text:
Removing extraneous factors in Einstein Action

I have an action of the form
$$\int \alpha \sqrt{g} R$$
where $\alpha$ is a field.
I want to do some kind of coordinate transformation, so that this term in the action becomes
$$\int \sqrt{g}R + \sqrt{g}(...)$$
where (...) consists of terms containing \alpha.

So basically, I want to remove $\alpha$ from the coefficient of R and move it to other terms in the lagrangian. Now, I know that this can be done! (I remember this from long ago), but for heaven's sake can't figure out how!

Can any one tell me? Just a reference would be fine.


text:
Demo Engine

This is an important introductory tutorial about how to use the SolidWorks CAD as a preprocessing tool, thanks to the Chrono::SolidWorks add-in.

You will learn how to model a 3D assembly using SolidWorks, how to export it as a .py file containing the Chrono::Engine system description, how to run a Python simulation of the exported system using PyChrono, and how to render the simulation using POVray.

Please follow all steps of the tutorial, once you understood all these steps, you will be able to create a lot of complex mechanisms that can be exported from SolidWorks to Chrono::Engine.

# Prerequisites:

• you must have a SolidWorks CAD license.
• the Chrono::SolidWorks add-in must be installed in SolidWorks. Optionally, for the last steps of the tutorial (rendering the animation) these tools must be installed:
• the PyChrono python module must be installed in your Python environment,
• the POVray rendering software must be installed; not mandatory if you want to use the Irrlicht realtime visualization;
• the VirtualDub video editing tool must be installed; not mandatory if you want to use the Irrlicht realtime visualization;
The files for this demo can be found in the directory C:/[install path]/chrono_solidworks/examples/engine. The directory contains all the parts needed for this assembly.

# Create a subassembly

• First, start SolidWorks.
• Use menu: File/New... and choose Assembly.
• If SolidWorks prompts you with a dialog about creating a layout, just close it; we do not need layouts in this tutorial.
• Use menu: Insert/Component/Existing Part... and select the Piston.sldprt file from the directory of demo files listed above. The piston can be placed where you prefer, in the 3D view:

• Use menu: Insert/Component/Existing Part... and select the Conrod.sldprt file from the directory of demo files. The piston can be placed where you prefer, in the 3D view:

• The Feature Manager view, at the left, should look like this:

• Use the Mate tool to create constraints between the piston and the conrod. This tool can be accessed easily by pushing this button:
• The mating tool should be familiar to all users of SolidWorks, it consists in selecting a pair of surfaces and setting which type of constraint should be enforced between them; press the green checkmark button when done:

• In detail, you must create a Concentric mating between the rod and the piston, like in this figure...

• ...and a Coincident mating between the rod and the piston, like in this figure:

• Created constraints can now be seen in the Feature Manager view, at the left:

• Use menu: File/Save As... and save this assembly as piston_rod.sldasm, in the directory that you prefer. This subassembly (piston + rod) will be used later as a building block for making a larger assembly of the entire motor.

# Create the root assembly

• Use menu: File/New... and choose Assembly. Now we will make the root assembly, that contains the entire motor. An assembly can contain parts or sub-assemblies too.
• Again, if prompted for the creation of a layout, just skip it by closing the layout dialog.
• Select the front plane by selecting it in the Feature Manager and clicking Sketch in the resulting popup menu, so that we can create a few reference axes on it.
• In the sketch, create the following scheme with an horizontal axis and four perpendicular axes, and add dimensions so that you are sure that they are separated by 89.3 mm:

• Use menu: Insert/Component/Part... and select the Crankshaft.sldprt file from the directory of demo files.

• Note that in the Feature Manager view, at the left, the name is shown as
{(f)}
where the (f) label means that this part is fixed. By default, the first part that is inserted in SolidWorks is fixed; but in our case we want it to be free because we will take care of constraining it to a revolute joint, so use the popup menu in the Feature Manager and change from *Fixed* to *Floating*, so you will see
(-) Crankshaft <1
with the (-) label.
• Use the Mate tool to create constraints between the crankshaft and the horizontal axis. Use the button:

• The constraint should be of Concentric type:

• Use menu: Insert/Component/Assembly... and select the piston_rod.sldprt file that you saved previously (or use the one provided in the demos directory). Place the subassembly in the 3D view.
• Note! the subassembly, by default, will be handled as a rigid assembly (that is, the piston and the conrod cannot be articulated) and this is not the desired behavior. The fact that the subassembly is rigid can be seen by looking at its icon:
• So, now use the popup menu and use the Properties button, shown in this picture:

• Now, in the window that opens, switch from Solve as.. Rigid to Solve as.. Flexible:

• Press Ok and close. You will see that the icon has changed into:

• Use the Mate tool to create constraints between the crankshaft and the conrod. Use the button:
• You must create a Concentric constraint and a Coincident constraint:
• Use the Mate tool to create constraints between the piston and the vertical axis that you created in the sketch. Use the button:
• You must create a Concentric constraint:

• Note that, finally, the hierarchy of your assembly should look like this:

• Use menu: File/Save As... and save this assembly as engine.sldasm, in the directory that you prefer. (note that a ready-to-use assembly with this name is already given for your convenience if you want to skip the previous steps of the tutorial).

# Export the assembly as a Chrono::Engine system

• Open the tab of the Chrono::Engine exporter, at the right of the view window:

• Check the Save test.py button (this will generate an example Python script that computes a simple animation and outputs data that can be postprocessed for rendering).
• Press the Save as Python... button and save as engine.py in an empty directory.

This operation will save the entire root assembly as a Python file, called engine.py, that contains lot of statements that, once executed as a Python module, will create many ChBody objects for all the moving parts in SolidWorks, and many ChLink objects for each SolidWorks mating constraint. Use an editor to inspect the engine.py file so you can get an idea of what happened.

Aside to the engine.py exported file, there will be a directory called engine_shapes. Look into that directory: you will find three .obj files, each corresponding to a 3D mesh with the visualization shape of the moving parts. These shapes will be used later by the Chrono::Engine postprocessing system if you want to do a rendering of the simulation.

You can also modify, edit or simplify these .obj meshes using 3D tools such as Blender

# Run a Chrono::Engine simulation in an interactive 3D view

Since you enabled the Save test.py option, a test.py program that loads and simulate the engine was automatically generated for you. You can run it directly from SoliWorks for a quick test, do this:

• Check the Visualization Irrlicht option, to use a 3D realtime view
• Press the Run test button.

At this point the test.py will load the engine.py model and show this interactive view, that you can pan&zoom with the mouse while the simulation is running:

That is!

Note, do not expect to see a lot of movement in this animation: the only motion will be caused by the weight of the piston that will make the crankshaft swing a bit. If you want to impose a continuous rotation, you need to follow the following tutorial,Demo_engine advanced.

# How is this working?

What is happening under the hood? When you press Run test:

1. SolidWorks calls the Python interpreter that will run the test.py file, based on PyChrono;
2. on its turn, the test.py file will:
3. load the engine.py file that you exported from SolidWorks
4. put all objects of engine.py into a ChSystem object
5. setup a simple visualization system based on Irrlicht
6. run the simulation loop

Note that on some systems the step 1 could fail if the Python interpreter is not available on a global path. If so, you can start test.py by hand, by opening for example an Anaconda shell and launching it with:

{run_test.py}
The test.py program, in fact, uses the mandatory -f parameter to specify the model to load.
Other flags can be used too, as the command line syntax is
run_test.py (-f|--filename)<model_file> [-d|--timestep]<dt> [-T|--Tlength]<max_simulated_time> [-v|--visualization]<irrlicht|pov> [-p|--datapath]<path_to_data_folder

Anyway: the test.py file is just a simple stub. You can use it as an inspiration and modify it for creating a more sophisticated Python program. We strongly suggest to open it in your Python IDE editor and see how it is working.

# Run a Chrono::Engine simulation using POVray

• Check the Visualization pov option, to use POVray as a postprocessing rendering tool
• Press the Run test button.

The simulation will be computed with no interactive view: wait few seconds. At the end of the simulation, you will find that the directory of engine.py has been populated with additional files; some of them are generated by the postprocessing system of Chrono::Engine. Proceed as follows:

• Open POVray.
• Drag and drop the rendering_frames.pov.ini file into POVray window.
• Press the Run button in POVray.

• A series of .bmp images will be generated with raytracing, and saved in the anim directory:

• To generate an animation starting from a sequence of .bmp files: just start the VirtualDub tool and load the first image of the sequence, all others will be automatically added:

• It is suggested that you choose a proper compression codec, in VirtualDub. To do so, use the menu Video/Compression... and configure the codec. We suggest a Mpeg4 / DivX codec. Maybe you need to install a codec pack if the Mpeg4 is not yet installed on your system.
• Also, remember that you might need to set the frame rate, that usually is 25 or 30 frames per second. Just use the menu Video/Frame rate...
• Finally Save as AVI.

# Proceed to a more complex system

Optionally, you can repeat this exercise by adding more cylinders, to create a complete four cylinder engine:

• Add three other piston_rod.sldasm subassemblies and create the proper constraints (avoiding creating redundant constraints); you should get this:

• Save the entire assembly as engine4c.sldasm
• Export the Chrono::Engine system as engine4c.py

This done, now you can proceed with the following Demo_engine advanced where you will learn advanced topics about how to add additional constraints (ex. a motor constraints that imposes the rotation of the crank), how to change the rendering look of the shapes, etc.

format: plain


text:
Limited access

Calcium has a work function of $2.69\text{ eV}$.

Which of the following statements can we conclude about the threshold frequency and wavelength for calcium?

A

This energy corresponds to a threshold frequency of $6.5\times { 10 }^{ 14 }\text{ Hertz}$ and a wavelength of $460\text{ nm}$.

B

This energy corresponds to a threshold frequency of $4.06\times { 10 }^{ 14 }\text{ Hertz}$ and a wavelength of $739\text{ nm}$.

C

This energy corresponds to a threshold frequency of $1.54\times { 10 }^{ -15 }\text{ Hertz}$ and a wavelength of $460\text{ nm}$.

D

This energy corresponds to a threshold frequency of $4.06\times { 10 }^{ 14 }\text{ Hertz}$ and a wavelength of $195\text{ nm}$.

Select an assignment template


text:
Investment

Feel free to send your offer to webmaster.

Geodesic

[[File:Spherical triangle.svg|thumb|right|150px|A geodesic triangle on the sphere. The geodesics are great circle arcs.]] In differential geometry, a geodesic (}}}}) is a generalization of the notion of a " straight line" to " curved spaces". The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are ( locally) the shortest path between points in the space. Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

Introduction

The shortest path between two points in a curved space can be found by writing the equation for the length of a curve (a function f from an open interval of R to the manifold), and then minimizing this length using the calculus of variations. This has some minor technical problems, because there is an infinite dimensional space of different ways to parameterize the shortest path. It is simpler to demand not only that the curve locally minimizes length but also that it is parameterized "with constant velocity", meaning that the distance from f(s) to f(t) along the geodesic is proportional to |st|. Equivalently, a different quantity may be defined, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimisation). Intuitively, one can understand this second formulation by noting that an elastic band stretched between two points will contract its length, and in so doing will minimize its energy. The resulting shape of the band is a geodesic. In Riemannian geometry geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only locally the shortest distance between points, and are parameterized with "constant velocity". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t2 from the unit interval to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant. Geodesics are commonly seen in the study of Riemannian geometry and more generally metric geometry. In general relativity, geodesics describe the motion of point particles under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting satellite, or the shape of a planetary orbit are all geodesics in curved space-time. More generally, the topic of sub-Riemannian geometry deals with the paths that objects may take when they are not free, and their movement is constrained in various ways. This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of Riemannian and pseudo-Riemannian manifolds. The article geodesic (general relativity) discusses the special case of general relativity in greater detail.

Examples

[[File:Transpolar geodesic on a triaxial ellipsoid case A.svg|thumb|right|200px| A geodesic on a triaxial ellipsoid.]] The most familiar examples are the straight lines in Euclidean geometry. On a sphere, the images of geodesics are the great circles. The shortest path from point A to point B on a sphere is given by the shorter arc of the great circle passing through A and B. If A and B are antipodal points, then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

Metric geometry

In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve from an interval I of the reals to the metric space M is a geodesic if there is a constant such that for any there is a neighborhood J of t in I such that for any we have d(\gamma(t_1),\gamma(t_2)) = v \left| t_1 - t_2 \right| . This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization, i.e. in the above identity v = 1 and d(\gamma(t_1),\gamma(t_2)) = \left| t_1 - t_2 \right| . If the last equality is satisfied for all , the geodesic is called a minimizing geodesic or shortest path. In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a length metric space are joined by a minimizing sequence of rectifiable paths, although this minimizing sequence need not converge to a geodesic.

Riemannian geometry

In a Riemannian manifold M with metric tensor g, the length of a continuously differentiable curve γ :  a,b → M is defined by L(\gamma)=\int_a^b \sqrt{ g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t)) }\,dt. The distance d(p, q) between two points p and q of M is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ :  a,b → M such that γ(a) = p and γ(b) = q. With this definition of distance, geodesics in a Riemannian manifold are then the locally distance-minimizing paths. The minimizing curves of L in a small enough open set of M can be obtained by techniques of calculus of variations. Typically, one introduces the following action or energy functional E(\gamma)=\frac{1}{2}\int_a^b g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))\,dt. For a piecewise C^1 curve (more generally, a W^{1,2} curve), the Cauchy–Schwarz inequality gives L(\gamma)^2 \le 2(b-a)E(\gamma) with equality if and only if g(\gamma',\gamma') is equal to a constant a.e. It happens that minimizers of E(\gamma) also minimize L(\gamma), because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of \gamma, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of the functional L(\gamma) are generally not very regular, because arbitrary reparameterizations are allowed. The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0, where \Gamma^\lambda_{\mu\nu} are the Christoffel symbols of the metric. This is the geodesic equation, discussed below.

Calculus of variations

Techniques of the classical calculus of variations can be applied to examine the energy functional E. The first variation of energy is defined in local coordinates by \delta E(\gamma)(\varphi) = \left.\frac{\partial}{\partial t}\right|_{t=0} E(\gamma + t\varphi). The critical points of the first variation are precisely the geodesics. The second variation is defined by \delta^2 E(\gamma)(\varphi,\psi) = \left.\frac{\partial^2}{\partial s \, \partial t} \right|_{s=t=0} E(\gamma + t\varphi + s\psi). In an appropriate sense, zeros of the second variation along a geodesic γ arise along Jacobi fields. Jacobi fields are thus regarded as variations through geodesics. By applying variational techniques from classical mechanics, one can also regard geodesics as Hamiltonian flows. They are solutions of the associated Hamilton equations, with (pseudo-)Riemannian metric taken as Hamiltonian.

Affine geodesics

A geodesic on a smooth manifold M with an affine connection ∇ is defined as a curve γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so }} at each point along the curve, where \dot\gamma is the derivative with respect to t. More precisely, in order to define the covariant derivative of \dot\gamma it is necessary first to extend \dot\gamma to a continuously differentiable vector field in an open set. However, the resulting value of () is independent of the choice of extension. Using local coordinates on M, we can write the geodesic equation (using the summation convention) as \frac{d^2\gamma^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{d\gamma^\mu }{dt}\frac{d\gamma^\nu }{dt} = 0\ , where \gamma^\mu = x^\mu \circ \gamma (t) are the coordinates of the curve γ(t) and \Gamma^{\lambda }_{\mu \nu } are the Christoffel symbols of the connection ∇. This is an ordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of classical mechanics, geodesics can be thought of as trajectories of free particles in a manifold. Indeed, the equation \nabla_{\dot\gamma} \dot\gamma= 0 means that the acceleration of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by the gravity.

Existence and uniqueness

The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique. More precisely: For any point p in M and for any vector V in TpM (the tangent space to M at p) there exists a unique geodesic \gamma \, : IM such that :\gamma(0) = p \, and :\dot\gamma(0) = V, where I is a maximal open interval in R containing 0. In general, I may not be all of R as for example for an open disc in R2. The proof of this theorem follows from the theory of ordinary differential equations, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the Picard–Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and V.

Geodesic flow

Geodesic flow is a local R- action on the tangent bundle TM of a manifold M defined in the following way G^t(V)=\dot\gamma_V(t) where t ∈ R, V ∈ TM and \gamma_V denotes the geodesic with initial data \dot\gamma_V(0)=V. Thus, Gt(V) = exp(tV) is the exponential map of the vector tV. A closed orbit of the geodesic flow corresponds to a closed geodesic on M. On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical one-form. In particular the flow preserves the (pseudo-)Riemannian metric g, i.e. g(G^t(V),G^t(V))=g(V,V). \, In particular, when V is a unit vector, \gamma_V remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

Geodesic spray

The geodesic flow defines a family of curves in the tangent bundle. The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray. More precisely, an affine connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles: TTM = H\oplus V. The geodesic spray is the unique horizontal vector field W satisfying \pi_* W_v = v\, at each point v ∈ TM; here π∗ : TTM → TM denotes the pushforward (differential) along the projection π : TM → M associated to the tangent bundle. More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf. Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy H_{\lambda X} = d(S_\lambda)_X H_X\, for every X ∈ TM \ {0} and λ > 0. Here d(Sλ) is the pushforward along the scalar homothety S_\lambda: X\mapsto \lambda X. A particular case of a non-linear connection arising in this manner is that associated to a Finsler manifold.

Affine and projective geodesics

Equation () is invariant under affine reparameterizations; that is, parameterizations of the form t\mapsto at+b where a and b are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of () are called geodesics with affine parameter. An affine connection is determined by its family of affinely parameterized geodesics, up to torsion . The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if \nabla, \bar{\nabla} are two connections such that the difference tensor D(X,Y) = \nabla_XY-\bar{\nabla}_XY is skew-symmetric, then \nabla and \bar{\nabla} have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as \nabla, but with vanishing torsion. Geodesics without a particular parameterization are described by a projective connection.

Computational methods

Efficient solvers for the minimal geodesic problem on surfaces posed as Eikonal equations can be found in R. Kimmel, A. Amir, and A. M. Bruckstein. Finding shortest paths on surfaces using level sets propagation. IEEE Trans. on Pattern Analysis and Machine Intelligence (PAMI), 17(6):635–640, 1995. R. Kimmel and J. A. Sethian. Computing Geodesic Paths on Manifolds in the Proceedings of National Academy of Sciences, 95(15):8431–8435, July, 1998.

Applications

Geodesics serve as the basis to calculate:

References

• . See chapter 2.
• . See section 2.7.
• . See section 1.4.
• .
• . See section 87.
• . Note especially pages 7 and 10.
• .
• . See chapter 3.


text:
# Tag Info

6

HINT: $$2y=-\sqrt{x^2+y^2}$$ But $\displaystyle\sqrt{x^2+y^2}\ge0\implies y\le0$

6

Absolute convergence in the right half-plane (standard for all Dirichlet series, but if you're unsure, you can compare via $$\zeta(\sigma)<\zeta(2)\quad\forall \sigma >2$$ by just comparing term-by-term) Then you take the limit inside, giving you $$\lim_{\sigma\to\infty}\zeta(\sigma)=\sum_{n=1}^\infty ... 5 Hint:$$ \frac{d}{dt}\tan^{-1}(t) = \frac{1}{1+t^2}  d f(x,y) = \frac{\partial f(x,y)}{\partial x} dx + \frac{\partial f(x,y)}{\partial y} dy $$5 For real x>1+\frac{1}{n-1} then$$x^n = (1+(x-1))^n> 1+n(x-1) = x+(n-1)(x-1)> x+1$$Now if z^n+z+1=0 then \|z\|^n = \|z+1\| \leq \|z\|+1. So, with x=\|z\|, we have x^n\leq x+1, so \|z\|=x\leq 1+\frac 1{n-1}. 5 Proof of (*) Adding the four finite sums,$$\sum_{k=1}^{n}H_{k}=(n+1)H_{n}-n,\sum_{k=1}^{n}\ln{n}=\ln{n!},\sum_{k=1}^{n}\gamma=\gamma\,n,\sum_{k=1}^{n}\frac{1}{2k}=\frac12H_{n},$$gives us a representation of the n-th partial sum for the infinite series. Writing the infinite series as the limit of partial sums, we get:$$\begin{align} ...

4

Let $w_1, w_2 \in \Bbb C - \gamma$. We have \begin{align} \left|H(w_2) - H(w_1)\right| &= \left|\int_\gamma \frac{h(z)(z - w_1) - h(z)(z - w_2)}{(z - w_2)(z - w_1)}dz\right| \\ &= \left|\int_\gamma \frac{h(z)(w_2 - w_1)}{(z - w_2)(z - w_1)}dz\right| \\ &= \left|\int_\gamma \frac{h(z)}{(z - w_2)(z - w_1)}dz\right| \left|w_2 - w_1\right|. \tag{*} ...

4

Let $w = \frac{z+1}{z-1}$. Then you have a power series in $w$, centered at $0$. Find its radius of convergence, call that $R$. Then find which $z$ correspond to $\lvert w\rvert < R$. The map $z \mapsto \frac{z+1}{z-1}$ can be explicitly inverted.

3

Think of a limit $$\lim_{n\to \infty} \sum_{k=1}^{\infty}\frac1{k^n}=\lim_{n\to \infty}\left(1+\frac1{2^{n}}+\frac{1}{3^n}+\ldots\right)=1$$ Each member of the series goes to $0$ except the first one.

3

In the big Montel theorem, and often otherwise, a normal family is a family $\mathcal{F}$ of meromorphic or holomorphic functions, such that every sequence $(f_n)$ of functions in $\mathcal{F}$ has a subsequence that converges uniformly on compact sets to a meromorphic/holomorphic function, or to the constant function $z \mapsto \infty$. Your $(f_n)$ ...

3

No, there is no such hope. For a simple example, take $f(z)=1/z$. In fact, an annulus is a domain of existence for $H^\infty$ as well as $A$, i.e. there exists bounded resp. continuous up to the boundary holomorphic functions on the annulus which can't be extended holomorphically across any boundary point. More concretely, let $g$ be your favorite ...

3

Notation first: $\mathfrak{S}_0(\mathbf{f}) \neq \mathfrak{G}_0(\mathbf{f})$. It's $S_0$, not $G_0$. Not that it matters, of course. What matters is that Ahlfors here isn't concerned with abstract Riemann surfaces, but with very special Riemann surfaces that come with a nice projection to $\mathbb{C}$. What does he mean, saying $\theta: G_0(f) = ... 3 Hint: Consider$w=z-1$. The integral becomes $$\oint_{|w-i|=5/4}\frac{\log(1+w)}{w^2}\mathrm{d}w$$ Recall the power series for$\log(1+w)=w-\frac{w^2}{2}+\frac{w^3}{3}-\dots$Since$w=0$is inside the contour, we can apply the Residue Theorem. Comment: There is nothing wrong with your approach using Cauchy's Integral Formula. We are both actually ... 3 By the root test we have $$\left|2^n\frac{(4z-8)^n}n\right|^{1/n}\xrightarrow{n\to\infty}2|4z-8|<1\iff|z-2|<\frac18\iff z\in B\left(2,\frac18\right)$$ so the radius is$\frac18$. 3 Let$ \displaystyle f(z) = \frac{\log z}{z^{2}+z+1}$and integrate around a keyhole contour where the branch cut for$\log z$is placed on the positive real axis. As the radius of the little circle goes to$0$and the radius of the big circle goes to$\infty$,$ \int f(z) \ dz$will vanish along both circles. You can use the ML inequality to show this. So ... 3 Look at$e^{1/z}$around$z = 0$. This function approaches every point in the complex plane (take the limit along the real axis, positive or negative or along the imaginary axis). Dividing by and adding a meromorphic function does not change this, so there is no singularity that can be characterized in the normal way with Laurent series, residues, etc. This ... 2 Here are a couple of tricks and general plans of approach I know: If$x\in\Bbb R$and for all$\epsilon>0$we have that$|x|\leq\epsilon$, then$x=0$. I think of this fact as being Real Analysis in a nutshell. Never forget that if$A\subseteq\Bbb R$is bounded above and$M=\sup A$, then for all$\epsilon>0$there is a$y\in A$such that$M< ...

2

\begin{align} \frac{y}{x^2+y^2} \, dx - \frac{x}{x^2+y^2} \, dy &=\frac{1}{(\frac{x}{y})^2+1}\frac{1}{y}dx+ \frac{1}{(\frac{x}{y})^2+1}(-\frac{x}{y^2}) dy\\ &=\frac{1}{(\frac{x}{y})^2+1}(\frac{1}{y}dx+(-\frac{x}{y^2})dy)\\ &=\frac{1}{(\frac{x}{y})^2+1}d(\frac{x}{y})\\ &= d\left(\tan^{-1}\left(\frac{x}{y}\right)\right) \end{align}

2

Your ratio is incorrect. You should have: $$\frac{c_n}{c_{n+1}}=\frac{2^{3n}(n+1)}{2^{3(n+1)}n}=\frac{2^{3n}(n+1)}{2^{3n+3}n}=\frac{n+1}{2^3n}$$

2

Some hints that might help you along the way: Start off by writing the expression as $$\oint_{C} \frac{z^4}{1 + z^8}dz$$ where $C$ is the contour. Now find the residues: $$1 + z_n^8 = 0, z_n \in C$$ Then the integral value is $$2 \pi i \sum Res_{z=z_n}f(z)$$

2

You might be interested in an alternative method using the Beta function. Note that the integrand is even, so \begin{align} \int_{-\infty}^{\infty}\frac{x^4}{1+x^8}dx &=2\int_{0}^{\infty}\frac{x^4}{1+x^8}dx\\ \end{align} Use the substitution $x=u^{1/8}$, then $dx=\frac{1}{8}u^{-7/8}du$. The integral becomes \begin{align} ...

2

Hint: For the radius of convergence, a useful criterion here is the "up to the first singularity" criterion.

2

What you are missing is that $$f''(a) = \oint_{\partial C_1(0)} \frac{\sin^2 z}{(z-a)^3}\,dz$$ only gives you the second and higher derivatives of $f$, you have $$(f(z) - \pi i\sin^2 z)'' \equiv 0.$$ So the difference is a polynomial of degree $\leqslant 1$, $$f(z) = \pi i\sin^2 z + a + bz.$$ The initial conditions yield $f(0) = a$ and $f'(0) = b$. ...

2

Because $f$ is continuous on $D$, $f$ is bounded near $p$ and hence $p$ is a removeable singularity (see here for instance: http://en.wikipedia.org/wiki/Removable_singularity). Because of this, Goursat's theorem and the Residue Theorem give the same result.

2

The assumption implies that $\operatorname{Im}(f)$ is zero on the bounding curve, and therefore vanishes on the interior by the maximum modulus principle. From this it follows that $f$ is constant. At worst you have a technical hiccup with your particular curve being only piecewise smooth.

2

I don't know if that is Cauchy's method, but one way to establish $g\equiv 0$ without differentiating is to use a functional equation: $$2\pi \cot (2\pi z) = \pi \frac{\cos^2(\pi z) - \sin^2(\pi z)}{\sin (\pi z)\cos (\pi z)} = \pi \cot (\pi z) + \pi\cot \left(\pi \left(z+\tfrac{1}{2}\right)\right),$$ and with $$s(z) = \frac{1}{z} + \sum_{n\neq 0} ... 1 Okay, I think I've got it. The solution to your integral is likely$$f(z)=\sqrt{\pi}e^{z^2}(\operatorname{sgn}(\operatorname{Re}(z)) - \operatorname{erf}(z) ).$$To see this, let first \operatorname{Im}(z)=:v> 0 and denote z=u+iv. First, note that$$\int_{-\infty}^\infty \frac{e^{-t^2}dt}{t+z}= e^{-z^2} \int_{-\infty+iv}^{\infty+iv} ...

1

The last step (before the limit $r\to1$ is taken) of the proof shows $$\sum_{n=0}^\infty nr^{2n}|a_n|^2\leq\frac{1}{r^2}$$ for each $r\in(0,1)$. Now for the left hand side we have $$\sum_{n=0}^Nnr^{2n}|a_n|^2\leq\sum_{n=0}^\infty nr^{2n}|a_n|^2\leq\frac{1}{r^2}$$ for each $N\in\mathbb N$, since the coefficients of the sum are non-negative. For fixed ...

1

Use that $$|\operatorname{Log}z|=|\log|z|+i\operatorname{arg}z|\le\sqrt{(\log|z|)^2+\pi^2}.$$ I am assuming that $-\pi<\operatorname{arg}z\le\pi$.

1

Note that $$\sqrt{(b-x)(x-a)}=\sqrt{\left(\frac{b-a}{2}\right)^2-\left(x-\frac{b+a}{2}\right)^2}\tag{1}$$ Therefore, we can substitute $$x=\frac{b+a}{2}-\frac{b-a}{2}\cos(\theta)\quad\text{so that}\quad\sqrt{(b-x)(x-a)}=\frac{b-a}{2}\sin(\theta)\tag{2}$$ Furthermore, by the definition of $y$, $$... 1 P_N(z) converges uniformly towards e^z over any compact subset of \mathbb{C}, and the Residue Theorem grants that:$$\oint_{|z|=2}\frac{dz}{P_N^3(z)-1}=2\pi i\cdot\sum_{\xi\in Z} \operatorname{Res}\left(\frac{1}{P_N^3(z)-1},z=\xi\right), where $Z$ is the set of zeroes of $P_N^3(z)-1$ that lie inside the disk $|z|\leq 2$. Since the zeroes of ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
# How do you factor y= x^3 + 9x^2 + 27x + 27 ?

Dec 2, 2015

$y = {x}^{3} + 9 {x}^{2} + 27 x + 27 = {\left(x + 3\right)}^{3}$

#### Explanation:

Notice that ${x}^{3}$ is a perfect cube and $27 = {3}^{3}$ is also a perfect cube.

So try ${\left(x + 3\right)}^{3}$ to see if the middle two terms match.

In general ${\left(a + b\right)}^{3} = {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + {b}^{3}$, so:

${\left(x + 3\right)}^{3} = \left({x}^{3}\right) + 3 \left({x}^{2}\right) \left(3\right) + 3 \left(x\right) \left({3}^{2}\right) + \left({3}^{3}\right)$

$= {x}^{3} + 9 {x}^{2} + 27 x + 27$


text:
# Generate points in circle uniformly

## Introduction

This chapter’s problem concerns uniformly generating a (potentially large) number of random points in a circle of a certain radius. Despite its simplicity the problem poses some unexpected challenges. We will discuss the best approach to this problem as well as one solution that many candidates provide which, whilst initially appearing correct actually fails in one crucial aspect (spoiler: it does not distribute points uniformly).

## Problem statement

Write a function that, given a circle of radius $$r$$ and centered at $$(x,y)$$ where $$r,x,y \in \mathcal{R}$$ returns a uniformly distributed point in the circle.

## Clarification Questions

What exactly does it mean for the point to be uniformly distributed?

It means that every point of the circle has the same probability of being picked/generated by the function.

## Discussion

Before discussing solutions it is worth mentioning that the fact that the circle is centered at $$(x,y)$$ makes very little difference and we can continue our discussion as if it were centered at $$(0,0)$$. This is the case because all the points we generate can then be translated to $$(x,y)$$ by simply adding $$x$$ and $$y$$ to the $$x$$-coordinate and $$y$$-coordinate of the generated point.

### Polar Coordinates - The wrong approach

Let’s start by discussing an intuituve, but ultimately incorrect, approach. One might think that in order to pick a point in the circle it is sufficient to

1. Pick a random angle $$\theta \in [0, 2\pi[$$

2. Pick a random radius $$\overline{r} \in [0,r]$$

3. Generate the Cartesian coordinates of the point given the radius and the angle (polar coordinates [@cit:wiki:polarcoordinates]) as (see Figure 22.1): $\begin{gathered} x=\overline{r}\sin(\theta) \\ y=\overline{r}\cos(\theta) \end{gathered}$

[fig:random_points_in_cirle:polar_coordinates]

Unfortunately, despite its appealing simplicity, this approach is wrong as it fails to produce points that are distributed uniformly in the circle. Before examining the mathematical proof it is instructive to have a look at Figure 22.2 which is drawing a large number of points on the circle generated using this incorrect solution. As you can see, the points are not generated uniformly as their density is higher towards the center. The bottom line is, do not use this solution in an interview. A possible matlab implementation of this buggy approach is shown at [list:random_points_in_circle:buggy]

Listing 1: Non-uniform random point in a circle generation using Matlab

function [px, py] = buggy_random_point(radius, x,y)
theta = rand()*2*pi;
px = r * sin(theta);
py = r *cos(theta);
endfunction

[fig:random_points_in_cirle:buggy]

### Loop approach

A good way to ensure that the point density is uniform across on the surface of the circle is to pick a point randomly in an enclosing square and make sure that we discard all the points that lie outside the circle. In other words, we keep asking for a random point $$(p_x=\text{rand()}, p_y=\text{rand()})$$ in the enclosing square until the following is true: $$p_x^2 + p_y^2 \leq r$$. In this way we are guaranteed to generate uniformly distributed points because we pick those points from a set of points that are already uniformly distributed in a square, and we exclude those which are not inside the circle. This method is also known as the exclusion method. Figure 22.3 depicts a large number of points generated with this method.

The downside here is that we might need to generate a number of points in the square before getting lucky and picking one lying in the circle. We need to make on average $$\approx 1.2732$$ tries before getting a point in the circle. This number is the ratio between the are of enclosing square and the area of the enclosed circle i.e. $$\frac{(2r)^2}{\pi r^2} = \frac{4}{\pi}$$.

[fig:random_points_in_cirle:loop]

A Matlab implementation of this approach is shown in Listing [list:random_points_in_circle:loop].

Listing 2: Random point in a circle generation using the exclusion method.

function [px, py, t] = random_point_loop(radius, x,y)
px = 100;
py = 100;
t=0;
while (px*px + py*py > 1)
signx = 2*randi([0,1])-1;
signy = 2*randi([0,1])-1;
t=t+1;
endwhile

endfunction

### Polar Coordinates - The right approach

In order for the points to be distributed uniformly it is necessary that the average distance between the points be the same regardless of how far they lie from the center of the circle. This means that, looking at the points generated on a circumference of radius $$2$$, there have to be twice as many points as the the number of points on a circumference of radius $$1$$. A circumference that is twice as long translates to needing twice as many points to maintain the same density. Another intuitive way to understand why simply picking a random angle and a random radius is not enough would be to think about having to distribute $$10$$ points at random on a circle of radius $$1$$ and $$2$$. It is clear that the circumference of radius $$2$$ would look emptier than the one with radius $$1$$ simply because there is more circumference to be filled but a constant amount of points.

The fundamental problem with the appraoch described in Section 22.3.1 is that the area of the circle is not uniformly covered. The random radius cuts through the area of the circle and this is the only parameter that affects how the points are going to be distributed across the full area of the circle. Therefore we should focus our attention how we can pick a better radius by making sure that larger radii are picked more often to accommodate for the larger area they define. In other words, we need to ensure that our random function for picking the radius takes the area of our circle into account.

Consider the area $$A$$ of a circle of radius $$r$$ i.e. $$A = \pi r^2$$. We can rearrange the formula so that $$r = \sqrt{\frac{A}{\pi}}$$. What this formula is really telling us is that the radius is proportional to $$\sqrt{A}$$. Now we have a way of choosing the radius that depends on the area of the circle. We can simply pick an area at random and then calculate the radius accordingly. This will make sure that the radius is picked taking into consideration the area of the circle. Figure 22.4 shows many points generated using this method. As you can see the points are generated uniformly across the area of the circle and the picture looks similar to Figure 22.3.

A C++ implementation of this method is shown in Listing [list:random_points_in_circle:sqrtcpp]. Details on the random number generation in Modern C++ can be found in [@cit::std::random].

Listing 3: C++ implementation of the function for generating a random point in a circle described in Section \ref{random_points_in_circle:sec:polar_sqrt}

auto generate_random_point_in_circle()
{
static std::uniform_real_distribution<double> dist_angle(0, 2 * M_PI);
const auto theta = dist_angle(rnd);
const auto x     = r * cos(theta);
const auto y     = r * sin(theta);
return std::make_pair{x + x_center, y + y_center};
}

[fig:random_points_in_cirle:polar_sqrt]

A Matlab implementation of this approach is shown in Listing [list:random_points_in_circle:sqrt].

Listing 4: Random point in a circle generation using polar coordinates and the $\approx \sqrt{A}$ dependency of the radius on the area of the circle.

function [px, py] = random_sqrt_area(radius, x,y)
r = sqrt(area/pi);
theta = rand()*2*pi;
px = r * sin(theta);
py = r *cos(theta);
endfunction

### Conclusion

For both the viable methods for generating random points withing a circle which we have discussed the time and space complexity is constant although the one presented in Section 22.3.3 will probably have better performance when included in a hot path i.e. in a loop for the generation of many random points.

All the code used to generate the Figures in this chapter is shown in Listing [list:random_points_in_circle:drivercode].

Listing 5: Matlab driver code for the generation of all figures in Chapter \ref{ch:random_points_in_circle}

function draw_points(n)
clf(1);
% n is the total number of points
px = zeros(1,n);
py = zeros(1,n);

tries = 0;
for i =0:n
%[x,y] = buggy_random_point(1,0,0);
% [x,y,t] = random_point_loop(1,0,0);
[x,y] = random_sqrt_area(1,0,0);
% tries = tries + t;
px(i+1) = x;
py(i+1) = y;
endfor

average = tries/n

% Plot a circle.
angles = linspace(0, 2*pi, n);
xCenter = 0;
yCenter = 0;
cx = radius * cos(angles) + xCenter;
cy = radius * sin(angles) + yCenter;
% Plot circle.
plot(cx, cy, 'b-', 'LineWidth', 2);
% Plot center.
hold on;
plot(xCenter, yCenter, 'k+', 'LineWidth', 2, 'MarkerSize', 16);
grid on;
axis equal;
xlabel('X', 'FontSize', 16);
ylabel('Y', 'FontSize', 16);

% Plot random points.
plot(px, py, 'r.', 'MarkerSize', 1);

rectangle('Position',[-1 -1 2 2], 'LineWidth',3,  'EdgeColor' , [0 .5 .5])

endfunction


text:
# The crystal field stabilization energy (CFSE) ${\text{[Fe(}}{{\text{H}}_2}{\text{O}}{{\text{)}}_6}]{\text{C}}{{\text{l}}_2}$ and ${{\text{K}}_2}{\text{[NiC}}{{\text{l}}_4}]$, respectively, are:a.) $\text{-0}\text{.4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{0}}}\text{ and -0}\text{.8}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$b.) $\text{-0}\text{.4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{0}}}\text{ and -1}\text{.2}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$c.) $\text{-2}\text{.4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{0}}}\text{ and -1}\text{.2}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$d.) $\text{-0}\text{.6}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{0}}}\text{ and -0}\text{.8}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$

Verified
140.7k+ views
Hint: Crystal field theory states that there is a difference of ${\text{10}}{{\text{D}}_{\text{q}}}$ between ${{\text{e}}_{\text{g}}}$ and ${{\text{t}}_{{\text{2g}}}}$ energy levels. When an electron goes into ${{\text{t}}_{{\text{2g}}}}$ which is a lower energy level, it stabilizes the system by an amount of $\text{-4}{{\text{D}}_{\text{q}}}$ and when an electron goes into ${{\text{e}}_{\text{g}}}$ level, it destabilizes the system by $\text{+6}{{\text{D}}_{\text{q}}}$.

The stability which results due to placing of a transition metal ion into a field that is caused by the set of ligands which surround it, is called Crystal field stabilization energy. In the complex, $\text{ }\!\![\!\!\text{ Fe(}{{\text{H}}_{2}}\text{O}{{\text{)}}_{6}}]\text{C}{{\text{l}}_{2}}$, Fe forms a +2 ion i.e. ${{\text{ }\!\![\!\!\text{ Fe(}{{\text{H}}_{2}}\text{O}{{\text{)}}_{6}}]}^{+2}}$.
The electronic configuration of Iron in +2 becomes $\text{ }\!\![\!\!\text{ Ar }\!\!]\!\!\text{ 3}{{\text{d}}^{3}}$.
is a weak field ligand and hence pairing of electrons does not occur. Electrons in ${{\text{t}}_{\text{2g}}}$ are given as $\text{t}_{\text{2g}}^{\text{2, 1, 1}}$.

Formula for calculating crystal field stabilization energy is $\Delta \text{ = no}\text{. of }{{\text{e}}^{-}}\text{ in }{{\text{t}}_{\text{2g}}}\text{ }\times \text{ (-0}\text{.4) + no}\text{. of }{{\text{e}}^{-}}\text{ in }{{\text{e}}_{g}}\text{ }\times \text{ (0}\text{.6)}$

Thus, for this complex - $\Delta \text{ = 4}{{\Delta }_{0}}\times \text{ (-0}\text{.4) + 2}{{\Delta }_{0}}\text{ }\times \text{ (0}\text{.6)}$
$\Delta \text{ = - 0}\text{.4}{{\Delta }_{0}}$
In the complex, ${{\text{K}}_{2}}\text{ }\!\![\!\!\text{ NiC}{{\text{l}}_{4}}]$ charge on Nickel is +2 as it forms ${{\text{ }\!\![\!\!\text{ NiC}{{\text{l}}_{4}}]}^{-2}}$.

The electronic configuration of Nickel in +2 is $\text{ }\!\![\!\!\text{ Ar }\!\!]\!\!\text{ 3}{{\text{d}}^{8}}$.

is also a weak field ligand and hence pairing does not occur and the complex formed has tetrahedral geometry. Electrons in ${{\text{e}}_{\text{g}}}$ are $\text{e}_{\text{g}}^{\text{2, 2}}$.
Putting these values in the formula we get, $\text{ }\!\!\Delta\!\!\text{ = -0}\text{.6 }\!\!\times\!\!\text{ 4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}\text{ + 0}\text{.4 }\!\!\times\!\!\text{ 4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$
$\text{ = -2}\text{.4}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}\text{ + 1}\text{.6}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$
$\text{ }\!\!\Delta\!\!\text{ = -0}\text{.8}{{\text{ }\!\!\Delta\!\!\text{ }}_{\text{t}}}$

So, the correct answer is “Option A”.

Note: Complexes which have a higher number of unpaired electrons are called as high spin complexes and the ones which have low number of unpaired electrons are called as low spin complexes. Most of the time, high spin complexes have weak field ligands and hence their splitting energy has lower value. Conversely, low spin complexes have strong field ligands and hence have higher value of splitting energy.


text:
## Package: sphinx / 3.2.1-1

Package Version Patches format
sphinx 3.2.1-1 3.0 (quilt)

### Patch series

view the series file
Patch File delta Description

tests/test_theming.py | 1 1 + 0 - 0 !
1 file changed, 1 insertion(+)

 disable a test that checks versions of js libraries

Sphinx.egg-info/requires.txt | 1 0 + 1 - 0 !
setup.py | 1 0 + 1 - 0 !
sphinx/search/da.py | 4 2 + 2 - 0 !
sphinx/search/de.py | 4 2 + 2 - 0 !
sphinx/search/es.py | 4 2 + 2 - 0 !
sphinx/search/fi.py | 4 2 + 2 - 0 !
sphinx/search/fr.py | 4 2 + 2 - 0 !
sphinx/search/hu.py | 4 2 + 2 - 0 !
sphinx/search/it.py | 4 2 + 2 - 0 !
sphinx/search/nl.py | 4 2 + 2 - 0 !
sphinx/search/no.py | 4 2 + 2 - 0 !
sphinx/search/pt.py | 4 2 + 2 - 0 !
sphinx/search/ro.py | 4 2 + 2 - 0 !
sphinx/search/ru.py | 4 2 + 2 - 0 !
sphinx/search/sv.py | 4 2 + 2 - 0 !
sphinx/search/tr.py | 4 2 + 2 - 0 !
16 files changed, 28 insertions(+), 30 deletions(-)

 replace snowballstemmer usage with pystemmer

The snowballstemmer module uses PyStemmer (bindings for libstemmer_c)
when it is available, and falls back to its own pure Python implementation
otherwise.

To avoid an extra dependency we can use PyStemmer directly.

tests/test_build_latex.py | 1 1 + 0 - 0 !
tests/test_build_linkcheck.py | 3 3 + 0 - 0 !
2 files changed, 4 insertions(+)

 skip linkcheck tests that require network access

doc/_themes/sphinx13/layout.html | 2 0 + 2 - 0 !
sphinx/themes/pyramid/layout.html | 2 0 + 2 - 0 !
2 files changed, 4 deletions(-)

 don't use google fonts to avoid privacy breach.

setup.py | 1 0 + 1 - 0 !
1 file changed, 1 deletion(-)

 remove setuptools from install_requires

It was added there because the entry points use pkg_resources module,
but it is provided in a separate package in Debian, pulling setuptools
itself is not needed.

doc/_templates/indexsidebar.html | 1 0 + 1 - 0 !
1 file changed, 1 deletion(-)

 do not use external image in the documentation template

doc/usage/builders/index.rst | 24 0 + 24 - 0 !
sphinx/application.py | 6 0 + 6 - 0 !
sphinx/builders/html/__init__.py | 1 0 + 1 - 0 !
3 files changed, 31 deletions(-)

 remove references to sphinxcontrib modules

They are not packaged yet, and when they are, we would want to avoid
dependency loops anyway.

tests/test_build_html.py | 3 3 + 0 - 0 !
1 file changed, 3 insertions(+)

 skip tests that require sphinxcontrib.jsmath module

doc/conf.py | 1 1 + 0 - 0 !
1 file changed, 1 insertion(+)

 use packaged mathjax (for our own documentation)

tests/test_api_translator.py | 2 2 + 0 - 0 !
1 file changed, 2 insertions(+)

 skip tests that require sphinxcontrib.serializinghtml module

setup.py | 14 8 + 6 - 0 !
1 file changed, 8 insertions(+), 6 deletions(-)

 move sphinxcontrib modules to extras_require

 use local intersphinx inventory


text:
# Impact of Fermi Surface Shape Engineering on Calculated Electronic Transport Properties of Bi-Sb-Te

## Article information

Korean J. Met. Mater.. 2021;59(1):54-60
Publication date (electronic) : 2021 January 5
doi : https://doi.org/10.3365/KJMM.2021.59.1.54
1Department of Materials Science and Engineering, University of Seoul, Seoul 02504 Republic of Korea
2Department of Materials Science and Engineering, Hongik University, Seoul 04066, Republic of Korea
김상일1, 임종찬2, 양희선,2, 김현식2,
1서울시립대학교, 신소재공학과
2홍익대학교, 신소재공학과
*Corresponding Author: Heesun Yang Tel: +82-2-320-3039, E-mail: hyang@hongik.ac.kr
*Corresponding Author: Hyun-Sik Kim Tel: +82-2-320-1698, E-mail: hyunsik.kim@hongik.ac.kr

- 임종찬: 학사과정, 김상일 · 양희선 · 김현식: 교수

Received 2020 October 7; Accepted 2020 November 14.

## Trans Abstract

Using thermoelectric refrigerators can address climate change because they do not utilize harmful greenhouse gases as refrigerants. To compete with current vapor compression cycle refrigerators, the thermoelectric performance of materials needs to be improved. However, improving thermoelectric performance is challenging because of the trade-off relationship between the Seebeck coefficient and electrical conductivity. Here, we demonstrate that decreasing conductivity effective mass by engineering the shape of the Fermi surface pocket (non-parabolicity factor) can decouple electrical conductivity from the Seebeck coefficient. The effect of engineering the non-parabolicity factor was shown by calculating the electronic transport properties of a state-of-the-art Bi-Sb-Te ingot via two-band model with varying non-parabolicity. The power factor (the product of the Seebeck coefficient squared and electrical conductivity) was calculated to be improved because of enhanced electrical conductivity, with an approximately constant Seebeck coefficient, using a non-parabolicity factor other than unity. Engineering the non-parabolicity factor to achieve lighter conductivity effective mass can improve the electronic transport properties of thermoelectric materials because it only improves electrical conductivity without decreasing the Seebeck coefficient (which is directly proportional to the band mass of a single Fermi surface pocket and not to the conductivity effective mass). Theoretically, it is demonstrated that a thermoelectric figure-of-merit zT higher than 1.3 can be achieved with a Bi-Sb-Te ingot if the non-parabolicity factor is engineered to be 0.2. Engineering the non-parabolicity factor is another effective band engineering approach, similar to band convergence, to achieve an effective improvement in power factor.

## 1. Introduction

Replacing current vapor-compression refrigerators with thermoelectric refrigerators can reduce the rate of climate change. Unlike vapor-compression refrigerators, which utilize harmful greenhouse gases as refrigerants, thermoelectric refrigerators do not require any greenhouse gases to operate. Thermoelectric refrigerators can produce cooling using a two-dimensional array of alternating p- and n-type thermoelectric materials connected in series via the Peltier effect [1]. The efficiency of thermoelectric refrigerators depends on the performance of the thermoelectric materials, which is characterized by a dimensionless figure-of-merit, zT = S2σT/κ, where S, σ, κ, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity and the absolute temperature, respectively. However, currently the zT of known thermoelectric materials is not high enough to compete with vapor-compression refrigerators.

Considerable efforts have been devoted to improving zT. Band engineering is frequently adopted to maximize S2σ (power factor). Enhancing the power factor is challenging because of the trade-off between S and σ. If S is increased by making the band mass of a single Fermi surface pocket (mb*) heavier, the relevant σ will be decreased (band flattening). However, band convergence is an effective approach to enhance the power factor, as it enhances S (via increasing valley degeneracy (Nv) with a constant mb*) without deteriorating σ [2-4]. Identifying a band parameter that is not coupled to both S and σ is key to successful power factor improvement.

Although it has often been neglected to avoid complication, the types of band masses that determine S and σ are different. While S is directly proportional to the product of Nv2/3 and mb*, which is equivalent to density-of-states effective mass (md* = Nv2/3 mb*), σ is inversely proportional to conductivity effective mass (mc*) [5-7]. For a spherical Fermi surface pocket, its mc* is identical to mb*. However, for an ellipsoidal pocket, the mc* becomes smaller than the mb* [8]. Therefore, controlling Fermi surface nonparabolicity (K) can decouple σ from the S.

Here, we theoretically demonstrate that control of K is another effective band engineering approach to improve zT. The effects of engineering K on each of the electronic transport properties are calculated using a two-band model, where fitted band parameters of Bi-Sb-Te (the best candidate for thermoelectric refrigerators) are used. We find that K engineering enhances the power factor by increasing σ while keeping S constant.

## 2. Experimental Procedure

The two-band model (solutions of the Boltzmann transport equations) was fitted to the experimental S, and σ of a state-of-the-art p-type Bi-Sb-Te ingot measured as a function of T to obtain the band parameters (mb* and deformation potential, Ξ) of the valence and conduction bands in the Bi-Sb-Te (spherical Fermi surface pockets assumed) [9]. For 300 K, 0.3, and 0.6 me (electron rest mass) were adopted for mb*, corresponding to the valence and conduction bands, respectively. The Nv of both bands were set to be six, and an energy gap (Eg) of 0.19 eV was adopted in the two-band model [10-12]. It was assumed that acoustic phonon scattering limited the carrier mobility of both bands in the Bi-Sb-Te. Details on the two-band model fitting can be found in our previous works [4,13,14,15]. While keeping the aforementioned parameters (mb*, Ξ, Nv, and Eg) constant, electronic transport properties were calculated for different valence band K (= 0.2, 0.5, 1, 2.5, and 4) via the two-band model (K = 1 was assumed for the conduction band). Here, each electronic transport property was calculated as a function of the chemical carrier concentration (n) for T = 300 K. The effect of K engineering was evaluated by comparing the electronic transport properties calculated for K ≠ 1 to those obtained for K = 1 (where Fermi surface pockets are spherical).

## 3. Results and Discussion

### 3.1. Decoupling mc* from mb* with K ≠ 1

To theoretically demonstrate how changing the shape of Fermi surface pockets affects the thermoelectric performance of p-type Bi-Sb-Te, its electronic transport properties were calculated using varying valence band K with constant band parameters (mb*, Ξ, Nv, and Eg) under the acoustic phonon scattering assumption. The band parameters of Bi-Sb-Te were first obtained via the two-band model while assuming its Fermi surface pockets were spherical (K = 1 for both bands). The K is defined as a ratio of mb* along the longitudinal ellipsoid direction (mb,∥*) to that along the transverse ellipsoid direction (mb,⊥*;).

(1) K=mb, || *mb, *

In terms of K, the degree of decoupling mc* from mb* is determined as below.

(2) mc *mb*=3K2/32K+1

When K in Eq. (2) equals to unity, the mc* and mb* are identical, but for K ≠ 1, the mc* becomes lighter than mb* as presented in Fig 1(a) [8]. The solid black line in Fig 1(a) represents mc*/mb* as a function of K. The symbols indicate mc*/mb* for each K in the range from 0.2 to 4. The specific Ks were chosen so that their corresponding mc*/mb* ratios in the valence band were all different. Engineering K to be less than 1 (mb,∥* being heavier than mb,⊥*;) is more effective for decoupling mc* from mb* than changing K to be larger than 1, according to Fig 1(a). In other words, decoupling σ from S becomes possible for a K other than 1.

(a) Decoupling mc* from mb* as a function of K, (b) drift mobility (μd) as a function of n for different K at 300 K.

The μd of Bi-Sb-Te was calculated as a function of n for varying valence band K (0.2, 0.5, 1, 2.5, and 4) to evaluate the effect of engineering K in drift mobility (μd) as shown in Fig 1(b). Commonly, the μd was described to be inversely proportional to mb*5/2 assuming K = 1. However, the μd is more accurately described by a product of mc*-1 and mb*-3/2 for K ≠ 1, as given in Eq. (3).

(3) μd1mc*×mb*3/2

If the mc* in Eq. (3) is expressed in terms of mb* and K (as in Eq. (2)), Eq. (3) becomes

(4) μd1mb*5/2×(mc*mb*)-1 =1mb*5/2×(2K+13K2/3)

According to Eq. (4), the μd is inversely proportional to the mc*/mb* ratios, which depend on K (Fig 1(a)). In Fig 1(b), the μd calculated with K = 1 is the lowest for all ranges of n (green triangle and line). In contrast, the μd calculated with K = 0.2 is the highest (orange square and line). Because the mc*/mb* ratio for K = 0.2 (mc*/mb* = 0.73) is smaller than that for K = 1 (mc*/mb* = 1), the μd for K = 0.2 is calculated to be higher than that for K = 1. For other K, the corresponding μd also depends on the reciprocal of the mc*/mb* ratio (Eq. (4)). Therefore, a higher μd is expected with a K that gives a smaller mc*/mb* ratio.

### 3.2. Effect of varying mc*/mb* on σ via K engineering

To examine the effect of engineering K on σ, the σ of Bi-Sb-Te was calculated for varying valence band K (0.2 – 4), as shown in Fig 2(a). Because the σ is directly proportional to μd as given below (e is the electric charge),

(a) σ, (b) electronic thermal conductivity (κe), and Lorenz number (L) in the inset of (b) calculated for different K as a function of n at 300 K.

(5) σ = neμd

a similar trend was observed in μd for different K (Fig 1(b)), and this is also expected for σ. According to Fig 2(a), σ for K other than unity were calculated to be higher than that for K = 1. Again, it is the value of mc*/mb* determined from a given K, and not the K itself that is correlated to σ. The mc*/mb* ratios which correspond to K (0.2 – 4) are provided in Table 1 (Eq. (2)).

The mc*/mb* ratios and corresponding K.

From Table 1, the K arranged in order of decreasing mc*/mb* are: 1, 0.5, 2.5, 4, and 0.2. The calculated σ is also increased with decreasing mc*/mb* (K: 1 → 0.5 → 2.5 → 4 → 0.2) as shown in Fig 2(a). Engineering K to reduce mc*/mb* was found to be beneficial to increase σ.

A minor difference between σ and μd is that σ only depends on mc*. Because n is directly proportional to mb*3/2, σ becomes inversely proportional to mc* only as below.

(6) σnμd(mb*3/2)(1mc*×mb*3/2)=1mc*

In other words, σ is the electronic transport parameter, which can be separated from S most effectively with relevant K engineering.

The effect of varying mc*/mb* (via K engineering) on electronic thermal conductivity (κe) was also studied. The heat-carrying carrier conductions (κe) contribute to the total κ along with heat conductions via lattice vibrations (lattice thermal conductivity, κl) and bipolar conductions (bipolar thermal conductivity, κbp) as in Eq. (7).

(7) κ=κl+κe+κbp

Among κl, κe, and κbp, the κe is the most directly related to σ via the Wiedemann-Franz law, as below, where L is the Lorenz number [16].

(8) κe=LσT

This is because the L is independent of both mb* and mc*, the κe in Eq. (8) is directly proportional to σ at constant T. Therefore, a higher κe is expected with a K that corresponds to a lower mc*/mb* like σ, but the higher κe decreases zT. Fig 2(b) shows the κe for varying K as a function of n. The κe with K = 0.2 (the lowest mc*/mb*, orange square and line) was calculated to be the largest, and that with K = 1 (the maximum mc*/mb*, green triangle, and line) was calculated to be the smallest. The inset in Fig 2(b) shows the L calculated for varying K as a function of n. Because the L is not a function of mc*, the calculated L are all identical for different K. The discrepancy among κe for different K is calculated to increase at higher n. For the range of n where the thermoelectric performance of most materials is optimized (~1019 cm-3), the increase in κe due to K engineering is negligible. Although decreasing mc*/mb* increases both κe and σ, the improvement in σ outweighs the increase in κe.

### 3.3. Effect of varying mc*/mb* on S via K engineering

To examine how varying mc*/mb* (via K engineering) affects S, the S of Bi-Sb-Te was calculated for varying valence band K (0.2 – 4) at 300 K, as shown in Fig 3(a). The S for different K are almost identical for n≥ ~5 × 1018 cm-3, but they are different for n < ~5 × 1018 cm-3. Although S calculated from a single band is independent of mc* (but dependent on md* instead), the overall S calculated from two bands does depend on mc* via σ contributions from each band. The overall S is calculated by Eq. (9) below, where Si and σi are the partial contributions from each band (i = VB (valence band), CB (conduction band)).

(a) S, and (b) bipolar thermal conductivity (κbp) calculated for different K as a function of n at 300 K.

(9) S=σVBSVB+σCBSCBσVB+σCB

When σVB and σCB in Eq. (9) are comparable at low n, the difference in σVB due to varying valence band K is well captured in the overall S. For example, the σVB for K = 0.2 is the highest among K in the range from 0.2 to 4 (Fig 2(a)). Hence, the S for K = 0.2 is calculated to be the highest for n < ~5 × 1018 cm-3, as in Fig 3(a) (SVB, SCB, and σCB being constant with respect to K). In Fig 2(a), we simply stated that the overall σ increased because of the decrease in the mc*/mb* ratio. To be more specific, the increase in the overall σ was due to the increase in σVB originating from a decrease in the valence band mc*/mb* ratio (K of the conduction band was fixed). However, once the σVB becomes much greater than σCB at high n, the S in Eq. (9) converges to SVB, as shown below.

(10) SσVBSVBσVBSVB

Because SVB is independent of mc*, the S is calculated to be approximately invariant with respect to K at n ≥ ~5 × 1018 cm3, as in Fig 3(a). Given that the optimum properties of thermoelectric materials are frequently realized at n ~ 1 × 1019cm-3, it can be concluded that the S is not affected by varying K.

The effect of varying mc*/mb* (via K engineering) on κbp was also examined. Unlike κe, both Si and σi (i = VB, CB) are required to calculate κbp, as shown below.

(11) κbpσVBσCBσVB+σCB(SVB-SCB)2T

In Eq. (11), the SVB and SCB are independent of varying K, as they do not depend on mc*. And because we assume that the K of the conduction band is unity for simplicity, only the σVB is changed with varying K. Therefore, it is expected that the κbp with a low mc*/mb* will be high because of high σVB, as in the case of S. However as calculated in Fig 3(b), the discrepancy in κbp due to different K is negligible for all n. Hence, varying K does not affect κbp.

### 3.4. Effect of varying mc*/mb* on power factor via K engineering

To demonstrate the effect of varying mc*/mb* (via K engineering) on the power factor, the power factor of Bi-Sb-Te was calculated for varying valence band K (0.2 – 4) at 300 K as shown in Fig 4(a). Since suppressing mc*/mb* is beneficial to σ (Fig 2(a)), while causing a negligible change in S (Fig 3(a)), an improvement in the power factor is expected. According to Fig 4(a), the highest power factor was calculated for the K = 0.2 (orange square and line), which corresponds to the lowest mc*/mb* (Table 1), and the lowest power factor was calculated for K = 1 (green triangle and line) which corresponds to the maximum mc*/mb*. The calculated power factor increased with decreasing mc*/mb* (K: 1 → 0.5 → 2.5 → 4 → 0.2), as observed in σ in Fig 2(a). Therefore, engineering K to be other than unity will always improve the power factor.

(a) Power factor and (b) zT calculated for different K as a function of n at 300 K. κl of 0.8 W/m-K was employed for the zT calculation.

Lastly, to investigate the effect of varying mc*/mb* (via K engineering) on zT, the zT of Bi-Sb-Te was calculated for varying valence band K (0.2 – 4) at 300 K as shown in Fig 4(b). For the zT calculation, κl is required in addition to the κe and κbp calculated in Fig 2(b) and 3(b), respectively. Because the κl cannot be calculated using the two-band model, the experimental κl of a state-of-the-art Bi-Sb-Te ingot was adopted from the literature (κl = 0.8 W/m-K at 300 K) [9]. The highest zT was calculated for K = 0.2 for all n, as shown in Fig 4(b). Its optimum zT was achieved near n ~ 1019 cm-3. The sharp decrease in zT for n > 1019 cm-3 is attributed to the increasing κe and κbp at high n. This confirms that engineering K to decrease mc*/mb* is another effective approach to improve zT.

Engineering K is another effective band engineering approach to improve the power factor of thermoelectric materials. Making K other than unity decouples mc* from mb*, which enables us to break the trade-off between S and σ. As mc*/mb* decreases, σ is improved without deteriorating S. Although κe is also increased with decreasing mc*/mb*, the κe increase is negligible compared to the σ increase. Therefore, zT improvement is guaranteed if the mc*/mb* is reduced by engineering K.

We assumed that the K of both the valence and conduction bands were unity when obtaining the band parameters of the state-of-the-art Bi-Sb-Te ingot from the two-band model fitting. However, it has been reported that the valence band K of Bi-Sb-Te was optically determined to be approximately 2 [17]. Consequently, the determined band parameters of Bi-Sb-Te (mb*, Ξ) with K = 1 can be marginally different from what they should be (with K = 2). The discrepancy between the fitted Ξ and the actual Ξ will be higher than that expected between the fitted and actual mb* because K only affects mc*-related parameters. Nonetheless, the fact that engineering K improves power factor remains intact because the beneficial effect of decreasing mc*/mb* on power factor applies to any set of band parameters.

Both decreasing mc*/mb* by engineering K and decreasing Ξ have similar impacts on μd as the μd is inversely proportional to the product of Ξ2, mc* and mb*3/2. In addition, because S is independent of Ξ, the effect of suppressing Ξ on electronic transport properties is almost identical to the effect of engineering K on electronic transport properties. Therefore, an additional optical study that measures mb* at different principal axes is required to confirm that an improvement in power factor originates from K engineering.

No experimental approach to engineer only K has been suggested yet. Although alloying is known to alter K, it also changes other band parameters at the same time, making the evaluation of the effect of K change difficult [17]. As an alternative, materials with innately low mc*/mb* have been theoretically sought as potential candidates for new high-performance thermoelectric materials [18].

Although the effect of engineering K was investigated using narrow band gap Bi-Sb-Te, its effect will be even greater for large band gap materials since the potential increase in κbp due to K engineering will be minimized because of their substantially low κbp [19]. Moreover, since we know that K other than unity can significantly alter the electronic transport properties of thermoelectric materials, accurate K input (obtained via measurements or simulations) is important to accurately determine materials’ band parameters via band modeling.

## 4. Conclusions

A new band engineering approach that by-passes the trade-off between S and σ has been suggested. Engineering the shape of Fermi surface pockets (K, non-parabolicity factor) to be ellipsoidal was found to decouple conductivity effective mass (mc*) from the band mass of a single Fermi pocket (mb*). When K is engineered to lower the mc*/mb* ratio, the σ is calculated to increase while S remains constant. It was calculated that the thermoelectric figure-of-merit (zT) of a Bi-Sb-Te ingot can be improved to higher than 1.3 if the nonparabolicity factor is engineered to be 0.2. Although alloying is known to change K, it also simultaneously alters other band parameters. However, it is suspected that alloying, combined with carefully designed doping, might only change K so that it improves the power factor of materials effectively.

## Acknowledgements

This work was financially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education (No. 2015R1A6A1A03031833, and NRF-2019R1C1C1005632).

## References

1. Snyder G. J., Toberer E. S.. Nat. Mater 7:105. 2008;
2. Kim H.-S., Heinz N. A., Gibbs Z. M., Tang Y., Kang S. D., Snyder G. J.. Mater. Today 20:452. 2017;
3. Kim H.-S.. J. Korean Ceram. Soc 56:325. 2019;
4. Lee K. H., Kim S.-i., Kim H.-S., Kim S. W.. ACS Appl. Energy Mater 3:2214. 2020;
5. Wang H., Pei Y., LaLonde A. D., Snyder G. J.. Proc. Natl. Acad. Sci. U.S.A 109:9705. 2012;
6. Dixon J. R., Riedl H. R.. Phys. Rev. 138:A873. 1965;
7. Ravich Y. I., Efimova B. A., Smirnov I. A.. Semiconducting Lead Chalcogenides Springer US; New York: 1970.
8. Wang H.. High Temperature Transport Properties of Lead Chalcogenides and Their Alloys in Ph.D. Thesis. California Institute of Technology; Pasadena, U.S.A: 2014.
9. Poudel B., Hao Q., Ma Y., Lan Y., Minnich A., Yu B., Yan X., Wang D., Muto A., Vashaee D., Chen X., Liu J., Dresselhaus M. S., Chen G., Ren Z.. Science 320:634. 2008;
10. Jenkins J. O., Rayne J. A., Ure R. W.. Phys. Rev. B 5:3171. 1972;
11. Greenaway D. L., Harbeke G.. J. Phys. Chem. Solids 26:1585. 1965;
12. Sehr R., Testardi L. R.. J. Phys. Chem. Solids 23:1219. 1962;
13. Kim H.-S., Lee K. H., Yoo J., Shin W. H., Roh J. W., Hwang J.-Y., Kim S. W., Kim S.-i.. J. Alloys Compd 741:869. 2018;
14. Cho H.-J., Kim H.-S., Sohn W.-H., Kim S.-i.. Korean J. Met. Mater 58:439. 2020;
15. Cho H.-J., Kim H.-S., Kim S.-i.. Korean J. Met. Mater 57:673. 2019;
16. Kim H.-S., Gibbs Z. M., Tang Y., Wang H., Snyder G. J.. APL Mater 3:041506. 2015;
17. Stordeur M., Stölzer M., Sobotta H., Riede V.. Phys. Status Solidi B 150:165. 1988;
18. Gibbs Z. M., Ricci F., Li G., Zhu H., Persson K., Ceder G., Hautier G., Jain A., Snyder G. J.. Npj Comput. Mater 3:8. 2017;
19. Kim G., Shin H., Lee J., Lee W.. Met. Mater. Int 10.1007/s12540-019-00279-2. 2020;

## Article information Continued

### Fig. 1.

(a) Decoupling mc* from mb* as a function of K, (b) drift mobility (μd) as a function of n for different K at 300 K.

### Fig. 2.

(a) σ, (b) electronic thermal conductivity (κe), and Lorenz number (L) in the inset of (b) calculated for different K as a function of n at 300 K.

### Fig. 3.

(a) S, and (b) bipolar thermal conductivity (κbp) calculated for different K as a function of n at 300 K.

### Fig. 4.

(a) Power factor and (b) zT calculated for different K as a function of n at 300 K. κl of 0.8 W/m-K was employed for the zT calculation.

### Table 1.

The mc*/mb* ratios and corresponding K.

mc*/mb* K
1 1
0.94 0.5
0.92 2.5
0.83 4
0.73 0.2


text:
# Define Fluid-Structure-Interactions

Virtual Wind Tunnel for ultraFluidX supports the P-FSI approach available in AcuSolve (based on the eigenmodes of the structure). Each part can be defined as flexible and the corresponding op2-file (containing the eigenmodes of the structure) can be assigned to each part individually. OptiStruct can be used to compute the eigenmodes up front.

The eigenmodes computed in OptiStruct are based on the location and orientation of the structural model. If the location of the model in Virtual Wind Tunnel for ultraFluidX is not the same as the one used to compute the eigenmodes, the P-FSI job does not work correctly.

Certain functions are not supported yet via the Virtual Wind Tunnel for ultraFluidX workspace. If any changes in the .vpar file were required, the Virtual Wind Tunnel for ultraFluidX job has to be submitted manually via the command line. The .vpar file can be generated by clicking Export in the Run dialog.
• Change the number of used modes (use less modes than present) by entering the following line to the .vpar file: flex_comp_1_modes = 6
The number of modes considered for the simulation is set by default to the number of present modes in the eigenmode file.
This will cause the usage of the first six eigenmodes of the eigenmode file corresponding to flexible component 1. If the number of modes is set to a higher value than the present modes in the op2 file, then the maximum number of modes is chosen. The pressure loads onto the parts are ramped over several time steps, to avoid a “shock” of the pressure loads onto the structure. Per default the ramping is linear between the time steps eight and 16 from zero to the full value. This yields the following default entries in the .vpar file:
mult_lower 	= 8		// loads have no effect on the structure till here
mult_upper	= 16		// loads have full effect on the structure from here
• Change the default time step interval for ramping in case it is to large or if it starts too late by editing the .vpar file (the .vpar file can be generated by clicking Export in the Run dialog).
• Monitor results, for example displacement of the structure over time, by defining monitor points.
For each monitor point, add the following lines to the .vpar file:
time_history_1_name 	= “point name”	  // the name of monitor point
time_history_1_coord	= ( 3.1, 0.0, 0.223 )	  // the (x,y,z) coordinates
time_history_1_output	= ( “x_displacement” )	  // output type for the report

Define a second or third monitor point by replacing the one in the above commands by two or three. Multiple outputs can be chosen, separated by comma. The possible candidates are x_displacement, y_displacement, z_displacement, displacement, x_velocity, y_velocity, z_velocity, velocity and pressure. The displacement options refer to the mesh displacement at the given coordinate point.

• Create animations by editing the .vpar file:
animation		= True	      // activate animation
saved_sates	= 50	     // keep the last 50 output files
outputFrequency	= 2	     // write output at every 2nd iteration
Activating the animation mode will generate one animation of a cross section through the entire body with the velocity magnitude as contour plot. Additionally, for each flexible part, three separate animations will be generated: one animation for the displacement (surface contour plot), one for the pressure (surface contour plot) and one animation showing a cross section cut with the velocity magnitude.


text:
# Difference between revisions of "2018 AMC 10B Problems/Problem 23"

## Problem

How many ordered pairs $(a, b)$ of positive integers satisfy the equation $$a\cdot b + 63 = 20\cdot \text{lcm}(a, b) + 12\cdot\text{gcd}(a,b),$$ where $\text{gcd}(a,b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ denotes their least common multiple?

$\textbf{(A)} \text{ 0} \qquad \textbf{(B)} \text{ 2} \qquad \textbf{(C)} \text{ 4} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ 8}$

## Solution

Let $x = lcm(a, b)$, and $y = gcd(a, b)$. Therefore, $a\cdot b = lcm(a, b)\cdot gcd(a, b) = x\cdot y$. Thus, the equation becomes

$$x\cdot y + 63 = 20x + 12y$$ $$x\cdot y - 20x - 12y + 63 = 0$$

Using Simon's Favorite Factoring Trick, we rewrite this equation as

$$(x - 12)(y - 20) - 240 + 63 = 0$$ $$(x - 12)(y - 20) = 177$$

Since $177 = 3\cdot 59$ and $x > y$, we have $x - 12 = 59$ and $y - 20 = 3$, or $x - 12 = 177$ and $y - 20 = 1$. This gives us the solutions $(71, 23)$ and $(189, 21)$. Since the $\text{GCD}$ must be a divisor of the $\text{LCM}$, the first pair does not work. Assume $a>b$. We must have $a = 21 \cdot 9$ and $b = 21$, and we could then have $a, so there are $\boxed{2}$ solutions. (awesomeag)

Edited by IronicNinja, Firebolt360, and mprincess0229~


text:
# Period in simple harmonic motion [closed]

I would like to know why period in simple harmonic motion does not depend on the amplitude of oscillation.

## closed as off-topic by stafusa, Jon Custer, Daniel Griscom, Cosmas Zachos, Emilio PisantyNov 8 '17 at 13:21

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – stafusa, Jon Custer, Daniel Griscom, Cosmas Zachos
If this question can be reworded to fit the rules in the help center, please edit the question.

Simple harmonic motion obeys the differential equation $\ddot{x}+\omega^2 x=0$ for a constant $\omega>0$ with the units of frequency. Since this is a homogeneous linear equation, its solutions are closed under multiplication by constants. Doing this changes the amplitude, but not the frequency.

The amplitude increases with the speed. The period is proportional to the amplitude, but inverse proportional to the speed. So, because the amplitude is proportional to the speed, their influences on the period cancel out.

In the general case, there is a body connected with a spring, as in the picture:
Let's assume that there is no friction. The restoring force is the only force that is acting on the body and when we apply Newton's second law we get:

Then we get:

## $ma + kx = 0$

This equation can take the following form:

## $m\frac{d^2x}{dt^2} + kx = 0$ $(a = \frac{d^2x}{dt^2})$

As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t. This is just a more complicated way to say that acceleration shows the rate of change of position over time. Calculous is required to answer your question.

## $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

Now we can substitute $\frac{m}{k} = \omega_o^2$ where is angular frequency. There is one more formula regarding angular frequency: $T = \frac{2\pi}{\omega_o}$ and finaly we get that $T = 2\pi\sqrt{\frac{k}{m}}$. You can see that the period doesn't depend on the amplitude beacuse it is not connected with it over the whole proof.

Best regards,
Sinestro