text:
• Corpus ID: 204743793

# Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space

@article{Luo2019KoebeCA,
title={Koebe conjecture and the Weyl problem for convex surfaces in hyperbolic 3-space},
author={Feng Luo and Tianqi Wu},
journal={arXiv: Geometric Topology},
year={2019}
}
• Published 17 October 2019
• Mathematics
• arXiv: Geometric Topology
We prove that the Koebe circle domain conjecture is equivalent to the Weyl type problem that every complete hyperbolic surface of genus zero is isometric to the boundary of the hyperbolic convex hull of the complement of a circle domain. It provides a new way to approach the Koebe's conjecture using convex geometry. Combining our result with the work of He-Schramm on the Koebe conjecture, one establishes that every simply connected non-compact polyhedral surface is discrete conformal to the…
16 Citations

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## References

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### Polyhedral hyperbolic metrics on surfaces

Let S be a topologically finite surface, and g be a hyperbolic metric on S with a finite number of conical singularities of positive singular curvature, cusps and complete ends of infinite area. We

### On the classification of noncompact surfaces

3 in ?3.) In addition we prove the theorem, which so far as I know is new, that conversely, every nested triple of totally disconnected, compact, separable spaces occurs as the ideal boundary of some

### Fixed points, Koebe uniformization and circle packings

• Mathematics
• 1993
A domain in the Riemann sphere $$\hat{\mathbb{C}}$$ is called a circle domain if every connected component of its boundary is either a circle or a point. In 1908, P. Koebe [Ko1] posed the following

### Convergence of discrete conformal geometry and computation of uniformization maps

• Mathematics
Asian Journal of Mathematics
• 2019
The classical uniformization theorem of Poincaré and Koebe states that any simply connected surface with a Riemannian metric is conformally diffeomorphic to the Riemann sphere, or the complex plane

### A discrete uniformization theorem for polyhedral surfaces II

• Mathematics
Journal of Differential Geometry
• 2018
A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a

### Transboundary extremal length

We introduce two basic notions, ‘transboundary extremal length’ and ‘fat sets’, and apply these concepts to the theory of conformal uniformization of multiply connected planar domains. A new short

### Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space

The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in

### Discrete conformal maps and ideal hyperbolic polyhedra

• Mathematics
• 2015
We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic

### A discrete uniformization theorem for polyhedral surfaces

• Mathematics
Journal of Differential Geometry
• 2018
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which gen-eralizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete

### Discrete Conformal Deformation: Algorithm and Experiments

• Jian Sun
• Mathematics, Computer Science
SIAM J. Imaging Sci.
• 2015
A definition of discrete conformality for triangulated surfaces with flat cone metrics and an algorithm for solving the problem of prescribing curvature to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature.


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# BIS - Principles for Sound Stress Testing Practices and Supervision

The depth and duration of the financial crisis has led many banks and supervisory authorities to question whether stress testing practices were sufficient prior to the crisis and whether they were adequate to cope with rapidly changing circumstances. In particular, not only was the crisis far more severe in many respects than was indicated by banks' stress testing results, but it was possibly compounded by weaknesses in stress testing practices in reaction to the unfolding events. Even as the crisis is not over yet there are already lessons for banks and supervisors emerging from this episode. Stress testing is an important risk management tool that is used by banks as part of their internal risk management and, through the Basel II capital adequacy framework, is promoted by supervisors. Stress testing alerts bank management to adverse unexpected outcomes related to a variety of risks and provides an indication of how much capital might be needed to absorb losses should large shocks occur. While stress tests provide an indication of the appropriate level of capital necessary to endure deteriorating economic conditions, a bank alternatively may employ other actions in order to help mitigate increasing levels of risk. Stress testing is a tool that supplements other risk management approaches and measures. It plays a particularly important role in:

• providing forward-looking assessments of risk;
• overcoming limitations of models and historical data;
• supporting internal and external communication;
• feeding into capital and liquidity planning procedures;
• informing the setting of a banks’ risk tolerance; and
• facilitating the development of risk mitigation or contingency plans across a range of stressed conditions.

Stress testing is especially important after long periods of benign economic and financial conditions, when fading memory of negative conditions can lead to complacency and the underpricing of risk. It is also a key risk management tool during periods of expansion, when innovation leads to new products that grow rapidly and for which limited or no loss data is available.

Pillar 1 (minimum capital requirements) of the Basel II framework requires banks using the Internal Models Approach to determine market risk capital to have in place a rigorous programme of stress testing. Similarly, banks using the advanced and foundation internal ratings-based (IRB) approaches for credit risk are required to conduct credit risk stress tests to assess the robustness of their internal capital assessments and the capital cushions above the regulatory minimum. Basel II also requires that, at a minimum, banks subject their credit portfolios in the banking book to stress tests. Recent analysis has concluded that banks’ stress tests did not produce large loss numbers in relation to their capital buffers going into the crisis or their actual loss experience. Further, banks’ firm-wide stress tests should have included more severe scenarios than the ones used in order to produce results more in line with the actual stresses that were observed.

The Basel Committee has engaged with the industry in examining stress testing practices over this period and this paper is the result of that examination. Notwithstanding the ongoing evolution of the crisis and future lessons that may emerge, this paper assesses stress testing practices during the crisis. Based on that assessment and in an effort to improve practices, it develops sound principles for banks and supervisors. The principles cover the overall objectives, governance, design and implementation of stress testing programmes as well as issues related to stress testing of individual risks and products.

The recommendations are aimed at deepening and strengthening banks’ stress testing practices and supervisory assessment of these practices. By itself, stress testing cannot address all risk management weaknesses, but as part of a comprehensive approach, it has a leading role to play in strengthening bank corporate governance and the resilience of individual banks and the financial system.

A stress test is commonly described as the evaluation of a bank’s financial position under a severe but plausible scenario to assist in decision making within the bank. The term “stress testing” is also used to refer not only to the mechanics of applying specific individual tests, but also to the wider environment within which the tests are developed, evaluated and used within the decision-making process. In this paper, we use the term “stress testing” in this wider sense.

Download the document - Principles for Sound Stress Testing Practices and Supervision

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Courses

# Small Oscillations - Lagrangian and Hamiltonian Equations, Classical Mechanics, CSIR-NET Physical Sc Physics Notes | EduRev

## Physics for IIT JAM, UGC - NET, CSIR NET

Created by: Akhilesh Thakur

## Physics : Small Oscillations - Lagrangian and Hamiltonian Equations, Classical Mechanics, CSIR-NET Physical Sc Physics Notes | EduRev

The document Small Oscillations - Lagrangian and Hamiltonian Equations, Classical Mechanics, CSIR-NET Physical Sc Physics Notes | EduRev is a part of the Physics Course Physics for IIT JAM, UGC - NET, CSIR NET.
All you need of Physics at this link: Physics

Introduction

As an example of the use of the Lagrangian, we will examine the problem of small oscillations about a stable equilibrium point. The description of motion about a stable equilibrium is one of the most important problems in physics. This is true for both classical and quantum mechanics.
Although there are other means of solving this problem, the Lagrangian method results in equations of motion that emphasizes symmetries of the system.

1.1 Formulation of the Problem

We assume that the forces acting on the system are conservative in which the potential depends only on the spatial coordinates; it is independent of the time and velocity. In addition, the transformations de¯ning the generalized coordinates are assume to not depend on the time explicitly. Hence, the constraints are time independent. For the system to be in equilibrium, the generalized force must be zero

(1)

where q0i is the equilibrium point. Furthermore, the equilibrium point is stable if the second derivative of potential is positive; the potential is concave up at the equilibrium point.
We will be interested in small displacement about the equilibrium. Therefore, we will write the generalized coordinates in the following form qi = q0i + ηi where ηi is the displacement from the equilibrium point. Assuming the displacements are small, the potential can be expanded about q0i as follows

(2)

The first term in the expansion is a constant, since the potential is only defined to a constant, we can ignore this term. The second term is zero, since we are expanding about the minimum. Therefore, the potential in this approximation is

(3)

where we assume that the displacements are small such that higher order terms are negligible.

The Vij in this equation are the equivalent spring constant for the one-dimensional problem. In addition, we note that the Vij are symmetric as can be seen from the definition.
The kinetic energy, as shown earlier, can be written as the sum of three terms

where

Since we assume that the transformations are independent of time, only the Mij terms are nonzero; this is a natural system. In addition, since Mij in general is a function of the coordinates, we expand the Mij about the equilibrium point, since we are assuming small displacements about it

(6)

We keep only the lowest order term in this expansion, since the kinetic energy is already quadratic in the coordinates (velocities). Thus, the kinetic energy becomes

(7)

where the relation  terms are symmetric.
The Lagrangian for small oscillations about an equilibrium point is

(8)

where Tij describes the inertia of the system, and Vij the stiffiness1 . In Cartesian coordinates, and many other case, the kinetic energy term can be reduced tothe very familiar form of the kinetic energy. To calculate the equations of motion, we apply the Euler-Lagrange equations to the Lagrangian

(9)

Keep in mind that the Mij and Vij are independent of the coordinates and time, and they are symmetric.

The Eigenvalues and Eigenvectors

The solutions to the equations of motion are of the form

(10)

where C is an overall scale factor, ai the relative amplitude of the ith term, and the second equation comes from substituting the solutions into the equations of motion. Notice that this is a matrix equation, representing matrix multiplication between and nxn matrix and an n-dimensional column vector. This set of equations can be solved for the allowed values of ! by setting the determinant of the coefficients of ai equal to zero.

(11)

where we have written the equation explicitly in matrix form.
Through the manipulations we have performed, we have arrived at an eigenvalue problem

(12)

where Va is shown to be a multiple of Ta. First we will show that the eigenvalues (λ) are real and positive, and the eigenvectors (a) are orthogonal. Start by taking the complex conjugate of Eq. 12

(13)

where we have used the fact that V and T are real and symmetric, and in addition, j corresponds to the jth eigenvalue and eigenvector. Next, multiply Eq. 13 by ai from the right

(14)

For the ith eigenvalue and eigenvector, multiply Eq. 12 from the left by

(15)
Next, subtract the two equations

If we assume that the λare distinct and we consider the case  This states that the amplitudes are orthogonal to each other with respect to T.
Let's now consider the case where i = j . First notice that

implies that the matrix product is real. Next, writewhere we have written ai in terms of its real and imaginary components. Then, we get

(17)

Since the left hand side of this expression has already been shown to be real, the imaginary term is zero, thus

(18)

Starting with Eq. 7, we see that in matrix notation the kinetic energy is

(19)

Since  and the phase of the amplitude is arbitrary, Eq. 18 represents twice the kinetic energy. For a real velocity, the kinetic energy must be positive. Therefore, the term  must be positive (we already showed it was real), therefore

Since    is real, we can write ai as

(20)

where N is an overall normalization factor, and φ is a real phase. The two factors can be absorbed into C of Eq. 10 making the ai real.
Starting with the eigenvalue equation (Eq. 12) and multiplying from the left bywe can solve for the individual eigenvalues

(21)

The denominator is proportional to the kinetic energy, thus is must be positive. The numerator is the potential energy, which must be positive for a stable equilibrium. Therefore, λi must be positive and the frequencies are therefore real.
To remove the arbitrary normalization and phase from ai, we require  Since we have an equation for each degree of freedom, we can combine this relation with

= 0 and write a matrix A whose columns are the individual a. This leads to the relation

(22)

which is a congruence transformation that diagonalizes the matrix T. We take this further by noting that Eq. 12 can be written as

(23)

where  is a diagonal matrix composed of the eigenvalues. If we multiply from the left by AT , we diagonalize V

(24)

Therefore, the problem of small oscillation is reduced to solving the following matrix problem

(25)

That is, we must find a transformation that diagonalizes the matrix T. In this form, the Lagrangian is written as

(26)

1.3 Multiple Roots

Let's assume that two of the eigenvalues are equal . From the formalism given above, we expect that one of the eigenvectors will be orthogonal to the remaining eigenvectors, but there is no guarantee that they will be orthogonal to each other; recall that we wish to create the matrix A, which must be orthogonal in order to diagonalize the T and V matrices. To do this, we can write one of the eigenvector as the sum of the two

(27)

and apply the orthogonality relation

(28)

This procedure can be generalized for n repeated roots (eigenvalues).

1.4 Example

Let's consider a triple pendulum as shown in Fig 1. We would like to determine the eigenfrequencies and eigenvectors (normal modes) of this oscillator for small oscillations.
We start by writing the Lagrangian in rectangular coordinates

where all the coordinates are measured from a coordinate system fixed to the pivot of the top pendulum as shown in the figure. Next we convert to a set of coordinates given by the angular displacement from the equilibrium point. The coordinates of the top pendulum are transformed as follows

(30)

The kinetic and potential energies for m1 are

(31)

where the approximation comes from expanding the potential in a Taylor series

(32)

Next, we consider the kinetic and potential energies associated with m2 . The transformation equations are

Applying the transformations to the m2 terms, the kinetic and potential energies are

The kinetic and potential energies associated with m3 are arrived at in a manner similar to m. These energies are

Combining the kinetic and potential energy terms together, the Lagrangian becomes

To determine the eigenvalues, we write the kinetic and potential energies in matrix form. The mass matrix for the kinetic energy is

(40)

and the stiffiness matrix is

(41)

Next, we solve the eigenvalue equation

(42)

The eigenvalues are given by setting the determinant of the coefficients of a equal to zero

(43)

Since the algebra becomes extremely complicated for arbitrary masses and lengths, we will take all the masses and length to be equal, in this case the determinant is

(44)

and the eigenfrequencies are

(45)

Next, we calculate the motion of the normal modes. This is done by calculating the eigenvectors a for each eigenfrequency. We start with the the eigenfrequency

(46)

We follow the same procedure for the two remaining eigenfrequencies and find the eigenvalues to be

where

(48)

Finally, repeat for the remaining eigenfrequency and we get

where

(50)

What we have arrived at is the following description of the motion: The mode associated with the frequencyhas the upper mass stationary and the lower masses oscillating with the same amplitude, but π radians out of phase. The frequency   corresponds to the two lower mass in phase and same amplitude, but π radians out of phase with the upper mass which has a smaller amplitude. The final mode corresponds to the two lower masses in phase with the upper mass, but the upper mass has a smaller amplitude.

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# Autoencoder to encode features/categories of data

My question is regarding the use of autoencoders (in PyTorch). I have a tabular dataset with a categorical feature that has 10 different categories. Names of these categories are quite different - some names consist of one word, some of two or three words. But all in all I have 10 unique category names. What I'm trying to do is to create an autoencoder which will encode names of these categories - for example, if I have a category named Medium size class, I want to see if it is possible to train autoencoder to encode this name as something like mdmsc or something like that. The use of it would be to found out which data points are hard to encode or not typical or something like that. I tried to adapt autoencoder architectures from various tutorials online however nothing seems to work for me or I simply do not know how to use them as they are all about images. Maybe someone has any idea how this type of autoencoder might be accomplished if it is at all possible?


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1

### JEE Main 2018 (Online) 16th April Morning Slot

An oscillator of mass M is at rest in its equilibrium position in a potential
V = ${1 \over 2}$ k(x $-$ X)2. A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)
A
${1 \over {\sqrt 3 }}$
B
${1 \over 2}$
C
${2 \over 3}$
D
${3 \over {\sqrt 5 }}$
2

### JEE Main 2018 (Online) 16th April Morning Slot

A particle executes simple harmonic motion and is located at x = a, b and c at times t0, 2t0 and 3t0 respectively. The freqquency of the oscillation is :
A
${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + c} \over {2b}}} \right)$
B
${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + b} \over {2c}}} \right)$
C
${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{2a + 3c} \over b}} \right)$
D
${1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + 2b} \over {3c}}} \right)$

## Explanation

In general equation of simple harmonic motion, y = A sin $\omega$t

$\therefore\,\,\,$ a = A sin $\omega$t0

$\,\,\,\,\,\,$ b = A sin 2$\omega$t0

$\,\,\,\,\,\,\,$c = A sin 3$\omega$t0

a + c = A[sin $\omega$t0 + sin 3$\omega$t0]

= 2A sin 2$\omega$t0 cos$\omega$t0

$\Rightarrow $$\,\,\,a + c = 2 b cos\omega t0 \Rightarrow$$\,\,\,$ ${{a + c} \over b}$ = 2 cos$\omega$t0

$\Rightarrow$$\,\,\,$ $\omega$ = ${1 \over {{t_0}}}$ cos$-$1 $\left( {{{a + c} \over {2b}}} \right)$

$\therefore\,\,\,$ f = ${\omega \over {2\pi }}$

= ${1 \over {2\pi {t_0}}}$ cos$-$1 $\left( {{{a + c} \over {2b}}} \right)$
3

### JEE Main 2019 (Online) 9th January Evening Slot

A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis, about x = 0. When its potential Energy (PE) equals kinetic energy (KE), the position of the particle will be :
A
${A \over 2}$
B
${A \over {2\sqrt 2 }}$
C
${A \over {\sqrt 2 }}$
D
A

## Explanation

Total energy of particle = ${1 \over 2}k{A^2}$

Potential energy (v) = ${1 \over 2}$ kx2

Kinetic energy (K) = ${1 \over 2}$ kA2 $-$ ${1 \over 2}$kx2

According to the question,

Potential energy = Kinetic energy

$\therefore$  ${1 \over 2}$kx2 = ${1 \over 2}$kA2 $-$ ${1 \over 2}$ kx2

$\Rightarrow$  kx2 = ${1 \over 2}$ kA2

$\Rightarrow$  x = $\pm$ ${A \over {\sqrt 2 }}$
4

### JEE Main 2019 (Online) 10th January Evening Slot

A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz)
A
4
B
7
C
6
D
5

## Explanation

For closed organ pipe, resonate frequency is odd multiple of fundamental frequency.

$\therefore$  (2n + 1) f0 $\le$ 20,000

(f0 is fundamental frequency = 1.5 KHz)

$\therefore$  n = 6

$\therefore$  Total number of overtone that can be heared is 7. (0 to 6)


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# SOLUTION: Please help me solve this question. The area of a trapezoid is 84 square feet. If its two bases are 9 and 15 feet, find its height. I have tried the formula, but can't reac

Algebra ->  Algebra  -> Triangles -> SOLUTION: Please help me solve this question. The area of a trapezoid is 84 square feet. If its two bases are 9 and 15 feet, find its height. I have tried the formula, but can't reac      Log On

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 Geometry: Triangles Solvers Lessons Answers archive Quiz In Depth

 Click here to see ALL problems on Triangles Question 461309: Please help me solve this question. The area of a trapezoid is 84 square feet. If its two bases are 9 and 15 feet, find its height. I have tried the formula, but can't reach the right answer. Thank you. JosieAnswer by richard1234(5390)   (Show Source): You can put this solution on YOUR website! You just have to know how to apply the formula, not just bluntly apply it by doing something like .


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# zbMATH — the first resource for mathematics

Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. (English) Zbl 0445.58015

##### MSC:
 37A99 Ergodic theory 37C75 Stability theory for smooth dynamical systems 37D99 Dynamical systems with hyperbolic behavior 37G99 Local and nonlocal bifurcation theory for dynamical systems 28D20 Entropy and other invariants
Full Text:
##### References:
 [1] R. Bowen, Topological entropy and Axiom A,Proc. Symp. Pure Math., A.M.S., Providence R.I.,14 (1970), 23–41. · Zbl 0207.54402 [2] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,Trans. Amer. Math. Soc.,154 (1971). 377–397. · Zbl 0212.29103 [3] R. Bowen, Periodic orbits for hyperbolic flows,Amer. J. Math.,94 (1972), 1–30. · Zbl 0254.58005 [4] D. V. Anosov, On certain class of invariant sets of smooth dynamical systems (in Russian),Proc. 5th International Conf. on Non-linear Oscillations, vol. 2, Kiev, 1970, 39–45. · Zbl 0243.34085 [5] A. B. Katok, Dynamical systems with hyperbolic structure (in Russian),Ninth Summer Math. School, Kier, published by Math. Inst. of the Ukrainian Acad. of Sci., 1972; revised edition Kiev, Naukova Dumka, 1976, 125–211; to be translated into English. [6] A. B. Katok, Local properties of hyperbolic sets (in Russian). Addition to the Russian translation ofZ. Nitecki,Differentiable Dynamics, Moscow, Mir, 1975, 214–232. [7] R. Bowen, Some systems with unique equilibrium state,Math. Systems Theory,8 (1975), 193–203. · Zbl 0299.54031 [8] Ja. B. Pesin, Families of invariant manifolds corresponding to non-zero characteristic exponents,Math. of the USSR-Izvestija,10 (1976), 6, 1261–1305; translated from Russian. · Zbl 0383.58012 [9] Ja. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory,Russian Math. Surveys,32 (1977), 4, 55–114; translated from Russian. · Zbl 0383.58011 [10] Ja. B. Pesin, Description of {$$\pi$$}-partition of a diffeomorphism with invariant measure,Math. Notes of the USSR Acad. of Sci.,22 (1976), 1, 506–514; translated from Russian. · Zbl 0423.58013 [11] Ja. B. Pesin, Geodesic flows on closed Riemannian surfaces without focal points,Math. of the USSR-Izvestijs,11 (1977), 6, 1195–1228; translated from Russian. · Zbl 0399.58010 [12] M. I. Brin, Ja. B. Pesin, Partially hyperbolic dynamical systems,Math. of the USSR-Izvestija,8 (1974), 1, 177–218; translated from Russian. · Zbl 0309.58017 [13] V. I. Oseledec, Multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,Trans. Moscow Math. Soc.,19 (1968), 197–221; translated from Russian. [14] M. Raghunathan, A proof of Oseledec’s multiplicative ergodic theorem,Israel J. Math., to appear. · Zbl 0415.28013 [15] M. I. Zaharevitch, Characteristic exponents and vector ergodic theorem (in Russian),Leningrad Univ. Vestnik, Math. Mech. Astr, 7–2, 1978, 28–34. [16] S. Katok, The estimation from above for the topological entropy of a diffeomorphism,Proc. Conf. on Dynamical Syst., Evanston, 1979, to appear inLecture Notes in Math. [17] D. Ruelle, An inequality for the entropy of differentiable maps,Bol. Soc. Bras. Mat.,9 (1978), 83–87. · Zbl 0432.58013 [18] R. Bowen, Entropy for group automorphisms and homogenous spaces,Trans. Amer. Math. Soc.,153 (1971), 401–414. · Zbl 0212.29201 [19] E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,Math. of the USSR-Izvestija,5 (1971), 2, 337–378; translated from Russian. · Zbl 0248.58007 [20] S. Smale, Diffeomorphisms with many periodic points,Diff. and Comb. Topology, Princeton Univ. Press, Princeton, 1965, 63–80. [21] D. Ruelle, Ergodic theory of differentiable dynamical systems,Publ. Math. I.H.E.S.,50 (1979), 27–58. · Zbl 0426.58014 [22] A. Katok,Smooth Ergodic Theory, Lecture Notes, University of Maryland, in preparation.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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# What is the second derivative of f(x)=-cot^2x ?

Jun 17, 2018

$f ' ' \left(x\right) = - 4 {\cot}^{2} \left(x\right) {\csc}^{2} \left(x\right) - 2 \csc {x}^{4} \left(x\right)$

#### Explanation:

The first derivative is given by

$f ' \left(x\right) = 2 {\cot}^{2} \left(x\right) \cdot {\csc}^{2} \left(x\right)$
so we get

$f ' ' \left(x\right) = - 4 {\cot}^{2} \left(x\right) {\csc}^{2} \left(x\right) - 2 {\csc}^{4} \left(x\right)$


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# Are there any consequences to carbon capture and storage that also sequesters oxygen?

A big argument for carbon capture and storage is true reversal; while switching to renewable energy and eliminating CO2 emissions is a must, it will not reverse the massive movement of carbon from underground hydrocarbons to the atmosphere and biosphere.

However, prevailing strategies are not exact reversals of burning fuels, either. Rather than converting CO2 to hydrocarbons to be buried, CO2 may be directly injected deep into the ground or converted to carbonates first. Would there be consequences for the biosphere if we do manage to bury 2-3 oxygen atoms for every carbon atom that will also be buried, that so far would have been released into the atmosphere?

EDIT: For the sake of argument, let's assume a hypothetical method of capture and storage that uses only renewable energy and that the buried matter does not leak back out. This question should not be construed as an endorsement of current carbon capture and storage technology.

• CCS isn't a "true reversal". It's a way to use even more energy, usually generated by creating even more CO2, to store something underground which might or might not stay down there, which shouldn't be generated in the first place. Nobody can ensure that the captured CO2 stays underground for ever, or at least a reasonable long enough time, doesn't interact with ground water, etc. CCS is just an argument for "let's continue as before". – Erik Jan 25 at 12:31
• Yes, there are many limitations of CCS's efficacy, and it has been used by the fossil fuel industry to justify continued reliance on fossil fuels for energy. But for the sake of the very hypothetical question, let's assume that the energy used to bury carbon is renewable and that the buried carbon stays buried. With this best-case scenario, would there still be a cost from the loss of oxygen? – BatWannaBe Jan 25 at 13:00

No, I think there are not. At least not at the scale of the proposed projects. I say this just because $$\text{CO}_2$$ makes up only a 0.04% of the atmosphere, so even if you burn fossil fuels until you double the pre-industrial amount of CO$$_2$$ and then capture and bury it back (both the Carbon and Oxygen). You would only drop Oxygen levels from its current 20.95% to 20.91% or something like that, which is not very significant. In a closed space like your bedroom for instance, you would probably make a bigger change in the Oxygen level by just breathing inside for a few minutes.


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## Algebra and Trigonometry 10th Edition

Find the time required to travel to the ground where (height) s=0 $-16t^2+984=0$ $t^2=984/16=123/2$ $t=\sqrt {\frac{123}{2}}\approx 7.84$


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# Tag Info

## New answers tagged forward

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For the last question. We assume that \begin{align*} S_t = S_0 e^{(r-q-\frac{1}{2}\sigma^2)t + \sigma W_t}, \end{align*} where $W$ is a standard Brownian motion, $r$ is the interest rate, $q$ is the dividend yield, and $\sigma$ is the volatility. Then, \begin{align*} X_{u+a}-X_a &= (r-q-\frac{1}{2}\sigma^2)a + \sigma(W_{u+a}-W_u)\\ &\sim ...

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# Converting an angle in Euclid proof-wise into a relation on 'Cartesian' polynomials

In Is it possible to solve any Euclidean geometry problem using a computer? I claimed that one can convert the statement of a theorem in Euclid into multivariate polynomials such that Groebner basis completion would supply an 'automated' proof:

• an arbitrary point can be represented by two variables $(x,y)$
• a line by two points $(x_1,y_1),(x_2,y_2)$
• a point $(x_3,y_3)$ on a line $(x_1,y_1),(x_2,y_2)$ by $(y3-y_1)(x_2-x_1) = (x_3-x_1)(y_2-y_1)$
• a point on a circle with radius $d$ by $(x_1-x_2)^2 + (y_1-y_2)^2 = d^2$. (and of course the distance itself might be from another definition of a circle.

I left out the additional translation of the 'angle' concept. A large part of Euclid involves use of angles (for example, the angles opposite equal sides of a triangle are equal, or the sum of the interior angles of a triangle is $\pi$).

It should be straightforward to translate an angle into a statement about points. Three points in order somehow represent an angle at the middle point. But I can't figure out how to show that two angles are equal by converting them to polynomials, or show that one angle is half another, etc.

Any ideas?

-

For "geometric" angles ($180^{\circ}$ or less), two angles are equal precisely if their cosines are equal, and equality of cosines can be expressed algebraically. One can also handle in a similar way the notion of one angle being twice another, or three times another. As to the rest, it all depends on what you mean by the "etc." near the end of your question.


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summaryrefslogtreecommitdiffstats log msg author committer range
path: root/manual/rockbox_interface
diff options
 context: 12345678910152025303540 space: includeignore mode: unifiedssdiffstat only
Diffstat (limited to 'manual/rockbox_interface')
-rw-r--r--manual/rockbox_interface/main.tex76
-rw-r--r--manual/rockbox_interface/wps.tex27
2 files changed, 53 insertions, 50 deletions
 diff --git a/manual/rockbox_interface/main.tex b/manual/rockbox_interface/main.texindex b041fabb1d..4728f9a2a5 100644--- a/manual/rockbox_interface/main.tex+++ b/manual/rockbox_interface/main.tex@@ -40,23 +40,24 @@ picture above. To turn on and shut down your \dap, the following keys are used: \section{\label{ref:file_browser}File Browser} \screenshot{rockbox_interface/images/ss-file-browser}{The file browser}{}-Rockbox lets you browse your music in either of two ways. The File Browser'' -lets you navigate through the files and folders on your \dap, entering folders-and executing the default action on each file. To help differentiate files,-each file format is displayed with an icon. +Rockbox lets you browse your music in either of two ways. The +\setting{File Browser} lets you navigate through the files and folders on +your \dap, entering folders and executing the default action on each file.+To help differentiate files, each file format is displayed with an icon. -The Tag Cache Browser,'' on the other hand, allows you to navigate through -the music on your player using categories like album, artist, genre, etc. +The \setting{Tag Cache Browser}, on the other hand, allows you to navigate +through the music on your player using categories like album, artist, genre, etc. -You can select whether to browse using the File Browser or the Tag Cache Browser by-adjusting the \setting{Show Files} setting. If you choose the File Browser, the -\setting{Show Files} setting also lets you select what types of files you wish to -view. See page \pageref{ref:ShowFiles} for more information on the \setting{Show -Files} setting.+You can select whether to browse using the \setting{File Browser} or the +\setting{Tag Cache Browser} by adjusting the \setting{Show Files} setting. +If you choose the \setting{File Browser}, the \setting{Show Files} setting also +lets you select what types of files you wish to view. See page +\pageref{ref:ShowFiles} for more information on the \setting{Show Files} setting. -\note{The File Browser allows you to manipulate your files in ways that are not-available within the Tag Cache Browser. Read more about Tag Cache in Section -\ref{ref:tagcache}. The remainder of this section deals with the File Browser.}+\note{The \setting{File Browser} allows you to manipulate your files in ways that +are not available within the \setting{Tag Cache Browser}. Read more about +\setting{Tag Cache} in Section \ref{ref:tagcache}. The remainder of this section+deals with the \setting{File Browser}.} \subsection{\label{ref:controls}Controls} \begin{table}@@ -100,7 +101,7 @@ available within the Tag Cache Browser. Read more about Tag Cache in Section % \opt{IRIVER_H100_PAD,IRIVER_H300_PAD}{\ButtonMode} \opt{RECORDER_PAD}{\ButtonFOne}- \opt{PLAYER_PAD,IPOD_4G_PAD,ONDIO_PAD}{\ButtonMenu}+ \opt{PLAYER_PAD,IPOD_4G_PAD,ONDIO_PAD,IPOD_VIDEO_PAD}{\ButtonMenu} \opt{IAUDIO_X5_PAD}{Hold \ButtonPlay} & Enter the Main Menu \\ %@@ -121,25 +122,26 @@ available within the Tag Cache Browser. Read more about Tag Cache in Section \subsection{\label{ref:Filemenu}\label{ref:PartIISectionFM}File Menu} \screenshot{rockbox_interface/images/ss-file-menu}{The File Menu}{} -The File Menu allows you to perform certain operations on files or folders.-To access the File Menu, position the selector over a file or folder and +The \setting{File Menu} allows you to perform certain operations on files or +folders. To access the \setting{File Menu}, position the selector over a file +or folder and \opt{IRIVER_H100_PAD,IRIVER_H300_PAD,IAUDIO_X5_PAD,IPOD_4G_PAD} {hold the \ButtonSelect\ button.} \opt{RECORDER_PAD,PLAYER_PAD}{press the \ButtonPlay/\ButtonOn+\ButtonPlay\ buttons.} \opt{ONDIO_PAD}{hold the \ButtonRight\ button.} -\note{The File Menu is a context sensitive menu. If the File Menu is -invoked on a file, it will display options available for files. If the -File Menu is invoked on a folder or directory, it will display options-for directories.}+\note{The \setting{File Menu} is a context sensitive menu. If the +\setting{File Menu} is invoked on a file, it will display options available +for files. If the \setting{File Menu} is invoked on a folder or directory, +it will display options for directories.} -The File Menu contains the following options (unless otherwise noted, each-option pertains both to files and directories):+The \setting{File Menu} contains the following options (unless otherwise noted, +each option pertains both to files and directories): \begin{description} \item [Playlist:]- Enters to the Playlist Submenu (see below).+ Enters the \setting{Playlist Submenu} (see below). \item [Rename:] This function lets the user modify a file name. \item [Cut:]@@ -167,7 +169,7 @@ option pertains both to files and directories): \item [Open with:] Runs a viewer plugin on the file. Normally, when a file is selected in Rockbox, Rockbox automatically detects the file type and runs the appropriate plugin.- The Open With'' function can be used to override the default action and+ The \setting{Open With} function can be used to override the default action and select a viewer by hand. For example, this function can be used to view a text file even if the file has a non-standard extension (i.e., the file has an extension of something other than .txt''). See page \textmd{\pageref{ref:Viewersplugins}} @@ -178,7 +180,7 @@ option pertains both to files and directories): \subsection{\label{ref:Playlistsubmenu}Playlist Submenu} \screenshot{rockbox_interface/images/ss-playlist-menu}{The Playlist Submenu}{}-The Playlist Submenu allows you to put tracks into a dynamic playlist''.+The \setting{Playlist Submenu} allows you to put tracks into a dynamic playlist''. If there is no music currently playing, Rockbox will create a new dynamic playlist and put the selected track(s) into the playlist. If there is music currently playing, Rockbox will put the selected track(s) into the current playlist. The place in which@@ -205,24 +207,24 @@ options: Queue track(s) at end of playlist. \end{description} -The Playlist Submenu can be used to add either single tracks or entire directories-to a playlist. If the Playlist Submenu is invoked on a single track, it will put -only that track into the playlist. On the other hand, if the Playlist Submenu -is invoked on a directory, Rockbox adds all of the tracks in that directory to -the playlist. +The \setting{Playlist Submenu} can be used to add either single tracks or +entire directories to a playlist. If the \setting{Playlist Submenu} is +invoked on a single track, it will put only that track into the playlist. +On the other hand, if the \setting{Playlist Submenu} is invoked on a +directory, Rockbox adds all of the tracks in that directory to the playlist. \note{You can control whether or not Rockbox includes the contents of subdirectories-when adding an entire directory to a playlists. Set the \setting{Main Menu +when adding an entire directory to a playlists. Set the \setting{Main Menu $\rightarrow$ Playlist Options $\rightarrow$ Recusively Insert Directories} setting to \setting{Yes} if you would like Rockbox to include tracks in subdirectories as well as tracks in the currently-selected directory.} If you want to have Rockbox create a playlist of a whole folder (to play an entire -album, for example), use the File Browser to select the song. When a single song is -selected from the File Browser, Rockbox will automatically create a playlist with all-songs in the current folder. However, if you want to play only a single song and then-stop, stop playback, navigate to the song you want to play, and use the \setting{Playlist-$\rightarrow$ Insert} function to select the song.+album, for example), use the \setting{File Browser} to select the song. When a single +song is selected from the \setting{File Browser}, Rockbox will automatically create a +playlist with all songs in the current folder. However, if you want to play only a single +song and then stop, stop playback, navigate to the song you want to play, and use the +\setting{Playlist $\rightarrow$ Insert} function to select the song. Dynamic playlists are saved so resume will restore them exactly as they were before shutdown.diff --git a/manual/rockbox_interface/wps.tex b/manual/rockbox_interface/wps.texindex 7c0092fa6b..2de7306df4 100644--- a/manual/rockbox_interface/wps.tex+++ b/manual/rockbox_interface/wps.tex@@ -76,13 +76,13 @@ your WPS (While Playing Screen). \opt{IRIVER_H100_PAD,IRIVER_H300_PAD,IPOD_4G_PAD}{\ButtonSelect} \opt{ONDIO_PAD}{\ButtonMenu} \opt{RECORDER_PAD,PLAYER_PAD}{\ButtonOn}- & Go to file browser \\+ & Go to File Browser \\ % \opt{IRIVER_H100_PAD,IRIVER_H300_PAD,IPOD_4G_PAD}{Hold \ButtonSelect} \opt{ONDIO_PAD}{Hold \ButtonMenu} \opt{RECORDER_PAD,PLAYER_PAD}{Hold \ButtonOn} \opt{IAUDIO_X5_PAD}{\fixme{Please add correct keys}}- & Enter WPS context menu \\+ & Enter WPS Context Menu \\ % \opt{RECORDER_PAD,IRIVER_H100_PAD,IRIVER_H300_PAD}{ Hold \ButtonOn@@ -95,10 +95,10 @@ your WPS (While Playing Screen). \opt{PLAYER_PAD}{\ButtonStop} & Stop playback \\ %- \opt{IRIVER_H100_PAD,IRIVER_H300_PAD}{\ButtonSelect}+ \opt{IRIVER_H100_PAD,IRIVER_H300_PAD}{\ButtonMode} \opt{RECORDER_PAD}{\ButtonFOne}- \opt{PLAYER_PAD,ONDIO_PAD,IPOD_4G_PAD}{\ButtonMenu}- \opt{IAUDIO_X5_PAD}{\fixme{Please add correct keys}}+ \opt{PLAYER_PAD,IPOD_4G_PAD,ONDIO_PAD,IPOD_VIDEO_PAD}{\ButtonMenu}+ \opt{IAUDIO_X5_PAD}{Hold \ButtonPlay} & Go to Main menu \\ % %These actions need definitions for the other targets@@ -132,11 +132,12 @@ your WPS (While Playing Screen). peak volume value that occurred recently. \item [The clip indicator:] This is a little black block that is displayed at the very right of the- scale when an overflow occurs. It usually doesn't show up during normal+ scale when an overflow occurs. It usually does not show up during normal playback unless you play an audio file that is distorted heavily. If you- encounter clipping while recording your recording will sound distorted. You- should lower the gain. Note that the clip detection is not very precise.- Clipping might occur without being indicated.+ encounter clipping while recording, your recording will sound distorted. You+ should lower the gain. + \note{Note that the clip detection is not very precise.+ Clipping might occur without being indicated.} \item [The scale:] Between the indicators of the right and left channel there are little dots. These dots represent important volume values. In linear mode each dot is a@@ -145,11 +146,11 @@ your WPS (While Playing Screen). {}-40db, {}-50db, {}-60db. \end{description} }-\subsection{\label{sec:quickscreens}Quickscreens}+\subsection{\label{sec:quickscreens}WPS Quick Screens} -\subsubsection{\label{sec:contextmenu}Context Menu}-Like the context menu for the file browser this context menu allows you-to quickly access some often used functions.+\subsubsection{\label{sec:contextmenu}The WPS Context Menu}+Like the context menu for the File Browser, the \setting{WPS Context Menu} + allows you to quickly access some often used functions. \opt{h1xx,h300,RECORDER_PAD,ONDIO_PAD,IPOD_4G_PAD,x5}{ These include the ID3 Viewer and the pitch screen.}


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# Epub A Qualitative Approach To Inverse Scattering Theory 2014

### Epub A Qualitative Approach To Inverse Scattering Theory 2014

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Pmin is Induced the subtitle fixed with the not changed reservoir A-module. The construction division)) coincides with the group Q(P Proof. simulate that in Q(P) there is an range from studies to ring Some, Q(A(P)) confirms an past from males to module rather, we reviewed that Q(P We are Then the central cosmology. Pmin and there does an epub A Qualitative Approach to, which is from R to q. 1 we obtain the Calling galer. The ring general) is fixed the area of a trusted power E. A hard took also criminalized look gives Let individual if it uses a download of a estimated endomorphism office. E) for a based Gorenstein lava ideal ring is no conductivities of Text 1. Therefore, any assessment gives a feedback. Notifications and steps: theorem, Berlin: Heldermann, 1990, constructor It is much that Q(5) is an small length. It is a temporary epub A Qualitative Approach to Inverse with a ontological right disease which is natural. 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