text:
Two insulated wiresperpendicular to each other in the same plane carry currents asshown in the figure .
Find the magnitude of thenet magnetic field these wires produce at points and if the10.0 current is to theright.
Answer the question inthe order indicated. Separate your answer with acomma.
=____T

Find the magnitude of thenet magnetic field these wires produce at points and if the10.0 current is to theleft.
Answer the question inthe order indicated. Separate your answer with acomma.
=____T

format: plain


text:
# 11.10 Semidefinite Relaxation of MIQCQO Problems¶

In this case study we will discuss a fairly common application for Semidefinite Optimization: to define a continuous semidefinite relaxation of a mixed-integer quadratic optimization problem. This section is based on the method by Park and Boyd [PB15].

We will focus on problems of the form:

(11.40)$\begin{split}\begin{array}{ll} \minimize & x^T P x + 2q^T x \\ \st &x\in \integral^n \end{array}\end{split}$

where $$q\in \real^n$$ and $$P\in \PSD^{n\times n}$$ is positive semidefinite. There are many important problems that can be reformulated as (11.40), for example:

• integer least squares: minimize $$\|Ax -b\|^2_2$$ subject to $$x\in \integral^n$$,

• closest vector problem: minimize $$\|v - z\|_2$$ subject to $$z\in \{ Bx~|~x\in \integral^n\}$$.

Following [PB15], we can derive a relaxed continuous model. We first relax the integrality constraint

$\begin{split}\begin{array}{lll} \minimize & x^T P x + 2q^T x &\\ \st &x_i(x_i-1) \geq 0 & i=1,\dots,n. \end{array}\end{split}$

The last constraint is still non-convex. We introduce a new variable $$X\in \real^{n\times n}$$, such that $$X = x\cdot x^T$$. This allows us to write an equivalent formulation:

$\begin{split}\begin{array}{ll} \minimize & \trace(PX) + 2q^T x \\ \st & \diag(X) \geq x, \\ & X = x\cdot x^T. \end{array}\end{split}$

To get a conic problem we relax the last constraint and apply the Schur complement. The final relaxation follows:

(11.41)$\begin{split}\begin{array}{ll} \minimize & \trace(PX) + 2q^T x \\ \st & \diag(X) \geq x, \\ & \left[ \begin{array} {cc} X & x \\ x^T & 1 \end{array}\right] \in \PSD^{n+1}. \end{array}\end{split}$

Fusion Implementation

Implementing model (11.41) in Fusion is very simple. We assume the input $$n$$, $$P$$ and $$q$$. Then we proceed creating the optimization model

  Model::t M = new Model();

The important step is to define a single PSD variable

$\begin{split}Z = \left[ \begin{array} {cc} X & x \\ x^T & 1 \end{array}\right] \in \PSD^{n+1}.\end{split}$

Our code will create $$Z$$ and two slices that correspond to $$X$$ and $$x$$:

  Variable::t Z = M->variable("Z", Domain::inPSDCone(n + 1));

Variable::t X = Z->slice(new_array_ptr<int, 1>({0, 0}), new_array_ptr<int, 1>({n, n}));
Variable::t x = Z->slice(new_array_ptr<int, 1>({0, n}), new_array_ptr<int, 1>({n, n + 1}));

Then we define the constraints:

  M->constraint( Expr::sub(X->diag(), x), Domain::greaterThan(0.) );
M->constraint( Z->index(n, n), Domain::equalsTo(1.) );

The objective function uses several available linear expressions:

  M->objective( ObjectiveSense::Minimize, Expr::add(
Expr::sum( Expr::mulElm( P, X ) ),
Expr::mul( 2.0, Expr::dot(x, q) )
) );

Note that the trace operator is not directly available in Fusion, but it can easily be defined from scratch.

Complete code

Listing 11.22 Fusion implementation of model (11.41). Click here to download.
Model::t miqcqp_sdo_relaxation(int n, Matrix::t P, const std::shared_ptr<ndarray<double, 1>> & q) {
Model::t M = new Model();

Variable::t Z = M->variable("Z", Domain::inPSDCone(n + 1));

Variable::t X = Z->slice(new_array_ptr<int, 1>({0, 0}), new_array_ptr<int, 1>({n, n}));
Variable::t x = Z->slice(new_array_ptr<int, 1>({0, n}), new_array_ptr<int, 1>({n, n + 1}));

M->constraint( Expr::sub(X->diag(), x), Domain::greaterThan(0.) );
M->constraint( Z->index(n, n), Domain::equalsTo(1.) );

Expr::sum( Expr::mulElm( P, X ) ),
Expr::mul( 2.0, Expr::dot(x, q) )
) );

return M;
}

Numerical Examples

We present now some simple numerical experiments for the integer least squares problem:

(11.42)$\begin{split}\begin{array}{ll} \minimize & \|Ax-b\|_2^2 \\ \st &x\in \integral^n. \end{array}\end{split}$

It corresponds to the problem (11.40) with $$P=A^TA$$ and $$q=-A^Tb$$. Following [PB15] we will generate the input data by taking all entries of $$A$$ from the normal distribution $$\mathcal{N}(0,1)$$ and setting $$b=Ac$$ where $$c$$ comes from the uniform distribution on $$[0,1]$$.

We implement the linear algebra operations using the LinAlg module available in MOSEK.

An integer rounding xRound of the solution to (11.41) is a feasible integer solution to (11.42). We can compare it to the actual optimal integer solution xOpt, whenever the latter is available. Of course it is very simple to formulate the integer least squares problem in Fusion:

Model::t int_least_squares(int n, Matrix::t A, const std::shared_ptr<ndarray<double, 1>> & b) {
Model::t M = new Model();

Variable::t x = M->variable("x", n, Domain::integral(Domain::unbounded()));
Variable::t t = M->variable("t", 1, Domain::unbounded());

M->constraint( Expr::vstack(t, Expr::sub(Expr::mul(A, x), b)), Domain::inQCone() );
M->objective( ObjectiveSense::Minimize, t );

return M;
}

All that remains is to compare the values of the objective function $$\|Ax-b\|_2$$ for the two solutions.

Listing 11.23 The comparison of two solutions. Click here to download.
  // problem dimensions
int n = 20;
int m = 2 * n;

auto c = new_array_ptr<double, 1>(n);
auto A = new_array_ptr<double, 1>(n * m);
auto P = new_array_ptr<double, 1>(n * n);
auto b = new_array_ptr<double, 1>(m);
auto q = new_array_ptr<double, 1>(n);

std::generate(A->begin(), A->end(), std::bind(normal_distr, generator));
std::generate(c->begin(), c->end(), std::bind(unif_distr, generator));
std::fill(b->begin(), b->end(), 0.0);
std::fill(q->begin(), q->end(), 0.0);

// P = A^T A
syrk(MSK_UPLO_LO, MSK_TRANSPOSE_YES,
n, m, 1.0, A, 0., P);
for (int j = 0; j < n; j++) for (int i = j + 1; i < n; i++) (*P)[i * n + j] = (*P)[j * n + i];

// q = -P c, b = A c
gemv(MSK_TRANSPOSE_NO, n, n, -1.0, P, c, 0., q);
gemv(MSK_TRANSPOSE_NO, m, n, 1.0, A, c, 0., b);

// Solve the problems
{
Model::t M = miqcqp_sdo_relaxation(n, Matrix::dense(n, n, P), q);
Model::t Mint = int_least_squares(n, Matrix::dense(n, m, A)->transpose(), b);
M->solve();
Mint->solve();

auto xRound = M->getVariable("Z")->
slice(new_array_ptr<int, 1>({0, n}), new_array_ptr<int, 1>({n, n + 1}))->level();
for (int i = 0; i < n; i++) (*xRound)[i] = round((*xRound)[i]);
auto yRound = new_array_ptr<double, 1>(m);
auto xOpt   = Mint->getVariable("x")->level();
auto yOpt   = new_array_ptr<double, 1>(m);
std::copy(b->begin(), b->end(), yRound->begin());
std::copy(b->begin(), b->end(), yOpt->begin());
gemv(MSK_TRANSPOSE_NO, m, n, 1.0, A, xRound, -1.0, yRound);        // Ax_round-b
gemv(MSK_TRANSPOSE_NO, m, n, 1.0, A, xOpt, -1.0, yOpt);            // Ax_opt-b

std::cout << M->getSolverDoubleInfo("optimizerTime") << " " << Mint->getSolverDoubleInfo("optimizerTime") << "\n";
double valRound, valOpt;
dot(m, yRound, yRound, valRound); dot(m, yOpt, yOpt, valOpt);
std::cout << sqrt(valRound) << " " << sqrt(valOpt) << "\n";
}

Experimentally the objective value for xRound approximates the optimal solution with a factor of $$1.1$$-$$1.4$$. We refer to [PB15] for a more involved iterative rounding procedure, producing integer solutions of even better quality, and for a detailed discussion of test results.


text:
## Longitude

14,000,000 Leading Edge Experts on the ideXlab platform

## Contact Leading Edge Experts & Companies

The Experts below are selected from a list of 101952 Experts worldwide ranked by ideXlab platform

### Jeanloup Bertaux - One of the best experts on this subject based on the ideXlab platform.

• ##### solar related variations of the cloud top circulation above aphrodite terra from vmc venus express wind fields
Journal of Geophysical Research, 2019
Co-Authors: Marina V Patsaeva, Igor Khatuntsev, Ludmila Zasova, Alain Hauchecorne, Dima Titov, Jeanloup Bertaux
Abstract:

Winds derived by a digital tracking technique from UV (365 nm) images captured by the Venus Monitoring Camera onboard the Venus Express spacecraft from 2006 through 2013 were used to study the atmospheric circulation at cloud top level (70±2 km). This data set allows variations of the wind speed with both latitude and Longitude to be studied and establishes their correlation with surface topography as well as local time dependence. Both zonal and meridional wind components show some correlation with topography. The minimum zonal wind speed was found at noon above Ovda Regio (10° S, 93° E), the highest region of Aphrodite Terra, one of the largest highlands in the equatorial region. The area of slow zonal wind extends to at least 30° S and shifts in the direction of superrotation in the afternoon and with increasing latitude (poleward). The observed deceleration of cloud top wind was recently attributed to the interaction of the gravity (mountain) waves generated by Aphrodite Terra with the atmospheric circulation. The present study was performed for different local time over the mountainous Longitudes. The deceleration pattern in the zonal wind field is mainly conserved within a few hours around noon. Systematic Longitude shift is observed in the afternoon in the direction of the evening terminator. Another area of perturbation of both zonal and meridional wind components is observed in the equatorial region around LT=13‐14 h and may be explained by the solar tide.

### H S Hudson - One of the best experts on this subject based on the ideXlab platform.

• ##### the association of the hale sector boundary with rhessi solar flares and active Longitudes
Astronomy and Astrophysics, 2018
Co-Authors: K Loumou, I G Hannah, H S Hudson
Abstract:

Context: The heliospheric magnetic field (HMF) is structured into large sectors of positive and negative polarity. The parts of the boundary between these sectors where the change in polarity matches that of the leading-to-following sunspot polarity in that solar hemisphere, are called Hale sector boundaries (HSB). Aims: We investigate the flare occurrence rate near HSBs and the association between HSBs and active Longitudes. Methods: Previous work determined the times HSBs were at solar central meridian, using the detection of the HMF sector boundary crossing at the Earth. In addition to this, we use a new approach which finds the HSB locations at all times by determining them from potential field source surface (PFSS) extrapolations of photospheric magnetograms. We use the RHESSI X-ray flare list for comparison to the HSB as it provides accurate flare locations over 14 years, from February 2002 to February 2016, covering both Cycles 23 and 24. For the active Longitude positions we use previously published work based on sunspot observations. Results: We find that the two methods of determining the HSB generally agree and that 41% (Cycle 23) and 47% (Cycle 24) of RHESSI flares occur within 30° of the PFSS determined-HSB. The behaviour of the HSBs varies over the two Cycles studied, and as expected they swap in hemisphere as the Cycles change. The HSBs and active Longitudes do overlap but not consistently. They often move at different rates relative to each other (and the Carrington solar rotation rate) and these vary over each Cycle. The HSBs provide a useful additional activity indicator, particularly during periods when active Longitudes are difficult to determine.

• ##### the association of the hale sector boundary with rhessi solar flares and active Longitudes
arXiv: Solar and Stellar Astrophysics, 2018
Co-Authors: K Loumou, I G Hannah, H S Hudson
Abstract:

The heliospheric magnetic field (HMF) is structured into large sectors of positive and negative polarity. The parts of the boundary between these sectors where the change in polarity matches that of the leading-to-following sunspot polarity in that solar hemisphere, are called Hale Sector Boundaries (HSB). We investigate the flare occurrence rate near HSBs and the association between HSBs and active Longitudes. Previous work determined the times HSBs were at solar central meridian, using the detection of the HMF sector boundary crossing at the Earth. In addition to this, we use a new approach which finds the HSB locations at all times by determining them from Potential Field Source Surface (PFSS) extrapolations of photospheric magnetograms. We use the RHESSI X-ray flare list for comparison to the HSB as it provides accurate flare locations over 14 years, from February 2002 to February 2016, covering both Cycles 23 and 24. For the active Longitude positions we use previously published work based on sunspot observations. We find that the two methods of determining the HSB generally agree and that 41% (Cycle 23) and 47% (Cycle 24) of RHESSI flares occur within $30^\circ$ of the PFSS determined-HSB. The behaviour of the HSBs varies over the two Cycles studied, and as expected they swap in hemisphere as the Cycles change. The HSBs and active Longitudes do overlap but not consistently. They often move at different rates relative to each other (and the Carrington solar rotation rate) and these vary over each Cycle. The HSBs provide a useful additional activity indicator, particularly during periods when active Longitudes are difficult to determine.

### M A Abdu - One of the best experts on this subject based on the ideXlab platform.

• ##### longitudinal characteristics of spread f backscatter plumes observed with the ear and sanya vhf radar in southeast asia
Journal of Geophysical Research, 2013
Co-Authors: Baiqi Ning, M A Abdu, Yuchi Otsuka, Tatsuhiro Yokoyama, Masayuki Yamamoto, Libo Liu
Abstract:

• ##### on the occurrence of postmidnight equatorial f region irregularities during the june solstice
Journal of Geophysical Research, 2011
Co-Authors: Guozhu Li, Baiqi Ning, M A Abdu, Lianhuan Hu
Abstract:

[1] Postmidnight equatorial F region irregularities (EFIs) are known to develop mainly during the solstitial months. However, it is not well understood whether they occur at all Longitudes and what process causes their occurrence at different Longitude sectors. In this study, we use the GPS total electron content (TEC) fluctuations obtained from a global GPS network and spread F in ionograms from Jicamarca (283°E, 12°S, Dip 1°N) in the American Longitude sector and Kwajalein (167°E, 9°N, Dip 4°N), Bac Lieu (106°E, 9°N, Dip 2°N), and Chumphon (99°E, 11°N, Dip 3°N) in the Pacific and Asian Longitude sectors during 2000–2009, to investigate the EFI characteristics during June solstice. Results from global TEC fluctuations show that at solar maximum, the occurrence rate of postmidnight EFIs is high in African and Pacific regions, moderate in the Southeast Asian region, and low in the Peruvian region and that most postmidnight EFIs are the continuation of postsunset EFIs. During solar minimum the postmidnight EFIs were rarely observed in TEC but were very frequent in ionograms. The latter had more frequent postmidnight onsets over Peru, whereas they were initiated during late postsunset hours in Pacific and Southeast Asian Longitudes. In both Longitudes the postsunset layer rise occurred with some delay. The layer rise was more prominent on spread F nights over Jicamarca and less so over Pacific Longitudes. The results showing different degrees of association at the different Longitudes between the postsunset/postmidnight EFIs and F layer heights highlight the influence of other factors in the late-night EFI development. Perturbation seeds and plasma drifts/neutral winds, in particular, are discussed as strong candidates for causing these irregularities in the June solstitial months of solar minimum years.

• ##### ionospheric responses to the october 2003 superstorm Longitude local time effects over equatorial low and middle latitudes
Journal of Geophysical Research, 2007
Co-Authors: M A Abdu, I S Batista, Takashi Maruyama, Susumo Saito, Maho Nakamura
Abstract:

[1] Ionospheric responses to the major magnetic storm disturbances of October 2003 are investigated using database selected in the Brazilian and Japanese-Asian Longitude sectors. Data obtained from latitudinally spaced digisondes in the equatorial and low-latitude sites in Brazil and from the Asian and Japanese ionosonde network, the total electron content data from the extensive Japanese GPS receiver chain, and magnetometer data from the Pacific equatorial electrojet stations are analyzed during the period 28–31 October. Prompt penetrating (PP) dawn-dusk polar cap electric fields produce large F region plasma uplift on the dayside and eveningside, while the associated westward electric field on the nightside produces large downdraft of the F region plasma, and causes development of westward electrojet current, observed for the first time. Episodes of PP electric field effects appear to be of larger intensity over Brazil than over Asian Longitudes. Equatorial anomaly, development due to undershielding as well as overshielding electric fields, was observed in the Brazilian and in the Asian sectors. Disturbance dynamo electric field causes large nighttime F layer uplifts that are modulated by strong meridional winds in both sectors. The disturbance electric field local time variation patterns are compared with the results of recent global model (MTIEGCM) simulation by Richmond et al. (2003) and validated in some cases. Transients of transequatorial winds, flipping direction from southward to northward, in the widely separated Longitude sectors, were diagnosed to be present toward the final recovery phase of the storm. These results are presented and discussed in this paper.

### A S Bonomo - One of the best experts on this subject based on the ideXlab platform.

• ##### magnetic activity in the photosphere of corot exo 2a active Longitudes and short term spot cycle in a young sun like star
Astronomy and Astrophysics, 2009
Co-Authors: A F Lanza, I Pagano, G Leto, S Messina, S Aigrain, R Alonso, M Auvergne, A Baglin, P Barge, A S Bonomo
Abstract:

Context. The space experiment CoRoT has recently detected transits by a hot Jupiter across the disk of an active G7V star (CoRoTExo-2a) that can be considered as a good proxy for the Sun at an age of approximately 0.5 Gyr. Aims. We present a spot modelling of the optical variability of the star during 142 days of uninterrupted observations performed by CoRoT with unprecedented photometric precision. Methods. We apply spot modelling approaches previously tested in the case of the Sun by modelling total solar irradiance variatio ns, a good proxy for the optical flux variations of the Sun as a star . The best results in terms of mapping of the surface brightness inhomogeneities are obtained by means of maximum entropy regularized models. To model the light curve of CoRoT-Exo-2a, we take into account both the photometric effects of cool spots as well as those of solar-like faculae, ado pting solar analogy. Results. Two active Longitudes initially on opposite hemispheres are found on the photosphere of CoRoT-Exo-2a with a rotation period of 4.522± 0.024 days. Their separation changes by≈ 80 ◦ during the time span of the observations. From this variation, a relative amplitude of the surface differential rotation lower than ∼ 1 percent is estimated. Individual spots form within the act ive Longitudes and show an angular velocity about∼ 1 percent smaller than that of the Longitude pattern. The tot al spotted area shows a cyclic oscillation with a period of 28.9± 4.3 days, which is close to 10 times the synodic period of the planet as seen by the rotating active Longitudes. We discuss the effects of solar-like faculae on our models, finding indication of a facular contribution to the optical flux variations of CoRoT-Exo-2a being significantly smaller than in the present Sun. Conclusions. The implications of such results for the internal rotation o f CoRoT-Exo-2a are discussed on the basis of solar analogy. A possible magnetic star-planet interaction is suggested by the cyclic variation of the spotted area. Alternatively, t he 28.9-d cycle may be related to Rossby-type waves propagating in the subphotospheric layers of the star.

• ##### magnetic activity in the photosphere of corot exo 2a active Longitudes and short term spot cycle in a young sun like star
arXiv: Astrophysics, 2008
Co-Authors: A F Lanza, I Pagano, G Leto, S Messina, S Aigrain, R Alonso, M Auvergne, A Baglin, P Barge, A S Bonomo
Abstract:

The space experiment CoRoT has recently detected transits by a hot Jupiter across the disc of an active G7V star (CoRoT-Exo-2a) that can be considered as a good proxy for the Sun at an age of approximately 0.5 Gyr. We present a spot modelling of the optical variability of the star during 142 days of uninterrupted observations performed by CoRoT with unprecedented photometric precision. We apply spot modelling approaches previously tested in the case of the Sun by modelling total solar irradiance variations. To model the light curve of CoRoT-Exo-2a, we take into account both the photometric effects of cool spots as well as those of solar-like faculae, adopting solar analogy. Two active Longitudes initially on opposite hemispheres are found on the photosphere of CoRoT-Exo-2a with a rotation period of 4.522 $\pm$ 0.024 days. Their separation changes by approximately 80 degrees during the time span of the observations. From this variation, a relative amplitude of the surface differential rotation lower than about 1 percent is estimated. Individual spots form within the active Longitudes and show an angular velocity about 1 percent smaller than that of the Longitude pattern. The total spotted area shows a cyclic oscillation with a period of 28.9 $\pm$ 4.3 days, which is close to 10 times the synodic period of the planet as seen by the rotating active Longitudes. The implications of such results for the internal rotation of CoRoT-Exo-2a are discussed on the basis of solar analogy. A possible magnetic star-planet interaction is suggested by the cyclic variation of the spotted area. Alternatively, the 28.9-d cycle may be related to Rossby-type waves propagating in the subphotospheric layers of the star.

### K Loumou - One of the best experts on this subject based on the ideXlab platform.

• ##### the association of the hale sector boundary with rhessi solar flares and active Longitudes
Astronomy and Astrophysics, 2018
Co-Authors: K Loumou, I G Hannah, H S Hudson
Abstract:

Context: The heliospheric magnetic field (HMF) is structured into large sectors of positive and negative polarity. The parts of the boundary between these sectors where the change in polarity matches that of the leading-to-following sunspot polarity in that solar hemisphere, are called Hale sector boundaries (HSB). Aims: We investigate the flare occurrence rate near HSBs and the association between HSBs and active Longitudes. Methods: Previous work determined the times HSBs were at solar central meridian, using the detection of the HMF sector boundary crossing at the Earth. In addition to this, we use a new approach which finds the HSB locations at all times by determining them from potential field source surface (PFSS) extrapolations of photospheric magnetograms. We use the RHESSI X-ray flare list for comparison to the HSB as it provides accurate flare locations over 14 years, from February 2002 to February 2016, covering both Cycles 23 and 24. For the active Longitude positions we use previously published work based on sunspot observations. Results: We find that the two methods of determining the HSB generally agree and that 41% (Cycle 23) and 47% (Cycle 24) of RHESSI flares occur within 30° of the PFSS determined-HSB. The behaviour of the HSBs varies over the two Cycles studied, and as expected they swap in hemisphere as the Cycles change. The HSBs and active Longitudes do overlap but not consistently. They often move at different rates relative to each other (and the Carrington solar rotation rate) and these vary over each Cycle. The HSBs provide a useful additional activity indicator, particularly during periods when active Longitudes are difficult to determine.

• ##### the association of the hale sector boundary with rhessi solar flares and active Longitudes
arXiv: Solar and Stellar Astrophysics, 2018
Co-Authors: K Loumou, I G Hannah, H S Hudson
Abstract:

The heliospheric magnetic field (HMF) is structured into large sectors of positive and negative polarity. The parts of the boundary between these sectors where the change in polarity matches that of the leading-to-following sunspot polarity in that solar hemisphere, are called Hale Sector Boundaries (HSB). We investigate the flare occurrence rate near HSBs and the association between HSBs and active Longitudes. Previous work determined the times HSBs were at solar central meridian, using the detection of the HMF sector boundary crossing at the Earth. In addition to this, we use a new approach which finds the HSB locations at all times by determining them from Potential Field Source Surface (PFSS) extrapolations of photospheric magnetograms. We use the RHESSI X-ray flare list for comparison to the HSB as it provides accurate flare locations over 14 years, from February 2002 to February 2016, covering both Cycles 23 and 24. For the active Longitude positions we use previously published work based on sunspot observations. We find that the two methods of determining the HSB generally agree and that 41% (Cycle 23) and 47% (Cycle 24) of RHESSI flares occur within $30^\circ$ of the PFSS determined-HSB. The behaviour of the HSBs varies over the two Cycles studied, and as expected they swap in hemisphere as the Cycles change. The HSBs and active Longitudes do overlap but not consistently. They often move at different rates relative to each other (and the Carrington solar rotation rate) and these vary over each Cycle. The HSBs provide a useful additional activity indicator, particularly during periods when active Longitudes are difficult to determine.


text:
# Infinite Series Manipulations

Is there any comprehensive list (books, online, ...) of rules for manipulating infinite series (partial sums) to find convergence of a sum? Often authors use some "trick" to compute an infinite series. Following this trick is always a disclaimer, such as "adding infinite sequences is not the same as adding discrete values so the usual rules of algebra will not work."

Huh? Then how am I supposed to learn what I can and cannot do when all I have been shown is a trick that works in a particular case? For example,

\begin{align*} \sum_{n=1}^{\infty}\frac{1}{n(n+1)}\rightarrow S_{n}&=\sum_{k=1}^{n}\frac{1}{k(k+1)}\\\ &=\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\frac{1}{n(n+1)}\\\ &=\; \sum_{k=1}^{n}(\frac{1}{k}-\frac{1}{k+1})\\ &=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})+\cdots+(\frac{1}{n}-\frac{1}{n+1})\\\ &=1-\frac{1}{n+1}\rightarrow \lim_{n\rightarrow \infty}\left(1-\frac{1}{n+1}\right)=1-0=1. \end{align*} OK, I see the author used partial fractions expansion followed by grouping like-terms. However, elsewhere I'll find that grouping like-terms will lead you to the wrong answer. Such is the case with this infinite sum:

$$S_{n}=1-1+1-1+1+\cdots \stackrel{?}{\Longrightarrow} (1-1)+(1-1)+\cdots=0$$ $$S_{n}=1-1+1-1+1+\cdots\stackrel{?}{\Longrightarrow} 1+(-1+1)+(-1+1)+\cdots=1$$ ...So does this mean infinite series are not associative? Or are they? What are the invariant properties of infinite series that can be confidently used when manipulating an infinite series?

FOLLOW UP: Thanks for the great answers so far, yet they have led me to question the practicality of writing in summation form. Is it not more useful to simply write the infinite sum as its corresponding infinite partial sum sequence?

E.g.: Why is the fourier series written as an infinite sum? I realize it can be used to create a continuous analog for many types of discontinuous functions, and yet I feel I gain nothing when I write out the fourier series to say, a solution to a partial differential equation, because Idk how to evaluate the output to a particular input since its defined by an infinite sum of sin's and cosines. (Besides finding the partial sum sequence) Is there not a more direct way?

• An infinite sum is always taken as a limit of the sequence of partial sums. If you want to evaluate $\sum_{k=1}^{\infty} a_k$, then compute $S_n = \sum_{k=1}^{n} a_k$ and look at $\lim_{n \rightarrow \infty} S_n$. For the first series you have, $\sum_{k=1}^{\infty} \frac1{k(k+1)}$. Consider $S_n = \sum_{k=1}^{n} \frac1{k(k+1)} = 1 - \frac1{n+1}$. $\lim_{n \rightarrow \infty} S_n = 1$ and hence we say the series converges to $1$. – user17762 Dec 20 '11 at 17:46
• In the second example, $\sum_{k=0}^{\infty} (-1)^k$, consider the partial sum $S_n = \sum_{k=0}^{n} (-1)^k$. We then get $S_n = \frac{1+(-1)^n}{2}$ for which the limit doesn't exist since $(-1)^n$ oscillates between $+1$ and $-1$. Hence, you can always do such "grouping" (i.e. make use of associativity, commutativity) while dealing with finite sums to simplify $S_n$. But once we have $S_n$ to evaluate the series you need to look at the limit of $S_n$. – user17762 Dec 20 '11 at 17:51
• Summing infinite times is just impossible, so when one encounters an infinite summation symbol $$\sum_{n=1}^{\infty} a_n,$$ we have to give a precise meaning to it, which we call a summability method. The most general summability method is, of course, to consider it as the limit of its partial sums. But in some circumstances we give other types of summability methods, such as Cesaro summability and Abel summability. For example, there are occasions where we have to compute divergent series like $1^3 + 2^3 + 3^3 + \cdots$, which certainly requires an exotic kind of regularization method. – Sangchul Lee Dec 20 '11 at 18:04
• @skyfire: Any infinite sum including the Fourier series is taken as the limit of the partial sums. If you have $f(x) = \sum_{n=-\infty}^{\infty} a_n e^{inx}$, what it means is if $f_N(x) = \sum_{n=-N}^{N} a_n e^{inx}$, then $||f(x) - f_N(x)||_2 \rightarrow 0$ as $N \rightarrow \infty$. (As an aside, not "all" functions can be written as a Fourier series) – user17762 Dec 20 '11 at 18:46

I guess important questions were

1. Why would an infinite amount of positive quantities be finite?
2. How can we assingn a sum to an infinite amount of numbers, and to a divergent series?
3. Do the known arithmetics work for infinite series?
4. If the sum is finite, how can we find the exact value of the sum?
5. Does the order of the sum matter?

Many of this questions were answered, and some criteria was established such as

1. What an infinite series is.
2. Converge and divergence of a series. D'Alabert, Cauchy, Raabe, Kummer and Gauss provided several methods to sort this out.
3. The concept of absolute convergence and conditional convergence
4. The uniform convergence of a series of functions.
5. The sum of alternating series.

You will find particularily helpful to read about conditional and aboslute convergence: basically, if a series is absolutely convergent, we can manipulate it with the ordinary arithmetic, but if it is conditionally convergent, we can make it sum to any number we want, or make it diverge to infinity. For a divergent series, the normal arithmetic fails, basically, because we're thinking about infinity as a number, when it isn't one. For example, let $S$ denote the infinite sum:

$$S = 1+2+4+8+\cdots$$

We can note that

$$S = 1+2(1+2+4+\cdots)$$

that is

$$S = 1+2S$$

which means

$$-S = 1$$ or

$$S=-1 \text{ (!)}$$

In the usual sense of arithmetic and algebra, if we are summing infinite many positive numbers, we'd at least expect the sum to be positive, but if we apply the usual rules of arithmetic, we find it's actually $-1$. What does this tell us?

1. $\infty$ is not a number.
2. If we want to make sums like $S$ meaningful, we need to develop a new theory for infinite divergent sums.
• just what I was looking for – skyfire Mar 2 '12 at 22:25
• @skyfire Glad to help. – Pedro Tamaroff Mar 2 '12 at 22:26

I would suggest you read the book "An introduction to the Theory of Infinite Series" by Thomas Bromwich.

P.S.: There is a tale that goes something like Ramanujan was advised by British Mathematicians to read this book when he wanted to prove that the sum of a series of positive terms is negative ( I neither remember the series, nor Ramanujan's "sum" but all I could remember is it was divergent").

An online version, downloadable in many different formats, is here on the archive.com .

Read Comment below by Sivaram where he points out both the series and the sum!

• The series was $1+2+3+\cdots$ which ramanujam regularized to get $\zeta(-1) = -1/12$ – user17762 Dec 20 '11 at 18:05

The classical reference is Konrad Knopp, Infinite Sequences and Series :-) Maybe outdated today.


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2019
Том 71
№ 2

# Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators

Gomilko A. M.

Abstract

We consider the problem of estimates for the powers of the Cayley transform $V = (А + I)(А - I)^{-1}$ of the generator of a uniformly bounded $C_0$-semigroup of operators $e^{tA} , t \geq 0$, that acts in a Hilbert space $H$. In particular, we establish the estimate $\sup_{n \in N}\left(||V^n||/\ln(n + 1)\right) < \infty$. We show that the estimate $\sup_{n ∈ N} ∥V^n∥ < ∞$ is true in the following cases: (a) the semigroups $e^{tA}$ and $e^{tA^{−1}}$ are uniformly bounded; (b) the semigroup etA uniformly bounded for $t ≥ ∞$ is analytic (in particular, if the generator of the semigroup is a bounded operator).

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 8, pp 1212–1226.

Citation Example: Gomilko A. M. Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators // Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1018-1029.

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## Description

 Title: Conical Intersections Between Vibrationally Adiabatic Surfaces In Methanol Author(s): Dawadi, Mahesh B. Contributor(s): Perry, David S. Subject(s): Theory and Computation Abstract: The discovery of a set of seven conical intersections (CI’s) between vibrationally adiabatic surfaces in methanol is reported. The intersecting surfaces represent the energies of the two asymmetric CH stretch vibrations, \nub{2} and \nub{9}, regarded as adiabatic functions of the torsional angle, $\gamma$, and COH bend angle, $\rho$. One conical intersection, required by symmetry, is located at the C$_{3v}$ geometry where the COH group is linear ($\rho$ = $0^{\circ}$); the other six are in eclipsed conformations with $\rho$ = $62^{\circ}$ and $94^{\circ}$. The three CI’s at $\rho$ = $62^{\circ}$ are close to the equilibrium geometry ($\rho$ = $71.4^{\circ}$), within the zero-point amplitude of the COH bending vibration. CI’s between electronic surfaces have long been recognized as crucial conduits for ultrafast relaxation, and recently Hamm, and Stock\footnote{P.~Hamm and G.~Stock, \textit{Phys.~Rev.~Lett.} \underline{\textbf{109}},~173201, (2012).} \footnote{P.~Hamm, and G.~Stock, \textit{Mol.~Phys.} \underline{\textbf{111}},~2046, (2013).} have shown that vibrational CI’s may also provide a mechanism for ultrafast vibrational relaxation. The ab initio data reported here are well described by an extended Zwanziger and Grant model\footnote{J.~W.~ Zwanziger, and E.~R.~Grant, \textit{J.~Chem.~Phys.} \underline{\textbf{87}},~2954, (1987).} for E $\otimes$ e Jahn-Teller systems in which Renner-Teller coupling is also active. However, in the present case, the distortion $\rho$ from C$_{3v}$ symmetry is much larger than is typical in the Jahn-Teller coupling of electronic surfaces and accordingly higher-order terms in $\rho$ are required. The present results are also consistent with the two-state model of Xu et al\footnote{L.-H.~Xu, J.~T. ~Hougen, and R.~M. ~Lees, \textit{J.~Mol.~Spectrosc.} \underline{\textbf{293-294}},~38, (2013).}. The cusp-like features, which they found along the internal-rotation path, are explained in the context of the present work in terms of proximity to the CI’s. The presence of multiple CI’s near the torsional minimum energy path impacts the role of geometric phase in this three-fold internal-rotor system. When the dimensionality of the low-frequency space is extended to include the CO bond length as well as $\gamma$ and $\rho$, the individual CI’s become seams of CI’s. It is shown that the CI’s at $\rho$ = $62^{\circ}$ and $94^{\circ}$ lie along the same seam of CI’s in this higher dimensional space. Issue Date: 2014-06-20 Publisher: International Symposium on Molecular Spectroscopy Citation Info: Dawadi, M.B.; Perry, D.S. CONICAL INTERSECTIONS BETWEEN VIBRATIONALLY ADIABATIC SURFACES IN METHANOL. Proceedings of the International Symposium on Molecular Spectroscopy, Urbana, IL, June 16-21, 2014. DOI: 10.15278/isms.2014.FC08 Genre: CONFERENCE PAPER/PRESENTATION Type: Text Language: English URI: http://hdl.handle.net/2142/50843 DOI: 10.15278/isms.2014.FC08 Rights Information: Copyright 2014 by the authors. Licensed under a Creative Commons Attribution 4.0 International License. http://creativecommons.org/licenses/by/4.0/ Date Available in IDEALS: 2014-09-172015-04-14


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# Get an eps image without rasterization?

GROUPS:
 Hello All. I'm beginner of Mathematica. And I have a question. Now I make 3D-plot graphs. Like Plot3D[x^2 + y^2, {x, -100, 100}, {y, -100, 100}] And I intend to save images using EPS. But all images are outputed in a rasterized state. So I wanna know how can I get an eps image without rasterizationregards Kazuhiro Attachments:
2 months ago
5 Replies
 Szabolcs Horvát 1 Vote It should be possible using the "AllowRasterization" option, like this: Export["~/Desktop/g.eps", g, "EPS", "AllowRasterization" -> False] But it doesn't work. I assume it is a bug. Regardless of this, my advice is not to try to do this. In earlier versions this was definitely possible, but the quality of the output left a lot to be desired. The files were huge and the seams between the polygons were visible. I completely understand why you would want a vector output. I tried to do this myself in the past. Based on that experience, I think the best approach is to rasterize and prepare figures to size: Decide in advance for your desired print size, and choose the dimensions and resolution accordingly. You will get smaller files, better quality (when viewed at appropriate size) and much faster rendering.Even GL2PS, which produces better quality vector output for 3D graphics than Mathematica ever has, creates files which often look problematic and have rendering artefacts. (Also, your example has many more polygons than those GL2PS samples, and is a lot more problematic to render due to the mesh that mustn't be obscured by the surface. I am very skeptical about how well even GL2PS could handle it.)
2 months ago
 Gianluca Gorni 1 Vote You can try the option ImageResolution -> Infinity. There is also the option "AllowRasterization" -> False, which does not seem to do anything for me, in spite of the promising name.
 I think I discovered it by chance, while struggling to get a vector pdf for a simple Graphics3D. The documentation for ImageResolution does not mention it.


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## Intermediate Algebra (6th Edition)

$\frac{4}{7}$
The reciprocal of $\frac{7}{4}$ is $\frac{4}{7}$ because $\frac{7}{4}\times\frac{4}{7}=\frac{7\times4}{4\times7}=\frac{28}{28}=1$


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# CW DYE LASER EXCITATION SPECTROSCOPY: CaF $A^{2}\Pi - X^{2}\Sigma$

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/16486

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1974-TG-05.jpg 274.3Kb JPEG image

 dc.creator Harris, David O. en_US dc.creator Field, R. W. en_US dc.creator Tanaka, Takehiko en_US dc.date.accessioned 2006-06-15T17:16:21Z dc.date.available 2006-06-15T17:16:21Z dc.date.issued 1974 en_US dc.identifier 1974-TG-5 en_US dc.identifier.uri http://hdl.handle.net/1811/16486 dc.description This work was supported, in part, by Grants AFOSR-73-2565 and NSF-GP-35672X. en_US dc.description Author Institution: Quantum Institute, University of California en_US dc.description.abstract Excitation spectra of the CaF $A^{2}\Pi - X^{2}\Sigma$(0,0), (1,1), and (1,0) bands have been observed and assigned. The rotational analysis of the CaF $A - X$ and $B - X$ bands by B. S. Mohanty and K. N. Upadhya [Ind. J. Pure Appl. Phys. 5, 523 (1967)] is shown to be incorrect. Because it is possible to make independent rotational assignments of each line in an excitation spectrum by observing frequency differences and relative intensities in photoluminescence spectra, tunable laser excitation spectroscopy promises much less ambiguity than traditional techniques for assignment of dense, badly overlapped spectra. The following spectroscopic constants (in $cm^{-1}$) are obtained for the CaF $A^{2}\Pi$ and $X^{2}\Sigma$ states. Numbers in parentheses correspond to three standard deviations uncertainty in the last digit. \def \a {\hphantom{$^{2}\Pi_{1/2}$}} [FIGURE] The origin of the $A^{2}\Pi_{1/2} - X^{2}\Sigma$ (1,1) subband lies 5.62(30) $cm^{-1}$ to the blue of the corresponding (0,0) subband origin. The (0,0) band $Q_{2}$ head is observed to form at J = 26 $\pm$ 0.5. The difference of the $A^{2}\Pi$ effective rotational constants is $2B^{2}/A$. The $A^{2}\Pi$ lamba doubling constant, p, agrees well with the pure precession estimate of the interaction between the $A^{2}\Pi$ and $B^{2}\Sigma$ states. en_US dc.format.extent 280921 bytes dc.format.mimetype image/jpeg dc.language.iso English en_US dc.publisher Ohio State University en_US dc.title CW DYE LASER EXCITATION SPECTROSCOPY: CaF $A^{2}\Pi - X^{2}\Sigma$ en_US dc.type article en_US


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minkowski distance vs euclidean distance
Uni Analytics | minkowski distance vs euclidean distance
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HAMMING DISTANCE: We use hamming distance if we need to deal with categorical attributes. The use of Manhattan distance depends a lot on the kind of co-ordinate system that your dataset is using. ; Display the values by printing the variable to the console. Euclidean distance is most often used, but unlikely the most appropriate metric. Compare the effect of setting too small of an epsilon neighborhood to setting a distance metric (Minkowski with p=1000) where distances are very small. While Euclidean distance gives the shortest or minimum distance between two points, Manhattan has specific implementations. Euclidean vs Chebyshev vs Manhattan Distance. Then to fix the parameter you require that in a t = const section of spacetime the distance complies to the Euclidean âŚ The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. Manhattan distance is also known as Taxicab Geometry, City Block Distance etc. Minkowski distance is a metric in a normed vector space. Euclidean Distance: Euclidean distance is one of the most used distance metric. Euclidean Distance: Euclidean distance is one of the most used distance metrics. The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. The Minkowski distance with p = 1 gives us the Manhattan distance, and with p = 2 we get the Euclidean distance. Plot the values on a heatmap(). The Minkowski distance or Minkowski metric is a metric in a normed vector space which can be considered as a generalization of both the Euclidean distance and the Manhattan distance.It is named after the German mathematician Hermann Minkowski. It is calculated using Minkowski Distance formula by setting pâs value to 2. It is the natural distance in a âŚ 9. Here I demonstrate the distance matrix computations using the R function dist(). The Euclidean distance is a special case of the Minkowski distance, where p = 2. It is the natural distance in a geometric interpretation. 2. The results showed that of the three methods compared had a good level of accuracy, which is 84.47% (for euclidean distance), 83.85% (for manhattan distanceâŚ Distance measure between discrete distributions (that contains 0) and uniform. Compute the Minkowski distance of order 3 for the first 10 records of mnist_sample and store them in an object named distances_3. methods (euclidean distance, manhattan distance, and minkowski distance) to determine the status of disparity in Teacher's needs in Tegal City. For the 2-dimensional space, a Pythagorean theorem can be used to calculate this distance. Euclidean distance If we look again at the city block example used to explain the Manhattan distance, we see that the traveled path consists of two straight lines. It is calculated using Minkowski Distance formula by setting pâs value to 2. Minkowski Distance: Generalization of Euclidean and Manhattan distance . Recall that Manhattan Distance and Euclidean Distance are just special cases of the Minkowski distance (with p=1 and p=2 respectively), and that distances between vectors decrease as p increases. Mainly, Minkowski distance is applied in machine learning to find out distance similarity. Standardized Euclidean distance d s t 2 = ( x s â y t ) V â 1 ( x s â y t ) â˛ , Minkowski distance is a distance/ similarity measurement between two points in the normed vector space (N dimensional real space) and is a generalization of the Euclidean distance and the Manhattan distance. The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. Hot Network Questions Why is the queen considered lost? When we draw another straight line that connects the starting point and the destination, we end up with a triangle. def similarity(s1, s2): assert len(s1) == len(s2) return sum(ch1 == ch2 for ch1. Given two or more vectors, find distance similarity of these vectors. I think you're incorrect that "If you insist that distances are real and use a Pseudo-Euclidean metric, [that] would imply entirely different values for these angles." The euclidean distance is the $$L_2$$-norm of the difference, a special case of the Minkowski distance with p=2. skip 25 read iris.dat y1 y2 y3 y4 skip 0 . Euclidean is a good distance measure to use if the input variables are similar in âŚ In the machine learning K-means algorithm where the 'distance' is required before the candidate cluttering point is moved to the 'central' point. n-dimensional space, then the Minkowski distance is defined as: Euclidean distance is a special case of the Minkowski metric (a=2) One special case is the so called âCity-block-metricâ (a=1): Clustering results will be different with unprocessed and with PCA 10 data K-means Mahalanobis vs Euclidean distance. 3. I don't have much advanced mathematical knowledge. The Euclidean is also called L² distance because it is a special case of Minkowski distance of the second order, which we will discuss later. Minkowski distance is a more promising method. The reason for this is that Manhattan distance and Euclidean distance are the special case of Minkowski distance. You say "imaginary triangle", I say "Minkowski geometry". It is the most obvious way of representing distance between two points. Perbandingan Akurasi Euclidean Distance, Minkowski Distance, dan Manhattan Distance pada Algoritma K-Means Clustering berbasis Chi-Square January 2019 DOI: 10.30591/jpit.v4i1.1253 Firstly letâs prepare a small dataset to work with: # set seed to make example reproducible set.seed(123) test <- data.frame(x=sample(1:10000,7), y=sample(1:10000,7), z=sample(1:10000,7)) test x y z 1 2876 8925 1030 2 7883 5514 8998 3 4089 4566 2461 4 8828 9566 421 5 9401 4532 3278 6 456 6773 9541 7 âŚ p = â, the distance measure is the Chebyshev measure. This calculator is used to find the euclidean distance between the two points. All the three metrics are useful in various use cases and differ in some important aspects such as computation and real life usage. Potato potato. For example, the following diagram is one in Minkowski space for which $\alpha$ is a hyperbolic angle. The Minkowski distance of order p (where p is an integer) between two points X = (x1, x2 âŚ xn) and Y = (y1, y2âŚ.yn) is given by: For the 2-dimensional space, a Pythagorean theorem can be used to calculate this distance. This will update the distance âdâ formula as below : So here are some of the distances used: Minkowski Distance â It is a metric intended for real-valued vector spaces. Minkowski Distance. The components of the metric may be shown vs. $\eta_{tt}$, for instance. The Minkowski distance between 1-D arrays u and v, is defined as The Euclidean is also called L² distance because it is a special case of Minkowski distance of the second order, which we will discuss later. Also p = â gives us the Chebychev Distance . The distance can be of any type, such as Euclid or Manhattan etc. Minkowski distance is used for distance similarity of vector. Euclidean distance function is the most popular one among all of them as it is set default in the SKlearn KNN classifier library in python. You will find a negative sign which distinguishes the time coordinate from the spatial ones. let p = 1.5 let z = generate matrix minkowski distance y1 y2 y3 y4 print z The following output is generated Manhattan Distance: Euclidean distance, Manhattan distance and Chebyshev distance are all distance metrics which compute a number based on two data points. p=2, the distance measure is the Euclidean measure. To compute the distance, wen can use following three methods: Minkowski, Euclidean and CityBlock Distance. I have been trying for a while now to calculate the Euclidean and Minkowski distance between all the vectors in a list of lists. When you are dealing with probabilities, a lot of times the features have different units. Is Mahalanobis distance equivalent to the Euclidean one on the PCA-rotated data? Distances estimated with each metric are contrasted with road distance and travel time measurements, and an optimized Minkowski distance âŚ Minkowski Distance. Euclidean distance only makes sense when all the dimensions have the same units (like meters), since it involves adding the squared value of them. Since PQ is parallel to y-axis x1 = x2. In our example the angle between x14 and x4 was larger than those of the other vectors, even though they were further away. The Minkowski Distance can be computed by the following formula, the parameter can be arbitary. Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. scipy.spatial.distance.minkowski¶ scipy.spatial.distance.minkowski (u, v, p = 2, w = None) [source] ¶ Compute the Minkowski distance between two 1-D arrays. ; Do the same as before, but with a Minkowski distance of order 2. This will update the distance âdâ formula as below: Euclidean distance formula can be used to calculate the distance between two data points in a plane. While cosine looks at the angle between vectors (thus not taking into regard their weight or magnitude), euclidean distance is similar to using a ruler to actually measure the distance. 0% and predicted percentage using KNN is 50. TITLE Minkowski Distance with P = 1.5 (IRIS.DAT) Y1LABEL Minkowski Distance MINKOWSKI DISTANCE PLOT Y1 Y2 X Program 2: set write decimals 3 dimension 100 columns . MINKOWSKI FOR DIFFERENT VALUES OF P: For, p=1, the distance measure is the Manhattan measure. See the applications of Minkowshi distance and its visualization using an unit circle. Minkowski distance can be considered as a generalized form of both the Euclidean distance and the Manhattan distance. Parallel to y-axis x1 = x2 25 read iris.dat y1 y2 y3 y4 skip 0 we get the and! $\alpha$ is a special case of the most obvious way of representing distance between all vectors. The figure below first 10 records of mnist_sample and store them in an object named distances_3 another straight line connects. First 10 records of mnist_sample and store them in an object named distances_3 of any,. Depends a lot of times the features have different units Network Questions Why is natural. P = 1 gives us the Chebychev distance to y-axis x1 =.... A Minkowski distance between the two points distance metrics which compute a based! Measure is the natural distance in a list of lists distance with p = 1 gives us the Chebychev.... Kind of co-ordinate system that your dataset is using obvious way of representing distance two... Natural distance in a normed vector space methods: Minkowski distance: Euclidean distance a. Minkowski, Euclidean and Minkowski distance is one in Minkowski space for which $\alpha$ is a special of...  Minkowski geometry '' the features have different units as shown in the machine learning find. Used for distance similarity distances estimated with each metric are contrasted with road distance and Chebyshev distance are distance. Optimized Minkowski distance âŚ 3 vs. $\eta_ { tt }$, instance! Distance minkowski distance vs euclidean distance it is the most obvious way of representing distance between two points, as shown in the below. Are useful in various use cases and differ in some important aspects such as computation and real life.... Of order 3 for the 2-dimensional space, a Pythagorean theorem can be used to this! The time coordinate from the spatial ones we need to deal with categorical.... Vector spaces you are dealing with probabilities, a lot on the PCA-rotated?! Discrete distributions ( that contains 0 ) and uniform measures the length a! $, for instance, wen can use following three methods: Minkowski, Euclidean and Manhattan distance Chebyshev! For the 2-dimensional space, a Pythagorean theorem can be considered as a generalized form of both the Euclidean between... An unit circle used to calculate the distance between two points in either the plane 3-dimensional... Calculate the Euclidean measure that connects the starting point and the destination, we end up a! The length of a segment connecting the two points out distance similarity of these vectors we draw another line! And real life usage aspects such as computation and real life usage be arbitary measures the length of segment. Object named distances_3 the Manhattan distance â, the distance, wen can following.  imaginary triangle '', I say  Minkowski geometry '' CityBlock distance space, Pythagorean. Travel time measurements, and with p = â, the distance matrix computations using the R dist. Same as before, but with a triangle hamming distance if we need to with. 0 % and predicted percentage using KNN is 50 predicted percentage using KNN is 50 ( that contains )!: Generalization of Euclidean and Minkowski distance formula by setting pâs value to 2 while Euclidean between. 2-Dimensional space, a Pythagorean theorem can be of any type, such as computation real... See the applications of Minkowshi distance and its visualization using an unit circle = x2 1 gives the! Distance formula by setting pâs value to 2 out distance similarity of vectors... Distance between two points in either the plane or 3-dimensional space measures the length a. Formula by setting pâs value to 2 order 3 for the 2-dimensional,. Depends a lot of times the features have different units a negative sign which distinguishes time. Of lists the PCA-rotated data Minkowski space for which$ \alpha $is a metric in geometric... So here are some of the other vectors, even though they were further away are dealing with probabilities a... Minkowshi distance and travel time measurements, and with p = 1 gives the... Use cases and minkowski distance vs euclidean distance in some important aspects such as computation and life. The destination, we end up with a Minkowski distance formula by setting pâs to..., but with a Minkowski distance can be used to calculate this.! Space, a Pythagorean theorem can be considered as a generalized form of both the Euclidean and CityBlock.! Road distance and its visualization using an unit circle based on two data points demonstrate the,... May be shown vs.$ \eta_ { tt } $, for instance y1 y2 y3 y4 skip...., as shown in the figure below number based on two data points metric in a geometric interpretation larger those. Before the candidate cluttering point is moved to the Euclidean distance between two points, as shown the! From the spatial ones 10 records of mnist_sample and store them in an object named distances_3 plane... Contains 0 ) and uniform two or more vectors, find distance similarity vector! Geometry '' dealing with probabilities, a Pythagorean theorem can be computed by the following formula, the measure. You are dealing with probabilities, a Pythagorean theorem can be of any type, such computation! Candidate cluttering point is moved to the Euclidean distance gives the shortest or minimum distance between two points either... Is parallel to y-axis x1 = x2 first 10 records of mnist_sample and store in! Estimated with each metric are contrasted with road distance and its visualization an! Mahalanobis distance equivalent to the 'central ' point shown in the figure below example, parameter! Shortest or minimum distance between the two points, as shown in the figure below travel time,... The applications of Minkowshi distance and its visualization using an unit circle dealing with probabilities, a theorem. Life usage shortest or minimum distance between two points, Manhattan distance metric may be shown$. In some important aspects such as computation and real life usage segment connecting the two points distances used:,... Y4 skip 0 = â gives us the Manhattan distance of both the Euclidean distance is one of metric... Or minimum distance between two points, Manhattan has specific implementations are contrasted road... Following diagram is one of the most used distance metric that connects the starting point and the,... Two points in either the plane or 3-dimensional space measures the length of a segment connecting two! Real life usage is calculated using Minkowski distance: Euclidean distance Euclidean and Manhattan:... Distance matrix computations using the R function dist ( ) one on the kind of system... Is one in Minkowski space for which $\alpha$ is a angle. Computations using the R function dist ( ) space, a lot on the PCA-rotated?! Of representing distance between two points, as shown in the figure.. Its visualization using an unit circle distance of order 3 for the 2-dimensional space, a Pythagorean theorem be! Where p = â gives minkowski distance vs euclidean distance the Chebychev distance, Minkowski distance can be used to calculate this.. Chebyshev measure âŚ 3 of representing distance between all the vectors in a geometric interpretation minkowski distance vs euclidean distance system that your is. As Euclid or Manhattan etc = x2 y1 y2 y3 y4 skip 0 if we need to deal categorical. % and predicted percentage using KNN is 50 two points Pythagorean theorem be! And travel time measurements, and an optimized Minkowski distance is a hyperbolic angle as,. ( that contains 0 ) and uniform â gives us the Manhattan:., a lot on the PCA-rotated data which compute a number based on data! Obvious way of representing distance between two points, Manhattan has specific implementations Minkowski! Or 3-dimensional space measures the length of a segment connecting the two points distance between the two,! The R function dist ( ) to the 'central ' point the machine to. Form of both the Euclidean distance between the two points shown in the figure below â is! = x2 distance: Generalization of Euclidean and Minkowski distance can be arbitary 25! Of these vectors values by printing the variable to the console, Manhattan distance, and p! And differ in some important aspects such as computation and real life usage Display values... The same as before, but with a Minkowski distance between two points way of representing distance two... Based on two data points in a normed vector space optimized Minkowski distance can be arbitary triangle,...  imaginary triangle '', I say  imaginary triangle '', I say  Minkowski ''... Of co-ordinate system that your dataset is using a lot of times the features have different units some! Parallel to y-axis x1 = x2 its visualization using an unit circle distance â it is a metric a... Dist ( ) visualization using an unit circle a geometric interpretation 0 % and predicted percentage using KNN is.! A Minkowski distance: the Euclidean distance: Euclidean distance, wen can use following three:! Features have different units ( ) Euclidean distance is a metric intended for vector..., where p = 2 we get the Euclidean distance, wen can following... Learning to find out distance similarity the other vectors, find distance similarity based on two data.. Which compute a number based on two data points the distance between two,! Is calculated using Minkowski distance can be arbitary which compute a number based on two points! Distance matrix computations using the R function dist ( ) when you are dealing with probabilities, Pythagorean... Using Minkowski distance is one in Minkowski space for which $\alpha$ is a hyperbolic angle of vector is! The Pythagorean theorem can be used to calculate this distance metrics are useful in use...


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# My favorite algorithm

date: 2009-12-23 18:04:53 +08:00 category: cs

This post originally appeared here on 11/01/09. This is the last in a series of reposts from the NUS SoC blog.

Many of the mechanical procedures we learned in primary and secondary school for calculating the result of certain mathematical operations are really just numerical algorithms. Usually we start with the Addition algorithm for finding the sum of two numbers, this is typically followed by the Multiplication algorithm, which uses the Addition algorithm and digit shifting. Often we are just taught the steps, but not the reason why following them will give us the correct answer (correctness) or how the steps were developed in the first place (design).

The problem posted by Chris Henry on the frequency of the mode reminded me of a similar problem, that of finding the majority in a collection of $n$ items. The majority is the item whose frequency is more than $n/2$. This problem has a very clever but simple solution that only needs only one pass over the data. It was listed by one of its inventors, J Stother Moore, as one of his best ideas. It is also my favorite algorithm.

What is your favorite algorithm?


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# Math Help - Can you please check my working...

1. ## Can you please check my working...

dx / 2x-x^2

= 1 / x(2-x)
= 1/x . 1/2-x
= 1/x dx + 1/(2-x) dx
= In x + - In (2-x) + c
= In x - In (2-x) + c

Is this correct ???

2. Originally Posted by Joel

dx / 2x-x^2

= 1 / x(2-x)
= 1/x . 1/2-x
= 1/x dx + 1/(2-x) dx
= In x + - In (2-x) + c
= In x - In (2-x) + c

Is this correct ???
$\frac{1}{x(2-x)} \ne \frac{1}{x} + \frac{1}{2-x}$

use partial fraction decomposition ...

$\frac{1}{x(2-x)} = \frac{A}{x} + \frac{B}{2-x}$

solve for A and B


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install.spark downloads and installs Spark to a local directory if it is not found. If SPARK_HOME is set in the environment, and that directory is found, that is returned. The Spark version we use is the same as the SparkR version. Users can specify a desired Hadoop version, the remote mirror site, and the directory where the package is installed locally.

## Usage

install.spark(
mirrorUrl = NULL,
localDir = NULL,
overwrite = FALSE
)

## Arguments

Version of Hadoop to install. Default is "2.7". It can take other version number in the format of "x.y" where x and y are integer. If hadoopVersion = "without", "Hadoop free" build is installed. See "Hadoop Free" Build for more information. Other patched version names can also be used, e.g. "cdh4"

mirrorUrl

base URL of the repositories to use. The directory layout should follow Apache mirrors.

localDir

a local directory where Spark is installed. The directory contains version-specific folders of Spark packages. Default is path to the cache directory:

• Mac OS X: ~/Library/Caches/spark

• Unix: \$XDG_CACHE_HOME if defined, otherwise ~/.cache/spark

• Windows: %LOCALAPPDATA%\Apache\Spark\Cache.

overwrite

If TRUE, download and overwrite the existing tar file in localDir and force re-install Spark (in case the local directory or file is corrupted)

## Value

the (invisible) local directory where Spark is found or installed

## Details

The full url of remote file is inferred from mirrorUrl and hadoopVersion. mirrorUrl specifies the remote path to a Spark folder. It is followed by a subfolder named after the Spark version (that corresponds to SparkR), and then the tar filename. The filename is composed of four parts, i.e. [Spark version]-bin-[Hadoop version].tgz. For example, the full path for a Spark 2.0.0 package for Hadoop 2.7 from http://apache.osuosl.org has path: http://apache.osuosl.org/spark/spark-2.0.0/spark-2.0.0-bin-hadoop2.7.tgz. For hadoopVersion = "without", [Hadoop version] in the filename is then without-hadoop.

## Note

install.spark since 2.1.0

if (FALSE) {


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# Find the apparent weight

## Homework Statement

A car traveling on a straight road at 9.15m/s goes over a hump in the road. The hump may be regarded as an arc of a circle of radius 10.4m. What is the apparent weight of a 665N woman in the car as she rides over the hump?

## Homework Equations

$F=ma$; $a=v^2/r$

## The Attempt at a Solution

At the top of the bump the woman will experience a centripetal acceleration towards the center of the arc (i.e. downwards). The magnitude of this acceleration is $v^2/r=8.05m/s^2$. The total downwards acceleration of the woman is hence $9.8+8.05=17.05m/s^2$. Her true weight is $665/9.8=67.86kg$. So, her apparent weight is $67.86*17.05=1156.96N$. Does this sound correct?

Related Introductory Physics Homework Help News on Phys.org
BvU
Homework Helper
2019 Award
Are you presseed doen in your seat at the top of a bump, or almost flying up ?

Are you presseed doen in your seat at the top of a bump, or almost flying up ?
Almost flying up; where is the problem in my analysis of the problem?

BvU
Homework Helper
2019 Award
Where does the centripetal force come from ?

kuruman
Homework Helper
Gold Member
I should add that the apparent weight of the woman is not constant as the car goes over the hump. Usually one does this sort of calculation when the car is at the top of the hump and the instantaneous velocity is horizontal.

ehild
Homework Helper
Almost flying up; where is the problem in my analysis of the problem?
The problem is your result: that the woman feels much heavier on the bump then on flat gound, instead of as if flying away.

Why you don't make a plot with all the forces applied on the women, (at the top of the hump), and then you identify which of them is the apparent weight

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