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# The Hurwitz and Riemann ζ Functions

 Title: The Hurwitz and Riemann ζ Functions Author: Manuel Eberl Submission date: 2017-10-12 Abstract: This entry builds upon the results about formal and analytic Dirichlet series to define the Hurwitz ζ function ζ(a,s) and, based on that, the Riemann ζ function ζ(s). This is done by first defining them for ℜ(z) > 1 and then successively extending the domain to the left using the Euler–MacLaurin formula. Apart from the most basic facts such as analyticity, the following results are provided: the Stieltjes constants and the Laurent expansion of ζ(s) at s = 1 the non-vanishing of ζ(s) for ℜ(z) ≥ 1 the relationship between ζ(a,s) and Γ the special values at negative integers and positive even integers Hurwitz's formula and the reflection formula for ζ(s) the Hadjicostas–Chapman formula The entry also contains Euler's analytic proof of the infinitude of primes, based on the fact that ζ(s) has a pole at s = 1. BibTeX: @article{Zeta_Function-AFP, author = {Manuel Eberl}, title = {The Hurwitz and Riemann ζ Functions}, journal = {Archive of Formal Proofs}, month = oct, year = 2017, note = {\url{https://isa-afp.org/entries/Zeta_Function.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: Bernoulli, Dirichlet_Series, Euler_MacLaurin, Winding_Number_Eval Used by: Dirichlet_L, Prime_Distribution_Elementary, Prime_Number_Theorem


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# Cheap and good PCB etching solution.....

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#### Rolf

##### Member
The etching solution is made by adding 1 part Muriatic Acid (the common kind that's sold in hardware stores, about $5.00 gal.) to 2 parts Hydrogen Peroxide (the common 3% kind that's sold in dollar stores & drugstores for$0.50 a pint)

Warning!!!!!!!!
Mixing is a hazardous process (add acid to the peroxide, not the other way around!)

The board pictured below was etched in five minutes at room temp.

2.75x1.30"

http://www.pbase.com/sinoline/pcb_experiments

Question: To save Press-n-Peel someone talked about taping it to a regular sheet of paper. I have been reluctant to feed tape through my laser jet. Any comments?
.

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#### Hero999

##### Banned
I like your cheep etching solution.

Press-n-Peel is a waste of money, I just use ordinary magazine paper. Just print your circuit onto it, iron to a clean piece of copper clad board and wast it off with water.

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#### Rolf

##### Member
Hero999 said:
I like your cheep etching solution.

Press-n-Peel is a waste of money, I just use ordinary magazine paper. Just print your circuit onto it, iron to a clean piece of copper clad board and wast it off with water.
The etching solution is not really mine, I am just trying to get the word around. I should have mentioned that the Hydrogen Peroxide is only $0.50 a pint. That works out to about$ 13.00 for three gallons of etching solution. And no shipping because it is available locally, even in a little hick town were I live.

"Press-n-Peel is a waste of money". Not if you tried many of the others with very limited success and a lot of frustration. Maybe there are differences in the different manufactures toners?
No frustration is worth a lot to me.

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#### Hero999

##### Banned
Heck, I use photo resist board for real PCBs, it doesn't work out that expensive, it's probably cheaper than Press-n-Peel.

It's easy to build a UV exposure, the cheapest way is to lay your board with the artwork on top on a table held down with a peice of glass. Use compact black light for the UV source, the easiest way is to put it in an old desk lamp and direct the beam at your PCB.

http://www.maplin.co.uk/Module.aspx?ModuleNo=33124&criteria=UV&doy=25m8

I know there are probably cheaper places to buy these from and don't use the incandesant type they're crap and don't last for very long.

Look, here are some cheaper alternatives:
http://www.blacklight.com/items/FTBPESL13T-BLB
http://www.bulbman.com/index.php?main_page=product_bulb_info&cPath=4351_8839&products_id=11675 - needs a holder with ballast though.
http://www.doityourself.com/invt/0307207

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#### bobledoux

##### Member
I've had great luck with the Pulsar.gs paper. SMT's with .010 inch traces work all the time, when using a cheap heat laminator. I also use ferric chloride rubbed on the board with a small sponge for etching.

I have enough junk. I don't need more specialized equipment to do basic boards.

#### Oznog

##### Active Member
It's cheap and works, but muriatic acid is pretty darn strong even at 1/3rd concentration. Mixing is a hazardous process (add acid to the peroxide, not the other way around!).

Actually there is a guy on eBay who sells bulk ferric chloride chunks in 10 lb lots for $40, shipping included. Etching solution actually takes quite a bit of solid per liter, but$4/lb is pretty darn affordable. You might want to find somebody else in town to split this much material with.

http://search.ebay.com/search/search.dll?cgiurl=http%3A%2F%2Fcgi.ebay.com%2Fws%2F&fkr=1&from=R8&satitle=ferric+chloride+&category0=

#### Hero999

##### Banned
Oznog said:
add acid to the peroxide, not the other way around!
Wikipedia says otherwise.

http://en.wikipedia.org/wiki/Piranha_solution#Preparation

Wikipedia said:
One must always add the peroxide to the acid, not the other way around.
Which way is the safest, acid to peroxide or peroxide to acid?

Rolf said:
To save Press-n-Peel someone talked about taping it to a regular sheet of paper. I have been reluctant to feed tape through my laser jet. Any comments?
You could always glue it using a glue stick I suppose. By the way (I can't remember if it was you) said they have had problems printing on magazine paper. They's said it got jammed in the printer. Well you could always try glueing it to a nomal piece of A4 paper.

#### CRayD

##### New Member
I've been using this solution since Moses was in short pants, but I would add a few comments. The fumes are TOXIC!!! do the mixing and etching outside!! and do not breath the fumes. Heating the solution to about 100f will speed up the etching dramatically. Also using a bubbler will speed up the etch and even it out across the PCB. The solution saturates rapidly, figure 1 oz. solution to 1 sq.in. of 1 oz. copper. The solution can't be easily rejuvinated. The soution will turn a pale green (the dissolved copper) as it saturates. Discard the used solution diluted 1 to 5 or better in water with a couple tablespoons of BiCab down the toilet.

#### evandude

##### New Member
Rolf said:
Question: To save Press-n-Peel someone talked about taping it to a regular sheet of paper. I have been reluctant to feed tape through my laser jet. Any comments?
.
I'm one of the people who recommends that. I just use normal clear/scotch tape and it's never shown any signs of melting or anything. It certainly extends the use of a sheet of press-n-peel... personally I do a lot of small boards and I can often get dozens of boards out of a single sheet, and at around $1.50 a sheet that makes the cost of the press-n-peel practically negligible in my opinion. I've become interested in CuCl2 + HCl etching, as seen at: http://members.optusnet.com.au/~eseychell/PCB/etching_CuCl/index.html That formulation has the advantage of being regenerable by simply bubbling air through it, and at most occasionally adding some water or more (cheap) acid. The hard part, as I gather from that page, is making the solution in the first place, as you need to purchase or make the CuCl2, and it can be very expensive to purchase, at least from chemical supply places. However, I found both Copper (II) Chloride and Copper (II) Oxide for sale at a fireworks supply place (skylighter.com) very cheaply (less than$10 a pound). I would like to talk with someone who's a bit more knowledgeable about chemistry before getting into this type of etching (for safety guidelines, and to make sure I know exactly how to monitor and regenerate it, so I don't ruin it), but it definitely seems appealing never having to throw out my etching solution, instead just being able to keep regenerating it, even if it means I have to put in a little more effort to storing and regenerating it.

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#### Rolf

##### Member
evandude said:
I'm one of the people who recommends that. I just use normal clear/scotch tape and it's never shown any signs of melting or anything. It certainly extends the use of a sheet of press-n-peel... personally I do a lot of small boards and I can often get dozens of boards out of a single sheet, and at around $1.50 a sheet that makes the cost of the press-n-peel practically negligible in my opinion. Thanks, I'll try the tape, it will probably save me about 50% that is now going to waste. I agree that Press-n-Peel is a bargain, got mine for$ 1.00 / sheet but I had to buy 30. So now I have enough for a lifetime.

#### evandude

##### New Member
Another tip for the tape method:
the general method is to print the pattern on plain paper, then tape the PnP on, and print again; this way you can cut the PnP to just barely larger than the pattern and have very little waste. But, that means the first pass is a pretty big waste of toner. When I am doing designs that have a lot of copper area (and thus use a lot of toner) I usually open up the pattern in an image editor and change the color to a very light gray and print that the first time, or quickly draw an empty rectangle the size of the pattern and print that the first time.

#### Rolf

##### Member
Simpler......

evandude said:
Another tip for the tape method:
the general method is to print the pattern on plain paper, then tape the PnP on, and print again; this way you can cut the PnP to just barely larger than the pattern and have very little waste. But, that means the first pass is a pretty big waste of toner. When I am doing designs that have a lot of copper area (and thus use a lot of toner) I usually open up the pattern in an image editor and change the color to a very light gray and print that the first time, or quickly draw an empty rectangle the size of the pattern and print that the first time.
To save toner; it is simpler {for me (when using PCB123)} to just draw a rectangle around the traces, then delete the trace layout. Then print just that layer, no need to go to a separate program. Lastly re-load the trace file and print your p-n-p.
But for small board I don't even bother.

#### jolino

##### New Member
Hi Rolf,
I have been reluctant to feed tape through my laser jet
I use "acetate" (A4 transparent film) are years on my HP1100 and no problem until now.

Regards

#### jagrolet

##### New Member
I have been using regular "Scotch Tape" for years to get the most out of my Press & Peel (the same method as "evandude"). I have never had a piece stuck in my laser printer or even had a piece come loose from the paper and Press & Peel. Ill continue using this method, but Im definately going to try the Muriatic Acid & Peroxide instead of the expensive Radio Shack ferric chloride. Man are they proud of that stuff.

cool good etch

#### sparthacuz

##### New Member
many many thanks man!!

simple but yet effective! you just saved my ass man. my medical officer and our janitor to include my commanding officer said how the hell did i etch the circuit without the procured solution?! thanks....

#### Rolf

##### Member
sparthacuz said:
simple but yet effective! you just saved my ass man. my medical officer and our janitor to include my commanding officer said how the hell did i etch the circuit without the procured solution?! thanks....
There must be more to this story than you are telling us, sounds interesting, please fill us in.

#### Pommie

##### Well-Known Member
When diluting acid, the rule is "do as ya outa, add acid to water". An even simpler way to remember is to think how hard it would be to add swimming pool water to 5 liters of acid. BTW, this doesn't really matter with HCl but with H2SO4 it's a different matter, add water to conc sulphuric = steam explosion and you get to wear the acid.

Also, if the CuCl method works then the first time you use HCl + H2O2 the end product is CuCl. Isn't it?

Mike.

#### Pommie

##### Well-Known Member
After thinking about it, I'm very dubious about the copper chloride method. If the method worked then the HCl + H2O2 method would keep working as the spent solution contains CuCl2 which should get consumed to produce CuCl which is insoluble (1.5g/Ltr). People who have tried this report a green spent liquid which suggests CuCl2 does not react any further.

Mike.

#### evandude

##### New Member
http://members.optusnet.com.au/~eseychell/PCB/etching_CuCl/index.html

That site has about all the information I think you'd need on the process. I also came across a PDF file that was some sort of information sheet about this process which is used in industry. I'm not an expert on the subject, but he does mention other factors in the process that would affect the etching ability of the Cupric chloride etchant, perhaps there's something else going on that's keeping it from working.

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text:
How do you add sqrt45 + sqrt125?

The answer is $8 \sqrt{5}$
$\sqrt{45} + \sqrt{125} =$
$\sqrt{9 \cdot 5} + \sqrt{25 \cdot 5} =$
$\sqrt{{3}^{2} \cdot 5} + \sqrt{{5}^{2} \cdot 5} =$
$3 \sqrt{5} + 5 \sqrt{5} =$
$\sqrt{5} \left(3 + 5\right) = 8 \sqrt{5}$


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The spectral type distribution for the former sample ofbinaries is the same as the general distribution of all stars in theHipparcos catalog, whereas the latter sample is clearly dominated bysolar-type stars, with an obvious dearth of blue stars. We point outthat the latter set includes mostly binaries with long periods (longerthan about 6 yr). Rotational velocities of A-type stars in the northern hemisphere. II. Measurement of v sin iThis work is the second part of the set of measurements of v sin i forA-type stars, begun by Royer et al. (\cite{Ror_02a}). Spectra of 249 B8to F2-type stars brighter than V=7 have been collected at Observatoirede Haute-Provence (OHP). Fourier transforms of several line profiles inthe range 4200-4600 Å are used to derive v sin i from thefrequency of the first zero. Statistical analysis of the sampleindicates that measurement error mainly depends on v sin i and thisrelative error of the rotational velocity is found to be about 5% onaverage. The systematic shift with respect to standard values fromSlettebak et al. (\cite{Slk_75}), previously found in the first paper,is here confirmed. Comparisons with data from the literature agree withour findings: v sin i values from Slettebak et al. are underestimatedand the relation between both scales follows a linear law ensuremath vsin inew = 1.03 v sin iold+7.7. Finally, thesedata are combined with those from the previous paper (Royer et al.\cite{Ror_02a}), together with the catalogue of Abt & Morrell(\cite{AbtMol95}). The resulting sample includes some 2150 stars withhomogenized rotational velocities. Based on observations made atObservatoire de Haute Provence (CNRS), France. 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In this paper a list of thefluxes derived in this way, corrected for a small dependence on B-V, isgiven for all the 1215 stars in Johnson's 1975 catalog with completeentries. Radial velocities in selected B-G starsSpectroscopic observations of a sample of nearby B-G stars are reported.The observations were obtained at resolution 19 km/s in the spectralregion near the 448.1228-nm line of Mg II using a CCD-detectorspectrograph on the coude-feed telescope at KPNO on April 16-20, 1986.The data are presented in extensive tables and briefly characterized. Rotation velocities of metallic-line starsThe rotation velocities (V sin i) of 81 Am stars were determined usingspectra of dispersion 15 A/mm. The profiles of Fe I 4045 A and Sr II4215 A lines were compared with the computed profiles. The line widthand its ratio to the central depth are found to be most sensitive to therotation velocity. The hydrogen spectral types obtained from the H-gammaequivalent width are also given. It is noted that the extremal Am starHR 4646 has a relatively high rotation velocity of at least 70 km/s. Ultraviolet and visible photometric parameters for the AM starsSome attempts to detect Am stars photometrically are presented. Theydeal with both ground-based photometry (Geneva system) and UVspectrophotometry and photometry from the TD-1 satellite data. For thispurpose a reddening-free parameter m is defined in the Genevaphotometric system. Spectrophotometric features possibly characteristicof the Am phenomenon are presented. As a conclusion, based on thecorrelation between spectral classification and the various parametersand features as well as the Kurucz (1979) stellar model atmospheres, them parameter is found to be most reliable and able to select stars with ahigh probability of being Am. Observed and computed UV spectral distribution of A and F starsAn automatic and fast procedure was implemented to determineTe and log g from the comparison of the UV S 2/68 spectrawith the Kurucz models. The method has been applied to all A and F starswith (B-V)0 greater than or equal to 0 and luminosity classes III, IV,IV-V, and V, included in the Ultraviolet Bright-Star SpectrophotometricCatalogue. From the analysis, it appears that the models match theobserved fluxes fairly well and that the effective temperatures derivedfrom the UV data agree with those derived from Stromgren photometricindices within 200 K. The ultraviolet energy distributions of late A starsObserved late A star energy distributions for the wavelength range1400-2500 A are compared. No difference is found between energydistributions of Am stars and those of normal slowly rotating A stars.The fluxes of rapidly rotating stars, however, appear to be increasedfor wavelengths smaller than 1530 A; this cannot be understood as aneffect of pole heating or reduced gravity. In addition, the comparisonof the UV energy distributions with model atmosphere energydistributions of Kurucz indicates some problems with the theoretical SiI absorption edges at 1530 A. Catalog of profiles and equivalent widths of the CA II K line in the spectra of metallic-line starsProfiles of the Ca II K line for 87 bright Am, A, and F stars weremeasured on spectrograms with a dispersion of 15 A/mm. Halfwidths of theprofiles for fixed values of line depth, central depths, and equivalentwidths are presented. In contrast to the case of peculiar stars, theobserved K-line profiles in the metallic-line stars do not show anypeculiar structure. Secondary standards for BVRI photometryA thorough test of Fernie's instrumentation system for matching theJohnson BVRI photometric system has been made. From 1,526 observationsof 186 stars in the Arizona-Tonantzintla Catalogue, there were foundexcellent transformations, except for a slight curvature in the (B-V)transformation, which is easily removed, and a difficulty in (R-I) forhighly reddened stars. Photometric values and their uncertainties aregiven for all the stars, and these are of sufficient quality to serve assecondary standards in BVRI for small- to moderate-size telescopes(1-m). Spectral classification from the ultraviolet line features of S2/68 spectra. III - Early A-type starsAbstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1978A&AS...33...15C&db_key=AST Five-channel photometry of cepheids and supergiants in the southern Milky Way.Abstract image available at:http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1976A&AS...24..413P&db_key=AST
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## CCSS.MATH.CONTENT.HSS.IC.B.4: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

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# Simplify the Exponents: Multiply

##### This Simplify the Exponents: Multiply worksheet also includes:

Students multiply 10 sets of numbers with exponents. Three problems require students to multiply three terms. Seven problems multiply two exponential terms.

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# Hectars to square meters converter

Hectares to square meters converter helps you to calculate how many square meters are there in
hectare area of metric units. It can also be defined as a square with all sides which has 1 meter in
length. It changes details of hectares to square meters with online hectares square meters converter
calculator. There is a chart of hectares to square meters with help of tables and formulas.

You can even check for square meters to hectares

## Hectares To square Meter converter

 Hectares square Meter:

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# What is the equation of the line with slope  m=-4  that passes through  (5,5) ?

Apr 16, 2018

color(green)(4x + y = 20

#### Explanation:

To write equation of the line, given : $m = - 4 , \left({x}_{1} , {y}_{1}\right) = \left(5 , 5\right)$

$\text{Point-Slope Form equation is } \left(y - {y}_{1}\right) = m \cdot \left(x - {x}_{1}\right)$

$\left(y - 5\right) = - 4 \cdot \left(x - 5\right)$

$y - 5 = - 4 x + 20$

color(green)(4x + y = 20, " rearranging"


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Tag Info

0

The shortest way is the following one. Define $$f(\eta):= \exp{(\pm\frac{1}{2} \eta \sigma^3)}\:,\quad g(\eta):=\cosh(\frac{1}{2}\eta) \mathbb{I}_{2x2}\pm \sinh(\frac{1}{2}\eta)\sigma^3.$$ Now notice that $$f'(\eta) = \pm\frac{1}{2} f(\eta)\sigma_3$$ and $$g'(\eta) = \frac{1}{2} \sinh(\frac{1}{2}\eta) \mathbb{I}_{2x2}\pm\cosh(\frac{1}{2}\eta)\sigma^3 = \pm\... 1 You just need to use the series expansion and observe that$$(\sigma^3)^{2n} = \mathbb I_{2}$$and$$(\sigma^3)^{2n+1} = \sigma^3$$1 The first step is to expand the exponential as a Taylor series:$$ \exp \left( \pm \frac{1}{2} \eta \sigma \right) = I \pm \left( \eta \sigma/2 \right) + \frac{1}{2} \left( \eta \sigma / 2 \right)^2 \pm \frac{1}{3!} \left( \eta \sigma / 2 \right)^3 \dots$$Noting that \sigma^2 = I, this series can now be split into even and odd terms: the even terms sum to ... -1 It is not momentum conservation that forbids a zero mass particle to decay to two particles. It is the definition of "four vector", it's algebra, and it's "length", the invariant mass, the definition of inertial frames. Two particles will have an invariant mass whereas the incoming particle has mass zero. The "length" of the four vectors is invariant ... 0 Yes, there is a direct mechanical analogue in what engineers do with their fingers when they strain a cube of material by parallel-pipeding it. This is also the same transformation the "cross polarization" of a gravitational wave (GW) does. Suppose the edges of the cube are the x,y,z axes. You look down on the xy plane with z axis sticking in your eye. ... 1 I just want to compile here a curated list of all the things I've seen said about this question here on this site, both right and wrong. Question: Can a massless particle spontaneously split in two? Correct Answer 1:(if the particle is a photon) No, because this would violate conservation of angular momentum. The photon must have helicity \pm 1 (for ... 0 From a purely classical point of view: "Does this [classical mass invariance] stem from other basic principles of classical mechanics or is this an independent experimental fact?" No, this fact doesn't stem from other classical principles. Classically, this is an independent experimental observation. It boils down to Newton's 2nd Law for a ... 3 It's not strictly forbidden for a massless particle to split into two, but the "volume of available phase space" is zero. Remember that the rates for processes are computed by finding a so-called matrix element, and then integrating it over phase space. The allowed phase space here is of measure zero, because you have the additional constraint that the two ... 3 The idea from early treatments of special relativity that mass increases with velocity was superseded in general relativity and is better not used. It is a fundamental principle that the laws of physics are covariant - they are formulated using tensor (& vector & scalar invariant) quantities so as to be the same for all observers. Proper mass, or ... -2 This is a known misconception . There are 2 kinds of masses 1) inertial mass : Inertial mass is found in Newton's law of motion F=ma. This m expresses the resistance of an object to change its velocity. 2)gravitational mass: Gravitational mass is found in Newton's law of gravity. g = G*m/r^2.This m expresses the result of an object to gravity. At low ... 1 Here are two related abstracts that might be of interest to you. (I have not read the papers to comment.) Geometrical interpretation of optical absorption J. J. Monzón, A. G. Barriuso, L. L. Sánchez-Soto, and J. M. Montesinos-Amilibia Phys. Rev. A 84, 023830 – Published 17 August 2011 https://doi.org/10.1103/PhysRevA.84.023830 "We reinterpret the transfer ... 1 Very good question, which has on-going very active research. Short answer is YES, indeed quantum entanglement makes cause and effect indistinguishable, at least in a usual sense that we get used to in everyday. And this is already proven experimental fact,- Physicists demonstrate new way to violate local causality. So we can talk now only about does causal ... 0 I would say that you want a mathematical proof. You can see in Reign's answer a particular case, but here I will post a more general one. As you may know, a Lorentz transformation can be written as a 4-dimensional matrix, \Lambda, so that x' = \Lambda x. This matrix must fulfill this relation, \Lambda^T \eta \Lambda = \eta \end{... 2 [Disclaimer: Throughout this post I will be bracketing the ideas of general relativity because I think they needlessly complicate the story.] The invariance of the speed of the light is more of a statement about the geometry of the universe than it is a statement about light. Suppose I am in an inertial frame S with coordinates (t,x,y,z). If one event ... 0 There seem to be some fundamental mis-understandings. Put yourself inside the rocket. However fast you go with respect to any chosen other body in the universe, your rocket would work exactly the same way, irrespective of your (relative) volicity. What you will notice: the faster you go wrt this other body, the longer your rocket will need to use fuel (... 6 Yes it fires out normally from the ships perspective, because from ships perspective, the ship is at rest. No it does not go faster than light from the point of view of another observer, because velocities are not added the same way as in classical mechanics, but are added by relativistic velocity-addition formula. In classical mechanics, when ship fires ... 1 Let \{ \gamma_0, \gamma_1, \gamma_2, \gamma_3 \} be an orthonormal basis for Minkowski spacetime, with inner product$$\gamma_\mu \cdot \gamma_\nu = \eta_{\mu\nu}$$and timelike basis vector \gamma_0. Every lightlike vector can be expressed as a Lorentz transformation of \gamma_0 + \gamma_3, to which only two spacelike vectors are orthogonal: \... 2 He then goes on to state that there are three mutually orthogonal space-like vectors. Sure, the proof is by example:$$\epsilon_1^\mu = (0, 1, 0, 0), \quad \epsilon_2^\mu = (0, 0, 1, 0), \quad \epsilon_3^\mu = (0, 0, 0, 1)$$where the time component is on the left. At one point he makes the statement that for given light-like vector, there are only two ... 1 Speed of light is constant wrt to reference inertial frame You always need two things to define physical quantities like distance and speed. Like in your example you have three objects, the sun, the earth and the rocket. If you delete the Sun you still have two observers for these physical quantities to make sense aka measured. If you delete earth the ... 2 SPOILER ALERT: The following is probably not the answer you wanted, but as a recovering ex-engineer I can't help myself: In practical terms, the bridge will not collapse, because the residence time of the relativistic car on the bridge will be insufficient for the applied stresses (the weight of the car) to propagate throughout the bridge structure (this ... 0 Here is the explanation I received on presenting this question to a maths and physics professor. I think it clarifies the 'soul' of the question (ie the point about the conceptual status of speed rather than proving the speed of light which is what most of the other answers seem to be doing): All speeds are relative and without reference to another object ... 2 He can't see a Euclidean metric, except by restricting to space. I think the writer did not mean what he appears to say, as quoted in the comment. I think he only intended to give two examples of diagonal metrics, not to give two examples of metrics which an observer can actually see. 3 If speed is not an entity in itself, but only dependent on other constant factors, how can the speed of anything (let alone light) be a constant? Am I completely missing something here? What you are missing is that distance and time themselves are not constant. Both distance and time depend in part on the velocity of the observer. You are on Earth, and ... 8 how speed can be a constant It's not, it depends on selected reference frame. The only exception is light speed in vacuum, which in any reference frame is c. But what if the Earth itself now starts moving? Now the distance is still changing, but how the individual speeds be calculated? In a pre-relativity times there was a Galilean speed addition rule ... 1 In special relativity speed of light is constant with respect to any inertial frame of reference. That is an axiom of special relativity. Having well defined frame of reference is all you need. It makes no difference if in your particular frame of reference there are other objects like Earth, Sun or some other thing. when a rocket leaves Earth and heads ... 15 My question is: If speed is not an entity in itself, but only dependent on other constant factors, how can the speed of anything (let alone light) be a constant? Am I completely missing something here? The question you are really asking is which is the more fundamental, speed or distance? Think about the distances in space. How are we to measure them ... 2 One has to distinguish between, on the one hand, the physical reality that exists outside of our minds and independently of how we think about it, and, on the other hand, human-made ways to describe this physical reality. Specifically, how speed is dependent on other quantities, how we measure it, etc. are reflections of how we think of it and describe it. ... 41 Speed of light is actually a pretty special case compared to how we typically think of speed (as far as I understand it). Movement is always relative to some frame of reference. In the case of a single isolated object, it's hard to really think about how you could have any frame of reference without at least a second object to measure the speed relative to.... 13 It is counter intuitive, this question or variants of it get asked a lot. Surely if you're travelling towards a beam of light it will appear to be travelling faster? The answer is it won't, every inertial observer measures the speed of light to be exactly c, regardless of their velocity. It is for this reason that the Galilean transformations break down ... 2 Maxwell's equations describe the electric and magnetic fields of a charge moving at any velocity, so long as you account for the fact that a moving charge acts like a current, and that changing electric & magnetic fields also can act like a source of magnetic and electric fields (through the \partial \bf{B}/\partial t and \partial \bf{E}/\partial t ... 0 Maxwell's Equations in their 19th C form are Lorentz covariant, which means they are valid after being boosted into a new frame of any boost-able velocity when transformed by a Lorentz transformation. (Note: v=c is not a boostable speed). It does take some effort to show they are unchanged, unless you put them in their "manifestly covariant" form. So for ... 0 That question doesn't make sense because only photons ( quanta of light ) or gluons ( quanta of strong interaction field ) can only move at lightspeed. And just to be clear Coulomb's law is not valid to describe chemistry because it is a classical theory which doesn't take in account the quantum effects.QED describes chemistry along with some other stuff. ... 0 The time dilation that you have described is how each observe the others time, so if they could "somehow" recede from each other at faster than light, they cannot observe each others time. So there is no meaningful answer. 5 Simple explanation: Its the only type of metric that conserves the speed of light. Mathematical explanation: Let us assume an inertial observer, O, measures the speed of light as dx/dt = c. Let us assume another inertial observer, O', measuring the speed of light in his coordinates, dx'/dt' = c. Naturally at this point one can ask, What is the ... 0 It's true that the specific set of coordinates that we choose is not Lorentz invariant, but when you integrate over all space, it doesn't matter. One however has to check that the measure is invariant. It's a nice excercise to show that, under a Lorentz transformation \Lambda the integral transforms are$$ \int \mathrm{d}^4x\, f(x) \mapsto \int \mathrm{d}^...

0

Action is constructed as Lorentz invariant dencity integrated over all Minkowski space. $\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ is Lorentz invariant dencity. $d^4x$ is Lorentz invariant measure of integration in rectangular coordinate system. If you wanna to choose some another coordinate system (curvilinear coordinate system), in general you need ...

-2

Time dilation is based on clock time. Clock time is the time we measure with when doing experiments. It is also the time according to our eyes. In clock time, the speed of light is constant. Most of Physics study is done using clock time. Things age, and happen more slowly when the local rate of Clock Time is slower that the rate of Clock Time it is ...

1

How would this observer explain the tensile stress force he observes on the crystal which someone standing next to the crystal will not detect? Stress is the space-space components of the stress energy tensor. For the stationary observer the stress energy tensor is $$\left( \begin{array}{cccc} \rho & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 ... 0 The moving observer won't see any stress. This is essentially Bell's spaceship paradox. http://math.ucr.edu/home/baez/physics/Relativity/SR/BellSpaceships/spaceship_puzzle.html From the point of view of the moving observer it is the cube that is moving. In detail, at the atomic level, from the point of view of the observer, if you solve for the required ... 2 Let us simplify the example without loosing the essential elements. Suppose instead that the gates shoot a laser towards the center once the door is closed, and that at the center you have a bomb with a coincidence detector so that if both lasers hit simultaneously the bomb explodes destroying everything inside the gates (and the gates too, why not?). In ... 1 By little group scaling, the three-particle MHV amplitude for three massless spin-1 particles is given by$$ \mathcal{M}^{--+} =\frac{\langle 12 \rangle^3}{\langle 13 \rangle\langle 32 \rangle}~.$$This is not permutation invariant in the sense that if I exchange legs one and two I pick up a minus sign. The equivalent graviton amplitude doesn't pick up a ... 1 Scattering amplitudes should obey little group scaling, i.e. a transformation on the momentum that leaves it invariant, which for four-momentum is p_i^\mu = \lambda^\mu_{~~\nu} p_i^\nu. In spinor helicity notation, and for massless particles, this transformation acts on the spinors as \lambda_i \rightarrow e^{i\theta}\lambda_i. The scattering ... 1 When doing a complex shift in order to enact the BCFW transformation, we take, as you pointed out,$$ \hat{p}_1 = p_1 - zq~. $$Choosing to do a \langle 12] shift means we make a clever choice for q and work with spinors, which gives$$\hat{p}_1 = |1\rangle|1] - z|2\rangle|1] = (|1\rangle - z|2\rangle)|1] = |\hat{1}\rangle|1].$$The choice of q is ... 2 You are doing it correctly. In reference to your comment, to find the minkowski metric in cylindrical coordinates, it is easiest to transform the line element so:$$ ds^2=g_{\mu\nu}dx^\alpha dx^\beta \tag{1}, $$Where usually the minkowski metric is denoted by \eta_{\mu\nu} Which is$$ ds^2 = -c^2dt^2+dx^2+dy^2+dz^2 \tag{2} $$In Cartesian. If you ... 1 A little over a hundred years ago this was a red-hot topic. The Michelson-Morley experiment sought to detect any variation in the speed of light from different directions and found none. Subsequent refinements have produced the same result. 1 The result that light's speed in vacuum doesn't change due to relative motion among the light source, the light receiver, or a hypothetical background medium was a major discovery at the beginning of the twentieth century. You are correct that it's a non-trivial problem, but you're incorrect that a non-electromagnetic observation is needed. Most books on ... 2 Yes, the proper length, the spacial distance between two points measured in the rest frame. 1 Assume you have \mathbf p = p^{\alpha'} \mathbf e_\alpha. Then$$ \mathbf p\cdot\mathbf e_\beta = p^{\alpha'} \mathbf e_\alpha \cdot\mathbf e_\beta = p^{\alpha'}\eta_{\alpha\beta}$$so$$ p^{\alpha'} = \eta^{\alpha\beta}\mathbf p\cdot\mathbf e_\beta$$In particular$$ p^{0'} = \eta^{00} \mathbf p\cdot\mathbf e_0 = -\mathbf p\cdot\mathbf e_0$$And yes, you ... 1 It is correct that moving with the charge, there will no longer be an observed magnetic field. In fact, it hints on something much deeper. Instead of a point charge, consider an infinite line of current. From Ampere's law, we can calculate the magnetic field$$ \vec{B} = \frac{\mu_0 I}{2 \pi r}\hat{\phi}  If we then stop the line current, we have an ...

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# Lagrangian Method for Christoffel Symbol and (non-)holonomic basis

I rencently learned about the lagrangian/variational method for computing Christoffel symbols.

Let $\mathcal{M}$ be a $m-$dimensional manifold with $g_{ij}$ being the metric tensor components and let $\gamma(\lambda) = (x^1(\lambda)\dots x^m(\lambda))$ a geodesic in $\mathcal{M}$. By assigning the following Lagrangian to the metric tensor $g_{ij}$ : $$\mathcal{L} = g_{ij} \dot{x}^i \dot{x}^j,$$ with $\dot{x}^i = \dfrac{dx^i}{d\lambda}$ and using the least action principle one gets the Euler-Lagrange equations : $$\dfrac{d}{d\lambda}\dfrac{\partial \mathcal{L}}{\partial \dot{x}^i} - \dfrac{\partial\mathcal{L}}{\partial x^i} = 0 \tag{1}$$ Remebering the geodesic equations written by means of Christoffel symbols : $$\ddot{x}^k + \Gamma^k_{\ ij} \dot{x}^i \dot{x}^j = 0,\tag{2}$$ Christoffel symbols are obtained comparing $(1)$ and $(2)$.

How does the fact the basis one uses is holonomic or not is taken into account ?

$\frac{d}{dt}\frac{\partial L}{\partial \dot{x}^i} -\frac{\partial L}{\partial x^i}=\lambda_j a^j_i$,
where $\lambda_j$ - lagrange multipliers defining nonholonomic constrains


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# Problem: The barometeric pressure was 749 mmHg when the open end manometer attached to a flask holding a gas read 13.0 mmHg. (mercury level highest on the side connected to the flask.) What is the pressure of the gas in the flask in kilopascals? 1.00 atm = 101.3 kPa

🤓 Based on our data, we think this question is relevant for Professor Ernst's class at UTAH.

###### FREE Expert Solution

We’re being asked to determine the pressure of a gas (in kPa) using an open-end manometer.

A manometer allows you to determine the pressure of a gas (Pgas) sample by using the atmospheric pressure (Patm) and the difference in the height of mercury (h).

Different conditions for an open-end manometer are illustrated below:

When Pgas is pushing the same force as Patm

$\overline{){{\mathbf{P}}}_{\mathbf{g}\mathbf{a}\mathbf{s}}{\mathbf{=}}{{\mathbf{P}}}_{\mathbf{a}\mathbf{t}\mathbf{m}}}$

When Pgas is pushing harder than Patm

###### Problem Details

The barometeric pressure was 749 mmHg when the open end manometer attached to a flask holding a gas read 13.0 mmHg. (mercury level highest on the side connected to the flask.) What is the pressure of the gas in the flask in kilopascals? 1.00 atm = 101.3 kPa


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Is the job market good for astro physic majors or general physic majors?

1. Oct 6, 2008

jinjishu

I am considering changing my major to Philosophy and Physics with a astrophysics minor, since my school doesn't offer astronomy and astrophysics as a major. My current major is biotechnology. I was reading some article that was a little scary, it is about how the job market for physicist isn't as great as it seems, so I am a little concerned.

2. Oct 6, 2008

mgb_phys

The job market for smart people with a science degree from a good university is as good as ever. There aren't necessarily that many jobs with physics in the title but most physicists aren't working in those anyway,

3. Oct 6, 2008

jinjishu

Where are they working at than?

4. Oct 6, 2008

mgb_phys

Engineering, technical consultancy, software, Wall St (ha ha)

In most countries there is also a huge demand for science teachers.

5. Oct 6, 2008

You can get paid $500 USD in China if you decided to teach overseas. Its not much but over there with that type of money coming in a month you'll be living a moderate life style. 6. Oct 6, 2008 arunma From what I can see, there are jobs for physics majors, but you should supplement your physics degree with a second major in engineering. As long as you don't mind working in a field that has nothing to do with physics (i.e. programming, engineering, etc.), you'll be able to find a job. If you want to actually do physics for a living though, you need to get your PhD. 7. Oct 6, 2008 jinjishu Yeah I planned on going for the PhD. So instead of doing a double with philosophy, I should do it with engineering? Does it matter which kind of engineering? Cause I always wanted to do philosophy, but I'll drop it if I have to or try to squeeze in a third major. 8. Oct 6, 2008 Choppy Jinjishu, You may want to be careful and make sure that a "philosophy and physics" degree actually qualifies you to get into graduate school if you really want a Ph.D. Some schools can be particular about the degrees that you apply with. 9. Oct 7, 2008 thewhills So you basically want to study metaphysics? Remember the best philosophers have no degree...They just sit and think. 10. Oct 11, 2008 jinjishu No, I don't want to do metaphysics, and also I don't want to do philosophy as a career choice. I just wanted to learn more about it, because I am interested. I would drop the philosophy major though if I have to. Would general physics PhD level doubled with mechanical engineer PhD with a astrophysics minor be adequate? 11. Oct 11, 2008 kingdomof It's definitely not worth anything more than a casual glance, in my opinion, at least. This is coming from someone who studied philosophy by himself for about 2 years. It's subject to arbitrary fashion and is pretty silly about it, much like art. Basically, stick with the science, not the bull. 12. Oct 11, 2008 Locrian Getting a PhD in both those subjects will be adequate if your goal is to be a poor graduate student for 8-12 years and end up in the same place you'd have been if you'd just gotten 1 PhD - or maybe worse. Is that what you're looking to do? Good luck with it. 13. Oct 12, 2008 jinjishu I don't know, it's why I am asking you, but I don't see having more knowledge as being worst. So there's no point in getting the PhD in engineering is what I am gathering, but is it still a good idea to do it in undergrad? Physics doubled with engineering 14. Oct 12, 2008 kingdomof When you get to the PhD level, you are extremely specialized in the field you are in. Being super specialized in two fields is just stupid, as you can only dedicate yourself to research in just one, realistically. 15. Oct 12, 2008 russ_watters Staff: Mentor When you are young, the point of college is to learn something that will be useful as a career. If you can do a double-major without taking any more time or lowering your grades in your "real" major, then it won't hurt you - but that would be unlikely. At least you are here asking the questions, though: too many people put no thought into how what they are doing in college will help them once they get out. Starbucks is littered with college educated coffee pourers who took art history because they felt like it. 16. Oct 13, 2008 jinjishu Okay so, it doesn't matter if I do engineering with physics? As long as I just get my PhD. in physics, I should be able to find a place in the job market? 17. Oct 13, 2008 Locrian Yes, you will find a place. What you might not find is a job worth having. . . but yes there will probably be a chair to sit in - or a classroom to stand in front of. Depending on what you get your PhD in, how well you do, and the job market at the time, that place could be valued at anything from$26k a year to \$90k a year.

Choose wisely.

18. Oct 13, 2008

jinjishu

Hm.. that's a little depressing, so even with a Ph.D., I could still wind up behind a desk or just teaching a classroom? (If that includes college, then it's not so bad.) Well taking that into account, maybe I should pick up that second major in engineering, just in case I don't like where I end up in physics.


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# E763 with vim and spell checker

I intend to add germand spell checking to vim. So I set

:set spl=de_de spell

after that there is a discussion, where I can create a directory site called .vim/spell. And afterwards there is an additional inquiry where I can download and install a .spl documents. And also once more for a .sug documents. Yet when I download and install both files I get a mistake

Fehler beim Ausführen von "~/.vim/spell/de.utf-8.spl":
E763: 'Word Characters' unterscheiden sich zwischen Rechtschreibwörterbüchern

But regardless of the mistake I can make use of the spell checking and also also the pointer functions.

Additionally I can not :mksp my very own .spl - documents from myspell, create it would not read the .dic - documents. And also I removed the spelldirectory and also retried it. Absolutely nothing functioned.

5
2022-06-07 14:32:47
Source Share

Debian wheezy:

sudo apt-get install vim-scripts # Needed for spell downloading script
sudo rm /usr/share/vim/vim73/spell/*.sug /usr/share/vim/vim73/spell/*.spl
sudo vim

currently cause downloading and install in vim:

set spell spelllang=en,de,OR_WHATEVER_LANGUAGE_YOU_WANT

et voilà, you will certainly get flawlessly functioning spell apply for all users.

(If you just desire them for the existing customer, miss the sudo prior to calling vim or, transform the installation directory site to /home/YOUR_USER/.vim/spell.)

3
2022-07-22 14:40:01
Source

I had the very same trouble with a self - assembled Vim 7.3.729 on Windows/x64. As the spell files at http://ftp.vim.org/vim/runtime/spell/de/ have not been upgraded yet, I addressed this in the adhering to way:

1. download and install the Arch Linux plan stated in skunk is solution from https://www.archlinux.org/packages/extra/any/vim-spell-de/
2. Extract the spell files from the vim-spell-de-20120506-1-any.pkg.tar.xz documents ( 7Zip can unbox them)
3. Overwrite the 4 files de.{latin1,utf-8}.{spl.sug} in ~/.vim/spell/
1
2022-07-05 22:42:09
Source

I had a comparable concern (very same errormessage) and also I presume, it resulted from a compatibility - trouble. The spell files I made use of (mounted in.vim/ spell) were from an older installment of vim and also when I upgraded vim, I obtained this mistake. I addressed it by utilizing the vim spell files given by my circulation - - in my instance Arch Linux

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Document Type : Research Paper

Authors

Department of Mathematics, Micheal Okpara University of Agriculture, Umudike, Umuahia, Abia State Nigeria.

Abstract

ln this paper, we introduce a new class of mapping called asymptotically demicontractive multivalued mapping in the setting of a real Hilbert space. Furthermore, a new iteration scheme was constructed, and it was proved that our algorithm converges strongly to the common element of solutions of an equilibrium problem and the set of common fixed points of two finite families of type-one asymptotically demicontractive multivalued mappings without any sum conditions imposed on the finite family of the control sequences. Also, we provided a numerical example to demonstrate
the implementablity of our proposed iteration technique.  Our results improve, extend and generalize many recently announced results in the current

Keywords

###### ##### References
[1] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35(1972), pp.171-174.
[2] I.K. Agwu and D.I. Igbokwe, Approximation of common fixed points of finite family of mixed-type total asymptotically quasi-pseudocontractive-type mappings in uniformly convex Banach spaces, Advances in Inequalities and Applications, 2020(2020), pp.4-16.
[3] F.E. Browder, Convergence theorem for sequence of nonlinear operators in Banach space, Math. Z., 74(1967), pp.201-225 . doi.10.1007/BF01109805.
[4] F.E. Browder W.V. and Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20(1967), pp.197-228.
[5] C.E. Chidume, Geometric properties of Banach space and nonlinear iterations, Springer Verlag, Califoria, (2009).
[6] G. Marino and H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert space, J. Math. Anal. Appl., 318(2006), pp.143-152.
[7] A. Moudafi, Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241(2000), pp.46-55.
[8] H.K. Xu, Iterative algorithm for nonlinear operators, J. London. Math. Soc., 66(2)(2002), pp.240-256.
[9] H.K. Xu, Viscosity approximation method for nonexpansive mappings, J. Math. Anal. Appl., 298(2004), pp.279-291.
[10] S. Kitahara and W. Takahashi, Image recovery by convex combination of sunny nonexpansive retractions, Topol. Mathods Nonlinear Anal., 2(2)(1993), pp.333-342.
[11] X. Qin and Y.C. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl., (2010), pp.2963-2972.
[12] G.S. Saluja, Convergence to common fixed point of two asymptotically quasi-nonexpansive mappings in the intermediate sense in Banch spaces, Mathematica Morvica, 19(2015), pp.33-48.
[13] A.N. Iusem and A.R. De Pierro, On the convergence of Hans method for convex programming quadratic objective, Math. Program., 52(1991), pp.265-284.
[14] P. Kumama, N. Petrot and R. Wangkeeree, A hybrid iterative scheme for equilibrium problems and fixed point problems for asymptotically $k$-strictly pseudocontractions, J. Comput. Math. Appl., 233(2010), pp.2013-2026.
[15] T.M.M. Sow, et al, Algorithms for a system of variational inequality problems and fixed point problems with demicontractive mappings, J. Nig. Math. Soc., 38(2019), pp.341-361.
[16] F.O. Isiogugu, Demiclosedness principle and approximation theorem for certain class of multivalued mappings in Hilbert spaces, Fixed Point Theory Appl., (2013), Airticle ID 61(2013).
[17] F.O. Isiogugu, On the approximation of fixed points for multivalued pseudocontractive mappings in Hilbert spaces,Fixed Point Theory Appl., 2016(2016), 59 pages.
[18] S.H. Khan and W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn, 53(1)(2001), pp.143-148.
[19] S.H. Khan and L. Yildifim, Fixed points of multivalued nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 73(2012), pp.1687-1812.
[20] W.A. Kirk and A. Massa, Remarks on asymptotic and Chybeshev centers, Houston. J. Math., 16(3)(1990), pp.179-182.
[21] T.C. Lim, A fixed point theorem for weakly inward multivalued contractions, J. Math. Anal. Appl., 249(2000), pp.323-327.
[22] J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc., 38(1973), pp.547-547.
[23] J.T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc., 74(1968), pp.639-640.
[24] S.B. Nadler, Multivalued mappings, Pac. J. Math., 30(1969), pp.475-488.
[25] P.L. Combettes and S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6(2005), pp.117-136.
[26] S. Takahashi and W. Takahashi, Viscosity approximation method for equilibrium problem and fixed problem in Hilbert spaces, J. Math. Anal. Appl.,331(2007), pp.505-515.
[27] S. Plubtieng and R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336(2007), pp.455-469.
[28] A. Tada and W. Takahashi, Weak and strong convergence theorems for nonexpansive mappings and an equilibrium problem, J. Optim. Theory. Appl., 113(2007), pp.359-370.
[29] H.K. Xu, An iterative approach to quadratic optimisation, J. Optim. Theory. Appl., 116(2003), pp.659-678.
[30] H. Zegeye and N. Shahzad, Convergence of Mann's type iteration for generalised asymptotically nonexpansive mappings, Comput. Math. Appl., 62 (2011), pp.4007-4014.
[31] P.E. Mainge, Strong convergent of projected subgradient for nonsmooth and nonstrictly convex minimization, Set~-~Valued Anal. , 16(2008), pp.899-912.
[32] M. De la sen, On some convergence properties of the modified lshikawa scheme for asymptotically demicontractive mappings with metricial parameterizing sequences, Hindawi J. Math., 2018(2018), 13 pages.
[33] Y. Shehu, Strong convergence theorems for fixed point problems, varietional inequality problems and systems of generalised equilibrium problems, Math. Comput. Model., 54(2011), pp.1510-1522.
[34] A. Abkar and M. Tavakkoli, Anew algorithm for two finite families of demicontractive mappings and equilibrium problems, Appl. Math. Comput., 266(2015), pp.491-500.
[35] B. Ali and L. Umar, Approximation of solutions of generalised mixed equilibrium problems and fixed points of multivalued asymptotically
quasi-nonexpansive mappings, J. Nig. Math. Soc., 38(2019), pp.569-592.
[36] N. Djitte and M. Sene, Convegence theorems for fixed points of multivalued
mappings in Hilbert space, lnt. J. Anal. ,2014(2014), pp.7-17.
[37] A. Banyawat and S. Suantai, Common fixed points of a countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces,
[38] T. Hu, J. Huang andE. Rhoades, A general principle for lshikawa iteration for for multivalued mappings, Indian J. Pure Appl. Math., 28(1997), pp. 1691-1098.
[39] M.O. Osilike and S.C. Aniagbosor, Fixed points of asymptotically demicontractive mappings in certain Banach spaces, Indian J. Pure Math. Appl., 32(10)(2001), pp.1519-1537.
[40] F. O. lsogugu, C. Izuchukwu and C.C. Okeke, New iteration scheme for approximating a common fixed point od finite family of mappings, Hindawi J. Math., 2020(2020), 4 pages.
[41] G.C. Ugwunnadi, O.T. Mewomo and C. lzuchukwu, Convergence theorem for finite family of asymptotically demicontractive multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl., 26(2020), pp.117-130.
[42] F. Zhang, H. Zhang and Y. Zhang, New iterative algoritm for two infinite families of multivalued quasi-nonexpansive mappings in uniformly convex Banach spaces, J. Appl. Math., 2013(2013), 7 pages.
[43] l.K. Agwu, D.I. Igbokwe and A. E. Ofem, Approximation of common solutions of fixed point problem for $\alpha$-hemicontractive mapping, split equilibrium problem and variational inequality problem, Advances. Fixed Point Theory, (2021),11 pages.
[44] E. Blum and W. Oetti, From optimization and variational inequality to equilibrium problems, Math. Student, 63(1994), pp.123-145.
[45] M. Eslamian, Hybrid method for equilibrium problems of finite families of nonexpansive semigroups, Revista de la Real Academia de Ciecias Exactas Fisicas aturales , 2012(2012), pp.30-45.
[46] M. Zhaoli and L. Wang, Demiclosedness principle and convergence theorems for asymptotically strictly pseudononspreading mappings and mixed equilibrium problems, Fixed Point Theory Appl., 2014(2014), pp.104-145.
[47] L. Qihou, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal., 26(1996), pp.1835-1842.
[48] M.O. Osilike and E.E. Chima, Mixed equilibrium and multipleset split feasiblity problems for asymptotically pseudonospreading mappings, Commun. Optimiz. Theory, 2020 (2020), pp.113-140.
[49] M.O. Osilike, et al, Demiclosedness principle and convergence theorems for $k$-strictly asymptotically pseudocontractive maps, J. Math. Anal. Appl.,326(2007), pp.1334-1345.
[50] U. Singthong, and S. Suantai, Equilibrium problems and fixed point problems for nonspreading-type mappings in Hilbert spaces, Int. J. Nonliear. Anal. Appl. ,2(2011), pp.51-61.
[51] M.E. Samei, Some Fixed point results on intuitionistic fuzzy metric spaces with a graph, Sahand Commun. Math. Anal., 13(2019), pp.141-152.
[52] Z. Baitiche, et al, Monotone iterative method for $\Psi$-Caputo fractional differential equation with nonlinear boundary conditions, Fractal and Fract., 5(2021), pp. 123-143.
[53] M.E. Samei, Convergence of an iterative scheme for multifunctions on fuzzy metric spaces, Sahand Commun. Math. Anal., 15(2019), pp.91-106.
[54] J. Alzabut et al, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-Local initial conditions, Symmetry, 13(2021), pp.473.
[55] I.K. Agwu and D.I. Igbokwe, A Modified proximal point algorithm for finite families of minimization problems and fixed point problems of asymptotically quasi-nonexpansive multivalued mappings Punjab University Journal of Mathematics ,54(8)(2022), pp.495-522.


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2014
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# Of The 2013 GCE O Level Results Aftermath – GVGT

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Tuition given in the topic of Miss Loi the Tutor from the desk of at 12:38 am (Singapore time)

Amidst deluge of calls and scheduling that typically follows a new semester, some students have been asking Miss Loi “”Hey how come this year got no O Level results video like this one to ‘motivate’ us?!”.

So in order to give everyone some kind of 交代, here’s a quick attempt of a video dedicated to The Temple‘s GCE O Level 2013 batch of students …

### Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

• ### Latest News

With the smell of Dettol hanging in the air, The Temple has cautiously reopened for physical joss sticks math tuition sessions in Novena.

Please check our revised secondary / O level maths group tuition schedule for updates throughout this uncertain period.

The 2018 exam papers are here! Miss Loi's 2018 Maths Exam Papers are finally here!

* Please refer to the relevant Ten-Year Series for the questions of these suggested solutions.


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Let $f:E\to D^*$ be a family of complex elliptic curves parametrized by the punctured open disk $D^*.$ Assume that the monodromy on $H^1$ is trivial (i.e. $R^1f_*\mathbb Z$ is a constant sheaf on $D^*$). Does this imply that $f$ extends to a family of elliptic curves over the full disk $D?$
Here's an attempt, which I'm not sure if it works: maybe there is an equivalence (Riemann-Deligne?) between families of elliptic curves over a (smooth) base over $\mathbb C$ and variations of $\mathbb Z$-Hodge structures satisfying Griffiths transversality, of rank 2 and weight 1 on the same base. The constant local system certainly extends to $D.$


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# Four balls with different colors in a box, how many times do I need to pick to see all four colors?

I have one white ball, one yellow ball, one red ball, one black ball. I put the four balls in a nontransparent box. I pick a ball from the box to see its color and put it back to the box.

Assuming picking is random, how many times on average do I need to pick in order to see all four colors?

If I'm lucky, I only need to pick four times. If I'm out of luck, I may get red, red, red, red, red, red,....

But what is it on average?

• mathematical boxes are always non-transparent for future reference – Jorge Dec 22 '14 at 20:54
• @JorgeFernández Also, the colors white, yellow, red, and black exist only on the balls in the box, nowhere else. You cannot "see all four colors" unless you see all four balls. – KSmarts Dec 22 '14 at 21:00
• lol, I had assumed that part. – Jorge Dec 22 '14 at 21:01
• Wait, did you not see the balls as you were putting them in the box? If not, how do you know what color they are? – KSmarts Dec 22 '14 at 21:01
• If I'm out of luck, I may get red, red, red, red, red, red,.... If such a sequence continues long enough, it'll really mess up an 'average'. – user2338816 Dec 23 '14 at 0:10

This is known as the coupon collector's problem.

The average number of tries needed to see each of 4 colors is $$\frac44+\frac43+\frac42+\frac41 = 8\frac{1}{3}$$

The terms are $4/4$ for the time to take one ball; then $4/3$ for the average time it takes after the first ball until you see another code; $4/2$ for the average time it takes after the first time you see the second color until you see a third, and finally $4/1$ for the time you then have to wait until you see the last color. (These can just be added due to the additivity of expectations).

• It is interesting to notice that by putting together this proof and mine, we get a nice identity for $n H_n$ involving sign-alternating binomial coefficients. – Jack D'Aurizio Dec 22 '14 at 21:09
• +1 for pointing out the name of the problem. Give me some time to understand the solution. – Gqqnbig Dec 22 '14 at 21:40
• Same result as the other three answers, but so much more intuitive – Evorlor Dec 23 '14 at 2:54
• That the average number of tries needed to hit one out of $k$ (remaining) among $n$ equal $n/k$ follows from solving a simple equation. But can it also be deduced from some form of additivity of expectations? – Marc van Leeuwen Mar 13 '15 at 20:01
• @Marc: The reference to additivity was for adding the fractions, not for finding them. (In hindsight I'm not sure whether is muddies matters more than it clarifies). – hmakholm left over Monica Mar 13 '15 at 20:47

Denote by $E(n)$ the expected number of additional draws when $n$ of the four balls have not been drawn so far. Then $E(0)=0$, and $$E(n)=1+{n\over4} E(n-1)+{4-n\over 4}E(n)\qquad(n\geq1)\ .$$ This immediately implies $$E(n)=E(n-1)+{4\over n}\qquad(n\geq1)\ ,$$ from which we get $E(4)={25\over3}$ by going through the motions.

Obviously, we stop picking when we take a ball having a color that never occurred before, so we have to count how many strings of length $n$ have only occurrences of $A,B,C$ (no $D$), with at least one occurence of $A$, at least one occurrence of $B$, at least one occurrence of $C$. By the inclusion-exclusion principle, they are: $$3^n - 3\cdot 2^n + 3$$ so the wanted expected value is: $$\sum_{n=4}^{+\infty}\frac{4n}{4^n}\left(3^{n-1}-3\cdot 2^{n-1}+3\right)=\color{red}{\frac{25}{3}}.$$

What is the probability you see all colors after exactly $n$ turns, how many sequences of length $n-1$ contain exactly three colors? first select the three colors in $4$ ways.

When using three colors there are $3^{n-1}$ sequences.

$3\cdot2^{n-1}-3$ use $2$ or $1$ colors.

Therefore $3^{n-1}-3\cdot2^{n-1}+3=3(3^{n-2}-2^{n-1}+1)$ use exactly three colors when those three colors are fixed.

Therefore there are $12(3^{n-2}-2^{n-1}+1)$ sequences.

Then the probability you see three colors for the first time in the $n$'th draw is $\frac{1}{4}\frac{12(3^{n-2}-2^{n-1}+1)}{4^{n-1}}=3\frac{(3^{n-2}-2^{n-1}+1)}{4^{n-1}}$.

Hence you want $3(\frac{1}{4}\sum\limits_{n=2}^\infty (n+2)\frac{3}{4}^n-\sum\limits_{n=3}^\infty (n+1)\frac{1}{2}^n+\sum\limits_{n=3}^\infty\frac{n+1}{4^n})=3(\frac{63}{16}-\frac{5}{4}+\frac{13}{144})=\frac{25}{3}$


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# Why is the map $\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2) \rightarrow \mathbb{H} : A \mapsto Ai$ injective?

First some notation:

$$\mathrm{SL}_2(\mathbb{R})=\left\lbrace\begin{pmatrix} a&b \\ c &d \end{pmatrix}\;\Biggm|\;a,b,c,d \in \mathbb{R} ,ad-bc=1 \right\rbrace$$

$$\mathrm{SO}(2)=\lbrace K\in \mathrm{SL}_2(\mathbb{R}) : A^TA=AA^T=I \rbrace$$.

$$\mathbb{H}$$ denotes the upper half plane.

The map $$\mathrm{SL}_2(\mathbb{R})/\mathrm{SO}(2) \rightarrow \mathbb{H} : A \mapsto Ai$$

is bijective.

My problem is to understand why the map is injective.

My idea is to take $$N\in \mathrm{SO}(2)$$ and consider $$Mi=Ni$$.

Since $$\mathrm{SO}(2)$$ is the stabilizer of $$i$$ it follows that $$M=N$$.

Thanks for the help.

• How do you mean $Ai$? (I guess, $i$ is the imaginary unit in $\Bbb H\subseteq \Bbb C$.) Jul 31 '19 at 19:20
• sorry , the operation should be the moebius transformation . Jul 31 '19 at 19:32
• Yes , i is the imaginary unit Jul 31 '19 at 19:34

If $$X$$ is any set and $$G$$ is a group acting transitively on it then we can identify $$X$$ with $$G/G_x$$ where $$G_x$$ is the stabilizer of some fixed point $$x \in X$$.
The upper triangular matrices act transitively on $$\Bbb H$$, so $$\mathrm{SL}_2(\Bbb R)$$ does too, and $$\mathrm{SO}(2)$$ is the stabilizer of $$i$$.
$$Ai = Bi \iff AB^{-1} \in \mathrm{SO}(2) \iff A \cdot \mathrm{SO}(2) = B \cdot \mathrm{SO}(2)$$, showing that $$A=B$$ modulo $$\mathrm{SO}(2)$$ as required.
$$\gamma . i = \beta . i \\\implies \beta^{-1} \gamma.i =\beta^{-1}.\beta.i= i\\ \beta^{-1}\gamma. i = \frac{ai+b}{ci+d}= i \implies ai+b = di-c \implies (c,d) = (-b,a)\\ \implies \beta^{-1} \gamma \in SO_2(\Bbb{R})\\ \implies \beta^{-1} \gamma SO_2(\Bbb{R}) =SO_2(\Bbb{R}) \\ \implies \gamma SO_2(\Bbb{R})=\beta SO_2(\Bbb{R})$$


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## Growth of carbon nanotubes and microfibers over stainless steel mesh by cracking of methane

The ${La}_{2}{NiO}_{4}$ film was synthesized on the 304 stainless steel (SS) mesh. The hydrogen reduction of La2NiO4 generated homogeneous nanocatalyst particles (probably ${Ni}/{La}_2{O}_{3}$) over which methane was cracked, producing carbon notubes/microfibers and hydrogen. The carbon nanotubes/microfibers were strongly bonded to the SS mesh. It was observed that the methane conversion always reached its maximum at the cracking temperature of 750 °C regardless of its concentration varying from 5% to 100%. The cracking of 5% methane diluted in nitrogen generated multiwalled carbon nanotubes while the cracking of 10–100% methane resulted in the formation of carbon solid microfibers together with globular carbon particles. Higher concentration of methane created thicker carbon fibers and a 30% concentration of methane resulted in the highest deposits of carbon on the mesh. After a compressed air blow and ultrasonic treatment, the carbon deposits were still strongly adhered to the mesh. As a result of the carbon tubes/fibers coverage, the specific surface area of the SS mesh was enhanced significantly from 0.03 ${m}^{2}/g$ to 21–45 ${m}^{2}/g$. XRD, HRTEM and Raman studies confirmed that the carbon products were of graphitic structure. Such advanced mesh material would have great application potential in industrial catalysis and other areas.

Published in:
Surface & Coatings Technology, 202, 3029-3042
Year:
2008
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