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To simplify the discussion below, I will first consider the case that all $\lambda_i > 0$, and then show how to deal with some unpenalized predictors. Part 1: All predictors are penalized ($\lambda_i > 0$ for all $i$) This case indeed works in exactly the way you described in your question. Let $\Lambda = \text{Diag}(\lambda_1,\dotsc, \lambda_p)$ be the diagonal matrix, where $\lambda_i$ is the penalty you want applied to the i-th predictor. Then you can write the LASSO problem (design matrix $X$, response $Y$), as follows: $$ \min_{\beta \in \mathbb R^p} \|\| Y - X\beta\|\|^2_2 + \|\| \Lambda \beta \|\|_1 $$ Now note that multiplying $X$ by $\Lambda^{-1}$ from the right means you multiply the i-th column by $1/\lambda_i$ and note: $$ \|\|Y - X\beta\|\|_2^2 + \|\| \Lambda \beta\|\|_1 = \|\|Y - X\Lambda^{-1} \Lambda\beta\|\|_2^2 + \|\| \Lambda \beta\|\|_1= \|\|Y - X\Lambda^{-1} \tilde{\beta}\|\|_2^2 + \|\| \tilde{\beta}\|\|_1 $$ In the last step I defined $\tilde{\beta}= \Lambda \beta$. Hence the original LASSO problem must be equivalent to: $$ \min_{\tilde{\beta} \in \mathbb R^p} \|\| Y - X\Lambda^{-1} \tilde{\beta}\|\|^2_2 + \|\| \tilde{\beta} \|\|_1 $$ This is a LASSO problem in which everyone gets a penalty of $1$. It is trivial to extend this so that everyone gets a penalty of $\lambda$ (then the $\Lambda$ entries would not represent the penalization of the i-th predictor, but the relative penalization); you might want this if you want to nest the above within a cross-validation based tuning of the regularization parameter. When you predict afterwards you need to remember what scaling you used though! E.g. if you use the original $X$, then use $\beta = \Lambda^{-1} \tilde{\beta}$! Part 2: Some unpenalized predictors (i.e. some $\lambda_i$ = 0) Let's say you now want to solve a LASSO problem, in which some predictors, let's call them $Z$ are not penalized, i.e.: $$ \min_{\beta, \gamma} \|\| Y - X\beta - Z\gamma\|\|^2_2 + \|\| \Lambda \beta \|\|_1 $$ (Here I just split the full design matrix into two parts $X$ and $Z$ corresponding to penalized or unpenalized predictors.) If your LASSO solver does not support unpenalized predictors, then as you mention in your comment, you could just use the technique from part 1 in which you essentially use a $\lambda_i$ really close to $0$ for unpenalized predictors. This would probably kind of work, except that it would be really bad from a numerical perspective, since some parts of $X\Lambda^{-1}$ would "blow" up. Instead, there is a better way to do this by orthogonalization. You could proceed in the following steps: Regress $Y \sim Z$, call the resulting coefficient $\tilde{\gamma}$ and also let $\tilde{Y}$ the residuals from this regression (i.e. $\tilde{Y} = Y - Z\tilde{\gamma}$). Regress $X \sim Z$: For each column of $X$, say the $i$-th column, run the regression $X_i \sim Z$. Then call $\tilde{X}$ the design matrix whose $i$-th column is the residual from the $i$-th regression. Run the following LASSO to get the fitted coefficient $\hat{\beta}$ (For this you will need the technique from Part 1.): $$ \min_{\beta \in \mathbb R^p} \|\| \tilde{Y} - \tilde{X}\beta\|\|^2_2 + \|\| \Lambda \beta \|\|_1 $$ Finally let $\hat{\gamma} = \tilde{\gamma} - (Z^TZ)^{-1}Z^TX\hat{\beta}$. Then $(\hat{\beta}, \hat{\gamma})$ will be the solutions to the full LASSO problem. Why does this work? This is a standard orthogonalization argument, similar to how the QR decomposition can be used to do linear regression. Essentially orthogonality (via the Pythagorean theorem -- I leave out the exact arguments as they are standard) allows us to split as follows (with $\hat{Y} = Y-\tilde{Y}$, $\hat{X} = X-\tilde{X}$): $$ \|\| Y-X\beta - Z\gamma\|\|^2 = \|\|\tilde{Y} - \tilde{X}\beta\|\|^2 + \|\| \hat{Y} - \hat{X}\beta - Z\gamma\|\|^2$$ So we want to solve: $$ \min_{\beta, \gamma} \{ \|\|\tilde{Y} - \tilde{X}\beta\|\|^2 + \|\| \hat{Y} - \hat{X}\beta - Z\gamma\|\|^2 + \|\| \Lambda \beta \|\|_1 \}$$ Now if we optimize for fixed $\beta$ over $\gamma$ we will get the expression from Part 4 of the above procedure and furthermore we get rid of the $\beta$ appearing in the 2nd square above. What remains is only the LASSO from part 3. Putting everything together gives us the procedure outlined above.
The Ellipse in mathematics is a curve in a place surrounded by two focal points where the sum of distances between two focal points is always constant. Ellipse is the generalization of a circle or we can call it as the special type of Ellipse containing two focal points at similar locations. Is it possible to define the shape of Ellipse? Yes, it is possible and it is done through eccentricity whose value lies between 0 and 1. If you would analyze analytically then Ellipse is a set of points that is defined as the ratio of the distance from any particular point to the curve whose sum is a constant number. This ratio in mathematical terms is popular as eccentricity. So, Ellipse or circle has the same definition or they are different? In case of a circle, the plane is parallel to the base of the come but in case of Ellipse, this is not parallel. So, they are different from each other. Ellipse was discovered and studied first by Menaechmus. It was Euclid who wrote about Ellipse formula and equations. In 1602, scientists told that Mars was oval and earth is round based on Ellipse concept and the focal point. In 1705, there were some interesting facts discovered about Sun like it has elliptical orbit or its shape etc. The Ellipse is the conic section that is closed and formed by the intersection of a cone by plane. They can be named as hyperbola or parabola and there are special formulas or equation to solve the tough Ellipse problems. Ellipse is the cross-section of a cylinder and parallel to the axis of the cylinder. With the help of basic Ellipse formulas, a lot of complex problems around the universe were possible to solve quickly. \[\large Area\;of\;the\;Ellipse=\pi r_{1}r_{2}\] \[\large Perimeter\;of\;the\;Ellipse=2\pi \sqrt{\frac{r_{1}^{2}+r_{2}^{2}}{2}}\] Where, r 1 is the semi major axis of the ellipse. r 2 is the semi minor axis of the ellipse. \[\large =2\pi \sqrt{\frac{10^{2}+5^{2}}{2}}\] \[\large =2\pi \sqrt{\frac{100+25}{2}}\] \[\large =2\pi \sqrt{\frac{125}{2}}\] Perimeter of the ellipse = 49.64 cm Ellipse is a set of points where two focal points together are named as Foci and with the help of those points, Ellipse can be defined. There are special equations in mathematics where you need to put Ellipse formulas and calculate the focal points to derive an equation. The sum of two focal points would always be a constant. Here is an example of the figure for clear understanding, what we meant by Ellipse and focal points exactly. Ellipse is used widely for Physics and Engineering. It is used to calculate the orbit of solar system inside a planet based on the concept of focal points. The same concept is valid for the moon orbiting planets having two astronomical bodies. If you remember, the shape of stars or planets is also defined by ellipsoids. Ellipse is the circle image under parallel projection and bounded case of perspective projection which is a simple intersection of the cone within the plane of projection. It can also be presented on the graph at horizontal or vertical axis with similar frequency. Further, it is used to understand the polarization effect in optical physics.
The linear homogeneous differential equation of the $n$th order with constant coefficients can be written as \[{ {y^{\left( n \right)}}\left( x \right) + {a_1}{y^{\left( {n – 1} \right)}}\left( x \right) + \cdots } + {{a_{n – 1}}y’\left( x \right) + {a_n}y\left( x \right) }={ 0,} \] where ${a_1},{a_2}, \ldots ,{a_n}$ are constants which may be real or complex. Using the linear differential operator $L\left( D \right),$ this equation can be represented as \[L\left( D \right)y\left( x \right) = 0,\] where \[{L\left( D \right) }={ {D^n} + {a_1}{D^{n – 1}} + \cdots }+{ {a_{n – 1}}D + {a_n}.}\] For each differential operator with constant coefficients, we can introduce the characteristic polynomial \[{L\left( \lambda \right) }={ {\lambda ^n} + {a_1}{\lambda ^{n – 1}} + \cdots }+{ {a_{n – 1}}\lambda + {a_n}.}\] The algebraic equation \[{L\left( \lambda \right) }={ {\lambda ^n} + {a_1}{\lambda ^{n – 1}} + \cdots }+{ {a_{n – 1}}\lambda }+{ {a_n} = 0}\] is called the characteristic equation of the differential equation. According to the fundamental theorem of algebra, a polynomial of degree $n$ has exactly $n$ roots, counting multiplicity. In this case the roots can be both real and complex (even if all the coefficients of ${a_1},{a_2}, \ldots ,{a_n}$ are real). Let us consider in more detail the different cases of the roots of the characteristic equation and the corresponding formulas for the general solution of differential equations. Case $1.$ All Roots of the Characteristic Equation are Real and Distinct We assume that the characteristic equation $L\left( \lambda \right) = 0$ has $n$ roots ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}.$ In this case the general solution of the differential equation is written in a simple form: \[{y\left( x \right) }={ {C_1}{e^{{\lambda _1}x}} + {C_2}{e^{{\lambda _2}x}} + \cdots }+{ {C_n}{e^{{\lambda _n}x}},}\] where ${C_1},{C_2}, \ldots ,{C_n}$ are constants depending on initial conditions. Case $2.$ The Roots of the Characteristic Equation are Real and Multiple Let the characteristic equation $L\left( \lambda \right) = 0$ of degree $n$ have $m$ roots ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _m},$ the multiplicity of which, respectively, is equal to ${k_1},{k_2}, \ldots ,{k_m}.$ It is clear that the following condition holds: \[{k_1} + {k_2} + \cdots + {k_m} = n.\] Then the general solution of the homogeneous differential equations with constant coefficients has the form \[ {y\left( x \right) }={ {C_1}{e^{{\lambda _1}x}} + {C_2}x{e^{{\lambda _1}x}} + \cdots } + {{C_{{k_1}}}{x^{{k_1} – 1}}{e^{{\lambda _1}x}} + \cdots } + {{C_{n – {k_m} + 1}}{e^{{\lambda _m}x}} }+{ {C_{n – {k_m} + 2}}x{e^{{\lambda _m}x}} + \cdots } + {{C_n}{x^{{k_m} – 1}}{e^{{\lambda _m}x}}.} \] It is seen that the formula of the general solution has exactly ${k_i}$ terms corresponding to each root ${\lambda _i}$ of multiplicity ${k_i}.$ These terms are formed by multiplying $x$ to a certain degree by the exponential function ${e^{{\lambda _i}x}}.$ The degree of $x$ varies in the range from $0$ to ${k_i} – 1,$ where ${k_i}$ is the multiplicity of the root ${\lambda _i}.$ Case $3.$ The Roots of the Characteristic Equation are Complex and Distinct If the coefficients of the differential equation are real numbers, the complex roots of the characteristic equation will be presented in the form of conjugate pairs of complex numbers: \[{{\lambda _{1,2}} = \alpha \pm i\beta ,\;\;}\kern-0.3pt{{\lambda _{3,4}} = \gamma \pm i\delta ,\; \ldots }\] In this case the general solution is written as \[ {y\left( x \right) }={ {e^{\alpha x}}\left( {{C_1}\cos \beta x + {C_2}\sin \beta x} \right) } + {{e^{\gamma x}}\left( {{C_3}\cos \delta x + {C_4}\sin \delta x} \right) + \cdots } \] Case $4.$ The Roots of the Characteristic Equation are Complex and Multiple Here, each pair of complex conjugate roots $\alpha \pm i\beta $ of multiplicity $k$ produces $2k$ particular solutions: \[ {{e^{\alpha x}}\cos \beta x,\;}\kern-0.3pt{{e^{\alpha x}}\sin\beta x,\;}\kern-0.3pt {{e^{\alpha x}}x\cos \beta x,\;}\kern-0.3pt{{e^{\alpha x}}x\sin \beta x, \ldots ,\;}\kern-0.3pt {{e^{\alpha x}}{x^{k – 1}}\cos \beta x,\;}\kern-0.3pt{{e^{\alpha x}}{x^{k – 1}}\sin\beta x.} \] Then the part of the general solution of the differential equation corresponding to a given pair of complex conjugate roots is constructed as follows: \[ {y\left( x \right) }={ {e^{\alpha x}}\left( {{C_1}\cos \beta x + {C_2}\sin \beta x} \right) } + {{x{e^{\alpha x}}\left( {{C_3}\cos \beta x }\right.}+{\left.{ {C_4}\sin \beta x} \right) + \cdots }} + {{{x^{k – 1}}{e^{\alpha x}}\left( {{C_{2k – 1}}\cos \beta x }\right.}+{\left.{ {C_{2k}}\sin \beta x} \right).}} \] In general, when the characteristic equation has both real and complex roots of arbitrary multiplicity, the general solution is constructed as the sum of the above solutions of the form $1-4.$ Solved Problems Click a problem to see the solution. Example 1Solve the differential equation $y^{\prime\prime\prime} + 2y^{\prime\prime} -$ $ y’ – 2y = 0.$ Example 2Solve the equation ${y^{\prime\prime\prime} – 7y^{\prime\prime} }+{ 11y’ – 5y = 0.}$ Example 3Solve the equation ${y^{IV}} – y^{\prime\prime\prime} + 2y’ = 0.$ Example 4Solve the equation ${y^V} + 18y^{\prime\prime\prime} + 81y’ = 0.$ Example 5Solve the differential equation ${y^{IV}} – 4y^{\prime\prime\prime} + 5y^{\prime\prime} $ $-\; 4y’ + 4y = 0.$ Example 1.Solve the differential equation $y^{\prime\prime\prime} + 2y^{\prime\prime} -$ $ y’ – 2y = 0.$ Solution. Write the corresponding characteristic equation: \[{\lambda ^3} + 2{\lambda ^2} – \lambda – 2 = 0.\] Solving it, we find the roots: \[ {{\lambda ^2}\left( {\lambda + 2} \right) – \left( {\lambda + 2} \right) = 0,\;\;}\Rightarrow {\left( {\lambda + 2} \right)\left( {{\lambda ^2} – 1} \right) = 0,\;\;}\Rightarrow {{\left( {\lambda + 2} \right)\left( {\lambda – 1} \right)\left( {\lambda + 1} \right) = 0,\;\;}}\Rightarrow {{\lambda _1} = – 2,\;\;}\kern-0.3pt{{\lambda _2} = 1,\;\;}\kern-0.3pt{{\lambda _3} = – 1.} \] It is seen that all three roots are real. Therefore, the general solution of the differential equations can be written as \[{y\left( x \right) }={ {C_1}{e^{ – 2x}} + {C_2}{e^x} }+{ {C_3}{e^{ – x}},}\] where ${C_1},$ ${C_2},$ ${C_3}$ are arbitrary constants.
When water freezes continuous translational symmetry is broken. When a metal becomes superconducting, what is the symmetry that gets broken? In most of the textbooks discussing this point, you should find something like : superconductors breaks the U(1)-gauge symmetry down to $\mathbb{Z}_{2}$. Fine, but what does it mean ? To explain it, let me be a bit outside the main stream discussion. What I'll discuss below is more a personal reflexion than something clearly stated in any book. Clearly, the origin of superconductivity -- as explained by Bardeen, Cooper and Schrieffer (BCS) -- is the instability of the Fermi surface due to the Cooper pairing. So the first question to ask is: why is the Fermi surface stable? You can find more details in the previous link, so let me give you the ultimate answer: the Fermi surface is stable since it is a topological concept. In short, the Fermi surface can be defined as a quantity which is not perturbed by some interactions. You can add impurities in your solid and/or various interactions between your electrons, the Fermi surface will not be so much deformed. Of course, an other arrangement of atoms in the solid gives an other Fermi surface, but the stability of this new one is still verified. Over the years, this concept of the stability of the Fermi surface has been refined, down to the work by Horava, reproduced in the book by Volovik. There you will see the topological invariant responsible for the stability of the Fermi surface (chapter 8), and the reason why it is a U(1) stability (well, it has to be Abelian for simple reasons you can guess easily, and the Fermi surface has a volume in the energy-space, so it can be reduced to a circle). The point is: the Fermi surface is stable with respect to almost all interactions, with the exception of the Cooper pairing. The reason is simple to understand: most of the interaction conserve the number of particles, but the Cooper pairing transmute the particles. In short, the vacuum of the Cooper pair is no more a Fermi gas/liquid, but a Bose gas/liquid. Then the volume of the Fermi surface (i.e. the number of fermions) is no more conserved. In other words, the topological protection ensuring the stability of the Fermi surface is no more at work when the Cooper pairing enters the stage. Now let us picturesquely understand why it is a $\text{U}\left(1\right)\rightarrow\mathbb{Z}_{2}$ breaking. The disappearance of the electrons at the Fermi surface creates a gap, reminiscent of the physics of semi-conductors. There, you know that there are 2-bands (conduction and valence). This is the first hint why only $\mathbb{Z}_{2}$. The second ingredient is that the Bose gas/liquid has no Fermi surface (tautology !), so it is not stable at all with respect to any interactions (re-tautology !), and so in principle the breaking should be from U(1) to nothing. But you still have two species of bosons: the hole-like and the particle-like, hence the doubled $\mathbb{Z}_{2}$ symmetry. Of course, all the arguments above are sketchy, so a more precise definition is still warm welcome. So, let us go back to the mainstream argument: the BCS-interaction reads $$H_{\text{BCS}}\sim c^{\dagger}\left(x\right)c^{\dagger}\left(x\right)c\left(x\right)c\left(x\right)$$ in a simplified form. To this Hamiltonian you can apply the transform $$c\left(x\right)\rightarrow e^{\mathbf{i}\varphi\left(x\right)}c\left(x\right)\;\;;\;\; c^{\dagger}\left(x\right)\rightarrow e^{-\mathbf{i}\varphi\left(x\right)}c^{\dagger}\left(x\right)$$ such that $H_{\text{int}}\rightarrow H_{\text{int}}$ and so $H_{\text{int}}$ is invariant with respect to a U(1)-gauge-transformation, since any real phase $\varphi$ is allowed and the group of multiplication by a phase $e^{\mathbf{i}\varphi\left(x\right)}$ is the group U(1). The mean-field counterpart of $H_{\text{int}}$ in the Cooper channel reads $$\tilde{H}_{\text{BCS}}\sim\Delta\left(x\right)c^{\dagger}\left(x\right)c^{\dagger}\left(x\right)+\Delta^{\dagger}\left(x\right)c\left(x\right)c\left(x\right)$$ and so it is only invariant when we choose $\varphi\in\left\{ 0,\pi\right\} $ in the above gauge-transformation. So there is only two possibilities if one wants to change the phase of the operators and keeps the mean-field Hamiltonian invariant. The group with only two elements is called $\mathbb{Z}_{2}$. That's the microscopic origin of the $\text{U}\left(1\right)\rightarrow\mathbb{Z}_{2}$ gauge-symmetry breaking.
Zeno mourns that his calculus students can’t read their own handwriting. Not only do their 2s become zeds, but their thetas become phis and their phis become rhos: [tex]\theta \rightarrow \phi,\ \varphi \rightarrow \rho.[/tex] Personally, it was the xi which always gave me trouble. That stupid little [tex]\xi[/tex] never came out right — and I know I’m not alone here. In that introductory string theory course of sainted memory, Prof. Zwiebach astonished our whole class by writing them freehand on the blackboard. You know, in retrospect, I wish my elementary school had skipped the cursive lessons and taught me how to write Greek letters. Oh, and “blackboard bold” characters too, the funky symbols with extra lines like [tex]\mathbb{R}[/tex] and [tex]\mathbb{C}[/tex] (these particular ones are used to stand for the real and complex number sets, respectively). I use Greek letters every day, but when was the last time I had to write in cursive? My signature doesn’t count. That’s not writing; that’s a mad scribble. I don’t know if Mom ever wanted me to be a doctor, but I’ve lived up to that in one respect at least. My autograph starts with a smushed B, continues with a series of wiggles interrupted by a figure that’s more an ampersand than an S, followed by another brood of squiggles. The cross on the t slashes across my entire name like the mark of Zorro. And nobody cares! The bank has never returned a rent check to me with a red stamp saying, “D minus for penmanship.” I can only recall one occasion in the past ten years in which I was actually obligated to write in cursive, and that was the GRE. When you take the Graduate Record Examination, you have to copy out an anti-cheating pledge, reproducing the words printed in one box in another box just below it, and you have to do it in cursive. Why? Buggered if I’ve got a clue. My “printing” is as distinctive as my “manuscript” would be. Are words somehow magically more honest if their letters are joined up? On black- and whiteboards, I use a “small caps” typeface (or is the proper word write-face?) which, if lacking in elegance, is at least legible from a distance. Lowercase letters are reserved for symbols and equations. If I didn’t take these measures, I’d be this guy:
Tagged: group Problem 343 Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$. Let $\Aut(N)$ be the group of automorphisms of $G$. Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime. Then prove that $N$ is contained in the center of $G$. Problem 332 Let $G=\GL(n, \R)$ be the general linear group of degree $n$, that is, the group of all $n\times n$ invertible matrices. Consider the subset of $G$ defined by \[\SL(n, \R)=\{X\in \GL(n,\R) \mid \det(X)=1\}.\] Prove that $\SL(n, \R)$ is a subgroup of $G$. Furthermore, prove that $\SL(n,\R)$ is a normal subgroup of $G$. The subgroup $\SL(n,\R)$ is called special linear group Problem 322 Let $\R=(\R, +)$ be the additive group of real numbers and let $\R^{\times}=(\R\setminus\{0\}, \cdot)$ be the multiplicative group of real numbers. (a) Prove that the map $\exp:\R \to \R^{\times}$ defined by \[\exp(x)=e^x\] is an injective group homomorphism. Add to solve later (b) Prove that the additive group $\R$ is isomorphic to the multiplicative group \[\R^{+}=\{x \in \R \mid x > 0\}.\]
The sceptic here is Hans Jelbring. He looks at a simple problem, two concentric spheres without heat sources, and checks their radiation balance to find what the temperature difference should be. Consensus science says, of course, that there should be none, but he found one, and then spent time working out the resulting perpetual motion machine. I'm not sure what was the point of that, but Trenberth was mentioned. I have sometimes done these analyses myself, being intrigued when what looks like a problem determined by geometry turns out to have a solution constrained by the Second Law of Thermodynamics (2LoT). Given the complexity, that can look like a miracle. The concentric problem.When you have concentric convex shapes, the radiation from the inner one ends up on the outer one. A really rookie mistake you can make is to expect the converse. Since the outer surface is larger, the inner body ends up being very hot. Hans managed to make that error here (case II, pipes). But he avoided it in the main post, which concerned spheres. There he noted, correctly, that he needed to calculate how much of the radiation from any one outer point impinged on the central body, and how much missed. So he drew a plot which you can see there, but which I'll modify to the one at right. There is a small surface element at dA on S2, the outer sphere. Some of its emission, in a cone angle α impinges on S1. The rest misses and ends up back on S2. I've shown dA at the bottom, but all locations are equivalent, and the total incident on S1 is got by summing the various dA's. I'll assume the spheres are black. Some trig relations: R1 = R2 sin α, r = R2 cos α The total emitted by dA is given by the Stefan Boltzmann relation $$F= dA\ \sigma \ T_2^4 $$ To get the fraction within the impinging cone, we want the part of that that would impinge on the surface S (if S1 wasn't there). Hans made here a common error of assuming that the radiation from dA is as for a sphere, uniform in all directions. Then you can just divide the area of S by the area of the hemisphere of which it is part. But dA is flat, and does not radiate uniformly. In fact, in its own plane it doesn't radiate at all (think of seeing a disc side on). There is a Law and theory applicable here. It's called Lambert's cosine law, and says that the intensity of radiation is proportional to cos θ, the angle from the normal. So that tells how the radiation incident on S should be summed. An integral is needed. You can imagine a ring element formed by an increment in θ. Its surface area will be $dS = 2\pi\ r^2\ sin \theta d\theta$. And if the impinging radiance is $I(\theta) = I_0\ cos \theta$ W/m2, then the total on S will be $$ I_0\int_\alpha^0 cos\theta dS = 2\pi\ r^2\ I_0 \int_\alpha^0 cos\theta\ sin\theta\ d\theta = \pi\ r^2\ I_0\ (1 - cos^2 \alpha) = \pi\ r^2\ I_0\ sin^2 \alpha$$ We can relate I 0to F by using this formula for the hemisphere, which catches all the radiation. The lower integration limit is then not α, but π/2. So $$ F = \pi r^2 I_0$$ So the power dP transferred from dA to S1 is $$dP = F\ sin^2 \alpha = dA\ \sigma\ T_2^4\ sin^2 \alpha$$ Integration over dA is just summation, so the total power P2 from S2 to S1 is $$ P2 = 4\pi R2^2\ T_2^4 sin^2 \alpha = 4\pi R1^2 T_2^4 $$ which exactly matches the Stefan-Boltzmann emission from S1 if $T_2=T_1$. Heat sourcesThe discussion on the Tallbloke thread was quite interesting, though Hans seems to have dropped out. DocMartyn posed the problem - what if S2 was a conducting shell in space and S1 had a heat source (Pu) generating 300W. He simplified with S2 as 2 sq m and S1 as 1 sq m. He asked what the temperature of S1 would be. It's actually enough to work out the fluxes from each body. The above reasoning is useful here. We can say that 300W has to be radiated out to space, and the 150 W/m2 sets the temperature of S2. It means also 150W/m2 is radiated inward. An amount P of this is absorbed by S1, which then radiates in total 300+P W. To get P, imagine that the 300 W was now generated within S2 rather than S1. Surprisingly perhaps, this does not change the temperature of S2. It still radiates 300W outward and inward, of which P arrives at S1. But now we know that S1, with no source, is at the same temperature as S2. So it radiates 150 W outward (its area is 1 sq m). And P is what comes in, so P=150. So the answer to the original problem is that the flux from S1 (with source) is 300+150=450 W/m2, and so the temperature is T1=298.48 °K. If T2 is the temperature of the shell, and T0 the temperature corresponding to 300 W emission only (ie S1 without shell), then $T1^4=T0^4+T2^4$ An interesting aspect of this reasoning is that nothing was said about spheres. If you allow the bodies to be conductive enough to keep their temperature uniform, they could have been any reasonable shape, though you have to account carefully if S1 isn't convex. I think DocMartyn chose his numbers with a common shell model of the greenhouse effect in mind. Update There's a fairly simple generalization which doesn't require the bodies to be spherical or concentric (though they need to be convex). Each has to be at a uniform temperature, which will generally require perfect conductivity. Suppose that S1 and S2 have respective areas A1 and A2, and there is a power source P watts. Start by assuming that is on S2. Then it must radiate P outwards, creating an emittance P/A2 W/m2. That forces (S-B) a temperature T0. All this is steady-state, black-body. Then S1 must also be at T0, and so also emits P/A2 W/m2. For power balance, this is also what it receives. Now suppose P shifts to body S1. S2 still has to emit P W, so is still at the same temperature. So the environment of S1 hasn't changed, and it still receives P/A2 W/m2, or PA1/A2 W. But with the extra source, it must now emit P(1+A1/A2) W, or emittance P*(1/A1+1/A2)W/m2. That determines its temperature.
Diagram of broach : Let, n = Total number of teeth in the broach L = Effective length of a broach in mm P - Tool pitch in mm L = Length to be broached in mm t - Rise per tooth in mm $n_s$ = Number of safety teeth $n_f$ Number of finishing teeth ( Range, 3 to 6 ) 1) Tool pitch (p) : p = 1.75 $\sqrt{l}$ 2) Rise per tooth (t) : Total rise = No. of teeth in broach x rise per tooth = n x t 3) Total number of teeth in the broach (n) n = (Roughing teeth) + (finishing teeth) n = ( $\frac{depth \ of \ cut}{cut \ per \ tooth}$ ) + ( $n_s$ + $n_f$) n = ( $\frac{total \ rise}{rise \ per \ tooth}$ ) + ( $n_s$ + $n_f$) 4) Effective length (L) L = No. of teeth in broach x pitch = n x p 5) Total load on a broach (F) a) for round holes f = Hole circumference x N X t x K = $\pi$ d x N X t x K b) For square holes f = Hole perimeter x N X t x K = 4 H x N x t x K Where, d = finished hole dia in mm N = Maximum no. of teeth cutting at a time t = Rise per tooth in mm K = Force required to cut 1 mm and 2 mm of metal at a given rise per tooth H = Finished length of one side of square hole in mm 6) Total effective length ( $L_T$ ) $L_T$ = Length of roughing teeth + Length of finishing teeth = No. of roughing teeth x pitch + No. of finishing teeth x $\frac{pitch}{2}$ $L_T$ = $n_r$ x p + $n_f$ x $\frac{p}{2}$
I recently came across this in a textbook (NCERT class 12 , chapter: wave optics , pg:367 , example 10.4(d)) of mine while studying the Young's double slit experiment. It says a condition for the formation of interference pattern is$$\frac{s}{S} < \frac{\lambda}{d}$$Where $s$ is the size of ... The accepted answer is clearly wrong. The OP's textbook referes to 's' as "size of source" and then gives a relation involving it. But the accepted answer conveniently assumes 's' to be "fringe-width" and proves the relation. One of the unaccepted answers is the correct one. I have flagged the answer for mod attention. This answer wastes time, because I naturally looked at it first ( it being an accepted answer) only to realise it proved something entirely different and trivial. This question was considered a duplicate because of a previous question titled "Height of Water 'Splashing'". However, the previous question only considers the height of the splash, whereas answers to the later question may consider a lot of different effects on the body of water, such as height ... I was trying to figure out the cross section $\frac{d\sigma}{d\Omega}$ for spinless $e^{-}\gamma\rightarrow e^{-}$ scattering. First I wrote the terms associated with each component.Vertex:$$ie(P_A+P_B)^{\mu}$$External Boson: $1$Photon: $\epsilon_{\mu}$Multiplying these will give the inv... As I am now studying on the history of discovery of electricity so I am searching on each scientists on Google but I am not getting a good answers on some scientists.So I want to ask you to provide a good app for studying on the history of scientists? I am working on correlation in quantum systems.Consider for an arbitrary finite dimensional bipartite system $A$ with elements $A_{1}$ and $A_{2}$ and a bipartite system $B$ with elements $B_{1}$ and $B_{2}$ under the assumption which fulfilled continuity.My question is that would it be possib... @EmilioPisanty Sup. I finished Part I of Q is for Quantum. I'm a little confused why a black ball turns into a misty of white and minus black, and not into white and black? Is it like a little trick so the second PETE box can cancel out the contrary states? Also I really like that the book avoids words like quantum, superposition, etc. Is this correct? "The closer you get hovering (as opposed to falling) to a black hole, the further away you see the black hole from you. You would need an impossible rope of an infinite length to reach the event horizon from a hovering ship". From physics.stackexchange.com/questions/480767/… You can't make a system go to a lower state than its zero point, so you can't do work with ZPE. Similarly, to run a hydroelectric generator you not only need water, you need a height difference so you can make the water run downhill. — PM 2Ring3 hours ago So in Q is for Quantum there's a box called PETE that has 50% chance of changing the color of a black or white ball. When two PETE boxes are connected, an input white ball will always come out white and the same with a black ball. @ACuriousMind There is also a NOT box that changes the color of the ball. In the book it's described that each ball has a misty (possible outcomes I suppose). For example a white ball coming into a PETE box will have output misty of WB (it can come out as white or black). But the misty of a black ball is W-B or -WB. (the black ball comes out with a minus). I understand that with the minus the math works out, but what is that minus and why? @AbhasKumarSinha intriguing/ impressive! would like to hear more! :) am very interested in using physics simulation systems for fluid dynamics vs particle dynamics experiments, alas very few in the world are thinking along the same lines right now, even as the technology improves substantially... @vzn for physics/simulation, you may use Blender, that is very accurate. If you want to experiment lens and optics, the you may use Mistibushi Renderer, those are made for accurate scientific purposes. @RyanUnger physics.stackexchange.com/q/27700/50583 is about QFT for mathematicians, which overlaps in the sense that you can't really do string theory without first doing QFT. I think the canonical recommendation is indeed Deligne et al's Quantum Fields and Strings: A Course For Mathematicians , but I haven't read it myself @AbhasKumarSinha when you say you were there, did you work at some kind of Godot facilities/ headquarters? where? dont see something relevant on google yet on "mitsubishi renderer" do you have a link for that? @ACuriousMind thats exactly how DZA presents it. understand the idea of "not tying it to any particular physical implementation" but that kind of gets stretched thin because the point is that there are "devices from our reality" that match the description and theyre all part of the mystery/ complexity/ inscrutability of QM. actually its QM experts that dont fully grasp the idea because (on deep research) it seems possible classical components exist that fulfill the descriptions... When I say "the basics of string theory haven't changed", I basically mean the story of string theory up to (but excluding) compactifications, branes and what not. It is the latter that has rapidly evolved, not the former. @RyanUnger Yes, it's where the actual model building happens. But there's a lot of things to work out independently of that And that is what I mean by "the basics". Yes, with mirror symmetry and all that jazz, there's been a lot of things happening in string theory, but I think that's still comparatively "fresh" research where the best you'll find are some survey papers @RyanUnger trying to think of an adjective for it... nihilistic? :P ps have you seen this? think youll like it, thought of you when found it... Kurzgesagt optimistic nihilismyoutube.com/watch?v=MBRqu0YOH14 The knuckle mnemonic is a mnemonic device for remembering the number of days in the months of the Julian and Gregorian calendars.== Method ===== One handed ===One form of the mnemonic is done by counting on the knuckles of one's hand to remember the numbers of days of the months.Count knuckles as 31 days, depressions between knuckles as 30 (or 28/29) days. Start with the little finger knuckle as January, and count one finger or depression at a time towards the index finger knuckle (July), saying the months while doing so. Then return to the little finger knuckle (now August) and continue for... @vzn I dont want to go to uni nor college. I prefer to dive into the depths of life early. I'm 16 (2 more years and I graduate). I'm interested in business, physics, neuroscience, philosophy, biology, engineering and other stuff and technologies. I just have constant hunger to widen my view on the world. @Slereah It's like the brain has a limited capacity on math skills it can store. @NovaliumCompany btw think either way is acceptable, relate to the feeling of low enthusiasm to submitting to "the higher establishment," but for many, universities are indeed "diving into the depths of life" I think you should go if you want to learn, but I'd also argue that waiting a couple years could be a sensible option. I know a number of people who went to college because they were told that it was what they should do and ended up wasting a bunch of time/money It does give you more of a sense of who actually knows what they're talking about and who doesn't though. While there's a lot of information available these days, it isn't all good information and it can be a very difficult thing to judge without some background knowledge Hello people, does anyone have a suggestion for some good lecture notes on what surface codes are and how are they used for quantum error correction? I just want to have an overview as I might have the possibility of doing a master thesis on the subject. I looked around a bit and it sounds cool but "it sounds cool" doesn't sound like a good enough motivation for devoting 6 months of my life to it
I have a wide table which exceeds the \textwidth. I want to let the left side of table align the left margin. How should I do? Note: This page is at left-hand side. MWE \documentclass[12pt,a4paper, twoside]{book}\usepackage[a4paper, textwidth=120mm, lmargin=25mm, marginparwidth=5mm, marginpar=55mm]{geometry}\usepackage[font=footnotesize, labelfont=bf]{caption}\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}\begin{document} some text here ............\newpage % To make this page left-hand side. We choose 6-hourly MRI-CGCM3, MPI-ESM-LR, GFDL CM3 models for this studies. However, it remains a restriction, that is, the lack of lots of variables. The Vitart algorithm (Knutson 2007) requires environmental variables such as vorticity at 850 hPa, temperature, geopotential, wind speeds on various pressure levels, and sea level pressure. But the CMIP models doesn't provide geopotential data and wind speed at sea level, we are going to skip the thickness criteria in the algorithm (in Sec. A.2.2) and replace surface wind by 850-hPa wind. \\\footnotesize \captionof{table}{CMIP5 models we used to analysis in our study. (Taylor et al. 2012, Camargo 2013)} \begin{tabular}{ccccc} \hline \hlineAcronym & Model name & Number {\footnote{The model number here follows which described in Camargo 2013.}} & Modeling center & Resolution \\ \hlineGFDL CM3 & \tabincell{c}{Geophysical Fluid Dynamics Laboratory \\ Climate Model, version 3} & M5 & \tabincell{c}{NOAA/Geophysical Fluid \\ Dynamics Laboratory} & $2.5^{\circ} \times 2.0^{\circ}$ \\[0.5cm]MPI-ESM-LR & \tabincell{c}{Max Planck Institute Earth \\ System Model, low resolution} & M12 & \tabincell{c}{Max Planck Institute \\ for Meteorology } & $1.9^{\circ } \times 1.9^{\circ }$ \\[0.5cm]MRI-CGCM3 & \tabincell{c}{Meteorological Research Institute \\ Coupled Atmospheric-Ocean General \\ Circulation Model, version 3} & M13 & Meteorological Research Institute & $1.1^{\circ }\times 1.2^{\circ }$ \\[0.5cm]\hline\end{tabular} \\[0.2cm]\end{document} Thanks!
Let\[\mathbf{v}_{1}=\begin{bmatrix}1 \\ 1\end{bmatrix},\;\mathbf{v}_{2}=\begin{bmatrix}1 \\ -1\end{bmatrix}.\]Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$\[ \Fun ( S , V ) = \{ f : S \rightarrow V \} . \] For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by\[ (f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S . \] (a) Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element? (b) Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism. (c) Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$. (d) Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as\[ V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} . \] (e) Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$? (f) Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$. Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis\[B = \left\{ 1+x , 1+x^2 , x – x^2 + 2x^3 , 1 – x – x^2 \right\}.\]Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis $B$. Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b – d & -3c \\ 2b – c & -3a \end{bmatrix}.\] For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,\[T (f) (x) = x f(x).\] Prove that $T$ is a linear transformation, and find its range and nullspace. Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$.Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$. Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$. (a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$. (b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$. Let $V$ be a vector space and $B$ be a basis for $V$.Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$. After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form\[\begin{bmatrix}1 & 0 & 2 & 1 & 0 \\0 & 1 & 3 & 0 & 1 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{bmatrix}.\] (a) What is the dimension of $V$? (b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$? (The Ohio State University, Linear Algebra Midterm) Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.Let\[W=\left\{\, A\in V \quad \middle \| \quad A=\begin{bmatrix}a & b\\c& -a\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Find the dimension of $W$. (The Ohio State University, Linear Algebra Midterm) Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]be a subset in $C[-1, 1]$. (a) Prove that $V$ is a subspace of $C[-1, 1]$. (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$. (c) Prove that\[B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}\]is a basis for $V$.
I wanted to better understand dfa. I wanted to build upon a previous question:Creating a DFA that only accepts number of a's that are multiples of 3But I wanted to go a bit further. Is there any way we can have a DFA that accepts number of a's that are multiples of 3 but does NOT have the sub... Let $X$ be a measurable space and $Y$ a topological space. I am trying to show that if $f_n : X \to Y$ is measurable for each $n$, and the pointwise limit of $\{f_n\}$ exists, then $f(x) = \lim_{n \to \infty} f_n(x)$ is a measurable function. Let $V$ be some open set in $Y$. I was able to show th... I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd... Consider a non-UFD that only has 2 units ( $-1,1$ ) and the min difference between 2 elements is $1$. Also there are only a finite amount of elements for any given fixed norm. ( Maybe that follows from the other 2 conditions ? )I wonder about counting the irreducible elements bounded by a lower... How would you make a regex for this? L = {w $\in$ {0, 1}* : w is 0-alternating}, where 0-alternating is either all the symbols in odd positions within w are 0's, or all the symbols in even positions within w are 0's, or both. I want to construct a nfa from this, but I'm struggling with the regex part
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away. Proving that a free particle moves with a constant velocity in an inertial frame of reference($\S$3. Galileo's relativity principle). The proof begins with explaining that the Lagrangian must only depend on the speed of the particle ($v^2={\bf v}^2$): $$L=L(v^2).$$ Hence the Lagrance's equations will be $$\frac{d}{dt}\left(\frac{\partial L}{\partial {\bf v}}\right)=0,$$ so $$\frac{\partial L}{\partial {\bf v}}=\text{constant}.$$ And this is where the authors say Since $\partial L/\partial \bf v$ is a function of the velocity only, it follows that $${\bf v}=\text{constant}.$$ Why so?I can put $L=\\|{\bf v}\\|=\sqrt{v^2_x+v^2_y+v^2_z}$. Then $$\frac{\partial L}{\partial {\bf v}}=\frac{2}{\sqrt{v^2_x+v^2_y+v^2_z}}\begin{pmatrix} v_x \\ v_y \\ v_z \end{pmatrix},$$ which will remain a constant vector $\begin{pmatrix} 2 \\ 0 \\ 0 \end{pmatrix}$ as the particle moves with an arbitrary non-constant positive $v_x$ and $v_y=v_z=0$. Where am I wrong here? If I am, how does one prove the quoted statement? Proving that$L=\frac{m v^2}2$ ($\S$4. The Lagrangian for a free particle). The authors consider an inertial frame of reference $K$ moving with a velocity ${\bf\epsilon}$ relative to another frame of reference $K'$, so ${\bf v'=v+\epsilon}$. Here is what troubles me: Since the equations of motion must have same formin every frame, the Lagrangian $L(v^2)$ must be converted by this transformation into a function $L'$ which differs from $L(v^2)$, if at all, only by the total time derivative of a function of coordinates and time(see the end of $\S$2). First of all, what does same formmean? I think the equations should be the same, but if I'm right, why wouldn't the authors write so? Second, it was shown in $\S$2 that adding a total derivative will not change the equations. There was nothing about total derivatives of time and coordinates being the only functions, adding which does not change the equations (or their form, whatever it means). Where am I wrong now? If I'm not, how does one prove the quoted statement and why haven't the authors done it? P. S. Could you recommend any textbooks on analytical mechanics? I'm not very excited with this one. Seems to hard for me.
As part of the design of an experiment, I am trying to model the magnetic fields inside a hollow rectangular waveguide cavity. I have not had any problems calculating the cavity physical dimensions for the desired mode and resonant frequency, and I have obtained equations for the electric and magnetic fields. $$ \mathrm{For\;reference\;:\;}\begin{cases}f_c=2.87\;\mathrm{[GHz]}\\\mathrm{Cavity\;dimensions\;}(a\cdot b\cdot d)=7.39\cdot5\cdot7.39\;\mathrm{[cm]}\\\mathrm{Mode\;TE_{101}}\end{cases}$$ $$\mathrm{Fields\;\;:\;}\begin{cases}E_x=0\\E_y=-2 j f_c \mu a H_0\sin\left(\frac{\pi x}{a}\right)\sin\left(\frac{\pi z}{a}\right) \\E_z=0\\H_x=-H_0 \sin\left(\frac{\pi x}{a}\right)\cos\left(\frac{\pi z}{a}\right)\\H_y=0\\H_z=H_0\cos\left(\frac{\pi x}{a}\right)\sin\left(\frac{\pi z}{a}\right)\end{cases}\;\;\;\;\mathrm{with}\;\;h^2\equiv \left(\frac{\pi}{a}\right)^2+\left(\frac{\pi}{b}\right)^2$$ My problem is that these equations have the field amplitude $H_0$ as a free parameter, where I would like it to be determined by the power injected into the cavity. I imagine that the assumption of perfectly conductive walls has to be discarded, to account for the fact that power is being continuously injected and dissipated. Is there an analytical method to calculate the field amplitude inside a resonant cavity from the injected power, or would this require numerical simulation using HFSS or similar finite-element modeling software? I am looking for ballpark, order-of-magnitude values, not exact results.
A paper Computing the Entropy of a Large Matrix by Thomas P. Wihler, Bänz Bessire, André Stefanov suggests approximating $x \lg x$ with a polynomial. Then you can use the trace of powers of the matrix to sum the results of applying that polynomial to each of the eigenvalues. (The polynomial-via-power-and-trace thing works because density matrices have orthogonal eigenvectors. Raising the matrix to a power will raise its eigenvalues to that power. Then the trace gives you the sum of the eigenvalues at that power.) Sounds fun. Not going to let the paper spoil it. The derivatives of $f(x) = x \ln x$ are: $$\begin{align}f^0(x) &= x \ln x\\f^1(x) &= 1 + \ln x\\f^2(x) &= x^{-1} \\f^3(x) &= -x^{-2}\\f^4(x) &= 2 x^{-3}\\\f^5(x) &= -6 x^{-4}\\\...\\f^k(x) &= (-1)^k \cdot (k-2)! \cdot x^{1-k}\end{align}$$ The derivatives don't always exist at $x=0$, but they do at $x=1$, so we can make a Taylor series from there: $$\begin{align}x \ln x &= \sum_{k=0}^\infty \frac{(x-1)^k}{k!} f^k(1)\\&=(1 \ln 1)\frac{(x-1)^0}{0!} + (1 + \ln 1)\frac{(x-1)^1}{1!} + \sum_{k=2}^\infty \frac{(x-1)^k}{k!} (-1)^k 1^{1-k}(k-2)!\\&=x - 1 + \sum_{k=2}^\infty \frac{(1-x)^k}{k(k-1)}\end{align}$$ So now we have a nice series $\frac{(1-x)^1}{-1} + \frac{(1-x)^2}{2} + \frac{(1-x)^3}{6} + \frac{(1-x)^4}{12} + \frac{(1-x)^5}{24} + ...$ that we can approximate by cutting off at some index $c$. Except that would converge like $O(c^{-1})$ when $x=0$, so we should probably try to account for the excess when the numerator is stuck at $1$: $\begin{align} x \ln x&=x - 1 + \sum_{k=2}^\infty \frac{(1-x)^k}{k(k-1)}\\&=x - 1 + \sum_{k=2}^{c-1} \frac{(1-x)^k}{k(k-1)} + \sum_{k=c}^\infty \frac{(1-x)^k}{k(k-1)}\\&\approxx - 1 + \sum_{k=2}^{c-1} \frac{(1-x)^k}{k(k-1)} + \sum_{k=c}^\infty \frac{(1-x)^{\Huge c}}{k(k-1)}\\&=x - 1 + \left( \sum_{k=2}^c \frac{(1-x)^k}{k(k-1)} \right) + \frac{(1-x)^c}{c-1}\end{align}$ Here's a graph of the difference between $x \ln x$ and the approximation when $c=10$: And voila, just write code to evaluate that polynomial over the eigenvalues by using traces of the matrix powers: import math import numpy def von_neumann_entropy(density_matrix, cutoff=10): x = numpy.mat(density_matrix) one = numpy.identity(x.shape[0]) base = one - x power = basebase result = numpy.trace(base) for k in range(2, cutoff): result -= numpy.trace(power) / (kk - k) power = power.dot(base) result -= numpy.trace(power) / (cutoff - 1) return result / math.log(2) # convert from nats to bits The convergence is still kinda terrible near $x \approx 0$, though. I tried replacing the last correction with some doubling steps like this: import math import numpy def von_neumann_entropy(density_matrix, cutoff=10): x = numpy.mat(density_matrix) one = numpy.identity(x.shape[0]) base = one - x power = basebase result = numpy.trace(base) for k in range(2, cutoff): result -= numpy.trace(power) / (kk - k) power = power.dot(base) # Twiddly hacky magic. a = cutoff for k in range(3): d = (a+1) / (4a(a-1)) result -= numpy.trace(power) * d power = power.dot(power) result -= numpy.trace(power) * d a = 2 result -= numpy.trace(power) / (a-1) 0.75 return result / math.log(2) # convert from nats to bits And that improved things a bit more, so that the maximum offset vs $-x \ln x$ across the [0, 1] range was $3.5 \cdot 10^{-3}$ instead of $2.9 \cdot 10^{-2}$ (for the given cutoff of 10). (The plot is reversed.) Of course, the paper I linked at the start of the answer probably has a much better polynomial. For example, my error analysis should have been in terms of the total entropy instead of the individual value. And I should have used tricks like computing the trace of $AB$ from $A$ and $B$ in $O(n^2)$ time without doing the multiplication. And I should have thrown some simulated annealing at the coefficients.
Let $(M,\omega)$ be a Quantized closed Kaehler manifold then by Koderia embedding theorem , $M$ must be algebraicly projective i.e, we have the embedding $$\phi: (M,\omega)\to (\mathbb CP^N, \omega_{FS})$$ So $$\phi^*\omega_{FS}=\omega+\frac{i}{2\pi}\partial\bar \partial \epsilon $$ where $\epsilon$ is a smooth function and is defined as follows: Definition of $\epsilon$ function: Let $\pi:(L,h)\to (M,\omega)$ be a prequantum line bundle and let $x\in M$ and $q\in L^+$ such that $\pi(q)=x$ and $H$ is the Hilbert space of global holomorphic sections ($h$ is hermitian metric). Then we can write $s(x)=\delta_q(s)q$ where $\delta_q:H\to \mathbb C$ is a linear continous functional of $s$ and by Riesz theorem $\delta_q(s)=\langle s,e_q \rangle_h$ where $e_q\in H$ and thus $s(x)= \langle s,e_q\rangle_hq$ and we can define the real valued function on $M$ by the formula $$\epsilon(x)=h(q,q)\left \\| e_q \right \\|_h^2$$ Now the conjectureis that, if $\epsilon$ be constant then $M$ is homogeneous space? Is there any counterexample or proof for it? This question is known as Andrea Loi's conjecture in his doctoral thesis Peter Crooks gave a counterexample and I removed the part simply connected, I want to see this conjecture still is conjecture :)
The following procedure is here. Consider a cantilever fixed at one end and loaded at the other one. In cartesian coordinates (if $y$ is horizontal and $x$ vertical, meaning that the load acts parallel to the $x$ axis) the equation of the curvature is: $$\frac{\frac{d^2x}{dy^2}}{\left[1+\left(\frac{dx}{dy}\right)^2\right]^{3/2}}=\frac{M(y)}{EI},$$ where $M(y)$ is the moment of bending, $I$ the moment of inertia of the cross-sectional area of the beam, and $E$ is the Young's modulus of the beam's material. Considering the variable $z=dx/dy$ and the lenght of the beam: $$s(l)=\int_0^l\sqrt{1+\left(\frac{dx}{dy}\right)^2}dy,$$ where $l$ is the projection of the beam onto the $y$ axis. Then we can write: $$\frac{z}{\sqrt{1+z^2}}=\int_0^y\frac{M(y)}{EI}dy = G(y).$$ The bending moment is $M(y)=P(l-y)$, where $P$ is the applied force parallel to the $x$ axis. Therefore $$\frac{ds}{dy}=\left(1-\frac{P^2}{E^2I^2\left[ly-\frac{y^2}{2}\right]^2}\right)^{-1/2}.$$ After this introduction I present my question: I carried out an experiment of bending spaghettis by placing them horizontally with a fixed end, and loaded at the other end. The aim was to find the spaghettis' Young's modulus. Unfortunately the theory developed for these deflections is mainly broad for small deflections (they take some approximations in the first equation I wrote). However I measured $l,P,I$ to be $0.17, 0.1, 5.75\times 10^{-12}$, in mks. My idea is to integrate numerically the last equation so that $s(l,E)=L$, where $L$ is the lenght of the spaghetti: $0.2$m. This means that the integration is made over $y$, from $0$ to $l$. Then, since $l$ is known, $s$ really depends on $E$. So there must be some $E$ for which the integral is $L$. However I need a really large $E$ (~$1\times 10^9$ Pa) so that the expression in the square root is positive, and this doesn't make sense because the spaghettis shouldn't have such a big $E$. What could be wrong with this approach?
The Annals of Probability Ann. Probab. Volume 22, Number 2 (1994), 659-679. A Law of the Iterated Logarithm for Stochastic Processes Defined by Differential Equations with a Small Parameter Abstract Consider the following random ordinary differential equation: $\dot{X}^\epsilon(\tau) = F(X^\epsilon(\tau), \tau/\epsilon, \omega) \text{subject to} X^\epsilon(0) = x_0$, where $\{F(x, t, \omega), t \geq 0\}$ are stochastic processes indexed by $x$ in $\mathfrak{R}^d$, and the dependence on $x$ is sufficiently regular to ensure that the equation has a unique solution $X^\epsilon(\tau, \omega)$ over the interval $0 \leq \tau \leq 1$ for each $\epsilon > 0$. Under rather general conditions one can associate with the preceding equation a nonrandom averaged equation: $\dot{x}^0(\tau) = \overline{F}(x^0(\tau)) \text{subject to} x^0(0) = x_0,$ such that $\lim_{\epsilon\rightarrow 0} \sup_{0\leq\tau\leq 1}E\|X^\epsilon(\tau) - x^0(\tau)\| = 0$. In this article we show that as $\epsilon \rightarrow 0$ the random function $(X^\epsilon(\cdot) - x^0(\cdot))/\sqrt{2\epsilon\log\log\epsilon^{-1}}$ almost surely converges to and clusters throughout a compact set $K$ of $C\lbrack 0, 1\rbrack$. Article information Source Ann. Probab., Volume 22, Number 2 (1994), 659-679. Dates First available in Project Euclid: 19 April 2007 Permanent link to this document https://projecteuclid.org/euclid.aop/1176988724 Digital Object Identifier doi:10.1214/aop/1176988724 Mathematical Reviews number (MathSciNet) MR1288126 Zentralblatt MATH identifier 0806.60017 JSTOR links.jstor.org Subjects Primary: 60F15: Strong theorems Secondary: 60F17: Functional limit theorems; invariance principles 93E03: Stochastic systems, general Citation Kouritzin, M. A.; Heunis, A. J. A Law of the Iterated Logarithm for Stochastic Processes Defined by Differential Equations with a Small Parameter. Ann. Probab. 22 (1994), no. 2, 659--679. doi:10.1214/aop/1176988724. https://projecteuclid.org/euclid.aop/1176988724
Communications in Mathematical Analysis Commun. Math. Anal. Volume 16, Number 2 (2014), 9-18. On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials Abstract Let $\{Q_{n,\lambda}^{(\alpha)}\}_{n\ge 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$\langle f,g\rangle s:=\int_{-1}^1 f(x)g(x)(1-x^2)^{\alpha-\frac{1}{2}} dx+\lambda \int_{-1}^1 f'(x)g'(x)(1-x^2)^{\alpha-\frac{1}{2}}dx,$$ where $\alpha \gt -\frac{1}{2}$ and $\lambda \ge 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov [3], in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{dx^k}$ in $[-M_{n,\lambda},M_{n,\lambda}]$, where $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial $Q_{n,\lambda}^{(\alpha)}$ are contained in $[-M_{n,\lambda},M_{n,\lambda}]$ and the function $\left\|Q_{n,\lambda}^{(\alpha)}\right\|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented. Article information Source Commun. Math. Anal., Volume 16, Number 2 (2014), 9-18. Dates First available in Project Euclid: 20 October 2014 Permanent link to this document https://projecteuclid.org/euclid.cma/1413810435 Mathematical Reviews number (MathSciNet) MR3270574 Zentralblatt MATH identifier 1321.33014 Subjects Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol s kii-type inequalities) Citation Paschoa, V. G.; Pérez, D.; Qintana, Y. On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials. Commun. Math. Anal. 16 (2014), no. 2, 9--18. https://projecteuclid.org/euclid.cma/1413810435
If a newform $f \in \mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N),\varepsilon)$ has an inner twist by some $\sigma \in \operatorname{Aut}(\mathbb{C})$, then $f^{\sigma}$ is a newform of the same level as $f$. Moreover, if $\varepsilon$ is trivial, then so is the nebentypus of $f^{\sigma}$ (see (3.8) of Ribet's paper), and so any inner twist must arise from a quadratic Dirichlet character. Ribet's paper actually shows (see (3.9)) that if $N$ is squarefree, then there are no nontrivial inner twists. More precisely, he shows that if $N$ is squarefree and $\chi$ is a quadratic Dirichlet character, then the twist $f \otimes \chi$ cannot have level $N$ and trivial nebentypus. This is not hard to see by a local argument: if $N$ is squarefree, then for any $p \mid N$, the local component $\pi_p$ of the associated automorphic representation must be isomorphic to $\mathrm{St}$, the Steinberg representation of conductor $p$ associated to the trivial character. Any nontrivial twist of $f$ leaving the nebentypus unchanged must necessarily be quadratic, but any twist of $\mathrm{St}$ by a ramified quadratic character has conductor at least $p^2$, not $p$ (and in fact exactly $p^2$ if $p$ is odd). Similarly, at any unramified place, the twist of an unramified principal series representation by a ramified quadratic character has conductor at least $p^2$. So the level of $f \otimes \chi$ is strictly greater than $N$. But if $N$ is not squarefree, then this is no longer the case! Indeed, in a recent paper (Theorem 6.4), I proved the following result (well, technically I proved it for Maaß cusp forms, but it generalises in an obvious way to holomorphic cusp forms). Fix a nonsquarefree odd integer $N$, and let $N' > 1$ be squarefree and such that $N'^2 \mid N$. Let $\mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N))_{\mathrm{nonmon}(\varepsilon_{\mathrm{quad}(N')})}$ denote the vector space spanned by newforms $f$ of weight $k$, level $N$, and trivial nebentypus such that $f$ does not have CM by the quadratic Dirichlet character $\varepsilon_{\mathrm{quad}(N')}$ modulo $N'$, but that the twist $f \otimes \varepsilon_{\mathrm{quad}(N')}$ is also a newform of level $N$ and trivial nebentypus. Then \[\frac{\dim \mathcal{S}_{k}^{\mathrm{new}} (\Gamma_0(N))_{\mathrm{nonmon} (\varepsilon _{\mathrm{quad}(N')} )} }{\dim \mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N))} \sim \prod_{\substack{p \mid N' \\ p^2 \parallel N}} \left(1 - \frac{p}{p^2 - p - 1}\right)\] as $k$ tends to infinity over the even integers. Furthermore, this still holds if we replace $\mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N))_{\mathrm{nonmon}(\varepsilon_{\mathrm{quad}(N')})}$ with \[\bigcap_{\substack{N^* \mid N' \\ N^* > 1}} \mathcal{S}_k^{\mathrm{new}}(\Gamma_0(N))_{\mathrm{nonmon}(\varepsilon _{\mathrm{quad}(N^*)})}.\] A similar result holds even when $N$ is squarefree if $\varepsilon$ is nontrivial; see Proposition 6.5 of my paper. That inner twists occur in abundance when $N$ is not squarefree was observed as far back as 1977 by Ribet; at the bottom of page 48 of this paper, Ribet writes It would be of interest to give an a priori construction of forms with extra twists. If the level $N$ is divisible by a high power of a prime, these forms seem to be more the rule than the exception.
For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by$$\gamma(K) := \sup \|f'(\infty)\|$$where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ in the complement of $K$ :$f \in H^{\infty}(\mathbb{C}_ {\infty} \setminus K)$, $\\|f\\|_{\infty} \leq 1$. Here $$f'(\infty) = \lim_{z \rightarrow \infty} z(f(z)-f(\infty)).$$ A theorem due to Ahlfors states that for each compact $K$, there always exists a unique (in the unbounded component of the complement of $K$) function $F$, called the Ahlfors function of $K$, such that $F \in H^{\infty}(\mathbb{C}_ {\infty} \setminus K)$, $\\|F\\|_{\infty} \leq 1$, and $F'(\infty)=\gamma(K)$. It's not hard to show that $\gamma$ is outer-regular, in the sense that if $(K_n)$ is a decreasing sequence of compact sets, then$$\gamma(\cap_n K_n) = \lim_{n\rightarrow \infty} \gamma(K_n).$$This essentially follows from Montel's theorem and the fact that $\gamma(E) \subseteq \gamma(F)$ whenever $E \subseteq F$. Question: Is analytic capacity inner regular? More precisely, if $(K_n)$ is a sequence of compact sets such that$$K_1 \subseteq K_2 \subseteq K_3 \subseteq \dots$$and such that $K:=\cup_n K_n$ is compact, then is it true that$\gamma(K) = \lim_{n \rightarrow \infty} \gamma(K_n)?$ I could not find anything in the litterature. Thank you, Malik EDIT As pointed out by Fedja in the comments, analytic capacity is comparableto a quantity which is continuous from below, see the article "Painleve's problem and the semiadditivity of analytic capacity" by Xavier Tolsa. The answer is yes if the compact sets $K_n$ and $K$ are connected. Indeed, for connected compact sets, analytic capacity is equal to logarithmic capacity, and logarithmic capacity is inner regular. The answer is yes if $K$ is a compact set whose boundary consists of a finite number of analytic and pairwise disjoint Jordan curves, provided we replace the condition $K:=\cup_n K_n$ by the condition that each compact subset of the interior of $K$ is eventually contained in some $K_n$. This easily follows from the fact that in this case, the Ahlfors function of $K$ extends analytically across the boundary of $K$. EDITI contacted Xavier Tolsa, and according to him, it's an open problem, related to the so called capacitability problem. It's not known if the Borel sets are capacitable. I'll leave the question open though, because I'd be very interested to hear about sufficient conditions or similar results.
Moving Across the Coordinate Plane May 13, 2011 Posted by Billy in : calculus , add a comment Albert is standing at the origin of the Cartesian plane, desperately in need of cake. Looking around, he spots some delicious chocolate cake at the point $(100,100)$. Albert immediately departs for the cake. He knows that if he goes outside the square with corners $(0,0)$, $(100,0)$, $(100,100)$, and $(0,100)$ the cake will disappear and he will starve. When Albert is at the point $(x,y)$, the maximum speed he can move is given by $v(x,y) = 5+\frac{y}{20}$. What is the minimum time required for Albert to reach the cake? A Tricky Exponent March 25, 2011 Posted by Saketh in : calculus , 1 comment so far Determine the exact value of $$\int^{\frac{\pi}{2}}_0 \frac{1}{1+(\tan{x})^{\sqrt{2}}} \,dx$$ An Integral March 25, 2011 Posted by Arjun in : calculus , 3 comments Find $$\int^{\frac{\pi}{2}}_0 \frac{1}{\sqrt{\tan{x}}} \,dx$$
Construction of 3-designs using $(1,\sigma)$-resolution 1. Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany Mathematics Subject Classification:05B0. Citation:Tran van Trung. Construction of 3-designs using $(1,\sigma)$-resolution. Advances in Mathematics of Communications, 2016, 10 (3) : 511-524. doi: 10.3934/amc.2016022 References: [1] R. D. Baker, Partitioning the planes of $AG_{2m}(2)$ into 2-designs,, 15 (1976), 205. Google Scholar [2] Z. Baranyai, On the factorization of the complete uniform hypergraph,, [3] T. Beth, D. Jungnickel and H. Lenz, [4] J. Bierbrauer, Some friends of Alltop's designs $4-(2^f+1,5,5)$,, 36 (2001), 43. Google Scholar [5] J. Bierbrauer and T. van Trung, Shadow and shade of designs $4-(2^f+1,6,10)$,, [6] R. C. Bose, A note on the resolvability of balanced incomplete block designs,, 6 (1942), 105. Google Scholar [7] [8] L. H. M. E. Driessen, [9] M. Jimbo, Y. Kunihara, R. Laue and M. Sawa, Unifying some known infinite families of combinatorial 3-designs,, 118 (2011), 1072. doi: 10.1016/j.jcta.2010.10.007. Google Scholar [10] [11] [12] [13] S. S. Shrikhande and D. Raghavarao, A method of construction of incomplete block designs,, 25 (1963), 399. Google Scholar [14] S. S. Shrikhande and D. Raghavarao, Affine $\alpha$-resolvable incomplete block designs,, [15] D. R. Stinson, C. M. Swanson and T. van Trung, A new look at an old construction: Constructing (simple) 3-designs from resolvable 2-designs,, 325 (2014), 23. doi: 10.1016/j.disc.2014.02.009. Google Scholar [16] [17] show all references References: [1] R. D. Baker, Partitioning the planes of $AG_{2m}(2)$ into 2-designs,, 15 (1976), 205. Google Scholar [2] Z. Baranyai, On the factorization of the complete uniform hypergraph,, [3] T. Beth, D. Jungnickel and H. Lenz, [4] J. Bierbrauer, Some friends of Alltop's designs $4-(2^f+1,5,5)$,, 36 (2001), 43. Google Scholar [5] J. Bierbrauer and T. van Trung, Shadow and shade of designs $4-(2^f+1,6,10)$,, [6] R. C. Bose, A note on the resolvability of balanced incomplete block designs,, 6 (1942), 105. Google Scholar [7] [8] L. H. M. E. Driessen, [9] M. Jimbo, Y. Kunihara, R. Laue and M. Sawa, Unifying some known infinite families of combinatorial 3-designs,, 118 (2011), 1072. doi: 10.1016/j.jcta.2010.10.007. Google Scholar [10] [11] [12] [13] S. S. Shrikhande and D. Raghavarao, A method of construction of incomplete block designs,, 25 (1963), 399. Google Scholar [14] S. S. Shrikhande and D. Raghavarao, Affine $\alpha$-resolvable incomplete block designs,, [15] D. R. Stinson, C. M. Swanson and T. van Trung, A new look at an old construction: Constructing (simple) 3-designs from resolvable 2-designs,, 325 (2014), 23. doi: 10.1016/j.disc.2014.02.009. Google Scholar [16] [17] [1] Josselin Garnier, George Papanicolaou. Resolution enhancement from scattering in passive sensor imaging with cross correlations. [2] Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. [3] Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. [4] Daniel Guo, John Drake. A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution. [5] [6] Amine Laghrib, Abdelkrim Chakib, Aissam Hadri, Abdelilah Hakim. A nonlinear fourth-order PDE for multi-frame image super-resolution enhancement. [7] [8] [9] [10] Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. [11] Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. [12] Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic. [13] H. T. Banks, D. Rubio, N. Saintier, M. I. Troparevsky. Optimal design for parameter estimation in EEG problems in a 3D multilayered domain. [14] Jae Man Park, Gang Uk Hwang, Boo Geum Jung. Design and analysis of an adaptive guard channel based CAC scheme in a 3G-WLAN integrated network. [15] Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. [16] [17] Vadim Azhmyakov, Alex Poznyak, Omar Gonzalez. On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach. [18] [19] [20] 2018 Impact Factor: 0.879 Tools Metrics Other articles by authors [Back to Top]
Suppose there are two charges (4uC each) fixed in the horizontal axis. One is in x=0 and the other in x=8m. I've obtained the electric field: $E=-k\cdot4\mu C \cdot [\frac{1}{x^2}+\frac{1}{(x-8m)^2}], x<0$ $E=k\cdot4\mu C \cdot [\frac{1}{x^2}-\frac{1}{(x-8m)^2}], 0<x<8m$ $E=k\cdot4\mu C \cdot [\frac{1}{x^2}+\frac{1}{(x-8m)^2}], x>8$ Now I had to find the points in space where the field is 0. So I've solved each part and obtain x=4m (the middle point between the two charges). But looking on the given answers it says that $x=\pm\infty$ is also a solution. I know that the limit is 0, but I'm not sure how to arrive to that solution and if it has a physical meaning.
Tagged: determinant of a matrix Problem 718 Let \[ A= \begin{bmatrix} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{bmatrix} . \] Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square. Compute the determinant of $A$.Add to solve later Problem 686 In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , \lambda=-i$, and $\lambda=0$. Add to solve later (b) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues $\lambda=i , \lambda=-i$, and $\lambda=-1$. Problem 582 A square matrix $A$ is called nilpotent if some power of $A$ is the zero matrix. Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix. Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$. Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample. Problem 571 The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017. There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold). The time limit was 55 minutes. This post is Part 2 and contains Problem 4, 5, and 6. Check out Part 1 and Part 3 for the rest of the exam problems. Problem 4. Let \[\mathbf{a}_1=\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \mathbf{a}_2=\begin{bmatrix} 2 \\ -1 \\ 4 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 0 \\ a \\ 2 \end{bmatrix}.\] Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. Problem 5. Find the inverse matrix of \[A=\begin{bmatrix} 0 & 0 & 2 & 0 \\ 0 &1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason. Problem 6. Consider the system of linear equations \begin{align} 3x_1+2x_2&=1\\ 5x_1+3x_2&=2. \end{align} (a) Find the coefficient matrix $A$ of the system. (b) Find the inverse matrix of the coefficient matrix $A$. (c) Using the inverse matrix of $A$, find the solution of the system. ( Linear Algebra Midterm Exam 1, the Ohio State University) Read solution Problem 546 Let $A$ be an $n\times n$ matrix. The $(i, j)$ cofactor $C_{ij}$ of $A$ is defined to be \[C_{ij}=(-1)^{ij}\det(M_{ij}),\] where $M_{ij}$ is the $(i,j)$ minor matrix obtained from $A$ removing the $i$-th row and $j$-th column. Then consider the $n\times n$ matrix $C=(C_{ij})$, and define the $n\times n$ matrix $\Adj(A)=C^{\trans}$. The matrix $\Adj(A)$ is called the adjoint matrix of $A$. When $A$ is invertible, then its inverse can be obtained by the formula For each of the following matrices, determine whether it is invertible, and if so, then find the invertible matrix using the above formula. (a) $A=\begin{bmatrix} 1 & 5 & 2 \\ 0 &-1 &2 \\ 0 & 0 & 1 \end{bmatrix}$. (b) $B=\begin{bmatrix} 1 & 0 & 2 \\ 0 &1 &4 \\ 3 & 0 & 1 \end{bmatrix}$. Problem 509 Using the numbers appearing in \[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& 9 \end{bmatrix}.\] Prove that the matrix $A$ is nonsingular.Add to solve later Problem 505 Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate the inverse matrix of $\begin{bmatrix} 2 & 1\\ 1& 2 \end{bmatrix}$. Problem 486 Determine whether there exists a nonsingular matrix $A$ if \[A^4=ABA^2+2A^3,\] where $B$ is the following matrix. \[B=\begin{bmatrix} -1 & 1 & -1 \\ 0 &-1 &0 \\ 2 & 1 & -4 \end{bmatrix}.\] If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$. ( The Ohio State University, Linear Algebra Final Exam Problem) Read solution Eigenvalues of Orthogonal Matrices Have Length 1. Every $3\times 3$ Orthogonal Matrix Has 1 as an Eigenvalue Problem 419 (a) Let $A$ be a real orthogonal $n\times n$ matrix. Prove that the length (magnitude) of each eigenvalue of $A$ is $1$. Add to solve later (b) Let $A$ be a real orthogonal $3\times 3$ matrix and suppose that the determinant of $A$ is $1$. Then prove that $A$ has $1$ as an eigenvalue.
2019-10-14 17:21 Performance of VELO clustering and VELO pattern recognition on FPGA/LHCb Collaboration This document contains plots and tables showing the performance obtained on VELO clustering and VELO pattern recognition using algorithms implementable on FPGA. The data used are simulated with LHCb Upgrade conditions.. LHCB-FIGURE-2019-011.- Geneva : CERN, 2019 - 16. Record dettagliato - Record simili 2019-10-11 14:20 TURBO stream animation /LHCb Collaboration An animation illustrating the TURBO stream is provided. It shows events discarded by the trigger in quick sequence, followed by an event that is kept but stripped of all data except four tracks [...] LHCB-FIGURE-2019-010.- Geneva : CERN, 2019 - 3. Record dettagliato - Record simili 2019-10-10 15:48 Record dettagliato - Record simili 2019-09-12 16:43 Pending/LHCb Collaboration Pending LHCB-FIGURE-2019-008.- Geneva : CERN, 10 Record dettagliato - Record simili 2019-09-10 11:06 Smog2 Velo tracking efficiency/LHCb Collaboration LHCb fixed-target programme is facing a major upgrade (Smog2) for Run3 data taking consisting in the installation of a confinement cell for the gas covering $z \in [-500, -300] \, mm $. Such a displacement for the $pgas$ collisions with respect to the nominal $pp$ interaction point requires a detailed study of the reconstruction performances. [...] LHCB-FIGURE-2019-007.- Geneva : CERN, 10 - 4. Fulltext: LHCb-FIGURE-2019-007_2 - PDF; LHCb-FIGURE-2019-007 - PDF; Record dettagliato - Record simili 2019-09-09 14:37 Background rejection study in the search for $\Lambda^0 \rightarrow p^+ \mu^- \overline{\nu}$/LHCb Collaboration A background rejection study has been made using LHCb Simulation in order to investigate the capacity of the experiment to distinguish between $\Lambda^0 \rightarrow p^+ \mu^- \overline{\nu}$ and its main background $\Lambda^0 \rightarrow p^+ \pi^-$. Two variables were explored, and their rejection power was estimated applying a selection criteria. [...] LHCB-FIGURE-2019-006.- Geneva : CERN, 09 - 4. Fulltext: PDF; Record dettagliato - Record simili 2019-09-06 14:56 Tracking efficiencies prior to alignment corrections from 1st Data challenges/LHCb Collaboration These plots show the ﬁrst outcoming results on tracking efficiencies, before appli- cation of alignment corrections, as obtained from the 1st data challenges tests. In this challenge, several tracking detectors (the VELO, SciFi and Muon) have been misaligned and the effects on the tracking efficiencies are studied. [...] LHCB-FIGURE-2019-005.- Geneva : CERN, 2019 - 5. Fulltext: PDF; Record dettagliato - Record simili 2019-09-06 11:34 Record dettagliato - Record simili 2019-09-02 15:30 First study of the VELO pixel 2 half alignment/LHCb Collaboration A first look into the 2 half alignment for the Run 3 Vertex Locator (VELO) has been made. The alignment procedure has been run on a minimum bias Monte Carlo Run 3 sample in order to investigate its functionality [...] LHCB-FIGURE-2019-003.- Geneva : CERN, 02 - 4. Fulltext: VP_alignment_approval - TAR; VELO_plot_approvals_VPAlignment_v3 - PDF; Record dettagliato - Record simili 2019-07-29 14:20 Record dettagliato - Record simili
There are numerous mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory proposed by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg)[2] and wave mechanics (invented by Erwin Schrödinger). In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom). Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" can be roughly translated from German as inherent or as a characteristic). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, quantum mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time; rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for (a) the state of something having an uncertainty relation and (b) a state that has a definite value. The latter is called the "eigenstate" of the property being measured. For example, consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wave function. The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. However, one can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and almost zero everywhere else. If one performs a position measurement on such a wavefunction, the result x will be obtained with almost 100% probability. In other words, the position of the free particle will almost be known. This is called an eigenstate of position (mathematically more precise: a generalized eigenstate (eigendistribution) ). If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. It can be shown that the wavelength is equal to h/p, where h is Planck's constant and p is the momentum of the eigenstate. If the particle is in an eigenstate of momentum then its position is completely blurred out. Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if one measures the observable, the wavefunction will instantaneously be an eigenstate (or generalized eigenstate) of that observable. This process is known as wavefunction collapse. It involves expanding the system under study to include the measurement device, so that a detailed quantum calculation would no longer be feasible and a classical description must be used. If one knows the corresponding wave function at the instant before the measurement, one will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in the previous example will usually have a wavefunction that is a wave packet centered around some mean position x0, neither an eigenstate of position nor of momentum. When one measures the position of the particle, it is impossible to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x0, where the amplitude of the wave function is large. After the measurement is performed, having obtained some result x, the wave function collapses into a position eigenstate centered at x. Wave functions can change as time progresses. An equation known as the Schrödinger equation describes how wave functions change in time, a role similar to Newton's second law in classical mechanics. The Schrödinger equation, applied to the aforementioned example of the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates. Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucleus (Fig. 1). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric). The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. [edit] Mathematical formulation Main article: Mathematical formulation of quantum mechanics See also: Quantum logic In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues. The time evolution of a quantum state is described by the Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution. The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states. It turns out that analytic solutions of Schrödinger's equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen-molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. An alternative formulation of quantum mechanics is Feynman's path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics. [edit] Interactions with other scientific theories The fundamental rules of quantum mechanics are very broad. They assert that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators, or if it even exists. These must be chosen appropriately in order to obtain a quantitative description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system moves to higher energies or equivalently, larger quantum numbers. In other words, classic mechanics is simply a quantum mechanics of large systems. This "high energy" limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse, give rise to the reality we perceive? When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator. Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical -\frac{e^2}{4 \pi\ \epsilon_0\ } \frac{1}{r} Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory. It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity. [edit] Derivation of quantization The particle in a 1-dimensional potential energy box is the most simple example where restraints lead to the quantization of energy levels. The box is defined as zero potential energy inside a certain interval and infinite everywhere outside that interval. For the 1-dimensional case in the x direction, the time-independent Schrödinger equation can be written as[3]: - \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi. The general solutions are: \psi = A e^{ikx} + B e ^{-ikx} \;\;\;\;\;\; E = \frac{k^2 \hbar^2}{2m} \psi = C \sin kx + D \cos kx \; (exponential rewrite) The presence of the walls of the box restricts the acceptable solutions to the wavefunction. At each wall : \psi = 0 \; \mathrm{at} \;\; x = 0,\; x = L Consider x = 0 * sin 0 = 0, cos 0 = 1. To satisfy \psi = 0 \; D = 0 (cos term is removed) Now Consider: \psi = C \sin kx \; * at X = L, \psi = C \sin kL \; * If C = 0 then \psi =0 \; for all x and would conflict with Born interpretation * therefore sin kL must be satisfied by kL = n \pi \;\;\;\; n = 1,2,3,4,5 \;
Let's say that you are pushing a wagon by applying a force $\vec{F}$ from behind, as such: Now, according to Newton's second law, the center of mass of the wagon should accelerate, because the sum of the external forces is not $\vec{0}\text{ N}$. If we assume that the coefficient of friction between the wheels and the ground is $0$, then I fully understand what's going on - $\vec{F}$ is the only applied force (or the sum of all the forces) on the particle system, which causes the CM to accelerate. However, the problem arises when I take into account the cases where the coefficient of friction isn't $0$. In these cases, the wheels will presumably begin to rotate. Since this rotation is due to friction, it seems likely that there is a frictional force (let's call it $\vec{f}$) on the wheels from the ground. However, if such a force is present, and the initial force remains unchanged, then according to Newton's second law, the acceleration of the wagon should be given by: $$\vec{a} = \frac{1}{m_{\text{wagon}}}(\vec{F} + \vec{f})$$ So if this is the case, wouldn't that make the acceleration different from the case where the coefficient of friction between the surfaces is $0$? In picture form, this is pretty much what I mean is the case generating the second equation: Moreover, I don't even know whether the direction of the frictional forces in the above picture is correct or not (i.e. whether $k_1, k_2 > 0$), which kind of goes to show that I'm fairly confused about this whole thing. Intuitively, it feels like $k_1 < 0 \wedge k_2 < 0$, since the ground "holds the wheels back", but in that case, won't friction "slow the wagon down" rather than helping it roll? I'll summarize my questions: When $\mu \neq 0$, which forces are present on the object? If there is a frictional force, which direction does it have? How does this force affect the acceleration of the wagon's CM?
Talk II on Bourguignon-Lawson's 1978 paper The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold. We show $C^\infty$ local rigidity for a broad class of new examples of solvable algebraic partially hyperbolic actions on ${\mathbb G}=\mathbb{G}_1\times\cdots\times \mathbb{G}_k/\Gamma$, where $\mathbb{G}_1$ is of the following type: $SL(n, {\mathbb R})$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, $n\geq3$, $m\geq 4$. These examples include rank-one partially hyperbolic actions. The method of proof is a combination of KAM type iteration scheme and representation theory. The principal difference with previous work that used KAM scheme is very general nature of the proof: no specific information about unitary representations of ${\mathbb G}$ or ${\mathbb G}_1$ is required. This is a continuation of the last talk. A classical problem in knot theory is determining whether or not a given 2-dimensional diagram represents the unknot. The UNKNOTTING PROBLEM was proven to be in NP by Hass, Lagarias, and Pippenger. A generalization of this decision problem is the GENUS PROBLEM. We will discuss the basics of computational complexity, knot genus, and normal surface theory in order to present an algorithm (from HLP) to explicitly compute the genus of a knot. We will then show that this algorithm is in PSPACE and discuss more recent results and implications in the field. We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong. There is a close analogy between function fields over finite fields and number fields. In this analogy $\text{Spec } \mathbb{Z}$ corresponds to an algebraic curve over a finite field. However, this analogy often fails. For example, $\text{Spec } \mathbb{Z} \times \text{Spec } \mathbb{Z} $ (which should correspond to a surface) is $\text{Spec } \mathbb{Z}$ (which corresponds to a curve). In many cases, the Fargues-Fontaine curve is the natural analogue for algebraic curves. In this first talk, we will give the construction of the Fargues-Fontaine curve. Consider a collection of particles in a fluid that is subject to a standing acoustic wave. In some situations, the particles tend to cluster about the nodes of the wave. We study the problem of finding a standing acoustic wave that can position particles in desired locations, i.e. whose nodal set is as close as possible to desired curves or surfaces. We show that in certain situations we can expect to reproduce patterns up to the diffraction limit. For periodic particle patterns, we show that there are limitations on the unit cell and that the possible patterns in dimension d can be determined from an eigendecomposition of a 2d x 2d matrix. Department of Mathematics Michigan State University 619 Red Cedar Road C212 Wells Hall East Lansing, MI 48824 Phone: (517) 353-0844 Fax: (517) 432-1562 College of Natural Science
When two distinct vectors are directed from one to point to another, then it is called as the vector. They are usually differentiated in terms of speed or velocity. Mostly, we don’t get any clue about the direction here in which direction the object is moving. SO, we need a formula here to calculate the direction of a vector. In physics, both magnitude or direction are given as the vector. Take an example of the rock, where it is moving at the speed of 5meters per second and direction is headed towards West then this is an example of the vector. So, let us have a quick discussion on vectors first. They are used to represent themagnitude and direction both. For the things with quantities that are given by force or velocity, they could also be represented in the form of vectors. Both of them also have the direction or magnitude. Let us consider for some seconds, a force of about 5 Newtons is applied to a given direction at any point in the space. This is the point where force does not change itself for the applied force. It signifies that forces are independent of any point of application. To apply the force in the right way, you should always know the magnitude and the direction. If x is the horizontal movement and y is the vertical movement, then the formula of direction is \[\LARGE \theta =\tan^{-1}\frac{y}{x}\] If (x1,y1) is the starting point and ends with (x2,y2), then the formula for direction is \[\LARGE \theta =\tan^{-1}\frac{(y_{2}-y_{1})}{(x_{2}-x_{1})}\]
I am new to seismic data processing and I really have no understanding of the term 'phase'. So, could anybody give me a simple explaination of the term? Thank you I think wikipedia can more than adequately answer your question. However, in brief, the phase term describes the relationship between a waveform and a fixed reference point in time. For example the sinusoid $sin(\omega t)$ is zero at $t = 0$ whereas $sin(\omega t - \phi)$ is zero at $t = \phi$. $\phi$ could be referred to here as the phase offset. Phase is a property of sinusoidal waves and describes how far along the wave is in it's cycle. A simple wave may be described as $$\sin(\theta)$$ Where $\theta$ is it's phase. It's usually measured in radians and at every multiple of 2$\pi$, the wave goes back to the beginning of its cycle and starts again. If the wave is varying in time, it may be described as: $$\sin( \omega t )$$ The wave will repeat its cycle every time $\omega t$ reaches $2\pi$, or every $2\pi/\omega$ seconds, which is the wave's period. The term, $\omega t$ is the wave's phase. In seismology, the wave will travel through time and space. A wave in space and time will look like: $$\sin( kx - \omega t )$$ Here, $k$ is the wave-vector describing the wavelength and direction of propagation of the wave in space. $kx - \omega t$ is the wave's phase. Every time, $kx - \omega t$ reaches $2\pi$, the wave will repeat its cycle. The wave's wavelength is $2\pi/k$ and it's period is $2\pi/\omega$. In a linear system, you can split any signal into a sum on sinusoidal signals at different frequencies, calculate how the individual sinusoidal signals propagate, then recombine them. In many case it is easier to do calculations with sines and phase than it is to handle the propagation of the individual waves themselves. In seismic data processing the term phase is used/referred to differently in depending on the context. The answer from @tobassit is fundamentally correct but the way in which geoscientists and data processors refer to phase can cause some confusion. Firstly, in terms of early stage seismic processing, where raw signal/receiver data is processed, the term "phase" is can be used to refer to the degree of phase rotation in the seismic wavelet in a processed dataset. During the seismic processing workflow a deconvolution and phase correction step is normally applied and the "phase" of the data will be set as either zero phase or mixed phase. Understanding the "phase" of a seismic dataset is critical in its subsequent use as it affects the relationship between peaks-troughs in the data and underlying the reflectivity series. Secondly, in terms of post stack seismic data analysis, seismic traces are often represented by an Analytic Signal Model for the purposes of analysis and interpretation. The analytical seismic trace is created by computing the Hilbert Transform of the real trace to produce quadrature trace (90 degree phase shifted version). This together with the original provides a complex trace from which number of useful quantities can be derived. These are known as the complex trace attributes and include Instantaneous Phase (see link on wikipedia page above for the Analytic Signal) which is often referred to as 'phase' by seismic interpreters. The phase attribute is useful due to its directly relationship to subsurface structure independent of reflection strength.
The Warsaw circle $W$ http://en.wikipedia.org/wiki/Continuum_%28topology%29 is a counterexample for quite a number of too naive statements. The Warsaw circle can be defined as the subspace of the plane $R^2$ consisting of the graph of $y = \sin(1/x)$, for $x\in(0,1]$, the segment $[−1,1]$ in the $y$ axis, and an arc connecting $(1,\sin(1))$ and $(0,0)$ (which is otherwise disjoint from the graph and the segment). Some observations: $W$ is weakly contractible (because a map from a locally path connected space cannot ''go over the bad point''). Let $I$ denote the segment $[−1,1]$ in the $y$ axis. Then $W/I\cong S^1$ is just the usual circle, and thus we have a natural projection map $g:W \to S^1$. The point-preimages of $g$ are either points or, for a single point on $S^1$, a closed interval. Thus the assumptions of the Vietoris-Begle mapping theorem hold for $g$, proving that $g$ induces an isomorphism in Cech cohomology. Thus the Cech cohomology of $W$ is that of $S^1$, but it has the singular homology of a point, by Hurewicz. Since $I\to W$ is an embedding of compact Hausdorff spaces, we have an induced long exact sequence in (reduced) topological $K$-theory (see, for example, Atiyah's $K$-theory Proposition 2.4.4). Since $I$ is contractible, we get that $W$ and $S^1$ also have the same topological $K$-theory. Note that the Warsaw circle is a compact metrizable space, being a bounded closed subspace of $R^2$. By looking on points on $I$ one sees that $W$ is not locally path-connected (and, in particular, not locally contractible). The above observations imply: A map with contractible point-inverses does not need to be a weak homotopy equivalence, even if both, source and target, are compact metric spaces. Assuming that the base and the preimages are finite CW complexes does not help. The Vietoris-Begle Theorem is false for singular cohomology (in particular, the wikipedia version of that Theorem is not quite correct). The embedding $I\to W$ cannot be a cofibration in any model structure on $Top$, where the weak equivalences are the weak homotopy equivalences and the interval $I$ is cofibrant. Because then we would have a cofiber sequence $I\to W\to S^1$ and thus also a long exact sequence in singular cohomology. $W$ does not have the homotopy type of a CW complex (since it is not contractible). Even though the map $g$ is trivial on fundamental groups, it does not lift to the universal cover $p: \mathbb{R} \to S^1$, because $g$ cannot be nullhomotopic. Thus the assumption of local path connectivity in the lifting theorem is necessary.
Surely there are many: these are all polynomials in one variable, so every two of them are algebraically dependent because of the transcendence degree argument :-) However, I am sure that this is not what you wanted to hear, so here you are a nice argument showing how to guess your formula and obtain other formulas somewhat similar to it. Note that there is a remarkable symmetry property $P_k(-1-N)=(-1)^{k+1} P_k(N)$ for $k>0$. (Basically, for $k>0$ the polynomial $P_k(x)$ is the only polynomial of degree $k+1$ solving the functional equation $f(x)-f(x-1)=x^k$ together with the condition $f(0)=0$, and then you can show that $Q_k(x)=(-1)^{k+1} P_k(-1-x)$ satisfied exactly the same conditions, which proves the symmetry property without any annoying computations.) If we re-define $P_0(N)=N+\frac12$ (and assume $P_{-1}=1$), this symmetry will hold in general. Now, the polynomial $P_1^2$, as a polynomial of degree $4$, should be a rational combination of $P_0$, $P_1$, $P_2$ and $P_3$ (and such a combination is clearly unique - you yourself observed that they form a basis), and because of the type of symmetry it possesses, it is actually a combination of $P_1$ and $P_3$ (because other polynomials change sign under the symmetry $N\mapsto -1-N$, and this would contradict the linear independence), and looking at it carefully we observe that the $P_1$-coefficient is equal to zero, and the $P_3$-coefficient is equal to~$1$, which is your formula. For the same reason, the product $P_mP_n$ is expressed as a linear combination of $P_l$ where $l\le m+n+1$, $l\equiv m+n+1\pmod{2}$, - half of the terms disappear for free! (And, because of vanishing at~$0$, the redefined $P_0=N+\frac12$ and $P_{-1}=1$ will not show up in such a combination if $m+n>0$.) Some examples: $6P_1P_2=5P_4+P_2$, $3P_2^2=2P_5+P_3$, $12P_2P_3=7P_6+5P_4$, $2P_3^2=P_7+P_5$, $60P_3P_4=27P_8+35P_6-2P_4$ (this last one is a bit disappointing!) etc.
Let $a \in \Bbb{R}$. Let $\Bbb{Z}$ act on $S^1$ via $(n,z) \mapsto ze^{2 \pi i \cdot an}$. Claim: The is action is not free if and only if $a \Bbb{Q}$. Here's an attempt at the forward direction: If the action is not free, there is some nonzero $n$ and $z \in S^1$ such that $ze^{2 \pi i \cdot an} = 1$. Note $z = e^{2 \pi i \theta}$ for some $\theta \in [0,1)$. Then the equation becomes $e^{2 \pi i(\theta + an)} = 1$, which holds if and only if $2\pi (\theta + an) = 2 \pi k$ for some $k \in \Bbb{Z}$. Solving for $a$ gives $a = \frac{k-\theta}{n}$... What if $\theta$ is irrational...what did I do wrong? 'cause I understand that second one but I'm having a hard time explaining it in words (Re: the first one: a matrix transpose "looks" like the equation $Ax\cdot y=x\cdot A^\top y$. Which implies several things, like how $A^\top x$ is perpendicular to $A^{-1}x^\top$ where $x^\top$ is the vector space perpendicular to $x$.) DogAteMy: I looked at the link. You're writing garbage with regard to the transpose stuff. Why should a linear map from $\Bbb R^n$ to $\Bbb R^m$ have an inverse in the first place? And for goodness sake don't use $x^\top$ to mean the orthogonal complement when it already means something. he based much of his success on principles like this I cant believe ive forgotten it it's basically saying that it's a waste of time to throw a parade for a scholar or win he or she over with compliments and awards etc but this is the biggest source of sense of purpose in the non scholar yeah there is this thing called the internet and well yes there are better books than others you can study from provided they are not stolen from you by drug dealers you should buy a text book that they base university courses on if you can save for one I was working from "Problems in Analytic Number Theory" Second Edition, by M.Ram Murty prior to the idiots robbing me and taking that with them which was a fantastic book to self learn from one of the best ive had actually Yeah I wasn't happy about it either it was more than $200 usd actually well look if you want my honest opinion self study doesn't exist, you are still being taught something by Euclid if you read his works despite him having died a few thousand years ago but he is as much a teacher as you'll get, and if you don't plan on reading the works of others, to maintain some sort of purity in the word self study, well, no you have failed in life and should give up entirely. but that is a very good book regardless of you attending Princeton university or not yeah me neither you are the only one I remember talking to on it but I have been well and truly banned from this IP address for that forum now, which, which was as you might have guessed for being too polite and sensitive to delicate religious sensibilities but no it's not my forum I just remembered it was one of the first I started talking math on, and it was a long road for someone like me being receptive to constructive criticism, especially from a kid a third my age which according to your profile at the time you were i have a chronological disability that prevents me from accurately recalling exactly when this was, don't worry about it well yeah it said you were 10, so it was a troubling thought to be getting advice from a ten year old at the time i think i was still holding on to some sort of hopes of a career in non stupidity related fields which was at some point abandoned @TedShifrin thanks for that in bookmarking all of these under 3500, is there a 101 i should start with and find my way into four digits? what level of expertise is required for all of these is a more clear way of asking Well, there are various math sources all over the web, including Khan Academy, etc. My particular course was intended for people seriously interested in mathematics (i.e., proofs as well as computations and applications). The students in there were about half first-year students who had taken BC AP calculus in high school and gotten the top score, about half second-year students who'd taken various first-year calculus paths in college. long time ago tho even the credits have expired not the student debt though so i think they are trying to hint i should go back a start from first year and double said debt but im a terrible student it really wasn't worth while the first time round considering my rate of attendance then and how unlikely that would be different going back now @BalarkaSen yeah from the number theory i got into in my most recent years it's bizarre how i almost became allergic to calculus i loved it back then and for some reason not quite so when i began focusing on prime numbers What do you all think of this theorem: The number of ways to write $n$ as a sum of four squares is equal to $8$ times the sum of divisors of $n$ if $n$ is odd and $24$ times sum of odd divisors of $n$ if $n$ is even A proof of this uses (basically) Fourier analysis Even though it looks rather innocuous albeit surprising result in pure number theory @BalarkaSen well because it was what Wikipedia deemed my interests to be categorized as i have simply told myself that is what i am studying, it really starting with me horsing around not even knowing what category of math you call it. actually, ill show you the exact subject you and i discussed on mmf that reminds me you were actually right, i don't know if i would have taken it well at the time tho yeah looks like i deleted the stack exchange question on it anyway i had found a discrete Fourier transform for $\lfloor \frac{n}{m} \rfloor$ and you attempted to explain to me that is what it was that's all i remember lol @BalarkaSen oh and when it comes to transcripts involving me on the internet, don't worry, the younger version of you most definitely will be seen in a positive light, and just contemplating all the possibilities of things said by someone as insane as me, agree that pulling up said past conversations isn't productive absolutely me too but would we have it any other way? i mean i know im like a dog chasing a car as far as any real "purpose" in learning is concerned i think id be terrified if something didnt unfold into a myriad of new things I'm clueless about @Daminark They key thing if I remember correctly was that if you look at the subgroup $\Gamma$ of $\text{PSL}_2(\Bbb Z)$ generated by (1, 2\|0, 1) and (0, -1\|1, 0), then any holomorphic function $f : \Bbb H^2 \to \Bbb C$ invariant under $\Gamma$ (in the sense that $f(z + 2) = f(z)$ and $f(-1/z) = z^{2k} f(z)$, $2k$ is called the weight) such that the Fourier expansion of $f$ at infinity and $-1$ having no constant coefficients is called a cusp form (on $\Bbb H^2/\Gamma$). The $r_4(n)$ thing follows as an immediate corollary of the fact that the only weight $2$ cusp form is identically zero. I can try to recall more if you're interested. It's insightful to look at the picture of $\Bbb H^2/\Gamma$... it's like, take the line $\Re[z] = 1$, the semicircle $\|z\| = 1, z > 0$, and the line $\Re[z] = -1$. This gives a certain region in the upper half plane Paste those two lines, and paste half of the semicircle (from -1 to i, and then from i to 1) to the other half by folding along i Yup, that $E_4$ and $E_6$ generates the space of modular forms, that type of things I think in general if you start thinking about modular forms as eigenfunctions of a Laplacian, the space generated by the Eisenstein series are orthogonal to the space of cusp forms - there's a general story I don't quite know Cusp forms vanish at the cusp (those are the $-1$ and $\infty$ points in the quotient $\Bbb H^2/\Gamma$ picture I described above, where the hyperbolic metric gets coned off), whereas given any values on the cusps you can make a linear combination of Eisenstein series which takes those specific values on the cusps So it sort of makes sense Regarding that particular result, saying it's a weight 2 cusp form is like specifying a strong decay rate of the cusp form towards the cusp. Indeed, one basically argues like the maximum value theorem in complex analysis @BalarkaSen no you didn't come across as pretentious at all, i can only imagine being so young and having the mind you have would have resulted in many accusing you of such, but really, my experience in life is diverse to say the least, and I've met know it all types that are in everyway detestable, you shouldn't be so hard on your character you are very humble considering your calibre You probably don't realise how low the bar drops when it comes to integrity of character is concerned, trust me, you wouldn't have come as far as you clearly have if you were a know it all it was actually the best thing for me to have met a 10 year old at the age of 30 that was well beyond what ill ever realistically become as far as math is concerned someone like you is going to be accused of arrogance simply because you intimidate many ignore the good majority of that mate
I have some conceptual trouble understanding what the different field operators in QED do. According to Wikipedia, the field operators are given by $\mathbf A(\mathbf{r} )=\sum _{\mathbf {k} ,\mu }{\sqrt {\frac {\hbar }{2\omega V\epsilon _{0}}}}\left\{\mathbf {e} ^{(\mu )}a^{(\mu )}(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }+{\bar {\mathbf {e} }}^{(\mu )}{a^{\dagger }}^{(\mu )}(\mathbf {k} )e^{-i\mathbf {k} \cdot \mathbf {r} }\right\}\\\mathbf {E} (\mathbf {r} )=i\sum _{\mathbf {k} ,\mu }{\sqrt {\frac {\hbar \omega }{2V\epsilon _{0}}}}\left\{\mathbf {e} ^{(\mu )}a^{(\mu )}(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }-{\bar {\mathbf {e} }}^{(\mu )}{a^{\dagger }}^{(\mu )}(\mathbf {k} )e^{-i\mathbf {k} \cdot \mathbf {r} }\right\}\\\mathbf {B} (\mathbf {r} )=i\sum _{\mathbf {k} ,\mu }{\sqrt {\frac {\hbar }{2\omega V\epsilon _{0}}}}\left\{\left(\mathbf {k} \times \mathbf {e} ^{(\mu )}\right)a^{(\mu )}(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }-\left(\mathbf {k} \times {\bar {\mathbf {e} }}^{(\mu )}\right){a^{\dagger }}^{(\mu )}(\mathbf {k} )e^{-i\mathbf {k} \cdot \mathbf {r} }\right\}$ Now as I understand this, each of these operators can act on the vacuum and create a particle. My first question is: is this particle in each case a photon, which differs, however, by polarization? Second, I wonder if a classical B-field is then "made up" by a coherent superpsition of (virtual) photons with the "B-field-type" polarization? I am interested in these conceptual questions in the context of the conversion of an axion into a photon within an external B-field. This is described by the term $-g_{a\gamma\gamma}\mathbf E\cdot\mathbf B\,a$. So can I actually understand this equation as an axion interacting with one of the virtual "B-field-type" polarized photons present due to an external B-field to give then an "E-field-type" polarized photon? (By the clumsy expressions "B/"E-field-type" I mean photons that would be created by the respective operators given above) In summary, I think my confusion comes from the fact that in QED, one usually deals with $A_\mu$, so in my mind, photons were associated with $A_\mu$, like in the Feynman rule saying that $\overbracket{A_\mu\|\mathrm p\rangle}$ gives an external photon line. But in the axion coupling term, there are actually the E and B fields, and hence the question in the title: what do these field operators actually create? Any clarification is very much appreciated, thank you very much in advance!
Join our Whatsapp Notifications and Newsletters touch here COURTESY OF ATIKA SCHOOL WITHOUT USING MATHEMATICAL TABLES OR A CALCULATOR, EVALUATE; \[\large \frac{\sqrt[3]{675\times 135}}{\sqrt{2025}}\] Step 1: find the prime factors of each number. this step will help find the cube root of the nominator and the square root of the denominator \[\large \frac{\sqrt[3]{3^{3}\times 5^{2}\times 3^{3}\times 5}}{\sqrt{3^{4}\times 5^{2}}}\] Step 2: Get the cube root and the square root,then compute the answer \[\large \frac{\sqrt{3^{2}\times 5}}{\sqrt{3^{2}\times 5}}\] \[\large = 1\] Join our Whatsapp Notifications and Newsletters touch here COURTESY OF ATIKA SCHOOL wHAT ARE nATURAL NUMBERS? Natural numbers also called counting numbers are numbers ranging from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Place Value This is the position of a digit in a number. Importance of place values Place value helps students understand values of numbers in a series of numbers, such as computing the total value of a number in a group of numbers, rounding off numbers, changing numbers from figures to words vis a vis and operations on numbers. This is very essential in the counting of numbers as applied in science, real life situations, mathematics, business and accounting etc. example 1. What is the position of 6 in the number 346789 PROCEDURE FOR FINDING A PLACE VALUE exercises What is the place value of 5 in the number 524239 Hundred Thousands What is the place value of 1 in the number 721 Ones state the place values of digit 8 in each of the following numbers (a) 1689 Tens (b) 4008772 Thousands (c) 2847246 Hundred Thousands (d) 184392649 Ten Millions (e) 281199300505 Ten billions Details Author Archives Categories Join our Whatsapp Notifications and Newsletters touch here COURTESY OF ATIKA SCHOOL All
A density meter, also known as a densimeter, is a device that measures the density. Density is usually abbreviated as either {\displaystyle \rho } \rho or {\displaystyle D} D. Typically, density either has the units of {\displaystyle kg/m^{3}} {\displaystyle kg/m^{3}} or {\displaystyle lb/ft^{3}} {\displaystyle lb/ft^{3}}. The most basic principle of how density is calculated is by the formula: {\displaystyle \rho ={\frac {m}{V}}} {\displaystyle \rho ={\frac {m}{V}}} Where: {\displaystyle \rho } \rho = the density of the sample. {\displaystyle m} m = the mass of the sample. {\displaystyle V} V = the volume of the sample. Many density meters can measure both the wet portion and the dry portion of a sample. The wet portion comprises the density from all liquids present in the sample. The dry solids comprise solely of the density of the solids present in the sample. A density meter does not measure the specific gravity of a sample directly. However, the specific gravity can be inferred from a density meter. The specific gravity is defined as the density of a sample compared to the density of a reference. The reference density is typically of that of water. The specific gravity is found by the following equation: {\displaystyle SG_{s}={\frac {\rho _{s}}{\rho _{r}}}} {\displaystyle SG_{s}={\frac {\rho _{s}}{\rho _{r}}}} Where: {\displaystyle SG_{s}} {\displaystyle SG_{s}} = the specific gravity of the sample. {\displaystyle \rho _{s}} \rho_s = the density of the sample that needs to be measured. {\displaystyle \rho _{r}} {\displaystyle \rho _{r}} = the density of the reference material (usually water). Density meters come in many varieties. Different types include: nuclear, coriolis, ultrasound, microwave, and gravitic. Each type measures the density differently. Each type has its advantages and drawbacks. Density meters have many applications in various parts of various industries. Density meters are used to measure slurries, sludges, and other liquids that flow through the pipeline. Industries such as mining, dredging, wastewater treatment, paper, oil, and gas all have uses for density meters at various points during their respective processes.
MWG state the continuity axiom as follows: C1. $\succsim$ is continuous if for all $L,L',L''$, the two sets below are both closed: \begin{align} S&=\{\alpha\in[0,1]:\alpha L+(1-\alpha)L''\succsim L'\}\\ T&=\{\alpha\in[0,1]:L'\succsim \alpha L+(1-\alpha)L''\} \end{align} Other authors (e.g. Kreps, Rubinstein, Levin) use a somewhat different formulation: C2. $\succsim$ is continuous if for all $L,L',L''$ with $L\succsim L'\succsim L''$, there exists an $\alpha\in[0,1]$ such that \begin{equation} L'\sim \alpha L+(1-\alpha)L'' \end{equation} Are the two formulations equivalent? It's easy to see how C1 implies C2 (just take $\alpha\in S\cap T$). But I'm not sure how C2 implies C1. The proofs using C2 usually first establish that, with independence, we have \begin{equation}\beta L+(1-\beta)L'\succ \alpha L+(1-\alpha)L'\end{equation}whenever $L\succ L'$ and $1>\beta>\alpha>0$. Hence, C2 and independence together imply C1. But does C2 imply C1 without independence?
Summarising the state of the art of heat transfer knowledge isn't easy, considering small libraries have been filled on this subject alone but I'll give it a shot. Heat transfer occurs following three quite distinct mechanisms (or modes, if you prefer): 1. Radiative heat transfer: Jim's answer to this question deals adequately with this so I don't have to. 2. Heat conduction: When temperature gradients (generally $\nabla T$) exist in an object then by Fourier's law, heat conduction will strive to minimise these gradients and uniformise the temperature throughout the object. 3. Convection: Convection combines radiative and conductive heat transfer with mass transport of heat: a domestic radiator (for instance) heats up the air surrounding it, which then rises because its density has been lowered. Convection can also be forced by means of ventilators that force the air (or a liquid) to flow over the hot surface. Real world heat transfer almost always combines the three modes although usually with emphasis on one of them (the predominant mode). So, to answer the OP's question, which model (equation) to choose? Let's look at a hot sphere to focus attention (ignore the 'dent' due to poor drawing skills): We're looking at a cut straight through the centre of a hot sphere (hotter than the surrounding temperature $T_{\infty}$). The left hand side is a solid sphere and due to heat conduction the boundary of the sphere will be cooler than the core, as schematised by the temperature distribution curve $T(r,t)$, here at a specific time $t$. The flow of heat inside the sphere is governed by Fourier's heat equation: $$\frac{\partial T}{\partial t}=\frac{k}{\rho c_p}\nabla^2T$$ Where $\frac{k}{\rho c_p}=\kappa$ is the thermal diffusivity of the material. Heat flow from the boundary of the sphere to the surrounding medium is then either through convection, radiation or a combination of both. The right hand side is a hollow sphere filled with a liquid (or a fluid, more generally) that is constantly stirred. Due to this stirring there are no temperature gradients and thus no internal heat conduction: $$\nabla^2T=0$$ Heat flow from the boundary of the sphere to the surrounding medium is then either through convection, radiation or a combination of both. Usually Newton's cooling law can be used to model that situation. Choice of model: The scientist/engineer will have to choose to model his real world system according to which of the two options, $\nabla^2T=0$ or $\nabla^2T\neq0$, best describes his system. That choice will also be influenced by mathematical considerations: models that require use of Fourier's equation tend to be mathematically more demanding. Analytical solutions may not exist or be difficult to use. Some examples of using the Fourier equation for 1D heat transfer problems can be found in that link and illustrate the (relative) difficulty in obtaining analytical solutions. Numerical (computer) solutions may be preferred in many cases.
Perfect Square Trinomials are commonly introduced in the algebra course and it is entitled as Special Product in the Mathematics. This is named so because polynomials are grouped together in a unique way while factoring them. Here, in this post, we will discuss everything about perfect square Trinomials and how are they calculated. You must be wondering what is Square Footage exactly? This is the total area occupied within a room or building. A room may be comprised of different shapes like square, rectangle, Triangle etc. Square Footage is the special unit to express the area of a room or building. Here is given Square Footage Formula for Square or Triangular area, \[\large Square\;Footage=Length\times Breadth\] The Square Footage Formula for a triangular area is, \[\large Square\;Footage=\frac{Breadth\times Length}{2}\] With the basic understanding of Mathematics concepts, you must be sure what is a Square? When a number is multiplied by itself then it is named as Square. But it may sound difficult to calculate the Square Root for any particular number. It may be quite a lengthy process when numbers are given in decimals. Calculating square of nine is easy i.e. 3 but how will you calculate the square root for 5? Here, you need to follow the lengthy process and complicated too. Don’t forget to consider the properties while calculating square roots for different numbers. With square root property formulas, it becomes easy to calculate the value with simple steps without even need for a calculator. There are multiple square root properties and we have listed only a few of them. \[\ \sqrt{a}\cdot \sqrt{b}=\sqrt{a \times b}\] \[\ \sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\] \[\ \sqrt{n^{2}\cdot a}=n\sqrt{a}\] \[\ \sqrt{a}+\sqrt{b}\neq \sqrt{a+b}\] \[\ \sqrt{a}-\sqrt{b}\neq \sqrt{a-b}\] Based on the above properties and square root symbol, n is the index and a, b are vertices. Perfect squares in mathematics are the group of polynomials that are factored further in a convenient manner. It is also useful in solving tough mathematical equations. \[\large \left(a+b\right)^{2}=a^{2}+2ab+b^{2}\] Before you understand the concept of special products – perfect square trinomials, let us first discuss some basic terminologies. A perfect square is simple that is multiplied by itself. For binomial expressions, there are only two terms are available i.e. x + 5. When there is some algebraic expression containing more three terms then it will be named as Trinomial. For example – 5x 2 + 5x + 4 At the same time, perfect square trinomials are special algebraic expressions that are generated when binomial is usually multiplied by itself. For example – \[\large \left(3x+2y\right)^{2}=9x^{2}+12xy+4y^{2}\] Once you are sure on Trinomial expressions then solving tough mathematical problems would be simpler. They are also useful in graphical problems too. As we discussed, how to convert a binomial expression in a Trinomial equation. In the same way, any trinomial equation can be reversed and converted to binomial expressions too. When you write trinomial equations then there needs to be a positive and negative version of the expression, if this is not the case then you don’t have perfect square Trinomials.The Perfect Square Trinomial Formula in mathematics is given as, \[\large \left(ax+b\right)^{2}=(ax)^{2}+2abx+b^{2}\] \[\large \left(ax-b\right)^{2}=(ax)^{2}-2abx+b^{2}\]
These equations have the form \[{{y^{\left( n \right)}}\left( x \right) + {a_1}{y^{\left( {n – 1} \right)}}\left( x \right) + \cdots }+{ {a_{n – 1}}y’\left( x \right) + {a_n}y\left( x \right) }={ f\left( x \right),}\] where ${a_1},{a_2}, \ldots ,{a_n}$ are real or complex numbers, and the right-hand side $f\left( x \right)$ is a continuous function on some interval $\left[ {a,b} \right].$ Using the linear differential operator $L\left( D \right)$ equal to \[{L\left( D \right) }={ {D^n} + {a_1}{D^{n – 1}} + \cdots }+{ {a_{n – 1}}D + {a_n},}\] the nonhomogeneous differential equation can be written as \[L\left( D \right)y\left( x \right) = f\left( x \right).\] The general solution $y\left( x \right)$ of the nonhomogeneous equation is the sum of the general solution ${y_0}\left( x \right)$ of the corresponding homogeneous equation and a particular solution ${y_1}\left( x \right)$ of the nonhomogeneous equation: \[y\left( x \right) = {y_0}\left( x \right) + {y_1}\left( x \right).\] For an arbitrary right side $f\left( x \right)$, the general solution of the nonhomogeneous equation can be found using the method of variation of parameters. If the right-hand side is the product of a polynomial and exponential functions, it is more convenient to seek a particular solution by the method of undetermined coefficients. Method of Variation of Parameters We assume that the general solution of the homogeneous differential equation of the $n$th order is known and given by \[ {{y_0}\left( x \right) }={ {C_1}{Y_1}\left( x \right) }+{ {C_2}{Y_2}\left( x \right) + \cdots } + {{C_n}{Y_n}\left( x \right).} \] According to the method of variation of constants (or Lagrange method), we consider the functions ${C_1}\left( x \right),$ ${C_2}\left( x \right), \ldots ,$ ${C_n}\left( x \right)$ instead of the regular numbers ${C_1},$ ${C_2}, \ldots ,$ ${C_n}.$ These functions are chosen so that the solution \[ {y = {C_1}\left( x \right){Y_1}\left( x \right) }+{ {C_2}\left( x \right){Y_2}\left( x \right) + \cdots } + {{C_n}\left( x \right){Y_n}\left( x \right)} \] satisfies the original nonhomogeneous equation. The derivatives of $n$ unknown functions ${C_1}\left( x \right),$ ${C_2}\left( x \right), \ldots ,$ ${C_n}\left( x \right)$ are determined from the system of $n$ equations: \[\left\{ \begin{array}{l} {{C’_1}\left( x \right){Y_1}\left( x \right) }+{ {C’_2}\left( x \right){Y_2}\left( x \right) + \cdots }+{ {C’_n}\left( x \right){Y_n}\left( x \right) = 0}\\ {{C’_1}\left( x \right){Y’_1}\left( x \right) }+{ {C’_2}\left( x \right){Y’_2}\left( x \right) + \cdots }+{ {C’_n}\left( x \right){Y’_n}\left( x \right) = 0}\\ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\ {{C’_1}\left( x \right)Y_1^{\left( {n – 1} \right)}\left( x \right) }+{ {C’_2}\left( x \right)Y_2^{\left( {n – 1} \right)}\left( x \right) + \cdots }+{ {C’_n}\left( x \right)Y_n^{\left( {n – 1} \right)}\left( x \right) }={ f\left( x \right)} \end{array} \right.\] The determinant of this system is the Wronskian of ${Y_1},$ ${Y_2}, \ldots ,$ ${Y_n}$ forming a fundamental system of solutions. By the linear independence of these functions, the determinant is not zero and the system is uniquely solvable. The final expressions for the functions ${C_1}\left( x \right),$ ${C_2}\left( x \right), \ldots ,$ ${C_n}\left( x \right)$ can be found by integration. Method of Undetermined Coefficients If the right-hand side $f\left( x \right)$ of the differential equation is a function of the form \[ {{P_n}\left( x \right){e^{\alpha x}}\;\;\text{or}\;\;}\kern-0.3pt {\left[ {{P_n}\left( x \right)\cos \beta x }\right.}+{\left.{ {Q_m}\left( x \right)\sin\beta x} \right]{e^{\alpha x}},} \] where ${P_n}\left( x \right),$ ${Q_m}\left( x \right)$ are polynomials of degree $n$ and $m,$ respectively, then the method of undetermined coefficients may be used to find a particular solution. In this case, we seek a particular solution in the form corresponding to the structure of the right-hand side of the equation. For example, if the function has the form \[f\left( x \right) = {P_n}\left( x \right){e^{\alpha x}},\] the particular solution is given by \[{y_1}\left( x \right) = {x^s}{A_n}\left( x \right){e^{\alpha x}},\] where ${A_n}\left( x \right)$ is a polynomial of the same degree $n$ as ${P_n}\left( x \right).$ The coefficients of the polynomial ${A_n}\left( x \right)$ are determined by direct substitution of the trial solution ${y_1}\left( x \right)$ in the nonhomogeneous differential equation. In the so-called resonance case, when the number of $\alpha$ in the exponential function coincides with a root of the characteristic equation, an additional factor ${x^s},$ where s is the multiplicity of the root, appears in the particular solution. In the non-resonance case, we set $s = 0.$ The same algorithm is used when the right-hand side of the equation is given in the form \[{f\left( x \right) }={ \left[ {{P_n}\left( x \right)\cos \beta x }\right.}+{\left.{ {Q_m}\left( x \right)\sin\beta x} \right]{e^{\alpha x}}.}\] Here the particular solution has a similar structure and can be written as \[{{y_1}\left( x \right) }={ {x^s}\left[ {{A_n}\left( x \right)\cos \beta x }\right.}+{\left.{ {B_n}\left( x \right)\sin\beta x} \right]{e^{\alpha x}},}\] where ${{A_n}\left( x \right)},$ ${{B_n}\left( x \right)}$ are polynomials of degree $n$ (for $n \ge m$), and the degree $s$ in the additional factor ${x^s}$ is equal to the multiplicity of the complex root $\alpha \pm \beta i$ in the resonance case (i.e. when the numbers $\alpha$ and $\beta$ coincide with the complex root of the characteristic equation), and accordingly, $s = 0$ in the non-resonance case. Superposition Principle The superposition principle is stated as follows. Let the right-hand side $f\left( x \right)$ be the sum of two functions: \[f\left( x \right) = {f_1}\left( x \right) + {f_2}\left( x \right).\] Suppose that ${y_1}\left( x \right)$ is a solution of the equation \[L\left( D \right)y\left( x \right) = {f_1}\left( x \right),\] and the function ${y_2}\left( x \right)$ is, accordingly, a solution of the second equation \[L\left( D \right)y\left( x \right) = {f_2}\left( x \right).\] Then the sum of the functions \[y\left( x \right) = {y_1}\left( x \right) + {y_2}\left( x \right)\] will be a solution of the linear nonhomogeneous equation \[ {L\left( D \right)y\left( x \right) = f\left( x \right) } = {{f_1}\left( x \right) + {f_2}\left( x \right).} \] Solved Problems Click a problem to see the solution.
Consider a action functional $S(g_{\mu \nu}, \psi)$ with spacetime metric $g_{\mu \nu}$ and fermionic matter fields $\psi$, which is invariant under diffeomorphisms/ coordinate transformations. However, in the partition function $Z = \int D[g_{\mu \nu}] D[\psi] \mu e^{iS}$ the path integration measure $\mu$ is NOT diffeomorphism invariant and therefore, the covariant divergence of the energy-momentum tensor $T_{\mu \nu} = \frac{2 \delta S}{\sqrt{g} \delta g^{\mu \nu}}$ does not vanish. Now I assume the following model: There are only distributions of the metric field allowed which correspond to triangulations of spacetime. That means the quantum gravitational theory should have an isomorphism of categories: $c: Riem \rightarrow Tri$. Here, $Riem$ is the category of Riemannian manifolds and $Tri$ the category of discrete manifolds that arise from triangulation of Riemannian manifolds. Hence, $\mu$ must be an indicator function that has the value 1 if a distribution of metric fields $g_{\mu \nu}(x)$ can be mapped functorially to category $Tri$ and has value 0 if this is not possible. Mathematically, the map $c$ must be a forgetful functor that forgets some kind of metric field distributions. Question: Is that a plausible theory with diffeomorphism invariant action, but diffeomorphism non-invariant measure (a theory of gravitational anomaly)? Can on this way be constructed a quantum gravity theory (with category theoretic assumptions), e.g. causal triangulation theory?
Maybe I have this wrong, but I think that in $\mathcal{DK}$, filtered colimits at least are constructed in the following way. Let $I$ be a $\lambda$-directed poset for simplicity, and let $G: I \to \mathcal{DK}$, $i \mapsto G_i: D_i \to \mathcal{K}$ be a functor, with transition maps $(D_{ii'}: D_i \to D_{i'}, \gamma_{ii'}: G_i \Rightarrow G_{i'} \circ D_{ii'})$ for $i\leq i'$. Let $G_\infty : D_\infty \to \mathcal{K}$ denote the colimit of $G$. Then we should have $D_\infty = \varinjlim_i D_i$, so an object of $\varinjlim_{i \in I} D_i$ is an equivalence class $[(i,d)]$ where $i \in I$ and $d \in D_i$. And $G_\infty$ should be given by $G_\infty([(i,d)]) = \varinjlim_{i \leq i'} G_{i'}(D_{ii'}(d))$ (where the colimit is of a diagram constructed using the $\gamma$ maps; this colimit is $\lambda$-directed). Now suppose that $F: C \to \mathcal{K}$ is such that $C$ is $\lambda$-presentable, and $F$ takes values in the $\lambda$-presentable objects of $\mathcal{K}$. A map $F \to G_\infty$ will consist first of a functor $f: C \to D_\infty$; this data commutes with the colimit because $C$ is $\lambda$-presentable. Secondly, there will be a natural transformation $\mu: F \Rightarrow G_\infty \circ f$. In components this consists of maps $\mu_c : Fc \to G_\infty(fc)$; the data of this component commutes with the colimit because $Fc$ is $\lambda$-presentable. The data of the whole natural transformation then commutes with the colimit because $C$ is $\lambda$-presentable; I'm not sure how to argue this conceptually, but it follows from the idea that a $\lambda$-presentable category is generated by $<\lambda$-many morphisms subject to $<\lambda$-many relations. That is, $F$ is indeed $\lambda$-presentable, with slightly weaker hypotheses than requested. It's interesting that standard results about presentability and fibrations like Makkai-Paré Theorem 5.3.4 don't seem to quite apply here.
The Annals of Statistics Ann. Statist. Volume 1, Number 4 (1973), 780-785. Estimation of the Covariance Function of a Homogeneous Process on the Sphere Abstract A homogeneous random process on the sphere $\{X(P): P \in S_2\}$ is a process whose mean is zero and whose covariance function depends only on the angular distance $\theta$ between the two points, i.e. $E\lbrack X(P)\rbrack \equiv 0$ and $E\lbrack X(P)X(Q)\rbrack = R(\theta)$. Given $T$ independent realizations of a Gaussian homogeneous process $X(P)$, we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate $R^{(T)}(\theta)$ of the covariance function $R(\theta)$ is proposed. Exact expressions for its first- and second-order moments are derived and it is shown that the sequence of processes $\{T^{\frac{1}{2}}\lbrack R^{(T)}(\theta) - R(\theta)\rbrack\}^\infty_{T=1}$ converges weakly in $C\lbrack 0, \pi\rbrack$ to a given Gaussian process. Article information Source Ann. Statist., Volume 1, Number 4 (1973), 780-785. Dates First available in Project Euclid: 12 April 2007 Permanent link to this document https://projecteuclid.org/euclid.aos/1176342475 Digital Object Identifier doi:10.1214/aos/1176342475 Mathematical Reviews number (MathSciNet) MR334443 Zentralblatt MATH identifier 0263.62052 JSTOR links.jstor.org Citation Roy, Roch. Estimation of the Covariance Function of a Homogeneous Process on the Sphere. Ann. Statist. 1 (1973), no. 4, 780--785. doi:10.1214/aos/1176342475. https://projecteuclid.org/euclid.aos/1176342475
Let\[\mathbf{v}_{1}=\begin{bmatrix}1 \\ 1\end{bmatrix},\;\mathbf{v}_{2}=\begin{bmatrix}1 \\ -1\end{bmatrix}.\]Let $V=\Span(\mathbf{v}_{1},\mathbf{v}_{2})$. Do $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$ form an orthonormal basis for $V$? For a set $S$ and a vector space $V$ over a scalar field $\K$, define the set of all functions from $S$ to $V$\[ \Fun ( S , V ) = \{ f : S \rightarrow V \} . \] For $f, g \in \Fun(S, V)$, $z \in \K$, addition and scalar multiplication can be defined by\[ (f+g)(s) = f(s) + g(s) \, \mbox{ and } (cf)(s) = c (f(s)) \, \mbox{ for all } s \in S . \] (a) Prove that $\Fun(S, V)$ is a vector space over $\K$. What is the zero element? (b) Let $S_1 = \{ s \}$ be a set consisting of one element. Find an isomorphism between $\Fun(S_1 , V)$ and $V$ itself. Prove that the map you find is actually a linear isomorpism. (c) Suppose that $B = \{ e_1 , e_2 , \cdots , e_n \}$ is a basis of $V$. Use $B$ to construct a basis of $\Fun(S_1 , V)$. (d) Let $S = \{ s_1 , s_2 , \cdots , s_m \}$. Construct a linear isomorphism between $\Fun(S, V)$ and the vector space of $n$-tuples of $V$, defined as\[ V^m = \{ (v_1 , v_2 , \cdots , v_m ) \mid v_i \in V \mbox{ for all } 1 \leq i \leq m \} . \] (e) Use the basis $B$ of $V$ to constract a basis of $\Fun(S, V)$ for an arbitrary finite set $S$. What is the dimension of $\Fun(S, V)$? (f) Let $W \subseteq V$ be a subspace. Prove that $\Fun(S, W)$ is a subspace of $\Fun(S, V)$. Let $\mathrm{P}_3$ denote the set of polynomials of degree $3$ or less with real coefficients. Consider the ordered basis\[B = \left\{ 1+x , 1+x^2 , x – x^2 + 2x^3 , 1 – x – x^2 \right\}.\]Write the coordinate vector for the polynomial $f(x) = -3 + 2x^3$ in terms of the basis $B$. Let $V$ denote the vector space of $2 \times 2$ matrices, and $W$ the vector space of $3 \times 2$ matrices. Define the linear transformation $T : V \rightarrow W$ by\[T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = \begin{bmatrix} a+b & 2d \\ 2b – d & -3c \\ 2b – c & -3a \end{bmatrix}.\] For an integer $n > 0$, let $\mathrm{P}_n$ be the vector space of polynomials of degree at most $n$. The set $B = \{ 1 , x , x^2 , \cdots , x^n \}$ is a basis of $\mathrm{P}_n$, called the standard basis. Let $T : \mathrm{P}_n \rightarrow \mathrm{P}_{n+1}$ be the map defined by, for $f \in \mathrm{P}_n$,\[T (f) (x) = x f(x).\] Prove that $T$ is a linear transformation, and find its range and nullspace. Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$.Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$. Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$. (a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$. (b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$. Let $V$ be a vector space and $B$ be a basis for $V$.Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$. After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form\[\begin{bmatrix}1 & 0 & 2 & 1 & 0 \\0 & 1 & 3 & 0 & 1 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{bmatrix}.\] (a) What is the dimension of $V$? (b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$? (The Ohio State University, Linear Algebra Midterm) Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.Let\[W=\left\{\, A\in V \quad \middle \| \quad A=\begin{bmatrix}a & b\\c& -a\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Find the dimension of $W$. (The Ohio State University, Linear Algebra Midterm) Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]be a subset in $C[-1, 1]$. (a) Prove that $V$ is a subspace of $C[-1, 1]$. (b) Prove that the set $B=\{e^x, e^{2x}, e^{3x}\}$ is a basis of $V$. (c) Prove that\[B’=\{e^x-2e^{3x}, e^x+e^{2x}+2e^{3x}, 3e^{2x}+e^{3x}\}\]is a basis for $V$.
Given the population model by the following linear first order PDE in $u(a,t)$ with constants $b$ and $\mu$ : $$u_a + u_t = -\mu t u\,\,\,\,\,a,t>0$$ $$u(a,0)=u_0(a)\,\,\,a≥0$$ $$u(0,t)=F(t)=b\int_0^\infty u(a,t)\,\mathrm{d}a$$ We can split the integral in two with our non-local boundary data: $$F(t)=b\int_0^t u(a,t)\,\mathrm{d}a+b\int_t^\infty u(a,t)\,\mathrm{d}a$$ Choosing the characteristic coordinates $(\xi, \tau)$ and re-arranging the expression to form the normal to the solution surface we have the following equation with initial conditions: $$\bigl(u_a, u_t, -1\bigr) \bullet \bigl(1, 1, -\mu t u \bigr)=0$$ $$x(0)=\xi, \,\,\,t(0)= 0,\,\,\, u(0)=u_0(\xi)$$ Characteristic equations: $$\frac{\mathrm{d}a}{\mathrm{d}\tau}=1, \,\,\,\frac{\mathrm{d}t}{\mathrm{d}\tau}=1, \,\,\,\frac{\mathrm{d}u}{\mathrm{d}\tau}=-\mu tu$$ Solving each of these ODE's in $\tau$ gives the following: $$(1)\int \mathrm{d}a=\int \mathrm{d}\tau \,\,\,\,\,\,\,\,\,\,(2)\int \mathrm{d}t=\int \mathrm{d}\tau\,\,\,\,\,\,\,\,\,\,(3)\int \mathrm{d}u=-\int \mu tu\,\mathrm{d}\tau$$ $$a = \tau + F(\xi)\,\,\,\,\,\,\,\,\,\,t=\tau + F(\xi)$$ $$\therefore a=\tau + \xi \,\,\,\,\,\,\,\,\,\,\therefore t=\tau$$ $$\int \mathrm{d}u=-\int \mu \tau u\,\mathrm{d}\tau$$ $$\int \frac{1}{u}\,\mathrm{d}u=-\int \mu \tau \,\mathrm{d}\tau$$ $$\ln u = - \frac{1}{2}\mu \tau ^2+ F(\xi)$$ $$u = G(\xi)\,e^{-\frac{1}{2}\mu \tau^2}$$ $$\therefore u = u_0(\xi)\,e^{-\frac{1}{2}\mu \tau^2}$$ Substituting back the original coordinates we can re-write this expression with a coordinate change: $$\xi = a-t \,\,\,\,\,\,\,\,\,\,\tau = t$$ $$\therefore u(a,t)=u_0(a-t)\,e^{-\frac{1}{2} t^2}$$ Now this is where I get stuck, how do I use the boundary data to come up with a well-posed solution? $$u(0,t)=u_0(-t)\,e^{-\frac{1}{2}\mu t^2}=b\int_0^t u(a,t)\,\mathrm{d}a+b\int_t^\infty u(a,t)\,\mathrm{d}a$$ Very unsure how I can evaluate this integral...
I don't know how to solve the following: Let $\alpha$ be a real root of $f(x)=x^4+3x-3\in Q[x]$. Is $\alpha$ a constructible number? Any help is welcome. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community I tried to get some info about the Galois group $G$ of $f$ (over $\mathbb Q$). It turns out that $f$ has a root $-1$ in $\mathbb F_5$ and $f(x)/(x+1)$ is irreducible in $\mathbb F_5[x]$. The group $G\subset S_4$ thus contains a $3$-cycle (Frobenius at $p=5$), in particular the order of $G$ is not a power of $2$, so the roots of $f$ are not constructible.
I am attempting to compute the (integral) cohomology ring structure of the 3 configuration space of $\mathbb{R}^m$ and have run into a few doubts. Using a result of Fadell and Neuwirth, we have that conf($\mathbb{R}^m$, 3) is the total space of the fibration: p: conf($\mathbb{R}^m$, 3) $\rightarrow$ conf($\mathbb{R}^m$, 2) given by the obvious map onto the first 2 factors with fibre homeomorphic to $\mathbb{R}^m - Q$ where $Q$ is two points. Since $\mathbb{R}^m - Q $ is homotopy equivalent to a wedge of two $m-1$ spheres, it has cohomology groups $\mathbb{Z}$ and $\mathbb{Z}\bigoplus \mathbb{Z}$ in dimensions $0$ and $m-1$ respectfully (Mayer-Vietoris). Similarly conf($\mathbb{R}^m$, 2) is homotopy equivalent to $S^{m-1}$ where a map is $(x,y) \rightarrow \frac{(x-y)}{\|x-y]}$. Assuming trival local coefficients (where the only problem could arise when $m=2$ since otherwise the base is simply connected) we can apply the Serre spectral sequence with $E_2^{p,q} = H^p(S^{m-1}, H^q(\vee_2 S^{m-1} )) $. An application of the UCT (since everything is free) gives that $E_2^{p,q} = H^p(S^{m-1}) \otimes H^q(\vee_2 S^{m-1} ) $ We then have that $E_2 = E_{\infty}$ since we only have cohomology in degree $0$ and $m-1$ for the base and fibre and hence there can be no non-trivial differentials. This leads me to a few questions. In general the $E_{\infty}$ product structure doesn't determine the product structure on the total space (see for example p 29 SSAT ) but things work out in this instance since everything is a free $\mathbb{Z}$ module and of finite type? What is a necessary and sufficient condition on the $E_{\infty}$ page structure to guarantee that the product structures coincide? I know that $H^(S^{m-1}) = \mathbb{Z}[a_1]/(a_1^2)$ where $\|a_1\| = m-1$ and that $H^(S^{m-1} \vee S^{m-1}) = \mathbb{Z}[a_2]/(a_2^2) \times \mathbb{Z}[a_3]/(a_3^2)$ where $\|a_2\|=\|a_3\| = m-1$. Does this simply imply that $H^*($conf($\mathbb{R}^m$, 3) ) $\simeq \mathbb{Z}[a_1]/(a_1^2)$ $\otimes \mathbb{Z}[a_2]/(a_2^2) \times \mathbb{Z}[a_3]/(a_3^2)\simeq \mathbb{Z}[a_1,a_2,a_3]/(a_1^2=a_2^2=a_3^2)$? Am I missing something?
Given that $\cos{160^{\circ}} = -q$, express $\cos70^{\circ}$ in terms of $q$. No example in the book, don't know how to do it?? I need a complete explanation. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community $$\cos(160^{\circ})=-q$$ $$\cos(180^{\circ}-20^{\circ})=-q$$ $$-\cos(20^{\circ})=-q$$ Because $\cos(180^{\circ}-x)=-\cos(x)$ $$\cos(20^{\circ})=q$$ $$\cos(70^{\circ})=\cos(90^{\circ}-20^{\circ})=\sin(20^{\circ})=\sqrt{1-q^2}$$ First because $\cos(90^{\circ}-x)=\sin(x)$ and second $\cos^2(x)+\sin^2(x)=1$ and the fact $20 ^{\circ}$ is in the first quadrant (and then is positive) Hint: $\cos 70^\circ=\cos\, (160^\circ - 90^\circ)=\sin 160^\circ$.
Mertens' Theorem says: $$\lim_{n \rightarrow \infty} \ \frac{1}{\log p_n} \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p_k}}} = e^{\gamma}.$$ Euler's product formula for the $\zeta$ function and his evaluation of $\zeta(2) = \pi^2/6$ says that $$\zeta(2) = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{1 - \displaystyle{\frac{1}{p^2_k}}} = \lim_{n \rightarrow \infty} \ \prod_{k=1}^{n} \frac{1}{\left(1 - \displaystyle{\frac{1}{p_k}}\right)\left(1 + \displaystyle{\frac{1}{p_k}}\right)} = \frac{\pi^2}{6}.$$ Your result is the second limit divided by the first. The derivation of this is so immediate that I doubt you will find it as a "result" in the literature. The correct people to cite are Mertens and Euler.
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem. Yeah it does seem unreasonable to expect a finite presentation Let (V, b) be an n-dimensional, non-degenerate symmetric bilinear space over a field with characteristic not equal to 2. Then, every element of the orthogonal group O(V, b) is a composition of at most n reflections. How does the Evolute of an Involute of a curve $\Gamma$ is $\Gamma$ itself?Definition from wiki:-The evolute of a curve is the locus of all its centres of curvature. That is to say that when the centre of curvature of each point on a curve is drawn, the resultant shape will be the evolute of th... Player $A$ places $6$ bishops wherever he/she wants on the chessboard with infinite number of rows and columns. Player $B$ places one knight wherever he/she wants.Then $A$ makes a move, then $B$, and so on...The goal of $A$ is to checkmate $B$, that is, to attack knight of $B$ with bishop in ... Player $A$ chooses two queens and an arbitrary finite number of bishops on $\infty \times \infty$ chessboard and places them wherever he/she wants. Then player $B$ chooses one knight and places him wherever he/she wants (but of course, knight cannot be placed on the fields which are under attack ... The invariant formula for the exterior product, why would someone come up with something like that. I mean it looks really similar to the formula of the covariant derivative along a vector field for a tensor but otherwise I don't see why would it be something natural to come up with. The only places I have used it is deriving the poisson bracket of two one forms This means starting at a point $p$, flowing along $X$ for time $\sqrt{t}$, then along $Y$ for time $\sqrt{t}$, then backwards along $X$ for the same time, backwards along $Y$ for the same time, leads you at a place different from $p$. And upto second order, flowing along $[X, Y]$ for time $t$ from $p$ will lead you to that place. Think of evaluating $\omega$ on the edges of the truncated square and doing a signed sum of the values. You'll get value of $\omega$ on the two $X$ edges, whose difference (after taking a limit) is $Y\omega(X)$, the value of $\omega$ on the two $Y$ edges, whose difference (again after taking a limit) is $X \omega(Y)$ and on the truncation edge it's $\omega([X, Y])$ Gently taking caring of the signs, the total value is $X\omega(Y) - Y\omega(X) - \omega([X, Y])$ So value of $d\omega$ on the Lie square spanned by $X$ and $Y$ = signed sum of values of $\omega$ on the boundary of the Lie square spanned by $X$ and $Y$ Infinitisimal version of $\int_M d\omega = \int_{\partial M} \omega$ But I believe you can actually write down a proof like this, by doing $\int_{I^2} d\omega = \int_{\partial I^2} \omega$ where $I$ is the little truncated square I described and taking $\text{vol}(I) \to 0$ For the general case $d\omega(X_1, \cdots, X_{n+1}) = \sum_i (-1)^{i+1} X_i \omega(X_1, \cdots, \hat{X_i}, \cdots, X_{n+1}) + \sum_{i < j} (-1)^{i+j} \omega([X_i, X_j], X_1, \cdots, \hat{X_i}, \cdots, \hat{X_j}, \cdots, X_{n+1})$ says the same thing, but on a big truncated Lie cube Let's do bullshit generality. $E$ be a vector bundle on $M$ and $\nabla$ be a connection $E$. Remember that this means it's an $\Bbb R$-bilinear operator $\nabla : \Gamma(TM) \times \Gamma(E) \to \Gamma(E)$ denoted as $(X, s) \mapsto \nabla_X s$ which is (a) $C^\infty(M)$-linear in the first factor (b) $C^\infty(M)$-Leibniz in the second factor. Explicitly, (b) is $\nabla_X (fs) = X(f)s + f\nabla_X s$ You can verify that this in particular means it's a pointwise defined on the first factor. This means to evaluate $\nabla_X s(p)$ you only need $X(p) \in T_p M$ not the full vector field. That makes sense, right? You can take directional derivative of a function at a point in the direction of a single vector at that point Suppose that $G$ is a group acting freely on a tree $T$ via graph automorphisms; let $T'$ be the associated spanning tree. Call an edge $e = \{u,v\}$ in $T$ essential if $e$ doesn't belong to $T'$. Note: it is easy to prove that if $u \in T'$, then $v \notin T"$ (this follows from uniqueness of paths between vertices). Now, let $e = \{u,v\}$ be an essential edge with $u \in T'$. I am reading through a proof and the author claims that there is a $g \in G$ such that $g \cdot v \in T'$? My thought was to try to show that $orb(u) \neq orb (v)$ and then use the fact that the spanning tree contains exactly vertex from each orbit. But I can't seem to prove that orb(u) \neq orb(v)... @Albas Right, more or less. So it defines an operator $d^\nabla : \Gamma(E) \to \Gamma(E \otimes T^M)$, which takes a section $s$ of $E$ and spits out $d^\nabla(s)$ which is a section of $E \otimes T^M$, which is the same as a bundle homomorphism $TM \to E$ ($V \otimes W^* \cong \text{Hom}(W, V)$ for vector spaces). So what is this homomorphism $d^\nabla(s) : TM \to E$? Just $d^\nabla(s)(X) = \nabla_X s$. This might be complicated to grok first but basically think of it as currying. Making a billinear map a linear one, like in linear algebra. You can replace $E \otimes T^*M$ just by the Hom-bundle $\text{Hom}(TM, E)$ in your head if you want. Nothing is lost. I'll use the latter notation consistently if that's what you're comfortable with (Technical point: Note how contracting $X$ in $\nabla_X s$ made a bundle-homomorphsm $TM \to E$ but contracting $s$ in $\nabla s$ only gave as a map $\Gamma(E) \to \Gamma(\text{Hom}(TM, E))$ at the level of space of sections, not a bundle-homomorphism $E \to \text{Hom}(TM, E)$. This is because $\nabla_X s$ is pointwise defined on $X$ and not $s$) @Albas So this fella is called the exterior covariant derivative. Denote $\Omega^0(M; E) = \Gamma(E)$ ($0$-forms with values on $E$ aka functions on $M$ with values on $E$ aka sections of $E \to M$), denote $\Omega^1(M; E) = \Gamma(\text{Hom}(TM, E))$ ($1$-forms with values on $E$ aka bundle-homs $TM \to E$) Then this is the 0 level exterior derivative $d : \Omega^0(M; E) \to \Omega^1(M; E)$. That's what a connection is, a 0-level exterior derivative of a bundle-valued theory of differential forms So what's $\Omega^k(M; E)$ for higher $k$? Define it as $\Omega^k(M; E) = \Gamma(\text{Hom}(TM^{\wedge k}, E))$. Just space of alternating multilinear bundle homomorphisms $TM \times \cdots \times TM \to E$. Note how if $E$ is the trivial bundle $M \times \Bbb R$ of rank 1, then $\Omega^k(M; E) = \Omega^k(M)$, the usual space of differential forms. That's what taking derivative of a section of $E$ wrt a vector field on $M$ means, taking the connection Alright so to verify that $d^2 \neq 0$ indeed, let's just do the computation: $(d^2s)(X, Y) = d(ds)(X, Y) = \nabla_X ds(Y) - \nabla_Y ds(X) - ds([X, Y]) = \nabla_X \nabla_Y s - \nabla_Y \nabla_X s - \nabla_{[X, Y]} s$ Voila, Riemann curvature tensor Well, that's what it is called when $E = TM$ so that $s = Z$ is some vector field on $M$. This is the bundle curvature Here's a point. What is $d\omega$ for $\omega \in \Omega^k(M; E)$ "really"? What would, for example, having $d\omega = 0$ mean? Well, the point is, $d : \Omega^k(M; E) \to \Omega^{k+1}(M; E)$ is a connection operator on $E$-valued $k$-forms on $E$. So $d\omega = 0$ would mean that the form $\omega$ is parallel with respect to the connection $\nabla$. Let $V$ be a finite dimensional real vector space, $q$ a quadratic form on $V$ and $Cl(V,q)$ the associated Clifford algebra, with the $\Bbb Z/2\Bbb Z$-grading $Cl(V,q)=Cl(V,q)^0\oplus Cl(V,q)^1$. We define $P(V,q)$ as the group of elements of $Cl(V,q)$ with $q(v)\neq 0$ (under the identification $V\hookrightarrow Cl(V,q)$) and $\mathrm{Pin}(V)$ as the subgroup of $P(V,q)$ with $q(v)=\pm 1$. We define $\mathrm{Spin}(V)$ as $\mathrm{Pin}(V)\cap Cl(V,q)^0$. Is $\mathrm{Spin}(V)$ the set of elements with $q(v)=1$? Torsion only makes sense on the tangent bundle, so take $E = TM$ from the start. Consider the identity bundle homomorphism $TM \to TM$... you can think of this as an element of $\Omega^1(M; TM)$. This is called the "soldering form", comes tautologically when you work with the tangent bundle. You'll also see this thing appearing in symplectic geometry. I think they call it the tautological 1-form (The cotangent bundle is naturally a symplectic manifold) Yeah So let's give this guy a name, $\theta \in \Omega^1(M; TM)$. It's exterior covariant derivative $d\theta$ is a $TM$-valued $2$-form on $M$, explicitly $d\theta(X, Y) = \nabla_X \theta(Y) - \nabla_Y \theta(X) - \theta([X, Y])$. But $\theta$ is the identity operator, so this is $\nabla_X Y - \nabla_Y X - [X, Y]$. Torsion tensor!! So I was reading this thing called the Poisson bracket. With the poisson bracket you can give the space of all smooth functions on a symplectic manifold a Lie algebra structure. And then you can show that a symplectomorphism must also preserve the Poisson structure. I would like to calculate the Poisson Lie algebra for something like $S^2$. Something cool might pop up If someone has the time to quickly check my result, I would appreciate. Let $X_{i},....,X_{n} \sim \Gamma(2,\,\frac{2}{\lambda})$ Is $\mathbb{E}([\frac{(\frac{X_{1}+...+X_{n}}{n})^2}{2}] = \frac{1}{n^2\lambda^2}+\frac{2}{\lambda^2}$ ? Uh apparenty there are metrizable Baire spaces $X$ such that $X^2$ not only is not Baire, but it has a countable family $D_\alpha$ of dense open sets such that $\bigcap_{\alpha<\omega}D_\alpha$ is empty @Ultradark I don't know what you mean, but you seem down in the dumps champ. Remember, girls are not as significant as you might think, design an attachment for a cordless drill and a flesh light that oscillates perpendicular to the drill's rotation and your done. Even better than the natural method I am trying to show that if $d$ divides $24$, then $S_4$ has a subgroup of order $d$. The only proof I could come up with is a brute force proof. It actually was too bad. E.g., orders $2$, $3$, and $4$ are easy (just take the subgroup generated by a 2 cycle, 3 cycle, and 4 cycle, respectively); $d=8$ is Sylow's theorem; $d=12$, take $d=24$, take $S_4$. The only case that presented a semblance of trouble was $d=6$. But the group generated by $(1,2)$ and $(1,2,3)$ does the job. My only quibble with this solution is that it doesn't seen very elegant. Is there a better way? In fact, the action of $S_4$ on these three 2-Sylows by conjugation gives a surjective homomorphism $S_4 \to S_3$ whose kernel is a $V_4$. This $V_4$ can be thought as the sub-symmetries of the cube which act on the three pairs of faces {{top, bottom}, {right, left}, {front, back}}. Clearly these are 180 degree rotations along the $x$, $y$ and the $z$-axis. But composing the 180 rotation along the $x$ with a 180 rotation along the $y$ gives you a 180 rotation along the $z$, indicative of the $ab = c$ relation in Klein's 4-group Everything about $S_4$ is encoded in the cube, in a way The same can be said of $A_5$ and the dodecahedron, say
It was not hard to google that the simpler sum $\displaystyle \sum_{p < N} \frac{1}{p} = c + \log \log N $ so it is divergent. The sum of reciprocals of primes squared also converges $\displaystyle \sum_p \frac{1}{p^2} = 0.4522\dots$ Does $\displaystyle \sum_{p} \frac{1}{p} \left\{ \frac{N}{p} \right\} $ stay bounded as $N$ gets large? What about $\displaystyle \sum_{p,q } \frac{1}{pq} \left\{ \frac{N}{pq} \right\} $ for large $N$? Maybe it's necessary to say $p,q < N$. This is related to my earlier question on the Euler $\phi$-function
I'm sure there must be an easy way to do this, but given the Fourier transform of an isotropic filter kernel, $\hat{f}(\mathbf{u}) = \mathcal{F}f(\mathbf{z})$, can one calculate the value of the kernel at $\mathbf{z} = 0$? Since $$f(\mathbf{z})=\int_{\mathbf{R}^n}\hat{f}(\mathbf{u})e^{2\pi i\mathbf{z}\cdot\mathbf{u}}\;d\mathbf{u}$$ $$f(\mathbf{0})=\int_{\mathbf{R}^n}\hat{f}(\mathbf{u})\;d\mathbf{u}$$ So you simply integrate (or sum in the discrete case) over $\hat{f}(\mathbf{u})$.
I second Willk's answer: gravity doesn't really matter at all for plasma containment, so trying to build a tokamak in orbit would be a huge complication for basically no gain. However, I just took a class on plasma physics so I would be remiss if I didn't cram a bunch more math down peoples' throats. Now, the main equation that will be governing plasma movement in a tokomak on Earth is $$m_j n_j \frac{D\mathbf{v_j}}{Dt}=q_j n_j (\mathbf{E+v_j \times B})-\nabla p + m_j n_j \mathbf{g}$$ where $m$ is the mass per particle, $n$ the number density, $q$ the particle charge, $\mathbf{v}$ the fluid velocity, $p$ the pressure, $\mathbf{E}$ the electric field, $\mathbf{B}$ the magnetic field, $\mathbf{g}$ the gravitational field, and the subscript denoting which species we are talking about (normally ion vs electron). Now I know that was a whole bunch to dump at once, but I have a very simple goal here: to show you that that the term involving $\mathbf{g}$ (the gravitational force term) is much smaller than the other forces at play. You see, the monstrous equation I gave is really nothing more than a dressed up version of Newton's second law: $\mathbf{F} = m \mathbf{a}$. The left hand side is called the convective derivative and decribes how the plasma is being pushed around (analogous to $\mathbf{a}$), while the right hand side lists the forces acting on the plasma. So, let's get a rough sense of the orders of magnitude that we have for the forces. First off, we will ignore the electric field, since that tends to be approximately zero in steady state plasmas due to a phenomenon called Debye shielding. I'm also going to ignore the term involving the magnetic field because that's the thing we want to adjust. So, we want to analyze the approximate magnitude of the term $$\nabla p$$which for thermodynamics reasons is equivalent to$$\gamma kT \nabla n$$ The plasma recombines at the walls of the vessel, so $n=0$ there. Meanwhile, at the center of a typical fusion plasma we have a typical value of $n=10^{19} m^{-3}$, and a cross sectional radius of maybe $1 m$, giving us an approximate gradient of $10^{19} m^{-4}$. Using the approximation of an isothermal plasma with $\gamma = 1$, and ITER's projected temperature of $kT = 8 keV$, we obtain $$\nabla p \approx 13 \times 10^3 N/m^3$$ Now, to compare this to the gravitational term. Using the heaviest particle mass (that of tritium ions in the case of ITER) and $n=10^{19} m^{-3}$, we get$$m n \mathbf{g} \approx 5 \times 10^{-7}N/m^3$$which is over 10 orders of magnitude less than the force felt due to pressure gradients! So, when you're designing the magnetic field topology, you can pretty safely ignore gravity. As for an intuitive reason: fusion plasmas are hot. This means particles are bouncing around incredibly quickly, and they bounce off each other so frequently that gravity has basically no time to alter their trajectory in any noticeable way. This is much like how you don't really need to worry about gravity when you're shooting at a target 10 feet away-- the bullet moves so fast that it doesn't really matter.
A Circle is defined as the collection of points where each point is targeted at equal distance from the center. There are a number of terminologies necessary to understand to get an in-depth learning of Circle. The surface area of a circle is the total space defined within boundaries of a circle. Since, the circle is a two-dimensional figure, in most of the cases area and surface area would be the same. This is easy to calculate the surface area of a circle when either radius, diameter, or circumference is known. For example, a circle of radius r has a total surface area of A square units and it can be calculated by using the following formula – \[\large Surface\;Area\;of\;a\;circle: A=\pi \times r^{2}\] Hence, this is clear that surface area of a circle directly depends on the radius r and value of π is fixed. As we discussed earlier, the circle is a two-dimensional figure, in most of the cases area and surface area would be the same. For example, the area A of a circle is the multiplication value of π R2. \[\large Area\;of\;a\;circle =\pi r^{2}=\frac{{\pi}d^{2}}{4}=\frac{{C} \times {r}}{2}\] \[\large Area\;of\;a\;half\;circle =\frac{{\pi}r^{2}}{2}\] \[\large Area\;of\;a\;Quarter\;circle =\frac{{\pi}r^{2}}{4}\] Where, r is the radius of the circle. d is the diameter of the circle. C is the circumference of the circle. Where r is the radius, and the area A of a circle is π times the squared radius, and π = 3.14. Sectors are the parts of a circle that are enclosed by an arc, two radii drawn to the extremities of the arc is named as the sector. The total space defined within boundaries of a sector is named as the area of sector of a circle and it is measured in terms of square units. \[\large Area\;of\;Sector\;of\;a\;circle: A=\frac{θ}{360}\pi r^{2}\] \[\large Length\;of\;an\;arc\;of\;a\;sector: Arc=\frac{θ}{360}{2 \pi r}\] Where, θ is the Angle of Sector of the circle. r is the Radius of the circle. This is clear from the diagram that each segment is bounded by two radium and arc. So, the perimeter of a segment would be defined as the length of arcs (major and minor) plus the sum of both the radius. And the area of the segment is generally defined in radians or degrees. A circle whose central angle is measured in termed of θ degrees then the area of a segment would be – \[\large Area\;of\;Radius:=\frac{1}{2} r^{2}(θ – Sinθ)\] Where, θ is the Angle of Sector of the circle. r is the Radius of the circle. A circle whose central angle is measured in termed of radians then the area of a segment would be – Area of sector = lr2 Where r is the radius of a circle and l is the length of the arc.
(a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions. Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$.If\[A\begin{bmatrix}1 \\3 \\5\end{bmatrix}=B\begin{bmatrix}2 \\6 \\10\end{bmatrix},\]then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. Let $A, B, C$ be $n\times n$ invertible matrices. When you simplify the expression\[C^{-1}(AB^{-1})^{-1}(CA^{-1})^{-1}C^2,\]which matrix do you get?(a) $A$(b) $C^{-1}A^{-1}BC^{-1}AC^2$(c) $B$(d) $C^2$(e) $C^{-1}BC$(f) $C$ Let $\calP_3$ be the vector space of all polynomials of degree $3$ or less.Let\[S=\{p_1(x), p_2(x), p_3(x), p_4(x)\},\]where\begin{align}p_1(x)&=1+3x+2x^2-x^3 & p_2(x)&=x+x^3\\p_3(x)&=x+x^2-x^3 & p_4(x)&=3+8x+8x^3.\end{align} (a) Find a basis $Q$ of the span $\Span(S)$ consisting of polynomials in $S$. (b) For each polynomial in $S$ that is not in $Q$, find the coordinate vector with respect to the basis $Q$. (The Ohio State University, Linear Algebra Midterm) Let $V$ be a vector space and $B$ be a basis for $V$.Let $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ be vectors in $V$.Suppose that $A$ is the matrix whose columns are the coordinate vectors of $\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5$ with respect to the basis $B$. After applying the elementary row operations to $A$, we obtain the following matrix in reduced row echelon form\[\begin{bmatrix}1 & 0 & 2 & 1 & 0 \\0 & 1 & 3 & 0 & 1 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{bmatrix}.\] (a) What is the dimension of $V$? (b) What is the dimension of $\Span\{\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4, \mathbf{w}_5\}$? (The Ohio State University, Linear Algebra Midterm) Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.Let\[W=\left\{\, A\in V \quad \middle \| \quad A=\begin{bmatrix}a & b\\c& -a\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of $V$. (b) Find a basis of $W$. (c) Find the dimension of $W$. (The Ohio State University, Linear Algebra Midterm) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).The time limit was 55 minutes. This post is Part 3 and contains Problem 7, 8, and 9.Check out Part 1 and Part 2 for the rest of the exam problems. Problem 7. Let $A=\begin{bmatrix}-3 & -4\\8& 9\end{bmatrix}$ and $\mathbf{v}=\begin{bmatrix}-1 \\2\end{bmatrix}$. (a) Calculate $A\mathbf{v}$ and find the number $\lambda$ such that $A\mathbf{v}=\lambda \mathbf{v}$. (b) Without forming $A^3$, calculate the vector $A^3\mathbf{v}$. Problem 8. Prove that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular. Problem 9.Determine whether each of the following sentences is true or false. (a) There is a $3\times 3$ homogeneous system that has exactly three solutions. (b) If $A$ and $B$ are $n\times n$ symmetric matrices, then the sum $A+B$ is also symmetric. (c) If $n$-dimensional vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ are linearly dependent, then the vectors $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4$ is also linearly dependent for any $n$-dimensional vector $\mathbf{v}_4$. (d) If the coefficient matrix of a system of linear equations is singular, then the system is inconsistent. The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).The time limit was 55 minutes. This post is Part 2 and contains Problem 4, 5, and 6.Check out Part 1 and Part 3 for the rest of the exam problems. Problem 4. Let\[\mathbf{a}_1=\begin{bmatrix}1 \\2 \\3\end{bmatrix}, \mathbf{a}_2=\begin{bmatrix}2 \\-1 \\4\end{bmatrix}, \mathbf{b}=\begin{bmatrix}0 \\a \\2\end{bmatrix}.\] Find all the values for $a$ so that the vector $\mathbf{b}$ is a linear combination of vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. Problem 5.Find the inverse matrix of\[A=\begin{bmatrix}0 & 0 & 2 & 0 \\0 &1 & 0 & 0 \\1 & 0 & 0 & 0 \\1 & 0 & 0 & 1\end{bmatrix}\]if it exists. If you think there is no inverse matrix of $A$, then give a reason. Problem 6.Consider the system of linear equations\begin{align}3x_1+2x_2&=1\\5x_1+3x_2&=2.\end{align} (a) Find the coefficient matrix $A$ of the system. (b) Find the inverse matrix of the coefficient matrix $A$. (c) Using the inverse matrix of $A$, find the solution of the system. (Linear Algebra Midterm Exam 1, the Ohio State University) The following problems are Midterm 1 problems of Linear Algebra (Math 2568) at the Ohio State University in Autumn 2017.There were 9 problems that covered Chapter 1 of our textbook (Johnson, Riess, Arnold).The time limit was 55 minutes. This post is Part 1 and contains the first three problems.Check out Part 2 and Part 3 for the rest of the exam problems. Problem 1. Determine all possibilities for the number of solutions of each of the systems of linear equations described below. (a) A consistent system of $5$ equations in $3$ unknowns and the rank of the system is $1$. (b) A homogeneous system of $5$ equations in $4$ unknowns and it has a solution $x_1=1$, $x_2=2$, $x_3=3$, $x_4=4$. Problem 2. Consider the homogeneous system of linear equations whose coefficient matrix is given by the following matrix $A$. Find the vector form for the general solution of the system.\[A=\begin{bmatrix}1 & 0 & -1 & -2 \\2 &1 & -2 & -7 \\3 & 0 & -3 & -6 \\0 & 1 & 0 & -3\end{bmatrix}.\] Problem 3. Let $A$ be the following invertible matrix.\[A=\begin{bmatrix}-1 & 2 & 3 & 4 & 5\\6 & -7 & 8& 9& 10\\11 & 12 & -13 & 14 & 15\\16 & 17 & 18& -19 & 20\\21 & 22 & 23 & 24 & -25\end{bmatrix}\]Let $I$ be the $5\times 5$ identity matrix and let $B$ be a $5\times 5$ matrix.Suppose that $ABA^{-1}=I$.Then determine the matrix $B$. (Linear Algebra Midterm Exam 1, the Ohio State University)
I have a data about customers and their activity on a website for a two-year period. Also, I have a customer support work evaluation data for a shorter period of that two-year period. The question is:... Suppose I have a set of units $n_i, i = 1,...,N$. One unit was receiving a treatment $n_j = 1$ at times $t=1$ to $t=\tau$. All other units never received the treatment so $\forall t, n_i=0,i\neq j$. ... There is a long literature on stock return predicabtility.When I read papers I see lots of authors still using granger causality tests. It is still prominently taught in undergrad and grad classess.... I have a data set of car accidents. It includes information on accidents (like date and time), vehicles (like make and year), and drivers (like age and gender).The goal is to estimate whether a new ... Considering that markets open in different time zones, in general the opening and closing times are different.I suppose that a stock market opening later (in the same day) receive information on the ... In some sense this is a follow-up toCorrelation between salary level and housing prices in a townThere I was reminded to ask for causality, not correlation. I am interested in the causal effect of ... I'm planning on using a regression discontinuity design (RDD) to estimate the impact of a sector-specific law on the firms in that sector. I need to create a control group and I have read somewhere in ... I'm a statistician/machine learning scientist more familiar with molecular bio than economics. Trying to find out if an issue I perceive in bio modeling also occurs in econometric modeling.A common ... The Partial least square structural equation modeling (PLS-PM, PLS-SEM) method to structural equation modeling allows estimating complex cause-effect relationship models with latent variables. This ... I have a problem regarding causation and effect. I have a linear model$ wage=\beta_0+\beta_1educ+\beta_2female+\beta_3hourseworked+errorterm$When I need to estimate the causal effect for educated ... It's a common refrain that economists still don't have a solid grasp of the causes of business cycles (in particular, the Great Depression), despite decades of serious study. What would it take to say ... I'm an undergraduate student I have an empirical question regarding Regression Discontinuity and Diff-in-Diff methods. I'm currently evaluating the impact on fertility of a Conditional Cash Transfer ... Elyase İskender posted in his blog an interesting plot correlating Professors Salary and Alumni Earnings Relation in US Universities. Is there a good way to establish causality here?Did the highest ...
This question originates from reading the [proof of Gell-Mann Low thoerem](https://arxiv.org/abs/math-ph/0612030v1). Let $\|\psi_0\rangle$ be an eigenstate of $H_0$ with eigenvalue $E_0$, and consider the state vector $$\|\psi^{(-)}_\epsilon\rangle=\frac{U_{\epsilon,I}(0,-\infty)\|\psi_0\rangle}{\langle \psi_0\| U_{\epsilon,I}(0,-\infty)\|\psi_0\rangle}$$ Gell-Mann and Low's theorem: If the $\|\psi^{(-)} \rangle :=\lim_{\epsilon\rightarrow 0^{+}}\|\psi^{(-)}_\epsilon\rangle$ exist, then $\|\psi^{(-)} \rangle$ must be an eigenstate of $H$ with eigenvalue $E$. And the eigenvalue $E$ is decided by following equation: $$\Delta E= E-E_0=-\lim_{\epsilon\rightarrow 0^+} i\epsilon g\frac{\partial}{\partial g}\ln \langle\psi_0\| U_{\epsilon,I}(0,-\infty)\|\psi_0\rangle$$ However we learn in scattering theory, $$U_I(0,-\infty) = \lim_{t\rightarrow -\infty} U_{full}(0,t)U_0(t,0) = \Omega_{+}$$ where $\Omega_{+}$ is the Møller operator. We can prove the identity $H\Omega_{+}=H_0 \Omega_{+}$ for the Møller operator. It says that the energy of a scattering state will not change when you turn on the interaction adiabatically. My questions: 1.The only way to avoid these contradiction is to prove that $\Delta E$ for scattering state of $H_0$ must be zero. How to prove this? In general, it should be that for a scattering state there will be no energy shift, for discrete state there will be some energy shift. But the Gell-Mann Low theorem do not tell me the result.es 2.How to tackle this explicit case? $$H_0 = \frac{\mathbf{p}^2}{2}- \frac{1}{\|\mathbf{r}\|}, \ H_I= \frac{1}{\|\mathbf{r}\|}$$ then $H=H_0+H_I=\frac{\mathbf{p}^2}{2}$. If we start with $\|\psi_0\rangle$ is the ground state of $H_0$, i.e. the ground state of Hydrogen atom, and evolve by $ U_{ I}(0,-\infty)\|\psi_0\rangle $, what is the result of this state? Although in this case the system experience some level crossing, the theorem tells us if the state exists, then it must be some eigenstate of $H$. In this case the adiabatic theorem cannot be used, but the Gell-mann Low theorem still works. 3.The existence of $\lim_{\epsilon\rightarrow 0^{+}}\|\psi^{(-)}_\epsilon\rangle$ is annoying. Is there some criterion of existence of $\lim_{\epsilon\rightarrow 0^{+}}\|\psi^{(-)}_\epsilon\rangle$? Or give me an explicit example in which this does exixt. 4.It seems that the Gell-Mann Low theorem is a generalized adiabatic theorem, which can be used for discrete spectrum or continuous spectrum. How to prove Gell-Mann Low theorem reduces to the adiabatic theorem under the condition of the adiabatic theorem?
I have a very fundamental question regarding simulation of DRIFTED geometric brownian motion. We have the standard Blackos Scholes model: $dS(t)=r S(t)dt+\sigma S(t) dW^{\mathbb{P}}(t)$, where $W^{\mathbb{P}}(t)$ is the standard Wiener process under probability measure $\mathbb{P}$. If we want to simulate this, using constant $\Delta t$, we use the recursive formula: $S_{t+1}=S_te^{(r-\frac{\sigma^2}{2})\Delta t+\sigma \sqrt{\Delta t} Z_t }$, where $Z_t \sim N(0,1)$. Now assume that we want to change the drift such that: $W^{\mathbb{Q}}(t) = W^{\mathbb{P}}(t) - \int_0^t \theta_s ds$ is a brownian motion under $\mathbb{Q}$ such that: $dS(t)=(r + \sigma \theta )S(t)dt+\sigma S(t) dW^{\mathbb{Q}}(t)$ Now this is where I become unsure. If I want to simulate the drifted process, is it just fine to use the similar method as: $S_{t+1}=S_te^{(r+\sigma \theta-\frac{\sigma^2}{2})\Delta t+\sigma \sqrt{\Delta t} Z_t }$, where $Z_t \sim N(0,1)$. OR is it that I have to use $Z_t \sim N(\theta, 1)$? Im not that strong in the change of measure part, so thats why I'm a bit unsure. Would appreciate for help. Thanks
I'm reading through my first textbook on linear optimization. The book states a theorem without proof and I'd like to understand why it's true. Glossary of Terms: Definition 1 The problem Maximize $f(x_1,x_2,\cdots,x_n)=c_1x_1+c_2x_2+\cdots+c_nx_n$ Subject to $a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\leq b_1$ $a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n \leq b_2$ $\cdots$ $a_{m1}x_1+a_{m2}x_2+ \cdots + a_{mn}x_n \leq b_n$ $x_1,x_2, \dots, x_n \geq 0$ is said to be a canonical maximization linear programming problem. The definition for minimization is analogous. Definition 2 Let $x= (x_1,x_2,\cdots ,x_n), y=(y_1,y_2,\cdots ,y_n)\in$ R$^n$. Then $tx+(1-t)y$ for $0\leq t\leq 1$ is said to be the line segment between $x$ and $y$ inclusive. Definition 3 The set of all points $(x_1,x_2, \cdots, x_n)$ satisfying the constraints of the canonical maximization problem is said to be the constraint set Definition 4 Let $S$ be a subset of R$^n$. $S$ is said to be convex if, whenever $x$=$(x_1,x_2,\cdots,x_n)$, $y$$=(y_1,y_2,\cdots,y_n)\in S$, then $tx+(1-t)y \in S$ for $0\leq t \leq 1$ The Theorem: The constraint set of a canonical maximization or canonical minimization linear programming problem is convex. My Work I know I must show that $(x_1,x_2,\cdots ,x_n)$ i.e. the constraint set satisfies the requirements for convexity, I have no idea how to do this though. The question isn't homework, we were told to accept it, but I want to understand.
Before we determine the order of matrix, we should first understand what matrices are. Matrices are defined as a rectangular array of numbers or functions. Since it is rectangular array, it is 2-dimensional. Basically, a two dimensional matrix consist of number of rows (m) and number of columns (n). Let us understand more with some examples. $ A =\left[ \begin{matrix} 3 & 4 & 9\cr 12 & 11 & 35 \cr \end{matrix} \right] $ $ B =\left[ \begin{matrix} 2 & -6 & 13\cr 32 & -7 & -23 \cr -9 & 9 & 15\cr 8 & 25 & 7\cr \end{matrix} \right] $ The two matrices shown above A and B. The general notation of a matrix is given as: $ A = [a_{ij}]_{m × n} $, where $ 1 ≤ i ≤ m , 1 ≤ j ≤ n $ and $i , j \in N $ You can see that the matrix is denoted by an upper case letter and its elements are denoted by the same letter in the lower case. $ a_{ij} $ represents any element of matrix which is in $ i^{th}$ row and $ j^{th} $ column. Similarly,$ b_{ij} $ represents any element of matrix B. So, in the matrices given above, the element $ a_{21} $ represents the element which is in the $ 2^{nd} $row and the $ 1^{st} $ column of matrix A. ∴a 21 = 12 Similarly, $ b_{32} = 9 , b_{13} = 13 $ and so on. Can you write the notation of 15 for matrix B ? Since it is in $ 3^{rd} $ row and $ 4^{th} $ column, it will be denoted by $ b_{34} $. If the matrix has $ m $ rows and $ n $ columns, it is said to be a matrix of the order $m × n$. We call this an m by n matrix. So, A is a 2 × 3 matrix and B is a 4 × 3 matrix. The more appropriate notation for A and B respectively will be: $ A =\left[ \begin{matrix} 3 & 4 & 9\cr 12 & 11 & 35 \cr \end{matrix} \right]_{2 × 3} $ $ B =\left[ \begin{matrix} 2 & -6 & 13\cr 32 & -7 & -23 \cr -9 & 9 & 15\cr 8 & 25 & 7\cr \end{matrix} \right]_{4 × 3} $ So, if you have to find the order of the matrix, count the number or its rows and columns and there you have it! It is quite fascinating that the order of matrix shares a relation with the number of elements present in a matrix. The order of a matrix is denoted by $a \times b$, and the number of element in a matrix will be equal to the product of a and b. In the above examples, A is of the order 2 × 3 .Therefore, the number of elements present in a matrix will also be 2 times 3, i.e. 6. Similarly, the other matrix is of the order 4 × 3, thus the number of elements present will be 12 i.e. 4 times 3. This gives us an important insight that if we know the order of matrix, we can easily determing the total number of elements, that the matrix has. The conclusion hence is: If a matrix is of m × n order, it will have mn elements. But is the converse of the previous statement true? The converse says that: If the number of element is mn, so the order would be $m \times n$. This is definitely not true. It is because the product of mn can be obtained by more than 1 ways, some of them are listed below: $mn × 1$ $1 × mn$ $m × n$ $n × m$ For example- Consider the number of elements present in a matrix to be 12. Thus the order of a matrix can be either of the one listed below: $12 \times 1$, or $1 \times 12$, or $6 \times 2$, or $ 2 \times 6$, or $4 \times 3$, or $3 \times 4$. Thus, we have 6 different ways to write the order of a matrix, for the given number of elements. Let us now look at a way to create a matrix for a given funciton: For $ P_{ij} = i-2j $ , let us construct a 3 × 2 matrix. So, this matrix will have 6 elements as following: $ P =\left[ \begin{matrix} P_{11} & P_{12}\cr P_{21} & P_{22} \cr P_{31} & P_{32} \cr \end{matrix} \right] $ Now, we will calculate the values of the elements one by one. To calculate the value of $ p_{11} $ , substitute $ i = 1 \space and \space j=1 \space in \space p_{ij} = i – 2j $ . $ P_{11} = 1 – (2 × 1) = -1 $ $ P_{12} = 1 – (2 × 2) = -3 $ $ P_{21} = 2 – (2 × 1) = 0 $ $ P_{22} = 2 – (2 × 2) = -2 $ $ P_{31} = 3 – (2 × 1) = 1 $ $ P_{32} = 3 – (2 × 2) = -1 $ Hence, $ P =\left[ \begin{matrix} -1 & -3\cr 0 & -2 \cr 1 & -1 \cr \end{matrix} \right]_{3 × 2} $ There you go! You now know what order of matrix is, and how to find the order of a matrix. To know more, download BYJU’S-The Learning App and study in an innovative way.
Talk II on Bourguignon-Lawson's 1978 paper The stable parametrized h-cobordism theorem provides a critical link in the chain of homotopy theoretic constructions that show up in the classification of manifolds and their diffeomorphisms. For a compact smooth manifold M it gives a decomposition of Waldhausen's A(M) into QM_+ and a delooping of the stable h-cobordism space of M. I will talk about joint work with Malkiewich on this story when M is a smooth compact G-manifold. We show $C^\infty$ local rigidity for a broad class of new examples of solvable algebraic partially hyperbolic actions on ${\mathbb G}=\mathbb{G}_1\times\cdots\times \mathbb{G}_k/\Gamma$, where $\mathbb{G}_1$ is of the following type: $SL(n, {\mathbb R})$, $SO_o(m,m)$, $E_{6(6)}$, $E_{7(7)}$ and $E_{8(8)}$, $n\geq3$, $m\geq 4$. These examples include rank-one partially hyperbolic actions. The method of proof is a combination of KAM type iteration scheme and representation theory. The principal difference with previous work that used KAM scheme is very general nature of the proof: no specific information about unitary representations of ${\mathbb G}$ or ${\mathbb G}_1$ is required. This is a continuation of the last talk. A classical problem in knot theory is determining whether or not a given 2-dimensional diagram represents the unknot. The UNKNOTTING PROBLEM was proven to be in NP by Hass, Lagarias, and Pippenger. A generalization of this decision problem is the GENUS PROBLEM. We will discuss the basics of computational complexity, knot genus, and normal surface theory in order to present an algorithm (from HLP) to explicitly compute the genus of a knot. We will then show that this algorithm is in PSPACE and discuss more recent results and implications in the field. We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains an infinite-rank subgroup and a Z-summand. The proof relies on the involutive Heegaard Floer homology package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is joint work with I. Dai, M. Stoffregen, and L. Truong. There is a close analogy between function fields over finite fields and number fields. In this analogy $\text{Spec } \mathbb{Z}$ corresponds to an algebraic curve over a finite field. However, this analogy often fails. For example, $\text{Spec } \mathbb{Z} \times \text{Spec } \mathbb{Z} $ (which should correspond to a surface) is $\text{Spec } \mathbb{Z}$ (which corresponds to a curve). In many cases, the Fargues-Fontaine curve is the natural analogue for algebraic curves. In this first talk, we will give the construction of the Fargues-Fontaine curve. Consider a collection of particles in a fluid that is subject to a standing acoustic wave. In some situations, the particles tend to cluster about the nodes of the wave. We study the problem of finding a standing acoustic wave that can position particles in desired locations, i.e. whose nodal set is as close as possible to desired curves or surfaces. We show that in certain situations we can expect to reproduce patterns up to the diffraction limit. For periodic particle patterns, we show that there are limitations on the unit cell and that the possible patterns in dimension d can be determined from an eigendecomposition of a 2d x 2d matrix. Department of Mathematics Michigan State University 619 Red Cedar Road C212 Wells Hall East Lansing, MI 48824 Phone: (517) 353-0844 Fax: (517) 432-1562 College of Natural Science
A friend asked me the following interesting question: Let $$A = \begin{bmatrix} R \\ \xi{\rm I} \end{bmatrix},$$ where $R \in \mathbb{R}^{n \times n}$ is an upper triangular and ${\rm I}$ is an identity matrix, both of order $n$, and $\xi \in \mathbb{R}$ is a scalar. Is there an efficient way to compute a QR factorization of $A$? I have found this question with a very nice answer, but I'd like to avoid doing the SVD because it is computationally expensive and my $R$ is not a constant like $W$ in that other question. Also, my $R$ is already triangular, which I hope can somehow be used. Edit: There was a comment (turned into an answer while I was writing this edit) on using Givens rotations. Since this is a logical first idea, I'd like to explain why I don't like it. We could use Givens rotations to cancel out the elements of $\xi{\rm I}$, but each Givens rotation is computing two linear combinations of two rows. That means that if I cancel out the first element of $\xi{\rm I}$, I will also introduce a bunch of non-zeros to the rest of that row. This means that I would need to go through the whole upper triangle of the bottom block, same as I'd have to do if $\xi{\rm I}$ was a general upper triangular matrix. Given that it is a diagonal matrix (with all its diagonal elements being the same, although I suspect this doesn't help much), I am hoping to get more efficient than that.
I was looking at the Draper & Smith's 'Applied Regression Analysis', as did the person who asked:this other question on CrossValidated In short - the variance-covariance matrix of the residuals in regression is given by $(I - H)\sigma^2$, where $H$ is the 'Hat Matrix'. So in general we must assume the residuals are not independent. Yet whenever I read about the assumptions of regression it says the error terms should be independent (as well as having zero expectation $E[\epsilon_i] = 0$ and equal variance $Var(\epsilon_i) = \sigma^2 \forall i$. I am missing something, or using a loose definition - my naive, inexperienced reading of this looks contradictory. Thank you, Chris
Zurich Instruments introduces the first functional unit for arithmetic operations on the results of lock-in amplifier and Boxcar measurements. The UHF Arithmetic Unit is included with every UHF Instrument but of course the functionality increases with further upgrades like UHF-BOX and UHF-PID installed. 4개의 연산유닛 The four arithmetic units (AU) allow real-time operations using the measurement results of demodulators, Boxcar units and auxiliary inputs connectors as input parameters. As an example the UHFLI, with its two signal inputs and a total of eight dual phase demodulators provides the chance to combine more than 50 different input parameters. The possible operations include addition, subtraction, multiplication, division, scaling, plus absolute value and phase angle calculation of complex numbers (e.g. X 1 + iY 2). LabOne 통합 A dedicated tab inside the LabOne user interface accommodates the four AU units. This unique integration enables quick graphical definition of the arithmetic operations and real-time monitoring of the results. These results are indeed streamed at a configurable rate to the host computer where they can be saved to the hard disk or analyzed in one of the time domain and frequency domain tools of LabOne: The use of the AU is particularly attractive for applications where the result of the arithmetic operation is directly used to provide feedback to an experimental setup; the results can be directly used as an input for the PID controllers (requires UHF-PID option) and are also directly available on any of the auxiliary outputs. Since they are available on the auxiliary outputs, they can also be selected as demodulator inputs, permitting a second lock-in step such as tandem demodulation. 폭넓은 어플리케이션 Balanced detection: noise suppression by differential measurement $c_0\cdot\{R_2,X_2,Y_2,\Theta_2\} - c_1\cdot\{R_1,X_1,Y_1,\Theta_1\}$ Differential signal measurement is a powerful method for cancelling out certain noise components in order to improve signal to noise ratio. The method is quite universal and applied in a wide range of situations, for instance in the optical domain where laser spectroscopy and imaging setups are limited by laser amplitude noise. In order to overcome such limitations one can split the laser beam into a probe beam, which passes through the actual setup, and a reference beam which does not, the beams being captured by separate photodiodes (PD). Subtracting the electrical PD signals cancels out the noise components which are common to both beams, and for measurements of periodic signals (pulsed lasers, heterodyne, etc.) also removes unwanted DC components which can help improve further signal analysis down stream. In situations where it is not possible to place the two PDs close enough to directly subtract their photo currents, one can connect them to the two inputs of the lock-in amplifier where they are measured with two demodulators with identical settings, in particular referenced to the same oscillator and with the same filter settings. Subtracting the demodulator outputs reduces the coherent noise from the signal of interest and increases the SNR. One limitation of this scheme is that the intensity of the two light beams needs to be carefully matched (assuming a symmetric detection setup) in order to maximize noise suppression. This process can be tedious, and for setups where the transmission of the probe beam experiences significant changes an auto-balancing approach can be useful in order maintain noise suppression over the course of the measurement. This can be achieved by introducing a slowly varying scaling parameter with lower bandwidth than the actual signal (LP, low-pass filtered), for instance g = LP(Rsig) / LP(Rref). The resulting signal would then be derived as c0 Rsig - g * Rref, c0 being an adjustable scaling factor to maximize performance. Normalization and relative measurement $\frac{R_1}{R_2}$ For measurements where tiny differences between two samples need to be detected while the signals themselves can change over many orders of magnitude - not uncommon for instance for impedance and optical transmission measurements - a relative measurement by dividing the two signals can help to properly keep track of the relevant measurement quantity. This also easily allows boosting of the signal by numerical scaling independent of the actual signal levels, which comes in handy when the signal is further processed with a PID controller to provide feedback to an experimental setup. In such situations the controller parameters will not have to be readjusted every time the signal levels change. This could be used, for example, in a spectroscopy setup where the frequency of a laser is adjusted to obtain a defined optical transmission of a gas cell. Modulation parameter output for AM and FM signals The analysis of signals with multiple frequencies, such as amplitude, frequency or phase modulated signals, can conveniently be performed with direct sideband extraction. The UHF-MOD option provides the demodulated outputs X,Y for the carrier (demodulator 1) and the sidebands independently (demodulator 2 and 3). Whenever an experiment requires the entire signal contained in the two first sidebands this can now be easily determined, even normalized to the carrier. The sum (for AM), or respectively the difference (for FM) of the sidebands provides a factor √2 SNR improvement compared to measurement of one sideband only. Dual Frequency Resonance Tracking (DFRT) $R_1-R_2 $ Reliable resonance frequency tracking often relies on phase-locked loops that exploit the phase information to provide fast feedback. However, there are case where the SNR ratio of the phase signal is insufficient for the feedback or ambiguous due to physical properties, for example in near-field scanning of polar materials where the phase flips by 180 degrees at the domain boundaries. Instead of relying on the phase signal one can probe the resonance with two frequencies detuned in order to hit the steepest slopes right and left of the resonance center frequency. The difference of the two amplitude signals provides an excellent means of providing an error signal to a PID controller (requires UHF-PID option). The DFRT Method blog article provides more details of the actual implementation.
4.1.2. Confidence areas Let $u = (u_1,…,u_d)^\top$ be a random vector with uniform margins on $(0, 1)$ and let $y = (y_1,…, y_d)^\top \in \mathbb{R}^d$ be defined by: $y_j = G(u_j)$, $j \in \{1,…,d\}$ where $G$ is a continuous increasing map defined between 0 and 1. Let us assume that the multivariate c.d.f of $u$ is a Gaussian copula with parameter $R_g$. Taking the quantile function of $N_1(0, 1)$ as $G$, then $y$ is distributed as $N_d(0, R_g)$ and the random variable $y^\top R_g^{-1} y$ as $\chi^2_d$. Given $\alpha \in (0, 1)$, the latter variable $y^\top R_g^{-1} y$ is less than $\chi^2_{d,\alpha}$ – the $\alpha$ order quantile of $\chi^2_d$ – with probability $\alpha$. Hence, $$\Gamma_g(\alpha) = \{ v= (v_1 ,…,v_d )^\top \in [0, 1]^d :\\ (G(v_1),…, G(v_d)) R_g^{-1}(G(v_1),…, G(v_d))^\top \leq\chi^2_{d,\alpha}\}$$ is a $d$-dimensional compact confidence area inside (resp. outside) of which $u$ falls with probability $\alpha$ (resp. $1-\alpha$). $R_g$ is a $d × d$ correlation matrix. Question. I have statistical questions about distribution. How to statiscally check whether variable $y^\top R_g^{-1} y$ is distributed as $\chi^2_d$? It's known that mean $M(\chi^2_d)=d$, and variance $D(\chi^2_d)=2d$, where d is degreee of freedom. Edit. Is it possible to use the Kolmogorov-Smirnov test here?
In elementary algebra, the quadratic formula would be the solution for quadratic equation. There are two possible techniques for solving the quadratic equation instead you put values in quadratic equation directly. These techniques are factoring, completing the square or graphing etc. Out of all these, working with quadratic formula is always the most convenient option. The general form of quadratic equation could be given as – \[\large ax^{2} + bx + c =0 \] Here x is the unknown variable, while a, b, and c are constants that could not be equal to the zero. This is easy to verify the quadratic formula by satisfying the quadratic equations and inserting the former values to the latter. Hence, the standard quadratic formula in mathematics is given as below – \[\large x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\] Here, the solution given y the quadratic formula is called as roots of quadratic equation. Geometrically, roots are used to give values on the parabola where y crosses the x-axis. If there is some formula whose yield is calculated equal to zero then this is easy to find immediately how many real zeros a particular quadratic equation has. The Greek mathematicians are using many geometrical methods to solve quadratic equations. There is one algebraic technique that is more popular than geometric techniques of Euclid. It mostly gives only one root as the output or sometimes both roots are also positive. One Indian mathematician defined this formula explicitly and it could be written as given below. \[\large x=\frac{\sqrt{b^{2}-4ac} -b }{2a}\] With these formulas, you can solve or find roots for almost any quadratic equation by plugging the values directly into formula.
In mathematics, Trigonometry shows the relationship between multiple sides and angles of a triangle. Trigonometry is used throughout the geometry where shapes are broken down into a collection of triangles. Basically, Trigonometry is the study of triangles, angles, and different dimensions. Although the definition may sound simpler yet it is vital for modern engineering, complex mathematics study, architecture, logarithms, calculus, and other fields. The word Trigonometry was derived from Greek words triangle (trigōnon) and measure (matron) during the 16 th century. So, this is not a new concept but into the existence of centuries and played an important role in the discovery of various mathematical and scientific theories. One of the biggest benefits of this mathematic technique was realized by the astronautical science and Indian astronomers. Most of Trigonometry formulas revolves around ratios and extremely handy to solve complex problems in Trigonometry. If you want to appear for any competitive exams after your school then hands-on knowledge of different Trigonometry formulas is essential. The basic of any Trigonometry formula is a Trigonometry Identity. So, you must be curious to know about Trigonometric identities, let us discuss the same in the next section. Also, $\tan(\frac{x}{2}) = \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))(1-\cos(x))}{(1+\cos(x))(1-\cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{1-\cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-\cos(x))^{2}}{\sin^{2}(x)}}\\ \\ \\ =\frac{1-\cos(x)}{\sin(x)}$ So, $\tan(\frac{x}{2}) =\frac{1-\cos(x)}{\sin(x)}$ Sin 2x + Cos 2x = 1 1 + tan 2x = sec 2x 1 + cot 2x = cosec 2x sinx = ∓√ 1– cos2x tanx = ∓√ sec2x- 1 cosx = ∓√ 1– sin2x Also called negative angle identities Sin(-x)=-sin x cos(-x)=-cos x tan(-x)=-tan x cot(-x)=-cot x sec(-x)=-sec x cosec(-x)=-cosec x Sinθ = Cosθ X Tanθ Cosθ = Sinθ X Cotθ Tanθ = Sinθ ⁄ Cosθ Cotθ = Cosθ⁄ Sinθ Trigonometry identities are Trigonometric functions of one or more angles where equality is defined for both sides. The identities are used to solve any complex Trigonometric equations or expressions. One of the most popular applications of Trigonometric identities is the integration of non-trigonometric functions. $\sin \theta = \frac{Opposite}{Hypotenuse}$ $\sec \theta = \frac{Hypotenuse}{Adjacent}$ $\cos\theta = \frac{Adjacent}{Hypotenuse}$ $\tan \theta =\frac{Opposite}{Adjacent}$ $csc \theta = \frac{Hypotenuse}{Opposite}$ $cot \theta = \frac{Adjacent}{Opposite}$ The Reciprocal Identities are given as: $cosec\theta =\frac{1}{\sin\theta }$ $sec\theta =\frac{1}{\cos\theta }$ $cot\theta =\frac{1}{\tan\theta }$ $sin\theta =\frac{1}{csc\theta }$ $cos\theta =\frac{1}{\sec\theta }$ $tan\theta =\frac{1}{cot\theta }$ A Trigonometry equation is an expression that may hold true or false for any angle. If it holds true then it is a Trigonometry identity otherwise they are termed as conditional equations. These equations can be solved with the help of basic Trigonometric formulas and identities. There are only a few equations that can be solved manually otherwise you need a calculator and special skills to find the solution to any Trigonometry problem. As discussed earlier, Trigonometry is the study of triangles, angles, and different dimensions. Although the definition may sound simpler yet it is vital for modern engineering, complex mathematics study, architecture, and other fields. With a deep understanding of Trigonometry, students would be able to work on precise angle of different sides of a triangle, calculation of distance among different triangle points and more important details that can be used for a variety of settings. Further, Trigonometry skills may help you in exploring wider job options too including engineering, architecture, aeronautical study etc. So, this is really important for the student to learn Trigonometry who are planning to enter in the field of scientific science or engineering. With the right mathematical skills, even the most complex problems can be understood in little time only without putting hard efforts. So, this would not be saying wrong that Trigonometry is one of most valuable branches of mathematics and online programs can make the learning even easier and more interesting to master the Trigonometry skills.
The Concept of Eigenvalues and Eigenvectors Consider a linear homogeneous system of $n$ differential equations with constant coefficients, which can be written in matrix form as \[\mathbf{X’}\left( t \right) = A\mathbf{X}\left( t \right),\] where the following notation is used: \[ {\mathbf{X}\left( t \right) = \left[ {\begin{array}{{20}{c}} {{x_1}\left( t \right)}\\ {{x_2}\left( t \right)}\\ \vdots \\ {{x_n}\left( t \right)} \end{array}} \right],\;\;}\kern-0.3pt {\mathbf{X’}\left( t \right) = \left[ {\begin{array}{{20}{c}} {{x’_1}\left( t \right)}\\ {{x’_2}\left( t \right)}\\ \vdots \\ {{x’_n}\left( t \right)} \end{array}} \right],\;\;}\kern-0.3pt {A = \left[ {\begin{array}{{20}{c}} {{a_{11}}}&{{a_{12}}}& \cdots &{{a_{1n}}}\\ {{a_{21}}}&{{a_{22}}}& \cdots &{{a_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{a_{n1}}}&{{a_{n2}}}& \cdots &{{a_{nn}}} \end{array}} \right].} \] We look for non-trivial solutions of the homogeneous system in the form of \[\mathbf{X}\left( t \right) = {e^{\lambda t}}\mathbf{V},\] where $\mathbf{V} \ne 0$ is a constant $n$-dimensional vector, which will be defined later. Substituting the above expression for $\mathbf{X}\left( t \right)$ into the system of equations, we obtain: \[{\lambda {e^{\lambda t}}\mathbf{V} = A{e^{\lambda t}}\mathbf{V},\;\; }\Rightarrow {A\mathbf{V} = \lambda \mathbf{V}.}\] This equation means that under the action of a linear operator $A$ the vector $\mathbf{V}$ is converted to a collinear vector $\lambda \mathbf{V}.$ Any vector with this property is called an eigenvector of the linear transformation $A,$ and the number $\lambda$ is called an eigenvalue. Thus, we conclude that in order the vector function $\mathbf{X}\left( t \right) = {e^{\lambda t}}\mathbf{V}$ be a solution of the homogeneous linear system, it is necessary and sufficient that the number $\lambda$ be an eigenvalue of the matrix $A,$ and the vector $\mathbf{V}$ be the corresponding eigenvector of this matrix. As it can be seen, the solution of a linear system of equations can be constructed by an algebraic method. Therefore, we provide some necessary information on linear algebra. Finding Eigenvalues and Eigenvectors of a Linear Transformation Let’s go back to the matrix-vector equation obtained above: \[A\mathbf{V} = \lambda \mathbf{V}.\] It can be rewritten as \[A\mathbf{V} – \lambda \mathbf{V} = \mathbf{0},\] where $\mathbf{0}$ is the zero vector. Recall that the product of the identity matrix $I$ of order $n$ and $n$-dimensional vector $\mathbf{V}$ is equal to the vector itself: \[I\mathbf{V} = \mathbf{V}.\] Therefore, our equation becomes \[ {A\mathbf{V} – \lambda I\mathbf{V} = \mathbf{0}\;\;\;}\kern-0.3pt {\text{or}\;\;\;\left( {A – \lambda I} \right)\mathbf{V} = \mathbf{0}.} \] It follows from this relationship that the determinant of ${A – \lambda I}$ is zero: \[\det \left( {A – \lambda I} \right) = 0.\] Indeed, if we assume that $\det \left( {A – \lambda I} \right) \ne 0,$ then the matrix will have the inverse matrix ${\left( {A – \lambda I} \right)^{ – 1}}.$ Multiplying on the left both sides of the equation by the inverse matrix ${\left( {A – \lambda I} \right)^{ – 1}},$ we get: \[ {{{\left( {A – \lambda I} \right)^{ – 1}}\left( {A – \lambda I} \right)\mathbf{V} }}={{ {\left( {A – \lambda I} \right)^{ – 1}} \cdot \mathbf{0},\;\;}}\Rightarrow {I\mathbf{V} = \mathbf{0},\;\;} \Rightarrow {\mathbf{V} = \mathbf{0}.} \] This, however, contradicts to the definition of the eigenvector, which must be different from zero. Consequently, the eigenvalues $\lambda$ must satisfy the equation \[\det \left( {A – \lambda I} \right) = 0,\] which is called the auxiliary or characteristic equation of the linear transformation $A.$ The polynomial on the left side of the equation is called the characteristic polynomial of the linear transformation (or linear operator) $A.$ The set of all eigenvalues ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}$ forms the spectrum of the operator $A.$ So the first step in finding the solution of a system of linear differential equations is solving the auxiliary equation and finding all eigenvalues ${\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}.$ Next, substituting each eigenvalue ${\lambda _i}$ in the system of equations \[\left( {A – \lambda I} \right)\mathbf{V} = \mathbf{0}\] and solving it, we find the eigenvectors corresponding to the given eigenvalue ${\lambda _i}.$ Note that after the substitution of the eigenvalues the system becomes singular, i.e. some of the equations will be the same. This follows from the fact that the determinant of the system is zero. As a result, the system of equations will have an infinite set of solutions, i.e. eigenvectors can be determined only to within a constant factor. Fundamental System of Solutions of a Linear Homogeneous System Expanding the determinant of the characteristic equation of the $n$th order, we have, in general, the following equation: \[{\left( { – 1} \right)^n}{\left( {\lambda – {\lambda _1}} \right)^{{k_1}}}{\left( {\lambda – {\lambda _2}} \right)^{{k_2}}} \cdots \kern0pt {\left( {\lambda – {\lambda _m}} \right)^{{k_m}}} = 0,\] where \[{k_1} + {k_2} + \cdots + {k_m} = n.\] Here the number ${k_i}$ is called the algebraic multiplicity of the eigenvalue ${\lambda_i}.$ For each such eigenvalue, there exists ${s_i}$ linearly independent eigenvectors. The number ${s_i}$ is called the geometric multiplicity of the eigenvalue ${\lambda_i}.$ It is proved in linear algebra that the geometric multiplicity ${s_i}$ does not exceed the algebraic multiplicity ${k_i},$ i.e. the following relation holds: \[0 \lt {s_i} \le {k_i}.\] It turns out that the general solution of the homogeneous system essentially depends on the multiplicity of the eigenvalues. Consider the possible cases that arise here. $1.$ Case ${s_i} = {k_i} = 1.$ All Roots of the Auxiliary Equation are Real and Distinct. In this simplest case, each eigenvalue ${\lambda _i}$ has one associated eigenvector ${\mathbf{V}_i}.$ These vectors form a set of linearly independent solutions \[ {{\mathbf{X}_1} = {e^{{\lambda _1}t}}{\mathbf{V}_1},\;\;}\kern-0.3pt{{\mathbf{X}_2} = {e^{{\lambda _2}t}}{\mathbf{V}_2}, \ldots ,\;}\kern-0.3pt {{\mathbf{X}_n} = {e^{{\lambda _n}t}}{\mathbf{V}_n},} \] that is, a fundamental system of solutions of the homogeneous system. By the linear independence of the eigenvectors the corresponding Wronskian is different from zero: \[ {{W_{\left[ {{\mathbf{X}_1},{\mathbf{X}_2}, \ldots ,{\mathbf{X}_n}} \right]}}\left( t \right) \text{ = }}\kern0pt {\left\| {\begin{array}{{20}{c}} {{x_{11}}\left( t \right)}&{{x_{12}}\left( t \right)}& \cdots &{{x_{1n}}\left( t \right)}\\ {{x_{21}}\left( t \right)}&{{x_{22}}\left( t \right)}& \cdots &{{x_{2n}}\left( t \right)}\\ \cdots & \cdots & \cdots & \cdots \\ {{x_{n1}}\left( t \right)}&{{x_{n2}}\left( t \right)}& \cdots &{{x_{nn}}\left( t \right)} \end{array}} \right\| } = {\left\| {\begin{array}{{20}{c}} {{e^{{\lambda _1}t}}{V_{11}}}&{{e^{{\lambda _2}t}}{V_{12}}}& \cdots &{{e^{{\lambda _n}t}}{V_{1n}}}\\ {{e^{{\lambda _1}t}}{V_{21}}}&{{e^{{\lambda _2}t}}{V_{22}}}& \cdots &{{e^{{\lambda _n}t}}{V_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{e^{{\lambda _1}t}}{V_{n1}}}&{{e^{{\lambda _2}t}}{V_{n2}}}& \cdots &{{e^{{\lambda _n}t}}{V_{nn}}} \end{array}} \right\| } = {{e^{\left( {{\lambda _1} + {\lambda _2} + \cdots + {\lambda _n}} \right)t}} }\kern0pt{\left\| {\begin{array}{{20}{c}} {{V_{11}}}&{{V_{12}}}& \cdots &{{V_{1n}}}\\ {{V_{21}}}&{{V_{22}}}& \cdots &{{V_{2n}}}\\ \cdots & \cdots & \cdots & \cdots \\ {{V_{n1}}}&{{V_{n2}}}& \cdots &{{V_{nn}}} \end{array}} \right\| }\ne{ 0.} \] The general solution is given by \[ {\mathbf{X}\left( t \right) }={ {C_1}{e^{{\lambda _1}t}}{\mathbf{V}_1} }+{ {C_2}{e^{{\lambda _2}t}}{\mathbf{V}_2} + \cdots } + {{C_n}{e^{{\lambda _n}t}}{\mathbf{V}_n},} \] where ${C_1},$ ${C_2}, \ldots ,$ ${C_n}$ are arbitrary constants. The auxiliary equation may have complex roots. If all the entries of the matrix $A$ are real, then the complex roots always appear in pairs of complex conjugate numbers. Suppose that we have a pair of complex eigenvalues ${\lambda _i} = \alpha \pm \beta i.$ This pair of complex conjugate numbers is associated to a pair of linearly independent real solutions of the form \[ {{\mathbf{X}_1} = \text{Re} \left[ {{e^{\left( {\alpha \pm \beta i} \right)t}}{\mathbf{V}_i}} \right],\;\;}\kern-0.3pt {{\mathbf{X}_2} = \text{Im} \left[ {{e^{\left( {\alpha \pm \beta i} \right)t}}{\mathbf{V}_i}} \right].} \] Thus, the real and imaginary parts of the complex solution form a pair of real solutions. $2.$ Case ${s_i} = {k_i} \gt 1.$ The Auxiliary Equation Has Multiple Roots, Whose Geometric and Algebraic Multiplicities are Equal. This case is similar to the previous one. Despite the existence of eigenvalues of multiplicity greater than $1,$ we can define $n$ linearly independent eigenvectors. In particular, any symmetric matrix with real entries that has $n$ eigenvalues, will have $n$ eigenvectors. Similarly, a unitary matrix has the same properties. In general, a square matrix of size $n \times n$ must be diagonalizable in order to have $n$ eigenvectors. The general solution of the system of $n$ differential equations can be represented as \[ {\mathbf{X}\left( t \right) \text{ = }}\kern0pt{ \underbrace {{{C_{11}}{e^{{\lambda _1}t}}\mathbf{V}_1^{\left( 1 \right)} }+{ {C_{12}}{e^{{\lambda _1}t}}\mathbf{V}_1^{\left( 2 \right)} + \cdots }+{ {C_{1{k_1}}}{e^{{\lambda _1}t}}\mathbf{V}_1^{\left( {{k_1}} \right)}}}_{{k_1}\;\text{terms}} } + {\underbrace {{{C_{21}}{e^{{\lambda _2}t}}\mathbf{V}_2^{\left( 1 \right)} }+{ {C_{22}}{e^{{\lambda _2}t}}\mathbf{V}_2^{\left( 2 \right)} + \cdots }+{ {C_{2{k_2}}}{e^{{\lambda _2}t}}\mathbf{V}_2^{\left( {{k_2}} \right)}}}_{{k_2}\;\text{terms}} }\kern0pt{\text{ + } \cdots } \] Here the total number of terms is $n,$ ${C_{ij}}$ are arbitrary constants. $3.$ Case ${s_i} \lt {k_i}.$ The Auxiliary Equation Has Multiple Roots, Whose Geometric Multiplicity is Less Than the Algebraic Multiplicity. In some matrices $A$ (such matrices are called defective), an eigenvalue ${\lambda_i}$ of multiplicity ${k_i}$ may have fewer than ${k_i}$ linearly independent eigenvectors. In this case, instead of missing eigenvectors we can find so-called generalized eigenvectors, so as to get a set of $n$ linearly independent vectors and construct the corresponding fundamental system of solution. Two ways are usually used for this purpose: Construction of the General Solution of a System of Equations Using the Method of Undetermined Coefficients; Construction of the General Solution of a System of Equations Using the Jordan Form. A detailed description of these methods is presented separately on the specified web pages. Below we consider examples of systems of differential equations corresponding to Cases $1$ and $2.$ Solved Problems Click a problem to see the solution.
I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference frame (RF) with coordinates $\bar x^\mu$, in flat spacetime $\eta_{\mu \nu}$ (Minkowski metric). If I have well understood, on one hand, I can go to an accelerated RF by change of coordinates $x^\mu(\bar x)$. The metric is given by: $$\tag{1}g_{\mu \nu}(x) = \frac{\partial \bar x^{\alpha}}{\partial x^{\mu}} \frac{\partial \bar x^{\beta}}{\partial x^{\nu}} \eta_{\alpha \beta}$$ On the other hand, I know that a curved space-time with metric $q_{\mu \nu}$ cannot be transformed to Minkowski $\eta_{\mu \nu}$ by coordinate transformation. In other words there does NOT exist any coordinate $x^\mu(\bar x)$ such that (in the whole coordinate patch): $$\tag{2}q_{\mu \nu}(x) = \frac{\partial \bar x^{\alpha}}{\partial x^{\mu}} \frac{\partial \bar x^{\beta}}{\partial x^{\nu}} \eta_{\alpha \beta}\qquad \leftarrow \text{(does not exists in curved space)}$$ So far, everything is more or less ok... But my question is: What is the difference between $q_{\mu \nu}$ and $g_{\mu \nu}$? I mean, in both cases a particle would "feel" some fictitious forces (in which I include the weight force due to the equivalence principle). What physical situation can $q_{\mu \nu}$ describe and $g_{\mu \nu}$ cannot? I additionally know that by change of coordinates $q_{\mu \nu}$ is locally Minkowski. But still, I can't see clearly the difference.
I've a 2D sensor which provides a range $r$ and a bearing $\phi$ to a landmark. In my 2D EKF-SLAM simulation, the sensor has the following specifications $$ \sigma_{r} = 0.01 \text{m} \ \ ,\sigma_{\phi} = 0.5 \ \text{deg} $$ The location of the landmark in x-axis is 30. EKF imposes the Gaussian noise, therefore the location of the landmark is represented via two quantities namely the mean $\mu_{x}$ and the variance $\sigma_{x}$. In the following graph The green is the mean $\mu_{x}$ which is very close to the true location (i.e. 30). The black is the measurements and red is $\mu_{x} \pm 3 \sigma_{x}$. I don't understand why the uncertainty is big while I'm using rather accurate sensor. The process noise for the robot's pose is $\sigma_{v} = 0.001$ which is small noise. I'm using C++. Edit: for people who ask about the measurements, this is my code $$ r = \sqrt{ (m_{j,y} - y)^{2} + (m_{j,x} - x)^{2}} + \mathcal{N}(0, \sigma_{r}^{2}) \\ \phi = \text{atan2} \left( \frac{m_{j,y} - y}{m_{j,x} - x} \right) + \mathcal{N}(0, \sigma_{\phi}^{2}) $$ std::vector<double> Robot::observe( const std::vector<Beacon>& map ){ std::vector<double> Zobs; for (unsigned int i(0); i < map.size(); ++i) { double range, bearing; range = sqrt( pow(map[i].getX() - x,2) + pow(map[i].getY() - y,2) ); // add noise to range range += sigma_rNormalized_Gaussain_Noise_Generator(); bearing = atan2( map[i].getY() - y, map[i].getX() - x) - a; // add noise to bearing bearing += sigma_pNormalized_Gaussain_Noise_Generator(); bearing = this->wrapAngle(bearing); if ( range < 1000 ){ // store measurements (range, angle) for each landmark. Zobs.push_back(range); Zobs.push_back(bearing); //std::cout << range << " " << bearing << std::endl; } } return Zobs;} where Normalized_Gaussain_Noise_Generator() is ( i.e. $\mathcal{N}(0, 1) )$ double Robot::Normalized_Gaussain_Noise_Generator(){ double noise; std::normal_distribution<double> distribution; noise = distribution(generator); return noise;} For the measurements (i.e. the black color), I'm using the inverse measurement function given the estimate of the robot's pose and the true measurement in polar coordinates to get the location of a landmark. The actual approach is as follows $$ \bar{\mu}_{j,x} = \bar{\mu}_{x} + r \cos(\phi + \bar{\mu}_{\theta}) \\ \bar{\mu}_{j,y} = \bar{\mu}_{y} + r \sin(\phi + \bar{\mu}_{\theta}) $$ This is how it is stated in the Probabilistic Robotics book. This means that the measurements in the above graph are indeed the predicted measurements not the true ones. Now under same conditions, the true measurements can be obtained as follows $$ \text{m}_{j,x} = x + r \cos(\phi + \theta) \\ \text{m}_{j,y} = y + r \sin(\phi + \theta) $$ The result is in the graph below, which means there is no correlations between the true measurements and the robot's estimate. This leads me to the same question - why the uncertainty behaves like that?
Suppose $\vec a$ = [4, 6] and $\vec b$ = [1, 2]. Determine: a) A vector with unit length in the opposite direction to $\vec b$ For this question I understand I would have to use the $\vec a$ = k ($\vec b$) equation since we are a talking about opposite direction which I would consider collinear and from there using the magnitude equation to equal $1$ since that is the unit length and I would substitute the result of $\vec a$ = k ($\vec b$) like so.. $$1=\sqrt (k^2+2k^2)$$ $$1=5k^2$$ $${ 1 \over\sqrt 5} = k$$ But now I have no idea what to do next because the final answer comes to [$-\sqrt 5 \over 5$,$-2\sqrt 5 \over 5$]. Have I done everything correct so far? What do I need to do next? b) The components of a vector with the same magnitude as $\vec a$ making an angle of $60^\circ$ with the positive x-axis. I have no idea how to do this question but I feel like I would have to use the dot product for it
On the Lerch zeta-function, Lith. Math. J. 36 (1996), 337-346 (with A. Laurinčikas). The universality theorem with weight for the Lerch zeta-function, in: New Trends in Probability and Statistics. V.4: Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second Intern. Conf. in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996. Eds. A.Laurinčikas, E.Manstavičius and V.Stakenas, Vilnius: TEV, Utrecht: VSP, (1997), 59-67. An explicit form of the limit distribution with weight for the Lerch zeta-function in the space of analytic functions, Lith. Math. J. 37 (1997), 230-242. On one Hilbert's problem for the Lerch zeta-function, Publ. Inst. Math., 65 (79) (1999), 63-69 (with A. Laurinčikas). On zeros of the Lerch zeta-function,in: Number Theory and Its Applications, S.Kanemitsu and K.Gyory (eds.),Kluwer Academic Publishers, (1999), 129-143 (with A. Laurinčikas).PDF On zeros of the Lerch zeta-function.II, in: Probability Theory and Mathematical Statistics, Proceedings of the Seventh Vilnius Conf. 1998, B.Grigelionis et al. (Eds.), TEV/Vilnius, VSP/Utrecht, (1999), 267-276.PDF On zeros of the Lerch zeta function. III,Scient. works of Lith. Math. Soc.: supl. to "Liet. Matem. Rink.",Vilnius: Technika, 1999, pp. 24-30.PDF A note on the Riemann $\xi$-function, Liet.Matem. Rink., 40 (Special Issue) (2000), 18-20 (Lithuanian). The Lerch zeta-function, Integral Transforms and Special Functions,10 (2000), 211-226 (with A. Laurinčikas). A note on the zeros of the Lerch zeta-function, Liet.Matem. Rink. 41 (Special Issue) (2001), 53-57 (Lithuanian). Twists of Lerch zeta-functions, Liet. matem. rink. 41 (2001), 172-182 (with J. Steuding).PDF On the zero distributions of Lerch zeta-functions, Analysis 22 (2002), 1-12 (with J. Steuding).PDF On a positivity property ofthe Riemann $\xi$-function, Liet. matem. rink. 42 (2002), 179-184.PDF On the universality of Estermann zeta-functions, Analysis 22 (2002), 285-296 (with A. Laurinčikas, R. Šleževičienė and J. Steuding).PDF On some inequalities concerning $\pi (x)$, Exp. Math. 11 (2002), 297-301.PDF Do Lerch zeta-functions satisfy the Lindelof hypothesis?,in: Analytic and Probabilistic Methods in NumberTheory, Proceedings of the Third Intern. Conf. in Honour ofJ. Kubilius, Palanga, Lithuania, 24-28 September 2001,(eds. A. Dubickas, A. Laurinčikas and E. Manstavičius),TEV, Vilnius, (2002), 61-74 (with J. Steuding).PDF On the mean square of Lerch zeta-functions, Arch. Math. 80 (2003), 47-60 (with A. Laurinčikas and J. Steuding).PDF Onthe Chebyshev function $\psi(x)$, Liet. matem. rink.43 (2003), 487-496 = Lith. Math. J. 401-409. The effective universality theorem for the Riemann zetafunction, in: Proceedings of the session in analytic numbertheory and Diophantine equations, MPI-Bonn, January - June 2002,Ed. by D. R. Heath-Brown, B. Z. Moroz, Bonner mathematischeSchriften, 360 (2003), 21 pp.PDF An approximation of the Hurwitz zeta-function by afinite sum, Liet.Matem. Rink. 43 (Special Issue)(2003), 32-34 (Lithuanian). On the Voronin's universality theorem for the Riemannzeta-function, Proceedings of Scientific Seminar of theFaculty of Physics and Mathematics, Šiauliai University 6 (2003), 29-33.PDF An approximate functional equation for the Lerchzeta-function, Math. Notes 74 (2003), 469-476 (with A. Laurinčikas and J.Steuding).PDF Approximation of the Lerch zeta-function, Liet. matem. rink.44 (2004), 176-180 = Lith. Math. J. 140-144.PDF Universality of Dirichlet L-functions with shifted characters, Liet.Matem. Rink. 44 (Special Issue)(2004), 48-50. Growth of the Lerchzeta-function, Liet. matem. rink.45 (2005), 45-56 = Lith. Math. J. 34-43.PDF Note on the zeros of the Hurwitz zeta-function, in: Voronoi's impact on modern science. Book 3: proceedings of the third Voronoi Conference on Number Theory and Spatial Tessellations. Mathematics and its Applications, 55 (2005), 10-12.PDF Simple zeros and discrete moments of the derivative of the Riemann zeta-function, J. Number Theory 115 (2005), 310-321 (with J. Steuding).PDF On the distribution of zeros of the Hurwitz zeta-function, Math. Comp.76 (2007), 323-337 (with J. Steuding).PDF On the Backlund equivalent for the Lindelof hypothesis, Adv. Stud. Pure Math. 49 (2007), 91-104.PDF Sum of the periodic zeta-function over the nontrivial zeros of the Riemann zeta-function, Analysis, München, 28 (2008), 209-217 (with J. Kalpokas).PDF Note on zeros of the derivative of the Selberg zeta-function, Arch. Math. 91 (2008), 238-246. PDF . Corrigendum, Arch. Math. 93 (2009), page 143. PDF Selberg's Central Limit Theorem on the Critical Line and the Lerch Zeta-Function, in: Proceedings of the conference "New Directions in the Theory of Universal Zeta- and L-Functions", Würzburg, Germany, October 6-10, 2008, Shaker Verlag, (2009), 57-64 (with A. Grigutis and A. Laurinčikas). PDF Effective uniform approximation by the Riemann zeta-function, Publ. Mat. 54 (2010), 209-219 (with A. Laurinčikas, K. Matsumoto, J. Steuding and R. Steuding). PDF Sum of the Dirichlet L-function over nontrivial zeros of another Dirichlet L-function, Acta Math. Hungar., 128 (2010), 287-298 (with J. Kalpokas and J. Steuding). PDF Self-approximation of Dirichlet L-functions, J. Number Theory, 131(7) (2011), 1286-1295. arXiv:1006.1507 Questions around the nontrivial zeros of the Riemann zeta-function - computations and classifications, Math. Model. Anal., 16(1) (2011), 72-81. (with J. Steuding). PDF Uniqueness theorems for L-functions, Comment. Math. Univ. St. Pauli, 60, No. 1,2 (2011), 15-35. (with J. Grahl and J. Steuding). PDF Zeros of the Lerch transcendent function, Mathematical Modelling and Analysis, 17, No. 2 (2012), 245-250. (with A. Grigutis). PDF The a-values of the Selberg zeta-function, Lith. Math. J., 52, No. 2 (2012), 145-154. (with R. Šimenas). PDF Zeros of the periodic zeta-function, Šiauliai Mathematical Seminar, 8(16) (2013), 49-62. (with R. Tamošiūnas). PDF Zeros of the Estermann zeta-function, Journal of the Australian Mathematical Society 94 (2013), 38-49,doi:10.1017/S1446788712000419 (with A. Dubickas, J.Steuding, and R.Steuding). PDF Complex B-splines and Hurwitz zeta functions, LMS Journal of Computation and Mathematics 16 (2013), 61-77, (with B. Forster, P. Massopust, and J. Steuding). PDF The discrete mean square of the Dirichlet L-function at nontrivial zerosof another Dirichlet L-function, International Journal of Number Theory 9(4) (2013), 945-963, (with J. Kalpokas). PDF Universality of the Selberg zeta-function for the modular group, Forum Mathematicum 25(3) (2013), 533-564, (with P. Drungilas and A. Kačėnas). PDF On the roots of the equation $\zeta(s)=a$, Abh. Math. Semin. Univ. Hambg. 84 (2014), 1-15, (with J. Steuding). arXiv:1011.5339 Self-approximation of Hurwitz zeta-functions, Functiones et Approximatio 51(1) (2014), 181-188, (with E. Karikovas). PDF The a-points ofthe Selberg zeta-function are uniformly distributed modulo one, Illinois J. Math. 58(1) (2014), 207–218, (with J. Steuding and R. Šimėnas). PDF On the Speiser equivalent for the Riemann hypothesis, European Journal of Mathematics 1 (2015), 337-350, (with R. Šimėnas). PDF The size of the Selberg zeta-functionat places symmetric with respectto the line Re(s)= 1/2, Results. Math. 70(1) (2016), 271–281, (with A. Grigutis). PDF Sum of the Lerch zeta-function over nontrivial zeros of the Dirichlet L-function, From arithmetic to zeta-functions. Number theory in memory of Wolfgang Schwarz. Cham: Springer. (2016), 141–153, (with J. Kalpokas). PDF On the distribution of the a-values of the Selbergzeta-function associated to finite volume Riemann surfaces, J. Number Theory 173 (2017), 64–86, (with R. Šimėnas). PDF Zeros of the Riemann zeta-function and its universality, Acta Arith. 181(2) (2017), 127-142 (with A. Laurinčikas and R. Macaitienė). Symmetry of zeros of Lerch zeta-function for equal parameters, Lith. Math. J. 57(4) (2017), 433-440 (with R. Tamošiūnas). PDF Discrete mean square of the Riemann zeta-function over imaginary parts of its zeros, Period. Math. Hungar. 76 (2018), 217-228 (with A. Laurinčikas). arXiv:1608.08493 The Riemann hypothesis and universality of the Riemann zeta-function, Math. Slovaca 68(4) (2018), 741-748 (with A. Laurinčikas). PDF Growth of the Selberg zeta-function, Kyushu J. Math. 72 (2018), 441-447. PDF Zero-free regions for derivatives of the Selberg zeta-function, Publ. Math. Debrecen 93 (2018), 369-385. PDF Asymptotic distribution of Beurling integers, Int. J. Number Theory 14(10) (2018), 2555-2569 (with L. Kaziulytė). PDF The size of the Lerch zeta-function at places symmetric with respect to the line Re(s)=1/2, Czechoslovak Math. J., 69 (2019), 25-37 (with A. Grigutis). PDF On the vertical distribution of the a-points of the Selberg zeta-function attached to a finite volume Riemann surface, Lith. Math. J., 59 (2019), 143-155 (with R. Šimėnas). PDF Second moment of the Beurling zeta-function, Lith. Math. J., 59 (2019), 317-337 (with P. Drungilas and A. Novikas). PDF Zeros of the Lerch zeta-function and of its derivative for equal parameters, to appear in Bull. Math. Soc. Sci. Math. Roumanie (with R. Tamošiūnas). arXiv:1902.03064 On primeness of the Selberg zeta-function, to appear in Hokkaido Math. J. (with J. Steuding). arXiv:1908.03108 Zeros of the extended Selberg class zeta-functions and of their derivatives, arXiv:1904.03123.
Naïvely this is what happens and it obviously is not helpful! In[7]:= Conjugate[SphericalHarmonicY[1, 1, θ, ϕ]]Out[7]= -(1/2) E^(-I Conjugate[ϕ]) Sqrt[3/(2 π)] Conjugate[Sin[θ]] So, I tried stating initially that $\theta$ and $\phi$ are reals but still that doesn't seem to have helped any bit, In[8]:= θ ∈ Reals; ϕ ∈ Reals;In[9]:= SphericalHarmonicY[1, 1, θ, ϕ]Out[9]= -(1/2) E^(I ϕ) Sqrt[3/(2 π)] Sin[θ]In[10]:= Conjugate[SphericalHarmonicY[1, 1, θ, ϕ]]Out[10]= -(1/2) E^(-I Conjugate[ϕ]) Sqrt[3/(2 π)] Conjugate[Sin[θ]] Kindly tell me how to do this? (I want to calculate sums like $\sum\limits_{m=-\ell}^{\ell}\left\| Y_{l,m} (\theta,\phi)\right\|^2$.)
X Search Filters Format Subjects Library Location Language Publication Date Click on a bar to filter by decade Slide to change publication date range BMC health services research, 07/2016, Volume 16 Suppl 3, p. 200 Journal Article BMC Health Services Research, ISSN 1472-6963, 07/2016, Volume 16, Issue S3 Journal Article 3. Search for heavy particles decaying into top-quark pairs using lepton-plus-jets events in proton–proton collisions at $\sqrt{s} = 13$ $\text {TeV}$ with the ATLAS detector European Physical Journal. C, Particles and Fields, ISSN 1434-6044, 07/2018, Volume 78, Issue 7 Here, a search for new heavy particles that decay into top-quark pairs is performed using data collected from proton–proton collisions at a centre-of-mass... PHYSICS OF ELEMENTARY PARTICLES AND FIELDS PHYSICS OF ELEMENTARY PARTICLES AND FIELDS Journal Article 4. Observation of Higgs boson production in association with a top quark pair at the LHC with the ATLAS detector Physics Letters B, ISSN 0370-2693, 09/2018, Volume 784, Issue C, pp. 173 - 191 The observation of Higgs boson production in association with a top quark pair ( ), based on the analysis of proton–proton collision data at a centre-of-mass... PHYSICS, NUCLEAR \| SEARCH \| ASTRONOMY & ASTROPHYSICS \| PHYSICS, PARTICLES & FIELDS \| Physics - High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Fysik \| Subatomär fysik \| Physical Sciences \| Subatomic Physics \| Naturvetenskap \| Natural Sciences PHYSICS, NUCLEAR \| SEARCH \| ASTRONOMY & ASTROPHYSICS \| PHYSICS, PARTICLES & FIELDS \| Physics - High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Fysik \| Subatomär fysik \| Physical Sciences \| Subatomic Physics \| Naturvetenskap \| Natural Sciences Journal Article 5. Search for pair production of up-type vector-like quarks and for four-top-quark events in final states with multiple b-jets with the ATLAS detector Journal of High Energy Physics, ISSN 1126-6708, 7/2018, Volume 2018, Issue 7, pp. 1 - 68 A search for pair production of up-type vector-like quarks (T ) with a significant branching ratio into a top quark and either a Standard Model Higgs boson or... vectorlike quarks \| Beyond Standard Model \| Hadron-Hadron scattering (experiments) \| Quantum Physics \| Quantum Field Theories, String Theory \| Classical and Quantum Gravitation, Relativity Theory \| Physics \| Elementary Particles, Quantum Field Theory \| BREAKING \| PARTON DISTRIBUTIONS \| BOSON \| GLUON \| vector-like quarks \| PLUS PLUS \| PROGRAM \| PHYSICS, PARTICLES & FIELDS \| Standard model (particle physics) \| Large Hadron Collider \| Leptons \| Luminosity \| Higgs bosons \| Quarks \| Searching \| Decay \| Transverse momentum \| Signal processing \| Pair production \| Jets \| Field theory \| Cross sections \| Bosons \| Physics - High Energy Physics - Experiment \| Fysik \| Physical Sciences \| Naturvetenskap \| Natural Sciences vectorlike quarks \| Beyond Standard Model \| Hadron-Hadron scattering (experiments) \| Quantum Physics \| Quantum Field Theories, String Theory \| Classical and Quantum Gravitation, Relativity Theory \| Physics \| Elementary Particles, Quantum Field Theory \| BREAKING \| PARTON DISTRIBUTIONS \| BOSON \| GLUON \| vector-like quarks \| PLUS PLUS \| PROGRAM \| PHYSICS, PARTICLES & FIELDS \| Standard model (particle physics) \| Large Hadron Collider \| Leptons \| Luminosity \| Higgs bosons \| Quarks \| Searching \| Decay \| Transverse momentum \| Signal processing \| Pair production \| Jets \| Field theory \| Cross sections \| Bosons \| Physics - High Energy Physics - Experiment \| Fysik \| Physical Sciences \| Naturvetenskap \| Natural Sciences Journal Article 02/2018 Phys. Rev. Lett. 120 (2018) 202007 A search for the narrow structure, $X(5568)$, reported by the D0 Collaboration in the decay sequence $X \to B^0_s \pi^\pm$,... Physics - High Energy Physics - Experiment Physics - High Energy Physics - Experiment Journal Article 7. Search for High-Mass Resonances Decaying to $\tau\nu$ in $pp$ Collisions at $\sqrt{s}$ = 13 TeV with the ATLAS Detector 01/2018 Phys. Rev. Lett. 120, 161802 (2018) A search for high-mass resonances decaying to $\tau\nu$ using proton-proton collisions at $\sqrt{s}$ = 13 TeV produced by... Physics - High Energy Physics - Experiment Physics - High Energy Physics - Experiment Journal Article 8. Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC Physics Letters B, ISSN 0370-2693, 09/2012, Volume 716, Issue 1, pp. 1 - 29 A search for the Standard Model Higgs boson in proton–proton collisions with the ATLAS detector at the LHC is presented. The datasets used correspond to... TRANSVERSE-MOMENTUM \| BROKEN SYMMETRIES \| PARTON DISTRIBUTIONS \| MASSES \| DECAY \| ASTRONOMY & ASTROPHYSICS \| QCD CORRECTIONS \| TAU \| PHYSICS, NUCLEAR \| COLLIDERS \| PHYSICS, PARTICLES & FIELDS \| Collisions (Nuclear physics) \| Analysis \| Standards \| Detectors \| Searching \| Decay \| Elementary particles \| Higgs bosons \| Standard deviation \| Channels \| Physics - High Energy Physics - Experiment \| Physics \| High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Qcd Corrections \| Hadron Colliders \| Transverse-Momentum \| Massless Particles \| Gauge Fields \| Proton-Proton Collisions \| Fysik \| Broken Symmetries \| Physical Sciences \| Naturvetenskap \| Parton Distributions \| Root-S=7 Tev \| Natural Sciences \| Cross-Sections TRANSVERSE-MOMENTUM \| BROKEN SYMMETRIES \| PARTON DISTRIBUTIONS \| MASSES \| DECAY \| ASTRONOMY & ASTROPHYSICS \| QCD CORRECTIONS \| TAU \| PHYSICS, NUCLEAR \| COLLIDERS \| PHYSICS, PARTICLES & FIELDS \| Collisions (Nuclear physics) \| Analysis \| Standards \| Detectors \| Searching \| Decay \| Elementary particles \| Higgs bosons \| Standard deviation \| Channels \| Physics - High Energy Physics - Experiment \| Physics \| High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Qcd Corrections \| Hadron Colliders \| Transverse-Momentum \| Massless Particles \| Gauge Fields \| Proton-Proton Collisions \| Fysik \| Broken Symmetries \| Physical Sciences \| Naturvetenskap \| Parton Distributions \| Root-S=7 Tev \| Natural Sciences \| Cross-Sections Journal Article 9. Search for doubly charged scalar bosons decaying into same-sign W boson pairs with the ATLAS detector The European Physical Journal C, ISSN 1434-6044, 1/2019, Volume 79, Issue 1, pp. 1 - 30 A search for doubly charged scalar bosons decaying into W boson pairs is presented. It uses a data sample from proton–proton collisions corresponding to an... Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology \| DISTRIBUTIONS \| PHYSICS \| PHYSICS, PARTICLES & FIELDS \| Analysis \| Detectors \| Collisions (Nuclear physics) \| Phenomenology \| Protons \| Confidence intervals \| Large Hadron Collider \| Leptons \| Particle collisions \| Searching \| Decay \| Luminosity \| Bosons \| Physics - High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Regular - Experimental Physics Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology \| DISTRIBUTIONS \| PHYSICS \| PHYSICS, PARTICLES & FIELDS \| Analysis \| Detectors \| Collisions (Nuclear physics) \| Phenomenology \| Protons \| Confidence intervals \| Large Hadron Collider \| Leptons \| Particle collisions \| Searching \| Decay \| Luminosity \| Bosons \| Physics - High Energy Physics - Experiment \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS \| Regular - Experimental Physics Journal Article 10. Measurement of the photon identification efficiencies with the ATLAS detector using LHC Run 2 data collected in 2015 and 2016 The European Physical Journal C, ISSN 1434-6044, 3/2019, Volume 79, Issue 3, pp. 1 - 41 The efficiency of the photon identification criteria in the ATLAS detector is measured using $$36.1\hbox { fb}^1$$ 36.1fb1 to $$36.7\hbox { fb}^1$$ 36.7fb1 of... Nuclear Physics, Heavy Ions, Hadrons \| Nuclear Physics, Heavy Ions, Hadrons \|
X Search Filters Format Subjects Library Location Language Publication Date Click on a bar to filter by decade Slide to change publication date range 1. Observation of a peaking structure in the J/psi phi mass spectrum from B-+/- -> J/psi phi K-+/- decays PHYSICS LETTERS B, ISSN 0370-2693, 06/2014, Volume 734, Issue 370-2693 0370-2693, pp. 261 - 281 A peaking structure in the J/psi phi mass spectrum near threshold is observed in B-+/- -> J/psi phi K-+/- decays, produced in pp collisions at root s = 7 TeV... PHYSICS, NUCLEAR \| ASTRONOMY & ASTROPHYSICS \| PHYSICS, PARTICLES & FIELDS \| Physics - High Energy Physics - Experiment \| Physics \| High Energy Physics - Experiment \| scattering [p p] \| J/psi --> muon+ muon \| experimental results \| Particle Physics - Experiment \| Nuclear and High Energy Physics \| Phi --> K+ K \| vertex [track data analysis] \| CERN LHC Coll \| B+ --> J/psi Phi K \| Peaking structure \| hadronic decay [B] \| Integrated luminosity \| Data sample \| final state [dimuon] \| mass enhancement \| width [resonance] \| (J/psi Phi) [mass spectrum] \| Breit-Wigner [resonance] \| 7000 GeV-cms \| leptonic decay [J/psi] \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS PHYSICS, NUCLEAR \| ASTRONOMY & ASTROPHYSICS \| PHYSICS, PARTICLES & FIELDS \| Physics - High Energy Physics - Experiment \| Physics \| High Energy Physics - Experiment \| scattering [p p] \| J/psi --> muon+ muon \| experimental results \| Particle Physics - Experiment \| Nuclear and High Energy Physics \| Phi --> K+ K \| vertex [track data analysis] \| CERN LHC Coll \| B+ --> J/psi Phi K \| Peaking structure \| hadronic decay [B] \| Integrated luminosity \| Data sample \| final state [dimuon] \| mass enhancement \| width [resonance] \| (J/psi Phi) [mass spectrum] \| Breit-Wigner [resonance] \| 7000 GeV-cms \| leptonic decay [J/psi] \| PHYSICS OF ELEMENTARY PARTICLES AND FIELDS Journal Article 2. Measurement of the ratio of the production cross sections times branching fractions of B c ± → J/ψπ ± and B± → J/ψK ± and ℬ B c ± → J / ψ π ± π ± π ∓ / ℬ B c ± → J / ψ π ± $$ \mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm }{\pi}^{\pm }{\pi}^{\mp}\right)/\mathrm{\mathcal{B}}\left({\mathrm{B}}_{\mathrm{c}}^{\pm}\to \mathrm{J}/\psi {\pi}^{\pm}\right) $$ in pp collisions at s = 7 $$ \sqrt{s}=7 $$ TeV Journal of High Energy Physics, ISSN 1029-8479, 1/2015, Volume 2015, Issue 1, pp. 1 - 30 The ratio of the production cross sections times branching fractions σ B c ± ℬ B c ± → J / ψ π ± / σ B ± ℬ B ± → J / ψ K ± $$ \left(\sigma... B physics \| Branching fraction \| Hadron-Hadron Scattering \| Quantum Physics \| Quantum Field Theories, String Theory \| Classical and Quantum Gravitation, Relativity Theory \| Physics \| Elementary Particles, Quantum Field Theory B physics \| Branching fraction \| Hadron-Hadron Scattering \| Quantum Physics \| Quantum Field Theories, String Theory \| Classical and Quantum Gravitation, Relativity Theory \| Physics \| Elementary Particles, Quantum Field Theory Journal Article Physical Review Letters, ISSN 0031-9007, 06/2013, Volume 110, Issue 25, p. 252002 The cross section for ee+ e- → π+ π- J/ψ between 3.8 and 5.5 GeV is measured with a 967 fb(-1) data sample collected by the Belle detector at or near the... Journal Article Physics Letters B, ISSN 0370-2693, 03/2017, Volume 766, Issue C, pp. 212 - 224 Journal Article Physics Letters B, ISSN 0370-2693, 05/2016, Volume 756, Issue C, pp. 84 - 102 A measurement of the ratio of the branching fractions of the meson to and to is presented. The , , and are observed through their decays to , , and ,... scattering [p p] \| pair production [pi] \| statistical \| Physics, Nuclear \| 114 Physical sciences \| Phi --> K+ K \| Astronomy & Astrophysics \| LHC, CMS, B physics, Nuclear and High Energy Physics \| f0 --> pi+ pi \| High Energy Physics - Experiment \| Compact Muon Solenoid \| pair production [K] \| Science & Technology \| mass spectrum [K+ K-] \| Ratio B \| Large Hadron Collider (LHC) \| Nuclear & Particles Physics \| 7000 GeV-cms \| leptonic decay [J/psi] \| J/psi --> muon+ muon \| experimental results \| Nuclear and High Energy Physics \| Physics and Astronomy \| branching ratio [B/s0] \| CERN LHC Coll \| B/s0 --> J/psi Phi \| CMS collaboration ; proton-proton collisions ; CMS ; B physics \| Physics \| Física \| Physical Sciences \| hadronic decay [f0] \| [PHYS.HEXP]Physics [physics]/High Energy Physics - Experiment [hep-ex] \| Physics, Particles & Fields \| 0202 Atomic, Molecular, Nuclear, Particle And Plasma Physics \| colliding beams [p p] \| hadronic decay [Phi] \| mass spectrum [pi+ pi-] \| B/s0 --> J/psi f0 scattering [p p] \| pair production [pi] \| statistical \| Physics, Nuclear \| 114 Physical sciences \| Phi --> K+ K \| Astronomy & Astrophysics \| LHC, CMS, B physics, Nuclear and High Energy Physics \| f0 --> pi+ pi \| High Energy Physics - Experiment \| Compact Muon Solenoid \| pair production [K] \| Science & Technology \| mass spectrum [K+ K-] \| Ratio B \| Large Hadron Collider (LHC) \| Nuclear & Particles Physics \| 7000 GeV-cms \| leptonic decay [J/psi] \| J/psi --> muon+ muon \| experimental results \| Nuclear and High Energy Physics \| Physics and Astronomy \| branching ratio [B/s0] \| CERN LHC Coll \| B/s0 --> J/psi Phi \| CMS collaboration ; proton-proton collisions ; CMS ; B physics \| Physics \| Física \| Physical Sciences \| hadronic decay [f0] \| [PHYS.HEXP]Physics [physics]/High Energy Physics - Experiment [hep-ex] \| Physics, Particles & Fields \| 0202 Atomic, Molecular, Nuclear, Particle And Plasma Physics \| colliding beams [p p] \| hadronic decay [Phi] \| mass spectrum [pi+ pi-] \| B/s0 --> J/psi f0 Journal Article Physical Review Letters, ISSN 0031-9007, 12/2017, Volume 119, Issue 24 We report a precise measurement of the J/ψ elliptic flow in Pb-Pb collisions at sNN=5.02 TeV with the ALICE detector at the LHC. The J/ψ mesons are... Transport modeling \| Subatomic Physics \| Forward rapidity \| Semicentral collisions \| Thermalization \| Transverse momenta \| Elliptic flows \| Momentum \| Lead compounds \| Pb-Pb collisions \| Binary alloys \| Fysik \| Physical Sciences \| Naturvetenskap \| Lead \| Tellurium compounds \| Lead alloys \| Subatomär fysik \| Natural Sciences \| Precise measurements Transport modeling \| Subatomic Physics \| Forward rapidity \| Semicentral collisions \| Thermalization \| Transverse momenta \| Elliptic flows \| Momentum \| Lead compounds \| Pb-Pb collisions \| Binary alloys \| Fysik \| Physical Sciences \| Naturvetenskap \| Lead \| Tellurium compounds \| Lead alloys \| Subatomär fysik \| Natural Sciences \| Precise measurements Journal Article European Physical Journal C, ISSN 1434-6044, 07/2018, Volume 78, Issue 7 We report on the measurement of the inclusive J/ψ polarization parameters in pp collisions at a center of mass energy √s=8 TeV with the ALICE detector at the... Engineering (miscellaneous); Physics and Astronomy (miscellaneous) \| Astrophysics \| J/psi: hadroproduction \| 114 Physical sciences \| J/psi: leptonic decay \| High Energy Physics - Experiment \| High Energy Physics \| Nuclear Experiment \| Engineering (miscellaneous) \| p p: colliding beams \| [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex] \| Physics and Astronomy (miscellaneous), Relativistic Heavy-Ion collisions \| Physics and Astronomy (miscellaneous) \| muon: pair production \| experimental results \| Experiment \| Nuclear and particle physics. Atomic energy. Radioactivity \| CERN LHC Coll \| 8000 GeV-cms \| J/psi: polarization \| helicity \| [PHYS.HEXP]Physics [physics]/High Energy Physics - Experiment [hep-ex] \| rapidity dependence \| transverse momentum dependence \| p p: scattering \| Fysik \| Physical Sciences \| Naturvetenskap \| Natural Sciences Engineering (miscellaneous); Physics and Astronomy (miscellaneous) \| Astrophysics \| J/psi: hadroproduction \| 114 Physical sciences \| J/psi: leptonic decay \| High Energy Physics - Experiment \| High Energy Physics \| Nuclear Experiment \| Engineering (miscellaneous) \| p p: colliding beams \| [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex] \| Physics and Astronomy (miscellaneous), Relativistic Heavy-Ion collisions \| Physics and Astronomy (miscellaneous) \| muon: pair production \| experimental results \| Experiment \| Nuclear and particle physics. Atomic energy. Radioactivity \| CERN LHC Coll \| 8000 GeV-cms \| J/psi: polarization \| helicity \| [PHYS.HEXP]Physics [physics]/High Energy Physics - Experiment [hep-ex] \| rapidity dependence \| transverse momentum dependence \| p p: scattering \| Fysik \| Physical Sciences \| Naturvetenskap \| Natural Sciences Journal Article Journal of High Energy Physics, ISSN 1126-6708, 2012, Volume 2012, Issue 5 Journal Article Physical Review Letters, ISSN 0031-9007, 2004, Volume 93, Issue 4 Journal Article Physical Review Letters, ISSN 0031-9007, 2008, Volume 101, Issue 8 Journal Article Physics Letters B, ISSN 0370-2693, 06/2014, Volume 734, pp. 261 - 281 A peaking structure in the mass spectrum near threshold is observed in decays, produced in pp collisions at collected with the CMS detector at the LHC. The... Journal Article 12. Prompt and non-prompt $$\hbox {J}/\psi $$ J/ψ production and nuclear modification at mid-rapidity in p–Pb collisions at $$\mathbf{\sqrt{{ s}_{\text {NN}}}= 5.02}$$ sNN=5.02 TeV The European Physical Journal C, ISSN 1434-6044, 6/2018, Volume 78, Issue 6, pp. 1 - 17 A measurement of beauty hadron production at mid-rapidity in proton-lead collisions at a nucleon–nucleon centre-of-mass energy $$\sqrt{s_\text {NN}}=5.02$$... Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Nuclear Physics, Heavy Ions, Hadrons \| Measurement Science and Instrumentation \| Nuclear Energy \| Quantum Field Theories, String Theory \| Physics \| Elementary Particles, Quantum Field Theory \| Astronomy, Astrophysics and Cosmology Journal Article
Difference between revisions of "Unitriangular matrix group:UT(3,p)" (→In coordinate form) m (→In coordinate form) (17 intermediate revisions by 2 users not shown) Line 2: Line 2: ==Definition== ==Definition== + + ===As a group of matrices=== ===As a group of matrices=== Line 9: Line 11: <math>\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \}</math> <math>\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \}</math> − The + The the + + + <math>= </math> + + + . + + <math> = </math> + + + + + + + + + + ,the [[]] , + − = + = − + <math>^-, </math> − <math>( + <math>(,,)</math> − + <math> + a_{12} + 0 a_{23} + + {}</math> ===Definition by presentation=== ===Definition by presentation=== Line 23: Line 48: The group can be defined by means of the following [[presentation]]: The group can be defined by means of the following [[presentation]]: − <math>\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy, x^p = y^p = z^p = + <math>\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy, x^p = y^p = z^p = \rangle</math> − where <math> + where <math></math> denotes the identity element. These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators <math>x,y,z</math> correspond to matrices: These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators <math>x,y,z</math> correspond to matrices: Line 43: Line 68: \end{pmatrix}</math> \end{pmatrix}</math> − === + + + ====== + + + + − + the <math>a_{12} </math> the <math></math>, the : − − <math> + <math>{ {} a_{13} a_{23} {} a_{13}, a_{23} {}}</math> − The + The <math>a_{13} a_{23} </math>: − + <math>\begin{pmatrix} 1 & a_{12} & \\ 0 & 1 & \\ 0 & 0 & 1 \\\end{pmatrix}</math> − − − − \end{pmatrix}</math> ==Families== ==Families== # These groups fall in the more general family <math>UT(n,p)</math> of [[unitriangular matrix group]]s. The unitriangular matrix group <math>UT(n,p)</math> can be described as the group of unipotent upper-triangular matrices in <math>GL(n,p)</math>, which is also a <math>p</math>-Sylow subgroup of the [[general linear group]] <math>GL(n,p)</math>. This further can be generalized to <math>UT(n,q)</math> where <math>q</math> is the power of a prime <math>p</math>. <math>UT(n,q)</math> is the <math>p</math>-Sylow subgroup of <math>GL(n,q)</math>. # These groups fall in the more general family <math>UT(n,p)</math> of [[unitriangular matrix group]]s. The unitriangular matrix group <math>UT(n,p)</math> can be described as the group of unipotent upper-triangular matrices in <math>GL(n,p)</math>, which is also a <math>p</math>-Sylow subgroup of the [[general linear group]] <math>GL(n,p)</math>. This further can be generalized to <math>UT(n,q)</math> where <math>q</math> is the power of a prime <math>p</math>. <math>UT(n,q)</math> is the <math>p</math>-Sylow subgroup of <math>GL(n,q)</math>. − # These groups also fall into the general family of [[extraspecial group]]s. + # These groups also fall into the general family of [[extraspecial group]]s. ==Elements== ==Elements== Line 123: Line 149: ==Subgroups== ==Subgroups== {{further\|[[Subgroup structure of unitriangular matrix group:UT(3,p)]]}} {{further\|[[Subgroup structure of unitriangular matrix group:UT(3,p)]]}} + + {{#lst:subgroup structure of unitriangular matrix group:UT(3,p)\|summary}} {{#lst:subgroup structure of unitriangular matrix group:UT(3,p)\|summary}} Line 132: Line 160: {{#lst:linear representation theory of unitriangular matrix group:UT(3,p)\|summary}} {{#lst:linear representation theory of unitriangular matrix group:UT(3,p)\|summary}} − == + ==== − + == − − − − − − − − − − − + of order the center, the . − − − − − − − − − − ==GAP implementation== ==GAP implementation== Line 184: Line 192: <tt>ExtraspecialGroup(5^3,5)</tt> <tt>ExtraspecialGroup(5^3,5)</tt> − == + ==== − == + == − + − + + + + + + + + <>p</> , + − − ==External links == ==External links == − * {{wp\| + * {{wp\|}} Latest revision as of 11:21, 22 August 2014 This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group. View other such prime-parametrized groups Contents 1 Definition 2 Families 3 Elements 4 Arithmetic functions 5 Subgroups 6 Linear representation theory 7 Endomorphisms 8 GAP implementation 9 External links Definition Note that the case , where the group becomes dihedral group:D8, behaves somewhat differently from the general case. We note on the page all the places where the discussion does not apply to . As a group of matrices The multiplication of matrices and gives the matrix where: The identity element is the identity matrix. The inverse of a matrix is the matrix where: Note that all addition and multiplication in these definitions is happening over the field . In coordinate form We may define the group as set of triples over the prime field , with the multiplication law given by: , . The matrix corresponding to triple is: Definition by presentation The group can be defined by means of the following presentation: where denotes the identity element. These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators correspond to matrices: Note that in the above presentation, the generator is redundant, and the presentation can thus be rewritten as a presentation with only two generators and . As a semidirect product This group of order can also be described as a semidirect product of the elementary abelian group of order by the cyclic group of order , with the following action. Denote the base of the semidirect product as ordered pairs of elements from . The action of the generator of the acting group is as follows: In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order , i.e., the elementary abelian subgroup of order is the subgroup: The acting subgroup of order can be taken as the subgroup with , i.e., the subgroup: Families These groups fall in the more general family of unitriangular matrix groups. The unitriangular matrix group can be described as the group of unipotent upper-triangular matrices in , which is also a -Sylow subgroup of the general linear group . This further can be generalized to where is the power of a prime . is the -Sylow subgroup of . These groups also fall into the general family of extraspecial groups. For any number of the form , there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. is an extraspecial group of order and "+" type. The other type of extraspecial group of order , i.e., the extraspecial group of order and "-" type, is semidirect product of cyclic group of prime-square order and cyclic group of prime order. Elements Further information: element structure of unitriangular matrix group:UT(3,p) Summary Item Value number of conjugacy classes order Agrees with general order formula for : conjugacy class size statistics size 1 ( times), size ( times) orbits under automorphism group Case : size 1 (1 conjugacy class of size 1), size 1 (1 conjugacy class of size 1), size 2 (1 conjugacy class of size 2), size 4 (2 conjugacy classes of size 2 each) Case odd : size 1 (1 conjugacy class of size 1), size ( conjugacy classes of size 1 each), size ( conjugacy classes of size each) number of orbits under automorphism group 4 if 3 if is odd order statistics Case : order 1 (1 element), order 2 (5 elements), order 4 (2 elements) Case odd: order 1 (1 element), order ( elements) exponent 4 if if odd Conjugacy class structure Note that the characteristic polynomial of all elements in this group is , hence we do not devote a column to the characteristic polynomial. For reference, we consider matrices of the form: Nature of conjugacy class Jordan block size decomposition Minimal polynomial Size of conjugacy class Number of such conjugacy classes Total number of elements Order of elements in each such conjugacy class Type of matrix identity element 1 + 1 + 1 + 1 1 1 1 1 non-identity element, but central (has Jordan blocks of size one and two respectively) 2 + 1 1 , non-central, has Jordan blocks of size one and two respectively 2 + 1 , but not both and are zero non-central, has Jordan block of size three 3 if odd 4 if both and are nonzero Total (--) -- -- -- -- -- Arithmetic functions Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions For some of these, the function values are different when and/or when . These are clearly indicated below. Arithmetic functions taking values between 0 and 3 Function Value Explanation prime-base logarithm of order 3 the order is prime-base logarithm of exponent 1 the exponent is . Exception when , where the exponent is . nilpotency class 2 derived length 2 Frattini length 2 minimum size of generating set 2 subgroup rank 2 rank as p-group 2 normal rank as p-group 2 characteristic rank as p-group 1 Arithmetic functions of a counting nature Function Value Explanation number of conjugacy classes elements in the center, and each other conjugacy class has size number of subgroups when , when See subgroup structure of unitriangular matrix group:UT(3,p) number of normal subgroups See subgroup structure of unitriangular matrix group:UT(3,p) number of conjugacy classes of subgroups for , for See subgroup structure of unitriangular matrix group:UT(3,p) Subgroups Further information: Subgroup structure of unitriangular matrix group:UT(3,p) Note that the analysis here specifically does not apply to the case . For , see subgroup structure of dihedral group:D8. Table classifying subgroups up to automorphisms Automorphism class of subgroups Representative Isomorphism class Order of subgroups Index of subgroups Number of conjugacy classes Size of each conjugacy class Number of subgroups Isomorphism class of quotient (if exists) Subnormal depth (if subnormal) trivial subgroup trivial group 1 1 1 1 prime-cube order group:U(3,p) 1 center of unitriangular matrix group:UT(3,p) ; equivalently, given by . group of prime order 1 1 1 elementary abelian group of prime-square order 1 non-central subgroups of prime order in unitriangular matrix group:UT(3,p) Subgroup generated by any element with at least one of the entries nonzero group of prime order -- 2 elementary abelian subgroups of prime-square order in unitriangular matrix group:UT(3,p) join of center and any non-central subgroup of prime order elementary abelian group of prime-square order 1 group of prime order 1 whole group all elements unitriangular matrix group:UT(3,p) 1 1 1 1 trivial group 0 Total (5 rows) -- -- -- -- -- -- -- Tables classifying isomorphism types of subgroups Group name GAP ID Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup Trivial group 1 1 1 1 Group of prime order 1 1 Elementary abelian group of prime-square order 0 Prime-cube order group:U3p 1 1 1 1 Total -- Table listing number of subgroups by order Group order Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup 1 1 1 1 1 1 0 1 1 1 1 Total Linear representation theory Further information: linear representation theory of unitriangular matrix group:UT(3,p) Item Value number of conjugacy classes (equals number of irreducible representations over a splitting field) . See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree three over a finite field degrees of irreducible representations over a splitting field (such as or ) 1 (occurs times), (occurs times) sum of squares of degrees of irreducible representations (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group lcm of degrees of irreducible representations condition for a field (characteristic not equal to ) to be a splitting field The polynomial should split completely. For a finite field of size , this is equivalent to . field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) Field where is a primitive root of unity. This is a degree extension of the rationals. unique minimal splitting field (characteristic ) The field of size where is the order of mod . degrees of irreducible representations over the rational numbers 1 (1 time), ( times), (1 time) Orbits over a splitting field under the action of the automorphism group Case : Orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation) Case odd : Orbit sizes: 1 (degree 1 representation), (degree 1 representations), (degree representations) number: 4 (for ), 3 (for odd ) Orbits over a splitting field under the multiplicative action of one-dimensional representations Orbit sizes: (degree 1 representations), and orbits of size 1 (degree representations) Endomorphisms Automorphisms The automorphisms essentially permute the subgroups of order containing the center, while leaving the center itself unmoved. GAP implementation GAP ID For any prime , this group is the third group among the groups of order . Thus, for instance, if , the group is described using GAP's SmallGroup function as: SmallGroup(343,3) Note that we don't need to compute ; we can also write this as: SmallGroup(7^3,3) As an extraspecial group For any prime , we can define this group using GAP's ExtraspecialGroup function as: ExtraspecialGroup(p^3,'+') For , it can also be constructed as: ExtraspecialGroup(p^3,p) where the argument indicates that it is the extraspecial group of exponent . For instance, for : ExtraspecialGroup(5^3,5) Other descriptions Description Functions used SylowSubgroup(GL(3,p),p) SylowSubgroup, GL SylowSubgroup(SL(3,p),p) SylowSubgroup, SL SylowSubgroup(PGL(3,p),p) SylowSubgroup, PGL SylowSubgroup(PSL(3,p),p) SylowSubgroup, PSL
closed as no longer relevant by Robin Chapman, Akhil Mathew, Yemon Choi, Qiaochu Yuan, Pete L. Clark Aug 22 '10 at 9:00 This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. $e^{\pi i} + 1 = 0$ Stokes' Theorem Trivial as this is, it has amazed me for decades: $(1+2+3+...+n)^2=(1^3+2^3+3^3+...+n^3)$ $$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left\| \frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}\right\| dt\\ = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$where $\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is aconjecture! It is a rare example of a conjectured explicit identitybetween real numbers that can be checked to arbitrary accuracy.This identity has been verified to over 20,000 decimal places.See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,2004 (pages 90-91). There are many, but here is one. $d^2=0$ Mine is definitely $$1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}+\cdots=\frac{\pi^2}{6},$$ an amazing relation between integers and pi. There's lots to choose from. Riemann-Roch and various other formulas from cohomology are pretty neat. But I think I'll go with $$\sum\limits_{n=1}^{\infty} n^{-s} = \prod\limits_{p \text{ prime}} \left( 1 - p^{-s}\right)^{-1}$$ 1+2+3+4+5+... = -1/12 Once suitably regularised of course :-) $$\frac{1}{1-z} = (1+z)(1+z^2)(1+z^4)(1+z^8)...$$ Both sides as formal power series work out to $1 + z + z^2 + z^3 + ...$, where all the coefficients are 1. This is an analytic version of the fact that every positive integer can be written in exactly one way as a sum of distinct powers of two, i. e. that binary expansions are unique. $V - E + F = 2$ Euler's characteristic for connected planar graphs. I'm currently obsessed with the identity $\det (\mathbf{I} - \mathbf{A}t)^{-1} = \exp \text{tr } \log (\mathbf{I} - \mathbf{A}t)^{-1}$. It's straightforward to prove algebraically, but its combinatorial meaning is very interesting. $196884 = 196883 + 1$ For a triangle with angles a, b, c $$\tan a + \tan b + \tan c = (\tan a) (\tan b) (\tan c)$$ Given a square matrix $M \in SO_n$ decomposed as illustrated with square blocks $A,D$ and rectangular blocks $B,C,$ $$M = \left( \begin{array}{cc} A & B \\\ C & D \end{array} \right) ,$$ then $\det A = \det D.$ What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality. But the proof is a single line: $$ \left( \begin{array}{cc} A & B \\\ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \\\ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \\\ B^t & D^t \end{array} \right). $$ It's too hard to pick just one formula, so here's another: the Cauchy-Schwarz inequality: \|\|x\|\| \|\|y\|\| >= \|(x.y)\|, with equality iff x&y are parallel. Simple, yet incredibly useful. It has many nice generalizations (like Holder's inequality), but here's a cute generalization to three vectors in a real inner product space: \|\|x\|\| 2\|\|y\|\| 2\|\|z\|\| 2+ 2(x.y)(y.z)(z.x) >= \|\|x\|\| 2(y.z) 2+ \|\|y\|\| 2(z.x) 2+ \|\|z\|\| 2(x.y) 2, with equality iff one of x,y,z is in the span of the others. There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. All of the inequalities, including Cauchy-Schwarz, are actually just generalizations of the 1-dimensional inequality: \|\|x\|\| >= 0, with equality iff x = 0, or rather, instantiations of it in the 2 nd, 3 rd, etc. exterior powers of the vector space. I always thought this one was really funny: $1 = 0!$ I think that Weyl's character formula is pretty awesome! It's a generating function for the dimensions of the weight spaces in a finite dimensional irreducible highest weight module of a semisimple Lie algebra. $2^n>n $ It has to be the ergodic theorem, $$\frac{1}{n}\sum_{k=0}^{n-1}f(T^kx) \to \int f\:d\mu,\;\;\mu\text{-a.e.}\;x,$$ the central principle which holds together pretty much my entire research existence. Gauss-Bonnet, even though I am not a geometer. Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις. That is, In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. The formula $\displaystyle \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+1} dx = \frac{\pi}{e}$. It is astounding in that we can retrieve $e$ from a formula involving the cosine. It is not surprising if we know the formula $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$, yet this integral is of a purely real-valued function. It shows how complex analysis actually underlies even the real numbers. It may be trivial, but I've always found $\sqrt{\pi}=\int_{-\infty}^{\infty}e^{-x^{2}}dx$ to be particularly beautiful. For X a based smooth manifold, the category of finite covers over X is equivalent to the category of actions of the fundamental group of X on based finite sets: \pi-sets === et/X The same statement for number fields essentially describes the Galois theory. Now the ideathat those should be somehow unifiedwas one of the reasons in the development of abstract schemes, a very fruitful topic that is studied in the amazing area of mathematics called the abstract algebraic geometry. Also, note that "actions on sets" is very close to "representations on vector spaces" and this moves us in the direction of representation theory. Now you see, this simple line actually somehow relates number theory and representation theory. How exactly? Well, if I knew, I would write about that, but I'm just starting to learn about those things. (Of course, one of the specific relations hinted here should be the Langlands conjectures, since we're so close to having L-functions and representations here!) E[X+Y]=E[X]+E[Y] for any 2 random varibles X and Y $\prod_{n=1}^{\infty} (1-x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2}$ $ D_A\star F = 0 $ Yang-Mills $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \frac{q-1}{2}}$. My favorite is the Koike-Norton-Zagier product identity for the j-function (which classifies complex elliptic curves): j(p) - j(q) = p -1 \prod m>0,n>-1 (1-p mq n) c(mn), where j(q)-744 = \sum n >-2 c(n) q n = q -1 + 196884q + 21493760q 2 + ... The left side is a difference of power series pure in p and q, so all of the mixed terms on the right cancel out. This yields infinitely many identities relating the coefficients of j. It is also the Weyl denominator formula for the monster Lie algebra.
Substitute $x = 2.5$ and $y =10$ into each equation and rearrange to find the values of the constants. For each equation substitue a series of $x$ values into the equation and tabulate the corresponding $y$ values. Then match the equation to the curve which gives the correct trajectory of points. Note: one point should be sufficient to match the equation to the curve. Curve 1 $y = 4x$ (as this is the only linear plot) Curve 2 $y = \frac{8}{5} x^2$ (at $x = 2.25$ , $y = 1.6 \times 2.25^2 = 8.1$ which matches the pink curve) Curve 3 $y = \frac{24}{125} x^4 + x$ ( at $x =1$, $y = 1 + 0.192 = 1.192$ which matches the black curve) Curve 4 $y = \frac{16}{25} x^3$ (at $x =1$, $y = 0.64$ which matches the red curve) The volume generated by rotation about the y axis can be found by the method of shells or the method of discs, here we use the method of shells. The method of shells is based upon filling the solid of revolution with an infinite number of thin cylindrical shells. The volume of each shell is equal to its circumference ($2 \pi x$) multiplied by its height $\left[ y = f(x) \right]$, by allowing the thickness of each shell to approach zero and summing all shells we obtain the definite integral for the volume of revolution $V_{rev}$$$V_{rev} = \int 2 \pi \ x \ y \mathrm{\ d}x $$ The volume calculated via the method of shells will be the volume between $x$-axis, the curve and the line $x=2.5$ (when we rotate the curve about $y$). The volume of the vessel $V_{vessel}$ can be found by subtracting the volume of revolution $V_{rev}$ from the volume of a cylinder of height $10 \mathrm{\ cm}$ and radius $2.5 \mathrm{\ cm}$.$$V_{vessel} = (10 \times 2.5^2 \pi) - V_{rev} = \frac{125\pi}{2}- V_{rev}$$ Volume generated by rotation about $y$$$V_{rev} = \int_0^{2.5} 2\pi (x)(4x) \ \mathrm{d}x = 8\pi \left[\frac{1}{3}x^3 \right]^{2.5}_0 = \frac{8 \pi}{3} \times 2.5^3 =\frac{125 \pi}{3}$$ $$\Rightarrow V_{vessel} = \frac{125\pi}{2} - \frac{125 \pi}{3} = \frac{125 \pi}{6} = 65.45 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ Volume generated by rotation about $y$$$V_{rev} = \int_0^{2.5} 2\pi (x) \left( \frac{8}{5}x^2 \right) \ \mathrm{d}x = \frac{16\pi}{5} \left[ \frac{1}{4}x^4 \right]^{2.5}_0 = \frac{4 \pi}{5} \times 2.5^4 = \frac{125 \pi}{4}$$ $$\Rightarrow V_{vessel} = \frac{125\pi}{2} - \frac{125 \pi}{4} = \frac{125 \pi}{4} = 98.17 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ Volume generated by rotation about $y$$$V_{rev} = \int_0^{2.5} 2\pi (x) \left( \frac{24}{125}x^4 + x \right) \ \mathrm{d}x = 2\pi \left[ \frac{24}{125} \frac{1}{6}x^6 + \frac{1}{3}x^3 \right]^{2.5}_0 = \left(\frac{8}{125} \times 2.5^6 + \frac{2}{3} \times 2.5^3 \right) \pi = \frac{625 \pi}{24}$$ $$\Rightarrow V_{vessel} = \frac{125\pi}{2} - \frac{625 \pi}{24} = \frac{875 \pi}{24} =114.5 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ Volume generated by rotation about $y$$$V_{rev} = \int_0^{2.5} 2\pi (x) \left( \frac{16}{25}x^3 \right) \ \mathrm{d}x = \frac{32\pi}{25} \left[ \frac{1}{5}x^5 \right]^{2.5}_0 = \frac{32 \pi}{125} \times 2.5^5 = 25 \pi$$ $$\Rightarrow V_{vessel} = \frac{125\pi}{2} - 25 \pi = \frac{75 \pi}{2} = 117.8 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ We can find the volume of revolution at some general height $h$, set this volume $V(h)$ to half the volume of the vessel and solve for $H$ the height at this volume. From above $V_{vessel} = \frac{125 \pi}{6}$$$\Rightarrow \frac{\pi}{48} H^3 = \frac{1}{2} \frac{125 \pi}{6}$$ $$\Rightarrow H = = \sqrt[3]{48 \frac{125}{12}} = \sqrt[3]{500} = 7.937 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ From above $V_{vessel} = \frac{125 \pi}{4}$$$\Rightarrow \frac{5\pi}{16} H^3 = \frac{1}{2} \frac{125 \pi}{4}$$ $$\Rightarrow H = \sqrt{\frac{16}{5} \frac{125}{8}} = \sqrt{50} = 7.071 \mathrm{\ cm^3 \quad (4\ s.f.)}$$ In order to evaluate the volume by the method of discs I would first need to arrange the function into the form $x = f(y)$, this may be quite difficult. For this reason, we shall evaluate the volume using the method of shells. The method of shells will not give the volume of the vessel directly; instead it gives the volume to the right of the curve rather than left. To find the volume of the vessel we must subtract the volume of revolution from the volume of a cylinder of radius $x$ and height $y$.$$V(x) = \pi x^2 y - \int_0^x 2 \pi x y \ \mathrm{d}x$$ We need to get this equation in terms of one unknown $x$, we can eliminate $y$$$y = \frac{24}{125}x^4 + x$$ $$V(x) = \pi x^2 \left( \frac{24}{125}x^4 + x \right) -2 \pi \int_0^x \frac{24}{125} x^5 + x^2 \ \mathrm{d}x = \pi \left( \frac{24}{125}x^6 + x^3 - 2 \left[\frac{4}{125}x^6 + \frac{1}{3}x^3 \right] ^x_0 \right) = \pi \left( \frac{16}{125}x^6 + \frac{1}{3}x^3 \right)$$ From above $V_{vessel} = \frac{875 \pi}{24}$, calling the radius of the vessel at the half full point $R$$$\Rightarrow \pi \left( \frac{16}{125}R^6 + \frac{1}{3}R^3 \right) = \frac{1}{2} \frac{875 \pi}{24}$$ $$\Rightarrow \frac{16}{125}(R^3)^2 + \frac{1}{3}R^3 - \frac{875}{48} = 0$$ Quadratic in $R^3$, solving gives one positive solution:$$R^3 = 10.703 \Rightarrow R = \sqrt[3]{10.703} = 2.204$$ Using equation of curve to find corresponding depth $H$:$$H = \frac{24}{125}R^4 + R = 6.732\mathrm{\ cm^3 \quad (4\ s.f.)}$$ From above $V_{vessel} = \frac{75 \pi}{2}$$$\Rightarrow \frac{3\sqrt[3]{5}\pi}{4\sqrt[3]{4}} H^\frac{5}{3} = \frac{1}{2} \frac{75 \pi}{2}$$ $$\Rightarrow H = \left( \frac{4\sqrt[3]{4}}{3\sqrt[3]{5}\pi} \frac{75}{4} \right)^\frac{3}{5} = \left( \frac{25\sqrt[3]{4}}{\sqrt[3]{5}} \right)^\frac{3}{5} = \sqrt[5]{12500} = 5\sqrt[5]{4} = 6.598\mathrm{\ cm^3 \quad (4\ s.f.)}$$
Chapters Balbharati SSC Class 10 Mathematics 2 Chapter 6: Trigonometry Chapter 6: Trigonometry solutions [Page 131] If \[\sin\theta = \frac{7}{25}\], find the values of cosθ and tanθ. If \[\tan \theta = \frac{3}{4}\], find the values of secθ and cosθ If \[\cot\theta = \frac{40}{9}\], find the values of cosecθ and sinθ. If 5 secθ – 12 cosecθ = 0, find the values of secθ, cosθ and sinθ. If tanθ = 1 them, find the values of Prave that: \[\frac{\sin^2 \theta}{\cos\theta} + \cos\theta = \sec\theta\] Prave that: \[\cos^2 \theta\left( 1 + \tan^2 \theta \right) = 1\] Prave that: Prave that: Prave that: Prave that: Prave that: \[\sec^4 \theta - \cos^4 \theta = 1 - 2 \cos^2 \theta\] Prave that: Prave that: If \[\tan\theta + \frac{1}{\tan\theta} = 2\], then show that \[\tan^2 \theta + \frac{1}{\tan^2 \theta} = 2\] Prove that: Prove that: Prove that: Chapter 6: Trigonometry solutions [Page 137] A person is standing at a distance of 80 m from a church looking at its top. The angle of elevation is of 45°. Find the height of the church. From the top of a lighthouse, an observer looking at a ship makes angle of depression of 60°. If the height of the lighthouse is 90 metre, then find how far the ship is from the lighthouse. Two buildings are facing each other on a road of width 12 metre. From the top of the first building, which is 10 metre high, the angle of elevation of the top of the second is found to be 60°. What is the height of the second building ? Two poles of heights 18 metre and 7 metre are erected on a ground. The length of the wire fastened at their tops in 22 metre. Find the angle made by the wire with the horizontal. A storm broke a tree and the treetop rested 20 m from the base of the tree, making an angle of 60° with the horizontal. Find the height of the tree. A kite is flying at a height of 60 m above the ground. The string attached to the kite is tied at the ground. It makes an angle of 60° with the ground. Assuming that the string is straight, find the length of the string. Chapter 6: Trigonometry solutions [Pages 138 - 139] Choose the correct alternative answer for the following question. sin\[\theta\] cosec\[\theta\]= ? (A) 1 (B) 0 (C)\[\frac{1}{2}\] (D)\[\sqrt{2}\] Choose the correct alternative answer for the following question. cosec 45 ° =? (A)\[\frac{1}{2}\] (B) \[\sqrt{2}\] (C)\[\frac{\sqrt{3}}{2}\] (D)\[\frac{2}{\sqrt{3}}\] Choose the correct alternative answer for the following question. 1 + tan 2 \[\theta\] = ? (A) cot 2θ (B) cosec 2θ (C) sec 2θ (D) tan 2θ Choose the correct alternative answer for the following question. (B) angle of depression. (C) 0 (D) straight angle. If \[\sin\theta = \frac{11}{61}\], find the values of cosθ using trigonometric identity. If tanθ = 2, find the values of other trigonometric ratios. If \[\sec\theta = \frac{13}{12}\], find the values of other trigonometric ratios. Prove the following. sec θ (1 – sin θ) (sec θ + tan θ) = 1 Prove the following. (sec θ + tan θ) (1 – sin θ) = cos θ Prove the following. sec 2 θ + cosec 2 θ = sec 2 θ × cosec 2 θ Prove the following. cot 2θ – tan 2θ = cosec 2θ – sec 2θ Prove the following. tan 4θ + tan 2θ = sec 4θ - sec 2θ Prove the following. Prove the following. sec 6x – tan 6x = 1 + 3sec 2x × tan 2x Prove the following. \[\frac{\tan\theta}{sec\theta + 1} = \frac{sec\theta - 1}{\tan\theta}\] Prove the following. Prove the following. \[\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac{1}{\sin\theta - \tan\theta}\] A boy standing at a distance of 48 meters from a building observes the top of the building and makes an angle of elevation of 30°. Find the height of the building. From the top of the light house, an observer looks at a ship and finds the angle of depression to be 30°. If the height of the light-house is 100 meters, then find how far the ship is from the light-house. Two buildings are in front of each other on a road of width 15 meters. From the top of the first building, having a height of 12 meter, the angle of elevation of the top of the second building is 30°.What is the height of the second building? A ladder on the platform of a fire brigade van can be elevated at an angle of 70° to the maximum. The length of the ladder can be extended upto 20 m. If the platform is 2m above the ground, find the maximum height from the ground upto which the ladder can reach. (sin 70° = 0.94) While landing at an airport, a pilot made an angle of depression of 20°. Average speed of the plane was 200 km/hr. The plane reached the ground after 54 seconds. Find the height at which the plane was when it started landing. (sin 20° = 0.342) Chapter 6: Trigonometry Balbharati SSC Class 10 Mathematics 2 Textbook solutions for Class 10th Board Exam Balbharati solutions for Class 10th Board Exam Geometry chapter 6 - Trigonometry Balbharati solutions for Class 10th Board Exam Geometry chapter 6 (Trigonometry) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Textbook for SSC Class 10 Mathematics 2 solutions in a manner that help students grasp basic concepts better and faster. Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. Balbharati textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students. Concepts covered in Class 10th Board Exam Geometry chapter 6 Trigonometry are Trigonometry Ratio of Zero Degree and Negative Angles, Application of Trigonometry, Heights and Distances, Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Trigonometric Ratios in Terms of Coordinates of Point, Angles in Standard Position. Using Balbharati Class 10th Board Exam solutions Trigonometry exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Balbharati Solutions are important questions that can be asked in the final exam. Maximum students of Maharashtra State Board Class 10th Board Exam prefer Balbharati Textbook Solutions to score more in exam. Get the free view of chapter 6 Trigonometry Class 10th Board Exam extra questions for Geometry and can use Shaalaa.com to keep it handy for your exam preparation
Stokes Theorem (also known as Generalized Stoke’s Theorem) is a declaration about the integration of differential forms on manifolds, which both generalizes and simplifies several theorems from vector calculus. As per this theorem, a line integral is related to a surface integral of vector fields. Learn the stokes law here in detail with formula and proof. Stokes’ Theorem Formula The Stoke’s theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.” $\oint _{C} \vec{F}.\vec{dr} = \iint_{S}(\bigtriangledown \times \vec{F}). \vec{dS}$ Where, C = A closed curve. S = Any surface bounded by C. F = A vector field whose components have continuous derivatives in an open region of R3 containing S. This classical declaration, along with the classical divergence theorem, fundamental theorem of calculus, and Green’s theorem are basically special cases of the general formulation specified above. This means that: If you walk in the positive direction around C with your head pointing in the direction of n, the surface will always be on your left. S is an oriented smooth surface bounded by a simple, closed smooth-boundary curve C with positive orientation. Gauss Divergence theorem The Gauss divergence theorem states that the vector’s outward flux through a closed surface is equal to the volume integral of the divergence over the area within the surface. Put differently, the sum of all sources subtracted by the sum of every sink results in the net flow of an area. Gauss divergence theorem is a result that describes the flow of a vector field by a surface to the behaviour of the vector field within the surface. Stokes’ Theorem Proof We assume that the equation of S is Z = g(x,y), (x,y)D Where g has a continuous second-order partial derivative. D is a simple plain region whose boundary curve C1 corresponds to C. This can be easily explained with 3D air projection being used at Byju’s – the learning app where all concepts like the same are explained in great detail. For Stokes’ theorem examples and Stokes’ theorem problems stay tuned with BYJU’S.
"Fully Homomorphic Encryption over the Integer" described a simple FHE scheme based on the GACD assumption. Its encryption function (on page 6) has the form $c \leftarrow (m + 2r + 2\sum_{i \in S} x_i) \mod x_0$, where $\sum_{i \in S} x_i$ can be viewed as some ciphertexts of plaintext 0 (which can be divided by the secret key $p \in 2\mathbb{Z}+1$), and $r$, as the paper informed us, is randomly chosen from $(-2^{\rho'}, 2^{\rho'})$. Decryption is taken by $m' = (c \mod p) \mod 2$. It seems that "mod p" is used to drop $2\sum_{i \in S} x_i$ and "mod 2" is used to drop $2r$. If $r > 0$, the item $2r \mod p$ is even. It will not change the parity of $c \mod p$. However, since $p$ is odd, for any $r < 0$, the item $2r \mod p$ in fact equals $p + 2r \mod p$, which is odd and will change the parity of $(c \mod p)$. I have programed for this algorithm and the result matchs the observation above. What's the mistake of my reasoning and/or my understanding?
In my population genetics book (see reference at bottom) they define them as: Nucleotide polymorphism (θ): proportion of nucleotide sites that are expected to be polymorphic in any sample of size 5 from this region. Of the genome. $\hat{θ}$ equals the proportion of nucleotide polymorphism observes in the sample (S) divided by $a_1 = \sum_{i=1}^{n-1} \frac{1}{I}$. If n= 5, $a_1 = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}=2.083$ Nucleotide diversity is the average proportion of nucleotide differences between all possible pairs of sequences in the sample. In R, I came up with that code which is in accordance with what is in the book. data # Only polymorphisms total.snp # This is the total number of sites that were looked (e.g. 16 might be polymorphic over the 500 sites that we've looked at. So 484 sites are monomorphic) n = nrow(data) # Number of samples pwcomp = n(n-1)/2 # Number of pairwise comparisons for(i in 1:n.col){ # Compute the number of differences in the samples that are polymorphic t.v = as.vector(table(data[,i])) z = outer(t.v,t.v,'') temp= c(temp,sum(z[lower.tri(z)])) } pi.hat = sum(temp)/(pwcomptotal.snp) # Nb of different pairwise comparison/(nb all pairwise comparison total nb of loci (polymorphic or not)) In the book, they say On the other hand, there is a theoretical relation between θ and π that is expected under simplifying assumption that the alleles are invisible to natural selection. In summary θ = π with this assumption and large sample sizes. So what is the difference in the Why are we calculating both (what could they tell us)? Hartl, D. L., & Clark, A. G. (1997). Principles of Population Genetics (3rd ed.). Sinauer Associates Incorporated.
My original GSOC proposal was to implement modify Mamba.jl to enable it to fit Crosscat, a general-purpose Bayesian model which fits tabular data using row-wise Dirichlet cluster models nested inside a column-wise Dirichlet cluster. This model is in itself broadly useful, but the real reason I chose this project was to work on something even more general: improving the tools for doing MCMC on models with a mix of discrete and continuous parameters. In the end, I was unable to complete the full original plan. However, I did implement a simple Dirichlet 1D Gaussian mixture model in Mamba. Though this model itself is extremely basic, it did require successfully reworking Mamba to enable variable numbers of parameters — a significant amount of work. Based on that work, more-sophisticated Dirichlet mixture models involving multiple dimensions and/or Dirichlet processes would be almost trivial, and even something more heavy-duty such as Crosscat (or improved versions thereof) would be far easier to implement in the future. I estimate that, while the practical usefulness of the demo model I’m delivering is a small fraction of a full-blown Crosscat, the actual work I’ve done is about 75% of what it would take to get there. Bayesian MCMC is a powerful all-purpose tool in the toolkit of statistics, and thus of almost all science. It allows one to flexibly build models which capture the interplay of known dynamics and unknown parameters, of measurable data and uncertain noise, thus extracting meaning from data. The great power of this idea lies in its flexibility. If you can write a likelihood model, you can at least attempt to do MCMC. Of course, issues of computation and convergence might make things hard in practice, but at least in theory, the idea is straightforward enough to be susceptible to automation. Currently, the outstanding tool in this regard is Stan. Stan’s sampler is NUTS (the “no U-turn sampler”), which relies on HMC (that is, automatic differentiation) to be able to efficiently explore posteriors, even if those posteriors lie along medium-dimensional manifolds in a high-dimensional parameter space — something that would have been effectively impossible for older, pre-HMC samplers without clever problem-specific tricks to make separate dimensions quasi-independent. And NUTS does this without even the hand tuning that many of its HMC siblings need. However, Stan’s fundamental design means it has certain weaknesses that are unlikely to be solved. First off, it uses a proprietary language for model definition, with all the limitations and friction that implies. End users are almost certainly not going to want to dig into Stan’s C++ code to add a feature. Second, because the NUTS sampler is built in at its foundation, it will probably continue to be at best a struggle to use it on models that mix discrete and continuous parameters. Julia, and specifically Mamba.jl, offer ways beyond those weaknesses. Though currently not nearly as mature as Stan, Mamba.jl does have basic functionality for building models using a flexible syntax that reflects the way statisticians think about them. Mamba also already has many sampler options including NUTS. Models are defined expressively and flexibly using general-purpose Julia code, not a single-purpose language; and various samplers can be combined, so that models can include discrete and continuous parameters. But before this project, Mamba was limited to models with fixed numbers of parameters. This closed the door to many useful kinds of models. For instance, in Dirichlet mixture models and other cluster models, the number of parameters depends on the number of latent clusters the fitted model finds in the data. That’s the gap my project was intended to fill. I faced several unexpected hurdles in carrying out this project. Firstly, there’s this (warning, blood). That’s me making a silly face in the emergency room after my broken arm; it was 3 days before I got out of the hospital and another week before I was off the pain meds and could type again. All-told, that accident (the classic fool-opening-a-car-door-while-I-was-passing-on-my-bike, with a side of rainstorm) probably cost me 2 weeks of work. Also, refactoring Mamba proved to be tougher than I’d expected. My plan was to add parameters to many of the basic Mamba types, to be able to switch between storing parameters in the existing fixed-sized array structures or in my newly-designed flexible-size structures. While I was at it, I also added type parameters to loosen up the hard-coded dependence on Float64 model parameters, so as to be able to use autodifferentiation numbers for HMC. This was pretty advanced for my starting level of expertise on both Julia in general and the Mamba package in particular; it took me a lot of error messages to really get my head around some stuff. (Of course, now that I do understand it, it seems trivial; but it was a struggle, because of course the issues did not show up as cleanly as I present them below.) For instance: Turns out Julia types aren’t covariant even when you really want them to be. For instance, even though VecDictVariateVal and SymDictVariateVal are trivially-different subtypes of my general-purpose abstract type DictVariateVal, it isn’t true that VecDictVariateVal{Float64} <: DictVariateVal{Float64}. This is especially confusing (at least, to me as a relative beginner) because, using where clauses, UnionAll types can be covariant. You can write SomeType{<:Real} or SomeType{T} where T<:Real, but never SomeType{T<:Real} where T; that last thing is just SomeType{false}, because it’s a category error; the type variable itself is never a subtype of real. This seems kinda obvious in this simplified minimal example, but believe me, there cases where it was far harder to see. In the coming days, I’ll be posting some Julia issues (youtube link, sorry) with suggestions for how to make both the syntax itself, and the error messages/warnings for when you get it wrong, better. The Mamba control flow is a bit tough to understand. One good trick for exploring a big existing package like this I found is to run the graphical profile browser on a working example; that gives you a useful picture of what calls what. All in all, it took me over 6 weeks to finish this refactor, when I’d optimistically planned that I would be able to do it as I went along by spending less than 1/3 of the first month on it. My work is at https://github.com/jamesonquinn/Mamba.jl; primarily in this branch. Aside from the overall refactoring, key files include the Dirichlet process distribution; the reversible jump sampler (based mainly on Richardson and Green 1997, but with some simplifications as well as some changes so as to base it on a Dirichlet process rather than separate Dirichlet distributions of weights for each number of clusters); and the demo example model. Here’s the model I used in that example: $y_i \stackrel{iid}{\sim} \mathcal{N}(\mu_{T_i},\sigma^2_{T_i})$ for $i\in {1..N}$ $T\sim DP(\alpha)$, a vector with dimension $N$ of integers (cluster indices). $\mu_t\stackrel{iid}{\sim} \mathcal{N}(\mu_0,\tau^2)$ for any $t$ $1/\sigma_t^2\stackrel{iid}{\sim} Gamma(\alpha,\beta)$ for any $t$ $\beta\stackrel{iid}{\sim} Gamma(\theta,\phi)$ $\alpha = 0.1$ $\mu_0 = \bar{y}$ $\tau = 2s_y$ $\theta=\phi=.1$ Here are some results for 45 simulated data points in two clusters with SD 3 and mean ±5: As you can see, both chains spend most time with at least one cluster each around the “correct” values, but occasionally they go wrong. This code, while it works for the example model, is not yet ready to be checked in to the main branch of Mamba. There were several cleanup steps for which I did not ultimately have time. I have updated the “slice” and “slicesimplex” samplers to work with the new data structures. However, the other samplers which I did not use in my work are currently broken in the gsocMNVP branch; they still try to use the old data structures. Updating them, along the same lines as the slice and slicesimplex samplers, would be a more-or-less routine task - an hour or so of work per sampler. The diagnostics and plots, aside from the traceplot shown above, are also not updated. Fixing this is a less trivial task, as, due to the “labelling problem”, most diagnostics need to be rethought in some way in order to apply to dirichlet models. Once that cleanup is done — a few days’ work - and the merge is complete, implementing the full Crosscat model as in the original plan should not be too difficult. Optimistically, I feel it would take 1-2 weeks… which means that realistically, probably 4-6 is more realistic. In any case, the new data structures I’ve implemented would make this job primarily a matter of just implementing the statistical algorithms; the data and model infrastructure is all well in place. With a combination of NUTS and discrete capabilities, I believe that Mamba will begin to actually be superior to Stan for some tasks. It has a long way to go to catch up to Stan’s maturity, but in solving the “two language problem”, it gives a strong incentive for me and others to continue on this work. I want to thank my GSoC mentor Benjamin Deonovic for his help and understanding in what has been a difficult but fun project.
The Annals of Probability Ann. Probab. Volume 8, Number 1 (1980), 1-67. Occupation Densities Abstract This is a survey article about occupation densities for both random and nonrandom vector fields $X: T \rightarrow \mathbb{R}^d$ where $T \subset \mathbb{R}^N$. For $N = d = 1$ this has previously been called the "local time" of $X$, and, in general, it is the Lebesgue density $\alpha(x)$ of the occupation measure $\mu(\Gamma) =$ Lebesgue measure $\{t\in T: X(t)\in \Gamma\}$. If we restrict $X$ to a subset $A$ of $T$ we get a corresponding density $\alpha(x, A)$ and we will be interested in its behavior both in the space variable $x$ and the set variable $A$. The first part of the paper deals entirely with nonrandom, nondifferentiable vector fields, focusing on the connection between the smoothness of the occupation density and the level sets and local growth of $X$. The other two parts are concerned, respectively, with Markov processes $(N = 1)$ and Gaussian random fields. Here the emphasis is on the interplay between the probabilistic and real-variable aspects of the subject. Special attention is given to Markov local times (in the sense of Blumenthal and Getoor) as occupation densities, and to the role of local nondeterminism in the Gaussian case. Article information Source Ann. Probab., Volume 8, Number 1 (1980), 1-67. Dates First available in Project Euclid: 19 April 2007 Permanent link to this document https://projecteuclid.org/euclid.aop/1176994824 Digital Object Identifier doi:10.1214/aop/1176994824 Mathematical Reviews number (MathSciNet) MR556414 Zentralblatt MATH identifier 0499.60081 JSTOR links.jstor.org Subjects Primary: 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives Secondary: 60G15: Gaussian processes 60G17: Sample path properties 60J55: Local time and additive functionals Citation Geman, Donald; Horowitz, Joseph. Occupation Densities. Ann. Probab. 8 (1980), no. 1, 1--67. doi:10.1214/aop/1176994824. https://projecteuclid.org/euclid.aop/1176994824
(I am assuming choice.) Suppose that ${\mathbb P}=(P,{\lt})$ is a partially ordered set (a poset), and that $\kappa\le\lambda$ are ordinals. The notation $$ {\mathbb P}\to(\kappa)^1_\lambda $$ means that whenever $f:P\to\lambda$, we can find some $i\in\lambda$ and some subset $H$ of $P$ such that $(H,{\lt})$ is order-isomorphic to $\kappa$ and $f(a)=i$ for all $a\in H$. Theorem.Suppose that $P$ is a poset and $P\to(\kappa)^1_\kappa$, where $\kappa$ is an infinite cardinal. Then $P\to(\alpha)^1_\kappa$ for all $\alpha\lt\kappa^+$. This is due to Galvin (unpublished). A nice combinatorial argument is presented in Stevo Todorcevic, "Partition relations for partially ordered sets", Acta Math. 155 (1985), no. 1-2, 1-25. In fact, Galvin result is that: For any poset $P$, $$ P\to(\kappa)^1_\lambda $$ implies $$ P\to(\alpha)^1_\lambda $$ whenever $\kappa\le\lambda$ are infinite cardinals and $\alpha\lt\kappa^+$. This can be proved by collapsing $\lambda$ to $\kappa$ with a $\kappa$-closed forcing, noting that in the extension $P\to(\kappa)^1_\kappa$, so (by the theorem) $P\to(\alpha)^1_\kappa$, and using the closure of the forcing to find such a homogeneous set of type $\alpha$ in the ground model. My question: How can we prove Galvin's result without appealing to a forcing argument? Very briefly, Stevo's proof of the theorem proceeds as follows: Given a poset $P$, let $\sigma'P$ be the collection of injective sequences $\tau$ whose domain is a successor ordinal and whose range is strictly increasing in the ordering of $P$. This is a poset under the "initial segment" ordering of sequences. Stevo proves two results: If $P\to(\kappa)^1_\kappa$ holds, then $\sigma' P\to(\kappa)^1_\kappa$ holds. If $\sigma' P\to(\alpha)^1_\gamma$ holds, then $P\to(\alpha)^1_\gamma$ holds. Item 2 is straightforward, and a more general result holds. Item 1 uses a delicate argument and I do not know of a more general statement. The general version of 2 says that many partition relations that hold for $\sigma' P$ must hold for $P$ as well. The combination of 1 and 2 is very powerful: It says that to prove partition results for posets $P$ satisfying $P\to(\kappa)^1_\kappa$ it suffices to prove the result for trees, for which the combinatorics tends to be much better understood than for arbitrary posets. For example, for trees $T$ it is essentially obvious that if $T\to(\kappa)^1_\kappa$, then $T\to(\alpha)^1_\kappa$ for all $\alpha\lt\kappa^+$, and the theorem follows.
Maybe this is something like what you have in mind: Let $V$ be a (complex) vector space of dimension $3$. It is easy to show that a generic subspace $P\subset S^2(V^)$ of dimension $4$ can be written as $$P = \mathrm{span}\{\ {x_1}^2,\ {x_2}^2,\ {x_3}^2,\ x_1x_2{+}x_2x_3{+}x_3x_1\ \}$$for some basis $x_1,x_2,x_3$ of $V^$. (There are only a finite number of other 'nongeneric' cases to be handled.) It seems that you are asking for a 'natural' basis for the kernel of the multiplication map$$\mu:P\otimes S^2(V^)\longrightarrow S^4(V^)$$Now, $P\subset S^2(V^)$ is invariant under permutations of the basis $x_1,x_2,x_3$, a group isomorphic to $S_3$. The action of $S_3$ on $S^2(V^)$ then has a natural complement to $P$, which is the space$$W = \mathrm{span}\{\ \ x_1x_2{-}x_2x_3,\ x_2x_3{-}x_3x_1\ \}.$$As representations of $S_3$, $W$ is irreducible and $P \simeq \mathbb{C}\oplus\mathbb{C}\oplus W$. Now, we have the natural decomposition$$P\otimes S^2(V^) = P\otimes (P \oplus W) = \Lambda^2(P)\oplus S^2(P)\oplus P\otimes W.$$Clearly, $\mu\bigl(\Lambda^2(P)\bigr) = 0$ (these are the relations you are calling the 'Koszul relations'), so we need to examine the other two pieces. It is easy to check that $\mu$ is surjective, so, indeed, there is a $3$-dimensional kernel of the map$$\mu: S^2(P)\oplus P\otimes W\to S^4(V^).$$It's obvious that $\mu$ is injective on $S^2(P)$ and it's easy to check that it's injective on $P\otimes W$. The two $\mu$-images intersect in a space of dimension $3$ that is invariant under $S_3$, and it is easy to see from this that the kernel $K$ splits as an $S_3$-module as a sum $K\simeq \mathbb{C}\oplus W$. Hence, there is one relation that is invariant under the action of $S_3$ and the complement is an irreducible $S_3$-module of dimension $2$. You can write them out without difficulty, but they all involve a significant number of terms.
Upcoming Seminars : every Tuesday and Thursday at 2:00 pm Room 210 TBA Past Seminars May 21 James Lutley Nuclear Dimension and the Toeplitz Algebra After reviewing classical Toeplitz matrices, we will briefly review the CPC approximation used on the Cuntz-Toeplitz algebras by Winter and Zacharias to study the Cuntz algebras. We will then show in full detail how these maps operate on the Toeplitz algebra itself. This calculation fails to determine the nuclear dimension of the Toeplitz algebra itself but we will show how these techniques can be extended to put this within reach. May 14 Dave Penneys Computing principal graphs part 2 Last week we looked at the Jones tower, the relative commutants, and the principal graph for some subfactors associated to finite groups. This week, we'll continue our analysis of the relative commutants and the principal. We will show how minimal projections in the relative commutants correspond to bimodules, and we'll discuss how we view the principal graph as the fusion graph associated to these bimodules, where fusion refers to Connes fusion of bimodules. We'll then compute a particular example of importance in the classification of subfactors at index 3+\sqrt{5}. May 9 Dave Penneys Computing principal graphs Subfactors are classified by their standard invariants, and standard invariants are classified by their principal graphs. I will give the appropriate definitions, and then I will compute some principal graphs. In particular, we will look at examples coming from groups and examples coming from compositions of subfactors. May 7 David Kerr Turbulence in automorphism groups of C-algebras Apr. 30 Nicola Watson Connes's Classification of Injective Factors Apr. 11 Greg Maloney Apr. 9 Danny Hay Apr. 4 Nicola Watson Mar. 28 Luis Santiago The $Cu^\sim$-semigroup of a C-algebra Mar. 26 Dave Penneys GJS C-algebras Guionnet-Jones-Shlyakhtenko (GJS) gave a diagrammatic proof of a result of Popa which reconstructs a subfactor from a subfactor planar algebra. In the process, certain canonical graded -algebras with traces appear. In the GJS papers, they show that the von Neumann algebras generated by the graded algebras are interpolated free group factors. In ongoing joint work with Hartglass, we look at the C * -algebras generated by the graded algebras. We are interested in a connection between subfactors and non-commutative geometry, and the first step in this process is to compute the K-theory of these C * -algebras. I will talk about the current state of our work. Mar. 21 Makoto Yamashita Deformation of algebras from group 2-cocycles Algebras with graded by a discrete can be deformed using 2-cocycles on the base group. We give a K-theoretic isomorphism of such deformations, generalizing the previously known cases of the theta-deformations and the reduced twisted group algebras. When we perturb the deformation parameter, the monodromy of the Gauss-Manin connection can be identified with the action of the group cohomology. Jan. 17 Zhiqiang Li Certain group actions on C-algebras We will discuss group actions of certain groups, mainly, discrete groups, for example, \mathbb{Z}^d, and finite groups, then look at several classifiable classes of such group actions, and finally we will give a classification of inductive limit actions of cyclic groups with prime orders on approximate finite dimensional C- algebras. Jan. 15 Ask Anything Seminar Jan 10 George Elliott Dec 20 Nadish de Silva Dec 18 James Lutley Dec 13 Dave Penneys Dec 11 George Elliott Dec 6 Greg Maloney A constructive approach to ultrasimplicial groups I will review the result of Riedel that says that every simple finitely generated dimension group with a unique state is ultrasimplicial. The proof involves explicitly constructing a sequence of positive integer matrices using a multidimensional continued fraction algorithm. This approach is similar to that used by Elliott and by Effros and Shen in their earlier results. Dec 4 Danny Hay Computing the decomposition rank of Z-stable AH algebras We will take a look at a recent paper of Tikuisis and Winter, in which it is shown that the decomposition rank of Z-stable AH algebras is at most 2. The result is important not only because establishing finite decomposition rank is significant for the classification program, but also because the computation is direct previous results of this type generally factor through classification theorems, and so shed no light on why finite dimensionality occurs. Nov 27 Zhiqiang Li (U of Toronto; Fields) Finite group action on C-algebra I am going to talk about some result of M. Izumi on finite group action on C-algebras. Mainy, there is a cohomology obstruction for C-algebra having finite group action with Rokhlin property. Nov 29 Mike Hartglass (Berkley) Rigid $C^{}$ tensor categories of bimodules over interpolated free group factors The notion of a fantastic (or factor) planar algebra will be presented and some examples will be given. I will then show how such an object can be used to diagrammatically describe a rigid, countably generated $C^{}$ tensor category $\mathcal{C}$. Following in the steps of Guionnet, Jones, and Shlyakhtenko, I will present a diagrammatic construction of a $II_{1}$ factor $M$ and a category of bimodules over $M$ which is equivalent to $\mathcal{C}$. Finally, I will show that the factor $M$ is an interpolated free group factor and can always be made to be isomorphic to $L(\mathbb{F}_{\infty})$. Therefore we will deduce that every rigid, countably generated $C^{}$ tensor category is equivalent to a category of bimodules over $L(\mathbb{F}_{\infty})$. This is joint work with Arnaud Brothier and David Penneys. Nov 22 Paul McKenney (Carnegie Mellon) Approximate -homomorphisms Abstract: I will discuss various notions of "approximate homomorphism", and show some averaging techniques that have been used to produce an actual homomorphism near a given approximate homomorphism. Nov 20 Brent Brenken (Univeristy of Calgary) Universal C-algebras of -semigroups and the C-algebra of a partial isometry Certain universal C-algebras for -semigroups will be introduced. Some basic examples, and ones that occur in describing the C-algebra of a partial isometry, will be discussed. The latter is a Cuntz- Pimsner C-algebra associated with a C-correspondence, and can be viewed as a form of crossed product C-algebra for an action by a completely positive map. The C-algebras involved occur as universal C-algebras associated with contractive -representations, and complete order -representations, of certain -semigroups. Nov 13 Nicola Watson (U of Toronto) Noncommutative covering dimension There have been many fruitful attempts to define noncommutative versions of the covering dimension of a topological space, ranging from the stable and real ranks to the decomposition rank. In 2010, Winter and Zacharias defined the nuclear dimension of a C-algebra, which has turned out to be a major development in the study of nuclear C-algebras. In this talk, we introduce nuclear dimension, discuss the differences between it and other dimension theories, and focus on why nuclear dimension is so important. (This is a practice for a talk I'm giving at Penn State, so it will be more formal than usual.) Nov 8 Danny Hay Nov 6 Greg Maloney Connes' fusion I'll give a basic introduction to Connes' fusion for bimodules over finite von Neumann algebras. Nov 6,8,13,15 Working seminars Octr 30 and Nov 1 "Wiki Week" Oct 25 Dave Penneys (U of Toronto) Infinite index subfactors and the GICAR algebra We will show how the GICAR algebra is the analog of the Temperley-Lieb algebra for infinite index subfactors. As a corollary, we will see that the centralizer algebra M_0'\cap M_{2n} is nonabelian for all n\geq 2. Oct 18 Greg Maloney Ultrasimplicial groups An ordered abelian group is called a dimension group if it is the inductive limit of a sequence of direct sums of copies of Z. Dimension groups are of interest in the study of operator algebras because they are the K0-groups of AF C-algebras. If, in addition, a dimension group admits such an inductive limit representation in which the maps are injective, then it is called an ultrasimplicial group. The question then arises: exactly which dimension groups are ultrasimplicial? There have been positive and negative results on this subject. Elliott showed that every totally ordered (countable) group is ultrasimplicial, and Riedel showed that a free simple dimension group of finite rank with a unique state is ultrasimplicial. Much later, Marra showed that every lattice ordered abelian group is ultrasimplicial. On the other hand, Elliott produced an example of a simple dimension group that is not ultrasimplicial, and later Riedel produced a collection of simple free dimension groups that are not ultrasimplicial. I will discuss the history of this subject and go through some calculations in detail. Oct 16 Martino Lupini (York) The complexity of the relation of unitary equivalence for automorphisms of separable unital C-algebras A classical result of Glimm from 1961 asserts that the irreducible representations of a given separable C-algebra A are classifiable by real numbers up to unitary equivalence if and only if A is type I. In 2008, Kerr-Li-Pichot and, independently, Farah proved that when A is not type I, then the irreducible representations are not even classifiable by countable structures. I will show that a similar dichotomy holds for classification of automorphisms up to unitary equivalence. Namely, the automorphisms of a given separable unital C- algebra A are classifiable by real numbers if and only if A has continuous trace, and not even classifiable by countable structures otherwise. Oct 11 Xin Li (University of Muenster) Semigroup C-algebras The goal of the talk is to give an overview of recent results about semigroup C-algebras. We discuss amenability, both in the semigroup and C-algebraic context, and explain how to compute K-theory for semigroup C-algebras. Oct 4 Zhi Qiang Li (U of Toronto; Fields) Finite group action on C-algebra I am going to talk about some result of M. Izumi on finite group action on C-algerbras. Mainy, there is a cohomology obstruction for C-algebra having finite group action with Rokhlin property. Sept. 18 Aaron Tikuisis Regularity for stably projectionless C-algebras There has been significant success recently in proving that unital simple C-algebras are Z-stable, under other regularity hypotheses. With certain new techniques (particularly concerning traces and algebraic simplicity), many of these results can be generalized to the nonunital setting. In particular, it can be shown that the following C-algebras are Z-stable: (i) (nonunital) ASH algebras with slow dimension growth (T-Toms); (ii) (nonunital) C-algebras with finite nuclear dimension (T); and (iii) (nonunital) C*-algebras with strict comparison and finitely many extreme traces (Nawata). I will discuss the proofs of these results, with emphasis on the innovations required for the nonunital setting.
Let us start with an example relying on a simple time-warp, based on the Hann window (code below). It consists in modifying time index $t\in [-1,1]$, here with function $t\mapsto 2 (t+1)^\alpha/2^\alpha-1$. There are many other ways. Asymmetrical windows, or non-symmetric windows, are a topic of interest, albeit somewhat marginally used. I have not seen a lot on literature on this topic. It however has been discussed here at SE.DSP in: They can be useful when the data is itself non-symmetric or skewed, when the processing requires symmetry imbalance, or when you have no other choice.Discrete finite-support dyadic wavelets fall into the latter and third category. Real-time analysis, or causal imbalance, are common for the second case. Exponentially weighted windows are an example, see What is the name of this digital low pass filter? or What is the name of these simple filter algorithms?. In the first case, you can find cases in analytical chemistry (chromatography), with trailing peaks or band broadening, in audio processing or spectral analysis. Options to create asymmetric window functions are described as follows: 0 - Zero: the exponential window, used in the "exponentially-weighted moving average filter" (EWMA, see Is there a technical term for this simple method of smoothing out a signal?) 1 - First, for "real-time" applications that can be bufferized, or offline needs, you can first perform an extension trick. It is common to preserve a little data before the time sample when you actually need the data. This happens with triggered frame acquisition: you start acquiring "useful data" after a threshold is crossed, but your have a custom buffer for the data before.Hence, you can extend the data frame buffer "to the left", with real data, or by symmetry, and then you can use longer (and more classical) windows (symmetric or not), so that they take off when your signal is not interesting, and "start" where you want to analyze. 2 - Second: for many models, you can combine a "left-sided window" with a certain up-rate, to a "right-sided window" with another down-rate. This can seem ad-hoc, but is in use in chemistry, for the shape of skewed peaks (two halves of a Gaussian for instance). The junction should be consistent. In What is the meaning of half window functions?, the window is non-continuous. In many applications, one often tries to have regular functions (continuous, differentiable). For discrete windows, one often choose that $w_l(0-)\approx w_r(0+)$, which is easily done on isotonic (increasing/decreasing) windows with $[0,1]$ normalization. Higher orders of regularization could be enfored as well. 3 - Third: you can build a closed-form skewed function. A lot of skewed probability distribution functions can be used (log-normal, beta, Weibull) as windows. 4 - Fourth: you can distort or warp the time axis of a symmetric window, to yield an asymmetric one. Some references: Matlab code: % Laurent Duval % 2019/08/06 % SeDsp59829 close all;clear all nSample = 65; timeUniform = linspace(-1,1,nSample); powerExp = 0.5; timeWarpPower = 2((timeUniform+1).^powerExp)/2^powerExp-1; hannWindow = @(time) 1/2(1+cos(pi*time)); hannWindowUniform = hannWindow(timeUniform); hannWarpPower = hannWindow(timeWarpPower); figure(1);clf;hold on plot(timeUniform,hannWindowUniform, 'ob') plot(timeUniform,hannWarpPower, 'gx') grid on
This talk is an exposition of Casson's invariant: we will cover the definition of Casson's invariant in terms of Heegaard decomposition and representation spaces, and show how it can be computed (and defined) in terms of the Alexander polynomial of knots. Local class field theory is about classifying all abelian extensions over a base local field (for instances Q_p , R, C). It turns out that this extrinsic data is completely determined by the intrinsic properties of the base field. We will start by reviewing some basic number theory facts. We will then discuss statements in local class field theory and see some interesting examples. We assume a basic knowledge of commutative algebra, infinite Galois theory and number theory. Given a von Neumann algebra M equipped with a trace, any self-adjoint operator in M can be thought of as a non-commutative random variable. For an n-tuple X of such operators, the free Stein information of X is a free probabilistic quantity defined by the behavior of a non-commutative Jacobian on the polynomial algebra generated by entries of X. It is a number in the interval [0,n] and its value can provide information about the entries of X as well as the von Neumann algebra they generate. In this talk, I will discuss these and other properties of the free Stein information and consider a few examples where it can be explicitly computed. This is based on joint work with Ian Charlesworth. Mathematics anxiety among elementary preservice teachers is a well–documented phenomenon that greatly affects their ability to engage in teacher preparation courses (e.g., Dutton, 1951; Gresham, 2007; Sloan, 2010). One way for instructors to engage with PSTs is to interact with them informally (Lamport, 1993). Informal conversations present an opportunity to increase students’ confidence and address their anxiety regarding mathematics content. A potential venue for informal conversations is office hours; however, college students often do not take advantage of office hours that are offered. This talk will describe preliminary results of a policy designed to increase instances of informal interactions between students and their instructors during office hours, by solely providing homework solutions to students during office hours. Initial evidence from surveys and course evaluations suggests that students who come into office hours engage with the instructor on topics they did not intend to discuss before coming to office hours, and suggests that these conversations have the potential to help reduce mathematics anxiety. We will review briefly 3 types of operators which are mapping spaces of real-valued functions which are defined on the real line equipped with standard normal probability measure. Those are the derivative, divergence and Ornstein-Uklenbeck operators. There are simple formulas that describe the relationships between those operators. Using those formulas the proofs of the following will be presented: 1. Poincare inequality : The variance of a function of N(0,1) is dominated by the second moment of its derivative. 2. An upper bound to the Wasserstein distance between the distribution of a function of N(0,1) (the function has mean 0 and standard deviation 1) and N(0,1) itself. This upper bound is (up to a constant) the multiplication of the L4 norm of the function derivative and the L4 norm of the function 2nd derivative. The material is based on Nourdin and Peccati book. Let $G$ be a Kac-Peterson group associated to a symmetrizable generalized Cartan matrix. Let $(b, d)$ be a pair of positive braids associated to the root system. We define the double Bott-Samelson cell associated to $G$ and $(b,d)$ to be the moduli space of configurations of flags satisfying certain relative position conditions. We prove that they are affine varieties and their coordinate rings are upper cluster algebras. We construct the Donaldson-Thomas transformation on double Bott-Samelson cells and show that it is a cluster transformation. In the cases where $G$ is semisimple and the positive braid $(b,d)$ satisfies a certain condition, we prove a periodicity result of the Donaldson-Thomas transformation, and as an application of our periodicity result, we obtain a new geometric proof of Zamolodchikov's periodicity conjecture in the cases of $D\otimes A_n$. This is joint work with Linhui Shen. In dynamical systems, one often encounters actions $\mathcal{A}\equiv \int_{\Omega}L(x, v(x))\rho dx$ which depend only on $v$, the velocity of the system and on $\rho$ the distribution of the particles. In this case, it is well–understood that convexity of $L(x, \cdot)$ is the right notion to study variational problems. In this talk, we consider a weaker notion of convexity which seems appropriate when the action depends on other quantities such as electro–magnetic fields. Thanks to the introduction of a gauge, we will argue why our problem reduces to understanding the relaxation of a functional defined on the set of differential forms (Joint work with B. Dacorogna). Department of Mathematics Michigan State University 619 Red Cedar Road C212 Wells Hall East Lansing, MI 48824 Phone: (517) 353-0844 Fax: (517) 432-1562 College of Natural Science
Proportional term: this controls how quickly to turn the steering when the heading is not at the set value.A low P will lead to sluggish steering, reacting only slowly to setheading changes. It may never reach the commanded value.A higher P will give a snappier response, ideally with the steeringturning rapidly and smoothly to follow commanded heading ... Are there any issues with this?The main issue with this is that while your proposed solution will instantaneously correct for a mismatch between the performance of the motors, it will not correct for accumulated error, let alone more complex errors in position such as Abbe error (see later).What is a better approach?There are several things you can ... At first, I did not go trough your code to check for errors in the formulas but from a high level perspective this seems ok. Therefore, your position controller is fine.What you lack is a lowlevel controller for the PWM signal.This controller should take the error of w_l - e.g. e_w_l - and w_r and provide a duty cycle accordingly.For that you should ... Your instincts are correct if you are talking about rotating both wheels forward (or reverse) with the same rotational speed. In that case, the robot would move linearly forward (or backward), just as you describe. The drawings at the link you provide seem to support this. However, this analysis is different from what the text describes.I think the ... A few things:I took a look at your data set. Did you make sure you used the time column correctly? The first entry is "1429481388546050050" without the decimal. To make it in seconds, it should be 1429481388.546050050.Your motion model is fine (I've used it before, for people who want to see it derived, it is very similar to this one). However, to avoid ... I would add a few lines after you check that theta is between +/- 2pi:meanDistance = (SL + SR)/2;posX = posX + meanDistancecos (theta);posY = posY + meanDistancesin(theta);This of course assumes theta is positive CCW starting from the +x-axis. This is similar but not the same as your code for X and Y, but your code appears to put the X origin on the ... OK. as drawn, ignoring mass and accelerations, the force $F_p$ will appear as a torque on your ball screw.However, the total force on the ball screw, and hence the torque, depends on the mass of the thing you're moving with the ball screw interacting with gravity (if it's being moved in anything other than a horizontal plane), and on whether or not the ... I have a bot with 2 independently driven wheels.I chose to use a gyro to keep it heading in the desired direction, bumps slippage and even picking it up and turning it around are of little consequence to it as it will just correct it's heading.I use a single PID, which adds/subtracts a correction to the desired current speed for each of the 2 motors in ... You aren't properly mapping your steering value to the wheel speeds. In fact, I don't think you're applying the PID correctly at all.From your code, my guess is that you're using get_segment_center to determine the adjustment that you need to make. My assumption is that this describes a distance measurement of how far the line sensor is off of the ... Pure pursuit is the standard method for following a trajectory with a differential drive (or ackerman steering) robot. It is a very simple technique. You should be able to search for it and find some (very old) papers describing it. At the foundation of PID control, there is the assumption that the quantity you are measuring (and using to compute your error) has a direct linear relationship with the quantity you are controlling.In practice, people frequently bend this rule without things going horribly wrong. In fact this resiliency to modeling error--when our assumptions (our "model")... In order to do this, you need to have something on the robot that can intercept your "single joystick" signal from the remote control and translate it to left/right wheel speeds. Your arduino might be able to serve this purpose, with the appropriate shield.For that calculation, check out this question on calculating left and right motor speeds based on ... To make compatible gears, you need to match the pitch and shape of the teeth.First, check out this wiki article about gears and especially the image about nomenclature, so you have a good idea of the names for things:You first need to determine the pitch of the gear you want to match, the 8-tooth gear. If it has a diameter of say 10 mm, then take the ... Kinematics of mobile robotsFor the figure on the left:I = Inertial frame;R = Robot frame;S = Steering frame;W = Wheel frame;$\beta$ = Steering angle;For the figure on the right:L = Distance between the wheels;r = radious of the wheel;Now we can derive some useful equations.Kinematics:$\hspace{2.5em}$ $\vec{v}_{IW} = \vec{v}_{IR} + \vec{\... Thanks for the update. Now it looks like $x_c$ and $y_c$ denote the origin/starting position, and $\theta$ is positive, measured CCW from the positive x-axis. Now I am even more concerned about the equations you're using. Consider just $x$. You have:$$x_k = x_{k-1} - \frac{v}{\omega} \sin{(\theta)} + \frac{v}{\omega} \sin{(\theta + \omega \Delta t)}$$... The short answer is that the equations/models for these different vehicle should be different but there is no value in using more accurate equations.All these equations are approximations that make assumptions about how the ground and wheels/tracks interact.If there is 2 wheels and no slipping of the wheels on the ground, then the equation is reasonably ... Your linear velocity should be the average of both wheel values. Assuming there's some wheel radius of WHEEL_RADIUS, as you've stated, then you should get each wheel speed as:left_velocity = left_rpm * (RPM_TO_RAD_PER_S * DIST_PER_RAD);right_velocity = right_rpm * (RPM_TO_RAD_PER_S * DIST_PER_RAD);linear_velocity = 0.5f * (left_velocity + right_velocity);... OK, I'm going to work on the assumption that you are trying to calculate the Instantaneous Centre of Curvature, and that the values of $R$ and $\theta$ that you have been given are the distance from the ICC to the mid-point of the wheel axle and the direction of travel relative to the x-axis.That should correspond with the diagram below, taken from ... The answer is very simple;Steer drive is what you know from a car where one motor powers both the wheels (either the front or the rear wheels) and then steering is achieved by turning the front wheels right or left.Differential drive means having two motors, one that powers all wheels (or the track) on the right side and one that powers all wheels on the ... You can alter the C code structure in the Controller, so that the code you want to test is indepencent from platfrom specific code. (You can only test the platform independent code using Software in the Loop anyways.) I am not sure what ways are available in C in order to achive this, but all inversion of control methods, e.g. Dependency injection, have ... 1) Load or import the data that is in your xlsx files into w1 and w2, by using importdata as an array of values.2) Write a for loop for extractting individual values from w1 and w2, to perform the necessary computation. The inputs needed for the function are objects of the structures R and ENC. The member variables are clearly specified in the description. Declare and initialize these values before calling the function. That should solve your problem.learn more about structures in Matlab and how to initialize one here: https://in.mathworks.com/help/matlab/ref/struct.html... Yes, I have experienced this. Wheel encoders are great for a second to second hint for an ekf predict step, but are generally awful for long term, long distance prediction. Odometery and imu can do better, but both are integrators and will accumulate error quickly. Add GPS, terrain features, or other global estimates for a real solution. You named a question "Standard equation for steering differential drive robot", so instead of going deep into your code, I'll try to give you a simple example, how can you steer a differential drive robot.Assumption: joystick steering - two input variables, throttle (forward speed) and steering. It would be easiest if they change from -255 (full revesre/... Well, if it is truly a caster wheel with two differential drives, then I'd just assume that the castor is not a constraint at all!It's a freely rotating wheel that should just follow the direction of motion induced by rotating the differential wheels. In that case, you can use this answer. Someone please check my math here, but the wheel encoder value $\theta$ is not a continuous random variable -- it can't have a variance.You might get a more reasonable continuous random variable in the form of the estimated total linear distance. In that case, the main source of variance in your encoder measurement is going to be the difference between ... To estimate the first three parameters (distance between wheels and distance per encoder tick of both left and right wheels) I made a calibration pattern on the ground as shown in the image:(this shape was chosen because it's easy to make using tape measure and a string)Then I will manually drive the robot along the path in a ABCDBCA pattern as precisely ...
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closed as no longer relevant by Robin Chapman, Akhil Mathew, Yemon Choi, Qiaochu Yuan, Pete L. Clark Aug 22 '10 at 9:00 This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center. If this question can be reworded to fit the rules in the help center, please edit the question. $e^{\pi i} + 1 = 0$ Stokes' Theorem Trivial as this is, it has amazed me for decades: $(1+2+3+...+n)^2=(1^3+2^3+3^3+...+n^3)$ $$ \frac{24}{7\sqrt{7}} \int_{\pi/3}^{\pi/2} \log \left\| \frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}\right\| dt\\ = \sum_{n\geq 1} \left(\frac n7\right)\frac{1}{n^2}, $$where $\left(\frac n7\right)$ denotes the Legendre symbol. Not really my favorite identity, but it has the interesting feature that it is aconjecture! It is a rare example of a conjectured explicit identitybetween real numbers that can be checked to arbitrary accuracy.This identity has been verified to over 20,000 decimal places.See J. M. Borwein and D. H. Bailey, Mathematics by Experiment: Plausible Reasoning in the 21st Century, A K Peters, Natick, MA,2004 (pages 90-91). There are many, but here is one. $d^2=0$ Mine is definitely $$1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}+\cdots=\frac{\pi^2}{6},$$ an amazing relation between integers and pi. There's lots to choose from. Riemann-Roch and various other formulas from cohomology are pretty neat. But I think I'll go with $$\sum\limits_{n=1}^{\infty} n^{-s} = \prod\limits_{p \text{ prime}} \left( 1 - p^{-s}\right)^{-1}$$ 1+2+3+4+5+... = -1/12 Once suitably regularised of course :-) $$\frac{1}{1-z} = (1+z)(1+z^2)(1+z^4)(1+z^8)...$$ Both sides as formal power series work out to $1 + z + z^2 + z^3 + ...$, where all the coefficients are 1. This is an analytic version of the fact that every positive integer can be written in exactly one way as a sum of distinct powers of two, i. e. that binary expansions are unique. $V - E + F = 2$ Euler's characteristic for connected planar graphs. I'm currently obsessed with the identity $\det (\mathbf{I} - \mathbf{A}t)^{-1} = \exp \text{tr } \log (\mathbf{I} - \mathbf{A}t)^{-1}$. It's straightforward to prove algebraically, but its combinatorial meaning is very interesting. $196884 = 196883 + 1$ For a triangle with angles a, b, c $$\tan a + \tan b + \tan c = (\tan a) (\tan b) (\tan c)$$ Given a square matrix $M \in SO_n$ decomposed as illustrated with square blocks $A,D$ and rectangular blocks $B,C,$ $$M = \left( \begin{array}{cc} A & B \\\ C & D \end{array} \right) ,$$ then $\det A = \det D.$ What this says is that, in Riemannian geometry with an orientable manifold, the Hodge star operator is an isometry, a fact that has relevance for Poincare duality. But the proof is a single line: $$ \left( \begin{array}{cc} A & B \\\ 0 & I \end{array} \right) \left( \begin{array}{cc} A^t & C^t \\\ B^t & D^t \end{array} \right) = \left( \begin{array}{cc} I & 0 \\\ B^t & D^t \end{array} \right). $$ It's too hard to pick just one formula, so here's another: the Cauchy-Schwarz inequality: \|\|x\|\| \|\|y\|\| >= \|(x.y)\|, with equality iff x&y are parallel. Simple, yet incredibly useful. It has many nice generalizations (like Holder's inequality), but here's a cute generalization to three vectors in a real inner product space: \|\|x\|\| 2\|\|y\|\| 2\|\|z\|\| 2+ 2(x.y)(y.z)(z.x) >= \|\|x\|\| 2(y.z) 2+ \|\|y\|\| 2(z.x) 2+ \|\|z\|\| 2(x.y) 2, with equality iff one of x,y,z is in the span of the others. There are corresponding inequalities for 4 vectors, 5 vectors, etc., but they get unwieldy after this one. All of the inequalities, including Cauchy-Schwarz, are actually just generalizations of the 1-dimensional inequality: \|\|x\|\| >= 0, with equality iff x = 0, or rather, instantiations of it in the 2 nd, 3 rd, etc. exterior powers of the vector space. I always thought this one was really funny: $1 = 0!$ I think that Weyl's character formula is pretty awesome! It's a generating function for the dimensions of the weight spaces in a finite dimensional irreducible highest weight module of a semisimple Lie algebra. $2^n>n $ It has to be the ergodic theorem, $$\frac{1}{n}\sum_{k=0}^{n-1}f(T^kx) \to \int f\:d\mu,\;\;\mu\text{-a.e.}\;x,$$ the central principle which holds together pretty much my entire research existence. Gauss-Bonnet, even though I am not a geometer. Ἐν τοῖς ὀρθογωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ὀρθὴν γωνίαν ὑποτεινούσης πλευρᾶς τετράγωνον ἴσον ἐστὶ τοῖς ἀπὸ τῶν τὴν ὀρθὴν γωνίαν περιεχουσῶν πλευρῶν τετραγώνοις. That is, In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. The formula $\displaystyle \int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+1} dx = \frac{\pi}{e}$. It is astounding in that we can retrieve $e$ from a formula involving the cosine. It is not surprising if we know the formula $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$, yet this integral is of a purely real-valued function. It shows how complex analysis actually underlies even the real numbers. It may be trivial, but I've always found $\sqrt{\pi}=\int_{-\infty}^{\infty}e^{-x^{2}}dx$ to be particularly beautiful. For X a based smooth manifold, the category of finite covers over X is equivalent to the category of actions of the fundamental group of X on based finite sets: \pi-sets === et/X The same statement for number fields essentially describes the Galois theory. Now the ideathat those should be somehow unifiedwas one of the reasons in the development of abstract schemes, a very fruitful topic that is studied in the amazing area of mathematics called the abstract algebraic geometry. Also, note that "actions on sets" is very close to "representations on vector spaces" and this moves us in the direction of representation theory. Now you see, this simple line actually somehow relates number theory and representation theory. How exactly? Well, if I knew, I would write about that, but I'm just starting to learn about those things. (Of course, one of the specific relations hinted here should be the Langlands conjectures, since we're so close to having L-functions and representations here!) E[X+Y]=E[X]+E[Y] for any 2 random varibles X and Y $\prod_{n=1}^{\infty} (1-x^n) = \sum_{k=-\infty}^{\infty} (-1)^k x^{k(3k-1)/2}$ $ D_A\star F = 0 $ Yang-Mills $\left(\frac{p}{q}\right) \left(\frac{q}{p}\right) = (-1)^{\frac{p-1}{2} \frac{q-1}{2}}$. My favorite is the Koike-Norton-Zagier product identity for the j-function (which classifies complex elliptic curves): j(p) - j(q) = p -1 \prod m>0,n>-1 (1-p mq n) c(mn), where j(q)-744 = \sum n >-2 c(n) q n = q -1 + 196884q + 21493760q 2 + ... The left side is a difference of power series pure in p and q, so all of the mixed terms on the right cancel out. This yields infinitely many identities relating the coefficients of j. It is also the Weyl denominator formula for the monster Lie algebra.
The Maxwell equations are relativistic. But what happens to them in an expanding space time? I assume that only the charge density $\rho$ is affected, i.e. only Gauss's law gets modified. Am I right with this assumption or are there further effects? The Wikipedia article "Maxwell's equations in curved spacetime" presents the equations in a particularly simple way that involves only partial derivatives rather than covariant derivatives. However, that obscures a simple procedure for making Lorentz-covariant equations generally covariant, which is to raise and lower all indices with a general metric tensor rather than the Minkowski tensor, and to replace all partial derivatives with covariant derivatives. The Lorentz-covariant Maxwell's equations are $$F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$$ $$\partial_\mu F^{\mu\nu}=j^\nu$$ and the generally-covariant Maxwell's equations are $$F_{\mu\nu}=D_\mu A_\nu-D_\nu A_\mu$$ $$D_\mu F^{\mu\nu}=j^\nu$$ where $D_\mu$ is the covariant derivative for the metric tensor $g_{\mu\nu}$. (In the first equation, the terms in the covariant derivative involving Christoffel symbols simply cancel, but in the second equation they are important.) If you prefer to start with the action, then you also have to make the 4-volume element generally invariant by the replacement $d^4x\rightarrow\sqrt{-g}\,d^4x,$ where $g$ is the determinant of the metric tensor. So $$S=\int\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+j^\mu A_\mu\right)d^4x$$ becomes $$S=\int\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+j^\mu A_\mu\right)\sqrt{-g}\,d^4x$$ Note: The Lagrangian density in the Wikipedia article has a $\sqrt{-g}$ in the first term but not in the second. This is because the article prefers to use $J^\mu$, a vector density of weight 1, rather than $j^\mu$, a vector, to represent the current density. Thank you very much for your explanation! In the meantime I also discovered a nice solution in the 3+1 formalism from THorne and Macdonald (Mon. Not. R. astr. Soc. (1982) 198, 339-343 and Microfiche MN 198/1): $\nabla\cdot E=4\pi\rho_e$ with $E$ the 4-vector of the EM-field. The classical result remains, that electric field lines terminate on electric charge. $\nabla\cdot B =0$ with $B$ the 4-vector of the EM-field. The classical result remains, there are no magnetic charges, i.e. the magnetic field lines never end. $D_\tau E +\frac{2}{3}\theta E-\sigma\cdot E = \alpha^{-1}\nabla\times\left(\alpha B\right) - 4\pi j$ with $\theta$ the expansion, $\sigma$ the shear, and the acceleration $a=\nabla\log\alpha$ implying that a fiducial observer, being in a perfectly conducting medium, would never experience an electric field (i.e. the field is frozen). Otherwise, the observed curl induced a time-changing magnetic field. $D_\tau B + \frac{2}{3}\theta B - \sigma\cdot B = -\alpha^{-1}\nabla\times\left(\alpha E \right)$ Everything is in much more detail in the stated paper, worth reading to anyone who'd like to use Maxwell's equations in the 3+1 formalism. :)
Let $A = \{ 1,2,3,4 \}$ Let $F$ be a set of all functions from $A \to A$. Let $S$ be a relation defined by : $\forall f,g \in F$ $fSg \iff f(i) = g(i)$ for some $i \in A$ Let $h: A \to A$ be the function $h(x) = 1 $ for all $x \in A$. How many functions $g \in F$ are there so that $gSh$ ? My solution : $gSh$ means that $g(i) = h(i)$ for some $i \in A$. So $g(i) = 1$. So number 1 always needs to connect to some x - 4 choices. Then we have $3$ left-over numbers that can connect to $4$ numbers each. So solution is $4 \cdot 4 \cdot 4 \cdot 4$. Is this correct at all? Thanks in advance !! :)
A metal (or otherwise, suitably elastic) circle is cut and the points are slid up and down a vertical axis as shown: How would one describe the resultant curves mathematically? Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up.Sign up to join this community This problem was first formulated by Leonhard Euler in 1744 WPlink: "That among all curves of the same length which not only pass through the points A and B, but are also tangent to given straight lines at these points, that curve be determined in which minimizes the value of \begin{equation} \int_A^B \frac{ds}{R^2} \end{equation} It is a problem of calculus of variations and the Euler-Lagrange equations WPlink allows to solve it as an ODE of the type: \begin{equation} \frac{dy}{dx} = \frac{a^2 - c^2 + x^2}{\sqrt{(c^2 - x^2)(2a^2 - c^2 + x^2)}} \end{equation} The physical meaning is this: the wire will take the shape that minimizes the total energy related to bending further in each point. This energy is similar to the the spring potential energy for deformations, but in this case the measure of the deformation is the curvature $k = \frac{1}{R}$. Since in your proposal, the elastic line was initially a circle, I would propose the integral to be: \begin{equation} \int_A^B (\frac{1}{R} - k_0)^2ds \end{equation} where $k_0$ would be the initial curvature which should be the rest one. I would set it constant here since for the circle it is the same in any point, but in general if you start with a different rest position, the rest curvature in each point will differ from point to point. Now let's analyze the curves, and group them. If we number them from left to right and from top to bottom we can make the 2 following groups: Fixed ends condition: The curves 1,2 and 6. This curves are only determined by fixating the extremes of the curve, or frontier conditions. This means that they are shapes that the curves will take naturally under no external forces on any point of it. Fixed ends + one fixed end angle conditions: The curves 3,4 and 5. It can be seen that 4 and 5 are the same. This curves need, apart from the fixed extremes condition, the fixing of one or two of the extremes' angle. A bending force there or some general external force acting on one r more points of it would cause them as well. If they did not have this extra condition, nothing would prevent them from falling back to form 2 or 6. Finally, here is a review of the solutions, with a great historical presentation of the problem: ElasticaHistory. But if you really want to get serious I recommend A Treatise on the Mathematical Theory of Elasticity.
(a) Solve the following system by transforming the augmented matrix to reduced echelon form (Gauss-Jordan elimination). Indicate the elementary row operations you performed.\begin{align}x_1+x_2-x_5&=1\\x_2+2x_3+x_4+3x_5&=1\\x_1-x_3+x_4+x_5&=0\end{align} (b) Determine all possibilities for the solution set of a homogeneous system of $2$ equations in $2$ unknowns that has a solution $x_1=1, x_2=5$. We say that two $m\times n$ matrices are row equivalent if one can be obtained from the other by a sequence of elementary row operations. Let $A$ and $I$ be $2\times 2$ matrices defined as follows.\[A=\begin{bmatrix}1 & b\\c& d\end{bmatrix}, \qquad I=\begin{bmatrix}1 & 0\\0& 1\end{bmatrix}.\]Prove that the matrix $A$ is row equivalent to the matrix $I$ if $d-cb \neq 0$. Find the value(s) of $h$ for which the following set of vectors\[\left \{ \mathbf{v}_1=\begin{bmatrix}1 \\0 \\0\end{bmatrix}, \mathbf{v}_2=\begin{bmatrix}h \\1 \\-h\end{bmatrix}, \mathbf{v}_3=\begin{bmatrix}1 \\2h \\3h+1\end{bmatrix}\right\}\]is linearly independent. (Boston College, Linear Algebra Midterm Exam Sample Problem) Let $P_2$ be the vector space of all polynomials of degree two or less.Consider the subset in $P_2$\[Q=\{ p_1(x), p_2(x), p_3(x), p_4(x)\},\]where\begin{align}&p_1(x)=x^2+2x+1, &p_2(x)=2x^2+3x+1, \\&p_3(x)=2x^2, &p_4(x)=2x^2+x+1.\end{align} (a) Use the basis $B=\{1, x, x^2\}$ of $P_2$, give the coordinate vectors of the vectors in $Q$. (b) Find a basis of the span $\Span(Q)$ consisting of vectors in $Q$. (c) For each vector in $Q$ which is not a basis vector you obtained in (b), express the vector as a linear combination of basis vectors. Let $V$ be the vector space of all $2\times 2$ matrices, and let the subset $S$ of $V$ be defined by $S=\{A_1, A_2, A_3, A_4\}$, where\begin{align}A_1=\begin{bmatrix}1 & 2 \\-1 & 3\end{bmatrix}, \quadA_2=\begin{bmatrix}0 & -1 \\1 & 4\end{bmatrix}, \quadA_3=\begin{bmatrix}-1 & 0 \\1 & -10\end{bmatrix}, \quadA_4=\begin{bmatrix}3 & 7 \\-2 & 6\end{bmatrix}.\end{align}Find a basis of the span $\Span(S)$ consisting of vectors in $S$ and find the dimension of $\Span(S)$.
Current browse context: math.AP Change to browse by: References & Citations Bookmark(what is this?) Mathematics > Analysis of PDEs Title: Layer potentials and boundary value problems for elliptic equations with complex $L^{\infty}$ coefficients satisfying the small Carleson measure norm condition (Submitted on 1 Nov 2013) Abstract: We consider divergence form elliptic equations $Lu:=\nabla\cdot(A\nabla u)=0$ in the half space $\mathbb{R}^{n+1}_+ :=\{(x,t)\in \mathbb{R}^n\times(0,\infty)\}$, whose coefficient matrix $A$ is complex elliptic, bounded and measurable. In addition, we suppose that $A$ satisfies some additional regularity in the direction transverse to the boundary, namely that the discrepancy $A(x,t) -A(x,0)$ satisfies a Carleson measure condition of Fefferman-Kenig-Pipher type, with small Carleson norm. Under these conditions, we establish a full range of boundedness results for double and single layer potentials in $L^p$, Hardy, Sobolev, BMO and H\"older spaces. Furthermore, we prove solvability of the Dirichlet problem for $L$, with data in $L^p(\mathbb{R}^n)$, $BMO(\mathbb{R}^n)$, and $C^\alpha(\mathbb{R}^n)$, and solvability of the Neumann and Regularity problems, with data in the spaces $L^p(\mathbb{R}^n)/H^p(\mathbb{R}^n)$ and $L^p_1(\mathbb{R}^n)/H^{1,p}(\mathbb{R}^n)$ respectively, with the appropriate restrictions on indices, assuming invertibility of layer potentials in for the $t$-independent operator $L_0:= -\nabla\cdot(A(\cdot,0)\nabla)$. Submission historyFrom: Mihalis Mourgoglou [view email] [v1]Fri, 1 Nov 2013 04:33:47 GMT (62kb)
Trigonometry is a branch of Mathematics, that involves the study of the relation between angles and lengths of triangles. It is a very important branch of Mathematics based on Statistics, Calculus and Linear Algebra. It is not only important in Mathematics but also plays a major role in Physics, Astronomy and Architectural Design. CBSE Class 12 mathematics contains Inverse Trigonometry Functions. This chapter includes definition, graphs and elementary properties of inverse trigonometric functions. Trigonometry formulas for class 12 play a critical role in these chapters. Hence all trigonometry formulas are provided here. Trigonometry formulas for class 12 contain all formulas in a single page for better understanding . We believe, Trigonometry Class 12 formulas provided here will help students to learn them and can have a quick glance when needed. Trigonometry formulas list Trigonometry Class 12 Formulas Definition $\theta = \sin^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \sin \theta$ $\theta = \cos^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \cos \theta$ $\theta = \tan^{-1}\left ( x \right )\, is\, equivalent\, to\, x = \tan\theta$ $\sin\left ( \sin^{-1}\left ( x \right ) \right ) = x$ $\cos\left ( \cos^{-1}\left ( x \right ) \right ) = x$ $\tan\left ( \tan^{-1}\left ( x \right ) \right ) = x$ $\sin^{-1}\left ( \sin\left ( \theta \right ) \right ) = \theta$ $\cos^{-1}\left ( \cos\left ( \theta \right ) \right ) = \theta$ $\tan^{-1}\left ( \tan\left ( \theta \right ) \right ) = \theta$ $\sin\left ( 2x \right ) = 2\, \sin\, x\, \cos\, x$ $\cos\left ( 2x \right ) = \cos^{2}x – \sin^{2}x$ $\tan\left ( 2x \right ) = \frac{2\, \tan\, x}{1 – \tan^{2}x}$ $\sin\frac{x}{2} = \pm \sqrt{\frac{1 – \cos x}{2}}$ $\cos\frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$ $\tan\frac{x}{2} = \frac{1- \cos\, x}{\sin\, x} = \frac{\sin\, x}{1 + \cos\, x}$< a’
I wanted to better understand dfa. I wanted to build upon a previous question:Creating a DFA that only accepts number of a's that are multiples of 3But I wanted to go a bit further. Is there any way we can have a DFA that accepts number of a's that are multiples of 3 but does NOT have the sub... Let $X$ be a measurable space and $Y$ a topological space. I am trying to show that if $f_n : X \to Y$ is measurable for each $n$, and the pointwise limit of $\{f_n\}$ exists, then $f(x) = \lim_{n \to \infty} f_n(x)$ is a measurable function. Let $V$ be some open set in $Y$. I was able to show th... I was wondering If it is easier to factor in a non-ufd then it is to factor in a ufd.I can come up with arguments for that , but I also have arguments in the opposite direction.For instance : It should be easier to factor When there are more possibilities ( multiple factorizations in a non-ufd... Consider a non-UFD that only has 2 units ( $-1,1$ ) and the min difference between 2 elements is $1$. Also there are only a finite amount of elements for any given fixed norm. ( Maybe that follows from the other 2 conditions ? )I wonder about counting the irreducible elements bounded by a lower... How would you make a regex for this? L = {w $\in$ {0, 1}* : w is 0-alternating}, where 0-alternating is either all the symbols in odd positions within w are 0's, or all the symbols in even positions within w are 0's, or both. I want to construct a nfa from this, but I'm struggling with the regex part

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