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# My Favorite Engineer, Physicist, Mathematician Joke

#### Ackbach

##### Indicium Physicus
Staff member
An engineer, a physicist, and a mathematician are each asked to compute the volume of a little red rubber ball. The mathematician find the equation of the surface, performs a triple integral, and computes the volume. The physicist dunks the ball in a pool of water, and measures the volume of the displaced water. The engineer looks up the value in his Book of Little Red Rubber Ball Volumes.

#### Krizalid

##### Active member
Haha this is funny, here in my country people tell a lot of these jokes.

1) Why the math book kill itself? - Because had too many problems.

2) A collection of functions is dancing and the exponential one is alone. Some function stands right to it and says: Why don't you integrate? - Doesn't matter, it's the same.

#### soroban

##### Well-known member

Why was the identity .$\sin2r \:=\:2\sin r$ .refused a loan?

. . He needed a $\cos r.$

#### Jameson

Staff member
Some corny ones here, but I just love Sheldon so much I had to post this. Hope the embedding works on this one...

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#### Alexmahone

##### Active member

Why was the identity .$\sin2r \:=\:2\sin r$ .refused a loan?

. . He needed a $\cos r.$
I don't get it.

#### Random Variable

##### Well-known member
MHB Math Helper
I don't get it.
cosigner

---------- Post added at 04:02 PM ---------- Previous post was at 03:59 PM ----------

It's not letting me make back-to-back posts.

Methods of Mathematical Proof

Proof by vigorous handwaving:
Works well in a classroom or seminar setting.

Proof by deferral:
"We'll prove this later in the course."

"As is quite clear, the elementary aforementioned statement is obviously valid."

Proof by omission:
"The other 253 cases are analogous."

Proof by seduction:
"Convince yourself that this is true."

Proof by reduction to the wrong problem:
"To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem."

Proof by reference to inaccessible literature:
The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.

Proof by vehement assertion:
It is useful to have some kind of authority relation to the audience.

Proof by cumbersome notation:
Best done with access to at least four alphabets and special symbols.

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#### soroban

##### Well-known member

What is the difference between a psychotic and a neurotic?

A psychotic thinks 2 +2 = 5.

A neurotic knows that 2 + 2 = 4,
but it worries him.

What is the difference between data and information?

Data
382436

Information
38-24-36

My two secrets for success:

1. Never tell anyone everything you know.

#### Random Variable

##### Well-known member
MHB Math Helper
More Methods of Mathematical Proof

Proof by imagination:
"Well, we'll pretend its true."

Proof by lack of interest:
"Does anyone really want to see this?"

Proof by lack of sufficient time:
"Because of the time constraint, I'll leave the proof to you."

Proof by lost reference:
"I know I saw it somewhere."

Proof by definition:
"We'll define it to be true."

Proof by clever variable choice:
"Let A be the number such that this proof works."

Proof by illegibility:
(scribble, scribble) QED

Proof by intimidation:
"Don't be stupid! Of course it's true!"

Proof by accident:
"Hey, what have we here?!"

Proof by Profanity:
(example omitted)

#### dwsmith

##### Well-known member
One day an engineer and a physicist decide to go for a hot air balloon ride. After some time, they decide to come down and figure out where they are. They see a person running and call down, "Hey, where are we?" After a few moments, the running responds, "You are in a hot air balloon." The engineer yells down, "You must be a mathematician!". The runner in surprise ask how he knows? The engineer responds, "your answer took to long, was 100% correct, and absolutely useless."

#### CaptainBlack

##### Well-known member
One day an engineer and a physicist decide to go for a hot air balloon ride. After some time, they decide to come down and figure out where they are. They see a person running and call down, "Hey, where are we?" After a few moments, the running responds, "You are in a hot air balloon." The engineer yells down, "You must be a mathematician!". The runner in surprise ask how he knows? The engineer responds, "your answer took to long, was 100% correct, and absolutely useless."
How about "We provided you with the tools to answer that yourself 200 years ago!"

CB

#### dwsmith

##### Well-known member
How about "We provided you with the tools to answer that yourself 200 years ago!"

CB
What?

#### CaptainBlack

##### Well-known member
Almost all the mathematical tools required for engineering have been around for most of the last 200 years.

CB

#### HallsofIvy

##### Well-known member
MHB Math Helper
A physicist, a chemist, and a MICROSOFT engineer are riding in a car when it suddenly quits running. The physicist suggests that they check the electrical system. The chemist suggest that they check the gasoline. The MICROSOFT engineer suggests that they close all of the windows then open them again.

#### CaptainBlack

##### Well-known member
A physicist, a chemist, and a MICROSOFT engineer are riding in a car when it suddenly quits running. The physicist suggests that they check the electrical system. The chemist suggest that they check the gasoline. The MICROSOFT engineer suggests that they close all of the windows then open them again.
But...but...but the engine control processor does not have a dispaly attached

CB

#### earboth

##### Active member
A helicopter was flying around above Seattle yesterday when an electrical malfunction disabled all of the aircraft's electronic navigation and communication equipment. Due to the clouds and haze the pilot could not determine his position or course to steer to the airport. The pilot saw a tall building, flew toward it, circled, drew a handwritten sign and held it in the helicopter's window. The sign said "WHERE AM I ?" in large letters.

People in the tall building quickly responded to the aircraft, drew a large sign and held it in a building window. Their sign said, "YOU ARE IN A HELICOPTER." The pilot smiled, waved, looked at his map and determine the course to steer to SEATAC (Seattle/Tacoma) airport and landed safely.

After they were on the ground, the co-pilot asked the pilot how the "YOU ARE IN A HELICOPTER" sign helped determine their position. The pilot responded, "I knew that had to be the MICROSOFT building because they gave me a technically correct but completely useless answer."

#### issacnewton

##### Member
An engineer, a mathematician, and a physicist are staying for the night in a hotel. Fortunately for this joke, a small fire breaks out in each room.The physicist awakes, sees the fire, makes some careful observations, and on the back of the hotel's wine list does some quick calculations. Grabbing the fire extinguisher, he puts out the fire with one, short, well placed burst, and then crawls back into bed and goes back to sleep.
The engineer awakes, sees the fire, makes some careful observations, and on the back of the hotel's room service list (pizza menu) does some quick calculations. Grabbing the fire extinguisher (and adding a factor of safety of 5), he puts out the fire by hosing down the entire room several times over, and then crawls into his soggy bed and goes back to sleep.
The mathematician awakes, sees the fire, makes some careful observations, and on a blackboard installed in the room, does some quick calculations. Jubliant, he exclaims "A solution exists!", and crawls into his dry bed and goes back to sleep.

Alternate Ending

The mathematician awakes, sees the cooling embers of the fire from one of his neighbors, fans it back into a roaring inferno, observes that "this reduces to a previously solved problem", crawls into his warm bed, and goes back to sleep

#### Deveno

##### Well-known member
MHB Math Scholar
my favorite is this one:

a topologist, an engineer, and a physicist are all locked in a room with a can of food, and no can opener, and left there for a week. after the week is up, the doors are unlocked. in the engineer's room, the walls are pock-marked with the dents he put in it to bash open the can by throwing it against the wall. in the physicist's room, they find she has crafted a crude can-opener from pieces of her cot. in the topologist's room, they just find the can. the physicist grabs her can-opener, and opens the can....the topologist crawls out of the can, and sheepishly says: "oops, sign error."

alternate ending (mathematician instead of topologist):

they open the mathematician's room, to find him dead on the floor. scrawled in blood on the floor is:

theorem: if i don't open the can, i will die of starvation

proof: assume the opposite, and derive a contradiction.

...

#### Chris L T521

##### Well-known member
Staff member
YES!! I've seen a bunch of these things floating around the internet, and I had yet to see one for mathematicians. Thanks for posting! XD

#### Alexmahone

##### Active member
What does the mathematical expression in your signature represent?

#### Also sprach Zarathustra

##### Member
$F(U)C(k)$

Fermat's $U$ number, $\Cos(k)$ in power series.

#### Deveno

##### Well-known member
MHB Math Scholar
if f is the identity transformation from the vector space of real-valued functions defined on all of $\mathbb{R}$ to itself, and we agree to make the substitution:

x = nln(u),

then certainly we can all agree that:

$\int e^x = f(u^n)$


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# Ruffini – abstract algebra in Java

Class inheritance and generics/templates makes it possible to write code with nice abstractions very similar to the abstractions used in math, particularly in abstract algebra, where a computation can be described for an entire class of algebraic structures sharing some similar properties (eg. groups and rings). This has been described nicely in the book “From Mathematics to Generic Programming” by Alexander A. Stepanov and Daniel E. Rose.

Inspired by this book, I have implemented a library for computations on abstract algebraic structures such as groups, rings and fields. The library is called Ruffini (named after the italian mathematician Paolo Ruffini) and is developed in Java using generics to achieve the same kind of abstraction as in abstract algebra, eg. that you do not specify what specific algebraic structure and elements are used, but only what abstract structure it has, eg. that it is a group or a ring.

Abstract algebraic structures are defined in Ruffini by a number of interfaces extending each other, each describing operations on elements of some set represented by a generic class E. The simplest such interface is a semigroup:

public interface Semigroup<E> {

/**
* Return the result of the product ab in this
* semigroup.
*
* @param a An element a in this algebraic structure.
* @param b Another element b in this algebraic
*          structure.
* @return The product of the two elements, ab.
*/
public E multiply(E a, E b);

}

The semigroup is extended by a monoid interface which adds a getIdentity() method and by a group interface which adds an invert(E) method. Continuing like this, we end up with interfaces for more complicated structures like rings, fields and vector spaces.

Now, each of these interfaces describes functions that can be applied to a generic type E (eg. that given two instances of E, the multiply method above returns a new instance of type E). As in abstract algebra, these functions can be used to describe complicated computations without specifying exactly what algebraic structure is being used.

Ruffini currently implements a number of algorithms that is defined and described for the abstract structures, including the discrete Fourier transform, the Euclidean algorithm, computing Gröbner bases, the Gram-Schmidt process and Gaussian elimination. The abstraction makes it possible to write the algorithms only once in a clear, while still being usable on any of the algebraic structures defined in the library, such as integers, rational numbers, integers modulo n, finite fields, polynomial rings and matrix rings. Below is the code for the Gram-Schmidt process:

public List<V> apply(List<V> vectors) {
List<V> U = new ArrayList<>();

Projection<V, S> projection =
new Projection<>(vectorSpace, innerProduct);

for (V v : vectors) {
for (V u : U) {
V p = projection.apply(v, u);
v = vectorSpace.subtract(v, p);
}
}
return U;
}

Here V and S are the generic types for resp. the vectors and scalars of the underlying vector space, and Projection is a class defining projection given a specific vector space and an inner product.

# Solving the 0-1 Knapsack problem with Excel

Given a list of items each with a value and a weight, the Knapsack problem seeks to find the set of items with the largest combined value within a given weight limit. There are a nice dynamic programming solution which I decided to implement in a spread sheet. I used Google Sheets but the solution is exported as an excel-sheet.

The solution builds a Knapsack table for round 0,1,…,limit. In each round r the solution for the problem with limit r is constructed as a column in the table, so the table has to be as wide as the maximum limit. Once the table is built, the solution can be found using backtracking. This is all described pretty well on Wikipedia, https://en.wikipedia.org/wiki/Knapsack_problem#0/1_knapsack_problem.

The main challenge was to translate this algorithm from procedural pseudocode to a spreadsheet. Building the table is simple enough (once you learn the OFFSET command in Excel which allows you to add or subtract a variable number of rows and columns from a given position), but the backtracking was a bit more tricky.

Assuming that the weights are stored in column A from row 3 and the corresponding values are stored in column B, the table starts in column D. The round numbers are stored in row 1 from columnD. Row 2 are just 0’s and all other entries are (the one below is from D3):

=IF($A3>D$1; D2; MAX(D2; OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));-1;-$A3) +$B3))

The table stops after round 40, so we find the solution with weight at most 40.

The backtracking to find the actual solution after building the table is done in column AT and AU. In the first column, the number of weight spent after 40 rounds are calculated from bottom to top row using the formula (from AT3):

=IF(OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));0;-AT23-2) <> OFFSET(INDIRECT(ADDRESS(ROW(); COLUMN()));-1;-AT23-2); A22 + AT23; AT23)

The solution is shown in column AU where for each row we simply check if the accumulated weight increased with this item.

# Gram-Schmidt process for integers

Given a set of linearly independent vectors, the Gram-Schmidt process produces a new set of vectors spanning the same subset, but the new set of vectors are now mutually orthogonal. The algorithm is usually stated as working over a vector space, but will in this post be altered to work over a module which is a vector space where the scalars are elements in a ring instead of a field. This could for instance be the case if we are given a set of integer vectors we wish to orthogonalise.

The original version can be described as follows: Given a set of vectors v_1, \ldots, v_n we define u_1 = v_1 and

u_k = v_k - \text{proj}_{u_1}(v_k) - \text{proj}_{u_2}(v_k) - \cdots - \text{proj}_{u_{k-1}}(v_k)

for k = 2,3,\ldots,n. The projection operator is defined as

\text{proj}_u(v) = \frac{\langle u, v \rangle}{\langle u, u \rangle} u.

If we say that all v_i have integer entries, we see that u_i must have rational entries, and simply scaling each vector with a multiple of all the denominators of the entries will give a vector parallel to the original vector but with integer entries. But what if we are interested in an algorithm that can only represent integers?

The algorithm is presented below. Note that the algorithm works over any module with inner product (aka a Hilbert Module).

# Given: Set of vectors V.
# Returns: A set of mutually orthogonal vectors spanning
# the same subspace as the vectors in V.

U := Ø
k := 1

while V ≠ Ø do
poll vector v from V
w := kv

for (u,m) in U do
w -= m〈v,u〉u

# Optional: Divide all entries in w by gcd of entries

if V ≠ Ø do
n :=〈w,w〉
for (u,m) in U do
m *= n
Put (w,k) in U
k *= n
else do
# No more iterations after this one
Put (w,k) in U

return first coordinates of elements in U

When used on integers, the entries in the vectors grow very fast, but this may be avoided by dividing w by the greatest common divisor of the entries.

# DIY preamp for Leslie 760

I have recently bought a used Leslie 760 amplifier, but unfortunately it came without the pedal used to change the speed of the rotator and which also acts as preamp. A used pedal and the necessary cables is priced at about €400 so I decided to build one myself.

Luckily, some nice people has posted scanned versions of the old manuals for Leslie amps and put them online: http://www.captain-foldback.com/Leslie_sub/leslie_manuals.htm.

The amplifier does not have a normal jack-plug for connection with an instrument but has instead a 9-pole plug. At first glance, this seems a bit confusing, but it is quite simple: Pole 1 is ground and pole 2 is sound input. The rotator is activated by grounding pole 6 (slow) or pole 7 (fast).

With this information it was easy to build everything I needed: I built a small box with room for two jack plugs to be put on the amp. The first plug is mono for instrument connection, and is attached to pole 1 and 2. The other plug is for stereo which is connected to pole 6 and 7 (and ground). At the other end of this stereo cable I attached a switch pedal with two buttons: One for switching between grounding the two poles of the stereo jack and another for switching the connection on/off. I got everything for about €25 on musik-ding.de.

I still needed a preamp and found one by the brand ‘Art’ at 4Sound.dk for €35, so for about 60€ I got everything I needed to run the amp.

You are more than welcome to write me at mail@jonaslindstrom.dk if you have any questions.

format: plain


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# System of Linear Equations

A set of linear equations with two or more variables having degree one. Before we go in details it is also recommended to have a look on other form of linear equations like linear equations with one variable and with two variables and so on. System of linear equation is one the most prominent topics in algebra. In this section will learn more about system of linear equations with two variables and different methods to solve them such as substitution method and elimination methods etc.

## What is the System of Linear Equations

System of linear equations is a set of two or more linear equations working together and involving the same set of variables. A general system can contains m linear equations with n unknowns.

For Example:
System of linear equations with 2 variables is:

$a_1 x + b_1 y + c_1 = 0$ and
$a_2 x + b_2 y + c_2 = 0$.

System of linear equations with 3 variables:
$a_1 x + b_1 y + c_1 z + d_1= 0$
$a_2 x + b_2 y + c_2 z + d_2 = 0$ and
$a_3 x + b_3 y + c_3 z + d_3 = 0$

## General Form

For $x_1,x_2, x_3,.....x_n$ are the unknowns and $b_1, b_2,.... b_m$ are the constant terms and $a_{11}, a_{12},.......,a_{mn}$ are the coefficients of the system. A general system of m linear equations with n unknowns can be written as:

$a_{11}x_1+a_{12}x_2+....+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+....+a_{2n}x_n=b_2$
.
.
.
$a_{m1}x_1+a_{m2}x_2+....+a_{mn}x_n=b_m$

## Solving System of Linear Equations in Two Variables

Let us consider, $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$.

There can be different ways/methods to solve these homogeneous system of linear equations. Some are
1. Elimination Method
2. Substitution Method.
3. Matrix Method
4. Cross Multiplication Method

The solutions for the system of equations can be consistent or inconsistent. All based upon the ratios of the coefficients. For the above system of equations:

 System of Equations Condition Solution Type Consistent $\frac{a_{1}}{a_{2}}\: \neq \: \frac{b_{1}}{b_{2}}$ Unique solution Consistent $\frac{a_{1}}{a_{2}}\: = \: \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ Infinite solution Inconsistent $\frac{a_{1}}{a_{2}}\: = \: \frac{b_{1}}{b_{2}}\: \neq \: \frac{c_{1}}{c_{2}}$ Solution does not exists

## System of Linear Equations in Three Variables:

The general form of linear equation in three variables, x, y and z is
ax + by + cz +d =0, where a, b, c are real numbers and a, b, c not all equal to 0.
This represent the equation of a plane in three-dimensional co-ordinate system, where a, b, c are the direction ratios of the normal to the plane.
To solve the equation in three variables, we need to have three conditions (equations) relating the variables x, y and z.
Elimination method is the most suitable method to solve the equations.

## Linear System of Differential Equations:

A differential equation in which the dependent variable and its derivatives appear only in first degree is called a linear differential equation.
An ordinary linear differential equation of order n is of the form,
$\frac{\mathrm{d^n}y }{\mathrm{d} x^n}+P_{1}\frac{d^{n-1}y }{dx^{n-1}}+P_{2}\frac{d^{n-2}y }{dx^{n-2}}+...................P_{n}y=X$, where $P_{1}, P_{2},....................P_{n}$ and X are functions of x.

If X = 0, it is called homogeneous equation, otherwise it is a non-homogeneous equation.

First order linear differential equation:
The general form of a linear equation of the first order is,
$\frac{\mathrm{d} y}{\mathrm{d} x}+ Py = Q$, where P and Q are functions of x.
The solution to the above equation is given by,
y = $e^{-\int Pdx}\int Qe^{\int Pdx}dx$

## Examples

Example 1: Solve : 2x + 3y = 25  and  3x + 2y =25
Solution:
2x + 3y = 25   ------------(1)
3x + 2y = 25   ------------(2)
Multiplying Equation (1) by 3,
3 ( 2x + 3y ) = 3 ( 25)
=> 6x + 9y = 75   -----------(3)
Multiplying equation (2) by 2,
2 ( 3x + 2y ) = 2 ( 25)
=> 6x + 4y = 50    ------(4)

Solve equation 3 and equation 4,
Subtracting (4) from  (3)
5y = 25
=> y = 25 /5 = 5
Substituting y=5 in (1) we get,
2x + 3(5) = 25
=>  2x + 15 = 25
=>  2x   = 25 - 15  = 10
=> x = 10/2 = 5
Therefore, the two equations intersect at the point (5, 5).

Example 2: Solve for x and y. x + 3y = 8 ;  3x + 9y = 24
Solution:
Let x + 3y = 8  --------(1)
3x + 9y = 24    ------- (2)
Substituting x = 8 - 3y in (2),
3 ( 8 - 3y ) + 9y = 24
=>  24 - 9y + 9y = 24
This condition is true for all values of y.
we get 24 = 24 which is true.
Since $\frac{a_{1}}{a_{2}}\: = \: \frac{b_{1}}{b_{2}}\: \neq \: \frac{c_{1}}{c_{2}}$ = 1/3
So solution does not exist for this particular example.

## Word Problems

Follow below steps to solve word problems.
Step 1: Read and understand the problem carefully.
Step 2: Identify the unknown quantities. Represent with the variable.
Step 3: Formulated the equations.
Step 4: Simplify the equations, using any of the methods.

Example: 4 chairs and 3 tables cost Rs. 1400 and 5 chairs and 2 tables cost  Rs. 1400. Find the cost of a chair and a table.
Solution:
Let the cost of a chair be Rs. x , and the cost of a table be Rs. y.
 Chairs Tables Total Cost Equation No. of Items 4 3 1400 Cost 4x 3y 4x + 3y 4x + 3y = 1400 No. of items 5 2 1400 Cost 5x 2y 5x + 2y 5x + 2y = 1400

Let us solve linear system of equations we have framed here.
Hence we have the equations,
4x + 3y = 1400    -----------(1)
5x + 2y = 1400      ---------(2)
Multiplying (1) by 5,
5( 4x + 3y ) = 5( 1400)
=>     20 x + 15 y = 7000  -----------(3)
Multiplying (2) by 4, we get
4 ( 5x + 2y ) = 4 ( 1400)
=> 20 x + 8 y = 5600   -------(4)

20 x + 15 y = 7000 ----------------------(3)

Subtracting (3) from (4),
- 7 y   = -1400
=>   y = -1400/-7 = 200
Substituting y= 200 in Equation (1), we get,
4x + 3 ( 200) = 1400
=>    4x + 600     = 1400
=>    4x = 1400 - 600 = 800
=>    x = 800/4 = 200
Therefore cost of a Chair is Rs. 200 and cost of a Table is Rs. 200.


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text:
User:

Documento sin título
 Pre-algebra Arithmetics Integers Divisibility Decimals Fractions Exponents Percentages Proportional reasoning Radical expressions Graphs Algebra Monomials Polynomials Factoring Linear Equations Graphs of linear equations Rectangular Coordinate System Midpoint Formula Definition of Slope Positive and negative slope Determine the slope of a line Equations of lines Equation of lines (from graph) Applications of linear equations Inequalities Quadratic equations Graphs of quadratic equations Absolute Value Radical expressions Exponential equations Logarithmic equations System of equations Graphs and functions Plotting points and naming quadrants Interpreting Graphs Relations and Functions Function Notation Writing a Linear Equation from a Table Writing a Linear Equation to describe a Graph Direct Variation Indirect Variation Domain and range Sequences and series Matrices Inverse of a matrix Determinants Inner product Geometry Triangles Polygons 2-D Shapes 3-D Shapes Areas Volume Pythagorean Theorem Angles Building Blocks Geometry Transformations Parallel, coincident and intersepting lines Distances in the plane Lines in space Plane in space Angles in the space Distances in the space Similarity Precalculus Sequences and series Graphs Graphs Definition of slope Positive or negative slope Determine the slope of a line Equation of a line (slope-intercept form) Equation of a line (point slope form) Equation of a line from graph Domain and range Quadratic function Limits (approaches a constant) Limits (approaches infinity) Asymptotes Continuity and discontinuities Parallel, coincident and intersepting lines Introduction to Functions Limits Continuity Asymptotes Trigonometry Trigonometric ratios The reciprocal trigonometric ratios Trigonometric ratios of related angles Trigonometric identities Solving right angles Law of sines Law of cosines Domain of trigonometric functions Statistics Mean Median Mode Quartiles Deciles Percentiles Mean deviation Variance Standard Deviation Coefficient of variation Skewness kurtosis Frequency distribution Graphing statistics & Data Factorial Variations without repetition Variations with repetition Permutations without repetition Permutation with repetition Circular permutation Binomial coefficient Combinations without repetition Combinations with repetition

 Polynomials Polynomial Vocabulary Simplifying expressions Addition Subtraction Multiplication (FOIL) Multiplication Special products Long division Synthetic division Remainder theorem Roots and factors of a polynomial Polynomial Function Polynomial Vocabulary A monomial is a number, variable, or product with non-fractional exponents. 3, 4xy, -8x, -$\frac{1}{3}$x100 are monomials 4-10, x-3=$\frac{1}{x^3}$ are not monomials A polynomial is the sum or difference of monomials. 3 is a monomial 4x+2y is a binomial 4xy-5x+8 is a trinomial The degree of a monomial is the sum of the exponents of the variables. The degree of 4xy is 2 The degree of 4x2y3 is 5 The degree of 3 is 0 The degree of a polynomial is the degree of the term with the largest degree. The degree of the polynomial 4x3+x2+x8 is 8 Standard Form of a polynomial is the list of the monomials in order from largest to smaller degree. The polynomial x8+4x3+x2 is written in its Standard Form Leading coefficient is the coefficient of the term with the largest degree. The leading coefficient of the polynimial x8+4x3+x2 is 1.


text:
2 arguments->algorithms

Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation distance is 0, 1, 2, 3, many, then I am happy...

I know there are arguments algorithms for computing distances IN the curve complex, but this is not quite the same...

1

Translation distance in the curve complex

Given a map $\psi: S\rightarrow S,$ for $S$ a closed surface, is there any algorithm to compute its translation distance in the curve complex? I should say that I mostly care about checking that the translation distance is/is not very small. That is, if the algorithm can pick among the possibilities: translation distance is 0, 1, 2, 3, many, then I am happy...

I know there are arguments for computing distances IN the curve complex, but this is not quite the same...


text:
# Solving differential equation

What is the general method for solving a differential equation of
the form

\frac{\partial^{2}z}{\partial x^{2}}+\frac{\partial^{2}z}{\partial y\partial x}=C

where C is a constant.

You can define $$w=\frac{\partial z}{\partial x}$$That simplifies your equation. Defining a=x+y and b=x-y and rewriting the equation in terms of those should make it easier again. Looks like there is a lot of freedom in the solution with just this condition. z can be recovered by integration later.


text:
## Chemistry 10th Edition

b) $LiNO_3$, c) $NaI$ and e) $KClO_4$
Analyzing the base of the cation and the acid of the anion, we obtain: a) Strong base, but weak acid $(HP{O_4}^{2-})$ b) Strong base and strong acid. c) Strong base and strong acid. d) Strong base, but weak acid $(HC{O_3}^-)$ e) Strong base and strong acid.