text:
# Subring

Last updated

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R). The latter gives a strictly weaker condition, even for rings that do have a multiplicative identity, so that for instance all ideals become subrings (and they may have a multiplicative identity that differs from the one of R). With definition requiring a multiplicative identity (which is used in this article), the only ideal of R that is a subring of R is R itself.

## Definition

A subring of a ring (R, +, ∗, 0, 1) is a subset S of R that preserves the structure of the ring, i.e. a ring (S, +, ∗, 0, 1) with SR. Equivalently, it is both a subgroup of (R, +, 0) and a submonoid of (R, ∗, 1).

## Examples

The ring ${\displaystyle \mathbb {Z} }$ and its quotients ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ have no subrings (with multiplicative identity) other than the full ring.

Every ring has a unique smallest subring, isomorphic to some ring ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ with n a nonnegative integer (see characteristic). The integers ${\displaystyle \mathbb {Z} }$ correspond to n = 0 in this statement, since ${\displaystyle \mathbb {Z} }$ is isomorphic to ${\displaystyle \mathbb {Z} /0\mathbb {Z} }$.

## Subring test

The subring test is a theorem that states that for any ring R, a subset S of R is a subring if and only if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.

As an example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X].

## Ring extensions

If S is a subring of a ring R, then equivalently R is said to be a ring extension of S, written as R/S in similar notation to that for field extensions.

## Subring generated by a set

Let R be a ring. Any intersection of subrings of R is again a subring of R. Therefore, if X is any subset of R, the intersection of all subrings of R containing X is a subring S of R. S is the smallest subring of R containing X. ("Smallest" means that if T is any other subring of R containing X, then S is contained in T.) S is said to be the subring of R generated by X. If S = R, we may say that the ring R is generated by X.

## Relation to ideals

Proper ideals are subrings (without unity) that are closed under both left and right multiplication by elements of R.

If one omits the requirement that rings have a unity element, then subrings need only be non-empty and otherwise conform to the ring structure, and ideals become subrings. Ideals may or may not have their own multiplicative identity (distinct from the identity of the ring):

• The ideal I = {(z,0) | z in Z} of the ring Z × Z = {(x,y) | x,y in Z} with componentwise addition and multiplication has the identity (1,0), which is different from the identity (1,1) of the ring. So I is a ring with unity, and a "subring-without-unity", but not a "subring-with-unity" of Z × Z.
• The proper ideals of Z have no multiplicative identity.

If I is a prime ideal of a commutative ring R, then the intersection of I with any subring S of R remains prime in S. In this case one says that Ilies overI  S. The situation is more complicated when R is not commutative.

## Profile by commutative subrings

A ring may be profiled[ clarification needed ] by the variety of commutative subrings that it hosts:

## Related Research Articles

In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group.

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, more specifically ring theory, the Jacobson radical of a ring is the ideal consisting of those elements in that annihilate all simple right -modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left-right symmetric. The Jacobson radical of a ring is frequently denoted by or ; the former notation will be preferred in this article, because it avoids confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules.

In mathematics, an algebra over a field is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear".

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In the branch of abstract algebra known as ring theory, a unit of a ring is any element that has a multiplicative inverse in : an element such that

In abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set of all n × n matrices with entries in R is a matrix ring denoted Mn(R). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs.

In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice.

In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced.

In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D.

In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, it is the following assertion:

If , then or .

In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings. In commutative algebra, semiprime ideals are also called radical ideals.

In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that

This is a glossary of commutative algebra.

## References

• Iain T. Adamson (1972). Elementary rings and modules. University Mathematical Texts. Oliver and Boyd. pp. 14–16. ISBN   0-05-002192-3.
• Page 84 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN   978-0-201-55540-0, Zbl   0848.13001
• David Sharpe (1987). . Cambridge University Press. pp.  15–17. ISBN   0-521-33718-6.


text:
Find the matrix that is the result of the addition or subtraction of two given matrices.

### Problem

$\Large{\left[ \begin{array}{cc} 3 & 1 \\ 0 & -2 \end{array} \right]+\left[ \begin{array}{cc} 4 & 0 \\ 3 & -1 \end{array} \right]= }$
Get first 1 correct, or 3 in a row.

format: plain


text:
# pandas.Series.name¶

property Series.name

Return the name of the Series.

The name of a Series becomes its index or column name if it is used to form a DataFrame. It is also used whenever displaying the Series using the interpreter.

Returns
label (hashable object)

The name of the Series, also the column name if part of a DataFrame.

Series.rename

Sets the Series name when given a scalar input.

Index.name

Corresponding Index property.

Examples

The Series name can be set initially when calling the constructor.

>>> s = pd.Series([1, 2, 3], dtype=np.int64, name='Numbers')
>>> s
0    1
1    2
2    3
Name: Numbers, dtype: int64
>>> s.name = "Integers"
>>> s
0    1
1    2
2    3
Name: Integers, dtype: int64

The name of a Series within a DataFrame is its column name.

>>> df = pd.DataFrame([[1, 2], [3, 4], [5, 6]],
...                   columns=["Odd Numbers", "Even Numbers"])
>>> df
Odd Numbers  Even Numbers
0            1             2
1            3             4
2            5             6
>>> df["Even Numbers"].name
'Even Numbers'


text:
2013
11-12

Find the Clones

Doubleville, a small town in Texas, was attacked by the aliens. They have abducted some of the residents and taken them to the a spaceship orbiting around earth. After some (quite unpleasant) human experiments, the aliens cloned the victims, and released multiple copies of them back in Doubleville. So now it might happen that there are 6 identical person named Hugh F. Bumblebee: the original person and its 5 copies. The Federal Bureau of Unauthorized Cloning (FBUC) charged you with the task of determining how many copies were made from each person. To help you in your task, FBUC have collected a DNA sample from each person. All copies of the same person have the same DNA sequence, and different people have different sequences (we know that there are no identical twins in the town, this is not an issue).

The input contains several blocks of test cases. Each case begins with a line containing two integers: the number 1 ≤ n ≤ 20000 people, and the length 1 ≤ m ≤ 20 of the DNA sequences. The next n lines contain the DNA sequences: each line contains a sequence of m characters, where each character is either A’, C’, G’ or T’.

The input is terminated by a block with n = m = 0 .

For each test case, you have to output n lines, each line containing a single integer. The first line contains the number of different people that were not copied. The second line contains the number of people that were copied only once (i.e., there are two identical copies for each such person.) The third line contains the number of people that are present in three identical copies, and so on: the i -th line contains the number of persons that are present in i identical copies. For example, if there are 11 samples, one of them is from John Smith, and all the others are from copies of Joe Foobar, then you have to print 1′ in the first andthe tenth lines, and 0′ in all the other lines.

9 6
AAAAAA
ACACAC
GTTTTG
ACACAC
GTTTTG
ACACAC
ACACAC
TCCCCC
TCCCCC
0 0

1
2
0
1
0
0
0
0
0

Huge input file, ‘scanf’ recommended to avoid TLE.

import java.util.HashMap;
import java.util.Scanner;

public class Main {

public static void main(String[] args) {
Scanner sc = new Scanner(System.in);
while (sc.hasNext()) {
int n = sc.nextInt();
int m = sc.nextInt();
if (n == 0 && m == 0)
break;
sc.nextLine();
HashMap< String, Integer> map = new HashMap< String, Integer>();
int[] dispear = new int[n];
for (int i = 0; i < n; ++i) {
String dna = sc.nextLine();
// /Integer hasCode = dna.hashCode();
if (map.get(dna) != null) {
map.put(dna, map.get(dna) + 1);
dispear[map.get(dna) - 1]--;
dispear[map.get(dna)]++;
} else {
map.put(dna, 0);
dispear[0]++;
}
}
for (int i = 0; i < n; ++i) {
System.out.println(dispear[i]);
}
}
}
}


text:
# Thread: Help with elimination-substitution

1. ## Help with elimination-substitution

The Jones's and the Johnsons want to see who sleeps more, women or men. The Jones family has 1 female and 3 males. On the night of the test the Jones family slept a total of 43 hours. The Johnson family has 3 males and 4 females and their total sleep was 60.72 hours.
How many hours does each male and female sleep, assuming that all males sleep the same amount of hours and all females sleep the same amount of hours.

Ok, cool.
So...
3x+y=43
3x+4y=60.72
Then...
5y=103.72
then...
y=20.74

Then when I substitute it it comes out wrong. Have I already made a mistake?

Could someone please just help me through this?

2. ## Re: Help with elimination-substitution

Originally Posted by Russellmuscle387
The Jones's and the Johnsons want to see who sleeps more, women or men. The Jones family has 1 female and 3 males. On the night of the test the Jones family slept a total of 43 hours. The Johnson family has 3 males and 4 females and their total sleep was 60.72 hours.
How many hours does each male and female sleep, assuming that all males sleep the same amount of hours and all females sleep the same amount of hours.

Ok, cool.
So...
3x+y=43
3x+4y=60.72
Then...
5y=103.72
then...
y=20.74

Then when I substitute it it comes out wrong. Have I already made a mistake?

Could someone please just help me through this?
First you need to ask yourself if that answer makes sense. Do you know many people that sleep 20.74 hours per day?

2nd you need to be careful.

You system of equations is correct

$\begin{matrix}3x & + & y &=& 43 \\ 3x & + & 4y & = & 60.72 \end{matrix}$

Now you need to multiply the top equation by -1 and add it to the bottom equation (or vice versa). This gives

$\begin{matrix}-3x & - & y &=& -43 \\ 3x & + & 4y & = & 60.72 \\ \hline & & 3y & = & 17.72 \end{matrix}$


text:
# Celebratio Mathematica

## Shiing-Shen Chern

### Students

 Katsumi Nomizu :Invariant affine connections on homogeneous spaces University of Chicago, Ph.D. 1953 Louis Auslander :Contribution to the curvature theory of Finsler spaces University of Chicago, Ph.D. 1954 San Dao Liao :On the theory of obstructions of fiber bundles University of Chicago, Ph.D. 1955 Jerome Spanier :Contributions to the theory of almost-complex manifolds University of Chicago, Ph.D. 1955 Alexandre Rodrigues :Characteristic classes of homogeneous spaces University of Chicago, Ph.D. 1957 Harold I. Levine :Contributions to the theory of analytic maps of complex manifolds into projective spaces University of Chicago, Ph.D. 1957 Haruo Suzuki :On the realization of Stiefel–Whitney characteristic classes by submanifolds University of Chicago, Ph.D. 1957 Joseph Albert Wolf :On the manifolds covered by a given compact, connected Riemannian homogeneous manifolds University of Chicago, Ph.D. 1959 Nicholas Petridis :Quasiconformal mappings and pseudomeromorphic curves University of Chicago, Ph.D. 1961 William Pohl :Differential geometry of higher order University of California, Berkeley, Ph.D. 1961 Manfredo P. do Carmo :The cohomology ring of certain Kählerian manifolds University of California, Berkeley, Ph.D. 1963 Thomas Banchoff :Tightly embedded two-dimensional polyhedral manifolds University of California, Berkeley, Ph.D. 1964 Leo Amaral :Hypersurfaces in non-Euclidean spaces University of California, Berkeley, Ph.D. 1964 Howard Garland :On the cohomology of lattices in Lie groups University of California, Berkeley, Ph.D. 1964 Robert Brown Gardner :Differential geometric methods in partial differential equations University of California, Berkeley, Ph.D. 1965 Alan Weinstein :The cut locus and conjugate locus of a Riemannian manifold University of California, Berkeley, Ph.D. 1967 Bernard Shiffman :On the removal of singularities in several complex variables University of California, Berkeley, Ph.D. 1968 Robert Alan Wolf :Some integral formulas related to the volume of tubes University of California, Berkeley, Ph.D. 1968 Donald Pelles :Intrinsic measures on complex manifolds and holomorphic mappings University of California, Berkeley, Ph.D. 1969 Robeert Reilly :The Gauss map in the study of submanifolds of spheres University of California, Berkeley, Ph.D. 1969 Steven Jordan :Some invariants for complex manifolds University of California, Berkeley, Ph.D. 1970 Dominic Leung :Deformations of integrals of exterior differential systems University of California, Berkeley, Ph.D. 1970 Hon-Fei Lai :Characteristic classes of real manifolds immersed in complex manifolds University of California, Berkeley, Ph.D. 1971 Shing-Tung Yau :On the fundamental group of compact manifolds of non-positive curvature University of California, Berkeley, Ph.D. 1971 João Barbosa :On the minimal Immersions of $$S^2$$ in $$S^{2m}$$ University of California, Berkeley, Ph.D. 1972 John James Millson :Chern–Simons invariants of constant-curvature manifolds University of California, Berkeley, Ph.D. 1973 David Bleecker :Contributions to the theory of surfaces University of California, Berkeley, Ph.D. 1973 Plinio Simoes :A class of minimal cones in $$R^n$$, $$n > 8$$, that minimize area University of California, Berkeley, Ph.D. 1973 Harold Gerard Donnelly :Chern–Simons invariants of reductive homogeneous spaces University of California, Berkeley, Ph.D. 1974 Shiu-Yuen Cheng :Spectrum of the Laplacian and its applications to differential geometry University of California, Berkeley, Ph.D. 1974 Sidney Martin Webster :Real hypersurfaces in complex space University of California, Berkeley, Ph.D. 1975 Douglas Dunham :Holomorphic and meromorphic vector fields on compact Hermitian symmetric spaces University of California, Berkeley, Ph.D. 1975 Chen Han Sung :Contributions to holomorphic curves in complex manifolds University of California, Berkeley, Ph.D. 1975 James Faran :Segre families and real hypersurfaces University of California, Berkeley, Ph.D. 1978 Peter Li :On the first eigenvalue and eigenfunctions of the Laplace operator on a compact Riemannian manifold and their geometric applications University of California, Berkeley, Ph.D. 1979 Theodore Shifrin :The kinematic formula in complex integral geometry University of California, Berkeley, Ph.D. 1979 Stuart Smith :Contributions to the eigenvalue problem for the Laplacian University of California, Berkeley, Ph.D. 1979 Ai-Nung Wang :Contributions to differential geometry University of California, Berkeley, Ph.D. 1981 Jon Gordon Wolfson :Minimal surfaces in complex manifolds University of California, Berkeley, Ph.D. 1982


text:
Question
The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.

$0.322 \Omega$

Solution Video


text:
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# Duke Fuqua 2012 - Calling All Applicants

Author Message
Manager
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

29 Mar 2012, 12:14
Was WL'ed in R2, but received an interview invitation right now.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

29 Mar 2012, 12:22
ulm wrote:
Was WL'ed in R2, but received an interview invitation right now.

Hey congratulations. waitlist can really be a nightmare(waitlisted at tepper). Invited for fuqua interview though R3 applicant
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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29 Mar 2012, 17:16
_________________

Extraordinary claims require extraordinary evidence

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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

29 Mar 2012, 18:22
navra wrote:

I got a call on Wednesday from Sharon, one of the admissions directors, regarding a merit scholarship. They may still be calling people. Hope you hear good news!
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

30 Mar 2012, 06:05
travisz wrote:

I just think that if they have a round 3, they should allocate at least 5-10% of spots there to make it meaningful. In addition, do they want best candidates or candidates who apply early. It is interesting that they gave so much advantage to early applicants.

Well, anyways, I will try my best to ace the interview.

Maybe it has something to do with the competition between all those backlogged as Wait-lists from EA, R1, R2 etc and the nature of the competition greatly impacts how many WL-ers get admitted vs. direct R3 applicants.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

30 Mar 2012, 10:02
1
KUDOS
Honestly, I think if you're applying to round 3, it has to be in your mind that the likelihood of acceptance is going to be small unless you're clearly above the grade in most of the categories of your application. Even then, there's no guarantee.

As other people have suggested, I think the choices in round 3 are much more about filling out the class than merely taking qualified candidates. I'm not sure if that means "we need more int'l kids" or "we've got too many accountants" or "our average GMAT is low, just take some really high GMAT kids" or what, but it seems to be fairly acknowledged that Round 3 is not approached the same way as previous rounds.

If Duke is looking for 450 kids, and they've got 420 deposits by the time interviews for Round 3 are finished, they're really only going to be looking for 30 kids between the waitlists and the Round 3 applicants, whether there are 50 round 3 applicants or 500.

My understanding is that's probably true for most top schools.

It's an unfortunate reality of Round 3, but a fairly well-documented one. It almost surprises me that there isn't a "PROCEED AT YOUR OWN RISK" pop-up when you select round 3 app.

That being said, of course, good luck to the round 3'ers! If it doesn't work out, go EA next year and you'll have a much better shot!
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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30 Mar 2012, 10:33
navra wrote:

I received a call last Monday with good news about a scholarship. I was told they were pulling together admit packs to be sent end of week / early next week. Now I just need to decide between Booth and Fuqua w/ . Glad to finally be in this position after dealing w/ two WLs that didn't work out last time around.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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30 Mar 2012, 13:03
So I just booked my annoyingly expensive ticket from California for BDW and get in Thursday April 12th in the am. Seeing as I have no friends there, nor have I ever been to NC is anyone in a similar situation that will be there for BDW that would want to get a meal or do some sight seeing or something on Thursday? I like making friends! PM me if interested
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

### Show Tags

31 Mar 2012, 08:29
inspired12 wrote:
So I just booked my annoyingly expensive ticket from California for BDW and get in Thursday April 12th in the am. Seeing as I have no friends there, nor have I ever been to NC is anyone in a similar situation that will be there for BDW that would want to get a meal or do some sight seeing or something on Thursday? I like making friends! PM me if interested
There will be plenty of activities during BDW, you should worry about not having stuff to do.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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31 Mar 2012, 12:55
Do you guys know if Fuqua scholarships are merit-based or need-based? Perhaps both?
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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31 Mar 2012, 13:25
Politico2012 wrote:
Do you guys know if Fuqua scholarships are merit-based or need-based? Perhaps both?

Scholarships are all merit based
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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31 Mar 2012, 14:29
What is the difference between 'Not Invited' and 'Wait listed'. Does the former imply you have zero chance of being interviewed and are basically awaiting a formal rejection? Does the latter only apply to post-interview status or pre-interview status as well? If so, wouldn't it be the same as being 'Not Invited'?

This is a weird system, hence all the questions!
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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31 Mar 2012, 16:30
asimov wrote:
inspired12 wrote:
So I just booked my annoyingly expensive ticket from California for BDW and get in Thursday April 12th in the am. Seeing as I have no friends there, nor have I ever been to NC is anyone in a similar situation that will be there for BDW that would want to get a meal or do some sight seeing or something on Thursday? I like making friends! PM me if interested
There will be plenty of activities during BDW, you should worry about not having stuff to do.

Im getting there on Thursday morning the day before BDW, just wanted to see if anyone is in the same situation and wants to meet up for a meal/drink on Thursday!
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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31 Mar 2012, 21:51
packfan1 wrote:
Politico2012 wrote:
Do you guys know if Fuqua scholarships are merit-based or need-based? Perhaps both?

Scholarships are all merit based

catakiosk wrote:
What is the difference between 'Not Invited' and 'Wait listed'. Does the former imply you have zero chance of being interviewed and are basically awaiting a formal rejection? Does the latter only apply to post-interview status or pre-interview status as well? If so, wouldn't it be the same as being 'Not Invited'?

This is a weird system, hence all the questions!

Applicants can only officially get waitlisted on the decision date, whether or not they are invited to interview.

inspired12 wrote:
Im getting there on Thursday morning the day before BDW, just wanted to see if anyone is in the same situation and wants to meet up for a meal/drink on Thursday!

Try posting this on the FB group wall. You'll probably get more responses over there.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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01 Apr 2012, 19:00
inspired12 wrote:
asimov wrote:
inspired12 wrote:
So I just booked my annoyingly expensive ticket from California for BDW and get in Thursday April 12th in the am. Seeing as I have no friends there, nor have I ever been to NC is anyone in a similar situation that will be there for BDW that would want to get a meal or do some sight seeing or something on Thursday? I like making friends! PM me if interested
There will be plenty of activities during BDW, you should worry about not having stuff to do.
Im getting there on Thursday morning the day before BDW, just wanted to see if anyone is in the same situation and wants to meet up for a meal/drink on Thursday!

Hi inspired12 or any other people who are coming early to BDW, feel free to go to some of the campus events that day, or see if you can sit in on a class. It will get you a feel for Fuqua student life.

To see a list of all the events going on at Fuqua: https://fuqua.campusgroups.com/calendar

On Thursday (April 12th), it looks like Nissan is doing a talk about how they recovered from the Tsunami, sounds very interesting.
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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02 Apr 2012, 09:50
More great accolades for Fuqua Alum Tim Cook:

http://www.eweek.com/c/a/IT-Management/ ... EO-736704/
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Re: Duke Fuqua 2012 - Calling All Applicants [#permalink]

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02 Apr 2012, 19:00
Anyone received their admit packs? There is a message in FB that some have already received. So i'm pondering whether some of those who live outside NC have received either
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02 Apr 2012, 23:39
navra wrote:
Anyone received their admit packs? There is a message in FB that some have already received. So i'm pondering whether some of those who live outside NC have received either
I got it more than 24 hrs ago and I live in India
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03 Apr 2012, 09:11
For those WLers who are interested, the decision will be rendered with the R3 applicants (4th of May).
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03 Apr 2012, 17:02
Ok, now let me rephrase it: is there anyone who hasn't yet received their packs? Surely, except for me
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Re: Duke Fuqua 2012 - Calling All Applicants   [#permalink] 03 Apr 2012, 17:02

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# Duke Fuqua 2012 - Calling All Applicants

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# Perform network meta-analysis

## Introduction

This vignette illustrates how to perform a one-stage Bayesian random-effects network meta-analysis with consistency equation using the run_model function. This function includes arguments to handle aggregate missing participant outcome data (MOD) in each arm of every trial via the pattern-mixture model.

## Example on a binary outcome

We will use the network on pharmacologic interventions for chronic obstructive pulmonary disease (COPD) from the systematic review of Baker et al. (2009). This network comprises 21 trials comparing seven pharmacological interventions with each other and placebo. The exacerbation of COPD (harmful outcome) is the analysed binary outcome (see ?nma.baker2009).

## Perform Bayesian random-effects network meta-analysis

### Run the model

run_model calls the jags function from the R2jags package to perform the Bayesian analysis using the BUGS code of Dias and colleagues (2013).

run_model(data = nma.baker2009,
measure = "OR",
heter_prior = list("halfnormal", 0, 1),
D = 0,
n_chains = 3,
n_iter = 10000,
n_burnin = 1000,
n_thin = 1)

With only the minimum required arguments, the function adjusts MOD under the missing-at-random assumption (MAR; by default) via the informative missingness odds ratio (IMOR) in the logarithmic scale (White et al. (2008)): The minimum required arguments of run_model include specifying:

• the dataset (a data-frame with one-trial-per-row format) in data (see ?data_preparation);
• the effect measure in measure (see ‘Arguments’ in ?run_model):
• the prior distribution for the heterogeneity parameter in heter_prior (see ?heterogeneity_param_prior);
• the direction of the outcome in D (here, D = 0 because the outcome is harmful; see, ‘Arguments’ in ?run_model)
• the number of chains in n_chains (see ‘Arguments’ in ?run_model – also for the subsequent arguments);
• the number of iterations in n_iter;
• the number of burn-in in n_burnin, and
• the thinning in n_thin.

### Using all arguments

Suppose we decide to use an empirically-based prior distribution for the between-trial variance that aligns with the outcome and interventions under investigation. We also consider a hierarchical structure for the prior normal distribution of the log IMOR that is specific to the interventions in the network (assumption = "HIE-ARM") (Turner et al., 2015a; Spineli, 2019). We still assume MAR on average with variance of log IMOR equal to 1 (var_misspar = 1) which is also the default argument. In this case, run_model must be specified as follows:

run_model(data = nma.baker2009,
measure = "OR",
model = "RE",
assumption = "HIE-ARM",
heter_prior = list("lognormal", -2.06, 0.438),
mean_misspar = c(0, 0),
var_misspar = 1,
D = 0,
n_chains = 3,
n_iter = 10000,
n_burnin = 1000,
n_thin = 1)

The argument model = "RE" refers to the random-effects model. For the fixed-effect model, use model = "FE".

heter_prior = list("lognormal", -2.06, 0.438) refers to ‘symptoms reflecting the continuation of condition’ for the ‘pharmacological versus placebo’ comparison-type as elicited by Turner and colleagues (2015b).

In the argument mean_misspar = c(0, 0), the first and second element of the vector refers to the mean log IMOR in the non-reference interventions and the reference intervention of the network, respectively – the latter is always the intervention with identifier equal to 1. Hence, for all non-reference interventions we can consider the same mean log IMOR. See ‘Details’ in ?missingness_param_prior

### The output

run_model returns a list of R2jags output on the summaries of the posterior distribution, and the Gelman-Rubin convergence diagnostic of the monitored parameters (see ‘Value’ in ?run_model). The output is used as an S3 object by other functions of the package to be processed further and provide an end-user-ready output. See, for example, the function ?league_heatmap that creates the league table with the effect sizes of all possible comparisons in the network.

### No or partially extracted missing participant outcome data

run_model can also handle a dataset where MOD have not be extracted or MOD have been extracted for some trials or trial-arms. For illustrative purposes, we removed the item m from nma.baker2009 to indicate that MOD were not extracted for this outcome:

                  study t1 t2 t3 t4 r1 r2 r3 r4  n1  n2 n3 n4
1 Llewellyn-Jones, 1996  1  4 NA NA  3  0 NA NA   8   8 NA NA
2        Paggiaro, 1998  1  4 NA NA 51 45 NA NA 139 142 NA NA
3          Mahler, 1999  1  7 NA NA 47 28 NA NA 143 135 NA NA
4        Casaburi, 2000  1  8 NA NA 41 45 NA NA 191 279 NA NA
5       van Noord, 2000  1  7 NA NA 18 11 NA NA  50  47 NA NA
6         Rennard, 2001  1  7 NA NA 41 38 NA NA 135 132 NA NA

Using the minimum required arguments, run_model will run and provide results.

run_model calls the data_preparation function. The latter creates a pseudo-data-frame for the item m (see ‘Value’ in ?data_preparation) that assigns NA to all trial-arms. data_preparation also creates the pseudo-data-frame I that has the same dimension with the other items in the dataset, and assigns the zero value to all trial-arms to indicate that no MOD have been extracted. Both pseudo-data-frames aim to retain in the dataset the trials without information on MOD; otherwise, these trials would have been excluded from the analysis. See ‘Details’ in ?data_preparation and ?run_model.

## References

Dias S, Sutton AJ, Ades AE, Welton NJ. Evidence synthesis for decision making 2: a generalized linear modeling framework for pairwise and network meta-analysis of randomized controlled trials. Med Decis Making 2013;33(5):607–617. doi: 10.1177/0272989X12458724

White IR, Higgins JP, Wood AM. Allowing for uncertainty due to missing data in meta-analysis–part 1: two-stage methods. Stat Med 2008;27(5):711–27. doi: 10.1002/sim.3008

Turner NL, Dias S, Ades AE, Welton NJ. A Bayesian framework to account for uncertainty due to missing binary outcome data in pairwise meta-analysis. Stat Med 2015a;34(12):2062–80. doi: 10.1002/sim.6475

Spineli LM. An empirical comparison of Bayesian modelling strategies for missing binary outcome data in network meta- analysis. BMC Med Res Methodol 2019;19(1):86. doi: 10.1186/s12874-019-0731-y

Turner RM, Jackson D, Wei Y, Thompson SG, Higgins JPT. Predictive distributions for between-study heterogeneity and simple methods for their application in Bayesian meta-analysis. Stat Med 2015b;34(6):984–98. doi: 10.1002/sim.6381


text:
# Smooth dynamical systems as infinitesimal discrete dynamical systems

Here I am working with nonstandard analysis in the sense of Robinson, taking an ultrapower of the real numbers and building things out of that. But in general I am going to be a bit sloppy and not worry too much about the details.

Recall that a standard function is differentiable if, for every standard $$x$$ and for every infinitesimal $$\epsilon$$, $$f(x+\epsilon)-f(x)/\epsilon \approx a$$, where $$a$$ is also standard.

Fix an infinitesimal $$h$$. Let $$M$$ be a standard differentiable manifold, fix $$p \in M$$, and consider the tangent space $$T_pM$$. Recall that the elements of this vector space (may be taken to be) smooth paths $$\gamma: I \to M$$, with $$0 \in I \subset \mathbb{R}$$ some open interval, up to the equivalence relation of having the same first derivative at $$p$$ (which may be checked in any choice of local coordinates, all coordinates giving the same answer). In particular we may compute this derivative, letting $$\phi: U \to \mathbb{R}$$, $$p \in U \subset M$$ being the local coordinates of choice, as the standard part of $$(\phi(\gamma(h))-\phi(p))/h$$. Having fixed $$h$$, we may thus replace each curve with just the choice of $$\gamma(h)$$, the only condition on this being that in any choice of local coordinates, the above quotient has a standard part (eg it is not unbounded). But since the change-of-coordinate maps are in particular differentiable, it suffices to verify this for one such choice. We may call this property “being at $$O(h)$$-order infinitesimal distance from from $$p$$”. For example, suppose that $$M = \mathbb{R}$$, $$p=0$$ and $$\phi = 1_\mathbb{R}$$. Then if $$h = (1/n)$$, it will not do to take $$\gamma(h) = (1/\sqrt{n})$$, for in that case we will get the unbounded nonstandard real $$(\sqrt{n})$$ for the difference quotient.

Let $$\widetilde{D_pM}$$ be the set of such points in $$M$$. We don’t quite have a map $$T_pM \to \widetilde{D_pM}$$, because e.g the path $$\gamma(t)=t^2$$, which has local derivative zero, has $$\gamma(h) = h^2 \neq 0$$. In order to make this map well-defined, we need to consider a quotient by the relation of “being at distance $$o(h)$$”. More precisely, let $$v \approx v’$$ if, if any coordiante chart, $$d(v,v’)/h$$ is infinitesimal. (The local differentiability ensures this does not depend on the choice of coordinates.) Then let $$D_pM = \widetilde{D_pM}/\approx$$ - now the map $$T_pM \to D_pM$$ is well-defined.

We can also define a vector space structure on $$D_pM$$ that makes this map linear. We simply lift the addition and scalar multiplication from any coordinate chart. Local differentiability imply that $$\psi(h(v+v’)) = \psi’(0) h(v+v’) + h\epsilon$$, wher $$\psi$$ is a change-of-coordinate map, so that addition is well-defined up to an infinitesimal times $$h$$, which is quotiented out by in any case. Scalar multiplication works similarly. Note that we are defining a vector space over $$\mathbb{R}$$, not over the full $$*\mathbb{R}$$. Multiplication by a number of order $$o(h)$$ is not invertible (in fact such products are always zero), and similarly scaling by an unbounded number may take you out of the coordinate patch, and is thus not well-defined.

In fact with this, we have an isomorphism $$T_pM \to D_pM$$ - any $$v \in D_pM$$ is the image of $$\gamma(t) = p + tv/h$$, and two curves have the same derivative exactly if $$\gamma(h) \approx \gamma’(h)$$.

Recall that smooth dynamical system on $$M$$ is a smoth section $$M \to TM$$. We can exploit our isomorphism above, by observing that each $$D_p$$ is actually a quotient of a subset of $$M$$. Hence we may ask for a function $$s: *M \to *M$$ so that $$s(p) \in \widetilde{D_p} \subset *M$$ - this induces a smooth dynamical system. Every (standard) smooth dynamical system has this form, and $$s,s’$$ induce the same system if $$s(p) \approx s’(p)$$ for each $$p$$.

What’s cool about this is that these are essentially discrete dynamical systems, albeit nonstandard ones. A discrete dynamical system is a set with an “advance one timestep” function $$s: X \to X$$. So a smooth dynamical system is a manifold with an “advance time $$h$$” function $$s: *M \to *M$$, subject to the condition that $$s(p)$$ is $$h$$-close to $$p$$, and up to a certain equivalence relation. This suggests a way to use the same conceptual tools to study smooth and discrete dynamical systems.

The paper Differential geomtry via infinitesimal displacements, by Nowik and Katz, provides a more in-depth analysis of this idea.


text:
# zbMATH — the first resource for mathematics

The complexity of propositional proofs. (English) Zbl 0845.03025
“The classical propositional calculus has an undeserved reputation among logicians as being essentially trivial. I hope to convince the reader that it presents some of the most challenging and interesting problems in modern logic.” So begins this survey article, whose main theme is to compare which proof system can(not) simulate which others in polynomial time. The systems considered include Frege systems, Gentzen systems, tableaux, resolutions, and truth tables, and the author provides a full page chart showing their relations. Not merely citing results, he reproduces proofs and thus presents “some of the techniques that have proved successful in deriving lower bounds on the complexity of proofs”. An instance is $$k$$-valuation that is employed by Pudlák et al. in giving exponential lower bounds on proofs of the pigeonhole principle in bounded-depth Frege systems. One area left out is bounded arithmetic and its connection with complexity theory. (The reader is referred to J. Krajíček’s recent monograph [Bounded arithmetic, propositional logic, and complexity theory (1995; Zbl 0835.03025)].) With a number of open problems and references to an extensive bibliography, this is a survey worthy of its name. Whether the kinds of problems referred to are “most challenging and interesting…in modern logic” is a decision for the reader to make.

##### MSC:
 03F20 Complexity of proofs 03B05 Classical propositional logic 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
Full Text:


text:
### Author Topic: Q3 TUT 0601  (Read 1823 times)

#### Victor Ivrii

• Elder Member
• Posts: 2563
• Karma: 0
##### Q3 TUT 0601
« on: October 12, 2018, 06:05:54 PM »
Find a differential equation whose general solution is
$$y=c_1e^{2t}+c_2e^{-3t}.$$

#### Nick Callow

• Jr. Member
• Posts: 9
• Karma: 9
##### Re: Q3 TUT 0601
« Reply #1 on: October 12, 2018, 06:06:11 PM »
Find a differential equation with general solution $c_1e^{2t} + c_2e^{-3t}$.

Given that $r = -3, 2$ we know the characteristic equation of this general solution will be of the form $(r-2)(r+3)$. Expanding this, we get that the differential equation has the form $r^2 + r - 6$. Therefore, a differential equation with the above general solution will be $y''(t) + y'(t) - 6y(t) = 0$.

We can check this by working in reverse. Suppose we have a differential equation $y''(t) + y'(t) - 6y(t) = 0$. Then we can try a solution of the form $y(t) = e^{rt}$. Consequently, $y'(t) = re^{rt}$ and $y''(t) = r^2e^{rt}$. Subbing this into the differential equation we have, $r^2e^{rt} + re^{rt} - 6e^{rt} = 0$. We can factor out an $e^{rt}$ and the characteristic equation becomes $r^2 + r - 6$. Factoring this, we have $(r-2)(r+3)$. Since our $r = -3,2$ we know two particular solutions are $y_1(t) = e^{2t}$ and $y_2(t) = e^{-3t}$. The general form of this will be $c_1e^{2t} + c_2e^{-3t}$. This matches what the question was asking, so we are finished.

#### Keyue Xie

• Newbie
• Posts: 2
• Karma: 1
##### Re: Q3 TUT 0601
« Reply #2 on: October 12, 2018, 06:53:18 PM »
$$y=c_1e^{2t} + c_2e^{-3t}$$
$$r_1 = 2, r_2 = -3$$
$$(r-2)(r+3) = 0$$
$$r^2+ r - 6 = 0$$
$$y''(t) + y'(t) - 6y(t) = 0$$

#### Shengying Yang

• Jr. Member
• Posts: 10
• Karma: 17
##### Re: Q3 TUT 0601
« Reply #3 on: October 12, 2018, 07:04:07 PM »
I agree with Nick's answer. Here is just a version without word explanation which is easier to see.
\begin{align*} ∵ r=2 , r=-3 \end{align*}
\begin{align*} ∴(r-2)(r+3)=0 \end{align*}
\begin{align*} ∴ r^2-2r+3r-6=0 \end{align*}
\begin{align*} ∴r^2+r-6=0 \end{align*}

\begin{align*} ∴ y''+y'-6y=0 \end{align*}


text:
# Quantum Dots in Gated Nanowires and Nanotubes

 Title: Quantum Dots in Gated Nanowires and Nanotubes Author: Churchill, Hugh Olen Hill Full Text & Related Files: Churchill_gsas.harvard_0084L_10412.pdf (22.06Mb; PDF) Abstract: This thesis describes experiments on quantum dots made by locally gating one-dimensional quantum wires. The first experiment studies a double quantum dot device formed in a Ge/Si core/shell nanowire. In addition to measuring transport through the double dot, we detect changes in the charge occupancy of the double dot by capacitively coupling it to a third quantum dot on a separate nanowire using a floating gate. We demonstrate tunable tunnel coupling of the double dot and quantify the strength of the tunneling using the charge sensor. The second set of experiments concerns carbon nanotube double quantum dots. In the first nanotube experiment, spin-dependent transport through the double dot is compared in two sets of devices. The first set is made with carbon containing the natural abundance of $$^{12}C$$ (99%) and $$^{13}C$$ (1%), the second set with the 99% $$^{13}C$$ and 1% $$^{12}C$$. In the devices with predominantly $$^{13}C$$, we find evidence in spin-dependent transport of the interaction between the electron spins and the $$^{13}C$$ nuclear spins that was much stronger than expected and not present in the $$^{12}C$$ devices. In the second nanotube experiment, pulsed gate experiments are used to measure the timescales of spin relaxation and dephasing in a two-electron double quantum dot. The relaxation time is longest at zero magnetic field and goes through a minimum at higher field, consistent with the spin-orbit-modified electronic spectrum of carbon nanotubes. We measure a short dephasing time consistent with the anomalously strong electron-nuclear interaction inferred from the first nanotube experiment. Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:9414563


text:
# Resources tagged with: Multiplication & division

Filter by: Content type:
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Challenge level:

### There are 141 results

Broad Topics > Calculations and Numerical Methods > Multiplication & division

### A Chance to Win?

##### Age 11 to 14 Challenge Level:

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

### What Two ...?

##### Age 7 to 11 Short Challenge Level:

56 406 is the product of two consecutive numbers. What are these two numbers?

### Tom's Number

##### Age 7 to 11 Challenge Level:

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

### Clever Santa

##### Age 7 to 11 Challenge Level:

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

### Jumping

##### Age 7 to 11 Challenge Level:

After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

### Ones Only

##### Age 11 to 14 Challenge Level:

Find the smallest whole number which, when mutiplied by 7, gives a product consisting entirely of ones.

### The Remainders Game

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Play this game and see if you can figure out the computer's chosen number.

### Learning Times Tables

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In November, Liz was interviewed for an article on a parents' website about learning times tables. Read the article here.

### Being Resilient - Primary Number

##### Age 5 to 11 Challenge Level:

Number problems at primary level that may require resilience.

### 3388

##### Age 11 to 14 Challenge Level:

Using some or all of the operations of addition, subtraction, multiplication and division and using the digits 3, 3, 8 and 8 each once and only once make an expression equal to 24.

### Fair Feast

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Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

### One O Five

##### Age 11 to 14 Challenge Level:

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

### What Is Ziffle?

##### Age 7 to 11 Challenge Level:

Can you work out what a ziffle is on the planet Zargon?

### Countdown

##### Age 7 to 14 Challenge Level:

Here is a chance to play a version of the classic Countdown Game.

### Integrated Product Sudoku

##### Age 11 to 16 Challenge Level:

This Sudoku puzzle can be solved with the help of small clue-numbers on the border lines between pairs of neighbouring squares of the grid.

### Journeys on the Gattegno Tens Chart

##### Age 5 to 11

Alf describes how the Gattegno chart helped a class of 7-9 year olds gain an awareness of place value and of the inverse relationship between multiplication and division.

### A First Product Sudoku

##### Age 11 to 14 Challenge Level:

Given the products of adjacent cells, can you complete this Sudoku?

### Looking at Lego

##### Age 7 to 11 Challenge Level:

This task offers an opportunity to explore all sorts of number relationships, but particularly multiplication.

### 1, 2, 3, 4, 5

##### Age 7 to 11 Challenge Level:

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

### Ducking and Dividing

##### Age 7 to 11 Challenge Level:

Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

### Difficulties with Division

##### Age 5 to 11

This article for teachers looks at how teachers can use problems from the NRICH site to help them teach division.

### Divide it Out

##### Age 7 to 11 Challenge Level:

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

### Multiples Sudoku

##### Age 11 to 14 Challenge Level:

Each clue in this Sudoku is the product of the two numbers in adjacent cells.

### More Magic Potting Sheds

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

### Super Shapes

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The value of the circle changes in each of the following problems. Can you discover its value in each problem?

### What Is the Question?

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These pictures and answers leave the viewer with the problem "What is the Question". Can you give the question and how the answer follows?

### Oh! Hidden Inside?

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Find the number which has 8 divisors, such that the product of the divisors is 331776.

### Book Codes

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Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

##### Age 11 to 14 Challenge Level:

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

### Amy's Dominoes

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Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

### One Million to Seven

##### Age 7 to 11 Challenge Level:

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

### Multiplication Series: Illustrating Number Properties with Arrays

##### Age 5 to 11

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

### Asteroid Blast

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A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.

### Odds and Threes

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A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

### Domino Numbers

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Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

### Magic Potting Sheds

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Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

##### Age 11 to 14 Challenge Level:

Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

##### Age 11 to 14 Challenge Level:

If you take a three by three square on a 1-10 addition square and multiply the diagonally opposite numbers together, what is the difference between these products. Why?

### Repeaters

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Two Many

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What is the least square number which commences with six two's?

### Numbers Numbers Everywhere!

##### Age 5 to 11

Bernard Bagnall recommends some primary school problems which use numbers from the environment around us, from clocks to house numbers.

### Skeleton

##### Age 11 to 14 Challenge Level:

Amazing as it may seem the three fives remaining in the following `skeleton' are sufficient to reconstruct the entire long division sum.

### Long Multiplication

##### Age 11 to 14 Challenge Level:

A 3 digit number is multiplied by a 2 digit number and the calculation is written out as shown with a digit in place of each of the *'s. Complete the whole multiplication sum.

### Surprising Split

##### Age 7 to 11 Challenge Level:

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

### Like Powers

##### Age 11 to 14 Challenge Level:

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n$ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

### Four Go for Two

##### Age 7 to 11 Challenge Level:

Four Go game for an adult and child. Will you be the first to have four numbers in a row on the number line?

### The Clockmaker's Birthday Cake

##### Age 7 to 11 Challenge Level:

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

### Times Right

##### Age 11 to 16 Challenge Level:

Using the digits 1, 2, 3, 4, 5, 6, 7 and 8, mulitply a two two digit numbers are multiplied to give a four digit number, so that the expression is correct. How many different solutions can you find?

### Square Subtraction

##### Age 7 to 11 Challenge Level:

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

### Tug Harder!

##### Age 7 to 11 Challenge Level:

In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first?


text:
Lifetimes are pretty much what makes Rust Rust.

Easy concurrency, straightforward memory allocation, and overall data safety would not be possible without explicit lifetimes.

But they are also tricky, and this is aimed at helping people understand the concepts and syntax.

Rust is a unique language in that it deallocates memory on the heap without requiring the writer to call free, while at the same time having no need for a garbage collector. Rust knows when it’s okay to use a reference by keeping track of its lifetime.

Each time a reference is returned by or passed into a function, Rust checks at compile time to make sure it fulfills the lifetime requirement specified in the type signature.

So every reference in Rust (i.e. pointer) has a lifetime. A lifetime is part of the type signature for any reference. Sometimes they can be elided and the compiler can infer them. Nonetheless, you cannot program Rust without knowing how to specify lifetimes.

Lifetimes fulfill two roles for Rust:

1. To know when it’s safe to dereference a pointer

2. To allow data to be shared safely

Now, you are not in charge of defining lifetimes.

Sometimes, however, you will have to give names to existing lifetimes in order to change the default behavior of the compiler.

# Where do Lifetimes come from?

Lifetimes are always named with the same syntax. They look like <'a>, <'b>, or <'c>, and they are generally one letter prefaced by an apostrophe.

When you use one, though, there is more syntax involved. Depending on the context they can look like one of: something<'a>, Box<something + 'a> or &'a something

Note well: You will only ever write a lifetime within a type declaration.

So if you don’t define lifetimes, what does?

# Functions

This is a biggie. A function can define a lifetime that can be used in it’s type declarations.

This makes sense: If you want to take a reference to something on the function stack, you have to be prepared for it to disappear when the function is over. And if not, you need to make sure it has an appropriate lifetime.

In Go, you can do this:

func example_function() *int {
b := 3
return &b
}

func main() {
fmt.Println(*example_function());
}

However, you cannot do that in Rust:

fn example_function<'a>() -> &'a i32 {
let b = 3;
&b // this does not compile
}

fn main() {
println!("{}", example_function());
}

The reason is that &b does not live long enough to be dereferenced outside of the function that made it.

Why is this so important? Because this is what can happen in C:

int* example_function(void) {
int a = 2;
return &a;
}

void do_something(int b) {
// lies, don't do anything
}

int main(void) {
int* c = example_function();
do_something(3);
int d = 4;
printf("%i\n", *c);
}

In C this is Undefined Behavior. When I run this, it just prints “3”.

That’s right, the data can just change under your nose. I was honestly scared when I ran that. But Rust prevents crazy things like that from taking place!

So just to explicitly point out the syntax:

example_function<'a> is saying “for any lifetime called 'a…“. You can then go on to use this lifetime in the remainder of the type definition.

&'a i32 says “This is a reference to an integer that has lifetime 'a”, which means it has to last as long as “any lifetime”. However, b happens to only have a lifetime that lasts as long as the function’s scope, so Rust complains.

(Now at this point you might be asking how you actually would return a reference to 3 in Rust… that’s a more complicated question and the answer is to put it on the heap. Look up the Box type to learn more.)

And now on to bigger fish.

# Structs

If a struct includes a reference to something, then that reference damn sure better last as long as the struct.

Here’s how you can make sure of that:

## Wrong example:

struct Sheep {
age: &i32,
}

fn main() {
let a = 3;
let s = Sheep { age: &a };
println!("{};", s)
}

This gives the following error:

error: missing lifetime specifier [E0106]
age: &i32,
^~~~

Rust is mad. In the struct definition, you haven’t told it how long the reference to the i32 is allowed to stay around. And yet, in order for your code to be safe, it has to stick around for at least as long as the struct.

## Right Example

struct Sheep<'c> {
age: &'c i32,
}

fn main() {
let a = 3;
let s = Sheep { age: &a };
println!("{};", s)
}

You’ll notice that not much here has changed. But now Rust has pronounced your code safe — Hurray!

Here are the changes:

1. Sheep<'c> instead of Sheep
This is making a parameter for the struct in Rust — It’s lifetime might have to depend on the lifetimes of its fields, so now you are able to say how and which ones. (Note: you can do things like Sheep<'c, 'd> if you need more than one lifetime.)

2. age: &'c i32 instead of age: &i32
Now this says that the integer has to live for as long as the struct.

This is insanely impressive. With these small additions, Rust will now tell you if there is any chance of you having invalid data, even across threads. AND all of this happens at compile time without affecting the efficiency of your code.

# Implementations

There is one other place where a lifetime can be defined — Implementations.

It’s very similar to structs. Each implementation is an implementation of a certain trait for a struct. So if the struct requires an explicit lifetime, you need to have one to give it.

## Wrong Example

struct Sheep<'c> {
age: &'c mut i32,
}

impl Sheep {
fn grow_old(&mut self)  {
*self.age += 100
}
}

This fails with the following:

error: wrong number of lifetime parameters: expected 1, found 0 [E0107]
impl Sheep {
^~~~~

## Right Example

What’s the problem?

Sheep takes a lifetime parameter now, so one must be supplied:

struct Sheep<'c> {
age: &'c mut i32,
}

impl<'c> Sheep<'c> {
fn grow_old(&mut self) {
*self.age += 100
}
}

Note that impl<'c> names a new lifetime and Sheep<'c> says that all self’s in this implementation have at least the lifetime 'c.

This is a lot like an extenuation of the lifetimes with structs, but there is strange syntax with the impl<'c> Sheep<'c> so I wanted to point it out.


text:
Run a Test

run_test(
test,
envir = parent.frame(),
local = TRUE,
args = list(),
defaults = list(),
output = "stdout+stderr",
timeout = getOption("future.tests.timeout", 30)
)

## Arguments

test A Test. The environment where tests are run. Should tests be evaluated in a local environment or not. Arguments used in this test. (optional) Named list with default argument values. If TRUE, standard output is captured, otherwise not. Maximum time allowed for evaluation before a timeout error is produced.

## Value

Value of test expression and benchmark information.


text:
# Work is force into distance or displacement

1. Oct 29, 2013

### emmfranklin

Respected all,

If i push an object on the floor and drag it say 2 m in a straight line with force y. then the work done is force times displacement.
that is 2y.

first doubt

which formula is correct one
work = force multiplied by displacement
or
work = force multiplied by distance.

since most websites say ... work = force multiplied by displacement

hence

in my given example both the displacement and distance is the same. so ill get the same answer.

but

now if i push my box in a circle track and bring it back to the starting point . the distance travelled might be say 20 meters
but the displacement will be 0 meters.

so what will be my work done.

according to the above formulas

it can be either
20 y joules

or

0 joules

which is correct .

is it work done or not

2. Oct 29, 2013

### arildno

For a force of constant magnitude, working constantly in the strict direction of the velocity of the object (however that changes), the work done by that force is magnitude of force times distance travelled.

I'm not sure where you've read anything else (I'm not doubting you!), but I will at least inform you that English Wikipedia, in the section Mathematical Calculation, has the correct relationship.

Last edited: Oct 29, 2013
3. Oct 29, 2013

### sophiecentaur

This should be clear cut - but I don't think it is. If you are considering what we call the Useful Work Done On an object then the Force times Displacement formula will tell you. This is fine when you are dealing with a conservative field - like when you are taking an ideal mass up to the top of an ideal hill or moving a charge from one object to another, through space.
But, where there is not a conservative field, the work done should be calculated by integrating along the path of the motion. The work done on a real car, getting it to the top of a real hill, will very much depend upon the route taken.
I would advise not getting too tied up with which of the two methods is 'correct' and consider the context of the actual problem before choosing which to apply. Most real world problems are concerned more with Work Done By but Work Done On will give a good idea of the energy you might get back from the object you just lifted up.

4. Oct 29, 2013

### arildno

sophiecentaur is giving the proper formulation for the GENERAL case.
What I wrote is valid for the SPECIAL case that I outlined:
The work done by a force that a) Has CONSTANT magnitude and b) Remains strictly tangential to the particle's path, that work equals (magnitude of) force times distance.

In integral form:
$$W=\int_{t_{0}}^{t_{1}}\vec{F}\cdot\vec{v}dt=\int_{t_{0}}^{t_{1}} F\frac{\vec{v}}{||\vec{v}||}\cdot\vec{v}dt=\int_{t_{0}}^{t_{1}}F||\vec{v}||dt=F\int_{t_{0}}^{t_{1}}||\vec{v}||dt$$

The integral of the SPEED over the time interval equals the distance travelled.
(If the force is always antiparallell to the velocity (say, as in kinetic friction), but constant in magnitude, add a minus sign in the expression.)

Last edited: Oct 29, 2013
5. Oct 29, 2013

### sophiecentaur

I think the OP was asking why there seem to be two answers to the same simple question. I was (we are) just pointing out that the answer depends upon what you actually want to know in each case and that you should not slavishly follow one or the other approach.
Many students write in, asking for a general rule about lots of things. It is very risky to try to apply the same rule in every case. Half the time, you could be wrong! That doesn't mean basic Physics is a matter of choice - it means that you often need to dig deeper than the surface to find how to apply it.

6. Oct 29, 2013

### arildno

Sure!
That's why I felt it important in the second post to bring about explictly that MY case is a special case of what YOU wrote, by deriving my result from the general definition of work.

7. Oct 29, 2013


text:
# Proving a Cauchy Schwarz Inequality [closed]

Let $a_1,a_2,\cdots,a_n ;b_1,b_2,\cdots,b_n \in \mathbb{R}^{+}$. Prove that:

$$\sqrt{a_1^2+b_1^2}+\sqrt{a_2^2+b_2^2}+\cdots +\sqrt{a_n^2+b_n^2}\geq \sqrt{(a_1+a_2+\cdots+a_n)^2+(b_1+b_2+\cdots+b_n)^2 }$$

## closed as off-topic by Davide Giraudo, Leucippus, pjs36, Shailesh, Claude LeiboviciApr 18 '17 at 6:39

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

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Davide Giraudo, Leucippus, pjs36, Shailesh, Claude Leibovici
If this question can be reworded to fit the rules in the help center, please edit the question.

• Hint: prove for $n=2$ and then use induction. – ntt Apr 17 '17 at 15:36
• It's just Minkowski – Michael Rozenberg Apr 17 '17 at 17:49

By C-S $$\sum_{i=1}^n\sqrt{a_i^2+b_i^2}=\sqrt{\sum_{i=1}^n(a_i^2+b_i^2)+2\sum_{1\leq i<j\leq n}\sqrt{(a_i^2+b_i^2)(a_j^2+b_j^2)}}\geq$$ $$\geq\sqrt{\sum_{i=1}^n(a_i^2+b_i^2)+2\sum_{1\leq i<j\leq n}(a_ia_j+b_ib_j)}=$$
$$=\sqrt{\sum_{i=1}^n a_i^2+2\sum_{1\leq i<j\leq n}a_ia_j+\sum_{i=1}^nb_i^2+2\sum_{1\leq i<j\leq n}b_ib_j}=$$ $$=\sqrt{(a_1+a_2+...+a_n)^2+(b_1+b_2+...+b_n)^2}$$ and we are done!


text:
Linear Yang-Mills theory as a homotopy AQFT

Marco Benini, Simen Bruinsma, Alexander Schenkel
June 03, 2019
It is observed that the shifted Poisson structure (antibracket) on the solution complex of Klein-Gordon and linear Yang-Mills theory on globally hyperbolic Lorentzian manifolds admits retarded/advanced trivializations (analogs of retarded/advanced Green's operators). Quantization of the associated unshifted Poisson structure determines a unique (up to equivalence) homotopy algebraic quantum field theory (AQFT), i.e. a functor that assigns differential graded $\ast$-algebras of observables and fulfills homotopical analogs of the AQFT axioms. For Klein-Gordon theory the construction is equivalent to the standard one, while for linear Yang-Mills it is richer and reproduces the BRST/BV field content (gauge fields, ghosts and antifields).

Keywords:
algebraic quantum field theory, locally covariant quantum field theory, Gauge theory, derived critical locus, homotopical algebra, chain complexes, BRST/BV formalism


text:
English Információ A lap Pontverseny Cikkek Hírek Fórum

Rendelje meg a KöMaL-t!

VersenyVizsga portál

Kísérletek.hu

Matematika oktatási portál

B. 4867. The sum of the real numbers $\displaystyle a$, $\displaystyle b$, $\displaystyle c$ and $\displaystyle d$ is $\displaystyle 0$. Let $\displaystyle M=ab+bc+cd$ and $\displaystyle N=ac+ad+bd$. Prove that at least one of the sums $\displaystyle 20M+17N$ and $\displaystyle 20N+17M$ is non-positive.

(Bulgarian problem)

(4 points)

Deadline expired on 10 May 2017.

Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás. A $\displaystyle 20M+17N$ és $\displaystyle 20N+17M$ számok összege

$\displaystyle 37(M+N)=37(ab+ac+ad+bc+bd+cd)=$

$\displaystyle =37\cdot \frac{(a+b+c+d)^2-(a^2+b^2+c^2+d^2)}{2}=-\frac{37}{2}(a^2+b^2+c^2+d^2)\leq 0,$

így legalább az egyikük nem pozitív.

Statistics on problem B. 4867.
 86 students sent a solution. 4 points: 78 students. 3 points: 2 students. 2 points: 1 student. 1 point: 1 student. Unfair, not evaluated: 2 solutions. Unfair, not evaluated: 2 solutions.

• Problems in Mathematics of KöMaL, April 2017

•  Támogatóink: Morgan Stanley


text:
## Prealgebra (7th Edition)

what percent of 40 is 8 ? what percent is percent 40 is base 8 is amount $\frac{8}{40}$=$\frac{p}{100}$ set cross product equal 40$\times$ p = 8 $\times$ 100 Multiply 40p= 800 divide both sides by 40 $\frac{40p}{40}$ = $\frac{800}{40}$ p= 20 8 is 20 % of 40.


text:
Dataset

Business tendency indicators in construction by kind of activity (balances, %)

Table ID
KR070m
Most recent data period
Most recent data release
Next data release
Confidence indicators in construction

Confidence indicator in construction is a composite indicator characterising the general development of construction sector in timeseries.

It is calculated as net balance of the answers to two questions: level of current order books and employment expectations in next 3 months. Balance is calculated as difference between positive and negative answers, expressed in per cent.


text:
# An Example Setup for Using Krita with an Eye Tracker¶

Attention

This is not a reference document. It is based on the experiences of only one user. The information might not be as applicable when using different eye tracker devices or different control software.

Eye tracker devices are becoming more affordable and they are finding their way into more computer setups. Although these devices are used by various types of users, we will mainly focus on users who have physical disabilities and can only use their eyes to interact with the computer.

If you don’t already have experience with such a case, here are a few things you’ll need to know before you start:

• The eye tracker needs to be properly calibrated such that the pointer will be very close to the point where the user is looking at. This might be difficult to achieve, especially if the positioning of the eye tracker with respect to the user can not be fixed between different sessions.

• The lack of accuracy in control makes it nearly impossible to hit small areas on the screen such as small buttons or menu items. Corners and edges of the screen might be difficult to reach too. You also don’t want to put interface elements close to one another since it increases the chances of selecting the wrong element accidentally.

• Mouse operations like single click, double click, right click, drag and drop, etc. all demand extra effort in the form of switching modes in the program that controls the device. You will want to keep these switches to a minimum so that the work will not be interrupted frequently.

• Switching the mode doesn’t automatically start the operation. You need an extra action for that. In our case, this action is “dwelling”. For example, to start a program, you switch to the left double click mode and then dwell on the icon for the application to activate the double click. Adjusting the dwell time is an important tradeoff: shorter dwell times allow for faster work but are also more error-prone.

## Requirements¶

Besides the obvious requirement of having an eye tracker device, you will also need a control program that will let you interact with the device. When you obtain the device, such a program will most probably be provided to you but that program might not be sufficient for using the device with Krita.

One of the basic functionalities of these programs is to emulate mouse clicks. In our case, the program provides a hovering menu which includes large buttons for switching modes between left/right mouse buttons and single/double clicks. After selecting the mode, the hovering menu can be collapsed so that it will leave more screen space for the application.

In order to make them easier to configure and use, some programs include only basic modes like single clicks. This is sufficient for many popular applications like e-mail agents and browsers, but for Krita you need the drag and drop mode to be able to draw. If the provided control software doesn’t support this mode (usually called “mouse emulation”), you can contact the manufacturer of the device for assistance, or look for open source options.

## Starting Krita¶

Basically, setting the control program to left double click mode and dwelling on the Krita icon on the desktop would be enough to start up Krita but there are some issues with this:

• On startup, Krita asks you to choose a template. It’s likely that you don’t want to go through this setting every time and just want to start with a blank template.

• Later, saving the document will require interacting with the file save dialog which is not very friendly for this type of use.

A workaround for these issues could be creating and saving a blank template and running a script that will copy this template under a new name and send it to Krita. Here’s an example script for Windows which uses a timestamp suffix to make sure that each file will have a different name (replace USERNAME with the actual user name):

@echo off
for /f "tokens=2 delims==" %%a in ('wmic OS Get localdatetime /value') do set "dt=%%a"
set "YY=%dt:~2,2%" & set "YYYY=%dt:~0,4%" & set "MM=%dt:~4,2%" & set "DD=%dt:~6,2%"
set "HH=%dt:~8,2%" & set "Min=%dt:~10,2%" & set "Sec=%dt:~12,2%"
set "datestamp=%YYYY%%MM%%DD%" & set "timestamp=%HH%%Min%%Sec%"
set "fullstamp=%YYYY%-%MM%-%DD%_%HH%-%Min%-%Sec%"
start "C:\Program Files\Krita (x64)\bin\krita.exe" "%filename%"

Double clicking on this script will create a new Krita file in the same folder as the script file. Since the file already has a name, the file save dialog will be avoided. Combined with autosaving, this can be an efficient way to save your work.

Tip

Storing these files directly on a cloud storage service will be even safer.

You might also deal with some timing issues when starting Krita:

• After the icon for Krita or for the script is double clicked and Krita starts loading, lingering on the icon will start a second instance.

• Similarly, after double clicking, if another window is accidentally brought to the foreground, Krita might start up partially visible behind that window.

To prevent these problems, it will help if the users train themselves to look at some harmless spot (like an empty space on the desktop) until Krita is loaded.

## Layout¶

Since the interface elements need to be large, you have to use the screen area economically. Running in full-screen mode and getting rid of unused menus and toolbars are the first steps that you can take. Here’s the screenshot of our layout:

You will want to put everything you need somewhere you can easily access. For our drawings, the essential items are brushes and colors. So we’ve decided to place permanent dockers for these.

Krita features many brushes but the docker has to contain a limited number of those so that the brush icons can be large enough. We recommend that you create a custom brush preset to your own liking.

There are various tools for selecting color but most of them are not easily usable since they require quite a high level of mouse control. The Python Palette Docker is the simplest to use where you select from a set of predefined colors, similar to brush presets. Again, similarly to brush selection, it will help to create a custom set of favorite colors.

Once you are happy with your layout, another feature that will help you is to lock the dockers. It’s possible to accidentally close or move dockers. For example, in drag and drop mode you can accidentally grab a docker and drag it across the screen. To prevent this, put the following setting in the kritarc file:

LockAllDockerPanels=true

(Check the Krita FAQ for how to find the configuration kritarc file on your system.)

If you’re using a hovering mouse control menu like we do, you also have to figure out where to place it when it’s collapsed. Put it somewhere where it will be easily accessible but where it will not interfere with Krita. On the screenshot you can see it at the left edge of the screen.

## Summary¶

In summary, we work as explained below.

To start Krita:

1. On the desktop, pull up the hovering mouse menu and select left double click mode.

2. Double click on the new drawing creation script. Look away at some harmless spot until Krita loads.

Drawing with Krita:

1. Switch to left single click mode.

2. Select a brush and/or color using the dockers.

3. Switch to drag and drop mode. You’re ready to draw.

4. Go to the point where you want to start a stroke and dwell until dragging starts (this emulates pressing and holding your finger on the mouse button).

5. Draw.

6. When you want to finish the current stroke, dwell at the ending point until you get out of dragging (this emulates lifting your finger from the mouse button).

7. Repeat the whole process.

Finishing:

1. Switch to left single click mode.

2. Click on the button for closing the window.

3. When warned about unsaved changes, click the button for saving the file.


text:
## STPM - L^p Concentration of Semiclassical Quasimodes

Melissa Tacy
September 30, 2010

## STPM - Torsion-Freeness of Certain Cohomology Groups of PEL-Type Shimura Varieties

Junecue Suh
September 30, 2010

## STPM - Sparce Approximation of PSD Matrices

Nikhil Srivastava
September 30, 2010

I will discuss the problem of approximating a given positive semidefinite matrix A , written as a sum of outer products $vv^T$ , by a much shorter weighted sum in the same outer products. I will then mention an application to sparsification of finite undirected graphs.

## STPM - Some Questions Regarding Arithmetic Circuits

Srikanth Srinivasan


text:
# Tag Info

1

A more general theorem in Folland's Real Analysis can answer this question:

1

Assuming we've shown that $\mu$ is a measure, and that we're only concerned with bounded sets: Say $B$ is a bounded closed set and $E\subset B$ is measurable. First, note that $$\mu(E)=\inf\{\mu(V):E\subset V\subset B, V\text{ relatively open}\}.$$ (Here "relatively open" means open relative to $B$; that is, $V=B\cap W$ where $W$ is an open subset of $\Bbb ... 3 This is a general fact about seminormed spaces. You have a family of seminorms$\|\cdot\|_{\alpha,\beta}$. The sets$U_{\alpha,\beta,r} = \{\phi : \|\phi\|_{\alpha,\beta}<r\}$form a sub-basis of neighborhoods of zero, meaning that their finite intersections yield an basis of such neighborhoods. This is precisely the topology for which the convergence ... 1 This is indeed correct. You might want to expand on why these (basic) open sets are indeed disjoint (triangle inequality argument). Abstractly, this argument works because the point-evaluations$e_v$, where$v \in V$, form a family of functions (to a Hausdorff space, namely the underlying field) that separates points, in this case basically by definition ... 2 For$(1)$, to prove that$\mathcal D(U)$with$\{\| \cdot \|\}_{m \in \mathbb N}$is a locally convex topological vector space, you need to show that$\{\|\cdot\|_m\}_{m \in \mathbb N}$is a separating family of semi-norms on$\mathcal D(U)$. The question already says "norms" in it, so maybe you just have to show that it's separating, i.e., given$\phi \in ...

1

My question is: if I permit partially ordered countable families of Fréchet spaces, but still require the inductive limit to be strict (as defined above), do I wind up with the same class of LF-spaces, or is this definition actually more general? Fact: The limit $E$ of a countable inductive system $(E_i)_{i \in I}$ of Fréchet spaces is already an ...

1

Hint: ${s\over{s+t}}x+{t\over{s+t}}y\in A$ if $x,y\in A$ and $A$ is convex, thus $(s+t)({s\over{s+t}}x+{t\over{s+t}}y)=sx+ty\in (s+t)A$

Top 50 recent answers are included


text:
# Tag Info

9

Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html), $G$ has an abelian normal subgroup of index at most $(d+1)!$, which must be trivial since $G$ is ...

8

Based on Sean Eberhard's comment, it follows that if $f(n)$ is the number of non-isomorphic abelian subgroups of $S_n$, then $f(n)\geq p_{\mathbb{P}}(n)$, where $p_{\mathbb{P}}(n)$ is the number of partitions of $n$ into prime parts. It was proved by Roth and Szekeres that $$\log p_{\mathbb{P}}(n) = \frac{2 \pi}{\sqrt{3}}\Big(\frac{n}{\log n}\Big)^{1/2}\Big(... 7 Claim 1: (same as my comment) Let q_1, \dots, q_k > 1 be prime powers. Then G = C_{q_1} \times \cdots \times C_{q_k} is isomorphic to a subgroup of S_n if and only if q_1 + \cdots + q_k \leq n. Proof: If q_1 + \cdots + q_k \leq n we can choose disjoint cycles of lengths q_1, \dots, q_k, so the condition is clearly sufficient. Conversely ... 6 It is a standard fact that the automorphism group of G=\mathrm{SL}_2(\mathbf{R}) as topological group equals \mathrm{PGL}_2(\mathbf{R}), which is also the automorphism group of the Lie algebra viewed as \mathbf{R}-algebra. In particular, the outer automorphism group as topological group is cyclic of order 2. I claim that this is the same as abstract ... 6 The answer is yes: \text{Rad}(kG)^s is generated as an ideal by (g-1)^s for G an elementary abelian p-group and s \leq p-1. Lemma: Let V be a k-vector space and let s \leq p-1. Then \text{Sym}^s(V) is spanned by the elements v^s for v \in V. Proof: \text{Sym}^s(V) is clearly spanned by products of the form v_1 v_2 \cdots v_s. We ... 3 Under these assumptions, note that \mathrm{Rad}(kG) coincides with the augmentation ideal \omega(kG) of kG. Now, as remarked by Jeremy Rickard, the question is equivalent to whether the elements (g-1)^s with g\in G generate \mathrm{Rad}^s(kG) as an ideal of kG. Theorem 3.7 in Section 5.3 of the book [D.S. Passman: The algebraic structure of ... 3 Yes, there is a combinatorial way to see this. Firstly, it is much easier to show a baby version of this type of phenomenon. A while ago I read a nice blog post by Qiaochu explaining the identity$$\int_{0}^1 (2\cos \pi x)^n (2\sin^2\pi x)dx=\begin{cases} 0 &\text{ if n is odd} \\ C_{n/2} & \text{ if n is even} \end{cases} by interpreting this ...

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This was not originally an answer to the question you asked but to the converse question about geodesic paths in $X$ mapping to quasigeodesic paths in $\widehat{X}$. But you asked for a reference for that in a comment, so here it is. But note that this result involves extending the given generating set of $G$ to a larger finite generating set by adjoining ...

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This is a (cw) post to illustrate Claim 2 from the answer by Sean Eberhard. Below is the plot of the list $(\frac1n,\frac{\log f(n)}{\pi\sqrt{2/3}(n/\log n)^{1/2}})$ for $n$ from $1000$ to $10000$. I used the list of values from OEIS A023893. As you see after $n$ about 3000 the sequence sort of changes behavior; at about $n=5000$ (to be precise, at $5226$) ...

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We denote by $\mathbb{Z}/m$ the cycle group of order $m$. Some examples: The finite groups $\mathbb{Z}/{p_1^{a_1}}\oplus \cdots \oplus \mathbb{Z}/p_k^{a_k}$, where the $p_i$ are pairwise distinct primes and the $a_i$ are positive integers such that $\sum_i p_i^{a_i}\leq n$, can be clearly embedded into $S_n$. Edit: I also thought to be true that whenever $n=... 2 Theorem (Elek–Szabo). Let$G$be an infinite residually finite hyperbolic group with Property (T). Then$P_G\$ is a finitely generated sofic group that is not residually amenable. That's Theorem 3 of Elek, Gábor; Szabó, Endre, On sofic groups., J. Group Theory 9, No. 2, 161-171 (2006). ZBL1153.20040 arXiv:math/0305352. The construction is also somewhat ...

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