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http://www.emtmadeeasy.com/	## Pages ### Antenna Theory - Solved Numerical's / Problems - ElectroMagnetic Theory... 1) An electric field strength of 10 µV/m is to be measured at an observation point θ = π/2, 500 km from a λ/4 monopole antenna operating in air at 50 MHz. • (a) What is the length of the dipole? • (b) Calculate the current that must be fed to the antenna. • (c) Find the average power radiated by the antenna. • (d) If a transmission line with Zo = 75 Ω is connected to the antenna, determine the standing wave ratio. --- For Solution CLICK HERE. 2) Calculate the directivity of • The Hertzian Monopole. 3) A certain antenna with an efficiency of 95% has maximum radiation intensity of 0.5 W/sr. Calculate its directivity when (a) The input power is 0.4 W (b) The radiated power is 0.3 W 4) Evaluate the directivity of an antenna with normalized radiation intensity 5) Determine the maximum effective area of a Hertzian dipole of length 10 cm operating at 10 MHz. If the antenna receives 3 µW of power, what is the power density of the incident wave? --- For Solution CLICK HERE. 6) An antenna in air radiates a total power of 100 kW so that maximum radiated electric field strength of 12 mV/m is measured 20 km from the antenna. Find its: (a) Directivity in dB (b) Maximum power gain if ηr = 98%. 7) A C-band radar with an antenna 1.8 m in radius transmits 60 kW at a frequency of 6000 MHz. If the minimum detectable power is 0.26 mW, for a target cross section of 5 m2, calculate the maximum range in nautical miles and the signal power density at half this range. Assume unity efficiency and that the effective area of the antenna is 70% of the actual area. --- For Solution CLICK HERE. 8) The magnetic vector potential at point P(r, θ, φ) due to a small antenna located at the origin is given by where r2 = x2 + y2 + z2. Find E(r, θ, φ, t) and H(r, θ, φ, t) at the far field. --- For Solution CLICK HERE. 9) A Hertzian dipole at the origin in free space has dl = 20 cm and I = 10 cos2π107t A, find \|Eθs\| at the distant point (100, 0, 0). --- For Solution CLICK HERE. 10) A 2-A source operating at 300 MHz feeds a Hertzian dipole of length 5 mm situated at the origin. Find Es and Hs at (10, 30°, 90°). --- For Solution CLICK HERE. 11) An antenna can be modeled as an electric dipole of length 5 m at 3 MHz. Find the radiation resistance of the antenna assuming a uniform current over its length. --- For Solution CLICK HERE. 12) A half-wave dipole fed by a 50 Ω transmission line, calculate the reflection coefficient and the standing wave ratio. --- For Solution CLICK HERE. 13) A 1-m-long car radio antenna operates in the AM frequency of 1.5 MHz. How much current is required to transmit 4 W of power? --- For Solution CLICK HERE. 14) An antenna located on the surface of a flat earth transmits an average power of 200 kW. Assuming that all the power is radiated uniformly over the surface of a hemisphere with the antenna at the center, calculate (a) The time-average Poynting vector at 50 km, and (b) The maximum electric field at that location. 15) A 100-turn loop antenna of radius 20 cm operating at 10 MHz in air is to give a 50 mV/m field strength at a distance 3 m from the loop. Determine (a) The current that must be fed to the antenna (b) The average power radiated by the antenna 16) Sketch the normalized E-field and H-field patterns for a. A half-wave dipole b. A quarter-wave monopole. 17) In free space, an antenna has a far zone field given by 18) At the far field, the electric field produced by antenna is Sketch the vertical pattern of the antenna. Your plot should include as many points as possible. 19) At the far field, an antenna produces Calculate the directive gain and the directivity of the antenna?	{"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": 0.9332233667373657, "perplexity": 1430.4949375038345}, "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-2015-14/segments/1427132827069.83/warc/CC-MAIN-20150323174707-00173-ip-10-168-14-71.ec2.internal.warc.gz"}
http://pixel-druid.com/primitive-element-theorem.html	§ Primitive element theorem • Let $E/k$ be a finite extension. We will characterize when a primitive element exists, and show that this will always happen for separable extensions. § Forward: Finitely many intermediate subfields implies primitive • If $k$ is a finite field, then $E$ is a finite extension and $E^\times$ is a cyclic group. The generator of $E^\times$ is the primitive element. • So suppose $k$ is an infinite field. Let $E/k$ have many intermediate fields. • Pick non-zero $\alpha, \beta \in E$. As $c$ varies in $k$, the extension $k(\alpha + c\beta)$ varies amongst the extensions of $E$. • Since $E$ only has finitely many extensions while $k$ is infinite, pigeonhole tells us that there are two $c_1 \neq c_2$ in $E$ such that $k(\alpha + c_1 \beta) = k(\alpha + c_2 \beta)$. • Define $L \equiv k(\alpha + c_1 \beta)$. We claim that $L = E$, which shows that $\alpha + c_1 \beta$ is a primitive element for $E$. • Since $k(\alpha + c_2 \beta) = k(\alpha + c_1 \beta) = L$, this implies that $\alpha + c_2 \beta \in L$. • Thus, we find that $\alpha + c_1 \beta \in L$ and $\alpha + c_2 \beta \in L$. Thus, $(c_1 - c_2) \beta in L$. Since $c_1, c_2 \in k$, we have $(c_1 - c_2)^{-1} \in K$, and thus $\beta \in L$, which implies $\alpha \in L$. • Thus $L = k(\alpha, \beta) = k(\alpha + c_1 \beta)$. • Done. Prove for more generators by recursion. § Backward: primitive implies finitely many intermediate subfields • Let $E = k(\alpha)$ be a simple field (field generated by a primitive element). We need to show that $E/k$ only has finitely many subfields. • Let $a_k(x) \in k[x]$ be the minimal polynomial for $\alpha$ in $k$. By definition, $a$ is irreducible. • For any intermediate field $k \subseteq F \subseteq E$, define $a_F(x) \in F[x]$ to be the minimal polynomial of $\alpha$ in $F$. • Since $a_k$ is also a member of $F[x]$ and $a_k, a_F$ share a common root $\alpha$ and $a_F$ is irreducible in $F$, this means that $a_F$ divides $a_k$. • Proof sketch that irreducible polynomial divides any polynomial it shares a root with (Also written in another blog post): The GCD $gcd(a_F, a_k) \in F[x]$ must be non constant since $a_F, a_k$ share a root). But the irreducible polynomial $a_F$cannot have a smaller polynomial ( $gcd(a_F, a_k)$) as divisor. Thus the GCD itself is the irreducible polynomial $a_F$. This implies that $a_F$ divides $a_k$since GCD must divide $a_k$. • Since $a_k$ is a polynomial, it only has finitely many divisors (upto rescaling, which does not give us new intermediate fields). • Thus, there are only finitely many intermediate fields if a field is primitive. § Interlude: finite extension with infinitely many subfields • Let $F = F_p(t, u)$ where $t, u$ are independent variables. This has finite degree since it has a vector space basis $\{ t^iu^j \}$. • Let $\alpha, \beta$ be roots of $x^p - t$ and $x^p - u$. Define $L \equiv F(\alpha, \beta)$. • Consider intermediate subfields $F_\lambda \equiv F(\alpha + \lambda \beta)$ for $\lambda \in F$. • Suppose $\lambda \neq \mu$ for two elements in $F$. We want to show that $F_\lambda \neq F_\mu$. This gives us infinitely many subfields as $F$ has infinitely many elements. TODO § Part 2: If $E/k$ is finite and separable then it has a primitive element • Let $K = F(\alpha, \beta)$ be separable for $\alpha, \beta \in K$. Then we will show that there exists a primitive element $\theta \in K$ such that $K = F(\theta)$. • By repeated application, this shows that for any number of generators $K = F(\alpha_1, \dots, \alpha_n)$, we can find a primitive element. • If $K$ is a finite field, then the generator of the cyclic group $K^\times$ is a primitive element. • So from now on, suppose $K$ is infinite, and $K = F(\alpha, \beta)$ for $\alpha, \beta \in F$. • Let $g$ be the minimal polynomial for $\alpha$, and $h$ the minimal polynomial for $\beta$. Since the field is separable, $g, h$ have unique roots. • Let the unique roots of $g$ be $\alpha_i$ such that $\alpha = \alpha_1$, and similarly let the unique roots of $h$ be $\beta_i$ such that $\beta = beta_1$. • Now consider the equations $\alpha_1 + f_{i, j} \beta_1 = \alpha_i + f_{i, j} \beta_j$ for $i \in [1, deg(g)]$ and $j \in [1, deg(h)]$. • Rearranging, we get $(\alpha_1 - \alpha_j) = f_{i, j} (\beta_j - \beta_1)$. Since $\beta_j \neq \beta_1$ and $\alpha_1 \neq \alpha_j$, this shows that there is a unique $f_{i, j} \equiv (\alpha_1 - \alpha_j)/(\beta_j - \beta_1)$ that solves the above equation. • Since the extension $F$ is infinite, we can pick a $f_$ which avoids the finite number of $f_{i, j}$. • Thus, once we choose such an $f_$, let $\theta \equiv a_1 + f b_1$. Such a $\theta$ can never be equal to $\alpha_i + f \beta_j$ for any $f$, since the only choices of $f$that make $\alpha_1 + f \beta_1 = \alpha_i + f \beta_j$ true are the $f_{i, j}$, and $f_$ was chosen to be different from these! • Now let $F_\theta \equiv F(\theta)$. Since $\theta \in K$, $E$ is a subfield of $K$. • See that $K = F(\alpha, \beta) = F(\alpha, \beta, \alpha + f \beta) = F(\beta, \alpha + f \beta) = F(\theta, \beta) = F_\theta(\beta)$. • We will prove that $K = F_\theta$. • Let $p(x)$ denote the minimal polynomial for $\beta$ over $F_\theta$. Since $K = F_\theta(\beta)$, if $p(x)$ is trivial, the $K = F_\theta$. • By definition, $\beta$ is a root of $h(x)$. Since $p(x)$ is an irreducible over $F_\theta$, we have that $p(x)$ divides $h(x)$[proof sketch: irreducible polynomial $p(x)$ shares a root with $h(x)$. Thus, $gcd(p(x), h(x))$ must be linear or higher. Since $gcd$ divides $p(x)$, we must have $gcd = p(x)$ as $p(x)$ is irreducible and cannot have divisors. Thus, $p(x)$, being the GCD, also divides $h(x)$]. • Thus, the roots of $p(x)$ must be a subset of the roots $\{ \beta_j \}$ of $h(x)$. • Consider the polynomial $k(x) = g(\theta - f_ \cdot x)$. $\beta$ is also a root of the polynomial $k(x)$, since $k(\beta) = g(\theta - f_* \beta)$, which is equal to $g((\alpha + f_* \beta) - f_* \beta) = g(\alpha) = 0$. [since $\alpha$ is a root of $g$]. • Thus, we must have $p(x)$ divides $k(x)$. • We will show that $\beta_j$ is not a root of $k(x)$ for $j \neq 2$. $k(\beta_j) = 0$ implies $g(\theta - f_* \beta_j) = 0$, which implies $\theta - f_* \beta_j = \alpha_i$since the roots of $g$ are $\alpha_i$. But then we would have $\theta = \alpha_i + f_* \beta_j$, a contradiction as $\theta$ was chosen precisely to avoid this case! • Thus, every root of $p(x)$ must come from $\{ \beta_j \}$. Also, the roots of $p(x)$ must come from the roots of $k(x)$. But $k(x)$ only shares the root $\beta_1$with the set of roots $\beta_2, \dots, \beta_j$. Also, $p(x)$ does not have multiple roots since it is separable. Thus, $p(x)$ is linear, and the degree of the field extension is 1. Therefore, $K = E = F(\theta)$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 171, "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": 0.9990069270133972, "perplexity": 110.4981268709653}, "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-2022-21/segments/1652662584398.89/warc/CC-MAIN-20220525085552-20220525115552-00326.warc.gz"}
http://www.ck12.org/book/CK-12-Chemistry---Second-Edition/r13/section/18.2/	<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" /> # 18.2: Collision Theory Difficulty Level: At Grade Created by: CK-12 ## Lesson Objectives The student will: • define the collision theory. • describe the conditions for successful collisions. • explain how the kinetic molecular theory applies to the collision theory. • describe the rate in terms of the conditions of successful collisions. ## Vocabulary • activated complex • activation energy • collision frequency • collision theory • kinetic molecular theory • threshold energy ## Introduction Consider the chemical reaction CH4+2 O2CO2+2 H2O\begin{align}\mathrm{CH}_4 + 2 \ \mathrm{O}_2 \rightarrow \mathrm{CO}_2 + 2 \ \mathrm{H}_2\mathrm{O}\end{align}. In the reactants, the carbon atoms are bonded to hydrogen atoms, and the oxygen atoms are bonded to other oxygen atoms. Each atom in the reactants is bonded to its full capacity and cannot form any more bonds. In the products, the carbon atoms are bonded to oxygen atoms, and the hydrogen atoms are bonded to oxygen atoms. The bonds that are present in the products cannot form unless the bonds in the reactants are first broken, which requires an input of energy. The energy to break the old bonds comes from the kinetic energy of the reactant particles. The reactant particles are moving around at random with an average kinetic energy related to the temperature. If a reaction is to occur, the kinetic energy of the reactants must be high enough that when the reactant particles collide, the collision is forceful enough to break the old bonds. Once the old bonds are broken, the atoms in the reactants would be available to form new bonds. At that point, the new bonds of the products could be formed. When the new bonds are formed, potential energy is released. The potential energy that is released becomes kinetic energy that is absorbed by the surroundings (primarily the products, the solvent solution if there is one, and the reaction vessel). Chemists have chosen to give a name to the group of particles that exist for the split second just after the reactant bonds have been broken and just before the product bonds form. This group of un-bonded particles is called the activated complex. The activated part comes from the fact that these atoms are ready to form bonds, and the complex part comes from the fact that the group of particles is a jumble of particles from all the reactant molecules. A successful collision would proceed as follows: Reactantsinput\ of\ energyactivated\ complexoutput\ of\ energyproducts\begin{align}\text{Reactants} \rightarrow \text{input\ of\ energy} \rightarrow \text{activated\ complex} \rightarrow \text{output\ of\ energy} \rightarrow \text{products}\end{align} The reactants, the activated complex, and the products all have a precise amount of potential energy in their bonds. The potential energy of the activated complex is called the threshold energy. This threshold energy is the minimum potential energy that must be reached in order for a reaction to occur. The input of energy that is necessary to raise the potential energy of the reactants to this threshold energy is called the activation energy. The activation energy must be provided from the kinetic energy of the reactant particles during the collision. In those cases where the reactants do not collide with enough energy to break the old bonds, the reactant particles will simply bounce off each other and remain reactant particles. ## How Reactions Occur We know that a chemical system can be made up of atoms (H2, N2, K)\begin{align}(\mathrm{H}_2, \ \mathrm{N}_2, \ \mathrm{K})\end{align}, ions (NO3, Cl, Na+\begin{align}(\mathrm{NO}_3^-, \ \mathrm{Cl}^-, \ \mathrm{Na}^+\end{align}), or molecules (H2O, CH3CH3, C12H22O11)\begin{align}(\mathrm{H}_2\mathrm{O}, \ \mathrm{CH}_3\mathrm{CH}_3, \ \mathrm{C}_{12}\mathrm{H}_{22}\mathrm{O}_{11})\end{align}. We also know that in a chemical system, these particles are moving around at random. The collision theory explains why reactions occur between these atoms, ions, and/or molecules at the particle level. The collision theory provides us with the ability to predict what conditions are necessary for a successful reaction to take place. These conditions include: 1. the particles must collide with each other 2. the particles must have proper orientation 3. the particles must collide with sufficient energy to break the old bonds A chemical reaction involves breaking bonds in the reactants, re-arranging the atoms into new groupings (the products), and the formation of new bonds in the products. Therefore, not only must a collision occur between reactant particles, but the collision has to have sufficient energy to break all the reactant bonds that need to be broken in order to form the products. Some collision geometries need less collision energy than others, and the optimal collision geometry requires the smallest amount of particle kinetic energy for the reaction to occur. If the reactant particles collide with less than the activation energy, the particles will rebound (bounce off each other), and no reaction will occur. ## The Kinetic Molecular Theory The kinetic molecular theory provides the foundation for the collision theory. Part of the kinetic molecular theory maintains that the collision between particles are “perfectly elastic.” The term “perfectly elastic” is a term from physics meaning that kinetic energy in conserved in the collision. That is, if no bonds are broken, the colliding particles simply rebound, and the total kinetic energy before and after the collision is exactly the same. The kinetic molecular theory states that gas molecules consist of particles that are moving in random motion. This random motion is always in a straight line, and the particles only deviate when there is a collision with the walls of a container or with another particle. The only collisions of any consequence, however, are those between other particles. In the chapter on kinetic-molecular theory, it was discussed that the particles in a sample of material are not all at exactly the same temperature. The particles of the substance actually have a distribution of kinetic energies, and the temperature of the substance is an expression of the average kinetic energy. As a result, some of the particles have more than the average kinetic energy and some have less. Therefore, some of the reactant particles will have sufficient kinetic energy to react, and some of the reactant particles will not. In a slow reaction, the majority of molecules do not have the minimum amount of energy necessary for a reaction to take place. In the figure below, the graph illustrates the number of molecules in the system versus the kinetic energy of these molecules. The area under the curve represents the total number of particles. The area shaded in red shows the number of molecules that do have sufficient energy for a successful collision. If the temperature is increased, the average kinetic energy of the particles increases, and the number of molecules with sufficient kinetic energy for a successful collision will also increase. The figure below shows the changes due to the increased temperature. At the higher temperature (T2)\begin{align}(\mathrm{T}_2)\end{align}, the number of molecules with energy greater than the activation energy increases. Therefore, the number of molecules with enough kinetic energy to have successful collisions increases with increasing average kinetic energy. ## Reactions May Occur When Particles Collide Looking back at the three conditions introduced in the first section, consider the following reaction: F2+NO2FNO2+F\begin{align}\mathrm{F}_2 + \mathrm{NO}_2 \rightarrow \mathrm{FNO}_2 + \mathrm{F}\end{align} If there is not enough energy, the particles will simply rebound off each other and bonds will not be broken, as illustrated below. The original reactants will remain. In order to have a successful collision, the particles must collide with enough energy and with the correct geometry to break the F2\begin{align}\mathrm{F}_2\end{align} and NO2\begin{align}\mathrm{NO}_2\end{align} bonds and form the FNO2\begin{align}\mathrm{FNO}_2\end{align} and F\begin{align}\mathrm{F}\end{align} products, as seen below. The F\begin{align}\mathrm{F}\end{align} would then further react with another element as it is not normally found un-reacted as just F\begin{align}\mathrm{F}\end{align}. ## Rate of Reaction Dependent On Various Factors As stated earlier, there are three conditions that must occur in order for a successful collision to occur. First, the reactant particles must collide. The total number of collisions per second is known as the collision frequency, regardless of whether these collisions are successful or not. The collision frequency depends on the concentration of the particles in the container, the temperature of the reaction, and the size of the particles themselves. Second, the particles must collide with the proper orientation. Third, the particles must collide with sufficient energy. From this knowledge, we can conclude that the rate of the reaction depends on the fraction of molecules that have enough energy and that collide with the proper orientation. The rate depends on the collision frequency itself. Putting this all together we get the following: Rate=collision\ frequency×collision\ energy×collision\ geometric\ orientation\begin{align}\text{Rate} = \text{collision\ frequency} \times \text{collision\ energy} \times \text{collision\ geometric\ orientation}\end{align} ## Lesson Summary • The collision theory explains why reactions occur between atoms, ions, and/or molecules and allows us to predict what conditions are necessary for a successful reaction to take place. • The kinetic molecular theory provides the foundation for the collision theory on the molecular level. • The minimum amount of energy necessary for a reaction to take place is known as the threshold energy. • With increasing temperature, the kinetic energy of the particles and the number of particles with energy greater than the activation energy increases. • The total number of collisions per second is known as the collision frequency, regardless of whether these collisions are successful or not. • Reaction rate=collision frequency × collision energy × collision geometric orientation.\begin{align}\mathrm{Reaction\ rate} = \mathrm{collision\ frequency}\ \times \ \mathrm{collision\ energy}\ \times \ \mathrm{collision\ geometric\ orientation.}\end{align} ## Review Questions 1. According to the collision theory, it is not enough for particles to collide in order to have a successful reaction to produce products. Explain 2. Due to the number of requirements for a successful collision, according to the collision theory, the percentage of successful collisions is extremely small. Yet, chemical reactions are still observed at room temperature and some at very reasonable rates. Explain. 3. What is a basic assumption of the kinetic molecular theory? 1. All particles will lose energy as the velocity increases 2. All particles will lose energy as the temperature increases 3. All particles will increase velocity as the temperature decreases 4. All particles are in random motion 4. According to the collision theory, which of the following must happen in order for a reaction to be successful: i. particles must collide, ii. particles must have proper geometric orientation, iii. particles must have collisions with enough energy? 1. i, ii 2. i, iii 3. ii, iii 4. i, ii, iii 5. What would happen in a collision between two particles if there was insufficient kinetic energy and improper geometric orientation? 1. The particles would rebound and there would be no reaction. 2. The particles would keep bouncing off each other until they eventually react, therefore the rate would be slow. 3. The particles would still collide but the byproducts would form. 4. The temperature of the reaction vessel would increase. 6. Illustrate the successful collision that would occur between the following: \begin{align}2 \ \mathrm{NO} + 2 \ \mathrm{H}_2 \rightarrow \mathrm{N}_2 + 2 \ \mathrm{H}_2\mathrm{O}\end{align}. ### Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes Show Hide Details Description Tags: Subjects:	{"extraction_info": {"found_math": true, "script_math_tex": 16, "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": 0.8303160667419434, "perplexity": 615.8684869476426}, "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-2017-04/segments/1484560280410.21/warc/CC-MAIN-20170116095120-00549-ip-10-171-10-70.ec2.internal.warc.gz"}
https://www.ideals.illinois.edu/browse?value=Faber%2C+Richard+George&type=author	Browse by Author "Faber, Richard George" • (1995) We prove linear and non-linear lifting theorems for locally convex subspaces of $L\sb0,$ and we give a characterization for locally bounded subspaces of $L\sb0.$ For every closed locally convex subspace E of $L\sb0$ and ... application/pdf PDF (1Mb)	{"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": 0.8954411149024963, "perplexity": 1675.1506971558117}, "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-2015-18/segments/1429246643088.10/warc/CC-MAIN-20150417045723-00023-ip-10-235-10-82.ec2.internal.warc.gz"}
http://lt-jds.jinr.ru/search?f=author&p=Shmatov%2C%20S.V.&ln=en	JINR Document Server 36 records found 1 - 15 jump to record: Search took 0.01 seconds. 1 CMS experiment at the LHC: Results and outlook / Golutvin, I.A. ; Shmatov, S.V. e-proceeding Detailed record 2 Physics of the standard model with the CMS experiment at the LHC / Shmatov, S.V. e-proceeding Detailed record 3 JINR Tier-1 centre for the CMS experiment at LHC / Astakhov, N.S. ; Baginyan, A.S. ; Belov, S.D. ; Dolbilov, A.G. ; Golunov, A.O. ; Gorbunov, I.N. ; Gromova, N.I. ; Kadochnikov, I.S. ; Kashunin, I.A. ; Korenkov, V.V. ; et al e-proceeding Detailed record 4 Cross Section for the Drell–Yan Process in Proton–Proton Collisions at the Large Hadron Collider (LHC) / Gavrilenko, M.G. ; Konoplyanikov, V.F. ; Savina, M.V. ; Shulga, S.G. ; Shmatov, S.V. e-proceeding Detailed record 5 Investigation of Drell–Yan processes in the CMS at the LHC / Gorbunov, I.N. ; Shmatov, S.V. e-proceeding Detailed record 6 Overview of physics results from the CMS experiment at the LHC / Shmatov, S.V. e-proceeding Detailed record 7 Uncertainties of Drell-Yan production cross section in pp collisions at the LHC / Konoplyanikov, V.F. ; Savina, M.V. ; Shmatov, S.V. ; Shulga, S.G. e-proceeding Detailed record 8 Measurement of the forward-backward asymmetry of $\mu^+ \mu^-$ pairs in CMS / Gorbunov, I.N. ; Shmatov, S.V. e-proceeding Detailed record 9 Measurement of forward-backward asymmetry $A_{FB}$ and of the weak mixing angle in processes of dilepton production in proton-proton collisions at $\sqrt{s}$ = 7 TeV in the CMS experiment at the LHC / Gorbunov, I.N. ; Shmatov, S.V. e-proceeding Detailed record 10 Searches for physics beyond the standard model in proton-proton interactions at $\sqrt{s}$ = 7 TeV in the CMS experiment at the LHC / Shmatov, S.V. e-proceeding Detailed record 11 CMS remote center at JINR / Golunov, A.O. ; Gorbunov, N.V. ; Korenkov, V.V. ; Shmatov, S.V. ; Zarubin, A.V. e-proceeding Detailed record 12 Systematic shifts in jet energy scale from the process in CMS detector / Altsybeev, I.G. ; Zarubin, A.V. ; Konoplyanikov, V.F. ; Tumasyan, A.R. ; Shmatov, S.V. e-proceeding Detailed record 13 Searches for extra dimensions in the CMS experiment at the Large Hadron Collider (LHC) / Shmatov, S.V. e-proceeding 14 Measurement of space resolution of endcap hadronic calorimeter CMS using beam testing of CMS HCAL prototype in 2003 / Golutvin, I.A. ; Zarubin, A.V. ; Konoplyanikov, V.F. ; Moisenz, P.V. ; Shmatov, S.V. e-proceeding 15 Setting the jet energy scale in the CMS calorimeter using events with direct photons / Golutvin, I.A. ; Zarubin, A.V. ; Kodolova, O.L. ; Konoplyanikov, V.F. ; Ulyanov, A.L. ; Shmatov, S.V. e-proceeding JINR Document Server : 36 records found 1 - 15 jump to record: 2 Shmatov, S V 41 Shmatov, S. 3 Shmatov, Sergei 629 Shmatov, Sergey Interested in being notified about new results for this query? Set up a personal email alert or subscribe to the RSS feed. Haven't found what you were looking for? Try your search on other servers: Shmatov, S.V. in Amazon Shmatov, S.V. in CERN EDMS Shmatov, S.V. in CERN Indico Shmatov, S.V. in CERN Intranet Shmatov, S.V. in CiteSeer	{"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": 0.9336666464805603, "perplexity": 27518.850235760037}, "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/1555578636101.74/warc/CC-MAIN-20190424074540-20190424100540-00334.warc.gz"}
https://en.wanweibaike.com/wiki-Donald_Rumsfeld	# Donald Rumsfeld Donald Rumsfeld Official portrait, 2001 13th and 21st United States Secretary of Defense In office January 20, 2001 – December 18, 2006[1] PresidentGeorge W. Bush Deputy Preceded byWilliam Cohen Succeeded byRobert Gates In office November 20, 1975 – January 20, 1977 PresidentGerald Ford DeputyBill Clements Preceded byJames Schlesinger Succeeded byHarold Brown 6th White House Chief of Staff In office September 21, 1974 – November 20, 1975 PresidentGerald Ford Preceded byAlexander Haig Succeeded byDick Cheney 9th United States Ambassador to NATO In office February 2, 1973 – September 21, 1974 PresidentRichard Nixon Gerald Ford Preceded byDavid Kennedy Succeeded byDavid Bruce Director of the Cost of Living Council In office October 15, 1971 – February 2, 1973 Preceded byPosition established Succeeded byPosition abolished Counselor to the President In office December 11, 1970 – October 15, 1971 Served with Robert Finch PresidentRichard Nixon Preceded byBryce Harlow Pat Moynihan Succeeded byRobert Finch 3rd Director of the Office of Economic Opportunity In office May 27, 1969 – December 11, 1970 PresidentRichard Nixon Preceded byBertrand Harding Succeeded byFrank Carlucci Member of the U.S. House of Representatives from Illinois's 13th district In office January 3, 1963 – March 20, 1969 Preceded byMarguerite Church Succeeded byPhil Crane Personal details Born Donald Henry Rumsfeld July 9, 1932 (age 88) Chicago, Illinois, U.S. Political partyRepublican Spouse(s) Joyce Pierson (m. after 1954) Children3 EducationPrinceton University (AB) Georgetown University Law Center Signature WebsiteLibrary website Military service Nickname(s)"Rummy" Allegiance United States Branch/service United States Navy Years of service1954–1957 (active) 1957–1975 (Reserve) Rank Captain Donald Henry Rumsfeld (born July 9, 1932) is a retired American politician. Rumsfeld served as Secretary of Defense from 1975 to 1977 under Gerald Ford, and again from January 2001 to December 2006 under George W. Bush.[2] He is both the youngest and the second-oldest person to have served as Secretary of Defense. Additionally, Rumsfeld was a three-term U.S. Congressman from Illinois (1963–69), director of the Office of Economic Opportunity (1969–70), counsellor to the president (1969–73), the United States Permanent Representative to NATO (1973–74), and White House Chief of Staff (1974–75). Between his terms as Secretary of Defense, he served as the CEO and chairman of several companies. Born in Illinois, Rumsfeld attended Princeton University, graduating in 1954 with a degree in political science. After serving in the Navy for three years, he mounted a campaign for Congress in Illinois's 13th Congressional District, winning in 1962 at the age of 30. While in Congress, he was a leading co-sponsor of the Freedom of Information Act. Rumsfeld accepted an appointment by President Richard Nixon to head the Office of Economic Opportunity in 1969; appointed counsellor by Nixon and entitled to Cabinet-level status, he would also head up the Economic Stabilization Program before being appointed ambassador to NATO. Called back to Washington in August 1974, Rumsfeld was appointed chief of staff by President Ford. Rumsfeld recruited a young one-time staffer of his, Dick Cheney, to succeed him when Ford nominated him to be Secretary of Defense in 1975. When Ford lost the 1976 election, Rumsfeld returned to private business and financial life, and was named president and CEO of the pharmaceutical corporation G. D. Searle & Company. He was later named CEO of General Instrument from 1990 to 1993 and chairman of Gilead Sciences from 1997 to 2001. Rumsfeld was appointed Secretary of Defense for a second time in January 2001 by President George W. Bush. As Secretary of Defense, Rumsfeld played a central role in the invasion of Afghanistan and invasion of Iraq. Before and during the Iraq War, he claimed that Iraq had an active weapons of mass destruction program; yet no stockpiles were ever found.[3][4] A Pentagon Inspector General report found that Rumsfeld's top policy aide "developed, produced, and then disseminated alternative intelligence assessments on the Iraq and al Qaida relationship, which included some conclusions that were inconsistent with the consensus of the Intelligence Community, to senior decision-makers".[5] Rumsfeld's tenure was controversial for its use of torture and the Abu Ghraib torture and prisoner abuse scandal.[6] Rumsfeld gradually lost political support and resigned in late 2006. In his retirement years, he published an autobiography Known and Unknown: A Memoir as well as Rumsfeld's Rules: Leadership Lessons in Business, Politics, War, and Life. ## Early life and education Rumsfeld's 1954 yearbook portrait from Princeton Donald Henry Rumsfeld was born on July 9, 1932, in Chicago, Illinois, the son of Jeannette Kearsley (née Husted) and George Donald Rumsfeld.[7] His father came from a German family that had emigrated in the 1870s from Weyhe in Lower Saxony,[8][9][10]:15–16 but young Donald was sometimes ribbed about looking like a "tough German".[10]:16 and 31 Growing up in Winnetka, Illinois, Rumsfeld became an Eagle Scout in 1949 and is the recipient of both the Distinguished Eagle Scout Award from the Boy Scouts of America[11] and its Silver Buffalo Award in 2006. Living in Winnetka, his family attended a Congregational church.[12] From 1943 to 1945, Rumsfeld lived in Coronado, California, while his father was stationed on an aircraft carrier in the Pacific in World War II.[13] He was a ranger at Philmont Scout Ranch in 1949.[14] Rumsfeld attended Baker Demonstration School,[15] and later graduated[16] from New Trier High School. He attended Princeton University on academic and NROTC partial scholarships. He graduated in 1954 with an A.B. in politics after completing a senior thesis titled "The Steel Seizure Case of 1952 and Its Effects on Presidential Powers".[17][18] During his time at Princeton, he was an accomplished amateur wrestler, becoming captain of the varsity wrestling team, and captain of the Lightweight Football team playing defensive back. While at Princeton he was friends with another future Secretary of Defense, Frank Carlucci. Rumsfeld married Joyce P. Pierson on December 27, 1954. They have three children, six grandchildren, and one great-grandchild. He attended Case Western Reserve University School of Law and Georgetown University Law Center, but did not take a degree from either institution. ### Naval service Lieutenant Donald Rumsfeld (right, standing) during his service within the United States Navy, in front of a Grumman S-2 Tracker aircraft in 1957. As a Command Pilot and Naval Aviator, Rumsfeld flew the Grumman S-2 Tracker during his service within the United States Navy. Rumsfeld served in the United States Navy from 1954 to 1957, as a naval aviator and flight instructor. His initial training was in the North American SNJ Texan basic trainer after which he transitioned to the T-28 advanced trainer. In 1957, he transferred to the Naval Reserve and continued his naval service in flying and administrative assignments as a drilling reservist. On July 1, 1958, he was assigned to Anti-submarine Squadron 662 at Naval Air Station Anacostia, District of Columbia, as a selective reservist.[19] Rumsfeld was designated aircraft commander of Anti-submarine Squadron 731 on October 1, 1960, at Naval Air Station Grosse Ile, Michigan, where he flew the S2F Tracker.[19] He transferred to the Individual Ready Reserve when he became Secretary of Defense in 1975 and retired with the rank of captain in 1989.[20] ## Career in government (1962–1977) ### Member of Congress Rumsfeld during his time in Congress In 1957, during the Dwight D. Eisenhower administration, Rumsfeld served as administrative assistant to David S. Dennison Jr., a Congressman representing the 11th district of Ohio. In 1959, he moved on to become a staff assistant to Congressman Robert P. Griffin of Michigan.[21] Engaging in a two-year stint with an investment banking firm, A. G. Becker & Co., from 1960 to 1962,[22] Rumsfeld would instead set his sights on becoming a member of Congress. He was elected to the United States House of Representatives for Illinois's 13th congressional district in 1962, at the age of 30, and was re-elected by large majorities in 1964, 1966, and 1968.[23] While in Congress, he served on the Joint Economic Committee, the Committee on Science and Aeronautics, and the Government Operations Committee, as well as on the Subcommittees on Military and Foreign Operations. He was also a co-founder of the Japanese-American Inter-Parliamentary Council[24] in addition to being a leading cosponsor of the Freedom of Information Act.[25] In 1965, following the defeat of Barry Goldwater by Lyndon B. Johnson in the 1964 presidential election, which also led to the Republicans losing many seats in the House of Representatives, Rumsfeld proposed new leadership for the Republicans in the House, suggesting that representative Gerald Ford from Michigan's 5th congressional district was the most suited candidate to replace Charles A. Halleck as Republican leader.[26] Rumsfeld, along with other members of the Republican caucus, then urged Gerald Ford to run for Republican leader. Ford eventually defeated Halleck and became House Minority Leader in 1965. The group of Republicans that encouraged Ford to run for the Republican leadership was later known as the "Young Turks". Rumsfeld would later serve during Ford's presidency as his chief of staff in 1974, and was chosen by Ford to succeed James Schlesinger as United States Secretary of Defense in 1975.[26] Congressman Donald Rumsfeld in 1967 During Rumsfeld's tenure as member of the U.S. House of Representatives, he voiced concerns about U.S. ability in the Vietnam War, saying that President Johnson and his national security team was too overconfident with how the war was being conducted. On one occasion Rumsfeld joined with other members of the House and traveled to Vietnam for a fact-finding mission to see for themselves how the war was going. The trip led to Rumsfeld believing that the South Vietnamese government was much too dependent on the United States. Rumsfeld was also unsatisfied when he received a briefing about war planning from the commander of the U.S. troops in Vietnam, General William Westmoreland.[26] The trip led Rumsfeld to cosponsor a resolution to bring the conduct of the war to the House floor for further debate and discussion about the mismanagement which ultimately would decide the fate of the war. However under constant pressure from the Johnson administration, the Democrats, who at that time held the majority at the House of Representatives, blocked the resolution from consideration.[26] As a young Congressman, Rumsfeld attended seminars at the University of Chicago, an experience he credits with introducing him to the idea of an all volunteer military, and to the economist Milton Friedman and the Chicago School of Economics.[27] He would later take part in Friedman's PBS series Free to Choose.[28] During his tenure in the House, Rumsfeld voted in favor of the Civil Rights Acts of 1964 and 1968,[29][30] and the Voting Rights Act of 1965.[31] Rumsfeld with his son, Nick, in the Oval Office with President Nixon, 1973 Chief of Staff Rumsfeld (left) and Deputy-Chief of Staff Dick Cheney (right) meet with President Ford, April 1975 Rumsfeld resigned from Congress in 1969 – his fourth term – to serve President Richard Nixon in his administration, and would serve in a variety of executive branch positions throughout the Nixon presidency. In 1969, Nixon sought to reform and reorganize the United States Office of Economic Opportunity, an organization created during the Kennedy administration and greatly expanded as a part of Lyndon Johnson's Great Society programs, rather than eliminate it outright. He appointed Rumsfeld director of the organization with Cabinet rank.[32] Rumsfeld had voted against the creation of OEO when he was in Congress, and initially rejected Nixon's offer, citing his own inherent belief that the OEO did more harm than good, and he felt that he was not the right person for the job.[33]:119–121 He only accepted after personal pleas from the president. As director, Rumsfeld sought to reorganize the Office to serve as "a laboratory for experimental programs".[33]:125 Several beneficial anti-poverty programs were saved by allocating funds to them from other less-successful government programs. During this time, he hired Frank Carlucci and Dick Cheney to serve under him. He was the subject of one of legendary writer Jack Anderson's columns, alleging that "anti-poverty czar" Rumsfeld had cut programs to aid the poor while spending thousands to redecorate his office. Rumsfeld dictated a four-page response to Anderson, labeling the accusations as falsehoods, and invited Anderson to tour his office. Despite the tour, Anderson did not retract his claims, and would only much later admit that his column was a mistake.[33]:125 When he left OEO in December 1970, Nixon named Rumsfeld Counselor to the President, a general advisory position; in this role, he retained Cabinet status.[10]:75 He was given an office in the West Wing in 1969 and regularly interacted with the Nixon administration hierarchy. He was named director of the Economic Stabilization Program in 1970 as well, and later headed up the Cost of Living Council. In March 1971 Nixon was recorded saying about Rumsfeld "at least Rummy is tough enough" and "He's a ruthless little bastard. You can be sure of that."[34][35][36][37][38] In February 1973, Rumsfeld left Washington to serve as U.S. Ambassador to the North Atlantic Treaty Organization (NATO) in Brussels, Belgium. He served as the United States' Permanent Representative to the North Atlantic Council and the Defense Planning Committee, and the Nuclear Planning Group. In this capacity, he represented the United States in wide-ranging military and diplomatic matters, and was asked to help mediate a conflict on behalf of the United States between Cyprus and Turkey.[33]:157 U.S. Secretary of Defense Donald Rumsfeld following a test flight on a brand new Strategic Bomber aircraft Rockwell B-1 Lancer, April 1976. In August 1974, after Nixon resigned as president in the aftermath of the Watergate scandal, Rumsfeld was called back to Washington to serve as transition chairman for the new president, Gerald Ford. He had been Ford's confidant since their days in the House, before Ford was House minority leader. As the new president became settled in, Ford appointed Rumsfeld White House Chief of Staff, where he served from 1974 to 1975. ### Secretary of Defense (1975–1977) Secretary of Defense Rumsfeld shares a laugh with President Ford in a Cabinet meeting, 1975 In October 1975, Ford reshuffled his cabinet in the Halloween Massacre. He named Rumsfeld to become the 13th U.S. Secretary of Defense; George H. W. Bush became Director of Central Intelligence. According to Bob Woodward's 2002 book Bush at War, a rivalry developed between the two men and "Bush senior was convinced that Rumsfeld was pushing him out to the CIA to end his political career."[39] At the Pentagon, Rumsfeld oversaw the transition to an all-volunteer military. He sought to reverse the gradual decline in the defense budget and to build up U.S. strategic and conventional forces, undermining Secretary of State Henry Kissinger at the SALT talks.[40] He asserted, along with Team B (which he helped to set up),[41] that trends in comparative U.S.-Soviet military strength had not favored the United States for 15 to 20 years and that, if continued, they "would have the effect of injecting a fundamental instability in the world".[20] For this reason, he oversaw the development of cruise missiles, the B-1 bomber, and a major naval shipbuilding program.[40] In 1977, Rumsfeld was awarded the nation's highest civilian award, the Presidential Medal of Freedom.[1] Kissinger, his bureaucratic adversary, would later pay him a different sort of compliment, pronouncing him "a special Washington phenomenon: the skilled full-time politician-bureaucrat in whom ambition, ability, and substance fuse seamlessly".[42] In early 1977 Rumsfeld briefly lectured at Princeton's Woodrow Wilson School and Northwestern's Kellogg School of Management. His sights instead turned to business, and from 1977 to 1985 Rumsfeld served as chief executive officer, president, and then chairman of G. D. Searle & Company, a worldwide pharmaceutical company based in Skokie, Illinois. During his tenure at Searle, Rumsfeld led the company's financial turnaround, thereby earning awards as the Outstanding Chief Executive Officer in the Pharmaceutical Industry from the Wall Street Transcript (1980) and Financial World (1981). In 1985, Searle was sold to Monsanto Company. Rumsfeld served as chairman and chief executive officer of General Instrument Corporation from 1990 to 1993. A leader in broadband transmission, distribution, and access control technologies for cable, satellite and terrestrial broadcasting applications, the company pioneered the development of the first all-digital high-definition television (HDTV) technology. After taking the company public and returning it to profitability, Rumsfeld returned to private business in late 1993. From January 1997 until being sworn in as the 21st Secretary of Defense in January 2001, Rumsfeld served as chairman of Gilead Sciences, Inc. Gilead is the developer of Tamiflu (Oseltamivir), which is used in the treatment of bird flu.[43] As a result, Rumsfeld's holdings in the company grew significantly when avian flu became a subject of popular anxiety during his later term as Secretary of Defense. Following standard practice, Rumsfeld recused himself from any decisions involving Gilead, and he directed the Pentagon's General Counsel to issue instructions outlining what he could and could not be involved in if there were an avian flu pandemic and the Pentagon had to respond.[44][45] ### Part-time public service During his business career, Rumsfeld continued part-time public service in various posts. In November 1983, Rumsfeld was appointed Special Envoy to the Middle East by President Ronald Reagan, at a turbulent time in modern Middle Eastern history when Iraq was fighting Iran in the Iran–Iraq War. The United States wished for Iraq to win the conflict, and Rumsfeld was sent to the Middle East to serve as a mediator on behalf of the President. As President Reagan's Special Envoy to the Middle East, Rumsfeld met with Saddam Hussein during a visit to Baghdad in December 1983, during the Iran–Iraq War (see video here). When Rumsfeld visited Baghdad on December 20, 1983, he met Saddam Hussein at Saddam's palace and had a 90-minute discussion. They largely agreed on opposing Syria's occupation of Lebanon; preventing Syrian and Iranian expansion; and preventing arms sales to Iran. Rumsfeld suggested that if U.S.-Iraq relations could improve the U.S. might support a new oil pipeline across Jordan, which Iraq had opposed but was now willing to reconsider. Rumsfeld also informed Iraqi Deputy Prime Minister and Foreign Minister Tariq Aziz that "Our efforts to assist were inhibited by certain things that made it difficult for us ... citing the use of chemical weapons."[10]:159–60 Rumsfeld wrote in his memoir Known and Unknown that his meeting with Hussein "has been the subject of gossip, rumors, and crackpot conspiracy theories for more than a quarter of a century ... Supposedly I had been sent to see Saddam by President Reagan either to negotiate a secret oil deal, to help arm Iraq, or to make Iraq an American client state. The truth is that our encounter was more straightforward and less dramatic."[33]:6 In addition to taking the position of Middle East envoy, Rumsfeld served as a member of the President's General Advisory Committee on Arms Control (1982–1986); President Reagan's Special Envoy on the Law of the Sea Treaty (1982–1983); a senior adviser to President Reagan's Panel on Strategic Systems (1983–1984); a member of the Joint Advisory Commission on U.S./Japan Relations (1983–1984); a member of the National Commission on the Public Service (1987–1990); a member of the National Economic Commission (1988–1989); a member of the board of visitors of the National Defense University (1988–1992); a member of the FCC's High Definition Television Advisory Committee (1992–1993); a member of the U.S. Trade Deficit Review Commission (1999–2000); a member of the Council on Foreign Relations; and chairman of the U.S. Commission to Assess National Security Space Management and Organization (2000). Among his most noteworthy positions was chairman of the nine-member Commission to Assess the Ballistic Missile Threat to the United States from January to July 1998. In its findings, the commission concluded that Iraq, Iran, and North Korea could develop intercontinental ballistic missile capabilities in five to ten years and that U.S. intelligence would have little warning before such systems were deployed.[46] During the 1980s, Rumsfeld became a member of the National Academy of Public Administration, and was named a member of the boards of trustees of the Gerald R. Ford Foundation, the Eisenhower Exchange Fellowships, the Hoover Institution at Stanford University and the National Park Foundation. He was also a member of the U.S./Russia Business Forum and chairman of the Congressional Leadership's National Security Advisory Group.[47] Rumsfeld was a member of the Project for the New American Century, a think-tank dedicated to maintaining U.S. primacy. In addition, he was asked to serve the U.S. State Department as a foreign policy consultant from 1990 to 1993. He also sat on European engineering giant Asea Brown Boveri's board from 1990 to 2001, a company that sold two light-water nuclear reactors to the Korean Peninsula Energy Development Organization for installation in North Korea, as part of the 1994 agreed framework reached under President Bill Clinton. Rumsfeld's office said that he did not "recall it being brought before the board at any time" though Fortune magazine reported that "board members were informed about this project".[48] ### Presidential and vice-presidential aspirations During the 1976 Republican National Convention, Rumsfeld received one vote for Vice President of the United States, although he did not seek the office, and the nomination was easily won by Ford's choice, Senator Bob Dole.[49] During the 1980 Republican National Convention he again received one vote for vice president.[50] Economist Milton Friedman later noted that he, Friedman, regarded Reagan's pick of Bush as "the worst decision not only of his campaign but of his presidency", and that Rumsfeld was instead his preference. "Had he been chosen," Friedman said, "I believe he would have succeeded Reagan as president and the sorry Bush-Clinton period would never have occurred."[51] Rumsfeld briefly sought the presidential nomination in 1988, but withdrew from the race before primaries began.[52] During the 1996 election season, he initially formed a presidential exploratory committee, but declined to formally enter the race. He was instead named national chairman for Republican nominee Bob Dole's campaign.[53] ## Secretary of Defense (2001–2006) Rumsfeld is administered the oath of office as the 21st Secretary of Defense on January 20, 2001 by Director of Administration and Management David O. Cooke (left), as Joyce Rumsfeld holds the Bible in a ceremony at the Eisenhower Executive Office Building Rumsfeld was named Secretary of Defense soon after President George W. Bush took office in 2001 despite Rumsfeld's past rivalry with the previous President Bush. Bush's first choice, FedEx founder Fred Smith, was unavailable and Vice President-elect Cheney recommended Rumsfeld for the job.[54] Rumsfeld's second tenure as Secretary of Defense cemented him as the most powerful Pentagon chief since Robert McNamara and one of the most influential Cabinet members in the Bush administration.[55] His tenure would prove to be a pivotal and rocky one that led the United States military into the 21st century. Following the September 11 attacks, Rumsfeld led the military planning and execution of the U.S. invasion of Afghanistan and the subsequent 2003 invasion of Iraq. He pushed hard to send as small a force as soon as possible to both conflicts, a concept codified as the Rumsfeld Doctrine.[56] Throughout his time as defense secretary, Rumsfeld was noted for his candor and quick wit when giving weekly press conferences or speaking with the press.[57] U.S. News & World Report called him "a straight-talking Midwesterner" who "routinely has the press corps doubled over in fits of laughter".[57] By the same token, his leadership was exposed to much criticism through provocative books covering the Iraq conflict, like Bob Woodward's State of Denial, Thomas E. Ricks' Fiasco, and Seymour Hersh's Chain of Command. ### September 11, 2001 attacks "The Pentagon is functioning" was the message Rumsfeld stressed during a press conference in the Pentagon briefing room barely eight hours after terrorists crashed a hijacked commercial jetliner into the Pentagon. Rumsfeld is flanked, left to right, by Secretary of the Army Tom White, Chairman of the Joint Chiefs of Staff General Hugh Shelton, and Senators John Warner (R-VA), and Carl Levin (D-MI), the Ranking Member and Chairman of the Senate Armed Services Committee. A day after Rumsfeld announced that the Department of Defense could not account for about $2.3 trillion worth of transactions,[58] al-Qaeda terrorists hijacked commercial airliners and crashed them in coordinated strikes into both towers of the World Trade Center in Lower Manhattan, New York City, and the Pentagon in Washington, D.C. The fourth plane crashed into a field in Shanksville, Pennsylvania, and its target was likely a prominent building in Washington, D.C., most probably either the Capitol Building or the White House.[59] Within three hours of the start of the first hijacking and two hours after American Airlines Flight 11 struck the World Trade Center, Rumsfeld raised the defense condition signaling of the United States offensive readiness to DEFCON 3, the highest it had been since the Arab-Israeli war in 1973.[60] Rumsfeld addressed the nation in a press conference at the Pentagon, just eight hours after the attacks and stated, "It's an indication that the United States government is functioning in the face of this terrible act against our country. I should add that the briefing here is taking place in the Pentagon. The Pentagon's functioning. It will be in business tomorrow."[61] ### Military decisions in the wake of 9/11 Rumsfeld and New York City Mayor Rudy Giuliani speak at the site of the World Trade Center attacks in Lower Manhattan on November 14, 2001 On the afternoon of September 11, Rumsfeld issued rapid orders to his aides to look for evidence of possible Iraqi involvement in regard to what had just occurred, according to notes taken by senior policy official Stephen Cambone. "Best info fast. Judge whether good enough hit S.H." – meaning Saddam Hussein – "at same time. Not only UBL" (Osama bin Laden), Cambone's notes quoted Rumsfeld as saying. "Need to move swiftly – Near term target needs – go massive – sweep it all up. Things related and not."[62][63] In the first emergency meeting of the National Security Council on the day of the attacks, Rumsfeld asked, "Why shouldn't we go against Iraq, not just al-Qaeda?" with his deputy Paul Wolfowitz adding that Iraq was a "brittle, oppressive regime that might break easily—it was doable," and, according to John Kampfner, "from that moment on, he and Wolfowitz used every available opportunity to press the case."[64] The idea was initially rejected at the behest of Secretary of State Colin Powell, but, according to Kampfner, "Undeterred Rumsfeld and Wolfowitz held secret meetings about opening up a second front—against Saddam. Powell was excluded." In such meetings they created a policy that would later be dubbed the Bush Doctrine, centering on "pre-emption" and the war on Iraq, which the PNAC had advocated in their earlier letters.[65] Richard Clarke, the White House counter-terrorism coordinator at the time, has revealed details of another National Security Council meeting the day after the attacks, during which officials considered the U.S. response. Already, he said, they were certain al-Qa'ida was to blame and there was no hint of Iraqi involvement. "Rumsfeld was saying we needed to bomb Iraq," according to Clarke. Clarke then stated, "We all said, 'No, no, al-Qa'ida is in Afghanistan.'" Clarke also revealed that Rumsfeld complained in the meeting, "there aren't any good targets in Afghanistan and there are lots of good targets in Iraq."[66] Rumsfeld wrote in Known and Unknown, "Much has been written about the Bush administration's focus on Iraq after 9/11. Commentators have suggested that it was strange or obsessive for the President and his advisers to have raised questions about whether Saddam Hussein was somehow behind the attack. I have never understood the controversy. I had no idea if Iraq was or was not involved, but it would have been irresponsible for any administration not to have asked the question."[33]:347 Excerpt from Donald Rumsfeld memo dated November 27, 2001[67] A memo written by Rumsfeld dated November 27, 2001 considers an Iraq war. One section of the memo questions "How start?", listing multiple possible justifications for a US-Iraq War.[67][68] ### War in Afghanistan Secretary of Defense Donald Rumsfeld with Chairman of the Joint Chiefs of Staff General Richard B. Myers, U.S. Air Force during a press conference at The Pentagon on January 24, 2002. Both Rumsfeld and General Myers played major role in a decision making for War on Terrorism following the September 11 Attacks. Rumsfeld routinely gave a briefing to the press regarding the progress of War on Terrorism. Secretary of Defense Donald Rumsfeld converse with U.S. Ambassador to Afghanistan Dr. Zalmay Khalilzad during a visit to Kandahar, Afghanistan on February 26, 2004, accompanied by Lieutenant General David Barno and Brigadier General Lloyd Austin whom later will also served as Secretary of Defense in 2021 during the administration of President Joe Biden. Rumsfeld directed the planning for the War in Afghanistan after the September 11 attacks.[56] On September 21, 2001, USCENTCOM Commander General Tommy Franks, briefed the President on a plan to destroy al Qaeda in Afghanistan and remove the Taliban government. General Franks, also initially proposed to Rumsfeld that the U.S. invade Afghanistan using a conventional force of 60,000 troops, preceded by six months of preparation. Rumsfeld, however feared that a conventional invasion of Afghanistan could bog down as had happened to the Soviets and the British.[69] Rumsfeld rejected Franks's plan, saying "I want men on the ground now!" Franks returned the next day with a plan utilizing U.S. Special Forces.[70][71] Despite air and missile attacks against al Qaeda in Afghanistan, USCENTCOM had no pre-existing plans for conducting ground operations there.[71] The September 21, 2001 plan emerged after extensive dialogue, but Secretary Rumsfeld also asked for broader plans that looked beyond Afghanistan.[72] On October 7, 2001, just hours after the 2001 invasion of Afghanistan was launched, Rumsfeld addressed the nation in a press conference at the Pentagon stating "While our raids today focus on the Taliban and the foreign terrorists in Afghanistan, our aim remains much broader. Our objective is to defeat those who use terrorism and those who house or support them. The world stands united in this effort".[73] Rumsfeld also stated "the only way to deal with these terrorist threats is to go at them where they exist. You cannot defend at every place at every time against every conceivable, imaginable, even unimaginable terrorist attack. And the only way to deal with it is to take the battle to where they are and to root them out and to starve them out by seeing that those countries and those organizations and those non-governmental organizations and those individuals that are supporting and harboring and facilitating these networks stop doing it and find that there's a penalty for doing it".[73] Rumsfeld in another press conference at the Pentagon on October 29, 2001 stated "As the first weeks of this effort proceed, it bears repeating that our goal is not to reduce or simply contain terrorist acts, but our goal is to deal with it comprehensively. And we do not intend to stop until we've rooted out terrorist networks and put them out of business, not just in the case of the Taliban and the Al Qaeda in Afghanistan, but other networks as well. And as I've mentioned, the Al Qaeda network crosses some 40, 50-plus countries."[74] Rumsfeld announced in November 2001, that he received "authoritative reports" that Al-Qaeda's number three Mohammed Atef, bin Laden's primary military chief and a planner of the September 11 attacks on America, was killed by a U.S. airstrike.[75][76][77] "He was very, very senior," Rumsfeld said. "We obviously have been seeking [him] out."[75] In a press conference at the Pentagon on November 19, 2001, Rumsfeld described the role of U.S. ground forces in Afghanistan as firstly In the north, American troops are "embedded in Northern Alliance" elements, helping arrange food and medical supplies and pinpointing airstrikes and In the south, commandos and other troops are operating more independently, raiding compounds, monitoring roadblocks and searching vehicles in the hope of developing more information about al-Qaeda and Taliban leaders.[77][75] On December 16, 2001, Rumsfeld visited U.S. troops in Afghanistan at Bagram Air Base.[78] On May 1, 2003, Rumsfeld during a visit to Afghanistan meeting with U.S. troops stationed in Kabul told the press "General Franks and I have been looking at the progress that's being made in this country and have concluded that we are at a point where we clearly have moved from major combat activity to a period of stability and stabilization and reconstruction and activities." "I should underline however, that there are still dangers, there are still pockets of resistance in certain parts of the country and General McNeal and General Franks and their, the cooperation they have with the President Karzai's government and leadership and Marshall Fayheems assistance. We will be continuing as a country to work with the Afghan government and the new Afghan National Army to see that the any areas where there is resistance to this government and to the coalition forces will be dealt with promptly and efficiently."[79] There was also controversy between the Pentagon and the CIA over who had the authority to fire Hellfire missiles from Predator drones.[80] Even though the drones were not ready for deployment until 2002,[80] Daniel Benjamin and Steven Simon have argued that "these quarrels kept the Predator from being used against al Qaeda ... One anonymous individual who was at the center of the action called this episode 'typical' and complained that 'Rumsfeld never missed an opportunity to fail to cooperate. The fact is, the Secretary of Defense is an obstacle. He has helped the terrorists.'[81] ### Iraq War Secretary of Defense Donald H. Rumsfeld and Chairman of the Joint Chiefs of Staff General Richard B. Myers inspecting the joint services honor guard during the opening ceremonies of the Joint Service Open House at Andrews Air Forces Base, Maryland, May 17, 2002. Secretary of Defense Donald Rumsfeld accompanied by Chairman of the Joint Chiefs of Staff General Richard B. Myers and joined by military representatives from 29 countries of the worldwide coalition on the war against terrorism, while speaking to the reporter outside The Pentagon on March 11, 2002. Secretary of Defense Donald H. Rumsfeld (left) and the Commander of U.S. Central Command General Tommy Franks, listen to a question at the close of a Pentagon press conference on March 5, 2003. Rumsfeld and Franks gave reporters an operational update and fielded questions on the possible conflict in Iraq. Before and during the Iraq War, Rumsfeld claimed that Iraq had an active weapons of mass destruction program; in particular during his famous phrase There are known knowns in a press conference at the Pentagon on February 12, 2002[82] no stockpiles were ever found.[3][4] Bush administration officials also claimed that there was an operational relationship between Al Qaeda and Saddam Hussein. A Pentagon Inspector General report found that Rumsfeld's top policy aide, Douglas J. Feith, "developed, produced, and then disseminated alternative intelligence assessments on the Iraq and al Qaida relationship, which included some conclusions that were inconsistent with the consensus of the Intelligence Community, to senior decision-makers".[5] The job of finding WMD and providing justification for the attack would fall to the intelligence services, but, according to Kampfner, "Rumsfeld and Wolfowitz believed that, while the established security services had a role, they were too bureaucratic and too traditional in their thinking." As a result, "they set up what came to be known as the 'cabal', a cell of eight or nine analysts in a new Office of Special Plans (OSP) based in the U.S. Defense Department." According to an unnamed Pentagon source quoted by Hersh, the OSP "was created in order to find evidence of what Wolfowitz and his boss, Defense Secretary Rumsfeld, believed to be true—that Saddam Hussein had close ties to Al Qaeda, and that Iraq had an enormous arsenal of chemical, biological, and possibly even nuclear weapons that threatened the region and, potentially, the United States".[65] On January 22, 2003, after the German and French governments voiced opposition to invading Iraq, Rumsfeld labeled these countries as part of "Old Europe", implying that countries that supported the war were part of a newer, modern Europe.[83] After the war in Afghanistan was launched, Rumsfeld participated in a meeting in regard to the review of the Department of Defense's Contingency Plan in the event of a war with Iraq. The plan, as it was then conceived, contemplated troop levels of up to 500,000, which Rumsfeld felt was far too many. Gordon and Trainor wrote: As [General] Newbold outlined the plan ... it was clear that Rumsfeld was growing increasingly irritated. For Rumsfeld, the plan required too many troops and supplies and took far too long to execute. It was, Rumsfeld declared, the "product of old thinking and the embodiment of everything that was wrong with the military".[84] Rumsfeld in a press conference at the Pentagon on February 27, 2003, told reporters after being asked a question that Army Chief of Staff General Eric Shinseki suggested it would take several hundred thousand troops on the ground to secure Iraq and provide stability. Is he wrong?. Rumsfeld replied "the idea that it would take several hundred thousand U.S. forces I think is far from the mark. The reality is that we already have a number of countries that have offered to participate with their forces in stabilization activities, in the event force has to be used."[85] Rumsfeld addressed the nation in a press conference at the Pentagon on March 20, 2003, just hours after the launch of the 2003 Invasion of Iraq, where he announced the first strike of the war to liberate Iraq and that "The days of the Saddam Hussein regime are numbered," and "We continue to feel there is no need for a broader conflict if the Iraqi leaders act to save themselves and act to prevent such a conflict."[86] Rumsfeld's role in directing the Iraq War included a plan that was the Shock and Awe campaign,[87] which resulted in a lightning invasion with 145,000 soldiers on the ground that took Baghdad in well under a month with very few American casualties.[87] Many government buildings, plus major museums, electrical generation infrastructure, and even oil equipment were looted and vandalized during the transition from the fall of Saddam Hussein's regime to the establishment of the Coalition Provisional Authority. A violent insurrection began shortly after the military operation started. On March 30, 2003, Rumsfeld in an interview with George Stephanopoulos on ABC's This Week program, answered a question by Stephanopoulos about finding weapons of mass destruction in Iraq, Rumsfeld stated "We know where they are. They're in the area around Tikrit and Baghdad and east, west, south and north somewhat."[88] On April 9, 2003, at a press conference at the Pentagon, Rumsfeld addressed reporters during the Fall of Baghdad, and stated "The scenes of free Iraqis celebrating in the streets, riding American tanks, tearing down the statues of Saddam Hussein in the center of Baghdad are breathtaking."[89] After the Iraq invasion, U.S. troops were criticized for not protecting the historical artifacts and treasures located at the National Museum of Iraq. On April 11, 2003, at a press conference at the Pentagon, when asked at the time why U.S. troops did not actively seek to stop the lawlessness, Rumsfeld replied, "Stuff happens ... and it's untidy and freedom's untidy, and free people are free to make mistakes and commit crimes and do bad things. They're also free to live their lives and do wonderful things. And that's what's going to happen here."[90] He further commented that, "The images you are seeing on television you are seeing over, and over, and over, and it's the same picture of some person walking out of some building with a vase, and you see it 20 times, and you think, "My goodness, were there that many vases?"[90] On July 24, 2003, at a press conference at the Pentagon, Rumsfeld commented on the release of photographs of the deceased Uday Hussein and Qusay Hussein. "It is not a practice that the United States engages in on a normal basis," Rumsfeld said. "I honestly believe that these two are particularly bad characters and that it's important for the Iraqi people to see them, to know they're gone, to know they're dead, and to know they're not coming back." Rumsfeld also said, "I feel it was the right decision, and I'm glad I made it."[91][92][93] In October 2003, Rumsfeld approved a secret Pentagon "roadmap" on public relations, calling for "boundaries" between information operations abroad and the news media at home. The Roadmap advances a policy according to which as long as the U.S. government does not intentionally target the American public, it does not matter that psychological operations reach the American public.[94] On December 14, 2003, Rumsfeld in an interview with journalist Lesley Stahl on 60 Minutes after U.S. forces captured Saddam Hussein in Operation Red Dawn, stated, "Here was a man who was photographed hundreds of times shooting off rifles and showing how tough he was, and in fact, he wasn't very tough, he was cowering in a hole in the ground, and had a pistol and didn't use it, and certainly did not put up any fight at all. I think that ... he resulted in the death of an awful lot of Iraqi people, In the last analysis, he seemed not terribly brave."[95] As Secretary of Defense, Rumsfeld was deliberate in crafting the public message from the Department of Defense. People will "rally" to the word "sacrifice", Rumsfeld noted after a meeting. "They are looking for leadership. Sacrifice = Victory." In May 2004, Rumsfeld considered whether to redefine the war on terrorism as a fight against "worldwide insurgency". He advised aides "to test what the results could be" if the war on terrorism were renamed.[96] Rumsfeld also ordered specific public Pentagon attacks on and responses to U.S. newspaper columns that reported the negative aspects of the war. During Rumsfeld's tenure, he would regularly visit U.S. troops stationed in Iraq.[97] The Australia Broadcasting Corporation reported that though Rumsfeld didn't specify a withdrawal date for troops in Iraq, "He says it would be unrealistic to wait for Iraq to be peaceful before removing US led forces from the country, adding that Iraq had never been peaceful and perfect."[98] On August 2, 2006, at a press conference at the Pentagon, Rumsfeld commented on the Sectarian violence in Iraq where he stated "there's sectarian violence; people are being killed. Sunnis are killing Shi'a and Shi'a are killing Sunnis. Kurds seem not to be involved. It's unfortunate, and they need a reconciliation process."[99] On October 26, 2006 at a press conference at the Pentagon after the failure of Operation Together Forward in Iraq, Rumsfeld stated "Would defeat in Iraq be so bad?" Well, the answer is: Yes, it would be. Those who are fighting against the Iraqi government want to seize power so that they can establish a new sanctuary and a base of operations for terrorists and any idea that U.S. military leaders are rigidly refusing to make adjustments in their approaches is just flat wrong. the military is continuing to adapt and to adjust as required. Yes, there are difficulties and problems to be sure."[100] As a result, Rumsfeld stirred controversy as to whether the forces that did invade Iraq were enough in size.[84] In 2006, Rumsfeld responded to a question by Brit Hume of Fox News as to whether he pressed General Tommy Franks to lower his request for 400,000 troops for the war: Absolutely not. That's a mythology. This town [Washington, D.C] is filled with this kind of nonsense. The people who decide the levels of forces on the ground are not the Secretary of Defense or the President. We hear recommendations, but the recommendations are made by the combatant commanders and by members of the Joint Chiefs of Staff and there hasn't been a minute in the last six years when we have not had the number of troops that the combatant commanders have requested.[101] Rumsfeld told Hume that Franks ultimately decided against such a troop level.[102] President George W. Bush, Defense Secretary Rumsfeld, and Deputy Secretary Wolfowitz in March 2003 Rumsfeld with Russian Minister of Defense Sergei Ivanov on March 13, 2002. Russia actively supported the American war against terrorism. Rumsfeld with Uzbek Defense Minister Kadyr Gulyamov. Uzbekistan was a key ally in the War on Terror. Throughout his tenure, Rumsfeld sought to remind the American people of the 9/11 attacks and threats against Americans, noting at one time in a 2006 memo to "[m]ake the American people realize they are surrounded in the world by violent extremists".[103][96] According to The Guardian report, Rumsfeld was allegedly including biblical quotes in top secret briefing papers to appeal George W Bush, known for his devout religious beliefs, to invade Iraq as more like "holy war" or "a religious crusade" against Muslims.[104] In a September 2007 interview with The Daily Telegraph, General Mike Jackson, the head of the British army during the invasion, criticized Rumsfeld's plans for the invasion of Iraq as "intellectually bankrupt", adding that Rumsfeld is "one of those most responsible for the current situation in Iraq", and that he felt that "the US approach to combating global terrorism is 'inadequate' and too focused on military might rather than nation building and diplomacy."[105] Secretary Rumsfeld responds to a reporter's question during a Pentagon press briefing. Rumsfeld and General Richard Myers, Chairman of the Joint Chiefs of Staff, gave reporters an operational update on Operation Iraqi Freedom on October 2, 2003. Rumsfeld with Indonesia Minister of Defense Juwono Sudarsono in Jakarta, Indonesia June 7, 2006. ### Condolence letters In December 2004, Rumsfeld was heavily criticized for using a signing machine instead of personally signing over 1000 letters of condolence to the families of soldiers killed in action in Iraq and Afghanistan. He promised to personally sign all letters in the future.[106] ### Prisoner abuse and torture concerns Comment from Rumsfeld: "I stand for 8–10 hours a day. Why is standing [by prisoners] limited to 4 hours?" The Department of Defense's preliminary concerns for holding, housing, and interrogating captured prisoners on the battlefield were raised during the military build-up prior to the Iraq War. Because Saddam Hussein's military forces surrendered when faced with military action, many within the DOD, including Rumsfeld and United States Central Command General Tommy Franks, decided it was in the best interest of all to hand these prisoners over to their respective countries. Additionally, it was determined that maintaining a large holding facility was, at the time, unrealistic. Instead, the use of many facilities such as Abu Ghraib would be used to house prisoners of interest prior to handing them over, and Rumsfeld defended the Bush administration's decision to detain enemy combatants. Because of this, critics, including members of the U.S. Senate Armed Services Committee, would hold Rumsfeld responsible for the ensuing Abu Ghraib torture and prisoner abuse scandal. Rumsfeld himself said: "These events occurred on my watch as Secretary of Defense. I am accountable for them."[107] He offered his resignation to President Bush in the wake of the scandal, but it was not accepted.[108] Rumsfeld poses with Marines during one of his trips to Camp Fallujah, Iraq, on Christmas Eve 2004. In a memo read by Rumsfeld detailing how Guantanamo interrogators would induce stress in prisoners by forcing them to remain standing in one position for a maximum of four hours, Rumsfeld scrawled a handwritten note on the memo reading: "I stand for 8–10 hours a day. Why is standing [by prisoners] limited to 4 hours? D.R."[109] Various organizations, such as Human Rights Watch, have called for investigations of Rumsfeld regarding his involvement in managing the Iraq War and his support of the Bush administration's policies of "enhanced interrogation techniques", which are widely regarded as torture.[110][111] Scholars have argued that Rumsfeld "might be held criminally responsible if [he] would be prosecuted by the ICC".[112] In 2005 the ACLU and Human Rights First filed a lawsuit against Rumsfeld and other top government officials, "on behalf of eight men who they say were subjected to torture and abuse by U.S. forces under the command of Defense Secretary Donald Rumsfeld".[113] In 2005, a suit was filed against Rumsfeld by several human rights organizations for allegedly violating U.S. and international law that prohibits "torture and cruel, inhuman, or degrading punishment".[113] Donald Vance and Nathan Ertel filed suit against the U.S. government and Rumsfeld on similar grounds, alleging that they were tortured and their rights of habeas corpus were violated.[114][115][116][117] In 2007, U.S. District Judge Thomas F. Hogan ruled that Rumsfeld could not "be held personally responsible for actions taken in connection with his government job".[118] The ACLU tried to revive the case in 2011 with no success.[119] In 2004 German prosecutor Wolfgang Kaleck filed a criminal complaint charging Donald Rumsfeld and 11 other US officials as war criminals who either ordered the torture of prisoners or drafted laws that legitimated its use. The charges based on breaches of the UN Convention against Torture and the German Code of Crimes against International Law.[120] ### Resignation Rumsfeld with former British Prime Minister Margaret Thatcher alongside the Chairman of the Joint Chiefs of Staff General Peter Pace, 2006 Eight U.S. and other NATO-member retired generals and admirals called for Rumsfeld to resign in early 2006 in what was called the "Generals Revolt", accusing him of "abysmal" military planning and lack of strategic competence.[121][122][123] Commentator Pat Buchanan reported at the time that Washington Post columnist David Ignatius, who traveled often to Iraq and supported the war, said the generals "mirror the views of 75 percent of the officers in the field, and probably more".[124] Rumsfeld rebuffed these criticisms, stating, "out of thousands and thousands of admirals and generals, if every time two or three people disagreed we changed the secretary of defense of the United States, it would be like a merry-go-round."[125] Bush defended Rumsfeld throughout and responded by stating that Rumsfeld is "exactly what is needed".[126] Rumsfeld shakes President Bush's hand as he announces his resignation, November 8, 2006. On November 1, 2006, Bush stated he would stand by Rumsfeld as defense secretary for the length of his term as president.[127] Rumsfeld wrote a resignation letter dated November 6, 2006 and, per the stamp on the letter, Bush saw it on Election Day, November 7, 2006.[128] In the elections, the House and the Senate shifted to Democratic control. After the elections on November 8, 2006, Bush announced Rumsfeld would resign his position as Secretary of Defense. Many Republicans were unhappy with the delay, believing they would have won more votes if voters had known Rumsfeld was resigning.[128] Bush nominated Robert Gates to succeed Rumsfeld.[129][130][131] On December 18, 2006, Rumsfeld's resignation took effect. ## Retirement and later life (2006–present) Rumsfeld shares a laugh with his successor, Robert Gates, at a ceremony to unveil his official portrait as Secretary of Defense, June 25, 2010 Dedication ceremony of the Pentagon Memorial in 2008 Rumsfeld greeting President George W. Bush in 2019 In the months after his resignation, Rumsfeld toured the New York City publishing houses in preparation for a potential memoir.[132] After receiving what one industry source labeled "big bids",[citation needed] he reached an agreement with the Penguin Group to publish the book under its Sentinel HC imprint.[citation needed] Rumsfeld declined to accept an advance for the publication of his memoir, and has said he is donating all proceeds from the work to veterans groups.[133] His book, entitled Known and Unknown: A Memoir, was released on February 8, 2011.[134] In conjunction with the publication of Known and Unknown, Rumsfeld established "The Rumsfeld Papers", a website with documents "related to the endnotes" of the book and his service during the George W. Bush administration;[135] during the months that followed the book's publication, the website was expanded to include over 4,000 documents from his archive. As of June 2011,[needs update] the topics include his Congressional voting record, the Nixon administration, documents and memos of meetings while he was part of the Ford, Reagan, and George W. Bush administrations, private sector documents, and NATO documents, among others.[135] In 2007, Rumsfeld established The Rumsfeld Foundation, which focuses on encouraging public service in the United States and supporting the growth of free political and free economic systems abroad. The educational foundation provides fellowships to talented individuals from the private sector who want to serve for some time in government.[133] Rumsfeld personally financed the foundation.[136] As of January 2014, the foundation has sponsored over 90 fellows from Central Asia, provided over$1.2 million in tuition and stipend support for graduate students, awarded over $3 million in microfinance grants, and donated over$2.4 million to charities for veterans affairs.[137] Rumsfeld was awarded the "Defender of the Constitution Award" at the 2011 Conservative Political Action Conference in Washington, D.C., on February 10, 2011. After his retirement from government, Rumsfeld criticized former fellow Cabinet member Condoleezza Rice, Secretary of State, in his memoir, asserting that she was basically unfit for office. In 2011, she responded, saying that Rumsfeld "doesn't know what he's talking about. The reader may imagine what can be correct about the conflicted matter."[138] In February 2011, Rumsfeld endorsed the repeal of the military's "Don't ask, don't tell" policy, saying that allowing gays and lesbians to openly serve "is an idea whose time has come".[139] In March 2011, Rumsfeld spoke out on the 2011 military intervention in Libya, telling ABC News Senior White House Correspondent Jake Tapper that the Obama administration should "recognize the mission has to determine the coalition. The coalition ought not determine the mission." Rumsfeld also used the word "confusion" six times to describe the United Nations-backed military effort in Libya.[140] In October 2011, Rumsfeld conducted an interview with Al Jazeera's Washington, D.C. bureau chief Abderrahim Foukara. Foukara asked Rumsfeld whether, in hindsight, the Bush administration had sent enough troops into Iraq to secure the borders of the country, and whether that made the United States culpable in the death of innocent Iraqis. Foukara said people in the Pentagon told Rumsfeld the number of troops sent into Iraq was insufficient. Rumsfeld said, "You keep making assertions which are fundamentally false. No one in the Pentagon said they were not enough." Foukara pressed Rumsfeld repeatedly. Rumsfeld then asked, "Do you want to yell or do you want to have an interview?" Foukara then asked, "Do you think the numbers that you went to Iraq with did absolve you from the responsibility of tens, maybe hundreds of thousands of innocent Iraqis killed by the Coalition and those criminals that you talked about?" Rumsfeld called the question "pejorative" and said Foukara was "not being respectful" (Foukara disagreed) and was "just talking over, and over, and over again".[141][142] Rumsfeld was the subject of the 2013 Errol Morris documentary The Unknown Known, the title a reference to his response to a question at a February 2002 press conference. In the film Rumsfeld "discusses his career in Washington D.C. from his days as a congressman in the early 1960s to planning the invasion of Iraq in 2003".[143] In January 2016, in partnership with the literary and creative agency Javelin, which handled design and development,[144] Rumsfeld released a mobile app game of solitaire called Churchill Solitaire, emulating a variant of the card game as played by Winston Churchill.[145] Rumsfeld and the Churchill family said that profits from the game would be donated to charity.[146][147] In June 2016, Rumsfeld announced that he would vote for Donald Trump in the 2016 presidential election.[148] ## Electoral history Rumsfeld gives the command at the 2005 Pepsi 400, which he served as the grand marshal[149] During the four elections during which he ran to represent Illinois's 13th congressional district, Rumsfeld received shares of the popular vote that ranged from 58% (in 1964) to 76% (in 1966). In 1975 and 2001, Rumsfeld was overwhelmingly confirmed by the U.S. Senate after presidents Gerald Ford and George W. Bush, respectively, appointed him as U.S. Secretary of Defense. ## Awards and reputation Donald H. Rumsfeld Rumsfeld has been awarded 11 honorary degrees.[150] Following his years as CEO, president, and later chairman of G. D. Searle & Company, he was recognized as Outstanding CEO in the pharmaceutical industry by the Wall Street Transcript (1980) and Financial World (1981).[151] Some of his other awards include: Secretary of State Henry Kissinger described Rumsfeld as "the most ruthless man" he knew.[161] ## Affiliation history Soviet leader Leonid Brezhnev, President Ford and Rumsfeld in Vladivostok, Soviet Union, November 1974 Rumsfeld and Victoria Nuland at the NATO-Ukraine consultations in Vilnius, Lithuania, on October 24, 2005 ## Works • Rumsfeld, Donald (1998). Strategic imperatives in East Asia. Heritage lectures, no. 605. Washington, D.C.: Heritage Foundation. Speech given March 3, 1998, in Washington, D.C. • Rumsfeld, Donald (2011). Known and Unknown: A Memoir. Sentinel. ISBN 978-1-59523-067-6. • Rumsfeld, Donald (2013). Rumsfeld's Rules. Broadside Books. ISBN 9780062272867. • Rumsfeld, Donald (2018). When the Center Held: Gerald Ford and the Rescue of the American Presidency. ISBN 978-1501172939. ## References 1. ^ a b "Donald H. Rumsfeld – George W. Bush Administration". Office of the Secretary of Defense – Historical Office. 2. ^ "Donald H. Rumsfeld – Gerald Ford Administration". Office of the Secretary of Defense – Historical Office. 3. ^ a b "Truth, War And Consequences: Why War? – In Their Own Words – Who Said What When". Frontline. PBS. Retrieved May 28, 2019. 4. ^ a b Jackson, Brooks (September 2, 2005). "Anti-war Ad Says Bush, Cheney, Rumsfeld & Rice "Lied" About Iraq". FactCheck.org. Retrieved May 28, 2019. 5. ^ a b Landay, Jonathan S. (February 8, 2007). 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Retrieved November 11, 2013. 161. ^ BBC News, November 8, 2006 "Profile Donald Rumsfeld	{"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": 0.30519330501556396, "perplexity": 16580.171703965832}, "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-2021-25/segments/1623488262046.80/warc/CC-MAIN-20210621025359-20210621055359-00592.warc.gz"}
http://clay6.com/qa/49356/a-die-has-two-faces-each-with-number-1-three-faces-each-with-number-2-and-o	Browse Questions # A die has two faces each with number 1,three faces each with number 2 and one face with number 3.If die is rolled once,determine P(1 or 3) $\begin{array}{1 1}(A)\;\large\frac{1}{2}\\(B)\;\large\frac{1}{4}\\(C)\;\large\frac{1}{8}\\(D)\;\large\frac{1}{6}\end{array}$ Toolbox: • Required probability =$\large\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$ Step 1: Given A die is rolled $\therefore$ Total number of outcomes n(S)=6 The die has 2 faces with number 1 The die has 3 faces with number 2 The die has 1 face with number 3 Step 2: Let E be the event of getting a number 1 or 3 $\therefore n(E)=2+1=3$ $\therefore$ Required probability =$\large\frac{n(E)}{n(S)}$ $\Rightarrow \large\frac{3}{6}$ $\Rightarrow \large\frac{1}{2}$ Hence (A) is the correct answer.	{"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": 0.19244696199893951, "perplexity": 2494.19059628433}, "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/1492917123632.58/warc/CC-MAIN-20170423031203-00502-ip-10-145-167-34.ec2.internal.warc.gz"}
https://www.arxiv-vanity.com/papers/1009.5633/	arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org. # Densities of Minor-Closed Graph Families David Eppstein Computer Science Department University of California, Irvine Irvine, California, USA ###### Abstract We define the limiting density of a minor-closed family of simple graphs to be the smallest number such that every -vertex graph in has at most edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least . It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, and . For multigraphs, the only possible limiting densities are the integers and the superparticular ratios . ## 1 Introduction Planar simple graphs with vertices have at most edges. Outerplanar graphs have at most edges. Friendship graphs have edges. Forests have at most edges. Matchings have at most edges. Where do the coefficients , , , , and of the leading terms in these bounds come from? Planar graphs, forests, outerplanar graphs, and matchings all form instances of minor-closed families of simple graphs, families of the graphs with the property that any minor of a graph in the family (a simple graph formed from by contracting edges and removing edges and vertices) remains in the family. The friendship graphs (graphs in the form of triangles sharing a common vertex) are not minor-closed, but are the maximal graphs in another minor-closed family, the graphs formed by adding a single vertex to a matching. For any minor-closed family of simple graphs there exists a number such that every -vertex graph in has at most edges [7, 18, 19]. We define the limiting density of to be the smallest number with this property. In other words, it is the coefficient of the leading linear term in the extremal function of , the function that maps a number to the maximum number of edges in an -vertex graph in . We may rephrase our question more formally, then, as: which numbers can be limiting densities of minor-closed families? As we show, the set of limiting densities is countable, well-ordered, and topologically closed (Theorem 19). Additionally, the set of limiting densities of minor-closed graph families is the closure of the set of densities of a certain family of finite graphs, the density-minimal graphs for which no minor has a greater ratio of edges to vertices (Theorem 20). To prove this we use a separator theorem for minor-closed families [1] to find density-minimal graphs that belong to a given minor-closed family, have a repetitive structure that allows them to be made arbitrarily large, and are close to maximally dense. By analyzing the structure of density-minimal graphs, we can identify the smallest two cluster points of the set of limiting densities, the numbers and , and all of the other possible limiting densities that are at most . Specifically, the limiting densities below 1 are the superparticular ratios for , and the corresponding density-minimal graphs are the -vertex trees. The limiting densities between 1 and are the rational numbers of the form , , and for ; the corresponding density-minimal graphs include the friendship graphs and small modifications of these graphs. Beyond the pattern is less clear, but each number is a cluster point in the set of limiting densities, and is a cluster point of cluster points. Similarly the number is a cluster point of cluster points of cluster points, etc. We summarize this structure in Theorem 22. One may also apply the theory of graph minors to families of multigraphs allowing multiple edges between the same pair of vertices as well as multiple self-loops connecting a single vertex to itself. In this case the theory of limiting densities and density-minimal graphs is simpler: the only possible limiting densities for minor-closed families of multigraphs are the integers and the superparticular ratios, and the only limit point of the set of limiting densities is the number (Theorem 23). ## 2 Related work ### 2.1 Limiting densities of minor-closed graph families A number of researchers have investigated the limiting densities of minor-closed graphs. It is known that every minor-closed family has bounded limiting density [11] and that this density is at most for graphs with an -vertex forbidden minor [7, 18]. This asymptotic growth rate is tight: -free graphs have limiting density , and the constant factor hidden in the -notation above is known [19]. There have also been similar investigations into the dependence of the limiting density of -minor-free graphs on the number of vertices in when is not a clique [8, 12]. However these works have a different focus than ours: they concern either the asymptotic growth rate of the limiting density as a function of the forbidden minor size or, in some cases, the limiting densities or extremal functions of specific minor-closed families [4, 9, 15, 17] rather than, as here, the structure of the set of possible limiting densities. ### 2.2 Growth rates of minor-closed graph families Bernardi, Noy, and Welsh [3] investigate a different set of real numbers defined from minor-closed graph families, their growth rates. The growth rate of a family of graphs is a number such that the number of -vertex graphs in the family, with vertices labeled by a permutation of the numbers from to , grows asymptotically as [14]. Bernardi et al. investigate the topological structure of the set of growth rates, show that this set is closed under the doubling operation, and determine all growth rates that are at most 2.25159. ### 2.3 Upper density of infinite graphs For arbitrary graphs, not belonging to a minor-closed family, the density is often defined differently, as the ratio \|E\|/(\|V\|2)=2\|E\|\|V\|(\|V\|−1) of the number of edges that are present in the graph to the number of positions where an edge could exist. (This definition is not useful for minor-closed families: for any nontrivial minor-closed family, the density defined in this way necessarily approaches zero in the limit of large .) This definition of density can be extended to infinite graphs as the upper density: the upper density of an infinite graph is the supremum of numbers with the property that contains arbitrarily large subgraphs with density . Although defined in a very different way to our results here, the set of possible upper densities is again limited to a well-ordered countable set, consisting of , , and the superparticular ratios [5, Exercise 12, p. 189]. ## 3 Density-minimal graphs We define the density of a simple graph , with edges and vertices, to be the ratio . We say that is density-minimal if no proper minor of has equal or greater density. Equivalently, a connected graph is density-minimal if there is no way of contracting some of the edges of , compressing multiple adjacencies to a single edge, and removing self-loops, that produces a smaller graph with greater density: edge removals other than the ones necessary to form a simple graph are not helpful in producing dense minors. The rank of a connected graph , again with edges and vertices, is the number of independent cycles in ; we say that is rank-minimal if no proper minor of has the same rank. Every density-minimal graph is also rank-minimal, because a smaller graph with equal rank would have greater density. ### 3.1 Examples of density-minimal graphs Every tree is density-minimal: a tree with edges (such as the path ) has density , and contracting edges in a tree can only produce a smaller tree with a smaller density. Additional examples of density-minimal graphs are provided by the friendship graphs formed from a set of triangles by identifying one vertex from each triangle into a single supervertex; these graphs are famous as the finite graphs in which every two vertices have exactly one common neighbor [6], but they also have the property that is density-minimal with density . For, if we contract edges in to form a minor , then must itself have the form of a friendship graph together possibly with some additional degree-one vertices connected to the central vertex. Removing the degree-one vertices can only increase the density of , but once this is done must itself be a friendship graph with fewer triangles than and smaller density. This shows that every minor of has smaller density, so is density-minimal. Let be formed from the friendship graph by adding one more vertex, whose two neighbors are the two endpoints of any edge in . Then a similar argument shows that is density-minimal with density . If a second vertex is added in the same way to produce a graph , then is density-minimal with density . However, adding a third vertex in the same way does not generally produce a density-minimal graph: its density is , and (if ) one of the triangles of the friendship graph from which it was formed can be removed leaving a smaller minor with the same density. These are not the only density-minimal graphs with these densities—a set of triangles can be connected together at shared vertices to form cactus trees other than the friendship graphs with the same density—but as we now show, their densities are the only possible densities of density-minimal graphs in this numerical range. ### 3.2 Bounding the rank of low-density density-minimal graphs In order to determine the possible densities of density-minimal graphs, it is helpful to have the following technical lemma, which allows us to restrict our attention to graphs of low rank. ###### Lemma 1. Every biconnected graph of rank four or higher contains a minor of density at least . ###### Proof. Let be biconnected with rank four or higher. We perform an open ear decomposition of the graph (a partition of the edges into a sequence of subgraphs, the first of which is a cycle and the rest of which are simple paths, where the endpoints of each path belong to previous components of the decomposition) [20]. The first four ears of this decomposition form a biconnected subgraph of with rank exactly four, so by replacing with this subgraph we may assume without loss of generality that the rank of is four. We may also assume without loss of generality that has no edges that could be contracted in a way that preserves both its rank and its biconnectivity, because otherwise we could replace with the graph formed by performing these contractions; in particular, this implies that every degree-two vertex of is part of a triangle. Additionally, no two degree-two vertices can be adjacent, for if they were then the third vertex of their triangle would be an articulation point, contradicting biconnectivity. We now assert that has at most six vertices, and therefore has density at least . For, suppose that had vertices with , and edges. In this case, since the number of edges is less than , there must be a vertex with degree two. Removing leaves a smaller graph with vertices and edges; again, the number of edges is less than three-halves of the number of vertices, so there must be a vertex that has degree two in ( cannot have degree one, because if it did then would have two adjacent degree-two vertices). If is not adjacent to in , then the two neighbors of in form a triangle as they do in , for the same reason that the neighbors of form a triangle; if on the other hand is adjacent to , then its two remaining neighbors in must again form a triangle or else the edge where is not adjacent to could be contracted preserving rank and biconnectivity. Removing from leaves a second smaller graph with rank two. Additionally, since both and belonged to triangles of , their removal cannot create any articulation points in , so is biconnected. But (by ear decomposition again) the only possible structure for a biconnected rank-two graph such as is a theta graph , in which two degree-three vertices are connected by three paths of lengths , , and respectively. If has seven or more vertices, then has five or more vertices, and . If one of the paths of the theta graph has length three or more, then only two of the edges of this path can be part of triangles containing and , and the third edge of the path can be contracted in to produce a smaller biconnected rank-four graph, contradicting our assumption that no such contraction exists; this case is shown in Figure 2(left). In the remaining case, ; only two of the six edges of can be part of triangles involving and , and any one of the four remaining edges can be contracted in to produce a smaller biconnected rank-four graph, again contradicting our assumption. These contradictions show that the number of vertices in is at most six; since its rank is four, its density must be at least . ∎ Increasing the rank past four without increasing the connectivity does not increase the density past the threshold of Lemma 1: there exist biconnected graphs of arbitrarily high rank in which the densest minors have density , namely the cycles of triangles shown in Figure 2(right). A simple case analysis based on ear decompositions shows that there are exactly five rank-minimal biconnected graphs of rank between one and three: the triangle , the diamond graph with four vertices and rank two, the complete graph with four vertices and rank three, and two different 2-trees with five vertices and rank three (Figure 3). ### 3.3 Classification of density-minimal graphs with low density ###### Lemma 2. Let a graph be density-minimal, with density . Then Δ∈{ii+1 ∣∣ i≥0}∪{3i+2j2i+j+1 ∣∣ i≥1,j∈{0,1,2}} and for each number in this set there exists a density-minimal graph with density . ###### Proof. First, suppose that . Then must be acyclic and connected, for if it had a cycle it would have a triangle minor with density and not be density-minimal, and if it were disconnected then the densest of its components would be a minor with density at least as great as that of itself. But an acyclic connected graph is a tree, and has density where is its number of edges. In the remaining cases, . must be bridgeless, because any bridge could be contracted producing a denser graph. By Lemma 1, each block (biconnected component) of must have rank at most three, for otherwise that block by itself would have a minor with density at least , contradicting the assumption that is density-minimal. Additionally, each block must be rank-minimal, for otherwise itself would not be rank-minimal. Therefore, each block must be a triangle, the diamond graph, or one of the two rank-three 2-trees. If there are two rank-three blocks, three rank-two blocks, or one rank-three block and one rank-two block, then those blocks alone (with the remaining blocks contracted away) would again form a minor with density at least . The only remaining cases are a graph in which the blocks consist of triangles, with density , a graph in which the blocks consist of triangles and one rank-two block, with density , a graph in which the blocks consist of triangles and one rank-three block, with density , or a graph in which the blocks consist of triangles and two rank-two blocks, with density . Each of these densities belongs to the set specified in the lemma. To show that each in the set of densities stated in the lemma is the density of some density-minimal graph , we need only recall the path graphs , the friendship graphs , and the graphs ad formed by adding one or two degree-two vertices to a friendship graph. As we have already argued, these graphs are density-minimal, and together they cover all the densities in the given set. ∎ The set of achievable densities allowed by Lemma 2, in numerical order up to the limit point , is 0,12,23,34,45,56,…,1,65,54,97,43,1511,118,1813,75,2417,1712,2719,107,…32. Figure 1 shows density-minimal graphs achieving some of these densities. ## 4 Fans of graphs We now introduce a notation for constructing large graphs with a repetitive structure from a smaller model graph. Given a graph , a proper subset of the vertices of , and a positive integer , we define the graph to be the union of copies of , all sharing the same copies of the vertices in and having distinct copies of the vertices in . For instance: • If is a triangle , then is the friendship graph • For the same triangle , is a 2-tree formed by triangles sharing a common edge. Three of the graphs in Figure 3 take this form, for . • The complete bipartite graph is can be represented in multiple different ways as a fan: it is isomorphic to where is any divisor of and is the -vertex side of the bipartition of , and symmetrically there is a representation as a fan for any divisor of . • Two more examples are shown in Figure 4. ### 4.1 Basic observations about fans If has vertices and has vertices (where as we require to be a proper subset of the vertices), then has vertices. is the disconnected graph formed by disjoint copies of ; however, if is connected and is nonempty then is also connected. ###### Lemma 3. Every graph has at least vertices. ###### Proof. This follows immediately from the requirement that be a proper subset of the vertices of ; there is at least one vertex that does not belong to , and that is replicated times in . ∎ As the following lemma shows, it is not very restrictive to consider only those fans in which the central subset forms a clique. The advantage of restricting in this way is that, when considering minors of the fan, we do not have to consider the ways in which such a minor might add edges between vertices of . ###### Lemma 4. Let be a graph, let be a subset of the vertices of such that is connected and every vertex in has a neighbor in , and let be the graph formed from by adding edges between every pair of vertices in . Then there exists a constant (depending on and ) with the property that, for every , is a minor of . ###### Proof. Let be a maximum clique in the subgraph induced in by , and let . To form as a minor of , simply contract one of the copies of onto each vertex in . ∎ For instance, in Figure 4(left), the three central vertices do not form a clique, but they can be made into a clique by contracting one of the six copies of the outer subgraph into the rightmost of the three central vertices. Thus, in this example, we may take . ### 4.2 Densest minors of fans The following technical lemma will be used to compare the densities of minors of fans in which different copies of are contracted differently from each other. ###### Lemma 5. Let , , , , , and be any six non-negative numbers, with , , and positive. Then , with equality only in the case that the two terms in the maximum are equal. ###### Proof. is a weighted average of the other two fractions with weights and respectively. As a weighted average with positive weights, it is not larger than the maximum of the two averaged values, and can be equal only when the two averaged values are equal. ∎ With the assumption that is a clique, any fan has a density-minimal minor that is also a fan with equal or greater density: ###### Lemma 6. Let be a connected graph, let be a proper subset of that induces a clique in , let every vertex in be adjacent to at least one vertex in , and let induce a connected subgraph of . Then, for every , there exists a densest minor of that is isomorphic to , where is some minor of (that may depend on and is the image of in . In addition if is nonempty then can be chosen so that is density-minimal; if is empty then can be chosen to be itself density-minimal. ###### Proof. Let be a densest minor of , and let be the image of in . Note that it is not possible for all copies of to be contracted onto in , for (with ) the density that would arise from this possibility is less than the density s(s−1)/2+kss+k=ss+k⋅s−12+ks+ks of the graph in which, in each copy of , the vertices that are not part of are contracted into a single vertex: as the formula shows, this latter graph’s density is a weighted average of and , and therefore exceeds . Suppose that the copies of in the fan are transformed in into two or more different minors, and let and be two of these minors. Define the integers , , and respectively to be the number of edges contributed to by minors of type , the number of edges contributed to by minors of type , and the number of remaining edges (including edges connecting vertices in . Similarly, define , , and respectively to be the number of vertices contributed by minors of type , the number of vertices contributed by minors of type , and the remaining number of vertices (including vertices in ). Then the density of is , the density of the minor of formed by replacing all copies of by is , and the density of the minor formed by replacing all copies of by is . By Lemma 5, one of these two replacements provides a minor of that is at least as dense as but that has one fewer different type of minor of . By induction on the number of types of minors of appearing in , there is a densest minor of of the form where is a minor of . Among all choices of leading to densest minors of this form, choose to be minimal in the minor ordering. If , must be density-minimal, for any minor of with equal or smaller density would have been chosen in place of If , must be density-minimal, for (again using Lemma 5 to reduce the number of different types of minor in the graph) substituting some copies of for smaller minors could only produce a graph of equal density if these substitute minors could all be chosen to be isomorphic copies of a single minor , but then (because of the choice of as giving the densest minor of this form) would have equal density to , contradicting the choice of as a minor-minimal graph whose fan has this density. ∎ ## 5 Limiting density from density-minimal graphs As we now show, the density-minimal graphs constructed in the previous two sections provide examples of minor-closed families, with the limiting density of the minor-closed family equal to the density of the density-minimal graph. ### 5.1 Minor-closed families with the density of a given density-minimal graph For any graph , define to be the family of minors of graphs of the form for some positive . The graphs in are characterized by the property that every connected component is a minor of . Clearly, is minor-closed. ###### Lemma 7. If is density-minimal, then the limiting density of equals the density of . ###### Proof. contains arbitrarily large graphs with this density, namely the graphs . Therefore its limiting density is at least equal to the density of . But the fact that its limiting density is no more than the density of is immediate, for if it contained any denser graph then that graph would have a dense component forming a minor of and contradicting the assumed density-minimality of . ∎ ###### Corollary 8. The set of limiting densities of minor-closed graph families is a superset of the set of densities of density-minimal graphs. ### 5.2 Gaps in the densities of density-minimal graphs The idea of designing a minor-closed family to achieve a given density comes up in a different way in the following lemma, which is the basis for our proof that the limiting densities form a well-ordered set. ###### Lemma 9. For every there exists a number such that the open interval does not contain the density of any density-minimal graph. ###### Proof. Let be the minor-closed family of graphs that do not contain any minor that is more dense than . By the Robertson–Seymour theorem, can be characterized by a finite set of minimal forbidden minors; a minimal forbidden minor for is a graph whose density is greater than but all of whose proper minors have density less than or equal to . Thus, any minimal forbidden minor of must have density greater than ; let be the smallest density among these finitely many forbidden minors. Now suppose that is a density-minimal graph whose density is greater than . is not in , so it has as a minor one of the forbidden minors of , all of which have density at least . Because is density-minimal, its density is at least as great as that of its minor, and therefore is also at least . Therefore, cannot have density within the interval . ∎ ## 6 Separators in minor-closed graph families We will use separators as a tool to find dense fans in any minor-closed family, so in this section we formalize the mathematics of separators in the form that we need. If is any graph, define a separation of to be a collection of subgraphs that partition the edges of . We call these subgraphs the separation components of a separation. Define the separator of a separation to be the set of vertices that belong to more than one separation component. The separator theorem of Alon, Seymour, and Thomas [1] can be rephrased in terms of separations: ###### Lemma 10 (Alon, Seymour, and Thomas [1]). For any minor-closed graph family there exists a constant such that every -vertex graph in has a separation of into two subgraphs and , each having at least vertices, with at most vertices in the separator. For our purposes we need a form of separator theorem that produces much smaller separation components, with only constant size. This can be achieved by repeatedly applying the separator theorem of Lemma 10 to larger components until they are small enough. A finer separation theorem of this type for planar graphs, proved in the same way from the planar separator theorem, has long been known [10, Theorem 3] and similar methods have been used as well in the context of minor-closed graphs (e.g. see [16, Lemma 3.4]). However, in order to control the density of the components, we need to be careful about how we measure the size of the separator, since a single separator vertex may appear in many components. We define the separator multiplicity of a vertex in a separation of to be one less than the number of separation components containing , so that a vertex has nonzero separator multiplicity if and only if it belongs to the separator, and we define the total multiplicity of a separation to be the sum of the separator multiplicities of the vertices. If a vertex has separator multiplicity zero we say that it is an internal vertex of its separation component. ###### Lemma 11. For any minor closed graph family there exists a constant such that, for every and every -vertex graph , there is a separation of into subgraphs, each having at most vertices, with total multiplicity at most . ###### Proof. Set X=2σF√2/3+√1/3−1~{}and~{}ρF=3X2. We prove by induction on a stronger version of the lemma: whenever , there exists a separation in which the size of each separation component is as specified and for which the total multiplicity is at most For the induction does not go through because is negative, but in this case the lemma itself follows trivially since we may choose a separation with a single separation component (all of ) and no vertices in the separator; the total multiplicity is , which is is at most as desired. As a base case for the induction, whenever , we again choose a separation with a single separation component and no vertices in the separator; for at least this large, which again exceeds the zero value of the total multiplicity. For larger values of , apply Lemma 10 to find two separation components and of , with separator . Each separation component will have at least vertices, so we may apply the induction hypothesis to them, separating them into components with at most vertices per component. Our separation of is the union of the sets of components in these two separations. The separator for this separation consists of the union of the separators in the two subgraphs and , together with the vertices in . Let and be the numbers of vertices in and , respectively. When we combine the two separations, the separator multiplicity of vertices outside remains unchanged, and the multiplicity of vertices in increases by exactly one. Therefore, by the induction hypothesis, the combined total multiplicity is at most The worst case, for the terms and appearing in this expression, is that is and is only a small amount larger than ; for, in that case, the sum of these two terms is as small as possible. If and are closer together, the sum of these terms is larger and a larger amount is subtracted from the total expression. Substituting and for this worst case and using the assumption that to bound allows this upper bound on the total multiplicity of the separator to be regrouped as ϵn−(X√2/3+X√1/3−2σF)√n. Due to our choice of , this further simplifies to as required by the induction. ∎ ## 7 Density-minimal graphs from limiting density As we now show, the limiting density of a minor-closed graph family may be approximated by the densities of a subfamily of the density-minimal graphs that it contains. Some care is needed, though, as it is possible for a minor-closed graph family to contain some density-minimal graphs whose densities are strictly larger than the limiting density of the family. The main idea of the sequence of lemmas in this section is to show that contains large dense graphs with a structure that is progressively more uniform, eventually leading to the result that it contains large dense density-minimal fans. ###### Lemma 12. Let be a minor-closed family with limiting density . Then for any and any integer , contains a graph , such that has a separation into or more separation components, each of which has size (where the constant hidden in the -notation depends only on ), such that the disjoint union of the separation components forms a graph with density at least . ###### Proof. Let . Because has limiting density , it contains arbitrarily large graphs of density at least . By Lemma 11, any such graph with vertices has a separation into separation components of size , in which the total multiplicity of the separation is at most . For sufficiently large , the number of components is at least . The density of the disjoint union of the separation components can be calculated by replacing the denominator, , in the definition of the density of by the new denominator ; therefore, the density of the disjoint union is at most . ∎ A simple argument related to Chebyshev’s inequality lets us replace the density of the disjoint union of the separation components (a weighted average of their individual densities) by a lower bound on the density of each separation component. ###### Lemma 13. Let be a minor-closed family with limiting density . Then for any and any integer , contains a graph , such that has a separation into separation components of size in which each component has density at least . As in Lemma 12, the hidden constant in the -notation depends only on . ###### Proof. By Lemma 12, for any there is graph in with a separation into separation components of size , such that the density of the disjoint union of the components is at least . But the density of the disjoint union is a weighted average of the densities of its separation components, weighted by the number of vertices in each component. The weights of any two components differ by at most a factor of . Therefore, if denotes the largest density of any graph of size in (necessarily bounded since there are only finitely many graphs of that size) then, in order to achieve a weighted average of , every separation components with density less than (at least units below the average) must be balanced by at least one component with density greater than that threshold (and at most units above the average). By setting sufficiently large we may find a a graph in which at least of the separation components have density at least , and by deleting the lower-density components of we can find a graph with dense separation components. As is a subgraph of , it must belong to . ∎ We define two separation components to be isomorphic if they are isomorphic as labeled graphs, with a labeling of each vertex according to whether it is an internal vertex of the component or a separator vertex. ###### Lemma 14. Let be a minor-closed family with limiting density . Then for any and any integer , contains a graph , such that has a separation into isomorphic separation components of size in which each component’s density is at least . The hidden constant in the -notation depends only on . ###### Proof. Let be the bound on the component size for and given by Lemma 13, and let be the number of isomorphism classes of -vertex labeled graphs in in which each vertex is labeled as an internal vertex or a separator vertex; note that, by Lemma 13, . By Lemma 13, there is a graph in with a separation into components of size at most , all of which have density at least . At least of these components must be isomorphic to each other; let be the union of those isomorphic components, with all the other components removed. Then has the stated properties, and as a subgraph of it belongs to . ∎ The labeling used in Lemma 14 is not refined enough for our purposes. What we need is that, if is a vertex of the separator of , then every separation component uses in the same way. More formally, we define a separation of a graph to be uniform if every separation component in it is isomorphic to the same -vertex graph (for some ), and if the vertices of can be labeled with the integers from to in such a way that all components are isomorphic as labeled graphs. Equivalently, a separation is uniform if (for some ) the separation vertices may be given labels from to in such a way that the separation vertices within each separation component have distinct labels, and all pairs of separation components have an isomorphism that respects the labels. To achieve this, we use color coding, a variant of the probabilistic method developed for solving subgraph isomorphism problems [2]. ###### Lemma 15. Let be a minor-closed family with limiting density . Then for any and any integer , contains a graph , such that has a uniform separation into separation components of size each of which has density at least . The hidden constant in the -notation depends only on . ###### Proof. Let be the size of the components produced by Lemmas 13 and 14. By Lemma 14, there exists a graph with a separation into isomorphic separation components of size at most in which each component has density at least . Suppose that, for each separation component of , of the vertices are separator vertices, and the remaining vertices are internal to the component. We then choose, independently and uniformly at random for each vertex of the separator of , a label from to , and we also choose (arbitrarily rather than randomly) a labeling of a single isomorphic copy of the separation component (separate from ) that places distinct labels from to on its separator vertices. For each separation component of , the probability is at least that it has an isomorphism to such that the labels of the component in match the labels in , so the expected number of separation components with label-preserving isomorphisms of this type is at least . Therefore, there exists a labeling of that matches or exceeds this expected value, and allows at least of the separation components of to be given matching labels on their separator vertices. By keeping these matching components and deleting the rest, we find a subgraph of that belongs to and has a uniform separation as required. ∎ Observe that, in a uniform separation of a simple graph, there can be no edges in which both endpoints belong to the separator, because that would result in the graph being a multigraph. Therefore, if the graph is connected, each separation component must have at least one non-separator vertex. In the labeling defining a uniform separation of a graph , we say that the label of a separator vertex is singular if there is only one vertex in with that label, and plural otherwise. ###### Lemma 16. Let be a minor-closed family with limiting density . Then for any and any integer , contains a graph , such that has a uniform separation into separation components of size in which each component has density at least and in which each separator vertex label is singular. The hidden constant in the -notation depends only on . ###### Proof. Let be the maximum size of the separation components produced by Lemma 15 for and . By Lemma 15, there exists a graph in with a uniform separation into separation components of size at most in which each component has density at least . We then consider each label of a separator vertex in in turn; for each such label, we either make it singular or make it not be a separator vertex any more, at the expense of reducing the number of separation components to the square root of its former value. Our choice of separation components at the start of this process ensures that there will be at least separation components when we have completed the process. So, suppose that we have separation components in which label is plural, and we wish to find some subset of of the components such that, for the graph induced by that subset, is either singular or not a separator vertex. We consider the number of different separator vertices that have the label . If this number is at least , then we may choose a single separation component for each different vertex with that label, ending up with at least separation components overall; the graph induced by these separation components remains uniformly separated and, in its separation, the vertices labeled are no longer separation vertices because each belongs to a single separation component. Making these vertices into internal vertices of the separation components does not change the density of the components. In the other case, there are fewer than separator vertices with label . Therefore, because there are separation components, one of the vertices labeled must be shared by at least separation components. The graph induced by these separation components remains uniformly separated and in it is singular. Repeating this refinement process once for each of the labels of separator vertices produces a uniform separation in which all separator vertex labels are singular, as required. ∎ A graph with a uniform separation in which each separator vertex label is singular is just a more complicated way of describing a fan, and the density of the fan is at least as large as the density of the separation components it is formed from. However, in order to apply Lemma 6 we need the fan to have some more structure: the shared vertices of the fan should form a clique, the repeated parts of the fan should be connected, and each shared vertex should be adjacent to a repeated vertex. ###### Lemma 17. Let be a minor-closed family with limiting density . Then for any there exists a graph , and a proper subset of the vertices of such that for all positive , such that is connected and forms a clique, such that each vertex of is adjacent to a vertex in , and such that for all sufficiently large the density of is at least . ###### Proof. By Lemma 16 we can find, for any , a graph with vertices and a proper subset such that has density at least and belongs to . By choosing appropriately, we can additionally guarantee that also belongs to for all ; for, if each of the finitely many choices for had a fixed bound on the number of times it could be repeated in a fan, this would contradict Lemma 16 for values of that exceeded the maximum of these bounds. Because and come from a uniform separation, there are no edges in the subgraph induced by . Let be the number of edges in , be the number of vertices in , be the number of vertices in , and and be the number of edges incident to and vertices in each connected component of . Then and is a weighted average of the quantities , so if is the component of maximizing then . Let be the set of vertices in incident to , and let have as its vertex set and have as its edges all the edges in that are incident to together with an edge between every two vertices in . By Lemma 4, belongs to for all . As required, forms a clique, is connected, and every vertex in has a neighbor that is not in . As grows larger, the density of converges to , so for all sufficiently large it is larger than . ∎ Dense fans in can be used to generate density-minimal graphs whose densities approximate the limiting density of . ###### Lemma 18. Let be a minor-closed family with limiting density . Then for any there exists a density-minimal graph in whose density belongs to the closed interval . ###### Proof. By Lemma 17 we can find , , and such that for all positive , such that meets the conditions of Lemma 6, and such that the density of is at least for . By Lemma 9 there exists such that there are no density-minimal graphs with densities in the open interval . may contain graphs with density or larger, but only finitely many of them; let be the number of vertices in the largest such graph, and set . By Lemma 6 there exist and such that is a densest minor of , and therefore also has density at least ; since has at least vertices (Lemma 3), it has density less than and therefore its density is at most . According to Lemma 6, and can be chosen so that either is density-minimal, or is density-minimal and has the same density as the fan; in either case we have found a density-minimal graph in whose density lies in the desired range. ∎ ## 8 Main results ###### Theorem 19. The set of limiting densities of minor-closed graph families is countable, well-ordered, and topologically closed. ###### Proof. By the Robertson–Seymour theorem [13], according to which every minor-closed graph family can be characterized by a finite set of forbidden minors, there are countable many minor-closed families and therefore at most countably many limiting densities. There can be no infinite descending sequence of limiting densities of minor-closed families, because if there were, there would be an infinite descending subsequence , , , that converged to some limit . But then, for every , there would be a limiting density within of , and (by Lemma 18) a density-minimal graph with density within of , contradicting Lemma 9 according to which there is an open interval that does not contain the density of any density-minimal graph. Therefore, the set of limiting densities is well-ordered. By well-ordering, any cluster point of the set of limiting densities must be the limit of an increasing subsequence of limiting densities of minor-closed families , , . But then is also minor-closed and has as its limiting density; hence the set of limiting densities contains all its cluster points and is topologically closed. ∎ ###### Theorem 20. The set of limiting densities of minor-closed graph families is the topological closure of the set of densities of density-minimal graphs. ###### Proof. By Corollary 8, the set of limiting densities of minor-closed graph families contains the set of densities of density-minimal graphs, and by Theorem 19 it contains the closure of this set. By Lemma 18 every limiting density of a minor-closed graph family belongs to the closure of the set of densities of density-minimal graphs. ∎ ###### Theorem 21. Let be the limiting density of a minor-closed graph family . Then is a cluster point in the set of limiting densities. ###### Proof. Let be defined as in the proof of Lemma 9 as the family of graphs with no minor denser than . Define to be the family of graphs with the following two properties: • Each connected component of has at most vertices, and • Within each connected component of , one can find a vertex (the apex of the component) such that . is minor-closed, and as we show below, the sequence of graph families (for fixed and variable ) has a sequence of limiting densities that converges to but that does not ever actually reach . Because the density of any graph in is at most , any connected component of a graph in that has vertices has at most edges that are not incident to , and another edges incident to . Therefore, its density is at most , and the limiting density of is at most . In particular, it cannot be the case that any family has limiting density exactly equal to . For any there exists a density-minimal graph in with density in the interval , by Lemma 18. Because is density-minimal and has density at most , it belongs to . Let have edges and vertices, and form a graph by connecting each vertex in to a new vertex . The graphs have density , which approaches as becomes large, so for some sufficiently large the graph has density at least . The graphs belong to : each component of these graphs has vertices, meeting the limit on the component size in , and within each component of the vertex can be chosen as the apex. The family contains arbitrarily large graphs with density at least , so its limiting density is within the interval and the sequence of limiting densities of these families approaches . We have found a sequence of minor-closed families of graphs with limiting densities approaching but not equalling , so is a cluster point in the set of limiting densities. ∎ We say that a number is an order-1 cluster point in the set of limiting densities if is any cluster point of the set, and that is an order- cluster point if is a cluster point of order- cluster points. ###### Theorem 22. The subset of limiting densities of minor-closed graph families that are less than is the set {ii+1 ∣∣ i≥0}∪{3i+2j2i+j+1 ∣∣ i≥1,j∈{0,1,2}}. For the integers and , the numbers are order- cluster points in the set of limiting densities. ###### Proof. The description of the limiting densities below follows from Theorem 20 and Lemma 2. The characterizations of the numbers as order- cluster points follows by induction on using Theorem 21, with the description of the subset of limiting densities that are less than or equal to 1 as a base case for the induction. ∎ ## 9 Multigraphs Instead of using simple graphs, the theory of graph may be formulated in terms of multigraphs, graphs in which two vertices may be connected by a bond of two or more edges and in which a single vertex may have any number of self-loops, edges that have only that vertex as its endpoints. In some ways, the theory for multigraphs is simpler: for instance, the forbidden minor for forests in the simple graph world is a triangle, whereas for forests it is a smaller graph with one vertex and one self-loop. This simplicity carries over to the theory of limiting densities as well. ###### Theorem 23. The limiting density of a minor-closed family of multigraphs can only be an integer, a superparticular ratio for a positive integer , or unbounded. ###### Proof. If includes bonds with arbitrarily large numbers of edges, then its limiting density is unbounded. Otherwise, there can be at most finitely many different multigraphs in that have a fixed number of vertices, the only ingredient needed to make the counting arguments in our proofs go through. So, multigraphs in obey a separator theorem (obtained by applying the Alon–Seymour–Thomas separator theorem to the underlying simple graph of the multigraph), they contain dense fans of multigraphs, and their densities are approximated by the densities of finite density-minimal multigraphs. But if a connected multigraph contains a cycle, self-loop, or bond, then its densest minor is just a single vertex with density equal to its rank, because in the multigraph world edge contractions do not lead to the removal of any other edges, and therefore can only increase the density of a graph with at least as many edges as vertices. Therefore, the only density-minimal multigraphs are the trees and the single-vertex multigraphs, and the only possible densities that can be obtained from them are the ones in the statement of the theorem. ∎ More specifically, if can have a single connected component with arbitrarily high rank, its limiting density is unbounded. Otherwise, let be the largest number such that graphs in can have an unbounded number of rank- connected components. If is nonzero, the limiting density of is . If is zero, and can have a single connected component with unbounded size, its limiting density is one. And if is zero and all components of have bounded size, the limiting density of is , where is the number of edges in the largest tree such that for all . ## 10 Conclusions We have investigated the structure of the set of limiting densities of minor-closed graph families; this set is topologically closed and well-ordered, contains cluster points of cluster points, and its exact members are known up to the threshold . However, our results open many additional questions for investigation: • Are all limiting densities rational? • Is every limiting density equal to the density of a density-minimal graph, or are there cluster points in the set of limiting densities that are not themselves densities of density-minimal graphs? A positive answer to this question would also imply that all limiting densities are rational. • If is a limiting density, where and are integers, can the size of the gap between and the next larger limiting density be lower-bounded by a nonzero closed-form expression in terms of and ? Due to the existence of cluster points, some dependence on seems to be necessary: a monotonic function of alone would have to approach zero as approaches the first cluster point, and could not give a nontrivial lower bound on the gaps for higher values of . • Are there any other cluster points between and other than the ones of the form that we have already identified? • What is the order type of the set of limiting densities? Since the limiting densities contain cluster points of any finite order, the order type is at least ; is this the exact order type? • What is the smallest cluster point of order for each integer ? Is it the number itself? • If is a cluster point of limiting densities, must be a limiting density? If so, it would follow that all limiting densities are rational, that is the smallest cluster point of order , and that the order type of the set of limiting densities is . ### Acknowledgements This work was supported in part by NSF grant 0830403 and by the Office of Naval Research under grant N00014-08-1-1015. We thank Éric Fusy for calling our attention to reference [3], and an anonymous referee for many helpful suggestions.	{"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": 0.892920732498169, "perplexity": 347.6765023804124}, "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-2020-45/segments/1603107896048.53/warc/CC-MAIN-20201028014458-20201028044458-00031.warc.gz"}
https://www.genealogy.math.ndsu.nodak.edu/id.php?id=46417	Eric H. Grosse Ph.D. Stanford University 1981 Dissertation: Approximation and Optimization of Electron Density Maps	{"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": 0.9975324273109436, "perplexity": 14327.840841330999}, "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-2022-40/segments/1664030334332.96/warc/CC-MAIN-20220925004536-20220925034536-00434.warc.gz"}
https://en.wikiversity.org/wiki/QB/AstroMirandaTitan	# QB/AstroMirandaTitan < QB • Quizbank now resides on MyOpenMath at https://www.myopenmath.com (although I hope Wikiversity can play an important role in helping students and teachers use these questions!) • At the moment, most of the physics questions have already been transferred. To see them, join myopenmath.com as a student, and "enroll" in one or both of the following courses: • Quizbank physics 1 (id 60675) • Quizbank physics 2 (id 61712) • Quizbank astronomy (id 63705) The enrollment key for each course is 123. They are all is set to practice mode, giving students unlimited attempts at each question. Instructors can also print out copies of the quiz for classroom use. If you have any problems leave a message at user talk:Guy vandegrift. See special:permalink/1863359 for a wikitext version of this quiz. ### LaTexMarkup begin %[[File:Quizbankqb_{{SUBPAGENAME}}.pdf\|thumb\|See[[:File:Quizbankqb_{{SUBPAGENAME}}.pdf]]]] %CurrentID: {{REVISIONID}} %'''PDF: [[:File:Quizbankqb_{{SUBPAGENAME}}.pdf]]''' %Required images: [[file:Wikiversity-logo-en.svg\|45px]][[File:PIA12481 Titan specular reflection.jpg\|45px]][[File:Titan dunes crop.png\|45px]] %This code creates both the question and answer key using \newcommand\mytest %%% EDIT QUIZ INFO HERE %%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\quizname}{QB/AstroMirandaTitan} \newcommand{\quiztype}{conceptual}%[[Category:QB/conceptual]] %%%%% PREAMBLE%%%%%%%%%%%% \newif\ifkey %estabkishes Boolean ifkey to turn on and off endnotes \documentclass[11pt]{exam} \RequirePackage{amssymb, amsfonts, amsmath, latexsym, verbatim, xspace, setspace,datetime} \RequirePackage{tikz, pgflibraryplotmarks, hyperref} \usepackage[left=.5in, right=.5in, bottom=.5in, top=.75in]{geometry} \usepackage{endnotes, multicol,textgreek} % \usepackage{graphicx} % \singlespacing %OR \onehalfspacing OR \doublespacing \parindent 0ex % Turns off paragraph indentation % BEGIN DOCUMENT \begin{document} \title{AstroMirandaTitan} \author{The LaTex code that creates this quiz is released to the Public Domain\\ Attribution for each question is documented in the Appendix} \maketitle \begin{center} \includegraphics[width=0.15\textwidth]{666px-Wikiversity-logo-en.png} \\Latex markup at\\ \end{center} \begin{frame}{} \begin{multicols}{3} \tableofcontents \end{multicols} \end{frame} \pagebreak\section{Quiz} \keytrue \begin{questions}\keytrue \question The 1982 Voyager flyby of Miranda (a moon of Uranus) established that \_\_\_\_\_ \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice Miranda has the largest active volcano in the solar system \choice Miranda has geysers. \choice Miranda probably has an iron core \choice Two other answers are correct (making this the only true answer). \CorrectChoice inspired a theory a previous incarnation was destroyed by a collision \end{choices} \question It has been suggested that Miranda's "racetrack" \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice is antipodal to an impact crater \CorrectChoice Two other answers are correct (making this the only true answer). \choice is associated with tidal heating \choice is an impact crater \choice is a series of rifts created by an upwelling of warm ice \end{choices} \question According to Wikipedia, the largest lakes on Titan are probably fed by \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice rivers from the highlands \choice methane rain \choice geysers \choice liquid water rain \CorrectChoice underground aquifers \end{choices} \question \includegraphics[width=0.24\textwidth]{PIA12481-Titan-specular-reflection.png} The bright spot on Saturn's moon Titan is \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice a volcano \choice lightening \choice aurora borealis (northern lights) \CorrectChoice a lake \choice solar wind particles striking the atmosphere \end{choices} \question One "year" on Saturn's largest moon Titan lasts \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice 3 hours \choice 3 years \choice 30 hours \CorrectChoice 30 years \choice 300 days \end{choices} \question \includegraphics[width=0.5\textwidth]{Titan-dunes-crop-rotate.png} The photographs compare \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice summer windstorms and winter doldrums \choice northern and southern hemispheres \choice winter windstorms and summer doldrums \CorrectChoice Titan and Earth \choice wet and dry seasons \end{choices} \question The liquid water ocean of Saturn's largest moon Titan, \ifkey\endnote{ placed in Public Domain by Guy Vandegrift: {\url{https://en.wikiversity.org/wiki/special:permalink/1863359}}}\fi \begin{choices} \choice is less than one meter in depth \CorrectChoice explains how the elevation of a smooth planet seems to rise and fall \choice is postulated to cover 15-30% of its surface \choice is known to contain life \end{choices} \end{questions} \newpage	{"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, 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https://mathoverflow.net/users/5701/vaughn-climenhaga	# Vaughn Climenhaga less info reputation 11135 bio website math.uh.edu/~climenha location Houston, TX age 32 member for 4 years, 11 months seen 18 hours ago profile views 3,690 I'm an assistant professor at University of Houston. I'm interested in dynamical systems, ergodic theory, thermodynamic formalism, dimension theory, multifractal analysis, non-uniform hyperbolicity, and things along those lines. 67 Proofs without words 36 Is the boundary $\partial S$ analogous to a derivative? 16 Birkhoff ergodic theorem and the measure of the bad points 16 Connection between properties of Dynamical and Ergodic Systems 16 If you were to axiomatize the notion of entropy … # 4,708 Reputation This user has no recent positive reputation changes # 17 Questions 21 Analogues of Luzin's theorem 17 Generic points and local entropies 16 Growth rate of number of loops in a graph 11 Generalisation of Lebesgue decomposition theorem 10 Connectedness of space of ergodic measures # 63 Tags 193 ds.dynamical-systems × 46 36 pr.probability × 8 132 ergodic-theory × 35 36 ho.history-overview × 2 83 reference-request × 14 32 measure-theory × 7 46 gn.general-topology × 3 29 fractals × 5 41 mathematics-education × 2 26 symbolic-dynamics × 6 # 3 Accounts MathOverflow 4,708 rep 11135 Mathematics 473 rep 312 TeX - LaTeX 136 rep 3	{"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": 0.8959107398986816, "perplexity": 4532.468255330197}, "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-2015-14/segments/1427131296456.82/warc/CC-MAIN-20150323172136-00168-ip-10-168-14-71.ec2.internal.warc.gz"}
https://codegolf.stackexchange.com/questions/103571/an-array-of-challenges-1-alternating-arrays/103758	# Alternating Arrays An alternating array is a list of any length in which two (not necessarily different) values are alternating. That is to say, all even-indexed items are equal, and all odd-indexed items are equal. Your task is to write a program or function which, when given a list of positive integers, outputs/returns truthy if it is alternating and falsy otherwise. This is , so the shortest code (in bytes) wins! Edge Cases: [] -> True [1] -> True [1,1] -> True [1,2,1] -> True Other Test Cases: [1,2,1,2] -> True [3,4,3] -> True [10,5,10,5,10] -> True [10,11] -> True [9,9,9,9,9] -> True [5,4,3,5,4,3] -> False [3,2,1,2,1,2] -> False [1,2,1,2,1,1,2] -> False [2,2,3,3] -> False [2,3,3,2] -> False # Example Here is an example you can test your solution against, written in Python 3 (not golfed): def is_alternating(array): for i in range(len(array)): if array[i] != array[i%2]: return False return True • What are the possible values of the elements of the array? – Robert Hickman Dec 19 '16 at 21:06 • @RobertHickman a list of positive integers, within your language's standard int size – FlipTack Dec 19 '16 at 21:12 • oh I see that in the question now. Oops and thanks. – Robert Hickman Dec 19 '16 at 21:12 # Jelly, 6 bytes My first Jelly program! I know I'll never beat Dennis, but I'm awfully proud of this regardless. s2ZE€Ạ // Main link: Argument A (array) s2 // Split A into chunks of 2 // [1,2,1,2] -> [[1,2],[1,2]] Z // Transpose A // [[1,2],[1,2]] -> [[1,1],[2,2]] E€ // Map an equality check to each sublist of A // [[1,1],[2,2]] -> [1,1] Ạ // Any: return 0 if A contains any falsy values, else return 1 // [1,1] -> 1 Try it here! # PHP, 52 bytes Run with -r. foreach($argv as$i=>$n)$i<3\|$n==$argv[$i-2]?:die(1); loops from 3rd to last argument. Continue while argument equals that two positions earlier, else exit with code 1 (error). Exit with 0 (ok) after loop finishes. or for 53 bytes: while(++$i<$argc-3&$f=$argv[$i]==$argv[2+$i]);echo$f; loops through all arguments but the last two. Continue while current argument equals that two positions later. Print 1 for true, nothing for false (string representations of true and false). # Oracle SQL, 73 Bytes select count()from(select a,lag(a,2)over(order by 1)b from t)where a!=b; ## Output: True = 0 False != 0 ### True Example: create table t (a number); truncate table t; insert into t values (10); insert into t values (5); insert into t values (10); insert into t values (5); insert into t values (10); commit; select count()from(select a,lag(a,2)over(order by 1)b from t)where a!=b; COUNT() ---------- 0 ### False Example: create table t (a number); truncate table t; insert into t values (1); insert into t values (2); insert into t values (3); insert into t values (3); insert into t values (1); insert into t values (2); commit; select count() from(select a,lag(a,2)over(order by 1)b from t)where a!=b; COUNT() ---------- 4 # Perl, 27 bytes Includes +1 for p perl -pE '$_=!/\b(\d+) \d+ (?!\1\b)/' <<< "1 2 1 2" # Add++ v5.1, 9 bytes L~,Ñ+B]B= Try it online! # Perl 6, 30 bytes 2>[(0,2...;1,3...)].all.Set Try it online! Anonymous Whatever lambda that takes a list and returns an all Junction that boolifies to True/False. ### Explanation: [( ; )] # Index from the list 0,2... # The even indexed elements 1,3...* # The odd indexed elements .all # Are both the lists .Set # When converted to a set 2> # The length is smaller than 2? # Common Lisp, 62 bytes (defun f(l)(or(not #1=(caddr l))(and(=(car l)#1#)(f(cdr l))))) Try it online! # C, 52 50 49 47 bytes Thanks to @ceilingcat for golfing two bytes! f(i,l)inti;{return--l<2?1:i-i[2]?0:f(++i,l);} Outputs 1 if the array alternates, 0 otherwise. Try it online! # 05AB1E, 5 bytes 2∍s∍Q Try it online! Explanation: 2∍s∍Q //Full Program 2∍ //extend/shorten input to length 2 e.g. [1,2,1,2,2] -> [1,2] s∍ //extend/shorten to length of input e.g. [1,2] -> [1,2,1,2,1] Q //is equal to input e.g. [1,2,1,2,1] != [1,2,1,2,2] • Why is this non-competing? – pppery Oct 1 '19 at 1:49 • @pppery It's non-competing because it uses the latest version of 05AB1E which was released after this challenge was posted – Cowabunghole Oct 1 '19 at 21:14 • Such answers no longer need to be marked non-competing – pppery Oct 1 '19 at 21:26 • Huh, TIL. Thanks for the update! Maybe now that I'm competing my 1 year old answer can win this 3 year old challenge :P – Cowabunghole Oct 3 '19 at 20:31 • – pppery Oct 3 '19 at 20:31	{"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": 0.15257036685943604, "perplexity": 10860.881157645346}, "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-2020-16/segments/1585370507738.45/warc/CC-MAIN-20200402173940-20200402203940-00384.warc.gz"}
https://www.physicsforums.com/threads/escape-velocity-confusion.958564/	# Escape Velocity Confusion • I • Start date • Tags • #1 2 0 Hey there, If body 1, mass M1 has escape velocity V_e1 = (2GM1/r).5 but M2 is more massive than M1 is this relation still valid? In this case, the subordinate body really isn't the subordinate body so does this still hold? And r (distance b/t the two) changes not only due to the motion of M2 but the motion of M1 being dragged by M2 and I'm not sure this equation accounts for that change. I guess my question is whether or not escape velocity accounts for the acceleration of the body being escaped from? If it makes any difference, this question arose as I'm programming a simulation which when using Cowell's method (which accounts for the acceleration of both masses) yields an escape velocity much higher than when using Kepler's (which yields the accepted escape velocity, but the Kepler method also assumes one body to be of negligible mass, namely that the body at the center doesn't move). • #2 44 6 Hey there, If body 1, mass M1 has escape velocity V_e1 = (2GM1/r).5 but M2 is more massive than M1 is this relation still valid? In this case, the subordinate body really isn't the subordinate body so does this still hold? And r (distance b/t the two) changes not only due to the motion of M2 but the motion of M1 being dragged by M2 and I'm not sure this equation accounts for that change. I guess my question is whether or not escape velocity accounts for the acceleration of the body being escaped from? If it makes any difference, this question arose as I'm programming a simulation which when using Cowell's method (which accounts for the acceleration of both masses) yields an escape velocity much higher than when using Kepler's (which yields the accepted escape velocity, but the Kepler method also assumes one body to be of negligible mass, namely that the body at the center doesn't move). Yes, you're right. Kepler's method assumes that the body escaping is far less massive than the body it escapes. So, it appears you've answered your own question. Kepler's method is inappropriate because it assumes the lesser mass is the escaping mass. This is clearly stated in the assumptions of Kepler's laws. We don't say the earth orbits the moon but we say the moon orbits the earth, and the earth orbits the sun. So the velocity needed for the sun to escape the earth gravitation is definitely not equal to the above formula. wherever the sun goes, the earth follows. • #3 Janus Staff Emeritus Gold Member 3,641 1,514 Hey there, If body 1, mass M1 has escape velocity V_e1 = (2GM1/r)**.5 but M2 is more massive than M1 is this relation still valid? In this case, the subordinate body really isn't the subordinate body so does this still hold? And r (distance b/t the two) changes not only due to the motion of M2 but the motion of M1 being dragged by M2 and I'm not sure this equation accounts for that change. I guess my question is whether or not escape velocity accounts for the acceleration of the body being escaped from? If it makes any difference, this question arose as I'm programming a simulation which when using Cowell's method (which accounts for the acceleration of both masses) yields an escape velocity much higher than when using Kepler's (which yields the accepted escape velocity, but the Kepler method also assumes one body to be of negligible mass, namely that the body at the center doesn't move). The more general escape velocity equation is $$V_e= \sqrt{ \frac{2G(m_1+m_2){r}}$$ In most real problems m2 is really small compared to m1, and m1 dominates the equation. If the two masses are more comparable in size, then you need to use the general equation and if m2 is much much more massive than m1 you can just use it in the equation. Ve is just the velocity that the two masses would have to be moving relative to each other. • #4 2 0 Ah! That clears things up nicely. Thank you both! • #5 FactChecker Gold Member 6,182 2,389 Adding missing '}' to @Janus' LaTex: $$V_e= \sqrt{ \frac{2G(m_1+m_2)}{r} }$$ • Last Post Replies 6 Views 2K • Last Post Replies 6 Views 1K • Last Post Replies 4 Views 962 • Last Post Replies 16 Views 7K • Last Post Replies 1 Views 3K • Last Post Replies 5 Views 2K • Last Post Replies 5 Views 4K • Last Post Replies 2 Views 1K • Last Post Replies 4 Views 6K • Last Post Replies 8 Views 1K	{"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": 0.9295263290405273, "perplexity": 927.7129591382378}, "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-2021-17/segments/1618038878326.67/warc/CC-MAIN-20210419045820-20210419075820-00412.warc.gz"}
http://piping-designer.com/index.php/mathematics/geometry/846-apothem	# Apothem Written by Jerry Ratzlaff on . Posted in Geometry • Apothem is the length of a line perpendicular from the center to the midpoint of the side of a regular polygon. • A irregular polygon has no center so there can be no apothem. $$A= r cos {\left( \frac {180} {n} \right)}$$ Where: $$A$$ = apothem $$r$$ = radius $$cos$$ = cosine in degrees $$n$$ = number of sides ## Given Edge formula $$A=\frac {s} {2 tan \left( \frac {180} {n} \right) }$$ Where: $$A$$ = apothem $$s$$ = edge (side length) $$tan$$ = tangent in degrees $$n$$ = number of sides	{"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": 0.7165043950080872, "perplexity": 1907.3010317489336}, "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/1529267867493.99/warc/CC-MAIN-20180625053151-20180625073151-00120.warc.gz"}
https://webot.org/info/en/?search=Fair_coin	A fair coin, when tossed, should have an equal chance of landing either side up. In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin. In theoretical studies, the assumption that a coin is fair is often made by referring to an ideal coin. John Edmund Kerrich performed experiments in coin flipping and found that a coin made from a wooden disk about the size of a crown and coated on one side with lead landed heads (wooden side up) 679 times out of 1000. [1] In this experiment the coin was tossed by balancing it on the forefinger, flipping it using the thumb so that it spun through the air for about a foot before landing on a flat cloth spread over a table. Edwin Thompson Jaynes claimed that when a coin is caught in the hand, instead of being allowed to bounce, the physical bias in the coin is insignificant compared to the method of the toss, where with sufficient practice a coin can be made to land heads 100% of the time. [2] Exploring the problem of checking whether a coin is fair is a well-established pedagogical tool in teaching statistics. Role in statistical teaching and theory The probabilistic and statistical properties of coin-tossing games are often used as examples in both introductory and advanced text books and these are mainly based in assuming that a coin is fair or "ideal". For example, Feller uses this basis to introduce both the idea of random walks and to develop tests for homogeneity within a sequence of observations by looking at the properties of the runs of identical values within a sequence. [3] The latter leads on to a runs test. A time-series consisting of the result from tossing a fair coin is called a Bernoulli process. Fair results from a biased coin If a cheat has altered a coin to prefer one side over another (a biased coin), the coin can still be used for fair results by changing the game slightly. John von Neumann gave the following procedure: [4] 1. Toss the coin twice. 2. If the results match, start over, forgetting both results. 3. If the results differ, use the first result, forgetting the second. The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips and the two flips are independent. This works only if getting one result on a trial doesn't change the bias on subsequent trials, which is the case for most non- malleable coins (but not for processes such as the Pólya urn). By excluding the events of two heads and two tails by repeating the procedure, the coin flipper is left with the only two remaining outcomes having equivalent probability. This procedure only works if the tosses are paired properly; if part of a pair is reused in another pair, the fairness may be ruined. Also, the coin must not be so biased that one side has a probability of zero. This method may be extended by also considering sequences of four tosses. That is, if the coin is flipped twice but the results match, and the coin is flipped twice again but the results match now for the opposite side, then the first result can be used. This is because HHTT and TTHH are equally likely. This can be extended to any power of 2. The expected value of flips at the n game ${\displaystyle E(F_{n})}$ is not hard to calculate, first notice that in step 3 whatever the event ${\displaystyle HT}$ or ${\displaystyle TH}$ we have flipped the coin twice so ${\displaystyle E(F_{n}\|HT,TH)=2}$ but in step 2 (${\displaystyle TT}$ or ${\displaystyle HH}$) we also have to redo things so we will have 2 flips plus the expected value of flips of the next game that is ${\displaystyle E(F_{n}\|TT,HH)=2+E(F_{n+1})}$ but as we start over the expected value of the next game is the same as the value of the previous game or any other game so it doesn't really depend on n thus ${\displaystyle E(F)=E(F_{n})=E(F_{n+1})}$ (this can be understood the process being a martingale ${\displaystyle E(F_{n+1}\|F_{n},...,F_{1})=F_{n}}$ where taking the expectation again get us that ${\displaystyle E(E(F_{n+1}\|F_{n},...,X_{1}))=E(F_{n})}$ but because of the law of total expectation we get that ${\displaystyle E(F_{n+1})=E(E(F_{n+1}\|F_{n},...,F_{1}))=E(F_{n})}$) hence we have: Graph of ${\displaystyle {\frac {1}{P(H)(1-P(H))}}}$the further away ${\displaystyle P(H)}$ is from ${\displaystyle 0.5}$ the further expected number of flips before a successful result. {\displaystyle {\begin{aligned}E(F)&=E(F_{n})\\&=E(F_{n}\|TT,HH)P(TT,HH)+E(F_{n}\|HT,TH)P(HT,TH)\\&=(2+E(F_{n+1}))P(TT,HH)+2P(HT,TH)\\&=(2+E(F))P(TT,HH))+2P(HT,TH)\\&=(2+E(F))(P(TT)+P(HH))+2(P(HT)+P(TH))\\&=(2+E(F))(P(T)^{2}+P(H)^{2})+4P(H)P(T)\\&=(2+E(F))(1-2P(H)P(T))+4P(H)P(T)\\&=2+E(F)-2P(H)P(T)E(F)\\\end{aligned}}} ${\displaystyle \therefore E(F)=2+E(F)-2P(H)P(T)E(F)\Rightarrow E(F)={\frac {1}{P(H)P(T)}}={\frac {1}{P(H)(1-P(H))}}}$ The more biased our coin is, the more likely it is that we will have to perform a greater number of trials before a fair result. References 1. ^ Kerrich, John Edmund (1946). . E. Munksgaard. 2. ^ Jaynes, E.T. (2003). Probability Theory: The Logic of Science. Cambridge, UK: Cambridge University Press. p. 318. ISBN 9780521592710. Archived from the original on 2002-02-05. anyone familiar with the law of conservation of angular momentum can, after some practice, cheat at the usual coin-toss game and call his shots with 100 per cent accuracy. You can obtain any frequency of heads you want; and the bias of the coin has no influence at all on the results!{{ cite book}}: CS1 maint: bot: original URL status unknown ( link) 3. ^ Feller, W (1968). An Introduction to Probability Theory and Its Applications. Wiley. ISBN 978-0-471-25708-0. 4. ^ von Neumann, John (1951). "Various techniques used in connection with random digits". National Bureau of Standards Applied Math Series. 12: 36.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 16, "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": 0.8168520927429199, "perplexity": 455.0478503229217}, "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-2023-06/segments/1674764500356.92/warc/CC-MAIN-20230206145603-20230206175603-00817.warc.gz"}
https://dk.upce.cz/handle/10195/74255/browse?rpp=20&etal=-1&sort_by=-1&type=author&starts_with=L&order=ASC	# Browsing 25 (2019) Scientific papers, Series A by Author Order: Results: Sorry, there are no results for this browse.	{"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": 0.9345337152481079, "perplexity": 9755.647892661562}, "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-2020-05/segments/1579250613416.54/warc/CC-MAIN-20200123191130-20200123220130-00549.warc.gz"}
https://www.albert.io/ie/organic-chemistry-2/aromatic-syntheis-using-amines-2	? Free Version Moderate # Aromatic Syntheis Using Amines 2 ORGO2-9BMWN7 Which reaction sequence shown below would be the MOST effective way to convert benzene into 1,3,5-tribromobenzene? A $1)$ $\rm { HNO}_\rm {3}, \rm{H}_2SO_\rm{4}$ $2)$ $\rm { HCl, Sn}$ $3)$ $\rm{3\text{ }equivalents\text{ }of}\text{ }\rm{Br}_\rm {2} + \rm{FeBr}_3$ $4)$ $\rm {Aq. acid, 100\text{ } °C}$ B $\rm{3\text{ }equivalents\text{ }of}\text{ }\rm{Br}_\rm {2} + \rm{FeBr}_3$ C $1)$ $\rm { HNO}_\rm {3}, \rm{H}_2SO_\rm{4}$ $2)$ $\rm{2\text{ }equivalents\text{ }of}\text{ }\rm{Br}_\rm {2} + \rm{FeBr}_3$ $3)$ $\rm{HCl, Sn}$ $4)$ $\rm { NaNO}_\rm {2}, \rm {HCl, 0-5\text{ } °C}$ $5)$ $\rm{CuBr}$ D $1)$ $\rm { HNO}_\rm {3}, \rm{H}_2SO_\rm{4}$ $2)$ $\rm{HCl, Sn}$ $3)$ $\rm{3\text{ }equivalents\text{ }of}\text{ }\rm{Br}_\rm {2}$ $4)$ $\rm { NaNO}_\rm {2}, \rm {HCl, 0-5\text{ } °C}$ $5)$ $\rm{H}_3PO_\rm{2}$	{"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": 0.15396864712238312, "perplexity": 4450.681124745406}, "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/1480698542520.47/warc/CC-MAIN-20161202170902-00511-ip-10-31-129-80.ec2.internal.warc.gz"}
http://www.oscarfrias.com/work-life-balance-mark-watney-style	# Work-life balance, Mark Watney style Over the past months I got really excited about The Martian, both the movie and the book… But I mean, who hasn’t? It’s one of those super believable sci-fi stories with an awesome character in an incredible setup and lots of SCIENCE. What could be better than that? But ok, nerd side aside, what does this even have to do with achieving work-life balance? Let me give you a little bit of context: I recently moved to a new city, moved in with my wife into a new apartment (new for us at least) with a garden and started a new job. Also I have a super short commute (compared to my previous 3hr/day), so I suddenly found myself with way more time in my hands. I got excited thinking about all the things I was going to do with that time but sadly, what I’ve noticed is that, since my work is really engaging and interesting I let it took many more of those extra hours I thought I had available. Two months in and now I find myself barely having any time to come back home, have dinner and go to bed just to repeat it all over again. But this has to stop. And here’s when Mark Watney comes into play. I’m gonna science the shit out of it. I love the mental process that Mark Watney is supposed to have. Very engineer-y. Maybe too engineer-y, but it just makes a lot of sense and I love when things make a lot of sense. So, following this process, here’s an attempt of calculating my work-life balance. As I understand it, work life balance is achieved when time spent on things related to life is greater than the time spent on things related to work, or: $\sum_{i=0}^{n-1}&space;tl_{i}&space;-&space;\sum_{j=0}^{m-1}tw_{j}&space;\geq&space;0$ Where: $tl_{i}$ = Time spent in a life-related activity, and $tw_{j}$ = Time spent in a work-related activity So first I have to list all the things I “need to”, “have to” and “want to” do with my time every week. Then I can prioritize them and allocate some time to them. So, let’s see… Need to’s: • Work – 8hr/ day = 40hr/week • Eat – 2hr/day (all meals accounted) = 14hr/week • Sleep – 8hr/day = 40hr/week • Shower and bathroom – 1hr (total/day) = 7hr/week Have to’s: • Cook/Bake (I “have to” eat, right?) – 1hr = 7hr/week • House chores (cleaning, laundry, dishes, etc) – .5hr = 3.5hr/week • Personal finances – 1hr/ week • Commute – 1hr/day = 5hr/week Want to’s: • Wife time – As much as possible • Family time (parents, siblings) – 1hr/week • Friends time (drinks, anybody?) – 6hr/week • Gardening – 4hr/week • Exercise (Parkour, Martial arts, gym) – 1hr/day = 7hr/week • Tinkering (oh so many projects) – 5hr/week • Read – .5hr/day = 3.5hr/week • Travel So, all these together add up to 144hr/week plus “as much time as possible” shared with my wife, plus whatever time I have for short trips. Granted, most of the things I do at home I’m sharing them with my wife, but let’s say I want to spend a good whole hour with her every day as quality time in addition to whatever else we might be sharing. So all accounted for adds up to 151hr/week. A week has 168hr so I still have 17hr of free time every week, or 2.4hr every day in which I should find myself having nothing to do! Now, as you may have noticed, exercising is part of the “want to’s” since I honestly am slowly turning into the quintessential couch potato, so that means that I have an extra free hour a day, or 3.4hrs every day that I should have nothing to do. Where the hell are those hours? It’s almost a whole day (23.8hr) every week that I’m losing somewhere somehow! I could be traveling one day every week, I could be taking a class, I could be training so hard! But I just don’t have time for it. What am I missing? I barely use the internet at home, just maybe to watch an episode here and there or a movie on the weekends, but those hours fall into the “as much time as possible” with my wife… maybe we watch too much TV?? Some of it might just be how the schedule gets “forced” by the course of the sun (day/night) and how that forces some activities to happen at specific times, but still… a whole day a week? Come on! So, what I need to do now is really commit to my allocated times, especially the time allocated to the office, and see what happens. I might be able to put some extra hour here and there at the office, but can’t keep working 10, 12, 14hr days. I’m just going to burn out and I really like this job. Also, it’s going to be really cool to see me exercising seriously 1hr a day every day. Hooray for fitness! That being said, it’s time for bed (commitment to the schedule, remember?). I guess I should also allocate some time a week for writing posts. Oh I have so many drafts that I need to finish! (or is it a “want to”?) Am I a space pirate now? Cheers! 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http://www.olerogeberg.com/2011_02_01_archive.html	Friday, February 25, 2011 Fighting publication bias #1 Short version: By having a hierarchy of journals that accept work partly based on a prediction of how important/novel the work seems to be to a few referees and an editor, researchers will • try too hard to find results that will seem to be novel/important • try too hard to reproduce new results and show that they too have found this new novel/important thing • shelve their work (because it seems flawed or because will at best be publishable only in less interesting lower-tier journals) if they fail to reproduce the new novel/important things The current academic publishing system with peer-reviewed journals is an attempt to achieve a lot of different goals at the same time: • Facilitate scientific progress, by • ensuring quality of published research by weeding out work that is riddled with errors, poor methodology etc. through anonymous peer-review by relevant experts • assessing/predicting importance of research and thus how “high up” in the journal hierarchy it should be published, • making research results broadly accessible so that disciplines can build their way brick-by-brick to greater truths • promoting a convergence towards consensus by ensuring reproducibility of research and promoting academic dialogue and debate • Simplify the evaluation of individual researchers (given the above, the number of articles weighted by journal type is a proxy for the importance and quality of your research) • Generate huge profits for publishing houses (To quote an article from Journal of Economic Perspectives, “ • The six most-cited economics journals listed in the Social Science Citation Index are all nonproﬁt journals, and their library subscription prices average about $180 per year. Only ﬁve of the 20 most-cited journals are owned by commercial publishers, and the average price of these ﬁve journals is about$1660 per year. Now, clearly, not all of these goals are compatible – most obviously, it is hard to square rocketing subscription costs with the goal of making research results more accessible. However, the ranking of academics based on where in a hierarchy of journals they have published seems likely to lead to issues as well. If you want to get ahead as a researcher, you need to be published, preferably in good journals. If you want to be published in a good journal you need to do something surprising and interesting. You need to either show that something people think is smart is stupid, or that something people think is stupid is smart. As a result, you get a kind of publication bias that can be illustrated by a simple thought experiment: Imagine that the world is exactly as we think it is. If you drew a number of random samples, the estimates for various parameters of interest would tend to be distributed rather nicely around the true values. Only the researchers “lucky” enough to draw the outlier samples whose estimated parameters were surprising would be able to write rigorously done research that supported new (and false) models of the world that were in line with these (non-representative) results. This is actually not a very subtle point: One out of twenty samples will by definition have results that reject a true null hypothesis at 5% significance level. OK, so let us say ideological bias, fashions and trends in modeling approaches etc. are irrelevant, so the result is published. Right away, this becomes a hot new topic, and anyone else able to reproduce it (read: anyone else drawing random but non-representative samples) get published. And then, gradually, the pendulum shifts – and the interesting and novel thing is to disprove the new result. Now, clearly the above thought model is too simple. For one thing, we don’t know the truth. But the recent New Yorker essay on “The decline effect” sounds like this might be part of what’s going on: all sorts of well-established, multiply confirmed findings have started to look increasingly uncertain. It’s as if our facts were losing their truth: claims that have been enshrined in textbooks are suddenly unprovable. This phenomenon doesn’t yet have an official name, but it’s occurring across a wide range of fields, from psychology to ecology. In the field of medicine, the phenomenon seems extremely widespread, affecting not only antipsychotics but also therapies ranging from cardiac stents to Vitamin E and antidepressants The essay discusses a number of explanations (some of them sort of mystical and new-agish), but also notes the explanation above. When biologist Leigh Simmons failed to replicate a new interesting result, he failed to replicate it: “But the worst part was that when I submitted these null results I had difficulty getting them published. The journals only wanted confirming data. It was too exciting an idea to disprove, at least back then.” For Simmons, the steep rise and slow fall of fluctuating asymmetry is a clear example of a scientific paradigm, one of those intellectual fads that both guide and constrain research: after a new paradigm is proposed, the peer-review process is tilted toward positive results. But then, after a few years, the academic incentives shift—the paradigm has become entrenched—so that the most notable results are now those that disprove the theory. It seems to me that this is an almost unavoidable result of the current journal system, but not an unavoidable result of peer-reviewed journals as such. The problem seems to me to stem from the hierarchy of journals, and from the two tasks we give to referees (assess quality and assess importance/interest). The new open-access mega-journals (PLOS One, Sage Open, etc) that aim to publish all competently done research independently of how “important” it seems should at least mitigate the problem. Not necessarily by making it less important to have a “breakthrough” paper with a seemingly important result, but by making it easier to publish null-results. Monday, February 21, 2011 Rewards and incentives can be OK There’s a result from behavioral economics that increasing rewards and incentives for a behavior (e.g. to get kids to read more books) “crowds out” intrinsic motivation and leaves them less interested in books than before once the rewards dry out. Barking up the wrong tree notes a study that fails to find this when it comes to getting kids to eat vegetables. Good to know for those of us with kids. Liking and intake of the vegetable were assessed in a free-choice consumption task at preintervention, postintervention, 1 month after intervention, and 3 months after intervention. Liking increased more in the three intervention conditions than in the control condition, and there were no significant differences between the intervention conditions. These effects were maintained at follow-up. Children in both reward conditions increased consumption, and these effects were maintained for 3 months; however, the effects of exposure with no reward became nonsignificant by 3 months. These results indicate that external rewards do not necessarily produce negative effects and may be useful in promoting healthful eating. Sunday, February 20, 2011 Economists are usually in favor of free trade. I myself am both an economist and usually in favor of free trade. But I thought this post on “Kids prefer Cheese” which Mark Thoma recently re-blogged had a good and valid point that economists do well to remember: Even if trade benefits the trading partners, that does not mean that a large number of people in a country may not be hurt by allowing free trade. And even if the monetary gains of the winners are bigger in sum than the monetary losses of the losers, that isn’t always a big help for the losers since there is no redistribution automatically triggered making everyone at least as well off as before. Economists usually defend their stance on such issues by talking about Pareto efficiency, saying that making someone better off is always good provided someone else isn’t made worse off by it. Then they switch from talking about Pareto improvements (probably rare in actual policy) to talking about potential Pareto improvements, where the winners could compensate the losers and achieve a true Pareto improvement. Of course, they won’t do so in actuality, which makes the policy also have a redistributive element. A common reply is that redistribution should not be solved through trade measures, but through redistributive policies. But, guess what, most of those aren’t that popular amongst economists either: They distort incentives and reduce efficiency and involve moral hazard problems and, besides, inequality isn’t that horrible anyway. I may be completely wrong, but my guess is many of the economists most adamant about the glories of completely free trade are also amongst those staunchest in opposition to redistributive taxation and public welfare schemes. Though, being a guess, that may be just based on stereotypes and shouldn’t be given too much weight. Anyway, here’s an excerpt: People, the United States is not a person! Only in DSGE models do we assume that all individuals are identical! There is no "our" to which general statements can be attached. Yes, going from autarky to free trade will raise the GDPs of both nations, but that is a very far cry from saying that a large number of individuals will not be made worse off in the process. I figure that NGM is familiar with the Stolper-Samuelson theorem, so I guess he is assuming the political process always provides adequate compensation for the losers?? ROFLMAO, anyone? Individuals should be allowed to contract with whoever they wish, without government interference based solely on geography. Now, that is not much of an economic argument, but, to tell the ugly truth, THERE ISN'T MUCH OF AN ECONOMIC ARGUMENT. Once you factor in agent heterogeneity, imperfect competition, increasing returns, and an arbitrarily large number of traded goods, the welfare economics of free trade is murky at best. More good stuff making the same point here Friday, February 18, 2011 Manipulating maths for whose amusement? Amplify’d from www.technologyreview.com Q&A: The Experimenter Gary Loveman, the CEO of Caesars Entertainment, says there are three ways to get fired from the hotel and casino company: theft, sexual harassment, and running an experiment without a control group. Loveman, who has a PhD in economics from MIT and was a professor at Harvard Business School, has impressed the importance of data analysis on his employees, who are expected to quickly scale small tests into company-wide initiatives. For example, they might test which is likelier to get customers to spend more: a free meal or a free night in a hotel. When you got your economics PhD from MIT in 1989, subdisciplines like behavioral economics and experimental economics had a mixed reputation. Now—a couple of Nobel Prizes in the field later—they seem to be cornerstones of how many businesses and industries try to innovate. My impression is that when I got my PhD, we were really manipulating mathematics for our own amusement, and we weren't producing all that much to help real people make real decisions. That was dissatisfying to me and, frankly, frustrating. The notion that we could do experiments based on the central tenets of economics and have that make a real-world difference was exciting. Of course, with Freakonomics and Predictably Irrational these themes have become more popularized and accessible. It's a very heartening development, and it's increased my enthusiasm for my own discipline enormously. What do you like to tell your academic colleagues about the challenges of real-world experimentation and innovation? Honestly, my only surprise is that it is easier than I would have thought. I remember back in school how difficult it was to find rich data sets to work on. In our world, where we measure virtually everything we do, what has struck me is how easy it is to do this. I'm a little surprised more people don't do this.Read more at www.technologyreview.com Experimental evidence on infinitely repeated games??? Infinity is a loooong time! Surely an ongoing study by definition, reporting on some results from a work in progress. And from the most prestigious economics journal - the American Economic Review - no less. My own criticism of the predictions from the theory of infinitely repeated games would be more directed towards their lack of applicability in my (AFAIK) finite life. However, if I could gain immortality only by agreeing to spend it sitting in a laboratory playing prisoner's dilemma for ever - then I think I would pass. My guess is they have a sample selection problem. And yes, I know I'm being dumb. And no, I'm not being serious. Amplify’d from www.ingentaconnect.com The Evolution of Cooperation in Infinitely Repeated Games: Experimental Evidence Authors: Bó, Pedro Dal; Fréchette, Guillaume R. Source: The American Economic Review,Volume 101, Number 1, February 2011, pp. 411-429(19) Abstract: A usual criticism of the theory of infinitely repeated games is that it does not provide sharp predictions since there may be a multiplicity of equilibria. To address this issue, we present experimental evidence on the evolution of cooperation in infinitely repeated prisoner's dilemma games as subjects gain experience. We show that cooperation may prevail in infinitely repeated games, but the conditions under which this occurs are more stringent than the subgame perfect conditions usually considered or even a condition based on risk dominance. Read more at www.ingentaconnect.com Monday, February 14, 2011 Economists should not be unduly concerned with reality? It’s “Quotes out of context” day today. Here’s a couple of interesting quotes by prominent economists that I came across in a blog-post I stumbled onto. None of them really say anything factually wrong, but they seem (out of context, at least) indicative of an attitude valuing logically correct, sophisticated and elegant mathematical systems over pragmatically useful and informative, well-supported theories about the world. One danger of this is that if we use the word “economic theories” about both logical systems and theories-of-the-world, and if we also say that logical systems are correct or true when they are logically consistent and valued by economists, then it is only a small slip of the mind before we allow our views of the world to be colored and influenced by the logical systems that have yet to be related to reality. There’s one by Samuelson: Nobel Prizewinner Paul Samuelson's conclusion in his famous 1939 article on "The Gains from International Trade": "In pointing out the consequences of a set of abstract assumptions, one need not be committed unduly as to the relation between reality and these assumptions."[3] This attitude did not deter him from drawing policy conclusions affecting the material world in which real people live. And one from the textbook Microeconomics by William Vickery, winner of the 1997 Nobel Economics Prize: "Economic theory proper, indeed, is nothing more than a system of logical relations between certain sets of assumptions and the conclusions derived from them... The validity of a theory proper does not depend on the correspondence or lack of it between the assumptions of the theory or its conclusions and observations in the real world. A theory as an internally consistent system is valid if the conclusions follow logically from its premises, and the fact that neither the premises nor the conclusions correspond to reality may show that the theory is not very useful, but does not invalidate it. In any pure theory, all propositions are essentially tautological, in the sense that the results are implicit in the assumptions made."[4] Thursday, February 10, 2011 Should we see it coming? Michael Lewis has a wonderfully engaging, well-written (long) article about the Irish economic catastrophe in Vanity Fair. Worth reading for all sorts of reasons. Here, I just want to point out the simple arguments and observations used by an economics professor during the boom to argue that there was a housing bubble. It’s puzzling how something that seems obvious in retrospect, based on simple, big-picture statistics that were easily googled at the time, could be so ignored or downplayed or rejected by economists and others alike at the time. The sense that “this time is different,” “past cases don’t apply,” and that all sorts of more or less good “small” arguments are enough to (psychologically?) weaken the impact of the big-picture items. Sometimes, the difficult thing is to just keep pounding on the big, strong, clear argument instead of allowing yourself to get derailed into lots of smaller-scale discussions of all sorts of details that don’t really count for much in the big picture. (It seems to me, for instance on the basis of this graph, that Norwegian house prices are grossly inflated today (the red curve is Norway, the blue US, both in real terms and normalized to 1890 levels)– but when I present this graph to others I constantly get derailed into side-tracks like “building standards are more stringent now than in the past, which might have increased costs”) Morgan Kelly is a professor of economics at University College Dublin, […] Kelly saw house prices rising madly and heard young men in Irish finance to whom he had recently taught economics try to explain why the boom didn’t trouble them. And they troubled him. “Around the middle of 2006 all these former students of ours working for the banks started to appear on TV!” he says. “They were now all bank economists, and they were nice guys and all that. And they were all saying the same thing: ‘We’re going to have a soft landing.’ ” The statement struck him as absurd: real-estate bubbles never end with soft landings. A bubble is inflated by nothing firmer than expectations. The moment people cease to believe that house prices will rise forever, they will notice what a terrible long-term investment real estate has become and flee the market, and the market will crash. It was in the nature of real-estate booms to end with crashes—just as it was perhaps in Morgan Kelly’s nature to assume that, if his former students were cast on Irish TV as financial experts, something was amiss. “I just started Googling things,” he says. Googling things, Kelly learned that more than a fifth of the Irish workforce was employed building houses. The Irish construction industry had swollen to become nearly a quarter of the country’s G.D.P.—compared with less than 10 percent in a normal economy—and Ireland was building half as many new houses a year as the United Kingdom, which had almost 15 times as many people to house. He learned that since 1994 the average price for a Dublin home had risen more than 500 percent. In parts of the city, rents had fallen to less than 1 percent of the purchase price—that is, you could rent a million-dollar home for less than \$833 a month. The investment returns on Irish land were ridiculously low: it made no sense for capital to flow into Ireland to develop more of it. Irish home prices implied an economic growth rate that would leave Ireland, in 25 years, three times as rich as the United States. (“A price/earning ratio above Google’s,” as Kelly put it.) Where would this growth come from? Since 2000, Irish exports had stalled, and the economy had been consumed with building houses and offices and hotels. “Competitiveness didn’t matter,” says Kelly. “From now on we were going to get rich building houses for each other.” The endless flow of cheap foreign money had teased a new trait out of a nation. “We are sort of a hard, pessimistic people,” says Kelly. “We don’t look on the bright side.” Yet, since the year 2000, a lot of people had behaved as if each day would be sunnier than the last. The Irish had discovered optimism. Their real-estate boom had the flavor of a family lie: it was sustainable so long as it went unquestioned, and it went unquestioned so long as it appeared sustainable. After all, once the value of Irish real estate came untethered from rents there was no value for it that couldn’t be justified. The 35 million euros Irish entrepreneur Denis O’Brien paid for an impressive manor house on Dublin’s Shrewsbury Road sounded like a lot until a trust controlled by the real-estate developer Sean Dunne’s wife reportedly paid 58 million euros for a 4,000-square-foot fixer-upper just down the street. But the minute you compared the rise in prices to real-estate booms elsewhere and at other times, you re-anchored the conversation; you biffed the narrative. The comparisons that sprung to Morgan Kelly’s mind were with the housing bubbles in the Netherlands in the 1970s and Finland in the 1980s, but it almost didn’t matter which examples he picked: the mere idea that Ireland was not sui generis was the panic-making thought. “There is an iron law of house prices,” he wrote. “The more house prices rise relative to income and rents, the more they subsequently fall.” Tuesday, February 8, 2011 Why scientists are liberals – some speculative comments The Freakonomics blog discusses political bias in sciences, and quotes a social psychologist who estimated that 80% of attendees at a conference were liberals (he asked for a show of hands). Dubner seems worried that political views will shape research conclusions, and writes: How can it be that an academic field is so politically homogeneous? What kind of biases does such homogeneity produce? What sort of ideas get crowded out? And how homogeneous are other disciplines? I have to say that I was surprised at the overt political (leftward) bias exhibited by several prominent economists at the recent American Economics Association meetings, although my sample set was quite small. It is interesting — and sobering — that two fields, psychology and economics, that we rely upon to describe and amend bias in the world are themselves so susceptible to bias within the ranks of their practitioners. Krugman disagrees, implying that research conclusions probably push attitudes towards the liberal side, saying Biologists, physicists, and chemists are all predominantly liberal; does this reflect discrimination, or the tendency of people who actually know science to reject a political tendency that denies climate change and is broadly hostile to the theory of evolution? Now, I don’t mean to say that political bias in the academy is absent, although it’s not consistent: I can well imagine that it’s hard to be a conservative in some social sciences, but in economics, the obvious bias in things like acceptance of papers at major journals is towards, not against, a doctrinaire free-market view. But the point is that doing head counts is a terrible way to assess that bias. It might be that the most recent amusing statistical post on the blog for dating service OK Cupid has the answer. The post analyzes its database to identify the most unthreatening, innocent questions that best predict characteristics that you may not want to ask about directly (whether they’re religious, would have sex on a first date, their political ideology etc.). Based on their national US data they write that the question identifying politics is • Do you prefer the people in your life to be simple or complex? Because... We were very surprised to find that this one question very strongly predicts a person's ideas on these divisive issues: Should burning your country's flag be illegal? Should the death penalty be abolished? Should gay marriage be legal? Should Evolution and Creationism be taught side-by-side in schools? In each case, complexity-preferrers are 65-70% likely to give the Liberal answer. And those who prefer simplicity in others are 65-70% likely to give the Conservative one. Seems to me that this is pretty consistent with the “bias” in academia. Academia is often very much concerned with complexity – finding nuances in interpretations and methods, considering alternative explanations for patterns in data, etc. If you prefer simplicity as a general trait in people and thoughts you would probably be pretty frustrated as an academic. And a 2:1 ratio is roughly 66%, which isn’t that far away from the estimate of 80% that we started with. Sidenote: Interesting that Dubner is so worried by the left-wing attitudes of economists he encountered, given that his freakonomics podcast on how the world would look if it was driven by the gloriously rational economists basically said they would implement Milton Friedman’s pretty libertarian proposals. His (presumably unbiased and representative?) economist picked to answer on the behalf of the profession was Russ Roberts at George Mason University, who answered that his policy program would start with some obvious things. I would get rid of the Department of Commerce. The Department of Commerce doesn’t do anything except subsidize exports, which is just a way of saying it makes certain companies rich at the expense of the rest of us. So I don’t think the Department of Commerce does anything particularly useful, I would get rid of that. I’d get rid of the Department of Education. I don’t think that the Federal Government has any productive role to play in the school system. I’d get rid of all tariffs. I’d let people be free to buy whatever they wanted from all around the world. What else? I would get rid of the minimum wage law, which I think makes it hard for low-skilled people to find work; it makes them artificially expensive. I’d change the Federal Reserve. We spend a lot of time trying to find the right interest rate. That’s a fool’s game that has contributed to the current crisis. So I would change the Federal Reserve. I would certainly at a minimum require it to only care about price stability. Right now it cares about price stability, unemployment, the health of the stock market, Wall Street salaries, evidently. So I would get all of those things out. It’s going to be hard to do legislatively, so I would probably replace the the Fed with a Friedmanite fixed growth and money supply or just abolish it entirely and let private money emerge. I’m getting out of control here. You don’t say… Update: More links and discussion here from McArdle in the Atlantic. Her take seems to be that there is a bias, that it is amusing to see conservatives (usually dismissive of bias accusations) believe it and liberals (usually sympathetic to bias accusations) dismissive, that it is unsolvable, and that we should all just try harder to get along and see each other's point of view. Thursday, February 3, 2011 Why intuitive stories are important and dangerous Below are some excerpts from a blogpost on the importance of "simple" stories/models that Paul Krugman praised on his NYT blog. I fail to find a simple moral to the story as it seems (to me) to involve a lot of different views on this issue, such as (in my formulations): "Simple case-stories/thought-experiments are a necessary adjunct to sophisticated models/theories - because we cannot reason using models but need simplified versions that our brains can grasp" "Simple case-stories/thought-experiments are rhetorically convincing in discussions/debates" "Simple case-stories/thought-experiments trigger the psychological feeling of understanding/insight which is a better signal of truth than other kinds of evidence" "Formal/standard models in economic theory are accepted because other economists accept them (emperor's new clothes) but people who accept them don't really understand the mechanisms they involve" "Economists are confused concerning what it takes to evaluate claims about the real world" Here's part of Krugman's comment on the same post (http://krugman.blogs.nytimes.com/2011/02/02/models-plain-and-fancy/ ): "I have nothing against mathematical models and econometrics. But my experience is that many misunderstandings in economics come about because people don’t have in their minds any intuitive notion of what it is they’re supposed to be modeling. The whole notion of an economy-wide shortfall in demand is just hard to grasp — by famous economists as well as the lay public; quite a lot of our hopeless public debate reflects the fact that many people, some of them imagining themselves to be sophisticated about the issue, just can’t visualize what Keynesian ideas are about. But the baby-sitting coop offers a human-scale example, and makes the whole thing clear." Amplify’d from modeledbehavior.com more than any other analysis the baby-sitting coop story made me a confident Keynesian. Before then I could parrot the New Keynesian models and understood that this was more or less what a smart economist was supposed to say. However, I didn’t know how to counter the logic of Laizze Faire except to say, “well there are sticky prices and an Euler equation and so the household will adjust consumption . . . “ This is compelling to virtually no one – not even, on a deep level, to myself. When it really came down to it, I would have been left with “Great Depression! Want it to happen again? No? Then we need to spend more money or cut taxes! Why? Because I am very smart and I have a whiteboard. Do you have a whiteboard?” However, a simple story about baby-sitting and it all fell into placeRead more at modeledbehavior.com	{"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": 0.3522631824016571, "perplexity": 1894.7482365484343}, "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/1505818689373.65/warc/CC-MAIN-20170922220838-20170923000838-00544.warc.gz"}
http://web2.0calc.com/questions/12-1-2-4-2-5-8-3-4	+0 # 12 1/2 + 4 2/5 - 8 3/4 = 0 173 1 12 1/2 + 4 2/5 - 8 3/4 = Guest Apr 20, 2017 Sort: #1 +18777 0 12 1/2 + 4 2/5 - 8 3/4 = 1. $$\begin{array}{\|rcll\|} \hline && 12 \frac12 + 4 \frac25 - 8 \frac34 \\ &=& 12+4-8 + \frac12 + \frac25 - \frac34 \\ &=& 8 + 0.5 + 0.4 - 0.75 \\ &=& 8 + 0.15 \\ &=& 8.15 \\ &=& 8 \frac{15}{100} \\ &=& 8 \frac{3}{20} \\ \hline \end{array}$$ 2. $$\begin{array}{\|rcll\|} \hline && 12 \frac12 + 4 \frac25 - 8 \frac34 \\ &=& 12+4-8 + \frac12 + \frac25 - \frac34 \\ &=& 8 + \frac12\cdot \frac{10}{10} + \frac25\cdot \frac{4}{4} - \frac34\cdot \frac{5}{5} \\ &=& 8 + \frac{10}{20} + \frac{8}{20} - \frac{15}{20} \\ &=& 8 + \frac{10+8-15}{20} \\ &=& 8 \frac{3}{20} \\ \hline \end{array}$$ 3. $$\begin{array}{\|rcll\|} \hline && 12 \frac12 + 4 \frac25 - 8 \frac34 \\ &=& \frac{2\cdot 12+1}{2} + \frac{4\cdot 5+2}{5} + \frac{4\cdot 8+3}{4} \\ &=& \frac{25}{2} + \frac{22}{5} + \frac{35}{4} \\ &=& \frac{25}{2}\cdot \frac{10}{10} + \frac{22}{5}\cdot \frac{4}{4} - \frac{35}{4}\cdot \frac{5}{5} \\ &=& \frac{250}{20} + \frac{88}{20} - \frac{175}{20} \\ &=& \frac{250+88-175}{20} \\ &=& \frac{163}{20} \\ &=& 8 \frac{3}{20} \\ \hline \end{array}$$ heureka Apr 20, 2017 ### 19 Online Users We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners. See details	{"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": 0.9224396347999573, "perplexity": 6339.548458576233}, "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/1512948537139.36/warc/CC-MAIN-20171214020144-20171214040144-00782.warc.gz"}
https://www.sparrho.com/item/the-power-of-vertex-sparsifiers-in-dynamic-graph-algorithms/1a7c2f8/	The Power of Vertex Sparsifiers in Dynamic Graph Algorithms Research paper by Gramoz Goranci, Monika Henzinger, Pan Peng Indexed on: 18 Dec '17Published on: 18 Dec '17Published in: arXiv - Computer Science - Data Structures and Algorithms Abstract We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for $(1+\varepsilon)$-approximating the energy of electrical flows in $n$-vertex planar graphs with $\tilde{O}(r\varepsilon^{-2})$ worst-case update time and $\tilde{O}((r+\frac{n}{\sqrt{r}})\varepsilon^{-2})$ worst-case query time, for any $r$ larger than some constant. For $r=n^{2/3}$, this gives $\tilde{O}(n^{2/3}\varepsilon^{-2})$ update time and $\tilde{O}(n^{2/3}\varepsilon^{-2})$ query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.	{"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": 0.692121684551239, "perplexity": 769.0984143038747}, "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-2018-30/segments/1531676590866.65/warc/CC-MAIN-20180719105750-20180719125750-00198.warc.gz"}
https://www.gamedev.net/forums/topic/576296-smooth-camera-movement/	• 12 • 10 • 10 • 13 • 10 # Smooth camera movement This topic is 2786 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic. ## Recommended Posts I want to implement a camera similar to the one from Homeworld 2. For those of you who don't know it: It has some sort of focus point and you orbit around it using the mouse (and change the radius with the mouse wheel). Implementing that alone is a trivial task, however I've been thinking about how I can smooth the camera movement. Usually I would divide both positions into an actual position and target: The actual position will then need to change over time to reach the target. I could come up with a certain speed (based on the distance between the actual and target positions) that moves the camera. However this approach will not work if I change the target position in between: vector3 targetFocus, actualFocus;// Per updatevector3 delta = targetFocus - actualFocus;float dist = delta.length();float maxSpeed = ...;vecto3 speed = delta / length * std::min(maxSpeed, length); // Maybe this can be simplifiedactualFocus += speed; Permanently storing the speed of the camera and using acceleration instead might work, however I must ensure that the camera doesn't miss the target (ie. doesn't deccelerate too slow / accelerates too fast). But maybe there is a simple approach I don't think of: What else can I do to ensure a smooth movement of such a camera? ##### Share on other sites Maybe you could increase the speed by acceleration which is based on the distance between the actual target and a projected target which would be reached using the current speed. This way acceleration would become negative if the camera would miss the target. And is the sign of speed indicates the rotation direction the sign of the acceleration would indicate the same, i.e. you always accelerate in one direction and if that direction is opposite to the current speed the effect is decceleration (or negative acceleration). If I'm not completely wrong this approach could also help with situation where the new target is set to be on the opposite side of the current target, i.e. when the target is to the right, the camera rotates left and you now set the new target on the left of the camera which would mean the camera should stop smoothly and then begin to rotate right. I hope this is understandable, if not, please don't hesitate to reveal my poor skill of describing things in a way that others can understand [smile]. ##### Share on other sites I think I know what you mean, but how can I ensure that the camera never accelerates too much? Ideally I want to prevent the camera from missing the target, otherwise the camera could bounce around the target for several times. ##### Share on other sites This is off the top of my head so be careful: you'll probably want to fix a maximum acceleration (to prevent unacceptable changes in velocity?) and maximum velocity (to prevent unacceptable changes in position between frames). Given that, when a target is identified and current velocity is 0, calculate target_distance = target_pos - camera_pos. Then accelerate each frame until you reach max velocity. Save the distance traveled from the start position (accelDist) as the distance needed to decelerate at max deceleration. Continue at max velocity until you reach a position accelDist from the target. Then decelerate to the target. If you reach midpoint before reaching max velocity, just start decelerating. Note: that's an underdamped system to prevent overshoot and is the slowest way to get there. You may want to set the target used for calcs to something beyond the actual target and just set the camera to the target when you're within some delta. Note2: that doesn't address a change in the target when velocity > 0. Have to do a bit more thinking as max accelerations and max velocities have to be handled. It would have to do with calculating the extra distance to decelerate from current velocity to 0 at max deceleration ( 0 = v - 1/2att; solve for t. Calculate distance. ) EDIT: duh.. accelDist can be pre-calculated. ##### Share on other sites This is actually really easy to do. Just define your start and target coordinates as well as a desired travel time, and use a non-linear interpolator to move the camera between the two. Depending on the curve you select, you can get different acceleration/deceleration profiles and therefore different visual feels for the movement itself. Handling target point changes is also easy; just keep a history of the last, say, 3 target points. Each time you reach a target point, erase it from the history. Then, during interpolation, do a weighted sum of the history instead of just using the final interpolated value. By decreasing the weight of a historical target sharply over time, and subsequently increasing/blending up the weight of the new target, you can get a pretty convincing "oops, slow down and turn around" effect going. ##### Share on other sites Would that work for a moving target? I was recently trying some things for smooth transitions of my camera, rather than it's position just changing instantly, where each frame it moves a portion of the distance it needs to go. The trouble with that is it never actually gets there. It's a fine effect for some games, like the camera is on elastic behind the target, but in mine, a 2D solar system game, the targets for the camera are different planets, which are always moving, so the camera moves smoothly towards the target when its changed but never actually gets there. It settles a distance away such that its first step towards the target is just enough to keep up but never catch it. ##### Share on other sites I did this recently, if you look up B-Spline curves you can get very smooth and natural feeling movement. For a B-Spline curve of degree 3 you need 4 control points, the way I did it was by storing an internal list of 5 control points, and I'd change the fifth to whatever target position I wanted (then duplicate it whenever I passed a segment). I stored a scaling factor (called timeFactor) that adjusted the speed it traveled along a curve segment, with a value of 1 it takes exactly one second to travel along a segment, which means you will not notice a deviation from the course for 1 second after you tell it to move to a new position. void ElasticCamera::Update(float dt){ //ElasticCamera::t is the distance along the current curve, range [0,1) this->t += dt * this->timeFactor; if(this->t >= 1.0f) { while(this->t >= 1.0f) { this->t -= 1.0f; } this->points.erase(this->points.begin()); while(this->points.size() < 4) { this->points.push_back(this->points.back()); } } InterpolatePosition();}void ElasticCamera::MoveTo(const Vector3 &point){ // Only ever keep five points in the list. while(this->points.size() >= 5) { this->points.pop_back(); } this->points.push_back(point);}void ElasticCamera::SetPosition(const Vector3 &position){ this->points.assign(4, position);}void ElasticCamera::SetTimeFactor(float timeFactor){ if(timeFactor >= 0.0f) { this->timeFactor = timeFactor; }}void ElasticCamera::AdjustSpeed(float scalar){ if(scalar >= 0.0f) { this->timeFactor = scalar; }}float ElasticCamera::Basis(int i, float t){ // The performance of this function can be improved. switch (i) { case -2: return (((-t + 3.0f) t - 3.0f) * t + 1.0f) / 6.0f; case -1: return (((3.0f * t - 6.0f) * t) * t + 4.0f) / 6.0f; case 0: return (((-3.0f * t + 3.0f) * t + 3.0f) * t + 1.0f) / 6.0f; case 1: return (t * t * t) / 6.0f; default: return 0.0f; }}void ElasticCamera::InterpolatePosition(){ Vector3 position; std::list<Vector3>::const_iterator point = this->points.begin(); for (int i = -2; i <= 1; ++i) { position += point Basis(i, this->t); if(++point == this->points.end()) { //If we hit this on anything other than the last loop we've got problems. break; } } this->position = position;} Edit: You will never settle on the target point unless all four control points are equal, so this may not work like you want with moving targets. You will find the position is between the second and third control point, so that may be acceptable for you. ##### Share on other sites I just solved mine actually. One of those glorious moments of realization. I had been moving the camera after the target with this camera.x += (target.x - camera.x) * 5 * GameAccess.elapsed;camera.y += (target.y - camera.y) * 5 * GameAccess.elapsed; But it never caught it obviously enough. So I changed things a bit so the camera's position was relative to its target, so relative to the target it kept approaching what it considered a static point. Then converted the relative position back to absolute to render things properly. When changing target I set the camera's relative position to be its current absolutePos - target.absolutePos and use this to approach it. Quickly heads towards the target and gradually comes to a stop. _relativePosition.x += (0 - _relativePosition.x) * 5 * GameAccess.elapsed;_relativePosition.y += (0 - _relativePosition.y) * 5 * GameAccess.elapsed; Simple once I realised to do it relative. I may change the interpolation to more of a curve but that's look and feel stuff for later. I'll remember B-splines for when the time comes. Thanks	{"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": 0.15974389016628265, "perplexity": 1337.9173660352976}, "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-2018-13/segments/1521257647584.56/warc/CC-MAIN-20180321063114-20180321083114-00030.warc.gz"}
https://www.physicsforums.com/threads/peskin-and-schroeder-eq-2-52.344358/	# Peskin and Schroeder eq. (2.52) 1. Oct 9, 2009 ### sizzleiah Hi there. I've just finished reading chapter 2 of Peskin and Schroeder, and I managed to follow all of their calculations - with one exception: 1. The problem statement, all variables and given/known data I'm not sure how P&S arrive at the integral in equation (2.52) (page 27) from the previous step in the calculation of D(x-y). 2. Relevant equations We're trying to calculate $$D(x-y)=<0\|\phi (x) \phi (y) \|0>$$ for a real Klein-Gordon scalar field $$\phi$$, where $$x-y$$ is purely spatial. 3. The attempt at a solution Getting to the step right before eq. (2.52) is easy enough - it's just a standard integration in spherical coordinates. Then P&S make branch cuts to create a simply connected domain, so that they can apply path independence to the contour integration. I'm ok with all of that, but then they lose me when they write down the integral in eq. (2.52). It's confusing to me for a couple of reasons. One is that I'm not entirely sure how to deal with a contour that goes off to infinity in this way - where we can't restrict the variable of integration to be real (doesn't the complex plane only have one infinity?). Another is that it seems that for the lower limit of the integration to be valid, P&S are claiming that we have $$p=i m$$. Are they implying that we should be integrating along the branch cut? This seems very strange to me. I'm obviously no complex analyst, but I knew enough to be able to understand fairly easily what they did on the next few pages with the Feynman propagator. So...what am I missing? Thanks! 2. Oct 10, 2009 ### Ben Niehoff There IS only one infinity in the complex plane, which is precisely why this contour deformation works. The original contour is along the real axis. Instead, we push it up so that it is along the branch cut. That is, the first part of the contour comes down on the left side of the branch cut, then goes around the pole, and then back up the right side of the branch cut. So, splitting the contour up piecewise, there are really three integrals: $$\int_{i\infty - \varepsilon}^{im - \varepsilon} f(p) \; dp\; + \int_{\pi/4}^{9\pi/4} f(im + \varepsilon e^{i\theta}) \; d\theta \; + \int_{im + \varepsilon}^{i\infty + \varepsilon} f(p) \; dp$$ The middle integral vanishes. The third integral is equal to the first integral times a phase (this phase is incurred by going around the pole to get to the other side of the branch cut). Evidently, the phase is -1; thus giving us +1 when we reverse the limits of integration to match the first integral. Notice in P&S 2.52 that a factor of 2 has been canceled from the previous line; this is due to taking the sum of the first and third integral in my expression above. 3. Oct 10, 2009 ### sizzleiah Hey, thanks a lot for responding. It looks like my complex analysis is more rusty than I thought. Why are the limits of integration on the 2nd integral not $$-\pi$$ to 0? Naively, I would think that we'd just integrate from the left part of the contour over to the right part along a semicircle in this way. For the 3rd integral, after we've reversed the limits of integration, you're saying that we can change the $$+\epsilon$$ in the limits to $$-\epsilon$$ at the cost of a -1 phase? I don't see why that is. 4. Oct 10, 2009 ### Ben Niehoff You can, but it's easier to make a full circle, because then all the dependence on the pole is isolated in the middle integral. Look at f(p). Put in $p = im + \varepsilon e^{i\pi/4}$ and $p = im + \varepsilon e^{i9\pi/4}$. You should get a phase difference of -1. Also, note that we're taking the limit as $\varepsilon \rightarrow 0$. 5. Oct 10, 2009 ### sizzleiah Ok, I get it. Thanks a lot for the help.	{"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": 0.9553232192993164, "perplexity": 254.17520230877184}, "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-2018-13/segments/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00330.warc.gz"}
https://opus.bibliothek.uni-wuerzburg.de/frontdoor/index/index/year/2022/docId/26539	Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems Please always quote using this URN: urn:nbn:de:bvb:20-opus-265399 • In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping andIn this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation 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": 0.9968345165252686, "perplexity": 193.6603491242683}, "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/1679296949533.16/warc/CC-MAIN-20230331020535-20230331050535-00366.warc.gz"}
http://www.freesoftwaredwnload.com/tag/windows-nt2000	# Tag Archives: Windows NT/2000 ## Easy Office Recovery Easy Office Recovery, Recover erased Microsoft Office files and recover valid information with Easy Office Recovery. Working with all versions of Microsoft Office from Office 95 to 2010, Easy Office Recovery will fix Microsoft Word documents, recover Excel spreadsheets, repair PowerPoint documents, and recover Microsoft Outlook mail, notes and attachments. Its proprietary document recovery algorithms can extract valid data from documents corrupted badly enough to be rejected by Microsoft Office built-in recovery tools. Including modern, proprietary information recovery algorithms, Easy Office Recovery can often achieve outstanding results in recovering valid data out of severely corrupted documents. Tested on thousands of documents in various condition, Easy Office Recovery working in full auto mode was able to provide the results of the work of a qualified data recovery specialist 95% of the time. The ability of Easy Office Recovery to automatically find and recover missing and deleted Office documents from healthy, damaged, crashed and inaccessible hard drives makes it an easy choice for all types of users. Even if a document was partially overwritten or damaged beyond repair, Easy Office Recovery can still discover and extract usable data out of the file. Fully visual pre-recovery preview of all types of Office documents is available. The preview displays all formatting, recognizes and shows embedded pictures and objects. Working in all 32-bit and 64-bit versions of Windows, Easy Office Recovery can undelete and repair spreadsheets, presentations, messages, contacts and appointments created in Microsoft Word, Excel, PowerPoint, and Outlook versions 95 through 2010. The office recovery software can recover office files from hard disks, memory cards, and flash memory drives. Easy Office Recovery	{"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": 0.8903003931045532, "perplexity": 7838.054549976181}, "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/1508187822747.22/warc/CC-MAIN-20171018051631-20171018071631-00725.warc.gz"}
https://pub.uni-bielefeld.de/record/1784701	Exact equation of state for ideal relativistic quantum gases Karsch F, Miller D (1980) Physical Review, A 22(3): 1210-1219. Zeitschriftenaufsatz \| Veröffentlicht \| Englisch Autor*in Abstract / Bemerkung The use of analytical methods yields the fugacity of the ideal relativistic quantum gases in terms of the particle densities, pressures, and energy densities, which we compute numerically. The exact equation of state in the thermodynamic limit arising from these solutions may be computed and compared with their respective series expansions. Erscheinungsjahr 1980 Zeitschriftentitel Physical Review, A Band 22 Ausgabe 3 Seite(n) 1210-1219 ISSN 0556-2791 Page URI https://pub.uni-bielefeld.de/record/1784701 Zitieren Karsch F, Miller D. Exact equation of state for ideal relativistic quantum gases. Physical Review, A. 1980;22(3):1210-1219. Karsch, F., & Miller, D. (1980). Exact equation of state for ideal relativistic quantum gases. Physical Review, A, 22(3), 1210-1219. doi:10.1103/PhysRevA.22.1210 Karsch, F., and Miller, D. (1980). Exact equation of state for ideal relativistic quantum gases. Physical Review, A 22, 1210-1219. Karsch, F., & Miller, D., 1980. Exact equation of state for ideal relativistic quantum gases. Physical Review, A, 22(3), p 1210-1219. F. Karsch and D. Miller, “Exact equation of state for ideal relativistic quantum gases”, Physical Review, A, vol. 22, 1980, pp. 1210-1219. Karsch, F., Miller, D.: Exact equation of state for ideal relativistic quantum gases. Physical Review, A. 22, 1210-1219 (1980). Karsch, Frithjof, and Miller, David. “Exact equation of state for ideal relativistic quantum gases”. Physical Review, A 22.3 (1980): 1210-1219. Alle Dateien verfügbar unter der/den folgenden Lizenz(en): This Item is protected by copyright and/or related rights. [...] Volltext(e) Name Access Level Open Access	{"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": 0.941659152507782, "perplexity": 4870.568658269444}, "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-2020-45/segments/1603107902038.86/warc/CC-MAIN-20201028221148-20201029011148-00616.warc.gz"}
https://emplatform.eu/publications/details?id=520	Selected Publications ## Effects of photon statistics in wave mixing on a single qubit W. V. Pogosov, A. Yu. Dmitriev, O. V. Astafiev In quantum optomechanics, finding materials and strategies to limit losses has been crucial to the progress of the field. Recently, superfluid 4He was proposed as a promising mechanical element for quantum optomechanics. This quantum fluid shows highly desirable properties (e.g., extremely low acoustic loss) for a quantum optomechanical system. In current implementations, superfluid optomechanical systems suffer from external sources of loss, which spoils the quality factor of resonators. In this work, we propose an alternate implementation, exploiting nanofluidic confinement. Our approach, based on acoustic resonators formed within phononic nanostructures, aims at limiting radiation losses to preserve the intrinsic properties of superfluid 4He. In this work, we estimate the optomechanical system parameters. Using recent theory, we derive the expected quality factors for acoustic resonators in different thermodynamic conditions. We calculate the sources of loss induced by the phononic nanostructures with numerical simulations. Our results indicate the feasibility of the proposed approach in a broad range of parameters, which opens prospects for more complex geometries. Phys. Rev. A 104 (2021) 023703 doi: 10.1103/PhysRevA.104.023703 arxiv: https://arxiv.org/abs/2107.10769	{"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": 0.8141166567802429, "perplexity": 2275.969778926776}, "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-2021-39/segments/1631780056856.4/warc/CC-MAIN-20210919095911-20210919125911-00698.warc.gz"}
https://www.arxiv-vanity.com/papers/hep-th/9704079/	###### Abstract We study the asymptotic solutions of the Schrödinger equation for the color-singlet reggeon compound states in multi-color QCD. We show that in the leading order of asymptotic expansion, quasiclassical reggeon trajectories have a form of the soliton waves propagating on the 2-dimensional plane of transverse coordinates. Applying the methods of the finite-gap theory we construct their explicit form in terms of Riemann theta-functions and examine their properties. LPTHE–Orsay–97–13 hep-th/9704079 March, 1997 Solitons in high-energy QCD G. P. Korchemsky and I. M. Krichever Laboratoire de Physique Théorique et Hautes Energies***Laboratoire associé au Centre National de la Recherche Scientifique (URA D063) Université de Paris XI, Centre d’Orsay, bât. 211 91405 Orsay Cédex, France Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia Department of Mathematics, Columbia University, U.S.A. Landau Institute for Theoretical Physics, Kosygina str. 2, 117940 Moscow, Russia ## 1. Introduction It widely believed that the Regge asymptotics of hadronic scattering amplitudes in high-energy QCD should be described by an effective Regge theory. In this effective theory reggeized gluons, or reggeons, play the role of a new collective degrees of freedom. Reggeons form a color singlet compound states, QCD Pomerons and Odderons, which propagate in the channel between scattering hadrons and contribute to a power rise with energy of the physical cross-sections [1]. In the Bartels-Kwiecinski-Praszalowicz approach (BKP) [2], the color singlet reggeon compound states are built from a conserved number of reggeons. For a given number of reggeons, , the wave function of these states depends only on the transverse reggeon coordinates () and it satisfies the BKP equation HN\|χN⟩=εN\|χN⟩ (1.1) where is an effective QCD Hamiltonian acting on the transverse coordinates of interacting reggeons. QCD Pomeron and Odderon appear as the states in the spectrum of with the maximal energy . The BKP equation has a number of remarkable properties in multi-color limit of QCD, and . Firstly, introducing holomorphic and antiholomorphic coordinates on 2-dim plane of transverse reggeon coordinates, and , respectively, one can find that the wave function splits into the product of holomorphic and antiholomorphic parts [3] χN(x1,y1,...,xN,yN;x0,y0)=φN(z1−z0,...,zN−z0)⋅¯¯¯¯φN(¯z1−¯z0,...,¯zN−¯z0), where is the coordinate of the center-of-mass of the reggeon compound state. The effective hamiltonian is invariant under transformations zj→azj+bczj+d,¯zj→¯a¯zj+¯b¯c¯zj+¯d, (1.2) with and , while the wave function is transformed as χN→(cz0+d)2h(¯c¯z0+¯d)2¯hχN. (1.3) The conformal weights take the values and with corresponding to the principal value representation of the group. Secondly, the QCD effective Hamiltonian is closely related to the Hamiltonian of XXX Heisenberg magnet of noncompact spin and, as a consequence, the system of interacting reggeons possesses a large enough family of conserved charges in order for the Schrödinger equation (1.1) to be completely integrable [4, 5]. This implies that instead of solving the BKP equation one can define the holomorphic wave function as a common eigenfunction of the family of mutually commuting operators qk=∑1≤j1 with and . There are also additional constraints on (L3+h)φN(z1,...,zN)=0,L+φN(z1,...,zN)=0, (1.5) which follow from the transformation properties of the wave function (1.3). Here, the notation was introduced for the generators L3=iN∑k=1zkpk,L+=iN∑k=1z2kpk,L−=−iN∑k=1pk with . The conserved charges defined in (1.4) have the meaning of higher Casimir operators of the group those eigenvalues determine the quantum numbers of the reggeon compound states. In particular, q2=−∑j>kz2jkpjpk=−L3L3−12(L−L++L+L−) is a quadratic Casimir operator and its eigenvalue corresponding to the wave function is given by provided that conditions (1.5) are fulfilled. To find the holomorphic reggeon wave function one has to diagonalize simultaneously the remaining commuting hamiltonians , , on the space of functions satisfying (1.4). This leads to a system of coupled partial differential equations on the wave function . The same system can be also interpreted [6] as a set of Schrödinger equations, in which the conformal weight plays the role of the effective Planck constant, . This suggests to consider the quasiclassical limit of large and develop the WKB expansion for the holomorphic reggeon wave function φN(z1,...,zN)=exp(iS0+iS1+...), (1.6) where , , at large and . In a complete analogy with quantum mechanics, the leading term, , defines the action function for the system of reggeons. For a given set of quantum numbers , it satisfies the system of the Hamilton-Jacobi equations N∑k=1zk∂S0∂zk=ih,N∑k=1z2k∂S0∂zk=0,ˆqα(→z,∂S0∂→z)=qα,α=3,...,N, (1.7) where and stands for the symbol of the operator (1.4). For reggeon state, the BFKL Pomeron, the Hamilton-Jacobi equations (1.7) can be easily solved S0(z1,z2)=−ihln(1z1−1z2),φ2(z1,z2)=(z12z1z2)h×[1+O(h−1)]. The last expression defines the holomorphic wave function in the leading order of the WKB expansion. Comparing it with the well-known expression for the wave function of the BFKL Pomeron [7] we find that it is exact to all orders in . For reggeon states, the Hamilton-Jacobi equations (1.7) can be solved in the separated variables [8]. Their solutions describe the compound reggeon states as a system of classical particles moving along quasiperiodic trajectories. The corresponding action-angle variables were constructed in [8], where the close relation between classical reggeon trajectories and finite-gap soliton solutions [9] was established. The main goal of the present paper is to apply algebro-geometrical methods [10] in order to obtain the explicit form of the reggeon trajectories on the dimensional plane of transverse coordinates in terms of Riemann theta-functions [11]. ## 2. Hamiltonian flows In the leading order of the WKB expansion, we replace in (1.4) the momenta operators by the corresponding classical functions to get the family of classical Hamiltonians , , . For the system of reggeons with the coordinates , momenta and the only nontrivial Poisson bracket , these hamiltonians generate the hierarchy of the evolution equations ∂zn∂tα={zn,qα}=∂qα∂pn,∂pn∂tα={pn,qα}=−∂qα∂zn, (2.1) with being the corresponding evolution times and . Their solutions define the reggeon trajectories subject to the periodicity condition zk+N(t)=zk(t),pk+N(t)=pk(t). (2.2) The system of evolution equations (2.1) has a sufficient number of the integrals of motion to be completely integrable. It admits the Lax pair representation, which can be found using the equivalence of the system of reggeons and the XXX Heisenberg magnet of spin . Namely, for each reggeon we define the Lax operator as Lk=(1−EzkpkEpk−Ez2kpk1+Ezkpk)={1}\kern-2.5pt\hbox{l}+E(1zk)⊗(−zk,1)pk, (2.3) with being an arbitrary complex spectral parameter. Then, the evolution equations (2.1) are equivalent to the matrix relation ∂tαLk={Lk,qα}=A(α)k+1Lk−LkA(α)k, (2.4) where is a matrix depending on the choice of the hamiltonian. As an example, for the hamiltonian one can obtain A(N)k=EqNzk−1,k(1zk)⊗(zk−1,−1) (2.5) and the corresponding evolution equations look like ∂tNxk=qNpk,∂tNpk=qN(1zk−1,k−1zk,k+1). The explicit form of for the remaining hamiltonians can be deduced from the Yang-Baxter equation for the Lax operator and it will not be important in the sequel. The integration of the evolution equations (2.1) and (2.4) is based on the Baker-Akhiezer function . It is defined as a solution of the following system of matrix relations Lk(E)Ψk=Ψk+1,∂tαΨk=A(α)k(E)Ψk,Ψk=(ϕkχk), (2.6) where and are scalar components. We construct the monodromy matrix T(E)=LN(E)...L2(E)L1(E)=(A(E)C(E)B(E)D(E)) (2.7) and verify using (2.4) that its eigenvalues are integrals of motion due to . The monodromy matrix generates the shift and we impose the periodicity condition on the Baker-Akhiezer functions by requiring to be the Bloch-Floquet function Ψk+N(E)=eP(E)Ψk(E). (2.8) Here, is an eigenvalue of the monodromy matrix (2.7) and it satisfies the characteristic equation det(T(E)−eP(E))=e2P(E)−Λ(E)eP(E)+1=0, (2.9) where can be calculated using (2.7) and (2.3) to be a polynomial of degree in the spectral parameter with the coefficients given by the integrals of motion Λ(E)=2+q2E2+q3E3+...+qNEN. Introducing the complex function one rewrites (2.9) in the form of algebraic complex curve 111To make a correspondence with notations of [8], one has to redefine the local parameter on the curve as . ΓN:y2=E−2(Λ2(E)−4)=(q2+q3E+...+qNEN−2)(4+q2E2+q3E3+...+qNEN). (2.10) For any complex in general position the equation (2.9) has two solutions for , or equivalently in (2.10). Under appropriate boundary conditions on (to be discussed later) each of them defines a branch of the Baker-Akhiezer function, . Then, being a double-valued function on the complex plane, becomes a single-valued function on the Riemann surface corresponding to the complex curve . This surface is constructed by gluing together two copies of the complex plane along the cuts , , running between the branching points of the curve (see for detail Fig. 1), defined as simple roots of the equation Λ2(ej)=4,j=1,...,2N−2. (2.11) Their positions on the complex plane depend on the quantum numbers , , ., of the reggeon compound state. In general, thus defined Riemann surface has a genus and it depends on the number of reggeons, e.g. it is a sphere for state, or BFKL Pomeron, and a torus for state known as QCD Odderon. To distinguish points on the Riemann surface lying above on the upper and the lower sheets we assign them a sign , where the upper (+) sheet corresponds to the asymptotics as . In particular, one finds from (2.9) the behaviour of the Bloch multiplier on different sheets of for as e±P(E)=qNEN×(1+O(1/E)), (2.12) where sign corresponds to the choice of the sheet. ### 2.1. Boundary conditions Being written in components, the first condition (2.6) on the Baker-Akhiezer function looks like χn+1−χn=zn(φn+1−φn)=Eznpn(χn−znφn), (2.13) where and are single-valued function of on the Riemann surface . We notice that for the solutions to this equation do not depend on the reggeon number . Moreover, using (2.3) and (2.5) one can show that , and, therefore, there exist two special points on the Riemann surface , at which the Baker-Akhiezer function takes constant values, . The choice of the constants implies the normalization of the Baker-Akhiezer function. Let us show that the values are fixed by the constraints (1.5). At the vicinity of on one the sheets of the Baker-Akhiezer function can be expanded as Ψn(Q;{t})=Ψ(0)+EΨ(1)n({t})+O(E2), where coefficients depend on the choice of sheet and the leading term is a constant. Let us substitute this identity into relation and expand its both sides in powers of . Taking into account small- expansion of the monodromy matrix (2.7), T(E)={1}\kern-2.5pt\hbox{l}+iE(L3L−L+−L3)+O(E2), (2.14) one obtains that is an eigenvector of the following matrix (−hL−0h)Ψ(0)=−iP′(0)Ψ(0), where conditions (1.5) were imposed. We find two solutions for corresponding to , which fix (up to overall constant) the value of the Baker-Akhiezer functions at the points as Ψn(Q+0)=(10),Ψn(Q−0)=⎛⎝L−2h1⎞⎠. (2.15) Here, is the conformal weight of holomorphic wave function and . In the quantum case, the operator commutes with the hamiltonians , but it does not take any definite value on the subspace (1.5), since and . In the quasiclassical limit, has a zero Poisson bracket with and it does not depend on the evolution times . Its value is fixed by the initial conditions as L−=−iN∑k=1pk(0)≡−iPtot, with being the total holomorphic momentum of reggeons.222In hadronic scattering amplitudes, is defined as a holomorphic component of the total momentum transferred in the channel. We notice, however, that the relations (2.6) and (2.13) are invariant under transformations pk→pk(czk+d)2,Ψn→(dcba)Ψn with the coordinates transformed as in (1.2). Since this transformation changes the value of , one can choose its parameters in such a way that one puts and keeps the relations and unchanged.333These three conditions form a set of the second-class constraints for the reggeon system. Their quantization leads to the Baxter equation for the wave function of the reggeon compound state [12]. The relation between original and transformed reggeon coordinates looks as follows 1zn=−iPtot2h+1z(0)n,pn=p(0)n(1−iPtot2hz(0)n)2. (2.16) It allows to integrate the evolution equations (2.1) for to get expressions for and and then restore the physical solutions using (2.16). Therefore, in what follows we put and obtain the normalization conditions (2.15) for the Baker-Akhiezer function in the canonical form (up to overall constant factor) Ψ(0)n(Q+0)=(10),Ψ(0)n(Q−0)=(01). (2.17) It easy to see that for the solutions of the evolution equations (2.1) and (1.5) are invariant under rescaling of the coordinates z(0)n→ρz(0)n,p(0)n→ρ−1p(0)n. (2.18) For the Baker-Akhiezer function, this corresponds to a freedom in choosing an overall normalization factors in (2.17). One can further simplify analysis of the evolution equations (2.1) for and by noticing that and the dependence on the “lowest” time is given by , and similarly for , leading to z(0)n(t2,t3,...,tN)=z(0)n(0,t3,...,tN)e2iht2,p(0)n(t2,t3,...,tN)=p(0)n(0,t3,...,tN)e−2iht2. (2.19) Thus, one can forget for a moment about dependence of the reggeon coordinates and restore it in the final expressions using (2.19). ### 2.2. Singularities of the Baker-Akhiezer function Let us show that the components of the Baker-Akhiezer function are meromorphic functions on the Riemann surface having poles, the same number of zeros and essential singularities at two infinities situated on the upper and lower sheets of . Let us take the component of the monodromy matrix (2.7) and observe that is a polynomial of degree in the spectral parameter and as according to (2.14). Then, one defines the points , , as roots of this polynomial B(Ek)/Ek=0,k=1,...,N−1 and considers the relation at the points on the Riemann surface situated above (A(Ek)0C(Ek)D(Ek))(φ1(Qk)χ1(Qk))=eP(Ek)(φ1(Qk)χ1(Qk)). (2.20) Solving it one obtains the values of on two sheets of . If the point belongs to the same sheet of on which , then one gets from (2.20) that . Thus, the upper component of the Baker-Akhiezer function has zeros on above the points defined as roots of the polynomial . In similar way, zeros of the lower component of are related to the zeros of the polynomial defined by the component of the monodromy matrix. In general, two components of the Baker-Akhiezer function, , have different sets of zeros on and their positions on depend both on the reggeon number, , and the evolution times, . Relation (2.17) implies that one of the roots should be at the points . Indeed, it follows from (2.14) and (2.7) that the polynomials and vanish at for and , respectively, and therefore the components of the Baker-Akhiezer function vanish at the points in agreement with (2.13). The remarkable property of the roots and the corresponding values of is that they form the set of separated coordinates for the system of reggeons [13]. Namely, they belong to the spectral curve eP(Ek)+e−P(Ek)=Λ(Ek)=2+q2E2k+...+qNENk,k=1,...,N−1 (2.21) and have the following Poisson brackets {Ek,En}={P(Ek),P(En)}=0,{E−1k,P(En)}=δkn. Inverting the relations (2.21) one can express in terms of and then obtain the equations of motion for zeros of the Baker-Akhiezer function generated by hamiltonians in the following differential form [8] dtk=−N−1∑j=1dEjEk−2j√Λ2(Ej)−4,k=2,3,...,N. (2.22) These equations can be integrated by the Abel map and their solution describe linear reggeon trajectories on the Jacobian variety of the Riemann surface [8]. As we will show in Sect. 3, using the Baker-Akhiezer function one can perform the inverse Abel transformation and construct the explicit expressions for the reggeon trajectories on dim plane of transverse coordinates in terms of Riemann theta-functions. For the Baker-Akhiezer function to be a well defined meromorphic function on the Riemann surface the number of its zeros, , should match the number of simple poles, which we choose to be at the points , , in general position on . Moreover, considering the relations (2.6) and (2.13) at the points close to on one finds that the components of share the common set of poles and the positions of , , on do not depend neither on , nor on times . This set can be considered as part of the initial data for the evolution equations (2.1). Finally, let us examine the relation (2.13) in the neighborhood of punctures on two sheets of . Taking the limit one obtains from (2.13) that the solutions have the following asymptotics on, say, sheet, and . Let us require that the solutions to (2.13) should have a similar behaviour on the upper sheet, and . This behaviour is consistent with (2.13) provided that the r.h.s. of (2.13) scales as at large , or equivalently leading to the expression for the reggeon coordinates as a ratio of the components of the Baker-Akhiezer function at the point . Thus, having constructed the Baker-Akhiezer function satisfying desired asymptotic behaviour one will be able to determine the reggeon coordinates. ## 3. Construction of the Baker-Akhiezer function The existence of the Baker-Akhiezer function with the properties established in the previous section can be deduced from the following well-known fact in the theory of finite-gap soliton solutions [10]. For a smooth hyperelliptic algebraic curve of genus with 2 punctures at and a given set of points in general position there exists a unique function such that is a meromorphic outside the punctures and has simple poles at the points ; satisfies the normalization conditions (2.13) at the points ; At the vicinity of two punctures on the upper and lower sheets the components have the following asymptotics Ψαn(Q→Q±∞;{τ})=E∓ne±∑N−2j=1τjEj[ϕ±α(n,{τ})+O(1E)], (3.1) where is some independent function. The following comments are in order. In this definition, the Baker-Akhiezer function depends on the set of parameters , which are different, in general, from the evolution times entering (2.1). However, as a function of , satisfies the system of first-order linear matrix equations similar to (2.6) and, as we will show in Sect. 4.1, the evolution times corresponding to the hamiltonian flows are related to auxiliary parameters by a linear transformation, . Finally, substituting expression (3.1) into (2.13) and comparing the asymptotic behaviour of different sides of the relation (2.13) for we obtain the following consistency conditions z(0)n=ϕ+2(n,{τ})ϕ+1(n,{τ})=ϕ−2(n+1,{τ})ϕ−1(n+1,{τ}) (3.2) and p(0)n=1z(0)n,n+1ϕ+1(n,{τ})ϕ+1(n+1,{τ})=1z(0)n−1,nϕ−1(n+1,{τ})ϕ−1(n,{τ}). (3.3) Together with (2.16) these relations provide the solution to the hierarchy of the evolution equations for reggeon coordinates and momenta, (2.1), in terms of the Baker-Akhiezer function. ### 3.1. Basis of differentials To write down the expression for the Baker-Akhiezer function satisfying the above conditions one defines the canonical basis of cycles on as shown in Fig. 1 and constructs the basis of normalized differentials as follows [11]. The unique set of holomorphic differentials on , or differentials of the 1st kind, , is defined as dωk=N−2∑j=1UkjdEEj−1y(E)=N−2∑j=1UkjdEEj√Λ2(E)−4. (3.4) The coefficients are fixed by the normalization conditions ∮αjdωk=2πδjk,j,k=1,...,N−2 (3.5) and they can be calculated as an inverse to the following matrix (U−1)jk=12π∮αkdEEj√Λ2(E)−4. (3.6) The unique set of meromorphic differentials on of the 2nd kind, , with the th order pole at the punctures is defined as dΩ(j)=dE√Λ2(E)−4[jqNEN+j−1+jqN−1EN+j−2+...+O(E)]. The coefficients in front of the remaining powers of in the numerator are fixed by the normalization conditions ∮αkdΩ(j)=0,dΩ(j)∣∣∣Q→Q±∞E→∞=±(jEj−1+O(1/E2))dE. (3.7) The unique dipole differentials on , or differentials of the 3rd kind, and , having simple poles at the points and , respectively, and normalized by the conditions ∮αjdΩ∞=∮αjdΩ0=0,dΩ∞∣∣∣Q→Q±∞E→∞=∓dEE,dΩ0∣∣∣Q→Q±0E→0=±dEE (3.8) are defined as dΩ∞=−1NdEΛ′(E)√Λ2(E)−4=dE√Λ2(E)−4[−qNEN−1+...+O(E)] (3.9) and dΩ0=2ihdE√Λ2(E)−4+iN−2∑j=1(U0)jdEEj√Λ2(E)−4. (3.10) To satisfy the normalization conditions (3.8) the coefficients have to be equal to (U0)k=−hπN−2∑j=1Ujk∮αjdE√Λ2(E)−4 (3.11) with the matrix defined in (3.4) and (3.6). Let us also define the following dimensional vectors , and Vk=−i∮βkdΩ∞,W(j)k=−i∮βkdΩ(j),Ak(Q)=∫Qγ0dωk, (3.12) where , is an arbitrary reference point on and integration in goes along some path on between the points and . ### 3.2. Baker-Akhiezer function The components of the Baker-Akhiezer function, , satisfying the conditions formulated in the beginning of Sect. 3 can be expressed in terms of the theta-function defined by the Riemann surface (for definition see Appendix A) as follows [14] Ψαn(Q;{τ})=hα(Q)hα(˜Qα)θ(A(Q)+Vn+∑N−2j=1W(j)τj+Zα)θ(Z0)θ(A(Q)+Zα)θ(Vn+∑N−2j=1W(j)τj+Z0)exp(n∫Q˜QαdΩ∞+N−2∑j=1τj∫Q˜QαdΩ(j)). (3.13) Here, and are normalization points, the dimensional constant vectors and are given by Zα=Z0−A(˜Qα),Z0=A(˜Q1)+A(˜Q2)−g+1∑n=1A(γn)−K with being the vector of Riemann constants. The function is defined as hα(Q)=θ(A(Q)+Zα)θ(A(Q)−Tα)θ(A(Q)−T	{"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": 0.9751651287078857, "perplexity": 502.12144418023183}, "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/1610703514423.60/warc/CC-MAIN-20210118061434-20210118091434-00233.warc.gz"}
https://blender.stackexchange.com/questions/3391/is-it-possible-to-create-an-opalescence-shader-in-cycles/35973	# Is it Possible to Create an Opalescence Shader in Cycles? Is it possible to make an Opalescent shader in cycles? From Wikipedia: Opalescence is a type of dichroism seen in highly dispersed systems with little opacity. The material appears yellowish-red in transmitted light and blue in the scattered light perpendicular to the transmitted light. The phenomenon is named after the appearance of opals. I attempted to do something similar by mixing to glass shaders with IsCamera ray as the factor: However, the color of each glass shader is getting mixed, so the color of the object is not separate from the color of the caustics: How can I make a shader with separate color inputs for the caustics and the object itself? Or if it's possible, make a more physically based opalescence shader? Update: I managed to separate the caustics from the object with this node setup: However, I can't seem to mix them. Here are results with the Factor of the last mix node at 0, .5 and 1: I did not expect setting the factor to .5 to work, though I'm not sure why using an Add node blows it out: I could composite these together, but I would prefer to have it work in one material if possible. How can I accomplish this? • Can you change the title? Currently the title sounds like a tutorial request, but the question itself is more specific. – CharlesL Oct 19 '13 at 14:45	{"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": 0.23704218864440918, "perplexity": 953.3054320984505}, "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/1573496670987.78/warc/CC-MAIN-20191121204227-20191121232227-00354.warc.gz"}
https://www.physicsforums.com/threads/absolute-humidity.156090/	# Absolute humidity 1. Feb 13, 2007 ### Orbital 1. The problem statement, all variables and given/known data Estimate the mass of water vapor that is contained in $1 m^3$ of saturated air at a temperature of plus 20 degrees Celsius 2. The attempt at a solution I looked up the saturation pressure for water vapor at the given temperature. Can I just use the ideal gas-law for the mass? i.e. is the solution as simple as $$m_{H_2O} = \frac{P_{sat, H_2O}}{R_{H_2O}T}$$ or do I have to make another approach? 2. Feb 13, 2007 ### lalbatros yes this looks right, but -as usual- be careful with the units Last edited: Feb 13, 2007 3. Feb 13, 2007 ### Orbital Can you tell me why the phrasing of the problem is physically incorrect? Similar Discussions: Absolute humidity	{"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": 0.9483607411384583, "perplexity": 1468.0519107023983}, "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-2017-22/segments/1495463609409.62/warc/CC-MAIN-20170528024152-20170528044152-00101.warc.gz"}
http://mathoverflow.net/questions/126969/lower-bounds-for-petersson-inner-products-of-cuspforms-with-integral-fourier-coe/127123	# Lower bounds for Petersson inner products of cuspforms with integral Fourier coefficients Let $N\geq 1$ be an integer and let $S_2(\Gamma_0(N))$ be the cusp forms of weight 2 for the usual congruence subgroup $\Gamma_0(N)\subset SL_2(\mathbb Z)$. Let $a_n(f)$ denote the n-th Fourier coefficient of $f\in S_2(\Gamma_0(N))$, $n\geq 1$. Let $X_0(N)$ be a smooth projective model over $\mathbb Q$ of the modular curve associated to $\Gamma_0(N)$ and let $$(f,g)=\int_{X_0(N)}f(z)\overline{g(z)}dxdy,$$ $z=x+iy$, be the Petersson inner product of $f,g\in S_2(\Gamma_0(N))$. Question (a) Does there exist a constant $c>0$, depending at most on $\Gamma_0(N)$ (or $X_0(N)$), with the following property: Suppose $f\in S_2(\Gamma_0(N))$ is an eigenform with $a_1(f)=1$ which lies in the new part, $g\in S_2(\Gamma_0(N))$ and $a_n(f),a_n(g)\in\mathbb Z, n\geq 1$. If $(f,g)>0$, then $$(f,g)\geq c.$$ (b) Can one find such a constant $c>0$ with an explicit dependence on $N$. (c) Can one find such a constant $c>0$ which is absolute. My main interest is when $f$ in the above question is in addition a newform for $\Gamma_0(N)$ with $a_1(f)=1$, but I don't know if this assumption simplifies things. Further, David Loeffler mentions below that $(f,g)$ is related to a residue of a certain $L$-series at $s=2$. - I am a bit confused about your last remark. If you take two different newforms $f$ and $g$ satisfying your normalization, then $(f,g)=0$ by multiplicity one. – GH from MO Apr 9 '13 at 16:06 Dear GH, thanks for the remark. The assumptions of the question still hold: There I assume that (f,g)>0. I will edit the question to make this more clear. – ranicl Apr 9 '13 at 16:51 Given $f$, the possible $(f,g)$ form a subgroup of ${\bf R}$, which is either discrete or dense. Once the space of cuspforms has dimension at least $2$ one would expect it to be dense unless $f = 0$ (why should two or more "random" Petersson products be proportional?), and thus to contain arbitrarily small positive elements. Is there a further missing assumption? – Noam D. Elkies Apr 9 '13 at 17:24 If you assume that $f$ is a newform with integer coefficients, then the orthogonal of $f$ with respect to the Petersson scalar product admits a basis consisting of forms with integral coefficients, thus the image of the linear map on $S_2(\Gamma_0(N),\mathbf{Z})$ given by $g \mapsto (f,g)$ is a lattice of $\mathbf{R}$. Thus $c$ exists in this case, but it is not clear how to compute a lower bound in terms of $N$ because of the possible congruences between $f$ and other newforms as David explained. – François Brunault Apr 9 '13 at 21:39 Ah, I saw the "integer coefficients" part but didn't appreciate the significance of "newform" (implying not just in the "new" space but an actual Hecke eigenform). – Noam D. Elkies Apr 10 '13 at 2:50 It's not hard to see that the answer to (a) is yes. There is a basis of $S_2^{\textrm{new}}(\Gamma_0(N))$ consisting of newforms. These newforms come into Galois orbits $\{f^\sigma\}_{\sigma}$. Here $\sigma$ runs through the embeddings of $K_f$ into $\mathbf{C}$, where $K_f$ is the field generated by the Fourier coefficients of $f$. A basic fact is that the $\mathbf{C}$-span of a given Galois orbit is generated by cusp forms with integral coefficients. This follows from considering the forms $\sum_{\sigma} \sigma(a) f^\sigma$ where $a$ runs through a $\mathbf{Z}$-basis of the ring of integers of $K_f$. It follows that if the newform $f$ has integral coefficients, then its orthogonal complement $f^\perp$ is generated by cusp forms with integral coefficients. Now consider the linear map $\lambda_f : S_2(\Gamma_0(N),\mathbf{Z}) \to \mathbf{R}$ given by $\lambda_f(g)=(f,g)$. By the previous remark, the kernel of $\lambda_f$ has rank one less than the rank of $S_2(\Gamma_0(N),\mathbf{Z})$, which implies that the image of $\lambda_f$ is of the form $c_f \mathbf{Z}$ for some $c_f >0$. Thus we can take for $c$ the minimum of all the $c_f$. In fact $c_f = (f,f)/m_f$ for some integer $m_f$ measuring the congruences of $f$ with other cusp forms. - Dear Francois Brunault, thats a fantastic answer. Thank you very much!!! – ranicl Apr 11 '13 at 11:06 I typed a comment but the formatting wouldn't come out right, so here it is as an answer! I cannot work out why you expect the "Plancherel or Parseval type" formula to work. Does it not bother you a little that $a_n(f)$ and $a_n(g)$ are perfectly capable of being integers for all $n$, so your series is obviously divergent? Much better is to consider the series $$L(f, g, s) = \sum_{n \ge 1} a_n(f) \overline{a_n(g)} n^{-s},$$ which converges for $Re(s) > 2$ (this is not so easy to see, but it is easy to show that it converges for $Re(s) \gg 0$). This has meromorphic continuation to all $s \in \mathbb{C}$ with a pole at $s = 2$ at which the residue is (maybe up to a normalizing constant depending on $N$ that I have forgotten) the Petersson product $\langle f, g \rangle$. But this does not help you to get lower bounds on $\langle f, g \rangle$ as far as I can see. Some quite grotty things can happen, e.g. if $f$ is an eigenform and there is another newform $f'$ with $f = f'$ modulo some integer $N$, then one can take $g = (f - f')/N$, and this will be integral but its Petersson product with $f$ will be $\langle f, f \rangle / N$. So the issue of bounding Petersson products below is quite closely related to the issue of congruences between eigenforms. - Dear David Loeffler, thanks a lot for the helpful comment. I will edit my question according to your comment. – ranicl Apr 9 '13 at 13:09	{"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": 0.9871523380279541, "perplexity": 140.61222758349726}, "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-2015-40/segments/1443738006497.89/warc/CC-MAIN-20151001222006-00155-ip-10-137-6-227.ec2.internal.warc.gz"}
http://mathhelpforum.com/discrete-math/202123-structural-induction-problem.html	1. ## Structural Induction problem Hi, I am now struggling with structural induction, this cannot be proved so I needed to prove with stronger property but I am stuck... please help with proving.. so, this is based on Haskell property (tree structure) so 'Haskell' definition of binary tree is data Tree a = Nul \| Node a (Tree a) (Tree a) so basically, if the tree has no node then it is Nul or otherwise tree a has node a and left to it another tree that has less number than a and another tree on the right with the number greater than Node a. so the problem has this property. mult Nul = 1 ------ (M1) mult (Node a t1 t2) = a * mult t1 * mult t2 ------ (M2) multa t = multb t 1 ------ (MA) multb Nul acc = acc --------- (MB1) multb (Node a t1 t2) acc = multb t1 (a * (multb t2 acc)) ---- (MB2) and I need to prove mult t = multa t certain two cannot be equal hence must prove with stronger property... it seems that multb is acc times multiplication all nodes but I don't know how to prove from this point.. help me thank you very much for your support always.. 2. ## Re: Structural Induction problem Originally Posted by gomdohri mult (Node a u1 u2) = multa (Node a u1 u2) = a* mult u1 * mult u2 ----- (M1) = a * multa u1 * multa u2 ---(M2) = a * (multb u1, 1) * (multb u2 1) ----- IH1, IH2 = multa (Node a u1 u2) (RH) = multb (Node a u1 u2) 1 ---- (MA) = multb u1 (a * (mult b u2 acc) ) ---- (MB2) You are right; the explanations on the right do not make much sense. For example, M1 has to do with nult Nul, but there is no Nul here. The transition from the second to the third line is justified by IH1 and IH2, not M2, and so on. Originally Posted by gomdohri it seems that multb is acc times multiplication all nodes Yes, and acc * mult t = multb t acc is exactly the stronger induction hypothesis you need. 3. ## Re: Structural Induction problem Amazing! I must admit I didn't understand a freakin' word you said... Sorry but I have to ask what part of math is this and if you could recommend some texts about it. Curiosity will get me killed, I know. 4. ## Re: Structural Induction problem Haskell is a functional programming languages. Programs in Haskell are regular mathematical equations, so one can use them to replace equal expressions as in ubiquitous in mathematics. This is in contrast to imperative programming languages where expressions may have side effects like changing values of variables. So basically we have a term algebra generated by a unary functional symbol Nul and a ternary functional symbol Node. Then there are equations that define functions mult, multa and multb by recursion. Even though this setting is studied in universal algebra and category theory, the closest math area is logic, in particular, proof theory and complexity theory, which study syntactic expressions. Recently theories were developed that unite functional programming and proofs. They are based on Curry-Howard correspondence between proofs and programs. So, in a single theory one can write a functional program like mult and also construct a proof of its properties, like mult (Node a t1 t2) = mult (Node a t2 t1). This proof would also be a program that constructs evidence of the equality from given evidences like axioms and assumptions. One example of such theory is Calculus of Inductive Constructions. It is a descendant of Church's typed lambda calculus, and besides ideas from this calculus, it heavily uses induction principles for various term algebras. Edit: I'll think about good sources on this topic. 5. ## Re: Structural Induction problem Oh... my mistake. I put it in wrong place M1 should be M2 and M2 should be IH1 and IH2. Sorry about that. Let me try the rest and if I am still struggling with this I will reply this message again. Thank you 6. ## Re: Structural Induction problem Originally Posted by MathoMan Amazing! I must admit I didn't understand a freakin' word you said... Sorry but I have to ask what part of math is this and if you could recommend some texts about it. Curiosity will get me killed, I know.	{"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": 0.8694189786911011, "perplexity": 2671.8858046549467}, "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-2017-13/segments/1490218189462.47/warc/CC-MAIN-20170322212949-00654-ip-10-233-31-227.ec2.internal.warc.gz"}
https://mathoverflow.net/users/124995/elizeu-fran%C3%A7a?tab=topactivity	Elizeu França ### Questions (9) 4 Homological class of the fiber in the total space of the one circle bundle 3 Euler class and the real homological class of the fiber in an orientable sphere bundle 2 Characterization of uniformly convergent sequence of functions 2 Principal circle bundles that are smooth foliated 2 Does a compact leaf of the smooth transversaly orientable foliation have trivial normal bundle? ### Reputation (355) This user has no recent positive reputation changes 5 When is a closed differential form harmonic relative to some metric? 1 Ehresmann fibration theorem for manifolds with boundary 1 Whether fine topology and uniform topology on C(X,Y) coincide , when metric on Y is bounded ### Tags (15) 6 smooth-manifolds × 2 1 differential-topology 6 dg.differential-geometry × 2 0 foliations × 5 5 riemannian-geometry 0 fibre-bundles × 3 1 gn.general-topology × 2 0 limits-and-convergence × 2 1 gt.geometric-topology 0 homology ### Bookmarks (1) 33 When is a closed differential form harmonic relative to some metric?	{"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": 0.9069288969039917, "perplexity": 2531.6822397061806}, "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/1627046155458.35/warc/CC-MAIN-20210805063730-20210805093730-00650.warc.gz"}
https://everything.explained.today/Transpose/	# Transpose Explained In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).[1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.[2] In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. ## Transpose of a matrix ### Definition The transpose of a matrix, denoted by,[3],, A\intercal ,[4] [5],[6], or, may be constructed by any one of the following methods: 1. Reflect over its main diagonal (which runs from top-left to bottom-right) to obtain 2. Write the rows of as the columns of 3. Write the columns of as the rows of Formally, the -th row, -th column element of is the -th row, -th column element of : \operatorname{T}\right] \left[A ij =\left[A\right]ji. If is an matrix, then is an matrix. In the case of square matrices, may also denote the th power of the matrix . For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as . An advantage of this notation is that no parentheses are needed when exponents are involved: as, notation is not ambiguous. In this article this confusion is avoided by never using the symbol as a variable name. #### Matrix definitions involving transposition A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, is symmetric if A\operatorname{T } = \mathbf. A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, is skew-symmetric if A\operatorname{T } = -\mathbf. A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, is Hermitian if A\operatorname{T } = \overline. A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, is skew-Hermitian if A\operatorname{T } = -\overline. A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, is orthogonal if A\operatorname{T } = \mathbf^. A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, is unitary if A\operatorname{T } = \overline. ### Examples \begin{bmatrix} 1&2 \end{bmatrix}\operatorname{T } = \, \begin 1 \\ 2 \end \begin{bmatrix} 1&2\\ 3&4 \end{bmatrix}\operatorname{T } = \begin 1 & 3 \\ 2 & 4 \end \begin{bmatrix} 1&2\\ 3&4\\ 5&6 \end{bmatrix}\operatorname{T } = \begin 1 & 3 & 5\\ 2 & 4 & 6 \end ### Properties Let and be matrices and be a scalar. ### Products If is an matrix and is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: is and is . Furthermore, these products are symmetric matrices. Indeed, the matrix product has entries that are the inner product of a row of with a column of . But the columns of are the rows of, so the entry corresponds to the inner product of two rows of . If is the entry of the product, it is obtained from rows and in . The entry is also obtained from these rows, thus, and the product matrix is symmetric. Similarly, the product is a symmetric matrix. A quick proof of the symmetry of results from the fact that it is its own transpose: \left(AA\operatorname{T}\right)\operatorname{T}=\left(A\operatorname{T}\right)\operatorname{T}A\operatorname{T}=AA\operatorname{T}. [7] ### Implementation of matrix transposition on computers See also: In-place matrix transposition. On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement. However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing memory locality. Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an n × m matrix in-place, with O(1) additional storage or at most storage much less than mn. For n ≠ m, this involves a complicated permutation of the data elements that is non-trivial to implement in-place. Therefore, efficient in-place matrix transposition has been the subject of numerous research publications in computer science, starting in the late 1950s, and several algorithms have been developed. ## Transposes of linear maps and bilinear forms Recall that matrices can be placed into a one-to-one correspondence with linear operators. The transpose of a linear operator can be defined without any need to consider a matrix representation of it. This leads to a much more general definition of the transpose that can be applied to linear operators that cannot be represented by matrices (e.g. involving many infinite dimensional vector spaces). ### Transpose of a linear map See main article: Transpose of a linear map. Let denote the algebraic dual space of an -module . Let and be -modules. If is a linear map, then its algebraic adjoint or dual, is the map defined by . The resulting functional is called the pullback of by . The following relation characterizes the algebraic adjoint of for all and where is the natural pairing (i.e. defined by). This definition also applies unchanged to left modules and to vector spaces. The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint (below). The continuous dual space of a topological vector space (TVS) is denoted by . If and are TVSs then a linear map is weakly continuous if and only if, in which case we let denote the restriction of to . The map is called the transpose of . If the matrix describes a linear map with respect to bases of and, then the matrix describes the transpose of that linear map with respect to the dual bases. ### Transpose of a bilinear form See main article: Bilinear form. Every linear map to the dual space defines a bilinear form, with the relation . By defining the transpose of this bilinear form as the bilinear form defined by the transpose i.e., we find that . Here, is the natural homomorphism into the double dual. Transpose should not be confused with Hermitian adjoint. If the vector spaces and have respectively nondegenerate bilinear forms and, a concept known as the adjoint, which is closely related to the transpose, may be defined: If is a linear map between vector spaces and, we define as the adjoint of if satisfies BX(x,g(y))=BY(u(x),y) for all and . These bilinear forms define an isomorphism between and, and between and, resulting in an isomorphism between the transpose and adjoint of . The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors use the term transpose to refer to the adjoint as defined here. The adjoint allows us to consider whether is equal to . In particular, this allows the orthogonal group over a vector space with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps for which the adjoint equals the inverse. Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. The Hermitian adjoint of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal. • . • Book: Maruskin, Jared M. . [{{Google books \|plainurl=yes \|id=aOF3-hx3u1kC \|page=122 }} Essential Linear Algebra ]. San José . Solar Crest . 2012 . 978-0-9850627-3-6 . 122–132 . • Book: Schwartz, Jacob T. . [{{Google books \|plainurl=yes \|id=fMG3y1z9Jw0C \|page=126}} Introduction to Matrices and Vectors ]. Mineola . Dover . 2001 . 0-486-42000-0 . 126–132 .	{"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": 0.9893375635147095, "perplexity": 522.7543478896437}, "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-2022-27/segments/1656103984681.57/warc/CC-MAIN-20220702040603-20220702070603-00743.warc.gz"}
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=100&t=20374&p=56912	Reaction Profiles Krishil_Gandhi_1D Posts: 27 Joined: Fri Jul 22, 2016 3:00 am Reaction Profiles Hey guys, How are we supposed to draw the reaction profile from the reaction mechanism? I understand that you can determine the number of transition states/steps from the reaction mechanism, however, how can we deduce which transition state is higher in energy or whether the delta G is negative or positive based off the reaction mechanism? I'd appreciate it if someone can explain this. Thanks in advance. Krishil janavi_patel_2K Posts: 20 Joined: Wed Sep 21, 2016 2:59 pm Re: Reaction Profiles I know that if the exergonic reaction is favorable and K>1, then Gibbs free energy is negative. If the endergonic reaction if favorable and K<1, then Gibbs free energy is positive.	{"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": 0.9754533767700195, "perplexity": 1493.1937660096194}, "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-2019-35/segments/1566027313617.6/warc/CC-MAIN-20190818042813-20190818064813-00250.warc.gz"}
https://www.nature.com/articles/ncomms6156	Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. # Suppressing qubit dephasing using real-time Hamiltonian estimation ## Abstract Unwanted interaction between a quantum system and its fluctuating environment leads to decoherence and is the primary obstacle to establishing a scalable quantum information processing architecture. Strategies such as environmental and materials engineering, quantum error correction and dynamical decoupling can mitigate decoherence, but generally increase experimental complexity. Here we improve coherence in a qubit using real-time Hamiltonian parameter estimation. Using a rapidly converging Bayesian approach, we precisely measure the splitting in a singlet-triplet spin qubit faster than the surrounding nuclear bath fluctuates. We continuously adjust qubit control parameters based on this information, thereby improving the inhomogenously broadened coherence time from tens of nanoseconds to >2 μs. Because the technique demonstrated here is compatible with arbitrary qubit operations, it is a natural complement to quantum error correction and can be used to improve the performance of a wide variety of qubits in both meteorological and quantum information processing applications. ## Introduction Hamiltonian parameter estimation is a rich field of active experimental and theoretical research that enables precise characterization and control of quantum systems1. For example, magnetometry schemes employing Hamiltonian learning have demonstrated dynamic range and sensitivities exceeding those of standard methods2,3. Such applications focused on estimating parameters that are quasistatic on experimental timescales. However, the effectiveness of Hamiltonian learning also offers exciting prospects for estimating fluctuating parameters responsible for decoherence in quantum systems. The quantum system that we study is a singlet-triplet (ST0) qubit4,5 which is formed by two gate-defined lateral quantum dots (QDs) in a GaAs/AlGaAs heterostructure (Fig. 1a, Supplementary Fig. 1), similar to that of refs 6, 7. The qubit can be rapidly initialized in the singlet state \|S› in ≈20 ns and read out with 98% fidelity in ≈1 μs (refs 8, 9; Supplementary Fig. 2). Universal quantum control is provided by two distinct drives10: the exchange splitting, J, between \|S› and \|T0›, and the magnetic field gradient, ΔBz, due to the hyperfine interaction with host Ga and As nuclei. The Bloch sphere representation for this qubit can be seen in Fig. 1b. In this work, we focus on qubit evolution around ΔBz (Fig. 2a). Due to statistical fluctuations of the nuclei, ΔBz varies randomly in time, and consequently oscillations around this field gradient decay in a time (ref. 4). A nuclear feedback scheme relying on dynamic nuclear polarization11 can be employed to set the mean gradient, (gμBΔBz/h≈60 MHz in this work) as well as reduce the variance of the fluctuations. Here, g≈−0.44 is the effective gyromagnetic ratio in GaAs, μB is the Bohr magneton and h is Planck’s constant. In what follows, we adopt units where g*μB/h=1. The nuclear feedback relies on the avoided crossing between the \|S› and \|T+› states. When the electrons are brought adiabatically through this crossing, their total spin changes by Δms=±1, which is accompanied by a nuclear spin flip to conserve angular momentum. With the use of this feedback, the coherence time improves to (ref. 11; Fig. 2b), limited by the low nuclear pumping efficiency10. Crucially, the residual fluctuations are considerably slower than the timescale of qubit operations12. In this work we employ techniques from Hamiltonian estimation to prolong the coherence of a qubit by more than a factor of 30. Importantly, our estimation protocol, which is based on recent theoretical work13, requires relatively few measurements (≈100) which we perform rapidly enough (total time ≈100 μs) to resolve the qubit splitting faster than its characteristic fluctuation time. We adopt a paradigm in which we separate experiments into ‘estimation’ and ‘operation’ segments, and we use information from the former to optimize control parameters for the latter in real-time. Our method dramatically prolongs coherence without using complex pulse sequences such as those required for non-identity dynamically decoupled operations14. ## Results ### Rotating frame S−T0 qubit To take advantage of the slow nuclear dynamics, we introduce a method that measures the fluctuations and manipulates the qubit on the basis of precise knowledge but not precise control of the environment. We operate the qubit in the rotating frame of ΔBz, where qubit rotations are driven by modulating J at the frequency (refs 15, 16). This is in contrast to traditional modes of operation of the ST0 qubit, which rely on DC voltage pulses. To measure Rabi oscillations, the qubit is adiabatically prepared in the ground state of ΔBz (\|ψ›=\|↑↓›), and an oscillating J is switched on (Fig. 2e), causing the qubit to precess around J in the rotating frame. In addition, we perform a Ramsey experiment (Fig. 2c) to determine , and as expected, we observe the same decay (Fig. 2d) as Fig. 2b. More precisely, the data in Fig. 2d represent the average of 1,024 experimental repetitions of the same qubit operation sequence immediately following nuclear feedback. The feedback cycle resets ΔBz to its mean value (60 MHz) with residual fluctuations of between experimental repetitions. However, within a given experimental repetition, ΔBz is approximately constant. Therefore we present an adaptive control scheme where, following nuclear feedback, we quickly estimate ΔBz and tune to prolong qubit coherence (Fig. 3a). ### Bayesian estimation To estimate ΔBz , we repeatedly perform a series of single-shot measurements after allowing the qubit to evolve around ΔBz (using DC pulses) for some amount of time (Fig. 2a). Rather than fixing this evolution time to be constant for all trials, we make use of recent theoretical results in Hamiltonian parameter estimation13,16,17 and choose linearly increasing evolution times, tk=ktsamp, where k=1,2,…N. We choose the sampling time tsamp such that the estimation bandwidth is several times larger than the magnitude of the residual fluctuations in ΔBz, roughly 10 MHz. With a Bayesian approach to estimate ΔBz in real-time, the longer evolution times (large k) leverage the increased precision obtained from earlier measurements to provide improved sensitivity, allowing the estimate to outperform the standard limit associated with repeating measurements at a single evolution time. Denoting the outcome of the kth measurement as mk (either \|S› or \|T0›), we define P(mkBz) as the conditional probability for mk given a value ΔBz. We write where rk=1 (−1) for mk=\|S›(\|T0›), and α=0.25 and β=0.67 are parameters determined by the measurement error and axis of rotation on the Bloch sphere (see Methods). Since we assume that earlier measurement outcomes do not affect later ones (that is, that there is no measurement back-action), we write the conditional probability for ΔBz given the results of N measurements as: Using Bayes’ rule, that is, PBz\|mk)=P(mkBz)PBz)/P(mk), and equation (1), we can rewrite equation (3) as: where N is a normalization constant and P0Bz) is a prior distribution to which the algorithm is empirically insensitive, and which we take to be a constant over the estimation bandwidth. After the last measurement, we find the value of ΔBz that maximizes the posterior distribution PBz\|mN,mN−1,...m1). We implement this algorithm in real-time on a field-programmable gate array (FPGA), computing PBz) for 256 values of ΔBz between 50 and 70 MHz. With each measurement mk, the read-out signal is digitized and passed to the FPGA, which computes PBz) and updates an analogue voltage that tunes the frequency of a voltage controlled oscillator (Fig. 1a; Supplementary Note 1, Supplementary Fig. 3). Following the Nth sample, nearly matches ΔBz, and since the nuclear dynamics are slow, the qubit can be operated with long coherence without any additional complexity. To quantify how well the FPGA estimate matches ΔBz, we perform a Ramsey experiment (deliberately detuned to observe oscillations) with this real-time tracking of ΔBz and find optimal performance for N≈120, with a maximum experimental repetition rate, limited by the FPGA, of 250 kHz and a sampling time tsamp=12 ns. Under these conditions, and making a new estimate after every 42 Ramsey experiments, we observe , a 30-fold increase in coherence (Fig. 3b). We note that these data are taken with the same pulse sequence as those in Fig. 2d. To further compare qubit operations with and without this technique, we measure Ramsey fringes for ≈250 s (Fig. 3d), and histogram the observed Ramsey detunings. With adaptive control we observe a stark narrowing of the observed frequency distribution, consistent with this improved coherence (Fig. 3c). ## Discussion Although the estimation scheme employed here is theoretically predicted to improve monotonically with N (ref. 13), we find that there is an optimum (N≈120), after which slowly decreases with increasing N (Fig. 4a). A possible explanation for this trend is fluctuation of the nuclear gradient during the estimation period. To investigate this, we obtain time records of ΔBz using the Bayesian estimate and find that its variance increases linearly in time at the rate of (6.7±0.7 kHz)2 μs−1 (Fig. 4c). The observed linear behaviour suggests a model where the nuclear gradient diffuses, which can arise, for example, from dipolar coupling between adjacent nuclei. Using the measured diffusion of ΔBz, we simulate the performance of the Bayesian estimate as a function of N (see Methods). Given that the simulation has no free parameters, we find good agreement with the observed , indicating that indeed, diffusion limits the accuracy with which we can measure ΔBz (Fig. 4a). This model suggests that increasing the rate of measurements during estimation will improve the accuracy of the Bayesian estimate. Because our FPGA limits the repetition rate of qubit operations to 250 kHz, we demonstrate the effect of faster measurements through software post-processing with the same Bayesian estimate. To do so, we first use the same estimation sequence, but for the operation segment, we measure the outcome after evolving around ΔBz for a single evolution time, tevo, rather than performing a rotating frame Ramsey experiment, and we repeat this experiment a total of Ntot times. In processing, we perform the Bayesian estimate of each ΔBz,i, sort the data by adjusted time (for i=1,2,…Ntot), and average together points of similar τ to observe oscillations (see Methods). We fit the decay of these oscillations to extract and the precision of the Bayesian estimate, . For the same operation and estimation parameters, we find that extracted from software post-processing agrees with that extracted from adaptive control Supplementary Fig. 4, Supplementary Note 2. Using a repetition rate as high as 667 kHz, we show coherence times above 2,800 ns, corresponding to an error of σΔBz=80 kHz (Fig. 4d), indicating that improvements are easily attainable by using faster (commercially available) FPGAs. In addition, we use this post-processing to examine the effect of this technique on the duty cycle of experiments as well as the stability of the ΔBz estimate. To do so we introduce a delay Tdelay between the estimation of ΔBz and the single evolution measurement performed in place of the operation. We find , where c=0.99 (Fig. 4c), consistent with diffusion of ΔBz. Indeed, this dependence underscores the potential of adaptive control, since it demonstrates that after a single estimation sequence, the qubit can be operated for >1 ms with . Thus, adaptive control need not significantly reduce the experimental duty cycle. In this work, we have used real-time adaptive control on the basis of Hamiltonian parameter estimation of a ST0 spin qubit to prolong from 70 ns to >2 μs. Dephasing due to nuclear spins has long been considered a significant obstacle to quantum information processing using semiconductor spin qubits18, and elimination of nuclear spins is an active and fruitful area of research19,20,21. However, here we have shown that with a combination of nuclear feedback, rotating frame ST0 spin resonance, and real-time Hamiltonian estimation, we are able to achieve ratios of coherence times to operation times in excess of 200 without recourse to dynamical decoupling12,22,23. If the same adaptive control techniques were applied to gradients as high as 1 GHz (ref. 10), ratios exceeding 4,000 would be possible, and longer coherence times may be attainable with more sophisticated techniques13. Though the observed coherence times are still smaller than the Hahn echo time, (ref. 12), the method we have presented is straightforward to implement, compatible with arbitrary qubit operations, and general to all qubits that suffer from non-Markovian noise. Looking ahead, it is likely, therefore, to have a key role in realistic quantum error correction efforts24,25,26,27, where even modest improvements in baseline error rate greatly diminish experimental complexity and enhance prospects for a scalable quantum information processing architecture. ## Methods ### Bayesian estimate We wish to calculate the probability that the nuclear magnetic field gradient has a certain value, ΔBz, given a particular measurement record comprising N measurements. We follow the technique in Sergeevich et al.13 with slight modifications. Writing the outcome of the kth measurement as mk, we write this probability distribution as To arrive at an expression for this distribution, we will write down a model for the dynamics of the system, that is, P(mN,mN−1,...m1Bz). Using Bayes’ rule we can relate the two equations as First, we seek a model that can quantify P(mN,mN−1,...m1Bz) that accounts for realistic errors in the system, namely measurement error, imperfect state preparation and error in the axis of rotation around the Bloch sphere. For simplicity, we begin with a model that accounts only for measurement error. Denoting the error associated with measuring a \|S› (\|T0›) as ηS (ηT), we write We combine these two equations and write where rk=1 (−1) for mk=\|S›(\|T0›) and α and β are given by Next, we generalize the model to include the effects of imperfect state preparation, and the presence of non-zero J during evolution, which renders the initial state non-orthogonal to the axis of rotation around the Bloch sphere (see above). We assume that the angle of rotation around the Bloch sphere lies somewhere in the xz plane and makes an angle θ with the z axis. We define δ=cos2 (θ). Next, we include imperfect state preparation by writing the density matrix ρinit=(1−ε)\|S› ‹S\|)+ε\|T0› ‹T0\|. With this in hand, we can write down the model Using the same notation for rk=1 (−1) for mk=\|S›(\|T0›), we rewrite this in one equation as where we now have We find the best performance for α=0.25 and β=0.67, which is consistent with known values for qubit errors. We next turn our attention to implementing Bayes’ rule to turn this model into a probability distribution for ΔBz. First, we assume that all measurements are statistically independent, allowing us to write We next use Bayes rule (6) and rewrite this equation as Using our model (13) we can rewrite this as where N is a normalization constant, and P0Bz) is a prior distribution for ΔBz which we take to be a constant over the estimation bandwidth, and to which the estimator is empirically insensitive. With this formula, it is simple to see that the posterior distribution for ΔBz can be updated in real time with each successive measurement. After the Nth measurement, we choose the value for ΔBz, which maximizes the posterior distribution (18). ### Simulation with diffusion We simulate the performance of our software scaling and hardware (FPGA) estimates of ΔBz using the measured value of the diffusion rate. We assume that ΔBz obeys a random walk, but assume that during a single evolution time tk, ΔBz is static. This assumption is valid when , where is the diffusion rate of ΔBz. For an estimation of ΔBz with N different measurements, we generate a random walk of N different values for ΔBz (using the measured diffusion), simulate the outcome of each measurement, and compute the Bayesian estimate of ΔBz using the simulated outcomes. By repeating this procedure 4,096 times, and using the mean squared error, MSE=‹(ΔBz−ΔBzestimated)2› as a metric for performance, we can find the optimal number of measurements to perform. To include the entire error budget of the FPGA apparatus, we add to this MSE the error from the phase noise of the VCO, the measured voltage noise on the analogue output controlling the VCO, and the diffusion of ΔBz during the ‘operation’ period of the experiment. ### Software post-processing The estimate of ΔBz can be independently verified using software analysis. In this experiment, we use the same method to estimate ΔBz as in the adaptive control experiment, but in the operation segment perform oscillations around ΔBz for verification. We choose m different evolution times and measure each n times for a total of Ntot=m × n measurements of ΔBz. In the ith experiment (i=1,2,…Ntot), we evolve for a time tevo,i, accumulating phase φiBz,itevo,i. Because we make a precise measurement of ΔBz at the start of each experiment, we can employ it to rescale the time, tevo,i, so that the phase accumulated for a given time is constant using the equation, This sets φi(τi)=‹ΔBzτi, with residual error arising from inaccuracy in the estimate of ΔBz,i. The data are then sorted by τ, and points of similar τ are averaged using a Gaussian window with στ=0.5 nsT≈16 ns, where T is the period of the oscillations. How to cite this article: Shulman, M. D. et al. Suppressing qubit dephasing using real-time Hamiltonian estimation. Nat. Commun. 5:5156 doi: 10.1038/ncomms6156 (2014). ## References 1. Wiseman, Howard M. & Milburn, Gerard J. Quantum Measurement and Control Cambridge Univ. Press (2010). 2. Waldherr, G. et al. High-dynamic-range magnetometry with a single nuclear spin in diamond. Nat. Nanotechnol. 7, 105–108 (2012). 3. Nusran, N. M., Ummal, M. & Gurudev Dutt, M. V. High dynamic range magnetometry with a single electronic spin in diamond. 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Universal quantum control in two-electron spin quantum bits using dynamic nuclear polarization. Nat. Phys. 5, 903–908 (2009). 11. Bluhm, H., Foletti, S., Mahalu, D., Umansky, V. & Yacoby, A. Enhancing the coherence of a spin qubit by operating it as a feedback loop that controls its nuclear spin bath. Phys. Rev. Lett. 105, 216803 (2010). 12. Bluhm, H. et al. Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200 μs. Nat. Phys. 7, 109–113 (2011). 13. Sergeevich, A., Chandran, A., Combes, J., Bartlett, S. D. & Wiseman, H. M. Characterization of a qubit Hamiltonian using adaptive measurements in a fixed basis. Phys. Rev. A 84, 052315 (2011). 14. Kestner, J. P., Wang, X., Bishop, L. S., Barnes, E. & Das Sarma, S. Noise-resistant control for a spin qubit array. Phys. Rev. Lett. 110, 140502 (2013). 15. Coish, W. A. & Loss, D. Hyperfine interaction in a quantum dot: non-Markovian electron spin dynamics. Phys. Rev. B 70, 195340 (2004). 16. Klauser, D., Coish, W. A. & Loss, D. Nuclear spin state narrowing via gate-controlled Rabi oscillations in a double quantum dot. Phys. Rev. B 73, 205302 (2006). 17. Ferrie, C., Granade, C. E. & Cory, D. G. How to best sample a periodic probability distribution, or on the accuracy of Hamiltonian finding strategies. Quantum Inf. Process. 12, 611 (2013). 18. Schroer, M. D. & Petta, J. R. Quantum dots: time to get the nukes out. Nat. Phys. 4, 516–518 (2008). 19. Balasubramanian, G. et al. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 8, 383–387 (2009). 20. Wild, A. et al. Few electron double quantum dot in an isotopically purified 28Si quantum wel. Appl. Phys. Lett. 100, 143110 (2012). 21. Muhonen, J. T. et al. Storing quantum information for 30 seconds in a nanoelectronic device. Preprint at http://arxiv.org/abs/1402.7140 (2014). 22. Hahn, E. L. Spin echoes. Phys. Rev. 80, 580–594 (1950). 23. Uhrig, G. S. Exact results on dynamical decoupling by pi pulses in quantum inforamtion processes. New J. Phys. 10, 083024 (2008). 24. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information Cambridge Univ. Press (2000). 25. Reed, M. D. et al. Realization of a three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2011). 26. Waldherr, G. et al. Quantum error correction in a solid-state hybrid spin register. Nature 506, 204–207 (2014). 27. Taminiau, T. H., Cramer, J., Van der Sar, T., Dobrovitski, V. V. & Hanson, R. Universal control and error correction in multi-qubit spin registers in diamond. Nat. Nanotechnol. 9, 171–176 (2014). ## Acknowledgements We acknowledge James MacArthur for building the correlated double sampler. This research was funded by the United States Department of Defense, the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), and the Army Research Office grant (W911NF-11-1-0068 and W911NF-11-1-0068). All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies either expressly or implied of of IARPA, the ODNI, or the U.S. Government. S.P.H was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. A.C.D. acknowledges discussions with Matthew Wadrop regarding extracting diffusion constants. A.C.D. and S.D.B. acknowledge support from the ARC via the Centre of Excellence in Engineering Quantum Systems (EQuS) project number CE110001013. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award no. ECS-0335765. CNS is a part of Harvard University. ## Author information Authors ### Contributions V.U. prepared the crystal M.D.S. fabricated the sample, J.M.N. programmed the FPGA, M.D.S, S.P.H., J.M.N., S.D.B, A.C.D, and A.Y. carried out the experiment, analyzed the data, and wrote the paper. ### Corresponding authors Correspondence to M. D. Shulman or A. Yacoby. ## Ethics declarations ### Competing interests The authors declare no conflict of interest. ## Supplementary information ### Supplementary Information Supplementary Figures 1-4 and Supplementary Notes 1-2 (PDF 868 kb) ## Rights and permissions Reprints and Permissions Shulman, M., Harvey, S., Nichol, J. et al. Suppressing qubit dephasing using real-time Hamiltonian estimation. Nat Commun 5, 5156 (2014). https://doi.org/10.1038/ncomms6156 • Accepted: • Published: • DOI: https://doi.org/10.1038/ncomms6156 • ### Fast universal quantum gate above the fault-tolerance threshold in silicon • Akito Noiri • Kenta Takeda • Seigo Tarucha Nature (2022) • ### Probabilistic teleportation of a quantum dot spin qubit • Y. Kojima • T. Nakajima • S. Tarucha npj Quantum Information (2021) • ### A singlet-triplet hole spin qubit in planar Ge • Daniel Jirovec • Andrea Hofmann • Georgios Katsaros Nature Materials (2021) • ### Floquet-enhanced spin swaps • Haifeng Qiao • John M. Nichol Nature Communications (2021) • ### Adiabatic quantum state transfer in a semiconductor quantum-dot spin chain • Haifeng Qiao • John M. Nichol Nature Communications (2021) By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.	{"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": 0.9019132852554321, "perplexity": 2237.787759175454}, "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-2022-21/segments/1652662522284.20/warc/CC-MAIN-20220518151003-20220518181003-00665.warc.gz"}
https://philpapers.org/s/R.%20Schwarzschild	## Works by R. Schwarzschild 18 found Order: Disambiguations Roger Schwarzschild [17] R. Schwarzschild [1] 1. Pluralities.Roger Schwarzschild - 1996 - Springer. Precursors. 2.1 Introduction Thus far I have presented an approach to the semantics of plurals in the form of two rather similar grammars for a fragment of English. And I have given a few examples of the kinds of things one can say within this ... Export citation Bookmark 126 citations 2. Givenness, Avoidf and Other Constraints on the Placement of Accent.Roger Schwarzschild - 1999 - Natural Language Semantics 7 (2):141-177. This paper strives to characterize the relation between accent placement and discourse in terms of independent constraints operating at the interface between syntax and interpretation. The Givenness Constraint requires un-F-marked constituents to be given. Key here is our definition of givenness, which synthesizes insights from the literature on the semantics of focus with older views on information structure. AvoidF requires speakers to economize on F-marking. A third constraint requires a subset of F-markers to dominate accents.The characteristic prominence patterns of "novelty (...) Export citation Bookmark 55 citations 3. Quantifiers in Comparatives: A Semantics of Degree Based on Intervals. [REVIEW]Roger Schwarzschild & Karina Wilkinson - 2002 - Natural Language Semantics 10 (1):1-41. The sentence Irving was closer to me than he was to most of the others contains a quantifier, most of the other, in the scope a comparative. The first part of this paper explains the challenges presented by such cases to existing approaches to the semantics of the comparative. The second part presents a new analysis of comparatives based on intervals rather than points on a scale. This innovation is analogized to the move from moments to intervals in tense semantics. (...) Export citation Bookmark 55 citations 4. (1) is an example of an adjectival comparative. In it, the adjective important is flanked by more and a comparative clause headed by than. This article is a survey of recent ideas about the interpretation of comparatives, including (i) the underlying semantics based on the idea of a threshold; (ii) the interpretation of comparative clauses that include quantifiers (brighter than on many other days); (iii) remarks on differentials such as much in (1) above: what they do in the comparative and (...) Export citation Bookmark 34 citations 5. In the formation of extended noun phrases, expressions are used that describe some dimension. Weight is described by each of the prenominal expressions in heavy rock, too much ballast, 2 lb rock, 2 lbs of rocks. The central claim of this paper is that the position of these types of expressions within the noun phrase limits the kinds of dimensions they may describe. The limitations have to do with whether or not the dimension tracks relevant part-whole relations. An analogy is (...) Translate Export citation Bookmark 25 citations 6. Singleton Indefinites.R. Schwarzschild - 2002 - Journal of Semantics 19 (3):289-314. I investigate the possibility that the apparent unique scope‐taking abilities of indefinites can be explained in terms of quantifier domain restriction, without departing from the classical view of indefinites as existential quantifiers over individuals whose scope is syntactically constrained in the same way as other quantifiers. The key idea is that when the domain of a quantifier is reduced to a singleton set, it becomes effectively scopeless. Indefinites, on this view, are freer than other quantifiers to make use of this (...) Export citation Bookmark 47 citations 7. A Course in Semantics.Daniel Altshuler, Terence Parsons & Roger Schwarzschild - 2019 - Cambridge, MA, USA: MIT Press. An introductory text in linguistic semantics, uniquely balancing empirical coverage and formalism with development of intuition and methodology. -/- This introductory textbook in linguistic semantics for undergraduates features a unique balance between empirical coverage and formalism on the one hand and development of intuition and methodology on the other. It will equip students to form intuitions about a set of data, explain how well an analysis of the data accords with their intuitions, and extend the analysis or seek an alternative. (...) Export citation Bookmark 3 citations 8. Plurals, Presuppositions and the Sources of Distributivity.Roger Schwarzschild - 1993 - Natural Language Semantics 2 (3):201-248. This paper begins with a discussion ofcumulativity (e.g., ‘P(a) & P(b) implies P(a+b)’), formalized using a verb phrase operator. Next, the meanings of distributivity markers such aseach and non-distributivity indicators such astogether are considered. An existing analysis ofeach in terms of quantification over parts of a plurality is adopted. However,together is problematic, for it involves a cancellation or negation of the quantification associated witheach. (The four boys together owned exactly three cars could not be true if each of the boys (...) Export citation Bookmark 36 citations 9. This dissertation is based on the compositional model theoretic approach to natural language semantics that was initiated by Montague (1970) and developed by subsequent work. In this general approach, coordination and negation are treated following Keenan & Faltz (1978, 1985) using boolean algebras. As in Barwise & Cooper (1981) noun phrases uniformly denote objects in the boolean domain of generalized quanti®ers. These foundational assumptions, although elegant and minimalistic, are challenged by various phenomena of coordination, plurality and scope. The dissertation solves (...) Translate Export citation Bookmark 13 citations 10. Types of Plural Individuals.Roger Schwarzschild - 1992 - Linguistics and Philosophy 15 (6):641 - 675. Export citation Bookmark 18 citations 11. There are predicates that I call “stubbornly distributive” based on what happens when they are combined with plural count noun phrases. I will use these stubbornly distributive predicates to identify and analyze a certain subset of mass nouns which I call “multi-participant nouns”. Traffic and rubble are multi-participant nouns but furniture and luggage turn out not to be. Importantly, ‘typical’ mass nouns like water are multiparticipant nouns. Export citation Bookmark 8 citations 12. GIVENness, AVOIDF, and Constraints on the Placement of Focus.Roger Schwarzschild - forthcoming - Natural Language Semantics. Export citation Bookmark 4 citations 13. The association of only with focus is explained in terms of (a) a semantics for only which makes no mention of focus and (b) discourse appropriateness conditions on the use of focus and principles of quantifier domain selection. This account differs from previous ones in giving sufficient conditions for association with focus but without stipulating it in the meaning of lexical items. Detractors have contended that foci have different pragmatic import depending on whether or not they are associated with a (...) Export citation Bookmark 3 citations 14. This paper grew out of a reaction to Elisabeth Selkirk's contribution to the Handbook of Phonology (Goldsmith 1996). Section 1.2 of that article is concerned with syntactic and semantic aspects of the placement of pitch accents in English. As will be seen in the data to be presented below, the constellation of pitch accents in an utterance is determined in part by properties of the preceding discourse, including the distinction between new and old information. This means for example, that a (...) No categories Translate Export citation Bookmark 3 citations 15. Export citation Bookmark 16. I. Intervals & the Unique Witness Property (1) John is taller than Mary. (2) \!d \!dS John is d-tall ‚ (d B dS) ‚ Jill is d S-tall. (3) \!d \!dS ϕ(d) ‚ (d B dS) ‚ ψ(dS). (p-p) (4) SNEAKERS. Grant expresses interest in a pair of sneakers. I offer to buy them for him, having the impression that they cost somewhere in the $20-$30 range. We arrive at the store and to my horror I discover that the sneakers (...) Translate Export citation Bookmark 17. It is natural to think of comparisons in terms of points on a scale. Jack is taller than Jill if the point associated with Jack on the height scale is higher than Jill’s point. Jack is much taller than Jill is if Jack’s point is separated from Jill’s by a sizable amount. It is also natural to think of temporal discourse in terms of points on a time line. The analogy between the two is worth taking seriously. Export citation Bookmark 18. This paper strives to characterize the relation between accent placement and discourse in terms of independent constraints operating at the interface between syntax and interpretation. The GIVENness Constraint requires un-F-marked constituents to be GIVEN. Key here is our definition of GIVENness which synthesizes insights from the literature on the semantics of focus with older views on information structure. AvoidF requires speakers to economize on F-marking. A third constraint requires a subset of F-markers to dominate accents. 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https://www.physicsforums.com/threads/polarity-of-induced-emf-in-a-conducting-ring.586581/	# Polarity of Induced EMF in a Conducting Ring 1. Mar 13, 2012 ### samirgaliz I have a question regarding a conducting loop of radius r in a changing magnetic field B. I understand and can determine the direction of the induced current which implies the existing of an electric field that is tangential to the loop. Since this is a closed loop, I am having trouble determining the polarity of the induced emf(potential difference). How would it be possible to measure experimentally this voltage or even use it to power up a light bulb? Any help would be appreciated. Thanks 2. Mar 13, 2012 ### chrisbaird In a purely theoretical sense, ungrounded AC circuits do not have a stationary polarity. The emf, and therefore the induced current, oscillates in direction back and forth, and therefore the polarity continually switches as the system oscillates. In practice, something special is done to one leg of the circuit (such as grounding it) that is not done to the other leg of the circuit, and we therefore externally impose an effective stationary polarity on the AC system. 3. Mar 13, 2012 ### samirgaliz Thanks for your speedy reply. I forgot to mention that the loop is closed and stationary and the magnetic filed is continuously increasing in one direction perpendicular to the loop and thus the induced current does not change direction. 4. Mar 14, 2012 ### Philip Wood Unless I've missed something, the answer is very easy. The emf is in the same direction as the current. The conventional current is in the same direction as the electric field at each point in the ring. So work is done on the conventional (positive) charge carriers by the field. So the emf is in the same sense as the current. 5. Mar 14, 2012 ### samirgaliz Thanks Philip. The issue is with polarity. Typically a current flow from high to low potential. Is it because the electric field here is non-conservative (line integral along a closed path in this case is not = 0) and is created due to the change in magnetic field and not by charge separation as in the case of a battery? 6. Mar 14, 2012 ### Philip Wood As you say, the electric field is non-conservative. I would say that the concept of p.d. is inapplicable, though I stand to be corrected. We still have: emf = IR. 7. Mar 15, 2012 ### samirgaliz I agree with you and thanks for the clarification. Much appreciated. 8. Mar 15, 2012 ### vanhees71 An emf, by definition, has no current. It's based on Faradays Law, which is one of the fundamental Maxwell equations of electromagnetism. Using Heaviside-Lorentz units it reads $$\vec{\nabla} \times \vec{E}=-\frac{1}{c} \frac{\partial \vec{B}}{\partial t}.$$ This Law can be written in integral form by integrating over an arbitrary closed loop $\partial A$, encirceling an area $A$. With help of Stokes's Law, one gets $$\int_{\partial A} \mathrm{d} \vec{x} \cdot \vec{E}=-\frac{1}{c} \int_{A} \mathrm{d}^2 \vec{F} \cdot \partial_t \vec{B}.$$ Here, by definition, the boundary $\partial A$ of the surface $A$ is orientied positive relative to the surface by the right-hand rule: Directing the thumb of your right hand in direction of the surface-normal vectors (whose direction can be chosen arbitrarily!), the fingers point into the positively oriented tangent vectors of the boundary curve, $\partial A$. This uniquely defines the signs in Faraday's Law in integral form. 9. Mar 15, 2012 ### Philip Wood vanhees71, as always, is spot-on. And, of course, the magnitude and direction of the emf don't depend in any way on the presence of a current. The ring could be an insulator. Nonetheless, if you have been taught how to predict the sense of a current in a conducting ring, then you can use the same rule to give the sense of the emf in a ring whether conducting or not. This procedure is more convoluted than starting from Faraday's law, but it does deliver the goods. [Actually, the rule for current direction can be justified in terms of energy conservation, which is pretty fundamental!] 10. Mar 15, 2012 ### Hassan2 Actually I have a similar problem. When I use Faraday's law, I can't say ,from the result , that what is the direction of emf. I need to use "right hand lay" for that. But in a case of a moving conductor in static magnetic field, since the line integral has a direction , I can say the direction of the emf without using the right hand law. I have read that, the Faraday's law had no " - " sign at the beginning. The negative sign was proposed by Lenz. What does the negative sign tell us about the direction of the emf? 11. Mar 15, 2012 ### Philip Wood You're right about Lenz's contribution. Faraday himself didn't use algebra; his mathematics didn't go beyond arithmetic and the concept of proportionality (possibly some geometry, too, but I don't think we've much evidence on this). So he didn't write the law as an equation, let alone use a minus sign. 12. Mar 15, 2012 ### technician I agree with Philip Wood, Faraday did not use high level mathematics to convey his findings. He invented magnetic flux lines of force to explain what he discovered. The facts are easy to state (Faraday's laws) Whenever a conductor experiences a changing magnetic flux linkage and emf is induced. The induced emf = rate of change of flux linkage The emf opposes the changing flux linkage (usually known as Lenz's law) The - sign in an equation indicates 'opposition' this occurs in other aspects of physics (SHM etc) If there is a complete circuit the induced emf causes a (induced) current to flow. 13. Mar 15, 2012 ### cabraham The emf does not "cause" a current. In order to create the time-changing emf, a current takes place. The emf cannot exist w/o displacement current. Even if the loop is open, there is still a current. In the ac domain, an ac voltage cannot exist w/o an ac current. The 2 are mutual & inclusive. Neither exists alone. Neither is the cause nor the effect of the other. Does this help? BR. Claude 14. Mar 15, 2012 ### Philip Wood The only way I can make sense of this, and the rest of your post, is by assuming that you are using 'current' to mean not just conduction current but so-called displacement current as well. In my posts I have been using 'current' to mean ordinary conduction current. I like to think this is the normal usage. 15. Mar 15, 2012 ### technician If there is not a complete circuit there is no conventional current. I was of the understanding that, in physics, when we say current we mean conventional current 16. Mar 15, 2012 ### technician I would also go back to the original post......How is it possible to measure the voltage and power a light bulb...... It is easy, break the loop and connect a voltmeter or a lamp......it will Be easier to demonstrate the truth with a coil of several turns and a moving magnet. Standard, basic school demonstration of Faradays laws.......no maths needed at this level. 17. Mar 15, 2012 ### cabraham Per Maxwell, current is current is current. Displacement current is as real as is conduction current. Maxwell's equations includes both - ref Ampere Law. My point was that the emf is not the "cause" of the current. The only way the loop can have no displacement current is when it is a perfect short. But there is no emf here, yet there is a conduction current. Current in this case exists w/o emf. I was just clarifying what is happening. Current in the loop is not caused by the emf, but rather, both are dependent on Lorentz force. Claude 18. Mar 16, 2012 ### Philip Wood Whatever our views on displacement current, I think it is misleading to use the unqualified word 'current' to include both displacement current and ordinary conduction current. That's all. Last edited: Mar 16, 2012 19. Mar 16, 2012 ### cabraham I can't believe that there is no Lorentz force. For an emf to be induced, charges must be moved. Without Lorentz force, charges won't move in the 1st place. If charges can't move, there can be no induced emf. When moving a magnet, the magnetic field is accompanied by an electric field. Not to be offensive, but how extensively have you studied e/m fields? BS, MS, PhD, phy or EE? Just wondering what you base your statements on. You seem to be stating the following. Moving a magnet towards a loop results in NO Lorentz force, yet an emf is induced which results in a current if the path is closed. If that is your belief, then it does not agree with known laws. I have trouble understanding how electrons move w/o Lorentz force. Remember that under time changing conditions, E fields & H fields (or B if you prefer) cannot exist independently. A stationary magnet has only H/B, no D/E. But once in motion, E & H are both present. A stationary electron feels a force due to Lorentz force via the E field, F = qE. Although vXB = 0 since v=0, the E field still accounts for a Lorentz force. https://www.physicsforums.com/showthr...46#post2163546 [Broken] Please refer to the above link. In circuit terms I computed the relation between flux (internal & external), frequency, current, & voltage. If you disagree, then please provide the correction you feel is needed. BR. Claude Last edited by a moderator: May 5, 2017 20. Mar 16, 2012 ### cabraham Also, I forgot to mention that although the electrons are not initially moving, the magnet is moving wrt the electrons. There is relative motion between electron and B field, & I recall learning that relative motion is all that is needed for Lorentz force. An electron moving past a magnet, or a magnet moving past an electron, will both result in Lorentz force. Am i missing something? Claude Last edited: Mar 16, 2012 Similar Discussions: Polarity of Induced EMF in a Conducting Ring	{"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": 0.9086975455284119, "perplexity": 793.707878432613}, "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/1502886110485.9/warc/CC-MAIN-20170822065702-20170822085702-00090.warc.gz"}
http://math.stackexchange.com/questions/551825/given-only-the-graphical-representation-of-a-curve-how-can-i-find-the-function	# Given only the graphical representation of a curve, how can I find the function? I'm developing an application which will: A. identify curves in 2D images using the generalized Hough Transform B. determine from the graphical curve the mathematical function needed to generate it I'm not sure quite how to approach item B; I think that if the order $N$ of the function is known the problem can be solved by sampling $N+1$ points along the curve, and solving the quadratic/cubic/quartic... etc. formula to get the complete function: ex. Given a curve known to be quadratic (order $2$), we can sample $3$ points to obtain three equations to solve for the $a,b, \text{ and } c$ variables in $y=ax^2+bx+c$. The real question is how can I adapt this approach for curves whose function's polynomial order is not known-- how can I determine the polynomial order of the function needed to generate a given curve, with only the graphical representation of the curve to work with? thanks! - Trig functions can be incorporated by sampling for periodicity. – gt6989b Nov 4 '13 at 18:07 Take a bunch of points, if that doesn't work out well enough, take more points and repeat. Or read up on polynomial regression. There is hardly such thing as (uniquely) identifying a function from a picture, even for polynomials. – Karolis Juodelė Nov 4 '13 at 18:19	{"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": 0.8677113652229309, "perplexity": 374.60471549378843}, "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/1448398445080.12/warc/CC-MAIN-20151124205405-00119-ip-10-71-132-137.ec2.internal.warc.gz"}
https://daviddalpiaz.github.io/stat3202-au18/homework/pp-01-assign.html	# Exercise 1 Consider independent random variables $$X_1$$, $$X_2$$, and $$X_3$$ with • $$\text{E}[X_1] = 1$$, $$\text{Var}[X_1] = 4$$ • $$\text{E}[X_2] = 2$$, $$\text{SD}[X_1] = 3$$ • $$\text{E}[X_3] = 3$$, $$\text{SD}[X_1] = 5$$ (a) Calculate $$\text{E}[5 X_1 + 2]$$. (b) Calculate $$\text{E}[4 X_1 + 2 X_2 - 6 X_3]$$. (c) Calculate $$\text{Var}[5 X_1 + 2]$$. (d) Calculate $$\text{Var}[4 X_1 + 2 X_2 - 6 X_3]$$. # Exercise 2 Consider random variables $$H$$ and $$Q$$ with • $$\text{E}[H] = 3$$, $$\text{Var}[H] = 16$$ • $$\text{SD}[Q] = 4$$, $$\text{E}\left[\frac{Q^2}{5}\right] = 3.2$$ (a) Calculate $$\text{E}[5H^2 - 10]$$. (b) Calculate $$\text{E}[Q]$$. # Exercise 3 Consider a random variable $$S$$ with probability density function $f(s) = \frac{1}{9000}(2s + 10), \ \ 40 \leq s \leq 100.$ (a) Calculate $$\text{E}[S]$$. (b) Calculate $$\text{SD}[S]$$. # Exercise 4 Consider independent random variables $$X$$ and $$Y$$ with • $$X \sim N(\mu_X = 2, \sigma^2_X = 9)$$ • $$Y \sim N(\mu_Y = 5, \sigma^2_Y = 4)$$ (a) Calculate $$P[X > 5]$$. (b) Calculate $$P[X + 2Y > 5]$$. # Exercise 5 Consider random variables $$Y_1$$, $$Y_2$$, and $$Y_3$$ with • $$\text{E}[Y_1] = 1$$, $$\text{E}[Y_2] = -2$$, $$\text{E}[Y_3] = 3$$ • $$\text{Var}[Y_1] = 4$$, $$\text{Var}[Y_2] = 6$$, $$\text{Var}[Y_3] = 8$$ • $$\text{Cov}[Y_1, Y_2] = 1$$, $$\text{Cov}[Y_1, Y_3] = -1$$, $$\text{Cov}[Y_2, Y_3] = 0$$ (a) Calculate $$\text{Var}[3Y_1 - 2Y_2]$$. (b) Calculate $$\text{Var}[3Y_1 - 4Y_2 + 2Y_3]$$. # Exercise 6 Consider using $$\hat{\xi}$$ to estimate $$\xi$$. (a) If $$\text{Bias}[\hat{\xi}] = 5$$ and $$\text{Var}[\hat{\xi}] = 4$$, calculate $$\text{MSE}[\hat{\xi}]$$ (b) If $$\hat{\xi}$$ is unbiased, $$\xi = 6$$, and $$\text{MSE}[\hat{\xi}] = 30$$, calculate $$\text{E}\left[\hat{\xi}^2\right]$$ # Exercise 7 Using the identity $(\hat{\theta}-\theta) = \left(\hat{\theta}-\text{E}[\hat{\theta}]\right) + \left(\text{E}[\hat{\theta}] - \theta\right) = \left(\hat{\theta} - \text{E}[\hat{\theta}]\right) + \text{Bias}[\hat{\theta}]$ show that $\text{MSE}[\hat{\theta}] = \text{E}\left[(\hat{\theta} - \theta)^2\right] = \text{Var}[\hat{\theta}] + \left(\text{Bias}[\hat{\theta}]\right)^2$ # Exercise 8 Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a population with mean $$\mu$$ and variance $$\sigma^2$$. Consider three estimators of $$\mu$$: $\hat{\mu}_1 = \frac{X_1 + X_2 + X_3}{3}, ~~~\hat{\mu}_2 = \frac{X_1}{4} + \frac{X_2 + \cdots + X_{n - 1}}{2(n - 2)} + \frac{X_n}{4}, ~~~\hat{\mu}_3 = \bar{X},$ Calculate the mean squared error for each estimator. (It will be useful to first calculate their bias and variances.) # Exercise 9 Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a distribution with density $f(x) = \frac{1}{\theta}e^{-x/\theta}, ~~x > 0, \theta \geq 0$ Consider five estimators of $$\theta$$: $\hat{\theta}_1 = X_1, ~~~\hat{\theta}_2 = ~~~\frac{X_1 + X_2}{2}, ~~~\hat{\theta}_3 = ~~~\frac{X_1 + 2X_2}{3}, ~~~\hat{\theta}_4 = \bar{X}, ~~~\hat{\theta}_5 = 5$ Calculate the mean squared error for each estimator. (It will be useful to first calculate their bias and variances.) # Exercise 10 Suppose that $$\text{E}\left[\hat{\theta}_1\right] = \text{E}\left[\hat{\theta}_2\right] = \theta$$, $$\text{Var}\left[\hat{\theta}_1\right] = \sigma_1^2$$, $$\text{Var}\left[\hat{\theta}_2\right] = \sigma_2^2$$, and $$\text{Cov}\left[\hat{\theta}_1, \hat{\theta}_2\right] = \sigma_{12}$$. Consider the unbiased estimator $\hat{\theta}_3 = a\hat{\theta}_1 + (1-a)\hat{\theta}_2.$ If $$\hat{\theta}_1$$ and $$\hat{\theta}_2$$ are independent, what value should be chosen for the constant $$a$$ in order to minimize the variance and thus mean squared error of $$\hat{\theta}_3$$ as an estimator of $$\theta$$? # Exercise 11 Let $$Y$$ have a binomial distribution with parameters $$n$$ and $$p$$. Consider two estimators for $$p$$: $\hat{p}_1 = \frac{Y}{n}$ and $\hat{p}_2 = \frac{Y + 1}{n + 2}$ For what values of $$p$$ does $$\hat{p}_2$$ achieve a lower mean square error than $$\hat{p}_1$$? # Exercise 12 Let $$X_1, X_2, \ldots, X_n$$ denote a random sample from a population with mean $$\mu$$ and variance $$\sigma^2$$. Create an unbiased estimator for $$\mu^2$$. Hint: Start with $$\bar{X}^2$$. # Exercise 13 Let $$X_1, X_2, X_3, \ldots, X_n$$ be iid random variables form $$\text{U}(\theta, \theta + 2)$$. (That is, a uniform distribution with a minimum of $$\theta$$ and a maximum of $$\theta + 2$$.) Consider the estimator $\hat{\theta} = \frac{1}{n}\sum_{i = 1}^{n}X_i = \bar{X}$ (a) Calculate the bias of $$\hat{\theta}$$ when estimating $$\theta$$. (b) Calculate the variance of $$\hat{\theta}$$. (c) Calculate the mean squared error of $$\hat{\theta}$$ when estimating $$\theta$$.	{"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": 0.9983697533607483, "perplexity": 408.3348872339858}, "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-2021-43/segments/1634323587854.13/warc/CC-MAIN-20211026072759-20211026102759-00311.warc.gz"}
http://mathoverflow.net/revisions/71278/list	These three ensembles are hermitian matrices over a (finite dimensional real) field of numbers, and it is known that the only finite dimensional real fields are the real numbers, the complex numbers ($2$-dimensional) and the quaternionic numbers ($4$-dimensional). Octionions Octonions are not a field of number since you do not have associativity. The motivation in physics comes from the fact that an hermitian matrix represents a finite dimentional Hamiltonian (an Hermitian operator) in quantum mechanics (then you add randomness, in order to take in account the lack of information about your system, and you let the size of the matrix, that is the dimension of your state space where your Hamiltionian is acting on, going to infinity). In this setting, $N\times N$ quaternionic matrices have to be seen as subclasses of complex hermitian matrices (but of size $2N\times 2N$) and both real symmetric and quaternionic hermitian matrices are a subclass of complex hermitian matrices, with extra symmetries. Anyway, you may imagine many different matrix models relevant for studying (look for Wigner matrices, the answer of Beenakker about other symmetries in physics, the generalized $\beta$-ensemble of Edelman, etc ...) These three ensembles are hermitian matrices over a (finite dimensional real) field of numbers, and it is known that the only finite dimensional real fields are the real numbers, the complex numbers ($2$-dimensional) and the quaternionic numbers ($4$-dimensional). Octionions are not a field of number since you do not have associativity. The motivation in physics comes from the fact that an hermitian matrix represents a finite dimentional Hamiltonian (an Hermitian operator) in quantum mechanics (then you add randomness, in order to take in account the lack of information about your system, and you let the size of the matrix, that is the dimension of your state space where your Hamiltionian is acting on, going to infinity). In this setting, $N\times N$ quaternionic matrices have to be seen as subclasses of complex hermitian matrices (but of size $2N\times 2N$) and both real symmetric and quaternionic hermitian matrices are a subclass of complex hermitian matrices, with extra symmetries. Anyway, you may imagine many different matrix models relevant for studying (look for Wigner matrices, the answer of Beenakker about other symmetries in physics, the generalized $\beta$-ensemble of Edelman, 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": 0.9118813276290894, "perplexity": 342.3026815996184}, "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/1368702957608/warc/CC-MAIN-20130516111557-00096-ip-10-60-113-184.ec2.internal.warc.gz"}
https://education.ti.com/en/customer-support/knowledge-base/ti-83-84-plus-family/product-usage/11717	Education Technology # Knowledge Base ## Solution 11717: Calculating the Inverse Student-T (invT) Probability on the TI-84 Plus Family or TI-Nspire™ Handheld in TI-84 Plus Mode. ### How can I calculate the inverse student-T probability on the TI-84 Plus family or TI-Nspire handheld in TI-84 Plus mode with OS 2.30 and higher? The Inverse Student-T is usually used for determining probabilities of population means from sampled distributions when the standard deviation of the population is unknown. Specifically, t = (x-u)*sqrt(n)/s with n-1 degrees of freedom and where x is the sampled mean, u is the population mean, s is the sample standard deviation, n is the sample size, and the population is Gaussian. Example: The command for this function is under the DISTR menu. Please refer to the instructions provided below. 1) Press [2nd][DISTR] 2) Select 4:InvT( 3) Enter a scalar or list of values at which to evaluate the t inverse and a scalar value for degrees of freedom separated with a comma. ' We will key in the following example: InvT(0.68759,18) 4) Press [ENTER] to display the answer of 0.4975567034	{"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": 0.8321876525878906, "perplexity": 1297.6474273295596}, "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/1627046154302.46/warc/CC-MAIN-20210802012641-20210802042641-00407.warc.gz"}
http://mathhelpforum.com/pre-calculus/117799-exponential-equations-print.html	# exponential equations • Dec 1st 2009, 08:11 AM extraordinarymachine exponential equations help please, i'm not sure how to solve this question. i know i need to use change of base, but i'm still confused. solve equation, check for extraneous roots 2^x + 12(2)^-x = 7 • Dec 1st 2009, 08:35 AM Hello extraordinarymachine Quote: Originally Posted by extraordinarymachine help please, i'm not sure how to solve this question. i know i need to use change of base, but i'm still confused. solve equation, check for extraneous roots 2^x + 12(2)^-x = 7 Multiply both sides by $2^x$ and re-arrange: $(2^x)^2-7.2^x+12=0$ $\Rightarrow (2^x-3)(2^x-4)=0$ Can you complete it now? • Dec 1st 2009, 09:56 AM mrmohamed Quote: Originally Posted by extraordinarymachine help please, i'm not sure how to solve this question. i know i need to use change of base, but i'm still confused. Quote: Originally Posted by extraordinarymachine solve equation, check for extraneous roots 2^x + 12(2)^-x = 7	{"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": 0.9336404800415039, "perplexity": 1338.3332167789633}, "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/1518891811243.29/warc/CC-MAIN-20180218003946-20180218023946-00321.warc.gz"}
https://math.meta.stackexchange.com/questions/27359/maximum-number-of-closed-votes-queue-limit	# Maximum Number of Closed Votes Queue Limit I have been noticing that there are more frequent occurrences of the Closed Votes queue getting over 200-items (red-dot). I have not seen this on any of the other queues. Should consideration be given to allowing reviewers more than the current 20-reviews max per day on the Closed Votes queue only? Maybe statistics can be reviewed to see averages and increase the number to say $30-40$? This is allowing items in the close cycle to stay open longer, which almost seems to go against the purpose of the queue. • Great question! I've always just thrown up my hands and sighed in frustration when I see such numbers of close votes in the review queue, even after I've "put in my 20 reviews" in for any given day. – amWhy Nov 15 '17 at 22:07 • @amWhy: I agree, there are many excellent reviewers such as yourself that fill their quota of 20. I think these reviewers should be able to do twice as many and maybe there is a rule that the folks that can do this have a rep = ... in order to even qualify to do more reviews. – Moo Nov 15 '17 at 22:20 • If it ever gets over 1000, we'll get 40 reviews per day. – Glorfindel Nov 18 '17 at 10:01 • @Glorfindel: Is that an update as I don't recall it being the norm? – Moo Nov 18 '17 at 13:16 • It has been so for years on Stack Overflow (where the queue regularly reaches 10k entries). They’ve done some experimenting with other queues as well, e.g. for Triage the 40 reviews already kicks in at a queue size of 150. – Glorfindel Nov 18 '17 at 13:20 • @amWhy: I have seen that you reach 20/day frequently, Have you noticed the above behavior? – Moo Nov 18 '17 at 13:23 • @Moo I don't think Math.SE ever had more then $1000$ reviews pending for one queue. – kingW3 Nov 18 '17 at 15:03 • It happens at least once, when policy changed @kingW3 – user99914 Nov 18 '17 at 17:07 • @Moo The numbers do get high; I know that (only once in Jan. this year, when I went on a "reviewing strike" how crucial it is to have committed reviewers. It would be great if the "burden/maintenance" would be spread more widely, but some active answerers are too caught up in chasing rep and answering (often questions needing closure) that they take for granted the work behind the scenes that makes there rep chase possible, to begin with. So I believe that it makes sense to grant extra review privileges to those with a gold badge in a given queue. – amWhy Nov 18 '17 at 22:14 • But yes, there've been times when the numbers waiting in the queue become astronomical, virtually undermining the intention of review queues in the first place. – amWhy Nov 18 '17 at 22:16 • Well, you must strike often, @MartinArgerami, because I think half the votes to "leave open" absurd, and often cast by the answerers of proposed PSQs – amWhy Nov 18 '17 at 23:54 • Perhaps if more people reviewed, there'd be fewer closed questions? You are retreating from the fact that there are too many questions/answers in the review queue as compared to the number of users who actually care to review. I only occasionally see you, @MartinArgerami, in the review queue. Put your money where your mouth is, and spend more time. And get over the "idiotic" audits we all confront. – amWhy Nov 19 '17 at 0:00 • There you go, @MartinArgerami! If you don't want to be part of the solution, then don't post noise in this thread. – amWhy Nov 19 '17 at 0:36 • @amWhy: I offered you the other side of the problem (a huge number of frivolous closing votes) and you choose to berate me. If you want to negate that problem, then yes, I'll leave you alone to solve it. – Martin Argerami Nov 19 '17 at 0:38 • @MartinArgerami If you would not have negated the problem ("No thanks": Choosing not to be a part of the solution to the problem, and only simply a naysayer attacking those who want to be a part of the solution), I'd have no reason to comment as I did. You were the one who opted out, and berated yourself. When you're active in the review queue, regularly and vote as you see fit, that's how you offer the "other side of the problem." Empty words and claims have little value when you don't behave accordingly. So stop complaining, and start contributing, "close" or "leave open"... – amWhy Nov 19 '17 at 0:50	{"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": 0.26931244134902954, "perplexity": 1512.2507114533712}, "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-2020-29/segments/1593657147031.78/warc/CC-MAIN-20200713225620-20200714015620-00260.warc.gz"}
https://mathoverflow.net/questions/238686/is-there-a-complete-riemannian-manifold-with-infinite-volume-whose-the-time-one/238699	# Is there a complete Riemannian manifold with infinite volume whose the time-one map of the geodesic flow is recurrent? Let M be complete Riemannian manifold M with infinite volume, it is know that the geodesic flow, $\varphi^t:T^1M \rightarrow T^1M$ preserves the Liouville measure $\mu$, that is, $\mu(\varphi^t(A)) = \mu(A)$ for every $t \in \Bbb{R}$ and for all borelian set A. In particular, the time-one map $\varphi^1$ also preserves the measure $\mu$ because $\mu(\varphi^{-1}(A)) = \mu(A)$. We say that the map $\varphi^1$ is recurrent (or conservative) if all wandering set W for $\varphi^1$( meaning that $W \cap \varphi^{-n}(W) = \varnothing$ for $n \geq 1$) has necessarily $\mu(W) = 0$. An another equivalent definition is, if A is borelian set with $\mu(A) > 0$ then $$A \subset \displaystyle\bigcup_{n \geq 1} \varphi^{-n}(A) \ \ (mod \ \ \mu)$$ Question: Is there a complete Riemannian manifold M with infinite volume whose the time-one map of the geodesic flow is recurrent ? Take a compact, connected Riemannian manifold $M$ with negative sectional curvature. Then choose any cover $M'$ of $M$ which is connected, Galois, and whose group of deck transformations is $\mathbb{Z}$ or $\mathbb{Z}^2$. In dimension $2$ and for a $\mathbb{Z}$ cover, this can be done by taking $\mathbb{Z}$ copies of the manifold $M$, cutting along an essential curve (in dimension $2$). Then we get manifold with boundaries which can be indexed by $\mathbb{Z} \times \{0,1\}$. Glue $(p,0)$ with $(p+1,1)$. The geodesic flow on $T^1 M'$ behaves roughly like a random walk on $\mathbb{Z}$ (or $\mathbb{Z}^2$) with centered and bounded increments, so it is recurrent. There are other examples in the same fashion, for instance $\mathbb{C} \setminus \mathbb{Z}$ endowed with a $\mathbb{Z}$-invariant hyperbolic metric. See for instance: • J. Aaronson, M. Denker, Distributional limits for hyperbolic, infinite volume geodesic flows (Tr. Mat. Inst. Steklova 216 (1997), Din. Sist. i Smezhnye Vopr., 181--192) • J. Aaronson, M. Denker, The Poincar\'e series of $\mathbb{C}\setminus\mathbb{Z}$ (Ergodic Theory Dynam. Systems 19 (1999), no. 1, 1--20.)	{"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": 0.9877888560295105, "perplexity": 98.19247333518544}, "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-2020-10/segments/1581875146681.47/warc/CC-MAIN-20200227094720-20200227124720-00024.warc.gz"}
https://chemistry.stackexchange.com/questions/4506/what-does-under-stoichiometric-or-stoichiometric-mean	# What does under-stoichiometric or stoichiometric mean? When people talk about oxides like $\ce{SiO2}$ or $\ce{Al2O3}$, they use expressions like stoichiometric or under-stoichiometric. I understand that this refers to the relative composition of materials, but I don't really know what it exactly means. Say for $\ce{SiO2}$, does under-stoichiometric mean that it isn't exactly $\ce{SiO2}$, but more like $\ce{SiO_{1.8}}$? Does the term usually refer to the oxygen composition? Yes, the convention is to refer to the oxygen atoms when talking about over- and under-stoichiometric in metaloxides. Normally they also indicate this explicitly by writing $\ce {SiO}_x$ (like in the article here for example) or $\ce {Al_2O}_x$. Being explicit about it is a good habit because it avoids ambiguity.	{"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": 0.7922434210777283, "perplexity": 606.7585884858283}, "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/1614178365454.63/warc/CC-MAIN-20210303042832-20210303072832-00174.warc.gz"}
https://rd.springer.com/chapter/10.1007/978-3-319-45994-3_9	RP 2016: Reachability Problems pp 119-133 # Distributed Synthesis of State-Dependent Switching Control • Laurent Fribourg • Nicolas Markey • Florian De Vuyst • Ludovic Chamoin Conference paper Part of the Lecture Notes in Computer Science book series (LNCS, volume 9899) ## Abstract We present a correct-by-design method of state-dependent control synthesis for linear discrete-time switching systems. Given an objective region R of the state space, the method builds a capture set S and a control which steers any element of S into R. The method works by iterated backward reachability from R. More precisely, S is given as a parametric extension of R, and the maximum value of the parameter is solved by linear programming. The method can also be used to synthesize a stability control which maintains indefinitely within R all the states starting at R. We explain how the synthesis method can be performed in a distributed manner. The method has been implemented and successfully applied to the synthesis of a distributed control of a concrete floor heating system with 11 rooms and $$2^{11}=2048$$ switching modes. ## Keywords Heat Transfer Coefficient Control Synthesis Symbolic State Parametric Extension Control Pattern These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves. ## References 1. 1. Alur, R., Henzinger, T.A.: Reactive modules. Formal Methods Syst. Des. 15(1), 7–48 (1999) 2. 2. Asarin, E., Bournez, O., Dang, T., Maler, O., Pnueli, A.: Effective synthesis of switching controllers for linear systems. Proc. IEEE 88(7), 1011–1025 (2000) 3. 3. Fribourg, L., Kühne, U., Markey, N.: Game-based synthesis of distributed controllers for sampled switched systems. In: SynCoP 2015, OASIcs 44, pp. 48–62 (2015)Google Scholar 4. 4. Fribourg, L., Kühne, U., Soulat, R.: Finite controlled invariants for sampled switched systems. Formal Methods Syst. Des. 45(3), 303–329 (2014) 5. 5. Gillula, J.H., Hoffmann, G.M., Huang, H., Vitus, M.P., Tomlin, C.: Applications of hybrid reachability analysis to robotic aerial vehicles. Int. J. Rob. Res. 30(3), 335–354 (2011) 6. 6. Girard, A.: Low-complexity switching controllers for safety using symbolic models. In: ADHS 2012, pp. 82–87 (2012)Google Scholar 7. 7. Jaulin, L., Kieffer, M., Didrit, O., Walter, E.: Applied Interval Analysis. Springer, Berlin (2001) 8. 8. Kühne, U., Soulat, R.: Minimator 1.0 (2015). https://bitbucket.org/ukuehne/minimator/ 9. 9. Larsen, K.G., Mikučionis, M., Muñiz, M., Srba, J., Taankvist, J.H.: Online and compositional learning of controllers with application to floor heating. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 244–259. Springer, Heidelberg (2016). doi: 10. 10. Le Coënt, A., Fribourg, L., Markey, N., De Vuyst, F., Chamoin, L.: Distributed synthesis of state-dependent switching control. Technical report, March 2016. https://hal.archives-ouvertes.fr/hal-01295738 11. 11. Le Coent, A., Alexandre Dit Sandretto, J., Chapoutot, A., Fribourg, L.: Control of nonlinear switched systems based on validated simulation. In: SNR 2016. IEEE (2016)Google Scholar 12. 12. Liberzon, D.: Switching in Systems and Control. Springer, Berlin (2012) 13. 13. Meyer, P.-J., Girard, A., Witrant, E.: Safety control with performance guarantees of cooperative systems using compositional abstractions. In: ADHS 2015, pp. 317–322 (2015)Google Scholar 14. 14. Mitchell, I.M.: Comparing forward and backward reachability as tools for safety analysis. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 428–443. Springer, Heidelberg (2007) © Springer International Publishing Switzerland 2016 ## Authors and Affiliations • 1 • Laurent Fribourg • 2 • Nicolas Markey • 2 Email author • Florian De Vuyst • 1 • Ludovic Chamoin • 3 1. 1.CMLA, ENS Cachan, CNRS, Université Paris-SaclayCachan CedexFrance 2. 2.LSV, ENS Cachan, CNRS, Université Paris-SaclayCachan CedexFrance 3. 3.LMT, ENS Cachan, CNRS, Université Paris-SaclayCachan CedexFrance	{"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": 0.7431413531303406, "perplexity": 19835.480590111918}, "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-2019-43/segments/1570987803441.95/warc/CC-MAIN-20191022053647-20191022081147-00210.warc.gz"}
https://library.curriki.org/oer/G-SRT-Sine-and-Cosine-of-Complementary-Angles-198183	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: Suppose $0 \lt a \lt 90$ is the measure of an acute angle. Draw a picture and explain why $\sin{a} = \cos{(90 -a)}$ Are there any angle measures \$0 \lt... Non-profit Tax ID # 203478467	{"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": 0.5170547366142273, "perplexity": 380.4327751878852}, "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-2020-24/segments/1590347439928.61/warc/CC-MAIN-20200604094848-20200604124848-00270.warc.gz"}
https://math.stackexchange.com/questions/3191865/convergence-of-frechet-derivative-functional-related-to-palais-smale-sequence	# Convergence of Frechet Derivative Functional related to Palais-Smale Sequence Let $$\phi : H_{0}^{1}(\Omega)\to\mathbb{R}$$ be an energy functional for bounded domain $$\Omega$$ such that $$\phi'$$ be its Frechet derivative, and there exists a bounded sequence $$\{u_{n}\}_{n\in\mathbb{N}}$$ bounded in $$H_{0}^{1}(\Omega)$$ such that $$\phi(u_{n})\to d$$ and $$\phi'(u_{n})\to 0$$ as $$n\to\infty$$. Assume that $$u_{n}\to u$$ weakly in $$H_{0}^{1}(\Omega)$$ and define $$\langle\,\cdot\, , \,\cdot\,\rangle$$ as the inner dot product in $$H_{0}^{1}(\Omega)$$. Claim : $$\langle\phi'(u_{n})-\phi'(u),u_{n}-u\rangle\to 0$$ as $$n\to\infty$$. My question is how to show that claim. My attempt so far is to separate it into : $$\langle\phi'(u_{n}),u_{n}-u\rangle - \langle\phi'(u),u_{n}-u\rangle$$ The second term tends to zero as $$n\to\infty$$ by weak convergence of $$\{u_{n}\}_{n\in\mathbb{N}}$$ but I am not sure how to show that the first term tends to zero since $$u_{n}-u$$ still depends on $$n$$ so I do not know whether I could use the assumption that $$\phi'(u_{n})\to0$$ as $$n\to\infty$$. Any help is much appreciated! Thank you very much!	{"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": 21, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9943422079086304, "perplexity": 42.08007561550701}, "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/1560627999876.81/warc/CC-MAIN-20190625172832-20190625194832-00242.warc.gz"}
https://ftp.aimsciences.org/article/doi/10.3934/cpaa.2013.12.1307	# American Institute of Mathematical Sciences May 2013, 12(3): 1307-1319. doi: 10.3934/cpaa.2013.12.1307 ## The regularity for a class of singular differential equations 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China Received February 2012 Revised May 2012 Published September 2012 We find an iteration technique and thus prove the optimal global regularity for the boundary value problem of a class of singular differential equations with strongly singular lower terms at the boundary. As applications, we obtain the regularity for the radial solutions of Ginzburg-Landau equations and harmonic maps. Citation: Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307 ##### References: [1] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices," Birkhauser, 1993. doi: 10.1007/978-1-4612-0287-5. [2] K. Q. Chang, W. Y. Ding and R. G. Ye, Finite time blow-up of the heat flow of harmonic maps from spheres, J. Differential Geom., 36 (1992), 507-515. [3] T. R. Ding and C. Z. Li, "A Course on Ordinary Diferential Equations," Higher Educational Press, Beijing, 1991. [4] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Seconder Order," Springer-Verlag, Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0. [5] M. Guan, S. Gustafson and T. P. Tsai, Global existence and blow-up for harmonic map heat flow, J. Differential Equations, 246 (2009), 1-20. doi: 10.1016/j.jde.2008.09.011. [6] C. F. Gui, H. Y. Jian and H. J. Ju, Properties of translating solutions to mean curvature flow, Discrete Contin. Dyn. Syst., 28 (2010), 441-453. doi: 10.3934/dcds.2010.28.441. [7] R. M. Herve and M. Herve, Qualitative study of the real solutions of a differential equation associated with the Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 427-440. [8] H. Y. Jian, H. J. Ju and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975. [9] H. Y. Jian and Y. N. Liu, Ginzburg-Landau vortex and mean curvature flow with external force field, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1831-1842. doi: 10.1007/s10114-005-0698-y. [10] H. Y. Jian and B. Song, Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Differential Equations, 170 (2001), 123-141. doi: 10.1006/jdeq.2000.3822. [11] H. Y. Jian and X. J. Wang, Bernsterin theorem and regularity for a class of Monge Amp\ere equations, preprint, 2010. [12] H. Y. Jian and X. J. Wang, Global regularity for fully nonlinear singular elliptic equations, preprint, 2011. [13] H. Y. Jian and X. W. Xu, The vortex dynamics of a Ginzburg-Landau system under pinning effect, Science in China Ser. A, 46 (2003), 488-498. doi: 10.1007/BF02884020. [14] H. J. Ju, J. Lu and H. Y. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963. [15] P. Mironescu , On the stability of radial solutions of the Ginzburg-Landau equations, J. Funct. Anal., 130 (1995), 334-344. doi: 10.1006/jfan.1995.1073. [16] P. Raphael and R. Schweyer, Stable blow-up dynamics for 1-corotational enenrgy critical harmonic heat flow, preprint, arXiv:1106.0914 show all references ##### References: [1] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices," Birkhauser, 1993. doi: 10.1007/978-1-4612-0287-5. [2] K. Q. Chang, W. Y. Ding and R. G. Ye, Finite time blow-up of the heat flow of harmonic maps from spheres, J. Differential Geom., 36 (1992), 507-515. [3] T. R. Ding and C. Z. Li, "A Course on Ordinary Diferential Equations," Higher Educational Press, Beijing, 1991. [4] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Seconder Order," Springer-Verlag, Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0. [5] M. Guan, S. Gustafson and T. P. Tsai, Global existence and blow-up for harmonic map heat flow, J. Differential Equations, 246 (2009), 1-20. doi: 10.1016/j.jde.2008.09.011. [6] C. F. Gui, H. Y. Jian and H. J. Ju, Properties of translating solutions to mean curvature flow, Discrete Contin. Dyn. Syst., 28 (2010), 441-453. doi: 10.3934/dcds.2010.28.441. [7] R. M. Herve and M. Herve, Qualitative study of the real solutions of a differential equation associated with the Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 427-440. [8] H. Y. Jian, H. J. Ju and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975. [9] H. Y. Jian and Y. N. Liu, Ginzburg-Landau vortex and mean curvature flow with external force field, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1831-1842. doi: 10.1007/s10114-005-0698-y. [10] H. Y. Jian and B. Song, Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Differential Equations, 170 (2001), 123-141. doi: 10.1006/jdeq.2000.3822. [11] H. Y. Jian and X. J. Wang, Bernsterin theorem and regularity for a class of Monge Amp\ere equations, preprint, 2010. [12] H. Y. Jian and X. J. Wang, Global regularity for fully nonlinear singular elliptic equations, preprint, 2011. [13] H. Y. Jian and X. W. Xu, The vortex dynamics of a Ginzburg-Landau system under pinning effect, Science in China Ser. A, 46 (2003), 488-498. doi: 10.1007/BF02884020. [14] H. J. Ju, J. Lu and H. Y. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963. [15] P. Mironescu , On the stability of radial solutions of the Ginzburg-Landau equations, J. Funct. Anal., 130 (1995), 334-344. doi: 10.1006/jfan.1995.1073. [16] P. Raphael and R. Schweyer, Stable blow-up dynamics for 1-corotational enenrgy critical harmonic heat flow, preprint, arXiv:1106.0914 [1] Yuanzhen Shao. Continuous maximal regularity on singular manifolds and its applications. Evolution Equations and Control Theory, 2016, 5 (2) : 303-335. doi: 10.3934/eect.2016006 [2] Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142 [3] Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. 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https://www.physicsforums.com/threads/conservative-talk-show-host-waterboarded.316085/	Conservative talk show host waterboarded • #1 24 0 Main Question or Discussion Point In an effort to show waterboarding is not torture, the conservative talk show host "Mancow" agreed to put his money were his mouth is, and actually be waterboarded. He was not sleep deprived, confined, or in any way "prepared" for it, the way the US would, and he had the knowledge it would end whenever he wanted it to. The punch line? He lasted six seconds. Afterwords, he agreed, "Waterboarding is absolutely torture." Any opinions on this? (I'd like to see Hannity up next.) Related General Discussion News on Phys.org • #2 mgb_phys Homework Helper 7,774 12 And he knew it was an act and they wouldn't kill him. Suppose he was seized in the studio by foreign soldiers with machine guns, hooded, flown around the world and woke up in a Syrian prison - how long would he last then. • #3 24 0 And he knew it was an act and they wouldn't kill him. Suppose he was seized in the studio by foreign soldiers with machine guns, hooded, flown around the world and woke up in a Syrian prison - how long would he last then. That's what I meant when I said he had the knowledge it would end whenever he wanted it to. But a very valid point all the same. • #4 CRGreathouse Homework Helper 2,820 0 I was hoping that something like this would happen. Not having any knowledge of interrogation techniques or SERE, I haven't been able to hold an informed opinion about what is and isn't torture. But I always wondered why those who said it wasn't torture didn't try it. Sure, you won't be able to replicate it perfectly -- but you should be able to get some reasonable idea about it. I applaud this talk show host (about whom I know nothing beyond the contents of this thread) for 1. testing his ideas 2. having the courage to change his opinion. • #5 2,425 6 I haven't been able to hold an informed opinion about what is and isn't torture. Where is the line is irrelevant. What is relevant is to be as far as possible from the line, not as close as possible according to the official texts. • #6 299 1 He wasn't the first to do this. I think they're starting to do it for publicity now. • #7 CRGreathouse Homework Helper 2,820 0 Where is the line is irrelevant. What is relevant is to be as far as possible from the line, not as close as possible according to the official texts. I don't agree with that. For example, a country could be further from the line by releasing all criminals from prison. But that's probably a bad idea. Or consider the possibility that a rogue soldier/police officer/vigilante/etc. might torture a person (say, with probability p). Assigning three people to each prisoner instead of one might reduce the probability of turture to something like p^2. But if we want to reduce this as far as possible, we'll go much further, etc, etc. A country 'should' be far from the line, but not "as far as possible". More practically, I think the line should be found for judicial reasons. Actually, several lines: * Beyond some point, a person enacting a policy should be held legally liable for that decision. (This may or may not be Constitutionally possible in the US retroactively, but certainly it is possible for the future.) * Beyond another (further) point, a person carrying out such actions specifically allowed should be held liable for their actions. (Even if genocide were legal, it would not be permissible to carry it out.) * Beyond another (probably yet further) point, a person carrying out such actions even under specific orders to do so would be held liable for them. • #8 CRGreathouse Homework Helper 2,820 0 He wasn't the first to do this. I think they're starting to do it for publicity now. I really think that the senior Defense Department officials should have gone through it in the first place -- not just the brass, but the civvies too. There may be permissible techniques (contra humanino's sensible opinion, above), but if our leadership can't bring themselves to go through it it's probably too much. Honestly, I don't envy the job of the interrogators* and their commanders. In their position I would have spent many sleepless nights trying to balance the deontological against the practical, and the value of innocent lives against the harm to a (possibly also innocent) life. The difficulty of this situation is one of the best reasons that the decision should be made ahead of time by society. Is it permissible to imprison a suspected terrorist? Put him in solitary confinement? Question him for twenty hours straight? Etc. * Note: "interrogators" not "torturers". The need to get information from enemy soldiers, renegade militants, terrorists, etc. won't go away, but that doesn't mean society needs to condone torture in any way. In the nicest form imaginable an interrogation would be a debriefing. • #9 Gokul43201 Staff Emeritus Gold Member 7,051 17 qUkj9pjx3H0[/youtube] Christopher Hitchens did it last year. [url]4LPubUCJv58[/youtube] I find the Mancow story interesting from a purely political point of view. I don't think it sheds any additional light on the debate of whether waterboarding is torture. • #10 100 1 He wasn't the first to do this. I think they're starting to do it for publicity now. maybe people will actually pay to get waterboarded. this could be the biggest thing since bungee jumping, heck, maybe even mixed martial arts. reality game show, anyone? are YOU tougher than a 5th grade terrorist? • #11 LowlyPion Homework Helper 3,090 4 maybe people will actually pay to get waterboarded. this could be the biggest thing since bungee jumping, heck, maybe even mixed martial arts. reality game show, anyone? are YOU tougher than a 5th grade terrorist? Cheney could start a side business then franchising "EIT"eries. Their motto: "Anywhere else it would be torture." • #12 10 3 maybe people will actually pay to get waterboarded. this could be the biggest thing since bungee jumping, heck, maybe even mixed martial arts. reality game show, anyone? are YOU tougher than a 5th grade terrorist? And the backyard imitators will send a few idiots six feet under and the lawsuits will follow. Especially drunken fools. I'm also thinking adoption as a hazing ritual for clubs, frats, the boy scouts. • #13 LowlyPion Homework Helper 3,090 4 In an effort to show waterboarding is not torture, the conservative talk show host "Mancow" agreed to put his money were his mouth is, and actually be waterboarded. He was not sleep deprived, confined, or in any way "prepared" for it, the way the US would, and he had the knowledge it would end whenever he wanted it to. The punch line? He lasted six seconds. Afterwords, he agreed, "Waterboarding is absolutely torture." Any opinions on this? (I'd like to see Hannity up next.) Mancow ponying up got Sean Hannity off the hook with Kieth Olbermann who was preparing to offer up $1000 a second that Hannity lasted after Hannity was claiming it wasn't torture. • #14 turbo Gold Member 3,077 45 I'd love to see that pus-bag Limbaugh waterboarded right alongside the too-snide jerk Cheney and see which one of those creeps broke first. Torture is torture, and the scenery-chewing ravings of these loons does not mitigate that. It would be nice to see them maintain their composure while being tortured though. • #15 2,425 6 For example, a country could be further from the line by releasing all criminals from prison. People subject to torture are usually not criminals since they have not even been proven guilty. Your argument indicates that you missed more than my point. • #16 4,239 1 maybe people will actually pay to get waterboarded. this could be the biggest thing since bungee jumping, heck, maybe even mixed martial arts. reality game show, anyone? are YOU tougher than a 5th grade terrorist? Coming to you at your next county fair. • #17 What will the response be from Hannity, Limbaugh, etc ? Let's make some predictions. I predict that Hannity will say something like "You're asking me to be waterboarded? Why would I put myself through that, that's a tough/rough/unpleasant treatment we use on America's enemies / terrorists / enemy combatants...sure it's a little tough / rough /unpleasant but it's not torture, and it has saved American lives many times over!" I was talking to someone on the phone a few weeks ago when they broke the news to me that Arlen Spector was switching parties, and I told the person that if they switched the channel to fox news that they would be doing a slander piece on spector right then, and sure enough... • #18 1,838 7 You had people in Guantananamo who, accoding to the usual rules, were not required to cooporate with interrogations. It was decided that some of the people who did not cooporate would be waterboarded in order to make them so uncomfortable that they would choose to avoid it, which meant they had to decide to cooporate with interrogations. This alone almost surely implies that waterboarding is torture. If it were not torture, it would not have worked and some other technique would have been applied. Last edited: • #19 LowlyPion Homework Helper 3,090 4 I'd love to see that pus-bag Limbaugh waterboarded right alongside the too-snide jerk Cheney and see which one of those creeps broke first. Torture is torture, and the scenery-chewing ravings of these loons does not mitigate that. It would be nice to see them maintain their composure while being tortured though. I think you need to see Cheney in a broader perspective. Apparently these days he's shopping around a book deal to make some coin. (I think he's asking more than$2M.) In that context then, the more controversial he makes himself, then maybe no matter how reviled he was in office and even now for his absurd statements about how great torture is, he's just another entertainer like Limbaugh, and even this Mancow. They're playing from the same play book, praying at the same church - Greed. • #20 turbo Gold Member 3,077 45 People subject to torture are usually not criminals since they have not even been proven guilty. Your argument indicates that you missed more than my point. Most Americans (US, at least) have been glossing over that point for many years. It's not just the extraordinary renditions in foreign prisons, either. The "School of the Americas" has been training torturers for decades. Some time, the US voters have to be educated in the techniques that our "allies" have been trained in, and vote to STOP it. • #21 Evo Mentor 23,113 2,474 I applaud this talk show host (about whom I know nothing beyond the contents of this thread) for 1. testing his ideas 2. having the courage to change his opinion. This talk show host is a well known moron, idiot, and publicity hound. He will do anything for ratings and is disgusting. Just thought you might want to know. • #22 LowlyPion Homework Helper 3,090 4 This talk show host is a well known moron, idiot, and publicity hound. He will do anything for ratings and is disgusting. Just thought you might want to know. Which one? I've never heard Mancow in his natural habitat. The others mentioned here like Hannity and Limbaugh of course might also fit that description in some quarters. By the by, hope all is well. I haven't seen you about for awhile. • #23 Hans de Vries Gold Member 1,089 23 Pain and distress is just psychology.. It's all in the mind isn't it. I doesn't really exist, or does it? Torture without visible permanent damage can't be really torture, or can it? The reflex response in the brain due to an imminent dead by drawning is apparently that extreme and overwhelming. Instantaneous action is required otherwise you're dead, but you can't do anything because your From a seal who underwent waterboarding during his training to prepare him for the combat zone in Vietnam. Jesse Ventura: You Give Me a Water Board, Dick Cheney and One Hour... Regards, Hans. Last edited by a moderator: • #24 lisab Staff Emeritus Gold Member 1,887 616 Where is the line is irrelevant. What is relevant is to be as far as possible from the line, not as close as possible according to the official texts. This is a great guideline, humanino. If we're so close to the line that we have to question if we have crossed it...then we have crossed it. • #25 4,239 1 This is a great guideline, humanino. If we're so close to the line that we have to question if we have crossed it...then we have crossed it. So, I give up. These are lowly political hacks with no credibility, or we're seeing compelling videos displaying the actions of credible guys. I'm from Missouri. • Last Post Replies 10 Views 734 • Last Post Replies 9 Views 787 • Last Post Replies 34 Views 6K • Last Post Replies 1 Views 3K • Last Post Replies 158 Views 15K • Last Post Replies 1 Views 1K • Last Post Replies 4 Views 1K • Last Post Replies 28 Views 5K	{"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": 0.2283291071653366, "perplexity": 3938.2287533863027}, "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-2020-10/segments/1581875149238.97/warc/CC-MAIN-20200229114448-20200229144448-00480.warc.gz"}
https://cs.stackexchange.com/questions/71992/using-induction-to-prove-that-n3log4n-on4/72002	# Using Induction to prove that $n^3log^4n=O(n^4)$ I need to prove the following asymptotic relation for the purpose of cacluating a recurrence relation: $$n^3log^4n=O(n^4)$$ I tried and failed to do it with induction, which, if possible using basic Calculus 1 level math, I would like you to help me with. I am not required to do it with induciton or anything, I just wondered if I can do it. I was able to prove it though, in a differet manner: $$f \in o(g) \Rightarrow f \in O(g)$$ $$lim_{n\to \infty} \frac{n^3log^4n}{n^4}=0$$ Using L'Hôpital's rule 4 times, which proves that: $$n^3log^4n=o(n^4)$$ By the definitino of $o$. Therefore it also follows that $$n^3log^4n=O(n^4)$$ For the basis of the induction let $n=1$ and with $c=1$: $$f(1) = 1^3log^4(1) = 0 \leq 1=1^4 \cdot 1=c \cdot g(1)$$ From here I think that it would be enough to show that: $$log^4(n) \leq n$$ Although I am not sure how to continue from here. • I don't really see why you want to prove it by induction. I guess it's just curiosity but, still, what would you learn from an inductive proof? Probably it's possible but induction isn't necessarily a sensible way to prove every property of the natural numbers. (By the way, it's easier to use L'Hôpital once to prove that $\log n = o(n^c)$ for all $c>0$ and then you know that $\log^4 n = o((n^{1/4})^4) = o(n)$. Indeed, unless you're specifically asked to prove it, you can just use $\log n=o(n^c))$ for all $c$ as a canned fact.) – David Richerby Mar 24 '17 at 13:04 • Indeed only curiosity. Thanks for the generalization of the proof, I like it a lot! – Lumon Mar 24 '17 at 13:34 You showed that f (n) = o (g (n)). That's it. There is nothing else to prove. If you look at the definition of o (f (n)) and O (f (n)), they are almost identical except one says "for every eps > 0" and the other says "there is one c > 0". You can take every single eps of the little-o definition and use it as the c in the big-O definition. You already have your induction basis: $$f(1) = 1^3log^4(1) = 0 \leq 1=1^4 \cdot 1=c \cdot g(1)$$ Now you need to apply the induction step. Suppose $$n^3log^4n\leq c \cdot n^4$$ Then prove: $$(n+1)^3log^4(n+1)\leq c \cdot (n+1)^4$$ and your proof by induction will be completed. • Good luck with the induction step. If c = 1 and log is taken to be base 2, the statement is false for 3 ≤ n ≤ 65,535. – gnasher729 Mar 24 '17 at 22:20 • Yes, I can form the induction step, proving it though is the problem, maybe what @gnasher729 said was the problem, a bad choice for c – Lumon Mar 25 '17 at 10:42 • @Lumon: The problem is not that you cannot prove it, the problem is that for small n the induction step itself is actually false, independent of c. For small n, the left hand side grows faster than the right hand side. – gnasher729 Mar 25 '17 at 23:52	{"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": 0.868507444858551, "perplexity": 227.1423067782645}, "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/1627046150307.84/warc/CC-MAIN-20210724160723-20210724190723-00073.warc.gz"}
https://christoph.ruegg.name/blog/content-based-storage-in-the-cloud.html	One derivative of the NoSQL movement that rediscovers non-relational storage approaches lately is a content-based value store. Such a store is similar to a Key-Value store but uses a cryptographic hash of the value as key. ### An SHA-1 hash of the value is good enough to identify it The SHA-1 hash function is unique, meaning that for every value there's exactly one key that can be computed using SHA-1, hence value implies key. We can always compute the unique key of a value. The probability of an SHA-1 hash collision is extremely low. The most cited numbers show that you'd need 1024 values in order to cause a 50% chance of a collision. Even with a whopping 1018 distinct values the likelihood of at least one collision is already down at 10-9. Hence, a key refers to a single value with extremely high probability. While the SHA1 function is not strictly injective, it is approximatively injective enough for almost all practical applications. Note that this is different from common non-distributed Hash Tables where a very short hash function is used to directly jump fast to an inner data structure (bucket) containing all items sharing the same hash. The motivation for hashing in such hash tables is to be able to directly compute the position where an item is stored, avoiding long linear or binary searches. ### Verifiable Consistency A nice side effect of using SHA1 is that given the key, the value retrieved from the storage can be verified (detect data corruption or tampering) simply by recomputing its SHA-1 hash and comparing it to the key. You can even digitally sign a key and by that implicitly sign its value and all those referred by it. ### Keys are uniformly distributed When using the common hex string format, the 160bit long keys are always 40 characters long and look something like this: 1: d921970aadf03b3cf0e71becdaab3147ba71cdef We can safely treat them as if their characters were distributed uniformly (0-9, a-f). This brings some advantages especially when used in a distributed or cloud-like scenario, as simple prefix ranges (like 0-3, 4-7, 8-b, c-f) can be used for partitioning and distributed processing. On the other hand this means that you can't use other indexed keys or ordering out of the box without further logic or storage on top of it. ### The value of a key is fixed and can't be changed The value associated with a key can never change. If you store an updated value, you'll get a new key for it and update the reference to this new key. This has severe consequences on where this storage scheme can be used efficiently. For example, a typical relational data model with cyclic relations wouldn't fit at all to such a content-based data store. However, in practice in a cloud-like application this is often not that much of an issue. Even more so as soon as you realize that the existence of read-only stale yet still consistent data is not an issue either (see CQRS). Again this fits very well with distributed and cloud computing, as it becomes trivial to aggressively cache values locally. If a value is found in the local cache it is guaranteed to be up to date (since values can't change), so you don't even have to check for timestamps or whether it has been changed remotely. Since in Azure the instances come with a lot of local storage, a simple MRU cache for a few GB can save you a lot of downloads and roundtrips if you use only a relatively small number of instances or have managed to create some weak affinity between jobs and Azure instances. #### Example: Large Queue Messages Azure Queues have content size limitations, that's why Lokad.Cloud implements logic to let messages transparently overflow to blob storage. To do that it needs a way to store a value in a blob that it can retrieve later by some identifier. This identifier is then packed to the actual message. There's no need to access it in any other way, so it's a perfect candidate for a content-based value store. In my experience, in real life the probability that a message is processed on the same worker that originally put it there is often high or at least not negligible. In all these cases, a cached content-based value store would save you from having to download these blobs completely, but still work correctly otherwise. ### Implicit Value-Deduplication Since the same value leads to the same key, trying to store the same value twice means you get the same storage location and the value gets stored only once. The second trial can even be aborted early by provoking a precondition violation, or skipped completely if it is already in the local cache (depending on the deletion plan). #### Example: Daily Backup Snapshots I recently wrote a small service that periodically takes full snapshots of all tables and blobs of a set of Azure storage accounts to a separate account, keeps the last N snapshots each and removes the rest. Often only a small subset of blobs or table entities actually change in a day. Had I used content-based storage, I could have saved a lot of storage (and thus cost) by deduplication without having to implement complicated incremental or differential backups. Taking a snapshot would likely also have taken less time thanks to some saved uploads. ### Trivial Distribution and Replication Other than any classical relational databases and key value stores, replication and distribution of data in such a content-based value store is trivial since there can't be any conflicts. This is why the caching mentioned above works so well. Replication simply means to copy the values of all missing keys over to the target. A consequence of this is that for some scenarios there's no technical need for a single master database. A peer can synchronize with any other peer, resulting in full peer to peer support. Distributed hash tables (DHT) as used by most file sharing solutions including BitTorrent work similarly and turn out to be very efficient. ### History Consistency and Versioning Since values can't change, they remain consistent with each other even when they become stale. That's why this approach is used by most of the popular distributed version control systems like Git and Mercurial as well. The Git object model is nicely described in the git community book (the following two images are taken from there). In essence, all objects are stored just as described here. In addition to data blobs (i.e. source code files) there are also tree objects representing a folder simply by listing all the SHA-1 keys of its child elements, again stored by its hash: If a file changes in git in a new revision, it will get a new hash. The folder/tree containing it will update that hash in its list, and in turn will itself get a new hash. Both the old a new version are therefore still available completely and consistently simply by referring to the hash of the respective version of the tree. Historical consistency can be useful for all kind of applications. Note that this approach persists snapshots of values and content, not how they are changed. This is thus a dual counterpart to concepts like event sourcing where only the actions causing changes of the values are persisted but not the actual values. ### Append-Only Storage or Value Scavenging Unless you need an append-only storage, you need to be careful about deleting values in such a system. Since there is implicit deduplication, you can't just delete what you've just inserted since the same value could also be used in other places. There are several approaches how you can attack this, depending on your scenario: • Garbage Collection: If there is a hierarchy where all values are referenced by another value, you can follow the tree from time to time and then remove all values you haven't seen. This is used by all the distributed version control systems. Be careful about race conditions though. • Reference Tracking: Use metadata to list all keys or items referring a value. If you remove the last reference, remove it. This can be combined with garbage collection. You can also use reference counters, but they are difficult to handle correctly in an unreliable world like a cloud environment where instantaneous VM shutdowns without prior notice are to be expected. • Time-Based: You "touch" a value (update a timestamp in the metadata) whenever it is used, and from time to time remove all items that haven't been used for a while. Note that that causes a lot of round trips (although they could be performed asynchronously in the background). • Limited Lifetime: Sometimes its good enough to just define that a value can safely be removed after a day or a month.	{"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": 0.3170231580734253, "perplexity": 1028.7084395678473}, "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/1590347390758.21/warc/CC-MAIN-20200526112939-20200526142939-00399.warc.gz"}
https://www.physicsforums.com/threads/isomorphism-question-with-d4.405213/	# Homework Help: Isomorphism question with D4 1. May 23, 2010 ### tyrannosaurus 1. The problem statement, all variables and given/known data Let G=<x, y\|x^4=y^4=e, xyxy^–1=e>. Show that \|G\|≤16. Assuming \|G\|=16, show G/<y^2> is isomorphic to D4. 2. Relevant equations 3. The attempt at a solution Here is what I have: since xyxy^-1=e, we know that yxy^-1=x^-1=x^3, so we know that x is a conjugate and partions G. So G= <x> union y<x> and \|G\|<= 16. Lets assume that \|G\|=16. So \|G\|/\|<y^2>\|=8, thus \|G\|/\|<y^2>\|>= \|D4\|. D4=<a,b\|a^4=b^2=(ab)^2=e, or ab=ba^3>. Let w=x<y^2> , z=y<y^2> and q=e<y^2> (where q is are identity element in G/<y^2>. Need to show that G/<y^2>= <w,z\|w^4=z^2=(wz)^2=e>. 1. Since x and y are generates in G, then w and z are generates in G/<y^2>. I am not sure if this is right, could someone explain this to me? (I need to show that w and z generate G/<y^2>). 2. w^4=e since (x<y^2>)^4=x^4<y^2>=e<y^2>=q. 3. z^2=e since (y<y^2>)^2=y^2<y^2>=e<y^2>=q. 4. (wz)^2=(wz)(wz)=e. Since wz=(z^-1)(w^-1), then wz=xy<y^2> and z^-1w^-1=y^3x^3<y^2>=yx^3<y^2>. Thus xy<y^2>=yx^3<y^2>. Therefore, G/<y^2> is isomorphic to D4.	{"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": 0.9952040314674377, "perplexity": 6364.618347560168}, "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/1544376827175.38/warc/CC-MAIN-20181216003916-20181216025916-00371.warc.gz"}
https://brilliant.org/problems/number-of-0s-in-the-number-of-0s/	# Number of 0's in the Number of 0's Let $N$ be the number of 0's needed to write out $10^{100000}$. How many 0's are needed to write out the number $N$? ×	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 3, "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": 0.33657005429267883, "perplexity": 383.64311522568363}, "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-2020-16/segments/1585371612531.68/warc/CC-MAIN-20200406004220-20200406034720-00245.warc.gz"}
http://shapedbytruth.org/2017/01/13/for-one-is-your-master/	# for one is your Master The King James Bible is consistent: Matthew 23:10 KJB Neither be ye called masters: for one is your Master, even Christ. 1 Cor 12:28 KJB And God hath set some in the church, first apostles, secondarily prophets, thirdly teachers, after that miracles, then gifts of healings, helps, governments, diversities of tongues. Eph 4:11 KJB And he gave some, apostles; and some, prophets; and some, evangelists; and some, pastors and teachers; Now look at these verses in the New King James Version. Matthew 23:10 NKJV And do not be called teachers; for One is your Teacher, the Christ. 1 Cor 12:28 NKJV And God has appointed these in the church: first apostles, second prophets, third teachers, after that miracles, then gifts of healings, helps, administrations, varieties of tongues. Eph 4:11 NKJV And He Himself gave some to be apostles, some prophets, some evangelists, and some pastors and teachers, The NKJB has Jesus saying not to be called teachers, yet has God appointing teachers. The NKJV fabricates a contradiction where there should be not be one. This manufactured contradiction is in many of the new versions. Here is Matthew 23:10 in several other translations: AMP Do not let yourselves be called leaders or teachers; for One is your Leader (Teacher), the Christ. CEB Don’t be called teacher, because Christ is your one teacher. DARBY Neither be called instructors, for one is your instructor, the Christ. ESV Neither be called instructors, for you have one instructor, the Christ. ISV Nor are you to be called ‘Teachers,’ because you have only one teacher, the Messiah! LEB And do not be called teachers, because one is your teacher, the Christ. MSG “Don’t let people do that to you, put you on a pedestal like that. You all have a single Teacher, and you are all classmates. MEV Nor be called teachers, for you have one Teacher, the Christ. MOUNCE Nor should you be called ‘instructors,’ because Christ is your only instructor. · NET Nor are you to be called ‘teacher,’ for you have one teacher, the Christ. NIRV You shouldn’t be called ‘teacher.’ You have one Teacher, and he is the Messiah. NIV Nor are you to be called instructors, for you have one Instructor, the Messiah. NIVUK Nor are you to be called instructors, for you have one Instructor, the Messiah. NKJV And do not be called teachers; for One is your Teacher, the Christ. NLT And don’t let anyone call you ‘Teacher,’ for you have only one teacher, the Messiah. NRSV Nor are you to be called instructors, for you have one instructor, the Messiah. NRSVA Nor are you to be called instructors, for you have one instructor, the Messiah. NRSVACE Nor are you to be called instructors, for you have one instructor, the Messiah. NRSVCE Nor are you to be called instructors, for you have one instructor, the Messiah. NTE Nor should you be called “teacher”, because you have one teacher, the Messiah. TLV Nor are you to be called teachers; for One is your Teacher, the Messiah. The word of God is infallible, and without error. At least in the King James Bible. The modern versions introduce errors and contradictions that would not otherwise exist!	{"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": 0.8885498046875, "perplexity": 9244.68591989609}, "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-2019-09/segments/1550247488490.40/warc/CC-MAIN-20190218220415-20190219002415-00282.warc.gz"}
http://www.astroexplorer.org/details/apjlaa7cfbf3	Image Details Choose export citation format: On the Evolution of the Central Density of Quiescent Galaxies • Authors: Sandro Tacchella, C. Marcella Carollo, S. M. Faber, Anna Cibinel, Avishai Dekel, David C. Koo, Alvio Renzini, and Joanna Woo 2017 The Astrophysical Journal Letters 844 L1. • Provider: AAS Journals Caption: Figure 3. Breaking the age–metallicity degeneracy in the rest-frame (I − J) vs. (B − I) color–color plane. The colored lines represent quiescent Bruzual & Charlot (2003) τ-models with ﹩\tau =0.5,1,3,6,10,15\times {10}^{8}\,\mathrm{years}﹩ and ﹩\mathrm{sSFR}\lt {10}^{-11}\,{\mathrm{yr}}^{-1};﹩ the colors of the model lines refer to stellar mass-weighted age (as indicated in the color bar). The age–metallicity grid marked by the dashed lines is obtained from these τ-models. The geometry of the grid enables the identification of three, well-defined bins of mass-weighted stellar age for the QGs: ﹩\lesssim 2\,\mathrm{Gyr}﹩ (blue), ﹩\approx 2-4\,\mathrm{Gyr}﹩ (yellow) and ﹩\gtrsim 4\,\mathrm{Gyr}﹩ (red) marked by the three colored zones. The arrow in the top left corner shows a dust attenuation of ﹩E(B-V)=0.1;﹩ dust effects are roughly parallel to the lines of constant metallicity and do not significantly affect our age classification. The contours indicate the distribution of colors for our sample galaxies (enclosing 68% and 95% of the sample); the error bar in the bottom right corner shows the median uncertainty in the color estimates.	{"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": 0.8640384078025818, "perplexity": 8408.228776709406}, "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-2018-13/segments/1521257650262.65/warc/CC-MAIN-20180324112821-20180324132821-00342.warc.gz"}
https://mynote.dinhanhthi.com/machine-learning-coursera-4	### ML Coursera 4 - Week 4: Neural Networks - Representation Posted on 16/09/2018, in Machine Learning. This note was first taken when I learnt the machine learning course on Coursera. Lectures in this week: Lecture 8. settings_backup_restore ## Neural Networks • Algorithms try to mimic the brain. • 80s, 90s (very old) • activation function = sigmoid (logistic) $g(z)$ • (sometimes) $\theta$ called weights parameters • Neural network: first layer (input layer), intermidiate layers (hidden layer) and the last layer (output layer) • Notations: • $a_i^{(j)}$ = “activation” of unit $i$ in layer $j$ $$a_{\text{unit}}^{(\text{layer})}$$ • $\Theta^{(j)}$ = matrix of weights controlling function mapping from layer $j$ to layer $j+1$ • $\Theta \in \mathbb{R}^{m\times n+1}$ where $m$ = number of unit in current layer, $n$ = number of units in previous layer (not include unit 0). \begin{align} \Theta^{(\text{previous})} &: \text{previous layer} \to \text{current layer} \\ \Theta^{(\text{previous})} &\in \mathbb{R}^{\text{current} \times (\text{previous}+1)} \end{align} $$a_{k\in \text{current units}}^{(\text{current layer})} = g\left( \sum_{i\in \text{prev units}} \Theta_{ki}^{(\text{prev layer})} a_i^{(\text{prev layer})} \right)$$ ### Forward propagation: vectorized implementation \begin{align} z^{(2)} &= \Theta^{(1)} a^{(1)}\\ a^{(2)} &= g(z^{(2)}) \end{align} • Neural network is look like logistic regression, except that instead of using $x_1, x_2, \ldots$, it’s using $a^{(j)}_1, a^{(j)}_2, \ldots$ • Neural network learning its own features. • Architectures = how different neurons connected to each other. • Final \begin{align} h_{\theta}(x) = a^{(j+1)} = g(z^{(j+1)}) = g(\Theta^{(j)}a^{(j)}). \end{align} ## Exercise Programmation: Multi-class Classification and Neural Networks settings_backup_restore See again Multiclass classification: one-vs-all. • lrCostFunction.m This ex is the same with the one in the previous week. h = sigmoid(Xtheta); % hypothesis J = 1/m ( -y' * log(h) - (1-y)' * log(1-h) ) + lambda/(2m) sum(theta(2:end).^2); grad(1,1) = 1/m * X(:,1)' * (h-y); grad(2:end,1) = 1/m * X(:,2:end)' * (h-y) + lambda/m * theta(2:end,1); • oneVsAll.m initial_theta = zeros(n + 1, 1); options = optimset('GradObj', 'on', 'MaxIter', 50); for c = 1:num_labels [theta] = fmincg (@(t)(lrCostFunction(t, X, (y == c), lambda)), initial_theta, options); all_theta(c,:) = theta(:); end info fmincg works similarly to fminunc, but is more more efficient for dealing with a large number of parameters. • predictOneVsAll.m h = X * all_theta'; [~, p] = max(h,[],2); • predict.m : “you implemented multi-class logistic regression to recognize handwritten digits. However, logistic regression cannot form more complex hypotheses as it is only a linear classifier.” X = [ones(m, 1) X]; % add column 1 to a2 z2 = X * Theta1'; a2 = sigmoid(z2); a2 = [ones(m, 1) a2]; % add column 1 to a2 h = a2 * Theta2'; [~, p] = max(h,[],2); keyboard_arrow_right Next to Week 5. Top	{"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": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9608190059661865, "perplexity": 16067.432390231448}, "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/1544376826715.45/warc/CC-MAIN-20181215035757-20181215061757-00567.warc.gz"}
https://www.physicsforums.com/threads/high-symmetry-points.256170/	# High symmetry points 1. Sep 14, 2008 ### saray1360 hi, I have found a website in which it has described how to find high symmetry points for wurtzite structure. Here is its address: http://cmt.dur.ac.uk/sjc/thesis_mcg/node53.html But, as I know these points are being used for standard numbers of atoms for example in a wurtzite system we should have 4 atoms in the unit cell and if we change the number of atoms these points change! Please let me know how I can calculate the high symmetry points for a unit cell with 12 atoms in a hexagonal lattice such as wurtzite structure of ZnO nanowires? Thanks, Sara 2. Sep 14, 2008 ### JustinLevy The brillouin zone is determined by the lattice symmetry, not the placement (or number) of atoms in the unit cell. What they are describing there (which "paths in k-space" along which to calculate the bandstructure), there have appeared some standards for each lattice symmetry, but ultimately it is a just a convention: We'd like to have and represent the bandstructure in the entire brillouin zone, but it is far easier to represent slices of this. Which slices are important to you, or is enough to feel like you "covered enough" is up to you or just a convention. A similar problem you probably have seen is how one represents the electron orbitals for materials. It is just difficult to represent higher dimensional data. Usually we cut a plane (or line) through and show the densities on this surface, or show a 3-D model with the surface of a constant chosen density for the orbital. 3. Sep 16, 2008 ### saray1360 Dear all, It is trivial, but please let me know how I can clculate high symmetry points for hexagonal lattice. I know the points are gamma, A, H , L, M and K but I do not know how to calculate them. For example I have a k-point grid in reciprocal lattice as follows: 7.0 0.0 0.0 0.0 7.0 0.0 0.0 0.0 5.0 Please do help me, it is really urgent for me. Regards, sara 4. Nov 12, 2010 ### silveto Similar Discussions: High symmetry points	{"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": 0.8835915923118591, "perplexity": 673.6307488498119}, "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-2017-22/segments/1495463612553.95/warc/CC-MAIN-20170529203855-20170529223855-00451.warc.gz"}
http://www.ask.com/question/how-to-calculate-cubic-feet	# How do you calculate cubic feet? In dealing with cubic feet you must be dealing in volume. Therefore, you must have a length, width, and a height of something. If you have a length of 12, a width of 7, and a height of 4; all we have to do is multiply 12 x 7 x 4 = 336/12 = 28 Q&A Related to "How do you calculate cubic feet?" Cubic feet is calculated by measuring feet in three different directions like length, width and depth. Then you would multiply all three of these measurements together. http://www.ask.com/web-answers/Reference/Other/how... 1. Measure the length of your attic in feet. 2. Measure the width of your attic in feet. 3. Measure the height of your attic in feet. Make sure to measure in the center, at the attic's http://www.ehow.com/how_5782380_calculate-cubic-fe... 2ftx2ftx2t= 8ft. 3. http://wiki.answers.com/Q/How_can_calculate_cubic_... Calculating cubic feet, or volume is necessary for estimating concrete or attic http://www.chacha.com/question/how-to-calculate-cu...	{"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": 0.9539732336997986, "perplexity": 953.1200413613568}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "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-2014-41/segments/1410657132883.65/warc/CC-MAIN-20140914011212-00120-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
http://math.stackexchange.com/users/31769/justin	# Justin less info reputation 421 bio website location Houston, TX age 33 member for 2 years, 7 months seen Oct 4 at 0:35 profile views 224 BBA Finance/Economics, MSc Statistics, and currently pursuing PhD in Finance # 49 Questions 7 Simple question regarding continuous functions on $\mathbb{Q}$ 5 Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$ 5 If X is a random variable then so is \|X\| 4 Exponentials of stochastic processes and Brownian motions 4 Combinations of i.i.d Inverse Chi-Square RVs and their characteristic functions # 538 Reputation +5 Discontinuity points of a Distribution function +5 Existence/Finiteness in $\mathbb{R}^2_+$ using Fubini or Tonelli +5 Existence, finiteness of Lebesgue integral on $(0,\infty)$ of $f(x)=\frac{1-e^{-x}}{x}-\frac{1}{1+x}$ +5 Exponentials of stochastic processes and Brownian motions 2 Basics of probability - Independent Events. 2 Find nth term given the equation of a series 1 If $F_X(z)>F_Y(z)$ for all $z$, then $P[X0$. 1 Reduce a range of values 1 What is the distribution of a random variable that is the product of the two normal random variables ? # 35 Tags 5 probability × 12 1 mathematical-modeling 2 sequences-and-series × 2 0 measure-theory × 41 1 probability-theory × 33 0 convergence × 9 1 statistics × 6 0 real-analysis × 6 1 normal-distribution × 2 0 integration × 6 # 5 Accounts Mathematics 538 rep 421 Quantitative Finance 141 rep 2 Stack Overflow 138 rep 16 Cross Validated 118 rep 4 MathOverflow 101 rep 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": 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": 0.8224347233772278, "perplexity": 1604.317357032157}, "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-2014-52/segments/1418802770060.91/warc/CC-MAIN-20141217075250-00129-ip-10-231-17-201.ec2.internal.warc.gz"}
http://sci.renewable.media/Momentum	# Momentum Newton's Law of Motion No.1: An object at rest will remain at rest, and an object in motion will continue to move at constant velocity, unless a net force is applied. Consequences: • Conservation of Momentum in both elastic and inelastic collisions • Conservation of kinetic energy in elastic collisions but not inelastic. • Conservation of angular momentum. Examples to help visualise the concept: • In space, free of gravitational influence, an object will move at a constant velocity. Any change in velocity of a mass would have to conserve momentum. An astronaut who throws away his toolbox, would find himself accelerated in the opposite direction to the box. The velocity of the astronaut would be the velocity of the toolbox x (mass of toolbox)/(mass of astronaut). • $$v_a = -v_t⋅{m_t}/{m_a}$$ Newton's First Law of Motion introduces the concept of 'inertia'. Inertia is the tendency of objects to stay as they are, whether moving or standing still. To change the motion of an object, its inertia has to be overcome by a force, which changes the energy state of the object. Newton's First Law of Motion in action: there is insufficient friction between the coin and the card to transfer the impulse to the coin. The inertia of the coin causes it to remain in place until gravity suggests otherwise. The inertia of a moving object can be measured by its momentum, p = mv, where p is the momentum, m is the mass and v is the velocity of the object. Newton's first law explains why an impact is greater the heavier an object, and why a skater will spin faster when she pulls in her arms and legs. Billiard balls colliding provide a good example of the transfer of momentum according to the First Law. $$p = mv$$ where p is the momentum, m is the mass of an object moving at constant velocity v ### Impulse Both force and momentum are vectors. The force is the first derivative (change over time t) of linear momentum: $$F↖{→} = {Δp↖{→}}/{Δt}$$ The change in linear momentum is equal to the impulse: $$I = Δp = F⋅Δt$$ If the mass is constant: $F↖{→}_{net} = {Δp↖{→}}/{Δt} = {mv↖{→}_f - mv↖{→}_i}/{Δt} = m{v↖{→}_f - v↖{→}_i}/{Δt} = m{Δv↖{→}}/{Δt} = ma↖{→}$ ## Elastic Collisions Newton's First Law of Motion in action: the final two balls have momentums which sum to the momentum of the first ball. An elastic collision is one in which the kinetic energy and momentum are conserved. $E_k_1_{total} \text"(before collision)"$ $= E_k_2_{total} \text"(after collision)"$ Therefore, $½mv_{A_1}^2 + ½mv_{B_1}^2 = ½mv_{A_2}^2 + ½mv_{B_2}^2$ where A and B are two objects before (1) and after (2) a collision. Since momentum p = mv, it follows that the kinetic energy is: $½mv^2 = ½pv$ Therefore, if kinetic energy is conserved, momentum is also conserved. In large objects, some energy is always lost due to friction and internal heat. However, some collisions can be made 'more elastic' by the means of springs or more elastic materials, so an understanding of the theoretical perfect elastic collision has a role. In the case of atoms and molecules, an ideal gas is assumed to have all collisions being elastic. In other cases, perfect elasticity is not likely to be the case. ### A coin trick A consequence of Newton's first law is that in elastic collisions both momentum and kinetic energy is conserved. In inelastic collisions, only momentum is conserved. Try this: find four coins of the same type (if you are like me, this will only be possible on payday). Now, place three of the coins in a straight row, so they are all touching. Slide the fourth coin quickly so that it impacts the end of the row in a straight line. What do you observe? You should see that the first coin stops dead. None of the coins move except the one on the other end, which moves away at about the speed of the first coin. What is happening? Why doesn't the first coin's momentum become distributed amongst all the coins? This is a demonstration of an elastic collision. It will happen with any number of equal mass objects. The first object transfers nearly all of its kinetic energy and momentum to the next object. This energy is transferred to the next object and from that one to the next, until the last object. Since the last object has no object to stop it, it takes the energy of the previous coin and converts it to kinetic energy. Since momentuim is p = mv, kinetic energy can be expressed as $E_k = ½mv^2 = ½pv$ Therefore, if kinetic energy is conserved (same as before), the momentum must be conserved as well! ## Inelastic Collisions A ball bouncing experiences several energy transformations. An inelastic collision is one in which the kinetic energy is not conserved, but momentum is. The total momentum before the collision equals the total momentum after the collision: $$Σp_1 = Σp_2$$ $$mv_{A_1} + mv_{B_1} = mv_{A_2} + mv_{B_2}$$ Content © Renewable.Media. All rights reserved. Created : September 2, 2013 Last updated :February 27, 2016 ## Site Index ### Latest Item on Science Library: The most recent article is: Air Resistance and Terminal Velocity View this item in the topic: Mechanics and many more articles in the subject: ### Resources Science resources on ScienceLibrary.info. Games, puzzles, enigmas, internet resources, science fiction and fact, the weird and the wonderful things about the natural world. Have fun while learning Science with ScienceLibrary.info. ### Great Scientists #### John Nash 1928 - 2015 John Nash was an American mathematician whose work in fields like Game Theory was revolutionary. He was awarded the Nobel Prize for Economics in 1994.	{"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": 0.7301983833312988, "perplexity": 507.13994580782673}, "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-2021-31/segments/1627046151972.40/warc/CC-MAIN-20210726000859-20210726030859-00176.warc.gz"}
https://dsp.stackexchange.com/questions/68291/gsp-as-an-extenstion-of-dsp	# GSP as an extenstion of DSP I am a PhD. in pure mathematics. 1. Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory. 2. I am reading the interesting paper, but could not get how the Fourier transform is extended to the graph Fourier transform as illustrated on page 23. Indeed, why should the matrix V be considered as the extension of (discrete) Fourier transform? • Thanks, It will be done just right now. Should it be deleted? – Ali Bagheri May 20 at 11:37 • This might be better on the maths site than on dsp – Neil_UK May 20 at 13:06 • yeah, i kinda don't grok this question at all. – robert bristow-johnson Jun 13 at 5:01 • @robertbristow-johnson This is about "signal processing on graphs", a new area of SP. As I understand it, the idea is to extend SP concepts like frequency transforms, spectra, densities, convolution, etc and apply them to graphs instead of signals. The hypothesis is that this will allow people to find interesting things about the graphs (just like, for example, the DFT allows us to find interesting things about a signal). – MBaz Jun 13 at 14:20 • According to page 23, $\mathbf{V}$ diagonalizes $\mathbf{S}$. Applications employ Fourier series/transforms to diagonalize operations (such as differentiation) that involve translation. It allows the user to work with a simpler diagonal operator and then move back to the original basis when the work is complete. The eigenvectors of $\mathbf{V}$ seem to be viewed as orthogonal modes of oscillations on the graph, much as eigenfunctions of the Lapacian on $\mathbb{R}^n$ are interpreted as modes of oscillation of a field permeating $\mathbb{R}^n$. It is in that sense that they are Fourier-like. – Joe Mack Jun 21 at 16:08	{"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": 0.6595562696456909, "perplexity": 784.0030056843485}, "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-2020-34/segments/1596439735963.64/warc/CC-MAIN-20200805153603-20200805183603-00318.warc.gz"}
https://zid.univie.ac.at/en/latex/	# LaTeX The University of Vienna provides a web service to represent LaTeX code via https://latex.univie.ac.at ## Note Employees and students can also use the online LaTeX editor Overleaf via the ZID. ## Displaying LaTeX code 1. Enter an URL with the LaTeX code attached in the form https://latex.univie.ac.at/?latex-code in the address bar of your browser.. 2. The service will create a .png graphics, which you can view, save or embed as you wish. ### Examples URL Result https://latex.univie.ac.at/?x=1 https://latex.univie.ac.at/?\sqrt{4+2x} https://latex.univie.ac.at/?\sum_{n=1}^\infty1/{n^2}={\pi^2}/6 https://latex.univie.ac.at/?\int_0^\infty{t-ib\over t^2+b^2}e^{iat}\,dt=e^{ab}E_1\qquad https://latex.univie.ac.at/?\mathrm{Hallo~Welt}~\heartsuit	{"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": 0.4806135594844818, "perplexity": 9252.10773878731}, "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-2022-40/segments/1664030334620.49/warc/CC-MAIN-20220925225000-20220926015000-00208.warc.gz"}
https://search.r-project.org/CRAN/refmans/condvis/html/ceplot.html	ceplot {condvis} R Documentation ## Interactive conditional expectation plot ### Description Creates an interactive conditional expectation plot, which consists of two main parts. One part is a single plot depicting a section through a fitted model surface, or conditional expectation. The other part shows small data summaries which give the current condition, which can be altered by clicking with the mouse. ### Usage ceplot(data, model, response = NULL, sectionvars = NULL, conditionvars = NULL, threshold = NULL, lambda = NULL, distance = c("euclidean", "maxnorm"), type = c("default", "separate", "shiny"), view3d = FALSE, Corder = "default", selectortype = "minimal", conf = FALSE, probs = FALSE, col = "black", pch = NULL, residuals = FALSE, xsplotpar = NULL, modelpar = NULL, xcplotpar = NULL) ### Arguments data A dataframe containing the data to plot model A model object, or list of model objects response Character name of response in data sectionvars Character name of variable(s) from data on which to take a section, can be of length 1 or 2. conditionvars Character names of conditioning variables from data. These are the predictors which we can set to single values in order to produce a section. Can be a list of vectors of length 1 or 2. Can be a character vector, which is then paired up using arrangeC. If NULL, an attempt will be made to extract all variable names which are not response or sectionvars from model, and these will be arranged using arrangeC. threshold This is a threshold distance. Points further than threshold away from the current section will not be visible. Passed to similarityweight. lambda A constant to multiply by number of factor mismatches in constructing a general dissimilarity measure. If left NULL, behaves as though lambda is set greater than threshold, and so only observations whose factor levels match the current section are visible. Passed to similarityweight. distance A character vector describing the type of distance measure to use, either "euclidean" (default) or "maxnorm". type This specifies the type of interactive plot. "default" places everything on one device. "separate" places condition selectors on one device and the section on another. (These two options require XQuartz on OS X). "shiny" produces a Shiny application. view3d Logical; if TRUE plots a three-dimensional regression surface if possible. Corder Character name for method of ordering conditioning variables. See arrangeC. selectortype Type of condition selector plots to use. Must be "minimal" if type is "default". If type is "separate", can be "pcp" (see plotxc.pcp) or "full" (see plotxc.full). conf Logical; if TRUE plots confidence bounds (or equivalent) for models which provide this. probs Logical; if TRUE, shows predicted class probabilities instead of just predicted classes. Only available if S specifies two numeric predictors and the model's predict method provides this. col Colour for observed data. pch Plot symbols for observed data. residuals Logical; if TRUE, plots a residual versus predictor plot instead of the usual scale of raw response. xsplotpar Plotting parameters for section visualisation as a list, passed to plotxs. Can specify xlim, ylim. modelpar Plotting parameters for models as a list, passed to plotxs. Not used. xcplotpar Plotting parameters for condition selector plots as a list, passed to plotxc. Can specify col for highlighting current section, cex, and trim (see plotxc). ### References O'Connell M, Hurley CB and Domijan K (2017). “Conditional Visualization for Statistical Models: An Introduction to the condvis Package in R.”Journal of Statistical Software, 81(5), pp. 1-20. <URL:http://dx.doi.org/10.18637/jss.v081.i05>. condtour, similarityweight ### Examples ## Not run: ## Example 1: Multivariate regression, xs one continuous predictor mtcars$cyl <- as.factor(mtcars$cyl) library(mgcv) model1 <- list( quadratic = lm(mpg ~ cyl + hp + wt + I(wt^2), data = mtcars), additive = mgcv::gam(mpg ~ cyl + hp + s(wt), data = mtcars)) conditionvars1 <- list(c("cyl", "hp")) ceplot(data = mtcars, model = model1, response = "mpg", sectionvars = "wt", conditionvars = conditionvars1, threshold = 0.3, conf = T) ## Example 2: Binary classification, xs one categorical predictor mtcars$cyl <- as.factor(mtcars$cyl) mtcars$am <- as.factor(mtcars$am) library(e1071) model2 <- list( svm = svm(am ~ mpg + wt + cyl, data = mtcars, family = "binomial"), glm = glm(am ~ mpg + wt + cyl, data = mtcars, family = "binomial")) ceplot(data = mtcars, model = model2, sectionvars = "wt", threshold = 1, type = "shiny") ## Example 3: Multivariate regression, xs both continuous mtcars$cyl <- as.factor(mtcars$cyl) mtcars$gear <- as.factor(mtcars$gear) library(e1071) model3 <- list(svm(mpg ~ wt + qsec + cyl + hp + gear, data = mtcars, family = "binomial")) conditionvars3 <- list(c("cyl","gear"), "hp") ceplot(data = mtcars, model = model3, sectionvars = c("wt", "qsec"), threshold = 1, conditionvars = conditionvars3) ceplot(data = mtcars, model = model3, sectionvars = c("wt", "qsec"), threshold = 1, type = "separate", view3d = T) ## Example 4: Multi-class classification, xs both categorical mtcars$cyl <- as.factor(mtcars$cyl) mtcars$vs <- as.factor(mtcars$vs) mtcars$am <- as.factor(mtcars$am) mtcars$gear <- as.factor(mtcars$gear) mtcars$carb <- as.factor(mtcars$carb) library(e1071) model4 <- list(svm(carb ~ ., data = mtcars, family = "binomial")) ceplot(data = mtcars, model = model4, sectionvars = c("cyl", "gear"), threshold = 3) ## Example 5: Multi-class classification, xs both continuous data(wine) wine$Class <- as.factor(wine$Class) library(e1071) model5 <- list(svm(Class ~ ., data = wine, probability = TRUE)) ceplot(data = wine, model = model5, sectionvars = c("Hue", "Flavanoids"), threshold = 3, probs = TRUE) ceplot(data = wine, model = model5, sectionvars = c("Hue", "Flavanoids"), threshold = 3, type = "separate") ceplot(data = wine, model = model5, sectionvars = c("Hue", "Flavanoids"), threshold = 3, type = "separate", selectortype = "pcp") ## Example 6: Multi-class classification, xs with one categorical predictor, ## and one continuous predictor. mtcars$cyl <- as.factor(mtcars$cyl) mtcars$carb <- as.factor(mtcars$carb) library(e1071) model6 <- list(svm(cyl ~ carb + wt + hp, data = mtcars, family = "binomial")) ceplot(data = mtcars, model = model6, threshold = 1, sectionvars = c("carb", "wt"), conditionvars = "hp") ## End(Not run) [Package condvis version 0.5-1 Index]	{"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": 0.31726813316345215, "perplexity": 14597.399227094607}, "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/1652662538646.33/warc/CC-MAIN-20220521045616-20220521075616-00744.warc.gz"}
https://link.springer.com/article/10.1007/s00446-018-0323-9	# Contention resolution on a fading channel • Jeremy T. Fineman • Seth Gilbert • Fabian Kuhn • Calvin Newport Article ## Abstract In this paper, we study upper and lower bounds for contention resolution on a single hop fading channel; i.e., a channel where receive behavior is determined by a signal to interference and noise ratio equation. The best known previous solution solves the problem in this setting in $$O(\log ^2{n}/\log \log {n})$$ rounds, with high probability in the system size n. We describe and analyze an algorithm that solves the problem in $$O(\log {n} + \log {R})$$ rounds, where R is the ratio between the longest and shortest link, and is a value upper bounded by a polynomial in n for most feasible deployments. We complement this result with an $$\varOmega (\log {n})$$ lower bound that proves the bound tight for reasonable R. We note that in the classical radio network model (which does not include signal fading), high probability contention resolution requires $$\varOmega (\log ^2{n})$$ rounds. Our algorithm, therefore, affirms the conjecture that the spectrum reuse enabled by fading should allow distributed algorithms to achieve a significant improvement on this $$\log ^2{n}$$ speed limit. In addition, we argue that the new techniques required to prove our upper and lower bounds are of general use for analyzing other distributed algorithms in this increasingly well-studied fading channel setting. ## Keywords Contention resolution Leader election Wireless channel Wireless algorithms SINR model ## Notes ### Acknowledgements This research was support in part by the following grants: NSF CCF 1314633, NSF CCF 1320279, NUS FRC T1 251RES1404 and ERC Grant No. 336495 (ACDC). ## References 1. 1. Abramson, N.: The ALOHA system: another alternative for computer communications. In: Proceedings of the Fall Joint Computer Conference (1970)Google Scholar 2. 2. Bar-Yehuda, R., Goldreich, O., Itai, A.: On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization. In: Proceedings of the International Symposium on Principles of Distributed Computing (1987)Google Scholar 3. 3. Chlamtac, I., Kutten, S.: On broadcasting in radio networks: problem analysis and protocol design. In: Proceedings of the Conference on Computer Communication (1985)Google Scholar 4. 4. Daum, S., Gilbert, S., Kuhn, F., Newport, C.: Broadcast in the ad hoc SINR model. In: Proceedings of the International Symposium on Distributed Computing (2013)Google Scholar 5. 5. Daum, S., Kuhn, F., Newport, C.: Efficient symmetry breaking in multi-channel radio networks. In: Proceedings of the International Symposium on Distributed Computing (2012)Google Scholar 6. 6. Gallager, R.: A perspective on multiaccess channels. IEEE Trans. Inf. Theory 31(2), 124–142 (1985) 7. 7. Gasieniec, L., Pelc, A., Peleg, D.: The wakeup problem in synchronous broadcast systems. SIAM J. Discrete Math. 14(2), 207–222 (2001) 8. 8. Goussevskaia, O., Moscibroda, T., Wattenhofer, R.: Local broadcasting in the physical interference model. In: Proceedings of the International Workshop on the Foundations of Mobile Computing. ACM (2008)Google Scholar 9. 9. Greenberg, A., Winograd, S.: A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels. J. ACM 32(3), 589–596 (1985) 10. 10. Hajek, B., van Loon, T.: Decentralized dynamic control of a multiaccess broadcast channel. IEEE Trans. Autom. Control 27(3), 559–569 (1982) 11. 11. Halldorsson, M.M., Mitra, P.: Distributed connectivity of wireless networks. In: Proceedings of the ACM Conference on Distributed Computing (2012)Google Scholar 12. 12. Halldorsson, M.M., Mitra, P.: Towards tight bounds for local broadcasting. In: Proceedings of the International Workshop on the Foundations of Mobile Computing. ACM (2012)Google Scholar 13. 13. Jurdzinski, T., Kowalski, D.: Distributed backbone structure for algorithms in the SINR model of wireless networks. In: Proceedings of the International Conference on Distributed Computing (2012)Google Scholar 14. 14. Jurdzinski, T., Kowalski, D.R., Rozanski, M., Stachowiak, G.: On the impact of geometry on ad hoc communication in wireless networks. In: Proceedings of the ACM Conference on Distributed Computing (2014)Google Scholar 15. 15. Jurdziński, T., Stachowiak, G.: Probabilistic algorithms for the wakeup problem in single-hop radio networks. In: Bose, P., Morin, P. (eds.) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, BerlinGoogle Scholar 16. 16. Jurdzinski, T., Stachowiak, G.: The cost of synchronizing multiple-access channels. In: Proceedings of the ACM Conference on Distributed Computing (2015)Google Scholar 17. 17. Kaplan, M.: A sufficient condition for non-ergodicity of a Markov chain. IEEE Trans. Inf. Theory 25, 470–471 (1979) 18. 18. Komlos, J., Greenberg, A.: An asymptotically nonadaptive algorithm for conflict resolution in multiple-access channels. IEEE Trans. Inf. Theory 31(2), 302–306 (1985) 19. 19. Moscibroda, T., Wattenhofer, R.: The complexity of connectivity in wireless networks. In: Proceedings of the IEEE International Conference on Computer Communications (2006)Google Scholar 20. 20. Newport, C.: Radio network lower bounds made easy. In: Proceedings of the International Symposium on Distributed Computing (2014)Google Scholar 21. 21. Roberts, L.G.: ALOHA packet system with and without slots and capture. ACM SIGCOMM Comput. Commun. Rev. 5(2), 28–42 (1975) 22. 22. Scheideler, C., Richa, A., Santi, P.: An $$O(\log {n})$$ dominating set protocol for wireless ad-hoc networks under the physical interference model. In: Proceedings of the International Symposium on Mobile Ad Hoc Networking and Computing (2008)Google Scholar 23. 23. Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput. 15(2), 468–477 (1986) © Springer-Verlag GmbH Germany, part of Springer Nature 2018 ## Authors and Affiliations • Jeremy T. Fineman • 1 • Seth Gilbert • 2 • Fabian Kuhn • 3 • Calvin Newport • 1 1. 1.Georgetown UniversityWashingtonUSA 2. 2.National University of SingaporeSingaporeSingapore 3. 3.University of FreiburgFreiburgGermany	{"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": 0.9470534920692444, "perplexity": 4186.593928219176}, "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/1539583509336.11/warc/CC-MAIN-20181015163653-20181015185153-00127.warc.gz"}
http://math.stackexchange.com/users/19297/sgt-pepper	# sgt pepper Unregistered less info reputation 3 bio website location age member for 3 years, 1 month seen Nov 17 '11 at 9:14 profile views 8 # 3 Questions 4 Evaluation of the integral $\int_0^\infty \left(\frac{\pi^2}{4}-x^2\right)^{-2}\cdot\frac{\pi^2}{4}\cos^2 x\,dx$ 1 Duality and the Fourier transform 1 Integration error spotting # 31 Reputation This user has no recent positive reputation changes This user has not answered any questions # 6 Tags 0 integration 0 fourier-analysis 0 special-functions 0 definite-integrals 0 duality-theorems 0 contour-integration # 1 Account Mathematics 31 rep 3	{"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": 0.4488297998905182, "perplexity": 10996.267849240334}, "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-2014-52/segments/1418802770554.119/warc/CC-MAIN-20141217075250-00099-ip-10-231-17-201.ec2.internal.warc.gz"}
https://www.lessonplanet.com/teachers/initial-sounds-ch	# Initial Sounds: Ch In this word and letter recognition worksheet, students examine 6 words that are missing letters. Students fill in the initial letters in the words as instructed. Picture clues are included. Students also practice tracing the letter.	{"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": 0.8092445135116577, "perplexity": 6244.130540692052}, "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-2017-30/segments/1500549429485.0/warc/CC-MAIN-20170727202516-20170727222516-00652.warc.gz"}
https://www.lessonplanet.com/teachers/math-worksheet-advanced-multiplication-using-numbers-between-10-and-100-2	# Math Worksheet: Advanced Multiplication Using Numbers Between 10 and 100, #2 In this online interactive 2 digit multiplication worksheet, students practice their math skills as they solve 15 problems that require them to multiply numbers between 10 and 100. The answers may be typed in on the worksheet and submitted to be scored. Concepts Resource Details	{"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": 0.8783854246139526, "perplexity": 1403.325137935928}, "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/1505818687642.30/warc/CC-MAIN-20170921044627-20170921064627-00266.warc.gz"}
https://test.topocondmat.org/w12_manybody/w12_assignments.html	# Review assignment¶ ### Observation of the Fractional Quantum Hall Effect in Graphene (arXiv:0910.2763)¶ Kirill I. Bolotin, Fereshte Ghahari, Michael D. Shulman, Horst L. Stormer, Philip Kim When electrons are confined in two dimensions and subjected to strong magnetic fields, the Coulomb interactions between them become dominant and can lead to novel states of matter such as fractional quantum Hall liquids. In these liquids electrons linked to magnetic flux quanta form complex composite quasipartices, which are manifested in the quantization of the Hall conductivity as rational fractions of the conductance quantum. The recent experimental discovery of an anomalous integer quantum Hall effect in graphene has opened up a new avenue in the study of correlated 2D electronic systems, in which the interacting electron wavefunctions are those of massless chiral fermions. However, due to the prevailing disorder, graphene has thus far exhibited only weak signatures of correlated electron phenomena, despite concerted experimental efforts and intense theoretical interest. Here, we report the observation of the fractional quantum Hall effect in ultraclean suspended graphene, supporting the existence of strongly correlated electron states in the presence of a magnetic field. In addition, at low carrier density graphene becomes an insulator with an energy gap tunable by magnetic field. These newly discovered quantum states offer the opportunity to study a new state of matter of strongly correlated Dirac fermions in the presence of large magnetic fields. Hint: Fractional quantum Hall effect in graphene ### Exotic non-Abelian anyons from conventional fractional quantum Hall states (arXiv:1204.5479)¶ David J. Clarke, Jason Alicea, Kirill Shtengel Non-Abelian anyons--particles whose exchange noncommutatively transforms a system's quantum state--are widely sought for the exotic fundamental physics they harbor as well as for quantum computing applications. There now exist numerous blueprints for stabilizing the simplest type of non-Abelian anyon, defects binding Majorana modes, by judiciously interfacing widely available materials. Following this line of attack, we introduce a device fabricated from conventional fractional quantum Hall states and s-wave superconductors that supports exotic non-Abelian anyons that bind `parafermions', which can be viewed as fractionalized Majorana fermions. We show that these modes can be experimentally identified (and distinguished from Majoranas) using Josephson measurements. We also provide a practical recipe for braiding parafermions and show that they give rise to non-Abelian statistics. Interestingly, braiding in our setup produces a richer set of topologically protected qubit operations when compared to the Majorana case. As a byproduct, we establish a new, experimentally realistic Majorana platform in weakly spin-orbit-coupled materials such as GaAs. Hint: Fractional Majoranas in fractional quantum Hall edges ### High threshold universal quantum computation on the surface code (arXiv:0803.0272)¶ Austin G. Fowler, Ashley M. Stephens, Peter Groszkowski We present a comprehensive and self-contained simplified review of the quantum computing scheme of Phys. Rev. Lett. 98, 190504 (2007), which features a 2-D nearest neighbor coupled lattice of qubits, a threshold error rate approaching 1%, natural asymmetric and adjustable strength error correction and low overhead arbitrarily long-range logical gates. These features make it by far the best and most practical quantum computing scheme devised to date. We restrict the discussion to direct manipulation of the surface code using the stabilizer formalism, both of which we also briefly review, to make the scheme accessible to a broad audience. Hint: A scheme for quantum computation using the toric code ### Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states (arXiv:1502.01665)¶ Eran Sagi, Yuval Oreg, Ady Stern, Bertrand I. Halperin We consider an annular superconductor-insulator-superconductor Josephson-junction, with the insulator being a double layer of electron and holes at Abelian fractional quantum Hall states of identical fillings. When the two superconductors gap out the edge modes, the system has a topological ground state degeneracy in the thermodynamic limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional limit, where the width of the insulator becomes small, the ground state energies are split. We discuss several implications of the topological degeneracy that survive the crossover to the quasi-one-dimensional limit. In particular, the Josephson effect shows a $2\pi d$-periodicity, where $d$ is the ground state degeneracy in the 2 dimensional limit. We find that at special values of the relative phase between the two superconductors there are protected crossing points in which the degeneracy is not completely lifted. These features occur also if the insulator is a time-reversal-invariant fractional topological insulator. We describe the latter using a construction based on coupled wires. Furthermore, when the superconductors are replaced by systems with an appropriate magnetic order that gap the edges via a spin-flipping backscattering, the Josephson effect is replaced by a spin Josephson effect. Hint: Making a fractional quantum Hall effect by coupling wires ### Bonus: Find your own paper to review!¶ Do you know of another paper that fits into the topics of this week, and you think is good? Then you can get bonus points by reviewing that paper instead! MoocSelfAssessment description In the live version of the course, you would need to share your solution and grade yourself. Do you have questions about what you read? Would you like to suggest other papers? Tell us: Discussion Many-body topology is available in the EdX version of the course.	{"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": 0.8233678936958313, "perplexity": 1106.0688517580684}, "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-2020-29/segments/1593657172545.84/warc/CC-MAIN-20200716153247-20200716183247-00528.warc.gz"}
http://blog.sbtsu.org/i-love-tobxkr/how-much-topsoil-do-i-need-to-topdress-a-lawn%3F-8f514d	To calculate the weight of a cubic yard of soil, you simply have to multiply the volume by its density. A very light topdressing application can be completed more often if the amount added is shallow enough to be brushed into aeration holes. If you plan on overseeding, do this after the topdressing is down. 4 Ways To Topdress A Lawn Wikihow . Why top dress a lawn. When to apply top dressing. That can be done in the next step. REASONS FOR TOP-DRESSING LAWNS. Measuring in metres is simplest, as this will give you the volume of topsoil you need in cubic metres. Thus, if your lawn is in a state where you doubt the topdressing would be sufficient or ideal, you might need to look for more practical solutions instead. Hey Master Gardeners! Lawns based on good quality soil should not need top dressing every year although if you want a really top class lawn then you may wish to do so. Obviously the dips will get more but you must not smother the grass. My lawn is tiny. How much topsoil do I need is one of the most commonly asked questions by gardeners looking to buy topsoil. Dethatching will leave a lot of debris on the surface, so you will need to remove it before mowing or top dressing. How Much Topsoil Do I Need To Topdress A Lawn, 5 / 5 ( 1 votes ) Topdress level your lawn using sand leveling the lawn before reseeding part improve the soil in your lawn leveling out an uneven lawn top dressing a lawn the benefits and 4 ways to topdress a lawn wikihow. How to top dress your lawn Making your lawn top dressing . One of the first and most important steps is deciding what type of topdressing material you should use. This video above shows you how to topdress with our Soil3 Level Mix.. Topdress to level a warm season lawn when it's actively growing, after spring green-up. If the lawn is in poorer condition, overseeding alone will not be enough and it's better to start again with a new lawn or an alternative to grass. That moves material down to the soil surface. However we recommend that you leave it for a few months and most of those issues will take care of themselves. Here’s What Top Dressing Does for Your Lawn: It needs less water because the top dressing retains soil moisture, reduces surface evaporation and keeps the roots cool in the heat of summer. and total cost of the soil: Visit the Lawn Care Store now offering contactless delivery, Privacy Policy \| Copyright © 2020 Lawn Solutions Australia. There are two different cases that can help determine the amount of topdressing you need to spread. There are two reasons to top dress lawns. A slope of 1 to 4 percent is ideal. For example: A site has an area of 10m x 15m and needs a coverage of 150mm. Work out the size of your lawn before you buy; as a general guide a 2kg bag will be enough to top dress a square metre of lawn. We have removed most of the larger ones that were 3" or more in diameter from the surface. How much topsoil do I need for new lawn? From 1 to 8 pounds (½ to 4 kg) may be needed per square yard/meter (see below). Because of this you may want to run your mixture through a soil sieve (1/4” holes) before applying the topdressing. Topdress in lawns in the spring for warm season grasses and the fall for cool season grasses. This allows three or four mows before severe heat or cold sets in. Try to get the top dressing about a quarter of an inch thick on all areas of your lawn. Mow the lawn as low as possible without stressing the grass too much. How do I topdress the lawn? In this case, you’ll need to top dress more than once over the course of a few seasons. On a seeded lawn: After sowing lawn grass seeds, apply a thin layer — about 1/4-inch — of compost as top-dressing to help maintain consistent soil moisture while seeds germinate and tender grass seedlings get established. It has many benefits to the health of the lawn and the reasons for top dressing include: Maintaining a true and level lawn surface – Top dressing fills in all of the small hollows and undulations therefore creating a level lawn surface. To test how much topsoil is already in your lawn, use a garden fork. A soil test can help you determine if your existing soil will support good lawn … Any areas where it globs or mounds up should be corrected immediately. Ideally, do it in early fall or spring since you’ll want to give your grass time to grow through 3-4 more mowings before severe heat or cold, especially if you are overseeding. Calculating Depth. There are a lot of rocks - from 1/2 inch to 12". To fill in bare spots or reinvigorate a thin lawn, you can apply grass seed over an existing lawn, provided at least 50 percent of the lawn is in good condition. How many cubic yards of compost would I need to do this? Is that a decent amount to topdress with? If you are scarifying the lawn in autumn (September is a good month for this) then you should do this BEFORE you top dress the lawn. Get more lawn care tips - https://hubs.ly/H08vmz00 More information on The Lawn Guy - https://goo.gl/ZLuHSo Simply push it into the soil and if the prongs go right into the earth, the soil should be deep enough. Upon first thought, it seems somewhat absurd to spread a layer of compost or sand, your grass. Once you’re happy with the even level and spread of top dressing, it’s time to water it … Topsoil similar to the existing soil structure is acceptable and will help smooth out the ground, but doesn’t contain much organic material. Treat bare spots as needed and the entire lawn every few years. Water the lawn. Quantity of Top Dressing Soil If this is being done purely to smooth the lawn then you can apply at 3 to 4 kilos per square metre which will give about ¼” or just over half a centimetre in depth. Steps to Estimate Soil Volume This promotes a strong, healthy lawn, and a healthy lawn means less weeds and pests. How much topsoil do I need is one of the most commonly asked questions by gardeners looking to buy topsoil. To calculate how much topsoil you need, simply measure the dimensions of the area you wish to cover and multiply this by the depth required. Repeat the Process if Necessary Allow the topdress to settle. We live in a new development, so it is pretty barren in the backyard. Your lawn will become more tolerant of heavy use which is a good thing if you have children who play on your lawn. If you only add 5mm of topsoil at a time you should be able to top dress twice in the spring and once in the autumn. This is what topdressing is, though, and it’s a great thing to periodically do for your lawn. The first is to improve the level, fill in holes or fix the grade and the second is to add nutrient back into the soil as part of your annual or regular lawn care program. peakbagger66. One of these days I would like to topdress the lawn with about 1" of compost. After all, dirt is supposed to be under the grass. Fertile, rapidly growing turf and lawns that are more prone to thatch may require heavier topdressing rates. For new home seed lawn, calculate 4-6" depth. Using a higher proportion of organic material for sandy soils is a good idea. This process is used to add natural organic nutrients and minerals to the lawn and is highly recommended for all lawn owners to undertake every so often. Choosing the wrong material can create serious problems. How often a lawn needs topdressing depends on the situation. As far as the ideal amount of topsoil is concerned, it is generally recommended to use about 4/10 cubic yards of topsoil for every 1000 square feet of yard. Add organic matter to the topsoil, improving poor soil by enhancing its productive. ft. Topdressing with Level Mix to Level Existing Lawns: Approximately 1 Cubic Yard per 500 to 1,000 sq. Be sure to account for settling, 1/3 of topsoil will settle; To top dress your existing lawn area, calculate 1" depth. Topdressing your lawn can be a DIY project, but truthfully it is a very labor intensive endeavor. With an Australia wide network, there is a Lawn Solutions Australia Accredited grower or supplier near you. Topdressing is a simpler — albeit still challenging — way to see the following benefits in your lawn. Once you have successfully topdressed your yard, now it’s time to be patient, as it can take a few seasons to see the full results. To achieve that balance, it’s a good idea to mix soil, compost, and potting blends for the best plant growth. You’ll most likely need to add soil amendments, like Miracle-Gro Soil, to create the ideal growing environment. As mentioned this can consist of a mix of materials depending on the existing soil composition and health, such as sand, loam, topsoil or peat. Fertilise a few weeks prior to top dressing to ensure maximum growth at the time. Then, the topsoil calculator works out the total weight: $$Weight = Volume \times Density = 16.76\,ft^3 \times 1000\,lb/yd^3 = 0.31\,t$$. Rolawn Lawn Topdressing can be applied anytime, when the grass is actively growing. Use our free topsoil calculator to quickly identify how many tonnes or 25L/kg bags you need for a project. It helps to increase nutrient retention, improves drainage and increases disease and pest resistance. Aerate the lawn. If you are looking for a way to improve the health of your lawn or want to level it out then top dressing your lawn with compost is a very good idea. You need to replace the topsoil to work with your existing subsoil, but just how much do you need? It can be done all at once, or in stages. Recommended depths for topsoil • General gardening: for most planting, such as for a herbaceous border, you'll need around 8 inches (200mm) of topsoil. In most cases top dressing is done to correct poor preparation and lack of soil underneath or to fill in low spots and correct uneven areas in the lawn. Generally late spring to early summer is best. The depth of topsoil for growing vegetables will vary depending on what you plan to grow. Instead, a spade should be used to lift up the affected lawn area, and the top dressing sand placed underneath the turf to achieve most of the levelling. Many people also make the mistake of top dressing with sand to try to correct clay soil . the grass. Aerate or core your lawn, spread the mix evenly over the desired area, then rake, level lawn or broom it into the lawn profile. Upon first thought, it seems somewhat absurd to spread a layer of compost or sand over your grass. Should only be done during the growing season (this is when you need to mow your lawn weekly) and the earlier in the season the better. LawnStarter is a startup making lawn care easy affordable and reliable. The… (Sorry my math skills stink!) This video above shows you how to topdress with our Soil3 Level Mix. Shovel the loam or sand onto the lawn in even piles and then level with a garden leveller. For new home seed lawn, calculate 4-6" depth. Combine with other cultural practices like, Check the soil pH and adjust accordingly if it is. 6 Things to Look for in a Landscaping Maintenance Company, 14 Deer-Resistant Plants and Trees to Grow in Boston, The 12 Big Metro Economies That Are Ready to Bloom Through 2021, 2021’s Best and Worst States for Living Off the Grid, 10 Walkway Ideas for Your Yard and Garden, 14 Treehouse Ideas for Your Backyard Playhouse. As far as the ideal amount of topsoil is concerned, it is generally recommended to use about 4/10 cubic yards of topsoil for every 1000 square feet of yard. How does the calculator work out the cost? Anything less, or anything more is likely to hinder the level of success and the overall outcome you get. Check out my full article on how to level a lawn with topsoil. When you do cut it, make sure it’s not too short or too long, just a trim, leaving the length of the grass at 40–45mm. 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http://terrytao.wordpress.com/2013/05/	You are currently browsing the monthly archive for May 2013. A finite group ${G=(G,\cdot)}$ is said to be a Frobenius group if there is a non-trivial subgroup ${H}$ of ${G}$ (known as the Frobenius complement of ${G}$) such that the conjugates ${gHg^{-1}}$ of ${H}$ are “disjoint as possible” in the sense that ${H \cap gHg^{-1} = \{1\}}$ whenever ${g \not \in H}$. This gives a decomposition $\displaystyle G = \bigcup_{gH \in G/H} (gHg^{-1} \backslash \{1\}) \cup K \ \ \ \ \ (1)$ where the Frobenius kernel ${K}$ of ${G}$ is defined as the identity element ${1}$ together with all the non-identity elements that are not conjugate to any element of ${H}$. Taking cardinalities, we conclude that $\displaystyle \|G\| = \frac{\|G\|}{\|H\|} (\|H\| - 1) + \|K\|$ and hence $\displaystyle \|H\| \|K\| = \|G\|. \ \ \ \ \ (2)$ A remarkable theorem of Frobenius gives an unexpected amount of structure on ${K}$ and hence on ${G}$: Theorem 1 (Frobenius’ theorem) Let ${G}$ be a Frobenius group with Frobenius complement ${H}$ and Frobenius kernel ${K}$. Then ${K}$ is a normal subgroup of ${G}$, and hence (by (2) and the disjointness of ${H}$ and ${K}$ outside the identity) ${G}$ is the semidirect product ${K \rtimes H}$ of ${H}$ and ${K}$. I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group ${G}$, in particular relying on the fact that a character on a subgroup ${H}$ can induce a character on ${G}$, which can then be decomposed into irreducible characters with natural number coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement ${H}$ has even order or when ${H}$ is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.) I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ${{\bf C} G \equiv L^2(G)}$) with the Fourier analysis of class functions (i.e. the representation theory of the centre ${Z({\bf C} G) \equiv L^2(G)^G}$ of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here. The main ideas are as follows. The space ${L^2(G)^G}$ of class functions can be viewed as a commutative algebra with respect to the convolution operation ${}$; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions ${\phi}$ such that ${\phi\phi = \phi}$. In terms of characters, idempotents are nothing more than sums of the form ${\sum_{\chi \in \Sigma} \chi(1) \chi}$ for various collections ${\Sigma}$ of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that ${\phi(1)}$ is a natural number, and more generally we will show that ${{\bf E}_{(a,b) \in S} {\bf E}_{x \in G} \phi( a x b^{-1} x^{-1} )}$ is a natural number whenever ${S}$ is a subgroup of ${G \times G}$ (see Corollary 4 below). For instance, the quantity $\displaystyle \hbox{rank}(\phi) := {\bf E}_{a \in G} {\bf E}_{x \in G} \phi(a xa^{-1} x^{-1})$ is a natural number which we will call the rank of ${\phi}$ (as it is also the linear rank of the transformation ${f \mapsto f\phi}$ on ${L^2(G)}$). In the case that ${G}$ is a Frobenius group with kernel ${K}$, the above integrality properties can be used after some elementary manipulations to establish that for any idempotent ${\phi}$, the quantity $\displaystyle \frac{1}{\|G\|} \sum_{a \in K} {\bf E}_{x \in G} \phi( axa^{-1}x^{-1} ) - \frac{1}{\|G\| \|K\|} \sum_{a,b \in K} \phi(ab^{-1}) \ \ \ \ \ (3)$ is an integer. On the other hand, one can also show by elementary means that this quantity lies between ${0}$ and ${\hbox{rank}(\phi)}$. These two facts are not strong enough on their own to impose much further structure on ${\phi}$, unless one restricts attention to minimal idempotents ${\phi}$. In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that ${\phi}$ has rank one, and then the integrality gap comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent ${\phi}$ either preserves ${\frac{\|G\|}{\|K\|} 1_K}$ or annihilates it, which makes ${\frac{\|G\|}{\|K\|} 1_K}$ itself an idempotent, which makes ${K}$ normal. Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our joint paper “Multiple recurrence and convergence results associated to ${{\bf F}_{p}^{\omega}}$-actions“. This paper is primarily concerned with limit formulae in the theory of multiple recurrence in ergodic theory. Perhaps the most basic formula of this type is the mean ergodic theorem, which (among other things) asserts that if ${(X,{\mathcal X}, \mu,T)}$ is a measure-preserving ${{\bf Z}}$-system (which, in this post, means that ${(X,{\mathcal X}, \mu)}$ is a probability space and ${T: X \mapsto X}$ is measure-preserving and invertible, thus giving an action ${(T^n)_{n \in {\bf Z}}}$ of the integers), and ${f,g \in L^2(X,{\mathcal X}, \mu)}$ are functions, and ${X}$ is ergodic (which means that ${L^2(X,{\mathcal X}, \mu)}$ contains no ${T}$-invariant functions other than the constants (up to almost everywhere equivalence, of course)), then the average $\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x)\ d\mu \ \ \ \ \ (1)$ converges as ${N \rightarrow \infty}$ to the expression $\displaystyle (\int_X f(x)\ d\mu) (\int_X g(x)\ d\mu);$ see e.g. this previous blog post. Informally, one can interpret this limit formula as an equidistribution result: if ${x}$ is drawn at random from ${X}$ (using the probability measure ${\mu}$), and ${n}$ is drawn at random from ${\{1,\ldots,N\}}$ for some large ${N}$, then the pair ${(x, T^n x)}$ becomes uniformly distributed in the product space ${X \times X}$ (using product measure ${\mu \times \mu}$) in the limit as ${N \rightarrow \infty}$. If we allow ${(X,\mu)}$ to be non-ergodic, then we still have a limit formula, but it is a bit more complicated. Let ${{\mathcal X}^T}$ be the ${T}$-invariant measurable sets in ${{\mathcal X}}$; the ${{\bf Z}}$-system ${(X, {\mathcal X}^T, \mu, T)}$ can then be viewed as a factor of the original system ${(X, {\mathcal X}, \mu, T)}$, which is equivalent (in the sense of measure-preserving systems) to a trivial system ${(Z_0, {\mathcal Z}_0, \mu_{Z_0}, 1)}$ (known as the invariant factor) in which the shift is trivial. There is then a projection map ${\pi_0: X \rightarrow Z_0}$ to the invariant factor which is a factor map, and the average (1) converges in the limit to the expression $\displaystyle \int_{Z_0} (\pi_0)_ f(z) (\pi_0)_* g(z)\ d\mu_{Z_0}(x), \ \ \ \ \ (2)$ where ${(\pi_0)_: L^2(X,{\mathcal X},\mu) \rightarrow L^2(Z_0,{\mathcal Z}_0,\mu_{Z_0})}$ is the pushforward map associated to the map ${\pi_0: X \rightarrow Z_0}$; see e.g. this previous blog post. We can interpret this as an equidistribution result. If ${(x,T^n x)}$ is a pair as before, then we no longer expect complete equidistribution in ${X \times X}$ in the non-ergodic, because there are now non-trivial constraints relating ${x}$ with ${T^n x}$; indeed, for any ${T}$-invariant function ${f: X \rightarrow {\bf C}}$, we have the constraint ${f(x) = f(T^n x)}$; putting all these constraints together we see that ${\pi_0(x) = \pi_0(T^n x)}$ (for almost every ${x}$, at least). The limit (2) can be viewed as an assertion that this constraint ${\pi_0(x) = \pi_0(T^n x)}$ are in some sense the “only” constraints between ${x}$ and ${T^n x}$, and that the pair ${(x,T^n x)}$ is uniformly distributed relative to these constraints. Limit formulae are known for multiple ergodic averages as well, although the statement becomes more complicated. For instance, consider the expression $\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x) h(T^{2n} x)\ d\mu \ \ \ \ \ (3)$ for three functions ${f,g,h \in L^\infty(X, {\mathcal X}, \mu)}$; this is analogous to the combinatorial task of counting length three progressions in various sets. For simplicity we assume the system ${(X,{\mathcal X},\mu,T)}$ to be ergodic. Naively one might expect this limit to then converge to $\displaystyle (\int_X f\ d\mu) (\int_X g\ d\mu) (\int_X h\ d\mu)$ which would roughly speaking correspond to an assertion that the triplet ${(x,T^n x, T^{2n} x)}$ is asymptotically equidistributed in ${X \times X \times X}$. However, even in the ergodic case there can be additional constraints on this triplet that cannot be seen at the level of the individual pairs ${(x,T^n x)}$, ${(x, T^{2n} x)}$. The key obstruction here is that of eigenfunctions of the shift ${T: X \rightarrow X}$, that is to say non-trivial functions ${f: X \rightarrow S^1}$ that obey the eigenfunction equation ${Tf = \lambda f}$ almost everywhere for some constant (or ${T}$-invariant) ${\lambda}$. Each such eigenfunction generates a constraint $\displaystyle f(x) \overline{f(T^n x)}^2 f(T^{2n} x) = 1 \ \ \ \ \ (4)$ tying together ${x}$, ${T^n x}$, and ${T^{2n} x}$. However, it turns out that these are in some sense the only constraints on ${x,T^n x, T^{2n} x}$ that are relevant for the limit (3). More precisely, if one sets ${{\mathcal X}_1}$ to be the sub-algebra of ${{\mathcal X}}$ generated by the eigenfunctions of ${T}$, then it turns out that the factor ${(X, {\mathcal X}_1, \mu, T)}$ is isomorphic to a shift system ${(Z_1, {\mathcal Z}_1, \mu_{Z_1}, x \mapsto x+\alpha)}$ known as the Kronecker factor, for some compact abelian group ${Z_1 = (Z_1,+)}$ and some (irrational) shift ${\alpha \in Z_1}$; the factor map ${\pi_1: X \rightarrow Z_1}$ pushes eigenfunctions forward to (affine) characters on ${Z_1}$. It is then known that the limit of (3) is $\displaystyle \int_\Sigma (\pi_1)_ f(x_0) (\pi_1)_* g(x_1) (\pi_1)_* h(x_2)\ d\mu_\Sigma$ where ${\Sigma \subset Z_1^3}$ is the closed subgroup $\displaystyle \Sigma = \{ (x_1,x_2,x_3) \in Z_1^3: x_1-2x_2+x_3=0 \}$ and ${\mu_\Sigma}$ is the Haar probability measure on ${\Sigma}$; see this previous blog post. The equation ${x_1-2x_2+x_3=0}$ defining ${\Sigma}$ corresponds to the constraint (4) mentioned earlier. Among other things, this limit formula implies Roth’s theorem, which in the context of ergodic theory is the assertion that the limit (or at least the limit inferior) of (3) is positive when ${f=g=h}$ is non-negative and not identically vanishing. If one considers a quadruple average $\displaystyle \frac{1}{N} \sum_{n=1}^N \int_X f(x) g(T^n x) h(T^{2n} x) k(T^{3n} x)\ d\mu \ \ \ \ \ (5)$ (analogous to counting length four progressions) then the situation becomes more complicated still, even in the ergodic case. In addition to the (linear) eigenfunctions that already showed up in the computation of the triple average (3), a new type of constraint also arises from quadratic eigenfunctions ${f: X \rightarrow S^1}$, which obey an eigenfunction equation ${Tf = \lambda f}$ in which ${\lambda}$ is no longer constant, but is now a linear eigenfunction. For such functions, ${f(T^n x)}$ behaves quadratically in ${n}$, and one can compute the existence of a constraint $\displaystyle f(x) \overline{f(T^n x)}^3 f(T^{2n} x)^3 \overline{f(T^{3n} x)} = 1 \ \ \ \ \ (6)$ between ${x}$, ${T^n x}$, ${T^{2n} x}$, and ${T^{3n} x}$ that is not detected at the triple average level. As it turns out, this is not the only type of constraint relevant for (5); there is a more general class of constraint involving two-step nilsystems which we will not detail here, but see e.g. this previous blog post for more discussion. Nevertheless there is still a similar limit formula to previous examples, involving a special factor ${(Z_2, {\mathcal Z}_2, \mu_{Z_2}, S)}$ which turns out to be an inverse limit of two-step nilsystems; this limit theorem can be extracted from the structural theory in this paper of Host and Kra combined with a limit formula for nilsystems obtained by Lesigne, but will not be reproduced here. The pattern continues to higher averages (and higher step nilsystems); this was first done explicitly by Ziegler, and can also in principle be extracted from the structural theory of Host-Kra combined with nilsystem equidistribution results of Leibman. These sorts of limit formulae can lead to various recurrence results refining Roth’s theorem in various ways; see this paper of Bergelson, Host, and Kra for some examples of this. The above discussion was concerned with ${{\bf Z}}$-systems, but one can adapt much of the theory to measure-preserving ${G}$-systems for other discrete countable abelian groups ${G}$, in which one now has a family ${(T_g)_{g \in G}}$ of shifts indexed by ${G}$ rather than a single shift, obeying the compatibility relation ${T_{g+h}=T_g T_h}$. The role of the intervals ${\{1,\ldots,N\}}$ in this more general setting is replaced by that of Folner sequences. For arbitrary countable abelian ${G}$, the theory for double averages (1) and triple limits (3) is essentially identical to the ${{\bf Z}}$-system case. But when one turns to quadruple and higher limits, the situation becomes more complicated (and, for arbitrary ${G}$, still not fully understood). However one model case which is now well understood is the finite field case when ${G = {\bf F}_p^\omega = \bigcup_{n=1}^\infty {\bf F}_p^n}$ is an infinite-dimensional vector space over a finite field ${{\bf F}_p}$ (with the finite subspaces ${{\bf F}_p^n}$ then being a good choice for the Folner sequence). Here, the analogue of the structural theory of Host and Kra was worked out by Vitaly, Tamar, and myself in these previous papers (treating the high characteristic and low characteristic cases respectively). In the finite field setting, it turns out that nilsystems no longer appear, and one only needs to deal with linear, quadratic, and higher order eigenfunctions (known collectively as phase polynomials). It is then natural to look for a limit formula that asserts, roughly speaking, that if ${x}$ is drawn at random from a ${{\bf F}_p^\omega}$-system and ${n}$ drawn randomly from a large subspace of ${{\bf F}_p^\omega}$, then the only constraints between ${x, T^n x, \ldots, T^{(p-1)n} x}$ are those that arise from phase polynomials. The main theorem of this paper is to establish this limit formula (which, again, is a little complicated to state explicitly and will not be done here). In particular, we establish for the first time that the limit actually exists (a result which, for ${{\bf Z}}$-systems, was one of the main results of this paper of Host and Kra). As a consequence, we can recover finite field analogues of most of the results of Bergelson-Host-Kra, though interestingly some of the counterexamples demonstrating sharpness of their results for ${{\bf Z}}$-systems (based on Behrend set constructions) do not seem to be present in the finite field setting (cf. this previous blog post on the cap set problem). In particular, we are able to largely settle the question of when one has a Khintchine-type theorem that asserts that for any measurable set ${A}$ in an ergodic ${{\bf F}_p^\omega}$-system and any ${\epsilon>0}$, one has $\displaystyle \mu( T_{c_1 n} A \cap \ldots \cap T_{c_k n} A ) > \mu(A)^k - \epsilon$ for a syndetic set of ${n}$, where ${c_1,\ldots,c_k \in {\bf F}_p}$ are distinct residue classes. It turns out that Khintchine-type theorems always hold for ${k=1,2,3}$ (and for ${k=1,2}$ ergodicity is not required), and for ${k=4}$ it holds whenever ${c_1,c_2,c_3,c_4}$ form a parallelogram, but not otherwise (though the counterexample here was such a painful computation that we ended up removing it from the paper, and may end up putting it online somewhere instead), and for larger ${k}$ we could show that the Khintchine property failed for generic choices of ${c_1,\ldots,c_k}$, though the problem of determining exactly the tuples for which the Khintchine property failed looked to be rather messy and we did not completely settle it. [This guest post is authored by Ingrid Daubechies, who is the current president of the International Mathematical Union, and (as she describes below) is heavily involved in planning for a next-generation digital mathematical library that can go beyond the current network of preprint servers (such as the arXiv), journal web pages, article databases (such as MathSciNet), individual author web pages, and general web search engines to create a more integrated and useful mathematical resource. I have lightly edited the post for this blog, mostly by adding additional hyperlinks. - T.] This guest blog entry concerns the many roles a World Digital Mathematical Library (WDML) could play for the mathematical community worldwide. We seek input to help sketch how a WDML could be so much more than just a huge collection of digitally available mathematical documents. If this is of interest to you, please read on! The “we” seeking input are the Committee on Electronic Information and Communication (CEIC) of the International Mathematical Union (IMU), and a special committee of the US National Research Council (NRC), charged by the Sloan Foundation to look into this matter. In the US, mathematicians may know the Sloan Foundation best for the prestigious early-career fellowships it awards annually, but the foundation plays a prominent role in other disciplines as well. For instance, the Sloan Digital Sky Survey (SDSS) has had a profound impact on astronomy, serving researchers in many more ways than even its ambitious original setup foresaw. The report being commissioned by the Sloan Foundation from the NRC study group could possibly be the basis for an equally ambitious program funded by the Sloan Foundation for a WDML with the potential to change the practice of mathematical research as profoundly as the SDSS did in astronomy. But to get there, we must formulate a vision that, like the original SDSS proposal, imagines at least some of those impacts. The members of the NRC committee are extremely knowledgeable, and have been picked judiciously so as to span collectively a wide range of expertise and connections. As president of the IMU, I was asked to co-chair this committee, together with Clifford Lynch, of the Coalition for Networked InformationPeter Olver, chair of the IMU’s CEIC, is also a member of the committee. But each of us is at least a quarter century older than the originators of MathOverflow or the ArXiv when they started. We need you, internet-savvy, imaginative, social-networking, young mathematicians to help us formulate the vision that may inspire the creation of a truly revolutionary WDML! Some history first. Several years ago, an international initiative was started to create a World Digital Mathematical Library. The website for this library, hosted by the IMU, is now mostly a “ghost” website — nothing has been posted there for the last seven years. [It does provide useful links, however, to many sites that continue to be updated, such as the European Mathematical Information Service, which in turn links to many interesting journals, books and other websites featuring electronically available mathematical publications. So it is still worth exploring ...] Many of the efforts towards building (parts of) the WDML as originally envisaged have had to grapple with business interests, copyright agreements, search obstructions, metadata secrecy, … and many an enterprising, idealistic effort has been slowly ground down by this. We are still dealing with these frustrations — as witnessed by, e.g., the CostofKnowledge initiative. They are real, important issues, and will need to be addressed. Suppose that ${G = (G,\cdot)}$ is a finite group of even order, thus ${\|G\|}$ is a multiple of two. By Cauchy’s theorem, this implies that ${G}$ contains an involution: an element ${g}$ in ${G}$ of order two. (Indeed, if no such involution existed, then ${G}$ would be partitioned into doubletons ${\{g,g^{-1}\}}$ together with the identity, so that ${\|G\|}$ would be odd, a contradiction.) Of course, groups of odd order have no involutions ${g}$, thanks to Lagrange’s theorem (since ${G}$ cannot split into doubletons ${\{ h, hg \}}$). The classical Brauer-Fowler theorem asserts that if a group ${G}$ has many involutions, then it must have a large non-trivial subgroup: Theorem 1 (Brauer-Fowler theorem) Let ${G}$ be a finite group with at least ${\|G\|/n}$ involutions for some ${n > 1}$. Then ${G}$ contains a proper subgroup ${H}$ of index at most ${n^2}$. This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen ${n^2}$ slightly to ${n(n+2)/2}$) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if ${g}$ is a an involution of a group ${G}$, and the centraliser ${C_G(g) := \{ h \in G: gh = hg\}}$ has order ${n}$, then clearly ${n \geq 2}$ (as ${C_G(g)}$ contains ${1}$ and ${g}$) and the conjugacy class ${\{ aga^{-1}: a \in G \}}$ has order ${\|G\|/n}$ (since the map ${a \mapsto aga^{-1}}$ has preimages that are cosets of ${C_G(g)}$). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem ${G}$ contains a subgroup of order at least ${\max( n, \|G\|/n^2)}$. In particular, we can conclude that every group ${G}$ of even order contains a proper subgroup of order at least ${\|G\|^{1/3}}$. Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions: Corollary 2 (Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ has order at most ${(n^2)!}$. Indeed, by the previous discussion ${G}$ has a proper subgroup ${H}$ of index less than ${n^2}$, which then gives a non-trivial permutation action of ${G}$ on the coset space ${G/H}$. The kernel of this action is a proper normal subgroup of ${G}$ and is thus trivial, so the action is faithful, and the claim follows. If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample ${G}$ of minimal order ${\|G\|}$ to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution ${g}$. Take such an involution and consider its centraliser ${C_G(g)}$; this is a proper subgroup of ${G}$ of some order ${n < \|G\|}$. As ${G}$ is a minimal counterexample to the classification, one can in principle describe ${C_G(g)}$ in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of ${C_G(g)}$, that all the possible simple groups ${G}$ that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for ${C_G(g)}$, there are only finitely many ${G}$ that could generate this class for one of its centralisers, so this task should be doable in principle for any given isomorphism class for ${C_G(g)}$. That’s all one needs to do to prove the classification of finite simple groups! Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument is based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a ${p}$-group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of ${p}$-groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these ${p}$-groups in the key case when ${G}$ is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic ${p}$. See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy. The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of quasirandomness for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2: Corollary 3 (Weak Brauer-Fowler theorem) Let ${G}$ be a finite simple group with an involution ${g}$, and suppose that ${C_G(g)}$ has order ${n}$. Then ${G}$ can be identified with a subgroup of the unitary group ${U_{4n^3}({\bf C})}$. One can get an upper bound on ${\|G\|}$ from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!). Proof: Let ${A}$ be the set of all involutions in ${G}$, then as discussed above ${\|A\| \geq \|G\|/n}$. We may assume that ${G}$ has no non-trivial unitary representation of dimension less than ${4n^3}$ (since such representations are automatically faithful by the simplicity of ${G}$); thus, in the language of quasirandomness, ${G}$ is ${4n^3}$-quasirandom, and is also non-abelian. We have the basic convolution estimate $\displaystyle \\|1_A * 1_A * 1_A - \frac{\|A\|^3}{\|G\|} \\|_{\ell^\infty(G)} \leq (4n^3)^{-1/2} \|G\|^{1/2} \|A\|^{3/2}$ (see Exercise 10 from this previous blog post). In particular, $\displaystyle 1_A * 1_A * 1_A(0) \geq \frac{\|A\|^3}{\|G\|} - (4n^3)^{-1/2} \|G\|^{1/2} \|A\|^{3/2} \geq \frac{1}{2n^3} \|G\|^2$ and so there are at least ${\|G\|^2/2n^3}$ pairs ${(g,h) \in A \times A}$ such that ${gh \in A^{-1} = A}$, i.e. involutions ${g,h}$ whose product is also an involution. But any such involutions necessarily commute, since $\displaystyle g (gh) h = g^2 h^2 = 1 = (gh)^2 = g (hg) h.$ Thus there are at least ${\|G\|^2/2n^3}$ pairs ${(g,h) \in G \times G}$ of non-identity elements that commute, so by the pigeonhole principle there is a non-identity ${g \in G}$ whose centraliser ${C_G(g)}$ has order at least ${\|G\|/2n^3}$. This centraliser cannot be all of ${G}$ since this would make ${g}$ central which contradicts the non-abelian simple nature of ${G}$. But then the quasiregular representation of ${G}$ on ${G/C_G(g)}$ has dimension at most ${2n^3}$, contradicting the quasirandomness. $\Box$	{"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": 289, "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": 0.9194364547729492, "perplexity": 238.77155893792826}, "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/1398223206770.7/warc/CC-MAIN-20140423032006-00481-ip-10-147-4-33.ec2.internal.warc.gz"}
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=mm&paperid=3201&option_lang=eng	RUS ENG JOURNALS PEOPLE ORGANISATIONS CONFERENCES SEMINARS VIDEO LIBRARY PERSONAL OFFICE General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS Matem. Mod.: Year: Volume: Issue: Page: Find Matem. Mod., 2012, Volume 24, Number 1, Pages 109–128 (Mi mm3201) Numerical analysis of new model of metals cristallization processes, two-dimensional case V. T. Zhukov, N. A. Zaitsev, V. G. Lysov, Yu. G. Rykov, O. B. Feodoritova M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow Abstract: The paper is devoted to the numerical simulation of new model of metals crystallization processes. The difficulties of such simulation arise from the multiscale phenomenology of crystallization process. Nowadays the experimental researches establish the multiplicity of the details of crystallization process, however the general theoretical view of this process does not yet exist. The model which is used in the present paper is based on the description of a space occupied by the crystallizing alloy as the porous medium. The propagation of perturbations in such a medium is described by the equations of Biot’s type. The emergence of germs is described by modified Kahn-Hilliard equation. Previously the 1D numerical scheme has been constructed and its convergence property has been demonstrated. It is shown the possibility to model different crystallization regimes when changing the parameters of the model. In the presented paper 2D numerical model and results of computations are given. Due to multiscale phenomena the calculations require significant CPU time and hence the 2D model is based on explicit and explicit iteration algorithms which are implemented efficiently on multiprocessor computers. Keywords: numerical simulation, metals crystallization, grid discretization, explicit iterations, time integration. Full text: PDF file (776 kB) References: PDF file HTML file English version: Mathematical Models and Computer Simulations, 2012, 4:4, 440–453 Bibliographic databases: Document Type: Article UDC: 519.6 Citation: V. T. Zhukov, N. A. Zaitsev, V. G. Lysov, Yu. G. Rykov, O. B. Feodoritova, “Numerical analysis of new model of metals cristallization processes, two-dimensional case”, Matem. Mod., 24:1 (2012), 109–128; Math. Models Comput. Simul., 4:4 (2012), 440–453 Citation in format AMSBIB \Bibitem{ZhuZaiLys12} \by V.~T.~Zhukov, N.~A.~Zaitsev, V.~G.~Lysov, Yu.~G.~Rykov, O.~B.~Feodoritova \paper Numerical analysis of new model of metals cristallization processes, two-dimensional case \jour Matem. Mod. \yr 2012 \vol 24 \issue 1 \pages 109--128 \mathnet{http://mi.mathnet.ru/mm3201} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2978093} \transl \jour Math. Models Comput. Simul. \yr 2012 \vol 4 \issue 4 \pages 440--453 \crossref{https://doi.org/10.1134/S2070048212040096} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84928997620} • http://mi.mathnet.ru/eng/mm3201 • http://mi.mathnet.ru/eng/mm/v24/i1/p109 SHARE: Citing articles on Google Scholar: Russian citations, English citations Related articles on Google Scholar: Russian articles, English articles This publication is cited in the following articles: 1. E. A. Lukashev, E. V. Radkevich, N. N. Yakovlev, O. A. Vasileva, “Vvedenie v obobschennuyu teoriyu neravnovesnykh fazovykh perekhodov Kana—Khillarda (termodinamicheskii analiz zadach mekhaniki sploshnoi sredy)”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 21:3 (2017), 437–472 • Number of views: This page: 413 Full text: 115 References: 37 First page: 21	{"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": 0.22662420570850372, "perplexity": 9916.055365598448}, "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-2018-13/segments/1521257647681.81/warc/CC-MAIN-20180321180325-20180321200325-00734.warc.gz"}
http://nrich.maths.org/8095/solution	### Consecutive Numbers An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore. ### Tea Cups Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour. ### Exploring Wild & Wonderful Number Patterns EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules. # Round a Hexagon ##### Stage: 2 Challenge Level: This activity proved to be a little tricky for many pupils so maybe some more thoughts need to be given about external and internal angles. What also made it harder was asking for some suggestions about proving the solution offered. However, from Rebecca at the Baden Powell School in the UK we had this really good solution sent in. If she went round every angle once when she went round the last angle she would be facing the way she started. She could also do this with a square or rectangle. $90+90+90+90=360$ It also works with any triangle, because she always faces the same way when she gets back to the start. All shapes outer angles will add up to $360°$ because Brenda always faces the same way when she gets back to the start. From Sooyoung  who said they went to GCMS  which I assume is Gosforth Central Middle School in the U.K.  we had the following interesting suggestion: If you shrank the hexagon as small as a dot, the exterior angle would look like a rough circle which has 360 degrees altogether. Sophia  at Towcester C E Primary School send in this thought of hers: I think the answer is 360 degrees becuse the dot goes the whole way round and ended up facing the same way as it started. I added up $90, 90, 90$ and $90$ and it added up to $360$ degrees for a square a hexagon and a triangle. So, thank you for those and I hope that there wil be even more learning about angles and shapes for us all.	{"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": 0.21064649522304535, "perplexity": 1252.5131128599367}, "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-2014-41/segments/1410657137108.99/warc/CC-MAIN-20140914011217-00305-ip-10-234-18-248.ec2.internal.warc.gz"}
https://www.gradesaver.com/textbooks/math/calculus/calculus-8th-edition/chapter-1-functions-and-limits-1-1-four-ways-to-represent-a-function-1-1-exercises-page-21/31	Calculus 8th Edition { x \| x $\ne$ ± 3} Since division by 0 is not allowed, we must have: x² - 9 $\ne$ 0 x² $\ne$ 9 x $\ne$ ± $\sqrt 9$ x $\ne$ ± 3	{"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": 0.8574071526527405, "perplexity": 1657.4766404563127}, "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-2018-43/segments/1539583509196.33/warc/CC-MAIN-20181015121848-20181015143348-00537.warc.gz"}
http://mathhelpforum.com/trigonometry/33491-help-trig-identities.html	# Math Help - Help with trig identities 1. ## Help with trig identities verify the identity: tanxcos(squared)x=[(2tanxcos(squared)x-tanx)/(1-tan(squared)x] $\frac{sin(x)}{cos(x)}\cdot[cos(x)^2]=\frac{\frac{2sin(x)}{cos(x)^3}-\frac{sin(x)}{cos(x)}}{sec(x)^2-2}$..can you see whta to do from there 3. Originally Posted by kelsey3 verify the identity: tanxcos(squared)x=[(2tanxcos(squared)x-tanx)/(1-tan(squared)x] with identities, always start with the more complicated side (the right side in this case). change everything to sine and cosines and then simplify recommendations here: factor out the tangent first. remember your identities for cos(2x) 4. Originally Posted by Mathstud28 $\frac{sin(x)}{cos(x)}\cdot[cos(x)^2]=\frac{\frac{2sin(x)}{cos(x)^3}-\frac{sin(x)}{cos(x)}}{sec(x)^2-2}$..can you see whta to do from there it's actually not that bad a problem if you factor out tangent first... 5. ## Haha I think your secondary purpose on here is to help people with math...I think your primary purpose is to point out a highschooler(me)'s flaws..haha just kidding 6. Originally Posted by Mathstud28 I think your secondary purpose on here is to help people with math...I think your primary purpose is to point out a highschooler(me)'s flaws..haha just kidding hehe, i was thinking the same thing, haha. it seems i have been picking on you a lot tonight, huh? don't worry about it, when i just started here, the same thing happened to me. just look through some of my old posts, i was constantly making a fool of myself (i still do sometimes). it's tough love. i have nothing against you. trust me, it will help you in the long run. if you feel like someone's always there to critique you, you'll start to think things through more carefully. Power to you! 7. ya no kiddin....i dont really get how you got the sec(squared)x-2 on the bottom and where the cos(squared)x went. I don't know what identity you used. 8. ## Here you go $sec(x)^2=tan(x)^2+1$ and I made a little error...dont worry I know what I am doing the cos(x)^2 went to the bottom 9. man im retarded......i got (sinx/cosx)/(1/cos(squared)x-2) and then i brought the bottom up and crossed out a cosine but its still not right 10. ## I am sorry what exactly does that mean..show me your steps and I will critque you...use LaTeX if you can 11. how do i use that?....im new at this... 12. i just don't know how to get rid of the 2 on the bottom 13. ## To write it do this the most important thing you will need to know is the fraction $\frac{cos(x)}{sin(x)}=cot(x)$ is made by[ math]\frac{cos(x)}{sin(x)}=cot(x)[/math but i took off the ] after /math...look up LaTeX if you plan on using this site often it is a neccessity to learn 14. Dude.....that looks way too complicated. I just dont know how to get rid of the 2 on the bottom....i get the rest 15. Originally Posted by kelsey3 how do i use that?....im new at this... see the tutorial here for how to use LaTeX i will start you off. Consider the RHS: $\displaystyle \frac {2 \tan x \cos^2 x - \tan x}{1 - \tan^2 x} = \tan x \cdot \frac {2 \cos^2 x - 1}{1 - \frac {\sin^2 x}{\cos^2 x}}$ $\displaystyle = \tan x \cdot \frac {2 \cos^2 x - 1}{\frac {\cos^2 x - \sin^2 x}{\cos^2 x}}$ $= \frac {\sin x}{\cos x} \cdot {\color{red}2 \cos^2 x - 1} \cdot \frac {\cos^2 x}{{\color{red}\cos^2 x - \sin^2 x}}$ now what? (of course, i put the two things in red for a reason) Page 1 of 2 12 Last	{"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": 7, "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": 0.6129738688468933, "perplexity": 1801.0798347020582}, "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/1448398450559.94/warc/CC-MAIN-20151124205410-00182-ip-10-71-132-137.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/160905/understanding-torsion-from-a-presentation	# Understanding torsion from a presentation Let $F_2 = \langle a,b \rangle$ be the free group on two generators, and for each word $w \in F_2$, let $G(w) = \langle a, b \ \| \ w \rangle$. Is the following statement true? $G(w)$ is torsion free if and only if for all $k \geq 2$ and for all $v \in F_2$, $w \neq v^k$ In other words, is it true that $G(w)$ is torsion free unless there is an obvious reason why it is not? -	{"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": 0.8657646775245667, "perplexity": 46.62936732181889}, "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/1461863599979.27/warc/CC-MAIN-20160428171319-00221-ip-10-239-7-51.ec2.internal.warc.gz"}
http://mathhelpforum.com/discrete-math/153231-functions-proof.html	1. ## A functions proof Hi, I have a question I am having trouble starting: Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A). I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated. 2. Originally Posted by nzmathman Hi, I have a question I am having trouble starting: Let A be a nonempty set. Prove there is no function A onto P(A) (power set of A). I am not sure if it just means a general function f:A -> P(A) or if we are talking about a function that is onto. Any help will be greatly appreciated. Yes, the question is asking about a surjection. Otherwise, you could take the function to be $x\mapsto \{x\}$. To prove this result, you can use arguments based on the cardinality of $\mathcal{P}(A)$. Now, do you know about the rather handy relationship between $\|\mathcal{P}(A)\|$ and $\|A\|$ which states that $\|A\| < \|\mathcal{P}(A)\|$? 3. It was just the wording I was slightly confused by, but your example x->{x} made it very clear what the question wanted me to do, and I think I have proved it o.k. now. Thanks! 4. You may want to refer to this thread - I had the same question sometime back http://www.mathhelpforum.com/math-he...et-113875.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": 5, "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": 0.7454624176025391, "perplexity": 206.18968719655518}, "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-2017-26/segments/1498128323711.85/warc/CC-MAIN-20170628153051-20170628173051-00075.warc.gz"}
http://en.wikipedia.org/wiki/Darcy's_law	Darcy's law Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments[1] on the flow of water through beds of sand. It also forms the scientific basis of fluid permeability used in the earth sciences, particularly in hydrogeology. Background Although Darcy's law (an expression of conservation of momentum) was determined experimentally by Darcy, it has since been derived from the Navier-Stokes equations via homogenization. It is analogous to Fourier's law in the field of heat conduction, Ohm's law in the field of electrical networks, or Fick's law in diffusion theory. One application of Darcy's law is to water flow through an aquifer; Darcy's law along with the equation of conservation of mass are equivalent to the groundwater flow equation, one of the basic relationships of hydrogeology. Darcy's law is also used to describe oil, water, and gas flows through petroleum reservoirs. Description Diagram showing definitions and directions for Darcy's law. Darcy's law is a simple proportional relationship between the instantaneous discharge rate through a porous medium, the viscosity of the fluid and the pressure drop over a given distance. $Q=\frac{-kA}{\mu} \frac{(P_b - P_a)}{L}$' The total discharge, Q (units of volume per time, e.g., m3/s) is equal to the product of the intrinsic permeability of the medium, k (m2), the cross-sectional area to flow, A (units of area, e.g., m2), and the pressure drop (Pb - Pa), (Pascals), all divided by the viscosity, μ (Pa·s) and the length over which the pressure drop is taking place (m). The negative sign is needed because fluid flows from high pressure to low pressure. If the change in pressure is negative (where Pa > Pb), then the flow will be in the positive 'x' direction. Dividing both sides of the equation by the area and using more general notation leads $q=\frac{-k}{\mu} \nabla P$ where q is the flux (discharge per unit area, with units of length per time, m/s) and $\nabla P$ is the pressure gradient vector (Pa/m). This value of flux, often referred to as the Darcy flux, is not the velocity which the fluid traveling through the pores is experiencing. The fluid velocity (v) is related to the Darcy flux (q) by the porosity (n). The flux is divided by porosity to account for the fact that only a fraction of the total formation volume is available for flow. The fluid velocity would be the velocity a conservative tracer would experience if carried by the fluid through the formation. $v=\frac{q}{n}$ Darcy's law is a simple mathematical statement which neatly summarizes several familiar properties that groundwater flowing in aquifers exhibits, including: • if there is no pressure gradient over a distance, no flow occurs (these are hydrostatic conditions), • if there is a pressure gradient, flow will occur from high pressure towards low pressure (opposite the direction of increasing gradient - hence the negative sign in Darcy's law), • the greater the pressure gradient (through the same formation material), the greater the discharge rate, and • the discharge rate of fluid will often be different — through different formation materials (or even through the same material, in a different direction) — even if the same pressure gradient exists in both cases. A graphical illustration of the use of the steady-state groundwater flow equation (based on Darcy's law and the conservation of mass) is in the construction of flownets, to quantify the amount of groundwater flowing under a dam. Darcy's law is only valid for slow, viscous flow; fortunately, most groundwater flow cases fall in this category. Typically any flow with a Reynolds number less than one is clearly laminar, and it would be valid to apply Darcy's law. Experimental tests have shown that flow regimes with Reynolds numbers up to 10 may still be Darcian, as in the case of groundwater flow. The Reynolds number (a dimensionless parameter) for porous media flow is typically expressed as $Re = \frac{\rho v d_{30}}{\mu}$. where ρ is the density of water (units of mass per volume), v is the specific discharge (not the pore velocity — with units of length per time), d30 is a representative grain diameter for the porous media (often taken as the 30% passing size from a grain size analysis using sieves - with units of length), and μ is the viscosity of the fluid. Derivation For stationary, creeping, incompressible flow, i.e. $D\left(\rho u_i\right)/Dt\approx0$, the Navier-Stokes equation simplify to the Stokes equation: $\mu\nabla^2 u_i +\rho g_i -\partial_i P=0$, where $\mu$ is the viscosity, $u_i$ is the velocity in the i direction, $g_i$ is the gravity component in the i direction and P is the pressure. Assuming the viscous resisting force is linear with the velocity we may write: $-\left(k_{ij}\right)^{-1}\mu\phi u_j+\rho g_i-\partial_i P=0$, where $\phi$ is the porosity, and $k_{ij}$ is the second order permeability tensor. This gives the velocity in the $n$ direction, $k_{ni}\left(k_{ij}\right)^{-1} u_j= \delta_{nj} u_j = u_n = -\frac{k_{ni}}{\phi\mu}\left(\partial_i P-\rho g_i\right)$, which gives Darcy's law for the volumetric flux density in the $n$ direction, $q_n=-\frac{k_{ni}}{\mu}\left(\partial_i P -\rho g_i\right)$. In isotropic porous media the off-diagonal elements in the permeability tensor are zero, $k_{ij}=0$ for $i\neq j$ and the diagonal elements are identical, $k=k_{ii}$, and the common form is obtained $\boldsymbol{q}=-\frac{k}{\mu}\left(\boldsymbol{\nabla} P -\rho \boldsymbol{g}\right)$. Additional forms of Darcy's law For very short time scales, a time derivative of flux may be added to Darcy's law, which results in valid solutions at very small times (in heat transfer, this is called the modified form of Fourier's law), $\tau \frac{\partial q}{\partial t}+q=-K \nabla h$, where τ is a very small time constant which causes this equation to reduce to the normal form of Darcy's law at "normal" times (> nanoseconds). The main reason for doing this is that the regular groundwater flow equation (diffusion equation) leads to singularities at constant head boundaries at very small times. This form is more mathematically rigorous, but leads to a hyperbolic groundwater flow equation, which is more difficult to solve and is only useful at very small times, typically out of the realm of practical use. Another extension to the traditional form of Darcy's law is the Brinkman term, which is used to account for transitional flow between boundaries (introduced by Brinkman in 1949 [2]), $\beta \nabla^{2}q +q =-K \nabla P$, where β is an effective viscosity term. This correction term accounts for flow through medium where the grains of the media are porous themselves, but is difficult to use, and is typically neglected. Another derivation of Darcy's law is used extensively in petroleum engineering to determine the flow through permeable media - the most simple of which is for a one dimensional, homogeneous rock formation with a fluid of constant viscosity. $Q= \frac {k A}{\mu} \left( \frac{\partial P}{\partial L} \right)$, where Q is the flowrate of the formation (in units of volume per unit time), k is the relative permeability of the formation (typically in millidarcies), A is the cross-sectional area of the formation, μ is the viscosity of the fluid (typically in units of centipoise, and L is the length of the porous media the fluid will flow through. $\partial P/ \partial L$ represents the pressure change per unit length of the formation. This equation can also be solved for permeability, allowing for relative permeability to be calculated by forcing a fluid of known viscosity through a core of a known length and area, and measuring the pressure drop across the length of the core. For very high velocities in porous media, inertial effects can also become significant. Sometimes an inertial term is added to the Darcy's equation, known as Forchheimer term. This term is able to account for the non-linear behavior of the pressure difference vs velocity data.[3] $\nabla P=-\frac{\mu}{k}q-\frac{\rho}{k_1}q^2$, where the additional term $k_1$ is known as inertial permeability. Darcy's law is valid only for flow in continuum region. For a flow in transition region, where both viscous and Knudsen friction are present a new formulation is used, which is known as binary friction model [4] $\nabla P=-\left(\frac{k}{\mu}+D_K\right)^{-1}q$, where $D_K$ is the Knudsen diffusivity of the fluid in porous media.	{"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": 29, "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": 0.899308443069458, "perplexity": 521.3235161857035}, "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-2013-48/segments/1386164567400/warc/CC-MAIN-20131204134247-00041-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/dtft-question.196567/	# DTFT question 1. Nov 7, 2007 ### l46kok 1. The problem statement, all variables and given/known data Compute the DTFT of the following signal. $$x[n] = (0.8)^n u[n]$$ 2. Relevant equations Properties of DTFT 3. The attempt at a solution Well, my professor tells me to use the properties of DTFT to solve this. I'd love to - except I don't know what the DTFT of $$(0.8)^n$$ is. I've tried looking it up on the DTFT table, but couldn't find any, can someone tell me what it is? 2. Nov 7, 2007 ### learningphysics It's in the DTFT pairs here: http://www.neng.usu.edu/classes/ece/5630/notes_transforms.pdf [Broken] on page 20. but maybe the prof wants you to derive it from the properties you already have in your text? Last edited by a moderator: May 3, 2017 Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Similar Discussions: DTFT question	{"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": 0.6226995587348938, "perplexity": 1812.4948719817487}, "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/1502886117519.92/warc/CC-MAIN-20170823035753-20170823055753-00238.warc.gz"}
http://eaglesnestministries.org/prayer-blessing-upon-california-leaders/	Open/Close Menu The Official Ministry Website Of Apostle Gary L. Greenwald LORD, WE LIFT UP THESE LEADERS TO YOU EVERY ELECTED AND EVERY APPOINTED POSITION OVER THE STATE OF CALIFORNIA EVERY TEACHER, EVERY UNIVERSITY PRESIDENT AND ALL SCHOOL FACULTY IN CALIFORNIA ALL THE MILITARY FORCES ALL THE POLICE AND OTHER GOVERNMENTAL AGENCIES INCLUDING ICE EVERY GOD ORDAINED APOSTLE, PROPHET, EVANGELIST, TEACHER, PASTOR IN CALIFORNIA CHARITABLE, NON-PROFIT ORGANIZATIONS, DISASTER RELIEF EVERY HEAD OF THE HOME OF FAMILIES IN CALIFORNIA LORD WE DECREE YOUR BLESSING UPON ALL THE AUTHORITIES IN CALIFORNIA. WE PRAY THAT EACH LEADER WILL HAVE SOUND COUNSEL AND GUIDANCE WE PRAY LORD THAT YOU WOULD REVEAL THE ANSWERS FOR WHICH THEY ARE SEEKING ABOUT THE PROBLEMS THEY ARE FACING WE PRAY THAT YOU WOULD EXPOSE & REMOVE EVERY PERSON AROUND EACH LEADER THAT IS PURPOSELY WORKING IN WITCHCRAFT, MIND CONTROL AND/OR SEDITION REVEAL TO EACH LEADER THE “JOSEPH”, THE “DANIEL”, THE “ESTHER”, THE “MORDECAI” THAT YOU HAVE PUT IN THEIR MIDST TO HELP THEM. RAISE UP YOUR GODLY ANOINTED ONES AND EMPOWER THEM WITH THE STRATEGIES AND WISDOM FROM HEAVEN TO ASSIST THESE LEADERS WE COVER EVERY LEADER WITH THE BLOOD OF JESUS AND WE BIND EVERY WORK OF DARKNESS MANIPULATING AND CONTROLLING THEM BEHIND THE SCENES. WE BIND THE SPIRIT OF FEAR THAT TARGETS LEADERS FROM TAKING APPROPRIATE ACTION. WE BREAK OFF ALL WITCHCRAFT TARGETED AGAINST THEM THROUGH THE BLOOD AND NAME OF JESUS. WE COMMAND EVERY LEADER TO BE LOOSED FROM DEMONIC STRONGHOLDS AND OPPRESSION. LORD, WE ASK THAT EACH LEADER HEAR YOUR VOICE. WE PRAY THAT THE SEVENFOLD MINISTRY OF THE SPIRIT BE UPON THEM: THE SPIRIT OF THE LORD, THE SPIRIT OF WISDOM, OF UNDERSTANDING, OF COUNSEL, OF MIGHT, OF KNOWLEDGE AND OF FEAR OF THE LORD (ISAIAH 11:2-3). CALIFORNIA STATE GOVERNMENT: CALIFORNIA STATE GOVERNOR: GAVIN NEWSON Current members As of 3 January 2019, California is represented by the following elected officials:[1] · California 1Doug LaMalfa (R), since 2013 · California 2Jared Huffman (D), since 2013 · California 3John Garamendi (D), since 2009 · California 4Tom McClintock (R), since 2009 · California 5Mike Thompson (D), since 1999 · California 6Doris Matsui (D), since 2005 · California 7Ami Bera (D), since 2013 · California 8Paul Cook (R), since 2013 · California 9Jerry McNerney (D), since 2007 · California 10Josh Harder (D), since 2019 · California 11Mark DeSaulnier (D), since 2015 · California 12Nancy Pelosi (D), since 1987 · California 13Barbara Lee (D), since 1998 · California 14Jackie Speier (D), since 2008 · California 15Eric Swalwell (D), since 2013 · California 16Jim Costa (D), since 2005 · California 17Ro Khanna (D), since 2017 · California 18Anna Eshoo (D), since 1993 · California 19Zoe Lofgren (D), since 1995 · California 20Jimmy Panetta (D), since 2017 · California 21T. J. Cox (D), since 2019 · California 22Devin Nunes (R), since 2003 · California 23Kevin McCarthy (R), since 2007 · California 24Salud Carbajal (D), since 2017 · California 25Katie Hill (D), since 2019 · California 26Julia Brownley (D), since 2013 · California 27Judy Chu (D), since 2009 · California 28Adam Schiff (D), since 1997 · California 29Tony Cárdenas (D), since 2013 · California 30Brad Sherman (D), since 1997 · California 31Pete Aguilar (D), since 2015 · California 32Grace Napolitano (D), since 1999 · California 33Ted Lieu (D), since 2015 · California 34Jimmy Gomez (D), since 2017 · California 35Norma Torres (D), since 2015 · California 36Raul Ruiz (D), since 2013 · California 37Karen Bass (D), since 2011 · California 38Linda Sánchez (D), since 2003 · California 39Gil Cisneros (D), since 2019 · California 40Lucille Roybal-Allard (D), since 1993 · California 41Mark Takano (D), since 2013 · California 42Ken Calvert (R), since 1993 · California 43Maxine Waters (D), since 1991 · California 44Nanette Barragan (D), since 2017 · California 45Katie Porter (D), since 2019 · California 46Lou Correa (D), since 2017 · California 47Alan Lowenthal (D), since 2013 · California 48Harley Rouda (D), since 2019 · California 49Mike Levin (D), since 2019 · California 50Duncan D. Hunter (R), since 2009 · California 51Juan Vargas (D), since 2013 · California 52Scott Peters (D), since 2013 · California 53Susan Davis (D), since 2001	{"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": 0.9624690413475037, "perplexity": 4054.438328644154}, "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-2020-05/segments/1579250593994.14/warc/CC-MAIN-20200118221909-20200119005909-00331.warc.gz"}
https://math.stackexchange.com/questions/3196980/is-filling-in-grid-corners-extrapolation-or-interpolation	# Is filling in grid corners extrapolation or interpolation? Let's say I measure temperature at locations on a regular 3x5 grid but I am missing data at the 4 grid corners. If I use the available measurements to estimate the grid corner values, am I performing extrapolation or interpolation? It seems like it's extrapolation because at an given corner, I have measurements at only 3 of the 8 neighboring points, and no 2 neighboring points which have measurements can be connected with a line that goes through the grid corner. So does that mean it's extrapolation? • That's what I would say. Go for it. – Ethan Bolker Apr 22 at 12:03	{"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": 0.8147258162498474, "perplexity": 612.7218249769585}, "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-2019-26/segments/1560627999263.6/warc/CC-MAIN-20190620165805-20190620191805-00460.warc.gz"}
https://stats.stackexchange.com/questions/299806/choose-probability-distribution-for-times-to-execute-a-sequence-of-moves-on-a-ru	# Choose probability distribution for times to execute a sequence of moves on a Rubik's Cube I am writing a software to help me train my speedcubing skills (specifically I want to execute 800 different sequences of moves fast) and there is one subproblem I am struggling with: What kind of probability distribution can I use to model my times to execute a given fixed sequence of moves? It doesn't have to be perfect, but I want at least some probability distribution that seems reasonable. Here are my observations: • There is kind of a lower bound. Even if it goes well and I make no mistakes, there is just a limit on how fast my fingers can turn. • Most of the times are kind of close together, but sometimes I screw up and I am much worse than most of my "normal" times. • Sometimes (rarely) I screw up very badly and really takes 5 times as long as usually. • Sometimes it goes really well, but then it is only slightly better than most of my "normal" times. Does any reasonable distribution come into your minds that I could use to model this? Clarification: The fact that it is a sequence of moves is irrelevant. I consider each sequence as one individual unit and don't care which moves it consists of. (The reason is that there are so many ways to execute a move with your fingers and it really depends which moves come before and after and it's too complicated to model that. So one sequence of moves is just one unit) There are 800 such sequences and I assume that they should have the same distribution, but with different parameters. Note: I posted this on MathOverflow recently, but people told me it was the wrong place, so I came here. • How about a Gamma distribution (wikipedia.com/en/Gamma_distribution)? If you assume the time to complete each move is iid Gamma, the time to complete $n$ moves is $\textrm{Gamma}(nk,\theta)$, where $k$ and $\theta$ are parameters determining the distribution of times to complete a single move (you might estimate these from some data you collect). – Will Aug 25 '17 at 18:16 • Gammas have lower bounds of zero, so they wouldn't be the first choice to model these times. Why not use the distribution of times you have already observed? Is there any need to fit some mathematical formula to it? – whuber Aug 25 '17 at 19:03 • You slightly misunderstood my question, I added some clarification. I don't really care about the individual moves, one sequence can be considered one unit. And I imagine that the times for each of the 800 sequences uses the same type of distribution, but with different parameters. But apart for that, the Gamma distribution looks great! I will take a look at it. – Lykos Aug 25 '17 at 19:04 • why don't you repeatedly time how long it takes you to do a particular sequence of moves ... then you can plot the empirical pdf of time taken, and see what the distribution looks like, and fit a suitable theoretical model – wolfies Aug 25 '17 at 19:13 • My goal is to do the following: My software gets the total time it took me to execute lots of pairs and triples of these sequences (but not the individual times) and I want to estimate the median for each individual sequence in order to make my software give me the slower ones more frequently. I intended to use a maximum likelihood estimator to estimate the parameters of the distribution given this input data and that would give me the medians. – Lykos Aug 25 '17 at 19:24	{"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": 0.8455431461334229, "perplexity": 291.50695606919555}, "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-20190620085246-20190620111246-00489.warc.gz"}
https://cs.stackexchange.com/questions/45736/prove-or-disprove-that-nl-is-closed-under-polynomial-many-one-reductions	# Prove or disprove that $NL$ is closed under polynomial many-one reductions If $B \in NL$ and there exists a Karp reduction (polynomial-time many-one reduction) from $A$ to $B$, then $A \in NL$. Prove that the above claim is correct, incorrect, or equivalent to an open question. Trying to solve this, I was pretty sure that the statement is either incorrect, or at least equivalent to an open question. My reasoning was that the statement is correct if the reduction is promised to be a log-space reduction (uses only $log \ n$ space). Removing this demand seems to be too much, but I wasn't able to prove this. I was then told that this statement is equivalent to the open question $NL = P$. I thought of this, and tried proving that if the statement is correct, then one can define a Karp reduction from any $L \in P$ to a language in $NL$, therefore proving $L \in NL$ and $P \subseteq NL$. However I couldn't find such a reduction. The only reduction I could think of that could help would be to transform the operations of a deterministic polynomial Turing machine $M_L$ to a configurations graph (a directed graph with an edge $(u,v)$ iff there exist two consecutive configurations $u, v$ in $M_L$. However, this reduction would take exponential $2^{p(n)}$ time, and not polynomial. Any contribution would be appreciated. ## 1 Answer Let's prove the two directions separately. Suppose first that $\mathsf{NL}=\mathsf{P}$. In order to prove the statement, suppose that $B \in \mathsf{NL}$ and that there is a polytime many-one reduction from $A$ to $B$. Since $\mathsf{P}$ is closed under polytime many-one reductions and $B \in \mathsf{NL} = \mathsf{P}$, clearly $A \in \mathsf{P} = \mathsf{NL}$, proving the statement. Suppose next that the statement is correct. Let $B$ be any non-trivial language in $\mathsf{NL}$ (for example, $B = \{\epsilon\}$), and let $A \in \mathsf{P}$ be arbitrary. Since $A \in \mathsf{P}$ there is a polytime function $f$ computing it. We can easily modify $f$ to a polytime reduction from $A$ to $B$ by fixing some YES instance and NO instance in $B$ (here we need that $B$ be non-trivial). The statement implies that $A \in \mathsf{NL}$. Since $A$ was arbitrary, we deduce that $\mathsf{P} \subseteq \mathsf{NL}$ and so $\mathsf{NL} = \mathsf{P}$. • Thanks Yuval. That second direction eluded me, since I was trying to find a "meaningful" reduction from any $L \in P$ to $B \in NL$, I didn't think of building a simple artificial reduction such as you described. – Cauthon Aug 31 '15 at 18:27	{"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": 0.9516319036483765, "perplexity": 137.49492603969355}, "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-2019-26/segments/1560627998600.48/warc/CC-MAIN-20190618003227-20190618025227-00453.warc.gz"}
http://clay6.com/qa/21216/the-energy-e-and-the-momentum-p-of-photon-are-given-by-e-hn-and-p-h-lambda-	Browse Questions # The energy $E$ and the momentum $p$ of photon are given by $E=hn\: and\: p=h/\lambda$ respectively. The velocity of photon will be $\begin {array} {1 1} (a)\;E/p & \quad (b)\;Ep \\ (c)\;(E/p)^2 & \quad (d)\;(E/p)^{\large\frac{1}{2}} \end {array}$ Ans : (b) E/p velocity, $v = n\lambda, \lambda=h/p$ $\Rightarrow v=E/p$ edited Mar 13, 2014	{"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": 0.9888466596603394, "perplexity": 2648.974718793787}, "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-2017-26/segments/1498128320257.16/warc/CC-MAIN-20170624101204-20170624121204-00164.warc.gz"}
http://mathhelpforum.com/discrete-math/90715-solved-find-number-questions-print.html	# [SOLVED] Find number of questions? • May 27th 2009, 09:48 AM fardeen_gen [SOLVED] Find number of questions? In a certain test there are $n$ questions. In the test, $4^{n - k}$ students gave wrong answers to at least $k$ questions, where $k = 1,2,\mbox{...},n$. If the total number of wrong answers is 341, then find the value of n. • Jun 4th 2009, 04:29 AM the_doc Solution Here's how to do it: Spoiler: The question states that the number of students with at least $k$ questions wrong is $4^{n-k}$. Hence the number that actually get exactly $k$ wrong, $N_k$ , must be the number that get at least $k$ wrong minus those that get at least $(k+1)$ wrong. So we have $N_k = 4^{n-k} - 4^{n-k-1}$ . So if we let $N_q$ be the total number of wrongly answered questions then $N_q = n + \sum_{k=1}^{n-1} k N_k$, $= n + \sum_{k=1}^{n-1} k (4^{n-k} - 4^{n-k-1})$, $= n + \sum_{k=1}^{n-1} k 4^{n-k} - \sum_{k=1}^{n-1} k 4^{n-k-1}$, $= n + \sum_{k=1}^{n-1} k 4^{n-k} - \sum_{k=1}^{n-1} (k+1) 4^{n-(k+1)} + \sum_{k=1}^{n-1} 4^{n-(k+1)}$ , $= n + \sum_{k=1}^{n-1} k 4^{n-k} - \sum_{m=2}^{n} m 4^{n-m} + \sum_{k=1}^{n-1} 4^{n-(k+1)}$ (letting $m= k+1$), $= n + \sum_{k=1}^{n-1} k 4^{n-k} - \sum_{m=2}^{n} m 4^{n-m} + \sum_{k=1}^{n-1} 4^{n-(k+1)}$, $= n + 4^{n-1} - n + \sum_{k=1}^{n-1} 4^{n-(k+1)}$, $= \sum_{k=0}^{n-1} 4^{n-(k+1)}$, $= \sum_{p=0}^{n-1} 4^{p}$ (letting $p = n-(k+1)$), $= \frac{4^{n}-1}{4-1} = \frac{1}{3}(4^n -1)$. (I know I could have explicitly written out the sum, 'spot the pattern' - as I did on paper, but decided this was typographically easier than using an array). So we have that $341 = \frac{1}{3}(4^n -1)$ and solving gives us that $\color[rgb]{0,0,1} \boxed{n = 5}$.	{"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": 26, "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": 0.9689204096794128, "perplexity": 601.1150205688277}, "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-2016-44/segments/1476988721405.66/warc/CC-MAIN-20161020183841-00103-ip-10-171-6-4.ec2.internal.warc.gz"}
https://sealing-induction.com/featured_item_tag/botle-seal-wads/	Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 1276 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 1313 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 1317 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 1345 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 3536 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 3543 Warning: "continue" targeting switch is equivalent to "break". Did you mean to use "continue 2"? in /home/daroupac/sealing-induction.com/wp-content/plugins/gravityforms/common.php on line 3556 botle seal wads - Sealing+ ## Bottle Seal Bottle Seal The typical induction bottle seal liners have a multi-laminate structure. It is often consists of a layer of pulpboard, a layer of wax, a layer of aluminum foil, and a layer of polymer. This kind is generally called 2 piece bottle seal liner. There is another type of bottle seal wads in which […]	{"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": 0.9427807331085205, "perplexity": 10487.776070421389}, "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-2021-04/segments/1610703514121.8/warc/CC-MAIN-20210118030549-20210118060549-00694.warc.gz"}
https://fractalforums.org/fractal-mathematics-and-new-theories/28/functional-graph-of-modular-arithmetic/4752/msg34226	• May 22, 2022, 10:04:34 AM ### Author Topic: functional graph of modular arithmetic (Read 183 times) 0 Members and 1 Guest are viewing this topic. #### kh40tika • Fractal Freshman • Posts: 7 ##### functional graph of modular arithmetic « on: May 10, 2022, 06:11:57 AM » Let p be a prime number of form 3n+1, let w be a 3rd root of unity of prime field F_p. Let x be an element of F_p, define: reduce(x) = min(x, xw, xw^2) Now define a pseudo random iterated map within F_p: f(x) = reduce(x-1) Note all the above operations are done in F_p. For example, let p = 19. Solutions of w^3 == 1 (mod 19) are 1, 7, 11. Thus w can be either 7 or 11. Since they are 3rd roots of unity, it's trivial 7^2 == 11 (mod 19) && 11^2 == 7 (mod 19). Selecting either of them gives the same f(x). Assume x = 3, then f(x) = min(x, mod(x7, 19), mod(x11, 19)) = min(3,2,14) = 2 It's trivial to see iterated map f(f(f(x)))... would always converge to 1, since each iteration would make x smaller. If we put all elements of F_p into a directed graph, where edges are defined by the map f(x), we always obtain a tree. Turns out the structure of this tree is somewhat non-random while being not so trivial. Here's some short Mathematica code to give a plot: Code: [Select] (generate random prime of form 3n+1)range = 10000;irule[i_Integer] := With[{ir = reduce[i]}, ir -> reduce[ir - 1]];graphList = {};gridShape = {5, 5};genPrime[lo_Integer, hi_Integer] := Do[With[{ret = RandomPrime[{lo, hi}]}, If[Mod[ret, 3] == 1, Return@ret]], {\[Infinity]}]Do[ p = genPrime[range, 2range]; pr = PrimitiveRoot[p]; w = PowerMod[pr, (p - 1)/3, p]; (3rd root) reduce[x_Integer] := Min[Mod[{x, xw, xww}, p]]; cbrtlist = Table[PowerMod[pr, i, p], {i, 0, (p - 1)/3 - 1}]; graph = irule /@ cbrtlist; AppendTo[graphList, graph], {Times @@ gridShape}](* render & save )img = ImageAssemble@ArrayReshape[Image[GraphPlot[#, PlotStyle -> {Opacity[0.3]}], ImageSize -> {256, 256}] & /@ graphList, gridShape] Running the above code would give the attached file intro.png If we modify f(x) into f(x) = reduce(x + 1), then the iterated map no longer converges to 1 and the resulting graph would have cycles. Attached file intro2.png shows the case. #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #1 on: May 10, 2022, 06:57:45 AM » More plots coming. Using f(x) = reduce(x + 128) gives a more segmented graph (intro3.png), while using f(x) = reduce(2x + 1) gives a graph closer to random functional graph (intro4.png). #### Alef • Fractal Freak • Posts: 772 • a catalisator / z=zsinh(z)-c^2 ##### Re: functional graph of modular arithmetic « Reply #2 on: May 10, 2022, 07:22:48 AM » Is this from a number theory, do lines are connecting numbers? It looks like hand drawings with a pencil on paper. « Last Edit: May 10, 2022, 08:18:30 AM by Alef » by Edgar Malinovsky aka Edgars Malinovskis. #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #3 on: May 10, 2022, 08:41:15 AM » Is this from a number theory, do lines are connecting numbers? It looks like hand drawings with a pencil on paper. The object of interest is called functional graph. https://math.stackexchange.com/questions/280717/whats-the-meaning-of-functional-graph The function f(x) over F_p defines the topology of such a graph. To obtain a nice looking plot, one would need a graph embedding algorithm to assign 2D or 3D coordinates to all graph vertices. #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #4 on: May 10, 2022, 02:19:12 PM » The iterative map over finite elements would always converge to either a loop or a fixed point. Using attractor time as shader input produces more interesting render. The graph embedding algorithm has also been changed for the new render. Formula is the one giving fixed point attractor: f(x) = reduce(x-1) • Fractal Frankfurter • Posts: 641 ##### Re: functional graph of modular arithmetic « Reply #5 on: May 10, 2022, 03:12:12 PM » #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #6 on: May 11, 2022, 12:22:17 PM » What about 3d fractals? Apart from finding a graph embedding in with 3d coordinates, one may also use high degree field extensions. By joining functional graph from multiple reduction directions, fractal surfaces may be obtained. Consider field extension of degree 2. Let p > 3 be a prime number, let q = p^2. Since p^2 == 1 (mod 3), F_q would always have 3rd roots (now without requirement of being 3n+1 form). Let (x,y) be an element of F_q. if we define two iterative maps: u(x,y) = reduce(x-1, y) v(x,y) = reduce(x, y-1) Merging functional graph of u(.) and v(.) gives new interesting manifold-like structure. Code: [Select] Needs["FiniteFields"];POWER = 3;(power degree for reduction during iteration)EXTPWR = \2;(field extension degree)p = RandomPrime[{150, 210}];q = p^EXTPWR;Fq = GF[p, EXTPWR];nVert = (q - 1)/POWER; nEdge = nVert - 1;pr = Fq[{0, 1}];w = Power[pr, (q - 1)/POWER];pwrlist = Table[Power[w, i], {i, 0, POWER - 1}];pwrlist[[1]] = Fq[{1, 0}];(fix integer 1)Clear[reduce, ifn, fieldElemSize];(this will allow some unique ording among elements of F_q)fieldElemSize[1] := 1;fieldElemSize[GF[p_, _][{i_Integer, j_Integer}]] := i + jp;reduce[x_] := MinimalBy[xpwrlist, fieldElemSize][[1]];ifn[i_] := reduce[i - 1];ifn2[i_] := reduce[i - Fq[{0, 1}]];xlist = Table[reduce[Power[pr, i]], {i, 0, nEdge}];ixmap = Association@Table[xlist[[i]] -> i, {i, Length@xlist}];ixmap[0] = 0;graph = {};Do[ If[i[[1, 1]] != 0, AppendTo[graph, ixmap[i] -> ixmap[ifn[i]]]]; If[i[[1, 2]] != 0, AppendTo[graph, ixmap[i] -> ixmap[ifn2[i]]]] , {i, xlist}];graphMap = Association@graph;graph = graph /. Rule -> UndirectedEdge;Rasterize[ GraphPlot3D[graph, GraphLayout -> {"SpringElectricalEmbedding", "MaxIteration" -> 100000, "Tolerance" -> 0.0001}, Background -> Black], ImageSize -> 1000] #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #7 on: May 11, 2022, 01:54:07 PM » More complicated formula and more 3d graph embeddings. Graph embedding operation has gone pretty slow. Might consider GPU accelerated sparse system solver. Code: [Select] {U, V} = RandomInteger[{1, 5}, {2}];reduce[x_] := MinimalBy[xpwrlist, fieldElemSize][[1]];ifn[i_] := reduce[i - U];ifn2[i_] := reduce[i - Fq[{0, V}]];ifn3[i_] := reduce[i - Fq[{U, V}]];sparseFactor = RandomInteger[{4, 24}];xlist = Table[reduce[Power[pr, i]], {i, 0, nEdge}];ixmap = Association@Table[xlist[[i]] -> i, {i, Length@xlist}];ixmap[0] = 0;graph = {};Do[With[{xy = i[[1]]}, AppendTo[graph, ixmap[i] -> ixmap[ifn[i]]]; If[Mod[xy[[1]] + xy[[2]], sparseFactor] == 0, AppendTo[graph, ixmap[i] -> ixmap[ifn2[i]]] ] ] , {i, xlist}];` #### kh40tika • Fractal Freshman • Posts: 7 ##### Re: functional graph of modular arithmetic « Reply #8 on: May 12, 2022, 04:13:28 AM » What about elliptic curves? Let E be an elliptic curve of isogeny y^2 = x^3 + k where k != 0. Then we search for a prime p of form 3n+1 such that E[F_p] has a prime order q also of form 3n+1. Such curves has a nice property called complex multiplication. Let w be 3rd root of unity (sqrt(3)I - 1)/2 Let (x,y) be a point on C, we have: w . (x,y) == (x / w, y) Now let w_p and w_q be w reduced on F_p and F_q respectively. Let (x,y) be a point on E[F_p], we have: w_q . (x,y) == (x / w_p, y) We can now do a 3rd root "reduction" similar to the finite field version. Combined with the EC point negation law -(x, y) == (x, -y), it's possible to do a 6th root "reduction". reduce_3(x,y) = (min(x, xw_p, xw_p^2), y) mod p reduce_6(x,y) = (min(x, xw_p, x*w_p^2), min(y, -y)) mod p Fact: bitcoin uses a curve with above properties. Let G be a generator of E[F_p]. Plotting the functional graph with f(x,y) = reduce(x,y) + G gives a graph with many components. Such pseudo-random iterated function over finite Abelian groups are exploited in Pollard's Rho algorithm for discrete logarithm. It's believed in a "random enough" iteration within a group of order N, a cycle an be found within approx 0.6267 sqrt(N) steps in average. Finding non-trivial functional graph within E[F_p] may give insight on better discrete logarithm algorithms. ### Similar Topics ###### Synthclipse presets import functional on 2.2.3 ? Started by Michael_Z_Freeman on Synthclipse 2 Replies 314 Views March 15, 2019, 10:28:12 AM by Michael_Z_Freeman ###### CPU: order of evaluation of an arithmetic expression Started by marcm200 on Programming 5 Replies 394 Views February 12, 2020, 08:35:38 AM by hobold ###### A fractal description for geometric shapes using graph transformations Started by iRyanBell on Fractal Mathematics And New Theories 6 Replies 432 Views July 25, 2020, 06:39:00 PM by C0ryMcG	{"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": 0.6992824673652649, "perplexity": 12352.80064523807}, "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-2022-21/segments/1652662545326.51/warc/CC-MAIN-20220522094818-20220522124818-00647.warc.gz"}
http://www.physicsforums.com/showthread.php?t=368355	# permutation and combination by kschong Tags: combination, permutation P: 5 For this question, i already solve the part a and part b. For the part c, i try to solve it but i cant get the answer that given. Can someone explain to me? How to do the part c. Thanks! Attached Thumbnails PF Patron HW Helper Sci Advisor Thanks P: 25,474 Hi kschong! So there's 24 tanks, of which 6 are high viscosity, add 4 are high impurity: what is the probability that 4 tanks chosen at random include exactly one of each? Show us your full calculations, and then we can see what went wrong, and we'll know how to help. Related Discussions Calculus & Beyond Homework 4 Precalculus Mathematics Homework 5 Calculus & Beyond Homework 5 Precalculus Mathematics Homework 8 Introductory Physics Homework 4	{"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": 0.948776364326477, "perplexity": 1347.7121975470186}, "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/1386163051984/warc/CC-MAIN-20131204131731-00022-ip-10-33-133-15.ec2.internal.warc.gz"}
http://zbmath.org/?q=an:1050.82029	# zbMATH — the first resource for mathematics ##### Examples Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev \| Tschebyscheff The or-operator \| allows to search for Chebyshev or Tschebyscheff. "Quasi* map" py: 1989 The resulting documents have publication year 1989. so: Eur J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A \| 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used. ##### Operators a & b logic and a \| b logic or !ab logic not abc right wildcard "ab c" phrase (ab c) parentheses ##### Fields any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article) Diffusion and memory effects for stochastic processes and fractional Langevin equations. (English) Zbl 1050.82029 Summary: We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker-Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the nonlinear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker-Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative [defined according to M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna (1969)]. In the generic linear or nonlinear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution. ##### MSC: 82C31 Stochastic methods in time-dependent statistical mechanics 82C70 Transport processes (time-dependent statistical mechanics) 60H15 Stochastic partial differential equations	{"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": 0.8734848499298096, "perplexity": 3288.3129630574495}, "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-2014-15/segments/1397609540626.47/warc/CC-MAIN-20140416005220-00347-ip-10-147-4-33.ec2.internal.warc.gz"}
https://astarmathsandphysics.com/university-maths-notes/metric-spaces/1942-proof-that-a-convergent-sequence-in-a-metric-space-has-a-unique-limit.html?tmpl=component&print=1&page=	## Proof That a Convergent Sequence in a Metric Space Has a Unique Limit Theorem Ifis a convergent sequence in a metric spacewiththenis unique. Proof Suppose conversely thatis a convergent sequence with two distinct limits, so that and Since From the triangle inequality Sinceconverges to x there existssuch that for alland sinceconverges tothere existssuch that for all Takethen for alland Hence This is an obvious contradiction so every convergence sequence has a unique limit.	{"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": 0.9921059608459473, "perplexity": 1783.8885598055242}, "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/1512948529738.38/warc/CC-MAIN-20171213162804-20171213182804-00162.warc.gz"}
https://nodus.ligo.caltech.edu:8081/40m/16348	40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop 40m Log Not logged in Mon Sep 20 15:42:44 2021, Ian MacMillan, Summary, Computers, Quantization Code Summary Wed Sep 22 14:22:35 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Mon Sep 27 12:12:15 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Mon Sep 27 16:03:15 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Mon Sep 27 17:04:43 2021, rana, Summary, Computers, Quantization Noise Calculation Summary Thu Sep 30 11:46:33 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Wed Nov 24 11:02:23 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Wed Nov 24 13:44:19 2021, rana, Summary, Computers, Quantization Noise Calculation Summary Tue Dec 7 10:55:25 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Fri Dec 10 13:02:47 2021, Ian MacMillan, Summary, Computers, Quantization Noise Calculation Summary Message ID: 16348 Entry time: Mon Sep 20 15:42:44 2021 Reply to this: 16355 Author: Ian MacMillan Type: Summary Category: Computers Subject: Quantization Code Summary This post serves as a summary and description of code to run to test the impacts of quantization noise on a state-space implementation of the suspension model. Purpose: We want to use a state-space model in our suspension plant code. Before we can do this we want to test to see if the state-space model is prone to problems with quantization noise. We will compare two models one for a standard direct-ii filter and one with a state-space model and then compare the noise from both. Signal Generation: First I built a basic signal generator that can produce a sine wave for a specified amount of time then can produce a zero signal for a specified amount of time. This will let the model ring up with the sine wave then decay away with the zero signal. This input signal is generated at a sample rate of 2^16 samples per second then stored in a numpy array. I later feed this back into both models and record their results. State-space Model: The code can be seen here The state-space model takes in the list of excitation values and feeds them through a loop that calculates the next value in the output. Given that the state-space model follows the form $\dot{x}(t)=\textbf{A}x(t)+ \textbf{B}u(t)$ and $y(t)=\textbf{C}x(t)+ \textbf{D}u(t)$ , the model has three parts the first equation, an integration, and the second equation. 1. The first equation takes the input x and the excitation u and generates the x dot vector shown on the left-hand side of the first state-space equation. 2. The second part must integrate x to obtain the x that is found in the next equation. This uses the velocity and acceleration to integrate to the next x that will be plugged into the second equation 3. The second equation in the state space representation takes the x vector we just calculated and then multiplies it with the sensing matrix C. we don't have a D matrix so this gives us the next output in our system This system is the coded form of the block diagram of the state space representation shown in attachment 1 Direct-II Model: The direct form 2 filter works in a much simpler way. because it involves no integration and follows the block diagram shown in Attachment 2, we can use a single difference equation to find the next output. However, the only complication that comes into play is that we also have to keep track of the w(n) seen in the middle of the block diagram. We use these two equations to calculate the output value $y[n]=b_0 \omega [n]+b_1 \omega [n-1] +b_2 \omega [n-2]$, where w[n] is $\omega[n]=x[n] - a_1 \omega [n-1] -a_2 \omega[n-2]$ Bit length Control: To control the bit length of each of the models I typecast all the inputs using np.float then the bit length that I want. This simulates the computer using only a specified bit length. I have to go through the code and force the code to use 128 bit by default. Currently, the default is 64 bit which so at the moment I am limited to 64 bit for highest bit length. I also need to go in and examine how numpy truncates floats to make sure it isn't doing anything unexpected. Bode Plot: The bode plot at the bottom shows the transfer function for both the IIR model and the state-space model. I generated about 100 seconds of white noise then computed the transfer function as $G(f) = \frac{P_{csd}(f)}{P_{psd}(f)}$ which is the cross-spectral density divided by the power spectral density. We can see that they match pretty closely at 64 bits. The IIR direct II model seems to have more noise on the surface but we are going to examine that in the next elog Attachment 1: 472px-Typical_State_Space_model.svg.png 3 kB Uploaded Mon Sep 20 17:18:53 2021	{"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": 5, "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": 0.8259707689285278, "perplexity": 1463.7179799589712}, "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-2022-05/segments/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00112.warc.gz"}
http://www.researchgate.net/researcher/9304987_Peter_S_Ozsvath	# Peter Ozsvath Princeton University, Princeton, New Jersey, United States Are you Peter Ozsvath? ## Publications (68)37.6 Total impact • Source ##### Article: Concordance homomorphisms from knot Floer homology Peter Ozsvath, Andras Stipsicz, Zoltan Szabo [Hide abstract] ABSTRACT: We modify the construction of knot Floer homology to produce a one-parameter family of homologies for knots in the three-sphere. These invariants can be used to give homomorphisms from the smooth concordance group to the integers, giving bounds on the four-ball genus and the concordance genus of knots. We give some applications of these homomorphisms. 07/2014; • Source ##### Article: Bordered Floer homology and the spectral sequence of a branched double cover II: the spectral sequences agree [Hide abstract] ABSTRACT: Given a link in the three-sphere, Ozsv\'ath and Szab\'o showed that there is a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double cover. The aim of this paper is to explicitly calculate this spectral sequence in terms of bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of bimodules associated to Dehn twists, and a general pairing theorem for polygons. The previous part (arXiv:1011.0499) focuses on computing the bimodules; this part focuses on the pairing theorem for polygons, in order to prove that the spectral sequence constructed in the previous part agrees with the one constructed by Ozsv\'ath and Szab\'o. 04/2014; • Source ##### Article: Combinatorial Heegaard Floer homology and sign assignments [Hide abstract] ABSTRACT: We provide an intergral lift of the combinatorial definition of Heegaard Floer homology for nice diagrams, and show that the proof of independence using convenient diagrams adapts to this setting. Topology and its Applications 01/2013; · 0.56 Impact Factor • Source ##### Article: Relative Q-gradings from bordered Floer theory Robert Lipshitz, Peter Ozsváth, Dylan Thurston [Hide abstract] ABSTRACT: In this paper we show how to recover the relative Q-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology. 11/2012; • Source ##### Article: Notes on bordered Floer homology Robert Lipshitz, Peter Ozsváth, Dylan Thurston [Hide abstract] ABSTRACT: This is a survey of bordered Heegaard Floer homology, an extension of the Heegaard Floer invariant HF-hat to 3-manifolds with boundary. Emphasis is placed on how bordered Heegaard Floer homology can be used for computations. 11/2012; • Source ##### Article: Knots in lattice homology Peter Ozsváth, András Stipsicz, Zoltán Szabó [Hide abstract] ABSTRACT: Assume that \Gamma_{v_0} is a tree with vertex set Vert(\Gamma_{v_0})={v_0, v_1,..., v_n}, and with an integral framing (weight) attached to each vertex except v_0. Assume furthermore that the intersection matrix of G=\Gamma_{v_0}-{v_0} is negative definite. We define a filtration on the chain complex computing the lattice homology of G and show how to use this information in computing lattice homology groups of a negative definite graph we get by attaching some framing to v_0. As a simple application we produce families of graphs which have arbitrarily many bad vertices for which the lattice homology groups are shown to be isomorphic to the corresponding Heegaard Floer homology groups. 08/2012; • Source ##### Article: Knot lattice homology in L-spaces Peter Ozsváth, András Stipsicz, Zoltán Szabó [Hide abstract] ABSTRACT: We show that the knot lattice homology of a knot in an L-space is equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring Z/2Z [U]). Suppose that G is a negative definite plumbing tree which contains a vertex w such that G-w is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology HF(G)\$ is isomorphic to the Heegaard Floer homology HF^-(Y_G) of the corresponding rational homology sphere Y_G. 07/2012; • Source ##### Article: A spectral sequence on lattice homology [Hide abstract] ABSTRACT: Using the link surgery formula for Heegaard Floer homology we find a spectral sequence from the lattice homology of a plumbing tree to the Heegaard Floer homology of the corresponding 3-manifold. This spectral sequence shows that for graphs with at most two "bad" vertices, the lattice homology is isomorphic to the Heegaard Floer homology of the underlying 3-manifold. 06/2012; • Source ##### Article: Tour of bordered Floer theory. Robert Lipshitz, Peter S Ozsváth, Dylan P Thurston [Hide abstract] ABSTRACT: Heegaard Floer theory is a kind of topological quantum field theory (TQFT), assigning graded groups to closed, connected, oriented 3-manifolds and group homomorphisms to smooth, oriented four-dimensional cobordisms. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with extended-TQFT-type gluing properties. In this survey, we explain the formal structure and construction of bordered Floer homology and sketch how it can be used to compute some aspects of Heegaard Floer theory. Proceedings of the National Academy of Sciences 05/2011; 108(20):8085-92. · 9.81 Impact Factor • Source ##### Article: A faithful linear-categorical action of the mapping class group of a surface with boundary Robert Lipshitz, Peter S. Ozsváth, Dylan P. Thurston [Hide abstract] ABSTRACT: We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin^c-structure is faithful. The paper is self-contained, and in particular does not assume any background in Floer homology. Comment: 26 pages, 9 figures Journal of the European Mathematical Society 12/2010; · 1.88 Impact Factor • Source ##### Article: Heegaard Floer homology and integer surgeries on links Ciprian Manolescu, Peter Ozsvath [Hide abstract] ABSTRACT: Let L be a link in an integral homology three-sphere. We give a description of the Heegaard Floer homology of integral surgeries on L in terms of some data associated to L, which we call a complete system of hyperboxes for L. Roughly, a complete systems of hyperboxes consists of chain complexes for (some versions of) the link Floer homology of L and all its sublinks, together with several chain maps between these complexes. Further, we introduce a way of presenting closed four-manifolds with b_2^+ > 1 by four-colored framed links in the three-sphere. Given a link presentation of this kind for a four-manifold X, we then describe the Ozsvath-Szabo mixed invariants of X in terms of a complete system of hyperboxes for the link. Finally, we explain how a grid diagram produces a particular complete system of hyperboxes for the corresponding link. 11/2010; • Source ##### Article: Bordered Floer homology and the branched double cover I Robert Lipshitz, Peter S. Ozsváth, Dylan P. Thurston [Hide abstract] ABSTRACT: Given a link in the three-sphere, Z. Szab\'o and the second author constructed a spectral sequence starting at the Khovanov homology of the link and converging to the Heegaard Floer homology of its branched double-cover. The aim of this paper and its sequel is to explicitly calculate this spectral sequence, using bordered Floer homology. There are two primary ingredients in this computation: an explicit calculation of filtered bimodules associated to Dehn twists and a pairing theorem for polygons. In this paper we give the first ingredient, and so obtain a combinatorial spectral sequence from Khovanov homology to Heegaard Floer homology; in the sequel we show that this spectral sequence agrees with the previously known one. Comment: 43 pages, 16 figures 11/2010; • Source ##### Article: Computing HF^ by factoring mapping classes Robert Lipshitz, Peter S. Ozsváth, Dylan P. Thurston [Hide abstract] ABSTRACT: Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these bimodules for arc slides explicitly, and then use them to give a combinatorial description of HF^ of a closed three-manifold, as well as the bordered Floer homology of any 3-manifold with boundary. 10/2010; • Source ##### Article: Heegaard Floer homology as morphism spaces Robert Lipshitz, Peter S. Ozsváth, Dylan P. Thurston [Hide abstract] ABSTRACT: In this paper we prove another pairing theorem for bordered Floer homology. Unlike the original pairing theorem, this one is stated in terms of homomorphisms, not tensor products. The present formulation is closer in spirit to the usual TQFT framework, and allows a more direct comparison with Fukaya-categorical constructions. The result also leads to various dualities in bordered Floer homology. 05/2010; • Source ##### Article: Bimodules in bordered Heegaard Floer homology Robert Lipshitz, Peter S. Ozsvath, Dylan P. Thurston [Hide abstract] ABSTRACT: Bordered Heegaard Floer homology is a three-manifold invariant which associates to a surface F an algebra A(F) and to a three-manifold Y with boundary identified with F a module over A(F). In this paper, we establish naturality properties of this invariant. Changing the diffeomorphism between F and the boundary of Y tensors the bordered invariant with a suitable bimodule over A(F). These bimodules give an action of a suitably based mapping class group on the category of modules over A(F). The Hochschild homology of such a bimodule is identified with the knot Floer homology of the associated open book decomposition. In the course of establishing these results, we also calculate the homology of A(F). We also prove a duality theorem relating the two version of the 3-manifold invariant. Finally, in the case of a genus one surface, we calculate the mapping class group action explicitly. This completes the description of bordered Heegaard Floer homology for knot complements in terms of the knot Floer homology. 03/2010; • Source ##### Article: Combinatorial Heegaard Floer homology and nice Heegaard diagrams Peter S. Ozsvath, Andras I. Stipsicz, Zoltan Szabo [Hide abstract] ABSTRACT: We consider a stabilized version of hat Heegaard Floer homology of a 3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance properties. 12/2009; • Source ##### Article: Grid diagrams and Heegaard Floer invariants [Hide abstract] ABSTRACT: We give combinatorial descriptions of the Heegaard Floer homology groups for arbitrary three-manifolds (with coefficients in Z/2). The descriptions are based on presenting the three-manifold as an integer surgery on a link in the three-sphere, and then using a grid diagram for the link. We also give combinatorial descriptions of the mod 2 Ozsvath-Szabo mixed invariants of closed four-manifolds, in terms of grid diagrams. 10/2009; • Source ##### Article: A combinatorial description of the \$U^2=0\$ version of Heegaard Floer homology Peter Ozsvath, Andras Stipsicz, Zoltan Szabo [Hide abstract] ABSTRACT: We show that every 3--manifold admits a Heegaard diagram in which a truncated version of Heegaard Floer homology (when the holomorpic disks pass through the basepoints at most once) can be computed combinatorially. Comment: Fixed figures International Mathematics Research Notices 11/2008; · 1.12 Impact Factor • Source ##### Article: Bordered Heegaard Floer homology: Invariance and pairing [Hide abstract] ABSTRACT: We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A-infinity module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A-infinity tensor product of the type D module of one piece and the type A module from the other piece is HF^ of the glued manifold. As a special case of the construction, we specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for HF^. We relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling. 10/2008; • Source ##### Article: Slicing planar grid diagrams: a gentle introduction to bordered Heegaard Floer homology [Hide abstract] ABSTRACT: We describe some of the algebra underlying the decomposition of planar grid diagrams. This provides a useful toy model for an extension of Heegaard Floer homology to 3-manifolds with parametrized boundary. This paper is meant to serve as a gentle introduction to the subject, and does not itself have immediate topological applications. Comment: 25 pages, 9 figures 10/2008; #### Publication Stats 3k Citations 37.60 Total Impact Points #### Institutions • ###### Princeton University • Department of Mathematics Princeton, New Jersey, United States • ###### Massachusetts Institute of Technology • Department of Mathematics Cambridge, MA, United States • ###### Columbia University • Department of Mathematics New York City, NY, United States	{"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": 0.9477234482765198, "perplexity": 849.0113439443279}, "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-2014-42/segments/1414119648146.28/warc/CC-MAIN-20141024030048-00038-ip-10-16-133-185.ec2.internal.warc.gz"}
https://gmatclub.com/forum/3x-2-2x-5-ax-kx-n-what-is-the-value-of-a-n-k-218391.html	GMAT Question of the Day - Daily to your Mailbox; hard ones only It is currently 18 Oct 2019, 01:43 ### GMAT Club Daily Prep #### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email. Customized for You we will pick new questions that match your level based on your Timer History Track every week, we’ll send you an estimated GMAT score based on your performance Practice Pays we will pick new questions that match your level based on your Timer History # (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? Author Message TAGS: ### Hide Tags Math Expert Joined: 02 Sep 2009 Posts: 58435 (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? [#permalink] ### Show Tags 12 May 2016, 01:45 00:00 Difficulty: 5% (low) Question Stats: 87% (01:45) correct 13% (02:07) wrong based on 178 sessions ### HideShow timer Statistics (3x + 2) (2x - 5) = ax^2 + kx + n . What is the value of a - n + k ? A. 5 B. 8 C. 9 D. 10 E. 11 _________________ Math Expert Joined: 02 Aug 2009 Posts: 7977 Re: (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? [#permalink] ### Show Tags 12 May 2016, 08:04 Bunuel wrote: (3x + 2) (2x - 5) = ax^2 + kx + n . What is the value of a - n + k ? A. 5 B. 8 C. 9 D. 10 E. 11 Simplify LHS.. $$(3x+2)(2x-5) = 6x^2-11x-10 = ax^2-n+k$$... equate the coefficients... a=6 ; k = -11 and n = -10.. so $$a-n+k = 6-(-10)+(-11) = 6+10-11 = 5$$ A _________________ Intern Joined: 14 Sep 2015 Posts: 9 Re: (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? [#permalink] ### Show Tags 13 May 2016, 01:13 1 Bunuel wrote: (3x + 2) (2x - 5) = ax^2 + kx + n . What is the value of a - n + k ? A. 5 B. 8 C. 9 D. 10 E. 11 Expanding we have 6x^2 - 15x + 4x -10 6x^2 - 11x - 10 Taking coefficients, a = 6, k= -11, n = -10 Therefore a - n + k = 6 - (-10) - 11 = 16 - 11 = 5 Non-Human User Joined: 09 Sep 2013 Posts: 13241 Re: (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? [#permalink] ### Show Tags 02 Feb 2019, 18:47 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Re: (3x + 2) (2x - 5) = ax² + kx + n . What is the value of a - n + k ? [#permalink] 02 Feb 2019, 18:47 Display posts from previous: Sort by	{"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": 0.5751978754997253, "perplexity": 1965.7549887378927}, "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-2019-43/segments/1570986679439.48/warc/CC-MAIN-20191018081630-20191018105130-00518.warc.gz"}
http://latex-community.org/forum/viewtopic.php?f=46&t=13883	### LaTeX Books Limited discount 50% coupon code tDRet6Y ### LaTeX users online In total there are 8 users online :: 0 registered, 0 hidden and 8 guests (based on users active over the past 5 minutes) Most users ever online was 1327 on Tue Nov 05, 2013 7:11 pm Users browsing this forum: No registered users and 8 guests ### LaTeX Friends Sponsored by DANTE e.V.: The German speaking TeX Users Group ## Problem with Equation Reference ### LaTeX Forum: Math & Science Add tags Information and discussion about LaTeX's math and science related features (e.g. formulas, graphs). I tried to use the following commands Code: Select all • Open in writeLaTeX \begin{equation}\label{xx} 3+4=7 \end{equation}Referring to the equation \eqref{xx} or Code: Select all • Open in writeLaTeX \begin{equation}3+4=7 \label{xx}\end{equation}Referring to the equation \eqref{xx} but it did not work... it worked just if instead of \eqref{xx} I wrote \ref{xx}. Why is it so? Should I use a particular package? N.B I am using Texniccenter and Mixtek as a compiler beth87 Posts: 6 Joined: Sat Jun 11th, 2011 "It doesn't work" is an error description that is as good as useless. What is the exact error message from the log? As a random shot I would guess that you forgot to load the amsmath package. Thorsten LaTeX Community Moderator ¹ System: openSUSE 13.1 (Linux 3.11.10), TeX Live 2013 (vanilla), TeXworks 0.5 (r1351) ² Posting stopped indefinitely due to offenses localghost Site Moderator Posts: 9218 Joined: Fri Feb 2nd, 2007 Location: Braunschweig, Germany	{"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": 0.9177075624465942, "perplexity": 17209.498628067115}, "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/1448398452560.13/warc/CC-MAIN-20151124205412-00142-ip-10-71-132-137.ec2.internal.warc.gz"}
http://www-subatech.in2p3.fr/fr/?view=seminar&id=221	# Study of neutrino properties using nuclear reactors: status and prospects ## Héctor Gomez ### APC – AstroParticule et Cosmologie (Paris) Despite enormous progress during the last years in understanding neutrino properties, the answer to fundamental questions about this particle still remains unknown. Among the different experiments proposed for neutrino research, those based on the study of neutrinos coming from nuclear reactors stand out, due to the quantity of questions that can answer. Some of them, as the precise measurement of θ13 mixing angle, has been already achieved. But others, as the understanding of the reactor anti-neutrino anomaly and its relationship with the sterile-neutrino existence, or the determination of the neutrino mass hierarchy, are still open questions that should be addressed by upcoming projects. In this seminar, an overall review of different experiments based on reactor neutrino detection will be presented. For the current projects measuring θ13, more emphasis will be put on Double Chooz and its results. Moreover, experiments looking for neutrinos at short baselines, as SoLid and Stereo, and future projects with wider physics program, as JUNO, will be described together with their corresponding objectives.	{"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": 0.8546122312545776, "perplexity": 1920.3936212992287}, "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-2019-51/segments/1575540531917.10/warc/CC-MAIN-20191211131640-20191211155640-00395.warc.gz"}
http://www.ipm.ac.ir/ViewPaperInfo.jsp?PTID=8928&school=Physics	## “School of Physics” Back to Papers Home Back to Papers of School of Physics Paper IPM / P / 8928 School of Physics Title: Spin Liquid Phase of XXZ s=1/2 Heisenberg Chain in A Transverse Magnetic Field Author(s): 1 J. Abouie 2 A. Langari 3 M. Siahatgar Status: Preprint Journal: Year: 2008 Supported by: IPM Abstract: We have investigated the zero and finite temperature behavior of the XXZ spin-1/2 Heisenberg chain in the presence of a transverse magnetic field (h). The phase diagram of this model has been studied by exact diagonalization method. The standard order parameters (magnetization, staggered magnetization) and the entanglement estimators have been obtained at zero temperature. We have also studied the specific heat of the model versus temperature for different values of h. For an intermediate region (hf < h < hc) the specific heat has two maxima, a sharp and a broaden one. The broaden peak of the specific heat is the characteristic of a spin liquid phase where there is no long range order in the ground state. The entanglement estimators also justify the presence of spin liquid phase in the intermediate region.	{"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": 0.8165169954299927, "perplexity": 1405.309371690641}, "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/1652662577757.82/warc/CC-MAIN-20220524233716-20220525023716-00538.warc.gz"}
http://mathhelpforum.com/trigonometry/34258-couple-questions-thanks.html	# Thread: a couple of questions...thanks 1. ## a couple of questions...thanks I dont know how to find the exact value of something(trig)...e.g. sec(13Pi/6) I get a long decimal(1.31...) but how am I meant to get the exact value answer which in this case is 2/3 of the square root of 3. (which is 1.31 when u calculate it but when I see 1.31 how am I meant to know that its 2/3 the square root of 3? question 2: 3(tan^2 )theta - sec theta = 1 for 0 <= theta =< 2PI can anyone solve that for me? thanks oh and for the following graph sketches: y = sin3x y = \|sin3x\| y = sin\|3x\| so like I was confused to whether they amplitude was 3 times bigger or the frequency is like 3 times smaller for them. can someone tell me what the difference of those curves are from a normal sinx 2. Originally Posted by grammar I dont know how to find the exact value of something(trig)...e.g. sec(13Pi/6) I get a long decimal(1.31...) but how am I meant to get the exact value answer which in this case is 2/3 of the square root of 3. (which is 1.31 when u calculate it but when I see 1.31 how am I meant to know that its 2/3 the square root of 3? [snip] $\sec \left(\frac{13 \pi}{6} \right) = \sec \left( \frac{(12 + 1) \pi}{6} \right) = \sec \left(\frac{12 \pi}{6} + \frac{\pi}{6}\right)$ $= \sec \left(2 \pi + \frac{\pi}{6}\right) = \sec \left(\frac{\pi}{6}\right) = \frac{1}{\cos \left(\frac{\pi}{6}\right)} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$. 3. Originally Posted by grammar [snip] question 2: 3(tan^2 )theta - sec theta = 1 for 0 <= theta =< 2PI can anyone solve that for me? thanks [snip] Substitute (from the Pythagorean Identity) $\tan^2 \theta = \sec^2 \theta - 1$: $3 (\sec^2 \theta - 1) - \sec \theta = 1 \Rightarrow 3 \sec^2 \theta - \sec \theta - 4 = 0$. Let $x = \sec \theta$ to see how to factorise: $3x^2 - x - 4 = 0 \Rightarrow (3x - 4)(x + 1) = 0$. Therefore either $x = \frac{4}{3}$ or $x = -1$. Therefore either $\sec \theta = \frac{4}{3}$ or $\sec \theta = -1$. The latter can be exactly solved easily. The former can only be found as either a generic exact answer or a decimal approximation. 4. Originally Posted by grammar [snip] oh and for the following graph sketches: y = sin3x y = \|sin3x\| y = sin\|3x\| so like I was confused to whether they amplitude was 3 times bigger or the frequency is like 3 times smaller for them. can someone tell me what the difference of those curves are from a normal sinx Draw the graph of $y = \|\sin (3x)\|$ by reflecting around the x-axis the parts of the graph of $y = \sin (3x)$ that lie below the x-axis. So there are salient points at $(0, \, \pm n \pi)$ where n is an integer. Draw the graph of $y = \sin \|3x\|$ by reflecting around the x-axis the part of the graph of $y = \sin (3x)$ that lies to the left of the y-axis. So there's a salient point at (0, 0). Note: $\sin \|3x\| = \sin (3x)$ for $x \geq 0$ and $\sin \|3x\| = \sin (-3x) = -\sin (3x)$ for $x < 0$. Salient point: Salient Point -- from Wolfram MathWorld	{"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": 19, "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": 0.8964421153068542, "perplexity": 483.7519821413275}, "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-2016-50/segments/1480698542246.21/warc/CC-MAIN-20161202170902-00295-ip-10-31-129-80.ec2.internal.warc.gz"}
http://learning.maxtech4u.com/tag/architecture-of-hop-field-neural-network/	Introduction of HopField Neural Network / December 22, 2017 Human beings are constantly thinking since ages about the reasons for human capabilities and incapabilities. Successful attempts have been made to design and develop systems that emulate human capabilities or help overcome human incapabilities. The human brain, which has taken millions of years to evolve to its present architecture, excels at tasks such as vision, speech, information retrieval, complex pattern recognition, all of which are extremely difficult tasks for conventional computers. A number of mechanisms have been which seems to enable human brain to handle various problems. These mechanisms include association; generalization and self-organization. A brain similar computational technique namely HopField Neural Network is explained here. Working of Hop Field Neural Network A neural network (or more formally artificial neural network) is a mathematical model or computational model inspired by the structure and functional aspects of biological neural networks. It consists of an interconnected group of artificial neurons. The original inspiration for the term Artificial Neural Network came from examination of central nervous systems and their neurons, axons, dendrites and synapses which constitute the processing elements of biological neural networks. One of the milestones for the current renaissance in the field of neural networks was the associative model proposed by Hopfield at… Insert math as $${}$$	{"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": 0.28291064500808716, "perplexity": 1451.544094357431}, "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/1537267156192.24/warc/CC-MAIN-20180919102700-20180919122700-00087.warc.gz"}
https://space.stackexchange.com/questions/31268/what-eliminates-the-velocity-when-occupants-return-from-iss-to-earth-and-how-mu/31269	What eliminates the velocity when occupants return from ISS to earth, and how much? The ISS has an orbital velocity of ~28000 km/h; the velocity $$v$$ relative to the landing site of the descent module is probably even higher than that most of the time. Once the occupants have landed, their velocity relative to the landing site is zero. My first question is: what is it that eliminates the velocity between detaching from the ISS and arrival? Three things come to mind: 1. the spacecraft (Soyuz) engine, 2. the atmosphere: a. descent module only, b. descent module with parachutes deployed, 3. the earth itself (final impact). Anything else? My second question is: how much does each of these modes contribute (measured in $$\Delta v/v$$ or $$(\Delta v/v)^2$$)? For the sake of the occupants' prolonged joy in space travel, I figure 3. has the smallest impact (pun intended), but how do the others relate to each other? Nearly all the velocity is cancelled by atmospheric deceleration of the descent module, before its parachutes are deployed. ISS orbital velocity is around 7700 m/s. An initial retro-burn of the Soyuz engines, of something like 115 m/s magnitude, is sufficient to lower the perigee of orbit into the uppermost part of the atmosphere. The orbital module and service module are then separated from the descent module. Once the descent module starts to enter the atmosphere, air resistance slows it, which further lowers the orbit, bringing the capsule into denser atmosphere, which slows it further, and so on. The Soyuz parachutes deploy starting at ~240 m/s (first drogue chutes to bring the capsule down to ~90 m/s then the mains to reach a 6 m/s descent rate). Just before touchdown, small solid rockets are fired for the final deceleration, producing another 3 m/s of ∆v. Thus, of the 7700m/s initial velocity, only about 360 m/s is cancelled via parachutes, reentry burn, and final retrorockets; 7340m/s (95%) of the deceleration is done by the descent module moving through the atmosphere. (I shamelessly stole correct figures from Steve Linton's answer.) This breakdown applies generally for all crewed spacecraft, though American capsules didn't have the final braking rockets, and the space shuttle touched down at ~100 m/s horizontal velocity, without deploying parachutes while airborne; atmospheric deceleration of the airframe does almost all of the work, because it's "free" apart from the heat shielding. • This is also the fundamental reason why reentry heating is so much of a problem. The Soyuz reentry module is ~3000 kg unloaded, so aerobraking is converting more than 3000kg*(7700m/s)^2 = 177 gigajoules of kinetic energy into heat. (Wolfram Alpha helpfully informs me this is roughly as much energy as would be released by burning 5100 liters of jet fuel.) – zwol Oct 11 '18 at 18:22 • @zwol Kinetic energy is 0.5mv^2, so 88.9 GJ. – ArtOfCode Oct 12 '18 at 0:00 • @Beanluc: Earth escape velocity really has little to do with landing on the Moon or Mars. For the Moon, you wind up going pretty slow at the gravitational midpoint (I mean where the gravity of Earth and Moon are equal, which of course is much closer to the Moon), then accelerate under the Moon's gravity. So final speed is ~= lunar escape velocity + however fast you were going at the midpoint. Likewise for Mars: you spend most of the trip at whatever the Hohmann transfer orbit's velocity is, then gain Mars' escape velocity on the approach. – jamesqf Oct 12 '18 at 2:05 • @Beanluc: Not magically, but it does get converted to potential energy while climbing out of the gravity well. – Dave Tweed Oct 12 '18 at 17:17 • @Beanluc: Sorry, your intuition is simply wrong on this. Reread jamesqf's most recent comment. The Apollo LEM had no trouble landing on the moon, but it never could have landed on Earth, even if it had no atmosphere. You keep saying the minimum landing velocity is the Earth's escape velocity, but Earth's escape velocity has literally NOTHING to do with it. How would you explain our ability to rendezvous with smaller interplanetary bodies like comets and asteroids, or even the ISS itself? The same rules apply everywhere. – Dave Tweed Oct 12 '18 at 17:33 The process is described here, which answers nearly all of your question. The reentry burn removes about 120 m/s of velocity from the capsule (that's your 1) and the final impact is 15 miles per hour (about 6 m/s). That's your 3. That leaves about 7.5 km/s for part 2. The only remaining question is the split between 2a and 2b, ie the velocity when the parachute opens. This source gives that. Firstly 2b is given as 240 m/s, leaving 7.25 km/s for 2a. Finally, a set of small rockets fire just before landing and reduce the 6 m/s to 3 m/s.	{"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": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7814847230911255, "perplexity": 1707.3032855512838}, "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-2021-17/segments/1618038118762.49/warc/CC-MAIN-20210417071833-20210417101833-00513.warc.gz"}