text:
## Evaluating the oscillators using simmetries

Various technical topics such as parallelism and efficiency, netCDF problems, the Yambo code structure itself, are posted here.

### Evaluating the oscillators using simmetries

Dear Yambo developers,
we are studying the Yambo code. In particular we are interested in how Yambo calculates the oscillators using symmetries.
According to eq. (39) of ref. ( A. Marini et a., yambo: An ab initio tool for excited state calculations, Computer Physics Communications (2009) ) we would like to understand the following code-line (written in the file src/wf_and_fft/scatter_Bamp.F).

Code: Select all
 isc%rhotw(1:isc%ngrho)=  rho_tw_rs(fft_g_table(g_rot(qs,1:isc%ngrho),iGo),1)

As far as we know, g_rot(qs, 1:isc%ngrho) simply applies the simmetry qs to the G vectors, rotating them. We would like to know what fft_g_table (contained in src/wf_and_fft/fft_setup.F) does.
Thank you,
Sincerely
Marco Govoni and Ivan Marri
Marco Govoni
Physics Department, University of Modena and Reggio Emilia (Italy)
marco.govoni

Posts: 35
Joined: Thu May 21, 2009 3:46 pm

### Re: Evaluating the oscillators using simmetries

Dear Marco Govoni and Ivan Marri,

Well I am not really into these technical details (Andrea is but he cannot answer at this moment), but I will try to answer, I hope it helps.

The subroutine calculates
rhotw(G)=<ib ik_is i_sp| e^[I(Rs(G-G0)).r] | ob ok_os o_sp>

to do this, first it multiplies the two wfs in direct space, then fft transforms this product (resulting in rho_tw_rs) and finally (in the line you pointed out) applies the rotation and subtracts G0 using two mappings:
1. Given the indexes of the symmetry operation and a G vector g_rot gives the index corresponding to the rotated Rs G vector.
2. Then, my understanding is that fft_g_table maps the index corresponding to the (rotated) G vector and the index corresponding to the G0 vector into the index corresponding to the G-G0 vector.

Cheers,

m
Dr Myrta Grüning
Atomistic Simulation Centre
Queen's University Belfast - Northern Ireland

http://www.researcherid.com/rid/B-1515-2009

myrta gruning

Posts: 303
Joined: Tue Mar 17, 2009 11:38 am

### Re: Evaluating the oscillators using simmetries

Let me only add that when performing and FFT from real to reciprocal space, the G components are stored in the array used by the FFT using an order different from the one used in the code. The table

Code: Select all
fft_g_table(iG,iGo)

then, gives the index in the FFT ordering of the component relative to the RL vector (G_{iG}-G_{iGo}). In this way the oscillator that is returned by scatter_Bamp is ordered in the correct way.
Andrea MARINI
Istituto di Struttura della Materia, CNR, (Italy)

andrea marini

Posts: 447
Joined: Mon Mar 16, 2009 4:27 pm

### Re: Evaluating the oscillators using simmetries

Dear developers,

we are studying how the oscillators strengths are implemented in the yambo code.
As pointed out in your Com.Phys.Commun. (A2 appendix), the matrix elements are defined as:

rho_{nm} (k,q,G) = Int_dr psi*_{nk} ( r) e^{i (q+G).r } psi_{mk'} ( r)

where k' = k - q - G_0, k' and k are in the 1BZ, q is in 1QBZ. Int_dr denote an integration in the real space over the unit cell. psi are the Bloch wavefunctions:

psi_{nk} ( r) = e^{i k.r} u_{nk} ( r)

so that

rho_{nm} (k,q,G) = Int_dr u*_{nk} ( r) e^{i (G-G_0).r } u_{mk'} ( r)

by applying symmetries on k and k' : k= R k_{IBZ} and k' = R' k'_{IBZ}
the periodic function u transforms as : u_{nk} ( r) = u_{n R k_{IBZ}} ( r) = u_{n k_{IBZ}} ( R^{-1} r)

as a consequence we have

(*) rho_{nm} (k,q,G) = Int_dr u*_{n k_{IBZ} } ( R^{-1} r) e^{i (G-G_0).r } u_{m k'_{IBZ} } ( R'^{-1} r)

----------------------

The procedure is implemented in Scatter_Bamp.F as follows:

Step #1 :
Load the wavefunctions in real space: u_{n k_{IBZ} } ( r) and u_{m k'_{IBZ} } ( r)

Step #2 :
Use symmetries in real space to evaluate the product :
u*_{n k_{IBZ} } ( R^{-1} r) u_{m k'_{IBZ} } ( R'^{-1} r)
where complex conjugate operations can be removed or added depending on the presence of Time-Reversal (litr and lotr variables)

Step #3 :
Perform a Fourier transformation of this product

Step #4 :
Rotate the G-space using symmetry qs and shift the whole rotated space of a G_0 vector.

Code: Select all
       isc%rhotw(1:isc%ngrho)=  rho_tw_rs(fft_g_table(g_rot(qs,1:isc%ngrho),iGo),1)

Comparing this procedure with eq. (*), we don't understand the meaning of rotation qs, which as far as we know is defined as:

q_{QBZ} = R_{qs} q_{QIBZ}

Are we omitting something?
Thank you in advance.

Sincerely,

Marco Govoni and Ivan Marri
Marco Govoni
Physics Department, University of Modena and Reggio Emilia (Italy)
marco.govoni

Posts: 35
Joined: Thu May 21, 2009 3:46 pm

### Re: Evaluating the oscillators using simmetries

marco.govoni wrote:we are studying how the oscillators strengths are implemented in the yambo code.
As pointed out in your Com.Phys.Commun. (A2 appendix), the matrix elements are defined as:

rho_{nm} (k,q,G) = Int_dr psi*_{nk} ( r) e^{i (q+G).r } psi_{mk'} ( r)

In the appendix it is explained how to move the symmetry acting on k on the right, in order to reduce the number of operations to do in the evaluation of the BS kernel.

This procedure, however, is implemented in the routines the call scatter_Bamp.F and not in the routine itself, where it is calculated the most general oscillator of the type

Code: Select all
 ! ! rhotw(G)=<ib ik_is i_sp| e^[I(RsG-G0).r] | ob ok_os o_sp>    ! !  is(3)=(ib,ik,is,i_sp) --<--:--<-- os(3)=(ob,ok,os,o_sp) !                             : !                            /:\ iqs=(ig0,iq,qs) !                             : ! !                     \:/ ! REMEMBER that     ->-:->-  =  conjg [-<-:-<-] !                                        /:\ !  iq_is = ik_is-ok_os + Go

as a consequence we have
(*) rho_{nm} (k,q,G) = Int_dr u*_{n k_{IBZ} } ( R^{-1} r) e^{i (G-G_0).r } u_{m k'_{IBZ} } ( R'^{-1} r)

Comparing this procedure with eq. (*), we don't understand the meaning of rotation qs, which as far as we know is defined as:

q_{QBZ} = R_{qs} q_{QIBZ}

Are we omitting something?

The operation of symmetry acting on G is due to the fact that all summations on G are rotated by the inverse of the symmetry R_{qs}. For example
Code: Select all
\sum_R \sum_q \sum_G <exp(i(Rq+G))> v(Rq+G) = \sum_R \sum_q \sum_G' <exp(iR(q+G'))> v(R(q+G'))
Andrea MARINI
Istituto di Struttura della Materia, CNR, (Italy)

andrea marini

Posts: 447
Joined: Mon Mar 16, 2009 4:27 pm


text:
I have written about the Go programming language before. There, I made clear that I (among other things) disagree with Go’s choice to not include a templating mechanism or generic types.

Now, a few months later, I still stand by that but want to provide some evidence as to why I don’t like the choice. What follows is a part of a function from a Go code base that I am contributing to. Rest assured that once I saw this monstrosity I made room in my schedule to simplify it and clean it up (in fact, what you see below has been simplified a bit for this post).

// search the AI-inventory for the object representing the product
invSlotLoop:
for _, invSlot := range aiInv.Slots {
if invSlot.IsObject() {
objectID := invSlot.ObjectID()
// is it the right product?
product, isProduct := aiHan.pool.Product.TryGet(objectID)
if isProduct && product.BlueprintID == job.BlueprintID {
// are all the slots that the current job should install done?
for _, jobSlotIndex := range job.BlueprintSlotIndices {
if !product.IsSlotInstalled(jobSlotIndex) {
continue invSlotLoop
}
}
// all slots done! this is our product!
// get it to one of the output inventories.
productStorable := aiHan.pool.Storable.Get(objectID)

outInvLoop:
for _, outInvEnt := range workplace.OutputInventories {
outInv := aiHan.pool.Inventory.Get(outInvEnt)
for _, outInvSlot := range outInv.Slots {
if outInvSlot.IsEmpty() && outInvSlot.IsCompatible(objectID, productStorable) {
// we have found an inventory where we can put it! walk to the inventory!
hasToWalk, err := aiHan.walkToEntity(aiEnt, outInvEnt)
if err != nil {
// nah, we can't walk to it. have to find another inventory
continue outInvLoop
}
if hasToWalk {
// walking is triggered, wait until he gets there
return
}
// transfer the object
aiHan.invTransferMan.Transfer(aiEnt, outInvEnt, nil, objectID)
}
}
}
}
}
}

The code above should of course be simplified by introducing a few functions that break the nesting and make it easier to follow the flow; there also is some error handling that I do not agree with.

I count a full 7 (!) levels of nesting, including 2 loops with labels and label-targeted continues. Of course, Go does not force you to write your code like this, but it definitely makes it easier and much more attractive for people to do it.

Notice how the first continue targeting the outer most loop is essentially checking whether some predicate ( = !product.IsSlotInstalled) is true for any item in a slice. In another language, we might write something like this:

if job.BlueprintSlotIndices.Any(x => !product.IsSlotInstalled(x)) {
continue
}

In Go, we would have to define Any for each type separately, with its own name like AnyUInt32. Well, it turns out that people don’t really do that. They write out the loop each and every time, because - well - it’s only five lines. The piece of code above shows pretty clearly where this is heading.

Functions serve multiple purposes: Firstly, they should encapsulate commonly used code so that you only have to debug it once etc. Secondly, and more importantly, they should abstract away details. While the details of a simple for-loop should be quite familiar to anyone, there is no arguing that within a function such as the one above a call like xs.Contains(...) would be much easier to parse1 than the equivalent 5-line loop, because there are already so many things going on.

By now, I treat each labeled loop in Go as a pretty serious problem. Heck, even just nested loops are suspicious.

Oh, and this is my version of the function from above (with some additional functionality):

productEntity, ok := getCompatibleFinishedProductInInventory(job, aiInv)
if !ok {
aiHan.tryTakeProduct(aiEnt, job, job.Crafter, true /* take only finished products */)
return
}

result := aiHan.tryStoreObject(aiEnt, productEntity, workplace.OutputInventories)
if result == AIInventoryTransferResultNoSuitableInventory {
aiHan.setNoSpaceFailState(aiEnt, nil)
aiHan.blockWorker(aiEnt)
} else if result == AIInventoryTransferResultNoPath {
aiHan.setNoPathFailState(aiEnt, ecs.InvalidEntityID)
aiHan.blockWorker(aiEnt)
}

1. Also, it would be shorter. I am not convinced that brevity is always a virtue in programming, but some people like Cliff Click argue that the number of bugs created by a programmer is mostly proportional to the lines of code written, not the number of features implemented.

format: plain


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# infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs Answer

Hello dear visitor to our network We will offer you a solution to this question infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

## infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs

A hypergraph $$H$$ is a pair consisting of a set $$V$$ of vertices and a family of subsets of $$V$$ called edge. One class of examples is obtained by taking a graph $$G=(V,E)$$ with set of vertices $$V$$ and set of edges $$E$$ where $$H=(V,Cliq (G) )$$. Here $$e\in Cliq(G)$$ if and only if for all $$v,w\in e$$ look for that $$v\ne w$$ we have $$\{v,w\}\in E$$.

If $$H=(V,F)$$ is a hypergraph, a covers is a family $$C$$ of subsets of $$F$$ look for that $$\bigcup C=V$$. The cover is strongly minimal if for any cover $$D$$, $$|C\setminus D|\le|D\setminus C|$$.

Please give an example of a hypergraph stemming from the cliques of a graph as above that has no strongly minimal cover.

A related post with a nice example by domotorp is here.

we will offer you the solution to infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs question via our network which brings all the answers from multiple reliable sources.


text:
# Properties

 Label 1872.2.a.h Level $1872$ Weight $2$ Character orbit 1872.a Self dual yes Analytic conductor $14.948$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1872.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.9479952584$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 39) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} + 4 q^{7}+O(q^{10})$$ q - 2 * q^5 + 4 * q^7 $$q - 2 q^{5} + 4 q^{7} + 4 q^{11} + q^{13} - 2 q^{17} - q^{25} + 10 q^{29} - 4 q^{31} - 8 q^{35} - 2 q^{37} - 6 q^{41} + 12 q^{43} + 9 q^{49} - 6 q^{53} - 8 q^{55} + 12 q^{59} - 2 q^{61} - 2 q^{65} + 8 q^{67} + 2 q^{73} + 16 q^{77} - 8 q^{79} + 4 q^{83} + 4 q^{85} + 2 q^{89} + 4 q^{91} + 10 q^{97}+O(q^{100})$$ q - 2 * q^5 + 4 * q^7 + 4 * q^11 + q^13 - 2 * q^17 - q^25 + 10 * q^29 - 4 * q^31 - 8 * q^35 - 2 * q^37 - 6 * q^41 + 12 * q^43 + 9 * q^49 - 6 * q^53 - 8 * q^55 + 12 * q^59 - 2 * q^61 - 2 * q^65 + 8 * q^67 + 2 * q^73 + 16 * q^77 - 8 * q^79 + 4 * q^83 + 4 * q^85 + 2 * q^89 + 4 * q^91 + 10 * q^97

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 −2.00000 0 4.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.2.a.h 1
3.b odd 2 1 624.2.a.i 1
4.b odd 2 1 117.2.a.a 1
8.b even 2 1 7488.2.a.by 1
8.d odd 2 1 7488.2.a.bl 1
12.b even 2 1 39.2.a.a 1
20.d odd 2 1 2925.2.a.p 1
20.e even 4 2 2925.2.c.e 2
24.f even 2 1 2496.2.a.q 1
24.h odd 2 1 2496.2.a.e 1
28.d even 2 1 5733.2.a.e 1
36.f odd 6 2 1053.2.e.d 2
36.h even 6 2 1053.2.e.b 2
39.d odd 2 1 8112.2.a.s 1
52.b odd 2 1 1521.2.a.e 1
52.f even 4 2 1521.2.b.b 2
60.h even 2 1 975.2.a.f 1
60.l odd 4 2 975.2.c.f 2
84.h odd 2 1 1911.2.a.f 1
132.d odd 2 1 4719.2.a.c 1
156.h even 2 1 507.2.a.a 1
156.l odd 4 2 507.2.b.a 2
156.p even 6 2 507.2.e.a 2
156.r even 6 2 507.2.e.b 2
156.v odd 12 4 507.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 12.b even 2 1
117.2.a.a 1 4.b odd 2 1
507.2.a.a 1 156.h even 2 1
507.2.b.a 2 156.l odd 4 2
507.2.e.a 2 156.p even 6 2
507.2.e.b 2 156.r even 6 2
507.2.j.e 4 156.v odd 12 4
624.2.a.i 1 3.b odd 2 1
975.2.a.f 1 60.h even 2 1
975.2.c.f 2 60.l odd 4 2
1053.2.e.b 2 36.h even 6 2
1053.2.e.d 2 36.f odd 6 2
1521.2.a.e 1 52.b odd 2 1
1521.2.b.b 2 52.f even 4 2
1872.2.a.h 1 1.a even 1 1 trivial
1911.2.a.f 1 84.h odd 2 1
2496.2.a.e 1 24.h odd 2 1
2496.2.a.q 1 24.f even 2 1
2925.2.a.p 1 20.d odd 2 1
2925.2.c.e 2 20.e even 4 2
4719.2.a.c 1 132.d odd 2 1
5733.2.a.e 1 28.d even 2 1
7488.2.a.bl 1 8.d odd 2 1
7488.2.a.by 1 8.b even 2 1
8112.2.a.s 1 39.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1872))$$:

 $$T_{5} + 2$$ T5 + 2 $$T_{7} - 4$$ T7 - 4 $$T_{11} - 4$$ T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T - 4$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 10$$
$31$ $$T + 4$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 12$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T - 2$$
$97$ $$T - 10$$


text:
# What do the $j$ and $m$ stand for in $|j,m\rangle$ for angular momentum in quantum mechanics?

I'm assuming it is a jth state with m value as total angular momentum?

-
Wikipedia discusses quantization of angular momentum e.g. here and here. –  Qmechanic Jan 28 '13 at 1:45
Also here, given that these are essentially the same indices we index spherical harmonics with. –  Chris White Jan 28 '13 at 2:13
The states $|j,m\rangle$ are simultaneous eigenstates of the total angular momentum squared operator $\mathbf J^2$ and the $z$-component of the total angular momentum operator $J_z$. The letter $j$ is related to the eigenvalue of the operator $\mathbf J^2$, while the letter $m$ gives the eigenvalue of the operator $J_z$. Specifically $$\mathbf J^2|j,m\rangle =\hbar^2 j(j+1)|j,m\rangle, \qquad J_z|j,m\rangle = \hbar \,m|j,m\rangle$$ Given these considerations, $j$ is called the total angular momentum quantum number.
But does J here include spin or not? If it doesn't, isn't it notationally more common to denote the total ang. momentum by $\ell$ and not $j$? I thought $j$ was for the total ang. momentum including spin... –  daaxix Jan 29 '13 at 18:30
@daaxix The letter J is generically used for all forms of angular momentum in quantum. What $J_i$ denotes depends on the context. As long as the components of $\mathbf J$ satisfy the angular momentum commutation relations $[J_i, J_j] = i\hbar\epsilon_{ijk}J_k$, one can show that the analysis of its eigenvalues and eigenvectors leads to states labeled in this way. So $J_i$ could denote the components of orbital angular momentum, or spin, or any combination of both, and the notation will carry through. –  joshphysics Jan 29 '13 at 18:40


text:
# January 1985 Council Minutes

JANUARY 1985 COUNCIL MINUTES

The Council met in California Room E of the Anaheim Convention Center, on 8
January 1985, at 2:00 PM.  Members present were S. Armentrout, R.G. Bartle, H.
Bass, D.M. Burns, Jr., W.W. Comfort, M.G. Crandall, P.L. Duren, D. Eisenbud.
R.M. Fossum, S.J. Friedlander, W. Gautschi, P.R. Halmos, R.C. Hartshorne, M.
Hochster, M. Jerison, W.B. Johnson, I. Kaplansky, J.B. Keller, I. Kra, M.
Lowengrub, R.B. Melrose, T.K. Milnor, M.S. Montgomery, C.C. Moore, W.D.
Neumann, J.E. Osborn, D.S. Passman, F.P. Peterson, E. Pitcher, M.B. Pour-El, H.
Rossi, G.C. Rota, L.P. Rothschild, G.R. Sell, M. Shub, L.W. Small, J.  Smoller,
O. Taussky Todd, J.E. Taylor, A. Terras, J.J. Uhl, Jr., D.A. Vogan, Jr., and
R.O. Wells, Jr.  There were five persons present with the privilege of the
floor, namely Felix E. Browder, Patrick Gallagher, Kenneth M. Hoffman, Joshua
A.  Leslie,, and John C. Polking.  Also present were R. Hahn, W. J. LeVeque,
J. W. Maxwell, C. McConway, J.P. Mesirov, B. J. Janson, J. Kister, J. L.
Selfridge, and W.B. Woolf.  President Kaplansky was in the chair.

MINUTES

1.1 Minutes of 15 AUGUST 1984.  These minutes had been distributed by mail.
The Secretary observed one omission.

4.7 NOTICES EDITORIAL COMMITTEE:  The Nominating Committee
proposed a slate of three candidates for the Editorial
Committee of the NOTICES for a term of four years effective
1 January 1985, namely:

Ralph Boas
Mary Ellen Rudin
Steven W. Weintraub.

The Council elected the slate.

With this addition the minutes were approved.

1.2 MINUTES OF BUSINESS BY MAIL.   The minutes of Business by Mail dated 20
November 1984, are attached and were approved.

MEETINGS

3.1 COUNCIL MEETINGS.  The next three meetings of the Council are:

Worcester, Friday, 19 April 1985 at 1:00 PM
Laramie, Sunday 11 August 1985
New Orleans, Monday, 6 January 1986.

The first of these meetings was set by the Council, whereas, the second and

COMMITTEES AND BOARDS

4.1 ELECTION OF OFFICERS FOR 1985.  The tellers reported.  The composition of
the Council and the BT of 1985 is to be found in the attachment where names of
persons just elected are marked with an asterisk.

Also elected to the Committee to Monitor Problems in Communication, were Judith
D. Sally and Floyd L. Williams.

The complete report was distributed in executive session.

4.2 ELECTION TO THE NOMINATING COMMITTEE:  The Tellers reported that those
elected to the Nominating Committee of 1985 and 1986 were:

Vera S. Pless
Mary E. Rudin
Michael Shub
Raymond O. Wells, Jr.

The report was distributed in executive session.

4.3 REVIEW OF SOCIETY ACTIVITIES.  The report of the EC on its triennial review
of meetings is attached.

4.4 SUMMER INSTITUTES AND SPECIAL SYMPOSIA.  The EC recommended that the name
of the Committee on Summer Institutes be changed to the Committee on Summer
Institutes and Special Symposia and that its charge be enlarged to include
selecting a subject and an organizing committee for a symposium in the spring
of alternate years in pure mathematics.  The latter duty is a successor
enterprise to the series of symposia in pure mathematics that have been
undertaken by the Committee to Select Hour Speakers for Central Sectional
Meetings.  The changes in name and charge were approved.

4.5 MEETINGS ON SPECIAL TOPICS.  The EC recommended that from time to time the
invited lectures and special sessions at a regional meeting be devoted
exclusively to a single broad topic in mathematics.

The proposal was approved.  It was specifically noted that the restriction
would not apply to contributed papers.

4.6 SCIENCE POLICY COMMITTEE:  Felix Browder, Chairman, presented the attached
report.  In it he referred to a resolution by the National Science Board, which
Dr. John Polking, representing the NSF, made available to interested parties
and which is attached to Browder's report.

4.7 JPBM:  Kenneth M. Hoffman, Executive Secretary for National Affairs,
presented the attached report.

4.8 TRANSACTIONS EDITOR: R.O.  Wells, Jr., managing editor, proposed that one
of the positions on the editorial committee be reserved for a managing editor,
who is not himself the primary editor for a field, but who watches over the
wellbeing of the entire journal.  This is, of course, a purely internal matter
in the committee.  In order that this person be an editor with substantial
experience, it was argued that it should be a person who has already served as
a member of the editorial committee (or as an associate editor.)  It was
pointed out that when the bylaws were amended to change the term of
TRANSACTIONS and PROCEEDINGS editors from three to four years, the Council
endorsed the convention that an editor who had served a four year term
ordinarily not be nominated for a second term.  A letter from Wells is
attached.

Views were expressed that possibly the position of managing editor could be
handled by one of the editors with a lighter load of papers, or by one of
the editors as a sole duty in the last year or two of a term.  The council
referred the issue for study by the Editorial Committees of the TRANSACTIONS
and PROCEEDINGS.

4.9  MR-ZBL  Professor Steve Armentrout reported on the exploration of the
possibilities of cooperation between MR and ZBL, or even of merger of part
or all of the operation.  A report from the Executive Editor, who is a
member of the negotiating team, is attached as part of the attachment to
5.1.

4.10 SIZE OF THE COUNCIL;  The Council of 15 August 1984, considered a proposed
amendment to the bylaws with the effect of reducing the size of the Council.
The Council asked the Secretary to draft the amendment formally.  The draft can
be found in the attachment, which contains not only the proposed  amendments
but several that are further along in the process of ratification.

The proposed amendment consists of changes in Articles III and IV (except one
footnoted change in old Section 2 of Article III.)

The other indicated changes correspond to three amendments already approved
by the Council, and about to be presented to the Business Meeting.

The Long Range Planning Committee has reviewed the proposed amendment and
approves.

The Council recommended the amendment with the understanding that there would
be an additional section stating that Council members of 1 January 1985, would
serve their full terms notwithstanding.  Moreover, certain editorial changes
are required for consistency in later sections.

4.11 COMMITTEE TERMS:  The EC recommended that the terms on society committees
appointed by the President be changed to end on 31 January instead of the
previous 31 December, as at present.

The Council approved.  In so doing, they noted that there are already natural
exceptions to the 31 December date, and that exceptions to the 31 January date
may appear.

With respect to joint committees, coordination with MAA and SIAM on the same
lines should be proposed.  However, it is noted that on some joint committees
it is already the case that terms of representatives of various bodies may
differ.

4.12 CORPORATE MEMBERS.  The Council had already approved an amendment to the
bylaws that states:

Dues and privileges of corporate members shall be
established by the Council, subject to approval by
the Board of Trustees.

The amendment is one of several to be presented to the Business Meeting of
10 January 1985, for approval.

In anticipation of approval by the Business Meeting, the Council approved the
following recommendations of the EC:

1.  That dues for corporate members be set AT $2,000. For corporate members choosing certin privileges with respect to copyright, the dues be set at$3,000.

2.  That corporate members be entitled to send up to
five representatives to meetings with registration at members'
rates.

3.  That corporate members be entitled to use the
Employment Register and Employment Information in the
Mathematical Sciences (with a ceiling on the number of ads)
with no charge.

4.  That corporate members retain all privileges that
they now enjoy.

In so doing, the Council was aware of advance approval of these dues and
privileges by the BT.

4.13 PRIZES IN AUTOMATIC THEOREM PROVING:  In a letter to President Robinson,
W.W. Bledsoe, of the University of Texas at Austin, asked that the Society
explore the possibility of administering prizes in Automatic Theorem Proving
(ATP) which are funded by the Fredkin Foundation.  Professor Bledsoe proposed
that:

1.  AMS appoint and maintain an ATP Committee responsible for:

--awarding the Leibniz prize, when and if the established criterion is met.
--awarding Current and Milestone prizes when appropriate.

2.  AMS would provide for invited lectures at an Annual Meeting where these
prizes are awarded.

Att. #21 in the minutes of the EC and BT of 15-17 November 1984, contains
Professor Bledsoe's letter describing the prizes and the present method of
administration, a news release, a description of tentative criteria for the
Leibniz prize, a list of present members of the ATP  prize committee, an
announcement of the prizes, nomination statement and citations for previous
winners.  It should be noted that an arrangement such as is proposed in item
2 is already informally in effect through special sessions devoted to ATP.

The EC recommended approval of the proposal in item 1 that the AMS appoint
a committee responsible for awarding the Leibniz Prize.  The EC further
recommended that the same committee study the proposal with respect to the
Current and Milestone Prizes, and the conditions for them.  The Council
approved.

4.14 PUBLIC SERVICE AWARD.  The Joint Policy Board for Mathematics,
(AMS-MAA-SIAM) recommended the establishment of a public service award, to be
given at irregular intervals when a suitable candidate appears.  The recipient
may be a mathematician, but need not be.  The award would consist of a
certificate, a ceremony, and accompanying publicity.  The EC recommended that
this Society enter into a cooperative arrangement with SIAM and/or MAA to offer
such an award through JPBM.  The Council approved.

4.15 COMMITTEE ON HUMAN RIGHTS;  A report from Professor Joshua Leslie, a
member of the Committee, on the case of Sion Assidon is attached.

Professor Chandler Davis, Chairman, reported by letter to the Secretary that
then President Julia Robinson, following a request from the Council of 7 August
1983, persisted in her efforts to obtain an informative response about the
annulling of the doctorate of E.P. Gil'bo, a Leningrad mathematician.

Professor Herbert Clemens, a member of the Committee, had requested that the
Council consider the case of Ada Cam, Manuel Alarcon, and Douglas Fuenteseca
of the mathematics department of the University of Antofagasta in Chile.  The
first two were arrested on 29 November 1984, and the third on 1 December 1984.
As of 6 December, inquiries about the reasons for their arrest and their
on Human Rights, and the Acting President sent cables of inquiry, as did the
National Academy and other mathematical organizations.  It was learned that,
subsequently, two had been released and the third was being held for three
months in a detention center.

4.16 CONGRESSIONAL SCIENCE FELLOW;  The EC and BT recommended that  AMS
support of the fellowship be discontinued after the 1985-86 year.  This
is a position parallel to that of the MAA.  See EC/BT minute 6.1 of
November 1984.  The Council, however, referred the matter to the Science
Policy Committee with the understanding that the contribution be suspended
until the Committee reports.

OFFICERS

5.1 REPORTS OF OFFICERS.  The agenda and Budget Committee and EC had requested
annual reports from the Treasurer, Secretary, and Executive Director, for
publication in the NOTICES.  These are in draft form and are attached.

8.1 NOTICES AND ABSTRACTS.  The Council had specified that the Deputy
Executive Director shall be the managing editor of the NOTICES and of the
ABSTRACTS.  With the retirement of Lincoln Durst, the office with the title
of DED vanishes.  The Secretary proposed that the managing editor of the
NOTICES and the managing editor of the ABSTRACTS be designated by the
Secretary in consultation with the Executive Director.  The Council approved.

8.2 ABSTRACT FEES.  Professor Susan Friedlander in the attached letter
proposed the following motion:

THE \$15.00 FEE FOR ABSTRACTS OF TALKS AT AMS MEETINGS
should be abolished.

A letter from Barry Simon is attached.  The EC and BT considered the issue and
were unanimous in recommending that the abstract fee remain in place for two
years.  They proposed that the matter be examined by the EC/BT of November
1986.  However, the motion of Professor Friedlander, as a recommendation
to the Trustees, passed by a roll call vote of 14 11/30 to 12 19/30.

8.3 DISABILITY MEMBERSHIP;  The Society has a category of member, informally
known as "emeritus member", for ordinary members retired on account of age
after at least twenty years of membership.  Privileges are the same as for
ordinary member, with the exception that the BULLETIN is no longer a privilege.
Dues are zero.  Professor Meyer Jerison proposed that the same privilege be
extended to members with long term disability.  The Council recommended an
amendment to the bylaws by inserting the words "or on account of long term
disability" before the first comma in Article 8, Section 8.

8.4 DEATH OF EDWARD L. COOPERMAN:  Professor Cooperman was a physicist at
California State University in Fullerton.  He was recently killed.  Professor
Joel Smoller introduced the following resolution, presented here in the form
agreed upon after discussion.

The Council of the American Mathematical Society resolves:

1.  To express its deep regret over the killing of Professor
Cooperman;

2.  To express its support for the continuation of contact
and exchange with Vietnamese mathematicians and its hope that there
will be no further threats or violence against people involved;

3.  To recommend participation of mathematicians from Viet
Nam in the 1986 ICM in Berkeley and to encourage efforts by the ICM
organizers to insure the safety of foreign delegations.

Professor Patrick Gallagher was given the privilege of the floor to speak in
favor of the resolution.  An expanded version of his remarks is attached.  The
Secretary spoke in opposition.  His statement is attached.  The motion failed.

INFORMATION AND RECORD

2.1 MINUTES OF THE EC/BT:  The minutes of the EC and BT of 15-17 November 1984,

2.2 SUMMER OF 1987.  The EC/BT were unanimous in approving that there be a
Summer Meeting in 1987, jointly with the MAA.  This was a decision delegated
to them by the Council of 15 August 1984.

2.3 MAT. SBORNIK.  The Council of 7 August 1983, proposed that a slip be sent
with each copy of each issue of MAT. SBORNIK (as an insert or an additional
sheet in the wrapper) with the following wording:

The AMS has been cognizant of the mathematical distinction
of the journal Mat. Sbornik.  For this reason and as a service
to the mathematical community, it was happy to enter into a
contract with VAAP (All Union Agency for Authors' Rights) to
translate the journal and publish the translation.  At the same
time, the Society wishes to record the fact that it deplores
the antisemitic bias that has appeared to become part of the
editorial policy of the journal. The AMS hopes that this
aspect of policy in the management of the journal will change.

The Council proposed that the same wording appear in the catalog of
publications.  The Trustees delayed in carrying out this recommendation,
but put it into force effective 1 January 1985.  The matter is to be reviewed
in two years.

Professor A.T. Bharucha-Reid submitted a comment that was delivered to the
Secretary too late to be read to the Council.  It is attached.

2.4 INDIRA GANDHI;  With the approval of the Acting President, the Society was
one of 44 affiliates of AAAS who signed the attached telegram of condolence.

2.5 EDWARD E. DAVID, JR.  A letter from him responding to Council endorsement
of the David report, and of his personal effort is attached.

2.6 ETTORE E. INFANTE;   A letter from him responding to a Council resolution
of appreciation of his work at NSF is attached.

The council adjourned at 5:40 PM January 8, 1985.

Everett Pitcher, Secretary.


text:
# Understanding curvature

I'm trying to understand how the Frenet frame is formed from the normal and tangent on a curve in $\mathbb{R}^3$.

For a curve $\gamma (s)$ in $\mathbb{R}^3$ parameterised by arc length let $T(s) = \gamma' (s)$ be the unit tangent.

From a previous questions, I understand that from $T . T = 1$ we get $T . T' = 0$. Then this gives us that $T'(s) = \kappa (s) N(s)$ for $\kappa(s) \in \mathbb{R}$ for $N(s)$ the unit normal.

BUT surely in $\mathbb{R}^3$ there are infinitely many normals as the vector can just 'rotate' round the curve whilst remaining perpendicular to the tangent and so I can't my head around how we can deduce $T'(s) = \kappa N(s)$ just from that fact $T . T' = 0$ ?

Thanks!

-
By definition $N(s)=T'(s)/\Vert T'(s)\Vert$ –  user8268 Dec 17 '12 at 10:04
Then surely $\kappa$ is always $1$ if $T$ is the unit tangent? –  user53076 Dec 17 '12 at 10:18
no; $\kappa=\Vert T'\Vert$ –  user8268 Dec 18 '12 at 21:22

Let's briefly talk about the case of plane curves first. In that case, there are some "canonical" ways of getting normal vector $N(s)$ from a unit tangent vector $T(s)$ - one way would be rotating the tangent vector by 90 degrees anti-clockwisely. If we define normal vector $N(s)$ this way, then it did not a priori depend on $T'(s)$, but then a calculation would show that $T'(s)$ and $N(s)$ lies on the same line, and curvature can be expressed as the ratio of their lengths.
For space curves, like you said there's no "canonical" way of defining normal vector by rotation, since there are infinitely many directions to rotate. So we do NOT define normal vectors by this method. Instead, we use the clue in the case of plane curves, and define $N(s) = \frac{T'(s)}{\|T'(s)\|}$. This way we can at least get a unit vector that is always perpendicular to the unit tangent vector $T(s)$.
Would $\kappa$ be always 1 then? NO. It should actually be $\|T'(s)\|$, assuming that you started with arc length parametrization. Again this is just the ratio of lengths of $T'(s)$, and $N(s)$.
We do lose something in this case though. In the case of plane curves, $N(s)$ may not have the same sign as $T'(s)$, so the sign of curvature has meaning - whether it is locally convex or concave. This does not carry over to the case of space curves.
In the three-dimensional case there is a small proviso, namely the condition $T'(s)\ne0$ for the considered $s$ values. –  Christian Blatter Dec 17 '12 at 13:08


text:
[dev-context] tex-text (XeTeX) & trep, tlig (LuaTeX)

Hans Hagen pragma at wxs.nl
Fri Dec 7 20:01:59 CET 2007

Mojca Miklavec wrote:

> In LuaTeX you define two "features":
> - trep (3 replacements)
> - tlig (8 ligatures)
> In XeTeX, these 11 and 4 others are combined in "mapping=tex-text".
> And I would like the ligatures to be the same on both engines, so I
> would like to clone the behavior from LuaTeX in XeTeX (once you decide
> what the default should be - from your reply I understand that you
> plan to change the behaviour) and split "mapping=tex-text" into more
> separate "mappings".

well, let'd assume that hardly anyone uses tlig, so then the 8 ligs
(well, not really ligs) can be kept, saves us some some headache; and we
just don't make it a default -)

> That's fine by me, I would agree. Only: what about apostrophe (')?
> (I'm, you're, ..)

shouldn't that one be left untouched?

> I'm not sure about it, but how is it with the apostrophe in
>    I'm, don't, ... ?

I don't wanna know -)

> I don't care too much about
>     " -> right double quote
> and
>     ` -> left single quote
> esp. the first one should better be left that way, and I never use `
> (and have no idea what it's used for except in mysql and TeX source
> code)

how about a vote on the list ...

>>> U+0021 U+0060; <>;U+00A1;  ; !` -> inverted exclam
>>> U+003F U+0060; <>;U+00BF;  ; ?` -> inverted question
>>>
>>> U+003C U+003C; <>;U+00AB;  ; << -> LEFT POINTING GUILLEMET
>>> U+003E U+003E; <>;U+00BB;  ; >> -> RIGHT POINTING GUILLEMET
>> let's get rid of it
>
> Fine by me.

real weird things ... the french should use proper utf chars but well ...

>> i'd like to let caps and such go away completely for mkiv so maybe we
>> end up with xetex defs versus luatex defs;
>
> So ... perhaps Caps support needs to be rewritten in XeTeX as well :)
> I still don't understand how to get "bold italic sans caps" for
> example (just as I don't know how to get bold/bold italic math).

i think that caps (in xetex) can best be done by just defining an extra
typeface and then switch typeface (fast too)

> I use LuaTeX because:
> - sometimes Lua is really handy and you have a marvellous database for
> Unicode well integrated :), nice to inspect fonts etc.
> I use XeTeX because:
> - LuaTeX does't (or at least didn't) always work as it should. XeTeX
> "saved my life" in May because of some dirty LuaTeX bugs stopped the
> show in the middle (I already had a working copy - final PDF, and
> after minor modifications it stopped working)

that is because mkiv parses the text and if it becomes too weird,
abusive, nonsence, offending it will enter a special mode

> - others keep asking questions (and since I suggested or approved
> quite some bugs in XeTeX recently, I feel responsible for helping the
> victims :)

sure, also, for some apps xetex is more convenient (faster, and many
docs re not that demanding so default feature handling can do the job well)

> - to show off with Zapfino - I don't have it as OpenType :-)

hm, you mean that you're using the apple font model ... now who was

> In any case: what I really love about ConTeXt is that [more-or-less]
> no change is needed to compile the same (simple) document with either
> engine (and compare the result or to switch quickly if support in one
> engine is buggy).

so we should keep that property

> When using lua that is no longer true, but still, same definitions and
> similar results with both engines would be nice. (Esp. if both engines
> will be merged in the future :)

by then i'm retired and you'll have to do the coding ...

both engines fill in a niche and do that well so no problem to support
both; eventually mkii will be the xetex thing and mkiv the luatex thing

>>> The problem is that "features=default" implies "script=latn", which is
>>> not always desired. A copy of mapping=tex-text comes from tlig & trep
>>> substitution.
>> we can fall back to dflt which in practice boils down to latn
>
> That's probably better.

>>> I assume that script=latn;language=dflt;+liga;+kern; is always on by
>>> default (were needed), so basically mapping=tex-text is the only thing
>>> that really needs to be added.
>> well, i'd prefer ... only -- and --- and make anything else up to the
>> user, which means, redefining default in cont-sys if needed
>>
>>> [Iwona-Bold.otf]:script=latn;language=dflt;+liga;+kern;mapping=tex-text;mapping=tex-text;
>> two mappings?
>
> I mean: the the font is called with
>     mapping=tex-text;mapping=tex-text;
> whis doesn't make so much sense.

no, but let's not waste too much tex processing on it, doesn't hurt i guess

> I can create two new mappings, so that "tlig=yes" would then be
> "mapping=tlig" and "trep=yes" would be "mapping=trep" instead of
> "mapping=tex-text".

yes, can be part of the context zip

> We only need to agree which features are where (and you need to remove
> ?` from beginner's manual). Spanish users probably have it on keyboard
> and "others" either don't need it or will find it somehow.

ah .. manual rewriting ...

> << and >> are not needed either in my opinion.
>
>>> Some (non-latin) fonts complain when one requests non-existing features.
>> in xetex you mean?
>
> Yes. Well, LuaTeX probably "complains" as well (as in report >> load
> otf: warning: Warning: Glyph 1423 is named fi which should mean it is
> mapped to
>  Unicode U+FB01, but Glyph 207 already has that encoding. etc.) -
> complaining is a good thing in general, it only means "please do not
> use latin script for non-latin fonts". And "features=default" should
> prbably not force latin. (Then it should at least be
> "features=default-latin" or somothing similar.)

luatex does not care because it does nothing with features, it loads the
file ...

and, mkiv nicely checks for the features supported, so if you ask for
mkmk in latin modern it will just ignore it (part of setting up the
processing sequences and caches), mkiv does not even bother you with a
message

>>> Also, it might be handy to be able to define
>>>     \definetypeface[basic][rm][Xserif][whatever][script=arab,language=...]
>>> (for now forget that one, interface needs to be extended once and properly).
>>>
>>>
>>>     \definefontsynonym[a][file:Iwona-Bold.otf][mapping=tex-text]
>>> doesn't work, so the only way seems to be
>>>     \definefontfeature[xetex][mapping=tex-text]
>>>     \definefontsynonym[a][file:Iwona-Bold.otf][xetex]
>>>
>>> I have tried to use
>>>     \definefontfeature[xetex][mapping=tex-text]
>>>     \definefontfeature[caps][+smcp]
>>>     \definefontsynonym[a][file:Iwona-Bold.otf][features={xetex,caps}]
>>> but that didn't work.
>> indeed, handling comma separated lists is too slow there .. ok, we can
>> do it for xetex and in luatex use lua for it ... or i could hash the
>> commalist itself ... needs a bit of thinking but eventually we need to
>> be able to combine features (this even more points into a separate
>> definition file for xetex)
>
> That's up to you. I don't know anything about internals here.

you can't fool me ..

-----------------------------------------------------------------


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## Triangle of the Stirling Numbers of the Second Kind

The Stirling numbers of the second kind $\left\{\begin{array}{c}n\\r\end{array}\right\},$ where $n$ and $r$ are natural numbers, are named after James Stirling (1692 – 1770). According to the corresponding recursive formula, they form a triangular scheme, in analogy to the Pascal’s triangle for binomial coefficients. For the first $10$ values of $n$ this scheme is

$\begin{array}{r|rrrrrrrrrr} n&\left\{\begin{array}{c}n\\0\end{array}\right\}&\left\{\begin{array}{c}n\\1\end{array}\right\}&\left\{\begin{array}{c}n\\2\end{array}\right\}&\left\{\begin{array}{c}n\\3\end{array}\right\}&\left\{\begin{array}{c}n\\4\end{array}\right\}&\left\{\begin{array}{c}n\\5\end{array}\right\}&\left\{\begin{array}{c}n\\6\end{array}\right\}&\left\{\begin{array}{c}n\\7\end{array}\right\}&\left\{\begin{array}{c}n\\8\end{array}\right\}&\left\{\begin{array}{c}n\\9\end{array}\right\}&\left\{\begin{array}{c}n\\10\end{array}\right\}\\ \hline 0&1\\ 1&&1\\ 2&&1&1\\ 3&&1&3&1\\ 4&&1&7&6&1\\ 5&&1&15&25&10&1\\ 6&&1&31&90&65&15&1\\ 7&&1&63&301&350&140&21&1\\ 8&&1&127&966&1701&1050&266&28&1\\ 9&&1&255&3025&7770&6951&2646&462&36&1\\ 10&&1&511&9330&34105&42525&22827&5880&750&45&1\\ \end{array}$

Note that empty entries in this table are actually $$0$$’s.

| | | | created: 2020-04-04 12:29:03 | modified: 2020-04-10 18:36:53 | by: bookofproofs | references: [8404]


text:
# How do you use synthetic division to find the factors of f(x)= x^4 -x^3 -19x^2+49x-30?

Aug 30, 2015

If you can spot a factor of $f \left(x\right)$ then you can use synthetic division to divide by that factor to give a simpler polynomial to factor.

Hence we can find:

$f \left(x\right) = \left(x - 1\right) \left(x + 5\right) \left(x - 2\right) \left(x - 3\right)$

#### Explanation:

First note that $f \left(1\right) = 1 - 1 - 19 + 49 - 30 = 0$, so $\left(x - 1\right)$ is a factor of $f \left(x\right)$

Use synthetic division to divide $f \left(x\right)$ by $\left(x - 1\right)$:

So $f \left(x\right) = \left(x - 1\right) \left({x}^{3} - 19 x + 30\right)$

Notice that ${\left(- 5\right)}^{3} - 19 \cdot \left(- 5\right) + 30 = - 125 + 95 + 30 = 0$, so $\left(x + 5\right)$ is also a factor of $f \left(x\right)$.

Divide ${x}^{3} - 19 x + 30$ by $\left(x + 5\right)$ using synthetic division - not forgetting to specify the coefficient $0$ of the ${x}^{2}$ term:

So $f \left(x\right) = \left(x - 1\right) \left(x + 5\right) \left({x}^{2} - 5 + 6\right)$

By this stage you can probably spot that ${x}^{2} - 5 + 6 = \left(x - 2\right) \left(x - 3\right)$ to complete our factorisation:

$f \left(x\right) = \left(x - 1\right) \left(x + 5\right) \left(x - 2\right) \left(x - 3\right)$


text:
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# Calling all Tepper applicants for 2008

Author Message
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Director
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Re: Calling all Tepper applicants for 2008 [#permalink]  01 Jan 2008, 22:43
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Re: Calling all Tepper applicants for 2008 [#permalink]  02 Jan 2008, 13:33
So Tepper would like official transcripts, and the deadline is fast approaching. Most of the other schools i'm applying to didn't require "official ones". I was able to input my own transcript and pending admission an "offical" would need to be submitted to ensure they match up... So i don't have a bunch of "offical" transcripts lying around, sealed and ready to mail off. I might get lukcy and get one quick enough that I can overnight it to meet the upcoming deadline. Do you think this is a deal breaker for applying R2?? I really want to get my app off in R2 and not have to go through R3, which i hear is always much more difficult to gain admission. Has anybody talked with CMU about this at all, maybe somebody had this happen in RD1? I have an official one i could scan in, but i can't send that one, myself, it's not sealed and clearly shows the date i recieved it (back in 2003 for my current employeer to see). I have put a request for officals to go to CMU and to my home address from my school, but the office which supplies them only came back to work from the holidays today... Just not sure if it's enough time.

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Posts: 707
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Re: Calling all Tepper applicants for 2008 [#permalink]  02 Jan 2008, 13:53
thunderdogg wrote:
So Tepper would like official transcripts, and the deadline is fast approaching. Most of the other schools i'm applying to didn't require "official ones". I was able to input my own transcript and pending admission an "offical" would need to be submitted to ensure they match up... So i don't have a bunch of "offical" transcripts lying around, sealed and ready to mail off. I might get lukcy and get one quick enough that I can overnight it to meet the upcoming deadline. Do you think this is a deal breaker for applying R2?? I really want to get my app off in R2 and not have to go through R3, which i hear is always much more difficult to gain admission. Has anybody talked with CMU about this at all, maybe somebody had this happen in RD1? I have an official one i could scan in, but i can't send that one, myself, it's not sealed and clearly shows the date i recieved it (back in 2003 for my current employeer to see). I have put a request for officals to go to CMU and to my home address from my school, but the office which supplies them only came back to work from the holidays today... Just not sure if it's enough time.

Where do you work? And what employer wanted to see your transcript?

You should post this question on their message board. In the meantime, I would send out the official transcripts now. Most schools have an overnight delivery. It'll take them a week or so to match all the documents.
Intern
Joined: 11 Dec 2007
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Re: Calling all Tepper applicants for 2008 [#permalink]  02 Jan 2008, 14:21
gmatclb wrote:

Where do you work? And what employer wanted to see your transcript?

You should post this question on their message board. In the meantime, I would send out the official transcripts now. Most schools have an overnight delivery. It'll take them a week or so to match all the documents.

I don't have an official one, or else i would have mailed it quite some time ago. I work for the government. they have paybands that can be dependent (to start) on your collegiate GPA and they want to see an official transcript to reflect that.
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Re: Calling all Tepper applicants for 2008 [#permalink]  02 Jan 2008, 14:34
thunderdogg wrote:
gmatclb wrote:

Where do you work? And what employer wanted to see your transcript?

You should post this question on their message board. In the meantime, I would send out the official transcripts now. Most schools have an overnight delivery. It'll take them a week or so to match all the documents.

I don't have an official one, or else i would have mailed it quite some time ago. I work for the government. they have paybands that can be dependent (to start) on your collegiate GPA and they want to see an official transcript to reflect that.

Great advice, to check out the Tepper Online Community message boards. Looks like they've already answered my question and it would seem that i should be okay. So any others of you like me out there, you should be okay too. i've posted the link below and the Tepper Rep's response:

"Thank you for the posts. We understand that it will take some time for all of your supporting documents to arrive (just as I'm sure you all understand it takes us a bit of time to process those supporting documents ). You both would be just fine if your supporting documents reach us a couple days after the round 2 deadline, so long as your online application has been submitted by that time."
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Re: Calling all Tepper applicants for 2008 [#permalink]  06 Jan 2008, 09:55
Hi Guys,
R2 Applicant here. A rather late question. For essay 3, we have 2 questions to answer and the suggested length is 2 pages total. Is it double-spaced or single-spaced ?
Thanks,
Vij
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Re: Calling all Tepper applicants for 2008 [#permalink]  06 Jan 2008, 10:35
vij, I used the same format as the first two essays - 11pt. font, double spaced. Each essay was one page long and I started each on a new page. Good luck!
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Re: Calling all Tepper applicants for 2008 [#permalink]  07 Jan 2008, 12:21
Anybody else send a RD2 application off?
I officially submitted mine yesterday.
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Re: Calling all Tepper applicants for 2008 [#permalink]  07 Jan 2008, 12:42
Submitted for Tepper R2.

I realized at the very last minute from gmatclub postings that my essays needed to be double-spaced. Phew.
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Re: Calling all Tepper applicants for 2008 [#permalink]  07 Jan 2008, 14:16
thunderdogg, wisepig, and vij - Good Luck with R2
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Re: Calling all Tepper applicants for 2008 [#permalink]  07 Jan 2008, 18:50
I took the liberty of creating the list for R2, please feel free to make any updates and\or add the dates that you submitted the application.
R2
vij101 - Submitted
wisepig - Submitted
thunderdogg - Submitted
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Re: Calling all Tepper applicants for 2008 [#permalink]  07 Jan 2008, 20:11
Just finished up all my apps so I have some time to spend on this board I am a CMU alum (BS Engineering) and lived in Pittsburgh for a while. I might not be able to answer some specific questions about the MBA program but if you would like to know about Pittsburgh, weather, restaurants, and what to do there etc. then I will try to answer your questions. Just a way for me to give back to the forum and the people here.

Best of luck to those applying!
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 05:59
Thanks BCC! I'll be sure to ask if I end up at Tepper. For starters, what area would be best for a single guy to live in?
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 06:29
Is it possible to live far enouogh away from school to walk?

And how much rent would that be for a studio apartment? 2 bed 1 bath apart?
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 14:13
Now that I am almost done with my R2 apps, I have started thinking about what I should do to get off the CMU WL.

I tried to get some talk time with Gwen to get some idea of what material they might be looking for, but she sent an auto-response saying "no time to talk....just send additional stuff to the waitlist e-mail ID."

So I started looking around to see where I could have messed up. I am pretty sure it is my essays...since my GMAT is 710+, decent work experience and leadership opportunities, good interview...although I think I rambled a bit.

The question is, Do I just send a refined goals essay (short-term/long-term goals, why CMU)
alone or send updated essays for all the questions.

Sometime later, I am planning to send my TOEFL score (which I had not reported earlier) in which I scored a 30/30 in the writing section to, perhaps, offset my 4.5AWA.
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 14:26
I wouldn't send in updated essays. That would be too much and look so fake.

I also wouldn't change any goals. But I would send in another essay to reiterate your interest and say maybe that you found out some more stuff that is a great fit for your goals.

BTW, I was waitlisted at another school last year and ultimately didn't get in, so take my advice for what it's worth.
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 14:27
sperinko, I would only send the information that adds to the strength of your application. If you feel that it is only the goals essay that requires a little refinement then I would only send the goals essay. If all of them could be made better then I would send all in. Sending your TOEFL in will definitely help. Also, if you have any pertinent information that was not addressed in the essays, I would send that in.
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 16:52
Thanks guys.
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Re: Calling all Tepper applicants for 2008 [#permalink]  08 Jan 2008, 19:56
clobo wrote:
Thanks BCC! I'll be sure to ask if I end up at Tepper. For starters, what area would be best for a single guy to live in?

np For single guys/gals, I would say South Side and Shadyside are good locations. There are a lot of bars and restaurants in both areas and so there will be plenty of chances to meet new people.

The drawback about South Side though is it is not within walking distance to school. So you probably have to own a car. Shadyside (unlike the name, the place is really not shady at all!) is a more upscale area than South Side so in general the rent will be higher than that of South Side even though there are a lot of developments currently going on in South Side. You can walk to school from Shadyside.

Squirrel Hill (squirrel? ) is also a good area but maybe not as much action as South Side or Shadyside.

gmatclb wrote:
Is it possible to live far enouogh away from school to walk?

And how much rent would that be for a studio apartment? 2 bed 1 bath apart?

Yes. There are tons of apartments 10-20 mins by walking to the school. You probably don't want to live too far away for your commute if you don't own a car. You can take public transportation but I never did - not that it is bad but I rather walk / catch a ride / drive. Of course your rent will depend on how nice of a place you want. I would say you can get a decent studio for $600/month. For a decent 2 bed and 1 bath I would say about$1000/month. The rent there is not expensive.

For more economical places try Amberson Apartments (walking), Fifth Neville (walking), Shadyside Commons (walking)
For very nice places try South Side Lofts (not walking), The Lofts on Centre (semi walking), The Flats @ South Side Works (not walking) http://www.southsidepgh.com/live_and_work/rent.php

Hope this helps. Let me know if you guys have more questions.
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Re: Calling all Tepper applicants for 2008 [#permalink]  14 Jan 2008, 18:25
I got my Tepper package today. Neat!! They sent me a pen with my name and Tepper Business School engraved on it.
Re: Calling all Tepper applicants for 2008   [#permalink] 14 Jan 2008, 18:25

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text:
# Proof that Frechet-metric generates same topology as the semi-norms

Given a countable family of semi-norms $p_i$, we can define a metric

$d(f,g) = \sum \limits_{i=0}^{\infty} 2^{-i} \frac{ p_i(f-g) }{ 1 + p_i(f-g) }$

We have the locally convex topology induced by the semi-norms as above, as well as the topology induced by the metric.

How does the proof work to show their equality?

I am known to the proof of Rudin (Functional analysis), but utilizes a different metric:

$d(f,g) = \max \limits_{i \in \mathbb N} 2^{-i} \frac{ p_i(f-g) }{ 1 + p_i(f-g) }$

Whereas the proof for this metric is fairly easy - you can handle value of the sequence on its own - i do not see how a similar proof might work for the first metric.

One guess would be to show equivalence of both metrics, but I don't see even that, as on $l^1$, the sum-norm-topology is strictly finer than the max-norm-topology.

Can you help me?

-
@user3557: Isn't there a $1/2^i$ missing in the definition of $d$? – Jonas Teuwen Nov 16 '10 at 18:27
Your title got cut off. – Nate Eldredge Nov 16 '10 at 18:59
Thank you. Corrected title and formulae. – shuhalo Nov 16 '10 at 19:09

To prove equivalence of the topologies you can use the fact that both topologies are characterized by their convergent sequences. This is true in any first countable space. Metric spaces are first countable, and the topology induced by a countable family of seminorms is first countable. Here, since you have translation invariance, it is enough to check sequences converging to 0 (although this simplification is not essential). That is, you can show that if $x_1,x_2,\ldots$ is a sequence in your vector space, then $d(x_n,0)\to 0$ as $n\to\infty$ if and only if for all $i$, $p_i(x_n)\to0$ as $n\to\infty$. For the left to right direction, you can use the fact that $\frac{p_i}{1+p_i}\leq 2^id$. For the other direction you can first bound the tail, then work with the remaining finitely many terms.

There is no actual need to consider sequences. You basically want to show that the identity map is a homeomorphism, and this reduces to showing it is continuous at 0 in both directions, which in any case involves the same estimates you would make when working with sequences. (In other words, considering first countability a priori isn't necessary. If you really want to, you could instead work with nets.) Note that Jonas T's answer suggests a complementary approach (that was given around the same time), working directly with neighborhoods of $0$ instead of with sequences.

-
Yes, this works as well, I have added in the edit another way of showing this (one you did not mention). – Jonas Teuwen Nov 16 '10 at 19:00
Could you maybe give me a pointer how two first-countable topologies are equal if their convergent sequences coincide? – shuhalo Nov 17 '10 at 0:41
If $X$ is first countable and $A\subseteq X$, then $x\in X$ is in the closure of $A$ if and only if there is a sequence in $A$ that converges to $x$. This implies that closed sets (and therefore open sets) are determined by convergent sequences. – Jonas Meyer Nov 17 '10 at 1:09

Assuming that you forgot a $2^{-i}$ in the definition of $d$ we argue as follows:

We know that a Hausdorff TVS is metrizable iff zero has a countable neighborhood basis. If this is true then the topology is even generated by a translation invariant metric.

Another theorem states that a topology on a TVS is locally convex iff it is generated by a family of seminorms.

So, if we combine these and use that your metric is translation invariant, we obtain the result.

However, this is indirect, probably there is a direct proof.

Edit: Oh, I don't think the direct proof is so hard, I will see if I have time to add it. You should just observe that the neighborhood $\{x : p_n(x) \leq 2^{-k}\}$ contains the set $\{x : |x| \leq 2^{-m-k-1} \}$ and that the set $\{x : |x| \leq 2^{-k} \}$ contains the neighborhood $\{x : p_{k + 1}(x) \leq 2^{-k - 2} \}$. Where $|x| := d(x,0)$.

-
There isn't necessarily a norm that induces the topology, and I'm not sure what you mean by $|x|$. If you mean $|x|=d(x,0)$, then it is not a norm, and if not, then what is $|x|$? – Jonas Meyer Nov 16 '10 at 19:35
Meyer: Oops, yes $|x| = d(x,0)$, and it is indeed no norm, but I think it still works (you don't use anyway that it is a norm). – Jonas Teuwen Nov 16 '10 at 20:04
T: Thanks for clarifying about the "norm". In the last paragraph $m=n$, right? And I don't follow "$\{x:|x|\leq2^{-k}\}$ contains the neighborhood $\{x:p_{k+1}(x)\leq2^{-k-2}\}$." For the second set did you want something like $\{x : p_i(x) \leq 2^{-k - 2},1\leq i\leq k \}$? (I say "something like" because I'm not thinking through the details, just trying to give something related to what you have but with control on $|x|$.) – Jonas Meyer Nov 16 '10 at 23:10


text:
# If the Circumference of a Circle and the Perimeter of a Square Are Equal, Then - Mathematics

MCQ
Sum

If the circumference of a circle and the perimeter of a square are equal, then

#### Options

• area of the circle = area of the square

• (area of the circle) > (area of the square)

• (area of the circle) < (area of the square)

• none of these

#### Solution

Area of the circle > Area of the square

Let r be the radius of the circle.

We know:

Circumference of the circle

= 2πr

Now,

Let a be the side of the square.
We know:
Perimeter of the square = 4a

Now,

2pi"r"=4"a"

=>"r" = (4"a")/(2pi)

∴ Area of the circle = πr2

=pixx((4"a")/(2pi))^2

=pixx(16"a"^2)/(4pi^2)

=(4"a"^2)/pi

=(4xx7"a"^2)/22

=(14"a"^2)/11

Also,

Area of the square = a2

clearly,(14"a"^2)/11 > "a"^2

∴ Area of the circle > Area of the square

Is there an error in this question or solution?

#### APPEARS IN

RS Aggarwal Secondary School Class 10 Maths
Chapter 18 Area of Circle, Sector and Segment
Multiple Choice Questions | Q 13 | Page 847


text:
[ Original Paper ]
Journal of Advanced Marine Engineering and Technology - Vol. 46, No. 6, pp.378-386
ISSN: 2234-7925 (Print) 2765-4796 (Online)
Print publication date 31 Dec 2022
Received 23 Oct 2022 Revised 07 Nov 2022 Accepted 02 Dec 2022

# Design of a superconducting magnet system using a cryogenic magnetic coupler for contactless current charging

Young Jin Hwang1 ; Jong-Eon Park
1Associate Professor, Division of Electronics and Electrical Information Engineering, Korea Maritime & Ocean University, Tel: +82-51-410-4411 yjhwang@kmou.ac.kr

Correspondence to: Assistant Professor, Division of Navigation Convergence Studies, Korea Maritime & Ocean University, 727, Taejong-ro, Yeongdo-gu, Busan 49112, Korea, E-mail: jepark@kmou.ac.kr, Tel: +82-51-410-4244

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

## Abstract

General-purpose electromagnets are widely used to measure and analyze the magnetic properties of various materials. The performance of these electromagnets depends mainly on the magnitude and uniformity of the magnetic field. As the allowable current density of copper wires, which are generally used in these electromagnets, does not exceed 5 A/mm2 at room temperature, it is difficult to generate a high magnetic field, and water-cooling is required to mitigate the temperature rise at higher current densities. Hightemperature superconductors (HTSs) have significantly higher current-carrying capacity than copper; thus, general-purpose electromagnets based on HTS coils can generate higher magnetic fields and are more compact than conventional electromagnets that use copper coils. However, given that the use of a superconducting joint is not feasible in an HTS coil, it should always be physically connected to an external power source at room temperature for current charging. This places a large thermal load on the cryogenic cooling system for the HTS coil, which may decrease the operating stability of the coil over time. This study proposes a generalpurpose HTS electromagnet based on a contactless current-charging method. The proposed electromagnet is composed of a ferromagnetic iron yoke, HTS coils, cryogenic cooling system, and contactless charging system. The HTS electromagnet was designed and analyzed using finite element analysis coupled with electrical circuit analysis. The HTS electromagnet offers better field performance than a conventional general-purpose electromagnet even though it is only half the weight of the latter. We expect that the proposed HTS electromagnet can be applied in fields that require excellent magnetic field performance but also have size limitations.

## Keywords:

Contactless charging method, Ferromagnetic iron yoke, Finite element analysis, General-purpose electromagnet HTS coil

## 1. Introduction

General-purpose electromagnets are widely used to measure and analyze magnetic properties, such as magnetic hysteresis and magnetic susceptibility [1]-[4].

The performance of general-purpose electromagnets depends mainly on the magnitude and uniformity of the magnetic field in the sample space. Generally, copper coils are used for generating a magnetic field in conventional electromagnets. However, as the allowable current density of copper wires does not exceed 5 A/mm2 at room temperature (RT), conventional electromagnets are not feasible for generating a high magnetic field in a limited volume.

Superconductors are known to have very high current-carrying capacities compared to normal conductors, such as copper and aluminum. In particular, high-temperature superconductors (HTSs) have a very high current-carrying capacity compared to normal conductors, and they are thus widely studied for application in ultra-high field magnets for physical property measurements [5]-[7]. HTS-based electromagnet systems are expected to generate higher magnetic fields in a compact size compared to conventional electromagnets that use copper coils. Therefore, several studies have attempted to apply HTS coils to general-purpose electromagnets [7]-[9]. In these studies, the HTS coil is current-charged in a power-driven mode, where the HTS coil is physically connected to an external power supply at RT. This places a large thermal load on the cryogenic cooling system for the HTS coil, which can thereby decrease the operating stability of the coil. As superconducting joints cannot be realized using the HTS tape, HTS coils, in contrast to low-temperature superconductor coils, cannot be operated in constant current mode.

Furthermore, the HTS coil must be physically connected to the external power supply even after its operating current attains the target current.

This issue has been addressed by investigating techniques, such as flux pumps and joint-less HTS coils [10]-[11]. However, these are limited to small-capacity magnets; thus, a technique that can be applied universally needs to be developed.

The present study proposes a new type of general-purpose HTS electromagnet based on a contactless current-charging method using a magnetic coupler. The proposed model is composed of HTS coils, a ferromagnetic iron yoke, a cryogenic cooling system, and a magnetic coupler. The iron yoke has an Hframe configuration and uses cobalt iron, which has superior magnetic properties. The HTS coils are designed using IcB data for a commercial 2G HTS tape. To do this, the dependency of the critical current of the HTS tape on an external magnetic field is measured. A finite element analysis model coupled with an electrical circuit model is developed to evaluate the current-charging characteristics. The results demonstrate the feasibility of the proposed general-purpose HTS electromagnet.

## 2. Configuration

Figure 1 shows the proposed general-purpose HTS electromagnet. It consists mainly of four parts: a pair of HTS coils, cryogenic cooling system, H-frame iron yoke, and current charging system. Although the shape of the H-frame iron yoke is identical to that of a conventional electromagnet, the diameter of the excitation coil can be reduced because of the use of the HTS coils, meaning that the size of the iron yoke can be reduced. Each HTS coil consists of double-pancake coils, and the pole is surrounded by the HTS coils. The proposed configuration does not require a refrigerant and is compact.

Conceptual structure of a general-purpose HTS electromagnet

In the proposed model, contactless charging method is used instead of the conventional power-driven mode to charge current to the HTS coil. A schematic of a contactless current charging system using a magnetic coupler is shown in Figure 2. In the charging system, a compensation network is necessary to reduce the leakage inductance because the transmitter (TX) coil and receiver (RX) coil are loosely coupled. In this study, a series–series compensation network was used for the proposed electromagnet system, as this network facilitates power transfers even with a large air gap.

Schematic of a contactless current charging system for the HTS coil

## 3. HTS electromagnet and magnetic coupler design

This study aims to propose an HTS electromagnet model that is more compact than a conventional electromagnet and has equal or better performance. Therefore, we proceed with the initial design based on model 3474 of GMW Associates, a representative general-purpose electromagnet manufacturing company. The model 3474 electromagnet has the following specifications: a pole diameter of 250 mm, H-frame iron yoke dimensions of 920 mm width × 636 mm thickness × 864 mm height, a weight of 1,800 kg, and a maximum magnetic field density of 3.5 T with a 10 mm gap distance and 25 mm pole face. The pole diameter of the proposed HTS electromagnet equals that of the model 3474 electromagnet but has smaller coil and iron yoke because of the use of the HTS coil.

To design the HTS electromagnet, an electromagnetic analysis coupled with an electrical circuit analysis was performed using commercial finite element modeling software and an electromagnetic analysis tool for solving Maxwell equations to determine the magnetic field within the model. Figure 3 shows the 3D model that includes the HTS coils, H-frame iron yoke, the pair of poles, and the air domain. The cryocooler and cryostat for HTS coils, and the underframe were ignored for simplification of the model. In the case of air domain, a sub-air domain was built to analyze field distribution more precisely in a region between pole faces. For an H-frame iron yoke and poles, a ferromagnetic material with the magnetization curve shown in Figure 4 was used.

3D geometry of the proposed general-purpose HTS electromagnet model for the finite element analysis

Magnetization curve (B–H curve) of the ferromagnetic material for the proposed HTS electromagnet

In addition, mesh convergence tests were performed to enhance the reliability of this simulation model, and the results are shown in Figure 3. The magnetic energy and flux linkage in the coil according to the number of nodes in the analysis model were simulated, and these values converged to 171.48 J and 14.3 Wb, respectively, at the number of nodes greater than 810,000.

Mesh convergence test result: magnetic energy & flux linkage vs. the number of nodes

The simulation was iterated while maintaining the pole diameter and changing the dimensions of the iron yoke and the magnetomotive force (MMF) of the HTS coil. Based on the simulation results, the height, width, and thickness of the iron yoke were designed to be 670, 750, and 400 mm, respectively, and the required MMF of the HTS coil was found to be 9.6 kA·t. When the HTS coil is excited with the maximum MMF, the magnetic field density at a gap distance of 20 mm and a pole face of 20 mm exceeds 3.8 T. Specifications of the proposed electromagnet model and commercial model 3474 electromagnet are compared in Table 1.

Comparison of specifications between the proposed electromagnet and commercial model 3474 electromagnet

Compared to the model 3474 electromagnet, the proposed electromagnet model has a 40% smaller overall volume and 20% higher field density at a 20 mm gap distance. In addition, the target specification of the field uniformity was determined so that the value could be further improved over the specification presented in model 3474. Based on the designed specifications of the electromagnet, an HTS coil capable of generating a target MMF was designed.

To design the HTS coils, the critical current degradation characteristics must be parameterized with the external magnetic fields and operating temperature. Figure 6 shows the measured dependency of the magnetic field on the critical current of the 2G HTS wire. The test equipment shown in Figure 7 was used for this experiment. The test equipment is a cryogen-free superconducting property measurement system (SPMS) which was developed by the Korea Electrotechnology Research Institute in 2011 [12]-[13]. The system consists of a racetrack-type background magnet, HTS current lead, and cryocooler. The background magnet has a split-pair winding configuration, and RT bore of the magnet has dimensions of 38 mm (length) × 134 mm (width) × 360 mm (depth). In addition, the field uniformity of SPMS is lower than 1% within the virtual sample region of 15 mm (width) × 100 mm(depth). The specifications of the test equipment are summarized in Table 2 [12].

Dependency of the critical current on the operating temperature and external magnetic field

System used for measuring the critical current of the HTS tape according to the operating temperature and external magnetic field

Specifications of the test equipment

After installing the HTS tape in the sample holder of the SPMS, the sample holder was inserted into the RT bore of the SPMS. Using a cryocooler connected to the sample holder, HTS tape was cooled to the target temperature. After that, the critical current of the HTS tape was measured using the 4-point method while gradually increasing the current flowing through the HTS tape using the power supply. In this experiment, the magnitude of the external magnetic field and the cooling temperature of the HTS tape were used as variables, and the critical current was measured while increasing the external magnetic field from 0.5 to 3 T at the field orientation (θ) of 90°. The IcB curve data at 40 K in Figure 6 was used to determine the maximum operating current of the HTS coil [14]-[15]. The specifications of the designed HTS coil are shown in Table 3.

Specifications of the HTS Coil

In this study, a TX coil and an RX coil for contactless current charging of the HTS coil were designed. Considering the overall size of the HTS electromagnet system, planar circular-type geometries of the TX coil and RX coil were selected. The two coils were designed based on the coil geometry parameters and inductance. The fundamental geometric specifications of the two coils were roughly designed to enhance the coupling coefficient, and then the two coils were designed in more detail via a finite element analysis.

The compensation capacitances at the transmitter side and receiver side in the series–series compensation topology were calculated using Equation (1) [16]-[17].

 (1)

where f0 is the operating frequency, which was set to 10 kHz in this study. The gap distance between the TX and RX coils was determined to be 30 mm considering the multi-layer insulation, which is used for radiation shielding of the cryogenic system. Table 4 shows the specifications of the magnetic coupler, including the TX coil, RX coil, and compensation network.

Specifications of the magnetic coupler

The contactless current-charging method used in the proposed electromagnet model has lower power transfer efficiency than the conventional power-driven method. However, the purpose of the proposed method is to improve the thermal stability of the HTS magnet system, which may be achieved by reducing the thermal load caused by the current lead. Therefore, in this study, the thermal load of the HTS magnet system under the power-driven method was compared to that under the contactless currentcharging method. The thermal loads of the HTS magnet system were calculated using Equations (2)-(4) [18]-[19].

 ${Q}_{CL}={\left(\sqrt{\frac{{P}_{CL}{L}_{CL}{I}^{2}}{{A}_{CL}}}-\sqrt{\frac{{k}_{CL}{A}_{CL}\left({T}_{H}-{T}_{L}\right)}{{L}_{CL}}}\right)}^{2}+2I\sqrt{{\rho }_{CL}{k}_{CL}\left({T}_{H}-{T}_{L}\right)}$ (2)
 ${Q}_{K}=\frac{{N}_{L}{A}_{L}{K}_{L}\left({T}_{H}-{T}_{L}\right)}{{L}_{L}}$ (3)
 ${Q}_{R}=\sigma \left({T}_{H}^{4}-{T}_{L}^{4}\right)/\left(\frac{1-{\epsilon }_{H}}{{\epsilon }_{H}{A}_{H}}+\frac{\frac{1}{{\epsilon }_{L}}+\frac{2N}{{\epsilon }_{N}}-N}{{A}_{L}}\right)$ (4)

where, QCL and QK are the conduction thermal loads by a pair of current leads and the leading-in tubes for the signal lines, respectively; QR is the thermal load by the radiation heat transfer from the vessel wall. The thermal load calculated from the equations was 4.87 W for the power-driven method and 1.94 W for contactless-charging method. Therefore, it is expected that the thermal load of the HTS magnet system can be reduced by more than 60% using the proposed model. Although the eddy-current loss due to leakage flux between Tx and Rx coils was not considered in this calculation, the use of a non-metallic material, such as glass fiber reinforced polymer for a part of the cryostat between the Tx and Rx coils is expected to reduce the effect.

## 4. Results and discussion

In this study, we analyzed the charging characteristics of the HTS coil by the contactless current-charging method. The excitation current to the HTS coil was supplied through a full-wave rectifier and a smoothing capacitor. Figure 8 shows the circuit diagram, including the full-wave rectifier and compensation network. Figure 9 shows the simulation results for the charging current in the transmitter current condition of 30 A and 10 kHz.

Circuit diagram having the compensation network and full-wave rectifier

Simulation results for the current charging characteristics of the HTS coil by contactless current-charging method

The current in the HTS coil was charged with an operating current of 30 A for approximately 200 ms, which is much faster than the normal charging speed of an HTS coil. Therefore, to stably charge the HTS coil, the current must be gradually increased on the transmitter side. After the current-charging of the HTS coil, the magnetic field distribution in the sample space was analyzed. Figure 10 shows the simulation results for the magnetic field distribution in the HTS electromagnet. The results indicate that the H-frame iron yoke and pair of poles play a significant role in guiding and focusing the magnetic field. Given that the H-frame iron yoke and poles consist of a ferromagnetic material with very high magnetic permeability, the magnetic field can flow across both pole tips through the air gap.

Magnetic field distribution at a 20 mm gap distance, 75 mm pole face, and 2.4 kA·t

Figures 11(a) and (b) demonstrate that the HTS electromagnet provides spatial field uniformity of <2% over a 10 mm diameter of the spherical volume (DSV). The spatial field can be improved by increasing the area of the pole face. However, this weakens the concentration of the magnetic field and the air gap field density [18]. Compared to the conventional general-purpose electromagnet, the proposed HTS electromagnet model can generate a relatively high magnetic field while maintaining excellent spatial field uniformity.

Magnetic field distribution in the 10 mm DSV of the air gap at a 20 mm gap distance, with 25 mm pole face, and at 8 kA·t: (a) 3D view, and (b) 2D view of the center

Figure 12 shows the distribution of the magnetic field in the y-axis direction. The graphs plot the magnetic field density from y = -80 mm to y = 80 mm in steps of 0.16 mm. Cut lines of the magnetic field at different positions are given, showing relatively good field homogeneity at r < 10 mm. The inhomogeneity of the magnetic field can be reduced by increasing the diameter of the pole face. Figure 13 shows the air gap field densities with the MMF and diameter of the pole face. The magnetic field densities in the air gap increase linearly to 2.0 T as the operating current increases. However, the magnetic field density in the oversaturated region is significantly limited by the coil current. This occurs because the leakage flux in the saturation region increases as the MMF stemming from the coil current increases. In addition, the maximum magnetic field of the HTS electromagnet with a 25 mm pole face and 5 mm gap distance is limited to 4.35 T or less based on the critical current of the HTS coil.

1D distribution of the magnetic field in the air gap between (0, -80, 0) mm and (0, 80, 0) mm

Central magnetic field density in the air gap according to the diameter of the pole face, and the MMF of the HTS coil: (a) 1 mm gap distance, (b) 5 mm gap distance, (c) 10 mm gap distance, and (d) 20 mm gap distance

## 5. Conclusion

In this study, a general-purpose electromagnet utilizing a ferromagnetic iron yoke and HTS coils charged using a contactless charging method was designed, and the current charging characteristics of the HTS coil and the magnetic field performance were assessed using finite element analysis coupled with electrical circuit analysis. To design the HTS coil, the dependency of critical current on the external magnetic field of 2G HTS tape was experimentally evaluated. Based on the simulation results, we confirmed that the HTS coil in the proposed HTS electromagnet model can be excited using the contactless charging method. In addition, the designed HTS electromagnet provided variable magnetic fields of 0–3 T at a 20 mm gap distance and 100 mm pole face with field homogeneity of better than 2% within a 10 mm DSV. Compared to the conventional electromagnet, the proposed model is more compact and has higher field performance, implying greater suitability in situations affected by size limitations. Thus, this design is expected to be used in nuclear magnetic resonance equipment, superconducting synchronous motors for electric propulsion, and superconducting synchronous generators for offshore wind power as it can reduce the thermal load caused by current conductors. However, this study only considers simulation results, whereas the actual model has not yet been fabricated and assessed. Therefore, in future research, we plan to verify the proposed method through actual model fabrication and testing.

## Acknowledgments

This work was supported by the Korea Maritime and Ocean University Research Fund in 2021, and a grant from the National Research Foundation of Korea (NRF) funded by the Korean government (MSIT) (No. 2021R1F1A1052767).

### Author Contributions

Conceptualization, Y. J. Hwang and J. E. Park; Methodology, Y. J. Hwang; Software, Y. J. Hwang; Validation, Y. J. Hwang and J. E. Park; Formal Analysis, Y. J. Hwang; Investigation, Y. J. Hwang; Resources, Y. J. Hwang; Data Curation, Y. J. Hwang; Writing—Original Draft Preparation, Y. J. Hwang; Writing—Review & Editing, J. E. Park; Visualization, Y. J. Hwang; Supervision, J. E. Park; Project Administration, Y. J. Hwang; Funding Acquisition, Y. J. Hwang.

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### Figure 1:

Conceptual structure of a general-purpose HTS electromagnet

### Figure 2:

Schematic of a contactless current charging system for the HTS coil

### Figure 3:

3D geometry of the proposed general-purpose HTS electromagnet model for the finite element analysis

### Figure 4:

Magnetization curve (B–H curve) of the ferromagnetic material for the proposed HTS electromagnet

### Figure 5:

Mesh convergence test result: magnetic energy & flux linkage vs. the number of nodes

### Figure 6:

Dependency of the critical current on the operating temperature and external magnetic field

### Figure 7:

System used for measuring the critical current of the HTS tape according to the operating temperature and external magnetic field

### Figure 8:

Circuit diagram having the compensation network and full-wave rectifier

### Figure 9:

Simulation results for the current charging characteristics of the HTS coil by contactless current-charging method

### Figure 10:

Magnetic field distribution at a 20 mm gap distance, 75 mm pole face, and 2.4 kA·t

### Figure 11:

Magnetic field distribution in the 10 mm DSV of the air gap at a 20 mm gap distance, with 25 mm pole face, and at 8 kA·t: (a) 3D view, and (b) 2D view of the center

### Figure 12:

1D distribution of the magnetic field in the air gap between (0, -80, 0) mm and (0, 80, 0) mm

### Figure 13:

Central magnetic field density in the air gap according to the diameter of the pole face, and the MMF of the HTS coil: (a) 1 mm gap distance, (b) 5 mm gap distance, (c) 10 mm gap distance, and (d) 20 mm gap distance

### Table 1:

Comparison of specifications between the proposed electromagnet and commercial model 3474 electromagnet

Parameter Proposed model Model 3474
Maximum field density @ 20 mm gap distance > 3.0 T > 3.8 T
Field Uniformity @ 20 mm gap distance < 4 % < 1 %
Dimensions H 864 / W 920 / D 636 mm H 670 / W 750 / D 400 mm
Iron yoke type H-frame H-frame
Maximum diameter of pole face 250 mm 250 mm

### Table 2:

Specifications of the test equipment

Parameter Spec.
Central magnetic field density > 3 T
Field uniformity < 2 %
Minimum sample temperature < 20 K
Maximum test current > 300 A
Temperature variation during testing < 0.1 K
Operating current for 3T 290 A

### Table 3:

Specifications of the HTS Coil

Parameter Spec.
HTS wire 2G HTS wire
(Ic = 137 A@77 K, self-field)
Manufacturer Superpower Inc.
Winding structure Double Pancake × 2 ea.
I. D. 360 mm
Winding turns 400 turns × 2 ea.
Operating temperature 40 K
Estimated coil Ic 123.3 A

### Table 4:

Specifications of the magnetic coupler

Item Tx side Rx side
Coil I.D. (mm) 20 100
Coil O.D. (mm) 200 150
Coil Turns 180 50
Coil Inductance (mH) 0.691 0.494
Compensation Capacitance (nF) 756 713


text:
# Sets: Some Properties of Operation of Intersection and Important Equation (For CBSE, ICSE, IAS, NET, NRA 2022)

Get unlimited access to the best preparation resource for CDS Elementary-Mathematics: fully solved questions with step-by-step explanation- practice your way to success.

## What Are Sets?

• Sets are defined as a well-defined collection of objects.
• First, we specify a common property among “things” (we define this word later) and then we gather up all the “things” that have this common property.
• For example, the items you wear: hat, shirt, jacket, pants, and so on.
• This is known as a set.
• A set without any element is termed as an empty set.
• A set comprising of definite elements is termed as a finite set whereas if the set has an indefinite number of elements it is termed an infinite set.
• There is a simple notation for sets. We simply list each element (or “member” ) separated by a comma, and then put some curly brackets around the whole thing:
• At the start we used the word “things” in quotes.
• We call this the universal set. It՚s a set that contains everything. Well, not exactly everything. Everything that is relevant to our question.
• Two sets P and Q are equal if they have the same number of elements. A set P is a subset of a set Q if all the elements of P are also an element of Q.
• A power set [A (P) ] of a set P comprising of all subsets of P.
• The union of sets P and Q is a set comprising of all elements which are either in sets P or Q.
• Sets are the fundamental property of mathematics. Now as a word of warning, sets, by themselves, seem pointless
• The intersection of sets P and Q is a set comprising of all common elements of sets P and Q.
• Similarly, the difference of sets P and Q in the same order is a set comprising of elements belonging to P but not Q.

## Important Equations

• For any two sets and ,
• If P and Q are finite sets such that P ∩ Q = φ, then n (P ∪ Q) = n (P) + n (Q) .
• If P ∩ Q ≠ φ, then
• If P is a subset of set U (Universal Set) , then its complement (P′) is also a subset of Universal Set (U) .

• .

.

.


text:
On the Addition of Two Cubes of Units and Nonunits mod $p^{\alpha}$
Received:February 19, 2016  Revised:March 23, 2016
Key Word: residue classes   cubes of units   exponential sum
Fund ProjectL:Supported by the Research Culture Fundation of Anhui Normal University (Grant No.2014xmpy11) and the National Natural Science Foundation of China (Grant No.11471017).
 Author Name Affiliation Cuifang SUN School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China Zhi CHENG School of Mathematics and Computer Science, Anhui Normal University, Anhui 241003, P. R. China
Hits: 1336
Let $p\equiv 2\pmod{3}$ be an odd prime and $\alpha$ be a positive integer. In this paper, for any integer $c$, we obtain a formula for the number of solutions of the cubic congruence $x^{3}+y^{3}\equiv c~({\rm mod}\,p^{\alpha})$ with $x, y$ units, nonunits and mixed pairs, respectively. We resolve a problem posed by Yang and Tang.