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# multiplicity

1. ### Zero and Multiplicity

For the given polynomial function, list each zero and its multiplicity. f(x) = [x - (1/3)]^2*(x - 1)^3 The zero of a function is the value that makes the function zero, right? I am not too clear on the idea of multiplicity.
2. ### Question regarding multiplicity

Hi all, I'm hoping you can give me some homework help so I can better understand this question. I was asked, given a 0.05 significance level, how many of the 20 significance tests in the example I could expect to be significant by chance alone. I worked this out to be 1. (Correct?) In the...
3. ### Multiplicity of Functions

What is the multiplicity of functions in simple words?
4. ### roots with multiplicity > 1 polynomials

I was solving the problem listed here.. Polynomial x I understood the logic behind the first line which says that "A root is of multiplicity 1 ==> the polynomial has a root in common with it's own derivative. But I wasn't able to follow the next where the reply is like this. Thus the given...
5. ### Need Help. Polynomials and multiplicity

Can someone help me with the problem. I need to understand it. I'm having a hard time creating the polynomial. The roots of multiplicity and the x=2 and x=0 are throwing me off. Where are they getting these from??? Here is the question: The polynomial of degree 5, has leading coefficient...
6. ### Multiplicity Homework Problem

Hi, I was wondering if I could have help with the following problem: Let A be an n by n matrix and let lambda be an eigenvalue of A whose eigenspace has dimension k, where 1<k<n. Show that lambda is an eigenvalue of A with multiplicity at least k. I'm supposed to use the following theorem...
7. ### Determine multiplicity

Determine the multiplicity for the root z = 1 for f(z)=(1-cos2\pi z)^{30} How exactly?
8. ### eigen value geometric multiplicity problem

find the jordan form of A=\left(\begin{array}{cccc} 4 & 1 & 1 & 1\\ -1 & 2 & -1 & -1\\ 6 & 1 & -1 & 1\\ -6 & -1 & 4 & 2\end{array}\right) ? my latex around the matrices is not working and it meesed up the whole thing so i removed it |tI-A|=\left|\begin{array}{cccc} t-4 & -1 & -1 & -1\\ 1...
9. ### eigenvalues --- multiplicity

ImageShack® - Online Photo and Video Hosting I found the 3 eigenvalues to be 1, 2 and -2. But how do I find which of the eigenvalues have multiplicity greater than 1? The only way i know is to calculate it (A-lambdaI), but is there some other way? Thanks
10. ### Algebraic and Geometric multiplicity

let A be the matrix |4 1 0 0 0| |0 4 1 0 0| |0 0 4 1 0| |0 0 0 4 1| |0 0 0 0 4| I am asked to find the algebraic multiplicity and the geometric multiplicity I think as it is an upper triangular matrix eigenvalues are the diagonal entries i.e. all 4, 4^5 so Am is 5, im not sure on the geometric...
11. ### Eigenvalue with multiplicity of 2?

haha im an idiot, i got it figured out
12. ### Multiplicity Problem.

I've been trying to figure out how to do this, but all the book says is... "In mathematics, we distinguish the difference between the solutions to this last equation (x^3-4x^2-3x+18=0) and the equation x^2-x-6=0 by saying x=3 is a solution of multiplicity 2 in the equation x^3-4x^2-3x+18=0...
13. ### State the multiplicity of each zero

Find the zeros of the polynomial function and state the multiplicity of each zero: P(x)=x^2 (3 x+5)^2 So I know that x=-5/3 but sense things are squared I'm loosing the picture. Is it 0 (multiplicity of 2) and -5/3 (multiplicity of 2)?
14. ### Characteristic roots problem

I am almost completely lost with this problem. I say almost completely because i have a vague understanding of the principle but my professor rushed through this lecture and our text book doesnt cover this.(Crying) Here is the problem: Prove that every solution of the equation: y'' + a1y' +...
15. ### Multiplicity of a root

Let c be a fixed real number. Show that a root of the equation x(x+1)(x+2)...(x+2009)=c can have multiplicity at most 2. Determine the number of values of c for which the equation has a root of multiplicity 2. Can anybody tell me how to compute this?
16. ### factor polynomial and state the multiplicity

how do i factor this completely to find the zeros and then state the multiplcity of each zero?? p(x)= x^3-x^2+x
17. ### Estimating local maxima and minima and identifying multiplicity of factors. #2

estimate the local maxima and minima and identify the mulltiplicity of the factors 1) F(X)=(2X^2+2X-4)(X^2-6X+9)
18. ### Estimating local maxima and minima and identifying multiplicity of factors #1

estimate the local maxima and minima and identify the mulltiplicity of the factors. 1) F(X)=X^2(x+2)^2(X-5)
19. ### Euler's phi function: multiplicity

I am supposed to show that, for Euler's totient function: a.) If n is an odd integer, then \phi(2n)=\phi(n). and b.) If n is an even integer, then \phi(2n)=2\phi(n). I thought I would use the multiplicity of phi to establish this, i.e. \phi(2n)=\phi(2)\cdot\phi(n)=1\cdot\phi(n)=\phi(n)...
20. ### The arithmetic and geometric multiplicity of an eigenvector

I have a matrix and I am supposed to calculate the eigenvalue, eigenvector and then give arithmetic and algebraic multiplicity.I can calculate the eigenvalues and the eigenvectors but I don't know how to calculate the arithmetic and geometric multiplicity. Tried to google it but it was a bit...

format: plain


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# p-adic numbers are Taylor expansions

Look at $P(x) = x-p$.  Fixing $p$ fixes the P(x).

$P(x)$ is a prime in the ring $\mathbb{C}[x]$.  We can build any polynomial out of these irreducible components.

Fix a prime, and expand your function:

$$f(x) = f(p) + f'(p)(x-p) + \frac{f”(p)}{2!}(x-p)^2 + …$$

If first few functions in the taylor series of two functions agree, than the functions are close, i.e., two functions are close if their difference is divisible by high powers of the monomial.

Let’s look at the integers, replacing primes of polynomials with primes of integers, fix a prime, and expand your function:

$$n = n(p) + n'(p)p + \frac{n”(p)}{2!}p^2 + …$$

Where $n(p), n'(p), \frac{n”(p)}{2!}, …$ are formal symbols, corresponding to the last coefficient, second to last coefficient, third to last, and so on and so forth.

Two integers are close if their expansions are divisible by high powers of p. The ring of all such integer expansions has only one maximal ideal, $p$.

This is how Hensel originally came upon the definition of the p-adics!


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Department of

# Mathematics

Seminar Calendar
for PhD Reunion events the year of Wednesday, September 13, 2017.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2017           September 2017          October 2017
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5                   1  2    1  2  3  4  5  6  7
6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
13 14 15 16 17 18 19   10 11 12 13 14 15 16   15 16 17 18 19 20 21
20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
27 28 29 30 31         24 25 26 27 28 29 30   29 30 31

Friday, September 15, 2017

8:00 am in Altgeld Hall,Friday, September 15, 2017

#### PhD Reunion

Abstract: The Department of Mathematics will hold a PhD reunion September 15-16, 2017. Visit the reunion webpage for more information http://www.math.illinois.edu/reunion/


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# Office rent

For rent 1m2 of office space is paid €165 per year. How much is the annual rent for an office as big as a football field with dimensions 71.5 m and 102.5 m?

Result

x =  1209243.75 Eur

#### Solution:

$x=165 \cdot \ (71.5 \cdot \ 102.5)=1209243.75 \ \text{Eur}=1.209244\cdot 10^{ 6 } \ \text{Eur}$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

## Next similar math problems:

1. Tony works
Tony works as a welder for a construction company. His hourly wage is $15.51 and he works 40 hours per week. Tony gets a paid vacation each year. The construction company is doing well and has plenty of work for the next year. Tony can expect to work the e 2. Conference 2 9 people attended a conference on behalf of their company. The conference fee was £520 per person and the company paid a total of £856 in travel costs for the 9 people. How much did the conference cost the company altogether? 3. One frame 5 picture frames cost € 12 more than three frames. How much cost one frame? 4. Concert On a Concert were sold 150 tickets for CZK 360, 235 tickets for 240 CZK and 412 for 180 CZK. How much was the total revenues for tickets? 5. Land cost The land is worth 33295 CZK. 1st owner has 5/15, 2nd owner has 4/15, 3rd owner has 1/5 4th owner has 1/5. How much CZK each of them will receive? 6. Customer A customer purchase three (3) writing pads from a store. She receive k9.70 change from a k10.00 note . How much was it's writing pad cost? 7. Product Result of the product of the numbers 1, 2, 3, 1, 2, 0 is: 8. Value of teeth To know value to the healthy teeth we discover only when we take care about tooth caries followed by loss of teeth and think how to replace missing teeth... Calculate the value (cost of money) healthy teeth, if we assume that man has with 32 teeth and rep 9. Addition of Roman numbers Added together and write as decimal number: LXVII + MLXIV 10. Find the 6 Find the total cost of 10 computers at$ 2100 each and 7 boxes of diskettes at \$12 each
11. Soda
A pack of 12 sodas cost 5.04. What is the unit rate in dollars per soda?
12. The largest number
Find the largest integer such that: 1. No figures is not repeat, 2. multiplication of every two digits is odd, 3. addition all digits is odd.
13. Bathroom 2
A bathroom is 2.4 meters long and 1.8 meters wide. How many square tiles 1 dm on each side are to be used to cover it?
14. School trip
6.A class went on a trip. Train ticket cost 6 euros, cable car ticket 6 euro. How many euros must pay classroom teacher for 23 students?
15. Seeds
Of the 72 wet seeds 7/8 germinated. How many seeds germinate?
16. Strip
From 5.9 cm wide strip should be cut rhombus with area 28 cm2. How long will be its side?
17. Rectangle
The rectangle area is 182 dm2, its base is 14 dm. How long is the other side? Calculate its perimeter.


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# adjacencyMatrix -- Returns the adjacency matrix of a Graph or Digraph

## Synopsis

• Usage:
• Inputs:
• D, an instance of the type Digraph,
• G, an instance of the type Graph,
• Outputs:
• A, ,

## Description

The adjacency matrix is the n by n matrix (where n is the number of vertices in graph/digraph G) with rows and columns indexed by the vertices of G. Entry A_(u,v) is 1 if and only if {u,v} is an edge of G and 0 otherwise. It is easy to observe that if we just use a simple graph G, then its adjacency matrix must be symmetric, but if we use a digraph, then it is not necessarily symmetric.

 i1 : D = digraph({{1,2},{2,3},{3,4},{4,3}},EntryMode=>"edges"); i2 : adjacencyMatrix D o2 = | 0 1 0 0 | | 0 0 1 0 | | 0 0 0 1 | | 0 0 1 0 | 4 4 o2 : Matrix ZZ <--- ZZ i3 : G = graph({1,2,3,4}, {{1,2},{2,3},{3,4},{4,3}}) o3 = Graph{1 => {2} } 2 => {1, 3} 3 => {2, 4} 4 => {3} o3 : Graph i4 : adjacencyMatrix G o4 = | 0 1 0 0 | | 1 0 1 0 | | 0 1 0 1 | | 0 0 1 0 | 4 4 o4 : Matrix ZZ <--- ZZ


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# height of a cylinder

Altitude of a Cylinder
Height of a Cylinder

The distance between the bases of a cylinder. Formally, the shortest line segment between the (possibly extended) bases. Altitude also refers to the length of this segment.


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0
Research Papers: Natural and Mixed Convection

# Natural Convection Heat Transfer From a Short or Long, Solid or Hollow Horizontal Cylinder Suspended in Air or Placed on Ground

[+] Author and Article Information
Swastik Acharya

Department of Mechanical Engineering,
Indian Institute of Technology Kharagpur,
BRH, A—513,
Kharagpur 721 302, India
e-mail: swastik.acharya8@gmail.com

Sukanta K Dash

Professor
Department of Mechanical Engineering,
Indian Institute of Technology Kharagpur,
Kharagpur 721 302, India
e-mail: sdash@mech.iitkgp.ernet.in

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 7, 2016; final manuscript received January 28, 2017; published online March 21, 2017. Assoc. Editor: Zhixiong Guo.

J. Heat Transfer 139(7), 072501 (Mar 21, 2017) (13 pages) Paper No: HT-16-1792; doi: 10.1115/1.4035919 History: Received December 07, 2016; Revised January 28, 2017

## Abstract

Numerical simulations have been conducted to study natural convection heat transfer from solid or hollow cylinders in the laminar range of Ra spanning from 104 to 108 for L/D in the range of $0.05≤(L/D)≤20$. Interesting flow structures around the thin hollow cylinder have been observed for small and large L/D. It has been found that the average Nu for solid or hollow horizontal cylinders in air is marginally higher than when they are on ground for the entire range of L/D and Ra limited up to 107. Up to a Ra of 107 Nu for a solid cylinder in air is higher than that of Nu for a hollow cylinder in air but when Ra exceeds 107 Nu for a hollow cylinder is marginally higher than that of the solid cylinder until an L/D of 0.2. When, L/D rises beyond 0.2 the situation reveres causing Nu for a solid cylinder to be again higher than that of the hollow cylinder when suspended in air. A solid cylinder on ground has higher Nu compared to that of a hollow cylinder on ground up to a Ra of 106. However, for higher Ra of 108 a hollow cylinder on ground has higher Nu compared to that of a solid cylinder on ground until an L/D of 5 and after that the situation reverses again.

<>
Topics: Cylinders

## References

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Petavel, J. , 1898, “ On the Heat Dissipated by a Platinum Surface at High Temperatures,” Philos Trans. R. Soc. A, 191(0), pp. 501–524.
Kennelly, A. , Wright, C. , and Bylevelt, J. V. , 1909, “ The Convection of Heat From Small Copper Wires,” Trans. Am. Inst. Electr. Eng., 28(1), pp. 363–393.
Langmuir, I. , 1912, “ The Convection and Conduction of Heat in Gases,” Trans. Am. Inst. Electr. Eng., 34(6), pp. 1229–1240.
Davis, A. , 1922, “ Natural Convective Cooling in Fluids,” Philos. Mag., 44(263), pp. 920–940.
Nelson, R. , 1924, “ Free Convection Heat in Liquids,” Phys. Rev., 23(1), pp. 94–103.
Nusselt, W. , 1929, “ Heat Dissipation From Horizontal Tubes and Wires to Gases and Liquids,” Ver DeutIng, 73, pp. 1475–1478.
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Tsubouchi, T. , and Masuda, H. , 1966–1967, “ Natural Convection Heat Transfer From a Horizontal Circular Cylinder With Small Rectangular Grooves,” Sci. Rep. Res. Inst. Tohaku Univ. Ser. B, 18, pp. 211–242.
Penney, W. R. , and Jefferson, T. B. , 1966, “ Heat Transfer From an Oscillating Horizontal Wire to Water and Ethylene Glycol,” ASME J. Heat Trans., 88(4), pp. 359–366.
Saville, D. , and Churchill, S. , 1967, “ Laminar Free Convection in Boundary Layers Near Horizontal Cylinders and Vertical Axisymmetric Bodies,” J. Fluid Mech., 29(2), pp. 391–399.
Churchill, S. , and Chu, H. , 1975, “ Correlating Equations for Laminar and Turbulent Free Convection From a Horizontal Cylinder,” Int. J. Heat Mass Trans., 18(9), pp. 1049–1053.
Nakai, S. , and Okazaki, T. , 1975, “ Heat Transfer Form a Horizontal Circular Wire at Small Reynolds and Grashof Numbers-I,” Int. J. Heat Mass Trans., 18, pp. 387–396.
Merkin, J. , 1977, “ Free Convection Boundary Layers on Cylinders of Elliptic Cross Section,” ASME J. Heat Trans., 99(3), pp. 453–457.
Kuehn, T. , and Goldstein, R. , 1980, “ Numerical Solution to the Navier–Stokes Equations for Laminar Natural Convection About a Horizontal Isothermal Circular Cylinder,” Int. J. Heat Mass Trans., 23(7), pp. 971–979.
Farouk, B. , and Güçeri, S. I. , 1981, “ Natural Convection From a Horizontal Cylinder-Laminar Regime,” ASME J. Heat. Trans., 103(3), pp. 522–527.
De Socio, L. , 1983, “ Laminar Free Convection Around Horizontal Circular Cylinders,” Int. J. Heat Mass Trans., 26(11), pp. 1669–1677.
Al-Arabi, M. , and Khamis, M. , 1982, “ Natural Convection Heat Transfer From Inclined Cylinders,” Int. J. Heat Mass Trans., 25(1), pp. 3–15.
Wang, P. , Khawita, R. , and Nguyen, D. L. , 1990, “ Transient Laminar Natural Convection From Horizontal Cylinders,” Int. J. Heat Mass Transfer, 34(6), pp. 1429–1442.
Saitoh, T. , Sajiki, T. , and Maruhara, K. , 1993, “ Bench Mark Solutions to Natural Convection Heat Transfer Problem Around a Horizontal Circular Cylinder,” Int. J. Heat Mass Trans., 36(5), pp. 1251–1259.
Chouikh, T. , Guizani, A. , and Maalej, M. , 1997, “ Numerical Study of Laminar Natural Convection Flow Around Horizontal Isothermal Cylinder,” Renewable Energy, 13(1), pp. 71–78.
Molla, M. M. , Paul, S. C. , and Hossain, A. , 2008, “ Natural Convection Flow for a Horizontal Cylinder With Uniform Heat Flux in Presence of Heat Generation,” Appl. Math. Model., 33, pp. 3226–3236.
Molla, M. M. , Hossain, M. A. , and Gorla, R. S. R. , 2005, “ Natural Convection Flow From Isothermal Horizontal Circular Cylinder With Temperature Dependent Viscosity,” Heat Mass Transfer, 41(7), pp. 594–598.
Atayilmaz, S. O. , and Teke, I. , 2009, “ Experimental and Numerical Study of the Natural Convection From a Heated Horizontal Cylinder,” Int. Commun. Heat Mass Trans., 36(7), pp. 731–738.
Ashjaee, M. , Yazdani, S. , Bigham, S. , and Yousefi, T. , 2011, “ Experimental and Numerical Investigation on Free Convection From Horizontal Cylinder Located Above an Adiabatic Surface,” Heat Transfer Eng., 33(3), pp. 213–214.
Mehrizi, A. A. , Farhadi, M. , Afrouzi, H. H. , Shayamehr, S. , and Lotfizadeh, H. , 2013, “ Lattice Boltzmann Simulation of Natural Convection Flow Around a Horizontal Cylinder Located Beneath an Insulated Plate,” J. Theor. Appl. Phys., 51(3), pp. 729–739.
Kuehner, J. P. , Hamed, A. M. , and Mitchell, J. D. , 2015, “ Experimental Investigation of Free Convection Velocity Boundary Layer and Plume Formation Region for a Heated Horizontal Cylinder,” Int. J. Heat Mass Transfer, 82, pp. 78–97.
Sedaghat, M. H. , Yaghoubi, M. , and Mafhrebi, M. J. , 2015, “ Analysis of Natural Convection Heat Transfer From a Cylinder Enclosed in a Corner of Two Adiabatic Wall,” Exp. Therm. Fluid Sci., 62, pp. 78–97.
Tsung-Yen, N. , 1993, “ Effect of Wall Condition on Natural Convection Over a Vertical Slender Hollow Circular Cylinder,” Appl. Sci. Res., 54, pp. 39–50.
Chang, L. C. , 2008, “ Numerical Simulation of Natural Convection of Micro Polar Fluid Flow Along Slender Hollow Cylinder With Wall Conduction Effect,” Non Linear Sci. Numer. Simul., 13, pp. 9–20.
Mamun, M. M. H. , Rahman, M. M. , Billah, M. M. , and Saidur, R. , 2010, “ A Numerical Study on the Effect of a Hollow Cylinder on Mixed Convection in a Ventilated Cavity,” Int. Commun. Heat Mass Transfer, 37(9), pp. 1326–1334.
Billah, M. M. , Rahman, M. M. , Sharif, M. U. , Rahim, N. A. , Saidur, R. , and Hassanuzzaman, M. , 2011, “ Numerical Analysis of Fluid Flow Due to Mixed Convection in a Lid-Driven Cavity Having a Heated Circular Hollow Cylinder,” Int. Commun. Heat Mass Transfer, 38(8), pp. 1093–1103.
Rani, H. P. , and Reddy, G. J. , 2011, “ Conjugate Transient Free Convective Heat Transfer From a Vertical Slender Hollow Cylinder With Heat Generation Effect,” Appl. Math., 1(2), pp. 90–98.
Ansys, 2013, “ Ansys Fluent, Release 15.0, User Manual,” ANSYS, Inc., Canonsburg, PA.
Senapati, J. R. , Dash, S. K. , and Roy, S. , 2016, “ Numerical Investigation of Natural Convection Heat Transfer Over Annular Finned Horizontal Cylinder,” Int. J. Heat Mass Transfer, 96, pp. 330–345.
Senapati, J. R. , Dash, S. K. , and Roy, S. , 2016, “ 3D Numerical Study of the Effect of Eccentricity on Heat Transfer Characteristics Over Horizontal Cylinder Fitted With Annular Fins,” Int. J. Therm. Sci., 108, pp. 28–39.
Morgan, V. T. , 1975, The Overall Convective Heat Transfer From Smooth Circular Cylinders, Academic Press, New York.

## Figures

Fig. 2

(a) A solid or thin hollow cylinder in air with the computational domain around it, (b) solid or the thin hollow cylinder lying on ground with its computational domain (aschematic representation)

Fig. 1

Flow vector around a (a) hollow and (b) solid isothermal cylinder on ground, L/D = 1, Ra = 10

Fig. 3

Influence of domain height on average Nu for a solid cylinder in air

Fig. 4

(a) Cell arrangement for hollow cylinder in air, (b) cross-sectional view, and (c) blown up view near the cylinder wall

Fig. 5

Average Nu for a hollow cylinder in air as a function of number of cells

Fig. 6

Average Nu for a solid cylinder in air, a comparison with experimental correlation

Fig. 11

Comparison of Nu between a solid and hollow cylinder on ground

Fig. 8

Temperature plume near a hollow cylinder on (a) ground and in (b) air, Ra = 108, L/D = 5

Fig. 9

Average Nu for a solid cylinder in air or ground when end faces are isothermal

Fig. 10

Comparison of Nu between a solid and hollow cylinder in air

Fig. 19

Thermal buoyant plume around a solid and hollow cylinder in air or ground

Fig. 12

Comparison of heat loss from the outer and inner surface of a hollow cylinder when placed in air

Fig. 13

Comparison of heat loss from the outer and inner surface of a hollow cylinder when placed on ground

Fig. 14

Velocity vector around a hollow cylinder placed on ground for Ra = 106, L/D (a) 0.5, (c) 1, and (e) 2

Fig. 15

Temperature plume around a hollow cylinder at Ra = 106 and L/D (a) 1, (b) 2, and (c) 5 when the cylinder is in air

Fig. 16

Temperature plume around a hollow cylinder at Ra = 106 and L/D (a) 1, (b) 2, and (c) 5 when the cylinder is on ground

Fig. 17

Thermal plume around a short solid cylinder in air (a)–(b) and ground (c)(d), Ra = 106

Fig. 18

velocity vector around a short solid cylinder in air (a)–(b) and ground (c)–(d), Ra = 106

Fig. 7

(a) Average Nu for a solid cylinder in air or lying on ground as a function of L/D and (b) average Nu for a hollow cylinder in air or lying on ground as a function of L/D

Fig. 20

A comparison of predicted Nu with that of the computed value for a solid cylinder in air

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# Is the supercurrent gauge invariant?

If we consider $${\cal N}=1$$ renormalizeable chiral gauge theories, specifically discussed in section 27.4 of Weinberg's Quantum Theory of Fields, Supersymmetry book, should the supercurrent be gauge invariant?

Any observable of a gauge invariant theory must also be gauge invariant. However, the supercurrent is a Grassmann variable, and thus is not observable.

I ask this because in order to calculate the supercurrent, Weinberg argues to calculate it in a specific gauge to make the analysis easier. This is only possible if the supercurrent itself is gauge invariant.

• Can you not look at the expression Weinberg calculates and just check (admittedly in a potentially tedious calculation) whether it is gauge invariant? – ACuriousMind Mar 6 at 20:33
• Yes that is a good point. I checked and it does not appear to be gauge invariant, nor covariant. But that puts into question the whole procedure of working in a special gauge! – LucashWindowWasher Mar 6 at 21:07

It is true that the supercurrent in this case is not gauge invariant. Weinberg writes the supercurrent in the gauge where at the point $$x^{\mu}=X^{\mu}$$, we have that the gauge connection vanishes, so $$A_{\mu}^a(X)=0$$. In this gauge, one term in the supercurrent reads $$S^{\mu}\supset-\frac{1}{4}f^a_{\rho\sigma}[\gamma^{\rho},\gamma^{\sigma}]\gamma^{\mu}\lambda^a$$
Where $$f^a_{\rho\sigma}=\partial_{\rho}A_{\sigma}^a-\partial_{\sigma}A_{\rho}^a$$ is the non-abelian gauge curvature tensor (remember that we are working in the gauge that $$A_{\mu}^a(X)=0$$), and $$\lambda^a$$ is a Majorana fermion supersymmetric pair to the gauge field $$A^{a}_{\mu}$$. This term is enough to show that the supercurrent is not gauge invariant.


text:
Emacs as an Email Client

Emacs is a very powerful editor, therefore there are many benefits from using it as an email client, such as direct integration with org-mode for todo and task management or the amazing editing capabilities of Emacs to write emails.

However, Emacs cannot do this natively, but there is great integration with a tool called mu. This tool is an indexer for your emails, and keeps track of them so that they are easily and quickly searchable. The author of this tool also wrote an emacs-lisp file that queries mu and provides a user interface in emacs to better interact with it and use it to read emails.

mu requires the emails to already be on the computer though, so the first step is to download them using IMAP.

IMAP is a protocol that can be used to download a copy of your emails from the server. A great tool to use to download them using IMAP is mbsync. In arch linux, this tool can be downloaded from the official repository using

sudo pacman -S isync

This command line utility has to first be set up using a config file, which is usually located in ~/.mbsyncrc, so that it knows where to download the emails from and how to authenticate properly.

The most important parts to set up in the config file are

IMAPAccount gmail
Host imap.gmail.com
User [email protected]
SSLType IMAPS
CertificateFile /etc/ssl/certs/ca-certificates.crt

to setup the account, and then the following to setup the directories where it should download emails to

IMAPStore gmail-remote
Account gmail

MaildirStore gmail-local
Subfolders Verbatim
Path ~/.mail/gmail/
Inbox ~/.mail/gmail/Inbox

Channel gmail
Master :gmail-remote:
Slave :gmail-local:
Patterns *
Create Both
SyncState *

Once mbsync is configured, the emails can be downloaded using

mbsync -a

** Indexing the Emails

Once they are downloaded (in this case in the ~/.mail directory), they have to be indexed so that they can quickly be searched using Emacs. This is done by using the following shell command.

mu index --maildir=~/.mail

However, as mu also has an emacs-lisp plugin, the following will also work after it has been configured correctly in emacs.

emacsclient -e '(mu4e-update-index)'

*** Emacs Configuration

To use mu in emacs as well, one first has to load the emacs lisp file using

(require 'mu4e)

After that, mu4e can be configured with different things like the home directory, and shortcuts that should be used in emacs. The full configuration can be seen in my Emacs configuration, which is hosted on Github

** Sending Emails

Sending emails from Emacs requires a different protocol which is SMTP, however, that is already included in Emacs. The most basic setup can be seen below.

(smtpmail-smt-user            . "[email protected]")
(smtpmail-local-domain        . "gmail.com")
(smtpmail-default-smtp-server . "smtp.gmail.com")
(smtpmail-smtp-server         . "smtp.gmail.com")
(smtpmail-smtp-service        . 587)

** Conclusion

Emacs is now ready to be used as a full featured email client.


text:
# Linear Spline – Piecewise Interpolation [C PROGRAM]

Lagrange or Newton polynomial interpolations are useful interpolation techniques to have in your sleeves, but they don’t always give the best or desired result. As the degree of the polynomial increases, so do the wiggles.

Therefore, it is often advantageous to use piecewise interpolation, also known as spline interpolation.

A spline is simply a curve that connects two or more specific points.

Originally, spline was a term for elastic rulers that were bent to pass through a number of predefined points (“knots”). These were used to make technical drawings for shipbuilding and construction by hand.

In this post I will be sharing with you a C program that performs linear spline interpolation.
The user is asked to enter a set of x and y-axis data-points, and then each of these is joined by a straight line.
So the code would involve finding the equation of line connecting the two points.

You might have noticed this kind of interpolation in chart plotting softwares like Origin, Excel, Gnuplot where when you plot the data-set using the line chart, then the points are joined together by line segments as shown below.

So basically the code would need to generate a set of line segments for 2 adjacent points given by, $(x_i,y_i)$ and $(x_{i+1},y_{i+1})$.
The equation can be found by using the following formula:
$y=\frac{y_{i+1}-y_i}{x_{i+1}-x_i}(x-x_i)+y_i$

So, here’s what we’re gonna do in the program:
1. We’re gonna ask the user to enter the no. of data points.
2. Then, we’re gonna let the user enter these data-points by running a simple for loop.
3. Next, we will calculate the equation of line between 2 adjacent points, and use it to interpolate the values.
4. We will store the interpolated values at a desired interval of say, 0.01, in arrays.
5. Then we will store them in a file and scatter plot them using a suitable software.

### CODE:

/*********************************
**********LINEAR SPLINE***********
*********************************/
#include<stdio.h>
/*******
Function to perform piecewise linear spline interpolation
Parameters:
n: no. of data-points
x[n]: x-axis points
y[n]: y-axis points
N: size of the array storing the interpolated x and y points
X[n]: array that stores the interpolated x-axis points
Y[n]: array that stores the interpolated y-axis points
********/
void lSpline(int n, double x[n], double y[n], int N, double h, double X[N], double Y[N]){
int i;
int j=0;
for(i=0;i<n-1;i++){
//
double yn,xn;
for(xn=x[i];xn<x[i+1];xn=xn+h){
yn=(y[i+1]-y[i])*(xn-x[i])/(double)(x[i+1]-x[i])+y[i];
//yn=(xn-x[i+1])/(x[i]-x[i+1])+(xn-x[i])/(x[i+1]-x[i]);
Y[j]=yn;
X[j]=xn;
j++;
}
}
}
main(){
int n;
int N=0; //N is the no. of interpolated values
int i;
double h=0.01; //Space interval at which interpolated values are calculated
printf("Enter the no. of data-points: (n)\n");
scanf("%d",&n);
double x[n];
double y[n];
printf("Enter the x-axis data points:\n");
for(i=0;i<n;i++){
scanf("%lf",&x[i]);
}
printf("Enter the y-axis data points:\n");
for(i=0;i<n;i++){
scanf("%lf",&y[i]);
}
//The following procedure calculates N
for(i=0;i<n-1;i++){
N=N+(x[i+1]-x[i])/h;
}
//A little adjustment to get the correct value of N
N=N+2+(n-4);
printf("\nThe no. of interpolated values N= %d\n",N);
double Y[N];
double X[N];
//Perform piece-wise linear interpolation
lSpline(n,x,y,N,h,X,Y);
//Store the interpolated values in a File
FILE *fp = "NULL";
fp=fopen("linSpline1.txt","w");
for(i=0;i<N;i++){
fprintf(fp,"%lf\t%lf\n",X[i],Y[i]);
}
}

### Android App:

I’ve created a few data interpolation android apps, that you can check out.


text:
Cyclist

Cyclist goes uphill 10 km for 50 minutes and downhill minutes for 29 minutes, both applied to the pedals same force. How long he pass 10 km by plane?

Result

t =  36.7 min

Solution:

$\dfrac{1}{ 50}+\dfrac{1}{ 29} = \dfrac{2}{ t} \ \\ \ \\ t = \dfrac { 2\cdot 50\cdot 29}{ 50+29 } = 36.7 \ \text{min}$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

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At the same time, two cyclists left the towns A and B at constant speeds. The first one going from town A to town B, and the second one from town B to town A. At one point of the trip they met. After they met, the first cyclist arrived at town B in 36min,
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text:
# complex conjugate calculator

Mathematical function, suitable for both symbolic and numerical manipulation. To improve this 'Conjugate transpose (Hermitian transpose) Calculator', please fill in questionnaire. An easy to use calculator that converts a complex number to polar and exponential forms. Here, $$2+i$$ is the complex conjugate of $$2-i$$. You can use them to create complex numbers such as 2i+5. You can also determine the real and imaginary parts of complex numbers and compute other common values such as phase and angle. Note that there are several notations in common use for the complex conjugate. Active 4 years, 1 month ago. Polynomial Roots Calculator. These conjugate complex numbers are needed in the division, but also in other functions. The operation also negates the imaginary part of any complex numbers. Hello, @jtrantow gave a very good explanation. (iii) The complex conjugate of w is denoted by w*. can be entered as co, conj, or \[Conjugate]. It's really the same as this number-- or I should be a little bit more particular. But to divide two complex numbers, say $$\dfrac{1+i}{2-i}$$, we multiply and divide this fraction by $$2+i$$.. Conjugate a complex number. In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix with complex entries, is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of + being −, for real numbers and ).It is often denoted as or ∗.. For real matrices, the conjugate transpose is just the transpose, = Mathematica » The #1 tool for creating Demonstrations and anything technical. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. Now, let’s move to the actual calculations. How to Use the Dividing Complex Numbers Calculator? Find, in the form z a k, the equation of circle passing through S, T and the origin. 1) Computes the complex conjugate of z by reversing the sign of the imaginary part. Convert a Complex Number to Polar and Exponential Forms - Calculator. It also demonstrates elementary operations on complex numbers. Complex_conjugate function calculates conjugate of a complex number online. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social … Is there a way to compute for the complex conjugate, given an array with complex elements? complex conjugate of exp(i*x) Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. A1. The conjugate of a complex number is that number with the sign of the imaginary part reversed. or z gives the complex conjugate of the complex number z. Complex number calculator: complex_number. Find real and complex zeroes of a polynomial. But, imaginary part differs in the sign, with same coefficient. 2) Additional overloads are provided for float , double , long double , and all integer types, which are treated as complex numbers with zero imaginary component. We learn the theorem and illustrate how it can be used for finding a polynomial's zeros. Definition: Complex conjugate in mathematics, is a pair of complex numbers, which has same real part. Complex Conjugate of this Hot Network Questions Fantasy book set in medieval times, in which one of the characters wields a double-sided battleaxe Wolfram|Alpha » Explore anything with the first computational knowledge engine. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Worksheets on Complex Number. Online calculator. Viewed 6k times 4. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Calculation for multiplication and division: Index Complex numbers . Why we use conjugate of current than the original phasor in the calculation of Complex Power i-e S=VI* Ask Question Asked 4 years, 1 month ago. We're asked to find the conjugate of the complex number 7 minus 5i. The conjugate of a complex number $z$ is written $\overline{z}$ or $z^*$ and is formed of … Details. Figure 2: Complex conjugate representation in (a) Cartesian form and (b) polar form. The calculator displays complex number and its conjugate on the complex plane, evaluate complex number absolute value and principal value of the argument . Some more important notions such as the conjugation transformation are introduced in a second section. Example: type in (2-3i)*(1+i), and see the answer of 5-i. Complex Number Calculator. Every complex number has a so-called complex conjugate number. For Polynomials of degree less than or equal to 4, the exact value of any roots (zeros) of the polynomial are returned. Compute , Evaluate expressions involving Complex Numbers, Take the Square Root (Step by Step) , Find the Conjugate, Compute Arg(z), Modulus(z) Solve any Complex Equation; Read Basics on Complex Analysis and Identities involving Trigonometric , Logarithmic, Exponential and Polynomial Functions, DeMoivre Theorem The notion of complex numbers was introduced in mathematics, from the need of calculating negative quadratic roots. I will try to summarize it according to my understanding. BYJU’S online dividing complex numbers calculator tool performs the calculation faster and it displays the division of two complex numbers in a fraction of seconds. Functions. Complex numbers have the property to be solutions of some equations that cannot be solved by usual real numbers, such as the equation x 2 =-1. ', performs a transpose without conjugation. Examples open all close all. Tips . So the conjugate of this is going to have the exact same real part. Conjugate the English verb complete: indicative, past tense, participle, present perfect, gerund, conjugation models and irregular verbs. [3] 9. The complex number u is given by u = -1 + 4 3i (i) Without using a calculator and showing all your working. The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).. The calculator will show you the work and detailed explanation. Suppose that Z j 3 4 , then Z j* 3 4 (just flip the sign of the imaginary part). Instructions:: All Functions . This calculator extracts the square root, calculate the modulus, finds inverse, finds conjugate and transform complex number to polar form.The calculator will … SEE: Conjugate Transpose. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. The complex number calculator only accepts integers and decimals. Translate complete in context, with examples of … Instructions. Able to display the work process and the detailed explanation. Complex Number Calculator. Complex number concept was taken by a variety of engineering fields. Thanks! I have an array consisting of imaginary elements using fast Fourier transform (fft.fft) and would like to return a list with the complex conjugate of each element. B2 ( a + bi) Error: Incorrect input. Mathematically, if I 15 30 o, then I * 15 30 (just flip the sign on the angle). And what you're going to find in this video is finding the conjugate of a complex number is shockingly easy. The complex conjugate zeros, or roots, theorem, for polynomials, enables us to find a polynomial's complex zeros in pairs. It has the same real part. We know that to add or subtract complex numbers, we just add or subtract their real and imaginary parts.. We also know that we multiply complex numbers by considering them as binomials.. Are we sort of trying to get rid of the imaginary part here or something like that? Conjugate automatically threads over lists. The complex number calculator allows to perform calculations with complex numbers (calculations with i). Complex Number Lesson. Additionally, if the value is expressed in rectangular form, as is often the case for impedances, it’s just as easy. The nonconjugate transpose operator, A. As an example we take the number $$5+3i$$ . Calculate Complex Conjugate of Real / Imaginary Numbers - Tutorial, Definition, Formula and Example. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. The utility of the conjugate is that any complex number multiplied by its complex conjugate is a real number: This operation has … Create a 2-by-2 matrix with complex elements. The procedure to use the dividing complex numbers calculator is as follows: Step 1: Enter the coefficients of the complex numbers, such as a, b, c and d in the input field. abs: Absolute value and complex magnitude: angle: Phase angle: complex: Create complex array: conj: Complex conjugate: cplxpair: Sort complex numbers into complex conjugate pairs: i: … Find Complex Conjugate of Complex Values in Matrix. Keep in mind that Figure 2(a) and 2(b) are two different ways of describing the same point. Open Live Script. Hermitian Conjugate. I am wondering why when we calculate correlation of 2 complex number we always use the conjugate on one of them. Let's take a closer look at these figures. The complex number conjugated to $$5+3i$$ is $$5-3i$$. Wolfram Web Resources. Here is the complex conjugate calculator. Applied physics and engineering texts tend to prefer , while most modern math and … The complex_modulus function allows to calculate online the complex modulus. We also work through some typical exam style questions. The complex conjugate is implemented in the Wolfram Language as Conjugate[z].. Just type your formula into the top box. Complex modulus: complex_modulus. I was thinking of extracting the imaginary bits by using nparray.imag, but it creates a list without "j" to denote imaginary. HyperPhysics****HyperMath*****Algebra: R Nave: Go Back: Complex Conjugate. the complex conjugate of I is written I *. If a complex number is a zero then so is its complex conjugate. The complex numbers w and w* are represented in Argand diagram by the points S and T respectively. Find the two square roots of u. Tool for calculating the value of the conjugate of a complex number. The idea is to find the modulus r and the argument θ of the complex number such that z = a + i b = r ( cos(θ) + i sin(θ) ) , Polar form z = a + ib = r e iθ, Exponential form The calculation rules are presented in a third section. B1 ( a + bi) A2. This online calculator finds the roots of given polynomial. 5. 0 Reply 0 [ - ] Reply by sami_aldalahmah January 3, 2018. Allows to perform calculations with I ) Nave: Go Back: complex.! And exponential forms complete in context, with same coefficient I 15 30 ( just flip sign!, enables us to find a polynomial 's complex zeros in pairs real and imaginary of. Polynomials, enables us to find the conjugate of z by reversing the sign the... In mind that figure 2 ( b ) polar form then so is complex. I is written I * them to create complex numbers w and w * are represented in Argand diagram the! Mathematically, if I 15 30 o, then z j * 3 4, then I * on! I 15 30 o, then I * 15 30 ( just flip the of... Absolute value and principal value of the complex plane, evaluate complex number number absolute value principal... Anything with the sign of the conjugate of a complex number z diagram by points... Wolfram Language as conjugate [ z ] the number \ ( 2-i\ ) the points S and T respectively in! ( 1+i ), and see the answer of 5-i is finding the conjugate of /! Also in other functions * * * * * * * HyperMath * * * *... Rid of the complex conjugate of a complex number online numbers such as the transformation... Should be a little bit more particular the conjugate of w is denoted by w * + )!, or roots, theorem, for polynomials, enables us to find in this video is finding conjugate. To summarize it according to my understanding phase and angle then so is its complex,. Of given polynomial of a complex number absolute value and principal value of the part! Expressions in the form z a k, the equation of circle passing through S T! To display the work process and the origin \ ( 2-i\ ) the same.! Process and the origin find in this video is finding the conjugate of a complex number to polar exponential! Complex zeros in pairs note that there are several notations in common use for the complex.. I was thinking of extracting the imaginary part here or something like that asked find!, for polynomials, enables us to find a polynomial 's zeros any complex numbers 7 minus 5i anything the... So the conjugate of a complex number is a zero then so is its complex conjugate the. Real / imaginary numbers - Tutorial, Definition, Formula and example an array with complex elements Cartesian. Entered as co, conj, or roots, theorem complex conjugate calculator for polynomials, enables us find!: Go Back: complex conjugate 1+i ), and see the of... Polynomials, enables us to find in this video is finding the conjugate of \ ( 5-3i\.! As conjugate [ z ] polynomial 's complex zeros in pairs number.! Complex number mind that figure 2 ( b ) polar form more important notions such as 2i+5 zero then is. Number 7 minus 5i it 's really the same as this number -- or should. This online calculator finds the roots of given polynomial will try to summarize it according my... Passing through S, T and the detailed explanation and principal value of the imaginary part here or like. Have the exact same real part plane, evaluate complex number has a so-called complex conjugate complex,! Now, let ’ S move to the actual calculations was taken by a of. Is its complex conjugate in pairs, let ’ S move to the actual calculations form z a k the... Real part w is denoted by w * or roots, theorem, for,. Used for finding a polynomial 's zeros find a polynomial 's complex zeros pairs. Argand diagram by the points S and T respectively online calculator finds roots... Answer of 5-i ', please fill in questionnaire detailed explanation, @ jtrantow gave a very explanation. First computational knowledge engine as phase and angle other common values such as phase and angle (... And illustrate how it can be entered as co, conj, or \ [ conjugate ] I 15 (. Angle ) what you 're going to find a polynomial 's complex in! Expressions in the division, but also in other functions part ) z. Typical exam style questions * 15 30 ( just flip the sign of the imaginary part here or like... As co, conj, or \ [ conjugate ] * 3 4 ( just flip the sign of imaginary. Find, in the Wolfram Language as conjugate [ z ] conjugate representation (. \ [ conjugate ] calculator only accepts integers and decimals T and the detailed.... Find in this video is finding the conjugate of I is written I.! Number to polar and exponential forms form z a k, the equation of circle passing through S, and. The complex conjugate, given an array with complex numbers are needed in the Wolfram as. The Wolfram Language as conjugate [ z ] as 2i+5 a + bi ) Error: Incorrect input real... - Tutorial, Definition, Formula and example 're asked to find a polynomial 's zeros 5-3i\.... Argand diagram by the points S and T respectively to summarize it to! Of describing the same point by reversing the sign, with same coefficient by the S! Part differs in the Wolfram Language as conjugate [ z ] number calculator only accepts integers and decimals '' denote! Number -- or I should be a little bit more particular hyperphysics * * Algebra: Nave. In context, with same coefficient ( 2-i\ ) suitable for both symbolic and manipulation! To compute for the complex conjugate nparray.imag, but it creates a list without  j to... A little bit more particular of w is denoted by w * are represented in Argand diagram the... Such as the conjugation transformation are introduced in a third section mathematics, is zero. Polynomial 's complex zeros in pairs using nparray.imag, but it creates a without! ) and 2 ( a + bi ) Error: Incorrect input does basic arithmetic complex... Sort of trying to get rid of the imaginary part ) 4, then z j 4... ’ S move to the actual calculations Language as conjugate [ z..... A ) and 2 ( b ) polar form values such as 2i+5 these complex! Take the number \ ( 5-3i\ ) + bi ) Error: Incorrect input numbers ( calculations complex. Bi ) Error: Incorrect input translate complete in context, with examples of this... \ ( 5+3i\ ) is the complex modulus extracting the imaginary part differs in the form a. Several notations in common use for the complex number to polar and exponential forms the complex numbers exact..., let ’ S move to the actual calculations ) and 2 ( a ) Cartesian and! Real part in mathematics, is a zero then so is its conjugate. It can be used for finding a polynomial 's complex zeros in pairs )! Keep in mind that figure 2: complex conjugate of the imaginary part any... Then so is its complex conjugate zeros, or \ [ conjugate ] plane, evaluate complex number z 5+3i\. Really the same point to complex conjugate calculator actual calculations does basic arithmetic on complex.. It 's really the same as this number -- or I should be a bit! Polar and exponential forms, given an array with complex numbers such as the conjugation transformation are introduced a! Anything with the first computational knowledge engine introduced in a third section of any complex numbers compute. I * 15 30 ( just flip the sign on the complex number calculator only accepts integers and decimals create. W is denoted by w * are represented in Argand diagram by the points S T. Is denoted by w * or \ [ conjugate ] with I ) number value... And angle ’ S move to the actual calculations angle ) engineering fields number. 5+3I\ ) is \ ( 2-i\ ) expressions in the sign of the imaginary part reversed which same! Calculator allows to perform calculations with complex numbers such as the conjugation transformation are introduced a... And compute other common values such as phase and angle multiplication and division Index. Something like that rid of the imaginary part Error: Incorrect input this number or..., conj, or \ [ conjugate ] third section rules are presented in a third section absolute. R Nave: Go Back: complex conjugate evaluates expressions in the sign on the angle.... Work and detailed explanation of the imaginary part that converts a complex number and its conjugate on the complex is... O, then z j * 3 4 ( just flip the sign, same. ) polar form the conjugation transformation are introduced in a third section flip... Use calculator that converts a complex number conjugated to \ ( 2+i\ ) is the complex conjugate number I try! Use them to create complex numbers ( calculations with I ) and illustrate how it can be entered as,! Mathematical function, suitable for both symbolic and numerical manipulation I was thinking of extracting the imaginary part differs the! Find in this video is finding the conjugate of a complex number online Definition: complex number! Complex numbers is denoted by w * are represented in Argand diagram by the points S T., which has same real part 1 tool for creating Demonstrations and anything technical good explanation suitable! Numbers w and w * the conjugate of the complex modulus ( a Cartesian.


text:
# Time Complexity solution codechef

Time Complexity solution codechef – A sorting algorithm A is said to have more time complexity than a sorting algorithm B if it uses more number of comparisons for sorting the same array than algorithm B.

## Time Complexity solution codechef

Given that algorithm A uses X comparisons to sort an array and algorithm B uses Y comparisons to sort the same array, find whether algorithm A has more time complexity.

### Input Format

• The first line of input will contain a single integer T, denoting the number of test cases.
• Each test case consists of two space-separated integers X and Y — the number of comparisons used by algorithms A and B to sort the array respectively.

### Output Format

For each test case, output on a new line, YES, if the algorithm A has more time complexity than B and NO otherwise.

You may print each character of the string in uppercase or lowercase (for example, the strings YESyEsyes, and yeS will all be treated as identical).

## Time Complexity solution codechef

• 1 \leq T \leq 100
• 1 \leq X, Y \leq 100

### Sample 1:

Input

Output

4
9 9
15 7
10 19
21 20

NO
YES
NO
YES

## Time Complexity solution codechef

Test case 1: The number of comparisons used by algorithm A is 9 and that used by B is also 9. Since the number of comparisons used by A is not more than that of BA does not have more time complexity than B.

Test case 2: The number of comparisons used by algorithm A is 15 and that used by B is 7. Since the number of comparisons used by A is more than that of BA does have more time complexity than B.

Test case 3: The number of comparisons used by algorithm A is 10 and that used by B is 19. Since the number of comparisons used by A is not more than that of BA does not have more time complexity than B.

Test case 4: The number of comparisons used by algorithm A is 21 and that used by B is 20. Since the number of comparisons used by A is more than that of BA does have more time complexity than B.


text:
# Approximate ratio

#### onako

The following is an approximation problem. Given a ratio:
$$D =\frac{ \sum a_ib_i}{\sum b_ib_i}$$

I wonder what steps to follow to give a reasonable approximation.
This is an intuition
$$D =\frac{ \sum a_i}{\sum b_i}$$.
Clearly, given that all b_i terms are the same, the approximation is the correct solution.
But, this is not the case. The question is, under which conditions (assumptions on a_i or b_i) is the
above approximation accurate.
I guess, the lower the standard deviation of the b_i's, the more accurate the approximation. But, suppose the
b_i's are [1,2,3,4,5,6,7...]. What condition needs be meet for the accurate approximation.
In a sense, the ratio I'm trying to approximate is the weighted average, and in the approximation I'm discarding
the weights. If this interpretation makes it easier to further interpret, please use it.

Also, as for the conditions, I thought of
$$a_i>a_k, => b_i>b_k$$
How is this condition affecting the accuracy of the ratio approximation. Is the approximation more accurate with
this assumption.
Thanks
(topic in Number theory and Calculus subforum)

Related Linear and Abstract Algebra News on Phys.org

#### fresh_42

Mentor
2018 Award
There is no reasonable way to compare the two quantities, other than all $b_i$ are almost equal.

### Want to reply to this thread?

"Approximate ratio"

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text:
# Finding the limit of complex function

I am trying to check the continuity of this complex function at the origin.

$f(z)=\begin{cases} \operatorname{Im}( \frac{z}{1+|z|} ) \qquad &\mbox{when } z\neq0,\\ 0 \qquad &\mbox{when }z=0. \end{cases}$

According to my understanding (correct me if i am wrong), in order for a this function to be continuous at the origin, first, $f(0)$ must exists!(which it does) Then,the limit of $f(z)$ as it tends to 0 must exists too. And both has to be the same.

so,

$\lim_{z\to0} ( \operatorname{Im} ( \frac{z}{1+|z|} ) ) = \lim_{x\to0 \\ y\to0}(\operatorname{Im} ( \frac{x+iy}{1+\sqrt[]{x^2+y^2} } ))$

But now , if we take only the imaginary part of the function,(i.e $y$ from $iy$), wouldn't it leave only leave $\lim_{x\to0 \\ y\to0} y$ ?

Then if y approach 0 first, then the limit is zero. But if x approach 0 first, nothing happens.

Can someone kindly enlighten me where have I got this wrong? I am learning complex analysis by myself and English is not my first language so, I believe I've got some wrong ideas on how to approach this problem.

• It'd leave you with, $$\lim_{x,y\to 0}\frac{y}{1+\sqrt{x^2+y^2}}$$ – Prasun Biswas Apr 19 '15 at 3:51
• Note $|\text {Im}(u)|\le |u|$ for any complex $u.$ – zhw. Apr 19 '15 at 3:52

$$\frac{z}{1+|z|}=\frac{x+iy}{1+\sqrt{x^2+y^2}}=\frac{x}{1+\sqrt{x^2+y^2}}+\left(\frac{y}{1+\sqrt{x^2+y^2}}\right)i$$

Now, it is trivial that,

$$\Im\left(\frac{z}{1+|z|}\right)=\frac{y}{1+\sqrt{x^2+y^2}}$$

I hope you can do the rest.

Note: $\Im(z)=\textrm{Im}(z)$

From the definition of a limit, $f(z) \to L$ as $z \to z_0$, if for any $\epsilon >0$, there exists a number $\delta>0$ such that $|f-L|<\epsilon$ whenever $0<|z-z_0|<\delta$.

Here, $f=\text{Im}\left(\frac{z}{1+|z|}\right)=\frac{y}{1+\sqrt{x^2+y^2}}$ and $z_0=0$.

Let's show that we can find a $\delta>0$ so that $|f-0|<\epsilon$ whenever $0<|z|=\sqrt{x^2+y^2}<\delta$.

\begin{align} |f(z)|&=\left|\frac{y}{1+\sqrt{x^2+y^2}}\right|\\\\ &\le\frac{\sqrt{x^2+y^2}}{1+\sqrt{x^2+y^2}}\\\\ &\le\frac{\sqrt{x^2+y^2}}{1}\\\\ &=\sqrt{x^2+y^2}\\\\ &<\epsilon \end{align}

whenever $\sqrt{x^2+y^2}<\epsilon$.

Thus, given any $\epsilon$, choose $\delta \le \epsilon$ to guarantee that $|f(z)|<\epsilon$ whenever $0<|z|<\delta$.


text:
# PlotLegend not showing

## Summary:

I plotted a few curves for the wavefunction of Quantum Harmonic oscillator problem with certain energy values. The plotlegends do not seem to work in the final output.

## Main Question or Discussion Point

Here, are the parts of the plot function

plot I:
p1 = Plot[Normal[g*Exp[-x^2/2] /. solute[[1]] /. en -> 3], {x, -8, 8},
PlotStyle -> {Dashed, Gray}, PlotLegends -> Automatic];
(* Here, [B]g[/B]=1 - x^2 - x^4/6 - x^6/30 - x^8/168 - x^10/1080 , energy [B]en[/B] =3 and solute is the series solution of [B]g[/B] with unknown coefficients*)

plot II:
p2 = Plot[
Normal[g*Exp[-x^2/2] /. solute[[1]] /. en -> 10], {x, -8, 8},
PlotStyle -> {Dashed, Red}, PlotLegends -> Automatic];

(* Here, g=1 - (9 x^2)/2 + (15 x^4)/8 - x^6/16 - (3 x^8)/896 - x^10/3840 *)
plot III:
p3 = Plot[
Normal[g*Exp[-x^2/2] /. solute[[1]] /. en -> 17], {x, -8, 8},
PlotStyle -> {Thick, Black}, PlotLegends -> Automatic];

(* Here, g= 1 - 8 x^2 + 8 x^4 - (32 x^6)/15 + (16 x^8)/105 *)
and then the SHOW command as follows

Show command:
Show[{p1, p2, p3}, PlotRange -> All,
LabelStyle -> Directive[Blue, Italic, 12]]
What did I do wrong, here?

Related MATLAB, Maple, Mathematica, LaTeX News on Phys.org
jedishrfu
Mentor
Try plotting one at a time and see if the plot legends appear then add one plot and try again.

If they appear on the first plot but not the 2nd or 3rd then you know where your error might be.

You could also try plotting them in reverse order.

The basic idea is to see if its the way you specified plot legends in each plot call vs a mistake in a plot that somehow hides the plot legend.

Also have you searched via google to see if others have had the same issue?

Here's what I found about plotlegends:

https://reference.wolfram.com/language/PlotLegends/ref/PlotLegend.html

In one example they do two plots in one call so maybe that's what you have to do.

Thanks @jedishrfu
I shall try your suggestion and see if it works and then post the working solution here.

So, this is the way i have learned to do it for the SHOW command
(I checked the plots individually and the PlotLegend worked fine. However, I found the following way to be more clean than filling the plot command with most of the attributes.)

Use of Legended:
Legended[Show[p1, p2, p3, PlotRange -> All],
Placed[LineLegend[{Gray, Red,
Green}, {"\!$$\*SubscriptBox[\(\[Psi]$$, $$2$$]\)(x)",
"\!$$\*SubscriptBox[\(\[Psi]$$, $$6$$]\)(x)",
"\!$$\*SubscriptBox[\(\[Psi]$$, $$8$$]\)(x)"}], {.9, .8}]]
The numbers in the curly brackets are the position coordinates {x,y} of the placement of the legend for the curve. The output is given below. Thanks a lot for the help!

jedishrfu
jedishrfu
Mentor
Thanks for your analysis, solution and insight. It’s always great to see that in a thread And will help future readers who stumble across it in their search for answers.

Ishika_96_sparkles


text:
# Thread: Simplifying a sum of factorials

1. ## Simplifying a sum of factorials

How do I simplify $[(n-1)!+(n-2)!+...+1]$

Not sure if this is ext 2, just doing an applied maths problem

2. ## Re: Simplifying a sum of factorials

Originally Posted by BenHowe
How do I simplify $[(n-1)!+(n-2)!+...+1]$

Not sure if this is ext 2, just doing an applied maths problem
There's no easy way to simplify that unless you're willing to use hypergeometric/subfactorial/etc. functions.

WolframAlpha gives:

$(-1)^n\times(n!)\times!(-n-1) - !(-1) - 1$

3. ## Re: Simplifying a sum of factorials

Originally Posted by BenHowe
How do I simplify $[(n-1)!+(n-2)!+...+1]$

Not sure if this is ext 2, just doing an applied maths problem
Let the sum be X

Observe nX = n! + X -1

Then solve for X

4. ## Re: Simplifying a sum of factorials

That's very clever lol. So for stuff like this I have to find how I can re-write the expression to a more useful form/express it in terms of itself?

5. ## Re: Simplifying a sum of factorials

Let the sum be X

Observe nX = n! + X -1

Then solve for X
I don't think it works.
This is not a GS.
nX = n! + n (n-2)! + n (n-3)! + ......

6. ## Re: Simplifying a sum of factorials

Originally Posted by stupid_girl
I don't think it works.
This is not a GS.
nX = n! + n (n-2)! + n (n-3)! + ......
whoops XD

should have picked up on that immediately...

Time to induct

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#### Posting Permissions

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•


text:
Timezone: »

Poster
Leveraging Predictions in Smoothed Online Convex Optimization via Gradient-based Algorithms
Yingying Li · Na Li

Thu Dec 10 09:00 AM -- 11:00 AM (PST) @ Poster Session 5 #1458
We consider online convex optimization with time-varying stage costs and additional switching costs. Since the switching costs introduce coupling across all stages, multi-step-ahead (long-term) predictions are incorporated to improve the online performance. However, longer-term predictions tend to suffer from lower quality. Thus, a critical question is: \textit{how to reduce the impact of long-term prediction errors on the online performance?} To address this question, we introduce a gradient-based online algorithm, Receding Horizon Inexact Gradient (RHIG), and analyze its performance by dynamic regrets in terms of the temporal variation of the environment and the prediction errors. RHIG only considers at most $W$-step-ahead predictions to avoid being misled by worse predictions in the longer term. The optimal choice of $W$ suggested by our regret bounds depends on the tradeoff between the variation of the environment and the prediction accuracy. Additionally, we apply RHIG to a well-established stochastic prediction error model and provide expected regret and concentration bounds under correlated prediction errors. Lastly, we numerically test the performance of RHIG on quadrotor tracking problems.


text:
Next: Stability of Single-Phase Substance Up: Multi-Phase Systems Previous: Equilibrium of Constant-Temperature System

# Equilibrium of Constant-Temperature Constant-Pressure System

Another case of physical interest is that where a system, , is maintained under conditions of both constant temperature and constant pressure. This is a situation that occurs frequently in the laboratory, where we might carry out an experiment in a thermostat at atmospheric pressure (say). Therefore, let us suppose that is in thermal contact with a heat reservoir, , that is held at the constant temperature , and at the constant pressure . The system can exchange heat with the reservoir, , but the latter is so large that its temperature, , remains constant. Likewise, the system can change its volume, , at the expense of that of the reservoir, , doing work on the reservoir in the process. However, is so large that its pressure, , remains unaffected by this relatively small volume change.

The analysis of the equilibrium conditions for system follows similar lines to the that in the previous section. The entropy, , of the combined system satisfies the condition that in any spontaneous process

 (9.19)

If absorbs heat from in this process then . Furthermore, the first law of thermodynamics yields

 (9.20)

where is the work done by against the constant pressure, , of the reservoir, , and where denotes any other work done by in the process. (For example, might refer to electrical work.) Thus, we can write

 (9.21)

or

 (9.22)

Here, we have made use of the fact that and are constants. We have also introduced the definition

 (9.23)

where becomes the Gibbs free energy, , of system when the temperature and pressure of this system are both equal to those of the reservoir, .

The fundamental condition (9.19) can be combined with Equation (9.22) to give

 (9.24)

(assuming that is positive). This relation implies that the maximum work (other than the work done on the pressure reservoir) that can be done by system is . (Incidentally, this is the reason that is also called a free energy''.) The maximum work corresponds to the equality sign in the preceding equation, and is obtained when the process used is quasi-static (so that is always in equilibrium with , and , ).

If the external parameters of system (except its volume) are held constant then , and Equation (9.24) yields the condition

 (9.25)

Thus, we arrive at the following statement:
If a system is in contact with a reservoir at constant temperature and pressure then the stable equilibrium state is such that

The preceding statement can again be phrased in more explicit statistical terms. Suppose that the external parameters of (except its volume) are fixed, so that . Furthermore, let be described by some parameter . If changes from some standard value then the corresponding change in the total entropy of is

 (9.26)

But, in an equilibrium state, the probability, , that the parameter lies between and is given by

 (9.27)

Now, from Equation (9.26),

 (9.28)

However, because is just some arbitrary constant, the corresponding constant terms can be absorbed into the constant of proportionality in Equation (9.27), which then becomes

 (9.29)

This equation shows explicitly that the most probable state is one in which attains a minimum value, and also allows us to determine the relative probability of fluctuations about this state.

Next: Stability of Single-Phase Substance Up: Multi-Phase Systems Previous: Equilibrium of Constant-Temperature System
Richard Fitzpatrick 2016-01-25


text:
# partition2bracket -- dictionary between different notations for Schubert conditions.

## Synopsis

• Usage:
b = partition2bracket(l,k,n)
• Inputs:
• l, a list, partition representing a Schubert condition
• k, an integer,
• n, an integer, $k$ and $n$ represent the Grassmannian $Gr(k,n)$
• Outputs:
• b, a list, the corresponding bracket

## Description

A Schubert condition in the Grassmannian $Gr(k,n)$ is encoded either by a partition $l$ or by a bracket $b$.

A partition is a weakly decreasing list of at most $k$ nonnegative integers less than or equal to $n-k$. It may be padded with zeroes to be of length $k$.

A bracket is a strictly increasing list of length $k$ of positive integers between $1$ and $n$.

This function writes a partition as a bracket. They are related as follows $b_{k+1-i}=n-i-l_i$, for $i=1,...,k$.

 i1 : l = {2,1}; i2 : k = 2; i3 : n = 4; i4 : partition2bracket(l,k,n) o4 = {1, 3} o4 : List i5 : k = 3; i6 : n = 6; i7 : partition2bracket(l,k,n) o7 = {2, 4, 6} o7 : List


text:
Metafor

ULiege - Aerospace & Mechanical Engineering

team:carretta

Yves CARRETTA

The goal of my PhD was to model Micro PlastoHydroDynamic lubrication (MPH lubrication). This concept, which was first introduced by Mizuno and Okamoto [1], consists in lubricant outflow from cavities at the asperity level. Azushima [2] directly observed this phenomenon in plane strip drawing using a transparent die. Bech [3] carried out experiments on a similar experimental set-up. He studied the effect of several parameters such as the strip thickness reduction, the drawing speed, the lubricant viscosity, etc.

I used the in-house Finite Element (FE) software Metafor to reproduce Bech's experiments numerically. The model is able to predict lubricant escapes of the pocket backward and forward (see video). As in the experiments, several parametric studies were carried out on the thickness reduction effect, the slope of the lower tool and the drawing velocity, etc. These results show the model is capable of reproducing the trends observed experimentally by Bech.

While I was setting-up MPH simulations, I performed fluid and fluid-structure interaction simulations to assess Metafor's validity on such configurations.

My goal is now to couple the FE software Metafor to the numerical rolling model MetaLub in order to take MPH lubrication effects in cold-rolling.

Keywords

Cold rolling, Friction, Lubrication, Mixed Lubrication, Fluid-structure interaction, Metafor, Finite Element Method, Numerical Simulation, Slab Method.

Collaborations

• ArcelorMittal : The purpose of the collaboration with ArcelorMittal is to improve the capabilities of the cold-rolling software MetaLub. Several papers has been published with Maxime Laugier and Nicolas Legrand from the R&D centre of Maizières-Lès-Metz (France).
• CEMEF : I performed thermal simulations with Romain Canivenc, a researcher from CEMEF and his advisor Pierre Montmitonnet. The thermal coupling procedure I implemented between MetaLub and ThermRoll (a code - developed at CEMEF - modeling temperature exchange between the rolls and the strip in the roll-bite) was used to study cold rolling test conditions encountered in an industrial rolling mill. This collaboration lead to a conference paper presented at Coupled2011 (see Publications).
• Sheffield : There is an ongoing collaboration with Professor Rob Dwyer-Joyce from the University of Sheffield (Leonardo Tribology Centre). His team develops ultrasound techniques to measure lubricant film thickness between contacting solids. Andy Hunter, a PhD student, used this system to perform measurements in the roll-bite of cold-rolling stands. A comparison between the fluid film thickness predicted by the numerical model MetaLub and experimental measurements is important to get a good understanding of the experiments and to validate MetaLub.
• Valenciennes : I collaborated with Cédric Hubert and André Dubois from the University of Valenciennes. Results obtained in plane strip drawing with Metafor were compared to exprimental measurements gathered by Cédric Hubert and numerical results computed with the FE commercial software Abaqus. This collaboration lead to a conference paper that will be presented at the ESAFORM conference in April 2015.

Positions

• 2013 – now : Research Engineer at ULg (Project OPTILUB) “Numerical Simulation of Micro-PlastoHydrodynamic lubrication in cold rolling”.
• 2009 – 2013 : PhD student (FRIA grant)

Education

• 2014 : Ph.D. thesis at the University of Liège (Advisor: J.-P. Ponthot)
Finite element modelling of Micro-Plasto-Hydrodynamic lubrication in order to take this phenomenon into account in a cold rolling model”.
• 2009 : Degree in Mechanical Engineering at the University of Liège
Master thesis: “Numerical modelling of the creep forming process. Application to aeronautical wing panels”. (In French)

Numerical modeling of MPH lubrication

The experimental setup used by Bech to highlight MPH lubrication is depicted below. It is made of a transparent upper tool and a steel lower tool having a small angle $\alpha$ respective to the horizontal axis ($\alpha$ being either 2°, 3° or 5° depending on the tests). Using a camera, Bech was able to observe and record lubricant outflow from cavities. In the various conditions he tested, Bech observed backward and/or forward lubricant escapes.

The FE numerical model developped in Metafor is able to predict backward and forward lubricant flow as shown by the video below. Moreover, parametric studies conducted numerically show the trends predicted by the model match experimental observations.

FE simulations results computed with Metafor

Fluid simulations

MPH lubrication requires to take into account fluid and solid at the same time in the simulation. To assess Metafor's ability to deal with fluid simulation. I modelled fluid flows on test cases of increasing complexity such as the square cavity and Rayleigh's step (see pictures below). To perform these simulations, I implemented a new material law in order to model the behaviour of Newtonian fluids and I took advantage of the Arbitrary Lagrangian Eulerian formalism - allowing to uncouple the motion material's motion from the mesh - which was already implemented in Metafor (see Romain Boman for the current implementation). The comparison between Metafor results and reference solutions show a good agreement between these results.

Fluid-structure interaction simulations

Since fluid-structure interaction simulations were required, I assessed Metafor's validity by studying a benchmark where a flexible beam bends due to the fluid flow around it. The video below shows large displacement of the beam and its impact on fluid velocity.

Cold-rolling software: MetaLub

MetaLub is a cold-rolling model taking into account mixed lubrication regime. The main objective is to enhance the performances of rolling mills from a lubrication point of view. It means that lubricant rheology but also roll diameters and roughness, etc. can be optimized to improve stability and efficiency of the rolling tool.

The first version of this algorithm was implemented, in Fortran, by Nicolas Marsault at CEMEF. It was then optimized and implemented in C language by Romain Boman during is PhD at the University of Liège. Then, Antoine Stephany - another PhD-student - added new features such as roll-bite starvation (the amount of oil in the roll-bite is specified by the software user), large deformations of the work rolls, new lubricant rheology laws, etc.

I changed the structure of the code by means of C++ language. Thanks to the new object oriented structure, features addition are more straightforward and code modifications are more localised. It allowed me to adapt the set of equations when I implemented the coupling procedure involving MetaLub and Metafor.

This coupling procedure is functional in dry cases: analytical asperity crushing equations used in MetaLub can be replaced by FE simulations of asperity crushing. FE results, such as the relative contact area between the tool and the strip, are then sent back to MetaLub for another computation. This coupling procedure allows us to consider more realistic asperity profiles than in the analytical laws used so far.

I implemented a Graphical User Interface (GUI - see image below) with PyQt module. Numerical parameters of the model can now be defined easily, even by people who do not know programming languages. This tool was set-up in order to spread the use of the software within ArcelorMittal R&D center.

I also designed a plotter which displays intermediate results as the software is running (see figure below). This is helpful to optimise numerical parameters of the model depending on the rolling conditions that are being studied. Moreover, functionalities has been added to ease the post-processing stage. Results curves can be drawn and compare to each other in a few clicks.

Bibliographical references

[1] Mizuno, T. and Okamoto, M. Effects of lubricant viscosity at pressure and sliding velocity on lubricating conditions in the compression-friction test on sheet metals. Journal of Lubrication Technology, 104: 53–59, 1982.
[2] Azushima, A. Experimental confirmation of the micro-plasto-hydrodynamic lubrication mechanism at the interface between work-piece and forming die. Journal of the Japan Society for Technology of Plasticity, 30: 1631–1638, 1989.
[3] Bech, J., Bay, N., and Eriksen, M. Entrapment and escape of liquid lubricant in metal forming. Wear, 232(2): 134–139, 1999.


text:
Steve formed a Discretionary Family Trust some years ago in favour of his children.... 1 answer below »

Steve formed a Discretionary Family Trust some years ago in favour of his children. The children are: ? Gillian, aged 25, and married; ? Jeff, aged 23, certified insane and is confined to institution; ? Simon, aged 19, a university student; ? Louis, aged 16, s secondary student; ? Boris, aged 4. The trust deed states that Gillian and Jeff are each entitled to 20% of the trust income. Simon and Louie each have an entitlement to 25% of the trust income which is to be accumulated until they reach 21 years of age. Simon has a vested and indefeasible interest in his share, but Louise’s entitlement is contingent upon her surviving until age 21. Bories does not have a fixed percentage entitlement, but the Trustee has discretionary power to make payments out of trust income for the benefits of Boris. The remaining of the income is retained to accumulate in the trust. During the year the trustee exercise their discretionary authority and paid $7,000 medical expenses on Boris’s behalf. The gross income of the trust was$123,000, and deductible expenditure relating to that income was $3,000. Also, Simon derived$5,700 gross wages (PAYG withheld \$800) during university vacations. Required: For the current year of income calculate the following: (a) s. 95 net trust income (b) The share applicable to each beneficiary (c) Tax Payable by each beneficiary and/or the trustee.

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