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# zbMATH — the first resource for mathematics

Efficient spectral ultraspherical-Galerkin algorithms for the direct solution of 2nth-order linear differential equations. (English) Zbl 1205.65224
Summary: Some efficient and accurate algorithms based on the ultraspherical-Galerkin method are developed and implemented for solving $$2n$$th-order linear differential equations in one variable subject to homogeneous and nonhomogeneous boundary conditions using a spectral discretization. We extend the proposed algorithms to solve the two-dimensional $$2n$$th-order differential equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost and roundoff errors.

##### MSC:
 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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##### References:
 [1] Boutayeb, A.; Twizell, E., Numerical methods for the solution of special sixth-order boundary value problems, Int. J. comput. math., 45, 207-233, (1992) · Zbl 0773.65055 [2] Twizell, E.; Boutayeb, A., Numerical methods for the solution of special and general sixth-order boundary value problems with applications to benard layer eigenvalue problems, Proc. roy. soc. London A., 431, 433-450, (1990) · Zbl 0722.65042 [3] Baldwin, P., Asymptotic estimates of the eigenvalues of a sixth order boundary value problem obtained by using global phase-integral methods, Phil. trans. roy. soc. London A, 322, 281-305, (1987) · Zbl 0625.76043 [4] El-Gamel, M.; Cannon, J.R.; Zayed, A.I., Sinc-Galerkin method for solving linear sixth-order boundary-value problems, Math. comput., 73, 1325-1343, (2003) · Zbl 1054.65085 [5] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1988), Springer-Verlag New York-Heidelberg-Berlin · Zbl 0658.76001 [6] Coutsias, E.A.; Hagstrom, T.; Torres, D., An efficient spectral method for ordinary differential equations with rational function coefficients, Math. comput., 65, 611-635, (1996) · Zbl 0846.65037 [7] Doha, E.H., An accurate double Chebyshev spectral approximations for poisson’s equation, Ann. univ. sci. Budapest. sect. comp., 10, 243-275, (1990) · Zbl 0717.65090 [8] Doha, E.H., An accurate solution of parabolic equations by expansion in ultraspherical polynomials, J. comp. math. appl., 21, 4, 75-88, (1990) · Zbl 0706.65089 [9] Doha, E.H., The ultraspherical coefficients of the moments of a general – order derivative of an infinitely differentiable function, J. comp. appl. math., 89, 53-72, (1998) · Zbl 0909.33007 [10] Doha, E.H., On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations, J. comp. appl. math., 139, 275-298, (2002) · Zbl 0991.33003 [11] Doha, E.H.; Abd-Elhameed, W.M., Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method, J. comput. appl. math., 181, 24-45, (2005) · Zbl 1071.65136 [12] Doha, E.H.; Al-Kholi, F.M.R., An efficient double Legendre spectral method for parabolic and elliptic partial differential equations, Int. J. comput. math., 78, 413-432, (2001) · Zbl 1004.65101 [13] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia, 1977. · Zbl 0412.65058 [14] Heinrichs, W., Improved condition number for spectral methods, Math. comput., 53, 103-119, (1989) · Zbl 0676.65115 [15] Heinrichs, W., Spectral methods with sparse matrices, Numer. math., 56, 25-41, (1989) · Zbl 0661.65120 [16] Karageorghis, A., Chebyshev spectral methods for solving two-point boundary value problem arising in the heat transfer, J. comput. methods appl. mech. eng., 70, 103-121, (1988) · Zbl 0636.65083 [17] Doha, E.H.; Abd-Elhameed, W.M., Efficient spectral-Galerkin algorithms for direct solution of second-order equations using ultraspherical polynomials, SIAM J. sci. comput., 24, 2, 548-571, (2002) · Zbl 1020.65088 [18] E.H. Doha, W.M. Abd-Elhameed, Efficient spectral ultraspherical-Galerkin approximations of one-dimensional fourth-order problems, in: E.H. Doha, M.A. Helal (Eds.), in: Proceedings of the international conference on mathematics: Trends and Developments, Egyptian Math. Soc., Cairo 28-31 Dec. 2002, vol. III, Fluid Mechanics and Numerical Analysis, (2003), pp. 99-126. [19] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials, Numer. algor., 42, 137-164, (2006) · Zbl 1103.65119 [20] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of the integrated forms for second-order equations using ultraspherical polynomials, Anziam j., 48, 361-386, (2007) · Zbl 1138.65104 [21] Doha, E.H.; Bhrawy, A.H., Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. numer. math., 58, 1224-1244, (2008) · Zbl 1152.65112 [22] E.H. Doha, Spectral solutions of differential and difference equations with polynomial coefficients by using classical orthogonal polynomials, J. Comput. Appl. Math., in press. [23] Shen, J., Efficient spectral-Galerkin method I: direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. sci. comput., 15, 1489-1505, (1994) · Zbl 0811.65097 [24] Shen, J., Efficient spectral-Galerkin method II: direct solvers of second-and fourth-order equations using Chebyshev polynomials, SIAM J. sci. comput., 16, 74-87, (1995) · Zbl 0840.65113 [25] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge [26] Doha, E.H., The coefficients of differentiated expansions and derivatives of ultraspherical polynomials, J. comp. math. appl., 19, 2-3, 115-122, (1991) · Zbl 0723.33008 [27] E.H. Doha, Fathalla A. Rihan, W.M. Abd-Elhameed, Efficient ultraspherical spectral-Galerkin algorithms for the approximation of fourth-order equations, submitted for publication. [28] Doha, E.H., On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials, J. phys. A: math. gen., 37, 657-675, (2004) · Zbl 1055.33007 [29] Graham, A., Kronecker product and matrix calculus with applications, (1981), Ellis Horwood Ltd. London [30] Funaro, D., Polynomial approximation of differential equations, Lecturer notes in physics, (1992), Springer-Verlag Heidelberg, Berlin, New York [31] Luke, Y., The special functions and their approximations, vol. 1, (1969), Academic Press New York
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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Question

The probability of winning on an arcade game is 0.659. if you play the arcade game 30 times. What is the probability of winning exactly 21 times?

1. The probability of winning exactly 21 times is 0.14

### What is Binomial Probability?

Binomial probability refers to the probability of exactly ‘x’ successes on ‘n’ repeated trials in an experiment which has two possible outcomes (commonly called a binomial experiment).
Probability ‘P’ = ⁿCₓ (probability of 1st)ˣ x (1 – probability of 1st)ⁿ⁻ˣ
For example:
What is the probability of getting 6 heads, when you toss a coin 10 times?
In a coin-toss experiment, there are two outcomes: heads and tails. Assuming the coin is fair , the probability of getting a head is 1/2 or 0.5 .
The number of repeated trials: n=10
The number of success trials: x = 6
The probability of success on individual trial: p = 0.5
Use the formula for binomial probability.
¹⁰C₆ (0.5)⁶ x (1 – 0.5)¹⁰⁻⁶
Simplify.
≈0.205
Here, we have given that:
Probability of winning on an arcade game is 0.659
so, Probability of loosing on an arcade game is 1-0.659 = 0.341
Number of times game played = 30
Probability of winning exactly 21 times = ?
now, n = 30
x = 21
probability of 1st or probability of winning = 0.659
1 – probability of 1st or probability of loosing = 0.341
using, binomial probability formula
Probability of winning exactly 21 times = ³⁰C₂₁ (0.659)²¹ x (0.341)⁷
on solving,
Probability of winning exactly 21 times = 0.14
Hence,
The probability of winning exactly 21 times is 0.14

format: plain


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# norminv

Normal inverse cumulative distribution function

## Description

example

x = norminv(p) returns the inverse of the standard normal cumulative distribution function (cdf), evaluated at the probability values in p.

x = norminv(p,mu) returns the inverse of the normal cdf with mean mu and the unit standard deviation, evaluated at the probability values in p.

example

x = norminv(p,mu,sigma) returns the inverse of the normal cdf with mean mu and standard deviation sigma, evaluated at the probability values in p.

[x,xLo,xUp] = norminv(p,mu,sigma,pCov) also returns the 95% confidence bounds [xLo,xUp] of x when mu and sigma are estimates. pCov is the covariance matrix of the estimated parameters.

example

[x,xLo,xUp] = norminv(p,mu,sigma,pCov,alpha) specifies the confidence level for the confidence interval [xLo,xUp] to be 100(1–alpha)%.

## Examples

collapse all

Find an interval that contains 95% of the values from a standard normal distribution.

x = norminv([0.025 0.975])
x = 1×2

-1.9600    1.9600

Note that the interval x is not the only such interval, but it is the shortest. Find another interval.

xl = norminv([0.01 0.96])
xl = 1×2

-2.3263    1.7507

The interval x1 also contains 95% of the probability, but it is longer than x.

Compute the inverse of cdf values evaluated at the probability values in p for the normal distribution with mean mu and standard deviation sigma.

p = 0:0.25:1;
mu = 2;
sigma = 1;
x = norminv(p,mu,sigma)
x = 1×5

-Inf    1.3255    2.0000    2.6745       Inf

Compute the inverse of cdf values evaluated at 0.5 for various normal distributions with different mean parameters.

mu = [-2,-1,0,1,2];
sigma = 1;
x = norminv(0.5,mu,sigma)
x = 1×5

-2    -1     0     1     2

Find the maximum likelihood estimates (MLEs) of the normal distribution parameters, and then find the confidence interval of the corresponding inverse cdf value.

Generate 1000 normal random numbers from the normal distribution with mean 5 and standard deviation 2.

rng('default') % For reproducibility
n = 1000; % Number of samples
x = normrnd(5,2,[n,1]);

Find the MLEs for the distribution parameters (mean and standard deviation) by using mle.

phat = mle(x)
phat = 1×2

4.9347    1.9969

muHat = phat(1);
sigmaHat = phat(2);

Estimate the covariance of the distribution parameters by using normlike. The function normlike returns an approximation to the asymptotic covariance matrix if you pass the MLEs and the samples used to estimate the MLEs.

[~,pCov] = normlike([muHat,sigmaHat],x)
pCov = 2×2

0.0040   -0.0000
-0.0000    0.0020

Find the inverse cdf value at 0.5 and its 99% confidence interval.

[x,xLo,xUp] = norminv(0.5,muHat,sigmaHat,pCov,0.01)
x = 4.9347
xLo = 4.7721
xUp = 5.0974

x is the inverse cdf value using the normal distribution with the parameters muHat and sigmaHat. The interval [xLo,xUp] is the 99% confidence interval of the inverse cdf value evaluated at 0.5, considering the uncertainty of muHat and sigmaHat using pCov. The 99% confidence interval means the probability that [xLo,xUp] contains the true inverse cdf value is 0.99.

## Input Arguments

collapse all

Probability values at which to evaluate the inverse of the cdf (icdf), specified as a scalar value or an array of scalar values, where each element is in the range [0,1].

If you specify pCov to compute the confidence interval [xLo,xUp], then p must be a scalar value.

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, norminv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [0.1,0.5,0.9]

Data Types: single | double

Mean of the normal distribution, specified as a scalar value or an array of scalar values.

If you specify pCov to compute the confidence interval [xLo,xUp], then mu must be a scalar value.

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, norminv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [0 1 2; 0 1 2]

Data Types: single | double

Standard deviation of the normal distribution, specified as a positive scalar value or an array of positive scalar values.

If you specify pCov to compute the confidence interval [xLo,xUp], then sigma must be a scalar value.

To evaluate the icdf at multiple values, specify p using an array. To evaluate the icdfs of multiple distributions, specify mu and sigma using arrays. If one or more of the input arguments p, mu, and sigma are arrays, then the array sizes must be the same. In this case, norminv expands each scalar input into a constant array of the same size as the array inputs. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Example: [1 1 1; 2 2 2]

Data Types: single | double

Covariance of the estimates mu and sigma, specified as a 2-by-2 matrix.

If you specify pCov to compute the confidence interval [xLo,xUp], then p, mu, and sigma must be scalar values.

You can estimate mu and sigma by using mle, and estimate the covariance of mu and sigma by using normlike. For an example, see Confidence Interval of Inverse Normal cdf Value.

Data Types: single | double

Significance level for the confidence interval, specified as a scalar in the range (0,1). The confidence level is 100(1–alpha)%, where alpha is the probability that the confidence interval does not contain the true value.

Example: 0.01

Data Types: single | double

## Output Arguments

collapse all

icdf values, evaluated at the probability values in p, returned as a scalar value or an array of scalar values. x is the same size as p, mu, and sigma after any necessary scalar expansion. Each element in x is the icdf value of the distribution specified by the corresponding elements in mu and sigma, evaluated at the corresponding element in p.

Lower confidence bound for x, returned as a scalar value or an array of scalar values. xLo has the same size as x.

Upper confidence bound for x, returned as a scalar value or an array of scalar values. xUp has the same size as x.

collapse all

### Normal Distribution

The normal distribution is a two-parameter family of curves. The first parameter, µ, is the mean. The second parameter, σ, is the standard deviation.

The standard normal distribution has zero mean and unit standard deviation.

The normal inverse function is defined in terms of the normal cdf as

$x={F}^{-1}\left(p|\mu ,\sigma \right)=\left\{x:F\left(x|\mu ,\sigma \right)=p\right\},$

where

$p=F\left(x|\mu ,\sigma \right)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\infty }^{x}{e}^{\frac{-{\left(t-\mu \right)}^{2}}{2{\sigma }^{2}}}dt.$

The result x is the solution of the integral equation where you supply the desired probability p.

## Algorithms

• The norminv function uses the inverse complementary error function erfcinv. The relationship between norminv and erfcinv is

$\text{norminv}\left(p\right)=-\sqrt{2}\text{ }\text{erfcinv}\left(2p\right)$

The inverse complementary error function erfcinv(x) is defined as erfcinv(erfc(x))=x, and the complementary error function erfc(x) is defined as

$\text{erfc}\left(x\right)=1-\text{erf}\left(x\right)=\frac{2}{\sqrt{\pi }}{\int }_{x}^{\infty }{e}^{-{t}^{2}}dt.$

• The norminv function computes confidence bounds for x by using the delta method. norminv(p,mu,sigma) is equivalent to mu + sigma*norminv(p,0,1). Therefore, the norminv function estimates the variance of mu + sigma*norminv(p,0,1) using the covariance matrix of mu and sigma by the delta method, and finds the confidence bounds using the estimates of this variance. The computed bounds give approximately the desired confidence level when you estimate mu, sigma, and pCov from large samples.

## Alternative Functionality

• norminv is a function specific to normal distribution. Statistics and Machine Learning Toolbox™ also offers the generic function icdf, which supports various probability distributions. To use icdf, create a NormalDistribution probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function norminv is faster than the generic function icdf.

## References

[1] Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions. New York: Dover, 1964.

[2] Evans, M., N. Hastings, and B. Peacock. Statistical Distributions. 2nd ed. Hoboken, NJ: John Wiley & Sons, Inc., 1993.


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# Hep Corrections

HepCorrections represents a collaboration of academics, public health practitioners and advocates interested in the elimination of hepatitis C disease in the...


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User alastair irving - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:07:46Z http://mathoverflow.net/feeds/user/8084 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/123863/class-numbers-of-the-field-generated-by-21-n Class numbers of the field generated by 2^{1/n} Alastair Irving 2013-03-07T12:43:28Z 2013-03-07T12:43:28Z
I have used sage to compute the class number of the number field generated by the polynomial $x^n-2$ for small $n$. Specifically, setting proof=False, (which hopefully just means that GRH is assumed), the class number is $1$ for all $n\leq 40$. This seems slightly surprising.

I have two questions:

Does anyone know if these class numbers have been computed for larger $n$? If so do we know a $n$ for which it isn't $1$?

Heuristically, how should the class numbers behave? Presumably there should be infinitely many $n$ for which its $1$, but is there a heuristic justification of why I haven't found a single $n$ where its not $1$. Obviously I've only considered very small $n$ but with other families of fields, real quadratic for example, its quite easy to find examples with class number $>1$.

 http://mathoverflow.net/questions/100346/are-there-examples-of-sets-containing-no-primes-but-for-which-both-type-i-and-typ Are there examples of sets containing no primes but for which both Type I and Type II information can be proven? Alastair Irving 2012-06-22T11:10:54Z 2012-06-22T17:33:18Z
In Harman's book "Prime Detecting Sieves" he describes a method to prove that a set contains primes if we have enough Type I and Type II information for it. As shown by Selberg's example of the set of numbers with an even number of prime factors Type I information is not sufficient to detect primes. Harman's methods require the Type II information to be given on sufficiently long intervals so that the sums we cannot give assymptotic formulae for are sufficiently small. Is it possible to prove, by giving a suitable counterexample, that if we have Type I information but only a very small amount of Type II information then we can't detect primes?

Let $A$ be the set in which we are interested, and suppose that $A\subseteq B$, the set of integers in $[x/2,x)$. Then, for a suitably small error $E$ and arbitrary bounded coefficients $a_m,b_n$ a Type I estimate is of the form $\sum_{mn\in A,m\leq M}a_m=\lambda \sum_{mn\in B,m\leq M}a_m+O(E)$ and a Type II sum is of the form $\sum_{mn\in A,x^\alpha\leq m\leq x^{\lpha+\beta}}a_mb_n=\lambda \sum_{mn\in B,x^\alpha\leq m\leq x^{\lpha+\beta}}a_mb_n+O(e).$ I am then wondering whether there are examples of sets satisfying these conditions for some $M,\alpha$ and sufficiently small $\beta$ which contain no primes. For definiteness take $M=x^{3/4}$ and $\alpha=1/4$.
 http://mathoverflow.net/questions/92599/distribution-of-primes-in-small-intervals/92610#92610 Answer by Alastair Irving for Distribution of primes in small intervals Alastair Irving 2012-03-29T21:01:18Z 2012-03-29T21:01:18Z
A weaker question is to ask for which functions $f$ the interval $[x,x+f(x)]$ contains a prime for all sufficiently large $x$. The sharpest uncoditional result is then that $f(x)\geq x^{0.525}$ is sufficient. We are therefore a long way from being able to prove results about $f(x)=\log^c x$.
 http://mathoverflow.net/questions/47495/prime-factorization-theory/47548#47548 Answer by Alastair Irving for Prime factorization theory Alastair Irving 2010-11-27T23:03:24Z 2010-11-27T23:03:24Z
Suppose $p,q$ are prime. Thus $\omega,\Omega,\lambda$ are known constants at $p,q$ so we want to know if $\omega,\Omega(p+q)$ can be written as functions in $p,q$. To answer this you need to define what is meant by a function, as clearly $\omega(p+q)$ is a function in $p,q$.

Presumably you mean a function which is built from some class of elementary functions. Therefore, a possible approach would be to try and prove that certain classes of functions are not sufficient. For example, we cannot have $\omega(p+q)=Ap+Bq+C$, for constants $A,B,C$, as we can pick suitable combinations of primes to give a system of linear equations in $a,B,C$ with no solutions, (I haven't actually done this computation). It seems reasonable that for any degree $d$ we can find suitable pairs of primes to show that the function isn't a polynomial with degree $\leq d$, but I don't have a proof of this. Thus showing that the function isn't a polynomial might be a worthwhile first It will probably help to use that the function is symmetric in $p,q$ so if its a polynomial then its a polynomial in $p+q,pq$.

Once polynomials are dealt with, then try and extend the class of possible functions to something bigger.
 http://mathoverflow.net/questions/47436/is-there-an-elementary-proof-that-the-mertens-function-is-not-ox-theta-if Is there an elementary proof that the Mertens function is not $O(x^\theta)$ if $\theta <1/2$? Alastair Irving 2010-11-26T15:24:15Z 2010-11-26T18:47:53Z
The Mertens function is the partial sums of the Moebius function: $M(x)=\sum_{n\leq x}\mu(n)$ Since the zeta-function has a zero on the critical line it follows that $M(x)\ne O(x^\theta)$ for any $\theta<\frac 12$.

Does anyone know if there is an elementary proof of this statement? (By elementary I mean a proof which does not depend on complex analysis, in particular the existance of a zero of $\zeta$). even an elementary proof of $M(x)$ being unbounded would be interesting to me.
 http://mathoverflow.net/questions/123863/class-numbers-of-the-field-generated-by-21-n Comment by Alastair Irving Alastair Irving 2013-03-08T11:07:19Z 2013-03-08T11:07:19Z Yes, I did mean the field formed by adjoining a single root of $x^n-2$, so I agree this is a very similar question to mathoverflow.net/questions/88288/ and therefore it should be closed. http://mathoverflow.net/questions/119815/frequency-of-the-gaps-between-consecutive-primes Comment by Alastair Irving Alastair Irving 2013-01-25T09:18:51Z 2013-01-25T09:18:51Z assuming I understand what you're asking then I think this is a very uninteresting sum, by rearranging the terms we see that $\sum kF(k)$ is simply the length of the interval $[2,p_n]$, so your result follows trivially.


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JH

# If $a \neq 0$ and $n$ is a positive integer, find the partial fraction decomposition of$$f(x) = \frac{1}{x^n (x - a)}$$[Hint: First find the coefficient of $\frac{1}{(x - a)}$. Then subtract the resulting term and simplify what is left.]

## $$-\frac{1}{a^{n} x}-\frac{1}{a^{n-1} x^{2}}-\dots-\frac{1}{a x^{n}}+\frac{1}{a^{n}(x-a)}$$

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### Video Transcript

Let's find the partial fraction in opposition. And here let's not that we're given is non zero. So here we can rewrite this. So for the first term X event, let's use what the author calls case, too. This is when you have repeated linear factors. So then a two over X squared and we go all the way up xn and then we'LL have a case One. This is a non repeatedly in your factor X minus a with good and multiply And both sides by the denominator on the left So on the right hand side and so on and we'LL have all the way up to a and X minus a and then we'LL have be accident Now the next step would be to go ahead and combine depending on the power of X So flu pullout xn one a one plus me Pull out a xn minus one. You get a two minus a one and so on. And when we plotted X day and minus a a and minus one and then finally the constant term and a So now we have comparing the constant terms We have a one on the left. We have a negative and and the right So that gives us a N equals negative one over, eh? And for instance, we can use that to find this. We know this is equal to zero because there's no X on the left. So we have a end here, Let me write it this way. And minus one equals a A and over, eh equals negative. A one over a square. Similarly, a N minus two equals a and minus one. Over, eh? So here, negative one. Over a cute. That's a negative up there and continuing in this fashion level. Me Go on to the next. Actually, let me take a step back here. That should have been in minus one. So what I had. Okay, so let me go. The next page, a one is a two over, eh? A three over a square. Just using the same formula repeatedly. Negative one over eight, then and then be equals negative. A one, because a one plus zero zero zero and then this is positive. One over. And so now that we've found all of our coefficients for the partial fraction to composition, just plug a moment. Negative one. An ex those are capital that was a one and then a two and minus one X squared and so on. Great. And then finally negative. One of rain and exit and and then R B was won over eight of the end, and that's her room was X minus a. There it is. That's our final answer.

JH

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# Inverse Reasons

Consider the reasons .

Note that the product of these two ratios is 1, that is, .

In this case we can state that are inverse reasons.

Two reasons are inverse to each other when their product is equal to 1.

Example:
are inverse reasons because .

Note that, in inverse reasons, the antecedent of one is the consequence of the other, and vice versa.


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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
4 초 512 MB 0 0 0 0.000%

## 문제

There was some complicated legend about navigation in futuristic space roads, and large hubs where they intersect, called cosmic crossroads. It was decided later on to skip this part and leave it up to your imagination, so let’s imagine how awesome it could have been, and proceed to the formal statement immediately.

Best regards from the problemsetter

Consider two sets of n lines passing through the point (0; 0; 0) in Euclidean three-dimensional space. Instead of actual lines, you are given two sets, 2n points in each set: ri = (xi; yi; zi) and rj′ = (xj′; yj′; zj′). These are the points where the lines intersect the unit sphere (two opposite points for each line).

It is known that the second set was obtained from the first one by some rotation around the origin and shuffling the order of the points. You have to find a permutation π1, . . . , π2n and the rotation φ of three-dimensional space such that, for all i,

rπi − φ(ri′)| ≤ 10−6.

The rotation φ should be described by point e = (x; y; z) and angle θ. It means we rotate the space around the axis from the origin to point e by angle θ following the right-hand rule (see notes).

It is guaranteed that the directions of the first set of lines were chosen uniformly at random, independently from each other.

## 입력

The first line of input contains a single integer n (2 ≤ n ≤ 4 · 104).

Each of the next 2n lines contains three real numbers xi, yi and zi describing points from the first set.

Each of the next 2n lines contains three real numbers xi′, yi′ and zi′ describing points from the second set.

All real numbers are given with the precision of up to 12 digits after decimal point.

It also holds that if point ri = (xi; yi; zi) lies in one of sets then the point −ri lies in that set as well.

It is guaranteed that solution exists and would still exist even with precision of 10−9 instead of 10−6.

## 출력

On the first line, print a single real number θ (−π ≤ θ ≤ π) describing the rotation angle.

On the second line, print three real numbers x, y, z (10−3 ≤ |x| + |y| + |z| ≤ 103) describing the point e on the rotation axis.

On the third line, print 2n integers π1, . . . , π2n.

Assume that rotation occurs in the sense prescribed by the right-hand rule (see notes).

## 예제 입력 1

2
0.923879533 0.382683432 0
0.923879533 -0.382683432 0
-0.923879533 -0.382683432 0
-0.923879533 0.382683432 0
0.382683432 0.923879533 0
0.382683432 -0.923879533 0
-0.382683432 -0.923879533 0
-0.382683432 0.923879533 0

## 예제 출력 1

-1.570796327
0.000000000 0.000000000 1.000000000
2 3 4 1

## 힌트

Just so you remember, rotation is done as follows:

Example test case looks like this:

The first set of points is {1, 2, 3, 4}, and the second set of points is {1′, 2′, 3′, 4′}. After the rotation of the second set of lines by −π/2 around axis (0; 0; 1), we obtain the following matching: {(1′; 2),(2′; 3),(3′; 4),(4′, 1)}.


text:
# Category:Tschirnhaus Transformations

This category contains results about Tschirnhaus Transformations.

Let $\map f x$ be a polynomial over a field $k$:

$\map f x = a_n x^n + a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \cdots + a_1 x + a_0$

Then the Tschirnhaus transformation is the linear substitution $x = y - \dfrac {a_{n - 1} } {n a_n}$.

The Tschirnhaus transformation produces a resulting polynomial $\map {f'} y$ which is depressed, as shown on Tschirnhaus Transformation yields Depressed Polynomial.

This technique is used in the derivation of Cardano's Formula for the roots of the general cubic.

## Pages in category "Tschirnhaus Transformations"

This category contains only the following page.


text:
Tutorials

# Part 3: Full-Waveform Inversion (FWI)

Dr. Rhodri Nelson
# If running on colab, uncomment the line below and run this cell:
#!pip in stall devito

This notebook highlights various aspects of seismic inversion based on Devito operators. In this example we aim to illustrate the core ideas behind seismic inversion, where we create an image of the subsurface from field recorded data.

## #Algorithm outline and requirements

Seismic inversion relies on two known parameters:

• Field data - or also called recorded/observed data. This is a shot record corresponding to the true velocity model. In order to simplify this tutorial we will fake field data by modelling it with the true velocity model.

• Initial velocity model. This is a velocity model that has been obtained by processing the field data. This model is a rough and very smooth estimate of the velocity as an initial estimate for the inversion. This is a necessary requirement for any optimization (method).

# #Imaging with back-propagation

## #The Objective Function

• The central purpose of FWI is to find a physical model of the wave-transmitting medium that minimises the difference between an observed dataset and the same dataset as predicted by the model.
• A common way of doing this is to measure the $L_2-norm$ of the difference between our predicted data set p and the observed data set d.
$$$\phi(\mathbf{m})=\frac{1}{2}||\mathbf{p}-\mathbf{d}||^2=\frac{1}{2}||\delta\mathbf{d}||^2=\frac{1}{2}\delta\mathbf{d}^{T}\delta\mathbf{d}.$$$
(1)#
• We now want to minimize this functional with respect to the model $\mathbf{m}$ (which in the acoustic example here will be $1/c^2$).

## #Local Inversion

The model update we need, $\delta\mathbf{m}$, to minimize the objective function is:

$$$\delta\mathbf{m} \approx - \left(\frac{\partial^2 \phi}{\partial\mathbf{m}^2}\right)^{-1}\frac{\partial \phi}{\partial\mathbf{m}} \equiv -\mathbf{H}^{-1}\nabla_{\mathbf{m}}\phi. \tag{23}$$$
(2)#
• $\nabla_{\mathbf{m}}\phi$ is the gradient of the objective function $\phi$ with respect to the model parameters
• $\mathbf{H}$ is the Hessian matrix of second differentials, both evaluated at $\mathbf{m}_0$.

## #Steepest Descent

• The Hessian is in general not practical to compute. We could, e.g., simply replace it a scalar $\alpha$, the step length, and find a suitable value for this:
$$$\delta\mathbf{m} = -\alpha\frac{\partial \phi}{\partial \mathbf{m}} = -\alpha\nabla_{\mathbf{m}}\phi . \tag{24}$$$
(3)#

The method that uses this approach is called the method of steepest descent, and in its simplest form it consists of the following steps:

1. start from a model $\mathbf{m}_0$,
2. evaluate the gradient of the objective function, $\nabla_{\mathbf{m}}\phi$, for the current model,
3. find the step length $\alpha$,
4. subtract $\alpha$ times the gradient from the current model to obtain a new model,
5. iterate from step 2 using the new model until the objective function is sufficiently small (or we run out of patience).

To implement this, we need a method of calculating the local gradient.

Note:

Simply ignoring the Hessian is a gross simplification and while it is not normally possible to incorporate its effects fully, there are several possibilities for approximating its effects - L-BFGS is widely used, as are conjugate gradients.

In principle, we could find the gradient by perturbing each of the model parameters in turn, and calculating what happens to the objective function each time. For $n$ model parameters, that would require $n+1$ modelling runs, and this is not computationally feasible. Fortunately there is a faster way using a solution to the adjoint problem.

First, write the gradient in terms of the residual data $\delta\mathbf{d}=\mathbf{p}-\mathbf{d}$:

$$$\nabla_{\mathbf{m}}\phi=\frac{\partial \phi}{\partial \mathbf{m}}=\frac{\partial}{\partial \mathbf{m}}\left(\frac{1}{2}\delta\mathbf{d}^T\delta\mathbf{d}\right)=\frac{\partial (\mathbf{p}-\mathbf{d})^T}{\partial \mathbf{m}}\delta\mathbf{d}=\left(\frac{\partial \mathbf{p}}{\partial \mathbf{m}}\right)^T\delta\mathbf{d}. \tag{25}$$$
(4)#

After playing around with some maths (don't worry about the details!) we end up with

$$$\nabla_{\mathbf{m}}\phi=-\mathbf{u}^T\left(\frac{\partial \mathbf{A}}{\partial \mathbf{m}}\right)^T(\mathbf{A^{-1}})^T\mathbf{R}^T\delta\mathbf{d}. \tag{31}$$$
(5)#

That is, to find the gradient, we must

• Calculate the forward wavefield $\mathbf{u}$
• Differentiate the numerical operator $\mathbf{A}$ with respect to the model parameters (this is an operation that we can do analytically).
• We must compute $(\mathbf{A}^{-1})^T\mathbf{R}^T\delta\mathbf{d}$.

## #Interpreting the expression for the gradient

• The final term our definition of the gradient, $(\mathbf{A}^{-1})^T\mathbf{R}^T\delta\mathbf{d}$, turns out to correspond to what we call the residual wave field $\delta\mathbf{u}$.

• $\delta\mathbf{u}$ that is generated by a (virtual) source $\delta\mathbf{d}$ located at the receivers, and that is propagated by the operator $\mathbf{A}^T$ which is the adjoint of the operator in the original wave equation.

• For one source, the gradient calculated this way requires only two modelling runs rather than the $n+1$ modelling runs that direct methods require.

• For multi-source datasets, the full gradient is a sum over all sources.

• In practical applications with real datasets, the wave equation will nearly always be modified in various ways to include additional physics, but this does not change the underlying approach.

## #Notes on the operators

In Part 2 we described in detail the creation of a forward modelling operator -- here we will call a wrapper function to such a model instead. This wrapper contains all necessary operators for seismic modeling, imaging and inversion. Operators introduced for the first time in this part will be properly described.

import numpy as np
%matplotlib inline

from devito import configuration
configuration['log-level'] = 'WARNING'

## #Computational considerations

As we will see, FWI is extremely computationally demanding (more so than, e.g., RTM). To keep this tutorial as lightweight as possible we therefore again use a very small demonstration model. We also define here a few parameters for the final example runs that can be changed to modify the overall runtime of the tutorial.

nshots = 9  # Number of shots to create gradient from
fwi_iterations = 5  # Number of outer FWI iterations

# #True and smooth velocity models

We will use a very simple model domain, consisting of a circle within a 2D domain. We will again use the "true" model to generate our synthetic shot data and use a "smooth" model as our initial guess. In this case the smooth model is very smooth indeed - it is simply a constant background velocity without any features.

from examples.seismic import demo_model, plot_velocity, plot_perturbation

# Define true and initial model
shape = (101, 101)  # Number of grid point (nx, nz)
spacing = (10., 10.)  # Grid spacing in m. The domain size is now 1km by 1km
origin = (0., 0.)  # Need origin to define relative source and receiver locations

model = demo_model('circle-isotropic', vp_circle=3.0, vp_background=2.5,
origin=origin, shape=shape, spacing=spacing, nbl=40)

model0 = demo_model('circle-isotropic', vp_circle=2.5, vp_background=2.5,
origin=origin, shape=shape, spacing=spacing, nbl=40,
grid = model.grid)

plot_velocity(model)
plot_velocity(model0)

## #Acquisition geometry

In this tutorial, we will use the easiest case for inversion, namely a transmission experiment. The sources are located on one side of the model and the receivers on the other side. This allows to record most of the information necessary for inversion, as reflections usually lead to poor inversion results.

# Define acquisition geometry: source
from examples.seismic import AcquisitionGeometry

t0 = 0.
tn = 1000.
f0 = 0.010
# First, position source centrally in all dimensions, then set depth
src_coordinates = np.empty((1, 2))
src_coordinates[0, :] = np.array(model.domain_size) * .5
src_coordinates[0, 0] = 20.  # Depth is 20m

# Initialize receivers for synthetic and imaging data
rec_coordinates[:, 1] = np.linspace(0, model.domain_size[0], num=nreceivers)
rec_coordinates[:, 0] = 980.

# Geometry

geometry = AcquisitionGeometry(model, rec_coordinates, src_coordinates, t0, tn, f0=f0, src_type='Ricker')
# We can plot the time signature to see the wavelet
geometry.src.show()
# Plot acquisition geometry
plot_velocity(model, source=geometry.src_positions,
receiver=geometry.rec_positions[::4, :])

## #True and smooth data

We can generate shot records for the true and smoothed initial velocity models, since the difference between them will again form the basis of our imaging procedure.

# Compute synthetic data with forward operator
from examples.seismic.acoustic import AcousticWaveSolver

solver = AcousticWaveSolver(model, geometry, space_order=4)
true_d, _, _ = solver.forward(vp=model.vp)
# Compute initial data with forward operator
smooth_d, _, _ = solver.forward(vp=model0.vp)
from examples.seismic import plot_shotrecord

# Plot shot record for true and smooth velocity model and the difference
plot_shotrecord(true_d.data, model, t0, tn)
plot_shotrecord(smooth_d.data, model, t0, tn)
plot_shotrecord(smooth_d.data - true_d.data, model, t0, tn)

To compute a single gradient $\nabla\Phi_s(\mathbf{m})$ in our optimization workflow we again use solver.forward to compute the entire forward wavefield $\mathbf{u}$ and a similar pre-defined gradient operator to compute the adjoint wavefield v. The gradient operator provided by our solver utility also computes the correlation between the wavefields, allowing us to proceed as follows:

• Simulate the forward wavefield with the background velocity model to get the synthetic data and save the full wavefield $\mathbf{u}$
• Compute the data residual
• Back-propagate the data residual and compute on the fly the gradient contribution at each time step.

This procedure is applied to multiple source positions and summed to obtain a gradient image of the subsurface. We again prepare the source locations for each shot and visualize them, before defining a single gradient computation over a number of shots as a single function.

# Prepare the varying source locations sources
source_locations = np.empty((nshots, 2), dtype=np.float32)
source_locations[:, 0] = 30.
source_locations[:, 1] = np.linspace(0., 1000, num=nshots)

plot_velocity(model, source=source_locations)
from devito import Eq, Operator

# Computes the residual between observed and synthetic data into the residual
def compute_residual(residual, dobs, dsyn):
if residual.grid.distributor.is_parallel:
# If we run with MPI, we have to compute the residual via an operator
# First make sure we can take the difference and that receivers are at the
# same position
assert np.allclose(dobs.coordinates.data[:], dsyn.coordinates.data)
assert np.allclose(residual.coordinates.data[:], dsyn.coordinates.data)
# Create a difference operator
diff_eq = Eq(residual, dsyn.subs({dsyn.dimensions[-1]: residual.dimensions[-1]}) -
dobs.subs({dobs.dimensions[-1]: residual.dimensions[-1]}))
Operator(diff_eq)()
else:
# A simple data difference is enough in serial
residual.data[:] = dsyn.data[:] - dobs.data[:]

return residual
# Create FWI gradient kernel
from devito import Function, TimeFunction, norm

import scipy
# Create symbols to hold the gradient
# Create placeholders for the data residual and data
time_range=geometry.time_axis,
coordinates=geometry.rec_positions)
time_range=geometry.time_axis,
coordinates=geometry.rec_positions)
time_range=geometry.time_axis,
coordinates=geometry.rec_positions)
objective = 0.
for i in range(nshots):
# Update source location
geometry.src_positions[0, :] = source_locations[i, :]

# Generate synthetic data from true model
_, _, _ = solver.forward(vp=model.vp, rec=d_obs)

# Compute smooth data and full forward wavefield u0
_, u0, _ = solver.forward(vp=vp_in, save=True, rec=d_syn)

# Compute gradient from data residual and update objective function
compute_residual(residual, d_obs, d_syn)

objective += .5*norm(residual)**2

return objective, grad

Having defined our FWI gradient procedure we can compute the initial iteration from our starting model. This allows us to visualize the gradient alongside the model perturbation and the effect of the gradient update on the model.

# Compute gradient of initial model
assert np.isclose(ff, 57283, rtol=1e0)
from devito import mmax
from examples.seismic import plot_image

plot_image(-update.data, vmin=-1e4, vmax=1e4, cmap="jet")

# Plot the difference between the true and initial model.
# This is not known in practice as only the initial model is provided.
plot_image(model0.vp.data - model.vp.data, vmin=-1e-1, vmax=1e-1, cmap="jet")

# Show what the update does to the model
alpha = .5 / mmax(update)
plot_image(model0.vp.data + alpha*update.data, vmin=2.5, vmax=3.0, cmap="jet")

We see that the gradient and the true perturbation have the same sign, therefore, with an appropriate scaling factor, we will update the model in the correct direction.

from sympy import Min, Max
# Define bounding box constraints on the solution.
def update_with_box(vp, alpha, dm, vmin=2.0, vmax=3.5):
"""
Apply gradient update in-place to vp with box constraint

Notes:
------
For more advanced algorithm, one will need to gather the non-distributed
velocity array to apply constrains and such.
"""
update = vp + alpha * dm
update_eq = Eq(vp, Max(Min(update, vmax), vmin))
Operator(update_eq)()
from devito import mmax

# Run FWI with gradient descent
history = np.zeros((fwi_iterations, 1))
for i in range(0, fwi_iterations):
# Compute the functional value and gradient for the current
# model estimate

# Store the history of the functional values
history[i] = phi

# Artificial Step length for gradient descent
# In practice this would be replaced by a Linesearch (Wolfe, ...)
# that would guarantee functional decrease Phi(m-alpha g) <= epsilon Phi(m)
# where epsilon is a minimum decrease constant
alpha = .05 / mmax(direction)

# Update the model estimate and enforce minimum/maximum values
update_with_box(model0.vp , alpha , direction)

print('Objective value is %f at iteration %d' % (phi, i+1))
# Plot inverted velocity model
plot_velocity(model0)
import matplotlib.pyplot as plt

# Plot objective function decrease
plt.figure()
plt.loglog(history)
plt.xlabel('Iteration number')
plt.ylabel('Misift value Phi')
plt.title('Convergence')
plt.show()

## #Final remarks

• For further examples and tutorials see the examples folder in the main Devito repository located here. Implementations of different wave-equations, PML boundary conditions, etc., can be found there.
• Out of the box, Devito can generate MPI code, GPU code (OpenMP, OpenACC, Cuda) and is compatible with ARM architectures.
• Get in touch!


text:
Hello,

I'm manipulating large quantities of HTML files (about 2000). I'm new with Python, and would appreciate any help.

For example, I want to delete the lines that contain "embed_music" in all the files, or change all instances of the word "Paragraph" to "Absatz".

This is my pseudo-code:

open target folder of html files (/project/html/)
read in all html files
*do the stuff here:
check for lines containing "embed_music", if yes delete
string replace for words with "Paragraph" to "Absatz"
*
close folder

Is my logic correct? Would appreciate any help or feedback!

3
Contributors
3
Replies
4
Views
5 Years
Discussion Span
Last Post by ihatehippies

to get you started...

*untested

import os

for name in os.listdir(aDirectory):
if not name.lower().endswith('.html'):
continue
with open(os.path.join(aDirectory, name), 'rb') as fileobj:
output = [line.replace("Paragraph", "Absatz") for line in fileobj.readlines() if not "embed_music" in line]
with open(os.path.join(aDirectory, name), 'wb') as fileobj:
fileobj.write(''.join(output))

You may want to create a backup of the folder before you run the code....

Edited by ihatehippies: n/a

Nice ihatehippies, I only unified the with to produce lines to output immediately to avoid temporary list (if you do lot of processing it might still pay off to use one and output from that), changed the name aDirectory to follow PEP8 convetion of words joined by underscore and removed unnecessary readlines as we can use fileobj directly as generator of lines.

import os
html_directory = r'I:\python27\Lib\site-packages\pygame\docs\ref'
pre = 'absatz'
for name in os.listdir(html_directory):
if name.lower().endswith('.html') and not name.startswith(pre):
with open(os.path.join(html_directory, name), 'rb') as fileobj, open(os.path.join(html_directory, '%s_%s' % (pre, name)), 'wb') as output:
output.write(''.join(line.replace("Paragraph", "Absatz")
for line in fileobj if not "embed_music" in line))

Of course you never should use this kind of code for html file but use Beautifulsoup3/Beautifulsoup4 or similar to preserve the structure of files.

Edited by pyTony: n/a

yeah I wasn't sure if you could open a file for writing if it was currently being read

This topic has been dead for over six months. Start a new discussion instead.
Have something to contribute to this discussion? Please be thoughtful, detailed and courteous, and be sure to adhere to our posting rules.


text:
# Confusion Matrix and Metrics

#### 2021 03 31

Behold the aptly named “confusion matrix”:

Condition
Decision present (TRUE): absent (FALSE): Sum: (b) by decision:
positive (TRUE): hi fa dec_pos PPV = hi/dec_pos
negative (FALSE): mi cr dec_neg NPV = cr/dec_neg
Sum: cond_true cond_false N prev = cond_true/N
(a) by condition sens = hi/cond_true spec = cr/cond_false ppod = dec_pos/N acc = dec_cor/N = (hi+cr)/N

Most people, including medical experts and social scientists, struggle to understand the implications of this matrix. This is no surprise when considering explanations like the corresponding article on Wikipedia, which squeezes more than a dozen metrics out of four essential frequencies (hi, mi, fa, and cr). While each particular metric is quite simple, their abundance and inter-dependence can be overwhelming.

Fortunately, the key representational structure can also be understood as a 2x2 matrix (aka. 2-by-2 contingency table), which is actually quite simple, and rather straightforward in its implications. In the following, we identify the most important aspects and some key measures that a risk-literate person should know (see Neth et al., 2021, for a more detailed account).

## Basics

Condensed to its core, the 2x2 matrix cross-tabulates (or “confuses”) two binary dimensions and classifies each individual case into one of four possible categories that result from combining the two binary variables (e.g., the condition and decision of each case) with each other. This sounds more complicated than it is:

Condition
Decision present (TRUE): absent (FALSE):
positive (TRUE): hi fa
negative (FALSE): mi cr

Fortunately, this table is not so confusing any more: It shows four frequency counts (or “joint” frequencies) that result from cross-tabulating two binary dimensions. And, perhaps surprisingly, all other metrics of interest in various contexts and domains follow from this simple core in a straightforward way. In the following, we illustrate how the other metrics can be constructed from the four essential frequencies.

### Adopting two perspectives on a population

Essentially, the confusion matrix views a population of N individuals in different ways by adopting different perspectives. “Adopting a perspective” means that we can distinguish between individuals on the basis of some criterion. The two primary criteria used here are:

(a) each individual’s condition, which can either be present (TRUE) or absent (FALSE), and
(b) each individual’s decision, which can either be positive (TRUE) or negative (FALSE).

Numerically, the adoption of each of these two perspectives splits the population into two subgroups.1 Applying two different splits of a population into two subgroups results in $$2 \times 2 = 4$$ cases, which form the core of the confusion matrix:

1. hi represents hits (or true positives): condition present (TRUE) & decision positive (TRUE).
2. mi represents misses (or false negatives): condition present (TRUE) & decision negative (FALSE).
3. fa represents false alarms (or false positives): condition absent (FALSE) & decision positive (TRUE).
4. cr represents correct rejections (or true negatives): condition absent (FALSE) & decision negative (FALSE).

Importantly, all frequencies required to understand and compute various metrics are combinations of these four frequencies — which is why we refer to them as the four essential frequencies (see the vignette on Data formats). For instance, adding up the columns and rows of the matrix yields the frequencies of the two subgroups that result from adopting our two perspectives on the population N (or splitting N into subgroups by applying two binary criteria):

(a) by condition (cd) (corresponding to the two columns of the confusion matrix):

\begin{aligned} \texttt{N} \ &= \ \texttt{cond_true} & +\ \ \ \ \ &\texttt{cond_false} & \textrm{(a)} \\ \ &= \ (\texttt{hi} + \texttt{mi}) & +\ \ \ \ \ &(\texttt{fa} + \texttt{cr}) \\ \end{aligned}

(b) by decision (dc) (corresponding to the two rows of the confusion matrix):

\begin{aligned} \texttt{N} \ &= \ \texttt{dec_pos} & +\ \ \ \ \ &\texttt{dec_neg} & \ \ \ \ \textrm{(b)} \\ \ &= \ (\texttt{hi} + \texttt{fa}) & +\ \ \ \ \ &(\texttt{mi} + \texttt{cr}) \\ \end{aligned}

To reflect these two perspectives in the confusion matrix, we only need to add the sums of columns (i.e., by condition) and rows (by decision):

Condition
Decision present (TRUE): absent (FALSE): Sum:
positive (TRUE): hi fa dec_pos
negative (FALSE): mi cr dec_neg
Sum: cond_true cond_false N

An third perspective is provided by considering the diagonals of the 2x2 matrix. In many semantic domains, the diagonals denote the accuracy of the classification, or the correspondence between dimensions (see below).

#### Example

To view a 2x2 matrix (or confusion table) in riskyr, we can use the plot_tab() function (i.e., plot an existing scenario as type = "tab"):

## (1) Plot table from basic input parameters: -----
plot_tab(prev = .05, sens = .75, spec = .66, N = 1000,
p_lbl = "def") # show condition probabilies (by default)

### Accuracy as a third perspective

A third way of grouping the four essential frequencies results from asking the question: Which of the four essential frequencies are correct decisions and which are erroneous decisions? Crucially, this question about decision accuracy can neither be answered by only considering each individual’s condition (i.e., the columns of the matrix), nor can it be answered by only considering each individual’s decision (i.e., the rows of the matrix). Instead, answering the question about accuracy requires that the other dimensions have been determined and then considering the correspondence between condition and decision. Checking the correspondence between rows and columns for the four essential frequencies yields an important insight: The confusion matrix contains two types of correct decisions and two types of errors:

• A decision is correct, when it corresponds to the condition. This is the case for two cells in (or the "" diagonal of) the confusion matrix:

• hi: condition present (TRUE) & decision positive (TRUE)
• cr: condition absent (FALSE) & decision negative (FALSE)
• A decision is incorrect or erroneous, when it does not correspond to the condition. This also is the case for two cells in (or the “/” diagonal of) the confusion matrix:

• mi: condition present (TRUE) & decision negative (FALSE)
• fa: condition absent (FALSE) & decision positive (TRUE)

Splitting all N individuals into two subgroups of those with correct vs. those with erroneous decisions yields a third perspective on the population:

(c) by accuracy (ac) or the correspondence between decisions and conditions (corresponding to the two diagonals of the confusion matrix):

\begin{aligned} \texttt{N} \ &= \ \texttt{dec_cor} & +\ \ \ \ \ &\texttt{dec_err} & \ \ \textrm{(c)} \\ \ &= \ (\texttt{hi} + \texttt{cr}) & +\ \ \ \ \ &(\texttt{mi} + \texttt{fa}) \\ \end{aligned}

#### Example

Re-arranging the cells of the 2x2 matrix allows illustrating accuracy as a third perspective (e.g., by specifying the perspective by = "cdac"):

plot_tab(prev = .05, sens = .75, spec = .66, N = 1000,
by = "cdac", p_split = "h",
p_lbl = "def", title_lbl = "Scenario 2")
#> Argument 'title_lbl' is deprecated. Please use 'main' instead.

#### Avoiding common sources of confusion

It may be instructive to point out two possible sources of confusion, so that they can be deliberately avoided:

1. Beware of alternative terms for mi and cr:

• Misses mi are often called “false negatives” (FN), but are nevertheless cases for which the condition is TRUE (i.e., in the cond_true column of the confusion table).

• Correct rejections cr are often called “true negatives” (TN), but are nevertheless cases for which the condition is FALSE (i.e., in the cond_false column of the confusion table).

Thus, the terms “true” and “false” are sometimes ambiguous by switching their referents. When used to denote the four essential frequencies (e.g., describing mi as “false negatives” and cr as “true negatives”) the terms refer to the correspondence of a decision to the condition, rather than to their condition. To avoid this source of confusion, we prefer the terms mi and cr over “false negatives” (FN) and “true negatives” (TN), respectively, but offer both options as pre-defined lists of text labels (see txt_org and txt_TF).

1. Beware of alternative terms for dec_cor and dec_err:
Similarly, it may be tempting to refer to instances of dec_cor and dec_err as “true decisions” and “false decisions”. However, this would also invite conceptual confusion, as “true decisions” would include cond_false cases (cr or TN cases) and “false decisions” would include cond_true cases (mi or FN cases). Again, we prefer the less ambiguous terms “correct decisions” vs. “erroneous decisions”.

## Accuracy metrics

The perspective of accuracy raises an important question: How good is some decision process (e.g., a clinical judgment or some diagnostic test) in capturing the true state of the condition? Different accuracy metrics provide different answers to this question, but share a common goal — measuring decision performance by capturing the correspondence of decisions to conditions in some quantitative fashion.2

While all accuracy metrics quantify the relationship between correct and erroneous decisions, different metrics emphasize different aspects or serve different purposes. We distinguish between specific and general metrics.

### A. Specific metrics: Conditional probabilities

The goal of a specific accuracy metric is to quantify some particular aspect of decision performance. For instance, how accurate is our decision or diagnostic test in correctly detecting cond_true cases? How accurate is it in detecting cond_false cases?

As we are dealing with two types of correct decisions (hi and cr) and two perspectives (by columns vs. by rows), we can provide 4 answers to these questions. To obtain a numeric quantity, we divide the frequency of correct cases (either hi or cr) by

(a) column sums (cond_true vs. cond_false): This yields the decision’s sensitivity (sens) and specificity (spec):

\begin{aligned} \texttt{sens} \ &= \frac{\texttt{hi}}{\texttt{cond_true}} & \ \ \textrm{(a1)} \\ \ \\ \texttt{spec} \ &= \frac{\texttt{cr}}{\texttt{cond_false}} & \ \ \textrm{(a2)} \\ \end{aligned}

(b) row sums (dec_pos vs. dec_neg): This yields the decision’s positive predictive value (PPV) and negative predictive value (NPV):

\begin{aligned} \texttt{PPV} \ &= \frac{\texttt{hi}}{\texttt{dec_pos}} & \ \ \ \textrm{(b1)} \\ \ \\ \texttt{NPV} \ &= \frac{\texttt{cr}}{\texttt{dec_neg}} & \ \ \ \textrm{(b2)} \\ \end{aligned}

### B. General metrics: Measures of accuracy

In contrast to these specific metrics, general metrics of accuracy aim to capture overall performance (i.e., summarize the four essential frequencies of the confusion matrix) in a single quantity. riskyr currently computes four general metrics (which are contained in accu):

#### 1. Overall accuracy acc

Overall accuracy (acc) divides the number of correct decisions (i.e., all dec_cor cases or the "" diagonal of the confusion table) by the number N of all decisions (or individuals for which decisions have been made). Thus,

Accuracy acc := Proportion or percentage of cases correctly classified.

Numerically, overall accuracy acc is computed as:

\begin{aligned} \texttt{acc} &= \frac{\texttt{hi} + \texttt{cr}}{\texttt{hi} + \texttt{mi} + \texttt{fa} + \texttt{cr}} = \frac{\texttt{dec_cor}}{\texttt{dec_cor} + \texttt{dec_err}} = \frac{\texttt{dec_cor}}{\texttt{N}} \end{aligned}

#### 2. Weighted accuracy wacc

Whereas overall accuracy (acc) does not discriminate between different types of correct and incorrect cases, weighted accuracy (wacc) allows for taking into account the importance of errors. Essentially, wacc combines the sensitivity (sens) and specificity (spec), but multiplies sens by a weighting parameter w (ranging from 0 to 1) and spec by its complement (1 - w):

Weighted accuracy wacc := the average of sensitivity (sens) weighted by w, and specificity (spec), weighted by (1 - w).

\begin{aligned} \texttt{wacc} \ &= \texttt{w} \cdot \texttt{sens} \ + \ (1 - \texttt{w}) \cdot \texttt{spec} \\ \end{aligned}

Three cases can be distinguished, based on the value of the weighting parameter w:

1. If w = .5, sens and spec are weighted equally and wacc becomes balanced accuracy bacc.

2. If 0 <= w < .5, sens is less important than spec (i.e., instances of fa are considered more serious errors than instances of mi).

3. If .5 < w <= 1, sens is more important than spec (i.e., instances of mi are considered more serious errors than instances of fa).

#### 3. Matthews correlation coefficient mcc

The Matthews correlation coefficient (with values ranging from $$-1$$ to $$+1$$) is computed as:

\begin{aligned} \texttt{mcc} \ &= \frac{(\texttt{hi} \cdot \texttt{cr}) \ - \ (\texttt{fa} \cdot \texttt{mi})}{\sqrt{(\texttt{hi} + \texttt{fa}) \cdot (\texttt{hi} + \texttt{mi}) \cdot (\texttt{cr} + \texttt{fa}) \cdot (\texttt{cr} + \texttt{mi})}} \\ \end{aligned}

The mcc is a correlation coefficient specifying the correspondence between the actual and the predicted binary categories. A value of $$0$$ represents chance performance, a value of $$+1$$ represents perfect performance, and a value of $$−1$$ indicates complete disagreement between truth and predictions.

See Wikipedia: Matthews correlation coefficient for details.

#### 4. F1 score

For creatures who cannot live with only three general measures of accuracy, accu also provides the F1 score, which is the harmonic mean of PPV (aka. precision) and sens (aka. recall):

\begin{aligned} \texttt{f1s} \ &= 2 \cdot \frac{\texttt{PPV} \cdot \texttt{sens}}{\texttt{PPV} + \texttt{sens}} \\ \end{aligned}

See Wikipedia: F1 score for details.

For many more additional scientific metrics that are defined on the basis of a 2x2 matrix, see Section 4. Integration (e.g., Figure 6 and Table 3) of Neth et al. (2021).

## References

• Neth, H., Gradwohl, N., Streeb, D., Keim, D.A., & Gaissmaier, W. (2021). Perspectives on the 2x2 matrix: Solving semantically distinct problems based on a shared structure of binary contingencies. Frontiers in Psychology, 11, 567817. doi: 10.3389/fpsyg.2020.567817 (Available online)

• Phillips, N. D., Neth, H., Woike, J. K., & Gaissmaier, W. (2017). FFTrees: A toolbox to create, visualize, and evaluate fast-and-frugal decision trees. Judgment and Decision Making, 12, 344–368. (Available online: pdf | html | R package )

### Resources

The following resources and versions are currently available:

Type: Version: URL:
A. riskyr (R package): Release version https://CRAN.R-project.org/package=riskyr
Development version https://github.com/hneth/riskyr/
B. riskyrApp (R Shiny code): Online version https://riskyr.org/
Development version https://github.com/hneth/riskyrApp/
C. Online documentation: Release version https://hneth.github.io/riskyr/
Development version https://hneth.github.io/riskyr/dev/


text:
Question

# Column IColumn II(A) If the roots of the equation(P)7x3−9x2+26x−k=0 are positiveand in A.P., then k is equal to(B) If the roots of the equation(Q)11x3−14x2+kx−64=0 are positiveand in G.P., then k is equal to(C) If the roots of the equation(R)246x3−kx2+6x−1=0 are positiveand in H.P., then k is equal to(D)The harmonic mean for the roots of(S)26equation x3−11x2+3x−26=0 is(T)56 Which of the following is the only CORRECT combination?

A
(A)(Q),(B)(T),(C)(S),(D)(R)
B
(A)(R),(B)(S),(C)(Q),(D)(Q)
C
(A)(R),(B)(T),(C)(Q),(D)(S)
D
(A)(Q),(B)(P),(C)(R),(D)(S)

Solution

## The correct option is C (A)→(R),(B)→(T),(C)→(Q),(D)→(S)(A) Let the roots be a−d,a,a+d ⇒a−d+a+a+d=9 ⇒a=3 So 3 is the root of the equation x3−9x2+26x−k=0 ⇒33−9(3)2+26(3)−k=0 ⇒k=24 (A)→(R) (B) Let the roots be ar,a,ar ⇒(ar)(a)(ar)=64 ⇒a=4 So 4 is the root of the equation x3−14x2+k−64=0 ⇒43−14(4)2+k(4)−64=0 ⇒k=56 (B)→(T) (C) f(x)=6x3−kx2+6x−1=0, here roots are in H.P. f(1x)=x3−6x2+kx−6=0, here the roots will be in A.P. Let the roots be a−d,a,a+d ⇒a−d+a+a+d=6 ⇒a=2 So 2 is the root of the equation x3−6x2+kx−6=0 ⇒k=11 (C)→(Q) (D) x3−11x2+3x−26=0 Let the roots be α,β,γ H.M.=31α+1β+1γ=3αβγαβ+βγ+γα=26 (D)→(S)

Suggest corrections


text:
# Solutions of linear ODE with INTEGER coefficeints

A meromorphic linear ODE can be solved by Frobenius method at a regular singular point. I'm interested in the solutions with integer coefficients, i.e. $$f(x)=x^\alpha\sum_{n=0}^{\infty}a_nx^n,\ \ \alpha\in\mathbb{C},\ a_n\in\mathbb{Z}.$$ I would like know what kind of linear ODE has such solutions and what kind of linear ODE does not. Does anyone know criteria?

• Whenever the recurrence relation for coefficients you get when using the Frobenius method admits for the integer solutions you have the solution with integer coefficients? – Kiryl Pesotski Jun 8 '17 at 16:00
• That's right. But the recurrence relation is not always easy. So I want other criteria, if there are any. – user356126 Jun 9 '17 at 0:22


text:
# American Institute of Mathematical Sciences

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May  2015, 14(3): 825-842. doi: 10.3934/cpaa.2015.14.825

## Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations

 1 Department of Mathematical Sciences, KAIST, 335 Gwahangno, Yuseong-gu Daejeon, 305-701, South Korea 2 Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Road, Shanghai, China 3 Department of Mathematics, Okayama University, 3-1-1 Tsushima-naka, Okayama 700-8530, Japan

Received  April 2014 Revised  December 2014 Published  March 2015

For $k =1,\cdots,K,$ let $M_k$ be a $q_k$-dimensional smooth compact framed manifold in $R^N$ with $q_k \in \{1,\cdots,N-1\}$. We consider the equation $-\varepsilon^2\Delta u + V(x)u - u^p = 0$ in $R^N$ where for each $k \in \{1,\cdots,K\}$ and some $m_k > 0,$ $V(x)=|\textrm{dist}(x,M_k)|^{m_k}+O(|\textrm{dist}(x,M_k)|^{m_k+1})$ as $\textrm{dist}(x,M_k) \to 0$. For a sequence of $\varepsilon$ converging to zero, we will find a positive solution $u_{\varepsilon}$ of the equation which concentrates on $M_1\cup \dots \cup M_K$.
Citation: Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825
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D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar [6] V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar [7] J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar [8] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar [9] J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. 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Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [20] M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [21] M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar [22] M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar [23] A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397. doi: 10.1016/0022-1236(86)90096-0. Google Scholar [24] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, \emph{Comm. Math. Phys., 68 (1979), 209. Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd ed., (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [26] C. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Comm. Partial Differential Equations}, 21 (1996), 787. doi: 10.1080/03605309608821208. Google Scholar [27] Y. Kabeya and K. Tanaka, Uniqueness of positive solutions of semilinear elliptic equations in $R^N$ and Séré's non-degeneracy condition,, \emph{Comm. Partial Differential Equations, 24 (1999), 563. doi: 10.1080/03605309908821434. Google Scholar [28] X. Kang and J. Wei, On interacting bumps of semi-classical states of nonlinear Schrödinger equations,, \emph{Adv. Differential Equations}, 5 (2000), 899. Google Scholar [29] T. Kato, Perturbation Theory for Linear Operators,, second ed., (1976). Google Scholar [30] O. Kwon, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 139A (2009), 833. doi: 10.1017/S0308210508000309. Google Scholar [31] C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on unbounded domains,, \emph{Comm. Partial Differential Equations, 16 (1996), 585. doi: 10.1080/03605309108820770. Google Scholar [32] Y. Y. Li, On a singular perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar [33] F. Mahmoudi and A. Malchiodi, Concentration on minimal submainifolds for a singularly perturbed Neumann problem,, \emph{Adv. Math.}, 209 (2007), 460. doi: 10.1016/j.aim.2006.05.014. Google Scholar [34] F. Mahmoudi, R. Mazzeo and F. 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show all references

##### References:
 [1] A. Ambrosetti, A. Malchiodi and W.-M Ni, Singularly perturves elliptic equation with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427. doi: 10.1007/s00220-003-0811-y. Google Scholar [2] A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity,, \emph{J. d'Anayse Math.}, 98 (2006), 317. doi: 10.1007/BF02790279. Google Scholar [3] A. Ambrosetti, A. Malchiodi and A. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Rational Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152. Google Scholar [4] A. Ambrosetti and D. Ruiz, Radial solutions concentrating on sphere of nonlinear Schrödinger equations with vanishing potentials,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 136 (2006), 889. doi: 10.1017/S0308210500004789. Google Scholar [5] M. Badiale. and T. D'Aprile, Concentration around a sphere for a singulary perturbed Schrödinger equation,, \emph{Nonlinear Anal.}, 49 (2002), 947. doi: 10.1016/S0362-546X(01)00717-9. Google Scholar [6] V. Benci. and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential,, \emph{J. Diff. Equat.}, 184 (2002), 109. doi: 10.1006/jdeq.2001.4138. Google Scholar [7] J. Byeon, Standing Waves for nonlinear Schrödinger equations with a radial potential,, \emph{Nonlinear Anal.}, 50 (2002), 1135. doi: 10.1016/S0362-546X(01)00805-7. Google Scholar [8] J. Byeon and Y. Oshita, Existence of multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Comm. Partial Differential Equations}, 29 (2004), 1877. doi: 10.1081/PDE-200040205. Google Scholar [9] J. Byeon and Y. Oshita, Uniqueness of a standing wave for nonlinear Schrödinger equations,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 138A (2008), 975. doi: 10.1017/S0308210507000236. Google Scholar [10] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Rational Mech. Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [11] J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations. II,, \emph{Calc. Var. Partial Differential Equations}, 18 (2003), 207. doi: 10.1007/s00526-002-0191-8. Google Scholar [12] D. Cao, E. S. Noussair and S. Yan, Multiscale-bump standing waves with a critical frequency for nonlinear Schrödinger equations,, to appear in \emph{Trans. of AMS}., (). doi: 10.1090/S0002-9947-08-04348-1. Google Scholar [13] D. Cao and S. Peng, Multi-bump bound states of Schrödinger equations with a critical frequency,, \emph{Math. Ann}, 336 (2006), 925. doi: 10.1007/s00208-006-0021-y. Google Scholar [14] E. N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations,, \emph{J. Differential Equations}, 74 (1998), 120. doi: 10.1016/0022-0396(88)90021-6. Google Scholar [15] E. N. Dancer and S. Yan, On the existence of multipeak solutions for nonlinear field equations on $R^N$,, \emph{Discrete Contin. Dynam. Systems, 6 (2000), 39. Google Scholar [16] E. N. Dancer, K. Y. Lam and S. Yan, The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations,, \emph{Abstr. Apply. Anal.}, 3 (1998), 293. doi: 10.1155/S1085337598000578. Google Scholar [17] M. del Pino and P. L. Felmer, Local mountaion passes for semilinear elliptic problems in unbounded domains,, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121. doi: 10.1007/BF01189950. Google Scholar [18] M. del Pino and P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations,, \emph{J. Funct. Anal.}, 149 (1997), 245. doi: 10.1006/jfan.1996.3085. Google Scholar [19] M. del Pino and P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations,, \emph{Ann. Inst. Henri Poincar\'e}, 15 (1998), 127. doi: 10.1016/S0294-1449(97)89296-7. Google Scholar [20] M. del Pino and P. L. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1. doi: 10.1007/s002080200327. Google Scholar [21] M. del Pino, P. L. Felmer and O. H. Miyagaki, Existence of positive bound states of nonlinear Schrödinger equations with saddle-like potential,, \emph{Nonlinear Anal. TMA, 34 (1998), 979. doi: 10.1016/S0362-546X(97)00593-2. Google Scholar [22] M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pur. App. Math.}, LX (2007), 113. doi: 10.1002/cpa.20135. Google Scholar [23] A. Floer and A. Weinstein, Non spreading wave packets for the cubic Schrödinger equations with a bounded potential,, \emph{J. Funct. 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