium content of the diets by approximately 4 and 2 mg/day, respectively. The results also indicate that a considerable amount of aluminium must be introduced from other sources. PMID:1542992
Jorhem, L; Haegglund, G
1992-01-01
49
The migration of aluminium (Al) from aseptic laminated paperboard packages was studied by monitoring the Al content of orange juice (plain juice, no additives) filled into such packages and stored at ambient temperature (23°C) for up to 1 year. Analytical procedure includes MW-assisted digestion of orange juice with concentrated nitric acid followed by determination of Al concentration by double-focusing sector
Ilia Rodushkin; Astrid Magnusson
2005-01-01
50
NASA Astrophysics Data System (ADS)
In this work, we studied the effect of surface treatment on the aluminium surface and a coupling agent to improve adhesion between aluminium with organic polymer. Thermoplastic natural rubber (TPNR) matrix was prepared by melt blending of natural rubber (NR), liquid natural rubber (LNR) compatibilizer, linear low density polyethylene (LLDPE) and polyethylene grafted maleic anhydride (PE-g-MAH). The PEgMAH concentration used was varied from 0% - 25%. In addition, the aluminium surface was pre-treated with 3-glycidoxy propyl trimethoxy silane (3-GPS) to enhance the mechanical properties of laminated composite. It was found that the shear strength of single lap joint Al-TPNR laminated composite showing an increasing trend as a function of PE-g-MAH contents for the 3-GPS surface treated aluminium. Moreover, the scanning electron microscope (SEM) revealed that the strength improvement was associated with the chemical state of the compound involved.
Muzakkar, M. Z.; Ahmad, S.; Yarmo, M. A.; Jalar, A.; Bijarimi, M.
2013-04-01
51
NASA Astrophysics Data System (ADS)
The modifications brought to aluminium by addition of silicon and nickel are investigated after a laser cladding process by a Nd: YAG laser. Samples are characterized using microscopic examination, X-ray diffraction and micro-hardness measurement. The optimum composition (Al 22Si 27Ni (wt.%)) is selected for its fine and homogeneous microstructure: presence of equiaxis precipitates (Si) and AIN (average size of 20 and 5 ?m) and its high hardness (412 HV5). Les modifications apportées à l'aluminium par addition de silicium et de nickel ont été étudiées après rechargement laser réalisé grâce à un laser Nd : YAG. Les échantillons ont été caractérisés par microscopie optique, diffraction des rayons X et macro-indentation. La composition optimale (Al 22Si 27Ni (wt.%)) a été retenue pour sa microstructure fine et homogène : présence des précipités équiaxes (Si) et Al3Ni (dimensions moyennes de 20 et 5 ?m), ainsi que pour sa dureté élevée : 412 HV5.
Dubourg, L.; Hlawka, F.; Cornet, A.
2002-07-01
52
ALUMINIUM forging alloys used in aerospace technology have, on occasion, suffered catastrophic failure as a result of stress-corrosion cracking. Study of this problem by thin-film transmission electron microscopy has disclosed a fundamental relationship between those aluminium structures within the same alloy (achieved by heat treating) that are highly sensitive to stress-corrosion cracking and those that are immune.
A. J. Jacobs
1966-01-01
53
Estimating damping in structures made of different materials and processes still remains as one of the biggest challenges. Aluminium is one such pioneer material which is being used extensively in aerospace, automotive and the manufacturing industry. Aluminium components are mainly manufactured by traditional casting and powder metallurgy process. The main objective of this paper is to estimate the damping ratio
Abhinav Alva; Vijay Desai
54
National Technical Information Service (NTIS)
By chlorinating aluminium under pressure in a glass combustion vessel, a value of -168.8 kcal/mole was obtained for the standard heat of formation of aluminium chloride. The total error is considered to be less than plus or minus 0.20 kcal.
P. Gross C. Hayman
1967-01-01
55
The use of lightweight bridge decks made of FRP composites or aluminium alloys is particularly effective for replacing deteriorated bridge decks. Therefore a research program has been undertaken to develop and implement an innovative aluminium bridge deck system, which would be applicable and realizable in domestic conditions. Several service load and ultimate load tests have been carried out on the
Tomasz W. Siwowski
2009-01-01
56
A study is carried out on the feasibility of using warm forming at temperatures from 100 to 250°C in order to improve the makeability of aluminium sheet components. Drawing tests are performed on 1050, 5754 and 6016 series aluminium sheets. Both box shaped and conical rectangular products are made with a tool with heated die and blankholder. The effect of
P. J Bolt; N. A. P. M Lamboo; P. J. C. M Rozier
2001-01-01
57
NASA Astrophysics Data System (ADS)
By use of the recurrent relation approach (RRA) we study the microscopic dynamics of liquid aluminium at T = 973 K and develop a theoretical model which satisfies all the corresponding sum rules. The investigation covers the inelastic features as well as the crossover of our theory into the hydrodynamical and the free-particle regimes. A comparison between our theoretical results with those following from a generalized hydrodynamical approach is also presented. In addition to this we report the results of our molecular dynamics simulations for liquid aluminium, which are also discussed and compared to experimental data. The results obtained reveal (i) that the microscopical dynamics of density fluctuations is defined mainly by the first four even frequency moments of the dynamic structure factor, and (ii) the inherent relation of the high-frequency collective excitations observed in experimental spectra of dynamic structure factor S(k,?) with the two-, three- and four-particle correlations.
Mokshin, A. V.; Yulmetyev, R. M.; Khusnutdinoff, R. M.; Hänggi, P.
2007-01-01
58
NASA Astrophysics Data System (ADS)
Ni-P-fly ash coatings were produced on zincate coated 5083 wrought aluminium alloy substrates with the aid of an electroless deposition technique. Structural and chemical characterization of the produced coatings was performed with the aid of X-ray diffraction (XRD), scanning electron microscopy (SEM) and electron dispersive X-ray analysis (EDS) techniques. The Ni-P-fly ash coating was found to consist of an amorphous Ni-P matrix with dispersed fly ash particles. The wear resistance of the Ni-P-fly ash coating on zincate treated aluminium alloy was observed to be higher than that of the bare aluminium alloy, when sliding against a stainless steel counterface. In addition, the adhesion between the Ni-P-fly ash/zincate coating and the aluminium alloy substrate was also studied with a scratch testing apparatus. The adhesion strength of Ni-P-fly ash/zincate coating on the aluminium alloy substrate was observed to be higher in comparison to the Ni-P/zincate coating on the same aluminium alloy.
Panagopoulos, C. N.; Georgiou, E. P.
2009-04-01
59
PubMed
Aluminium is light weight and toxic metal present ubiquitously on earth which has gained considerable attention due to its neurotoxic effects. The widespread use of products made from or containing aluminium is ensuring its presence in our body. There is prolonged retention of a fraction of aluminium that enters the brain, suggesting its potential for accumulation with repeated exposures. There is no known biological role for aluminium within the body but adverse physiological effects of this metal have been observed in mammals. The generation of oxidative stress may be attributed to its toxic consequences in animals and humans. The oxidative stress has been implicated in pathogenesis of various neurodegenerative conditions including Alzheimer's disease and Parkinson's disease. Though it remains unclear whether oxidative stress is a major cause or merely a consequence of cellular dysfunction associated with neurodegenerative diseases, an accumulating body of evidence implicates that impaired mitochondrial energy production and increased mitochondrial oxidative damage is associated with the pathogenesis of neurodegenerative disorders. Being involved in the production of reactive oxygen species, aluminium may impair mitochondrial bioenergetics and may lead to the generation of oxidative stress. In this review, we have discussed the oxidative stress and mitochondrial dysfunctions occurring in Al neurotoxicity. In addition, the ameliorative measures undertaken in aluminium induced oxidative stress and mitochondrial dysfunctions have also been highlighted. PMID:24560992
Kumar, Vijay; Gill, Kiran Dip
2014-03-01
60
NASA Astrophysics Data System (ADS)
In this work, a nanocomposite (Cr,Al)xN1-x/Si3N4 coating system was deposited on H11 hot work tool steel, using the Lateral Arc Rotating Cathodes (LARC®) deposition system and modulating the chemical composition of the chromium and aluminium-silicon content. Structural characterizations were performed using scanning electron microscopy, equipped with energy dispersive spectroscopy probe, and applying x-ray diffraction, for the evaluation of phase constitution and crystallite size. In addition to the structural features, the coatings' resistance to cyclic immersions in molten aluminium alloy was evaluated. The deposited CrAlSiN coatings exhibited an fcc-Cr1-xAlxN type structure with different aluminium contents, which directly influence hardness and wear and fatigue resistance in cyclic immersion tests. The main failure modes that occurred on the coatings' surface were soldering and thermal fatigue cracks mainly in the form of heat checks. The aluminium rich coatings were able to withstand about 15 000 cycles, whereas the decrease in the aluminium content in the coatings results in a decrease in the resistance to the immersion in molten aluminium bath. It is worthwhile to note that uncoated H11, subjected to similar testing conditions, withstood at maximum 5000 cycles.
Torres, E.; Ugues, D.; Brytan, Z.; Perucca, M.
2009-05-01
61
PubMed Central
OBJECTIVES: The study attempts to define biological indicators of aluminium uptake and excretion in workers exposed to airborne aluminium compounds in a primary aluminium smelter. Also, this study defines the total and respirable aluminium dust fractions in two different potrooms, and correlates their concentrations with biological indicators in this group of workers. METHODS: Air was sampled at defined work sites. Non-destructive and conventional techniques were used to find total and respirable aluminium content of the dust. Blood and urine was collected from 84 volunteers employed at various work stations throughout the smelter and from two different cohorts of controls matched for sex, age, and socioeconomic status. Aluminium in serum samples and urine specimens was measured by flameless atomic absorption with a PE 4100 ZL spectrometer. RESULTS: The correlation of aluminium concentrations in serum and urine samples with the degree of exposure was assessed for three arbitrary exposure categories; low (0.036 mg Al/m3), medium (0.35 mg Al/m3) and high (1.47 mg Al/m3) as found in different areas of the smelter. At medium and high exposure, the ratio of respirable to total aluminium in the dust samples varied significantly. At high exposure, serum aluminium, although significantly raised, was still within the normal range of an unexposed population. The workers with low exposure excreted aluminium in urine at levels significantly higher than the controls, but still within the normal range of the population. However, potroom workers with medium and high exposure had significantly higher urinary aluminium than the normal range. CONCLUSIONS: It is concluded that only urinary aluminium constitutes a practical index of occupational exposure at or above 0.35 mg Al/m3, and that the respirable fraction of the dust may play a major role in the biological response to exposure to aluminium in a smelter environment.
Rollin, H B; Theodorou, P; Cantrell, A C
1996-01-01
62
A new means of assembling galvanised steel to aluminium involving a reaction between solid steel and liquid aluminium was developed, using laser and gas tungsten arc welding (GTAW) processes. A direct aluminium melting strategy was investigated with the laser process, whereas an aluminium-induced melting by steel heating and heat conduction through the steel was carried out with the GTAW process.
G. Sierra; P. Peyre; F. Deschaux Beaume; D. Stuart; G. Fras
2008-01-01
63
PubMed Central
The increased availability of aluminium in biological environments, due to human intervention in the last century, raises concerns on the effects that this so far “excluded from biology” metal might have on living organisms. Consequently, the bioinorganic chemistry of aluminium has emerged as a very active field of research. This review will focus on our contributions to this field, based on computational studies that can yield an understanding of the aluminum biochemistry at a molecular level. Aluminium can interact and be stabilized in biological environments by complexing with both low molecular mass chelants and high molecular mass peptides. The speciation of the metal is, nonetheless, dictated by the hydrolytic species dominant in each case and which vary according to the pH condition of the medium. In blood, citrate and serum transferrin are identified as the main low molecular mass and high molecular mass molecules interacting with aluminium. The complexation of aluminium to citrate and the subsequent changes exerted on the deprotonation pathways of its tritable groups will be discussed along with the mechanisms for the intake and release of aluminium in serum transferrin at two pH conditions, physiological neutral and endosomatic acidic. Aluminium can substitute other metals, in particular magnesium, in protein buried sites and trigger conformational disorder and alteration of the protonation states of the protein's sidechains. A detailed account of the interaction of aluminium with proteic sidechains will be given. Finally, it will be described how alumnium can exert oxidative stress by stabilizing superoxide radicals either as mononuclear aluminium or clustered in boehmite. The possibility of promotion of Fenton reaction, and production of hydroxyl radicals will also be discussed.
Mujika, Jon I; Rezabal, Elixabete; Mercero, Jose M; Ruiperez, Fernando; Costa, Dominique; Ugalde, Jesus M; Lopez, Xabier
2014-01-01
64
Goal, Scope and Background This case study describes the development and utilization of a global, quantitative model of current and projected aluminium\\u000a and life cycle inventory mass flows. The model and key results were developed to be shared with global aluminium industry\\u000a technical experts, executives, and external stakeholders to better understand potential paths to more global sustainable aluminium.\\u000a \\u000a \\u000a \\u000a \\u000a Methods The
Kenneth J. Martchek
2006-01-01
65
Summary Effects of aluminium on theTrifolium repens var Huia-Rhizobium trifolii strain HP3 symbiosis were studied using an axenic solution-culture system. With, 10 ?M phosphate, 50 ?M aluminium reduced or inhibited root elongation at pH<5.0, root hair formation at pH< 5.0–5.5, and Rhizobium multiplication\\u000a in the rhizosphere and nodule formation at pH<6.0. In the absence of aluminium, root elongation and root hair
M. Wood; J. E. Cooper; A. J. Holding
1984-01-01
66
PubMed
An increasing amount of recycled aluminium is going into the production of aluminium alloy used for automotive applications. In these applications, it is necessary to control and remove alloy impurities and inclusions. Cleaning and fluxing processes are widely used during processing of the alloys for removal of inclusions, hydrogen and excess of magnesium. These processes use salt fluxes based in the system NaCl-KCl, injection of chlorine or mixture of chlorine with an inert gas. The new systems include a graphite wand and a circulation device to force convection in the melt and permit the bubbling and dispersion of reactive and cleaning agents. This paper discusses the recycling of aluminium alloys in rotary and reverberatory industrial furnaces. It focuses on the removal of magnesium during the melting process. In rotary furnaces, the magnesium lost is mainly due to the oxidation process at high temperatures. The magnesium removal is carried out by the reaction between chlorine and magnesium, with its efficiency associated to kinetic factors such as concentration of magnesium, mixing, and temperature. These factors are also related to emissions generated during the demagging process. Improvements in the metallic yield can be reached in rotary furnaces if the process starts with a proper salt, with limits of addition, and avoiding long holding times. To improve throughput in reverberatories, start the charging with high magnesium content material and inject chlorine gas if the molten metal is at the right temperature. Removal of magnesium through modern technologies can be efficiently performed to prevent environmental problems. PMID:20837560
Velasco, Eulogio; Nino, Jose
2011-07-01
67
PubMed Central
Introduction Aluminium is a known neurotoxin and occupational exposure to aluminium has been implicated in neurological disease including Alzheimer’s disease. Here we present the first comprehensive and unequivocal data demonstrating significantly elevated brain aluminium content in an individual occupationally exposed to aluminium. Case presentation A 66-year-old Caucasian man who died with Alzheimer’s disease showed significantly elevated brain aluminium content, 2.98 (2.73) ?g/g dry weight, n?=?46, following occupational exposure to aluminium over a period of 8 years. Conclusions That the individual developed an early onset aggressive form of Alzheimer’s disease suggests a role for aluminium in disease aetiology. That the exposure to aluminium was through occupational exposure to aluminium dust suggests a prominent role for the olfactory system and lungs in the accumulation of aluminium in the brain.
2014-01-01
68
NASA Astrophysics Data System (ADS)
The current state of the art in joining of carbon-fibre reinforced composites (CFRP) to metals such as aluminium is - for the case of aircraft structures, e.g.- riveting or bolting. However, to reduce structural weight and improve structural performance, integral, load-bearing aluminium-CFRP-structures are desirable. To produce such structures, a novel joint configuration together with an appropriate thermal, laser-based joining process is suggested by the authors. In this paper, the joint configuration (based on CFRP-Ti-aluminium joints) and the laser beam conduction welding process will be presented, and first specimens obtained will be discussed with respect to their properties. It will be shown that the novel approach is in principle suitable to produce load-bearing CFRP-aluminium structures.
Möller, F.; Thomy, C.; Vollertsen, F.; Schiebel, P.; Hoffmeister, C.; Herrmann, A. S.
69
Thixoforming is a type of semi-solid metal processing, which involves the processing of alloys in the semi-solid state. Disagglomeration of solid globules during thixoforming of wrought aluminium alloy 2014 was investigated by electron back-scatter diffraction. The recrystallisation and partial melting route was used to produce the thixotropic feedstock for thixoforming. The starting material was commercially extruded wrought aluminium alloy 2014
D. Liu; H. V. Atkinson; R. L. Higginson
2005-01-01
70
Nanocrystalline aluminium powder has been prepared by high-energy ball milling of flaked micron-sized aluminium powder in the presence of 10 wt% of graphite under argon atmosphere. The structure and chemical composition of as-prepared nanocomposites and the their thermally induced changes are studied by X-ray diffraction (XRD), transmission electron microscopy (TEM), and simultaneous TG-DTA technique (SDT). TEM studies reveal that the
A. Pivkina; A. Streletskii; I. Kolbanev; P. Ul'yanova; Yu. Frolov; P. Butyagin; J. Schoonman
2004-01-01
71
Aluminium metal matrix composites (AlMMCs) offer several advantages relative to monolithic aluminium alloys such as high stiffness, strength, wear resistance, low thermal expansion coefficient, etc. However, despite considerable improvements in developing AlMMCs, the lack of reliable joining methods restrict their greater application. Fusion welding of AlMMCs has not proved successful because high temperature nature of the process normally causes unfavorable
Tapan Kumar Pal
2005-01-01
72
Possible improvements in the mechanical properties of surface-coated aluminium have been investigated. Aluminium plasma sprayed with iron powder was processed using two different laser-beam settings (integrated and defocused) with varying processing parameters. Mechanical and microstructural characterizations of the specimens revealed great differences between the two laser-beam settings. The defocused laser beam showed a clear improvement of hardness which was not
L. Gjønnes; A. Olsen
1994-01-01
73
In the present study, Aluminium quantification in immunobiologicals has been described using atomic absorption spectroscopy (AAS) technique. The assay was found to be linear in 25–125?g\\/ml Aluminium range. The procedure was found to be accurate for different vaccines with recoveries of external additions ranging between 93.26 and 103.41%. The mean Limit of Variation (L.V.) for both intra- and inter-assay precision
Arti Mishra; Sumir Rai Bhalla; Sameera Rawat; Vivek Bansal; Rakesh Sehgal; Sunil Kumar
2007-01-01
74
This paper describes an investigation of the dissimilar metal spot welding of aluminium and steel sheet with an aluminium clad steel sheet insert. The study relates to application of this system in automotive fabrication in an effort to reduce motor vehicle weight. To ascertain the mechanical properties of the intermetallic compounds formed at the steel and aluminium bond interface, measurements
M. Yasuyama; K. Ogawa; T. Taka
1996-01-01
75
NASA Astrophysics Data System (ADS)
To get some insight by conclusions of analogy into the drying process of alcoholic aluminium fluoride sol-gels [AlF 3/(ROH) x], the structures of ?- and ?-AlF 3?3H 2O as well as of the nonahydrate AlF 3?9H 2O are reinvestigated and discussed based on X-ray single crystal structural data. In addition, neutron diffraction experiments of the latter allowed the refinement of proton positions. In accordance with crystal structures, low-temperature solid state 27Al-, 1H- and 19F-MAS NMR spectra convincingly confirm the structural similarity between ?-AlF 3?3H 2O and AlF 3?9H 2O, while the ?-phase material is structurally different forming chain structures. Thermal analysis of AlF 3/(ROH) x gave evidence for discrete AlF 3:ROH ratios of only 1:0.45 and 1:0.1, and solution NMR showed some similarities between aqueous and alcoholic systems.
Kemnitz, E.; Groß, U.; Rüdiger, St.; Scholz, G.; Heidemann, D.; Troyanov, S. I.; Morosov, I. V.; Lemée-Cailleau, M.-H.
2006-12-01
76
NASA Astrophysics Data System (ADS)
Self healing is a key property of biological materials, examples being the autonomous repair of fractured bones or torn skin tissue, as is discussed in other chapters of this book. More than a century ago, principles were developed defining the reconstitution of fractured bone in vivo which stated that, in essence, material dissolved from where it was not required was redeposited to where it was required as a response to mechanical stimuli and damage. As far as metals and other inanimate materials are concerned, it is well known that damage to oxide films, which normally protect the surfaces of metals such as aluminium (Al) and titanium (Ti) from corrosion, can be repaired by reoxidation in air, which can be seen as a form of self-repair. Now attention is being directed to processes that may possibly heal defects, such as cracks, which can develop in the interior of materials during manufacture, or when they are in service. Such self healing processes may then allow failures to be averted and the useful lives of components and structures to be extended.
Lumley, Roger
77
PubMed
A method for the complexometric determination of aluminium in iron ore, sinter, concentrates and agglomerates encountered in international trade is described. The sample is fused in a zirconium crucible with a mixed flux of sodium carbonate and sodium peroxide. The fused mass is completely soluble in hydrochloric acid. The R(2)O(3) oxides are then precipitated with ammonia and redissolved in hydrochloric acid. Elements such as iron, titanium and zirconium are separated from aluminium by solvent extraction with cupferron and chloroform. After removal of traces of organic matter from the aqueous phase, the solution is treated with an excess of EDTA, which is then back-titrated with zinc solution (Xylenol Orange as indicator). Addition of ammonium fluoride then releases EDTA equivalent to the aluminium and this is titrated with zinc solution. The method is rapid. The precision and accuracy are excellent, and the results comparable with those obtained by the referee method. PMID:18962398
Bhargava, O P
1979-02-01
78
PubMed
The effect of benzaldehyde, 2-hydroxybenzoyl hydrazone derivatives on the corrosion of aluminium in hydrochloric acid has been investigated using thermometric and polarization techniques. The inhibitive efficiency ranking of these compounds from both techniques was found to be: 2>3>1>4. The inhibitors acted as mixed-type inhibitors but the cathode is more polarized. The relative inhibitive efficiency of these compounds has been explained on the basis of structure of the inhibitors and their mode of interaction at the surface. Results show that these additives are adsorbed on an aluminium surface according to the Langmuir isotherm. Polarization measurements indicated that the rate of corrosion of aluminium rapidly increases with temperature over the range 30-55 degrees C both in the absence and in the presence of inhibitors. Some thermodynamic data of the adsorption process are calculated and discussed. PMID:10823698
Fouda, A S; Gouda, M M; El-Rahman, S I
2000-05-01
79
Dissimilar metal joints of Zn-coated Galvannealed steel (GA steel) and commercially available pure aluminium (A1050) sheets were produced by changing the laser power and the roller pressure by the laser pressure welding method. By this method, the YAG laser beam was irradiated into a flare groove made by these dissimilar metal sheets. In addition, the laser beam was scanned at
Koji Nishimoto; Tomoki Harano; Yoshihiro Okumoto; Ken Atagi; Hiroo Fujii; Seiji Katayama
2009-01-01
80
Tea plant takes up a large quantity of aluminium (Al) and fluoride (F) from acidic soils. It has been known that fluorosis can be developed for people who consume a large quantity of tea made from brick tea, a low quality tea consisting mainly of old tea leaves in China. In addition, it has been claimed that Alzheimer's disease (AD)
M. H Wong; K. F Fung; H. P Carr
2003-01-01
81
Anodic spark coatings on aluminium alloy were prepared in aqueous electrolytes with sodium tungstate. The influence of boric acid addition in the electrolyte on the surface morphology, elemental and phase composition of the coatings was investigated. In both cases the coatings contained O, Al and W. The coatings obtained in electrolyte with boric acid and sodium tungstate contain also B
I. V. Lukiyanchuk; V. S. Rudnev; V. G. Kuryavyi; D. L. Boguta; S. B. Bulanova; P. S. Gordienko
2004-01-01
82
Aluminium powders with a mean particle size of around 1?m were compacted by cold isostatic pressing (CIP) and additional forging. The specimens are characterized by hot compression tests, dilatometry and metallography. A 3D interconnected structure of alumina films <5nm in thickness is observed by transmission electron microscopy and field emission gun scanning electron microscopy; it is associated with the natural
C. Poletti; M. Balog; F. Simancik; H. P. Degischer
2010-01-01
83
Technological interest in aluminium-silicon alloys derives, fundamentally, from their excellent moulding properties provided by the presence of silicon. In addition, they possess high thermal and electrical conductivities and good corrosion resistance. The possibility of modification of their cast structure by adding small quantities of alkaline and alkaline-earth elements makes their mechanical characteristics very competitive. Metallographic observations of the structures of
A. J. Criado; J. A. Martínez; R. Calabrés
1997-01-01
84
The applicability of an aluminium alloy containing scandium for laser additive manufacturing (LAM) is considered. Modified aluminium alloys with a scandium content beyond the eutectic point offer great potential to become a high prioritized aerospace material. Depending on other alloying elements like magnesium or zirconium, strongly required weight reduction, corrosion resistance and improved strength properties of metallic light weight alloys
K. Schmidtke; F. Palm; A. Hawkins; C. Emmelmann
2011-01-01
85
National Technical Information Service (NTIS)
The metallurgy of TIG weldment in aluminium alloy was examined based on physical metallurgy characteristics. The presence of weld defects have been detected by x-ray radiographic and metallographic methods. The lack of penetration in the aluminium weldmen...
Mohamad Harun Abdul Aziz Mohamad Zailee Dollah Azlan Samah Jasmin Baba
1986-01-01
86
A study has been conducted to identify the effects of fabrication temperatures pressures, times and other variables on the strengths of diffusion-bonded joints between alumina and BS321 stainless steel produced using aluminium foil interlayers. The strengths of the alumina-aluminium and steel-aluminium interfaces were found to be influenced differently by some fabrication parameters, thus increasing the fabrication temperature promoted alumina-aluminium bonding
M. G. Nicholas; R. M. Crispin
1982-01-01
87
The paper presents a brief overview on several researches developed in the Pressure Welding Laboratory, Robotics of Welding Department - Dunarea de Jos University of Galati, Romania. After the necessary Introduction in the subject, considerations on Joining Aluminium Processes Classifications are presented, including an overview on the main joining processes used for aluminium. Several aspects of Aluminium cold welding processes
Danut IORDACHESCU; Bogdan Georgescu; Mihaela IORDACHESCU
88
The use of aluminium cemented seals in vacuum systems has been further investigated to find the effect of the flange surface finish on the adhesion of the aluminium joint and the performance of such seals in bakeable apparatus. Aluminium wire clamped between stainless-steel flanges and baked at 300° C formed more adherent joints as the surface finish was decreased from
L. Elsworth; L. Holland; L. Laurenson
1960-01-01
89
New requirements of the automotive industry, concerning lightweight and non-corroding construction, demand new production methods. Due to this the hydroforming process of aluminium alloys are of special interest. The disadvantage of aluminium alloys is the poorer formability compared to steel. A method to increase the formability of the aluminium alloys during the hydroforming process is the enhancement of the forming
Michael Keigler; Herbert Bauer; David Harrison; Anjali K. M. De Silva
2005-01-01
90
Summary accompanying the thesis: Energy Absorption of Monolithic and Fibre Reinforced Aluminium Cylinders by Jens de Kanter This thesis presents the investigation of the crush behaviour of both monolithic aluminium cylinders and externally fibre reinforced aluminium cylinders. The research is based on analytical work, numerical work and experimental work, which together show a complete picture of the tube axial crush
J. L. C. G. De Kanter
2006-01-01
91
The chlorination of aluminium in coal ash was studied experimentally in a fluidized bed reactor. With carbon (fluid coke) and CO as reducing agents, about 25% of aluminium in ash could be chlorinated at above 900 C in 2 hours. Global reaction rate data for aluminium as well as iron, silicon, and titanium indicated two very distinct reaction regions. A
W. Y. Svrcek; A. K. Mehrotra; L. A. Behie; P. R. Bishnoi
1983-01-01
92
PubMed
Oxidation can drastically change mechanical properties of nanostructures that typically have large surface-to-volume ratios. However, the underlying mechanisms describing the effect oxidation has on the mechanical properties of nanostructures have yet to be characterized. Here we use reactive molecular dynamics and show that the oxidation enhances the aluminium nanowire ductility, and the oxide shell exhibits superplastic behaviour. The oxide shell decreases the aluminium dislocation nucleation stress by increasing the activation volume and the number of nucleation sites. Superplasticity of the amorphous oxide shell is due to viscous flow as a result of healing of the broken aluminium-oxygen bonds by oxygen diffusion, below a critical strain rate. The interplay between the strain rate and oxidation rate is not only essential for designing nanodevices in ambient environments, but also controls interface properties in large-scale deformation processes. PMID:24887649
Sen, Fatih G; Alpas, Ahmet T; van Duin, Adri C T; Qi, Yue
2014-01-01
93
SciTech Connect
Ceramic whisker or particulate reinforced aluminium alloy composites have a great potential for automobile engineering components, aerospace structures, semi-conductor packaging and so on, because of the composites ability to exhibit a high specific elastic modulus and specific tensile strength, excellent wear resistance and heat resistance, low thermal expansion and good dimensional stability. A serious problem involving practical application of ceramic whisker or particulate reinforced aluminium alloy composites is due to the low tensile ductility, fracture toughness at room temperature and, also, their hardness qualities that make it difficult to deform by conventional forming processing and machining by ordinary tools. It has been found, however, that aluminium alloy composites reinforced by SiC or Si[sub 3]N[sub 4] whiskers or particulates produce superplasticity at a high strain rate of about 0.1s[sup [minus]1]. Superplastic deformation mechanisms of the ceramic whisker or particulate reinforced aluminium alloy composites are fine grain boundary sliding, interfacial sliding at a liquid phase and dynamic recrystallization. An AlN particulate reinforced aluminium alloy composite exhibits a high elastic modulus and a high thermal conductivity, and their thermal expansion is similar to silicon in that the AlN particulate reinforced aluminum alloy composite is expected to apply to semi-conductor packaging in the aerospace structure. In addition, if the composite could produce superplasticity at high strain rates, the market of aerospace application for superplastic composites could be expanded. The purpose of this study is to make clear if an AlN particulate reinforced aluminium alloy composite can produce superplasticity at high strain rate and the superplastic characteristics.
Imai, T. (National Industrial Research Inst. of Nagoya (Japan)); L'Esperance, G.; Hong, B.D. (Ecole Polytechnique de Montreal, Quebec (Canada))
1994-08-01
94
PubMed
This paper explores the possibility of increasing the ballistic performance of gun propellant with the addition of inorganic additives viz. aluminium and ammonium perchlorate. Compositions based on propellant NQ containing additional aluminium and ammonium perchlorate in different parts were studied theoretically and experimentally. Performance in respect of ballistic parameters, sensitivity, thermal characteristics, thermal stability and mechanical properties are evaluated and compared with that of the conventional triple base propellant NQ. Experimental data on comparative study indicate that the compositions containing aluminium and ammonium perchlorate are superior to propellant NQ in respect of energy. PMID:18206299
Damse, R S; Sikder, A K
2008-06-15
95
PubMed
This report represents the conclusions of a Joint FAO/WHO Expert Committee convened to evaluate the safety of various food additives and a food contaminant with a view to concluding as to safety concerns and to preparing specifications for identity and purity. The first part of the report contains a general discussion of the principles governing the toxicological evaluation of and assessment of dietary exposure to food additives. A summary follows of the Committee's evaluations of technical, toxicological and dietary exposure data for seven food additives (advantame; glucoamylase from Trichoderma reesei expressed in Trichoderma reesei; glycerol ester of gum rosin; glycerol ester of tall oil rosin; glycerol ester of wood rosin; nisin; and octenyl succinic acid modified gum arabic) and an assessment of dietary exposure to cadmium from cocoa and cocoa products. Specifications for the following food additives were revised: annatto extracts (solvent-extracted bixin and solvent-extracted norbixin); Benzoe tonkinensis; food additives containing aluminium and/or silicon; mineral oil (medium viscosity); modified starches; paprika extract; phosphates (analytical methods for the determination of phosphorus and revision of specifications); 3-phytase from Aspergillus niger expressed in Aspergillus niger; potassium aluminium silicate; and potassium aluminium silicate-based pearlescent pigments. Annexed to the report are tables summarizing the Committee's recommendations for dietary exposures to and toxicological evaluations of the food additives and contaminant considered. PMID:24779311
2013-01-01
96
The research on an effect of aluminium contents on detonation characteristics of ammonium nitrate explosives was carried out. Measurements of detonation velocity were performed. Parameters of blast waves produced by charges of the investigated explosives detonating in an open space were measured by the use of piezoelectric gauges. Quasi-static pressure measurements were conducted in a steel chamber of 0.15 m
Józef PASZULA; Waldemar A. TRZCI?SKI
97
National Technical Information Service (NTIS)
The effect of electron conditioning on commercially aluminium alloys 1100 and 6063 were investigated. Contrary to the assumption that electron conditioning, if performed long enough, can reduce and stabilize the SEY to low values (= 1.3, value of many pur...
F. L. Pimpec F. King R. E. Kirby
2004-01-01
98
The through thickness residual stress distributions within three 120mm thick rectilinear forgings, made from the high strength aluminium alloy 7449 have been measured using both neutron diffraction and deep hole drilling. Neutron diffraction measurements were made on two instruments, one using a pulsed spallation neutron source, the other a steady state reactor source. Heat treatment of the forgings included a
J. S. Robinson; S. Hossain; C. E. Truman; A. M. Paradowska; D. J. Hughes; R. C. Wimpory; M. E. Fox
2010-01-01
99
Wear data obtained from single and multiple scratch passes, combined with the examination of the reinforcement-matrix interfacial structure by transmission electron microscopy (TEM) and toughness measurements made on the reinforcement phase, wereused to evaluate the wear behaviour of aluminium-based composite coatings. Reinforcement properties such as reinforcement volume fraction (Vf ), size, interfacial bonding and toughness appear to be important variables
R. L. Deuis; C. Subramanian; G. P. Cavallaro
1997-01-01
100
SciTech Connect
A demonstration is given of the feasibility of two-photon excitation of aluminium phthalocyanine and of the pharmaceutical preparation 'Fotosens', used in photodynamic therapy. The excitation source was an Nd:YAG laser emitting at the 1064 nm wavelength. The spectra of the two-photon-excited luminescence were obtained and the two-photon absorption cross sections were determined. (lasers in medicine)
Meshalkin, Yu P; Alfimov, E E; Makukha, V K [Novosibirsk State Technical University, Novosibirsk (Russian Federation); Vasil'ev, N E; Denisov, A N; Ogirenko, A P [Siberian Laser Medicine Centre, Novosibirsk (Russian Federation)
1999-12-31
101
This document is part of Subvolume A1 'Light Metal Systems. Part 1: Selected Systems from Ag-Al-Cu to Al-Cu-Er' of Volume 11 'Ternary Alloy Systems - Phase Diagrams, Crystallographic and Thermodynamic Data critically evaluated by MSIT®' of Landolt-Börnstein - Group IV Physical Chemistry. It provides the data for the ternary system Al-Cr-Nb (Aluminium - Chromium - Niobium).
2004-01-01
102
National Technical Information Service (NTIS)
The natural form of the silicon in the pure aluminiumsilicon eutectic at slow growth rates is continuous 'rods'. The silicon phase is ahead of the aluminium phase during the growth process. A remarkably simple explanation is given in terms of constitution...
J. A. E. Bell W. C. Winegard
1964-01-01
103
The gun technique of splat cooling is utilized to extend the solid solubility of manganese in aluminium by a factor of four above that at the eutectic temperature. The supersaturated solid solutions can be retained up to 250° C without any significant decomposition. Isochronal and isothermal studies of the variation of the lattice parameter of the 6.4 wt % manganese
S. P. Bhat; T. R. Ramachandran; A. K. Jena
1974-01-01
104
A physically based model for predicting recrystallization microstructures and textures after hot deformation of aluminium is presented. The modelling approach taken differs from similar models developed for steels. The present model is based on recent experimental investigations directed towards identifying the nature of the nucleation sites for recrystallized grains of different crystallographic orientations. Particle stimulated nucleation, nucleation from cube bands
H. E. Vatne; T. Furu; R. Ørsund; E. Nes
1996-01-01
105
PubMed
We are living in an "aluminium age" with increasing bioavailability of the metal for approximately 125 years, contributing significantly to the aluminium body burden of humans. Over the course of life, aluminium accumulates and is stored predominantly in the lungs, bones, liver, kidneys and brain. The toxicity of aluminium in humans is briefly summarised, highlighting links and possible causal relationships between a high aluminium body burden and a number of neurological disorders and disease states. Aluminium salts have been used as depot-adjuvants successfully in essential prophylactic vaccinations for almost 100 years, with a convincing positive benefit-risk assessment which remains unchanged. However, allergen-specific immunotherapy commonly consists of administering a long-course programme of subcutaneous injections using preparations of relevant allergens. Regulatory authorities currently set aluminium limits for vaccines per dose, rather than per treatment course. Unlike prophylactic vaccinations, numerous injections with higher proportions of aluminium-adjuvant per injection are applied in subcutaneous immunotherapy (SCIT) and will significantly contribute to a higher cumulative life dose of aluminium. While the human body may cope robustly with a daily aluminium overload from the environment, regulatory cumulative threshold values in immunotherapy need further addressing. Based on the current literature, predisposing an individual to an unusually high level of aluminium, such as through subcutaneous immunotherapy, has the potential to form focal accumulations in the body with the propensity to exert forms of toxicity. Particularly in relation to longer-term health effects, the safety of aluminium adjuvants in immunotherapy remains unchallenged by health authorities - evoking the need for more consideration, guidance, and transparency on what is known and not known about its safety in long-course therapy and what measures can be taken to prevent or minimise its risks. The possibility of providing an effective means of measuring aluminium accumulation in patients undergoing long-term SCIT treatment as well as reducing their aluminium body burden is discussed. PMID:24892252
Kramer, Matthias F; Heath, Matthew D
2014-07-16
106
NSDL National Science Digital Library
Math can be fun! Practice your addtion facts in one of these amazing games! Go on a mission with math! -- Addition Mission Count your apples! -- Apple Addition Be a math teacher! -- Chalkboard Addition ...
Staggs, Ms.
2008-04-01
107
PubMed
The toxicity of aluminium to fish is related to interactions between aluminium and the gill surface. We investigated the possible effect of water ionic strength on this interaction. The mortality of brown trout (Salmo trutta L.) exposed to three different degrees of Al polymerisation was compared in water with increased ionic strength (mean 7.31 x 10(-4) M) after additions of the base cations Ca2+, Mg2+, Na+ or K+, and in water with no such addition (mean ionic strength 5.58 x 10(-4) M). Only a very slight ameliorating effect of increased ionic strength was observed, while the degree of Al polymerisation was of major importance in fish mortality. In addition, it was observed that smaller fish survived the Al exposures for a longer time than larger fish. We hypothesise that this is because larger fish are more susceptible to hypoxia than smaller fish. PMID:15519464
Alstad, Nina E W; Kjelsberg, Birgitte M; Vøllestad, L Asbjørn; Lydersen, Espen; Poléo, Antonio B S
2005-01-01
108
NSDL National Science Digital Library
I want you to practice your awesome addition skills on the websites below. I know you will do beautifully! Start by going to this website first: Addition Fun!!! Once you finish that website you may continue onto this website: Alien Addition If you have extra time play Around the World Addition: Once you have clicked on the link, click on the button for addition.After that you can choose to play on easy or ...
Bolton, Ms.
2009-04-22
109
PubMed Central
The objective of the present study was to investigate the use of propranolol–magnesium aluminium silicate intercalated complexes as drug reservoirs in hydroxypropylmethylcellulose tablets. The matrix tablets containing the complexes were prepared and characterised with respect to propranolol release and were subsequently compared with those loading propranolol or a propranolol–magnesium aluminium silicate physical mixture. Additionally, the effects of varying viscosity grades of hydroxypropyl methylcellulose, compression pressures and calcium acetate incorporation on the drug release characteristics of the complex-loaded tablets were also examined. The results showed that the complex-loaded tablets have higher tablet hardness than those containing propranolol or a physical mixture. The drug release from the complex-loaded tablets followed a zero-order release kinetic, whereas an anomalous transport was found in the propranolol or physical mixture tablets. The drug release rate of the complex tablet significantly decreased with increasing hydroxypropylmethylcellulose viscosity grade. Increase in the compression pressure caused a decrease in the drug release rate of the tablets. Furthermore, the incorporation of calcium ions could accelerate propranolol release, particularly in acidic medium, because calcium ions could be exchanged with propranolol molecules intercalated in the silicate layers of magnesium aluminium silicate. These findings suggest that propranolol-magnesium aluminium silicate intercalated complexes show strong potential for use as drug reservoirs in matrix tablets intended for modifying drug release.
Pongjanyakul, T.; Rojtanatanya, S.
2012-01-01
110
MedlinePLUS
... appealing Direct additives may be man-made or natural. Natural additives include: Adding herbs or spices to foods ... appearance of foods. Many spices, as well as natural and man-made flavors, bring out the taste ...
111
The mode I fatigue crack propagation (FCP) response of the closed-cell aluminium alloy foams Alulight and Alporas have been measured for a relative density in the range 0.1 to 0.4. The validity of linear elastic fracture mechanics (LEFM) to characterise the fatigue crack propagation (FCP) response is demonstrated, and K-increasing and K-decreasing tests are used to determine the full shape
O. B. Olurin; K. Y. G. McCullough; N. A. Fleck; M.F. Ashby
2001-01-01
112
A technique to measure the force-penetration behaviour of projectiles penetrating metal targets has been developed and used to examine the penetration characteristics of projectiles impacting soft 2S aluminium.The maximum force measured during complete penetration of a plate of given thickness was found to be essentially independent of projectile geometry, but the geometry influenced the total energy required for the same
A L Wingrove
1972-01-01
113
The cross section for atomic displacement has been determined at 7.5°K by means of electrical resistivity measurements in electron irradiated aluminium up to transferred energies of 1100 eV. These data and those from the literature have been evaluated with respect to the displacement function (DF). Below 200 eV the DF could be derived from the data with sufficient accuracy by
J. Wurm; F. Dworschak; H. Schuster; H. Wollenberger
1970-01-01
114
Aluminium sheet drawing processes can be improved by manipulating local flow behaviour\\u000aby means of elevated temperatures and temperature gradients in the tooling. Forming tests\\u000ashowed that a substantial improvement is possible not only for 5xxx but also for 6xxx series\\u000aalloys. Finite element method simulations can be a powerful tool for the design of warm\\u000aforming processes and tooling.
P. J. Bolt; R. J. Werkhoven; Boogaard van den A. H
2003-01-01
115
The Kumar-Samarin technique of centrifuging was adopted to study the structure of aluminium-silicon alloys in terms of the concept of the existence of clusters in liquid state. The investigation shows that in hypo-eutectic alloys the clusters are denser than the monatomic matrix whilst in hyper-eutectic alloys they are lighter. It is suggested that this difference in the distribution of silicon
Manjit Singh; Rajendra Kumar
1973-01-01
116
A TiB2 particle (61 vol%, 4 µm mean size) reinforced aluminium fabricated by liquid-aluminium infiltration was subjected to unlubricated rolling wear and was found from the weight loss to be 1.5 times more wear resistant than 17-4 ph stainless steel, twice as wear resistant as 1020 steel, 7.5 times more wear resistant than 2024 aluminium, and 12.8 times more wear
A. V. Smith; D. D. L. Chung
1996-01-01
117
We have synthesised 11-Å tobermorite hydrothermally, both pure and with increasing isomorphic substitution of aluminium for silicon. The samples were analysed by X-ray photoelectron spectroscopy (XPS). Aluminium was found, on the basis of its Al 2p binding energies, to be tetrahedrally coordinated. We observed no changes in Ca\\/(Si+Al) ratio upon aluminium substitution, implying that charge balancing does not occur via
Leon Black; Andreas Stumm; Krassimir Garbev; Peter Stemmermann; Keith R. Hallam; Geoffrey C. Allen
2005-01-01
118
NASA Astrophysics Data System (ADS)
Lithium aluminium borate glasses doped with CeO2 at the expense of Al2O3 have been prepared by melt quench technique. An intense fluorescence has been recorded when excited at 350 nm giving a large Stoke's shift. This emission has been attributed to 5d ? 4f transitions of Ce3+. An increase in fluorescence intensity, shifting of UV absorption edge towards longer wavelength and a decrease in band gap with addition of CeO2 concentration has been observed. Moreover densification and stabilization of glass network has been observed which is due to conversion of BO3 units to BO4 units since CeO2 at low concentrations act as a network modifier.
Kaur, Parvinder; Singh, Gurinder Pal; Kaur, Simranpreet; Singh, D. P.
2012-08-01
119
NASA Astrophysics Data System (ADS)
This paper discusses the mechanical properties of Titanium Carbide (TiC) particulate reinforced aluminium-silicon alloy matrix composite. TiC particulate reinforced LM6 alloy matrix composites were fabricated by carbon dioxide sand molding process with different particulate weight fraction. Tensile strength, hardness and microstructure studies were conducted to determine the maximum load, tensile strength, modulus of elasticity and fracture surface analysis have been performed to characterize the morphological aspects of the test samples after tensile testing. Hardness values are measured for the TiC reinforced LM6 alloy composites and it has been found that it gradually increases with increased addition of the reinforcement phase. The tensile strength of the composites increased with the increase percentage of TiC particulate.
Sayuti, M.; Sulaiman, S.; Baharudin, B. T. H. T.; Arifin, M. K. A.; Suraya, S.; Vijayaram, T. R.
2011-01-01
120
NASA Astrophysics Data System (ADS)
Aluminium alloys are some of the predominant metals in industrial applications such as production of heat exchangers, heat pumps. They have high heat conductivity coupled with a low specific weight. In cold working conditions, there is a risk of frost formation on the surface of aluminium in the presence of water vapour, which can lead to the deterioration of equipment performance. This work addresses the methods of surface modification of aluminium and their effect of the underlying surface morphology and wettability, which are the important parameters for frost formation. Three groups of real-life aluminium surfaces of different morphology: unpolished aluminium, polished aluminium, and aluminium foil, were subjected to surface modification procedures which involved the formation of a layer of hydrophilic hyperbranched polyethyleneglycol via in situ polymerization, molecular vapour deposition of a monolayer of fluorinated silane, and a combination of those. The effect of these surface modification techniques on roughness and wettability of the aluminium surfaces was elucidated by ellipsometry, contact angle measurements and atomic force microscopy. We demonstrated that by employing different types of surface modifications the contact angle of water droplets on aluminium samples can be varied from 12° to more than 120°. A crossover from Cassie-Baxter to Wenzel regime upon changing the surface roughness was also observed.
Rahimi, M.; Fojan, P.; Gurevich, L.; Afshari, A.
2014-03-01
121
PubMed
The ability of aluminium to inhibit the (Na(+)/K(+))ATPase activity has been observed by several authors. During chronic dietary exposure to AlCl3, brain (Na(+)/K(+))ATPase activity drops, even if no alterations of catalytic subunit protein expression and of energy charge potential are observed. The aluminium effect on (Na(+)/K(+))ATPase activity seems to implicate the reduction of interacting protomers within the oligomeric ensemble of the membrane-bound (Na(+)/K(+))ATPase. The activity of (Na(+)/K(+))ATPase is altered by the microviscosity of lipid environment. We studied if aluminium inhibitory effect on (Na(+)/K(+))ATPase is modified by alterations in synaptosomal membrane cholesterol content. Adult male Wistar rats were submitted to chronic dietary AlCl3 exposure (0.03 g/day of AlCl3) and/or to colestipol, a hypolidaemic drug (0.31 g/day) during 4 months. The activity of (Na(+)/K(+))ATPase was studied in brain cortex synaptosomes with different cholesterol contents. Additionally, we incubate synaptosomes with methyl-?-cyclodextrin for both enrichment and depletion of membrane cholesterol content, with or without 300 ?M AlCl3. This enzyme activity was significantly reduced by micromolar AlCl3 added in vitro and when aluminium was orally administered to rats. The oral administration of colestipol reduced the cholesterol content and concomitantly inhibited the (Na(+)/K(+))ATPase. The aluminium inhibitory effect on synaptosomal (Na(+)/K(+))ATPase was reduced by cholesterol depletion both in vitro and in vivo. PMID:23829947
Silva, V S; Oliveira, L; Gonçalves, P P
2013-11-01
122
Aluminium alloys containing small additions of both tin (?0.1wt%) and gallium (?0.05wt%) are shown to dissolve anodically at high rates in sodium chloride media at room temperatures; current densities >0.2Acm?2 can be obtained at potentials close to the open circuit potential, ??1500mV versus SCE. The tin exists in the alloys as a second phase, typically as ?1?m inclusions (precipitates) distributed
Maria Nestoridi; Derek Pletcher; Robert J. K. Wood; Shuncai Wang; Richard L. Jones; Keith R. Stokes; Ian Wilcock
2008-01-01
123
To assess the risk of aluminium (Al) toxicity during the restoration of the eutrophic lake Tiefwarensee by hypolimnetic addition of NaAl(OH)4-solution (aluminate) the generally limnological monitoring was accompanied by fractionation of Al in water and using Al accumulation on fish gills as bioindicator. The concentration of reactive Al species in the alkaline water (pH 8) peaked at 2mgL?1 in parts
Gerlinde Wauer; Hans-Christian Teien
2010-01-01
124
Several methods of manufacturing ?-aluminium oxynitride spinel (ALON) powder are discussed in this paper. In the carbothermal\\u000a reduction process, ALON is an intermediate compound, and powders with a high ALON content may be produced in flowing nitrogen\\u000a below 1650 C if process parameters are carefully controlled. In addition, ALON may be produced when Al2O3\\/C mixtures are\\u000a heated in a coke
LI YAWEI; LI NAN; YUAN RUNZHANG
1997-01-01
125
. ?Strontium modified aluminium–silicon cast alloys are well known for their outstanding mechanical properties as they combine\\u000a excellent strength with good ductility that is due to a modification of brittle Si in the eutectic with traces of Sr (0.3–0.5?wt%).\\u000a Although the level of Sr addition is very low, formation of ternary AlSiSr phases with deleterious effects on the ductility\\u000a can take
Katharina Gammer; Erhard Ogris; Peter J. Uggowitzer; Herbert Hutter
2003-01-01
126
In recent years, a lot of process innovations are strongly required for cost and weight reductions in addition to upgrading driving performance and reducing greenhouse grasses in automotive industry. In case of manufacturing automotive parts, a number of materials are used to satisfy their high accuracy and cost reduction. Dissimilar metal joint such as combination of aluminium alloy and austenitic
Hideaki Shirai; Masahito Mochizuki; Masao Toyoda
2012-01-01
127
Ellipsometry was used to investigate the effect of polyaluminium chloride (PAC) formulations of different degrees of hydrolysation on an adsorbed mucin film. The results were compared to the effect of aluminium chloride (AlCl3) and ferric chloride. A compaction of the mucin film took place upon addition of the formulations and this occurred to different extents and at different concentrations for
Jildiz Hamit-Eminovski; Krister Eskilsson; Thomas Arnebrant
2010-01-01
128
PubMed Central
Background Beryllium exposure occurs in aluminium smelters from natural contamination of bauxite, the principal source of aluminium. Aims To characterize beryllium exposure in aluminium smelters and determine the prevalence rate of beryllium sensitization (BeS) among aluminium smelter workers. Methods A population of 3185 workers from nine aluminium smelters owned by four different aluminium-producing companies were determined to have significant beryllium exposure. Of these, 1932 workers participated in medical surveillance programmes that included the serum beryllium lymphocyte proliferation test (BeLPT), confirmation of sensitization by at least two abnormal BeLPT test results and further evaluation for chronic beryllium disease in workers with BeS. Results Personal beryllium samples obtained from the nine aluminium smelters showed a range of <0.01–13.00 ?g/m3 time-weighted average with an arithmetic mean of 0.25 ?g/m3 and geometric mean of 0.06 ?g/m3. Nine workers were diagnosed with BeS (prevalence rate of 0.47%, 95% confidence interval = 0.21–0.88%). Conclusions BeS can occur in aluminium smelter workers through natural beryllium contamination of the bauxite and further concentration during the refining and smelting processes. Exposure levels to beryllium observed in aluminium smelters are similar to those seen in other industries that utilize beryllium. However, compared with beryllium-exposed workers in other industries, the rate of BeS among aluminium smelter workers appears lower. This lower observed rate may be related to a more soluble form of beryllium found in the aluminium smelting work environment as well as the consistent use of respiratory protection.
Slade, M. D.; Cantley, L. F.; Kirsche, S. R.; Wesdock, J. C.; Cullen, M. R.
2010-01-01
129
PubMed Central
Background Infant formulas are sophisticated milk-based feeds for infants which are used as a substitute for breast milk. Historically they are known to be contaminated by aluminium and in the past this has raised health concerns for exposed infants. We have measured the aluminium content of a number of widely used infant formulas to determine if their contamination by aluminium and consequent issues of child health persists. Methods Samples of ready-made milks and powders used to make milks were prepared by microwave digestion of acid/peroxide mixtures and their aluminium content determined by THGA. Results The concentration of aluminium in ready-made milks varied from ca 176 to 700 ?g/L. The latter concentration was for a milk for preterm infants. The aluminium content of powders used to make milks varied from ca 2.4 to 4.3 ?g/g. The latter content was for a soya-based formula and equated to a ready-to-drink milk concentration of 629 ?g/L. Using the manufacturer's own guidelines of formula consumption the average daily ingestion of aluminium from infant formulas for a child of 6 months varied from ca 200 to 600 ?g of aluminium. Generally ingestion was higher from powdered as compared to ready-made formulas. Conclusions The aluminium content of a range of well known brands of infant formulas remains high and particularly so for a product designed for preterm infants and a soya-based product designed for infants with cow's milk intolerances and allergies. Recent research demonstrating the vulnerability of infants to early exposure to aluminium serves to highlight an urgent need to reduce the aluminium content of infant formulas to as low a level as is practically possible.
2010-01-01
130
PubMed
A cryopreservation procedure using an aluminium cryo-plate was successfully developed using in vitro-grown Dalmatian chrysanthemum (Tanacetum cinerariifolium) shoot tips. Shoot cultures were cold-hardened at 5 degree C on MS medium containing 0.5 M sucrose over a period of 20 to 40 days. Shoot tips with basal plate (1.0-1.5 x 1.0 mm) were dissected from shoot cultures and precultured at 5 degree C for 2 days on MS medium containing 0.5 M sucrose. Precultured shoot tips were placed on aluminium cryo-plates (7 mm x 37 mm x 0.5 mm) with 10 wells (diameter 1.5 mm, depth 0.75 mm) and embedded in alginate gel. Osmoprotection was performed by immersing the cryo-plates for 30 or 60 min in 25 ml pipetting reservoirs filled with loading solution (2 M glycerol + 1.4 M sucrose). For dehydration, the loading solution was replaced with PVS 7M vitrification solution (30 percent glycerol, 19.5 percent ethylene glycol and 0.6 M sucrose in liquid MS basal medium), which was applied for 40 min. After rapid immersion in liquid nitrogen, shoot tips attached to the cryo-plates were rewarmed by immersion in cryotubes containing 2 ml 1 M sucrose solution. Using this procedure, regrowth of cryopreserved shoot tips of line 28v-75 reached 77 degree. This protocol was successfully applied to six additional lines, with high regrowth percentages ranging from 65 to 90 percent. By contrast, the modified vitrification protocol tested as a reference produced only moderate regrowth percentages. This new method displays many advantages and will facilitate large scale cryostorage in genebank. PMID:21766155
Yamamoto, Shin-ichi; Rafique, Tariq; Priyantha, Wickramage Saman; Fukui, Kuniaki; Matsumoto, Toshikazu; Niino, Takao
2011-01-01
131
A TEM study of unstretched, heat-treated aluminium alloy 2219 was performed with the aim of correlating the microstructure with hardness (H) and eddy current conductivity (C) data. Recently Chihoski has developed a new method of evaluating heat treatments of aluminium alloys by constructing in an H-C field a “sail” formed by a network of curved coordinate lines of quenching and
M. Natan; R. A. Chihoski
1983-01-01
132
The most evident trend in electronics production is towards miniaturization. Regarding the materials involved, another trend can be observed: intelligent combinations of different materials. One example is the combination of copper and aluminium. Copper is the material of choice for electronic packaging applications due to its superior electrical and thermal conductivity. On the other hand, aluminium offers technical and economical
Gerd Esser; Ihor Mys; Michael H. Schmidt
2004-01-01
133
The effect of cold deformation on the lubricated wear of 5083 aluminium alloy was investigated. SAE 10W was selected as liquid lubricant. The aluminium alloy was submitted to a cold rolling process, until the average thickness of the specimens was reduced by 7% and 15% respectively. From the experimental results obtained, the Stribeck curves for the as received and cold
C. N. Panagopoulos; E. P. Georgiou
2010-01-01
134
NASA Astrophysics Data System (ADS)
We use X-ray photoelectron spectroscopy (XPS), ultraviolet photoelectron spectroscopy (UPS) and metastable impact electron spectroscopy (MIES) to investigate the corrosion of aluminium components. Clean aluminium films were prepared under ultrahigh vacuum (UHV) conditions and exposed to water and NaCl. We attempt to provide a model for the mechanism of this interaction and its effects on the durability of the components.
Frerichs, Martin; Voigts, Florian; Hollunder, Sven; Masendorf, Rainer; Esderts, Alfons; Maus-Friedrichs, Wolfgang
2005-09-01
135
The coefficient of friction for steel-aluminium contact surfaces has been determined. The test was conducted by using a testing machine active on the basis of the twist-compression test. A flat plate of aluminium was placed under pressure between two steel dies. One of the dies (the upper one) was capable of rotating while the other (the lower one) was stationary
M. JAVADI; M. TAJDARI
2006-01-01
136
New production technologies like the hydroforming of sheet metal pairs have to be qualified for both steel and aluminium alloy sheets. While good mechanical properties and the excellent recyclability make sheets from aluminium alloys a good choice of material for lightweight construction, the lower formability compared to steel sheets raises various challenges for part manufacturing. Forming problems in the hydroforming
S. Novotny; P. Hein
2001-01-01
137
This article provides an overview of energy loads associated with the application of powder coated aluminium profiles on the exterior wooden surface of double glazed windows. The entire powder coating process has been analysed to measure its energy and environmental contribution towards window cladding. Materials involved in aluminium clad windows have been analysed for their energy impacts. The energy embodied
M. Asif; A. Davidson; T. Muneer
2001-01-01
138
PubMed
Aluminium salt slag (also known as aluminium salt cake), which is produced by the secondary aluminium industry, is formed during aluminium scrap/dross melting and contains 15-30% aluminium oxide, 30-55% sodium chloride, 15-30% potassium chloride, 5-7% metallic aluminium and impurities (carbides, nitrides, sulphides and phosphides). Depending on the raw mix the amount of salt slag produced per tonne of secondary aluminium ranges from 200 to 500 kg. As salt slag has been classified as toxic and hazardous waste, it should be managed in compliance with the current legislation. Its landfill disposal is forbidden in most of the European countries and it should be recycled and processed in a proper way by taking the environmental impact into consideration. This paper presents a review of the aluminium salt slag chemical and mineralogical characteristics, as well as various processes for metal recovery, recycling of sodium and potassium chlorides content back to the smelting process and preparation of value added products from the final non metallic residue. PMID:22480708
Tsakiridis, P E
2012-05-30
139
Two novel automatic Gas Tungsten Arc Welding (GTAW) systems have been developed at National Synchrotron Radiation Research Center for welding Taiwan Photon Source (TPS) aluminium bending chambers. One automatic system is designed by using two welding torches to implement two parallel welds on each side of an aluminium chamber simultaneously. The other is to use an industrial 6-axis robot with
C L Chen; S N Hsu; C C Chang; C K Kuan; Y B Chen; C Y Yang; C K Chan; H P Hsueh; G. Y. Hsiung; J. R. Chen
140
An image sensing system for the TIG (tungsten inert-gas arc) welding process of aluminium alloy was established. The relationships between the image sensing system and the characteristic of welding current were discussed in detail. Front and back images of the weld pool were obtained with different welding parameters. In order to process the image, the characteristics of an aluminium alloy
J. J. Wang; T. Lin; S. B. Chen
2005-01-01
141
NASA Astrophysics Data System (ADS)
Dissolved aluminium (Al) occurs in a wide range of concentrations in the world oceans. The concentrations of Al in the Southern Ocean are among the lowest ever observed. An all-titanium CTD sampling system makes it possible to study complete deep ocean sections of Al and other trace elements with the same high vertical resolution of 24 depths as normal for traditional CTD/Rosette sampling. Overall, 470 new data points of Al are reported for 22 full depth stations and 24 surface sampling positions along one transect. This transect consisted of 18 stations on the zero meridian proper from 51°57' S until 69°24'S, and 4 stations somewhat to the northeast towards Cape Town from 42°20'S, 09°E to 50°17'S, 01°27'E. The actual concentrations of Al in the Southern Ocean were lower than previously reported. The concentration of Al in the upper 25 m was relatively elevated with an average concentration of 0.71 nM ( n=22; S.D.=0.43 nM), most likely due to atmospheric input by a suggested combination of direct atmospheric (wet and dry) input and indirect atmospheric input via melting sea ice. Below the surface waters there was a distinct Al minimum with an average concentration of 0.33 nM ( n=22; S.D.=0.13 nM) at an average depth of 120 m. In the deep southernmost Weddell Basin the concentration of Al increased with depth to ˜0.8 nM at 4000 m, and a higher concentration of ˜1.5 nM in the ˜4500-5200 m deep Weddell Sea Bottom Water. Over the Bouvet triple junction region, where three deep ocean ridges meet, the concentration of Al increased to ˜1.4 nM at about 2000 m depth over the ridge crest. In the deep basin north of the Bouvet region the concentration of Al increased to higher deep values of 4-6 nM due to influence of North Atlantic Deep Water. In general the intermediate and deep distribution of Al results from the mixing of water masses with different origins, the formation of deep water and additional input from sedimentary sources at sea floor elevations. No significant correlation between Al and silicate (Si) was observed. This is in contrast to some other ocean regions. In the Southern Ocean the supply of Al is extremely low and any signal from Al uptake and dissolution with biogenic silica is undetectable against the high dissolved Si and low dissolved Al concentrations. Here the Al-Si relation in the deep ocean is uncoupled. This is due to the scavenging and subsequent loss of the water column of Al, whereas the concentration of Si increases in the deep ocean due to its input from deep dissolution of biogenic diatom frustules settling from the surface layer.
Middag, R.; van Slooten, C.; de Baar, H. J. W.; Laan, P.
2011-12-01
142
SciTech Connect
Bulk aluminium has been produced by warm-rolling followed by cold-rolling of commercial purity (99% purity) aluminium foil. The bonding appeared perfect from observation with the naked eye, light and transmission electron microscopy. By comparison with bulk aluminium of similar purity (AA1200) rolled to a similar strain (90%RA), the roll-bonded metal showed a much higher density of high-angle grain boundaries, similar strength and improved thermal stability. This study has implications for a number of applications in relation to the processing of aluminium. Roll bonding is of interest as a method for grain size refinement; oxide-containing materials have increased strength, enhanced work-hardening behaviour, and exhibit alterations in recrystallisation behaviour. The behaviour of the hard oxide film is of interest in aluminium processing, and has been investigated by characterising the size and distribution of oxide particles in the roll-bonded samples.
Barlow, C.Y.; Nielsen, P.; Hansen, N
2004-08-02
143
PubMed
The changes in protein adsorption onto aluminium surfaces coated with different thicknesses of oxide layers were examined. The oxide layers on aluminium substrates were derived by the anodizing technique. Protein adsorption studies were conducted using 125I-labelled albumin and the amount of albumin adsorbed was estimated with the help of a gamma counter. An increase in albumin adsorption was observed on oxide layer coated aluminium surfaces. The effect of anti-Hageman factor on albumin and fibrinogen adsorption on to bare aluminium, oxide layer coated aluminium and bare polyether urethane urea surfaces was also investigated. It was observed that the presence of anti-Hageman factor increased the adsorption of albumin and fibrinogen on to all these substrates. PMID:2383620
Sharma, C P; Sunny, M C
1990-05-01
144
NASA Astrophysics Data System (ADS)
This study investigates the effect of atmospheric pressure plasma torch (APPT) treatments on the surface of aluminium alloys. The influence of torch-to-sample distance, speed of treatment and ageing time is analyzed in terms of contact angles and surface energy. Results show that APPT treatment strongly increases the surface energy and wettability of aluminium surfaces. This is related to the formation of polar groups, as Fourier transform infrared (FTIR) spectroscopy has confirmed. In all conditions, hydrophobic recovery of aluminium surfaces takes place. Finally, the compatibility of the APPT treated aluminium substrate with ?-methacryloxypropyltrimethoxysilane (MPS) has been evaluated through adhesion work and spread tension, showing that it is possible to achieve a spontaneous wetting process of silane on aluminium.
Díaz-Benito, B.; Velasco, F.
2013-12-01
145
NASA Astrophysics Data System (ADS)
The effective thermal conductivity (ETC) of multilayer thermal insulation materials was experimentally investigated as a function of temperature (0-25 °C). The materials consisted of binary/ternary glass wools or ternary expanded polystyrene foams reinforced with aluminium foil. The experimental measurements were performed using a guarded hot plate with temperature differences of 5, 10 and 15 °C. The results indicated that significant correlations exist between ETC and the characteristics of the materials with decreasing temperature. The ETC decreases with reinforcement with aluminium foil at the same temperature or with temperature differences of 5 and 15 °C. In addition, it was clearly observed that the ETC decreases sharply with decreased temperature. Consequently, reflective materials may reduce the ETC at low temperatures.
Yüksel, N.; Avc?, A.; K?l?ç, M.
2012-09-01
146
NASA Astrophysics Data System (ADS)
Ni-Co-fly ash coatings were deposited on zincate treated 5083 wrought aluminium alloy substrates with the aid of the electrodeposition technique. Structural and chemical characterization of the produced composite coatings was performed with the aid of X-ray diffraction (XRD), scanning electron microscopy (SEM) and electron dispersive X-ray analysis (EDS) techniques. The Ni-Co-fly ash coatings were found to consist of a crystalline Ni-Co solid solution with dispersed fly ash particles. In addition, chemical analysis of the Ni-Co matrix showed that it consisted of 80 wt.% Ni and 20 wt.% Co. The co-deposition of fly ash particles leads to a significant increase of the microhardness of the coating. The corrosion behaviour of the Ni-Co-fly ash/zincate coated aluminium alloy, in a 0.3 M NaCl solution (pH = 3.5), was studied by means of potentiodynamic corrosion experiments.
Panagopoulos, C. N.; Georgiou, E. P.; Tsopani, A.; Piperi, L.
2011-03-01
147
NASA Astrophysics Data System (ADS)
A determination of the phase constituents of ceramic coatings produced on Al-Cu-Mg alloy by microarc discharge in alkaline solution was performed using x-ray diffraction. The profiles of the hardness, H, and elastic modulus, E, across the ceramic coating were determined by means of nanoindentation. In addition, a study of the influence of microarc oxidation coatings on the tensile properties of the aluminium alloy was also carried out. The results show that the H-and E-profiles are similar, and both of them exhibit a maximum value at the same depth of coating. The distribution of the ?-Al2O3 phase content determines the H- and E-profiles of the coatings. The tensile properties of 2024 aluminium alloy show less change after the alloy has undergone microarc discharge surface treatment.
Xue, Wenbin; Wang, Chao; Deng, Zhiwei; Chen, Ruyi; Li, Yongliang; Zhang, Tonghe
2002-11-01
148
NASA Astrophysics Data System (ADS)
Recently several shots have been fired on the AWE HELEN laser in order to study the spall of aluminium at high strain rates. In the first shot a radiograph was taken which showed a spall layer had formed. Further shots were fired and the free surface velocity of the aluminium was measured using velocity interferometry. Five of these shots showed spall had occurred. This paper attempts to model these shots using the Johnson spall model. Previously determined spall parameters(Giles, A.R. Maw, J.R. In Shock Compression of Condensed Matter 1997. Eds. Schmidt, S.C. Dandekar, D.P. Forbes, J.W. (The American Institute of Physics, 1998)) are found not to model the spall well, so new spall parameters are determined that match the laser results. The model is then applied to some published low strain rate plate impact experiments(Kanel, G.I. Razorenov, S.V. Utkin, A.V. Baumung, K. Shock Wave Profile Data. Scientific Association IVTAN of Russian Academy of Sciences, 1996), and it is found that the original spall parameters provide the best fit. The mechanism that causes the spall layer to form is different for the two sets of parameters.
Robinson, C. M.
2001-06-01
149
It is established that weld metal saturation by aluminium occurs at welding steels coated by aluminium, and this decreases mechanical properties of the welded joints substantially. Application of the gaseous medium with high oxidizing ability at welding allows for refining weld metal from aluminium and for improving strength and plastic properties of the aluminium-coated steel welded joints.
T. V. Chermashentseva; A. I. Kovtunov
2011-01-01
150
The objective of this work is to study the crushing behavior and the influence of surface treatments on an aluminium alloy before bonding with a carbon\\/epoxy composite. Two geometries of aluminium circular tubes are used, allowing a stable crushing mode. Chemical etching and degreasing were applied to the aluminium tube to improve adhesion between composite and aluminium. To compare the
J Bouchet; E Jacquelin; P Hamelin
2000-01-01
151
SciTech Connect
In this project, a concept to improve the performance of aluminum production cells by introducing potlining additives was examined and tested. Boron oxide was added to cathode blocks, and titanium was dissolved in the metal pool; this resulted in the formation of titanium diboride and caused the molten aluminum to wet the carbonaceous cathode surface. Such wetting reportedly leads to operational improvements and extended cell life. In addition, boron oxide suppresses cyanide formation. This final report presents and discusses the results of this project. Substantial economic benefits for the practical implementation of the technology are projected, especially for modern cells with graphitized blocks. For example, with an energy savings of about 5% and an increase in pot life from 1500 to 2500 days, a cost savings of 0.023 per pound of aluminum produced is projected for a 200 kA pot. Rudolf Keller 2004-08-10 152 PubMed This report represents the conclusions of a Joint FAO/WHO Expert Committee convened to evaluate the safety of various food additives, with a view to recommending acceptable daily intakes (ADIs) and to preparing specifications for identity and purity. The Committee also evaluated the risk posed by two food contaminants, with the aim of deriving tolerable intakes where appropriate and advising on risk management options for the purpose of public health protection. The first part of the report contains a general discussion of the principles governing the toxicological evaluation of and assessment of dietary exposure to food additives and contaminants. A summary follows of the Committee's evaluations of technical, toxicological and dietary exposure data for certain food additives (aluminium-containing food additives, Benzoe Tonkinensis, glycerol ester of gum rosin, glycerol ester of tall oil rosin, glycerol ester of wood rosin, octenyl succinic acid modified gum arabic, polydimethyl siloxane, Ponceau 4R, pullulan, pullulanase from Bacillus deromificans expressed in Bacillus licheniformis, Quinoline Yellow and Sunset Yellow FCF) and two food contaminants (cyanogenic glycosides and fumonisins). Specifications for the following food additives were revised: aluminium lakes of colouring matters; beta-apo-8'-carotenal; beta-apo-8'-carotenoic acid ethyl ester; beta-carotene, synthetic; hydroxypropyl methyl cellulose; magnesium silicate, synthetic; modified starches; nitrous oxide; sodium carboxymethyl cellulose; and sucrose monoesters of lauric, palmitic or stearic acid. Annexed to the report are tables summarizing the Committee's recommendations for dietary exposures to and toxicological evaluations of the food additives and contaminants considered. PMID:22519244 2011-01-01 153 PubMed Central The present investigation envisages the toxic effects of aluminium on the cholinergic system of male albino rat brain. Aluminium toxicity (LD50/24 h) evaluated as per Probit method was found to be 700 mg/kg body weight. One-fifth of lethal dose was taken as the sublethal dose. For acute dose studies, rats were given a single lethal dose of aluminium acetate orally for one day only and for chronic dose studies, the rats were administered with sublethal dose of aluminium acetate once in a day for 25 days continuously. The two constituents of the cholinergic system viz. acetylcholine and acetylcholinesterase were determined in selected regions of rat brain such as cerebral cortex, hippocampus, hypothalamus, cerebellum, and pons-medulla at selected time intervals/days under acute and chronic treatment with aluminium. The results revealed that while acetylcholinesterase activity was inhibited, acetylcholine level was elevated differentially in all the above mentioned areas of brain under aluminium toxicity, exhibiting area-specific response. All these changes in the cholinergic system were subsequently manifested in the behavior of rat exhibiting the symptoms such as adipsia, aphagia, hypokinesia, fatigue, seizures, etc. Restoration of the cholinergic system and overt behavior of rat to the near normal levels under chronic treatment indicated the onset of either detoxification mechanisms or development of tolerance to aluminium toxicity in the animal which was not probably so efficient under acute treatment. Yellamma, K.; Saraswathamma, S.; Kumari, B. Nirmala 2010-01-01 154 PubMed Central Background In studies in Gothenburg, Sweden, in the 1990s of an aluminium hydroxide-adsorbed pertussis toxoid vaccine, 745 of ?76 000 vaccinated children developed long-lasting itchy subcutaneous nodules at the vaccination site. Of 495 children with itchy nodules patch tested for aluminium allergy, 376 (76%) were positive. Objectives To study the prognosis of the vaccine-induced aluminium allergy. Patients and methods Two hundred and forty-one children with demonstrated aluminium allergy in the previous study were patch tested again 5–9 years after the initial test, with the same procedure as used previously. Results Contact allergy to aluminium was no longer demonstrable in 186 of the retested 241 children (77%). A negative test result was more common in children who no longer had itching at the vaccination site; it was also related to the age of the child, the time after the first aluminium-adsorbed vaccine dose, and the strength of the reaction in the first test. Conclusions Patch test reactivity to aluminium seems to disappear or weaken with time. Gente Lidholm, Anette; Bergfors, Elisabet; Inerot, Annica; Blomgren, Ulla; Gillstedt, Martin; Trollfors, Birger 2013-01-01 155 NSDL National Science Digital Library The following sites will provide some review and practice adding fractions. These links will help with finding common denominators. Equivalent Fractions Fraction Addition-Common Denominators Do some practice with visuals. Add Fractions with Like Denominators Add Fractions with Unlike Denominators See how fast you can get through these problems! Race the Clock-Add with Like Denominators Race the Clock-Add with Unlike Denominators Race the Clock-Add with Mixed Numbers Do you need a \\"How to...\\" refresher? Look here! How Do You Do That? Add with Like Demoninators How Do You Do That? Add with Unlike Demoninators How Do You Do That? Add with Mixed Numbers ... Romney, Mrs. 2005-06-18 156 Microsoft Academic Search The diffusion welding of carbon steel to aluminium may be achieved when the temperature and compression of the weld metal\\u000a allows diffusion at the interface between both components; the degree of interdiffusion largely determines the quality of\\u000a the welded joint. Such diffusion processes can take place under vacuum. However, in this paper, we report that high quality\\u000a welds can be M. Kamal Karfoul; Gordon J. Tatlock; Robert T. Murray 2007-01-01 157 NASA Astrophysics Data System (ADS) De nouveaux gels hybrides "organique—inorganique" à base d'aluminium sont élaborés par voie sol—gel. Le précurseur moléculaire utilisé est le butylate secondaire d'aluminium Al(OBu S) 3 dissous dans CCl 4, modifié par le propane-1,2-diol, le butane-1, x-diol ( x = 2, 3), et le pentane-1,2-diol. Nous avons étudié l'influence des principaux facteurs (nature du solvant, nature et quantité du diol) sur l'obtention des gels monolithiques et transparents. Toutes les préparations sont réalisées sans ajout d'eau. L'étude par RMN MAS 27Al, RMN CP MAS 13C, et par spectroscopie IR montre qu'une réaction d'échange a lieu entre les groupements OBus de l'alcoxyde d'aluminium et les diols utilisés. Les gels obtenussont monolithiques et transparents, les atomes d'aluminium y sont liés entre eux par des ponts organiques selon la formule —Al—O—R —O—Al— où R est une chaîne hydrocarbonée provenant du diol. This paper presents an original method for elaboration of gels with mixed organic—inorganic networks of the general formula —Al—O—R—O—Al—, with R being a hydrocarbon chain. Starting materials used are aluminum tri-sec-butoxide Al(O-sec-C 4H 9) 3, propane-1,2-diol, butane-1,2-diol, butane-1,3-diol, and pentane-1,2-diol. The reactions were carried out with either CCl 4 or cyclohexane as solvent. The exchange reaction between the diols and Al(O-sec-C 4H 9) 3 occurs with the elimination of sec-butanol molecules. Gels are obtained without the addition of water. The monolithic and transparent gels obtained are characterized by IR and as well as 13C and 27Al RMN techniques. Touati, F.; Gharbi, N.; Zarrouk, H. 1995-11-01 158 PubMed Aluminium compounds have been used as adjuvants in practical vaccination for more than 60 years to induce an early, an efficient and a long lasting protective immunity and are at present the most widely used adjuvants in both veterinary and human vaccines. Although the last two decades of systematic research into the nature of these adjuvants has contributed significantly to understanding their nature and their limitations as Th2 stimulators the more detailed mode of action of these adjuvants is still not completely understood. We have a comprehensive record of their behaviour and performance in practical vaccination, but an empirical approach to optimising their use in new vaccine formulations is still to some extent a necessity. The aim of the present review is to put the recent findings into a broader perspective to facilitate the application of these adjuvants in general and experimental vaccinology. PMID:15315845 Lindblad, Erik B 2004-09-01 159 PubMed Central A new patterning method using Deoxyribose Nucleic Acid (DNA) strands capable of producing nanogaps of less than 100 nm is proposed and investigated in this work. DNA strands from Bosenbergia rotunda were used as the fundamental element in patterning DNA on thin films of aluminium (Al) metal without the need for any lithographic techniques. The DNA strands were applied in buffer solutions onto thin films of Al on silicon (Si) and the chemical interactions between the DNA strands and Al creates nanometer scale arbitrary patterning by direct transfer of the DNA strands onto the substrate. This simple and cost-effective method can be utilized in the fabrication of various components in electronic chips for microelectronics and Nano Electronic Mechanical System (NEMS) applications in general. Khatir, Nadia Mahmoudi; Banihashemian, Seyedeh Maryam; Periasamy, Vengadesh; Majid, Wan Haliza Abd; Rahman, Saadah Abdul; Shahhosseini, Fatemeh 2011-01-01 160 National Technical Information Service (NTIS) During hot forming of aluminium alloys, grain size control is of vital importance for the quality of the final product. A coarse grained material may result in orange peeling when bending and reduced strength and ductility, important mechanical properties... R. Oersund O. Lohne 1991-01-01 161 NSDL National Science Digital Library In this brochure the European Aluminium Association (EAA) outlines the advantages of aluminum use in road transport. These advantages include environmental, revenue, energy, material composition, recycling, sustainability, and safety. European Aluminium Association (EAA) 162 NASA Astrophysics Data System (ADS) Plasma route to nanofabrication has drawn much attention recently. The dense plasma focus (DPF) device is used for depositing aluminium nanoparticles on n-type Si (111) wafer. The plasma chamber is filled with argon gas and evacuated at a pressure of 80 Pa. The substrate is placed at distances 4.0 cm, 5.0 cm and 6.0 cm from the top of the central anode. The aluminium is deposited on Si wafer at room temperature with two focused DPF shots. The deposits on the substrate are examined for their morphological properties using atomic force microscopy (AFM). The AFM images have shown the formation of aluminium nanoparticles. From the AFM images, it is found that the size of aluminium nanoparticles increases with increase in distance between the top of anode and the substrate for same number of DPF shots. Bilasini Devi, Naorem; Roy, Savita; Srivastava, M. P. 2010-02-01 163 Microsoft Academic Search Industrial use of non-ferrous materials is significant in the electrical and the chemical engineering applications. Copper–aluminium\\u000a joints are inevitable for certain applications due to unique performances such as higher electric conductivity, heat conductivity,\\u000a corrosion resistance and mechanical properties. Friction welding is the most common method used due to material and energy\\u000a saving. In the present study, copper and aluminium materials Mumin Sahin 2010-01-01 164 Microsoft Academic Search Residual stresses are detrimental to the fatigue, fracture and corrosion resistance of welds. The literature on residual stress\\u000a measurements in aluminium alloy friction stir welds is reviewed. The results of a large number of longitudinal residual stress\\u000a measurements performed by the slitting method on friction stir welds in 2024-T3, 6082-T6 and 5754-H111 aluminium alloys are\\u000a compared and their origin discussed. Kevin Deplus; Aude Simar; Wim Van Haver; Bruno de Meester 165 Microsoft Academic Search A total of 256 individual food samples were collected in Hong Kong for aluminium testing. Most of food samples were analysed in ready-to-eat form. High aluminium levels were found in steamed bread\\/bun\\/cake (mean: 100–320 mg kg), some bakery products such as muffin, pancake\\/waffle, coconut tart and cake (mean: 250, 160, 120 and 91 mg kg, respectively), and jellyfish (ready-to-eat form) Waiky W. K. Wong; Stephen W. C. Chung; K. P. Kwong; Yuk Yin Ho; Ying Xiao 2010-01-01 166 Microsoft Academic Search Aluminium-silicon alloys and aluminium-based metal-matrix composites (MMCs) containing hard particles offer superior operating performance and resistance to wear. In industrial processes where abrasive slurries are transported by rotating paddles or impellers, clements fabricated from MMC materials provide higher abrasive resistance and therefore a longer service life compared to those made from iron or nickel-based alloys. Composites characterized by a hardness R. L. Deuis; C. Subramanian; J. M. Yellup 1996-01-01 167 Microsoft Academic Search A theoretical model for porous structure formation during electrochemical anodization of aluminium is developed taking into account anodic oxide growth at the oxide\\/electrolyte and oxide\\/aluminium interfaces, electrochemical and field-enhanced oxide dissolution, and a three-dimensional configuration of electric field and current in the scalloped barrier oxide layer at pore bottoms. A system of equations ensuring a general description of the porous V. P. Parkhutik; V. I. Shershulsky 1992-01-01 168 Microsoft Academic Search Improving the formability of aluminium alloy sheet metal by using warm or elevated temperature has become a valid approach.\\u000a In this paper, viscous pressure bulging (VPB) at warm temperature is proposed. The coupled thermo-mechanical finite element\\u000a method and experimental method were used to investigate the VPB of aluminium alloy AA3003 at warm temperature. The temperature\\u000a distributions of sheet metal and Jianguang Liu; Qiucai Peng; Yan Liu; Zhongjin Wang 2007-01-01 169 Microsoft Academic Search A metallographic investigation of as-cast LM-13 aluminium-silicon alloy, solidified at different cooling rates (using permanent moulds or a single-roll melt spinner), is presented with special reference to the modification of eutectic silicon. and the refinement of primary aluminium. The refinement of microstructure with the increase in cooling rate is mainly attributed to the limited growth kinetics of the nucleated phase S. Das; A. H. Yegneswaran; P. K. Rohatgi 1987-01-01 170 Microsoft Academic Search Partial for aluminium and integral enthalpies of mixing of liquid Al–Cu–Ge alloys have been measured by high-temperature isoperibolic calorimetry at T=(1380±5) K. Thermodynamic activities, partial for aluminium and integral Gibbs free energies of mixing have been determined by the electromotive force method (emf) over the range 1050?T\\/K?1250. Integral excess entropies of mixing have been calculated from calorimetric enthalpies and emf Dmitry S. Kanibolotsky; Nataliya V. Kotova; Olena A. Bieloborodova; Vladyslav V. Lisnyak 2003-01-01 171 PubMed Ferrous and non-ferrous metal scraps are increasingly recovered from municipal solid waste incineration bottom ash and used in the production of secondary steel and aluminium. However, during the incineration process, metal scraps contained in the waste undergo volatilisation and oxidation processes, which determine a loss of their recoverable mass. The present paper evaluates the behaviour of different types of aluminium packaging materials in a full-scale waste to energy plant during standard operation. Their partitioning and oxidation level in the residues of the incineration process are evaluated, together with the amount of potentially recoverable aluminium. About 80% of post-consumer cans, 51% of trays and 27% of foils can be recovered through an advanced treatment of bottom ash combined with a melting process in the saline furnace for the production of secondary aluminium. The residual amount of aluminium concentrates in the fly ash or in the fine fraction of the bottom ash and its recovery is virtually impossible using the current eddy current separation technology. The average oxidation levels of the aluminium in the residues of the incineration process is equal to 9.2% for cans, 17.4% for trays and 58.8% for foils. The differences between the tested packaging materials are related to their thickness, mechanical strength and to the alloy. PMID:22749723 Biganzoli, Laura; Gorla, Leopoldo; Nessi, Simone; Grosso, Mario 2012-12-01 172 PubMed Carbonyl sulfide is found as a major sulfur compound in anodic gases of commercial aluminium electrolysis. Recent spectroscopic measurements on industrial aluminium smelters found typical CO/COS ratios between 80 and 200. This results in specific COS emissions of between 1 and 7 kg/t(Al) if all COS is released into the atmosphere. In 1993 aluminium production would have been responsible for between 0.02 and 0.14 Tg of COS emissions. Currently, aluminium production does not seem to influence the total atmospheric COS budget to an extent beyond its natural variability. If recent growth rates of global aluminium production are sustained, however, COS emissions would quadruple until 2030. Together with increasing aircraft emissions into the stratosphere, an increase of the sulfate background aerosol is to be expected that could significantly enhance ozone depletion. The use of inert anodes is recommended to reduce aluminium production emissions of COS and CF4, C2F6, CO2, and CO at the same time. PMID:24234611 Harnisch, J; Borchers, R; Fabian, P; Kourtidis, K 1995-11-01 173 NASA Astrophysics Data System (ADS) Nanostructures of cubic aluminium nitride were synthesized by DC arc-plasma-induced melting of aluminium in a nitrogen-argon ambient. The material flux ejected from the molten aluminium surface was found to react with nitrogen under highly non-equilibrium conditions and subsequently condense on a water-cooled surface to yield a mixture of nanowires and nanoparticles of crystalline cubic aluminium nitride. Both x-ray diffraction and electron diffraction measurements revealed that the as-synthesized nitrides adopted the cubic phase. Fourier transform infrared spectroscopy was used to understand the bonding configuration. Microstructural features of the synthesized material were best studied by transmission electron microscopy. From these analyses cubic aluminium nitride was found to be the dominating phase for both nanowires and nanoparticles synthesized at low currents. The typical particle size distribution was found to range over 15-80 nm, whereas the wires varied from 30 to 100 nm in diameter and 500 to 700 nm in length, depending upon the process parameters such as arc current and the nitrogen pressure. The reaction products inside the plasma zone were also obtained theoretically by minimization of free energy and the favourable zone temperature necessary for the formation of aluminium nitride was found to be {\\sim } 6000 K. Results are discussed in view of the highly non-equilibrium conditions that prevail during the arc-plasma synthesis. Balasubramanian, C.; Godbole, V. P.; Rohatgi, V. K.; Das, A. K.; Bhoraskar, S. V. 2004-03-01 174 NASA Astrophysics Data System (ADS) Solute atom nanoscale precipitates are responsible for the favourable mechanical properties of heat treatable aluminium alloys such as Al-Mg-Si (6xxx). The shape, structure and strengthening properties of age-hardening precipitates depend on alloy composition and thermo-mechanical history. We seek an improved understanding of the physics related to nucleation and precipitation on the atomistic level in these alloys. Once these mechanisms are sufficiently well described and understood, the hope is that 'alloy design' simulations can assist tailoring of materials with desired properties. In pure Al-Mg-Si we have determined the structure of nearly all the known metastable precipitate phases, by combining advanced TEM techniques (such as high resolution TEM and nano-beam diffraction) with atom probe tomography and density functional theory. We are now studying effects of additions /substitutions of Cu, Ag and/or Ge that promote formation of more disordered precipitates, employing aberration corrected high angle annular dark field scanning TEM. We find that all metastable precipitates contain variations of a widely spaced 'Si/Ge network'. In spite of disorder or defects, this network is surprisingly well ordered, with hexagonal projected sub-cell dimensions a = b ? 0.4 nm and c (along the fully coherent precipitate main growth direction) equal to 0.405 nm or a multiple of it. Holmestad, Randi; Bjørge, Ruben; Ehlers, Flemming J. H.; Torsæter, Malin; Marioara, Calin D.; Andersen, Sigmund J. 2012-07-01 175 PubMed The wastewater produced after brightening and anodizing aluminium has high concentrations of phosphates and sulphates. The addition of MgO in a first physico-chemical wastewater treatment step makes the selective recovery of phosphates in the form of magnesium phosphates feasible, which may be reused as fertilizer. The proposed wastewater treatment process allows manufacturers to reduce more than 70% of the volume of the precipitate in the sedimentation reactor and more than 50% of the weight of the final disposal sludge. In this study, the use of an alternative low-grade MgO (LG-MgO) as a source of magnesium, which is cheaper than pure MgO, is investigated. The phosphate concentration and pH of the treated wastewater is controlled by the formation and precipitation of newberyite or bobierrite as a function of the magnesium source added. According to experimental data, a reaction mechanism is proposed. Although LG-MgO reacts more slowly than pure MgO and it is necessary to add 3-4 times the stoichiometric amount, this procedure has considerable economic and technical advantages. PMID:16343583 Chimenos, J M; Fernández, A I; Hernández, A; Haurie, L; Espiell, F; Ayora, C 2006-01-01 176 PubMed According to previous animal studies, aluminium phosphides (AlPs) may induce oxidative stress leading to generation of free radicals and alteration in antioxidant defense system. This study was conducted to evaluate the existence and degree of oxidative stress in patients with acute AlP ingestion. A total of 44 acute AlP ingested patients as well as 44 age- and sex-matched controls were included. All patients had acute poisoning symptoms with AlP at the time of presentation and had blood samples analyzed for lipid peroxidation, total antioxidant capacity and total thiol. Our findings showed that there is a significant increase in lipid peroxidation in AlP ingested group along with a reduction in total antioxidant capacity and total thiols groups. These clinical data confirm previous experimental models that showed AlP exposure might significantly augment lipoperoxidative damage with simultaneous alterations in the antioxidant defense system. Hence, our findings might justify use of antioxidants in treatment of acute AlP poisoning which needs to be clarified by additional clinical trials. PMID:22407514 Kariman, Hamid; Heydari, Kamran; Fakhri, Mohammad; Shahrami, Ali; Dolatabadi, Ali Arhami; Mohammadi, Hossein Ali; Gharibi, Morteza 2012-09-01 177 Microsoft Academic Search Sheets from commercial purity aluminium AA1050 and aluminium alloy AA6016 were processed by accumulative roll bonding to obtain\\u000a an ultrafine-grained microstructure. The accumulative roll bonded samples showed a significant increase in specific strength\\u000a paired with high ductility. Despite a strongly elongated grain structure, tensile testing of samples oriented 45° to the rolling\\u000a direction revealed considerable improvement in elongation to failure Irena Topic; Heinz Werner Höppel; Mathias Göken 2008-01-01 178 Microsoft Academic Search To identify and locate rye DNA sequences homologous to three wheat c-DNAs (wali1, wali2 and wali5) whose expression is induced by aluminium (Al) stress, we designed three pairs of specific primers. They were used in the\\u000a amplification of genomic DNA from wheat-rye disomic addition lines. The wali2 pair of primers amplified a 878-bp rye DNA fragment (rali2) located on chromosomes F. J. Gallego; E. López-Solanilla; A. M. Figueiras; C. Benito 1998-01-01 179 Microsoft Academic Search Barley plants were grown in nutrient solutions in two greenhouse pot experiments after addition of 0 (0.5), 2, 6 and 12 mg aluminium pr. liter as Al2(SO4)3·18H2O. The Altreatments were combined with three levels of Ca+Mg-salt applications. Low pH value is a prerequisite for keeping Al in solution, and attempts were therefore made to produce low pH nutrient solutions without Gotfred Uhlen 1985-01-01 180 Microsoft Academic Search The adsorption behaviour of triethanolamine (TEA) on aluminium, AlMgSi and AlZnMg alloys covered with naturally formed oxide films was investigated in 0.5 M NaCl solution, pH 1.3, by means of potentiodynamic and electrochemical impedance spectroscopy techniques. The results of polarization measurements show that in all cases the addition of TEA induces a decrease in the cathodic currents without affecting the M. Metikoš-Hukovi?; R. Babi?; Z. Gruba?; S. Brin? 1994-01-01 181 NASA Astrophysics Data System (ADS) Within a French/German research project the suitability of a new class of materials, so-called aluminium foam sandwiches (AFS), was tested for space applications. While classical sandwich materials require particular, cost and time intensive processing techniques, AFS are suitable for mass production technologies as they are known from car industry and the like. Thus, it is expected that such materials could essentially contribute to cost reduction in space. In order to test the principal applicability and to gain some first experience in how the manufacture of AFS space components could work, a cone 3936 as used in Ariane 5 was chosen as demonstrator. In the forefront, the material had been intensively tested considering mechanical and thermal properties. By means of FEM the results of these experiments were used to simulate the behaviour under load and to optimise the design accordingly. Using AFS suitable production methods, the cone was built and tested simulating the loads during launch. The test results were compared to the predictions made by FEM and showed good agreement. Schwingel, Dirk; Seeliger, Hans-Wolfgang; Vecchionacci, Claude; Alwes, Detlef; Dittrich, Jürgen 2007-06-01 182 NASA Astrophysics Data System (ADS) This work deals with multicriteria ranking of aluminium coating methods. The alternatives used are: sulfuric acid anodization, A1; oxalic acid anodization, A2; chromic acid anodization, A3; phosphoric acid anodization, A4; integral color anodizing, A5; chemical conversion coating, A6; electrostatic powder deposition, A7. The criteria used are: cost of production, f1; environmental friendliness of production process, f2; appearance (texture), f3; reflectivity, f4; response to coloring, f5; corrosion resistance, f6; abrasion resistance, f7; fatigue resistance, f8. Five experts coming from relevant industrial units set grades to the criteria vector and the preference matrix according to a properly modified Delphi method. Sensitivity analysis of the ranked first alternative A1 against the second best', which was A3 at low and A7 at high resolution levels proved that the solution is robust. The dependence of anodized products quality on upstream processes is presented and the impact of energy price increase on industrial cost is discussed. Batzias, A. F. 2007-12-01 183 SciTech Connect An analytical method has been developed for the fluorimetric determination of nanogram amounts of aluminium in solution. The method is based on the reaction of aluminium with 8-hydroxyquinoline-5-sulfonic acid presence of hexadeciltrimethylammonium bromide as a surfactant agent. Synchronous scanned first and second derivative fluorimetry has been employed to increase the sensitivity of the method. The influence of reaction variables as well as instrumental parameters is discussed. The interference of various foreign ions has also been examined and in some cases eliminated or reduced by addition of 1,10-phenanthroline. Salinas, F.; de la Pena, A.; Duran, M.S. 1988-08-01 184 PubMed The aluminium distribution in polycrystalline SiC hot-pressed with aluminium, boron and carbon additives was studied using X-ray energy-dispersive spectroscopy (EDS) and transmission electron microscopy (TEM). The Al excess in homophase SiC grain boundary films was determined, taking into account dissolved Al in the SiC lattice. In the spot-EDS analysis, an electron beam probe with a calibrated diameter was formed, and the total beam-specimen interaction volume was defined, taking the beam spreading through crystalline TEM foil into consideration. EDS spectra were collected from regions containing intergranular films and adjacent matrix grains, respectively. A theoretical treatment was presented and experimental errors were estimated, with a further discussion about the effects of foil thickness. Experimental examples are given, followed by statistical EDS analyses for grain boundary films in SiC samples hot-pressed with increased amounts of Al additions. The results demonstrated a substantial Al segregation in the nanometer-wide intergranular films in all samples. Al additions higher than 3 wt% saturated the Al concentrations in SiC grains and in grain boundary films. The effect of foil thickness, and the parameters for determining the optimum incident beam diameter in the EDS analysis are discussed. PMID:12135460 Zhang, X F; Yang, Q; De Jonghe, L C; Zhang, Z 2002-07-01 185 Microsoft Academic Search Cause analysis of porosity in aluminium argon-arc welded joints is carried out, particularly that of aluminium–lithium alloy 1424. Feasible production methods of reducing the quantity of voids in the joints are considered. M. M. Shtrikman; A. V. Pinskii; A. A. Filatov; V. V. Koshkin; E. A. Mezentseva; N. V. Guk 2011-01-01 186 PubMed The infectivity of viruses (Qbeta, MS2, T4, and P1) after dosing virus-contaminated water with 4 types of aluminium coagulant was investigated. The concentrations of infectious viruses were determined after dissolving aluminium hydroxide flocs in alkaline solution. The concentration of infectious viruses did not recover to the initial value after a short floc-dissolution time (5 s). Although the infectious virus concentration increased as the floc-dissolution time was extended to 5 h, it did not recover fully. Irreversible adhesion between virus particles and aluminium coagulant is responsible for the insufficient recovery. We interpret this phenomenon as a virucidal activity of the aluminium coagulant. All tested aluminium coagulants (PACl, alum, and reagent grade aluminium chloride and aluminium sulfate) inactivated all types of viruses tested. PACl had the highest virucidal activity. The virucidal activity of aluminium coagulants was lower in river water, presumably owing to the presence of natural organic matter. PMID:15686022 Matsushita, T; Matsui, Y; Inoue, T 2004-01-01 187 SciTech Connect Corrosion of aluminium metal in ordinary Portland cement (OPC) based pastes produces hydrogen gas and expansive reaction products causing problems for the encapsulation of aluminium containing nuclear wastes. Although corrosion of aluminium in cements has been long known, the extent of aluminium corrosion in the cement matrices and effects of such reaction on the cement phases are not well established. The present study investigates the corrosion reaction of aluminium in OPC, OPC-blast furnace slag (BFS) and calcium aluminate cement (CAC) based systems. The total amount of aluminium able to corrode in an OPC and 4:1 BFS:OPC system was determined, and the correlation between the amount of calcium hydroxide in the system and the reaction of aluminium obtained. It was also shown that a CAC-based system could offer a potential matrix to incorporate aluminium metal with a further reduction of pH by introduction of phosphate, producing a calcium phosphate cement. Kinoshita, Hajime, E-mail: h.kinoshita@sheffield.ac.uk [Department of Materials Science and Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD (United Kingdom)] [Department of Materials Science and Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD (United Kingdom); Swift, Paul; Utton, Claire [Department of Materials Science and Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD (United Kingdom)] [Department of Materials Science and Engineering, The University of Sheffield, Mappin Street, Sheffield, S1 3JD (United Kingdom); Carro-Mateo, Beatriz [The Public University of Navarra, C/Esquíroz, 30 trasera, Pamplona 31007 (Spain)] [The Public University of Navarra, C/Esquíroz, 30 trasera, Pamplona 31007 (Spain); Marchand, Geraldine [The National Institute of Applied Sciences (INSA) Lyon, 20 Avenue Albert Einstein 69621 Villeurbanne Cedex (France)] [The National Institute of Applied Sciences (INSA) Lyon, 20 Avenue Albert Einstein 69621 Villeurbanne Cedex (France); Collier, Nick [National Nuclear Laboratory, Chadwick House, Birchwood Park, Warrington, WA3 6AE (United Kingdom)] [National Nuclear Laboratory, Chadwick House, Birchwood Park, Warrington, WA3 6AE (United Kingdom); Milestone, Neil [Industrial Research Ltd., 69 Gracefield Road, Lower Hutt, 5040 (New Zealand)] [Industrial Research Ltd., 69 Gracefield Road, Lower Hutt, 5040 (New Zealand) 2013-08-15 188 NASA Astrophysics Data System (ADS) Age hardening, or precipitation hardening, is one of the most widely adopted techniques for strengthening of aluminium alloys. Although various age hardening models have been developed for aluminium alloys, from the large volume of literature reviewed, it appears that the bulk of the research has been concentrated on wrought aluminium alloys, only a few of the established precipitation models have been applied to the casting aluminium alloys. In the present work, there are two modelling methods that have been developed and applied to the casting aluminium alloys A356 and A357. One is based on the Shercliff-Ashby methodology to produce a process model, by which we mean a mathematical relationship between process variables (alloy composition, ageing temperature and time) and material properties (yield strength or hardness) through microstructure evolution (precipitate radius, volume fraction). The other method is based on the Kampmann and Wagner Numerical (KWN) model which deals with concomitant nucleation, growth and coarsening and is thus capable of predicting the full evolution of the particle size distribution and then a strength model is used to evaluate the resulting change in hardness or yield strength at room temperature by taking into account contributions from lattice resistance, solid solution hardening and precipitation hardening. Wu, Linda; Ferguson, W. George 2009-08-01 189 National Technical Information Service (NTIS) The use of sintering additives has been the most effective way of promoting grain growth of uranium dioxide. We have established a same mechanism for additives which belongs to corundum structure: chromium, aluminium, vanadium and titanium sesquioxides. S... L. Bourgeois 1992-01-01 190 Microsoft Academic Search Copper- and nickel-coated graphite particles can be successfully introduced into aluminium-base alloy melts as pellets to produce cast aluminium-graphite particle composites. The pellets were made by pressing mixtures of nickel- or copper-coated graphite particles and aluminium powders together at pressures varying between 2 and 20 kg mm-2. These pellets were dispersed in aluminium alloy melts by plunging and holding them B. C. Pai; P. K. Rohatgi 1978-01-01 191 NASA Astrophysics Data System (ADS) The applicability of an aluminium alloy containing scandium for laser additive manufacturing (LAM) is considered. Modified aluminium alloys with a scandium content beyond the eutectic point offer great potential to become a high prioritized aerospace material. Depending on other alloying elements like magnesium or zirconium, strongly required weight reduction, corrosion resistance and improved strength properties of metallic light weight alloys can be achieved. The development, production and testing of parts built up by a laser powder bed process will be presented with regard to the qualification of the new material concept “ScalmalloyRP®” for laser additive manufacturing. Schmidtke, K.; Palm, F.; Hawkins, A.; Emmelmann, C. 192 Microsoft Academic Search Comparative studies of the catalytic properties and thermal stability of a copper-zinc-aluminium catalyst and its components have revealed that the catalytic activity is determined by a solid solution of copper and aluminium in zinc oxide containing OH? and CO32? groups in its anion skeleton. The presence of aluminium in the solid solution ensures the increase of the catalyst thermal stability L. I. Kuznetsova; T. M. Yurieva; T. P. Minyukova; S. V. Ketchik; L. M. Plyasova; G. K. Boreskov 1982-01-01 193 Microsoft Academic Search Nano-aluminium particles are produced through the wire explosion process in different gas medium. The particles produced by wire explosion process, in helium medium are of smaller size compared to argon\\/nitrogen medium. The nano aluminium powder on reaction with water forms oxides having bayerite and boehmite structure. It is observed that nano aluminium on reaction with KOH solution at room temperature Ramanujam Sarathi; Binu Sankar; Satyanarayanan R. Chakravarthy 2010-01-01 194 SciTech Connect Aluminium salts are utilized to impede intestinal phosphate absorption in chronic renal failure. Toxic side effects include anemia, which could result from impaired formation or accelerated clearance of circulating erythrocytes. Erythrocytes may be cleared secondary to suicidal erythrocyte death or eryptosis, which is characterized by cell shrinkage and exposure of phosphatidylserine (PS) at the erythrocyte surface. As macrophages are equipped with PS receptors, they bind, engulf and degrade PS-exposing cells. The present experiments have been performed to explore whether Al{sup 3+} ions trigger eryptosis. The PS exposure was estimated from annexin binding and cell volume from forward scatter in FACS analysis. Exposure to Al{sup 3+} ions ({>=} 10 {mu}M Al{sup 3+} for 24 h) indeed significantly increased annexin binding, an effect paralleled by decrease of forward scatter at higher concentrations ({>=} 30 {mu}M Al{sup 3+}). According to Fluo3 fluorescence Al{sup 3+} ions ({>=} 30 {mu}M for 3 h) increased cytosolic Ca{sup 2+} activity. Al{sup 3+} ions ({>=} 10 {mu}M for 24 h) further decreased cytosolic ATP concentrations. Energy depletion by removal of glucose similarly triggered annexin binding, an effect not further enhanced by Al{sup 3+} ions. The eryptosis was paralleled by release of hemoglobin, pointing to loss of cell membrane integrity. In conclusion, Al{sup 3+} ions decrease cytosolic ATP leading to activation of Ca{sup 2+}-permeable cation channels, Ca{sup 2+} entry, stimulation of cell membrane scrambling and cell shrinkage. Moreover, Al{sup 3+} ions lead to loss of cellular hemoglobin, a feature of hemolysis. Both effects are expected to decrease the life span of circulating erythrocytes and presumably contribute to the development of anemia during Al{sup 3+} intoxication. Niemoeller, Olivier M. [Department of Physiology, University of Tuebingen (Germany); Kiedaisch, Valentin [Department of Physiology, University of Tuebingen (Germany); Dreischer, Peter [Department of Physiology, University of Tuebingen (Germany); Wieder, Thomas [Department of Physiology, University of Tuebingen (Germany); Lang, Florian [Department of Physiology, University of Tuebingen (Germany)]. E-mail: florian.lang@uni-tuebingen.de 2006-12-01 195 NASA Astrophysics Data System (ADS) Aluminium is a common fuel component in propellants and explosives. There is a wealth of literature on Aluminium combustion in gases at relatively low pressure but limited data on combustion at high pressure (as in explosive detonation products). In this work we have carried out and analysed cylinder tests with Aluminium loaded explosives with a view to assessing the applicability of low pressure burning rates in this regime. The analysis makes use of detailed numerical two phase flow modelling and a range of experiments used to validate other relevant aspects of the physics, such as drag laws. We conclude that the burning rate is significantly faster than that implied by extrapolating laws applicable at lower pressures. Evans, David J.; Milne, Alec M.; Softley, Ian 2002-07-01 196 NASA Astrophysics Data System (ADS) Anodic layer growth on 2024 aluminium alloy at 70 °C, under 40 V, during 60 min, in 50 g L -1 di-sodium tetraborate solution containing di-sodium molybdate from 0.1 to 0.5 M (pH 10) is examined. Anodising behaviours strongly depend on additive concentration. Development of anodic films is favoured with weak molybdate additions (<0.3-0.4 M). The film thicknesses increase and the porosity of anodic layers decreases. Molybdenum (+VI), detected by X-ray photoelectron spectroscopy (XPS) analysis, is present in the anodic films and the Mo incorporation, studied by energy dispersive spectroscopy (EDS) analysis, increases with molybdate concentration. However, for high molybdate concentrations (>0.4 M), anodising behaviour becomes complex with the formation of a blue molybdenum oxide at the cathode. The growth of aluminium oxide is hindered. As the anodic layers are thinner, the Mo(+VI) incorporation significantly decreases. These two configurations implicate different corrosion performances in 5% sodium chloride solution at 35 °C. As the alkaline anodic layer formed with 0.3 M molybdate species is the thickest and the Mo incorporation is the more pronounced, its corrosion resistance is the highest. The effect of morphology and composition of anodic films on pitting corrosion is also discussed. Moutarlier, V.; Pelletier, S.; Lallemand, F.; Gigandet, M. P.; Mekhalif, Z. 2005-12-01 197 NASA Astrophysics Data System (ADS) The corrosion behaviour of AA6061 aluminium alloy in tropical seawater was investigated using weight loss measurement and electrochemical polarization technique. The electrochemical measurements showed that the presence of natural honey as corrosion resistant significantly decrease the corrosion current densities (icorr) and corrosion rates. It was observed that the inhibition efficiencies increased with the increasing concentrations of the resistant. Potentiodyanamic curves suggested that natural honey suppressed both cathodic and anodic processes. A good fit to Langmuir adsorption isotherm was obtained between the degree of surface coverage and the concentration of natural honey. It can be concluded that natural honey is an excellent corrosion resistant for aluminium alloy immersed in tropical seawater. Rosliza, R.; Senin, H. B.; Muzathik, A. M.; Wan Nik, W. B. 2009-07-01 198 PubMed Paspalum notatum Flugge has been widely utilized for the purpose of ecological restoration of degraded land in the tropics and subtropics, where soil active aluminium (Al) is usually high as a result of acidification. Pot experiments were conducted to determine Al toxicity on P. notatum and to compare its potential to remove Al with another three plant species, Vetiveria zizanioides, Tristania conferta and Schima wallichii. In the Al addition experiment, the biomass of P. notatum and Al accumulation significantly decreased as the added Al concentration increased, but Al concentration in the plant markedly increased. A parallel experiment was conducted with the above four species, grown in lateritic soil and in oil shale waste containing high concentration of active Al. The biomasses of all four species were reduced obviously in the waste compared to in the soil. The effects of substrate on Al concentration, accumulation and translocation efficiency differed among species, and plants had significantly higher Al accumulation factors when grown in the soil than in the waste. Most of the Al taken up by P. notatum was transferred to above-ground parts; as a result, Al concentration in stems and leaves became quite high, over 1000 or even 3000 mg kg(-1); whereas for the other three species, Al concentration in shoots was much lower than in roots. Paspalum notatum was therefore much higher than the other three species with regard to Al translocation efficiency and therefore P. notatum may be regarded as both an effective Al hyper-accumulator and a potential Al hyper-remover. PMID:19423590 Huang, Juan; Xia, Hanping; Li, Zhi'an; Xiong, Yanmei; Kong, Guohui; Huang, Juan 2009-10-01 199 PubMed Alumina used in the production of primary aluminium contains Be which partly vaporises from the cryolite bath into the workroom atmosphere. Since Be may be toxic at lower exposure levels than previously thought, the personal exposure to Be among workers in 7 Norwegian primary smelters has been assessed. In total, 480 personal Respicon® virtual impactor full shift air samples have been collected during 2 sampling campaigns and analysed for water soluble Be, Al and Na using inductively coupled plasma optical emission spectrometry. In addition, water soluble F(-) has been measured by ion chromatography. The Be air concentrations in the inhalable, thoracic and respirable aerosol fractions have been calculated. The Be concentrations in the inhalable aerosol fraction vary between the different smelters. The highest GM concentration of Be in the inhalable fraction (122 ng m(-3), n = 30) was measured in the prebake pot room of a smelter using predominantly Jamaican alumina where also the highest individual air concentration of 270 ng m(-3) of Be was identified. The relative distribution of Be in the different aerosol fractions was fairly constant with the mean Be amount for the two sampling campaigns between 44-49% in the thoracic fraction expressed as % of the inhalable amount. Linear regression analysis shows a high correlation between water soluble Be, Al, F and Na describing an average measured chemical bulk composition of the water soluble thoracic fraction as Na(5.7)Al(3.1)F(18). Be is likely to be present as traces in this particulate matter by replacing Al atoms in the condensed fluorides and/or as a major element in a nanoparticle sized fluoride. Thus, the major amount of Be present in the work room atmosphere of Al smelter pot rooms will predominantly be present in combination with substantial amounts of water soluble Al, F and Na. PMID:21993554 Skaugset, Nils Petter; Ellingsen, Dag G; Dahl, Kari; Martinsen, Ivar; Jordbekken, Lars; Drabløs, Per Arne; Thomassen, Yngvar 2012-02-01 200 PubMed Central Background Aluminium (Al) toxicity was frequent in the 1980s in patients ingesting Al containing phosphate binders (Alucaps) whilst having HD using water potentially contaminated with Al. The aim of this study was to determine the risk of Al toxicity in HD patients receiving Alucaps but never exposed to contaminated dialysate water. Methods HD patients only treated with Reverse Osmosis(RO) treated dialysis water with either current or past exposure to Alucaps were given standardised DFO tests. Post-DFO serum Al level > 3.0 ?mol/L was defined to indicate toxic loads based on previous bone biopsy studies. Results 39 patients (34 anuric) were studied. Mean dose of Alucap was 3.5 capsules/d over 23.0 months. Pre-DFO Al levels were > 1.0 ?mol/L in only 2 patients and none were > 3.0 ?mol/L. No patients had a post DFO Al levels > 3.0 ?mol/L. There were no correlations between the serum Al concentrations (pre-, post- or the incremental rise after DFO administration) and the total amount of Al ingested. No patients had unexplained EPO resistance or biochemical evidence of adynamic bone. Conclusions Although this is a small study, oral aluminium exposure was considerable. Yet no patients undergoing HD with RO treated water had evidence of Al toxicity despite doses equivalent to 3.5 capsules of Alucap for 2 years. The relationship between the DFO-Al results and the total amount of Al ingested was weak (R2 = 0.07) and not statistically significant. In an era of financial prudence, and in view of the recognised risk of excess calcium loading in dialysis patients, perhaps we should re-evaluate the risk of using Al-based phosphate binders in HD patients who remain uric. 2011-01-01 201 PubMed The uptake of ingested aluminium (Al) from food items commonly consumed in a normal human diet was investigated by feeding five test diets to guinea pigs. Al concentrations were measured in the femur, brain, kidney and upper intestinal contents. Consumption of these diets did not lead to elevated Al levels in brain. Levels of Al in the bone were elevated in animals fed sponge cake with a permitted Al-containing additive, and the presence of citrate as orange juice enhanced bone deposition and increased kidney Al levels. Less than 1% of Al in the upper intestinal contents was found in the soluble fraction, and characterization by SEC-ICP-MS indicated that this Al was not present as Al-citrate. PMID:8070734 Owen, L M; Crews, H M; Bishop, N J; Massey, R C 1994-08-01 202 PubMed A new analytical procedure for the direct determination of metal impurities (Cr, Cu, Fe and V) in aluminium oxide ceramic powders by slurry sampling fluorination assisted electrothermal vaporization-inductively coupled plasma-atomic emission spectrometry (ETV-ICP-AES) is reported. A polytetrafluoroethylene (PTFE) emulsion was used as a fluorinating reagent to promote the vaporization of impurity elements in aluminium oxide ceramic powders from the graphite tube. A vaporization stage with a long ramp time and a short hold time provided the possibility of temporal analyte-matrix separation. The experimental results indicated that a 10 microL 1% m/v slurry of aluminium oxide could be destroyed and vaporized completely with 600 micrograms PTFE under the selected conditions. Two aluminium oxide ceramic powder samples were used without any additional pretreatment. Analytical results obtained by using standard addition method with aqueous standard solution were checked by comparison of the results with pneumatic nebulization (PN)-ICP-AES based on the wet-chemical decomposition and analyte-matrix separation. The limits of detection (LODs) between 0.30 microgram g-1 (Fe) and 0.08 microgram g-1 (Cu) were achieved, and, the repeatability of measurements was mainly better than 10%. PMID:11270230 Peng, T; Chang, G; Wang, L; Jiang, Z; Hu, B 2001-03-01 203 Microsoft Academic Search To develop steel aluminium-tailored hybrids in a butt joint for sheets in a thickness of about 1mm conventional Friction Stir Welding is not feasible due to a high distortion of the welded specimen. Contrary to Friction Stir Welding the tool used for Friction Stir Knead Welding has no pin wherefore higher welding speeds can be realised. Due to the fact M. Geiger; F. Micari; M. Merklein; L. Fratini; D. Contorno; A. Giera; D. Staud 2008-01-01 204 Microsoft Academic Search This paper is focused on the behaviour of aluminium alloy T-stub joints. The analysis is developed by means of finite element method simulation carried out with the non-linear code ABAQUS. The procedure has been accurately calibrated on the basis of some existing experimental results. The analysis, which has been referred to several behavioural parameters, has shown that, contrary to steel G De Matteis; A Mandara; F. M Mazzolani 2000-01-01 205 NASA Astrophysics Data System (ADS) A demonstration is given of the feasibility of two-photon excitation of aluminium phthalocyanine and of the pharmaceutical preparation 'Fotosens', used in photodynamic therapy. The excitation source was an Nd:YAG laser emitting at the 1064 nm wavelength. The spectra of the two-photon-excited luminescence were obtained and the two-photon absorption cross sections were determined. Meshalkin, Yu P.; Alfimov, E. E.; Vasil'ev, N. E.; Denisov, A. N.; Makukha, V. K.; Ogirenko, A. P. 1999-12-01 206 PubMed Aluminium hydride has been synthesized electrochemically, providing a synthetic route which closes a reversible cycle for regeneration of the material and bypasses expensive thermodynamic costs which have precluded AlH(3) from being considered as a H(2) storage material. PMID:19557259 Zidan, Ragaiy; Garcia-Diaz, Brenda L; Fewox, Christopher S; Stowe, Ashley C; Gray, Joshua R; Harter, Andrew G 2009-07-01 207 Microsoft Academic Search A destructive technique under development to determine residual stresses in thick-walled pressure vessels has been employed to determine quenching residual stresses in 7060 aluminium alloy gas cylinder necks. The gas cylinders were supplied with interference fit collars attached to the apex of the gas cylinder neck. The effect the elastic interference fit stresses have on the quenching residual stresses in D. J. Sharman; H. L. Stark; D. W. Kelly 1997-01-01 208 Microsoft Academic Search A demonstration is given of the feasibility of two-photon excitation of aluminium phthalocyanine and of the pharmaceutical preparation 'Fotosens', used in photodynamic therapy. The excitation source was an Nd:YAG laser emitting at the 1064 nm wavelength. The spectra of the two-photon-excited luminescence were obtained and the two-photon absorption cross sections were determined. Yu P. Meshalkin; E. E. Alfimov; N. E. Vasil'ev; A. N. Denisov; V. K. Makukha; A. P. Ogirenko 1999-01-01 209 Microsoft Academic Search The main reason of the surface modification of the components such as pistons and cylinder blocks made of cast aluminium alloys is to obtain high hardness, wear and corrosion resistance of the working surface for larger lifetime of the motor-car and aircraft engines. In that aspect, the new conception of creating — by hybrid method — surface layers containing manganese, W. Serbi?ski 2005-01-01 210 National Technical Information Service (NTIS) Tests are reported on 76 aluminium plates with controlled initial out of flatness and residual stress. Two alloys were studied, namely 5083-M and 6082-TF. Some specimens were transversely welded. Their sizes ranged from 20 < b/t < 85, corresponding to a n... D. S. Mofflin J. B. Dwight 1983-01-01 211 Microsoft Academic Search In this paper, we report the results obtained recently by analysing around 400 features measured from 23.000 regions segmented in 50 radioscopic images of cast aluminium wheels with defects. The extracted features are divided into two groups: geometric features (area, perimeter, height, width, roundness, Fourier descriptors, invariant moments, and other shape factors) and grey value features (mean grey value, mean Domingo Mery; Romeu R. da Silva; Luiz P. Calôba; João M. A. Rebello 2003-01-01 212 Microsoft Academic Search Carbonation of raw red mud produced by aluminium refineries and a chemically and physically neutralized red mud (Bauxsol™) has been carried out to study the capacity of these wastes to capture carbon dioxide. After only 5 min of carbonation of raw red mud, total alkalinity dropped 85%. Hydroxide alkalinity was almost totally consumed, carbonate alkalinity dropped by 88%, and bicarbonate Graham B Jones; Gargi Joshi; Malcolm D Clark; David M McConchie 2006-01-01 213 Microsoft Academic Search A critical review of the literature on Al toxicity in plants, animals and humans reveals a similar mode of Al action in all living organisms, namely interference with the secondary messenger system (phosphoinositide and cytosolic Ca2+ signalling pathways) and enhanced production of reactive oxygen species resulting in oxidative stress. Aluminium uptake by plants is relatively quick (across the intact plasma Z. Rengel 2004-01-01 214 Microsoft Academic Search Friction stir welding is a solid state welding process which involves joining similar or dissimilar metals using a rotating tool. Tool geometry and traverse speed and rotating speed of motion of the tool, tool axial force and tilt angle are some of the variables in this process. Many materials like Aluminium alloy 2000, 6000 and 7000 series have been joined M. Sivashanmugam; S. Ravikumar; T. Kumar; V. S. Rao; D. Muruganandam 2010-01-01 215 PubMed The suitability of high performance chelation ion chromatography (HPCIC) using postcolumn reaction for the separation and determination of dissolved aluminium in complex samples was investigated. Use of a chelating ion-exchanger allowed for differentiation between kinetically labile and kinetically stable species of aluminium. Separation through a combination of chelation and cation-exchange was achieved using a 200 x 4.0 mm id column packed with particles of silica functionalised with iminodiacetic acid, with nitric acid-potassium chloride eluents. A temperature anomaly causing a five-fold increase in column efficiency for aluminium is believed to be a result of localised temperature effects in the particular type of instrument used. Postcolumn reagents investigated for the photometric detection included Tiron, Pyrocatechol Violet, Chrome Azurol S, and Eriochrome Cyanine R. The lowest detection limit (2.7 microg/L for a 100 microL sample volume) was achieved using 0.25 mM Eriochrome Cyanine R in 0.2 M hexamine (pH 6.1) with 1 mM cetyltrimethylpyridium bromide (CTAB). The optimised HPCIC system was applied successfully to the quantification of labile aluminium in paper mill process water. PMID:18563745 Tria, Juliette; Haddad, Paul R; Nesterenko, Pavel N 2008-07-01 216 Microsoft Academic Search The hot and warm formability of 2618 aluminium alloy, in the as-solutioned condition, was investigated in extended ranges of temperature and strain rate by means of torsion testing. Precipitation was found to occur during deformation. The effect of the precipitation of second phase particles, occurring during deformation, on the flow curve shape and on the stress level was evaluated. At P Cavaliere 2002-01-01 217 Microsoft Academic Search Directional solidification of aluminium-silicon eutectic alloys were carried out in order to investigate the mechanism by which the quench modification takes place. For this purpose a new type of Bridgman furnace was designed which can attain a high temperature gradient and a high interfacial growth velocity up to 1000 µm-1. It is established that the fibrous structure is the result S. Khan; R. Elliott 1996-01-01 218 Microsoft Academic Search Low expansion aluminium–silicon eutectic alloys are cast to produce most of the automotive pistons. The structure and properties of these alloys are very much dependent on the cooling rate, composition, modification and heat treatment operations. In this study, locally available automotive ‘scrap pistons’ were used as basic raw materials and a natural gas fired crucible furnace was used for melting M. M. Haque; A. Sharif 2001-01-01 219 Microsoft Academic Search The severe wear of a near eutectic aluminium–silicon alloy is explored using a range of electron microscopic, spectroscopic and diffraction techniques to identify the residually strained and unstrained regions, microcracks and oxidized regions in the subsurface. In severe wear the contact pressure exceeds the elastic shakedown limit. Under this condition the primary and eutectic silicon particles fragment drastically. The fragments Anirban Mahato; Nisha Verma; Vikram Jayaram; S. K. Biswas 2011-01-01 220 Microsoft Academic Search This paper present results obtained on rapid solidification of aluminium-silicon alloys from the liquid state. It shows that the limit of primary solid solubility is extended almost to the eutectic composition and that the large supersaturation is relieved on raising the annealing temperature to the range 110 to 450° C. This conclusion is based on measurements of lattice parameter and S. K. Bose; R. Kumar 1973-01-01 221 Microsoft Academic Search High speed machining of aluminium silicon alloy castings has gained significant interest from automotive industry involved in the development of the new generation of lightweight vehicles. This paper investigates the influence of workpiece microstructure, namely the secondary dendritic arm spacing (SDAS), tool material and geometry on tool wear mechanisms, cutting forces and surface integrity when face milling at cutting speeds E-G Ng; D. Szablewski; M. Dumitrescu; M. A. Elbestawi; J. H. Sokolowski 2004-01-01 222 NASA Astrophysics Data System (ADS) The annealing behaviour of aluminium has been studied in single-crystal InP implanted with 40 and 120 keV 27Al + ions. The implantation doses were 1×10 15 and 1×10 16 cm-2. The aluminium concentration profiles were determined by two techniques, Secondary ion mass spectrometry (SIMS) and the nuclear resonance broadening technique (NRB) which was used for checking purposes. The usability of the SIMS technique for profiling Al rich layers was studied. Significant inconsistencies were observed in the SIMS profiles with the high dose implanted samples. The 120 keV, 1×10 16 cm -2 implanted samples were subject to annealing in argon atmosphere in the temperature range 380-600°C. Redistribution and a significant aluminium surface enrichment took place. The coefficients of Al migration were determined by fitting a concentration independent solution of the diffusion equation to the experimental depth profiles. The deduced activation energy and frequency factor for aluminium migration in implanted InP are 0.78 eV and 5.5×10 5 nm2/ s, respectively. The effect of defects on the Al migration is discussed. Rutherford backscattering (RBS) spectra measured along <1 0 0> channeling direction for 120 keV, 10 16 cm -2 implanted samples indicated a loss of crystalline structure near the surface both in the as-implanted and in the annealed samples. Kyllönen, V.; Räisänen, J.; Seppälä, A.; Ahlgren, T.; Likonen, J. 2000-03-01 223 Microsoft Academic Search Galvanic corrosion of laser welded AA6061 aluminium alloy, arising from the varying rest potentials of the various weldment regions, was examined. The weld fusion zone is found to be the most cathodic region of the weldment while the base material is the most anodic region. The rate of galvanic corrosion, controlled by the cathodic process at the weld fusion zone, A. B. M. Mujibur Rahman; S. Kumar; A. R. Gerson 2007-01-01 224 Microsoft Academic Search Crashworthiness, strength and vibrational features of a railroad passenger car, which is originally made of steel members and then converted to an aluminium design, are studied. The finite element (FE) method is utilised for the static analysis in compliance with various scenarios defined in UIC CODE OR 577, modal analysis and simulation of the crash into a rigid wall. Firstly, C. Baykasoglu; E. Sunbuloglu; S. E. Bozdag; F. Aruk; T. Toprak; A. Mugan 2012-01-01 225 Microsoft Academic Search We have tested the influence of water ionic strength on the toxicity of aluminium in fish by comparing the mortality of Atlantic salmon (Salmo salar) parr exposed to Al-rich water with additions of Ca2+ or Na+. The fish were exposed in parallel to Al-rich water (Al 500 µg l-1, pH 5.8) under non-steady state conditions, with and without the addition Espen LYDERSEN; Sigurd ØXNEVAD; Ronny A. ANDERSEN; Frode BJERKELY; Leif Asbjørn VØLLESTAD 2002-01-01 226 PubMed Silicon, in its various forms (silicic acid (Si(OH)4) through to hydrated amorphous silica (SiOn(OH)4-2n, n = 2-4) is important in geological and biological processes. Silicon is considered to be an 'essential' element for some plants and higher animals and when present as silica imparts structural, defensive and photosynthetic advantages to many plants. For the majority of essential elements specific molecular binding sites have been identified, but this is not the case for silicon. This has lead to the proposal that silicon may not act directly on biological systems but may exert its influence via interactions with biometals. Metals for which this may be a viable option include aluminium, iron and calcium, all of which can be found co-located with silicon in both minerals and living organisms. This article briefly reviews the pertinent solution and solid state chemistry of silicon in relation to aluminium, iron and calcium. Examples where silicon and an additional metal are found in solid state phases are described. The biological systems where these deposits are found are extremely complex and preliminary results from a model system designed to study silicon-biometal interactions are described. PMID:9629678 Perry, C C; Keeling-Tucker, T 1998-02-15 227 PubMed Ellipsometry was used to investigate the effect of polyaluminium chloride (PAC) formulations of different degrees of hydrolysation on an adsorbed mucin film. The results were compared to the effect of aluminium chloride (AlCl(3)) and ferric chloride. A compaction of the mucin film took place upon addition of the formulations and this occurred to different extents and at different concentrations for the different formulations. The compaction of PAC of a low degree of hydrolysis behaved similarly to AlCl(3). PAC of a high degree of hydrolysis showed a greater compaction effect than the other aluminium formulations. The initial compaction concentration was found to be 0.001 mM which is less than previously found for aluminium-mucin complex formation in bulk. The reversibility of the compaction was also investigated. The compaction of the mucin film was found to be partly reversible for AlCl(3) and PAC of low degree of hydrolysis. No reversibility was observed for the formulations of PAC of high hydrolysis grade or for ferric chloride. The results are consistent with previously observed effects of PAC of a low degree of hydrolysis on bacterial surfaces where a compaction of surface polymers was indicated by the reduced range of repulsive steric interactions. PMID:20432077 Hamit-Eminovski, Jildiz; Eskilsson, Krister; Arnebrant, Thomas 2010-07-01 228 NASA Astrophysics Data System (ADS) The paper undertakes the preparation of Aluminium oxide doped Lithium borate glasses and characterizes to investigate the physical and optical studies of these glasses using density and UV-Visible spectroscopy. Results had been obtained with the help of FTIR Spectra in our previous work which indicate that Al2O3 acts as a network modifier by converting three coordinated boron atoms [BO3] to four coordinated boron atoms [BO4]. The effect of aluminium content on the optical properties of the present glass system has been studied from the optical absorption spectra. It shows the shifting of the band edge towards the higher wavelength and that there is corresponding decrease in the band gap, which is due to the conversion of BO3 to BO4 units. The evaluation of the compact structure of prepared glass samples with the presence of [BO4] groups has been made in terms of the physical properties such as density and molar volume. It is also confirmed by average boron-boron separation results. The research findings confirm that the addition of Aluminium oxide increases the covalent value and corresponding decrease in ionicity behaviour. Pal Singh, Gurinder; Kaur, Parvinder; Kaur, Simranpreet; Arora, Deepa Wali; Sharma, Mohandeep; Singh, D. P. 229 PubMed In the present study, Aluminium quantification in immunobiologicals has been described using atomic absorption spectroscopy (AAS) technique. The assay was found to be linear in 25-125 microg/ml Aluminium range. The procedure was found to be accurate for different vaccines with recoveries of external additions ranging between 93.26 and 103.41%. The mean Limit of Variation (L.V.) for both intra- and inter-assay precision was calculated to be 1.62 and 2.22%, respectively. Further the procedure was found to be robust in relation to digestion temperature, alteration in acid (HNO(3) and H(2)SO(4)) ratio used for sample digestion and storage of digested vaccine samples up to a period of 15 days. After validation, AAS method was compared for its equivalency with routinely used complexometric titration method. On simultaneously applying on seven different groups of both bacterial and viral vaccines, viz., DPT, DT, TT, Hepatitis-A and B, Antirabies vaccine (cell culture) and tetravalent DPT-Hib, a high degree of positive correlation (+0.85-0.998) among AAS and titration methods was observed. Further AAS method was found to have an edge over complexometric titration method that a group of vaccines, viz., ARV (cell culture, adsorbed) and Hepatitis-A, in which Aluminium estimation is not feasible by pharmacopoeial approved complexometric titration method (possibly due to some interference in the sample matrix), this newly described and validated AAS assay procedure delivered accurate and reproducible results. PMID:17644407 Mishra, Arti; Bhalla, Sumir Rai; Rawat, Sameera; Bansal, Vivek; Sehgal, Rakesh; Kumar, Sunil 2007-10-01 230 PubMed Aluminium is a potent neurotoxin and has been associated with Alzheimer's disease (AD) causality for decades. Prolonged aluminium exposure induces oxidative stress and increases amyloid beta levels in vivo. Current treatment modalities for AD provide only symptomatic relief thus necessitating the development of new drugs with fewer side effects. The aim of the study was to demonstrate the protective effect of chronic curcumin administration against aluminium-induced cognitive dysfunction and oxidative damage in rats. Aluminium chloride (100 mg/kg, p.o.) was administered to rats daily for 6 weeks. Rats were concomitantly treated with curcumin (per se; 30 and 60 mg/kg, p.o.) daily for a period of 6 weeks. On the 21st and 42nd day of the study behavioral studies to evaluate memory (Morris water maze and elevated plus maze task paradigms) and locomotion (photoactometer) were done. The rats were sacrificed on 43rd day following the last behavioral test and various biochemical tests were performed to assess the extent of oxidative damage. Chronic aluminium chloride administration resulted in poor retention of memory in Morris water maze, elevated plus maze task paradigms and caused marked oxidative damage. It also caused a significant increase in the acetylcholinesterase activity and aluminium concentration in aluminium treated rats. Chronic administration of curcumin significantly improved memory retention in both tasks, attenuated oxidative damage, acetylcholinesterase activity and aluminium concentration in aluminium treated rats (P<0.05). Curcumin has neuroprotective effects against aluminium-induced cognitive dysfunction and oxidative damage. PMID:19616038 Kumar, Anil; Dogra, Samrita; Prakash, Atish 2009-12-28 231 Microsoft Academic Search Tandem beam brazing with aluminium filler metal (BA4047) was conducted in order to develop the fluxless laser brazing technique of aluminium alloy (AA6022) to galvanized steels (GA and GI steels). Laser powers of tandem beam and offset distance of preheating beam from the root to the steel base metal were varied. Sound braze beads could be obtained by optimizing the Kazuyoshi Saida; Haruki Ohnishi; Kazutoshi Nishimoto 2010-01-01 232 SciTech Connect An electrorefining process in molten chloride salts is being developed at ITU to reprocess the spent nuclear fuel. According to the thermochemical properties of the system, aluminium is the most promising electrode material for the separation of actinides (An) from lanthanides (Ln). The actinides are selectively reduced from the fission products and stabilized by the formation of solid and compact actinide-aluminium alloys with the reactive cathode material. In this work, the maximum loading of aluminium with actinides was investigated by potentiostatic and galvano-static electrorefining of U-Pu- Zr alloys. A very high aluminium capacity was achieved, as the average loading was 1.6 g of U and Pu into 1 g of aluminium and the maximum achieved loading was 2.3 g. For recovery of the actinides from aluminium, a process based on chlorination and a subsequent sublimation of AlCl{sub 3} is proposed. (authors) Malmbeck, R.; Mendes, E.; Serp, J.; Soucek, P.; Glatz, J.P. [European Commission, JRC, Institute for Transuranium Elements, Postfach 2340, 76125 Karlsruhe (Germany); Cassayre, L. [Laboratoire de Genie Chimique - LGC, Universite Paul Sabatier, UMR 5503, 118 route de Narbonne, 31062 Toulouse Cedex 04 (France) 2007-07-01 233 NASA Astrophysics Data System (ADS) The crystallography of eutectic silicon in an unmodified A1-12.7 wt% Si alloy has been studied by transmission electron microscopy and diffraction of thin films cut from bulk samples. Twinning mechanisms are described by which the silicon changes direction or adjusts interflake spacing at the solid-liquid interface. The branching mechanism observed by previous workers is confirmed but found to be infrequent at moderate freezing rates. A side-branching mechanism is described which is considered to be the primary mechanism for branching in the wheatsheaf configuration. Orientation relationships between the eutectic silicon and aluminium phases are described. These differ from those previously observed in thin films drawn from the melt. It is proposed that orientation relationships vary with freezing rate. Evidence is presented that the eutectic aluminium re-nucleates repeatedly during growth of a single eutectic grain. Shamsuzzoha, M.; Hogan, L. M. 1986-08-01 234 NASA Astrophysics Data System (ADS) An experiment on "dendritic growth in aluminium alloy welding" was performed by the UNIBO team during the 3rd Student Parabolic Flight Campaign and the 30th Professional Parabolic Flight Campaign organised by ESA. Its purpose was to achieve a better understanding of crystal growth during tungsten inert gas (TIG) welding of an aluminium alloy to define the main parameters affecting the process under microgravity condition. The experiment had 4 phases : The paper discusses different aspects of the research, paying particularly attention not only to the influence of gravity, but also to other factors influencing welding microstructure, such as the Marangoni effect and the thermal transfer from the electrode to the material. The paper conclude the dissertation of the results offering new perspectives for welding studies and proposing a new approach to the scientific community to investigate this materials processes for manufacturing. Ferretti, S.; Amadori, K.; Boccalatte, A.; Alessandrini, M.; Freddi, A.; Persiani, F.; Poli, G. 2002-01-01 235 NASA Astrophysics Data System (ADS) Aluminium tracer diffusivities were measured in polycrystalline mullite. The artificial aluminium isotope 26Al was used as tracer isotope. An advanced preparation technique for the 26Al2O3 tracer source allowed to apply secondary ion mass spectrometry (SIMS) in order to analyse 26Al depth distributions in the polycrystalline material. Pre-exponential factors and activation enthalpies were determined for compositions of 78 wt.% Al2O3, 22 wt.% SiO2 (high-alumina material) and of 72 wt.% Al2O3, 28 wt.% SiO2 (low-alumina material), respectively. A strong dependence of the 26Al grain boundary diffusivity on the composition is observed. The results are discussed in comparison to previous data on grain boundary diffusivities of oxygen in mullite samples from the same batch. Fielitz, P.; Borchardt, G.; Schmücker, M.; Schneider, H. 2007-08-01 236 Microsoft Academic Search The sliding wear response of several wrought aluminium alloys (2124, 3004, 5056 and 6092) against a high purity alumina (99.9%) counterface was investigated, at a fixed sliding speed of 1m\\/s and a load range of 23–140N. The counterface was chosen so as to minimise the chemically driven aspects of adhesive wear. Severe wear was observed at all loads, with specific M. J. Ghazali; W. M. Rainforth; H. Jones 2007-01-01 237 Microsoft Academic Search The galvanic corrosion behaviour of a rivet joint of two sheets of the aluminium alloys EN AW-6014-T4 and EN AW-6082-T5 joined by an electrogalvanized steel blind rivet was investigated. The potentiodynamic polarization curves in a 5 wt.% NaCl solution show a potential reversion of the anodic and cathodic regions of the rivet joint. The surface potential was investigated with a L. Krüger; M. Mandel 2011-01-01 238 Microsoft Academic Search Closed die forgings manufactured from 2014 aluminium alloys have been subject to both standard and non-standard heat treatments in order to reduce the as-quenched residual stress magnitudes. Warm water (60°C) and boiling water quenches are investigated. The influence of changing the surface finish of the forgings during boiling water quenching on the mechanical properties and residual stress has also been D. A. Tanner; J. S. Robinson 2008-01-01 239 Microsoft Academic Search Particle effects on friction and wear of 6061 aluminium (6061 Al) reinforced with silicon carbide (SiC) and alumina (Al2O3) particles were investigated by means of Vickers microhardness measurements and scratch tests. Unreinforced 6061 Al matrix alloy was also studied for comparison. To explore the effect of heat treatment, materials subjected to three different heat treatment conditions, i.e. under-aged, over-aged and Z. F. Zhang; L. C. Zhang; Y.-W. Mai 1995-01-01 240 Microsoft Academic Search This paper presents an experimental and numerical investigation on low velocity perforation (in the velocity range 3.5–15.8m\\/s) of AA5083-H116 aluminium plates. In the tests, square plates were mounted in a circular frame and penetrated by a cylindrical blunt-nosed projectile. The perforation process was then computer analysed using the nonlinear finite element code LS-DYNA in order to investigate the effects of F. Grytten; T. Børvik; O. S. Hopperstad; M. Langseth 2009-01-01 241 Microsoft Academic Search Dissemination of closed cell metal foam unique properties (low density, efficient energy absorption, high vibration\\/sound\\u000a attenuation) in real life products has often been difficult to realise. With advanced pore morphology (APM) aluminium foam–polymer\\u000a hybrids a new and simplified process route targeted at application in foam-filled structures (e.g. automotive A-pillar) has\\u000a been introduced. APM foams are made from spherical, small volume Karsten Stöbener; Gerald Rausch 2009-01-01 242 Microsoft Academic Search The growth kinetics of chromate\\/fluoride conversion coatings are examined for 99.99% aluminium, and Al–2.3at.%Cu, Al–1.9at.%Au and Al–20at.%Au alloys. The thickening of coatings and thinning of substrates, the latter deposited by magnetron sputtering, are determined by transmission electron microscopy. The results reveal consumption of substrates during the coating process, to the maximum immersion time of 24 min. Initial relatively rapid thinning Y. Liu; P. Skeldon; G. E. Thompson; H. Habazaki; K. Shimizu 2004-01-01 243 Microsoft Academic Search Hollow aluminium bats were introduced over 30 years ago to provide improved durability over wooden bats. Since their introduction,\\u000a however, interest in hollow bats has focused almost exclusively around their hitting performance. The aim of this study was\\u000a to take advantage of the progress that has been made in predicting bat performance using finite elements and apply it to describe Eric Biesen; Lloyd Smith 2007-01-01 244 Microsoft Academic Search Toxic damage of brain cells by aluminium (A1) is discussed as a possible factor in the development of neurodegenerative disorders in humans. To investigate neurotoxic effects of A1, serum-free cultures of mechanically dissociated embryonic chick (stage 28–29) forebrain, brain stem and optic tectum, and for comparison meningeal cells, were treated with A1 (0–1000 M) for 7 days. Effects of A1 Judith Pia Miiller; Arend Bruinink 1994-01-01 245 Microsoft Academic Search Aluminium alloy matrix reinforced with 15wt% SiC particles were prepared by powder metallurgy (PM) method. Wear behaviour of the composite was investigated to find out effects of operating variables and hardness in terms of the Taguchi approach, on a pin-on-disc machine and compared with the previous work on the composite produced by liquid metallurgy method [1]. Analysis of variance (ANOVA) Yusuf ?ahin 2010-01-01 246 Microsoft Academic Search The fact that aluminium (Al) and lead (Pb) are both toxic metals to living organisms, including human beings, was discovered\\u000a a long time ago. Even when Al and Pb can reach and accumulate in almost every organ in the human body, the central nervous\\u000a system is a particular target of the deleterious effects of both metals. Select human population can Sandra V. Verstraeten; Lucila Aimo; Patricia I. Oteiza 2008-01-01 247 Microsoft Academic Search Aluminium titanate (Al2TiO5) is a promising engineering material because of its low thermal expansion coefficient, excellent thermal shock resistance, good refractoriness and non-wetting with most metals. However, it is susceptible to thermal dissociation in the temperature range ˜1100 1300 °C which degrades its desirable properties. In this work, the effect of atmospheres (i.e., air, argon, 50% oxygen\\/50% argon) on the I. M. Low; Z. Oo; B. H. O'Connor 2006-01-01 248 Microsoft Academic Search Aluminium titanate (Al2TiO5) is a promising engineering material because of its low thermal expansion coefficient, excellent thermal shock resistance, good refractoriness and non-wetting with most metals. However, it is susceptible to thermal dissociation in the temperature range ?1100–1300°C which degrades its desirable properties. In this work, the effect of atmospheres (i.e., air, argon, 50% oxygen\\/50% argon) on the isothermal stability I. M. Low; Z. Oo; B. H. O’Connor 2006-01-01 249 Microsoft Academic Search This paper uses the finite element alternating technique to study the effect of multiple cracks, i.e. multi-site damage (MSD), in an aluminium alloy plate. This problem arose as a result of MSD seen in the aft bulkhead of F\\/A-18 aircraft in service with the Royal Australian Air force (RAAF). For the particular problem under investigation, it is shown that the R. Jones; S. Hammond; J. F. Williams; M. Heller 1995-01-01 250 Microsoft Academic Search The formability of Al–Mg sheet can be improved considerably, by increasing the temperature.\\u000aAt elevated temperatures, the mechanical response of the material becomes strain rate dependent. To accurately\\u000asimulate warm forming of aluminium sheet, a material model is required that incorporates the temperature\\u000aand strain-rate dependency. In this paper hardening is described succesfully with a physically based material\\u000amodel for L. van Haaren; Boogaard van den A. H; J. Huetink 2004-01-01 251 Microsoft Academic Search Ceramic particulate composites containing up to 25 wt% silica sand in commercially pure aluminium (LM-O) and its eutectic silicon alloy (LM-6) were prepared by liquid metallurgy techniques. Pre-treated sand particles of sizes ranging from -180 to +90µm were added to the alloy melts, followed by pouring the resulting mix into permanent moulds. Quantitative metallographic examination revealed that sand particles were A. K. Gupta; T. K. Dan; P. K. Rohatgi 1986-01-01 252 Microsoft Academic Search WHEN metallic sodium is added to a eutectic aluminium-silicon alloy, the structure of the eutectic silicon is modified by a mechanism which has been described by several authors1-4. Sodium also modifies the structure of primary silicon crystals in this system, and various workers1,5 have stated that a spheroidal form of silicon can be obtained by this means. Evidence that these M. G. Day 1968-01-01 253 Microsoft Academic Search Ceramic particulate composites containing up to 25 wt% silica sand in commercially pure aluminium (LM-O) and its eutectic\\u000a silicon alloy (LM-6) were prepared by liquid metallurgy techniques. Pre-treated sand particles of sizes ranging from ?180\\u000a to +90?m were added to the alloy melts, followed by pouring the resulting mix into permanent moulds. Quantitative metallographic\\u000a examination revealed that sand particles were A. K. Gupta; T. K. Dan; P. K. Rohatgi 1986-01-01 254 Microsoft Academic Search This paper present results obtained on rapid solidification of aluminium-silicon alloys from the liquid state. It shows that\\u000a the limit of primary solid solubility is extended almost to the eutectic composition and that the large supersaturation is\\u000a relieved on raising the annealing temperature to the range 110 to 450 C. This conclusion is based on measurements of lattice\\u000a parameter and S. K. Bose; R. Kumar 1973-01-01 255 Microsoft Academic Search The functionality of structural light alloy castings can be improved by inserting into them, upon moulding, local iron base reinforcements. To acquire a better knowledge of such bimetallic assemblies, samples were prepared by immersing a mild steel bar (5mm in diameter) in aluminium base Al–Si alloy melts held at 730°C. After melt solidification, the bimetallic samples were cut into 5mm O. Dezellus; B. Digonnet; M. Sacerdote-Peronnet; F. Bosselet; D. Rouby; J. C. Viala 2007-01-01 256 Microsoft Academic Search The main purpose of this work was to study the cementation of mercury (HgCl42) in a chloride medium using metallic zinc, iron and aluminium as reducing agents.The reaction order was 1.08±0.05 with respect to mercury concentration. The reaction efficiency is strongly dependent on pH. For each metal in study, an ideal pH was established: 4.0 to 5.0 for zinc, 3.0 A. L. Anacleto; J. R. Carvalho 1996-01-01 257 Microsoft Academic Search Digital video imaging of the plasma electrolytic oxidation (PEO) of aluminium has been performed, which allowed evaluation of both dimensional characteristics of individual microdischarges appearing at the oxide-electrolyte interface and their collective behaviour throughout the oxidation process. It has been shown that the microdischarge cross-sectional dimensions vary within the range 0.01-1.35 mm2. In the course of PEO processing, small localized A. L. Yerokhin; L. O. Snizhko; N. L. Gurevina; A. Leyland; A. Pilkington; A. Matthews 2003-01-01 258 Microsoft Academic Search We present results of an optical emission spectroscopic study of Plasma during Electrolytic Oxidation (PEO) of aluminium in citric acid and in sodium–tungstate water solutions. The line shape analysis of the first two hydrogen Balmer lines indicates presence of two PEO processes characterized by relatively low electron number densities Ne?0.8×1015cm?3 and at 2.5×1016cm?3. Apart from these two Ne values, Ne?6.0×1016cm?3 J. Jovovi?; S. Stojadinovi?; N. M. Šišovi?; N. Konjevi? 2011-01-01 259 Microsoft Academic Search Gravity casting and squeeze casting were carried out on an aluminium alloy with 13.5% silicon and a zinc alloy with 4.6% aluminium with different temperatures, 660, 690 and 720°C for the former and 440, 460 and 480°C for the latter. A top-loading crucible furnace was used to melt the alloys. The die-preheat temperatures used were 200–220°C for the aluminium alloy L. J. Yang 2003-01-01 260 Microsoft Academic Search The properties and microstructures of several high-strength and high-modulus carbon fibrereinforced aluminium or aluminium alloy matrix composites (abbreviated as HSCF\\/Al and HMCF\\/Al, respectively, for the two types of fibre) have been characterized. The composites evaluated were fabricated by pressure casting based on a hybridization method. It was found that the strength degradation of high-modulus carbon fibres after infiltration of aluminium H. M. Cheng; A. Kitahara; S. Akiyama; K. Kobayashi; Y. Uchiyama; B. L. Zhou 1994-01-01 261 Microsoft Academic Search With the wide application of aluminium alloys to the automotive, aerospace and other industries, laser welding has become a critical joining technique for aluminium alloys. In this paper, two techniques have been used to measure the non-uniform residual stresses in laser-welded aluminium alloys, i.e. the Lcr ultrasonic technique and the moiré interferometry incremental hole-drilling (MIIHD) method. The non-uniform residual stress M. Ya; P. Marquette; F. Belahcene; J. Lu 2004-01-01 262 Microsoft Academic Search The phonon-limited resistivity of aluminium has been calculated using a local, first-principles pseudopotential which has been useful in the calculation of other properties of aluminium. This pseudopotential is obtained from the induced electron density around an aluminium ion in an electron gas. From this pseudopotential, the interionic potential, the phonons (which are calculated by the self-consistent harmonic approximation) and finally G. J. Vazquez; L. F. Magana 1990-01-01 263 PubMed Central Bulk aluminium samples containing alumina particles have been produced by different severe plastic deformation methods. Aluminium foils with different initial foil thicknesses were cold rolled to different amounts of strain and aluminium powders were consolidated and deformed by high pressure torsion (HPT). During processing, alumina particles from the foil or particle surface are easily incorporated and dispersed in the bulk material. The influence of these alumina particles on the developing microstructures and the mechanical properties has been studied. Bachmaier, Andrea; Pippan, Reinhard 2011-01-01 264 Microsoft Academic Search Violet-blue photoluminescent emission has been observed in aluminium oxide films deposited by the ultrasonic spray pyrolysis process at atmospheric pressure, using solutions of 0.05M aluminium acetylacetonate in a mixture of two parts of deionised water and two parts of methanol and aluminium chloride in the same solvent mixture. The films were deposited at substrate temperatures in the range from 240 A. Ortiz; J. C Alonso; V Pankov; D Albarran 1999-01-01 265 Microsoft Academic Search \\u000a Summary A possibility to reduce weight in lightweight constructions is the usage of magnesium instead of the typically used leight-weight\\u000a material aluminium. Magnesium is more than 30 % lighter than aluminium. However, the biggest problem of magnesium is the bad\\u000a corrosion behaviour. One approach of corrosion protection is coating with aluminium by roll-bounding. This paper gives an\\u000a introduction into the assembly Carina Maria Schlögl; Clemens Planitzer; Otto Harrer; Bruno Buchmayr 2011-01-01 266 Microsoft Academic Search This paper describes the slurry erosion of a range of HVOF deposited aluminium-based claddings on steel by sand in water. Coatings, approximately 300?m thick, of commercially pure aluminium, eutectic aluminium\\/silicon alloy (12%) and of a novel composite incorporating alumina in this alloy have been tested, both as sprayed and as ground to remove surface roughness as far as possible. Angular A. J Speyer; R. J. K Wood; K. R Stokes 2001-01-01 267 NASA Astrophysics Data System (ADS) The effect of ultrasonic vibration on nanoscale interfacial structure of thermosonic copper wire bonding on aluminium pads was investigated. It was found that bonding strength was determined by the extent of fragmentation of a native aluminium oxide overlayer (5-10 nm thick) on aluminium pads, forming paths for formation of intermetallic compound CuAl2 in areas of direct contact of bonded metal surfaces. The degree of fracture of the oxide layer was strongly affected by a level of ultrasonic power. Xu, H.; Liu, C.; Silberschmidt, V. V.; Chen, Z.; Acoff, V. L. 2011-04-01 268 Microsoft Academic Search Vermicompost extract (VCE) demonstrated the ability to raise the measured pH level of an acid soil thus showing a potential to limit aluminium toxicity. The relatively high proportion of humic substances (60% of VCE solid matter) suggested the possibility of formation of stable chelates with aluminium ions. A rate?based colorimetric method was utilised measuring at 585 nm the aluminium?pyrocatechol violet D. Alter; A. Mitchell 1992-01-01 269 NASA Astrophysics Data System (ADS) AlN/Ti/TiN/DLC layers more than 1 µm thick have been prepared by plasma-based ion implantation sequentially with nitrogen/titanium/nitrogen and titanium/carbon on 2024 aluminium alloy. On the basis of our previous AlN/Ti/TiN/DLC layers, new AlN/Ti/TiN/DLC layers thicken by thickening the 'TiN' layer and the diamond-like (DLC) film. The structures of the new layers have been characterized by XPS, AFM and Raman spectrum. The new layers exhibit improved composition and structure gradients at the interface in addition to a thicker 'TiN' layer and DLC film than the previous ones and hence display significant improvements in surface hardness and tribological properties. Also, their surface topography, DLC structure, nanohardness, stress and tribological properties as a function of the DLC film thickness have been discussed. Liao, J. X.; Tian, Z.; Li, E. Q.; Xu, J.; Jin, L.; Yang, H. G. 2007-08-01 270 PubMed The amount of particulate fluorides evolved from aluminium electrolysis cells is not entirely accounted for by the fluorides entrained in the anode gas. The largest additional source of particulate fluoride formation is by direct evaporation of fluorides into the anode gas stream and subsequent condensation on the drops of electrolyte generated in the process of bubble burst. A theoretical model was used for the calculation of the main physical parameters responsible for the formation of particle nuclei when the hot anode-gas is mixed with ambient air. The results of these calculations are in agreement with experimental observations reported in the literature. In particular, the size distribution, composition and morphology of the nano-particles support the theory of a vapour condensation mechanism under conditions of extreme supersaturation, but further studies are necessary. PMID:15877162 L'vov, Boris V; Polzik, Leonid K; Weinbruch, Stephan; Ellingsen, Dag G; Thomassen, Yngvar 2005-05-01 271 SciTech Connect Technological interest in aluminium-silicon alloys derives, fundamentally, from their excellent moulding properties provided by the presence of silicon. In addition, they possess high thermal and electrical conductivities and good corrosion resistance. The possibility of modification of their cast structure by adding small quantities of alkaline and alkaline-earth elements makes their mechanical characteristics very competitive. Metallographic observations of the structures of primary and eutectic silicon crystals, and their possible synergistic influence, provide useful data on the critical stages of formation and growth of eutectic silicon phase. The nucleation and growth of eutectic silicon, removed from the cooperative precipitation zone, induce refinement in crystal size and, therefore, an improvement of the mechanical properties of the moulded structure. The aim of this investigation was to observe the influence of primary silicon crystals on the nucleation and growth of eutectic silicon. Criado, A.J.; Martinez, J.A.; Calabres, R. [Complutense Univ. of Madrid (Spain). Dept. of Materials Science and Metallurgical Engineering] [Complutense Univ. of Madrid (Spain). Dept. of Materials Science and Metallurgical Engineering 1997-01-01 272 NASA Astrophysics Data System (ADS) For solving the problem of shock-induced dissemination and burning of aluminium particles in air, we have developed a new solver based on equilibrium equation of states (EOS) of 2-phase reactive mixtures. The solver uses two pre-calculated tables; the larger one describes the equilibrium states of reaction products and the smaller one describes states of fresh air. Being linked with gas-dynamics equations, the solver finds iteratively the mixture pressure and temperature; in addition it furnishes a complete description of chemical and physical transformations. 2D numerical simulations give encouraging agreement with experimental pressure histories recorded on the chamber wall. It is shown that multiple blast wave reflections from the walls of the chamber strongly accelerate particle burning. The results demonstrate the advantages of the equilibrium EOS model and appeal for 3D AMR calculations on massively-parallel computers, which should better define the initial stage of turbulent particle dissemination. Khasainov, Boris; Kuhl, Allen; Victorov, Sergey; Neuwald, Peter 2005-07-01 273 NASA Astrophysics Data System (ADS) In situ laser depassivation of plasma electrolytically oxididized (PEO) coatings on aluminium was investigated with nanosecond pulses. Ultraviolet radiation of 266 nm was chosen in order to achieve a high absorption in the dielectric coating. The additive accumulation of laser-induced material defects (incubation) affected the depassivation processes. Incubation occurred only at the edges of the ablation craters irradiated by the outer region of the Gaussian beam profile, where the local fluence is below the ablation threshold. The ablation rate in the spot center did not exhibit an incubation effect. Repassivation was interpreted by a linear combination of a high-field and a point defect growth model. At low overpotentials, field gradients affect the process driving the oxide growth at the buried interface. At high fields, corrosion reactions dominate at the oxide/solution interface. Nagy, Tristan O.; Pacher, Ulrich; Giesriegl, Ariane; Soyka, Lukas; Trettenhahn, Günter; Kautek, Wolfgang 2014-05-01 274 NASA Astrophysics Data System (ADS) As the need for fuel efficient automobiles increases, car designers are investigating light-weight materials for automotive bodies that will reduce the overall automobile weight. Aluminium alloy tube is a desirable material to use in automotive bodies due to its light weight. However, aluminium suffers from lower formability than steel and its energy absorption ability in a crash event after a forming operation is largely unknown. As part of a larger study on the relationship between crashworthiness and forming processes, constitutive models for 3mm AA5754 aluminium tube were developed. A nominal strain rate of 100/s is often used to characterize overall automobile crash events, whereas strain rates on the order of 1000/s can occur locally. Therefore, tests were performed at quasi-static rates using an Instron test fixture and at strain rates of 500/s to 1500/s using a tensile split Hopkinson bar. High rate testing was then conducted at rates of 500/s, 1000/s and 1500/s at 21circC, 150circC and 300circC. The generated data was then used to determine the constitutive parameters for the Johnson-Cook and Zerilli-Armstrong material models. Salisbury, C. P.; Worswick, M. J.; Mayer, R. 2006-08-01 275 NASA Astrophysics Data System (ADS) Climate change mitigation in the materials sector faces a twin challenge: satisfying rapidly rising global demand for materials while significantly curbing greenhouse-gas emissions. Process efficiency improvement and recycling can contribute to reducing emissions per material output; however, long-term material demand and scrap availability for recycling depend fundamentally on the dynamics of societies' stocks of products in use, an issue that has been largely neglected in climate science. Here, we show that aluminium in-use stock patterns set essential boundary conditions for future emission pathways, which has significant implications for mitigation priority setting. If developing countries follow industrialized countries in their aluminium stock patterns, a 50% emission reduction by 2050 below 2000 levels cannot be reached even under very optimistic recycling and technology assumptions. The target can be reached only if future global per-capita aluminium stocks saturate at a level much lower than that in present major industrialized countries. As long as global in-use stocks are growing rapidly, radical new technologies in primary production (for example, inert anode and carbon capture and storage) have the greatest impact in emission reduction; however, their window of opportunity is closing once the stocks begin to saturate and the largest reduction potential shifts to post-consumer scrap recycling. Liu, Gang; Bangs, Colton E.; Müller, Daniel B. 2013-04-01 276 NASA Astrophysics Data System (ADS) Pulsed laser ablation of Aluminium (Al) in pure water rapidly forms a thin alumina (Al2O3) layer which drastically modifies surface plasmon resonance (SPR) absorption characteristics in deep-UV region. Initially, pure aluminium nanoparticles (NPs) are generated in water without any stabilizers or surfactants at low laser fluence which gradually transform to stable Al-Al2O3 core-shell nanostructure with increasing either residency time or fluence. The role of laser wavelength and fluence on the SPR properties and oxidation characteristics of Al NPs has been investigated in detail. We also present a one-step in situ synthesis of oxide-free stable Al NPs in biocompatible polymer solutions using laser ablation in liquid method. We have used nonionic polymers (PVP, PVA and PEG) and anionic surfactant (SDS) stabilizer to suppress the Al2O3 formation and studied the effect of polymer functional group, polymeric chain length, polymer concentration and anionic surfactant on the incipient embryonic aluminium particles and their sizes. The different functional groups of polymers resulted in different oxidation states of Al. PVP and PVA polymers resulted in pure Al NPs; however, PEG and SDS resulted in alumina-modified Al NPs. The Al nanoparticles capped with PVP, PVA, and PEG show a good correlation between nanoparticle stability and monomeric length of the polymer chain. Singh, Rina; Soni, R. K. 2014-05-01 277 PubMed Aluminium (Al) uptake and transport in the root tip of buckwheat is not yet completely understood. For localization of Al in root tips, fluorescent dyes and laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) were compared. The staining of Al with morin is an appropriate means to study qualitatively the radial distribution along the root tip axis of Al which is complexed by oxalate and citrate in buckwheat roots. The results compare well with the distribution of total Al determined by LA-ICP-MS which could be reliably calibrated to compare with Al contents by conventional total Al determination using graphite furnace atomic absorption spectrometry. The Al localization in root cross-sections along the root tip showed that in buckwheat Al is highly mobile in the radial direction. The root apex predominantly accumulated Al in the cortex. The subapical root section showed a homogenous Al distribution across the whole section. In the following root section Al was located particularly in the pericycle and the xylem parenchyma cells. With further increasing distance from the root apex Al could be detected only in individual xylem vessels. The results support the view that the 10 mm apical root tip is the main site of Al uptake into the symplast of the cortex, while the subapical 10-20 mm zone is the main site of xylem loading through the pericycle and xylem parenchyma cells. Progress in the better molecular understanding of Al transport in buckwheat will depend on the consideration of the tissue specificity of Al transport and complexation. PMID:21831842 Klug, Benjamin; Specht, André; Horst, Walter J 2011-11-01 278 NASA Astrophysics Data System (ADS) Aluminium alloys are heavily used to manufacture structural parts in the aeronautic industry because of its lightness and its corrosion resistance. These alloys are successfully used in other industrial fields too, such as railway, automotive and naval industries. The need to contrast the severe use conditions and the heavy stresses developing in aeronautic field implies to protect the surfaces of the structures in aluminium alloy by any deterioration. To preserve by deterioration, it is necessary to make aluminium more suitable to be coated by protective paint. In the aeronautic industry, a complex and critical process is used in order to enhance both wettability and adhesive properties of aluminium alloy surfaces. Cold plasma treatment represents an efficient, clean and economic alternative to activate aluminium surfaces. The present work deals with air cold plasma treatment of 2024 aluminium alloy surfaces. The influence of dc electrical discharge cold plasma parameters on wettability of 2024 aluminium alloy surfaces has been studied. A set of process variables (voltage, time and air flow rate) has been identified and used to conduct some experimental tests on the basis of design of experiment (DOE) techniques. The experimental results show that the proposed plasma process may considerably increase aluminium alloy wettability. These results represent the first step in trying to optimise the aluminium adhesion by means of this non-conventional manufacturing process. Polini, W.; Sorrentino, L. 2003-05-01 279 SciTech Connect An interesting form of lightweight material which has emerged in the past 2 decades is metallic foam. This paper deals with the basic concepts of making metallic foams and a detailed study of foams produced from Al-SiC. In addition, some aspects of cellular solids based on honeycomb structures are outlined including the concept of producing both two-phase foams and foams with composite walls. Prakash, O. [Indian Inst. of Tech., Kanpur (India); Embury, J.D.; Sinclair, C. [McMaster Univ., Hamilton, ON (Canada); Sang, H. [Queens Univ., Kingston, ON (Canada); Silvetti, P. [Cordoba Univ. Nacional (Argentina). Facultad de Ciencias Exactas, Fisicas y Naturales 1997-02-01 280 Microsoft Academic Search Whenever the base contacting is effected by overcompensating the rear side electrically active emitter, short-circuits occur between the emitter and the alloyed aluminium rear contact. Without additional costly processing steps for the avoidance of excessive leakage currents this transition is non-rectifying. The development of novel cell designs like POWER (POlycrystalline Wafer Engineering Result) or EWT (Emitter Wrap-Through) solar cells with K. Faika; R. Kühn; P. Fath; E. Bucher 281 NASA Astrophysics Data System (ADS) Metal powder propulsion systems have been addressed intermittently since the Second World War, initially in the field of underwater propulsion where research in the application of propelling torpedoes continues until this day. During the post war era, researchers attempted to utilise metal powders as a fuel for ram jet applications in missiles. The 1960's and 1970's saw additional interest in the use of pure powder' propellants, i.e. fluidised metal fuel and oxidiser, both in solid particulate form. Again the application was for employment in space-constrained missiles where the idea was to maximise the performance of high energy density powder propellants in order to enhance the missile's flight duration. Metal powder as possible fuel was investigated for in-situ resource utilisation propulsion systems post-1980's where the emphasis was on the use of gaseous oxygen or liquid oxygen combined with aluminium metal powder for use as a lunar soil propellant'' or carbon dioxide and magnesium metal powder as a `Martian propellant''.Albeit aluminium metal powder propellants are lower in performance than cryogenic and Earth storable propellants, the former does have an advantage inasmuch that the propulsion system is generic, i.e. it can be powered with chemicals mined and processed on Earth, the Moon and Mars. Thus, due to the potential refuelling capability, the lower performing aluminium metal powder propellant would effectively possess a much higher change in velocity (V) for multiple missions than the cryogenic or Earth storable propellant which is only suitable for one planet or one mission scenario, respectively.One of the principal limitations of long duration human spaceflight beyond cis-lunar orbit is the lack of refuelling capabilities on distant planets resulting in the reliance on con- ventional non-cryogenic, propellants produced on Earth. If one could develop a reliable propulsion system operating on pro- pellants derived entirely of ingredients found on nearby plan- etary bodies, then not only could mission duration be extended, larger amounts of payload could be ferried to and from the destination and eventually the cost of transporting propellant ingredients from Earth could be reduced, if not eliminated. Ismail, A. M.; Osborne, B.; Welch, C. S. 282 Microsoft Academic Search With a view to develop low temperature fine grained alumina–aluminium titanate composite, influence of alumina particle size on the temperature of formation of the aluminium titanate, sintering behaviour and microstructure development of alumina–aluminium titanate composite prepared through a sol–gel core shell approach is reported. The alumina matrix composite containing 20wt% aluminium titanate has been prepared from alumina powders having different M. Jayasankar; K. P. Hima; S. Ananthakumar; P. Mukundan; P. Krishna Pillai; K. G. K. Warrier 2010-01-01 283 National Technical Information Service (NTIS) This report analyzes the consequences for the aluminium production in Norway from the reforms in the former centrally planned economics. The report is organised in the following way. Chapter 2 presents the latest developments in the world aluminium market... M. D. Riiser 1995-01-01 284 Microsoft Academic Search We demonstrate a process of fabricating carbon nanotube field-effect transistors with self-aligned gate insulators by anodic oxidation of aluminium. We use rapid thermal annealed multi-walled carbon nanotubes (CNTs) as the active material overlaid across an aluminium gate. The porous nanotube bundles allowed moisture from the air to penetrate, and oxidise the aluminium by anodic oxidation to construct a self-assembled aluminium Jeff T. H. Tsai; Wei-Syun Wang; Szu-Hung Chen; Chia-Liang Sun 2011-01-01 285 Microsoft Academic Search This study explores a model (MIKADO) to analyse scenarios for the reduction of the environmental impact of an aluminium die casting plant. Our model calculates the potential to reduce emissions, and the costs associated with implementation of reduction options. In an earlier paper [Neto, B., Kroeze, C., Hordijk, L., Costa, C., 2008. Modelling the environmental impact of an aluminium pressure Belmira Neto; Carolien Kroeze; Leen Hordijk; Carlos Costa; Tinus Pulles 2009-01-01 286 Microsoft Academic Search Heat transfer coefficients during squeeze cast of commercial aluminium were determined, by using the solidification temperature versus time curves obtained for varying applied pressures during squeeze casting process. The steel mould\\/cast aluminium metal interface temperatures versus times curve obtained through polynomial curves fitting and extrapolation was compared with the numerically obtained temperatures versus times curve. Interfacial heat transfer coefficients were J. O. Aweda; M. B. Adeyemi 2009-01-01 287 Microsoft Academic Search Cyclic shear tests with various constant strain amplitudes have been carried out to characterize the microstructural development under cyclic straining of aluminium-3004 and aluminium-5182 alloys. Substructural instabilities that occur inside the microstructure, their morphologies and crystallographies are reported. The hypothetical influence of these inhomogeneities on the observed macroscopic behaviour is discussed. G. F. Dirras 1997-01-01 288 Microsoft Academic Search This paper presents results of accelerated corrosion tests in a salt spray chamber as well as microelectrochemical measurements of thermally joint steel–aluminium mixed materials. The focus was set on analysing the corrosion behaviour of the different metallic materials (brazed seam, intermetallic phases, aluminium and steel sheet) in or within the vicinity of the brazed seam.Both corrosion tests show that the J. Wloka; H. Laukant; U. Glatzel; S. Virtanen 2007-01-01 289 Microsoft Academic Search One way of making car bodies lighter is to introduce some aluminium parts in place of steel. Steel and aluminium can be joined by laser braze welding. As in other types of thermal joining, inter-metallic phases may weaken the joint. In laser braze welding, these appear as a thin layer of brittle compounds at the steel\\/seam interface. Their formation is Alexandre Mathieu; Simone Matteï; Alexis Deschamps; Bruno Martin; Dominique Grevey 2006-01-01 290 Microsoft Academic Search INTRODUCTON Bimetallic steel-aluminium inserts are widely used since 1960s in cell's anode holders for joining aluminium bar and steel rod. Cell's operating conditions specify a number of requirements to operating parameters of inserts: 1. Since general voltage drop of the cell is rather small, about 4.5 V, increase of efficiency requires minimization of loss in the insert - no more A. Z. Bogunov; A. A. Kuzovnikov; D. V. Kiselyov 2009-01-01 291 Microsoft Academic Search The dissolution of an 18Cr-10Ni stainless steel in liquid aluminium at 700 to 850 ° C was found by the rotating disc technique to be non-selective and diffusion controlled. Experimentally determined values of the parameters characterizing the dissolution run are presented. In the case of saturated aluminium melts two intermetallic layers were found to form between the steel and the V. I. Dybkov 1990-01-01 292 Microsoft Academic Search The main problem in joining steel to aluminium by a thermal process (e.g. TIG, MIG) is the formation of brittle intermetallic phases, that causes the mechanical performance of the assembly to be poor. A laser beam, however, can localize the fusion and inhibit the formation of these phases. This article presents a feasibility study for steel\\/aluminium assemblies with filler wire Alexandre Mathieu; Sébastien Pontevicci; Jean-claude Viala; Eugen Cicala; Simone Matteï; Dominique Grevey 2006-01-01 293 PubMed Aluminium oxyhydroxide, Al(OH)3 is one of few compounds approved as an adjuvant in human vaccines. However, the mechanism behind its immune stimulating properties is still poorly understood. In vitro co-culture of an aluminium adjuvant and the human monocytic cell line THP-1 resulted in reduced cell proliferation. Inhibition occurred at concentrations of adjuvant several times lower than would be found at the injection site using a vaccine formulation containing an aluminium adjuvant. Based on evaluation of the mitochondrial membrane potential, THP-1 cells showed no mitochondrial rupture after co-culture with the aluminium adjuvant, instead an increase in mitochondrial activity was seen. The THP-1 cells are phagocytosing cells and after co-culture with the aluminium adjuvant the phagosomal pathway was obstructed. Primary or early phagosomes mature into phagolysosomes with an internal pH of 4.5 - 5 and carry a wide variety of hydrolysing enzymes. Co-culture with the aluminium adjuvant yielded a reduced level of acidic vesicles and cathepsin L activity, a proteolytic enzyme of the phagolysosomes, was almost completely inhibited. THP-1 cells are an appropriate in vitro model in order to investigate the mechanism behind the induction of a phagocytosing antigen presenting cell into an inflammatory cell by aluminium adjuvants. Much information will be gained by investigating the phagosomal pathway and what occurs inside the phagosomes and to elucidate the ultimate fate of phagocytosed aluminium particles. PMID:23992993 Ohlsson, Lars; Exley, Christopher; Darabi, Anna; Sandén, Emma; Siesjö, Peter; Eriksson, Håkan 2013-11-01 294 National Technical Information Service (NTIS) An attempt was made to study some of the factors affecting certain properties of lead borate glass-aluminium metal composites. Lead borate glass was chosen because its softening point is between 500 and 600C which is similar to that of the aluminium metal... A. Bishay J. Zahar M. M. Farag 1972-01-01 295 Microsoft Academic Search The long-term durability of adhesively bonded aluminium, composite and dissimilar substrate joints exposed to humid environments has been investigated. Failure of the joints was modelled with a cohesive zone model (CZM) approach where the governing parameters were determined from fracture mechanics test specimens saturated in a range of humid environments. The reduction in residual strength of an aluminium single lap C. D. M. Liljedahl; A. D. Crocombe; M. A. Wahab; I. A. Ashcroft 2007-01-01 296 Microsoft Academic Search An experimental and numerical investigation into the shear strength behaviour of aluminium alloy adhesive lap joints was carried out in order to understand the effect of geometrical and manufacturing parameters on the strength of adhesive bonding joints, with the aim of optimizing shear strength. The adherend material used for the experimental tests was an aluminium alloy in the form of A. M. Pereira; J. M. Ferreira; F. V. Antunes; P. J. Bártolo 2010-01-01 297 Microsoft Academic Search Purpose – To investigate the inhibitive effect of gum arabic (GA) for the corrosion of aluminium in alkaline (NaOH) medium and determine its adsorption characteristics. The present work is another trial to find a cheap and environmentally safe inhibitor for aluminium corrosion. Design\\/methodology\\/approach – The inhibition efficiency (%I) has been evaluated using the hydrogen evolution (via the gasometric assembly) and S. A. Umoren; I. B. Obot; E. E. Ebenso; P. C. Okafor; O. Ogbobe; E. E. Oguzie 2006-01-01 298 Microsoft Academic Search In this study, 5083 aluminium alloys, which were exposed to severe plastic deformation, were joined with friction welding method and the variation in mechanical properties of the joints was experimentally investigated. Severe plastic deformation methods can be classified as equal channel angular pressing (ECAP) (in other words, equal cross section lateral extrusion – ECSLE) and cyclic extrusion-compression. Aluminium alloy as Mumin Sahin; H. Erol Akata; Kaan Ozel 2008-01-01 299 Microsoft Academic Search Taguchi approach was applied to determine the most influential control factors which will yield better tensile strength of the joints of friction stir welded RDE-40 aluminium alloy. In order to evaluate the effect of process parameters such as tool rotational speed, traverse speed and axial force on tensile strength of friction stir welded RDE-40 aluminium alloy, Taguchi parametric design and A. K. LAKSHMINARAYANAN; V. BALASUBRAMANIAN 2008-01-01 300 Microsoft Academic Search The heat treatable aluminium alloy AA2024 is used extensively in the aircraft industry because of its high strength to weight ratio and good ductility. The non-heat treatable aluminium alloy AA5083 possesses medium strength and high ductility and used typically in structural applications, marine, and automotive industries. When compared to fusion welding processes, friction stir welding (FSW) process is an emerging N. Shanmuga Sundaram; N. Murugan 2010-01-01 301 Microsoft Academic Search In the sinter process for alumina production, it is still difficult to produce sandy aluminium hydroxide by the carbonation of sodium aluminate solution. Therefore, a precisely controllable and feasible process for the precipitation of sandy aluminium hydroxide from the supersaturated sodium aluminate solution neutralized by sodium bicarbonate aqueous solution is proposed for the first time, in which the neutralization reagent Yan Li; Yifei Zhang; Chao Yang; Yi Zhang 2009-01-01 302 Microsoft Academic Search Aluminium foil is an important material in laminates and has wide application in food packaging. Its barrier function against the migration of moisture, oxygen and other gases, and volatile aroma, as well as against the impact of light is generally higher than any plastic laminate material. Therefore, aluminium foil is used in the laminates when insufficient barrier properties are the Manuela Lamberti; Felix Escher 2007-01-01 303 Microsoft Academic Search While the characterisation of intermetallic coverage and intermetallic phase (IP) growth in gold ball bonding on aluminium is quite well understood, there is relatively little literature concerning the morphology and growth of IP's between Cu balls bonded on aluminium pad metallisation. The difference between Cu-Al IP growth compared with the well known Au-Al IP's has been studied mainly of larger F. W. Wulff; C. D. Breach; D. Stephan; Saraswati; K. J. Dittmer 2004-01-01 304 Microsoft Academic Search Purpose – Several solid solution combinations of aluminium oxide and iron oxide, for the preparation of a new pigment, were investigated to study the effect of aluminium oxide to iron oxide ratio on various properties of the resulting pigments. Design\\/methodology\\/approach – The conditions for the preparation of the pigments via solid solution interaction at 1,000°C had been estimated. Characterisation of N. M. Ahmed; M. M. Selim 2005-01-01 305 Microsoft Academic Search Many studies have demonstrated a relationship between secondary dendrite arm spacing (SDAS) and the mechanical behaviour of cast aluminium–silicon alloys, both for tensile and fatigue strength. SDAS is related to the solidification time and can be predicted, with a good approximation, by finite-element simulation. However, other microstructural features can affect the tensile behaviour of cast aluminium alloys such as size L. Ceschini; Alessandro Morri; Andrea Morri; A. Gamberini; S. Messieri 2009-01-01 306 Microsoft Academic Search The microstructure of an alumina fibre reinforced Al-7wt% Si alloy has been investigated. It was shown that the Al-Si eutectic structure which characterized this alloy was markedly changed by the presence of the fibres, with coarsening of silicon particles and a reduction in primary aluminium grain size. The coarse silicon particles exhibited twinning but no orientation relationship with the aluminium. Ming Yang; V. D. Scott 1991-01-01 307 Microsoft Academic Search Silicon carbide fibres (Nicalon) were coated with pure aluminium and aluminium alloys containing silicon. The coated fibres were annealed to produce an interfacial reaction zone between the coated layer and the fibre. The effect of this reaction zone on the tensile strength of the fibre was investigated. During the early stages of growth the reaction zone of the fibre is H. Liu; U. Madaleno; T. Shinoda; Y. Mishima; T. Suzuki 1990-01-01 308 Microsoft Academic Search A precise 'pulse-echo method' is used to measure the sound velocity in liquid aluminium and aluminium-silicon alloys from the melting point up to 1000 degrees C. The adiabatic compressibility and static structure factors are evaluated. It is found that the compressibility and structure factors display anomalous variations around the eutectic composition. N. M. Keita; S. Steinemann 1978-01-01 309 Microsoft Academic Search The alloying of aluminium to silicon during the fabrication of silicon integrated circuits, as is well known, often results in 'spearing' at the contact windows and that this represents a possible failure mechanism. The morphology and kinetics of the spearing process are discussed, in particular, evidence for the existence of a liquid phase at alloying temperatures below the silicon-aluminium eutectic L. A. Berthoud 1977-01-01 310 Microsoft Academic Search Number of constructions, produced from eutectic silumins are continually increasing, therefore problem of their welding became very actual. Eutectic aluminium - silicon alloys are sensitive to changes of a welding thermal cycle. These cast aluminium alloys constructions after welding have large residual stresses, and mechanical properties of welded joint deteriorate. It is difficult to define welding regime in case of N. Višniakov; D. Š?ekaturovien?; O. ?ernaš?jus 311 PubMed Conditions for the direct complexometric determination of magnesium in the presence of uranium, iron and aluminium are described. The method is based on the masking of uranium by hydroxylamine, and of iron and aluminium by triethanolamine. Methylthymol Blue is used as indicator for the EDTA titration of magnesium at pH 10. PMID:18960663 Mareska, V 1969-11-01 312 Microsoft Academic Search Polyaniline films were electrodeposited at pure aluminium from a tosylic acid solution containing aniline. These polymer films exhibited similar characteristics as pure polyaniline electrosynthesized at an inert platinum electrode, when removed from their respective substrates and dissolved in NMP. Both polymers had similar molecular weights and similar UV-visible absorption spectra. However, the aluminium substrate had a considerable effect on the Kenneth G. Conroy; Carmel B. Breslin 2003-01-01 313 Microsoft Academic Search A process has been developed in which aluminium matrix composites, reinforced by either graphite or polycrystalline alumina (FP) fibres, are prepared by immersing the fibres sequentially in molten baths of sodium, tin, and aluminium. The experimental details of this process are presented, and the theory behind the process is explained. In essence, sodium wets the fibres, and is then reacted D. M. Goddard 1978-01-01 314 National Technical Information Service (NTIS) The steady-state flow stress at high strain-rate and at elevated temperatures of commercial purity aluminium and copper was analysed using the theory of the thermally activated deformation of metals. The experimental activation energy for aluminium was de... S. K. Samanta 1970-01-01 315 Microsoft Academic Search Aluminium corrosion is a significant concern in the aqueous chemical environment of the reactor containment building following a hypothetical loss-of-coolant accident (LOCA) at a nuclear power plant. Aluminium corrosion may lead to the formation of precipitates that can, in combination with insulation debris, block the recirculation sump screens. This study investigated aluminium corrosion experimentally at both bench and pilot scale Dong Chen; Kerry J. Howe; Jack Dallman; Bruce C. Letellier 2008-01-01 316 Microsoft Academic Search This paper is intended to give insights into the formation process of aluminium alloyed back surface fields (BSF) and rear contacts made by firing screen printed aluminium pastes. It is shown that a sufficient oxygen supply from the gas ambient is important to thicken and strengthen the oxide shells enclosing the liquid aluminium and silicon during firing. The alloying action F. Huster 317 National Technical Information Service (NTIS) The goal was to prepare high-purity, superconducting aluminium thin films and establish their characteristic properties with respect to the intended use. The work accomplished to this end included operation of a UHV vapor deposition device; system design ... M. Loidl 1995-01-01 318 NASA Astrophysics Data System (ADS) The aluminium nitride is a material which should replace alumina or beryllia as substrate for power electronic applications. The powder, prepared by carbothermal nitridation, is dispersed in a butanone-2-ethanol azeotrop solvent with a phosphate ester. Electrical conductivity, sedimentation, viscosity have been used to determine which phenomena take place in the defloculation of the suspension. The stability of AlN suspensions is due mainly to an electrostatic mechanism, with a steric contribution. The aluminium nitride tape-casting slip has been sintered at 1 850°C in a nitrogen atmosphere. Removing of the binder and plasticizer can be performed in the same oven at 650°C, due to their nature and low concentration. The thermal conductivity of the substrates has been measured and is in the 160 to 200W/m.K range. Le nitrure d'aluminium est le matériau appelé à remplacer l'alumine ou l'oxyde de béryllium en tant que substrat dans les circuits de puissance en microélectronique. La poudre utilisée préparée par carbonitruration a été mise en suspension dans le milieu azéotropique butanone-2-éthanol en présence d'un ester phosphorique. Conductivité, sédimentation, viscosité ont permis d'appréhender les phénomènes qui permettent de défloculer la suspension. La stabilité est assurée par un mécanisme électrostatique ayant une contribution stérique. Le nitrure est coulé en bande puis fritté à 1?850°C sous azote. La qualité et la faible teneur du liant permettent de l'éliminer dans le même four à 650°C. Les conductivités thermiques des substrats sont comprises entre 160 et 200W/m.K. Jarrige, J.; Mexmain, J.; Oumaloul, M.; Bachelard, R.; Disson, J. P. 1993-04-01 319 NASA Astrophysics Data System (ADS) Thermal treatment has recently been emerging as a promising environmental technology to stabilize heavy metal-containing industrial sludge. This study used x-ray absorption spectroscopy (XAS) to identify the species of copper contaminant contained in aluminium oxide that is one of the main compositions of sludge and soil. Results indicate that the originally loaded copper nitrate was transformed into Cu(OH)2 after its dissolution in the aluminium oxide slurry. Extended x-ray absorption fine structure (EXAFS) fitting indicates that the main copper species in the 105 °C dried Cu(NO3)2-loaded aluminium oxide is Cu(OH)2 which accounts for ca. 75% of the loaded copper. After thermal treatment at 500 °C for 1 h, both x-ray absorption near-edge structure (XANES) and EXAFS fitting results show that CuO became the prevailing copper species (about 85%); the rest of the copper consisted of {\\sim }15{%} Cu(OH)2 and a negligible amount of Cu(NO3)2. It was found that most Cu(OH)2 and Cu(NO3)2 decomposed into CuO at 500 °C. Further increase of the heating temperature from 500 to 900 °C resulted in more decomposition of Cu(OH)2 and Cu(NO3)2; therefore CuO remained as the main copper species. However, it was suggested that about 15% of the loaded copper formed CuAl2O4 through the chemical reaction between CuO and Al2O3 at 900 °C. Wei, Yu-Ling; Wang, Hsi-Chih; Yang, Yaw-Wen; Lee, Jyh-Fu 2004-08-01 320 PubMed Central OBJECTIVE: To determine whether long term potroom workers in an aluminium smelter are at increased risk of neurological disorders. METHODS: Cross sectional study of 63 current and former aluminium potroom workers first employed before 1970 and with at least 10 years of service. A group of 37 cast house and carbon plant workers with similar durations of employment and starting dates in the same smelter were used as controls. The prevalence of neurological symptoms was ascertained by questionnaire. Objective tests of tremor in both upper and lower limbs, postural stability, reaction time, and vocabulary were conducted. All subjects were examined by a neurologist. RESULTS: No significant differences in age, race, or education were found between the two groups. Although the potroom group had higher prevalences for all but one of the neurological symptoms, only three odds ratios (ORs) were significantly increased; for incoordination (OR 10.6), difficulty buttoning (OR 6.2), and depression (OR 6.2). Tests of arm or hand and leg tremor in both the visible and non-visible frequencies did not show any significant differences between the two groups. Testing of postural stability showed no definitive pattern of neurologically meaningful differences between the groups. There were no significant differences between the two groups in reaction time, vocabulary score, or clinical neurological assessment. CONCLUSIONS: The objective measures of neurological function provided little support for the finding of increased neurological symptom prevalences in the potroom workers, although increased symptoms may be an indicator of early, subtle neurological changes. The results provide no firm basis for concluding that neurological effects among long term potroom workers are related to the working environment, in particular aluminium exposure, in potrooms. These findings should be treated with caution due to the low participation of former workers and the possibility of information bias in the potroom group. Sim, M; Dick, R; Russo, J; Bernard, B; Grubb, P; Krieg, E; Mueller, C; McCammon, C 1997-01-01 321 PubMed The title compound, [Al(H(2)O)(6)](CH(3)SO(3))(3) (common name: aluminium methane-sulfonate hexa-hydrate), was crystallized from an aqueous solution prepared by the precipitation reaction of aluminium sulfate and barium methane-sulfonate. Its crystal structure is the first of the boron group methane-sulfonates to be determined. The characteristic building block is a centrosymmetric unit containing two hexa-aqua-aluminium cations that are connected to each other by two O atoms of the -SO(3) groups in an O-H?O?H-O sequence. Further O-H?O hydrogen bonding links these blocks in orthogonal directions - along [010] forming a double chain array, along [10-1] forming a layered arrangement of parallel chains and along [101] forming a three-dimensional network. As indicated by the O?O distances of 2.600?(3)-2.715?(3)?Å, the hydrogen bonds are from medium-strong to strong. A further structural feature is the arrangement of two and four methyl groups, respectively, establishing 'hydro-phobic islands' of different size, all positioned in a layer-like region perpendicular to [101]. The only other building block within this region is one of the -SO(3) groups giving a local connection between the hydro-philic structural regions on both sides of the 'hydro-phobic' one. Thermal analysis indicates that a stepwise dehydration process starts at about 413?K and proceeds via the respective penta- and dihydrate until the compound completely decomposes at about 688?K. PMID:22904782 Trella, Thomas; Frank, Walter 2012-08-01 322 NASA Astrophysics Data System (ADS) A colloidal suspension of hollow aluminium, cap-shaped nanoparticles ('nano-caps') can be conveniently produced by evaporation of aluminium onto a spin-coated layer of polystyrene nanoparticles (PSNPs), followed by sonication and dissolution of the polymer template. Although ordinary spherical aluminium nanoparticles have a plasmon resonance in the ultra-violet, the 'nano-caps' show plasmon absorption between 700 and 1200 nm due to their geometry. The position of their extinction peaks can be tuned by varying the thickness of the aluminium and the shape of the nano-cap. The optical properties of these shapes were modelled using the discrete dipole approximation method, which confirmed that the 'caps' have very significantly red-shifted absorbance and scattering compared to spheres. This finding suggests that aluminium nano-caps might compete with gold and silver nanoparticles in applications requiring absorption in the near infrared. Liu, J.; Cankurtaran, B.; McCredie, G.; Ford, M. J.; Wieczorek, L.; Cortie, M. B. 2005-12-01 323 NSDL National Science Digital Library In this brochure, the European Aluminium Association (EAA) evaluates the need for vehicle lightweighting to reduce CO2 emissions. Since the 70âs aluminum has been used for some car components (radiators, cylinder heads, and bumper beams), but now has grown to the average amount of 140 kg per car produced in Europe. Aluminum castings, extrusions, forgings and sheets can now be found nearly everywhere, including in car bodies, closures, chassis, suspensions and wheels. This resource explains why, now more than ever, reducing vehicle mass is necessary and how aluminum can be used to further improve the sustainability and the safety of future generations of cars. Association, European A. 324 Microsoft Academic Search The effects of the daily administration of aluminium (25 mg kg-1, orally), ethanol (10% v\\/v, in drinking water) or both to adult rats, for 6 weeks, on the amount of aluminium present in the tissues and the functioning of brain biogenic amines, hepatic and serum transaminases and some haematopoietic variables were investigated.Ethanol alone was seen to inhibit the activity of S. J. S. Flora; Mamta Dhawan; S. K. Tandon 1991-01-01 325 Microsoft Academic Search Highly oriented polycrystalline aluminium-doped (Al-doped) and ruthenium–aluminium (Ru–Al) co-doped zinc oxide are prepared on borosilicate glass and polycarbonate (PC) substrates by co-sputtering at room temperature. To investigate the effect of Ru doping, co-sputtering is achieved by varying the sputtering power of Ru target while keeping the sputtering target power of Al-doped zinc oxide unchanged. Atomic force microscopy (AFM) data shows L. M. Wong; S. J. Wang; W. K. Chim 2010-01-01 326 NASA Astrophysics Data System (ADS) In this study, we investigated numerically the effect of aluminium concentration on the resonant tunnelling time (RTT) and the laser wavelength of random trimer barrier AlxGa1-xAs superlattices (RTBSL). Such systems consist of two different structures randomly distributed along the growth direction, with the additional constraint that the barriers of one kind appear in triply. An explicit formula is given for evaluating the transmission coefficient of superlattices (SL’s) with intentional correlated disorder. The method is based on Airy function formalism and the transfer-matrix technique. We discuss the impact of the aluminium concentration associated to the structural profile of the SL’s on the RTT and the laser wavelengths. Bendahma, F.; Bentata, S.; Djelti, R.; Aziz, Z. 2014-09-01 327 Microsoft Academic Search This paper describes an investigation of the properties of surfacing weld metals produced by the MIG and TIG processes using newly developed 1.2 mm dia. Al?Cu seamless cored electrode wires. Three types of Al?Cu cored wires with Cu contents of 40%, 45%, and 70% were tested on type A1050P (Al), A5083P (Mn?Mg?Al), A7075P (Cu?Mg?Zn?Al), and AC4B (Si?Cu?Zn, cast) aluminium alloy Y. Kanbe; Y. Nakada; S. Kurihana; H. Koike; T. Miyake 1994-01-01 328 PubMed The present work contributes to establishing the role of hydrogenation and of the substrates in the aluminium-induced crystallization process of amorphous germanium layers. For such a purpose, four series of a-Ge(Al) samples, deposited under identical nominal conditions, were studied: hydrogenated samples, H-free samples, and samples deposited on crystalline silicon and on glass substrates, respectively. On purpose, the impurity concentration was kept at a doping level (10??<[Al/Ge]<2 × 10?³). Furthermore, the films were submitted to isochronal cumulative thermal annealing in the 200-550?°C range. Raman scattering spectroscopy was used to characterize the crystallization process. The role of Al impurity as a precursor seed for the crystallization of a-Ge:H has been clearly established, confirming that the metal-induced crystallization (MIC) phenomenon occurs at an atomic level. Moreover, it has been found that hydrogenation and the periodic nature of the substrate play a fundamental role in the appearance of crystal seeds at low temperatures. The evolution of crystallization with annealing temperature and the analysis of the distribution of crystallite sizes indicate that the formation of crystal seeds occurs at the amorphous film-substrate interface. The importance of fourfold-coordinated aluminium as the embryo of nanocrystal formation is discussed. PMID:22251593 Muniz, L R; Ribeiro, C T M; Zanatta, A R; Chambouleyron, I 2007-02-21 329 NASA Astrophysics Data System (ADS) This paper reports an original inverse perforation tests on foam core sandwich panels under impact loading. The key point is the use of an instrumented Hopkinson pressure bar as a perforator and at the same time a measuring device. It aims at a high quality piercing force record during the whole perforation process, which is a lack of common free-flying projectile - target testing schemes. This new testing arrangement allows for the measurement of piercing force-displacement curves under quasi-static and impact loadings of sandwich samples, which is made of 40?mm AlSi7Mg0.5 Cymat foam cores and 0.8?mm thick 2024 T3 aluminium sheet as top and bottom skins. Compared with quasi-static top skin peak loads (the maximal load before the perforation of top skins) obtained under same geometric and clamping conditions and even in the case that the used foam core (Cymat) and aluminium skin sheet are known and have been confirmed rate insensitive, a significant enhancement under impact loading (25%) of the top skin peak load is found. Elnasri, I.; Zhao, H.; Girard, Y. 2006-08-01 330 PubMed In order to transport and cryopreserve human tissues, it is essential to have an easy-to-use recipient where tissues can be kept in sterile conditions. Here we show the results obtained by using Macopharma's tissue freezing bags, an aluminium-polyethylene multilayer bag, in our tissue bank of the Centro Comunitario de Sangre y Tejidos de Asturias. Five hundred and twenty-seven cancellous bone homografts were obtained from hospitals located 120 km around our Bank. The homografts were submitted to bacteriological controls and sent to our bank in these bags. They were stored at -70 degrees C and sent in dry ice to about 50 hospitals, where the tissue was bacteriologically controlled and grafted. Furthermore, the behaviour of these bags at -140 degrees C (vapour nitrogen) or -196 degrees C (liquid nitrogen) was tested. Our results indicate that Macopharma aluminium-polyethylene bags are suitable for the transporting and cryopreserving of cancellous bone homografts. These bags could also be used for keeping tissues in nitrogen containers. PMID:16933042 Meana, A; Martinez, R; Cañal, P; Arriaga, M J; Román, F San; Llames, S; Orós, C; Moreno, A; Fernandez, C 2006-01-01 331 NASA Astrophysics Data System (ADS) Split Hopkinson Pressure Bar system (SHPB) with large-diameter and Nylon bars introducing a shear-compression loading device is used in order to investigate the dynamic behaviour of aluminium honeycomb under multiaxial loadings conditions. All shear-compression configurations including the loading angle variation from 0? to 60? are performed with an impact velocity of about 15m/s. The adapted SHPB system with the device are validated numerically and a phenomenon of separation between the input bar and the input beveled bar is observed. Numerical results suggest that this phenomenon provides a cutting of the reflected wave. An electro optical extensometer is employed in experiments. A good agreement between the numerical elastic waves and the experimental ones is obtained. Experimental results show a significant effect of the loading angle on the apparent stress-strain curves. The initial peak value and the plateau stress decrease with the increase of the loading angle. The combined shear-compression device with an enhancement at the alignment set-up provides efficient results for samples dynamically loaded. This device will be used to investigate the influence of the in-plane orientation angle on the deformation mechanisms and multiaxial behaviour of aluminium honeycomb under dynamic and quasi-static loading conditions. Tounsi, R.; Zouari, B.; Chaari, F.; Haugou, G.; Markiewicz, E.; Dammak, F. 2012-08-01 332 PubMed Central A study of reflective interference spectroscopy [RIfS] properties of nanoporous anodic aluminium oxide [AAO] with the aim to develop a reliable substrate for label-free optical biosensing is presented. The influence of structural parameters of AAO including pore diameters, inter-pore distance, pore length, and surface modification by deposition of Au, Ag, Cr, Pt, Ni, and TiO2 on the RIfS signal (Fabry-Perot fringe) was explored. AAO with controlled pore dimensions was prepared by electrochemical anodization of aluminium using 0.3 M oxalic acid at different voltages (30 to 70 V) and anodization times (10 to 60 min). Results show the strong influence of pore structures and surface modifications on the interference signal and indicate the importance of optimisation of AAO pore structures for RIfS sensing. The pore length/pore diameter aspect ratio of AAO was identified as a suitable parameter to tune interferometric properties of AAO. Finally, the application of AAO with optimised pore structures for sensing of a surface binding reaction of alkanethiols (mercaptoundecanoic acid) on gold surface is demonstrated. 2012-01-01 333 NASA Astrophysics Data System (ADS) The response of aluminium foams to impact can be categorised by the impact velocity. Tests are reported ranging from quasi-static to impact velocities greater than the speed of sound in the foam. The techniques used ranging from drop-hammer and pneumatic launcher tests, to plate impact at velocities greater than 1000 m s-1. The quasi-static compression behaviour was elastic, perfectly-plastic, locking. For static and dynamic compression at low impact velocities, post-impact examination of partially crushed specimens showed that deformation was through the cumulative multiplication of crush bands. If the impact velocity is less than the velocity of sound, but above a certain critical impact velocity, the plastic compression occurs in a shock-like manner and the specimens deform by progressive cell crushing. At higher impact velocities the compaction front is not preceded by an elastic wave. Laboratory X-ray microtomography has been employed to acquire tomographic datasets of aluminium foams before and after tests. The morphology of the underformed foam was input as the input dataset to an Eulerian code. Hydrocode simulations were then carried out on real microstructure. These simulations provide insight to mechanisms associated with the localization of deformation. Harrigan, J. J.; Millett, J. C. F.; Milne, A. M. 2005-07-01 334 NASA Astrophysics Data System (ADS) The response of aluminium foams to impact can be categorised according to the impact velocity. Tests have been carried out at a range of impact velocities from quasi-static to velocities approaching the speed of sound in the foam. Various experimental arrangements have been employed including pneumatic launcher tests and plate impact experimants at velocities greater than 1000 m s-1. The quasi-static compression behaviour was approximately elastic, perfectly-plastic, locking. For static and dynamic compression at low impact velocities the deformation pattern was through the cumulative multiplication of discrete, non-contiguous crush bands. Selected impact tests are presented here for which the impact velocity is less than the velocity of sound, but above a certain critical impact velocity so that the plastic compression occurs in a shock-like manner and the specimens deform by progressive cell crushing. Laboratory X-ray microtomography has been employed to acquire tomographic datasets of aluminium foams before and after tests. The morphology of the underformed foam was used as the input dataset to an Eulerian code. Hydrocode simulations were then carried out on a real microstructure. These simulations provide insight to mechanisms associated with the localization of deformation. Harrigan, J. J.; Hung, Y.-C.; Tan, P. J.; Bourne, N. K.; Withers, P. J.; Reid, S. R.; Millett, J. C. F.; Milne, A. M. 2006-07-01 335 NASA Astrophysics Data System (ADS) A new catalytic spectrophotometric method is described for the determination of trace amounts of Al(III). The methods based on catalytic action of Al(III) on the oxidation of indigo carmine (IC) by ammonium persulfate in hexamethylene tetramine-hydrochloric acid ((CH 2) 6N 4-HCl) buffer medium (pH 5.4) and in the presence of surfactant—TritonX-100. The effects of some factors on the reaction speed were investigated. Aluminium concentration is linear for 0-1.2 × 10 -7 g/ml in this method. The detection limit of the proposed method is 1.96 × 10 -8 g/ml. Most of the foreign ions except for Cu(II), Fe(III) do not interfere with the determination, and the interference of Cu(II) and Fe(III) in this method can be removed by extraction with sodium diethyldithiocarbamate-carbon tetrachloride (DDTC-CCl 4). This system is a quasi-zero-order reaction for Al(III), but it is a quasi-first-order reaction for IC. The apparent rate constant is 2.62 × 10 -5 s -1 and the apparent activation energy is 6.60 kJ/mol in the system. The proposed method was applied to the determination of trace aluminium(III) in real samples with satisfactory results. Zheng, Huai-Li; Xiong, Wen-Qiang; Gong, Ying-Kun; Peng, De-Jun; Li, Ling-Chun 2007-04-01 336 SciTech Connect The response of aluminium foams to impact can be categorised according to the impact velocity. Tests have been carried out at a range of impact velocities from quasi-static to velocities approaching the speed of sound in the foam. Various experimental arrangements have been employed including pneumatic launcher tests and plate impact experimants at velocities greater than 1000 m s-1. The quasi-static compression behaviour was approximately elastic, perfectly-plastic, locking. For static and dynamic compression at low impact velocities the deformation pattern was through the cumulative multiplication of discrete, non-contiguous crush bands. Selected impact tests are presented here for which the impact velocity is less than the velocity of sound, but above a certain critical impact velocity so that the plastic compression occurs in a shock-like manner and the specimens deform by progressive cell crushing. Laboratory X-ray microtomography has been employed to acquire tomographic datasets of aluminium foams before and after tests. The morphology of the underformed foam was used as the input dataset to an Eulerian code. Hydrocode simulations were then carried out on a real microstructure. These simulations provide insight to mechanisms associated with the localization of deformation. Harrigan, J. J.; Hung, Y.-C.; Tan, P. J.; Bourne, N. K.; Withers, P. J.; Reid, S. R. [University of Manchester, PO Box 88, Sackville Street, Manchester M60 1QD (United Kingdom); Millett, J. C. F. [Defence Academy of the UK, Cranfield University, Shrivenham, Swindon, SN6 8LA (United Kingdom); Milne, A. M. [Fluid Gravity Engineering, 83 Market Street, St. Andrews, Fife, KY16 9NX (United Kingdom) 2006-07-28 337 NASA Astrophysics Data System (ADS) This work develops a facile and environment-friendly method for preparing the superhydrophobic aluminium alloy surface with excellent corrosion resistance. The superhydrophobic aluminium alloy surface is fabricated by the boiling water treatment and stearic acid (STA) modification. Results show that the boiling water treatment endows the aluminium alloy surface with a porous and rough structure, while STA modification chemically grafts the long hydrophobic alkyl chains onto the aluminium alloy surface. Just grounded on the micro- and nano-scale hierarchical structure along with the hydrophobic chemical composition, the superhydrophobic aluminium alloy surface is endued the excellent corrosion resistance. Feng, Libang; Che, Yanhui; Liu, Yanhua; Qiang, Xiaohu; Wang, Yanping 2013-10-01 338 US Patent & Trademark Office Database An asphalt enhancing additive, an asphalt cement and an asphalt mixture containing the additive, wherein the additive causes the mixture to be more easily workable so that the asphalt mixture can be manufactured at a temperature lower than the conventional ones, wherein the additive comprises the mixture of an aminic compound, a lubricant and a flux. 2011-09-13 339 NASA Astrophysics Data System (ADS) Elevated aluminum levels in rivers is known to be toxic for aquatic species, in particular Salmo salar; however it was only recently aluminium has been identified as a potential threat to Salmo salar populations in South Western Nova Scotia, Canada (SWNS) (Dennis and Clair 2012). Previously, it was thought SWNS rivers contained enough DOC to render the aluminium in rivers inactive. A key remaining question is whether aluminium levels are declining following atmospheric pollution reductions. Here we make a first assessment of long term (1980-2011) aluminium concentration trends in three watersheds located in SWNS, as measured by weekly grab samples. Our results show that total aluminium levels have significantly increased from 1980-2011 in all three sites. Estimates of ionic aluminium levels indicate that the ionic aluminum concentration frequently exceeds the threshold for the level of aquatic health determined by the European Inland Fisheries Advisory Commission (Howells et al. 1990). Data also indicates that calcium levels have yet to recover even with declining concentrations of riverine sulfate. This new knowledge that aluminium is at toxic levels and is worsening will have implications for policy on acidification mitigation in SWNS; this is an urgent issue as the local salmon population numbers currently are declining to near extirpation levels. Minichiello, Jeff; Sterling, Shannon; Ambrose, Sarah; Clair, Tom 2014-05-01 340 PubMed This study focuses on the effect of acidic water and aqueous aluminium on the monogenean ectoparasite Gyrodactylus salaris, infecting Atlantic salmon (Salmo salar) parr. G. salaris-infected salmon were exposed to various combinations of acidity and aluminium concentrations. The most pronounced effect was the elimination of parasites after 4 days when 202 micrograms Al/l was added to the water. The effect of aluminium was concentration dependent, but was relatively independent of pH (5.2, 5.6 and 5.9). At the lowest pH of 5.0 the effect of aluminium was enhanced. Acidic aluminium-poor water had no or minor effects on the G. salaris infections except at pH 5.0 where all parasites were eliminated within 9 days. The G. salaris populations increased exponentially in untreated control water. The results show for the first time that aqueous aluminium can, to a limited extent, have a positive effect on fish health. This study emphasizes that basic knowledge about abiotic environmental factors is of importance in order to understand the population dynamics, range extension and dispersal of ectoparasites such as G. salaris. Finally, our results suggest that aluminium treatment could form an effective disinfection method against ectoparasites in hatcheries and laboratories, as well as complementing the controversial rotenone treatments used against natural populations of G. salaris. PMID:10446701 Soleng, A; Poléo, A B; Alstad, N E; Bakke, T A 1999-07-01 341 PubMed Sub-cutaneous immunotherapy is an effective treatment for allergy. It works by helping to modify or re-balance an individual's immune response to allergens and its efficacy is greatly improved by the use of adjuvants, most commonly, aluminium hydroxide. Aluminium salts have been used in allergy therapy for many decades and are assumed to be safe with few established side-effects. This assumption belies their potency as adjuvants and their potential for biological reactivity both at injection sites and elsewhere in the body. There are very few data purporting to the safety of aluminium adjuvants in allergy immunotherapy and particularly so in relation to longer term health effects. There are, if only few, published reports of adverse events following allergy immunotherapy and aluminium adjuvants are the prime suspects in the majority of such incidents. Aluminium adjuvants are clearly capable of initiating unwanted side effects in recipients of immunotherapy and while there is as yet no evidence that such are commonplace it is complacent to consider aluminium salts as harmless constituents of allergy therapies. Future research should establish the safety of the use of aluminium adjuvants in sub-cutaneous allergy immunotherapy. PMID:24444186 Exley, Christopher 2014-01-01 342 PubMed Central OBJECTIVES--The purpose was to study the effects on the nervous system in welders exposed to aluminium and manganese. METHODS--The investigation included questionnaires on symptoms, psychological methods (simple reaction time, finger tapping speed and endurance, digit span, vocabulary, tracking, symbol digit, cylinders, olfactory threshold, Luria-Nebraska motor scale), neurophysiological methods (electroencephalography, event related auditory evoked potential (P-300), brainstem auditory evoked potential, and diadochokinesometry) and assessments of blood and urine concentrations of metals (aluminium, lead, and manganese). RESULTS--The welders exposed to aluminium (n = 38) reported more symptoms from the central nervous system than the control group (n = 39). They also had a decreased motor function in five tests. The effect was dose related in two of these five tests. The median exposure of aluminium welders was 7065 hours and they had about seven times higher concentrations of aluminium in urine than the controls. The welders exposed to manganese (n = 12) had a decreased motor function in five tests. An increased latency of event related auditory evoked potential was also found in this group. The median manganese exposure was 270 hours. These welders did not have higher concentrations of manganese in blood than the controls. CONCLUSIONS--The neurotoxic effects found in the groups of welders exposed to aluminium and manganese are probably caused by the aluminium and manganese exposure, respectively. These effects indicate a need for improvements in the work environments of these welders. Sjogren, B; Iregren, A; Frech, W; Hagman, M; Johansson, L; Tesarz, M; Wennberg, A 1996-01-01 343 NASA Astrophysics Data System (ADS) In this work, samples of aluminium alloys 5083-T0 and 6082-T6 have been welded under conduction regime, using a high power diode laser. The influence of experimental variables, as the laser power and the linear welding rate, on the sizes and properties of the butt weld beads has been studied. In addition to measure the depths and widths of the weld beads, their microstructure, microhardness profile and corrosion resistance have been analysed. The results obtained allow one to define the experimental conditions leading to good quality butt welds with higher penetration than those published in the recent literature under conduction regime. Maximum penetration values of 3 and 2.3 mm were obtained for 5083 and 6082, respectively. Additionally, a simple mathematical expression relating the weld depth ( d) with the laser power ( P) and the processing rate ( v) has been proposed: d=(P-bb)/(av)-(ba)/a, being a, a', b and b' constant values for each alloy and under the employed experimental conditions. The values of these coefficients have been estimated from the fitting to the experimental depth values of 5083 and 6082 butt welds generated under conduction regime. Sánchez-Amaya, J. M.; Delgado, T.; González-Rovira, L.; Botana, F. J. 2009-09-01 344 PubMed Aluminium concentration in samples of total parenteral nutrition solutions and samples of their individual components were analysed to know the exposure to this element. The median aluminium content obtained for the total parenteral nutrition solutions was 105.7 microg/L; for their individual components, 10% calcium gluconate and 1M monopotasic phosphate were the most contaminated, as well as the 1M sodium bicarbonate. The great variability found in the aluminium content of solutions suggests that contamination occurs during the manufacturing process. PMID:18039492 Alvarez, Laura; Rebollido, Maira; Fernandez-Lorenzo, José Ramon; Cocho, José Angel; Fraga, José Maria 2007-01-01 345 PubMed A new fluorometric method is reported for the determination of aluminium, based on the complex forming of 7-[(2,4-dicarboxymethyl-5-carboxybenezene)azo]-8-hydroxyquinoline-5-sulfonic (DCOBAQS) with aluminium. The complex was formed at pH 5.54 with lambda ex/lambda em = 510 nm/572 nm. A better line relationship curve was obtained over the range of 0-0.04 microgram.mL-1 Al(III), and the detection limit for Al(III) is 0.557 ng.mL-1. The method is applied to determination aluminium in tea with satisfactory results. PMID:12953588 Wu, F; Haung, J 2001-02-01 346 NASA Astrophysics Data System (ADS) Five minerals containing aluminium in different crystal configurations are studied. The different kinds of chemical bonding between aluminium and oxygen originate molecular orbitals with energy levels and transition probabilities varying from one compound to another. This effect appears as shifts and changes in relative intensities of K? emission lines and as modifications of the K? characteristic spectrum. In the present work, the aluminium K characteristic spectra obtained by means of an electron microprobe with a wavelength dispersive system are compared for topaz, albite, spodumene, biotite and corundum. Bonetto, Rita; Trincavelli, Jorge; Vasconcellos, Marcos 2005-11-01 347 PubMed Nano-sized layered aluminium or zinc-manganese oxides were synthesized and characterized by scanning electron microscopy, transmission electron microscopy, X-ray diffraction spectrometry, dynamic light scattering and atomic absorption spectroscopy. These oxides showed efficient water oxidizing activity in the presence of cerium(IV) ammonium nitrate as a non-oxo transfer oxidant. Amounts of dissolved manganese, zinc or aluminium, and water oxidation activities of these oxides were reported and compared with other manganese oxides. A mechanism for oxygen evolution and possible roles for zinc or aluminium ions are also proposed. PMID:22565665 Najafpour, Mohammad Mahdi; Pashaei, Babak; Nayeri, Sara 2012-06-21 348 PubMed Additives are used for improving food structure or preventing its spoilage, for example. Many substances used as additives are also naturally present in food. The safety of additives is evaluated according to commonly agreed principles. If high concentrations of an additive cause adverse health effects for humans, a limit of acceptable daily intake (ADI) is set for it. An additive is a risk only when ADI is exceeded. The healthiness of food is measured on the basis of nutrient density and scientifically proven effects. PMID:24772784 Heinonen, Marina 2014-01-01 349 NASA Technical Reports Server (NTRS) A method has been found for enhancing the melt flow of thermoplastic polyimides during processing. A high molecular weight 422 copoly(amic acid) or copolyimide was fused with approximately 0.05 to 5 pct by weight of a low molecular weight amic acid or imide additive, and this melt was studied by capillary rheometry. Excellent flow and improved composite properties on graphite resulted from the addition of a PMDA-aniline additive to LARC-TPI. Solution viscosity studies imply that amic acid additives temporarily lower molecular weight and, hence, enlarge the processing window. Thus, compositions containing the additive have a lower melt viscosity for a longer time than those unmodified. Pratt, J. R.; St. Clair, T. L.; Burks, H. D.; Stoakley, D. M. 1987-01-01 350 NASA Astrophysics Data System (ADS) The Hall-Heroult process for the production of Aluminium metal is some 125 years old. The process is energy constrained by the need to shed around half of the (electrical) energy supplied to the cell as waste heat. The molten cryolite electrolyte is sufficiently aggressive that the only reliable method of protecting the side wall of the cell is to maintain a frozen layer of electrolyte at the hot face of the sidewall. Thus the lack of a cryolite resistant sidewall is but one of several materials science constraints which still limit the energy efficiency of the process. An inert anode and non-consumable cathode are also significant challenges which limit cell life and energy efficiency. Thus there are major challenges in both materials development and new conceptual cell designs to improve the efficiency of this process. Metson, James; McIntosh, Grant; Etzion, Ronny 351 NASA Astrophysics Data System (ADS) This study presents the fabrication and temperature sensing properties of sensors based on aluminium phthalocyanine chloride (AlPcCl) thin films. To fabricate the sensors, 50-nm-thick electrodes with 50-?m gaps between them are deposited on glass substrates. AlPcCl thin films with thickness of 50-100 nm are deposited in the gap between electrodes by thermal evaporation. The resistance of the sensors decreases with increasing thickness and the annealing at 100 °C results in an increase in the initial resistance of sensors up to 24%. The sensing mechanism is based on the change in resistance with temperature. For temperature varying from 25 °C to 80 °C, the change in resistance is up to 60%. Simulation is carried out and results obtained coincide with experimental data with an error of ±1%. Muhammad Tariq Saeed, Chani; Abdullah, M. Asiri; Kh., S. Karimov; Atif, Khan Niaz; Sher Bhadar, Khan; Khalid., A. Alamry 2013-11-01 352 PubMed We present a model of charge transport in organic solids which explicitly considers the packing and electronic structure of individual molecules. We simulate the time-of-flight mobility measurement in crystalline and disordered films of tris(8-hydroxyquinoline) aluminium (Alq(3)). The morphology of disordered Alq(3) is modelled on a molecular scale, and density functional theory is used to determine the electronic couplings between molecules. Without any fitting parameters we predict electron mobilities in the crystalline and disordered phases of approximately 1 and approximately 10(-4) cm(2) V(-1) s(-1), respectively. In good agreement with experiment we find that electron mobilities are two orders of magnitude greater than those of holes. We explain this difference in terms of the spatial extent of the frontier orbitals. Our results suggest that charge transport in disordered Alq(3) is dominated by a few highly conducting pathways. PMID:18368176 Kwiatkowski, J J; Nelson, J; Li, H; Bredas, J L; Wenzel, W; Lennartz, C 2008-04-14 353 PubMed Central Of 30 workmen at risk 27 were examined and six found to have evidence of pulmonary fibrosis. In the two fatal cases the evidence for pulmonary fibrosis was conclusive and in three other cases it was sufficient; in one case it was suggestive. The clinical, radiological, and pathological features of these six cases are recorded. Results of respiratory function tests on the four non-fatal cases are given. The manufacturing process is described. An analysis of the powder is given, also the results of dust measurement in the contaminated atmosphere. The literature is reviewed and our own experience compared with reports from Germany and Canada. We concluded that the pulmonary fibrosis was caused by the dust inhaled at work and that the component responsible was finely divided aluminium. Images Mitchell, John; Manning, G. B.; Molyneux, M.; Lane, Ronald E. 1961-01-01 354 PubMed Central It has been proposed that aluminium toxicity may be mediated, at least in part, by free radical generation. We have investigated the effects of aluminium lactate administration on indices of hepatic oxidant stress, and the consequences of concomitant dietary vitamin E, in male albino Wistar rats. Aluminium lactate was administered for 4 weeks, by ip injection at 10 mg aluminium/kg body weight. Groups of animals received a chow diet containing 0, 5, 15, or 20 mg vitamin E/g of food. A control group of rats received a normal chow diet, without being injected with aluminium. The rats were killed after 4 weeks, and blood and liver tissue removed for the measurement of aluminium and markers of oxidative stress. Plasma and liver aluminium levels were increased in all groups of animals receiving aluminium lactate (P < 0.01), although these levels were significantly reduced in rats receiving concomitant vitamin E (P < 0.05). Aluminium treatment was associated with significantly increased levels of hepatic reactive oxygen species (ROS) (P < 0.01) that were attenuated by concomitant vitamin E (P < 0.05). Hepatic catalase and reduced glutathione levels were both reduced in animals treated with aluminium (P < 0.05). Abubakar, M G; Taylor, A; Ferns, G A A 2003-01-01 355 National Technical Information Service (NTIS) A study is made of stable and metastable aluminium-copper phase systems, hardening deposition mechanisms, electrical resistance, polycrystalline structures as well as the Guinier-Preston zone structure through small-angle neutron scattering methods. 50 re... P. A. Prieto Pulido 1982-01-01 356 PubMed Macro amounts of fluoride in aluminium reduction materials are successfully determined with a fluoride electrode. Except for anhydrous aluminium fluoride, which requires fusion with sodium hydroxide, samples are dissolved in aqueous media. Cryolite and sodium fluorosilicatc are dissolved in boiling sodium hydroxide solution. Other materials containing fluoride, such as fluorspar and the reduction cell bath and pot-lining, require dissolution in a hydrochloric acid solution of aluminium chloride. Potential interference from large amounts of aluminium (and calcium, if present) is eliminated and pH control attained by using ammoniacal sulphosalicylate (and EDTA). The procedures are reasonably rapid. Relative errors of less than 2% and a relative standard deviation of 1% are achieved. PMID:18961164 Palmer, T A 1972-10-01 357 PubMed The paper presents the results of investigations of the microstructures and properties of the aluminium coatings deposited by vacuum evaporation and magnetron sputtering. These coatings generally have a very refined microstructure with elongated nano-grains. However, the surface topography of the aluminium coating deposited by vacuum evaporation is more developed, its microstructure is less homogeneous and more porous. The residual tensile stresses in the aluminium coating deposited by magnetron sputtering are close to 130 MPa, and the texture is relatively pronounced. Vacuum evaporation does not induce residual stresses in the coatings and the texture is very weak. The results obtained indicate that the aluminium coatings produced by magnetron sputtering are more suitable for the diffusive Ti-Al intermetallic layers. PMID:20500421 Garbacz, H; Wieci?ski, P; Adamczyk-Cie?lak, B; Mizera, J; Kurzyd?owski, K J 2010-03-01 358 NASA Astrophysics Data System (ADS) Taking wollastonite as raw material, aluminium sulfate and water glass as coating modifier, a new type of wollastonite coated with nano-aluminium silicate powder-materials was obtained by chemical depositing method. By modificating this powder-materials with silane and filling it in PA6, the PA6 composite materials were obtained. The modificated powder-materials were characterized by SEM, BET, XRD, EDX, and FT-IR. The mechanism of this powder-materials modificated by silane was discussed. It was shown that the surface of wollastonite was coated with nano-aluminium silicate evenly, the average grain size of aluminium silicate was 54 nm, filling 30% of this powder-materials modified by silane in PA6, the mechanical properties and the heat distortion temperature of the PA6 composites filled by it were notably enhanced. Wang, Caili; Zheng, Shuilin; Liu, Guihua; Wang, Lijing; Huang, Peng; Wang, Zhaohua 359 Microsoft Academic Search The rate of release of sulphur oxides from basic aluminium potassium sulphate was studied in a hydrogen atmosphere under isothermal-isobaric conditions, and the kinetic parameters of the process were determined by the statistical criteria method. B. Pacewska 1990-01-01 360 National Technical Information Service (NTIS) The thermochemical oxidation of aluminium with air was successfully demonstrated in a relatively small burner. Although direct caloric measurements of the furnace efficiency could not be made, analyses of the combustion products showed that the powdered-a... W. Durisch O. Haas R. Muelli A. Tsukada H. R. Zumbrunnen 1985-01-01 361 National Technical Information Service (NTIS) Strontium and Antimony treated A356 aluminium alloy samples were metallographically characterized in the as cast and solution and aged conditions. Antimony treated alloy has shown slower spheroidizing kinetics of the Silicon particles during solution trea... R. Fuoco, E. R. Correa, A. V.O. Correa, M. Bocalini Junior 1992-01-01 362 NASA Astrophysics Data System (ADS) The preparation of polyfunctional organometallics is important in organic synthesis as these reagents are very popular nucleophiles. The preparation of functionalized aluminium reagents by direct insertion of aluminium powder is in general not possible. Such a reaction would be of special importance owing to the low price of aluminium compared with magnesium (it is half the price), the low toxicity of this metal and the chemoselectivity of the resultant organoaluminium reagents. We have now found that by adding catalytic amounts of selected metallic chlorides (TiCl4, BiCl3, InCl3 or PbCl2) in the presence of LiCl, aluminium powder inserts into various unsaturated iodides and bromides under mild conditions. These resulting new organoaluminium reagents undergo smooth Pd-catalysed cross-coupling and acylation reactions, as well as copper-catalysed allylic substitutions, affording various interesting products for pharmaceutical and material science applications. Blümke, Tobias; Chen, Yi-Hung; Peng, Zhihua; Knochel, Paul 2010-04-01 363 National Technical Information Service (NTIS) The use and the development of nuclear microanalysis has allowed the characterization of aluminium surfaces and oxidized phases obtained from industrial anodization procedures. The analytical performances of Rutherford backscattering of Li ions and alpha ... M. Fallavier 1977-01-01 364 National Technical Information Service (NTIS) The controlled fabrication of highly ordered anodic aluminium oxide (AAO) templates of unprecedented pore uniformity directly on Si, enabled by new advances on two fronts -- direct and timed anodisation of a high-purity Al film of unprecedented thickness ... A. Yin D. Cardimona J. Xu L. Guo M. Tzolov 2007-01-01 365 NASA Astrophysics Data System (ADS) The aluminium distribution between the major cell compartments of human neuroblastoma cells grown in culture has been determined using 21Al and accelerator mass spectrometry (AMS). Cells (IMR-32) were grown for eight days in a culture medium containing Al-EDTA (0.2mM) spiked with 26Al, harvested, and fractionated by standard biochemical techniques. 26Al in fractions after ashing to Al 2O 3 was determined by AMS using the 14UD accelerator at ANU Canberra. The cytoplasmic and nuclear cell compartments appeared to have reached diffusive equilibrium with the culture medium. Whilst 26Al was retained by the nuclear proteins and nuclear sap, 26Al did not appear to bind to the nucleic acids (DNA/RNA). King, S. J.; Templar, J.; Miller, R. V.; Day, J. P.; Dobson, C. B.; Itzhaki, R. F.; Fifield, L. K.; Allan, G. L. 1994-06-01 366 NSDL National Science Digital Library The Vector Addition model allows the user to practice vector addition of two vectors in two dimensions. You are given the magnitude and direction of the two vectors, and your goal is to fill in the nine values in the table (showing the x-component, y-component, and length) of the two vectors, and the resultant vector that is the sum of the first two vectors. The Vector Addition was created using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_bu_vector_addition.jar file will run the program if Java is installed. Duffy, Andrew 2010-04-25 367 Microsoft Academic Search Aluminium alloy 5083 is used in the fabrication of lightweight, high-speed marine vessels. The high cyclic service stresses on such structures render welded joints in them fatigue-critical. This paper presents experimental investigations on fatigue crack behaviour of welded 5083-H321 aluminium alloy plates. Crack propagation in the heat-affected zone of welded specimens was determined from tests conducted on single edge-notched tension Krishnakumar Shankar; Weidong Wu 2002-01-01 368 Microsoft Academic Search High cycle fatigue properties of the high-pressure die-cast magnesium alloys AZ91 hp, AS21 hp and AE42 hp and of the aluminium alloy AlSi9Cu3 are investigated at elevated temperatures. Fatigue tests are performed at ultrasonic cyclic frequency and load ratio R=?1. Compared with ambient air environment, the S–N curves determined in warm air of 125°C (magnesium alloys) and 150°C (aluminium alloy) H. Mayer; M. Papakyriacou; B. Zettl; S. Vacic 2005-01-01 369 Microsoft Academic Search A finite element model has been developed to predict the evolution of residual stress and distortion which takes into account the history-dependence of the yield stress–temperature response of heat-treatable aluminium alloys during welding. The model was applied to TIG welding of 2024-T3 aluminium alloy, and the residual strain predictions validated using high resolution X-ray synchrotron diffraction. The goal was to R. V. Preston; H. R.. Shercliff; P. J. Withers; S. Smith 2004-01-01 370 Microsoft Academic Search We report the heteroepitaxy of Al(111) on ?-Al2O3(0001) realized at 470°C by aluminium evaporation and characterized by Auger spectroscopy (AES) and low energy electron diffraction (LEED). This crystalline growth is preceded by suboxide formation due to the superficial oxygens of the sapphire substrate, and the nucleation of aluminium crystallites with a (111) surface randomly oriented around the (0001) axis. Epitaxial M. Vermeersch; F Malengreau; R Sporken; R Caudano 1995-01-01 371 Microsoft Academic Search The growth by RF reactive sputtering of aluminium nitride (AlN) at the surface of oxidised iron was studied by X-ray photoelectron spectroscopy (XPS). The observation of the substrate lines showed that the top layer of the iron oxide, constituted of Fe2O3 was first reduced into Fe3O4 by the incoming Al atoms to form an aluminium oxide. Moreover, an analysis of F. Malengreau; V. Hautier; M. Vermeersch; R. Sporken; R. Caudano 1995-01-01 372 Microsoft Academic Search The use of aluminium alloys in lightweight protective structures is increasing. Even so, the number of experimental and computational investigations that give detailed information on such problems is limited. In an earlier paper by some of the authors, perforation experiments were performed with 15–30mm thick AA5083-H116 aluminium plates and 20mm diameter, 98mm long, HRC 53 conical-nose hardened steel projectiles. In T. Børvik; M. J. Forrestal; O. S. Hopperstad; T. L. Warren; M. Langseth 2009-01-01 373 Microsoft Academic Search Wood\\/aluminium windows of a standard construction with different coating systems were installed in a facade facing south-east\\u000a on a building in Vienna. Over a period of three years the moisture content of the wooden frames and the climatic conditions\\u000a under the aluminium layers of the windows were measured. The results showed that humidity stress on the wooden frames of the Gerhard Grüll; Thomas Anderl; Irene Schweiger 2005-01-01 374 Microsoft Academic Search Tempering treatments T4, T5 and T6 have been applied after thixoforming to investigate the effects of heat treatment on microstructure and mechanical properties of thixoformed wrought aluminium alloy 2014, a ‘modified’ 2014 and high strength casting aluminium alloy 201. The ‘modified’ 2014 was a simplified version of 2014 with composition (wt%) Cu 3.87, Mg 0.6 and Si 0.14. The results D. Liu; H. V. Atkinson; P. Kapranos; H. Jones 2004-01-01 375 Microsoft Academic Search A new mean of assembling steel to aluminium was developed, following previous work by German workers [1]. In this new method, a laser-induced aluminium melt pool spreads and wets a solid steel, to generate, after solidification a sound and resistant interface layer.Joint properties were investigated, in terms of surface aspects, interface microstructures and mechanical resistances under tensile testing, for non-galvanized P. Peyre; G. Sierra; F. Deschaux-Beaume; D. Stuart; G. Fras 2007-01-01 376 Microsoft Academic Search Residual microstructures, including dynamic recrystallization and grain growth structures and a wide range of precipitation phenomena associated with a friction-stir-weld in a thin 6061-T6 aluminium plate have been systematically investigated utilizing light metallography and transmission electron microscopy. In this rather remarkable process, a hard steel head pin rotating at 400 r.p.m. was advanced into a solid 6061-aluminium plate at a L. E. MURR; G. LIU; J. C. McCLURE 1998-01-01 377 Microsoft Academic Search The electronic structure of an aluminium grain boundary with segregated sulphur impurity atoms has been calculated by a first-principles pseudo-potential method. It is found that a sulphur atom bonds to only one of the neighbouring aluminium atoms. This bond is a mixed-character metallic-covalent bond which is stronger than the metallic Al-Al bonds. Electrons that participate in forming this bond are Guang-Hong Lu; Masanori Kohyama; Ryoichi Yamamoto 2003-01-01 378 Microsoft Academic Search Steel aluminium Tailor Welded Hybrids are still mentioned to be difficult to be joint as intermetallic phases appear during\\u000a melting welding techniques. These phases are the reason for failure of the joint during loading or forming. As conventional\\u000a friction stir welding, a solid phase welding technology, is not feasible to join steel and aluminium, laser assistance for\\u000a preheating the steel M. Merklein; A. Giera 2008-01-01 379 Microsoft Academic Search A method for the determination of trace amounts of aluminium based on Solid-Phase Spectrophotometry has been developed. Aluminium reacts with Eriochrome Cyanine R to give a highly coloured complex, which is fixed on a dextran type anion-exchanger. The absorbance of the gel, at 590 and 750 nm, packed in a 1-mm cell, is measured directly. The calibration graph is linear J. M. Bosque-Sendra; M C. Valencia; S. Boudra 1994-01-01 380 Microsoft Academic Search AA6061 aluminium alloy (Al–Mg–Si alloy) has gathered wide acceptance in the fabrication of light weight structures requiring a high strength-to weight ratio and good corrosion resistance. Compared to the fusion welding processes that are routinely used for joining structural aluminium alloys, friction stir welding (FSW) process is an emerging solid state joining process in which the material that is being K. Elangovan; V. Balasubramanian; S. Babu 2009-01-01 381 Microsoft Academic Search This paper describes the development of a variable flow stress predictive machining theory for aluminium alloys. This theory is based on the Oxley's machining theory which allows for the high strain-rate\\/high temperature flow stress and thermal properties of the work materials and has so far been applied and tested for plain carbon steels.The developed predictive theory for aluminium has been B. Kristyanto; P. Mathew; J. A. Arsecularatne 2002-01-01 382 Microsoft Academic Search This elaboration presents the influence of casting method and anodic treatment parameters on thickness and structure of an anodic layer formed on aluminium alloys. As test materials it was employed the aluminium alloys AlSi12 and ALSi9Cu3, supplied by the Dutch company MIFA b.v., in which casting process and anodic treatment of samples were carried out also. It was presented herein J. Konieczny; L. A. Dobrza?ski; K. Labisz; J. Duszczyk 2004-01-01 383 Microsoft Academic Search Amorphous anodic oxide films have been formed at high efficiency on aluminium implanted with 0022-3727\\/31\\/17\\/002\\/img12 ions 0022-3727\\/31\\/17\\/002\\/img13 in order to study the behaviour of tungsten during film growth. The initial film is composed mainly of alumina because the outer layer of aluminium above the main implanted region of the substrate is oxidized. During this period, tungsten atoms, present in low J. C. S. Fernandes; M. G. S. Ferreira; J. C. Soares; C. M. Jesus; C. M. Rangel; P. Skeldon; G. E. Thompson; X. Zhou; H. Habazaki; K. Shimizu 1998-01-01 384 Microsoft Academic Search The effects of strontium on the solidification mode of hypereutectic aluminium-silicon alloys have been studied. Samples were prepared from an aluminium-17 wt% silicon-based alloy and strontium was added at several different concentrations. The development of the microstructure was investigated by cooling curve analysis, interrupted solidification experiments and optical and scanning electron microscopy. It was found that nucleation of primary silicon K. Nogita; S. D. McDonald; A. K. Dahle 2004-01-01 385 Microsoft Academic Search The effects of strontium on the solidification mode of hypereutectic aluminium–silicon alloys have been studied. Samples were prepared from an aluminium–17?wt% silicon-based alloy and strontium was added at several different concentrations. The development of the microstructure was investigated by cooling curve analysis, interrupted solidification experiments and optical and scanning electron microscopy. It was found that nucleation of primary silicon is K. Nogita; S. D. McDonald; A. K. Dahle 2004-01-01 386 Microsoft Academic Search The nano aluminium particles were produced in different ambience by the wire explosion process. The influence of pressure in the exploding wire chamber on the size of the particles was analyzed. Certain physico-chemical diagnostic studies, viz., wide angle X-ray diffraction (WAXD), thermo-gravimetric differential thermal analysis (TG–DTA) studies were carried out to characterize the produced nano aluminium powder. The compositions of R.. Sarathi; T. K. Sindhu; S. R. Chakravarthy 2007-01-01 387 Microsoft Academic Search Thermal neutron irradiation of aluminium or its alloys causes the production of silicon by transmutation. In aluminium–magnesium alloys, the transmutation-produced silicon reacts with magnesium and forms small precipitates. The precipitation in irradiated Al–Mg alloys is similar to the early stage of aging in thermally treated Al–Mg–Si alloys. This study evidences the simultaneous generation of two crystallographically different precipitate types. On M. Verwerft 2000-01-01 388 Microsoft Academic Search Concentrations of aluminium in drinking waters (tap water, still mineral water and sparkling mineral water), fruit juices and soft drinks were determined using graphite furnace atomic absorption spectrometry (GFAAS) of samples processed with a HNO3–V2O5 acid digestion pre-treatment. In water samples, aluminium was determined directly. We verified the sensitivity, accuracy and precision of the method and ruled out matrix interferences. Francisco F. López; Carmen Cabrera; M. Luisa Lorenzo; M. Carmen López 2002-01-01 389 Microsoft Academic Search The durability has been studied of aluminium-sealant joints immersed in jet-fuel, water and antifreeze. The sealants were a polysulfide, a fluorosilicone and a silicone, and durability was primarily controlled by the amount of fluid absorbed by the sealants. This gives polysulfide joints in antifreeze and silicone joints in jet-fuel very poor durability. The performance of polysulfide-aluminium joints was not significantly J. Comyn; J. Day; S. J. Shaw 1997-01-01 390 Microsoft Academic Search The present study deals with removal of various pollutants from a real wastewater by electrocoagulation treatment. Combined wastewater from one of the Delhi industrial areas was collected and treated by electrocoagulation process using iron and aluminium electrodes. Removal of Cr, Zn, Ni and Cu were achieved up to 100, 98.71, 69.22 and 48.08% respectively using aluminium electrode while Cr, Cu, Asheesh Kumar Yadav; Lakhvinder Singh; Ayusman Mohanty; Santosh Satya; T. R. Sreekrishnan 2012-01-01 391 Microsoft Academic Search The purpose of this paper is to propose an efficient and low cost defluoridation process based on electrocoagulation with aluminium bipolar electrodes. The performance of a pilot scale electrochemical reactor equipped with aluminium bipolar electrodes with an anode active area surface of about 1.6 m2 was studied. The pilot study yielded promising results, suggesting that further in-depth development studies are N. Mameri; H. Lounici; D. Belhocine; H. Grib; D. L. Piron; Y. Yahiat 2001-01-01 392 PubMed Aerosol particles with aerodynamic diameters between 0.18 and 10 microm were collected in the workroom air of two aluminium smelter potrooms with different production processes (Soderberg and Prebake processes). Size, morphology and chemical composition of more than 2000 individual particles were determined by high resolution scanning electron microscopy and energy-dispersive X-ray microanalysis. Based on chemical composition and morphology, particles were classified into different groups. Particle groups with a relative abundance above 1%(by number) include aluminium oxides, cryolite, aluminium oxides-cryolite mixtures, soot, silicates and sea salt. In both production halls, mixtures of aluminium oxides and cryolite are the dominant particle group. Many particles have fluoride-containing surface coatings or show agglomerations of nanometer-sized fluoride-containing particles on their surface. The phase composition of approximately 100 particles was studied by transmission electron microscopy. According to selected area electron diffraction, sodium beta-alumina (NaAl(11)O(17)) is the dominant aluminium oxide and cryolite (Na(3)AlF(6)) the only sodium aluminium fluoride present. Implications of our findings for assessment of adverse health effects are discussed. PMID:15877161 Hoflich, Burkard L W; Weinbruch, Stephan; Theissmann, Ralf; Gorzawski, Hauke; Ebert, Martin; Ortner, Hugo M; Skogstad, Asbjorn; Ellingsen, Dag G; Drablos, Per A; Thomassen, Yngvar 2005-05-01 393 PubMed The present study focuses on the relative sensitivity among freshwater fish species to aqueous aluminium. Seven common Scandinavian fish species were exposed to acidic Al-rich water, acidic Al-poor water, and approximately neutral water as a control. The relative sensitivity among the species to an acute aluminium challenge was documented, and was in the following order: Atlantic salmon, Salmo salar, as the most sensitive; then roach, Rutilus rutilus; minnow, Phoxinus phoxinus; perch, Perca fluviatilis; grayling, Thymallus thymallus; brown trout, Salmo trutta; and Arctic char, Salvelinus alpinus. Substantial mortality was observed in all species when exposed to the Al-rich medium. Some mortality was also observed in minnow, roach, and brown trout exposed to the acidic Al-poor medium and the control medium. A high resistance to aluminium was observed in Arctic char, while perch was found to be more sensitive to aluminium than expected and, for the first time, a toxic response to aqueous aluminium in grayling was documented. Through controlled experimental studies, the results confirm that aluminium is an important factor in the toxicity of acidified waters to freshwater fish species. PMID:15093412 Poléo, A B; ØStbye, K; Øxnevad, S A; Andersen, R A; Heibo, E; Vøllestad, L A 1997-01-01 394 PubMed Natural organic matter removal efficiency and characteristic by ferric chloride and aluminium sulphate were studied. Results showed that ferric chloride was effective in natural organic matter removal when coagulant dosage was higher than 15 mg/L, while aluminium was effective at lower dosage. The TOC of water was reduced to 4.19 mg/L and 9 mg/L at a dosage of 10 mg/L for aluminium sulphate and ferric chloride respectively, while TOC was reduced to 2.44 mg/L and 1.69 mg/L at the dosage of 20 mg/L. Ferric chloride decreased pH sharply than aluminium sulphate which made hydrolysate more positive and attachable for natural organic matter. UV254 and SUVA results showed that ferric chloride removed more conjugate structure materials and unsaturated band contents than aluminium. Ferric chloride was more effective in reducing lower molecular weight organic matter and hydrophilic substances than aluminium, when the dosage of coagulant was 20 mg/L, the removal efficiency of relative molecular weight below 10 000 was 16.4% and 6.1% respectively, while aluminum was more effective in high molecular weight matter removal than ferric chloride. PMID:18624177 Zhou, Ling-Ling; Zhang, Yong-Ji; Sun, Li-Hua; Li, Gui-Bai 2008-05-01 395 NASA Astrophysics Data System (ADS) The galvanic corrosion behaviour of aluminium 3004 and copper with different area ratios were studied in the tropical marine atmosphere at Tuticorin harbour over a period of 426 days. The area ratios of A Al: A Cu, studied were 1:1, 1:2, 1:4, 1:8, 2:1, 4:1 & 8:1. The galvanic corrosion behaviour of metals was studied in terms of relative increase in the corrosion rate of aluminium due to galvanic coupling with copper, relative decrease in the corrosion rate of copper due to galvanic coupling with aluminium, and the susceptibility of aluminium to pitting owing to galvanic coupling with copper. The galvanic potential and galvanic current of the system were monitored. Pits of different dimensions ranging from mild etchings to perforations were experienced on the borders and the surfaces of the interface of aluminium in contact with copper. The weathering parameters and the environmental pollutants which have a major role in influencing the galvanic corrosion of metals were also monitored. The corrosion products resulting from galvanic corrosion were analysed using XRD and the pitting on aluminium resulting from galvanic corrosion has been highlighted in terms of pit depth, size and density of pit, using a high resolution microscope. Subramanian, G.; Palraj, S.; Palanichamy, S. 2014-06-01 396 Microsoft Academic Search Similarity data can be represented by additive trees. In this model, objects are represented by the external nodes of a tree, and the dissimilarity between objects is the length of the path joining them. The additive tree is less restrictive than the ultrametric tree, commonly known as the hierarchical clustering scheme. The two representations are characterized and compared. A computer Shmuel Sattath; Amos Tversky 1977-01-01 397 SciTech Connect Compounds suitable as extreme pressure additives for lubricating oils are of the general formula CX3-S-S-R where X is a halogen and R is a hydrocarbyl. The performance of the compounds is improved by the addition of a stabilizer. Lenack, A.L. 1980-10-14 398 SciTech Connect Deposit control additives for internal combustion engines are provided which maintain cleanliness of intake systems without contributing to combustion chamber deposits. The additives are poly(oxyalkylene) carbamates comprising a hydrocarbyloxyterminated poly(Oxyalkylene) chain of 2-5 carbon oxyalkylene units bonded through an oxycarbonyl group to a nitrogen atom of ethylenediamine. Honnen, L.R.; Lewis, R.A. 1980-11-25 399 PubMed Approximately 30% of the world's total land area and over 50% of the world's potential arable lands are acidic. Furthermore, the acidity of the soils is gradually increasing as a result of the environmental problems including some farming practices and acid rain. At mildly acidic or neutral soils, aluminium (Al) occurs primarily as insoluble deposits and is essentially biologically inactive. However, in many acidic soils throughout the tropics and subtropics, Al toxicity is a major factor limiting crop productivity. The Al-induced secretion of organic acid (OA) anions, mainly citrate, oxalate, and malate, from roots is the best documented mechanism of Al tolerance in higher plants. Increasing evidence shows that the Al-induced secretion of OA anions may be related to the following several factors, including (a) anion channels or transporters, (b) internal concentrations of OA anions in plant tissues, (d) temperature, (e) root plasma membrane (PM) H(+)-ATPase, (f) magnesium (Mg), and (e) phosphorus (P). Genetically modified plants and cells with higher Al tolerance by overexpressing genes for the secretion and the biosynthesis of OA anions have been obtained. In addition, some aspects needed to be further studied are also discussed. PMID:23509687 Yang, Lin-Tong; Qi, Yi-Ping; Jiang, Huan-Xin; Chen, Li-Song 2013-01-01 400 NASA Astrophysics Data System (ADS) The growth mode, geometric and electronic structure of thin copper films deposited onto aluminium single crystal substrates have been studied by LEED, AES, angle-resolved UPS and work function changes. Growth occurs in a quasi-layer-by-layer fashion on Al(111). Temperature dependent interdiffusion leads to formation of a mixed Cu/Al interface upon which ordered epitaxial films eventually grow. The ordered structure is rotationally commensurate with the Al(111) substrate with an interatomic distance typical of bulk Cu. Electronic structure of films grown at 300 K converges to that of a bulk Cu single crystal, convergence being essentially complete at a coverage of 10 monolayers. In contrast, growth occurs in a long-range disordered fashion up to relatively high coverages on Al(100) probably in the form of densely packed Cu clusters above a mixed Cu/Al interfacial region. For submonolayer coverages the work function remains close to the clean Al values, increasing to 4.7 and 4.5 eV for (111) and (100) faces, respectively, in the range from 1 to 4 monolayers. Additional studies carried out at elevated (375 K) and low temperature (120 K) allow postulation of a fully coherent model of the complex growth and electronic properties of this particular metal-on-metal system. Barnes, C. J.; Asonen, H.; Salokatve, A.; Pessa, M. 1987-05-01 401 NASA Astrophysics Data System (ADS) The liquid metal flow behaviours in different runner system designs have important effects on the mechanical strength of aluminium alloy castings. In this paper, a new model has been developed which is a two-dimensional program using a finite difference technique and the Marker and Cell (MAC) method to simulate the flow of liquid metal during filling a mould. In the program the Eulerian method has been used for the liquid metal flow, while the Oxide Film Entrainment Tracking Algorithm (OFET) method (a Lagrangian method) has been used to simulate the movement of the oxide film on the liquid metal surface or in the liquid metal flow. Several examples have been simulated and tested and the relevant results were obtained. These results were compared with measured bending strengths. It was found that the completed program was capable of simulating effectively the filling processes of different runner systems. The simulation results are consistent with the experiment. In addition, the program is capable of providing clearer images for predicting the distribution of the oxide film defects generated during filling a mould. Dai, X.; Jolly, M.; Yang, X.; Campbell, J. 2012-07-01 402 NASA Astrophysics Data System (ADS) This study demonstrates and explains the effect of strontium modification on machinability of aluminium-silicon eutectic (LM-6 type) alloy. This alloy is known to have many favourable features including weight to strength ratio, high corrosion resistance and excellent castability. However, normal unmodified LM-6 alloy has poor machinability, which reduces its applications range. In this work, various samples of LM-6 alloy were cast using sand and metallic chill mould with and without strontium addition. Machining on each cast product, was carried out using recommended cutting parameters for Al-Si alloys. Strontium modified samples have recorded a reduction in average flank wear, an increase in shear plane angles and a reduction in chip thickness. The main reason for this improvement is the refining effect of strontium, which reduces the size of the hard silicon particles. As a result, their abrasive action on the tool face has reduced a lot. Dramatic reductions in tool wear rate were recorded when the microstructures were refined. On the other hand, when no refinement of microstructure occurs, tool wear rate becomes high. Chip analysis showed that strontium modified sample produced a thinner chip thickness with a larger shear plane angle, requiring less cutting forces. The tool wear depends not only on the phases present in the work material, but also on their sizes and distribution over entire structure. Thus, strontium modification has better effect on machinability of die cast alloy compared to that of the sand cast LM-6 alloy. Haque, M. M.; Khan, A. A.; Ismail, Ahmad F. 403 PubMed * Here we investigated the beneficial effect of aluminium (Al) on the development of the Al accumulating plant Melastoma malabathricum. * Seedlings of M. malabathricum were cultivated in a nutrient solution containing 0.5 mM Al and compared with barley (Hordeum vulgare). In addition, roots of M. malabathricum were divided into one part growing in a nutrient solution, and the other part growing in a calcium solution. Al (0.5 mM) was applied to either solution. * Al-induced improvements of the root activity contributed to a growth enhancement in M. malabathricum. Al exposure without nutrients did not increase root growth and Al accumulation in the leaves. The beneficial effect, however, was induced by the combination of Al and nutrients. * We suggest that without nutrients roots are not able to synthesize an adequate amount of citrate that is required for transporting Al to the leaves. High Al levels in the plant tissues and/or an interaction of Al with particular nutrient elements in the apoplast of root cells appear to be essential to exert the beneficial effect of Al. PMID:15720688 Watanabe, Toshihiro; Jansen, Steven; Osaki, Mitsuru 2005-03-01 404 PubMed Arbuscular mycorrhizal fungi alleviating the adverse Aluminium effects on growth and antioxidant activity was tested in Gmelina plants. Under greenhouse and aluminium stress condition, the mycorrhizal Gmelina plants showed good growth as compared to non mycorrhizal Gmelina plants. Mycorrhizal colonization in Gmelina was found not to be significantly influenced by aluminium concentrations. Results also indicate that symbiotic association was successfully established between Glomus intraradices and Gmelina plants and mycorrhizal colonization consequently increased the biomass of Gmelina. The root proline accumulation was found to increase in mycorrhizal Gmelina plants for osmotic adjustment of stress tissues under first and second level of Aluminium stress. It was observed that Mycorrhizal colonization increased the shoot root Peroxidase and Superoxide dismutase activities in mycorrhizal Gmelina under second level of Aluminium stress. Mycorrhizal fungi play a major role in phytostabilization by secreting one of the glycoprotein, i.e., Glomalin, which stabilizes the Aluminium in soil as well as in the roots of Gmelina plants. PMID:22908633 Dudhane, Mayura; Borde, Mahesh; Jite, Paramjit Kaur 2012-08-01 405 NSDL National Science Digital Library These games are fun to play and will test your addition skills! Click on the links to play the games! Play in the Kitten Match and test your math skills!! Pick the grade level you want to play on. Then pick numbers that add up to equal the number that they give you. Hurry, you're being timed! Use your addition skills to win a Diaper Derby!! First decide what you want your baby to look like! Then perform the addition problem that ... Smith, Ms. 2010-10-14 406 NASA Technical Reports Server (NTRS) A process for preparing polyimides having enhanced melt flow properties is described. The process consists of heating a mixture of a high molecular weight poly-(amic acid) or polyimide with a low molecular weight amic acid or imide additive in the range of 0.05 to 15 percent by weight of the additive. The polyimide powders so obtained show improved processability, as evidenced by lower melt viscosity by capillary rheometry. Likewise, films prepared from mixtures of polymers with additives show improved processability with earlier onset of stretching by TMA. Pratt, J. Richard (inventor); St.clair, Terry L. (inventor); Stoakley, Diane M. (inventor); Burks, Harold D. (inventor) 1993-01-01 407 NASA Technical Reports Server (NTRS) A process for preparing polyimides having enhanced melt flow properties is described. The process consists of heating a mixture of a high molecular weight poly-(amic acid) or polyimide with a low molecular weight amic acid or imide additive in the range of 0.05 to 15 percent by weight of additive. The polyimide powders so obtained show improved processability, as evidenced by lower melt viscosity by capillary rheometry. Likewise, films prepared from mixtures of polymers with additives show improved processability with earlier onset of stretching by TMA. Fletcher, James C. (inventor); Pratt, J. Richard (inventor); St.clair, Terry L. (inventor); Stoakley, Diane M. (inventor); Burks, Harold D. (inventor) 1992-01-01 408 NSDL National Science Digital Library This virtual manipulative provides students with practice adding positive and negative integers. Students are given an addition problem, and using one-to-one correspondence, the student is able to see what happens when adding negative integers. The addition problems can be computer generated or teacher generated and there is a free play mode which allows the student to practice with the chips and become familiar with the process of moving the chips around the page, and creating a visual representation of an addition problem with integers. University, Utah S. 2011-06-28 409 NSDL National Science Digital Library The Vector Addition Patterns model illustrates the tail-to-tip method of adding vectors. The table at the bottom shows the components and lengths of the vectors. You can also rotate the vectors and trace out some interesting patterns. The Vector Addition Patterns model was created using the Easy Java Simulations (EJS) modeling tool. It is distributed as a ready-to-run (compiled) Java archive. Double clicking the ejs_bu_vector_addition_patterns.jar file will run the program if Java is installed. Duffy, Andrew 2010-05-02 410 Microsoft Academic Search Aluminium Kalpha emission (1.5 keV) produced by an 8 J, 500 ps, Nd:glass laser incident at 45° onto a layered target of 0.8 µm thick aluminium (front side) and 1 µm thick iron (backside) has been used to probe the opacity of iron plasma. Source broadened spectroscopy and continuum emission analysis show that whole beam self-focusing within the aluminium plasma A. K. Rossall; L. M. R. Gartside; S. Chaurasia; S. Tripathi; D. S. Munda; N. K. Gupta; L. J. Dhareshwar; J. Gaffney; S. J. Rose; G. J. Tallents 2010-01-01 411 NASA Astrophysics Data System (ADS) AA2219 aluminium alloy square butt joints without filler metal addition were fabricated using gas tungsten arc welding (GTAW), electron beam welding (EBW), and friction stir welding (FSW) processes. The fabricated joints were post-weld aged at 175 °C for 12 h. The effect of three welding processes and post-weld aging (PWA) treatment on the fatigue properties is reported. Transverse tensile properties of the welded joints were evaluated. Microstructure analysis was also carried out using optical and electron microscopes. It was found that the post-weld aged FSW joints showed superior fatigue performance compared to EBW and GTAW joints. This was mainly due to the formation of very fine, dynamically recrystallized grains and uniform distribution of fine precipitates in the weld region. Malarvizhi, S.; Balasubramanian, V. 2011-04-01 412 ERIC Educational Resources Information Center The hypothesis that food additives are causally associated with hyperkinesis and learning disabilities in children is reviewed, and available data are summarized. Available from: American Medical Association 535 North Dearborn Street Chicago, Illinois 60610. (JG) Wender, Ester H. 1977-01-01 413 Cancer.gov While we tried to include potentially useful resources, this website is not exhaustive. New and additional resources may be available. Please let us know if there is a resource you would like to be included. 414 SciTech Connect A method is described of controlling, reducing or eliminating, ozone and related smog resulting from photochemical reactions between ozone and automotive or industrial gases comprising the addition of iodine or compounds of iodine to hydrocarbon-base fuels prior to or during combustion in an amount of about 1 part iodine per 240 to 10,000,000 parts fuel, by weight, to be accomplished by: (a) the addition of these inhibitors during or after the refining or manufacturing process of liquid fuels; (b) the production of these inhibitors for addition into fuel tanks, such as automotive or industrial tanks; or (c) the addition of these inhibitors into combustion chambers of equipment utilizing solid fuels for the purpose of reducing ozone. Lundby, W. 1993-06-29 415 NSF Publications Database ... EAM NSF Org: OD / OPP Date : January 31, 1991 File : opp93021 OFFICE OF SAFETY, ENVIRONMENT AND ... Palmer Station Boathouse Addition) To: Files (S.7 - Environment) On January 16, 1991, the civilian ... 416 PubMed In this study, aluminium oxide-manganese oxide (AOMO) composite material was synthesized, characterized, and tested for fluoride removal in batch experiments. AOMO was prepared from manganese(II) chloride and aluminium hydroxide. The surface area of AOMO was found to be 30.7m2/g and its specific density was determined as 2.78 g/cm3. Detailed investigation of the adsorbent by inductively coupled plasma-optical emission spectrometry, inductively coupled plasma-mass spectrometry, and ion chromatography (for sulphate only) showed that it is composed of Al, Mn, SO4, and Na as major components and Fe, Si, Ca, and Mg as minor components. Thermogravimetric analysis was used to study the thermal behaviour of AOMO. X-ray diffraction analysis showed that the adsorbent is poorly crystalline. The point of zero charge was determined as 9.54. Batch experiments (by varying the proportion of MnO, adsorbent dose, contact time, initial F concentration, and raw water pH) showed that fluoride removal efficiency ofAOMO varied significantly with percentage of MnO with an optimum value of about I11% of manganese oxide in the adsorbent. The optimum dose of the adsorbent was 4 g/L which corresponds to the equilibrium adsorption capacity of 4.8 mg F-/g. Both the removal efficiency and adsorption capacity showed an increasing trend with an increase in initial fluoride concentration of the water. The pH for optimum fluoride removal was found to be in the range between 5 and 7. The adsorption data were analysed using the Freundlich, Langmuir, and Dubinirn-Radushkevich models. The minimum adsorption capacity obtained from the non-linear Freundlich isotherm model was 4.94 mg F-/g and the maximum capacity from the Langmuir isotherm method was 19.2mg F-/g. The experimental data of fluoride adsorption on AOMO fitted well to the Freundlich isotherm model. Kinetic studies showed that the adsorption is well described by a non-linear pseudo-second-order reaction model with an average rate constant of 3.1 x 10(-2) g/min mg. It is concluded that AOMO is a highly promising adsorbent for the removal of excess fluoride from drinking water. PMID:24956783 Alemu, Sheta; Mulugeta, Eyobel; Zewge, Feleke; Chandravanshi, Bhagwan Singh 2014-08-01 417 PubMed The reduction of aluminium trichloride by lithium aluminium hydride in the presence of poly(vinylpyrrolidone) or poly(methylmethacrylate) in mesitylene yielded nano aluminium particles in the matrices of respective polymers. Solution phase synthesis methodology was used successfully to produce composites of various Al/polymer ratios. The composites were characterized by powder XRD patterns and 27Al-NMR with MAS spectroscopic study. The method was useful to produce up to 10 g of nano aluminium that were pure and stable. PMID:20648295 Ghanta, Sekher Reddy; Muralidharan, Krishnamurthi 2010-06-01 418 PubMed Aluminium (Al) has been measured in human breast tissue, nipple aspirate fluid and breast cyst fluid, and recent studies have shown that at tissue concentrations, aluminium can induce DNA damage and suspension growth in human breast epithelial cells. This paper demonstrates for the first time that exposure to aluminium can also increase migratory and invasive properties of MCF-7 human breast cancer cells. Long-term (32 weeks) but not short-term (1 week) exposure of MCF-7 cells to 10(-4) M aluminium chloride or 10(-4) M aluminium chlorohydrate increased motility of the cells as measured by live cell imaging (cumulative length moved by individual cells), by a wound healing assay and by migration in real time through 8 ?m pores of a membrane using xCELLigence technology. Long-term exposure (37 weeks) to 10(-4) M aluminium chloride or 10(-4) M aluminium chlorohydrate also increased the ability of MCF-7 cells to invade through a matrigel layer as measured in real time using the xCELLigence system. Although molecular mechanisms remain to be characterized, the ability of aluminium salts to increase migratory and invasive properties of MCF-7 cells suggests that the presence of aluminium in the human breast could influence metastatic processes. This is important because mortality from breast cancer arises mainly from tumour spread rather than from the presence of a primary tumour in the breast. PMID:23896199 Darbre, Philippa D; Bakir, Ayse; Iskakova, Elzira 2013-11-01 419 PubMed Central Aluminium is well known to inhibit plant elongation, but the role in this inhibition played by water relations remains unclear. To investigate this, tobacco (Nicotiana tabacum L.) suspension-cultured cells (line SL) was used, treating them with aluminium (50 ?M) in a medium containing calcium, sucrose, and MES (pH 5.0). Over an 18 h treatment period, aluminium inhibited the increase in fresh weight almost completely and decreased cellular osmolality and internal soluble sugar content substantially; however, aluminium did not affect the concentrations of major inorganic ions. In aluminium-treated cultures, fresh weight, soluble sugar content, and osmolality decreased over the first 6 h and remained constant thereafter, contrasting with their continued increases in the untreated cultures. The rate of sucrose uptake, measured by radio-tracer, was reduced by approximately 60% within 3 h of treatment. Aluminium also inhibited glucose uptake. In an aluminium-tolerant cell line (ALT301) isogenic to SL, all of the above-mentioned changes in water relations occurred and tolerance emerged only after 6 h and appeared to involve the suppression of reactive oxygen species. Further separating the effects of aluminium on elongation and cell survival, sucrose starvation for 18 h inhibited elongation and caused similar changes in cellular osmolality but stimulated the production of neither reactive oxygen species nor callose and did not cause cell death. We propose that the inhibition of sucrose uptake is a mechanism whereby aluminium inhibits elongation, but does not account for the induction of cell death. Abdel-Basset, Refat; Ozuka, Shotaro; Demiral, Tijen; Furuichi, Takuya; Sawatani, Ikuo; Baskin, Tobias I.; Matsumoto, Hideaki; Yamamoto, Yoko 2010-01-01 420 NASA Astrophysics Data System (ADS) Les besoins de reduction du poids se sont concretement traduits par l'introduction de nouvelles nuances plus legeres dans les structures automobiles. Ainsi, des alliages d'aluminium ont commence a etre integres dans les pieces de structure de plusieurs vehicules. La faible masse volumique des alliages d'aluminium (2,7g/cm3) permet d'alleger le poids du vehicule qui entraine une diminution de la consommation de carburant et, donc, des emissions de gaz a effet de serre. La striction et la rupture sont les principaux modes de defaillance qui entrainent le rebut systematique des pieces. C'est pourquoi, ameliorer la prediction d'apparition de ces defauts lors de la simulation va dans le sens d'une meilleure maitrise du procede. Dans le cadre de ce travail doctoral, deux modeles sont developpes pour simuler le comportement a grandes deformations d'alliages d'aluminium: un modele polycristallin de type Taylor et un modele a un ou plusieurs elements finis par grain. Les diagrammes limites de formage (DLF) pour les deux alliages d'aluminium AA5754 et AA6063 ont ete simules numeriquement en utilisant une formulation par elements finis pour les polycristaux basee sur l'hypothese de Taylor. Les DLF conventionnels et de l'hydroformage ont ete traces. L'effet des chemins de deformation sur la formabilite des alliages d'aluminium a aussi ete etudie. Finalement, des simulations numeriques avec les donnees de diffraction des electrons retrodiffuses (EBSD) pour 1'alliage d'aluminium AA5754 ont ete effectuees en utilisant le modele a un ou plusieurs elements par grain. Ces simulations sont executees avec differents modeles du durcissement (Asaro, Bassani et puissance). Mots-cles: Formabilite; Alliage d'aluminium; Hydroformage; Glissement cristallographique; Durcissement; Calcul parallele; Diagramme limite de formage (DLF); Diffraction electron. Eljaafari, Samira 421 NASA Technical Reports Server (NTRS) Phenylethynyl containing reactive additives were prepared from aromatic diamine, containing phenylethvnvl groups and various ratios of phthalic anhydride and 4-phenylethynviphthalic anhydride in glacial acetic acid to form the imide in one step or in N-methyl-2-pvrrolidinone to form the amide acid intermediate. The reactive additives were mixed in various amounts (10% to 90%) with oligomers containing either terminal or pendent phenylethynyl groups (or both) to reduce the melt viscosity and thereby enhance processability. Upon thermal cure, the additives react and become chemically incorporated into the matrix and effect an increase in crosslink density relative to that of the host resin. This resultant increase in crosslink density has advantageous consequences on the cured resin properties such as higher glass transition temperature and higher modulus as compared to that of the host resin. Connell, John W. (Inventor); Smith, Joseph G., Jr. (Inventor); Hergenrother, Paul M. (Inventor) 2002-01-01 422 NASA Technical Reports Server (NTRS) Phenylethynyl containing reactive additives were prepared from aromatic diamines containing phenylethynyl groups and various ratios of phthalic anhydride and 4-phenylethynylphthalic anhydride in glacial acetic acid to form the imide in one step or in N-methyl-2-pyrrolidi none to form the amide acid intermediate. The reactive additives were mixed in various amounts (10% to 90%) with oligomers containing either terminal or pendent phenylethynyl groups (or both) to reduce the melt viscosity and thereby enhance processability. Upon thermal cure, the additives react and become chemically incorporated into the matrix and effect an increase in crosslink density relative to that of the host resin. This resultant increase in crosslink density has advantageous consequences on the cured resin properties such as higher glass transition temperature and higher modulus as compared to that of the host resin. Connell, John W. (Inventor); Smith, Joseph G., Jr. (Inventor); Hergenrother, Paul M. (Inventor) 2002-01-01 423 PubMed An electro-coagulation laboratory scale system using aluminium plates electrodes was studied for the removal of organic and inorganic pollutants as a by-product from leather finishing industrial process. A fractional factorial 2(3) experimental design was applied in order to obtain optimal values of the system state variables. The electro-coagulation (EC) process efficiency was based on the chemical oxygen demand (COD), turbidity, total suspended solid, total fixed solid, total volatile solid, and chemical element concentration values. Analysis of variance (ANOVA) for final pH, total fixed solid (TFS), turbidity and Ca concentration have confirmed the predicted models by the experimental design within a 95% confidence level. The reactor working conditions close to real effluent pH (7.6) and electrolysis time in the range 30-45 min were enough to achieve the cost effective reduction factors of organic and inorganic pollutants' concentrations. An appreciable improvement in COD removal efficiency was obtained for electro-coagulation treatment. Finally, the technical-economical analysis results have clearly shown that the electro-coagulation method is very promising for industrial application. PMID:19844065 Espinoza-Quiñones, Fernando R; Fornari, Marilda M T; Módenes, Aparecido N; Palácio, Soraya M; Trigueros, Daniela E G; Borba, Fernando H; Kroumov, Alexander D 2009-01-01 424 PubMed The possible existence of the potentially toxic oxyanions of Al (Al(OH)(4)(-)), As (HAsO(4)(2-)), and Mo (MoO(4)(2-)) was examined in excessively limed lakes. In-situ dialysis (MWCO 1 kDa) was performed in the surface and bottom waters of two excessively limed lakes (pH 7.1-7.7) and one acidic lake (pH approximately 5.4). The dialysable metal concentrations were compared to the equilibrium distribution of species as calculated with the geochemical code Visual MINTEQ incorporating the CD-MUSIC and Stockholm Humic models for complexation onto colloidal ferrihydrite and dissolved organic matter. Arsenic and molybdenum in the excessively limed lakes were to a large extent present in the dialysable fraction (>79% and >92% respectively). They were calculated to exist as free or adsorbed oxyanions. Most of the Al was observed to reside in the colloidal fraction (51-82%). In agreement with this, model predictions indicated aluminium to be present mostly as colloids or bound to dissolved organic matter. Only a small fraction was modelled as Al(OH)(4)(-) ions. In most cases, modelled values were in agreement with the dialysis results. The free concentrations of the three oxyanions were mostly low compared to toxic levels. PMID:19540566 Sjöstedt, Carin; Wällstedt, Teresia; Gustafsson, Jon Petter; Borg, Hans 2009-09-01 425 PubMed Stoichiometric dicalcium silicate, Ca2SiO4, displays a well-known polymorphism with temperature. When this phase is doped by a range of elements, belite, one of the main phases of cements, is generated. Here, we thoroughly study the aluminum doping of dicalcium silicate. This type of study is important for cement characterization and also from a basic point of view. Ca2Si(1-2x)Al(2x)O(4-x)?(x) (x = 0, 0.010, 0.014, 0.03) has been prepared and studied by X-ray powder diffraction and the Rietveld method. The limiting composition has been established as Ca2Si0.972Al0.028O3.986?0.014. The (27)Al MAS NMR band located close to ~-70 ppm is ascribed to tetrahedral environments, in agreement with the proposed aliovalent Si/Al atomic substitution mechanism. Thermal analysis measurements under a wet atmosphere indirectly confirm the increase of oxygen vacancies as the amount of incorporated protons increases with the aluminium content. A thorough electrical characterization has been carried out including overall conductivity measurements under wet and dry atmospheres and conductivity as a function of the oxygen partial pressure. The samples show oxide anion conductivity with a small p-type electronic contribution under oxidizing conditions. These compounds display a very important proton contribution to the overall conductivities under humidified atmospheres. PMID:24292166 Cuesta, Ana; Aranda, Miguel A G; Sanz, Jesús; de la Torre, Angeles G; Losilla, Enrique R 2014-02-01 426 PubMed Calcium-aluminium-phosphate cements (CAPCs) for biomedical applications, mainly intended for applications in the dental field as non-resorbable fillers, were obtained by reacting Ca-aluminates compounds, i.e. CaO·Al(2)O(3) (CA) and CaO·2 Al(2)O(3) (CA(2)), with Al(H(2)PO(4))(3) aqueous solution. Hydroxyapatite was also introduced as a bioactive dispersed phase. Suitable elements like Sr and La were used to increase the radiopacity of the set yielded pastes towards X-ray wavelength used in clinical diagnostic radiographic equipments. La and Sr doped Ca-aluminates powders have been synthesized by solid state reaction at 1,400°C from a mixture of CaCO(3), Al(2)O(3), La(2)O(3) and SrCO(3). The characteristics of the obtained powders were analyzed and related to the starting compositions and synthesis procedures. The microstructure, setting time, radiopacity and compressive strength of the CAPCs have been investigated and discussed. PMID:21165760 Medri, V; Mazzocchi, M; Bellosi, A 2011-02-01 427 NASA Astrophysics Data System (ADS) Composites with an aluminium alloy matrix (AlMMC) exhibit several advantageous properties such as good strength, stiffness, low density, resistance and dimensional stability to elevated temperatures, good thermal expansion coefficient and particularly high resistance to friction wear. Therefore such composites are more and more used in modern engineering constructions. Composites reinforced with hard ceramic particles (Al2O3, SiC) are gradually being implemented into production in automotive or aircraft industries. Another application of AlMMC is in the electronics industry, where the dimensional stability and capacity to absorb and remove heat is used in radiators. However the main problems are still: a reduction of production costs, developing methods of composite material tests and final product quality assessment, standardisation, development of recycling and mechanical processing methods. AlMMC production technologies, based on liquid-phase methods, and the shaping of products by casting methods, belong to the cheapest production methods. Application of a suspension method for the production of composites with heterophase reinforcement may turn out to be a new material and technological solution. The article presents the material and technological aspects of the transfer procedures for the production of composite suspensions from laboratory scale to a semi-industrial scale. Dolata, A. J.; Dyzia, M. 2012-05-01 428 PubMed Central A new approach to the interpretation and analysis of coherent inelastic neutron scattering from polycrystals (poly-CINS) is presented. This article describes a simulation of the one-phonon coherent inelastic scattering from a lattice model of an arbitrary crystal system. The one-phonon component is characterized by sharp features, determined, for example, by boundaries of the (Q, ?) regions where one-phonon scattering is allowed. These features may be identified with the same features apparent in the measured total coherent inelastic cross section, the other components of which (multiphonon or multiple scattering) show no sharp features. The parameters of the model can then be relaxed to improve the fit between model and experiment. This method is of particular interest where no single crystals are available. To test the approach, the poly-CINS has been measured for polycrystalline aluminium using the MARI spectrometer (ISIS), because both lattice dynamical models and measured dispersion curves are available for this material. The models used include a simple Lennard-Jones model fitted to the elastic constants of this material plus a number of embedded atom method force fields. The agreement obtained suggests that the method demonstrated should be effective in developing models for other materials where single-crystal dispersion curves are not available. Roach, Daniel L.; Ross, D. Keith; Gale, Julian D.; Taylor, Jon W. 2013-01-01 429 PubMed Central Aluminium (Al) is highly abundant in the environment and can elicit a variety of toxic responses in biological systems. Here we characterize the effects of Al on Caenorhabditis elegans by identifying phenotypic abnormalities and disruption in whole-body metal homeostasis (metallostasis) following Al exposure in food. Widespread changes to the elemental content of adult nematodes were observed when chronically exposed to Al from the first larval stage (L1). Specifically, we saw increased barium, chromium, copper and iron content, and a reduction in calcium levels. Lifespan was decreased in worms exposed to low levels of Al, but unexpectedly increased when the Al concentration reached higher levels (4.8 mM). This bi-phasic phenotype was only observed when Al exposure occurred during development, as lifespan was unaffected by Al exposure during adulthood. Lower levels of Al slowed C. elegans developmental progression, and reduced hermaphrodite self-fertility and adult body size. Significant developmental delay was observed even when Al exposure was restricted to embryogenesis. Similar changes in Al have been noted in association with Al toxicity in humans and other mammals, suggesting that C. elegans may be of use as a model for understanding the mechanisms of Al toxicity in mammalian systems. Page, Kathryn E.; White, Keith N.; McCrohan, Catherine R. 2013-01-01 430 SciTech Connect At extreme pressures and temperatures, such as those inside planets and stars, common materials form new dense phases with compacted atomic arrangements and unusual physical properties. The synthesis and study of new phases of matter at pressures above 100 GPa and temperatures above 10{sup 4} K - warm dense matter - may reveal the functional details of planet and star interiors, and may lead to materials with extraordinary properties. Many phases have been predicted theoretically that may be realized once appropriate formation conditions are found. Here we report the synthesis of a superdense stable phase of body-centred-cubic aluminium, predicted by first-principles theories to exist at pressures above 380 GPa. The superdense Al phase was synthesized in the non-equilibrium conditions of an ultrafast laser-induced microexplosion confined inside sapphire ({alpha}-Al{sub 2}O{sub 3}). Confined microexplosions offer a strategy to create and recover high-density polymorphs, and a simple method for tabletop study of warm dense matter. Vailionis, Arturas; Gamaly, Eugene G.; Mizeikis, Vygantas; Yang, Wenge; Rode, Andrei V.; Juodkazis, Saulius (Swinburne); (Stanford); (CIW); (ANU); (Shinshu-MED) 2011-09-20 431 PubMed This study reports a work-environmental assessment and workers' exposure in a major prebake type aluminium smelter in India. Levels of known health hazards in and near the main smelting operations viz., the Potroom, the Carbon area, the Butt section, the Rodding shop, the Bath preparing area and the Casthouse were measured. Dustiness in general was high to excessively high. Mean levels of respirable dust (PM10) in air in the three dustiest areas were 24.07 mg/m3 in the Carbon areas, 27.57 mg/m3 in the Bath preparing and 4.44 mg/m3 in the Rodding shop. 40- 60% of the particles were less than 5 microm in size. 0.5- 2.82% particulate fluoride was obtained in the size fraction 0.4- 4.7 microm of the Potroom air. Naturally, exposures to total dusts were very high in these processes. The background levels of NOx and SO2 and fluorides (gaseous and particulate) were found to be within the prescribed Indian Standards. Higher exposures to gaseous and particulate fluoride, 3.85 and 6.53 mg/m3 respectively, were observed among the Rodding shop workers. The levels ofpolycyclic aromatic hydrocarbons (PAHs) were deemed to be excessive in the Carbon area. Measurements of heat stress were made in winter and were found to be lower than the prescribed limit. PMID:18783008 Mukherjee, Ashit K; Ravichandran, Beerappa; Bhattacharya, Sanat K; Roy, Sanjit K; Ahmed, Sabir; Thakur, Sridhar; Saiyed, Habibullah N 2008-09-01 432 NASA Astrophysics Data System (ADS) Non-linear enhancements of ultrasonic surface wave amplitude and frequency have been observed when an incident wave interacts with a surface defect. Previous measurements of surface wave interactions with defects have considered only those that are inclined normal to the surface. Here, the enhancement effects have been studied in aluminium samples with machined slots of fixed length and of varying angle to the horizontal; the degree of enhancement was studied as a function of defect angle using both a scanning laser source, and a scanning laser detector. An automated scanning system has been developed for use with the detector, an IOS two-wave mixer interferometer, capable of measuring the out-of-plane surface displacement on rough surfaces. B-scans, consisting of many A-scans stacked together, were used to identify wave modes present in the near field, the arrival times of which are dependent on the angle of the defect. The observed enhancement is caused by superposition of the incident Rayleigh wave with reflected and mode converted waves, thereby making it angle dependent. Clough, A. R.; Dutton, B.; Edwards, R. S. 2011-06-01 433 PubMed Central The polymers used in plastics are generally harmless. However, they are rarely used in pure form. In almost all commercial plastics, they are "compounded" with monomeric ingredients to improve their processing and end-use performance. In order of total volume used, these monomeric additives may be classified as follows: reinforcing fibers, fillers, and coupling agents; plasticizers; colorants; stabilizers (halogen stabilizers, antioxidants, ultraviolet absorbers, and biological preservatives); processing aids (lubricants, others, and flow controls); flame retardants, peroxides; and antistats. Some information is already available, and much more is needed, on potential toxicity and safe handling of these additives during processing and manufacture of plastics products. Deanin, R D 1975-01-01 434 PubMed A selective method has been developed for the extraction chromatography of aluminium (III) and its separation from several metal ions with a chromatographic column containing N-n-octylaniline (a liquid anion exchanger) coated on silanized silica gel as a stationary phase. The aluminium (III) was quantitatively extracted with the 0.065 mol/L N-n-octylaniline in 0.013 to 0.05 mol/L sodium succinate at a flow rate of 1.0 mL/min. The extracted metal ion has been recovered by eluting with 25.0 mL of 0.05 mol/L hydrochloric acid and estimated spectrophotometrically with aurintricarboxylic acid. The effects of the acid concentration, the reagent concentration, the flow rate and the eluting agents have been investigated. The log-log plots of distribution coefficient (Kd (Al (III))) versus N-n-octylaniline concentration in 0. 005 and 0. 007 mol/L sodium succinate gave the slopes of 0.5 and 0.7 respectively and showed the probable composition of the extracted species was 1:1 (metal to amine ratio) and the nature of extracted species was [RR'NH2(+), Al succinate2(-)]org. The extraction of aluminium(III) was carried out in the presence of various ions to ascertain the tolerance limit of individual ions. Aluminium(III) has been separated from multicomponent mixtures, pharmaceutical samples and synthetic mixtures corresponding to alloys. A scheme for mutual separation of aluminium (III), indium (III), gallium (III) and thallium (III) has been developed by using suitable masking agents. The method is fast, accurate and precise. PMID:23285976 Phule, Sachin R; Aher, Haribhau R; Kuchekar, Shashikant R; Han, Sung-H 2012-09-01 435 ERIC Educational Resources Information Center Traditionally, learning basic facts has focused on rote memorization of isolated facts, typically through the use of flash cards, repeated drilling, and timed testing. However, as many experienced teachers have seen, "drill alone does not develop mastery of single-digit combinations." In contrast, a fluency approach to learning basic addition Garza-Kling, Gina 2011-01-01 436 NSDL National Science Digital Library For manufacturing students learning the basics, this page from SnoCAMP would be a useful addition to in-class instruction. The webpage combines photographs of examples with descriptive text. This would be a good introduction for students learning the basics about manufacturing. 2013-07-23 437 PubMed We have reviewed evidence of adverse events after exposure to aluminium-containing vaccines against diphtheria, tetanus, and pertussis (DTP), alone or in combination, compared with identical vaccines, either without aluminium or containing aluminium in different concentrations. The study is a systematic review with meta-analysis. We searched the Cochrane Vaccines Field Register, the Cochrane Library, Medline, Embase, Biological Abstracts, Science Citation Index, and the Vaccine Adverse Event Reporting System website for relevant studies. Reference lists of retrieved articles were scanned for further studies. We included randomised and semi-randomised trials and comparative cohort studies if the report gave sufficient information for us to extract aluminium concentration, vaccine composition, and safety outcomes. Two reviewers extracted data in a standard way from all included studies and assessed the methodological quality of the studies. We identified 35 reports of studies and included three randomised trials, four semi-randomised trials, and one cohort study. We did a meta-analysis of data from five studies around two main comparisons (vaccines containing aluminium hydroxide vs no adjuvant in children aged up to 18 months and vaccines containing different types of aluminium vs no adjuvants in children aged 10-16 years). In young children, vaccines with aluminium hydroxide caused significantly more erythema and induration than plain vaccines (odds ratio 1.87 [95% CI 1.57-2.24]) and significantly fewer reactions of all types (0.21 [0.15-0.28]). The frequencies of local reactions of all types, collapse or convulsions, and persistent crying or screaming did not differ between the two cohorts of the trials. In older children, there was no association between exposure to aluminium-containing vaccines and onset of (local) induration, swelling, or a raised temperature, but there was an association with local pain lasting up to 14 days (2.05 [1.25-3.38]). We found no evidence that aluminium salts in vaccines cause any serious or long-lasting adverse events. Despite a lack of good-quality evidence we do not recommend that any further research on this topic is undertaken. PMID:14871632 Jefferson, Tom; Rudin, Melanie; Di Pietrantonj, Carlo 2004-02-01 438 SciTech Connect Aluminium doped barium and strontium hexaferrite nanoparticles BaAl {sub x}Fe{sub (12-x)}O{sub 19} and SrAl {sub x}Fe{sub (12-x)}O{sub 19} were synthesised via a sol-gel route using citric acid to complex the ions followed by an auto-combustion reaction. This method shows promise for the synthesis of complex ferrite powders with small particle size. It was found that around half of the iron could be substituted for aluminium in the barium ferrite with structure retention, whereas strontium aluminium ferrites could be produced with any aluminium content including total substitution of the iron. All synthesised materials consisted of particles smaller than 1 {mu}m, which is the size of a single magnetic domain, and various doping levels were achieved with the final elemental composition being within the bounds of experimental error. The materials show structural and morphological changes as they move from iron to aluminium ferrites. Such materials may be promising for imaging applications. Shirtcliffe, Neil J. [Nottingham Trent University, Clifton Lane, Clifton, Nottingham NG11 8NS (United Kingdom)]. E-mail: neil.shirtcliffe@ntu.ac.uk; Thompson, Simon [Nottingham Trent University, Clifton Lane, Clifton, Nottingham NG11 8NS (United Kingdom); O'Keefe, Eoin S. [QinetiQ, Farnborough, Hampshire (United Kingdom); Appleton, Steve [QinetiQ, Farnborough, Hampshire (United Kingdom); Perry, Carole C. [Nottingham Trent University, Clifton Lane, Clifton, Nottingham NG11 8NS (United Kingdom)]. E-mail: carole.perry@ntu.ac.uk 2007-02-15 439 NASA Astrophysics Data System (ADS) Asymmetric plate impact experiments are conducted on LY12 aluminium alloy in a pressure range of 85-131 GPa. The longitudinal sound speeds are obtained from the time-resolved particle speed profiles of the specimen measured with Velocity Interferometer System for Any Reflector (VISAR) technique, and they are shown to be good agreement with our previously reported data of this alloy in a pressure range of 20-70 GPa, and also with those of 2024 aluminium reported by McQueen. Using all of the longitudinal speeds and the corresponding bulk speeds calculated from the Gruneisen equation of state (EOS), shear moduli of LY12 aluminium alloy are obtained. A comparison of the shear moduli in the solid phase region with those estimated from the Steinberg model demonstrate that the latter are systematically lower than the measurements. By re-analysing the pressure effect on the shear modulus, a modified equation is proposed, in which the pressure term of P/?1/3 in the Steinberg model is replaced by a linear term. Good agreement between experiments and the modified equation is obtained, which implies that the shear modulus of LY12 aluminium varies linearly both with pressure and with temperature throughout the whole solid phase region. On the other hand, shear modulus of aluminium in a solid-liquid mixed phrase region decreases gradually and smoothly, a feature that is very different from the drastic dropping at the melting point under static conditions. Yu, Yu-Ying; Tan, Hua; Hu, Jian-Bo; Dai, Cheng-Da 2008-01-01 440 NASA Astrophysics Data System (ADS) A quick method for surface treatment of aluminium foil with environment-friendly and effective molybdate-based coating was developed in this study. Aluminium foil samples were treated with molybdate-based solution. The microstructure and composition of the resulting molybdate-based conversion coatings were explored by scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS) and X-ray photoelectric spectroscopy (XPS). We found that the molybdate-based conversion coating was composed mainly of MoO3, (MoO3)x(P2O5)y and Al2(MoO4)3 compounds. Furthermore, corrosion resistance of the treated aluminium foil was evaluated by electrochemical impedance spectroscopy (EIS) and potentiodynamic polarization. Our results show that all of the aluminium foils with molybdate-based conversion coatings have much better corrosion resistance than bare aluminium foil. Notably, the sample treated at 40 °C exhibited the best corrosion resistance. The new method is very suitable for continuous processing. Liang, Chang-Sheng; Lv, Zhong-Fei; Zhu, Ye-Ling; Xu, Shi-Ai; Wang, Hong 2014-01-01 441 PubMed Attention is devoted to the role of chelating agents in the treatment of aluminium related diseases. In fact, in spite of the efforts that have drastically reduced the occurrence of aluminium dialysis diseases, they so far constitute a cause of great medical concern. The use of chelating agents for iron and aluminium in different clinical applications has found increasing attention in the last thirty years. With the aim of designing new chelators, we synthesized a series of kojic acid derivatives containing two kojic units joined by different linkers. A huge advantage of these molecules is that they are cheap and easy to produce. Previous works on complex formation equilibria of a first group of these ligands with iron and aluminium highlighted extremely good pMe values and gave evidence of the ability to scavenge iron from inside cells. On these bases a second set of bis-kojic ligands, whose linkers between the kojic chelating moieties are differentiated both in terms of type and size, has been designed, synthesized and characterized. The aluminium(III) complex formation equilibria studied by potentiometry, electrospray ionization mass spectroscopy (ESI-MS), quantum-mechanical calculations and (1)H NMR spectroscopy are here described and discussed, and the structural characterization of one of these new ligands is presented. The in vivo studies show that these new bis-kojic derivatives induce faster clearance from main organs as compared with the monomeric analog. PMID:24200878 Toso, Leonardo; Crisponi, Guido; Nurchi, Valeria M; Crespo-Alonso, Miriam; Lachowicz, Joanna I; Mansoori, Delara; Arca, Massimiliano; Santos, M Amélia; Marques, Sérgio M; Gano, Lurdes; Niclós-Gutíerrez, Juan; González-Pérez, Josefa M; Domínguez-Martín, Alicia; Choquesillo-Lazarte, Duane; Szewczuk, Zbigniew 2014-01-01 442 NASA Technical Reports Server (NTRS) Polyimide resins (PMR) are generally useful where high strength and temperature capabilities are required (at temperatures up to about 700 F). Polyimide resins are particularly useful in applications such as jet engine compressor components, for example, blades, vanes, air seals, air splitters, and engine casing parts. Aromatic vinyl capped addition polyimides are obtained by reacting a diamine, an ester of tetracarboxylic acid, and an aromatic vinyl compound. Low void materials with improved oxidative stability when exposed to 700 F air may be fabricated as fiber reinforced high molecular weight capped polyimide composites. The aromatic vinyl capped polyimides are provided with a more aromatic nature and are more thermally stable than highly aliphatic, norbornenyl-type end-capped polyimides employed in PMR resins. The substitution of aromatic vinyl end-caps for norbornenyl end-caps in addition polyimides results in polymers with improved oxidative stability. Vannucci, Raymond D. (inventor); Malarik, Diane C. (inventor); Delvigs, Peter (inventor) 1991-01-01 443 Microsoft Academic Search Likelihood-based regression models such as the normal linear regression model and the linear logistic model, assume a linear (or some other parametric) form for the covariatesX_1, X_2, \\\\cdots, X_p$. We introduce the class of generalized additive models which replaces the linear form$\\\\sum \\\\beta_jX_j$by a sum of smooth functions$\\\\sum s_j(X_j)$. The$s_j(\\\\cdot)'s are unspecified functions that are Trevor Hastie; Robert Tibshirani 1986-01-01 444 Microsoft Academic Search \\u000a Metals are vital building blocks for MEMS. Pure metals and metal alloys are employed in microsystem design to achieve a wide\\u000a array of functionality. Common examples include electrical conductors, mechanical structures, magnetic elements, thermal conductors,\\u000a optical reflectors, and more. In this chapter, additive processes for metals are discussed in the context of their application\\u000a in MEMS. Particular attention is paid David P. Arnold; Monika Saumer; Yong-Kyu Yoon 445 NASA Astrophysics Data System (ADS) To establish the effect of reducing the temperature mismatch between the TiH2 decomposition temperature and the aluminium melting point on the foams morphological features and their mechanical compression behavior, a nickel coating on TiH2 powders was used as a hydrogen diffusion barrier and the size of TiH2 powders was controlled to modify the hydrogen evolution temperature. The nickel diffusion barrier was produced by an electroless deposition technique and the hydrogen evolution behavior of coated powders was investigated by thermogravimetrical analysis. The effect of particle size was determined with powders of five particle size fractions along with powders of different particle size obtained from a supplier. Foamable precursors were obtained by hot pressing a mix of aluminium powders with 1 wt.% of TiH2 powders and foams were fabricated at 750 and 800 °C. The foams mechanical strength was investigated by uni-axial compression on foam cylinders with and without outer skin. Coating produced a continuous and homogeneous deposit of 96.5 wt.% nickel and reduced the initial temperature mismatch by approximately 70°C. Additionally, the coating adhesion proved to be good enough to withstand the mixing and compaction processes. Nickel-coated TiH2 powders generated foams with a more homogeneous and reproducible pore structure than foams produced with powders in the as-received and passivated condition. On the other hand, the hydrogen evolution onset of TiH2 shifted towards higher temperatures as the particle size increased. The particle size influenced the foam expansion and the porosity features. Powders of larger particle size produced foams with a more uniform pore distribution and size. Finally, compression tests on skinless foams containing nickel displayed quasi-horizontal energy regimes with longer stroke lengths than the rest, however the final energy absorption efficiencies (above 7.2 kJ·kg-1) were not remarkably increased. Proa-Flores, Paula Mercedes 446 PubMed Central Aluminium (Al)-activated citrate secretion plays an important role in Al resistance in a number of plant species, such as rice bean (Vigna umbellata). This study further characterized the regulation of VuMATE1, an aluminium-activated citrate transporter. Al stress induced VuMATE1 expression, followed by the secretion of citrate. Citrate secretion was specific to Al stress, whereas VuMATE1 expression was not, which could be explained by a combined regulation of VuMATE1 expression and Al-specific activation of VuMATE1 protein. Pre-treatment with a protein translation inhibitor suppressed VuMATE1 expression, indicating that de novo biosynthesis of proteins is required for gene expression. Furthermore, post-treatment with a protein translation inhibitor inhibited citrate secretion, indicating that post-transcriptional regulation of VuMATE1 is critical for citrate secretion. Protein kinase and phosphatase inhibitor studies showed that reversible phosphorylation was important not only for transcriptional regulation of VuMATE1 expression but also for post-translational regulation of VuMATE1 protein activity. These results suggest that citrate secretion is dependent on both transcriptional and post-transcriptional regulation of VuMATE1. Additionally, VuMATE1 promoter–?-glucuronidase fusion lines revealed that VuMATE1 expression was restricted to the root apex and was entirely Al induced, indicating the presence of cis-acting elements regulating root tip-specific and Al-inducible gene expression, which will be an important resource for genetic improvement of plant Al resistance. Liu, Mei Ya; Chen, Wei Wei; Yang, Jian Li 2013-01-01 447 Microsoft Academic Search A comparison was made between high pressure die cast and wrought magnesium alloys and formed mild steel and aluminium in tensile, bending and buckling deformation. It was found that the energy absorption properties of magnesium alloys were particularly good in bending and buckling, absorbing up to 50% more energy than the aluminium and over 10 times more energy than the Mark Easton; Wei Qian Song; Trevor Abbott 2006-01-01 448 Microsoft Academic Search Multilayered aluminium-coated polymer laminates are the common barrier envelopes of vacuum insulation panels used as high performance thermal insulation in building applications. These laminates are made of up to ten distinct layers including three aluminium barrier layers applied by vacuum web coating. During their expected lifetime, which is required to last longer than 20 years, they will be subjected to heat S. Brunner; P. J. Tharian; H. Simmler; K. Ghazi Wakili 2008-01-01 449 Microsoft Academic Search As part of a programme to develop a high power density, Al\\/air battery with a NaCl brine electrolyte, the high rate dissolution of an aluminium alloy containing tin and gallium was investigated in a small volume cell. The objective was to define the factors that limit aluminium dissolution in condition that mimic a high power density battery. In a cell Maria Nestoridi; Derek Pletcher; Julian A. Wharton; Robert J. K. Wood 2009-01-01 450 NASA Astrophysics Data System (ADS) The reduction of aluminium trichloride by lithium aluminium hydride in the presence of poly(vinylpyrrolidone) or poly(methylmethacrylate) in mesitylene yielded nano aluminium particles in the matrices of respective polymers. Solution phase synthesis methodology was used successfully to produce composites of various Al/polymer ratios. The composites were charecterized by powder XRD patterns and 27Al-NMR with MAS spectroscopic study. The method was useful to produce up to 10 g of nano aluminium that were pure and stable.The reduction of aluminium trichloride by lithium aluminium hydride in the presence of poly(vinylpyrrolidone) or poly(methylmethacrylate) in mesitylene yielded nano aluminium particles in the matrices of respective polymers. Solution phase synthesis methodology was used successfully to produce composites of various Al/polymer ratios. The composites were charecterized by powder XRD patterns and 27Al-NMR with MAS spectroscopic study. The method was useful to produce up to 10 g of nano aluminium that were pure and stable. Electronic supplementary information (ESI) available: Supplementary figures. See DOI: 10.1039/b9nr00337a Ghanta, Sekher Reddy; Muralidharan, Krishnamurthi 2010-06-01 451 Microsoft Academic Search The friction and wear behaviour of Kevlar 49 sliding against aluminium under large contact pressures at low sliding velocities was examined. Kevlar 49 on aluminium friction is measured using yarns on capstans, and compared with literature results for Kevlar 49 on itself. A modified version of Howell's equation is presented in terms of stresses, ?=a??, to allow finite element modelling I. F Brown; C. J Burgoyne 1999-01-01 452 Microsoft Academic Search Cast aluminium alloys with eutectic composition have been frequently used in industry owing to their low cost, simple processing and light weight. Their mechanical properties strongly depend on their eutectic microstructure. Such methods as adding elements and rapid solidification have been widely used to refine the eutectic. A lot of work on the eutectic structure in cast aluminium alloys has H. Jiandong; L. Yongbing; L. Zhang 1990-01-01 453 Microsoft Academic Search Following a review of metal matrix composite (MMC) materials and production methods, the paper outlines the development and application of PCD cutting tools. Experimental data are presented for the drilling and single blade reaming of aluminium-silicon alloys containing 7% and 13% silicon and aluminium 2618 MMC alloy reinforced with 15 vol% silicon carbide (SiC) particulate. Though initially aimed only at R. T. Coelho; S. Yamada; D. K. Aspinwall; M. L. H. Wise 1995-01-01 454 Microsoft Academic Search The dry drilling of aluminium alloys (without using cutting fluids) is an environmentally friendly machining process but also an exceedingly difficult task due to aluminium's tendency to adhere to the drills made of conventional materials such as the high-speed steel (HSS). Diamond-like carbon (DLC) coatings improve the dry drilling performance due to their adhesion mitigating properties. In this work, improvements Sukanta Bhowmick; Ahmet T. Alpas 2008-01-01 455 Microsoft Academic Search The effects of aluminium foil and cling film on microbiological, chemical and sensory changes in wild sea bass (Dicentrarchus labrax) stored at chill temperature (4 °C) were studied. A quality assessment of wild sea bass stored in ice, in boxes without ice, wrapped in aluminium foil (WAF) and wrapped in cling film (WCF) at 4 °C was performed by monitoring sensory quality, F. Özogul; C. Gökbulut; G. Özyurt; Y. Özogul; M. Dural 2005-01-01 456 Microsoft Academic Search The objective of this study is to investigate the effect of variation of the tube length and the crosshead velocity on the amount of energy absorption of aluminium tubes (Al 6061) towards the inversion collapsed mode. The tests were performed on the Aluminium tubes using compression test apparatus according to ASTM E8 standard procedures. Two parameters that are included in Mohd Suhairil Meon; Hazran Husain; Muhamad Fauzi Othman; Muhammad Fairuz Remeli; Mohd Syahar Mohd Syawal 2011-01-01 457 Microsoft Academic Search A new mechanistic model has been established for the chemical and electrochemical mechanisms controlling nickel–aluminium bronze crevice corrosion. The crevice corrosion was initially confined to eutectoid regions with slight attack of the copper rich ?-phase within the ?+?III eutectoid. In the presence of high chloride concentrations, copper and aluminium complexes form and the hydrolysis of these complexes leads to the J. A. Wharton; K. R. Stokes 2008-01-01 458 PubMed Nanoscopic amorphous Lewis acidic aluminium fluorides, such as aluminium chlorofluoride (ACF) and high-surface aluminium fluoride (HS-AlF(3)), are capable of activating C-H bonds of aliphatic hydrocarbons. H/D exchange reactions are catalysed under mild conditions (40?°C). PMID:22125129 Prechtl, Martin H G; Teltewskoi, Michael; Dimitrov, Anton; Kemnitz, Erhard; Braun, Thomas 2011-12-16 459 Microsoft Academic Search The ¡§Aluminium Economy¡¨ is put forward as an attractive basis for an energy efficient community. As energy storage medium, aluminium batteries have high specific energy density and simple, safe construction. Aluminium is also demonstrating low-cost and high performance in energy related applications such as electric cable, light weight vehicle, building material, LED heat-sink, and solar reflector. It is particularly suited S. C. Mok 2011-01-01 460 PubMed For the first time, aluminium fluorides in liquid phase are available for optical applications. By modifying the conditions of the fluorolytic sol-gel synthesis of aluminium fluorides transparent sols with low viscosities were obtained. These sols consist mainly of small oligomeric or cluster units of aluminium fluoride which are not measurable by DLS, WAXS, SAXS and show unusual narrow signals in solid state NMR. Isolated particles with diameters up to five nanometers can be identified by TEM measurements and allow the use of their sols in optical and anti reflecting coatings. The sol particles were modified by trifluoroacetic acid to prevent agglomeration, and as a result, the obtained xerogels can be re-dispersed transparently in organic solvents. PMID:22885847 Fritz, C; Scholz, G; Feist, M; Kemnitz, E 2012-10-01 461 PubMed Ionic liquids have been shown to be highly effective lubricants for a steel on aluminium system. This work shows that the chemistry of the anion and cation are critical in achieving maximum wear protection. The performance of the ILs containing a diphenylphosphate (DPP) anion all showed low wear, as did some of the tris(pentafluoroethyl)trifluorophosphate (FAP) and bis(trifluoromethanesulfonyl)amide (NTf(2)) anion containing ILs. However, in the case of the FAP and NTf(2) based systems, a cation dependence was observed, with relatively poor wear resistance obtained in the case of an imidazolium FAP and two pyrrolidinium NTf(2) salts, probably due to tribocorrosion caused by the fluorine reaction with the aluminium substrate. The systems exhibiting poor performance generally had a lower viscosity, which also impacts on their tribological properties. Those ILs that exhibited low wear were shown to have formed protective tribofilms on the aluminium alloy surface. PMID:22555280 Somers, Anthony E; Biddulph, Shannon M; Howlett, Patrick C; Sun, Jiazeng; MacFarlane, Douglas R; Forsyth, Maria 2012-06-14 462 NASA Astrophysics Data System (ADS) At a time when greenhouse gas emissions must be reduced, the use of the aluminium alloys is expanding, in particular in the transportation industry. In order to extend the possibilities of aluminium assembly design, new Metal Inert Gas (MIG) welding processes have been conceived. They work at lower temperatures than usual arc processes (classic MIG or Tungsten Inert Gas). This study compares four arc welding processes, applied to the 6061 aluminium alloy. These four weld processes have been studied through the metallurgical analysis of the weld beads. Metallography, micro-hardness testings, X Ray radiography have been carried out on the produced weld beads. The processes are classified according to the quality of the beads like geometry of beads, size of the heat affected zone and presence of defects. Benoit, A.; Paillard, P.; Baudin, T.; Jobez, S.; Castagné, J.-F. 2011-01-01 463 NSDL National Science Digital Library This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Below is a table showing addition of numbers from1$through$5\$. In each column and each row of the table, even and odd numbers alternate. Explain wh...
464
SciTech Connect
This patent describes a method of making an additive comprising: (a) adding 2 parts by volume of 3% sodium hypochlorite to 45 parts by volume of diesel oil fuel to form a sulphur free fuel, (b) removing all water and foreign matter formed by the sodium hypochlorite, (c) blending 30 parts by volume of 24% lead naphthanate with 15 parts by volume of the sulphur free fuel, 15 parts by volume of light-weight material oil to form a blended mixture, and (d) heating the blended mixture slowly and uniformly to 152F.
Bishop, H.
1988-10-18
465
NASA Astrophysics Data System (ADS)
The number of contributions to the Symposium was so high that only the review and invited talks have found place, in the form of articles, in this volume. This Appendix lists all these additional contributions (oral and posters) which are not present as articles. The abstracts of all contributions were published in a booklet produced by the Local Organizing Committee and are available at the SAO/NASA Astrophysics Data System (ADS). More information on these contributions (PowerPoint presentations and/or articles) have been made public in the Internet web site of the conference (http://cab.inta-csic.es/molecular_universe/).
2011-12-01
466
PubMed
The present investigation was carried out to elucidate a possible molecular mechanism related to the effects of aluminium-induced oxidative stress on various mitochondrial respiratory complex subunits with special emphasis on the role of Peroxisome proliferator activated receptor gamma co-activator 1? (PGC-1?) and its downstream targets i.e. Nuclear respiratory factor-1(NRF-1), Nuclear respiratory factor-2(NRF-2) and Mitochondrial transcription factor A (Tfam) in mitochondrial biogenesis. Aluminium lactate (10mg/kgb.wt./day) was administered intragastrically to rats for 12 weeks. After 12 weeks of exposure, we found an increase in ROS levels, mitochondrial DNA oxidation and decrease in citrate synthase activity in the Hippocampus (HC) and Corpus striatum (CS) regions of rat brain. On the other hand, there was a decrease in the mRNA levels of the mitochondrial encoded subunits-NADH dehydrogenase (ND) subunits i.e. ND1, ND2, ND3, Cytochrome b (Cytb), Cytochrome oxidase (COX) subunits i.e. COX1, COX3, ATP synthase (ATPase) subunit 6 along with reduced expression of nuclear encoded subunits COX4, COX5A, COX5B of Electron transport chain (ETC). Besides, a decrease in mitochondrial DNA copy number and mitochondrial content in both regions of rat brain was observed. The PGC-1? was down-regulated in aluminium treated rats along with NRF-1, NRF-2 and Tfam, which act downstream from PGC-1? in aluminium treated rats. Electron microscopy results revealed a significant increase in the mitochondrial swelling, loss of cristae, chromatin condensation and decreases in mitochondrial number in case of aluminium treated rats as compared to control. So, PGC-1? seems to be a potent target for aluminium neurotoxicity, which makes it an almost ideal target to control or limit the damage that has been associated with the defective mitochondrial function seen in neurodegenerative diseases. PMID:24084166
Sharma, Deep Raj; Sunkaria, Aditya; Wani, Willayat Yousuf; Sharma, Reeta Kumari; Kandimalla, Ramesh J L; Bal, Amanjit; Gill, Kiran Dip
2013-12-01
467
Strontium is the most widely used and a very effective element for modifying the morphology of eutectic silicon, while Ti and B are commonly present in the commercial grain refiners used for Al–Si alloys. Systematic studies on the effects of combined additions of Sr and different AlTiB grain refiners on the Al+Si eutectic and primary aluminium solidification have been performed.
L. Lu; A. K. Dahle
2006-01-01
468
NASA Astrophysics Data System (ADS)
In the aeronautical and aerospace industries there is a great demand of materials with high resistance and low weight such as the aluminium alloys for the production of different elements that conforms the airships and aerospace vehicles. In this work, an experimental study was carried out in order to analyse the influence of the cutting parameters (feed rate, cutting speed and type of tool) on the forces generated during dry turning tests of UNS A92024-T3 aluminium bars. The main influents factors on the forces were the feed rate and the interaction between type of tool and the spindle speed.
de Agustina, B.; Rubio, E. M.
2012-04-01
469
The discharge characteristics of aluminium in inhibited and uninhibited 4 M KOH at 50°C have been explored. The performance of pure aluminium as a fuel is compared with that for two leading alloy fuels that had been evaluated in our previous work, Alloy BDW (Al-1Mg-0.1In-0.2Mn) and Alloy 21 (Al-0.2Ga-0.1In-0.1Tl). The inhibitors employed in this study, SnO32-, In(OH)3, BiO33-, Ga(OH)4-, MnO42-,
D. D. Macdonald; C. English
1990-01-01
470
NASA Astrophysics Data System (ADS)
ToF-SIMS has been used to analyse an aluminium surface treated with a vegetable oil using a mercapto silane as a coupling agent between the aluminium substrate and the vegetable oil. The coupling between the vegetable oil and the mercapto silane was obtained through a photoinduced thiol-ene reaction using UV-radiation. The ToF-SIMS results show that the desired thiol-ene reaction has taken place between the unsaturated parts of the vegetable oil and the thiol groups of the mercapto silane via the thiol-ene reaction forming a surface film.
Bexell, U.; Olsson, M.; Sundell, P.-E.; Johansson, M.; Carlsson, P.; Hellsing, M.
2004-06-01
471
PubMed
The end-point of the direct complexometric titration of Al(3+) in pH 4.6 solution can be determined by monitoring the fluorescence intensity of the aluminium-morin complex, by use of a bifurcated fibre-optic light guide. The method allows the determination of aluminium in the 1-800 ppm range with good precision. The procedure is applicable even when the solutions are strongly coloured or turbid, but because of the slow complexation kinetics requires a titration time of about 20 min. PMID:18964217
Wolfbeis, O S; Schaffar, B P; Chalmers, R A
1986-11-01
472
Fatigue behaviour of polycrystalline, isotropic graphite FU2590 (porosity 13vol%) and of FU2590 infiltrated with the aluminium alloy AlSi7Mg (FU2590\\/AlSi7Mg) is investigated. The interpenetrating graphite–aluminium composite is produced by squeeze casting infiltration, where the open porosity of FU2590 (10–11vol%) is infiltrated with AlSi7Mg. Fully reversed bending fatigue tests at cycling frequency 25Hz and fully reversed tension–compression experiments at 20kHz are performed.
H. Mayer; M. Papakyriacou
2006-01-01
473
PubMed
Compact high-efficiency ultraviolet solid-state light sources--such as light-emitting diodes (LEDs) and laser diodes--are of considerable technological interest as alternatives to large, toxic, low-efficiency gas lasers and mercury lamps. Microelectronic fabrication technologies and the environmental sciences both require light sources with shorter emission wavelengths: the former for improved resolution in photolithography and the latter for sensors that can detect minute hazardous particles. In addition, ultraviolet solid-state light sources are also attracting attention for potential applications in high-density optical data storage, biomedical research, water and air purification, and sterilization. Wide-bandgap materials, such as diamond and III-V nitride semiconductors (GaN, AlGaN and AlN; refs 3-10), are potential materials for ultraviolet LEDs and laser diodes, but suffer from difficulties in controlling electrical conduction. Here we report the successful control of both n-type and p-type doping in aluminium nitride (AlN), which has a very wide direct bandgap of 6 eV. This doping strategy allows us to develop an AlN PIN (p-type/intrinsic/n-type) homojunction LED with an emission wavelength of 210 nm, which is the shortest reported to date for any kind of LED. The emission is attributed to an exciton transition, and represents an important step towards achieving exciton-related light-emitting devices as well as replacing gas light sources with solid-state light sources. PMID:16710416
Taniyasu, Yoshitaka; Kasu, Makoto; Makimoto, Toshiki
2006-05-18
474
PubMed
Aluminium (Al) is used in water purification and is also present in several manufactured foods and medicines. Al is known to induce a broad range of physiological, biochemical and behavioural dysfunctions in laboratory animals and humans. This investigation was carried out to investigate the effects of subchronic exposure to Al (as AlCl?) in rats. Sprague-Dawley rats were randomly separated into two groups. Group 1 rats treated with sodium chloride served as the control, group 2 rats were treated with Al (as AlCl?, 5 mg/kg body weight) intraperitonally for 10 weeks. Animals were killed and blood samples were analyzed for blood serum alkaline phosphatase (ALP), aspartate aminotransferase (AST), alanine aminotransferase (ALT) and lactate dehydrogenase (LDH) enzyme activities and creatinine, urea (U) and uric acid (UA) levels for evaluating hepatotoxicity and nephrotoxicity. Blood parameters including red blood cells (RBCs), haemoglobin (Hb) concentration, haematocrit (Ht), platelets (PLTs) and white blood cells (WBCs) were compared between control and experimental group to assess haematoxicity. In order to determine the genotoxicity, the number of micronucleated hepatocytes (MNHEPs) was counted in isolated hepatocytes. In addition, histological alterations in liver and kidney samples were investigated. After exposure with Al, the enzymatic activities of ALP, AST, ALT and LDH, and the levels of U and UA significantly increased. RBC, WBC, PLT, Hb and Ht revealed significant decreases in experimental group compared to the control. AlCl? caused a significant increase in MNHEPs. Furthermore, severe pathological damages were established in both liver and kidney samples. Subchronic exposure to low doses of Al can produce serious dysfunctions in rat blood, liver and kidney, and exposure to this metal can result in greater damages. PMID:22421584
Geyikoglu, Fatime; Türkez, Hasan; Bakir, Tülay Ozhan; Cicek, Mustafa
2013-10-01
475
SciTech Connect
The dynamics of laser and X-ray radiation fields in experiments with cylindrical converter boxes (illuminators), which had earlier been carried out on the Iskra-5 laser facility (the second harmonic of iodine laser radiation, {lambda} = 0.66 {mu}m) was investigated in a sector approximation using the SND-LIRA numerical technique. In these experiments, the X-ray radiation temperature in the box was determined by measuring the velocity of the shock wave generated in the sample under investigation, which was located at the end of the cylindrical illuminator. Through simulations were made using the SND-LIRA code, which took into account the absorption of laser driver radiation at the box walls, the production of quasithermal radiation, as well as the formation and propagation of the shock wave in the sample under investigation. An analysis of the experiments permits determining the electron thermal flux limiter f: for f = 0.03 it is possible to match the experimental scaling data for X-ray in-box radiation temperature to the data of our simulations. The shock velocities obtained from the simulations are also consistent with experimental data. In particular, in the experiment with six laser beams (and a laser energy E{sub L} = 1380 J introduced into the box) the velocity of the shock front (determined from the position of a laser mark) after passage through a 50-{mu}m thick base aluminium layer was equal to 35{+-}1.6 km s{sup -1}, and in simulations to 36 km s{sup -1}. In the experiment with four laser beams (for E{sub L} = 850 J) the shock velocity (measured from the difference of transit times through the base aluminium layer and an additional thin aluminium platelet) was equal to 30{+-}3.6 km s{sup -1}, and in simulations to 30 km s{sup -1}. (interaction of laser radiation with matter)
Bondarenko, S V; Dolgoleva, G V; Novikova, E A
2013-07-31
476
NASA Technical Reports Server (NTRS)
Mainstream Engineering Corporation was awarded Phase I and Phase II contracts from Goddard Space Flight Center's Small Business Innovation Research (SBIR) program in early 1990. With support from the SBIR program, Mainstream Engineering Corporation has developed a unique low cost additive, QwikBoost (TM), that increases the performance of air conditioners, heat pumps, refrigerators, and freezers. Because of the energy and environmental benefits of QwikBoost, Mainstream received the Tibbetts Award at a White House Ceremony on October 16, 1997. QwikBoost was introduced at the 1998 International Air Conditioning, Heating, and Refrigeration Exposition. QwikBoost is packaged in a handy 3-ounce can (pressurized with R-134a) and will be available for automotive air conditioning systems in summer 1998.
1999-01-01
477
NASA Technical Reports Server (NTRS)
In an attempt to improve the thermal-oxidative stability (TOS) of PMR-type polymers, the use of 1,4-phenylenebis (phenylmaleic anhydride) PPMA, was evaluated. Two series of nadic end-capped addition curing polyimides were prepared by imidizing PPMA with either 4,4'-methylene dianiline or p-phenylenediamine. The first resulted in improved solubility and increased resin flow while the latter yielded a compression molded neat resin sample with a T(sub g) of 408 C, close to 70 C higher than PME-15. The performance of these materials in long term weight loss studies was below that of PMR-15, independent of post-cure conditions. These results can be rationalized in terms of the thermal lability of the pendant phenyl groups and the incomplete imidization of the sterically congested PPMA. The preparation of model compounds as well as future research directions are discussed.
Frimer, Aryeh A.; Cavano, Paul
1991-01-01
478
NASA Technical Reports Server (NTRS)
The additive is for a raw sewage treatment process of the type where settling tanks are used for the purpose of permitting the suspended matter in the raw sewage to be settled as well as to permit adsorption of the dissolved contaminants in the water of the sewage. The sludge, which settles down to the bottom of the settling tank is extracted, pyrolyzed and activated to form activated carbon and ash which is mixed with the sewage prior to its introduction into the settling tank. The sludge does not provide all of the activated carbon and ash required for adequate treatment of the raw sewage. It is necessary to add carbon to the process and instead of expensive commercial carbon, coal is used to provide the carbon supplement.
Kalvinskas, J. J.; Mueller, W. A.; Ingham, J. D. (inventors)
1980-01-01
479
This paper discusses the comparative study on machining hybrid aluminium metal matrix composites (Al6061-SiC-Al2O3) and carbon nano tubes (CNT) added hybrid MMCs. The ordinary hybrid composites were fabricated using the liquid metallurgy technique by stir casting method under which wt.6% of particulates SiC and Al2O3 were dispersed in the base matrix. The second CNT-hybrid composite is a woven alumina (Al2O3)
T. Sasimurugan; K. Palanikumar
2011-01-01
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A000292 Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6. (Formerly M3382 N1363) 622
%I M3382 N1363
%S 0,1,4,10,20,35,56,84,120,165,220,286,364,455,560,680,816,969,1140,
%T 1330,1540,1771,2024,2300,2600,2925,3276,3654,4060,4495,4960,5456,
%U 5984,6545,7140,7770,8436,9139,9880,10660,11480,12341,13244,14190,15180
%N Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.
%C a(n) = number of balls in a triangular pyramid in which each edge contains n balls.
%C One of the 5 Platonic polyhedral (tetrahedral, cube, octahedral, dodecahedral and icosahedral) numbers (cf. A053012).
%C Also (1/6)*(n^3 + 3*n^2 + 2*n) is the number of ways to color the vertices of a triangle using <= n colors, allowing rotations and reflections. Group is the dihedral group D_6 with cycle index (x1^3 + 2*x3 + 3*x1*x2)/6.
%C Also the convolution of the natural numbers with themselves. - Felix Goldberg (felixg(AT)tx.technion.ac.il), Feb 01 2001
%C Connected with the Eulerian numbers (1, 4, 1) via 1*a(x-2) + 4*a(x-1) + 1*a(x) = x^3. - _Gottfried Helms_, Apr 15 2002
%C a(n) = Sum_{1 <= i <= j <= n} |i - j|. - _Amarnath Murthy_, Aug 05 2002
%C a(n) = sum of all the possible products p*q where (p,q) are ordered pairs and p + q = n + 1. a(5) = 5 + 8 + 9 + 8 + 5 = 35. - _Amarnath Murthy_, May 29 2003
%C Number of labeled graphs on n+3 nodes that are triangles. - _Jon Perry_, Jun 14 2003
%C Number of permutations of n+3 which have exactly 1 descent and avoid the pattern 1324. - _Mike Zabrocki_, Nov 05 2004
%C Schlaefli symbol for this polyhedron: {3,3}.
%C Transform of n^2 under the Riordan array (1/(1-x^2), x). - _Paul Barry_, Apr 16 2005
%C a(n) = -A108299(n+5, 6) = A108299(n+6, 7). - _Reinhard Zumkeller_, Jun 01 2005
%C a(n) = -A110555(n+4, 3). - _Reinhard Zumkeller_, Jul 27 2005
%C a(n) is a perfect square only for n = {1, 2, 48}. a(48) = 19600 = 140^2. - _Alexander Adamchuk_, Nov 24 2006
%C a(n+1) is the number of terms in the expansion of (a_1 + a_2 + a_3 + a_4)^n. - _Sergio Falcon_, Feb 12 2007 [Corrected by _Graeme McRae_, Aug 28 2007]
%C a(n+1) is the number of terms in the complete homogeneous symmetric polynomial of degree n in 3 variables. - _Richard Barnes_, Sep 06 2017
%C This is also the average "permutation entropy", sum((pi(n)-n)^2)/n!, over the set of all possible n! permutations pi. - Jeff Boscole (jazzerciser(AT)hotmail.com), Mar 20 2007
%C a(n) = (d/dx)(S(n, x), x)|_{x = 2}. First derivative of Chebyshev S-polynomials evaluated at x = 2. See A049310. - _Wolfdieter Lang_, Apr 04 2007
%C If X is an n-set and Y a fixed (n-1)-subset of X then a(n-2) is equal to the number of 3-subsets of X intersecting Y. - _Milan Janjic_, Aug 15 2007
%C Complement of A145397; A023533(a(n))=1; A014306(a(n))=0. - _Reinhard Zumkeller_, Oct 14 2008
%C Equals row sums of triangle A152205. - _Gary W. Adamson_, Nov 29 2008
%C a(n) is the number of gifts received from the lyricist's true love up to and including day n in the song "The Twelve Days of Christmas". a(12) = 364, almost the number of days in the year. - Bernard Hill (bernard(AT)braeburn.co.uk), Dec 05 2008
%C Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF2 denominators of A156925. See A157703 for background information. - _Johannes W. Meijer_, Mar 07 2009
%C Starting with 1 = row sums of triangle A158823. - _Gary W. Adamson_, Mar 28 2009
%C Wiener index of the path graph P_n. - _Eric W. Weisstein_, Apr 30 2009
%C This is a 'Matryoshka doll' sequence with alpha=0, the multiplicative counterpart is A000178: seq(add(add(i,i=alpha..k),k=alpha..n),n=alpha..50). - _Peter Luschny_, Jul 14 2009
%C a(n) is the number of nondecreasing, three-element permutations of n distinct numbers. - Samuel Savitz, Sep 12 2009
%C a(n+4) = Number of different partitions of number n on sum of 4 elements a(6) = a(2+4) because we have 10 different partitions of 2 on sum of 4 elements 2 = 2 + 0 + 0 + 0 = 1 + 1 + 0 + 0 = 0 + 2 + 0 + 0 = 1 + 0 + 1 + 0 = 0 + 1 + 1 + 0 = 0 + 0 + 2 + 0 = 1 + 0 + 0 + 1 = 0 + 1 + 0 + 1 = 0 + 0 + 1 + 1 = 0 + 0 + 0 + 2. - _Artur Jasinski_, Nov 30 2009
%C a(n) corresponds to the total number of steps to memorize n verses by the technique described in A173564. - Ibrahima Faye (ifaye2001(AT)yahoo.fr), Feb 22 2010
%C The number of (n+2)-bit numbers which contain two runs of 1's in their binary expansion. - _Vladimir Shevelev_, Jul 30 2010
%C a(n) is also, starting at the second term, the number of triangles formed in n-gons by intersecting diagonals with three diagonal endpoints (see the first column of the table in Sommars link). - _Alexandre Wajnberg_, Aug 21 2010
%C Column sums of:
%C 1 4 9 16 25...
%C 1 4 9...
%C 1...
%C ..............
%C --------------
%C 1 4 10 20 35...
%C From _Johannes W. Meijer_, May 20 2011: (Start)
%C The Ca3, Ca4, Gi3 and Gi4 triangle sums (see A180662 for their definitions) of the Connell-Pol triangle A159797 are linear sums of shifted versions of the duplicated tetrahedral numbers, e.g., Gi3(n) = 17*a(n) + 19*a(n-1) and Gi4(n) = 5*a(n) + a(n-1).
%C Furthermore the Kn3, Kn4, Ca3, Ca4, Gi3 and Gi4 triangle sums of the Connell sequence A001614 as a triangle are also linear sums of shifted versions of the sequence given above. (End)
%C a(n-2)=N_0(n), n >= 1, with a(-1):=0, is the number of vertices of n planes in generic position in three-dimensional space. See a comment under A000125 for general arrangement. Comment to Arnold's problem 1990-11, see the Arnold reference, p. 506. - _Wolfdieter Lang_, May 27 2011
%C We consider optimal proper vertex colorings of a graph G. Assume that the labeling, i.e., coloring starts with 1. By optimality we mean that the maximum label used is the minimum of the maximum integer label used across all possible labelings of G. Let S=Sum of the differences |l(v) - l(u)|, the sum being over all edges uv of G and l(w) is the label associated with a vertex w of G. We say G admits unique labeling if all possible labelings of G is S-invariant and yields the same integer partition of S. With an offset this sequence gives the S-values for the complete graph on n vertices, n = 2, 3, ... . - _K.V.Iyer_, Jul 08 2011
%C Central term of commutator of transverse Virasoro operators in 4-D case for relativistic quantum open strings (ref. Zwiebach). - _Tom Copeland_, Sep 13 2011
%C Appears as a coefficient of a Sturm-Liouville operator in the Ovsienko reference on page 43. - _Tom Copeland_, Sep 13 2011
%C For n > 0: a(n) = number of triples (u,v,w) with 1 <= u <= v <= w <= n, cf. A200737. - _Reinhard Zumkeller_, Nov 21 2011
%C Regarding the second comment above by Amarnath Murthy (May 29 2003), see A181118 which gives the sequence of ordered pairs. - _L. Edson Jeffery_, Dec 17 2011
%C The dimension of the space spanned by the 3-form v[ijk] that couples to M2-brane worldsheets wrapping 3-cycles inside tori (ref. Green, Miller, Vanhove eq. 3.9). - _Stephen Crowley_, Jan 05 2012
%C a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = n. Also, a(n+1) is the number of 2 X 2 matrices with all terms in {0, 1, ..., n} and (sum of terms) = 3n. - _Clark Kimberling_, Mar 19 2012
%C a(n) = A004006(n) - n - 1. - _Reinhard Zumkeller_, Mar 31 2012
%C Using n + 4 consecutive triangular numbers t(1), t(2), ..., t(n+4), where n is the n-th term of this sequence, create a polygon by connecting points (t(1), (t(2)) to (t(2), t(3)), (t(2), t(3)) to (t(3), t(4)), ..., (t(1), t(2)) to (t(n+3), t(n+4)). The area of this polygon will be one-half of each term in this sequence. - _J. M. Bergot_, May 05 2012
%C Pisano period lengths: 1, 4, 9, 8, 5, 36, 7, 16, 27, 20, 11, 72, 13, 28, 45, 32, 17,108, 19, 40, ... . (The Pisano sequence modulo m is the auxiliary sequence p(n) = a(n) mod m, n >= 1, for some m. p(n) is periodic for all sequences with rational g.f., like this one, and others. The lengths of the period of p(n) are quoted here for m>=1.) - _R. J. Mathar_, Aug 10 2012
%C a(n) = the maximum possible number of rooted triplets consistent with any phylogenetic tree (level-0 phylogenetic network) containing exactly n+2 leaves. - _Jesper Jansson_, Sep 10 2012
%C For n > 0, the digital roots of this sequence A010888(a(n)) form the purely periodic 27-cycle {1, 4, 1, 2, 8, 2, 3, 3, 3, 4, 7, 4, 5, 2, 5, 6, 6, 6, 7, 1, 7, 8, 5, 8, 9, 9, 9}, which just rephrases the pisano period length above. - _Ant King_, Oct 18 2012
%C a(n) is the number of functions f from {1, 2, 3} to {1, 2, ..., n + 4} such that f(1) + 1 < f(2) and f(2) + 1 < f(3). - _Dennis P. Walsh_, Nov 27 2012
%C a(n) is the Szeged index of the path graph with n+1 vertices; see the Diudea et al. reference, p. 155, Eq. (5.8). - _Emeric Deutsch_, Aug 01 2013
%C Also the number of permutations of length n that can be sorted by a single block transposition. - _Vincent Vatter_, Aug 21 2013
%C From _J. M. Bergot_, Sep 10 2013: (Start)
%C a(n) is the 3 X 3 matrix determinant
%C | C(n,1) C(n,2) C(n,3) |
%C | C(n+1,1) C(n+1,2) C(n+1,3) |
%C | C(n+2,1) C(n+2,2) C(n+2,3) |
%C (End)
%C In physics, a(n)/2 is the trace of the spin operator S_z^2 for a particle with spin S=n/2. For example, when S=3/2, the S_z eigenvalues are -3/2, -1/2, +1/2, +3/2 and the sum of their squares is 10/2 = a(3)/2. - _Stanislav Sykora_, Nov 06 2013
%C a(n+1) = (n+1)*(n+2)*(n+3)/6 is also the dimension of the Hilbert space of homogeneous polynomials of degree n. - _L. Edson Jeffery_, Dec 12 2013
%C For n >= 4, a(n-3) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0111 (n-4 zeros), or, equivalently, a(n-3) is up-down coefficient {n,7} (see comment in A060351). - _Vladimir Shevelev_, Feb 15 2014
%C a(n) = one-half the area of the region created by plotting the points (n^2,(n+1)^2). A line connects points (n^2,(n+1)^2) and ((n+1)^2, (n+2)^2) and a line is drawn from (0,1) to each increasing point. From (0,1) to (4,9) the area is 2; from (0,1) to (9,16) the area is 8; further areas are 20,40,70,...,2*a(n). - _J. M. Bergot_, May 29 2014
%C Beukers and Top prove that no tetrahedral number > 1 equals a square pyramidal number A000330. - _Jonathan Sondow_, Jun 21 2014
%C a(n+1) is for n >= 1 the number of nondecreasing n-letter words over the alphabet [4] = {1, 2, 3, 4} (or any other four distinct numbers). a(2+1) = 10 from the words 11, 22, 33, 44, 12, 13, 14, 23, 24, 34; which is also the maximal number of distinct elements in a symmetric 4 X 4 matrix. Inspired by the Jul 20 2014 comment by _R. J. Cano_ on A000582. - _Wolfdieter Lang_, Jul 29 2014
%C Degree of the q-polynomial counting the orbits of plane partitions under the action of the symmetric group S3. Orbit-counting generating function is product_{i <= j <= k <= n} ( (1 - q^(i + j + k - 1))/(1 - q^(i + j + k - 2)) ). See q-TSPP reference. - _Olivier Gérard_, Feb 25 2015
%C Row lengths of tables A248141 and A248147. - _Reinhard Zumkeller_, Oct 02 2014
%C If n is even then a(n) = Sum_{k=1..n/2} (2k)^2. If n is odd then a(n) = Sum_{k=0..(n-1)/2} (1+2k)^2. This can be illustrated as stacking boxes inside a square pyramid on plateaus of edge lengths 2k or 2k+1, respectively. The largest k are the 2k X 2k or (2k+1) X (2k+1) base. - _R. K. Guy_, Feb 26 2015
%C Draw n lines in general position in the plane. Any three define a triangle, so in all we see C(n,3) = a(n-2) triangles (6 lines produce 4 triangles, and so on). - Terry Stickels, Jul 21 2015
%C a(n-2) = fallfac(n,3)/3!, n >= 3, is also the number of independent components of an antisymmetric tensor of rank 3 and dimension n. Here fallfac is the falling factorial. - _Wolfdieter Lang_, Dec 10 2015
%C Number of compositions (ordered partitions) of n+3 into exactly 4 parts. - _Juergen Will_, Jan 02 2016
%C Number of weak compositions (ordered weak partitions) of n-1 into exactly 4 parts. - _Juergen Will_, Jan 02 2016
%C For n >= 2 gives the number of multiplications of two nonzero matrix elements in calculating the product of two upper n X n triangular matrices. - _John M. Coffey_, Jun 23 2016
%C Terms a(4n+1), n >= 0, are odd, all others are even. The 2-adic valuation of the subsequence of every other term, a(2n+1), n >= 0, yields the ruler sequence A007814. Sequence A275019 gives the 2-adic valuation of a(n). - _M. F. Hasler_, Dec 05 2016
%C Does not satisfy Benford's law [Ross, 2012]. - _N. J. A. Sloane_, Feb 12 2017
%C C(n+2,3) is the number of ways to select 1 triplet among n+2 objects, thus a(n) is the coefficient of x1^(n-1)*x3 in exponential Bell polynomial B_{n+2}(x1,x2,...), hence its link with A050534 and A001296 (see formula). - _Cyril Damamme_, Feb 26 2018
%C a(n) is also the number of 3-cycles in the (n+4)-path complement graph. - _Eric W. Weisstein_, Apr 11 2018
%C a(n) is the general number of all geodetic graphs of diameter n homeomorphic to a complete graph K4. - _Carlos Enrique Frasser_, May 24 2018
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
%D V. I. Arnold (ed.), Arnold's Problems, Springer, 2004, comments on Problem 1990-11 (p. 75), pp. 503-510. Numbers N_0.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 194.
%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 83.
%D H. S. M. Coxeter, Polyhedral numbers, pp. 25-35 of R. S. Cohen, J. J. Stachel and M. W. Wartofsky, eds., For Dirk Struik: Scientific, historical and political essays in honor of Dirk J. Struik, Reidel, Dordrecht, 1974.
%D E. Deza and M. M. Deza, Figurate numbers, World Scientific Publishing (2012), page 93.
%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 4.
%D M. V. Diudea, I. Gutman, J. Lorentz, Molecular Topology, Nova Science, 2001, Huntington, N.Y. pp. 152-156.
%D J. C. P. Miller, editor, Table of Binomial Coefficients. Royal Society Mathematical Tables, Vol. 3, Cambridge Univ. Press, 1954.
%D V. Ovsienko and S. Tabachnikov, Projective Differential Geometry Old and New, Cambridge Tracts in Mathematics (no. 165), Cambridge Univ. Press, 2005.
%D Kenneth A Ross, First Digits of Squares and Cubes, Math. Mag. 85 (2012) 36-42. doi:10.4169/math.mag.85.1.36.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A. Szenes, The combinatorics of the Verlinde formulas (N.J. Hitchin et al., ed.), in Vector bundles in algebraic geometry, Cambridge, 1995.
%D D. Wells, The Penguin Dictionary of Curious and interesting Numbers, pp. 126-7 Penguin Books 1987.
%D B. Zwiebach, A First Course in String Theory, Cambridge, 2004; see p. 226.
%H N. J. A. Sloane, <a href="/A000292/b000292.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H O. Aichholzer and H. Krasser, <a href="http://www.ist.tugraz.at/publications/oaich/psfiles/ak-psotd-01.ps.gz">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H Luciano Ancora, <a href="https://upload.wikimedia.org/wikipedia/commons/9/9c/FigurateN.pdf">The Square Pyramidal Number and other figurate numbers</a>, ch. 5.
%H F. Beukers and J. Top, <a href="http://www.math.rug.nl/~top/oranges.pdf">On oranges and integral points on certain plane cubic curves</a>, Nieuw Arch. Wiskd., IV (1988), Ser. 6, No. 3, 203-210.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H C. E. Frasser and G. N. Vostrov, <a href="https://arxiv.org/abs/1611.01873">Geodetic Graphs Homeomorphic to a Given Geodetic Graph</a>, arXiv:1611.01873 [cs.DM], 2016. [p. 16, corollary 5]
%H Michael B. Green, Stephen D. Miller, and Pierre Vanhove, <a href="http://arxiv.org/abs/1111.2983">Small representations, string instantons, and Fourier modes of Eisenstein series</a>, arXiv:1111.2983 [hep-th], 2011-2013.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.NT/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.
%H A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 46. <a href="http://tohbook.info">Book's website</a>
%H Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.
%H C. Homberger, V. Vatter, <a href="http://arxiv.org/abs/1308.4946">On the effective and automatic enumeration of polynomial permutation classes</a>, arXiv preprint arXiv:1308.4946 [math.CO], 2013.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H Virginia Johnson, Charles K. Cook, <a href="http://arxiv.org/abs/1608.02420">Areas of Triangles and other Polygons with Vertices from Various Sequences</a>, arXiv:1608.02420 [math.CO], 2016.
%H R. Jovanovic, <a href="http://milan.milanovic.org/math/Math.php?akcija=SviTetra">First 2500 Tetrahedral numbers</a>
%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2002), 65-75.
%H C. Koutschan, M. Kauers, D. Zeilberger, <a href="http://dx.doi.org/10.1073/pnas.1019186108">A Proof Of George Andrews' and David Robbins' q-TSPP Conjecture</a>, Proc. Nat. Acad. Sc., vol. 108 no. 6 (2011), p2196-2199. See also <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/qtsppRig.html">Zeilberger's page for this article</a>.
%H T. Langley, J. Liese, J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Langley/langley2.html">Generating Functions for Wilf Equivalence Under Generalized Factor Order </a>, J. Int. Seq. 14 (2011) # 11.4.2
%H P. A. MacMahon, <a href="http://www.jstor.org/stable/90632">Memoir on the Theory of the Compositions of Numbers</a>, Phil. Trans. Royal Soc. London A, 184 (1893), 835-901. - _Juergen Will_, Jan 02 2016
%H T. P. Martin, <a href="http://dx.doi.org/10.1016/0370-1573(95)00083-6">Shells of atoms</a>, Phys. Reports, 273 (1996), 199-241, eq. (1).
%H Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
%H N. J. A. Sloane, <a href="/A000292/a000292.gif">Illustration of initial terms</a>
%H N. J. A. Sloane, <a href="/A000292/a000292a.jpg">Pyramid of 20 balls corresponding to a(3)=20.</a>
%H S. E. Sommars and T. Sommars, <a href="http://www.cs.uwaterloo.ca/journals/JIS/sommars/newtriangle.html">Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon</a>, J. Integer Sequences, 1 (1998), #98.1.5.
%H H. Stamm-Wilbrandt, <a href="https://www.ibm.com/developerworks/community/blogs/HermannSW/entry/sum_of_pascal_s_triangle_reciprocals10">Sum of Pascal's triangle reciprocals</a>
%H G. Villemin's Almanach of Numbers, <a href="http://villemin.gerard.free.fr/Wwwgvmm/Geometri/Tetraedr.htm">Nombres Tetraedriques</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCycle.html">Graph Cycle</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathComplementGraph.html">Path Complement Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PathGraph.html">Path Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral Number</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WienerIndex.html">Wiener Index</a>
%H A. F. Y. Zhao, <a href="http://www.emis.de/journals/JIS/VOL17/Zhao/zhao3.html">Pattern Popularity in Multiply Restricted Permutations</a>, Journal of Integer Sequences, 17 (2014), #14.10.3.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(n) = C(n+2,3) = n*(n+1)*(n+2)/6 (see the name).
%F G.f.: x / (1 - x)^4.
%F a(n) = -a(-4 - n) for all in Z.
%F a(n) = Sum_{k=0..n} A000217(k), partial sums of the triangular numbers.
%F a(n) = (n+3)*a(n-1)/n. - _Ralf Stephan_, Apr 26 2003
%F Sums of three consecutive terms give A006003. - _Ralf Stephan_, Apr 26 2003
%F a(n) = C(1, 2) + C(2, 2) + ... + C(n-1, 2) + C(n, 2); e.g., for n = 5: a(5) = 0 + 1 + 3 + 6 + 10 = 20. - _Labos Elemer_, May 09 2003
%F Determinant of the n X n symmetric Pascal matrix M_(i, j) = C(i+j+2, i). - _Benoit Cloitre_, Aug 19 2003
%F The sum of a series constructed by the products of the index and the length of the series (n) minus the index (i): a(n) = sum[i(n-i)]. Also the sum of n terms of A000217. - Martin Steven McCormick (mathseq(AT)wazer.net), Apr 06 2005
%F a(n) = Sum_{k=0..floor((n-1)/2)} (n-2k)^2 [offset 0]; a(n+1) = Sum_{k=0..n} k^2*(1-(-1)^(n+k-1))/2 [offset 0]. - _Paul Barry_, Apr 16 2005
%F Values of the Verlinde formula for SL_2, with g = 2: a(n) = Sum_{j=1..n-1} n/(2*sin^2(j*Pi/n)). - _Simone Severini_, Sep 25 2006
%F a(n) = Sum_{m=1..n} Sum_{k=1..m} k. - _Alexander Adamchuk_, Oct 28 2006
%F a(n) = Sum_{k=1..n} binomial(n*k+1, n*k-1), with a(0) = 0. - _Paolo P. Lava_, Apr 13 2007
%F a(n-1) = (1/(1!*2!))*Sum_{1 <= x_1, x_2 <= n} |det V(x_1, x_2)| = (1/2)*Sum_{1 <= i,j <= n} |i-j|, where V(x_1, x_2) is the Vandermonde matrix of order 2. Column 2 of A133112. - _Peter Bala_, Sep 13 2007
%F Starting with 1 = binomial transform of [1, 3, 3, 1, ...]; e.g., a(4) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - _Gary W. Adamson_, Nov 04 2007
%F a(n) = A006503(n) - A002378(n). - _Reinhard Zumkeller_, Sep 24 2008
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 4. - _Jaume Oliver Lafont_, Nov 18 2008
%F Sum_{n>=1} 1/a(n) = 3/2, case x = 1 in Gradstein-Ryshik 1.513.7. - _R. J. Mathar_, Jan 27 2009
%F E.g.f.:((x^3)/6 + x^2 + x)*exp(x). - _Geoffrey Critzer_, Feb 21 2009
%F Lim{n -> oo} A171973(n)/a(n) = sqrt(2)/2. - _Reinhard Zumkeller_, Jan 20 2010
%F With offset 1, a(n) = (1/6)*floor(n^5/(n^2 + 1)). - _Gary Detlefs_, Feb 14 2010
%F a(n) = Sum_{k = 1..n} k*(n-k+1). - _Vladimir Shevelev_, Jul 30 2010
%F a(n) = (3*n^2 + 6*n + 2)/(6*(h(n+2) - h(n-1))), n > 0, where h(n) is the n-th harmonic number. - _Gary Detlefs_, Jul 01 2011
%F a(n) = coefficient of x^2 in the Maclaurin expansion of 1 + 1/(x+1) + 1/(x+1)^2 + 1/(x+1)^3 + ... + 1/(x+1)^n. - _Francesco Daddi_, Aug 02 2011
%F a(n) = coefficient of x^4 in the Maclaurin expansion of sin(x)*exp((n+1)*x). - _Francesco Daddi_, Aug 04 2011
%F a(n) = 2*A002415(n+1)/(n+1). - _Tom Copeland_, Sep 13 2011
%F a(n) = (A007531(n) + A027480(n) + A007290(n))/11. - _J. M. Bergot_, May 28 2012
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 1. - _Ant King_, Oct 18 2012
%F G.f.: x*U(0) where U(k) = 1 + 2*x*(k+2)/( 2*k+1 - x*(2*k+1)*(2*k+5)/(x*(2*k+5)+(2*k+2)/U(k+1) )); (continued fraction, 3rd kind, 3-step). - _Sergei N. Gladkovskii_, Dec 01 2012
%F a(n) = Sum_{i=1..n} binomial(i+1,2). - _Enrique Pérez Herrero_, Feb 21 2013
%F a(n^2 - 1) = (1/2)*(a(n^2 - n - 2) + a(n^2 + n - 2)) and
%F a(n^2 + n - 2) - a(n^2 - 1) = a(n-1)*(3*n^2 - 2) = 10*A024166(n-1), by Berselli's formula in A222716. - _Jonathan Sondow_, Mar 04 2013
%F G.f.: x+4*x^2/(Q(0)-4*x) where Q(k) = 1 + k*(x+1) + 4*x - x*(k+1)*(k+5)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013
%F a(n+1) = det(C(i+3,j+2), 1 <= i,j <= n), where C(n,k) are binomial coefficients. - _Mircea Merca_, Apr 06 2013
%F a(n) = a(n-2) + n^2, for n > 1. - _Ivan N. Ianakiev_, Apr 16 2013
%F a(2n) = 4*(a(n-1) + a(n)), for n > 0. - _Ivan N. Ianakiev_, Apr 26 2013
%F G.f.: x*G(0)/2, where G(k)= 1 + 1/(1 - x/(x + (k+1)/(k+4)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 02 2013
%F a(n) = n + 2*a(n-1) - a(n-2), with a(0) = a(-1) = 0. - _Richard R. Forberg_, Jul 11 2013
%F a(n)*(m+1)^3 + a(m)*(n+1) = a(n*m + n + m), for any nonnegative integers m and n. This is a 3D analog of Euler's theorem about triangular numbers, namely t(n)*(2m+1)^2 + t(m) = t(2nm + n + m), where t(n) is the n-th triangular number. - _Ivan N. Ianakiev_, Aug 20 2013
%F Sum_{n>=0} a(n)/(n+1)! = 2*e/3 = 1.8121878856393... . Sum_{n>=1} a(n)/n! = 13*e/6 = 5.88961062832... . - _Richard R. Forberg_, Dec 25 2013
%F a(n+1) = A023855(n+1) + A023856(n). - _Wesley Ivan Hurt_, Sep 24 2013
%F For n >= 0, a(n)/2 = sqrt(((A005563(n))^3 + (A005563(n))^2)/144). Equivalently, a(n)/2 = (1/3)*sqrt((4*(s*(s + 1))^3 + (s*(s + 1))^2)) with s = n/2 (spin quantum number). s(s+1) = (2*Pi*S/h)^2 where S = Spin angular momentum and h = Planck's constant (see comments by Stanislav Sykora in this sequence and in A005563). - _Raphie Frank_, Jan 17 2014
%F a(n) = A024916(n) + A076664(n), n >= 1. - _Omar E. Pol_, Feb 11 2014
%F a(n) = A212560(n) - A059722(n). - _J. M. Bergot_, Mar 08 2014
%F Sum_{n>=1} (-1)^(n + 1)/a(n) = 12*log(2) - 15/2 = 0.8177661667... See A242024, A242023. - _Richard R. Forberg_, Aug 11 2014
%F 3/(Sum_{n>=m} 1/a(n)) = A002378(m), for m > 0. - _Richard R. Forberg_, Aug 12 2014
%F a(n) = Sum_{i=1..n} Sum_{j=i..n} min(i,j). - _Enrique Pérez Herrero_, Dec 03 2014
%F Arithmetic mean of Square pyramidal number and Triangular number: a(n) = (A000330(n) + A000217(n))/2. - _Luciano Ancora_, Mar 14 2015
%F a(k*n) = a(k)*a(n) + 4*a(k-1)*a(n-1) + a(k-2)*a(n-2). - _Robert Israel_, Apr 20 2015
%F Dirichlet g.f.: (zeta(s-3) + 3*zeta(s-2) + 2*zeta(s-1))/6. - _Ilya Gutkovskiy_, Jul 01 2016
%F a(n) = A080851(1,n-1) - _R. J. Mathar_, Jul 28 2016
%F a(n) = (A000578(n+1) - (n+1) ) / 6. - _Zhandos Mambetaliyev_, Nov 24 2016
%F G.f.: x/(1 - x)^4 = (x * r(x) * r(x^2) * r(x^4) * r(x^8) * ...), where r(x) = (1 + x)^4 = (1 + 4x + 6x^2 + 4x^3 + x^4); and x/(1 - x)^4 = (x * r(x) * r(x^3) * r(x^9) * r(x^27) * ...) where r(x) = (1 + x + x^2)^4. - _Gary W. Adamson_, Jan 23 2017
%F a(n) = A000332(n+3) - A000332(n+2). - _Bruce J. Nicholson_, Apr 08 2017
%F a(n) = A001296(n) - A050534(n+1). - _Cyril Damamme_, Feb 26 2018
%e a(2) = 3*4*5/6 = 10, the number of balls in a pyramid of 3 layers of balls, 6 in a triangle at the bottom, 3 in the middle layer and 1 on top.
%e Consider the square array
%e 1 2 3 4 5 6 ...
%e 2 4 6 8 10 12 ...
%e 3 6 9 12 16 20 ...
%e 4 8 12 16 20 24 ...
%e 5 10 15 20 25 30 ...
%e ...
%e then a(n) = sum of n-th antidiagonal. - _Amarnath Murthy_, Apr 06 2003
%e G.f. = x + 4*x^2 + 10*x^3 + 20*x^4 + 35*x^5 + 56*x^6 + 84*x^7 + 120*x^8 + 165*x^9 + ...
%e Example for a(3+1) = 20 nondecreasing 3-letter words over {1,2,3,4}: 111, 222, 333; 444, 112, 113, 114, 223, 224, 122, 224, 133, 233, 144, 244, 344; 123, 124, 134, 234. 4 + 4*3 + 4 = 20. - _Wolfdieter Lang_, Jul 29 2014
%e Example for a(4-2) = 4 independent components of a rank 3 antisymmetric tensor A of dimension 4: A(1,2,3), A(1,2,4), A(1,3,4) and A(2,3,4). - _Wolfdieter Lang_, Dec 10 2015
%p a:=n->n*(n+1)*(n+2)/6; seq(a(n), n=0..50);
%p A000292 := n->binomial(n+2,3); seq(A000292(n), n=0..50);
%t Table[Binomial[n + 2, 3], {n, 0, 20}] (* _Zerinvary Lajos_, Jan 31 2010 *)
%t FoldList[Plus, 0, Rest[FoldLtetrist[Plus, 0, Range[50]]]] (* corrected by _Robert G. Wilson v_, Jun 26 2013 *)
%t Accumulate[Accumulate[Range[0, 50]]] (* _Harvey P. Dale_, Dec 10 2011 *)
%t Table[n (n + 1)(n + 2)/6, {n,0,100}] (* _Wesley Ivan Hurt_, Sep 25 2013 *)
%t Nest[Accumulate, Range[0, 50], 2] (* _Harvey P. Dale_, May 24 2017 *)
%t Binomial[Range[20] + 1, 3] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t LinearRecurrence[{4, -6, 4, -1}, {0, 1, 4, 10}, 20] (* _Eric W. Weisstein_, Sep 08 2017 *)
%t CoefficientList[Series[x/(-1 + x)^4, {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 08 2017 *)
%o (PARI) a(n) = (n) * (n+1) * (n+2) / 6
%o (PARI) a=vector(10000);a[2]=1;for(i=3,#a,a[i]=a[i-2]+i*i); \\ _Stanislav Sykora_, Nov 07 2013
%o (PARI) is(n)=my(k=sqrtnint(6*n,3)); k*(k+1)*(k+2)==6*n \\ _Charles R Greathouse IV_, Dec 13 2016
%o (DERIVE) v(n):= [1, 2, 3, ..., n] w(n):= [n, ..., 3, 2, 1] a(n):= scalar product (v(n)w(n)) " From Roland Schroeder (florola(AT)gmx.de), Aug 14 2010 "
%o a000292 n = n * (n + 1) * (n + 2) div 6
%o a000292_list = scanl1 (+) a000217_list
%o -- _Reinhard Zumkeller_, Jun 16 2013, Feb 09 2012, Nov 21 2011
%o (Maxima) A000292(n):=n*(n+1)*(n+2)/6\$ makelist(A000292(n),n,0,60); /* _Martin Ettl_, Oct 24 2012 */
%o (MAGMA) [n*(n+1)*(n+2)/6: n in [0..50]]; // _Wesley Ivan Hurt_, Jun 03 2014
%o (GAP) a:=n->Binomial(n+2,3);; A000292:=List([0..50],n->a(n)); # _Muniru A Asiru_, Feb 28 2018
%Y Bisections give A000447 and A002492.
%Y Sums of 2 consecutive terms give A000330.
%Y a(3n-3) = A006566(n). A000447(n) = a(2n-2). A002492(n) = a(2n+1).
%Y Column 0 of triangle A094415.
%Y Cf. A000217 (first differences), A001044, (see above example), A061552, A040977, A133111, A133112, A152205, A158823, A156925, A157703, A173564, A058187, A190717, A190718, A100440, A181118, A222716.
%Y Partial sums are A000332. - _Jonathan Vos Post_, Mar 27 2011
%Y Cf. A216499 (the analogous sequence for level-1 phylogenetic networks).
%Y Cf. A068980 (partitions), A231303 (spin physics).
%Y Cf. similar sequences listed in A237616.
%Y Cf. A104712 (second column, if offset is 2).
%Y Cf. A145397 (non-tetrahedral numbers). - _Daniel Forgues_, Apr 11 2015
%Y Cf. A127324.
%Y Cf. A007814, A275019 (2-adic valuation).
%Y Cf. A000578 (cubes), A005900 (octahedral numbers), A006566 (dodecahedral numbers), A006564 (icosahedral numbers).
%Y Cf. A002817 (4-cycle count of \bar P_{n+4}), A060446 (5-cycle count of \bar P_{n+3}), A302695 (6-cycle count of \bar P_{n+5})
%K nonn,core,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E PARI program corrected by _Harry J. Smith_, Dec 22 2008
%E Corrected and edited by _Daniel Forgues_, May 14 2010
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Last modified August 19 05:13 EDT 2018. Contains 313844 sequences. (Running on oeis4.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7023220062255859, "perplexity": 6620.01824262602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-34/segments/1534221215075.58/warc/CC-MAIN-20180819090604-20180819110604-00361.warc.gz"} |
http://www.ma.utexas.edu/mediawiki/index.php?title=Nonlocal_porous_medium_equation&diff=398&oldid=397 | # Nonlocal porous medium equation
(Difference between revisions)
Revision as of 23:02, 3 June 2011 (view source)Nestor (Talk | contribs)m← Older edit Revision as of 23:03, 3 June 2011 (view source)Nestor (Talk | contribs) mNewer edit → Line 15: Line 15: This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained , there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. This means that the first model presents us with a [[free boundary problem]]. For this model global existence and Hölder continuity of weak solutions have been recently obtained , there is almost nothing known about the properties of its free boundary, making it a rich source of open questions. - For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years . + For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years . == References == == References ==
## Revision as of 23:03, 3 June 2011
The nonlocal porous medium equation of order $\sigma$ is the name currently given to two very different equations, namely
$u_t = \nabla \cdot \left ( u \nabla \mathcal{K_\alpha} (u) \right )$
$\mbox{ where } \mathcal{K}_\alpha(u) := C_{n,\alpha}\; u * |x|^{-n+\alpha},\;\; \alpha+2=\sigma$
and
$u_t +(-\Delta)^{s}(u^m) = 0$
These equations agree when $s=1$ and $m=2$, otherwise they are not only different superficially, they also exhibit extremely different behaviors. They are both fractional order Quasilinear equations.
The first of the two has the remarkable property (for nonlocal equations at least) that any initial data with compact support remains with compact support for all later times, the opposite is true of the second equation, for which instantaneous speed of propagation holds.
This means that the first model presents us with a free boundary problem. For this model global existence and Hölder continuity of weak solutions have been recently obtained [1], there is almost nothing known about the properties of its free boundary, making it a rich source of open questions.
For the second equation, both the Cauchy problem and long time behavior have been extensively studied in recent years [2].
## References
1. Caffarelli, Luis; Vazquez, Juan (2011), "Nonlinear Porous Medium Flow with Fractional Potential Pressure", Archive for Rational Mechanics and Analysis (Springer Berlin / Heidelberg): 1–29, ISSN 0003-9527
2. Pablo, Arturo de; Quirós, Fernando; Rodríguez, Ana; Vazquez, Juan Luis (2011), "A fractional porous medium equation", Advances in Mathematics 226 (2): 1378–1409, doi:DOI: 10.1016/j.aim.2010.07.017, ISSN 0001-8708 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8853145241737366, "perplexity": 1089.4424278409374}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267866965.84/warc/CC-MAIN-20180624141349-20180624161349-00466.warc.gz"} |
https://de.maplesoft.com/support/help/maple/view.aspx?path=Finance%2FFairRate | Finance - Maple Programming Help
Home : Support : Online Help : Mathematics : Finance : Interest Rates : Finance/FairRate
Finance
FairRate
return the fair rate of an interest rate swap
Calling Sequence FairRate(swap, discountrate)
Parameters
swap - swap data structure; interest rate swap discount - non-negative constant of a yield term structure; discount rate
Description
• The FairRate command returns the fair rate of the given interest rate swap.
Examples
> $\mathrm{with}\left(\mathrm{Finance}\right):$
> $\mathrm{SetEvaluationDate}\left("January 02, 2007"\right):$
> $\mathrm{EvaluationDate}\left(\right)$
${"January 2, 2007"}$ (1)
Consider two payment schedules. The first one consists of payments of 5% of the nominal every month between January 3, 2008 and January 3, 2018. The second one consists of payments of 3% of the nominal every quarter between January 3, 2010 and January 3, 2015.
> $\mathrm{Schedule1}≔\mathrm{Schedule}\left("January 03, 2008","January 03, 2018",\mathrm{Monthly}\right)$
${\mathrm{Schedule1}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (2)
> $\mathrm{Schedule2}≔\mathrm{Schedule}\left("January 03, 2010","January 03, 2015",\mathrm{Quarterly}\right)$
${\mathrm{Schedule2}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (3)
> $\mathrm{Rate1}≔0.05$
${\mathrm{Rate1}}{≔}{0.05}$ (4)
> $\mathrm{Rate2}≔\mathrm{BenchmarkRate}\left(0.03\right)$
${\mathrm{Rate2}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (5)
Consider two simple swaps that exchange the first set of payments with the second set.
> $\mathrm{Swap1}≔\mathrm{InterestRateSwap}\left(1000,\mathrm{Rate1},\mathrm{Schedule1},\mathrm{Rate2},\mathrm{Schedule2},0.03\right)$
${\mathrm{Swap1}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (6)
> $\mathrm{Swap2}≔\mathrm{InterestRateSwap}\left(1000,\mathrm{Rate2},\mathrm{Schedule2},\mathrm{Rate1},\mathrm{Schedule1},0.03\right)$
${\mathrm{Swap2}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (7)
> $\mathrm{DiscountRate}≔0.05$
${\mathrm{DiscountRate}}{≔}{0.05}$ (8)
> $\mathrm{NetPresentValue}\left(\mathrm{Swap1},\mathrm{DiscountRate}\right)$
${-146.0132438}$ (9)
> $\mathrm{NetPresentValue}\left(\mathrm{Swap2},\mathrm{DiscountRate}\right)$
${146.0132438}$ (10)
> $\mathrm{FairRate1}≔\mathrm{FairRate}\left(\mathrm{Swap1},\mathrm{DiscountRate}\right)$
${\mathrm{FairRate1}}{≔}{0.03045073683}$ (11)
> $\mathrm{FairRate2}≔\mathrm{FairRate}\left(\mathrm{Swap2},\mathrm{DiscountRate}\right)$
${\mathrm{FairRate2}}{≔}{0.03045073683}$ (12)
Consider the same simple swaps that use the fair rate.
> $\mathrm{Swap3}≔\mathrm{InterestRateSwap}\left(1000,\mathrm{FairRate1},\mathrm{Schedule1},\mathrm{Rate2},\mathrm{Schedule2},0.03\right)$
${\mathrm{Swap3}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (13)
> $\mathrm{Swap4}≔\mathrm{InterestRateSwap}\left(1000,\mathrm{Rate2},\mathrm{Schedule2},\mathrm{FairRate2},\mathrm{Schedule1},0.03\right)$
${\mathrm{Swap4}}{:=}{\mathbf{module}}\left({}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{\mathbf{end module}}$ (14)
> $\mathrm{NetPresentValue}\left(\mathrm{Swap3},\mathrm{DiscountRate}\right)$
${-1.451178377}{}{{10}}^{{-8}}$ (15)
> $\mathrm{NetPresentValue}\left(\mathrm{Swap4},\mathrm{DiscountRate}\right)$
${1.451178377}{}{{10}}^{{-8}}$ (16)
Compatibility
• The Finance[FairRate] command was introduced in Maple 15. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 34, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9083077907562256, "perplexity": 2412.1967844181686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703522242.73/warc/CC-MAIN-20210121035242-20210121065242-00401.warc.gz"} |
https://mathematica.stackexchange.com/questions/109620/can-i-get-step-by-step-calculation-of-results-obtained-in-mathematica/109622 | # Can i get step by step calculation of results obtained in mathematica? [duplicate]
In mathematica we generally obtain direct result instead of step by step analytical calculation. Can we somehow obtain these steps?.
• Does Trace / TracePrint do what you want? – Verbeia Mar 10 '16 at 5:27
WolframAlpha["Solve[a x^2 + b x + c == 0, x]", {{"Result", 4}, "Content"}, | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.585374653339386, "perplexity": 1692.7940582633366}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362741.28/warc/CC-MAIN-20210301151825-20210301181825-00291.warc.gz"} |
http://tex.stackexchange.com/questions/169957/correct-arrow-for-mathematical-limits | # Correct arrow for mathematical limits
I am currently using $$\phi(\varepsilon) \overrightarrow{ \varepsilon \rightarrow 0 }0$$ for writing down mathematical limits. But I am not very confident with this notation, as the text under the arrow is far too huge displayed. Does anybody here have a better suggestion
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For limits, it is advised to use \to ... in math mode of course. A little hint: Do not use $$...$$ but $...$ instead. – Christian Hupfer Apr 6 '14 at 15:40
ah, thank you. and how do you write some text under this arrow? – user48924 Apr 6 '14 at 15:42
\xrightarrow[under]{over} – Manuel Apr 6 '14 at 15:46
yes, this is it. thank you! – user48924 Apr 6 '14 at 15:47
@user180097 However I would define something like \tendsto{x\to\infty} (more “logical”) to be something like \xrightarrow[x\to\infty]{}. – Manuel Jun 22 '14 at 14:32
I have two versions, depending on what you perhaps need.
\documentclass{scrbook}
\usepackage{amsmath}
\begin{document}
$\phi(\varepsilon) \xrightarrow[\varepsilon \to 0]{} 0$
$\lim_{\varepsilon \to 0} \phi(\varepsilon) = 0$
\end{document}
I prefer the \lim style, it is mathematical cleaner, but that is also a matter of taste.
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@Manuel: I used an old template, but I know about the caveats of minimal ;-) – Christian Hupfer Apr 6 '14 at 15:51 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9416590332984924, "perplexity": 2041.859522383403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644064445.47/warc/CC-MAIN-20150827025424-00154-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://byjus.com/rs-aggarwal-solutions/rs-aggarwal-class-10-solutions-chapter-18-areas-of-circle-sector-and-segment-exercise-18-4/ | # RS Aggarwal Solutions Class 10 Ex 18D
## RS Aggarwal Class 10 Ex 18D Chapter 18
Q.1: The side of a square is 10 cm. Find
(i) The area of the inscribed circle, and
(ii) The area of the circumscribed circle. [Take $\pi = 3.14$ ]
Sol:
(i) If a circle is inscribed in a square, then the side of the square is equal to the diameter of the circle.
Side of the square = 10 cm
Side = Diameter = 10
$Therefore,$ Radius = 5 cm
Area of the inscribed circle = $\pi r^{2}$ = $3.14 \times 5 \times 5$ = 78.5 $cm^{2}$
(ii) If a circle is circumscribed in a square, then the diagonal of the square is equal to the diameter of the circle.
Diagonal of the square = $\sqrt{2} \times$ Side of the square
$= \sqrt{2} \times 10$
$= 10 \sqrt{2}$ cm
Diagonal = Diameter = $= 10 \sqrt{2}$ cm
$Therefore, r = 5 \sqrt{2}$ cm
Now, Area of the circumscribed circle = $\pi r^{2}$
$= 3.14 \times (5\sqrt{2})^{2}$ = $3.14 \times 50$ = 157 $cm^{2}$
Q.2: If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.
Sol:
If a square is inscribed in a circle, then the diagonals of the square are diameters of the circle.
Let the diagonal of the square be d cm.
Thus, we have:
Radius, r = $\frac{d}{2}$ cm
Area of the circle = $\pi r^{2}$ = $\pi \frac{d^{2}}{4} \; cm^{2}$
We know:
d = $\sqrt{2} \times Side$
$\Rightarrow Side = \frac{d}{\sqrt{2}} \; cm$
Area of the square = $( Side) ^{2}$
$= \left ( \frac{d}{\sqrt{2}} \right ) ^{2}$
$= \left ( \frac{d^{2}}{2} \right ) \; cm^{2}$
Ratio of the area of the circle to that of the square :
$= \frac{\pi \frac{d^{2}}{4}}{\frac{d^{2}}{2}}$ = $\frac{\pi}{2}$
Thus, the ratio of the area of the circle to that of the square is $\pi : 2$
Q.3: The area of a circle inscribed in an equilateral triangle is 154 $cm^{2}$. Find the perimeter of the triangle. [Take $\sqrt{3} = 1.73$ ]
Sol:
Let the radius of the inscribed circle be r cm.
Given:
Area of the circle = 154 $cm^{2}$
We know:
Area of the circle = $\pi r^{2}$
$\Rightarrow 154 = \frac{22}{7} r^{2}$
$\Rightarrow r^{2} = \frac{154 \times 7}{22}$
$\Rightarrow r^{2} = 49$
$\Rightarrow r = 7$
In a triangle, the centre of the inscribed circle is the point of intersection of the median and the altitudes of the triangle. The centroid divides the median of a triangle in the ratio 2:1.
Here, AO: OD = 2 : 1
Now, Let the altitude be h cm.
We have:
$\angle ADB = 90^{\circ}$
OD = $\frac{1}{3} AD$
$OD = \frac{h}{3}$
$\Rightarrow h = 3r$
$\Rightarrow h = 21$
Let each side of the triangle be a cm.
In the right- angled triangle ADB, we have:
$AB^{2} = AD^{2} + DB^{2}$
$a^{2} = h^{2} + \left ( \frac{a}{2} \right )^{2}$
$3a^{2} = 4h^{2}$
$a^{2} = \frac{ 4h^{2}}{3}$
$a= \frac{2h}{\sqrt{3}}$
$a= \frac{42}{\sqrt{3}}$
$Therefore,$ Perimeter of the triangle = 3a
$= 3 \times \frac{42}{\sqrt{3}}$
$= \sqrt{3} \times 42$ = 72.66 cm
Q.4: The radius of the wheels of a vehicle is 42 cm. How many revolutions will it complete in a 19.8 km long journey?
Sol:
Radius of the wheels = 42 cm
Circumference of the wheels = $2 \pi r$
$2 \times \frac{22}{7} \times 42$ = 264 cm = 2.64 m
Distance covered by the wheel in 1 revolution = 2.64 m
Total distance = 19.8 km or 19800 m
$Therefore,$ Number of revolutions taken by the wheel $= \frac{19800}{2.64} = 7500$
Q.5: The wheels of the locomotive of a train are 2.1 m in radius. They make 75 revolutions in one minute. Find the speed of the train in km per hour.
Sol:
Radius of the wheels = 2.1 m
Circumference of the wheels = $2 \pi r$
$2 \times \frac{22}{7} \times 2.1$ = 13.2 m
Distance covered by the wheels in 1 revolution = 13.2 m
Distance covered by the wheels in 75 revolutions $= \left ( 13.2 \times 75 \right ) = \left ( 990 \times \frac{1}{1000} \right ) km$
Distance covered by the wheel in 1 minute = Distance covered by the wheels in 75 revolutions $= \frac{990}{1000}km$
$Therefore,$ Distance covered by the wheels in 1 hour $= \frac{990}{1000} \times 60$ = 59.4 km/h
Q.6: The wheels of a car makes 2500 revolutions in covering a distance of 4.95 km. Find the diameter of a wheel.
Sol:
Distance = 4.95 km $= 4.95 \times 1000 \times 100$ cm
$Therefore,$ Distance covered by the wheels in 1 revolution = $\frac{Total \; distance \; covered }{Number \; of \; revolutions}$
$\frac{4.95 \times 1000 \times 100}{2500} = 198$ cm
Now,
Circumference of the wheel = 198 cm
$\Rightarrow 2 \pi r =198$
$\Rightarrow 2 \times \frac{22}{7} \times r =198$
$\Rightarrow r = \frac{198 \times 7}{44}$ = 31.5 cm
$Therefore,$ Diameter of the wheel = 2r = 2 (31.5) = 63 cm
Q.7: A boy is cycling in such a way that the wheels of his bicycle are making 14 revolutions per minute. If the diameter of a wheel is 60 cm, calculate the speed (in km/h) at which the boy is cycling.
Sol:
Diameter of the wheel = 60 cm
$Therefore,$ Radius of the wheels = 30 cm
Circumference of the wheels = $2 \pi r$
= $2 \times \frac{22}{7} \times 30$ = $\frac{1320}{7}$ cm
Distance covered by the wheel in 1 revolution = $\frac{1320}{7}$ cm
$Therefore,$ Distance covered by the wheel in 140 revolution = $\left ( \frac{1320}{7} \times 140 \times \frac{1}{100} \right )$ m
= $\left ( \frac{1320 \times 140}{7 \times 100} \times \frac{1}{1000} \right ) \; km = \frac{264}{1000} \; km$
Now,
Distance covered by the wheel in 1 minute = Distance covered by the wheel in 140 revolution $= \frac{264}{1000} \; km$
$Therefore,$ Distance covered by the wheel in 1 hour = $= \frac{264}{1000} \times 60 = 15.84 \; km/h$
Hence, the speed at which the boy is cycling is 15.84 km/h
Q.8: The diameter of the wheels of a bus is 140 cm. How many revolutions per minute do the wheels make when the bus is moving at a speed of 72.6 km/h.
Sol:
Diameter of the wheels = 140 cm
Circumference = $2 \pi r$
$= 2 \times \frac{22}{7} \times 70$ = 440 cm
Speed of the wheels = 72.6 km /h
Distance covered by the wheels in 1 minute = $= \frac{76.2 \times 1000 \times 100}{60} = 121000$ cm
Number of revolutions made by the wheels in 1 minute = $= \frac{Total \; distance \; covered}{Circumference }$
$= \frac{121000}{440}$ = 275
Hence, the wheel makes 275 revolutions per minute.
Q.9: The diameters of the front and rear wheels of a tractor are 80 cm and 2 m respectively. Find the number of revolutions that a rear wheel makes to cover the distance which the front wheel covers in 800 revolutions.
Sol:
Radius of front wheel = 40 cm = $\frac{2}{5}$ m
Circumference of the front wheel = $= \left ( 2 \pi \times \frac{2}{5} \right ) m = \frac{4 \pi}{5}$
Distance covered by the front wheel in 800 revolutions = $= \left ( \frac{4 \pi}{5}\times 800 \right ) m = \left (640 \pi \right ) m$
Radius of the rear wheel = 1 m
Circumference of the rear wheel $= \left (2 \pi \times 1\right ) = 2 \pi\; m$
$Therefore,$ Required number of revolution = $\frac{Distance \; covered \; by \; the \; front \; wheel \; in \; 800 \; revolutions}{Circumference \; of \; the \; rear \; wheel}$
$=\frac{640 \pi }{2 \pi}$ =320
Q.10: Four equal circles are described about the four corners of a square so that each touches woe of the others, as shown in the figure. Find the area of the shaded region, if each side of the square measures 14 cm.
Sol:
Side of the square = 14 cm
Radius of the circle = $= \frac{14}{2} = 7$ cm
Area of the quadrant of one circle $= \frac{1}{4} \times \frac{22}{7} \times 7 \times 7$ =38.5 $cm^{2}$
Area of the quadrant of four circles = $38.5 \times 4 = 154 cm^{2}$
Now,
Area of the square = $(Side)^{2}$ = $(14)^{2}$ = 196 $cm^{2}$
Area of the shaded region = Area of the square -Area of the quadrants of the four circles = 196 – 154 = 42 $cm^{2}$
Q.11: Four equal circles, each of radius 5 cm , touch each other, as shown in the figure. Find the area included between them. [Take $\pi = 3.14$ ]
Sol:
AB= BC= CD= AC = 10 cm
All sides are equal , so it is a square.
Area of a square = $Side^{2}$
Area of a square = $10^{2}= 100 \; cm^{2}$
Area of the quadrant of one circle = $= \frac{1}{4} \pi r^{2}$
= $= \frac{1}{4} \times \frac{22}{7} \times 5 \times 5$ = 19.64 $cm^{2}$
Area of the quadrant of four circles = $19.64 \times 4 = 78.57 \; cm^{2}$
Area of the shaded portion = Area of the square – Area of the quadrant of four circles
= 100 – 78.57 = 21.43 $cm^{2}$
Q.12: Four circles, each of radius a units, touch each other. Show that the area between them is $\left (\frac{6}{7} a^{2} \right )$ sq. units .
Sol:
When four circles touches each other, their centres from the vertices of a square. The sides of the square are 2a units.
Area of the square = $(2a)^{2}= 4a^{2}$ sq. units
Area occupied by the four sectors = $4 \times \frac{90}{360} \times \pi \times a^{2}$
= $\pi aa^{2}$ sq. units
Area between the circles = Area of the square – Area of the four sector
$= \left ( 4 – \frac{22}{7} \right ) a^{2}$
$= \left ( \frac{6}{7} \right ) a^{2}$ sq. units
Q.13: Three equal circles, each of radius 6 cm, touch one another as shown in the figure. Find the area enclosed between them. [$\pi = 3.14 \; and \; \sqrt{3} =1.732$]
Sol:
Join ABC, all sides are equal, so it is the equilateral triangle.
Now, Area of the equilateral triangle $= \frac{\sqrt{3}}{4} \times Side^{2}$
$= \frac{1.73}{4} \times 12 \times 12$
= 62.28 $cm^{2}$
Area of the arc of the circle = $= \frac{60}{360} \pi r^{2}$
$= \frac{1}{6} \times \frac{22}{7} \times 6 \times 6$ = 18.86 $cm^{2}$
Area of the three sectors $= 3 \times 18.86 = 56.57 \; cm^{2}$
Area of the shaded portion = Area of the triangle – Area of the three quadrant
= 62.28 – 56.57 = 5.71 $cm^{2}$
Q.14: If three circles of radius a each, are drawn such that each touches the other two, prove that the area included between them is equal to $\frac{4}{5} a^{2}$. [$\pi = 3.14 \; and \; \sqrt{3} =1.732$ ]
Sol:
When three circles touch each other, their centres from an equilateral triangle, with each side being 2a.
Area of the triangle = $\frac{\sqrt{3}}{4} \times 2a \times 2a = \sqrt{3}a^{2}$
Total area of the three sectors of circles $=3 \times \frac{60}{360} \times \frac{22}{7} \times a^{2} = \frac{1}{2} \times \frac{22}{7} a^{2} = \frac{11}{7} a^{2}$
Area of the region between the circles = Area of the triangle – Area of the three sectors
$= \left ( \sqrt{3} – \frac{11}{7} \right ) a^{2}$
$= \left ( 1.73 – 1.57 \right ) a^{2}$
$= 0.16 a^{2}$
$= \frac{4}{25} a^{2}$
Q.15: In the given figure, ABCD is a trapezium of area 24.5 $cm^{2}$. If $AD \parallel BC$, $\angle DAB = 90^{\circ}$, AD = 10 cm and ABE is quadrant of a circle then find the area of the shaded region.
Sol:
Area of trapezium = $\frac{1}{2} (AD + BC) \times AB$
$\Rightarrow 24.5 = \frac{1}{2} (10+4) \times AB$
$\Rightarrow AB = 3.5$ cm
Area of the shaded region = Area of trapezium ABCD – Area of quadrant ABE
= 24.5 – $\frac{1}{4} \pi (AB)^{2}$
$= 24.5 – \frac{1}{4} \times \frac{22}{7} \times (3.5)^{2}$
= 24.5 – 9.625 = 14.875 $cm^{2}$
Hence, the area of shaded region is 14.875 $cm^{2}$.
Q.16: ABCD is a field in the shape of trapezium $AD \parallel BC, \angle ABC = 90^{\circ} \;and \; \angle ADC = 60^{\circ}$. Four sectors are formed with centres A,B,C and D, as shown in the figure. The radius of each sector is 14 m. Find the following
(i) Total area of the four sectors,
(ii) Area of the remaining portion, given that AD = 55 m , BC = 45 m and AB = 30 m.
Sol:
(i) Area of four sectors = Area of sector having central angle $60^{\circ}$ + Area of sectors having central angle $90^{\circ}$ + Area of sectors having central angle $120^{\circ}$
$\frac{60^{\circ}}{360^{\circ}} .\pi (14)^{2} + \frac{90^{\circ}}{360^{\circ}}. \pi (14)^{2} + \frac{90^{\circ}}{360^{\circ}} .\pi (14)^{2} + \frac{120^{\circ}}{360^{\circ}} .\pi (14)^{2}$
$\left ( \frac{60^{\circ}}{360^{\circ}} + \frac{90^{\circ}}{360^{\circ}} + \frac{90^{\circ}}{360^{\circ}} + \frac{120^{\circ}}{360^{\circ}} \right ) \pi (14)^{2}$
$\left ( \frac{360^{\circ}}{360^{\circ}} \right ) .\pi (14)^{2}$
$= 616 m^{2}$
(ii) Area of the remaining portion = Area of trapezium ABCD – Area of four quadrants
$= \frac{1}{2}(AD+BC) \times AB – 616$
$= \frac{1}{2}(55+45) \times 30 – 616$
= 1500 – 616 = 884 $m^{2}$
Q.17: Find the area of the shaded region in the given figure, where a circulation arc of radius 6 cm has been drawn with the vertex of an equilateral triangle of side 12 cm as the centre and a sector of the circle of radius 6 cm with centre B is made. [ Use $\sqrt {3}=1.73 \; and \; \pi = 3.14$]
Sol:
In equilateral triangle all the angles are of $60^{\circ}$
$Therefore, \angle ABO = \angle AOB = 60^{\circ}$
Area of the shaded region = ( Area of triangle – Area of sector having central angle $60^{\circ}$ + Area of sector having central angle ( $360^{\circ} – 60^{\circ}$ )
= $\frac{\sqrt{3}}{4}(AB)^{2} – \frac{60}{360} . \pi (6)^{2} + \frac{300}{360}. \pi (6)^{2}$
$= \frac{\sqrt{1.73}}{4}(12)^{2} – \frac{1}{6}\times 3.14 (6)^{2} + \frac{5}{6} \times 3.14 (6)^{2}$
= 62.28 – 18.84 + 94.2
= 137.64 $cm^{2}$
Hence, the area of shaded region is 137.64 $cm^{2}$.
Q.18: In the given figure, ABCD is a rectangle with AB = 80 cm and BC = 70 cm,$\angle AED = 90^{\circ}$ and DE = 42 cm. A semicircle is drawn, taking BC as diameter. Find the area of the shaded region.
Sol:
We know that the opposite sides of a rectangle are equal.
In right triangle AED
$AE^{2}= AD^{2}- DE^{2}$
$AE^{2}= 70^{2}- 12^{2}$
$= 4900 – 1764$ = 3136
$\Rightarrow AE= 56$
Area of the shaded region = Area of rectangle – (Area of triangle + Area of semicircle )
= $AB \times BC – \left [ \frac{1}{2} \times AE \times DE +\frac{1}{2} \pi \left ( \frac{BC}{2} \right )^{2}\right ]$
$= 80 \times 70 – \left [ \frac{1}{2} \times 56 \times 42 +\frac{1}{2} \times \frac{22}{7} \times \left ( \frac{70}{2} \right )^{2} \right ]$
= 5600 – 3101 = 2499 $cm^{2}$
Hence, the area of shaded region is 2499 $cm^{2}$
#### Practise This Question
The great Himalayan National Park contains,within its reserved area,alpine meadows which were grazed by sheep in summer.Nomadic sheperds drove their flock up from the valleys every summer.When this national park was formed,this practise was put to an end.Now it is seen that wothout the regular grazing by sheep the grass grows very tall and then falls over preventing fresh growth.
Statement:
From the above paragraph we can successfully claim that the prejudice against traditional use of forests has a valid strong bias. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7270956039428711, "perplexity": 631.546549426584}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578530040.33/warc/CC-MAIN-20190420200802-20190420222802-00377.warc.gz"} |
https://larissasantos.com/2015/11/03/quadrant-asymmetry-in-the-angular-distribution-of-the-cosmic-microwave-background-in-the-planck-satellite-data/ | # Quadrant asymmetry in the angular distribution of the Cosmic Microwave Background in the Planck satellite data
Some peculiar features found in the angular distribution of the cosmic microwave background (CMB) measured by the Wilkinson Microwave Anisotropy Probe (WMAP) deserve further investigation. Among these peculiar features, is the quadrant asymmetry, which is likely related to the north-south asymmetry. In this paper, we aim to extend the analysis of the quadrant asymmetry in the ΛCDM framework to the Planck foreground-cleaned maps, using the mask provided by Planck team. We compute the two-point correlation function (TPCF) from each quadrant of the Planck CMB sky maps, and compare the result with 1000 Monte Carlo (MC) simulations generated assuming the ΛCDM best-fit power spectrum. We detect the presence of an excess of power in the southeastern quadrant (SEQ) and a significant lack of power in the northeastern quadrant (NEQ) in the Planck data. Asymmetries between the SEQ and the other three quadrants (southwestern quadrant (SWQ), northwestern quadrant (NWQ), and NEQ) are each in disagreement with an isotropic sky at a 95\% confidence level. Furthermore, by rotating the Planck temperature sky map with respect to z direction, we showed the rotation angle where the TPCF of the SEQ has its maximal power. Our analysis of the Planck foreground-cleaned maps shows that there is an excess of power in the TPCF in the SEQ and a significant lack of power in the NEQ when compared with simulations. This asymmetry is anomalous when considering the ΛCDM framework .
Acesse agora: http://arxiv.org/abs/1311.0714
### Escrito por Larissa Santos
Nascida na capital federal, morei nos EUA, Itália e atualmente me encontro no país mais populoso do mundo. Isso mesmo, estou morando na China, mais precisamente na cidade de Hefei. Sinta-se a vontade para procurar essa pequena cidade (para os parâmetros chineses) de mais de 7 milhões de habitantes no mapa. Sou formada em Física pela Universidade de Brasília e PhD em astrofísica pela universidade de Roma. Sempre ávida por dividir conhecimento, não somente no campo das ciências, mas também as minhas experiências pessoais , preferências literárias, etc. O leque é muito grande, então por simplicidade, digamos que gosto de debater sobre tudo e sobre nada, sobre qualquer coisa der na telha. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.836468517780304, "perplexity": 11183.840619337565}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154085.58/warc/CC-MAIN-20210731074335-20210731104335-00151.warc.gz"} |
https://www.kirellbenzi.com/blog/exploring-the-star-wars-expanded-universe | Exploring the Star Wars expanded universe (part 1)
January 7, 2016
In this post, I will try to give some insights on the Star Wars expanded universe. All the data come from the Wikipedia for Star Wars: Wookieepedia.
I have actually started to work on this project 2 years ago. With the release of the new movie The Force Awakens, it is the perfect time to share some fun facts about Star Wars that I didn’t know before (believe me when I say I’m a big fan).
The universe is huge and can be overwhelming to newcomers, especially with the new movie which altered the previously canon timeline. What happened in the books after the Return of the Jedi is now considered as “Legends”. If you want to have a look at the complete timeline, you can find it here but beware it’s insane…
So you claim to have exclusive info, prove it!
All right, all right! I can tell you the number of persons in the universe (on the 25th of December 2015). It is hard to give a correct number because the Wiki is ever changing and some characters are pure trash. However, I extracted: 21,647 characters (yes more than 20k). Once we remove characters whose name start with “Unidentified”, we reach 19,612. Did I tell you that the expanded universe was huge?
What are the most dominant species in the universe?
Let’s continue our investigation, it is known that the universe is vast and composed of different species. But let’s be more precise and plot the repartition of the top-10 species. To get more info on each species you can search Wookieepedia, for instance here is a Twi’lek.
Humans represent 78% of the total population!
By the way, where did you get these infos?
This is a technical part, an uninterested reader may skip it.
To extract these infos I have built a web scraper (a robot) designed to extract data from every character of Wookieepedia. All the data is stored on the fly in a graph database.
However, there is no common index to reach every character page on the site. To be sure that I was actually scrapping all the characters, I had to first fetch all the categories indexing the characters such as Category:Individuals.
From this root category, I created a network (a forest to be precise) of all subcategories under this root. Then for each category, a node in the category network, the robot fetches all the associated unique characters.
The second part of the scraping consists of creating the graph of the characters. Remember that we already fetched our characters so we just need to create links between them. To do that, our robot reads each page and looks for occurrences of other characters.
To resume, our graph database contains all the categories, all the characters and the relations between them (categories and characters). A character A is linked to another one B if its name appears on A’s associated wiki page.
This part was the longest to code and run: scraping all the data takes 2 days to complete.
What are the most connected characters?
Thanks to our newly graph we can extract find the most connected characters of the universe. Here we simply count the total number of edges for each node of our graph. With no surprise, the most connected characters are well-known: 14/15 characters are in the movies.
If you don’t know Revan, check him out. He is a very famous character from the Old Republic Era. Quoting Wookieepedia:
Revan … was a Human male who played pivotal roles as both Jedi and Sith in the Mandalorian Wars and the Jedi Civil War.
Character distribution timeline
Not everyone is familiar with the Star Wars expanded universe eras. Did you know that the timeline spans over 36,000 years?
To see a bit clearer, let’s create a timeline chart showing the distribution of characters across these eras. Some of the characters have been discarded (no era info). The extra colors highlight characters living in different eras. For instance Darth Vader both lives on the Rise of the Empire and in the Rebellion era (here in green).
We see that the most popular eras are from the films: Rise of the Empire and the Rebellion era. The Old Republic era is also popular (despite having fewer people) thanks to the MMORPG (video game) Star Wars: The Old Republic.
Filling missing eras for some characters
Thanks to our graph of characters, we can use some graph theory algorithms to the rescue. More specifically, we can use a label propagation algorithm. This idea is simple: imagine that I am Darth Malgus, if most of my neighbors on the graph belong to the Old Republic Era there is a good chance that I also belong to the same era.
The algorithm will iteratively propagate “labels”, here the era value on each node, to neighbors on the graph. To set a label on an empty node, we count the number of occurrences of each label from its neighbors and take the label with the max count (crudely).
A picture is worth a thousand words so let’s visualize our label propagation algorithm. In the following, nodes painted in black will be painted according to their most probable color using their neighbors.
How can we be sure that new labels are actually correct?
Well, we cannot be sure if we really don’t know the labels. But to test our algorithm and tune its parameters, we can extract a known subset of the network and remove the labels to simulate missing values. Since we know the ground truth, we can compute an accuracy score and assert the performance of the algorithm. In this particular example, with$40\%$ of missing data the accuracy is $0.8$. It means that we know only $60\%$ of the values, $\frac{8}{10}$ labels are correct. If it doesn’t look much, let me tell you that it’s actually very good (for a simple algorithm).
What’s next? Star Wars faction graph
A lot can be done with this dataset especially when we harvest the power of graph theory. For instance, let’s visualize the graph of the principal factions in the Star Wars universe. The node size is proportional to the number of characters in the faction. The color of each node follows the standard Star Wars color scheme: reds for the dark side, blues for the light side. Criminal organizations are in yellow. The links between factions summarize interactions between characters (edges of the character graph).
Your eyes can deceive you. Don’t trust them (Obi-Wan Kenobi, ep. 4)
At some point, I have to conclude this post. But this is not over, trust me. If you don’t, well, I find your lack of faith disturbing… Want to have a peek at what comes next?
Let me tell you that the universe has 464 Jedi and Sith masters so far!
You can read more about the intricacies of the Star Wars Expanded Universe in my next article : Exploring the Star Wars Expanded Universe (Part 2) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3303861618041992, "perplexity": 1002.3598843774162}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946445.46/warc/CC-MAIN-20230326173112-20230326203112-00070.warc.gz"} |