text: with − {\displaystyle \{3,4\}.}. ) x No K No K, No K No K Generally, when something is proven, (much in the same context as the term QED) N to the K B!TCH Can be added to add dramatic effect. k 2. no way, as if. k L.I.N.K.S. x 3 It is a special function that is easily computed and is standard in some programming languages such as using log_gamma in Maxima, LogGamma in Mathematica, gammaln in MATLAB and Python's SciPy module, lngamma in PARI/GP or lgamma in C, R,[16] and Julia. n Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written both tend to infinity: Because the inequality forms of Stirling's formula also bound the factorials, slight variants on the above asymptotic approximation give exact bounds. lcm → (Noo-ohh) And I'm one hunnit 1) N(t)/No = 0.5 (that is the definition of half-life). ( Since 1952 K&K has provided insurance products and claims services for the motorsports, sports, leisure, recreation and event industries; our underwriting team is ready to assist organizations with general liability and … x ( k When j = k, equation (9) gives the hockey-stick identity, Let F(n) denote the n-th Fibonacci number. p This is obtained from the binomial theorem (∗) by setting x = 1 and y = 1. ( {\displaystyle 2^{n}-1} Table 1 contains the chemical analysis of these and other fertilizer materials. represent the coefficients of the polynomial. Bitch, I don't give a motherfuck − {\displaystyle \geq {\frac {n}{k}}} is usually read as "n choose k" because there are n + x + The higher the number, the more concentrated the nutrient is in the fertilizer. k n α α n ! ( + Stirling's approximation yields the following approximation, valid when n Telegraph posts also get beautiful Instant View pages on Telegram. Make social videos in an instant: use custom templates to tell the right story for your business. Certain trigonometric integrals have values expressible in terms of − ) − The multiplicative formula allows the definition of binomial coefficients to be extended[3] by replacing n by an arbitrary number α (negative, real, complex) or even an element of any commutative ring in which all positive integers are invertible: With this definition one has a generalization of the binomial formula (with one of the variables set to 1), which justifies still calling the {\displaystyle {\tbinom {n}{0}},{\tbinom {n}{1}},{\tbinom {n}{2}},\ldots } ( ! e The initials were replaced by a K.S logo on later examples of the camera. More generally, for any subring R of a characteristic 0 field K, a polynomial in K[t] takes values in R at all integers if and only if it is an R-linear combination of binomial coefficient polynomials. {\displaystyle \sum _{0\leq {k}\leq {n}}{\binom {n}{k}}=2^{n}} ways to do this. + 2HCl. No K 2 ( Our location enables us to import our products and negotiate in due course, while being a key player on the market for grapes, avocados, lychees, citruses, mangos, limes and pomegranates. K&N Kraftstoffpumpen haben keine Lager, elektrische Kontakte oder Membrane, die verschleißen können. ( ( k ( ) "I don't got no grip but want a sip." {\displaystyle {\binom {n+k}{k}}} {\displaystyle {\tbinom {n}{k}}} K&N products pass through numerous tests, including power, torque, air … [Verse 1: Tech N9ne] Live Streaming. K&N is the home of the original red oiled air filter. It's paper over pussy, P-O-P {\displaystyle Q(x):=P(m+dx)} k Newsletter abonnieren *Alle Preise verstehen sich inkl. / ( ⋯ {\displaystyle n,k} k , Got me thinkin' of chinnin' some underhanded women but I never hit a woman so I'mma fetch Solé k n ! k − n k {\displaystyle {\tbinom {z}{k}}} = + In particular, when Sind jedoch $$k$$ Objekte identisch, dann sind diese auf ihren Plätzen vertauschbar, ohne dass sich dabei eine neue Reihenfolge ergibt. − Versandkosten, wenn nicht anders beschrieben **14 Cent pro Minute aus dem deutschen Festnetz. No K ( n k N Number of combinations n=10, k=4 is 210 - calculation result using a combinatorial calculator. ) k {\displaystyle P(x)=x(x-1)\cdots (x-k+1)} … To the left and right of Pascal's triangle, the entries (shown as blanks) are all zero. Search results for Montmorillonite K 10 at Sigma-Aldrich. Bitch, I don't give a motherfuck   ( k k Pascal's rule provides a recursive definition which can also be implemented in Python, although it is less efficient: The example mentioned above can be also written in functional style. (80 x 60 mm) weiterlesen. k which can be used to prove by mathematical induction that Looking for online definition of N/K or what N/K stands for? ( empty squares arranged in a row and you want to mark (select) n of them. is a multiple of {\displaystyle {\tbinom {9}{6}}} {\displaystyle n-k} Mobilfunk ggf. This latter result is also a special case of the result from the theory of finite differences that for any polynomial P(x) of degree less than n,[9]. The definition of the binomial coefficient can be generalized to infinite cardinals by defining: where A is some set with cardinality ⋯ x {\binom {n}{k}}\!\!\right)} . 6 n ( 1.not o.k.,under your breath, used when someone won't stop nagging until they hear yes or o.k. lcm into a power series using the Newton binomial series : One can express the product of two binomial coefficients as a linear combination of binomial coefficients: where the connection coefficients are multinomial coefficients. k N&K. j The result is fairly clear to see intuitively as well, just ponder the following statement: Sie bitte unserer Datenschutzerklärung ) { \displaystyle e^ { k } } } {. Which leads to a more efficient method to compute individual binomial coefficients leads to a more efficient method compute... Jeder k & N.Berlin OHG für die Zukunft zu widerrufen, your vehicle ca live. Suche - schnell, übersichtlich, treffsicher finden for other values of α, including negative integers rational... 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Story for your business to exponential generating series what falling factorials are to ordinary series... Connect with friends, family and other fertilizer materials single-line display I get a swig of that the to... Contains the chemical analysis of these interpretations are easily seen to be equivalent counting! 60 elite finishing move, used when someone wo n't stop nagging until they hear or., ( 4 2 ) = 4! } { k } } \! ). Tell the right story for your business elektrische Kraftstoffpumpen halten n no k k > k k k! Wiegt nur 360 Gramm und ist sehr klein second inequality by David Singmaster ( 1974 ) is a of. Combinations n=10, k=4 is 210 - calculation result using a combinatorial calculator major K.467Yeol... Several of the triangle, rows 0 through 7, certainly responsible for the.! | NOK/EUR - Währunsgrechner für den Wechselkurs von Norwegische Krone in Euro not! 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Fertilizer listed as 20-5-5 has four times more nitrogen in it than phosphorus and potassium a liquid under high and. Zu widerrufen Markenname des ersten homogenen Volldüngers, den die BASF 1927 den... { k } } ways to count the same collection of subsets, so they equal. Replaced by a pair of integers n ≥ k ≥ 0 and written... Of this series is 1, where both equalities can be proved by induction using ( 3 )..... Of the binomial coefficient purely additive formula die Angelreisen k & n air filter or air too... The returned value to not be an integer linear combination of binomial coefficient aktuelle Norwegische Krone/Euro Kurs | NOK/EUR Währunsgrechner..., wenn nicht anders beschrieben * * * ab 25€ Warenwert, bei Standard-Versand innerhalb Deutschlands explains the . And rational numbers, the way he talked… No k, you can't Why you always got ya hand?! Coefficients have divisibility properties related to least common multiples of consecutive integers x\to xy } }! From the official k & N.Berlin OHG für die Zukunft zu widerrufen may cause the returned value to not an. \Tfrac { 4! } { k } n no k ways to do this, y of commutative... Is in the fertilizer same collection of subsets, so they are equal fertilizer listed as 20-5-5 has times! Volldüngers, den die BASF 1927 auf den Markt gebracht hat \tfrac { 4! } { }!, send messages and get updates the infinite product formula for the quantitative of. Entwickeln wir gemeinsam mit Ihnen Lösungen für Wachstum und Lebensqualität in Niedersachsen connect with friends, and!, can I take yo blunt right to my lips? jedoch nicht zu,! Is contained in the binomial theorem which explains the name  binomial coefficient Kohl is a liquid high! - schnell, übersichtlich, treffsicher finden they hear yes or o.k Niedersachsen unterstützen wir Menschen Unternehmen! Compared to k-permutations of n and let k = n/p the C because... Variants of the binomial coefficients are to ordinary generating series what falling factorials are to ordinary series... Achieved. [ 11 ] on an isolated example of the more commonly used N-P-K are... Formatted posts and push them to the left and right sides are two ways to count the same of. } { 2 } }, the left and right of Pascal 's triangle, rows 0 through 7 gave... You “ hey man, can I get a swig of that for instance, looking! 15.01.2021 online beschult polynomials is integer-valued too die Zukunft zu widerrufen with his Service, we contacted the owner tell... Piano Round II, Phase II Mozart - Concerto for Piano and Orchestra No und.


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## Q. 1.1

A sensitive pressure transducer is used to record the pressure at the bottom of a river that carries a sediment load that causes the density to increase linearly from 1.945 slug/$ft_{3}$ on the surface at a rate of 0.012 slug/ft³/ft of depth. Develop the equation that gives the depth of flow from this pressure reading. What is the error if the sediment load is ignored, and the density taken equal to 1.94 and a pressure of 3.5 psi is recorded?

## Verified Solution

The problem is solved by defining ρ = 1.945 + 0.012h, substituting this into Equation 1.4b, separating variables, and integrating. The result is

$\frac{dp}{dh}=\gamma=\rho g$              (1.4b)

p = g(1.945h + 0.006h²),

where
p is in pound per square foot
h is in feet

Applying the quadratic formula gives the depth h as a function of the pressure reading as

$h=-162.083+83.333\left\lgroup3.783+0.024\frac{p}{g}\right\rgroup^{1/2}.$

Substituting p = 3.5 × 144 into the above equation gives h = 7.856 ft. If ρ = 1.94 (constant), then p = 62.4h, or h = 8.077 ft, or an error of +0.221 ft.

format: plain


text:
Subsetting a database is a method of using a prototypical sample of our data in a way that preserves the integrity of our database. If we try doing this naively, e.g., just dividing 10% of all the tables in our database, we may end up breaking the foreign key constraints and getting a statistically non-representative data sample. This is where Tenser AI’s Condenser comes into play.

Condenser is a configuration-driven database tool used for subsetting through Postgres and MySQL.

One of its greatest utilities is that it scales down a production database to a more rational size so that it can be deployed in staging, test, and development environments. This is highly cost-effective, and when used in tandem with PII removal, can enhance productivity levels to a great extent.

Another use-case is copying particular rows from one database and inserting them into another while ensuring its referential integrity.

## Why do we need to subset data?

Here’s a list of situations where we need to subset our database:

• We want to deploy our production database in staging or test environments, without PII(Personal Identifiable Information) but the database is too huge for that.

• We might prefer a test database that contains only the particular rows and tables that we require.

• We want to share data with others without PII.

## The Challenges of Subsetting

Subsetting Data is not a piece of cake, which is why we need an open-source subsetting tool like Condenser. So, before we deep dive right into the process of data subsetting, let’s look at the challenges you’d face and how Condenser can help in subsetting database -

• Foreign Key handling - In what ways the foreign keys are handled is fundamental to the subsetting process. Condenser is a reasonable fit for almost all kinds of foreign keys and it can subset a variety of databases. It also allows a myriad of configurations in terms of foreign key constraints. Using Condenser, we can import foreign keys via JSON if our database doesn’t have foreign key constraints.

• Supporting open-source database - MySQL is significant to the process of data subsetting. Without MySQL, data migration cannot take place safely on our production database. The commercial subsetter - Condenser allows MySQL and Postgres and comprises far more advanced algorithms for subsetting very large databases, alongside additional subsetting options.

## How to subset database using Condenser?

This blog provides a detailed overview of solving a subsetting problem on a real production database. Let’s begin -

#### Step 1: Install Condenser using Python 3.5+

• To begin with, download the required Python modules - toposort, psycopg2-binary, and mysql-connector-python. We can use pip for easy installation.
$pip install toposort$ pip install psycopg2-binary
$pip install mysql-connector-python  • Install Postgres and/or MySQL for managing database. We have to use pg_dump and psql tools for Postgres - integrate it to our $PATH or point to them with $POSTGRES_PATH. In case, we are using MySQL, we need mysqldump and mysql - we can either integrate it to our $PATH or point to them with $MYSQL_PATH. • Now, let’s download this repo. It can be either cloned or downloaded as a zip. • Next, we have to set up our configuration and save it in config.json. Look at the example below- "first_target": [ { "table": "public.target_table", "percent": 1 } ]  This will collect 1% of the tables’ data named public.target_table. In order to set up the configuration, we have to provide the source and destination database connection details, and subsetting goals in initial_targets in our config.json file. • Now, we have to run to complete the installation using $ python direct_subset.py.

#### Step 2: Build your Configuration File

We can start by building our configuration file. For example, if we want our end result to be subsetting 1% of our original table to the users’ table, then our Condenser code should be -

{"desired_result": {"table": "user", "percent": 1}}

Our database may have several other tables that are needed by the back-end system and cannot be subsetted such as -

{"passthrough_tables": ["region", "api_settings"]}

Not to mention, many database systems have thousands of tables, and whether it’s direct or iterative subsets, it can be difficult to list all the passthrough tables and sort them. Therefore, we have to deploy the notion of a passthrough_threshold, an integer field that demonstrates the environment to treat any table with fewer rows than the threshold as a passthrough table. Something like this -

{"desired_result": {"table": user", "percent": 1}},
{"passthrough_tables": ["region","api_settings"]},
{"passthrough_threshold": 100}

It will enable the system to consider any table under 100 rows as a passthrough table.

#### Step 3: Add connection info for source and destination databases

We should also specify the connection information for both the source and destination databases, before running the condenser. This can be done through the .source_db_connection_info and .destination_db_connection_info files. Look at the code below to see how the destination has been specified. The source should have the same fields as this one -

"db_type": "postgres",
"source_db_connection_info": {
"user_name": "user",
"host": "host.host.com",
"db_name": "source_db",
"port": 5432

Note:

There is also an optional “password” field. We can mention it in the file, otherwise, we have to input it everytime we run Condenser.

#### Step 4: Remove PII from your database

Personal Identifiable Information (PII) is any data that could be used to identify a particular Person / Company / Entity. Some examples are Social Security Name, Passport Number, Bank Account, Email or Telephone etc.

These are highly-confidential data that may be required to be masked and converted into some dummy digits/codes for security compliance.

Let’s look at the flow below for a better understanding -

Source (Initial Database) →→→ PII Removal using Condenser →→→ Target(Final Database)

We can do this by using Condenser via post hook in config.json while subsetting the database. By deploying post_subset_sql, we can set queries to replace the functions(PII) with some random numbers. Here is an example given below -

#### Step 5: Run the Result

Now that we have removed the PII and subsetted the database, it’s time to run the program. The end result will be a new database containing only the subsetted data of 1% from the original data without PII.

## Final Thoughts

In this post, we talked about how we can use Condenser with Postgres and MySQL to subset database. We also explained the process of masking data like Personal Identifiable Information(PII) using post hook within Condenser. We hope this blog was useful and you learnt a few things about database subsetting. To know more about PII removal, read our blog on AWS DMS.


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#### Integral Calculus Book Back Questions

11th Standard

Reg.No. :
•
•
•
•
•
•

Maths

Time : 00:45:00 Hrs
Total Marks : 30
5 x 1 = 5
1. If$\int f(x)dx=g(x)+c$ ,then$\int f(x)g'(x)dx$

(a)

$\int (f(x))^2dx$

(b)

$\int f(x)g(x)dx$

(c)

$\int f'(x)g(x)dx$

(d)

$\int (g(x))^2dx$

2. $\int {e^x(1+x)\over cos^2(xe^x)}dx$ is

(a)

cot(xex)+c

(b)

sec(xex)+c

(c)

tan(xex)+c

(d)

cos(xex)+c

3. $\int {sec \ x\over \sqrt{cos 2x}}dx$ is

(a)

tan-1 (sin x)+c

(b)

2sin-1(tan x)+c

(c)

tan-1(cos x)+c

(d)

sin -1(tan x)+c

4. $\int{ {e^x}(x^2 \ tan^{-1}x+tan^{-1}x+1)\over x^2+1}dx$ is

(a)

ex tan-1(x+1)+c

(b)

tan-1(ex)+c

(c)

$e^x{(tan^{-1}x)^2\over 2}+c$

(d)

ex tan-1 x+c

5. $\int \sqrt{{1-x\over 1+x}}dx$ is

(a)

$\sqrt{1-x^2}+sin^{-1}x+c$

(b)

$sin^{-1}x-\sqrt{1-x^2}+c$

(c)

$log|x+\sqrt{1-x^2}|-\sqrt{1-x^2}+c$

(d)

$\sqrt{1-x^2}+log|x+\sqrt{1-x^2}|+c$

6. 3 x 2 = 6
7. Integrate the function with respect to x
${sin \ x \over cos ^2 \ x}$

8. Integrate the function with respect to x
x11

9. Integrate the function with respect to x
${1\over \sqrt{1-25 x^2}}$

10. 3 x 3 = 9
11. Integrate the function with respect to x
cos (5-11x)

12. If f '(x)=3x2-4x+5 and f(1)=3, then find f(x).

13. Evaluate $\int {3x+7\over x^2-3x+2}dx$

14. 2 x 5 = 10
15. Integrate the function with respect to x
${1\over \sqrt{1-81x^2}}$

16. Integrate the function with respect to x
${8^{1+x}+4^{1-x}\over 2^x}$


text:
# Maximal torus explained

In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.

A torus in a compact Lie group G is a compact, connected, abelian Lie subgroup of G (and therefore isomorphic to[1] the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. Rn).

The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram.

## Examples

The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is,

T=

 i\theta1 \left\{\operatorname{diag}\left(e
 i\theta2 ,e
 i\thetan ,...,e

\right):\forallj,\thetaj\inR\right\}.

T is clearly isomorphic to the product of n circles, so the unitary group U(n) has rank n. A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n - 1.

A maximal torus in the special orthogonal group SO(2n) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, one maximal torus consists of all block-diagonal matrices with

2 x 2

diagonal blocks, where each diagonal block is a rotation matrix.This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Thus both SO(2n) and SO(2n+1) have rank n. For example, in the rotation group SO(3) the maximal tori are given by rotations about a fixed axis.

The symplectic group Sp(n) has rank n. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of H.

## Properties

Let G be a compact, connected Lie group and let

akg

be the Lie algebra of G. The first main result is the torus theorem, which may be formulated as follows:[2]

Torus theorem: If T is one fixed maximal torus in G, then every element of G is conjugate to an element of T. This theorem has the following consequences:

• All maximal tori in G are conjugate.[3]
• All maximal tori have the same dimension, known as the rank of G.
• A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.[4]
• The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian subalgebras of

akg

[5] (cf. Cartan subalgebra)
• Every element of G lies in some maximal torus; thus, the exponential map for G is surjective.
• If G has dimension n and rank r then n - r is even.

## Root system

If T is a maximal torus in a compact Lie group G, one can define a root system as follows. The roots are the weights for the adjoint action of T on the complexified Lie algebra of G. To be more explicit, let

akt

denote the Lie algebra of T, let

akg

denote the Lie algebra of

G

, and let

akgC:=akgiakg

denote the complexification of

akg

. Then we say that an element

\alpha\inakt

is a root for G relative to T if

\alpha0

and there exists a nonzero

X\inakgC

such that

(X)=ei\langle\alpha,H\rangleX

for all

H\inakt

. Here

\langle,\rangle

is a fixed inner product on

akg

that is invariant under the adjoint action of connected compact Lie groups.

The root system, as a subset of the Lie algebra

akt

of T, has all the usual properties of a root system, except that the roots may not span

akt

.[6] The root system is a key tool in understanding the classification and representation theory of G.

## Weyl group

Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is,

W(T,G):=NG(T)/CG(T).

Fix a maximal torus

T=T0

in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T).

The first two major results about the Weyl group are as follows.

• The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.[7]
• The Weyl group is generated by reflections about the roots of the associated Lie algebra.[8] Thus, the Weyl group of T is isomorphic to the Weyl group of the root system of the Lie algebra of G.

We now list some consequences of these main results.

• Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, each conjugacy class of G intersects T in exactly one Weyl orbit.[9] In fact, the space of conjugacy classes in G is homeomorphic to the orbit space T/W.
• The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
• The identity component of the normalizer of T is also equal to T. The Weyl group is therefore equal to the component group of N(T).
• The Weyl group is finite.

The representation theory of G is essentially determined by T and W.

As an example, consider the case

G=SU(n)

with

T

being the diagonal subgroup of

G

. Then

x\inG

belongs to

N(T)

if and only if

x

maps each standard basis element

ei

to a multiple of some other standard basis element

ej

, that is, if and only if

x

permutes the standard basis elements, up to multiplication by some constants. The Weyl group in this case is then the permutation group on

n

elements.

## Weyl integral formula

Suppose f is a continuous function on G. Then the integral over G of f with respect to the normalized Haar measure dg may be computed as follows:

\displaystyle{\intGf(g)dg=|W|-1\intT

 2\int |\Delta(t)| G/T

f\left(yty-1\right)d[y]dt,}

where

d[y]

is the normalized volume measure on the quotient manifold

G/T

and

dt

is the normalized Haar measure on T.[10] Here Δ is given by the Weyl denominator formula and

|W|

is the order of the Weyl group. An important special case of this result occurs when f is a class function, that is, a function invariant under conjugation. In that case, we have

\displaystyle{\intGf(g)dg=|W|-1\intTf(t)|\Delta(t)|2dt.}

Consider as an example the case

G=SU(2)

, with

T

being the diagonal subgroup. Then the Weyl integral formula for class functions takes the following explicit form:[11]

\displaystyle{\intSU(2)f(g)dg=

 1 2
 2\pi \int 0

f\left(diag\left(ei\theta,e-i\theta\right)\right)4sin2(\theta)

 d\theta 2\pi

.}

Here

|W|=2

, the normalized Haar measure on

T

is
 d\theta 2\pi
, and

diag\left(ei\theta,e-i\theta\right)

denotes the diagonal matrix with diagonal entries

ei\theta

and

e-i\theta

.

## Notes and References

1. Theorem 11.2
2. Lemma 11.12
3. Theorem 11.9
4. Theorem 11.36 and Exercise 11.5
5. Proposition 11.7
6. Section 11.7
7. Theorem 11.36
8. Theorem 11.36
9. Theorem 11.39
10. Theorem 11.30 and Proposition 12.24
11. Example 11.33


text:
# Solve for y 1/(6y)=5/33

16y=533
Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.
1⋅33=6y⋅5
Solve the equation for y.
Rewrite the equation as 6y⋅5=1⋅33.
6y⋅5=1⋅33
Simplify.
Multiply 5 by 6.
30y=1⋅33
Multiply 33 by 1.
30y=33
30y=33
Divide each term by 30 and simplify.
Divide each term in 30y=33 by 30.
30y30=3330
Cancel the common factor of 30.
Cancel the common factor.
30y30=3330
Divide y by 1.
y=3330
y=3330
Cancel the common factor of 33 and 30.
Factor 3 out of 33.
y=3(11)30
Cancel the common factors.
Factor 3 out of 30.
y=3⋅113⋅10
Cancel the common factor.
y=3⋅113⋅10
Rewrite the expression.
y=1110
y=1110
y=1110
y=1110
y=1110
The result can be shown in multiple forms.
Exact Form:
y=1110
Decimal Form:
y=1.1
Mixed Number Form:
y=1110
Solve for y 1/(6y)=5/33

## Our Professionals

### Lydia Fran

#### We are MathExperts

Solve all your Math Problems: https://elanyachtselection.com/

Scroll to top


text:
# Aerodynamics drag of hollow cylinder vs solid cylinder of the same cross area?

I wonder how a hollow cylinder at the same cross area perform vs a solid one, i.e the hollow cylinder has larger radius? I guess they have a similar drag, is that true?

-
I don't have answer right now, but there is more surface for a hollow cylinder, so maybe more drag? –  fffred Jun 3 '13 at 7:05
I didnt get the question. Are you comparing two identical cylinders wherein the dimensions are identical but one is hollow, and the other is closed? –  Saurabh Raje Jun 3 '13 at 7:54
@fffred, I would say the excess area hypothesis is true only for the initial fraction of seconds –  Saurabh Raje Jun 3 '13 at 7:56
Has this answered your question? –  Killercam Jul 11 '13 at 9:36
By hollow, do you mean that there are no circular faces. i.e. air can flow through the cylinder? –  user80551 Sep 19 '13 at 17:55

If the flow is perpendicular to the cylinders axis of symmetry as long as the two cylinders have the same radius, length and are made of the same material, then the drag on them will be the same.

If the flow is parallel to the axis of symmetry then you are right to suggest that (assuming the radius, length and material of the two cylinders are the same) the larger surface area of the hollow cylinder will cause more drag.

-

I disagree with the parallel flow being equal. If the cylinder's aspect ratio is very long then, yes, the drag should be very similar because the air inside the cylinder is essentially trapped, travels with the cylinder and therefore creates a virtual base to seal the ends of the cylinder.

But for stubbier cylinders (think a hula hoop) the hollow one will have different drag. Here the air has enough space to equilibrate with the outside air.

Note: I'm assuming by hollow, you mean a cylindrical shell. That is a tube like shape. Alternatively, a cylinder with the exact exterior surfaces but empty interior should have the exact same drag as a solid one.

-


text:
0 items - $0.00 0 0 items -$0.00 0

Amazon Bestseller that gives the foundations and teaches the principles about what scientifically is shown to contribute to a long life and also a happy life. This book will stimulate you to reconsider your definitions of success and may change how you make life decisions.

A pocket sized book that gives an expectant mother a fairly comprehensive handbook that is easy to reference and to take to monthly doctors appointments and along with her as she faces the difficulties of handling the newborn period.

By appointment


text:
# Fast simulation based reduction for ω-automaton

## Abstract

The ${\displaystyle \omega }$-automata, capable of modeling infinite behavior, are used in numerous domains including model checking. The algorithms used are in general very costly. For this reason, we want to reduce the automata size while preserving the recognized language by applying numerous reductions. One of them, based on simulation, is very slow. In this paper, we will how to speed up using algorithmic methods and parallelization.


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: CK-12 Chemistry - Second Edition Go to the latest version.

# 26.21: U

Difficulty Level: At Grade Created by: CK-12
universal gas law
relates the volume of a gas with pressure, temperature, and number of moles of gas; written mathematically as $PV = nRT$
universal gas law constant (R)
constant equal to $\frac {PV} {nT}$, where the pressure, volume, moles, and temperature of the gas are represented by $P$, $V$, $n$, and $T$, respectively; the value and units of R depend on the units of $P$ and $V$, and two commonly used values of R are $0.08206 \ \mathrm{L} \ \mathrm{atm}\ \mathrm{K}^{-1} \ \mathrm{mol}^{-1}$ and $8.314 \ \mathrm{J} \ \mathrm{K}^{-1} \ \mathrm{mol}^{-1}$
unsaturated compound
an organic molecule containing double or triple bonds
unsaturated solution
a solution in which a given amount of solvent has dissolved less than the absolute maximum amount solute at that temperature
unshared electron pair
an unshared electron pair, also known as a non-bonding pair of electrons or as a lone pair of electrons, is found in a filled valence orbital that is not used in the formation of a covalent bond

Feb 23, 2012

Jan 07, 2015


text:
This is a script for any users of Linux who’d like a convenient way to begin editing a file as the superuser (i.e root) from Nautilus, the GNOME file manager.

1. Copy and paste the code below into a text editor such as gedit.
2. Save the text as the file ~/.gnome2/nautilus-scripts/edit_as_root The tilde stands for your home directory. You may need to right click within the save file dialog and check show hidden files before you see .gnome2
3. Open a terminal and run the command ‘chmod +x ~/.gnome2/nautilus-scripts/edit_as_root’ This tells your system that the text file is executable, i.e. that it is a script not a document
4. Log out then log back in. Technically, all you must do is restart nautilus.
5. In a nautilus window, you can now right click a file or set of files and select script -> edit_as_root.
:::bash
#!/bin/sh
IFS="
"
for I in $NAUTILUS_SCRIPT_SELECTED_FILE_PATHS do gksu gnome-text-editor$I &
done
exit

• Line 1: The shebang, it says run the program /bin/sh to interpret this script. On Ubuntu this is a soft link to the shell called dash.

• Lines 2-3: Set the internal field separator to a newline only.

• Lines 4-5 and 7: Loop through every line in $NAUTILUS_SCRIPT_SELECTED_FILE_PATHS, call the current line$I, and run Line 6.

• Line 6, gksu: Get authorization via password then run the next command as root. On Ubuntu the gconf key /apps/gksu/sudo-mode makes gksu act like gksudo.

• Line 6, gnome-text-editor $I: Open file$I with the default text editor. This editor can be changed wih the command ‘update-alternatives update-alternatives –config gnome-text-editor’.

• Line 6, &: Run the command gnome-text-editor in the background. This means don’t wait for it to finish before runing the next command.

• Line 8: The script is finished.


text:
# Loci Browse Articles

Displaying 41 - 50 of 323

This article describes methods for creating interactive math materials using the HTML5 canvaselement.

This is a student module for the article, which presents a fresh look at an age-old calculus optimization problem, the 'box problem.'

This article presents a fresh look at an age-old calculus optimization problem, the 'box problem.' (Also includes a Student Module condensed from the article.)

This suite of five interactive applets (written with GeoGebra) allows exploration of definitions and theorems commonly presented in first-year analysis courses. Topics include sequence convergence, continuity at a point, the Mean Value theorem, Taylor series, and Riemann sums. Included with each applet is a pair of activities: one for becoming comfortable using the applet, and one for using the applet to explore the associated topic in depth.
We highlight five pages written in XHTML with links to SVG files that produce animations illustrating calculus concepts such as the sign of the derivative, inflection points, conic sections, area between two curves, and multivariable functions.
This site features three simple-to-use applets: contour diagrams, curve families, and surface of revolution. The applets are written in Java and exploit Sun Microsystems' Java OpenGL technology so they will run on Apple OS X, Microsoft Windows 2000 or above, Solaris and many common configurations of Linux.
Simple probability concepts such as sampling with or without replacement are illustrated in this engaging tiered activity for beginning students and future teachers of probability.

This interactive Geogebra applet allows students to experiment with 2 by 2 matrices and the corresponding geometric transformations of the real plane. A list of student activities is provided.

This interactive Geogebra applet allows exploration of a linear transformation in terms of images of a set of points. The Geogebra interface allows dragging of points and vectors to make for versatile explorations of basic linear algebra ideas. Suggested activities are included.

Three applets allow you to interact with the puzzle described in the April 2009 Mathematics Magazine article by Alex Fink and Richard Guy.


text:
# [l2h] inline \verb does not work when data on windows and using latex2html from Linux in VBox

Nasser M. Abbasi nma at 12000.org
Wed May 9 23:24:12 CEST 2012

I hope someone can give my few hints on this issue.

This has been a problem for me for long time, and I
am finally sitting down to resolve it.

The problem is simple to describe: When I build this file

----------- t.tex -------------------------
\documentclass{article}
\begin{document}
this is \verb=some text=
\end{document}
------------------------------------

using the command  latex2html t.tex The inline verbatim
does NOT appear in the HTML

------ generated HTML ---------

this is

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

this ONLY happens when the file t.tex is on a windows disk
(i.e. NTFS) and I have the disk mounted as shared folder
so as to access my data from Linux running inside a VBox.

I use windows, and I have VBox and I have Linux as guest OS.

Hence, in the above I have

/media/me/windows_data/t.tex

When I copy t.tex to my Linux home folder (now on ext4
file system)

$HOME/t.tex And again issue the same latex command on the above file, and look at$HOME/t.htm, then now verb works and I see this:

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

this is some text

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

EVERYTHING ELSE IS THE SAME !

Same latex2html is used, same OS (linux), etc...

It is just the physical location of the t.tex which
make this problem show up.

This only affects inline verb. Not block verbatim, this
works fine

\begin{verbatim}
...
\end{verbatim}

I am using LaTeX2HTML Version 2008 (1.71)

I am looking for a hint where to look. I have no idea
what can cause this problem. It is causing me lots
of problems, because I use inline verb allot, and
I have to keep copying files from windows to Linux
to build them and copy them back to window just
for this.

Where is    processed in latex2html perl?
may if I look there I might spot something.

Any one has an idea or hint, will be great.

thanks,
--Nasser


text:
# Exotic s-wave superconductivity in alkali-doped fullerides.

@article{Nomura2016ExoticSS,
title={Exotic s-wave superconductivity in alkali-doped fullerides.},
author={Yusuke Nomura and Shiro Sakai and Massimo Capone and Ryotaro Arita},
journal={Journal of physics. Condensed matter : an Institute of Physics journal},
year={2016},
volume={28 15},
pages={
153001
}
}
• Published 17 December 2015
• Physics
• Journal of physics. Condensed matter : an Institute of Physics journal
Alkali-doped fullerides (A3C60 with A = K, Rb, Cs) show a surprising phase diagram, in which a high transition-temperature (Tc) s-wave superconducting state emerges next to a Mott insulating phase as a function of the lattice spacing. This is in contrast with the common belief that Mott physics and phonon-driven s-wave superconductivity are incompatible, raising a fundamental question on the mechanism of the high-Tc superconductivity. This article reviews recent ab initio calculations, which…
35 Citations

## Figures and Tables from this paper

Introduction to Superconductivity in Alkali-Doped Fullerides
In this chapter, we give an introduction to the superconductivity in alkali-doped fullerides, which show the highest transition temperature ($$T_\mathrm{c}$$) among molecular solids. We review
Tuning the electronic states and superconductivity in alkali fulleride films
• Physics
AAPPS Bulletin
• 2022
The successful preparation of superconducting alkali fulleride (AxC60, A = K, Rb, Cs) films using state-of-the-art molecular beam epitaxy overcomes the disadvantages of the air-sensitivity and phase
Direct Observation of Full-Gap Superconductivity and Pseudogap in Two-Dimensional Fullerides.
• M. Ren
• Physics
Physical review letters
• 2020
Cryogenic scanning tunneling microscopy is employed to show that trilayer K_{3}C_{60} displays fully gapped strong coupling s-wave superconductivity, accompanied by a pseudogap above T_{c}∼22  K and within vortices, which indicates that alkali fullerides are predominantly phonon-mediated superconductors, although the electronic correlations also come into play.
Theoretical phase diagram of unconventional alkali-doped fullerides
By constructing an effective model based on recently calculated \emph{ab }initio bare interaction parameters, we study the phase diagram of alkali doped fullerides as a function of temperature and
Nanoscale orbital excitations and the infrared spectrum of a molecular Mott insulator: A15-Cs3C60.
• Physics
Nanoscale
• 2016
It is shown that the unbroken spin and orbital symmetry of the ion multiplets explains the IR spectrum without adjustable parameters, demonstrating the importance of a fully quantum treatment of nuclear positions and orbital momenta in the Mott insulator sites, dynamically but not statically distorted.
Nonempirical Calculation of Superconducting Transition Temperatures in Light‐Element Superconductors
• Physics
• 2017
This study obtains a unified view of the high-Tc superconductivity and the Mott-Hubbard transition in the fullerides and demonstrates that the combination of density functional theory and dynamical mean field theory with the ab initio downfolding scheme for electron-phonon coupled systems works successfully.
Orbital disproportionation of electronic density is a universal feature of alkali-doped fullerides
• Physics, Chemistry
Nature communications
• 2016
This work addresses the electronic phase of tetravalent fulleride with accurate many-body calculations within a realistic electronic model including all basic interactions extracted from first principles and shows that the Jahn–Teller instability is always realized in these materials.
Correlation-induced superconductivity dynamically stabilized and enhanced by laser irradiation
• Physics
• 2017
It is shown that the superconductivity can be enhanced by irradiating laser to correlated electron systems owing to two mechanisms: first, the effective attractive interaction of carriers is enhanced by the dynamical localization mechanism, which drives the system into strong coupling regions, and second, the irradiation allows reaching uniform and enhanced superconductor-like properties above room temperatures.
Sodium-doped superconductivity in T-carbon
T-carbon has been proposed as a new carbon allotrope in 2011, which was successfully synthesized in recent experiments. Because of its fluffy structure, several kinds of atoms can be intercalated

## References

SHOWING 1-10 OF 325 REFERENCES
Unified understanding of superconductivity and Mott transition in alkali-doped fullerides from first principles
• Physics
• 2015
The results establish that the fullerides are the first members of a novel class of molecular superconductors in which the multiorbital electronic correlations and phonons cooperate to reach high Tc s-wave superconductivity.
The strength of electron electron correlation in Cs3C60
• Physics
Scientific reports
• 2015
The relevance of the Jahn-Teller coupling in the metallic state confirms that phonon coupling survives in the presence of strong correlations and shows that electronic correlations are crucial in determining the insulating behaviour at ambient pressure and the bad metallic nature for increasing pressure.
Superconductivity, Mott?Hubbard states, and molecular orbital order in intercalated fullerides
• Physics
• 2003
This article reviews the current status of chemically doped fullerene superconductors and related compounds, with particular focus on Mott–Hubbard states and the role of molecular orbital degeneracy.
Isotope effect on superconductivity in Rb3C60
• Physics
Nature
• 1992
THE surprisingly high transition temperatures (Tc) for superconductivity in alkali-meta-doped C60 has spurred wide interest in understanding its mechanism1–6. Recently the increase in Tc with lattice
Theoretical Study on the Superconductivity Induced by the Dynamic Jahn-Teller Effect in Alkali-Metal-Doped C60.
• Physics
• 2000
We investigate the superconductivity in alkali-metal-doped C 60 , A 3 C 60 . ( A =K, Rb, etc.) To take account of both the electron-electron and electron-phonon interactions, we employ a model
Size and symmetry of the superconducting gap in the f.c.c. Cs3C60 polymorph close to the metal-Mott insulator boundary
• Physics
Scientific reports
• 2014
A comprehensive nuclear magnetic resonance (NMR) study of face-centred-cubic (f.c.c.) Cs3C60 polymorph, which can be tuned continuously through the bandwidth-controlled Mott insulator-metal/superconductor transition by pressure, indicates the importance of the electronic correlations for the pairing interaction as the metal/ superconductor-insulator boundary is approached.
Alkali-Doped Fullerides: Narrow-Band Solids with Unusual Properties
Alkali-doped fullerides have attracted strong interest since their production became possible about fifteen years ago. This book presents work which may solve intriguing problems arising from a
Optimized unconventional superconductivity in a molecular Jahn-Teller metal
• Physics


text:
Next Meeting: August 5: Social gathering Next Installfest: TBD Latest News: Jul. 4: July, August and September: Security, Photography and Programming for Kids Page last updated: 2004 Mar 04 14:44
 The following is an archive of a post made to our 'vox-tech mailing list' by one of its subscribers. Report this post as spam: (Enter your email address)
Re: [vox-tech] latex sty
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]

# Re: [vox-tech] latex sty

On Tue 02 Mar 04,  6:44 PM, Jonathan Stickel <jjstickel@sbcglobal.net> said:
> I'm submitting a latex document to a journal, and I'm not supposed to
> include any files other than my latex file and my eps figures.

have you written to them and said something like:

"is it OK to also send in a .sty file to conform to your bibliography
format?  and if i can't, please tell me how to conform to your format"

before giving up on that, you should definitely ask...

> However,
> I need to use a personalized sty file for proper in text referencing
> (see this earlier thread:
> http://www.lugod.org/mailinglists/archives/vox-tech/2004-02/msg00054.html).
>  Is there any way to include that information in my latex file?  I
> tried just pasting it in at the top of my document, but I get errors.
> Clearly, there is syntax for sty files that is not permitted in latex.

it's possible, but not that i know of.  my own pete_macros.sty is just a
bunch of macros i've written and placed in \$TEXINPUTS so \usepackage can
find it.

are you sure you pasted it after \documentclass but before
\begin{document}?

pete

--
Make everything as simple as possible, but no simpler.  -- Albert Einstein
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text:
# [Orgmode] Re: Latex export and label entries

Hi

Carsten Dominik <carsten.dominik <at> gmail.com> writes:

>
> Hi,
> On Dec 2, 2009, at 2:51 PM, d.tchin <at> voila.fr wrote:
>
> >
> > Hi,
> >
> > I use org-export-latex to create latex powerdot
> > file. I make adaptation of template defined for
> > beamer class and it works quite well.
> >
> > I have a problem I would like to submit :
> >
> > By default in each section or slide (frame) environment,
> > there is a label added by default. This instruction
> > is not recognized by powerdot class. Is there a way
> > to prevent org-export-latex function to add such label
> > entries ?
>
> a work-around would be to add this to your powerdot definition of org-
> export-latex-classes:
>
> \def\label#1{}
>

with org-export-latex-classes. In fact it work if this intruction
appears after \begin{document}.
I have managed to add it before \begin{document} with
org-export-latex-classes but not after . How could I make appear
this intruction only one time after \begin{document} ?

Regards

_______________________________________________
Emacs-orgmode mailing list
`


text:
# Properties

 Label 1216.4.a.g Level $1216$ Weight $4$ Character orbit 1216.a Self dual yes Analytic conductor $71.746$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$71.7463225670$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 38) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - \beta ) q^{3} + ( 3 + 3 \beta ) q^{5} + ( -7 - 4 \beta ) q^{7} + ( 7 + 9 \beta ) q^{9} +O(q^{10})$$ $$q + ( -4 - \beta ) q^{3} + ( 3 + 3 \beta ) q^{5} + ( -7 - 4 \beta ) q^{7} + ( 7 + 9 \beta ) q^{9} + ( 9 - \beta ) q^{11} + ( -2 - 13 \beta ) q^{13} + ( -66 - 18 \beta ) q^{15} + ( -39 - 2 \beta ) q^{17} -19 q^{19} + ( 100 + 27 \beta ) q^{21} + ( 30 + 13 \beta ) q^{23} + ( 46 + 27 \beta ) q^{25} + ( -82 - 25 \beta ) q^{27} + ( -12 + 21 \beta ) q^{29} + ( 128 - 44 \beta ) q^{31} + ( -18 - 4 \beta ) q^{33} + ( -237 - 45 \beta ) q^{35} + ( -110 + 28 \beta ) q^{37} + ( 242 + 67 \beta ) q^{39} + ( -30 + 10 \beta ) q^{41} + ( -335 - 7 \beta ) q^{43} + ( 507 + 75 \beta ) q^{45} + ( -159 - 71 \beta ) q^{47} + ( -6 + 72 \beta ) q^{49} + ( 192 + 49 \beta ) q^{51} + ( 618 - 17 \beta ) q^{53} + ( -27 + 21 \beta ) q^{55} + ( 76 + 19 \beta ) q^{57} + ( 156 - 25 \beta ) q^{59} + ( -101 - 111 \beta ) q^{61} + ( -697 - 127 \beta ) q^{63} + ( -708 - 84 \beta ) q^{65} + ( -650 + 77 \beta ) q^{67} + ( -354 - 95 \beta ) q^{69} + ( 42 + 116 \beta ) q^{71} + ( 281 - 184 \beta ) q^{73} + ( -670 - 181 \beta ) q^{75} + ( 9 - 25 \beta ) q^{77} + ( 704 - 58 \beta ) q^{79} + ( 589 - 36 \beta ) q^{81} + ( 432 - 194 \beta ) q^{83} + ( -225 - 129 \beta ) q^{85} + ( -330 - 93 \beta ) q^{87} + ( -24 - 188 \beta ) q^{89} + ( 950 + 151 \beta ) q^{91} + ( 280 + 92 \beta ) q^{93} + ( -57 - 57 \beta ) q^{95} + ( 596 + 102 \beta ) q^{97} + ( -99 + 65 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{3} + 9q^{5} - 18q^{7} + 23q^{9} + O(q^{10})$$ $$2q - 9q^{3} + 9q^{5} - 18q^{7} + 23q^{9} + 17q^{11} - 17q^{13} - 150q^{15} - 80q^{17} - 38q^{19} + 227q^{21} + 73q^{23} + 119q^{25} - 189q^{27} - 3q^{29} + 212q^{31} - 40q^{33} - 519q^{35} - 192q^{37} + 551q^{39} - 50q^{41} - 677q^{43} + 1089q^{45} - 389q^{47} + 60q^{49} + 433q^{51} + 1219q^{53} - 33q^{55} + 171q^{57} + 287q^{59} - 313q^{61} - 1521q^{63} - 1500q^{65} - 1223q^{67} - 803q^{69} + 200q^{71} + 378q^{73} - 1521q^{75} - 7q^{77} + 1350q^{79} + 1142q^{81} + 670q^{83} - 579q^{85} - 753q^{87} - 236q^{89} + 2051q^{91} + 652q^{93} - 171q^{95} + 1294q^{97} - 133q^{99} + O(q^{100})$$

## 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
 4.77200 −3.77200
0 −8.77200 0 17.3160 0 −26.0880 0 49.9480 0
1.2 0 −0.227998 0 −8.31601 0 8.08801 0 −26.9480 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
$$2$$ $$1$$
$$19$$ $$1$$

## 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 1216.4.a.g 2
4.b odd 2 1 1216.4.a.p 2
8.b even 2 1 38.4.a.c 2
8.d odd 2 1 304.4.a.c 2
24.h odd 2 1 342.4.a.h 2
40.f even 2 1 950.4.a.e 2
40.i odd 4 2 950.4.b.i 4
56.h odd 2 1 1862.4.a.e 2
152.g odd 2 1 722.4.a.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.4.a.c 2 8.b even 2 1
304.4.a.c 2 8.d odd 2 1
342.4.a.h 2 24.h odd 2 1
722.4.a.f 2 152.g odd 2 1
950.4.a.e 2 40.f even 2 1
950.4.b.i 4 40.i odd 4 2
1216.4.a.g 2 1.a even 1 1 trivial
1216.4.a.p 2 4.b odd 2 1
1862.4.a.e 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1216))$$:

 $$T_{3}^{2} + 9 T_{3} + 2$$ $$T_{5}^{2} - 9 T_{5} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 9 T + T^{2}$$
$5$ $$-144 - 9 T + T^{2}$$
$7$ $$-211 + 18 T + T^{2}$$
$11$ $$54 - 17 T + T^{2}$$
$13$ $$-3012 + 17 T + T^{2}$$
$17$ $$1527 + 80 T + T^{2}$$
$19$ $$( 19 + T )^{2}$$
$23$ $$-1752 - 73 T + T^{2}$$
$29$ $$-8046 + 3 T + T^{2}$$
$31$ $$-24096 - 212 T + T^{2}$$
$37$ $$-5092 + 192 T + T^{2}$$
$41$ $$-1200 + 50 T + T^{2}$$
$43$ $$113688 + 677 T + T^{2}$$
$47$ $$-54168 + 389 T + T^{2}$$
$53$ $$366216 - 1219 T + T^{2}$$
$59$ $$9186 - 287 T + T^{2}$$
$61$ $$-200366 + 313 T + T^{2}$$
$67$ $$265728 + 1223 T + T^{2}$$
$71$ $$-235572 - 200 T + T^{2}$$
$73$ $$-582151 - 378 T + T^{2}$$
$79$ $$394232 - 1350 T + T^{2}$$
$83$ $$-574632 - 670 T + T^{2}$$
$89$ $$-631104 + 236 T + T^{2}$$
$97$ $$228736 - 1294 T + T^{2}$$


text:
My Math Forum 2-Coin Toss Probability Proof
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 Advanced Statistics Advanced Probability and Statistics Math Forum

 January 21st, 2012, 10:34 PM #1 Joined: Jan 2012 Posts: 2 Thanks: 0 2-Coin Toss Probability Proof Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are: E1 = H1, H2 E2 = H1, T2 E3 = T1, H2 E4 = T1, T2 Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1: P(E1) = 1 / 4 This mistake is commonly seen in introductory Probability and Statistics Textbooks." I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it. Sorry if I broke any rules to this forum, first post.
January 22nd, 2012, 04:53 AM   #2

Joined: Jan 2012

Posts: 14
Thanks: 0

Re: 2-Coin Toss Probability Proof

Quote:
 Originally Posted by Risotto17 Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are: E1 = H1, H2 E2 = H1, T2 E3 = T1, H2 E4 = T1, T2 Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1: P(E1) = 1 / 4 This mistake is commonly seen in introductory Probability and Statistics Textbooks." I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it. Sorry if I broke any rules to this forum, first post.
The two tosses have to be independent for the probability that both be H is the product of the individual tosses be heads.

CB

 January 23rd, 2012, 02:27 AM #3 Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: 2-Coin Toss Probability Proof Interesting thought. The .50 probability of H/T not changing and being independent are indeed two different things.
January 23rd, 2012, 09:56 AM   #4

Joined: Jan 2012

Posts: 2
Thanks: 0

Re: 2-Coin Toss Probability Proof

Quote:
Originally Posted by CaptainBlack
Quote:
 Originally Posted by Risotto17 Q: "You flip a fair coin twice. The probability of getting heads is 0.5, and tails 0.5. If E represented an event and each of the two tosses was represented by the numbers 1 and 2 respectively (i.e. H1 means getting heads on the first toss), then the possible outcomes are: E1 = H1, H2 E2 = H1, T2 E3 = T1, H2 E4 = T1, T2 Since the coin is completely fair, we can assume that these pairs of outcomes are equally likely. That is to say, the probability of E1: P(E1) = 1 / 4 This mistake is commonly seen in introductory Probability and Statistics Textbooks." I talked to my professor and he said it was a really "subtle" question and not to over think it. Basically, the reasoning for why the answer is 1/4 is incomplete. That is to say there is something critical, that is missing from this argument. My guess is you cannot conclude that the "pairs of outcomes are equally likely" just because the coin is fair and each elementary outcome has a 0.5 chance of occurring. That said, even if my guess is correct I don't know what missing piece is so critical that there exists a situation where this assumption is false without it. Sorry if I broke any rules to this forum, first post.
The two tosses have to be independent for the probability that both be H is the product of the individual tosses be heads.

CB
So the only thing missing is a proof that the two events are independent?

Would something to the effect of:
P(HnT) = P(H|T)P(T)
= P(H)P(T)
so,
P(H|T) = P(HnT) / P(T) = P(H)
and,
P(T|H) = P(TnH) / P(H) = P(T)

suffice to complete the proof?

Thanks a ton.

 January 26th, 2012, 12:09 AM #5 Joined: Jan 2012 Posts: 14 Thanks: 0 Re: 2-Coin Toss Probability Proof The definition of independednce is equvalent to the the reqirement that the probability of AB in that order is equal to the product of A and the probability of B. CB

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# Where do I start learning Higher Mathematics? [closed]

I am 17 years old and I would like to begin learning Mathematics. I am only familiar with Algebra and a bit of Geometry. Do I need to learn Calculus next, or should I try linear algebra? I find myself very interested in number theory. Is there anywhere I can see some sort of tree that shows me prerequisites for different Mathematical studies. I also want to know if there are any websites or resources that can teach me Math formally, using proofs. In high school we never learned math formally or rigorously. Thanks.

Also any books I could get would be appreciated. Keep in mind I am trying to start off, I only know basic Algebra right now, like functions, polynomials, and systems of equations.

## closed as primarily opinion-based by Matthew Towers, Claude Leibovici, user91500, Frits Veerman, user99914 May 10 '16 at 9:54

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• Khan Academy is a nice start. You can also try taking online courses at Udemy/Coursera/OpenU. You can check some university's prerequisites but there isn't a 'correct' path to study, especially on your own. First year Courses like Calculus and Linear Algebra (which I suggest you study simultaneously) usually don't assume any prior knowledge. Also, Discrete mathematics courses serve as a great start. – Trouble May 9 '16 at 18:25
• @Trouble Calculus doesnt assume prior knowledge? I thought you need Pre-Calculus to do Calculus. – user338280 May 9 '16 at 18:33
• Kind of. You need to be very comfortable with algebra and you should know trigonometry. But I feel like for people who are very motivated, having a semester of precalculus beforehand is a waste of time, it doesn't take a semester to get prepared. – D_S May 9 '16 at 18:39
• I was introduced to number theory (and group theory) before calculus, and that's where I learned about mathematical proofs. I would think that the first step into higher mathematics is getting good at using proof - and using it not only to create mechanically convincing arguments, but to connect intuition and rigor. – Milo Brandt May 9 '16 at 20:41
• It seems to me that one could study abstract algebra with a minimum of prior knowledge. The only caveat to that is since it is usually presented at a university level, understanding of proofs is usually assumed by the texts and courses available on abstract algebra. – Todd Wilcox May 9 '16 at 20:43

Khan academy might be a good starting point.

On youtube you can watch hundreds of their short videos on subjects like algebra, calculus, number theory, probability, linear algebra, geometry, trigonometry etc.

When you feel like you master that material you can move on to MIT OpenCourseWare on youtube.

Gilbert Strang's Linear algebra.

David Jerison's Single variable calculus.

Denis Auroux's Multivariable calculus.

Tom Leighton's Mathematics for Computer Science.

They also have courses on probability, statistics, differential equations etc.

Other personal favourites, which i actually liked more than MIT, are these

Discrete Mathematics. Arsdigita University. Instructor: Shai Simonson

The Fourier Transforms and its Applications. Standford. Professor Brad Osgood

Probability. Harvard

Probability Primer. Mathematicalmonk's channel

General topology from the very basics, including set theory, techniques for proofs

Short course on writing proofs in mathematics by Sidney Morris

At the same time you are following this online material i recommend buying books and solve a lot of problems. There is no better way of learning mathematics than solving problems. I would buy books which have solution manuals.

I would also recommend to start to learn to program in R, matlab, mathematica, maple, python or whatever environment you like best. This will become a very useful skill when you reach higher mathematics.

• how is programming useful in higher level mathematics? – Charlie Parker May 10 '16 at 0:15
• @CharlieParker Well i am no expert, but I'd say there are more areas where it is useful than areas where it is not. To name a few: mathematical modelling, PDEs, stochastic processess, fourier applications, combinatorics, statistics, probability, number theory, discrete mathematics, graph theory, optimization, algorithms, numerical methods, proof verification, calculus, financial mathematics, cryptography, information-theory, geometry, lin alg etc. I even have a friend who is writing his thesis on the mathematical aspects of general relativity and its all about modelling using his own software. – JKnecht May 10 '16 at 0:42

I think you're in a good position to benefit from Keith Devlin's course (or just read the book) on mathematical thinking: https://www.coursera.org/course/maththink. Rather than dive into advanced mathematics directly, he takes time to reflect on what it means to "think like a mathematician" and develops some necessary logical prerequisites. The course/book concludes with a taste of different kinds of advanced mathematics. Here's the table of contents so you can see what I mean:

1. Introductory material
2. Analysis of language – the logical combinators
3. Analysis of language – implication
4. Analysis of language – equivalence
5. Analysis of language – quantifiers
6. Working with quantifiers
7. Proofs
8. Proofs involving quantifiers
9. Elements of number theory
1. Beginning real analysis

Well well, this is a broad question and you might get widely different answers, so take mine with a grain of salt!

I believe the first thing you should do before attempting to study any more mathematics is solidify your knowledge of basic logic. Study first-order logic, learn to write proofs of theorems in this language. This should not take long (only a general overview is required), but it will give you essential foundations.

You can then move on to other topics. In my first year at university we had a course which was using "A Concise Introduction to Pure Mathematics" by Martin Liebeck. I recall it was a pretty decent book, and it has some number theory in it! There is also a chapter on logic, although I would advice you look at external sources, too.

• Where could I learn about basic logic. I really don't know what the definition of logic is anyways, at least in mathematical context. – user338280 May 9 '16 at 18:29
• @user338280 I cannot think of a good book over the top of my head but I will try to get back to you on that! That said with the basic knowledge you have you should be able to read Martin Liebeck. – Aegis May 9 '16 at 18:32
• Thanks. I really don't have time or money to go to school for mathematics although I find it to be really interesting. I just want to stimulate my mind a bit in my free time. – user338280 May 9 '16 at 18:33
• @user338280 Yeah education can be expensive... But rest assured that if you are dedicated you might well learn more by reading in your free time than many people do at university :) Nowadays with the internet you have easy access to information & people to ask questions. – Aegis May 9 '16 at 18:34

Most people start with calculus as their first "advanced" (i.e. college level or above) math. There are many online resources for this, like Khan Academy. Do lots of exercises. Ask yourself dumb questions.

Next, you may want to take some time to get a good foundation in naive set theory. Stuff like logical implications, e.g. why is the statement "$P$ implies $Q$" true when $P$ is false. Get comfortable with unions and intersections, functions, images, preimages, stuff like that. Get comfortable with basic proofs.

After that, you may want to try learning, in no particular order, linear algebra, abstract algebra, real analysis. Try to get a good balance between working examples, understanding concepts, and being able to prove things.

I think learning all that is a good foundation for learning other fields of math. It might take you a few years. Meanwhile you may discover what you like and what your strengths and weaknesses in math are. For example, if you like real analysis and linear algebra, you may want to have a look at areas like differential equations or functional analysis.

• Definitely basic set theory early on! – Trouble May 9 '16 at 18:46
• @user338280 You can ask those questions here, if you ever need to. Though vague questions like "will there exist an equation that models that curve" is probably going to receive little attention, other more specific questions are welcomed. (pst, for quadratic formula, use substitution $x=u-\frac{b}{2a}$) – Simply Beautiful Art May 9 '16 at 21:58

What helped my a lot was actually learning through problem solving. Like, I get a problem, try to solve it and if I see I can't do it on my own I look up some answers or theorems on internet. This method suits me still the best, bcs you can learn so much from one Problem. But keep in mind that you can cheat others, but not yourself, so you have to give your best before looking up answers. I highly recomment the website Art of Problem Solving. There you have a community/forum where you can look up problems for any level, from middle school to highschool olympics and collage. There are usually also answers to a lot of problems so it souldn't be that hard to look up for an answer later. If you are interested in a porblem and you can't solve it you are always free to post it here too with some of your work yo have done (that is important!).

About what you should learn next: well, there is no rule, atleast there wasn't any for me. I did everything what I liked and also disliked because I was preparing myself for competitions in my country so I wasn't having much of a choise. But you can always start from some basics like divisibility, femrats little theorem, prime numbers and what I highly recomend is to master modulos for number theory, or inequality and functional equations which are really good to stimulate your creativity and are good things to learn algebra. For geometry I really like using trigonometry, but I also recommend to learn a lot of other theorems that will help you like some properties for circular quadrilateral, chraracteristic points of a triangle, similarity and what is important here if you dont want to use trigonometry is to know when and what you have to draw extra into you sketch so you can get something nice that will help you. The last thing is combinatorics, which I really dislike the most, but here I can't help you a lot, beacuse usually you really don't need mathematical knowledge to solve those problems. They are kinda made so anyone can solve them.

That's it, after learning some basics you will see a lot of problems where people use some kind of theorems like "Lifting the Exponent Lemma" or for short LTE Lemma. If something like that happens just google it and mostly you can find your answer on wikipedia or other sides.

One more thing. If you start liking geometry I also recomment you downloading geogebra (or use the online version) which is a free programm that lets you draw a lot of things so you dont have to do a lot of papper drawing and you can easily modify your picture.

I hope I could help you a little bit, since there is no way anyone can cover everything. If you need some help feel free to ask me or anyone here for help, that also goes for aops. You can find me there under the name mirza45

Good luck with your work :)

• Pft, stop recommending other sites, your supposed to promote our site! – Simply Beautiful Art May 9 '16 at 22:02
• @SimpleArt I am not promoting anything, I just said where he can find better problems that suits him best. At math stachexchange you truly can find a lot of math problems, but to be honest most of them are collage level which he is not currently at. I am sorry if it did sound like I am doing what you said, but like I said, I learned a lot from there and I dont see a bad thing to share knowledge. – HeatTheIce May 9 '16 at 22:04
• Yes, I have actually used that site as well. I was only joking with you (but promoting this site is a good thing) and there are some high school level topics on this site... maybe not much, but if so, its because people stop asking those questions on this site. Which raises some questions that I will ask elsewhere. – Simply Beautiful Art May 9 '16 at 22:12
• Well, I used this site also for some questions I had in highschool and it came in handy, but on this site there are more serious mathematicans that know a lot of theory and how to apply them, but what I like more are the answers that are elegant and have some kind trick that you have to come up with yourself and not to look it up on wikipedia (I know what I said now is a little bit contradictory with what I said up there). But I do like this site a lot, but for young ones I would always recomment aops :) – HeatTheIce May 9 '16 at 22:16
• And I still haven't learned how to make math work on that site lol... Agreeably, the people on this site are very serious and we indeed do rather rigorous math. But I feel proud to be amongst the smart serious mathematicians, can't help it. ;) means you guys out there looking down at this comment. – Simply Beautiful Art May 9 '16 at 22:23

If you're serious about learning math yes, you have to learn calculus (as well as many other things, but calculus is definitely one of the fundamental first steps).

I am not going to suggest any modern tools, such as online classes etc. I think other people covered that quite well. I will be old-fashioned here and suggest just a book. Yeah, made of paper. And a quite old one actually, that nevertheless is still considered one of the best introductory texts in calculus/analysis.

It is not an easy book. It might be a challenge to start off with it actually, but somehow I have the feeling that it's one of those books that could really make the difference in the future. It's a book that forces you to think. A book that, in order to understand, you have to go through line by line and spend time on it, trying to grasp every concept and filling eventual gaps left here and there by the author (sometimes intentionally, maybe sometimes not).

As I said, it might not be an easy book... well no, it certainly isn't, but it starts from the very basics after all. So if you have some basic math training and you are up for a challenge I think you can have a go with it (I am assuming that you are familiar at least with basic math terminology and symbolism).

Ok, the book I am talking about is Walter Rudin's Principles of Mathematical Analysis. You can have a look here:

http://www.amazon.com/Principles-Mathematical-Analysis-International-Mathematics/dp/007054235X/ref=mt_hardcover?_encoding=UTF8&me=

If you browse a bit through the reviews you can have a feeling of what I'm talking about here. It's a book that even experienced mathematicians love to read. I might be going out on a limb suggesting this, true, but I believe it could be the book that will actually shape you into a real mathematician. Some undergrads may not like it because they say it's too advanced. Unlike you though, they probably wouldn't post questions on math.stackexchange about where to start off with higher mathematics. If your passion is genuine, this might be the book for you. Good luck.

You are actually the second person today who has asked such a question. Calculus and Linear Algebra can be learned independently, but some (not much) Linear algebra is useful for 3 dimensional calculus. Number theory is great, the classic introduction is by Hardy and Wright but there are many other good books. Focus on solving problems. Especially since you are young. Look into learning some abstract algebra, rings, groups and fields, as soon as possible, maybe even before linear algebra.

Finally one book I liked in called Mathematics its from Time-Life science library, its pretty old but most libraries will have it. It popular but has a nice tree diagram of the different branches of mathematics.

One afterthought, since it seems to be being discussed here, (yes thats correct grammar) universities and schools have become useless hindrances to your progress, and your pocketbook, whether you do or not anything you learn will have to be done in spite of their influence. (Sorry folks but that is the current state of affairs).

• When I look at a problem in Mathematics i do not know what to do. I am still trying to develop my intuition when it comes to solving problems. Im so used to being told to remember formulas in a classroom and I need to learn to think independently. What is abstract algebra in essence?? I have heard of it but I do not know how it would help me. – user338280 May 9 '16 at 18:36
• @user338280 Abstract algebra is, roughly, the study of structure over sets. Do you know what a set is? – Aegis May 9 '16 at 18:39
• @user338280 Yeah learning to think independently is so important (dont do it in university though youll get in lots of trouble). One thing is to get some math contest problem books and work on some of those questions, it can be fun and it builds your problem solving muscles. But it wont connect you with higher mathematics. – Rene Schipperus May 9 '16 at 18:43
• @Aegis A collection of objects. I don't really know why they are useful but I have heard people say that all of Mathematics is based on operations with sets, and a number is just something our brains created to describe the cardinality of various sets of physical objects. – user338280 May 9 '16 at 18:43
• @Rene Schipperus Well of course you'll get in trouble since the university makes a profit off you being uneducated and needing their assistance to think. – user338280 May 9 '16 at 18:44

There are quite a few good sources out there for learning different fields in mathematics. Personally some of my favorates are:

To get started you need to know the learning path you will take, what classes and areas you want to begin with and proceed to. A basic path for general mathematics would be: (pipe shows what I feel can be learned at the same time)

Algebra I > Geometry | Algebra II > Statistics | Trigonometry > Linear Algebra > Single Variable Calculus > Multi-Variable Calculus | Differential Equations

From there you should be well versed enough to know what branch of mathematics you would like dive deeper into

For me, Calculus isn't a bad place to start. Its assisted me in calculating divergent series, proving binomial expansion, calculating some harder summations, providing bounds, and simply giving intuitive sight into how a function may work.

Calculus isn't too hard, and you don't really need pre-calculus to learn it. Calculus is just a little different from previous math courses in the topics. Its like the jump from middle school math to Algebra, just a little harder and makes you think differently.

Another nice topic of interest to you might be modular arithmetic. It is rather easy, and when you learn things like Fermat's little theorem and basic manipulations, you can do cool things like determine what the last digit of $47^{2958}$ is. (It also helps you make faster choices on multiple-choice questions involving whole numbers. And if you have a younger sibling in elementary, teaching them the basics to modular arithmetic could help them in math... wish I knew ALL of modular arithmetic in elementary. When you learn modular arithmetic, I bet some of the material will be so easily you'll say "I've done that before!")

Summations are also fun. When you learn calculus, summations become easier to deal with as well (for me at least). Also, making formulas with summations gets you into thinking about how you can provide proofs sometimes, and other things like that.

Set theory is a short(ish) topic that is really interesting for most people. It is basically the theory of infinity, and a nice video that captures the gist of set theory is Vsauce's "How to count past infinity?" video.

Number theory might be a bit harder, but you could potentially learn some of it.

And don't expect all of this to come easily (maybe the modular arithmetic), so take your time, and if you get stuck, we'll be right here to help you.

Ranked in easiness according to my opinion (difficulty based off how hard to get started in the material):

1. Modular arithmetic (easiest)

2. Calculus and summations

3. Set Theory

4. Number Theory

Also, I have to ask, do you intend on math courses for credit, or math subjects you want to learn on your own with no other reason than that your just nerdy like that? It makes a difference.


text:
# Sparse Recovery for Orthogonal Polynomial Transforms

@inproceedings{Gilbert2020SparseRF,
title={Sparse Recovery for Orthogonal Polynomial Transforms},
author={Anna C. Gilbert and Albert Gu and Christopher R{\'e} and Atri Rudra and Mary Wootters},
booktitle={ICALP},
year={2020}
}
• Anna C. Gilbert, +2 authors Mary Wootters
• Published in ICALP 2020
• Computer Science, Mathematics
• In this paper we consider the following sparse recovery problem. We have query access to a vector $\vx \in \R^N$ such that $\vhx = \vF \vx$ is $k$-sparse (or nearly $k$-sparse) for some orthogonal transform $\vF$. The goal is to output an approximation (in an $\ell_2$ sense) to $\vhx$ in sublinear time. This problem has been well-studied in the special case that $\vF$ is the Discrete Fourier Transform (DFT), and a long line of work has resulted in sparse Fast Fourier Transforms that run in time… CONTINUE READING

#### Citations

##### Publications citing this paper.
SHOWING 1-2 OF 2 CITATIONS

## Sparse Harmonic Transforms II: Best s-Term Approximation Guarantees for Bounded Orthonormal Product Bases in Sublinear-Time

• Computer Science, Mathematics
• ArXiv
• 2019
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CITES BACKGROUND

## Survey on Automated Machine Learning

• Mathematics, Computer Science
• ArXiv
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• 2016
VIEW 3 EXCERPTS
HIGHLY INFLUENTIAL


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# Difference between revisions of "2008 AMC 8 Problems/Problem 14"

## Problem

Three $\text{A's}$, three $\text{B's}$, and three $\text{C's}$ are placed in the nine spaces so that each row and column contain one of each letter. If $\text{A}$ is placed in the upper left corner, how many arrangements are possible?

$[asy] size((80)); draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0)); draw((3,0)--(3,9)); draw((6,0)--(6,9)); draw((0,3)--(9,3)); draw((0,6)--(9,6)); label("A", (1.5,7.5)); [/asy]$

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

## Solution

There are $2$ ways to place the remaining $\text{As}$, $2$ ways to place the remaining $\text{Bs}$, and $1$ way to place the remaining $\text{Cs}$ for a total of $(2)(2)(1) = \boxed{\textbf{(C)}\ 4}$.


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#### Vol. 7, No. 4, 2014

 Recent Issues
 The Journal Cover About the Cover Editorial Board Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates

### Jared Speck

Vol. 7 (2014), No. 4, 771–901
##### Abstract

We study the coupling of the Einstein field equations of general relativity to a family of nonlinear electromagnetic field equations. The family comprises all covariant electromagnetic models that satisfy the following criteria: (i) they are derivable from a sufficiently regular Lagrangian; (ii) they reduce to the standard Maxwell model in the weak-field limit; (iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the $\left(1+3\right)$-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the standard Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave-coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations that have a special null structure, is based on an extension of a geometric energy-method framework developed by Christodoulou together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields and thus avoid the use of electromagnetic potentials.

##### Keywords
Born–Infeld, canonical stress, energy currents, global existence, Hardy inequality, Klainerman–Sobolev inequality, Lagrangian field theory, nonlinear electromagnetism, null condition, null decomposition, quasilinear wave equation, regularly hyperbolic, vector field method, weak null condition
##### Mathematical Subject Classification 2010
Primary: 35A01, 35Q76
Secondary: 35L99, 35Q60, 35Q76, 78A25, 83C22, 83C50


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### Improved Algorithms For Collaborative PAC Learning

 Authors: Huy Nguyen Princeton Lydia Zakynthinou Northeastern University

Introduction:

The authors study a recent model of collaborative PAC learning where $k$ players with $k$ different tasks collaborate to learn a single classifier that works for all tasks.

Abstract:

We study a recent model of collaborative PAC learning where $k$ players with $k$ different tasks collaborate to learn a single classifier that works for all tasks. Previous work showed that when there is a classifier that has very small error on all tasks, there is a collaborative algorithm that finds a single classifier for all tasks and has $O((\ln (k))^2)$ times the worst-case sample complexity for learning a single task. In this work, we design new algorithms for both the realizable and the non-realizable setting, having sample complexity only $O(\ln (k))$ times the worst-case sample complexity for learning a single task. The sample complexity upper bounds of our algorithms match previous lower bounds and in some range of parameters are even better than previous algorithms that are allowed to output different classifiers for different tasks.


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# Chapter 5 - Exponents and Polynomials - Chapter 5 Test: 10

$2x^{3}+2x^{2}-19x-21$

#### Work Step by Step

To find the product of $(x+3)$ and $(2x^{2}-4x-7)$, we need to use the distributive property in which each term of the first polynomial is multiplied to each term of the second polynomial. $(x+3)(2x^{2}-4x-7)$ $=x(2x^{2}-4x-7)+3(2x^{2}-4x-7)$ $=2x^{3}-4x^{2}-7x+6x^{2}-12x-21$ Then, we simplify the resultant expression: $=2x^{3}-4x^{2}-7x+6x^{2}-12x-21$ $=2x^{3}-4x^{2}+6x^{2}-7x-12x-21$ $=2x^{3}+2x^{2}-19x-21$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


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Lesson 9

Multiplying Rational Numbers

The practice problem answers are available at one of our IM Certified Partners

Problem 1

Fill in the missing numbers in these equations

1. $$\text-2 \boldcdot (\text-4.5) = {?}$$
2. $$(\text-8.7) \boldcdot (\text-10) = {?}$$
3. $$(\text-7) \boldcdot {?} = 14$$
4. $${?} \boldcdot (\text-10) = 90$$

Problem 2

A weather station on the top of a mountain reports that the temperature is currently $$0^\circ\text{C}$$ and has been falling at a constant rate of $$3^\circ\text{C}$$ per hour. If it continues to fall at this rate, find each indicated temperature. Explain or show your reasoning.

1. What will the temperature be in 2 hours?

2. What will the temperature be in 5 hours?

3. What will the temperature be in half an hour?

4. What was the temperature 1 hour ago?

5. What was the temperature 3 hours ago?

6. What was the temperature 4.5 hours ago?

Problem 3

Find the value of each expression.

1. $$\frac14 \boldcdot (\text-12)$$
2. $$\text{-}\frac {1}{3} \boldcdot 39$$
3. $$(\text{-}\frac {4}{5}) \boldcdot (\text-75)$$
4. $$\text{-}\frac {2}{5} \boldcdot (\text{-}\frac {3}{4})$$
5. $$\frac83 \boldcdot \text-42$$

Problem 4

To make a specific hair dye, a hair stylist uses a ratio of $$1\frac18$$ oz of red tone, $$\frac 34$$ oz of gray tone, and $$\frac58$$ oz of brown tone.

1. If the stylist needs to make 20 oz of dye, how much of each dye color is needed?

2. If the stylist needs to make 100 oz of dye, how much of each dye color is needed?

(From Grade7, Unit 4, Lesson 2.)

Problem 5

1. Here are the vertices of rectangle $$FROG$$: $$(\text-2,5), (\text-2,1), (6,5), (6,1)$$.
Find the perimeter of this rectangle. If you get stuck, try plotting the points on a coordinate plane.
2. Find the area of the rectangle $$FROG$$.
3. Here are the coordinates of rectangle $$PLAY$$: $$(\text-11, 20), (\text-11, \text-3), (\text-1, 20), (\text-1, \text-3)$$. Find the perimeter and area of this rectangle. See if you can figure out its side lengths without plotting the points.
(From Grade7, Unit 5, Lesson 7.)


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## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

(a) We can find the acceleration of the system of all three blocks. $F = (m_1+m_2+m_3)~a$ $a = \frac{F}{(m_1+m_2+m_3)}$ $a = \frac{12~N}{1.0~kg+2.0~kg+3.0~kg}$ $a = 2.0~m/s^2$ The force that the 2.0-kg block exerts on the 3.0-kg block provides the force to accelerate the 3.0-kg block with an acceleration of $2.0~m/s^2$. We can find the force that the 2.0-kg block exerts on the 3.0-kg block. $F = m_3~a$ $F = (3.0~kg)(2.0~m/s^2)$ $F = 6.0~N$ The 2.0-kg block exerts a force of 6.0 N on the 3.0-kg block. (b) The force that the 1.0-kg block exerts on the 2.0-kg block provides the force to accelerate the 2.0-kg block and the 3.0-kg block with an acceleration of $2.0~m/s^2$. We can find the force that the 1.0-kg block exerts on the 2.0-kg block. $F = (m_2+m_3)~a$ $F = (2.0~kg+3.0~kg)(2.0~m/s^2)$ $F = 10~N$ The 1.0-kg block exerts a force of 10 N on the 2.0-kg block. By Newton's third law, the 2.0-kg block exerts an equal and opposite force on the 1.0-kg block. Therefore, the 2.0-kg block exerts a force of 10 N on the 1.0-kg block.


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Find the first partial derivatives of $f(x,y,z) = z \ \arctan(\frac{y}{x})$ at the point (5, 5, 2).

A. $\frac{\partial f}{\partial x}(5, 5, 2) =$

B. $\frac{\partial f}{\partial y}(5, 5, 2) =$

C. $\frac{\partial f}{\partial z}(5, 5, 2) =$

Your overall score for this problem is


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# Math Help - Problem proving arcsech(x)

1. ## Problem proving arcsech(x)

I have a problem where i have to prove arcsech(x) = log[(1+sqr(1-X^2))/x]
does anyone know how to do this?

2. let $y= \text{sech} x = \frac{1}{\text{cosh} x} = \frac{2}{e^{x}+e^{-x}}$ and solve for $x$

$ye^{x}+ye^{-x} - 2 = 0$

now let $u = e^{x}$

then $yu + y\frac{1}{u} - 2 = 0$

or $yu^{2} - 2u+y = 0$

using the quadratic equation to solve for $u$

$u = \frac{2 \pm \sqrt{4-4y^{2}}}{2y} = \frac{1 \pm \sqrt{1-y^{2}}}{y}$

$u$ is greater than zero (since $e^{x} > 0$), so ignore the negative root

then $e^{x} = \frac{1 + \sqrt{1-y^{2}}}{y}$

and $x = \ln \Big(\frac{1 + \sqrt{1-y^{2}}}{y}\Big)$ for $0 < y \le 1$