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I'm currently trying to understand the concepts and theory behind some of the common proof verifiers out there, but am not quite sure on the exact nature and construction of the sort of systems/proof calculi they use. Are they essentially based on higher-order logics that use Henkin semantics, or is there something mor... |
What's the meaning of the double turnstile symbol in logic or mathematical notation? :
$\models$
|
Many years ago, I had read a book entitled "Think of a Number" by Malcolm E. Lines, and it was an eminently readable and thought provoking book. In the book, there were topics like Fibonacci numbers (along with the live examples from the nature) and Golden Section. Now I'm looking for a similar book. Can anyone recomme... |
I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$, the cardinality of the continuum. However, I read in the book "Metric spaces" by Ó Searcóid that set of all $[0, 1]\to\mathbb{R}$ continuous functions is greater than $\mathfrak c$:
"It is demonstrated in many textbooks that $\mathb... |
When I used to compete in Olympiad Competitions back in high school, a decent number of the easier geometry questions were solvable by what we called a geometry bash. Basically, you'd label every angle in the diagram with the variable then use a limited set of basic geometry operations to find relations between the ele... |
An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who supported such a system. I can see that the natural numbers and rational numbers can easily defined in a finitist system, by... |
Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the reals.
This isn't an obvious step, so why is it true?
|
I just came back from my Number Theory course, and during the lecture there was mention of the Collatz Conjecture.
I'm sure that everyone here is familiar with it; it describes an operation on a natural number – $n/2$ if it is even, $3n+1$ if it is odd.
The conjecture states that if this operation is repeated, all numb... |
Let $A$ be a commutative ring. Suppose $P \subset A$ is a minimal prime ideal. Then it is a theorem that $P$ consists of zero-divisors.
This can be proved using localization, when $A$ is noetherian: $A_P$ is local artinian, so every element of $PA_P$ is nilpotent. Hence every element of $P$ is a zero-divisor. (As Matt... |
The group:
$$
G = \left\langle x, y \; \left| \; x^2 = y^3 = (xy)^7 = 1\right. \right\rangle
$$
is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a finitely presented group, I could do a coset enumeration, but I don't see how this helps if I want to prove that it's infinite... |
Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling).
Would this answer be any different if similar but non-identical polygons were allowed? And if only if rotated/translated by rational coefficients?
Would this answer be any different if w... |
I want to find the least squares solution to $\boldsymbol{Ax}=\boldsymbol{b}$ where $\boldsymbol{A}$ is a highly sparse square matrix.
I found two methods that look like they might lead me to a solution: QR factorization, and singular value decomposition. Unfortunately, I haven't taken linear algebra yet, so I can't re... |
Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?
Here a division algebra is an associative algebra where every nonzero number is invertible (like a field, but without assuming commutativity of multiplication).
This is an old result proved... |
In Intro Number Theory a key lemma is that if $a$ and $b$ are relatively prime integers, then there exist integers $x$ and $y$ such that $ax+by=1$. In a more advanced course instead you would use the theorem that the integers are a PID, i.e. that all ideals are principal. Then the old lemma can be used to prove that ... |
I know there's something called "ultrafinitism" which is a very radical form of constructivism that I've heard said means people don't believe that really large integers actually exist. Could someone make this a little bit more precise? Are there good reasons for taking this point of view? Can you actually get math ... |
I know some references where I can find this, but they seem tedious. Both Hartshorne and Ueno cover this.
I am wondering if there is an elegant way to describe these. If this task is too difficult in general, how about just $\mathbb{P}^n$?
Thanks!
|
There is a little triviality that has been referred to as the "soft maximum" over on John Cook's Blog that I find to be fun, at the very least.
The idea is this: given a list of values, say $x_1,x_2,\ldots,x_n$ , the function
$g(x_1,x_2,\ldots,x_n) = \log(\exp(x_1) + \exp(x_2) + \cdots + \exp(x_n))$
returns a valu... |
(Diagram and setup from UCSMP Precaluclus and Discrete Mathematics, 3rd ed.)
Above is a partial plot of the zeros of $p_c(x)=4x^4+8x^3-3x^2-9x+c$. The text stops at showing the diagram and does not discuss the shape of the locus of the zeros or describe the resulting curves. Are the curves in the locus some specific... |
How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$?
Edit: note that while returning to the origin is guaranteed $(p = 1)$ in $1$ and $2$ dimensions... |
In laymen's terms, as much as possible: What is the Riemann-Zeta function, and why does it come up so often with relation to prime numbers?
|
By this I mean books with math puzzles and problems similar to the ones you would find in mathematical olympiads.
|
What is an intuitive explanation of Markov chains, and how they work? Please provide at least one practical example.
|
(This was asked due to the comments and downvotes on this Stackoverflow answer. I am not that good at maths, so was wondering if I had made any basic mistakes)
Ignoring limits, I would like to know if this is a valid explanation for why $\frac00$ is undefined:
$x = \frac00$
$x \cdot 0 = 0$
Hence There are an infi... |
We all love a good puzzle
To a certain extent, any piece of mathematics is a puzzle in some sense: whether we are classifying the homological intersection forms of four manifolds or calculating the optimum dimensions of a cylinder, it is an element of investigation and inherently puzzlish intrigue that drives us. Indee... |
You are standing on a cliff at a height $h$ above the sea. You are capable of throwing a stone with velocity $v$ at any angle $a$ between horizontal and vertical. What is the value of $a$ when the horizontal distance travelled $d$ is at a maximum?
On level ground, when $h$ is zero, it's easy to show that $a$ needs to b... |
So I know that given a chain complex we can define the $d$-th cohomology by taking $\ker{d}/\mathrm{im}_{d+1}$. But I don't know how this corresponds to the idea of holes in topological spaces (maybe this is homology, I'm a tad confused).
|
What does it mean (in a mathematically rigorous way) to claim something is "generic?" How does this coincide with the Zariski topology?
|
Given a curve how do you intuitively construct the picture of its projective dual? I know points --> lines, lines--> points but for something like the swallowtail this is not really obvious.
|
According to Wikipedia, Godel's incompleteness theorem states:
No consistent system of axioms whose
theorems can be listed by an
"effective procedure" (essentially, a
computer program) is capable of
proving all facts about the natural
numbers.
This obviously includes our current system. So has anyone propos... |
How do we know if a particular function can be represented as a power series? And once we have come up with a power series representation, how does one figure out its radius of convergence ?
|
When I look at the Taylor series for $e^x$ and the volume formula for oriented simplexes, it makes $e^x$ look like it is, at least almost, the sum of simplexes volumes from $n$ to $\infty$. Does anyone know of a stronger relationship beyond, "they sort of look similar"?
Here are some links:
Volume formula
http://en.w... |
It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian integers that lets you prove this quickly. Can anyone explain it?
|
I'm learning Linear Algebra using MIT's Open Courseware Course 18.06
Quite often, the professor says "... assuming that the matrix is invertible ...".
Somewhere in the lecture he says that using a determinant on an $n \times n$ matrix is on the order of $O(n!)$ operations, where an operation is a multiplication and a s... |
Given n distinct objects, there are $n!$ permutations of the objects and $n!/n$ "circular permutations" of the objects (orientation of the circle matters, but there is no starting point, so $1234$ and $2341$ are the same, but $4321$ is different).
Given $n$ objects of $k$ types (where the objects within each type are i... |
From a standard $52$-card deck, how many ways are there to pick a hand of $k$ cards that includes one card from all four suits?
I know that for any specific $k$, it's possible to break it up into cases based on the partitions of $k$ into $4$ parts. For example, if I want to choose a hand of six cards, I can break it up... |
Congruence transformations (isometries) and similarity transformations (isometries + dilations) should be constructable. What about other affine transformations? Other conformal mappings?
edit: by constructable, I mean given the defining information for the transformation in a geometric way (e.g. a dilation requires ... |
For a long time I've eschewed bulky and inelegant calculators for the use of my trusty trig/log-log slide rule. For those unfamiliar, here is a simple slide rule simulator using Javascript.
To demonstrate, find the $LL_3$ scale, which is on the back of the virtual one. Let's say we want to solve $3^n$.
First, you wo... |
Assuming I can play forever, what are my chances of coming out ahead in a coin flipping series?
Let's say I want "heads"...then if I flip once, and get heads, then I win, because I've reached a point where I have more heads than tails (1-0). If it was tails, I can flip again. If I'm lucky, and I get two heads in a ro... |
Are there any results from algebraic geometry that have led to an interesting "real world" application?
|
Question : A $10$-year loan of $\$500$ is repaid with payments at the end of each year. The lender charges interest at an annual effective rate of $10\%$. Each of the first ten payments is $150\%$ of the amount of interest due. Each of the last ten payments is $X$. Calculate $X$.
My Attempt
$\$ 500$ will earn $\$50$ in... |
Suppose I have a loan of M dollars. At the end of each year, I am charged interest at rate R and make a repayment of P. The loan is repaid after n years.
How long (n) does it take to repay the loan if I am given the other variables?
How much are the repayments of P if I am given the other variable?
Suppose that the pa... |
The Problem:
Two trains travel on the same track towards each other, each going at a speed of 50 kph. They start out 50km apart. A fly starts at the front of one train and flies at 75 kph to the front of the other; when it gets there, it turns around and flies back towards the first. It continues flying back and forth ... |
Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying it:
The character table of a group is packs a lot of information about... |
The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside.
This can be proved easily by considering a cone as a solid of revolution, but I would like to know if it can be proved or at least visual demonstrated witho... |
In my AP chemistry class, I often have to balance chemical equations like the following:
$$ \mathrm{Al} + \text O_2 \to \mathrm{Al}_2 \mathrm O_3 $$
The goal is to make both side of the arrow have the same amount of atoms by adding compounds in the equation to each side.
A solution:
$$ 4 \mathrm{Al} + 3 \mathrm{ O_2} \... |
When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is differentiated with respect to $r$, we get $4 \pi r^2$.
Is this just a coincidence, or is there some deep explanation for ... |
The $24$ game is as follows. Four numbers are drawn; the player's objective is to make $24$ from the four numbers using the four basic arithmetic operations (in any order) and parentheses however one pleases.
Consider the following generalization. Given $n+1$ numbers, determine whether the last one can be obtained from... |
I am working on computing phase diagrams for alloys. These are
blueprints for a material that show what phase, or combination of
phases, a material will exist in for a range of concentrations and
temperatures (see this
pdf presentation).
The crucial step in drawing the boundaries that separate one phase
from another... |
The knight's tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight's move. Tours can be cyclic, if the last square is a knight's move away from the first, and acyclic otherwise.
There are several symmetries among kni... |
I just got out from my Math and Logic class with my friend. During the lecture, a well-known math/logic puzzle was presented:
The King has $1000$ wines, $1$ of which is poisoned. He needs to identify the poisoned wine as soon as possible, and with the least resources, so he hires the protagonist, a Mathematician. T... |
When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from $\mathbb R ^2$ to $\mathbb R ^2$. Can anyone explain what is different about complex numbers?
|
Chicken is a famous game where two people drive on a collision course straight towards each other. Whoever swerves is considered a 'chicken' and loses, but if nobody swerves, they will both crash. So the payoff matrix looks something like this:
B swerves B straight
A swerves t... |
I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful.
Is there any good examples of its use outside academia?
|
I've learned that multiplying two generating functions $f(x)$ and $g(x)$ will give the result
\begin{equation*}
\sum_{k=0}^\infty\left(\sum_{j=0}^k a_j\,b_{k-j}\right)x^k.
\end{equation*}
I've used the result, but it was presented in my class without proof and I'm having some trouble tracking one down. Weak google-fo... |
One thing I have been wondering about lately is how to write an equation that describes a pattern of numbers. What I mean is:
x 0 1 2
y 1 5 9
If I have this, I can tell that an equation that describes this would be $y=4x+1$. In fact, I don't even need the third pair of numbers. It's very easy when the... |
Motivatation
I am presently looking at a structure that I am trying to pin down- my strategy being to pull the thing up into the greatest possible generality (based on the bits I'm sure about) and narrow it down from there.
The situation I have is somewhat similar to the dual space structure in vector spaces, though al... |
I just came back from my Introduction to Rotational Kinematics class, and one of the important concepts they described was Rotational Inertia, or Moment of Inertia.
It's basically the equivalent of mass in Netwon's $F = m a$ in linear motion. The equivalent rotational equation is $\tau = I \alpha$, where $\tau$ is rot... |
I have heard that one can think of stacks on a site as taking sheaves but instead of the restrictions being equal, we just loosen it to isomorphic, and treat the sheaf conditions with the "obvious" coherence relations.
How seriously can one take this analogy? Note that my background is stacks is feeble at best.
I hope ... |
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute... |
I just came back from a class on Probability in Game Theory, and was musing over something in my head.
Assuming, for the sake of the question:
Playing cards in their current state have been around for approximately eight centuries
A deck of playing cards is shuffled to a random configuration one billion times per day
... |
I have been using Lagrange multipliers in constrained optimization problems, but I don't see how they actually work to simultaneously satisfy the constraint and find the lowest possible value of an objective function.
|
Randomly break a stick (or a piece of dry spaghetti, etc.) in two places, forming three pieces. The probability that these three pieces can form a triangle is $\frac14$ (coordinatize the stick form $0$ to $1$, call the breaking points $x$ and $y$, consider the unit square of the coordinate plane, shade the areas that ... |
About two years ago, I started thinking about the following problem: You're given an $N$ and an $S$, positive integers. You start with an $N$-gon that has positive integer labels at each vertex, such that the labels sum to $S$. A "move" is replacing the label at some vertex with the positive difference of the labels at... |
Two mathematicians each come into a coffee shop at a random time between 8:00 a.m. and 9:00 a.m. each day. Each orders a cup of coffee then sits at a table, reading a newspaper for 20 minutes before leaving to go to work.
On any day, what is the probability that both mathematicians are at the coffee shop at the same ti... |
I want to randomly generate trees, i.e. undirected acyclic graphs with a single root, making sure that all possible trees with a fixed number of nodes n are equally likely.
|
Suppose that I am buying cakes for a party. There are $k$ different types and I intend to buy a total of $n$ cakes. How many different combinations of cakes could I possibly bring to the party?
|
Suppose we model traffic flow between two points with a directed graph. Each route has either a constant travel time or one that linearly increases with traffic. We assume that each driver wishes to minimise their own travel time and we assume that the drivers form a Nash equilibria. Can removing a route ever decrease ... |
I need $n$ cakes for a party. I go to the cake shop and there are $k$ different kinds of cake. For variety, I'd like to get at least one of each cake.
How many ways can I do this?
|
In least-squares approximations the normal equations act to project a vector existing in N-dimensional space onto a lower dimensional space, where our problem actually lies, thus providing the "best" solution we can hope for (the orthogonal projection of the N-vector onto our solution space). The "best" solution is th... |
Recently I discussed an experiment with a friend. Assume we start a random experiment. At first there is an array with size $100,000$, all set to $0$. We calculate at each round a random number modulo $2$ and select one random position in that array. If the number in the array is $1$, nothing is changed and otherwise t... |
I've just came back from my Mathematics of Packing and Shipping lecture, and I've run into a problem I've been trying to figure out.
Let's say I have a rectangle of length $l$ and width $w$.
Is there a simple equation that can be used to show me how many circles of radius $r$ can be packed into the rectangle, in the op... |
Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest?
For example, in linear algebra, a basis is sometimes defined as a set of n-dimensional vectors which:
Is linearly independent
Spans the vector space
Contains exactly n elements
However, it is then shown ... |
Given
A straight line of arbitrary length
The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude yields its own magnitude.
Is there a way to geometrically construct (using only a compass and straightedge) the a line wi... |
Say there are three jars, $j_1, j_2, j_3$ filled with different binary sequences of length two.
The distribution of the binary sequences in each of the jars is given by the $p_i^k(1-p_i)^{n-k}$, where
$p_i = \frac{i}{m + 1}$ where $m$ is the number of jars, $i$ is the jar index, $k $is number of 1$$'s and $n$ is the... |
I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form
$$y = a x^2 + b x + c$$
My teacher gave me the formula:
$$x = -\frac{b}{2a}$$
as the $x$ coordinate of the vertex.
I asked her why, and she told me not to tell her how to do her job.
My smart friend mumbled s... |
I read this question the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle.
Why doesn't the same thing happen for squares?
If we start with the area formula for squares, $l^2$, this differentiates to $2l$ which is sort o... |
The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc.
I know mathematically a tensor can be represented by a $3 \times 3$ matrix. But I can't grasp its geometrical picture — unlike scalar (a number) and vector (an arrow with direction and magnitude) ... |
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series
$$\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2}... |
Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
|
Is there an algorithm for working out the best way (i.e. fewest multiplications) of calculating $A^n$ in a structure where multiplication is associative?
For example, suppose $A$ is a square matrix. Matrix multiplication is associative, and I can compute $A^9$ with $4$ multiplications:
$$A^2 = A \cdot A$$
$$A^3 = A^2 \... |
I keep hearing questions about whether something is NP-complete, but they never really mention what it is. Why do people care so much about NP-complete problems?
|
In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the following class of morphisms:
$\Sigma=\left\lbrace f\in \hom_{\text{gr}R\text{-mod}}\left(A,B\right) \ | \ \ker\left(f\right)_... |
Can someone point me to a proof that the set of irrational numbers is uncountable? I know how to show that the set $\mathbb{Q}$ of rational numbers is countable, but how would you show that the irrationals are uncountable?
|
Are there any real online mathematics (applied math, statistics, ...) degree programs out there?
I'm full-time employed, thus not having the flexibility of attending an on campus program. I also already have a MSc in Computer Science. My motivation for a math degree is that I like learning and am interested in the subj... |
I was coming back from my Driver's Education class, and something mathsy really stuck out to me.
One of the essential properties of a car is its current speed. Or speed at a current time. For example, at a given point in time in my drive, I could be traveling 40 mph. But what does that mean?
From my basic algebra cl... |
What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples?
For instance, if $f'(a) = b$, then $f(a)$ exists, $f$ is continuous at $a$, $f$ is differentiable at $a$, but $f'$ need not be continuous at $a$. A function for whic... |
I asked this question on Stack Overflow but it was closed as "not programming related". So I think this is probably the best place for it...
I read over the wikipedia article, but it seems to be beyond my comprehension. It says it's for optimization, but how is it different than any other method for optimizing things?... |
You have sheets of $42$-cent stamps and
$29$-cent stamps, but you need at least
$\$3.20$ to mail a package. What is the
least amount you can make with the $42$-
and $29$-cent stamps that is sufficient
to mail the package?
A contest problem such as this is probably most easily solved by tabulating the possi... |
The absolute value of a $2 \times 2$ matrix determinant is the area of a corresponding parallelogram with the $2$ row vectors as sides.
The absolute value of a $3 \times 3$ matrix determinant is the volume of a corresponding parallelepiped with the $3$ row vectors as sides.
Can it be generalized to $n-D$? The absolute ... |
Here's one that popped into my mind when I was thinking about binary search.
I'm thinking of an integer between 1 and n. You have to guess my number. You win as soon as you guess the correct number. If your guess is not correct, I'll give you a hint by saying "too high" or "too low". What's your best strategy?
This is ... |
I have a $3$-D sphere of radius $R$, centered at the origin.
$(x_1,y_1,z_1)$ and
$(x_2,y_2,z_2)$ are two points on the sphere.
The Euclidean distance is easy to calculate, but what if I were to restrict myself to traveling on the surface of the sphere?
Two approaches come to mind: use arc-length in some way, or sim... |
Wind resistance -- upwards acceleration, typically varies either linearly or quadratically by the current velocity.
There is a constant downward acceleration due to gravity.
How can we model the velocity over time of a falling object, subject only to wind resistance and downwards gravity?
I don't have much experience w... |
From Rudin's Principles of mathematical analysis,
6.2 Definition
Let $\alpha$ be a monotonically increasing function on $[a,b]$. ... Corresponding to each partition $P$ of $[a,b]$, we write
$$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1}).$$
He then goes on to define the Riemann Stieltjes integral of $f$ with respec... |
RHS is a well known test for determining the congruency of triangles. It is easy enough to prove it works, simply use Pythagorus' theorem to reduce to SSS. I thought that it seems strange that this only works for an angle being 90 degrees - or does it? What if I tried changed the given angle to 89 degrees or 91 degrees... |
Pick's theorem says that given a square grid consisting of all points in the plane with integer coordinates, and a polygon without holes and non selt-intersecting whose vertices are grid points, its area is given by:
$$i + \frac{b}{2} - 1$$
where $i$ is the number of interior lattice points and $b$ is the number of po... |
I am trying to show the following:
\begin{equation*}
E[e^{-\gamma W}]=e^{-\gamma(E[W]-\frac{\gamma}{2}Var [W])}
\end{equation*}
but I really can't remember what I am supposed to do to get from the LHS to the RHS. I have tried using integration this way
\begin{equation*}
\int We^{-\gamma W}dW
\end{equation*}
and then u... |
There is a well known theorem often stated as the angle in a semi-circle being $90$ degrees. To be more accurate, any triangle with one of its sides being a diameter and all vertices on the circle has its angle opposite the diameter being $90$ degrees. The standard proof uses isosceles triangles and is worth having as ... |
Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural language utterances, the frequency of any word is roughly inversely proportional to its rank in the frequency table) are not th... |
In a category I have two objects $a$ and $b$ and a morphism $m$ from $a$ to $b$ and one $n$ from $b$ to $a$. Is this always an isomorphism? Why is it emphasized that this has to be true, too: $m \circ n = \mathrm{id}_b$ and $n \circ m = \mathrm{id}_a$?
I am looking for an example in which the id-part is not true and th... |
There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula:
$$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$
For cubic equations of the form $ax^3+bx^2+cx+d=0$, there is a set of three equations, one for each root.
Is there a general formula for solving eq... |
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