text:
# Unable to simulate any thing

#1

Unable to simulate any thing because of Some options are disabled like Source,Analysis, Monitor and EME ports in Mode solution.

Still I am unable to get the enabled options in Mode solution
#2

Kindly provide screenshot of the error? If possible, the simulation file that you are having issues with?
Thanks,
-Lito

#3

Hi,

same problem has been raised. I didn’t get any enabled options (like source,analyser,monitors…) in mode solution. Plz chk the attached screenshot.

Sonali (@spdash)

Disclaimer :- http://mail.bharatividyapeeth.edu/disclaimer.html

#6

I installed mode solution in trial version. But some options are still disabled like source,monitors,analyzers… Without these options how should I simulate any thing?

Plz chk the screenshot for your reference.

#7

I have combined your posts into one since its the same issue.

From the screenshot - you cannot simulate since you have not defined your Solver/Simulation.

If you use EME you will get the following:

If you use varFDTD you will have different buttons enabled:

Hope this clarifies everything.

Check our our Knowledge Base and Lumerical University for resources like videos, webinars and tutorials.

#8

Thank you so much

#9

You are welcome. I will close this as resolved.

closed #10

This topic was automatically closed 3 days after the last reply. New replies are no longer allowed.


text:
Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.

QUESTION

What is the formula for potassium sulfite?

K_2SO_3

Proper nomenclature says that the ion with one less O atom that the "ate" form (i.e. sulfate) is given the suffix "ite". Hence, sulfite ion is SO_3^2-.

We require two potassium ions, each with a +1 charge to form the electrically neutral compound.

format: plain


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+0

# help me with my maths

0
61
3

how does sinus cosinus and tangens works???

Jun 18, 2020
edited by Guest  Jun 18, 2020
edited by Guest  Jun 18, 2020

#1
+8336
0

I believe you meant $$\sin(x), \cos(x)\text{, and }\tan(x)$$.

We have a mnemonic for that, namely "SOHCAHTOA".

In the following diagram:

AB is opposite to x, so we call it "opposite side"

BC is the "hypotenuse".

The mnemonic "SOHCAHTOA" helps us remember the following relations:

$$\color{red}\text{s}\color{black}\text{in}\;x = \dfrac{\color{red}\text{o}\color{black}\text{pposite side}}{\color{red}\text{h}\color{black}\text{ypotenuse}}$$

$$\color{red}\text{c}\color{black}\text{os}\;x = \dfrac{\color{red}\text{a}\color{black}\text{djacent side}}{\color{red}\text{h}\color{black}\text{ypotenuse}}$$

$$\color{red}\text{t}\color{black}\text{an}\;x = \dfrac{\color{red}\text{o}\color{black}\text{pposite side}}{\color{red}\text{a}\color{black}\text{djacent side}}$$

Jun 18, 2020
#2
0

ohhh sound logical,

but how do i calculate a side like from a triangle ABC i know AB and AC but i want to know BC, for example AB is 5 metres and AC is 9 Metres...

thanks btw!

Guest Jun 18, 2020
#3
+8336
0

Hello Guest:

Is any angle given? I mean, are there any right angles?

If yes, then you don't need sin, cos, or tan. You can calculate it directly by Pythagorean theorem.

MaxWong  Jun 18, 2020
edited by MaxWong  Jun 18, 2020


text:
# permutation operator

Let $V$ be a vector space over a field. Let $\sigma\in S_{n}$, the symmetric group on $\{1,\ldots,n\}$ and define a multilinear map $\phi:V\times\cdots\times V\to V^{\otimes n}=\overbrace{V\otimes\cdots\otimes V% }^{n\text{ times}}$ by

 $\phi(v_{1},\ldots,v_{n})=v_{\sigma^{-1}(1)}\otimes\cdots\otimes v_{\sigma^{-1}% (n)}.$

Then by the universal factorization property (http://planetmath.org/TensorProduct) for a tensor product (http://planetmath.org/TensorProduct) there is a unique linear map $P(\sigma):V^{\otimes n}\to V^{\otimes n}$ such that $P(\sigma)\otimes=\phi$. Then of course,

 $P(\sigma)v_{1}\otimes\cdots\otimes v_{n}=v_{\sigma^{-1}(1)}\otimes\cdots% \otimes v_{\sigma^{-1}(n)}.$

$P(\sigma)$ is called the permutation operator associated with $\sigma$.

## 1 Properties

1. 1.

$P(\sigma\tau)=P(\sigma)P(\tau)$

2. 2.

$P(e)=I$ , where $I$ is the identity mapping on $V^{\otimes n}$

3. 3.

$P(\sigma)$ is nonsingular and $P(\sigma)^{-1}=P(\sigma^{-1})$

Title permutation operator PermutationOperator 2013-03-22 16:15:38 2013-03-22 16:15:38 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Definition msc 15A04


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### Home > INT3 > Chapter 10 > Lesson 10.2.1 > Problem10-106

10-106.

Use your triangle tools to solve the problems below. Homework Help ✎

1. What is $PR$ in the diagram above?

Use the Law of Cosines to solve for segment $PR$.
Relabel the parts of the triangle so they match the formula if you need more assistance.

Refer to the Math Notes box in Lesson 7.2.3 if you need help remembering what this is.

2. What is the perimeter of quadrilateral $PQRS$?

Use the Law of Sines to find the measure of segment PS.
If you need more assistance refer to Math Note Box in section 7.2.2.

$\frac{\text{sin }35^\circ}{PS}=\frac{\text{sin }110^\circ}{PR}$

Keep in mind $RS = PS$, because $ΔRSP$ is an isosceles triangle, because the base angles are equal at $35^\circ$ each.

Perimeter $= 25.55$ cm


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Structure of Forward pp and {\rm p\bar{p}} Elastic Amplitudes at Low Energies

# Structure of Forward pp and p¯p Elastic Amplitudes at Low Energies

E. Ferreira Corresponding author. Email: erasmo@if.ufrj.br    A. K. Kohara    J. Sesma Instituto de Física, Universidade Federal do Rio de Janeiro
C.P. 68528, Rio de Janeiro 21945-970, RJ, Brazil
Departamento de Física Teórica, Facultad de Ciencias, 50009, Zaragoza, Spain
###### Abstract

Exact analytical forms of solutions for Dispersion Relations for Amplitudes and Dispersion Relations for Slopes are applied in the analysis of pp and scattering data in the forward range at energies below . As inputs for the energy dependence of the imaginary part, use is made of analytic form for the total cross sections and for parameters of the dependence of the imaginary parts, with exponential and linear factors. A structure for the dependence of the real amplitude is written, with slopes and a linear factor that allows compatibility of the data with the predictions from dispersion relations for the derivatives of the real amplitude at the origin. A very precise description is made of all data, with regular energy dependence of all quantities. It is shown that a revision of previous calculations of total cross sections, slopes and parameters in the literatures is necessary, and stressed that only determinations based on data covering sufficient range using appropriate forms of amplitudes can be considered as valid.

###### pacs:
13.85.Dz,13.60.Hb,13.85.Lg

## I Introduction

In the scattering theory in quantum mechanics, the elastic differential cross sections are written in terms of a complex amplitude with independent imaginary and real parts, that are fiunctions of two variables (spin effects neglected). In the analysis of observables , besides the nuclear amplitude, account is taken for the contribution from the real Coulomb interaction. This is very basic and obvious, but we show in the present work that this structure is not usually obeyed in the treatments of the pp and pp̄ systems, where is written without due account for the properties of the amplitudes. We give a treatment of these elastic processes using theoretical constraints and appropriate forms of the input quantities, arriving at realistic amplitudes to connect measurements and theoretical dynamical models.

Determinations of , and other parameters of pp and pp̄ forward elastic scattering are not direct experimental measurements. Rather, they result from model-dependent analytical limiting procedures, performed with forms assumed for the imaginary and real parts of the complex elastic amplitude. The work done in the laboratory consists in measuring values of the number of event rates in intervals . With attention given to fluxes and densities (we are only concerned with unpolarized beams and targets), tables of distributions in differential cross sections are produced. We stress that the identification of the amplitudes and their parameters requires use of proper theoretical framework.

The differential cross section is written as a sum of absolute values

 dσdt=dσIdt+dσRdt=(ℏc)2(|TI|2+|TR|2) (1)

and the disentanglement required for the determination of the amplitudes and is not at all trivial. Help is brought from the interference of nuclear and Coulomb interactions and from dispersion relations connecting real and imaginary parts through general principles of causality and analyticity.

Besides the entanglement to be resolved, we have that production rates are not obtainable directly at the origin , or even very close to it, but rather in sets of points of an interval. The determination of , , slopes and other quantities require extrapolation of data in a range, using analytical expressions, and the results obviously depend on their forms. The mathematical structures of the amplitudes are mounted using parameters that must be found in confront with the observed distribution in . Regularity in the behaviour of all quantities with the energy is important consideration to obtain sensible descriptions of the elastic processes. Experiments at different energies must be analysed globally, since separate fitting procedures may lead to values that are useless as a step for the phenomenology of the area.

The range of the data at given energy must be sufficient for representation through assumed analytical forms. In the low energy range, up to , often these conditions are not satisfied, even suffering insecure normalization in the measurements of , and compilations of published values for typical parameters result scattered in plots, without coherence and regularity. We propose an investigation of this energy range, with emphasis on the identification of the amplitudes, searching to build a bridge between measurements and mathematical description, necessary to guide models of the dynamics of the processes.

In the interval of from 30 to 60 GeV, pp and pp̄ from ISR/Cern and Fermilab data cover large range with good precision, showing fast increase in , a forward peak and a marked dip in . These measurements led to the establishment of the successful Regge phenomenology Regge (); DL (), based on the exchange of particles (pomerons, reggeons) in the channel. Several theoretical models were developed to describe dynamically this region of data in the and variables models ().

Above , experiments data_HE () have large energy gaps, passing fast by SPS/CERN, Fermilab and reaching the TeV range of LHC data_LHC (). Ingredients of QCD dynamics enter with less or more detail in the interpretation of these data SVM (); us_LHC (). According to QCD expectations, as the energy increases the response of the gluon density in the hadrons increases and the hadronic interaction becomes determined by the vacuum structure Tel-Aviv (); CGC (). The interpretation of the forward scattering parameters in the LHC experiments at 7, 8 and 13 TeV is not trivial, and ambiguities and possible discrepancies are not clarified LHC_2017 (). The potential of crucial information REAL_PART () in the real part of the forward amplitude at high energies requires that doubts in the analysis of the data be properly solved.

In this paper we analyse forward pp, pp̄ data with from to , using forms for real and imaginary scattering amplitudes restricted to the forward regime and exploring fully the theoretical resources and constraints of dispersion relations treated exactly, in order to extract pure information on the forward quantities, as much as possible independently of peculiar microscopic models. This is the most difficult range of data for the analysis, for both reasons of insufficiency in the data and sophistication of the mathematical solutions of dispersion relations at low energies. Anyhow, we believe that in this sector we can learn about determination of amplitude parameters, and hope that this technical knowledge may be useful in the present difficulties encountered in the analysis of the recent LHC experiments.

We propose a treatment of pp and pp̄ forward elastic scattering analysing all data of differential cross sections that seem qualified (namely covering necessary range with regularity) for the extraction of the real and imaginary parts of the complex amplitude. We use the simplest and realistically possible analytical forms, treating coherently the Coulomb interference and the Coulomb phase, we use dispersion relations for the amplitudes (DRA) and for their slopes (DRS) EF2007 () with exact solutions for the Principal Value integrals, obtaining coherent energy dependence of all quantities. DRA and DRS predict algebraic values for the real amplitudes and their derivatives at , and our aim is to have sound proposals for the energy dependence of other parameters in order to reduce flexibility and choices by fittings, and produce a complete coherent description of all data in pp and pp̄ unpolarized elastic scattering . To eliminate fluctuations that are not meaningful, we account for normalization uncertainties (systematic errors), investigating a normalization factor in each experiment that adjust the total cross section to the parametrized prediction. These factors are always very close to 1.

The plots and numbers presented in the Review of Particle Properties PDG () of the Particle Data Group for forward scattering parameters in the low energy range, taken from the experimental papers, are scattered and misleading. The results of our work for all data that we analyse are regularly distributed, presented in numbers and very precise plots of in Sec.III, showing a way for rationalization of the phenomenological knowledge. However, the proposed solutions are not meant to be conclusive, unique or fully convincing. Alternatives are possible and may be looked for.

In elastic pp and pp̄ scattering in the forward direction, the dependences of the amplitudes are mainly characterized by exponential forms, with slopes and that are essentially independent quantities, essentially not equal to each other: the real and imaginary amplitudes do not run parallel along the axis. We take special care in the investigation of the behavior of the real part, that has structure deviating from a pure exponential form for values included in the forward range.

Once the total cross sections for pp and pp̄ are parameterized in the energy, the usual Dispersion Relations for the Amplitudes (DRA) determine the real amplitudes at the origin (namely the parameters). Similarly, if the derivatives of the imaginary parts of pp and pp̄ at are given as functions of the energy, the derivatives of the real parts at are determined by the Dispersion Relations for Slopes (DRS). At low energies it is essential that in both DRA and DRS calculations the exact solutions Exact () be used.

The imaginary part is positive at and decreases with an exponential form, which must be multiplied by a proper factor pointing to a zero, so that the well known dip may be created in . Actually, in our analysis the dip is located outside the examined range, but a linear factor pointing to a distant zero has influence in the shape of the imaginary amplitude and its extrapolation for to use the optical theorem. In the real part the effect of the structure ( dependence) beyond the exponential slope is present in the small region, and is essential in DRA and DRS for the determination of the parameter , and here also a factor (linear, in our case) must be introduced.

The solutions for the forward amplitudes can be obtained with high accuracy, minimum freedom of parameters, and with remarkable simplicity and regularity in the energy dependence of all quantities.

Trusting to propose a realistic assumption, we write for the pp or pp̄ elastic differential cross sections

 dσdt(pp,p¯p)(s,t) =π(ℏc)2{[σ(ρ−μRt)4π(ℏc)2 eBRt/2+FC(t)cos(αΦ)]2 +[σ(1−μIt)4π(ℏc)2 eBIt/2+FC(t)sin(αΦ)]2} , (2)

where and we call attention for the different values expected for the slopes and of the imaginary and real amplitudes and introduce factors with linear dependence in each amplitude. This expression is applied for pp an pp̄, and the energy dependent quantities , are specific for each case.

In a given normalization we write for the real and imaginary nuclear (upper label ) amplitudes

 TNR(s,t)=14√π(ℏc)2 σ(ρ−μRt) eBRt/2 (3)

and

 TNI(s,t)=14√π(ℏc)2 σ(1−μIt) eBIt/2 . (4)

The optical theorem is implicit in Eq.(4). At , we have the usual definition of the parameter

 ρ=TNR(s,t=0)TNI(s,t=0) , (5)

remarking that the value of obtained by fitting of data in a certain range depends on the analytical forms (3,4) of the amplitudes.

In Eq.(2), is the fine-structure constant, is the Coulomb phase and is related with the proton form factor

 FC(t) =(−/+) 2α|t| F2proton(t) , (6)

for the ppp collisions, where

 Fproton(t)=[Λ2/(Λ2+|t|)]2 , (7)

with GeV.

In the present work we follow the usual belief that the phase of the Coulomb-Nuclear interference is based on the superposition of amplitudes in the eikonal formalism phase (). In Appendix A we present the calculation of the Coulomb phase adequate for the amplitudes (3) and (4).

The expressions for the derivatives of the amplitudes are

 ddtTNR(s,t)∣∣t=0=14√π(ℏc)2 σ(ρBR2−μR) (8)

and

 ddtTNI(s,t)∣∣t=0=14√π(ℏc)2 σ(BI2−μI) . (9)

The combinations of parameters

 DI=BI2−μI (10)

and

 DR=ρBR2−μR , (11)

entering respectively as input and output in DRS, are directly related with data, and are crucial for the determination of , , .

It must be noted that the usual direct evaluation of the exponential behavior in using a straight line for the measurements, actually informs the combined average

 B =2DI+ρDR1+ρ2= (BI−2μI)+ρ(ρBR−2μR)1+ρ2 . (12)

In a complete analysis of data all quantities in this expression, and not only the average slope in , must be determined.

The forms written above for and are representations valid for small , of amplitudes for the full range, studied in several models models (); us_LHC () at ISR/Cern and higher energies that stress the peculiar properties of the real part of the elastic amplitude with common features of strong slope and a zero for small . In the low energy region here studied, there are not sufficient data for large , and the analysis is restricted to the forward forms of Eqs.(3,4), showing that all quantities (for pp and pp̄) in these expressions are necessary and sufficient for the description of data obeying constraints from DRA and DRS.

At very low energies, namely below GeV the description of elastic processes, are influenced by details of quark-quark and quark-antiquark interactions, with account for specific intermediate states, as for example in a framework of partial waves very_low_energies (). The measurement of polarized amplitudes polarization (), not considered here, depend on the precise values of non-polarized quantities, as we obtain in the present work. In the energy range of our study, gluonic interactions are present, with global dynamics that is describable by simple analytical forms in the variables .

With total cross sections, imaginary slopes and the linear terms written as analytical forms with powers and logarithms in the energy, both DRA and DRS require evaluation of Principal Value (PV) integrals with the generic structure

 I(n,λ,x)=P∫+∞1x′λlogn(x′)x′2−x2dx′ . (13)

In recent studies, we have obtained the analytic exact solution for these integrals in terms of the Lerch’s Transcendents Exact (), and these solutions are applied in the present work, with demonstration that they are of fundamental importance, particularly in the low-energy range.

The mathematical formalism of our work is presented in Sec. II and the energy dependent inputs of the imaginary parts are written, with forms that are shown to be valid up to LHC energies and also predict correctly the integrated elastic cross sections.

In Appendix A we calculate the phase of the Nuclear-Coulomb interference for real amplitude of the form of Eq.(3).

In Appendix B we present in explicit form the calculation of dispersion relations for the amplitudes (DRA) and for their derivatives (DRS) with the exact solutions in terms of Lerch’s Transcendents.

In Appendix C we present alternative equivalent formalism for the total cross section in the language of pomeron and reggeon trajectories.

With established energy dependent inputs (pp,pp̄), (pp,pp̄), (pp,pp̄) we give in Sec.III precise description of the forward range of elastic pp and scattering, with the essential identification and separation of the real and imaginary amplitudes, with coherence and regularity in the energy dependence all quantities.

In Sec.IV we present conclusions and summarize achievements of our effort.

## Ii Formalism and Inputs for Dispersion Relations

### ii.1 Inputs for Imaginary Part of Elastic Amplitude

In this section we introduce the forms of the imaginary part of the elastic amplitudes, and explain the determination of their parameters. We stress that we only use qualified data on that may be considered as able to allow reliable analysis in terms of amplitudes written in the analytical forms of Eqs.(3,4). The method of construction of our proposal is iteractive, with inputs and outputs nourishing each other. In a first free analysis, we obtain values of parameters for , and , while the quantities of the real part are left free. The extracted values are put in regular behaviour with the energy, leading to analytical forms with terms of powers and logarithms as described below. Adopting these representations for the imaginary amplitude, we use exact forms of dispersion relations for amplitudes (DRA) and for slopes (DRS) to obtain the quantities of the real part, and then we review the imaginary amplitude.

In the reported experiments at low energies the momentum in the lab system is more often used, while at high energies the use of the the center of mass energy is more common. For pp and scattering the connection with the lab energy

 E=√p2LAB+m2 (14)

is

 s=2mE+2m2 , (15)

where is the p/p̄ mass. To work with the dispersion relations, the most useful quantity is the dimensionless ratio

 x=E/m (16)

and then

 s2m2=x+1 . (17)

Approximate relations that are often used at high energies are obviously , , .

The usual parametrizations PDG () for the total cross sections of the pp and pp̄ interactions has the forms

 σ∓(s)=P+Hlog2(ss0)+R1(ss0)−η1±R2(ss0)−η2, (18)

with parameters constants given in milibarns, in , while are dimensionless. The upper and lower indices , refer to pp and pp̄ scattering respectively. The representation is considered to be adequate for all energies and is based on a large number of values of (pp, pp̄) found in experimental papers. We have the radical claim that these values often are not of good precision, for several reasons, and in general because the optical theorem is not applied to well identified imaginary amplitudes.

This form of amplitude is based on the Pomeron/Reggeon dynamics assumed for the strong interactions DL (), and refers only to purely elastic processes. Contributions of diffractive nature, as first studied by Gribov and later formulated by Good and Walker GW () are not included in this framework. Single diffractive, double diffractive and truly inelastic processes have not been measured in the energy range of our study, while theoretical Tel-Aviv () work based on the gluonic dynamics of QCD and measurements start at energies of the ISR/Cern and Fermilab experiments data_HE (), namely . We show indeed that our treatment describes well the integrated elastic and total cross sections at these higher energies.

Dispersion relations are defined with respect to the lab system energy, and, for low energies, terms like and appear preventing to obtain closed exact forms. We then obtain a representation for the total cross sections in terms of dimensionless variables , , with , writing

 σ∓(x)=P+Hlog2(xx0)+R1(xx0)−η1±R2(xx0)−η2, (19)

and analyse all data for from 4 to 500 GeV/c using the assumed structures of the amplitudes in Eqs.(3,4).

We keep the value suggested PDG () for Eq. (18), now appearing as

 x0 = s0/(2m2) = 9.0741 , (20)

where . With this choice, the parameters , , , remain the same. and numerical values are the same, given in Table 1. With

 (ℏc)2=0.38938 \nobreakGeV2 mb

we also need

 (ℏc)21m2=0.4423 mb .

In Fig.1 we show the comparison between Eqs. (18) and (19) for pp̄ written with the same parameters. The difference between the curves represents the deviation of Eq.(17) to the approximated form .

For use as inputs in dispersion relations DRS we write the slopes and in terms of the variable as

 B∓I(x)=b0+b1logx+b2log2x+b3 x−η3 ±b4 x−η4, (21)

again with symmetry in the coefficients for pp̄ and pp. Similarly to Eq.(18) and (19), for the slopes of Eq.(21) can be written with similar analytical forms in the variable .

In Eq.(4) we have in addition to the slope of the imaginary part, , a term which is linear on dependence. The inputs for both pp and pp̄ are determined by a controlled analysis. The result is that the difference between the values of for pp and pp̄ are not important, and we assume for both the form

 μ∓I(x)=μ0+μ1logx . (22)

The numerical values of the input parameters are given in Table 1 and in Fig.2 we show the quantities , and of Eqs.(19,21,22) with the points obtained in the examination of the data described in Sec.III. In the inset plots with the variable we show that the extrapolations up to LHC energies (7 and 8 TeV) are compatible with the predicted results us_LHC (); LHC_2017 ().

The integrated elastic cross section of the imaginary part is given by

 σelIσ(x)=1σ∫−∞0dσIdt (23)

Plots of and the ratio with as function of the energy are given in Fig.3. We recall that this expression gives the ratio of the purely elastic processes. The remaining part of the ratio gives diffractive plus inelastic processes. We also mark in the figure the experimental values of in the ISR range ( 30 to 60 GeV) data_HE () and our published calculations for 1.8 GeV and LHC energies us_LHC ().

The dimensionless Fourier transforms of the amplitude in Eqs.(3,4). with respect to the momentum transfer

 ~T(b;s)=~TR+i~TI (24)

are given by

 ~TR(b;s)=σ2πBR{ρ+μRBR(2−b2BR)}e−b2/2BR (25)

and

 ~TI(b;s)=σ2πBI{1+μIBI(2−b2BI)}e−b2/2BI. (26)

The profile corresponding to the imaginary forward amplitude is dominant over the real part for low values. However from GeV ( fm) the real part can be dominant and this effect is more pronounced due to the presence of the parameter in Eq.(25).

In Appendix C alternative forms are written for the total cross section, or , in terms of power instead of logarithm as in Donnachie-Landshoff formalism, with all accuracy.

### ii.2 Dispersion Relations for Amplitudes and Slopes

The well known dispersion relations for pp and pp̄ elastic scattering are written in terms of even and odd dimensionless amplitudes,

 ReF+(E,t)=K+2E2πP∫+∞mdE′ImF+(E′,t)E′(E′2−E2), (27)
 ReF−(E,t)=2EπP∫+∞mdE′ImF−(E′,t)(E′2−E2). (28)

With , the even and odd combinations of amplitudes are related to the pp and pp̄ systems through

 Fpp(x,t)=F+(x,t)−F−(x,t), Fp¯p(x,t)=F+(x,t)+F−(x,t) . (29)

The optical theorem informs the normalization of the amplitudes by

 σpp(x)=Im Fpp(x,t=0)2m2x (30)

and similarly for pp̄.

With the exponential and linear factors in the imaginary parts, we write the inputs

 Im Fpp(x,t)2m2x=σpp[1−μppI]exp(BppIt/2), (31) Im Fp¯p(x,t)2m2x=σp¯p[1−μp¯pI]exp(Bp¯pIt/2), (32)

with functions , and for pp and pp̄ given in Eqs.(19, 21, 22).

As explained in Sec. I, the real parts are written with exponential and linear factors

 Re Fpp(x,t)2m2x =σpp(x)[ρpp(x)−μppR(x)t] exp[BppR(x)t/2] (33)

and similarly for pp̄.

The parameters are then obtained from

 12m2xRe F+(x,0)=12[(σρ)(p¯p)+(σρ)(pp)] (34)

and

 12m2xReF−(x,0)=12[(σρ)(p¯p)−(σρ)(pp)]  , (35)

with the LHS given by dispersion relations (27,28).

Thus the parameter of the real part is defined by

 σpp(x)ρpp(x)=Re Fpp(x,t=0)2m2x, (36)

and similarly for pp̄.

The derivatives of the real amplitude at are written

 ∂ReFpp(x,t)∂t∣∣t=0=2m2xσpp(x)[ρppBppR2−μppR](x), (37)

and similarly for pp̄. These quantities are determined by DRS, that give the even and odd combinations

 12m2x∂ReF+(x,t)∂t∣∣t=0=12[σp¯pDp¯pR+σppDppR] (38)

and

 12m2x∂ReF−(x,t)∂t∣∣t=0=12[σp¯pDp¯pR−σppDppR] . (39)

The quantities and are combinations of amplitude parameters as in Eq.(11).

Substituting these expressions into Eqs. (27) and (28), written in terms of the dimensionless variable , we obtain

 ReF+(x,t)=K+2m2x2π P∫+∞11x′2−x2[σp¯p(x′)(1−μp¯pI(x′))exp[Bp¯pI(x′)t/2] +σpp(x′)(1−μppI(x′))exp[BppI(x′)t/2]]dx′ (40)

and

 ReF−(x,t)=2m2xπ P∫+∞1x′x′2−x2[σp¯p(x′)(1−μp¯pI(x′))exp[Bp¯pI(x′]t/2) −σpp(x′)(1−μppI(x′))exp[BppI(x′)t/2]]dx′. (41)

Taking t=0 we obtain the Dispersion Relations for the Amplitudes (DRA)

 ReF+(x,t=0)=2m2x[σp¯pρp¯p+σppρpp](x)=K +2m2x2πP∫+∞11x′2−x2[σp¯p+σpp](x′)dx′ (42)

and

 ReF−(x,t=0)=2m2x[σp¯pρp¯p−σppρpp](x) (43) =2m2xπP∫+∞1x′x′2−x2[σp¯p−σpp](x′)dx′.

The expressions in terms of PV integrals are given in Appendix B.

To obtain DRS, we take derivatives of Eqs. (40) and (41) with respect to , writing

 ∂ReF+(x,t)∂t=m2x2πP∫+∞11x′2−x2 [σp¯p(x′)[Bp¯pI−2μp¯pI](x′)exp[Bp¯pI(x′)t/2] (44) +σpp(x′)[BppI(x′)−2μppI(x′)]exp[BppI(x′)t/2]]dx′, ∂ReF−(x,t)∂t=m2xπP∫+∞1x′x′2−x2 [σp¯p(x′)[Bp¯pI−2μp¯pI](x′)exp[Bp¯pI(x′)t/2] (45) −σpp(x′)[BppI(x′)−2μppI(x′)]exp[BppI(x′)t/2]]dx′.

With t=0, these equations become the Dispersion Relations for Slopes (DRS) that we may write

 4[σp¯pDp¯pR+σppDppR](x) (46) =xπP∫+∞12x′2−x2[σp¯pDp¯pI+σppDppI(x′)](x′)dx′, 4[σp¯pDp¯pR−σppDppR](x) (47) =1πP∫+∞12x′x′2−x2[σp¯pDp¯pI−σppDppI](x′)dx′.

where we have introduced the parameterization of the real amplitudes.

In Eqs.(II.2) and (II.2) the terms and given by Eq.(10) keep analytical form similar to that of given by Eq.(21), since the parametrization of is linear in . The presence of the quantity as an input in dispersion relations for slopes does not change the algebra of Eqs.(77), (78) (which were first written Exact () assuming ) and the contribution of can be given by the change of the parameters

 b0→b′0=b0−2μ0 , b1→b′1=b1−2μ1 . (48)

Introducing analytical expressions for the terms in the imaginary parts, we fall in Principal Value integrations of the form (68) that we can solve exactly Exact (). The input forms in Eqs.(19), (21) and (22) taken into the expressions from DRA and DRS, with numbers given in Sec.III lead to values for and the coefficients of the derivatives of the real parts at the origin for pp and pp̄. In the low energy end, namely with up to 30 GeV, it is essential to use the exact solutions for the PV integrals that appear in DRA and DRS in the calculations of and . Illustrating plots are given in Appendix B.

From comparison of the results with the data, the value of the separation constant that appear in the expressions of DRA is determined. We obtain the interval

 K=from (−310) to (−287) . (49)

In the examples and plots of the present paper we use the value .

Given the , and inputs, the quantities and are determined by DRA and DRS. Since is a combination of and , they must be determined by the data. We obtain that presents very regular energy dependence for both pp and pp̄ systems. We introduce the forms

 μR(pp)=c0+c1 x−ν1+c2logx (50)

and

 μR(p¯p)=c3+c4 x−ν4+c5logx . (51)

Numerical values for the constants are given in Table 2 and plots are shown in Fig.5.

### ii.3 Output Quantities and Plots

We must compare the results that are given in Sec.III with the predictions from dispersion relations for the quantities of the real part , and for the pp and pp̄ systems. Fig.4 shows the energy dependence of the predictions from DRA and DRS for and for the derivative coefficient .

In Fig.5(a) we show the energy dependence obtained for the parameter for pp and in Fig.5 (b) we form the quocient using as points values shown in the large table with fit results, and the lines are calculated with Eqs.(50,51) and the analytical results for and (pp and pp̄) from DRA and DRS. Note that the points where pass by zero must coincide with the zero of the sum . This condition results naturally in our solution.

We thus have a closed coherent determination of all quantities describing forward scattering, with no free local parameter.

## Iii Analysis of data in forward direction

The data data_analysed () presented in Tables 3, 4 cover ranges accessible for the analysis, that requires a regular set of points with . Ideally, it would be nice to have good quality data from the very low and going up to 0.1 , but this is not always available in the low energy range.

The analysis covers all data of elastic pp and pp̄ scattering in the energy range of from 3 to 30 GeV. These data have been treated along the history with incomplete theoretical expressions for . In this energy range it is believed that the dynamics of forward scattering is mainly determined by gluonic interactions resulting in smooth energy dependence of all parameters. On the other hand, at very low energies below the direct quark-quark and quark-antiquark interactions may be more important. This seems to be particularly visible in the pp̄ case.

Above , ISR/Cern and Fermilab data, covering very wide ranges, deserves to be studied with analytical forms including the whole range. There are many models models (); us_LHC () for this purpose, and the forward scattering forms here studied are part of these full- descriptions.

An important feature of our analysis is the absorption of normalization errors that accompany the determinations of . Even when these normalization undeterminecies are small , their influence in the parameters is large. The experimental papers use criteria for normalization that we do not consider legal or correct, as ignoring existence of real part, ignoring realistic dependences in the amplitudes, and comparison with other experiments. Some experiments report values using data that are not qualified for the analysis (as insufficient range). These difficulties lead to fluctuations in values of parameters that do not represent physical effects and do not allow a regular global description.

After a smooth description has been achieved and parameters of Eq.(19) are determined, the values of are imposed at each energy. Thus we introduce a constant normalization factor for each data set, chosen so that the total cross section equals the value determined given by Eq. (19). We thus write

 dσdt=f×dσdt∣∣data . (52)

The value of for the dataset of each experiment is given in Tables 3, 4. Thus the central values of are assumed and is determined. The error bars in and other quantities represent sensitivities of the fit to each parameter individually, without freedom for correlations.

The resulting suggested parameter values from our analysis are collected in Tables 3, 4. The input energies for the data are written primarily in terms of , as has been more usual in the presentation of data in this range, but the table also includes values. In some cases, mainly at very low energies, where data are not rich, we combine information from different experiments of same or nearby energies, in the same numerical treatment and in plots. We observe good matching of data sets.

We show examples of the treatment of the data in many plots. More detailed information is given in the figure captions. The log horizontal scale helps to expand and exhibit the small behavior, and it is remarkable that often the descriptions work very well up to , beyond the strictly forward range that determines parameters. We interpret that this is so thanks to appropriate form assumed for the amplitudes, and to the control established by DRA and DRS.

Ranges where pass by zero are particularly delicate. The experiments at = 23.542 and 23.882 GeV ( = 294.4 and 303.1 GeV) give an example in which there is discrepancy in literature for the sign. Our treatment solves the discrepancy, namely we show that passes by zero in this region, and this is valid for both experiments. Parameters are in Table 3.

For pp̄ in the range of our analysis is always small and the data are poor. We are then strongly dependent on the predictions from DRA and DRS.

We inspect and analyse the data using a Cern Minuit program for the determination of , nominally with 6 parameters. We use the form for the total pp and cross sections in Eq.(19), and iteratively with observations of the behaviour of and , and use of dispersion relations DRA and DRS as guides for the real parts. Simultaneously we obtain a value for the subtraction constant . Determinations of , and the normalization factor are made simultaneously. , are well represented by the simple analytical forms of Eqs.(21,22) and Table 1. Once the solution for each dataset is obtained, the error bars are obtained relaxing the value of each quantity in the fitting code, so that they represent the sensitivity of the value, but in general do not include correlations.

Details of the datasets and of the calculation of the analytical representations are given in the figure captions.


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GARP.2011.PQ.P1FRM 2011 Practice exam part 1/exam 2 question 24 (garp11-p1-24)

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New Member

Question:

Sam Neil, the new quantitative analyst, has been asked by portfolio manager to calculate the portfolio 1-day VAR measure based on the past 100 trading days. What will this be if worst 5 losses in the past 100 trading days are 316m, 385m, 412m, 422m,485m in USD?

The question asks what is the 1 day 98% VAR based on past 100 trading days. To my understanding the answer is the 98 observation (ranked from best to worst observation) ==> 422m USD.

However, the answer sheet indicates that the answer is "422 is the 2nd worse loss corresponding to 98% VAR. 422/sqrt(100) = 42.2 is the 1-day var." is this the correct answer, if yes why do we need to factor the 98 observation by sqrt of the number of observations?

Thank very much

Tzvi

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi Tzvi,

I agree with you and so do several others ... we submitted this as an error to GARP (but it did not display on the errata. Uggh.) I think the question was somehow trying to require scaling down from 100-day VaR to a 1-day VaR, but the question phrasing simply does not set this up. There is no way I see to read it such that the /SQRT(100) is warranted. So, 422 is better

(actually, my full quibble is that, if we want to go by the assigned Down, then $412 is correct. For 98th %ile, we should use the 3rd worst; i.e., 2/100 = 2% in the discrete loss tail. But if we don't want to go by Dowd, then 412 is okay, 422 is okay, and any interpolation of between the two is okay). Hope that helps, David Ruolin New Member Hi David, you answered this question both here and at http://www.bionicturtle.com/forum/t...sk-var-definition-and-methods-valuation.4151/ By "the question phrasing simply does not set this up", do you mean that the question should have asked students to answer the 100-day VaR first, and then explicitly assume the i.i.d. and normal distribution of the portifolio loss, before question students what the 1-day VaR is? Because as your 2012 Practice Questions "P1.T4" Question 29.2, choice "d" says (which is a correct statement), once the underlying variables are i.i.d., normal, then the VaR can be scaled by time. =================================================================== David's 2012 Practice Question P1.T4 Question 29.2: "Your colleague reports a 95% one-day VaR of$1.4 mil for a equties portfolio. If we assume 250 trading days in a year, each of the following is a valid conclusion EXCEPT which of the following is FALSE?

a) If the VaR is accurate, we expect a daily loss in excess of $1.4 million to occur on about 12 or 13 days (12.5) during year b) If the return distribution is normal, then we can assume the VaR is sub-additive c) This is a parametric VaR and therefore cannot characterize a heavy-tailed distribution d) If the returns are i.i.d. normal, we can scale to a 10-day VaR with$1.4*SQRT (10)=\$4.3 mil 95% 10-day VaR

===================================================================

Scaling volatility by the square root of time, under an i.i.d. assumption, is particularly stated in your 2012 T2 study note (page 24), could you say something about the scaling of VaR (e.g. and i.i.d assumption AND a normal distribution)?

Additionally, about the choice b of Question 29.2 I quoted above, you said in the answer that "b is true". Nowhere in your T4 study note or T4 video said that VaR is sub-additive under a normal distribution (the T4 study note gives an example how VaR is not sub-additive), but is VaR being sub-additive under "normal distribution" something that we should just know/remember? (And by now you might have guessesd: I did not purchase the Kevin Dowd book to check about this).

Thanks.


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Question

# The distance travelled by the car, if a car travels $$4\ km$$ towards north at an angle of $$45^{\circ}$$ to the east and then travels a distance of $$2\ km$$ towards north at an angle of $$135^{\circ}$$ to the east, is

A
6 km
B
8 km
C
5 km
D
2 km

Solution

## The correct option is A $$6\ km$$The distance traveled by the car is the sum of the distances covered in all the direction.It is because is does not require any direction of motion.Distance =length of path traveled$$=4km+2km=6km$$ Physics

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# Current Affairs 6th Class

#### Time and Clock

Time and Clock   Learning Objectives
• Time and Clock
Time and clock Clock is a device which is used to measure and display the time. The face or dial of a clock is divided into 60 equal parts called minute space and it is numbered from 1 to 12, in such a way that each subsequent number is equidistant (5 minutes spaces apart) from the preceding number. A clock has two hands, the smaller one is the hour hand/ while larger one is called the minute hand. When hour hand travels five minutes spaces, the minute hand travels 60 minutes spaces. Therefore, in 60 minutes the minute hand gains 55 minute on the hour hand.     Properties of clock
• In 60 minutes/ the minute hand gains 55 minutes on the hour hand,
• A clock is said to be too fast, if it shows time more than that of shown by a correct clock.
• A clock is said to be too slow, if it shows time less than that of shown by a correct clock.
• In every hour, both the hands coincide once.
• When the two hands are at right angle, they are 15 minutes spaces apart.
• When the two hands are in opposite directions, they are 30 minutes spaces apart, this situation occurs once in a hour.
• The hands of a clock are in straight line when they are coincident or opposite to each other.
• Hour hand of a clock takes 12 hours to trace the angle of $360{}^\circ$.
• Minute hand of a clock takes 1 hour to trace the angle of $360{}^\circ$.
Example 1 Choose the time from the following options when the hands of a clock will be in the same straight line. (a) 4:00 p.m.                           (b) 8:00 p.m. (c) 7:00 p.m.                          (d) 6:00 p.m. (e) None of these   Answer (d) Explanation: The two hands of a clock are in the same straight line when they are either coincident or 30 minutes spaces apart from each other. At 4:00 p.m., the minute hand will be 20 minutes spaces apart from the hour hand. So option (a) is incorrect. At 8:00 p.m., the minute hand will be 20 minutes spaces apart from the hour hand. So option (b) is incorrect. At 7:00 p.m., the minute hand will be 25 minutes spaces apart from the hour hand. So option (c) is incorrect. At 6:00 p.m., the minute hand will be 30 minutes spaces apart from the hour hand. So option (d) is correct. Therefore, the correct answer is 6 : 00 p.m.   Between 2 O'clock to 10 O'clock, how many times the hands of a clock are at right angle? (a) 14                                      (b) 12 (c) 16                                       (d) 15 more...

#### Non-Verbal

Non-Verbal   Learning Objectives
• What is Non-Verbal Reasoning
• Types of Problems
What is Non-verbal Reasoning Non-Verbal reasoning is a figure based reasoning. It has no language at all. To solve non-verbal problems one has to find out the pattern of pictorial presentation in the given figure. To get a more clear concept about non-verbal reasoning. Let us see the types of problems coming before you.   Types of Problems Problems Based on Mirror Image In a mirror image, left part of an object becomes right part and right part becomes left part. Remember the rule given below, Left Hand Side (LH.S.) $\overset{{}}{leftrightarrows}$ Right hand Side (R.H.S.)   Example 1
Actual Figure Mirror Image
(i)
(ii)
(iii)
(iv) more...

#### Number System

Number System   Learning Objective
• Number System
• HCF
• LCM
Number System A system of naming or representing numbers.   Number A number is a mathematical object which is used to count, label and measure,   Example 1, 5, 19, 325   Main Type Natural numbers/whole-numbers, integers, rational numbers, irrational numbers, real numbers.   Natural Numbers Counting numbers 1, 2, 3, 4, 5, 6... are called natural numbers. These numbers are also referred to as the positive integers. Properties
• The set of natural numbers, commonly denoted by N.
• 1 is the smallest natural number.
• No largest natural number can be found because the set of natural numbers is infinite,
• The successor of a natural number is 1 more than the number.
• The predecessor of a natural number is 1 less than the number.
•               Whole Numbers The natural numbers along with zero form the collection of whole numbers.   Example 0, 1, 2, 3, 4, 5, 6 ....   Properties
• The set of whole numbers, commonly denoted by W.
• 0 is the smallest whole number.
• No largest whole number can be found because the set of whole numbers is infinite.
• The successor of a whole number is 1 more than the number
• The predecessor of a whole number is 1 less than the number.
•   Factors A factor of a number is an exact divisor of that number.   Example 1 Factors of $4=1,\,2,\,4$ Example 2 Factors of $15=1,\,3,\,5,\,15$   Properties
• 1 is the factor of every number.
• Every number is a factor of itself.
• The factors of a number are smaller or equal to the number.
• Numbers of factors of a given number are finite.
• Every factor of a number is an exact divisor of that number.
• A number for which sum of all of its factors is equal to twice of twice the number is called a perfect number.
• 1 is the only number which has exactly one factor, namely itself.
Multiple A multiple of any natural number is a number formed by multiplying that number by any whole number or a multiple is the product of any quantity and an integer or a number is said to be multiple of any of its factors.   Example 1 First three multiple of 4 are $4\times 1= 4,4\times 2 = 8$ and $4\times 3 =12$ Example 2 First three multiple of 19 are $19\times 1=19,19\times 2=38$ and $19\times 3=57$   Properties
• 0 is the multiple of everything.
• Every number is a multiple of itself,
• Every multiple is greater than or equal to that number
• The number of multiples of a given number is infinite.
• A number is a multiple of each of its factors,
•               Prime Numbers The numbers which have exactly two factors, 1 and the number itself, are called prime numbers. Example more...

#### Algebra

Algebra   Learning Objectives
• Algebra
• Ratio
• Proportion
Algebra Algebra is generalized arithmetic in which unknown or unspecified numbers are represented by using letters known as literals.   Constant and variable: A symbol having a fixed numerical value is called a constant and a symbol which takes on various numerical values is known as variables., Example: $8,-25,\text{ }6\frac{6}{11},\,3\frac{1}{2}$ are examples of constants whereas a, b, c, u, u x and y are examples of variables. Algebraic expression: An algebraic expression composed of arithmetic numbers, letters and signs of operation.   Example: $5x+8,\text{ }9y+3x$ and 8z are examples of algebraic expressions. Terms of an expression: Various parts of an algebraic expression separated by the signs plus (+) or minus (-) are known as the terms of the expression. Example: The algebraic expression $6x+8y+9$ have three terms, 6x, 8y and 9. Monomial: An algebraic expression having only one term is known as monomial. Example: $2,5x,6xy$ and$-89xyz$ are examples of monomials. Binomial: An algebraic expression having two terms is known as binomial. Example: $2x+y,3z+5y$ and $x+8$are examples of binomials, Trinomial: An algebraic expression having three terms is known as trinomial Example: $x+y-9,\text{ }5a+6b+c$ and $5x+xy+9$ are examples of trinomials. Polynomial: An algebraic expression containing two or more terms is called polynomial, Example: $5x+6,\text{ }7x+8y+9$and $15+xy+9y+8x$ are examples of polynomials, Factor: A factor is any one of two or more numbers that are multiplied together. Example: 5 and x are factors of 5x. Coefficients: In a product of numbers and literals, any of the factors is called the coefficient of the product of other factors. Example: In 7xy, the coefficient of x is 7y and the coefficient of y is 7x. Like terms: The terms which have the same literal factors are called like or similar terms. Example: 7x, 9x, x are examples of like terms. Unlike terms: The terms which have different literal factors are called unlike or dissimilar terms. Example: $5x,3xy,7xz$ are examples of unlike terms. Algebraic equation: An equation is a mathematical statement equating two quantities. Example: $x+9=15,\text{ }5x-8=3x$ and $3x-7=4$ are examples of algebraic equations. Solution of equation: The value of the variable in an equation which satisfies the equation is called solution of the equation. Example: $x+9=20$, here the value $~x=11$ satisfies the equation/ therefore $~x=11$ is the solution of the equation $x+9=20$.   Ratio   The ratio of two quantities of the same kind and in the same units is a fraction which shows how many times the one quantity is of other.   Example If a and b are two physical quantities of same kind and in the same units, then the fraction $\frac{a}{b}$is called the ratio of a to b and denoted as a : b. Here a and b are called terms of the ratio. The former 'a' is called the first term or antecedent and the latter b is called second term of consequent Note 1: A ratio is unchanged if the two more...

#### Geometry

Geometry   Learning Objectives
• Geometry
• Types of angles
• Parts of Circle
Geometry The branch of mathematics which deals with mathematical objects like points, lines, panes and space is called geometry. Point: A point is to be thought of as a location in space. In other words, a point determines location in a space. Line segment: Let A and B be two point on a plane. Then the straight path between points a and B is known as Sine segment AB. The above line segment is denoted as$\overline{AB}$.   Ray: A line segment extended endlessly in one direction Is. called a ray. The above ray Is denoted as $\overrightarrow{AB}$.   Line: A line is a straight path that extends on and on in both directions endlessly. The above ray Is denoted as $\overrightarrow{AB}$.   Line: A line is a straight path that extends on and on in both directions endlessly.
Parallel lines: If two or more lines do not meet each other however far they are extended, then they are called parallel lines
Intersecting lines: If two or more lines meet each other at one point they are called intersecting lines.
Concurrent lines: Three or more lines in a place are concurrent if all of them pass through the same point. The common point is called point of concurrence.
more...

#### Mensuration

Mensuration     Learning Objectives
• Mensuration
• Perimeter of Geometrical Shapes
• Area of Geometrical Shapes
Mensuration Mensuration is the branch of mathematics which deals with the measurement of lengths, area and volume of the plane and solid figures.   Perimeter of a plane figure: The distance all round a plane figure is called perimeter of the figure or the lengths of boundary of a plane figure is known as its perimeter                         \ Perimeter of the quadrilateral $ABCD=AB+BC+CD+DA$                                                 $=40\text{ }cm+10\text{ }cm+40\text{ }cm+10\text{ }cm$                                                 $=100\text{ }cm$ Perimeter of the hexagon $ABCDEF=AB+BC+CD+DE+EF+FA$                                                 $=100\text{ }m+120\text{ }m+90\text{ }m+45\text{ }m+60\text{ }m+80\text{ }m$                                                 $=495\text{ }m$   Perimeter of a scalene triangle = Sum of all the three sides of the triangle. Example: Find the perimeter of the triangle ABC. Perimeter of the triangle $ABC=AB+BC+CA$                                     $=4\text{ }cm+12\text{ }cm+8\text{ }cm$                                     $=24cm$   Perimeter of a rectangle $=2\times \left( length+breadth \right).$ Example: Find the perimeter of the following rectangle ABCD. Perimeter of rectangle $ABCD=2\times \left( AB+BC \right)$                                     $=2\times \left( 15cm+9cm \right)$                                     $=48\text{ }cm$   Perimeter of regular shapes Perimeter of an equilateral triangle $\text{=3 }\!\!\times\!\!\text{ length of one side}$. Example: Find the perimeter of the given triangle. Perimeter of the triangle $=3\times 4\text{ }cm$                                     $=12\text{ }cm$               Perimeter of a square: $\text{4 }\!\!\times\!\!\text{ length of one side}\text{.}$ Example: Find the perimeter of the given square. Perimeter of the square $=4\times 1\text{ }m$                                     $=4\text{ }m$   Perimeter of regular pentagon $=5\text{ }\times \text{ }length\text{ }of\text{ }one\text{ }side.$ Example: Find the perimeter of the given pentagon. Perimeter of the pentagon $=5\times 4\text{ }cm$                                     $=20\text{ }cm$   Perimeter of the regular hexagon $=6\times length\text{ }of\text{ }one\text{ }side.$ Perimeter of the hexagon $=6\times 5\text{ }cm$                                     $=30\text{ }cm.$   Area of a plane figure: The measurement of the region enclosed by a plane figure is called area of the figure or area is the amount of surface covered by the shape. Area of triangle $=\frac{1}{2}\times base\times height.$ Example: Find the area of the given triangle. Area of the triangle $=\frac{1}{2}\times 4cm\times 6cm=12c{{m}^{2}}$                         Area of rectangle: $length\times breadth.$ Example: Find the area of the given rectangle. Area of the rectangle $=8\text{ }cm\times 6\text{ }cm=48\text{ }c{{m}^{2}}$ more...

#### Data Handling

Data Handling   Learning Objectives
• Data
• Statistics
Data A collection of information in the form of numerical figures is called data. Raw data: Data obtained in the original form is called raw data. Tabulation of data: Arranging the data in a systematic form in the form of a table is known as tabulation of data.   Statistics The branch of mathematics which deals with the collection, presentation, analysis and interpretation of numerical data is called statistics.   Example A dice was thrown 30 times and the following outcomes were noted: 4, 3, 3, 2, 5, 4, 4, 6, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 1, 2, 3, 3, 4, 5, 6, 3, 2, 1, 3, 4. Represent the above data in the form of frequency distribution. Explanation: We may present the data as shown below:
Outcome Tally marks No. of outcomes
1 3
2 6
3 more...

#### Mental Ability

MENTAL ABILITY   FUNDAMENTALS Mental Ability: The power to learn or retain knowledge, in law, the ability to understand the fact and significance of your behavior.
•                      Mental ability develop logical thinking among students.
•                      Mental ability is based on some concepts.
Series completion
•                      Number series
•                     Alphabet series

•                Number Series: The three or more numbers having a sequence of pattern is given. The numbers follow a certain rule which relates the consecutive terms. Students should be able to recognize the rule or pattern. This will be help them to find the next term or number.
Look at the example given below 1.            Series of odd number 1, 3, 5, 7, 9, ___________ Ans.:    Next number is 11.   2.            Series of even number. more...

#### Mensuration Basics

MENSURATION BASICS   MENSURATION
•                    The branch of geometry that deals with the measurement of length, area, or volume.
PERIMETER
•                   The length of the sides enclosing the figure is called its perimeter.
AREA
•                   The amount of space in the boundary of plane figure.
•                    Unit of area is square unit.
Some important formula of Mensuration SQUARE:-
•                     Perimeter of square$=4\times side=4a$
•                    Area of square$={{(side)}^{2}}={{(a)}^{2}}$
•                    Diagonal of square $=\sqrt{2}\times$ side
RECTANGLE                         more...

#### Geometry

GEOMETRY   GEOMETRY Geometry is derived from two greek words "Geo" means "Earth" metron means "Measurement". That means measurement of Earth is called geometry.               Basics terms of geometry
•                  There are three basics undefined terms of geometry.
(i) Point       (ii) Line       (iii) Plane   Point: Point is a mark of position, it is made by sharp tip of pen, pencil and nail.
•           It is denoted by capital letter.
•           It is represented by
•           A point has no length, no breadth and no thickness.

•                   Line segment: The distance between two points in a same plane is called a line segment.
•           It is denoted by$\overline{AB}$. It is measured in 'cm' or 'inch'.
•           more...

#### Trending Current Affairs

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Remark 109.71.5. Proposition 109.71.4 contradicts [Theorem 1.1, AokiHomStacks]. The problem is the non-effectivity of formal objects for $\underline{\mathop{\mathrm{Mor}}\nolimits }_ S(X, [S/A])$. The same problem is mentioned in the Erratum to . Unfortunately, the Erratum goes on to assert that $\underline{\mathop{\mathrm{Mor}}\nolimits }_ S(\mathcal{Y}, \mathcal{Z})$ is algebraic if $\mathcal{Z}$ is separated, which also contradicts Proposition 109.71.4 as $[S/A]$ is separated.

Comment #4617 by Zhang on

"the assert" should be "to assert".

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# In the synchronous motor, the breakdown torque is

In the synchronous motor, the breakdown torque is

### Right Answer is: Directly proportional to applied voltage

#### SOLUTION

Breakdown torque is the maximum torque of the motor, which is produced at full rated voltage and frequency.

Power in the synchronous motor is given as

P = 2NT/60 V2

From the above equation, it is clear that torque T ∝ P.

The pull-out torque with excitation is supplied with a constant voltage source which varies directly with motor terminal voltage. Therefore as power varies as applied voltage V, the torque of a synchronous motor is also varies as applied voltage V.

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# Find the equations of the bisectors of the angles between the planes 2x-y+2z+3=0 and 3x-2y+6z+8=0 and specify the plane which bisects the acute angle and the plane which bisects the obtuse angle.

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Text Solution

Solution :
The given planes are 2x-y+2z+3=0 and 3x-2y+6z+8=0 , where d_(1), d_(2) gt 0 and a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)= 6+2+12gt 0.
 " "(a_(1)x+b_(1)y+c_(1)z+d_(1))/(sqrt(a_(1)^(2)+b_(1)^(2)+c_(1)^(2)))=-(a_(2)x+b_(2)y+c_(2)z+d_(2))/(sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))" " (obtuse angle bisector)
 " "(a_(1)x+b_(1)y+c_(1)z+d)/(sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))=(a_(2)x+b_(2)y+c_(2)z+d_(2))/(sqrt(a_(2)^(2)+b_(2)^(2)+c_(2)^(2)))" " (acute angle bisector)
 i.e., " "(2x-y+2z+3)/(sqrt(4+1+4))=pm(3x-2y+6z+8)/(sqrt(9+4+36))
 or " "(14x-7y+14z+21)=pm(9x-6y+18z+24)
 Taking the positive sign on the right hand side, we get
 " "5x-y-4z-3=0
 Taking the negative sign on the right hand side, we get
 " "23x-13y+32z+45=0" " (acute angle bisector)

Transcript

TimeTranscript
00:00 - 00:59hello guys welcome to daughter in today's question where to find the equation of the bisector of the angle between the Planes 2 x minus y + 2 Z + 3 equal to zero and 3 x minus 2 Y + Z + 1 equal to zero and we are to specify the plane which bisects the acute angle and the plane which bisect the angle so here in the given question D1 comedy 2012 + V1 V2 + 3132 is equal to 6 + 2 + 12 + greater than zero so we get even X + 1 Y + Z Plus D 1 divided by under root of a one square + b square + c square is
01:00 - 01:592 - of a 2 X + B2 bi plus Si To Z plus day 2 / under root of Hetu square + b square + c square so this is our angle equation for obtuse angle bisector of angle bisector now we have to find for acute angle bisector so we get for acute angle bisector SA1 X + 1 by 1 + 7 Z plus 32 / under root of a 1 square + b square + c square equal to 2 X + 2 Y + Z A to Z plus 2
02:00 - 02:59will come given here it will come due to under root of a two square + b square + c to square so it is equation for acute angle bisector acute angle bisector sunao aur equation will be come to x minus y + 2 Z plus 3 / under root of 4 + 1 + 4 equal to plus minus 3 X - 2y + 6 Z + 8 / under root of 9 + 4 + 36 so we get for 3 x minus 7 by + 14 Z + 21 equal to plus minus 9 effects
03:00 - 03:59minus 6 Y + 18 Z + 24 now will be taking positive site on both sides we get 23 x minus 13 by + 32 Oz + 45 equal to zero which is acute angle bisector acute angle bisector ank you guys I hope you like my video


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# 6.4: Parabolas and Analytic Geometry

Difficulty Level: At Grade Created by: CK-12

Brandyn is graphing parabolas as a part of his homework assignment. He is familiar with the standard form of a parabola: $y = ax^2$ , and the longer form: $y - k = a(x - h)^2$ .

On his third problem, he runs into a snag. He has simplified the equation significantly, and has been trying to get it to fix the standard form, but he keeps coming up with this: $x - 4 = 3(y - 3)^2$ . He can't understand why the y term is the squared one, instead of the x term.

What is going on here?

Embedded Video:

### Guidance

This is our second lesson on parabolas. In the initial lesson, we explored the parabola using the distance formula, and touched on the use of the focus and directrix. In this lesson, we first examine parabolas from the "analytic geometry" point of view, and then work a few examples with the focus and directrix of a parabola.

Finding the equation of a parabola using analytic geometry

Consider a cone oriented in space as pictured below:

If the cone opens at an angle such that at any point its radius to height ratio is $a$ , then the cone could be defined as the set of points such that the distance from the $z-$ axis is $a$ times the $z-$ coordinate. Or, in other words, the set of points $(x,y,z)$ satisfying:

$\sqrt{(x - 0)^2 + (y - 0)^2} = az$

Or:

$x^2 + y^2 = a^2 z^2$

This equation works for negative values of $x, y$ , and $z$ , giving the general equation for a two-sided cone.

To consider the intersection of this cone with a plane that is parallel to the line marked $l$ marked in the diagram above, it is most convenient to rotate the entire cone about the $y$ axis until the left side of the cone is vertical, then intersect it with a vertical plane perpendicular to the $x-$ axis. Such a rotation leaves the $y-$ variable unchanged. To see what it does to the $x$ and $z$ variables, let’s see what happens to the point $(x,z)$ on the $xz-$ plane when it is rotated by an angle of $\theta$ .

In the above diagram, $P(x,z)$ is rotated by an angle of $\theta$ to the point $P'$ . We have marked the side lengths $QP = x$ and $SQ = z$ . By the Pythagorean Theorem, $SP = \sqrt{x^2 + z^2}$ . We also have $SP' = \sqrt{x^2 + z^2}$ , since rotation leaves the distance from the origin unchanged. To find the $x-$ coordinates of our rotated point $P'$ , we can use the fact that $\cos(90 - \alpha - \theta) = \frac{SQ'}{\sqrt{x^2 + z^2}}$ . But by properties of cosine we have:

$\cos(90 - \alpha - \theta) = \sin(\alpha + \theta),$

and substituting with the sine addition formula gives us:

$\frac{SQ'}{\sqrt{x^2 + z^2}} = \sin(\alpha) \cos(\theta) + \cos(\alpha) \sin(\theta),$

which we can use our diagram to change to:

$\frac{SQ'}{\sqrt{x^2 + z^2}} = \frac{x}{\sqrt{x^2 + z^2}} \cos(\theta) + \frac{z}{\sqrt{x^2 + z^2}} \sin(\theta)$

which simplifies to:

$SQ' = x \cos(\theta) + z \sin(\theta)$

To find the $x-$ coordinates of our rotated point $P'$ , we can use the fact that $\sin(90 - \alpha - \theta) = \frac{P' Q'}{\sqrt{x^2 + z^2}}$ . But by properties of sine we have:

$\sin(90 - \alpha - \theta) = \cos(\alpha + \theta)$

and substituting with the cosine addition formula gives us:

$\frac{P' Q'}{\sqrt{x^2 + z^2}} = \cos(\alpha) \cos(\theta) - \sin(\alpha)\sin(\theta),$

which we can use our diagram to change to:

$\frac{P' Q'}{\sqrt{x^2 + z^2}} = \frac{z}{\sqrt{x^2 + z^2}} \cos(\theta) - \frac{x}{\sqrt{x^2 +z^2}} \sin(\theta)$

which simplifies to:

$P'Q'=z \cos(\theta) - x \sin(\theta)$

Looking back at the picture, this means that the coordinates of $P'$ are $(x \cos(\theta) + z \sin(\theta), \ z \cos(\theta) - x \sin(\theta))$ . In other words, in rotating from $P$ to $P'$ , the $x-$ coordinate changes to $x \cos(\theta) + z \sin(\theta)$ and the $z-$ coordinate changes to $z \cos(\theta) - x \sin(\theta)$ .

If this rotation happens to every point on the cone, we can substitute $x \cos(\theta) + z \sin(\theta)$ for $x$ and $z \cos(\theta) - x \sin(\theta)$ for $z$ into our equation of the cone, resulting in a new equation for the cone after rotating by $\theta$ .

$(x \cos(\theta) + z \sin(\theta))^2 + y^2 & = a^2(z \cos(\theta) - x \sin(\theta))^2 \\ x^2 \cos^2(\theta) + 2xz \cos(\theta) \sin(\theta) + z^2 \sin^2(\theta) + y^2 & = a^2(x^2 \sin^2(\theta) - 2xz \cos(\theta) \sin(\theta) + z^2 \cos^2(\theta)) \\x^2 \cos^2(\theta) + 2xz \cos(\theta) \sin(\theta) + z^2 \sin^2(\theta) + y^2 & = a^2 x^2 \sin^2(\theta) - 2a^2 xz \cos(\theta) \sin(\theta) + a^2 z^2 \cos^2(\theta)) \\x^2 \cos^2(\theta) + 2xz \cos(\theta) \sin(\theta) + z^2 \sin^2(\theta) + y^2 & = a^2 x^2 \sin^2(\theta) - 2a^2 xz \cos(\theta) \sin(\theta) + a^2 z^2 \cos^2(\theta))$

Now in the case of the tilted cone, we want to tilt the cone such that the left side becomes vertical. Since the factor $a$ determines how tilted the cone is, we can see from the triangle below that $\sin(\theta) = \frac{a}{\sqrt{1 + a^2}}$ and $\cos(\theta) = \frac{1}{\sqrt{1 + a^2}}$ .

So the equation becomes:

$x^2 \frac{1}{1 + a^2} + 2xz \frac{a}{1 + a^2} + z^2 \frac{a^2}{1 + a^2} + y^2 & = a^2 x^2 \frac{a^2}{1 + a^2} - 2a^2 xz \frac{a}{1 + a^2} + a^2 z^2 \frac{1}{1 + a^2} \\x^2 + 2xza + z^2 a^2 + y^2(1 + a^2) & = a^4 x^2 - 2a^3xz + a^2 z^2 \\x^2 + 2xza + y^2(1 + a^2) & = a^4 x^2 - 2a^3 xz$

Now that we have tilted our cone, to take a cross section that is parallel to the left side of the cone, we can simply cut it with a vertical plane. The equation of a vertical plane going through $(b,0,0)$ and perpendicular to the $x-$ axis is $x = b$ . Therefore, setting $x$ equal to the constant in the equation above will give us the intersection of the tilted cone and a plane parallel to one side of the cone. $b$ . Here is a picture of the rotation and the cross-section, which lies in an $xz-$ plane.

Setting $x$ equal to the constant $b$ , we have:

$b^2 + 2abz + y^2(1 + a^2) & = a^4 b^2 - 2a^3 bz \\z(2ab + 2a^3 b) & = -y^2(1 + a^2) +a^4 b^2 - b^2 \\z & =\left (\frac{-1 - a^2}{2ab + 2a^3 b}\right )y^2 + (a^4 b^2 - b^2)$

Although this coefficient and constant term seem complicated, $a$ and $b$ can be chosen so that the coefficient of the $y^2$ term can be equal to any number (you will explore this fact in an exercise). The constant term can be ignored since any parabola can be shifted vertically by any amount.

So the general form of a parabola is:

$z = Ay^2$

where $A$ is any constant.

Or, using the more standard $x-$ and $y-$ coordinates the form of a parabola is

$y = ax^2$

As before, this equation can be adapted to produce the shifted and horizontally oriented forms.

#### Example A

In the proof above, we greatly simplified the formula near the end by substituting A for $\frac{-1 -a^2}{2ab + 2a^3 b}$ .

Explain why this was permissible by showing that for any $A$ there exist constants $a$ and $b$ such that $A = \frac{-1 -a^2}{2ab + 2a^3 b}$ .

Solution

Solving for $b$ in terms of $A$ and $a$ , we have: $A(2ab + 2a^3 b) & = -1 - a^2 \\2Aab(1 + a^2) & = -(1 +a^2) \\2 Aab & = -1 \\2Aab & = -1 \\b & = -\frac{1}{2Aa}$ So we can set $b=2Aa$ and the relationship will hold.

#### Example B

Draw a sketch of the following parabola. Also identify its directrix and focus. $3x^2 + 6x - y = 0$

Solution

Factor and complete the square to get: $3(x + 1) = y + 3$ The vertex is at: $(-1, -3)$

The focus is $(-1, -2 \frac{11}{12})$ The directrix is the line $y = -3 \frac{1}{12}$

#### Example C

Find the equation for a parabola with directrix $y=-2$ and focus (3,8).

Solution

The vertex is vertically midway between the focus and directrix: $\frac {-2 + 8}{2} = 3$ , the same horizontally as the focus: $x = 3$ and therefore at: $(3, 3)$ .

Substituting those values into the formula gives:

$y - 3 = \frac{1}{20}(x -3)^2$

Concept question wrap-up Brandyn keeps coming up with a y 2 term because this is a sideways parabola.

### Vocabulary

Analytic Geometry is a branch of mathematics concerned with modeling and exploring shapes by using algebraic formulas and a coordinate system.

The directrix is the line representing all points the same distance from the focus and a point on a parabola.

The focus is the point that "anchors" a parabola. Any point on the parabola is exactly the same distance from the focus and from the line of the directrix.

### Guided Practice

Questions

1) Find the equation for a parabola with directrix $y = 3$ and focus (2, -1).

2) Find the equation for a parabola with directrix $y = -4$ and focus (-2, 4).

3) Describe the shape of a parabola as it relates to a cone or double cone.

4) Sketch the parabola, and identify the directrix and focus:

$4x^2 -3x +y = 7$

Solutions

1) To find the equation for a parabola with directrix $y = 3$ and focus (2, -1):

Using the vertex form of a parabola $(y - k)^2 = 4a(x - h)$
Recall that the vertex y-value $k$ is the midpoint of the directrix and the focus on the line perpendicular to the directrix and crossing the focus. Therefore the y-value of the vertex is $1$
Recall that the vertex x-value $h$ is the same as the focus, therefore the vertex x-value is $2$
Finally recall that $a$ the distance from the vertex to the focus or from the vertex to the directrix (which are the same): $\therefore a = 2$

Substituting gives: $(y - 1)^2 = 8(x - 2) \to y^2 +1 = 8x -16 \to y^2 = 8x -17$

2) To find the equation for a parabola with directrix $y = -4$ and focus (-2, 4).

Using the vertex form of a parabola $(y - k)^2 = 4a(x - h)$
The y-value of the vertex is midway between focus and directrix, so $k$ = $0$
The vertex x-value $h$ is the same as the focus: $-2$
Finally recall that $a$ the distance from the vertex to the focus or from the vertex to the directrix (which are the same): $\therefore a = 4$

Substituting gives: $(y - 0)^2 = 16(x - (-2)) \to y^2 = 16x +32$

3) The shape of a parabola as it relates to a cone or double cone, is that a parabola represents the revealed shape when a hollow cone is sliced through at an angle equal to the side of the cone. Particularly clear with a double cone is the fact that slicing through at a steeper angle will result in two curves (a hyperbola) and a shallower angle will result in an ellipse.

4) To sketch the parabola $y = -3x^2 + 6x -2$ and identify the vertex.

$y = ax^2 + bx + c$ ..... Recall the standard form of a parabola
$a = -3 | b = 6 | c = -2$ ..... Extract a, b, c
$x = \frac{-6}{2(-3)} \to x = 1$ ..... The x-coordinate of the vertex = $\frac{-b}{2a}$
$y = (-3)(1) + 6(1) -2$ ..... Substitute the calculated x-value to solve for y
$y = 1$ ...... The vertex = $(1, 1)$
$x = 1\pm\frac{1}{\sqrt3}$ ..... Identify the x-intercepts using the Quadratic Formula

### Practice

Use the image to identify the parts of the parabola:

1. The Focus
2. The Vertex
4. The Directrix
5. The Parabola

Use the image and given equation of the parabola to identify the following:

1. The coordinates of the focus
2. The equation of the directrix
3. The length of the focal radius
4. The equation of the axis of symmetry
5. The coordinates of the vertex

1. Find the equation for a parabola with directrix: $x = 2$ and focus: $(0, -2)$
2. Find the equation for a parabola with vertex: $(5, -2)$ and directrix: $y = -5$
3. Find the equation for a parabola with focus: $(3, 5)$ and vertex: $(3, 1)$

Use the image to identify the vertex, axis of symmetry and equation of the parabola:

### Vocabulary Language: English

Analytic Geometry

Analytic Geometry

Analytic geometry is a branch of mathematics concerned with modeling and exploring shapes by using algebraic formulas and a coordinate system.
axis of symmetry

axis of symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. The parabola is symmetrical about this line.
Conic

Conic

Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
directrix

directrix

The directrix of a parabola is the line that the parabola seems to curve away from. All points on a parabola are equidistant from the focus of the parabola and the directrix of the parabola.
focus

focus

The focus of a parabola is the point that "anchors" a parabola. Any point on the parabola is exactly the same distance from the focus as from the directrix.
Parabola

Parabola

A parabola is the set of points that are equidistant from a fixed point on the interior of the curve, called the '''focus''', and a line on the exterior, called the '''directrix'''. The directrix is vertical or horizontal, depending on the orientation of the parabola.

Nov 01, 2012

Feb 26, 2015

# We need you!

At the moment, we do not have exercises for Parabolas and Analytic Geometry.


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1. ## vector space example

let X be any non-empty set.
Let G be the set of all real valued functions defined on X.
Let vector addition be the usual addition of real valued function, and scalar multiplication be the usual multiplication of real valued function by real numbers.
G is a real vector space.

I don't really understand it.
Can you please give me an example of it? Thank you very much.

2. Originally Posted by happystudent
let X be any non-empty set.
Let G be the set of all real valued functions defined on X.
Let vector addition be the usual addition of real valued function, and scalar multiplication be the usual multiplication of real valued function by real numbers. G is a real vector space.
I don't really understand it.
What don’t you understand?
Can you give the set of requirements necessary to have a vector space?
Is the set of functions closed with respect to addition and scalar multiplication?
Can you finish off the list of requirements?


text:
# Optimizing this SVG animation for size [closed]

I'd like to code review generated animation, while this might stretch definition of this site, I really hope this is interesting and revelant topic, as this SVG file is really like any other programming language.

I'd like to display certain physical phenomenon, here is example of generated animation (2.5MB of SVG code, after download should start animating after 5 seconds). SVG seemed a good format since I just could dump every hit, whith coordinates in SI units and it just worked.

Every hit is displayed as a dot, like that: (... many more circles..) (... many more circles..)

Fill color and the like is stored in enclosig g tag.

When I animate this I just:

• Add fill-opacity to zero on g tag.
• Add a set element that sets opacity to 1 when this hit should appear.

So each hit looks like that:

 <circle cx="32.6938819885mm" cy="0.733043327928mm" r="0.2mm">
<set attributeName="fill-opacity" attributeType="XML" begin="6.80948s" to="1" />
</circle>

There is some more to this image (like axis) but this is unimportant.

Code that generated this image can be seen in ipython notebook, which can be downloaded from here.

To generate this image I use python and svgwrite, but I really need a hand in pure svg format.

My main question is:

• How can I optimize this svg image so it is smaller and it renders faster, without any significant loss in precision. For not generated image weights 2.5MB which is too much for me.

Other concerns:

• Is there is any other thing you notice in svg I'd be happy to hear it, as this is my first attempt in generating SVG without an application like Inkscape.
• Related question: how can I display label saying: "Current time: 1.2ns" that updates itself with current time in animation. See also my question on this matter on SO.
• How can (easily) start animation (for example) on key-press (it is non obvious, as events do not depend on themselves).

## closed as off-topic by Stephen Rauch, Heslacher, Toby Speight, Ludisposed, hjpotter92Jul 23 '18 at 8:23

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

• "Lacks concrete context: Code Review requires concrete code from a project, with sufficient context for reviewers to understand how that code is used. Pseudocode, stub code, hypothetical code, obfuscated code, and generic best practices are outside the scope of this site." – Stephen Rauch, Heslacher, Ludisposed, hjpotter92
If this question can be reworded to fit the rules in the help center, please edit the question.


text:
# An ion (Mn^(a+)) has the magnetic moment equal to 4.9 B.M What is the value of (a) :

30 views
An ion (Mn^(a+)) has the magnetic moment equal to 4.9 B.M What is the value of (a) :
A. 3
B. 4
C. 2
D. 5

by (120k points)
N/a


text:
This paper is built upon Elastic Weight Consolidation 1 (EWC), a common method to avoid catastrophic forgetting. In this setting, we want to learn the $$Kth$$ task without forgetting the $$K-1$$ previous tasks. To do so, EWC compute the optimal $$\theta_{:K}$$ given $$\theta_{:K-1}$$ such that it maximizes the posterior $$p_{:K} = p(\theta \vert D_{:K})$$ where $$D_{:K}$$ are all datasets up to task $$K$$.

This can be done using the Fisher matrix of previous task $$\bar F_{:K-1}$$ (See eq. 3).

An assumption that EWC requires is that the Fisher matrix must be block diagonal, which is hard to get in practice.

This paper aims to approximate a block diagonal Fisher matrix by doing a reparameterization of all weights in the network. I’ll only describe the process for the fully-connected layer, but the process is similar for CNNs.

Using eq. 7, we can approximate the Fisher matrix from two components which are available : the inputs and the gradients.

From those two components, we can find two new weights $$U_{1}, U_{2}$$. These weights will provide an approximate diagonalisation of $$W$$, thus preventing catastrophic forgetting. The learnable weight then become $$W' = U^{T}_{2}WU^{T}_{1}$$.

## Results

On multiple datasets, the authors show that their method works better than standard EWC.

1. Kirkpatrick, J., Pascanu, R., Rabinowitz, N., Veness, J., Desjardins, G., Rusu, A. A., … & Hassabis, D. (2017). Overcoming catastrophic forgetting in neural networks. Proceedings of the National Academy of Sciences, 114(13), 3521-3526.


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# tsboot

0th

Percentile

##### Bootstrapping of Time Series

Generate R bootstrap replicates of a statistic applied to a time series. The replicate time series can be generated using fixed or random block lengths or can be model based replicates.

Keywords
ts, nonparametric
##### Usage
tsboot(tseries, statistic, R, l = NULL, sim = "model",
endcorr = TRUE, n.sim = NROW(tseries), orig.t = TRUE,
ran.gen, ran.args = NULL, norm = TRUE, …,
parallel = c("no", "multicore", "snow"),
ncpus = getOption("boot.ncpus", 1L), cl = NULL)
##### Arguments
tseries

A univariate or multivariate time series.

statistic

A function which when applied to tseries returns a vector containing the statistic(s) of interest. Each time statistic is called it is passed a time series of length n.sim which is of the same class as the original tseries. Any other arguments which statistic takes must remain constant for each bootstrap replicate and should be supplied through the … argument to tsboot.

R

A positive integer giving the number of bootstrap replicates required.

sim

The type of simulation required to generate the replicate time series. The possible input values are "model" (model based resampling), "fixed" (block resampling with fixed block lengths of l), "geom" (block resampling with block lengths having a geometric distribution with mean l) or "scramble" (phase scrambling).

l

If sim is "fixed" then l is the fixed block length used in generating the replicate time series. If sim is "geom" then l is the mean of the geometric distribution used to generate the block lengths. l should be a positive integer less than the length of tseries. This argument is not required when sim is "model" but it is required for all other simulation types.

endcorr

A logical variable indicating whether end corrections are to be applied when sim is "fixed". When sim is "geom", endcorr is automatically set to TRUE; endcorr is not used when sim is "model" or "scramble".

n.sim

The length of the simulated time series. Typically this will be equal to the length of the original time series but there are situations when it will be larger. One obvious situation is if prediction is required. Another situation in which n.sim is larger than the original length is if tseries is a residual time series from fitting some model to the original time series. In this case, n.sim would usually be the length of the original time series.

orig.t

A logical variable which indicates whether statistic should be applied to tseries itself as well as the bootstrap replicate series. If statistic is expecting a longer time series than tseries or if applying statistic to tseries will not yield any useful information then orig.t should be set to FALSE.

ran.gen

This is a function of three arguments. The first argument is a time series. If sim is "model" then it will always be tseries that is passed. For other simulation types it is the result of selecting n.sim observations from tseries by some scheme and converting the result back into a time series of the same form as tseries (although of length n.sim). The second argument to ran.gen is always the value n.sim, and the third argument is ran.args, which is used to supply any other objects needed by ran.gen. If sim is "model" then the generation of the replicate time series will be done in ran.gen (for example through use of arima.sim). For the other simulation types ran.gen is used for ‘post-blackening’. The default is that the function simply returns the time series passed to it.

ran.args

This will be supplied to ran.gen each time it is called. If ran.gen needs any extra arguments then they should be supplied as components of ran.args. Multiple arguments may be passed by making ran.args a list. If ran.args is NULL then it should not be used within ran.gen but note that ran.gen must still have its third argument.

norm

A logical argument indicating whether normal margins should be used for phase scrambling. If norm is FALSE then margins corresponding to the exact empirical margins are used.

...

Extra named arguments to statistic may be supplied here. Beware of partial matching to the arguments of tsboot listed above.

parallel, ncpus, cl

See the help for boot.

##### Details

If sim is "fixed" then each replicate time series is found by taking blocks of length l, from the original time series and putting them end-to-end until a new series of length n.sim is created. When sim is "geom" a similar approach is taken except that now the block lengths are generated from a geometric distribution with mean l. Post-blackening can be carried out on these replicate time series by including the function ran.gen in the call to tsboot and having tseries as a time series of residuals.

Model based resampling is very similar to the parametric bootstrap and all simulation must be in one of the user specified functions. This avoids the complicated problem of choosing the block length but relies on an accurate model choice being made.

Phase scrambling is described in Section 8.2.4 of Davison and Hinkley (1997). The types of statistic for which this method produces reasonable results is very limited and the other methods seem to do better in most situations. Other types of resampling in the frequency domain can be accomplished using the function boot with the argument sim = "parametric".

##### Value

An object of class "boot" with the following components.

t0

If orig.t is TRUE then t0 is the result of statistic(tseries,…{}) otherwise it is NULL.

t

The results of applying statistic to the replicate time series.

R

The value of R as supplied to tsboot.

tseries

The original time series.

statistic

The function statistic as supplied.

sim

The simulation type used in generating the replicates.

endcorr

The value of endcorr used. The value is meaningful only when sim is "fixed"; it is ignored for model based simulation or phase scrambling and is always set to TRUE if sim is "geom".

n.sim

The value of n.sim used.

l

The value of l used for block based resampling. This will be NULL if block based resampling was not used.

ran.gen

The ran.gen function used for generating the series or for ‘post-blackening’.

ran.args

The extra arguments passed to ran.gen.

call

The original call to tsboot.

##### References

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Kunsch, H.R. (1989) The jackknife and the bootstrap for general stationary observations. Annals of Statistics, 17, 1217--1241.

Politis, D.N. and Romano, J.P. (1994) The stationary bootstrap. Journal of the American Statistical Association, 89, 1303--1313.

boot, arima.sim

• tsboot
• ts.return
##### Examples
# NOT RUN {
lynx.fun <- function(tsb) {
ar.fit <- ar(tsb, order.max = 25)
c(ar.fit$order, mean(tsb), tsb) } # the stationary bootstrap with mean block length 20 lynx.1 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "geom") # the fixed block bootstrap with length 20 lynx.2 <- tsboot(log(lynx), lynx.fun, R = 99, l = 20, sim = "fixed") # Now for model based resampling we need the original model # Note that for all of the bootstraps which use the residuals as their # data, we set orig.t to FALSE since the function applied to the residual # time series will be meaningless. lynx.ar <- ar(log(lynx)) lynx.model <- list(order = c(lynx.ar$order, 0, 0), ar = lynx.ar$ar) lynx.res <- lynx.ar$resid[!is.na(lynx.ar$resid)] lynx.res <- lynx.res - mean(lynx.res) lynx.sim <- function(res,n.sim, ran.args) { # random generation of replicate series using arima.sim rg1 <- function(n, res) sample(res, n, replace = TRUE) ts.orig <- ran.args$ts
ts.mod <- ran.args$model mean(ts.orig)+ts(arima.sim(model = ts.mod, n = n.sim, rand.gen = rg1, res = as.vector(res))) } lynx.3 <- tsboot(lynx.res, lynx.fun, R = 99, sim = "model", n.sim = 114, orig.t = FALSE, ran.gen = lynx.sim, ran.args = list(ts = log(lynx), model = lynx.model)) # For "post-blackening" we need to define another function lynx.black <- function(res, n.sim, ran.args) { ts.orig <- ran.args$ts
ts.mod <- ran.args$model mean(ts.orig) + ts(arima.sim(model = ts.mod,n = n.sim,innov = res)) } # Now we can run apply the two types of block resampling again but this # time applying post-blackening. lynx.1b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "fixed", n.sim = 114, orig.t = FALSE, ran.gen = lynx.black, ran.args = list(ts = log(lynx), model = lynx.model)) lynx.2b <- tsboot(lynx.res, lynx.fun, R = 99, l = 20, sim = "geom", n.sim = 114, orig.t = FALSE, ran.gen = lynx.black, ran.args = list(ts = log(lynx), model = lynx.model)) # To compare the observed order of the bootstrap replicates we # proceed as follows. table(lynx.1$t[, 1])
table(lynx.1b$t[, 1]) table(lynx.2$t[, 1])
table(lynx.2b$t[, 1]) table(lynx.3$t[, 1])
# Notice that the post-blackened and model-based bootstraps preserve
# the true order of the model (11) in many more cases than the others.
# }

Documentation reproduced from package boot, version 1.3-24, License: Unlimited

### Community examples

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text:
Magicode
0

# 03.03 Operations in Pandas

One of the essential pieces of NumPy is the ability to perform quick element-wise operations, both with basic arithmetic (addition, subtraction, multiplication, etc.) and with more sophisticated operations (trigonometric functions, exponential and logarithmic functions, etc.). Pandas inherits much of this functionality from NumPy, and the ufuncs that we introduced in Computation on NumPy Arrays: Universal Functions are key to this.
Pandas includes a couple useful twists, however: for unary operations like negation and trigonometric functions, these ufuncs will preserve index and column labels in the output, and for binary operations such as addition and multiplication, Pandas will automatically align indices when passing the objects to the ufunc. This means that keeping the context of data and combining data from different sources–both potentially error-prone tasks with raw NumPy arrays–become essentially foolproof ones with Pandas. We will additionally see that there are well-defined operations between one-dimensional Series structures and two-dimensional DataFrame structures.

### Ufuncs: Index Preservation

Because Pandas is designed to work with NumPy, any NumPy ufunc will work on Pandas Series and DataFrame objects. Let's start by defining a simple Series and DataFrame on which to demonstrate this:
import pandas as pd
import numpy as np
rng = np.random.RandomState(42)
ser = pd.Series(rng.randint(0, 10, 4))
ser
0 6 1 3 2 7 3 4 dtype: int64
df = pd.DataFrame(rng.randint(0, 10, (3, 4)),
columns=['A', 'B', 'C', 'D'])
df
If we apply a NumPy ufunc on either of these objects, the result will be another Pandas object with the indices preserved:
np.exp(ser)
0 403.428793 1 20.085537 2 1096.633158 3 54.598150 dtype: float64
Or, for a slightly more complex calculation:
np.sin(df * np.pi / 4)
Any of the ufuncs discussed in Computation on NumPy Arrays: Universal Functions can be used in a similar manner.

### UFuncs: Index Alignment

For binary operations on two Series or DataFrame objects, Pandas will align indices in the process of performing the operation. This is very convenient when working with incomplete data, as we'll see in some of the examples that follow.

#### Index alignment in Series

As an example, suppose we are combining two different data sources, and find only the top three US states by area and the top three US states by population:
area = pd.Series({'Alaska': 1723337, 'Texas': 695662,
'California': 423967}, name='area')
population = pd.Series({'California': 38332521, 'Texas': 26448193,
'New York': 19651127}, name='population')
Let's see what happens when we divide these to compute the population density:
population / area
Alaska NaN California 90.413926 New York NaN Texas 38.018740 dtype: float64
The resulting array contains the union of indices of the two input arrays, which could be determined using standard Python set arithmetic on these indices:
area.index | population.index
Index(['Alaska', 'California', 'New York', 'Texas'], dtype='object')
Any item for which one or the other does not have an entry is marked with NaN, or "Not a Number," which is how Pandas marks missing data (see further discussion of missing data in Handling Missing Data). This index matching is implemented this way for any of Python's built-in arithmetic expressions; any missing values are filled in with NaN by default:
A = pd.Series([2, 4, 6], index=[0, 1, 2])
B = pd.Series([1, 3, 5], index=[1, 2, 3])
A + B
0 NaN 1 5.0 2 9.0 3 NaN dtype: float64
If using NaN values is not the desired behavior, the fill value can be modified using appropriate object methods in place of the operators. For example, calling A.add(B) is equivalent to calling A + B, but allows optional explicit specification of the fill value for any elements in A or B that might be missing:
A.add(B, fill_value=0)
0 2.0 1 5.0 2 9.0 3 5.0 dtype: float64

#### Index alignment in DataFrame

A similar type of alignment takes place for both columns and indices when performing operations on DataFrames:
A = pd.DataFrame(rng.randint(0, 20, (2, 2)),
columns=list('AB'))
A
B = pd.DataFrame(rng.randint(0, 10, (3, 3)),
columns=list('BAC'))
B
A + B
Notice that indices are aligned correctly irrespective of their order in the two objects, and indices in the result are sorted. As was the case with Series, we can use the associated object's arithmetic method and pass any desired fill_value to be used in place of missing entries. Here we'll fill with the mean of all values in A (computed by first stacking the rows of A):
fill = A.stack().mean()
A.add(B, fill_value=fill)
The following table lists Python operators and their equivalent Pandas object methods:
Python OperatorPandas Method(s)
+add()
-sub(), subtract()
*mul(), multiply()
/truediv(), div(), divide()
//floordiv()
%mod()
**pow()

### Ufuncs: Operations Between DataFrame and Series

When performing operations between a DataFrame and a Series, the index and column alignment is similarly maintained. Operations between a DataFrame and a Series are similar to operations between a two-dimensional and one-dimensional NumPy array. Consider one common operation, where we find the difference of a two-dimensional array and one of its rows:
A = rng.randint(10, size=(3, 4))
A
array([[3, 8, 2, 4], [2, 6, 4, 8], [6, 1, 3, 8]])
A - A[0]
array([[ 0, 0, 0, 0], [-1, -2, 2, 4], [ 3, -7, 1, 4]])
According to NumPy's broadcasting rules (see Computation on Arrays: Broadcasting), subtraction between a two-dimensional array and one of its rows is applied row-wise.
In Pandas, the convention similarly operates row-wise by default:
df = pd.DataFrame(A, columns=list('QRST'))
df - df.iloc[0]
If you would instead like to operate column-wise, you can use the object methods mentioned earlier, while specifying the axis keyword:
df.subtract(df['R'], axis=0)
Note that these DataFrame/Series operations, like the operations discussed above, will automatically align indices between the two elements:
halfrow = df.iloc[0, ::2]
halfrow
Q 3 S 2 Name: 0, dtype: int64
df - halfrow


text:
# relation of boundary and connected

Let $$X$$ be a connected topological space, $$U$$, $$V\in X$$ two non-disjoint open subsets none of which contains the other one. Prove that if their boundaries $$Fr(U)$$ and $$Fr(V)$$ are connected, then $$Fr(U)\cap Fr(V)\neq\emptyset$$.

I've tried that since $$X$$ is connected, and $$U, V$$ are proper subsets of $$X$$, then $$Fr(U)$$ and $$Fr(V)$$ are not empty. And also since $$Fr(U)$$ and $$Fr(V)$$ are connected, then $$Cl(U)$$ and $$Cl(V)$$ are connected. Then $$Cl(U)\cap V\neq\emptyset$$, and $$Cl(U)\cap (X-V)\neq\emptyset$$, then $$Cl(U)\cap Fr(V)\neq\emptyset$$, but I can not see this goes anywhere, any help, thanks!

• what are $A$ and $B$? – ΑΘΩ Dec 21 '19 at 1:01
• Should be U and V, typed wrong – Cathy Dec 21 '19 at 1:02

After edit: Still false. Consider two points $$x,y\in X=\Bbb R^n$$, and then $$U=X\setminus\{x\}$$ and $$V=X\setminus\{y\}$$.
Before edit: False. Consider $$X=\Bbb R^2$$, $$U=\{(x,y)\in\Bbb R^2\,:\, x^2+y^2<1\}$$ and $$V=\{(x,y)\in\Bbb R^2\,:\, (x-1)^2+y^2<1\}$$.
• typed wrong, sorry, the question should be $Fr(U)\cap Fr(V)\neq\emptyset$ – Cathy Dec 21 '19 at 1:08


text:
# Lattice determination of the $K \to (ππ)_{I=2}$ Decay Amplitude $A_2$

T. Blum, P. A. Boyle, N. H. Christ, N. Garron, E. Goode, T. Izubuchi, C. Jung, C. Kelly, C. Lehner, M. Lightman, Q. Liu, A. T. Lytle, R. D. Mawhinney, C. T. Sachrajda, A. Soni, C. Sturm

Research output: Contribution to journalArticlepeer-review

## Abstract

We describe the computation of the amplitude A_2 for a kaon to decay into two pions with isospin I=2. The results presented in the letter Phys.Rev.Lett. 108 (2012) 141601 from an analysis of 63 gluon configurations are updated to 146 configurations giving Re$A_2=1.381(46)_{\textrm{stat}}(258)_{\textrm{syst}} 10^{-8}$ GeV and Im$A_2=-6.54(46)_{\textrm{stat}}(120)_{\textrm{syst}}10^{-13}$ GeV. Re$A_2$ is in good agreement with the experimental result, whereas the value of Im$A_2$ was hitherto unknown. We are also working towards a direct computation of the $K\to(\pi\pi)_{I=0}$ amplitude $A_0$ but, within the standard model, our result for Im$A_2$ can be combined with the experimental results for Re$A_0$, Re$A_2$ and $\epsilon^\prime/\epsilon$ to give Im$A_0/$Re$A_0= -1.61(28)\times 10^{-4}$ . Our result for Im\,$A_2$ implies that the electroweak penguin (EWP) contribution to $\epsilon^\prime/\epsilon$ is Re$(\epsilon^\prime/\epsilon)_{\mathrm{EWP}} = -(6.25 \pm 0.44_{\textrm{stat}} \pm 1.19_{\textrm{syst}}) \times 10^{-4}$.
Original language English Physical Review D Unpublished - 22 Jun 2012