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I heard about Yakov Perelman and his books. I just finished reading his two volumes of Physics for Entertainment. What a delightful read! What a splendid author. This is the exact book I've been searching for. I can use it to develop interest for science (math & physics) in my students.
His math books:
Mathematics Can... |
I have a set and its cardinality. How can I calculate the cardinality of its complementary set?
EDIT: Gotta apologize. I was talking about well-defined sets with a fairly low positive integer as cardinality. Qwirk comes pretty close when talking about subsets and a given superset.
So see it this way. I have a certain s... |
If in a $\displaystyle\bigtriangleup$ ABC, $\displaystyle\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $, then $\displaystyle\bigtriangleup$ ABC is of which type ?
|
This is extracted from my module :
In a $\displaystyle\bigtriangleup ABC$,
$$\displaystyle \frac{\sin (A) + \sin(B)}{\sin(A) -\sin(B)} = \frac{k \cdot(\sin (A) + \sin(B))}{k \cdot(\sin(A) -\sin(B))} = \frac{a+b}{a-b}$$
where $A,B,C$ are the angles and $a,b,c$ are the opposite sides,I can't figure out how this step ho... |
I claim that it is commonly believed that Mathematical objects can be seen as genuinely static, with no "Platonic" time in which they do genuinely evolve.
Nevertheless time has its place in mathematics:
An endomorphism of a set (seen as a set of states of a system) into itself can be seen as evolution of the system in... |
(1). I was wondering about what are the relation and differences between formal and informal logic?
What topics does each of them have?
For example, topics such as Meaning and Definition, Syllogistic Logic, Inductive Logic, Probabilistic/Statistical Reasoning, Deductive Logic
Is symbolic logic same as formal logic?
(2)... |
According to the wikipedia article: http://en.wikipedia.org/wiki/Levenberg_Marquardt
--
$S(\boldsymbol\beta+\boldsymbol\delta) \approx \|\mathbf{y} - \mathbf{f}(\boldsymbol\beta) - \mathbf{J}\boldsymbol\delta\|^2$
Taking the derivative with respect to δ and setting the result to zero gives:
$(J^{T}J)\boldsymbol \delta... |
Let $f$ be a distribution on $\mathbf{R}^n$ (in the Schwartz sense) such that
$$\frac{\partial f}{\partial x_i} = 0 \text{ for $i = 1, \ldots, n$.}$$
Then how to prove that $f$ is a constant? I had this exercise in a class last year but I couldn't find how to do the induction step (for $n = 1$ it is clear of course).
|
In 1D projective geometry,
I want to compute the 2x2 Homography matrix $H$ (in homogeneous coordinates), given 3 pairs of corresponding points.
i.e. I want to find H such that:
$$\left(\begin{array}{cc}
h_{11} & h_{12}\\
h_{21} & h_{22}\end{array}\right)\left(\begin{array}{ccc}
0 & a & a+b\\
1 & 1 & 1
\end{array}... |
Are there necessary and sufficient conditions for the existence of a circuit, or a disjoint set of circuits, that passes through each vertex once in a directed graph?
|
I've come up with the following variation of a logarithm, intended to work on values that can be 0, or can grow exponentially from zero in either positive or negative direction.
$$myLog(x) =
\begin{cases}
\log(x+1), & \mbox{if }x \geq 0 \\
-\log(-x+1), & \mbox{if }x \lt 0
\end{cases}
$$
Is there a name for this formu... |
What is the difference between:
1) $\frac{\partial (\textbf{x}^{T}A)}{\partial \textbf{x}}$
and
2) $\frac{\partial (A\textbf{x})}{\partial \textbf{x}}$
where A is a nxn matrix and x is a n sized column vector.
Using the definition of a Jacobian on wikipedia (http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinan... |
I need to prove that
$$P(A \backslash B) = P(A) - P(A\cap B)$$
and
$$P(A\cup B) = P(A) + P(B) - P(A\cap B)$$
using the axioms of probability but can't see where to start. Note: this is homework, so a hint only would be nice :)
|
A manifold $M$ is simply connected if for every pair of 1-cubes $c_1,c_2: [0,1]\rightarrow M$ with
$c_1(0) = c_1(1) = c_2(0) = c_2(1) = t$
there is a 2-cube $b$ such that
1) $b(1,0) = c_1$ and $b(1,1) = c_2$
2) for all $p$ in $[0,1]$, $b(2,0)(p) = b(2,1)(p) = t$
My question is, if I'm given a manifold M,
for example ... |
Given a countable family of semi-norms $p_i$, we can define a metric
$d(f,g) = \sum \limits_{i=0}^{\infty} 2^{-i} \frac{ p_i(f-g) }{ 1 + p_i(f-g) }$
We have the locally convex topology induced by the semi-norms as above, as well as the topology induced by the metric.
How does the proof work to show their equality?
I a... |
Let $p:E\rightarrow B$ be a Serre fibration of path connected spaces with fiber $F$. Are the connecting homomorphisms $\partial:\pi_{n+1}(B)\rightarrow \pi_{n}(F)$ in the long exact sequence of $p$ induced by a continuous map $\Omega B\rightarrow F$?
|
What do the finite-dimensional continuous complex representations of the additive group $\mathbb{Q}$ with the usual topology look like? With the discrete topology? Which representations are indecomposable? Irreducible?
The only ones I can think of are of the form $t \mapsto e^{tA}$ for some $A \in \mathcal{M}_n(\mat... |
I'm familiar with the notation $\mathrm{ord}_a(x)$, when $x$ is an integer. However, I'm reading a book where this notation is used with $x$ rational. I'm not sure how to interpret this?
|
I figured that I can create a deterministic "random" numbers generator by utilizing a bit of "magic" that I picked up from some cryptography. However I seem to have missed a detail.
Basically the function $\displaystyle f(n) = p^n \mod q$ should for $\displaystyle n$ in the range $\displaystyle [1,q-1]$ produce all the... |
How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?
It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and... |
Given an open set $U \subset \mathbb R ^n $, there exists an exhaustion by compact sets, i.e. a sequence of compact sets $K_i$, s.t.
$\cup _{i=0}^{\infty} K_i = U$ and $\forall i \in \mathbb N : K_i \subset K_{i+1} ^{\circ}$
We can imagine that different exhaustions by compact sets 'propagate' at different speeds for ... |
I'm working on a program that displays a bunch of thumbnails on the bottom of the screen and I'm using the mouse's position to slide the images from side to side.
The center of the screen is 0 on a number line. Every frame, the mouse's X position is either added or subtracted from the x position of each thumbnail. (So... |
Functions from $\mathbb{C} \rightarrow \mathbb{C}$ are hard to visualize because of their 4-dimensional nature. One nice way of looking at them is by what's called domain coloring. An example from the wiki article is shown below.
When we look at the graph of a real function ($\mathbb{R} \rightarrow \mathbb{R}$), we can... |
If we have a module $M$ over a ring $Z$, and we consider the the canonical module homomorphism $M→Q⊗M$ over $Z$, is it true that the kernel of this map is the torsion submodule of $M$? Why?
|
In the proof of Stone-Weierstrass theorem provided in Rudin's Principles of Mathematical Analysis, why do we only need to show that there exists a sequence of polynomials $P_n$ that converges uniformly to the continuous complex function on $[0,1]$?
|
So in algebraic geometry, the prime spectrum of the ring of polynomials over a field $k$ is isomorphic to the set of $k$-homomorphisms from the ring of polynomials to $k$, and we do this by using the fact that the residue field (localise w.r.t. some prime ideal, then mod out the ideal generated by this in the localisat... |
I am looking for the proof of the following theorem:
Let $(a_n)$ be a sequence of real
numbers. Then there exists a function
$f$ which is infinitely differentiable
at 0, and $$ \frac{d^nf}{dx^n}(0) =
a_n, \ \ \text{for all } n.$$
I would appreciate either a sketch of the proof or an online reference to it.... |
I've been trying to prove the following statement, but the converse has been giving me trouble.
A topological space $X$ is regular if and only if every closed subset $Z\subseteq X$ is the intersection of all open sets $U\subseteq X$ which contain it.
Here, the defintion of regular is that for any point $p\in X$ and ... |
I was reading Rudin's Principles of Mathematical Analysis, and I came across the definition 7.19, where it says that a sequence of functions $f_n(x)$ is pointwise bounded on E if there exists a finite-valued function $\phi$ defined on E such that
$$ |f_n(x)| < \phi(x) $$
for x element of E, n = 1, 2 ,3 ...
While $f_... |
When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers?
I'm looking for an answer deeper than "there are $6\times60^\circ=360^\circ$ in a circle, so you can picture it".
|
I am given two random variables $Y_1$ and $Y_2$ and:
$E(Y_1) = 4$
$E(Y_2) = -1$
$V(Y_1) = 2$
$V(Y_2) = 8$
I am asked to find $Cov(Y_1,Y_1)$
I know $Cov(Y_1,Y_2) = E(Y_1Y_2)-E(Y_1)E(Y_2)$
I'm not sure if it is a typo, but it says $Cov(Y_1,Y_1)$ NOT $Cov(Y_1,Y_2)$
The answer is $2$, but i'm not sure how to get $E(Y_1Y_... |
Suppose that there is a certain collected works of plays that is N symbols long in the following sense: a "symbol" is one of the 26 letters of the alphabet, a line break, period, space, or a colon; in other words there are 30 possible symbols.
If "a monkey" randomly "types" 1 of these 30 symbols at a rate of one per se... |
How to find the value of
$$\displaystyle\sqrt{3} \cdot \cot (20^{\circ}) - 4 \cdot \cos (20^{\circ})$$
manually ?
|
In a $\displaystyle\bigtriangleup$ ABC,R is circumradius and $\displaystyle 8R^2 = a^2 + b^2 + c^2 $ , then $\displaystyle\bigtriangleup$ ABC is of which type ?
|
I tried solving this, and I am pretty sure I am integrating this correctly, however, my solution manual shows -1 in the equation when doing this and I do not know why. The answer in the solution manual is correct.
Problem: Find the corresponding distribution function and use it to determine the probability that a rand... |
This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider
$$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$
The question asks:
(1) For what values of x does the series converge absolutely. We got that the series converges when x $\not=$ 0 & x $\not= -1/k^2 $ ... |
I have a question about a transformation of a matrix. Lets say we have the following matrix
$
M =
\left[ {\begin{array}{cc}
4 & 3 \\
4 & 3 \\
\end{array} } \right]
$
Then I want to transform all numbers in the matrix with the following function:
$f(x) = (x - 4)^2 + 2x -4$
What would be the correct notation... |
At Wikipedia I found that "according to a strengthened version of Frucht's theorem, there are infinitely many asymmetric cubic graphs".
Are there infinitely many asymmetric
planar cubic graphs, too?
If so, does it follow that there is an infinite asymmetric planar cubic graph?
If so, how could this graph be characteri... |
Reposting from stackoverflow :) - Told to go here :)
So I've got a issue with a math assignment, and I were hoping someone could at least point me to what I should be doing, because currently I'm just wasting my time, thats for sure. - Please don't solve it all, thats up for me to do so :)
The assignment sounds like th... |
$G$ is a finite abelian group. $A \times B$ can be embedded in $G$.
Does this mean there exist $C$, $D$ such that $G=C \times D$ and $A$ can be embedded in $C$ and $B$ can be embedded in $D$?
|
Define a subprobability measure to be a measure on Borel sigma algebra of the real line $\mathbb{R}$ with the measure for the whole real line less or equal than 1.
I was wondering about the definition of vague convergence of a sequence of subprobability measures $\{ \mu_n, n\geq 1 \}$ to another subprobability measure... |
Is it possible to define distribution function for a measure on a general measurable space, which generalizes the definition of distribution function for a probability measure on the extended real line?
If yes, how?
If no, how general/special can the measure and the measurable space be in order to define its distribu... |
YARFMO (Yet another reposting from Mathoverflow) ;-)
The more you know about math the more you find conceptions previously thought correct to be false:
1.) math is not as exact as many believe - in many cases, as Gowers points out in his amazing book "Mathematics. A very short intro", it is mainly about approximations,... |
I was wondering what is wrong in the following proof:
Proposition. Let $\{f_{n}\}$ be a sequence of integrable functions such that $f_{n}$ converges pointwise to a function $f$. Show that if:
$\lim \int |f_{n} - f| d\mu =0$ then $\int |f| d\mu = \lim \int |f_{n}| d\mu$.
Well I used the fact that $||f_{n}| - |f| | \leq ... |
We are suppose to compute the quaternion which performs 1/5 of the rotation of this quaternion:
[ 0.965 (0.149 -0.149 0.149)]
The answer provided is shown as below:
Picture of the answer
I'm stuck at the second last row of the ans. How do you change from the '0.998 q0 + 0.202 q' to [0.998 0.03 -0.03 0.03]
|
Let $f: A \rightarrow B$ be a ring homomorphism between commutative rings with identity. Then there exists an induced map $f' : Spec(B) \rightarrow Spec(A)$. If $f'$ is surjective, then clearly every prime ideal of $A$ is a contracted ideal. Now my question is, is the converse true?
|
If $\sin(A+B) + \sin(B+C) + \cos(C-A) = \frac{3}{2}$ show that, $$1.\sin A + \cos B + \sin C = 0$$ $$2. \cos A + \sin B +\cos C = 0$$
|
Given $1<q\leq2$ and $0\leq p\leq1$, let us consider the following function:
$$\phi\left(\alpha\right)=p\times\left|1-\alpha\right|^{q}+\left(1-p\right)\times\left|1+\alpha\right|^{q}$$
The minimum over $\mathbb{R}$ I have found is:
$$\alpha\left(p\right)=\frac{p^{\frac{1}{q-1}}-\left(1-p\right)^{\frac{1}{q-1}}}{p^{\f... |
Prove that X is a Random Variable IFF sigma field generated by X is countably generated.
|
Find modules $M,N$ and a ring $A$ s.t. $M\otimes_A N\not \cong M\otimes_{\mathbb{Z}}N$.
|
This is a problem in baby Rudin. Chapter 4, problem 15. I'm just looking at examples at the moment. It says that any continuous open map is monotone. Recall, an open map $f: X \to Y$ is such that any open set $U$ in $X$ will haves its image, $f(U)$ open in $Y$.
So,
Let $$ f: \mathbb{R} \to [-1, 1] $$ $$f: x \mapsto \si... |
Consider the following arithmetic function, $f(a,b) = (a+1, \lceil a- \frac{a}{b} \rceil , \lceil a- \frac{2a}{b} \rceil, \dots, \lceil \frac{a}{b} \rceil, 1)$, where all terms except the first and last are computed by the application of a ceiling function.
For any pair of positive integers $(a,b)$, the vector $f(a,b)... |
This question is based on the diagram
taken from this link. I don't understand why the areas of A6 and A5 adds up to $2bc\cos(A)$.
|
Let $K \subset R^n$ be compact, and let $C_0^{\infty}(K)$ be the space of smooth functions on with support in $K$.
For $p \in [1,\infty), \alpha$ multiindex, let $|f|_{\alpha,p} = (\int_K |D^{\alpha}f|^p)^{\frac{1}{p}}$, and $|f|_{\alpha,\infty} = sup_K |D^{\alpha}f|$ - as usual.
For $p \in [1,\infty]$, let $\tau_p$ be... |
I have to do this for an upcoming exam, but cannot find anywhere (in the textbook or online) how to do this.
I only really need to know a couple points to plot it... when x = 0, and then the earliest crossing of the x-axis.
|
In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It might be out of date in 2010. This raises its own question:
are there other real-life examples of geometries formed by ... |
After asking a few questions earlier, I think I've been able to describe my main issue. Could someone show me why this identity is true:
$$\frac{\partial \textbf{x}^{T} B \textbf{x}}{\partial \textbf{x}} = (B + B^{T})\textbf{x}$$
I've gone through things like Matrixcookbook which seem to just show this identity witho... |
Problem: Calculate limit of $\frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \frac{1}{\sqrt{n+3}} + \cdots + \frac{1}{\sqrt{n+n^2}}$ as $n$ approaches infinity.
Solution: Denote the above some as $X$, then we can bound it:
$$ \infty\longleftarrow\frac{1}{\sqrt{n+n^2}} \lt X \lt \frac{n^2}{\sqrt{n+1}} \lt \frac{n^2}{\sqrt... |
I was looking for alternate proofs of the theorem that "a vector space $V$ of dimension greater than $1$ over an infinite field $\mathbf{F}$ is not a union of fewer than $|\mathbf{F}|$ proper subspaces" and possible generalizations.
A simple measure-theoretic proof over $\mathbb{R}$ is as follows: By countable additivi... |
Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$:
$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$
Is this set a fractal, and If so, then what is it's dimension?
|
A commutative ring $R$ with unity is Hermite if for all $x,y\in R$ there exists $t,u,v\in R$ such that $x=tu$, $y=tv$ and $(u,v)=(1)$. Is there a finite commutative ring with unity that is not Hermite?
This characterisation is taken from theorem 3 of:
Some Remarks About Elementary Divisor Rings, Leonard Gillman and Mel... |
Let $f(x)$ be a Lipschitz real-valued function defined on a closed interval $I$. The derivative $f '(x)$ exists a.e. since $f$ is absolutely continuous.
My question is: Is $f '(x)$ necessarily square integrable, i.e. in $L^2(I)$ ?
|
How does one show that any open interval in n-dimensional Euclidean space is connected?
This is homework, and I have been stuck on it for a few hours now. Seems like it should be easy, but I can't get a proof that I am comfortable with.
|
Suppose $f$ is a real-valued function and $f(x) \geq 1$ for every real $x \in [0,1]$.
Why we can always find a unique positive integer $n$ such that $2^{n} \leq f(x) < 2^{n+1}$ ?
|
I'm reading Number Theory I, by Kato, et al. The authors claim that for $x$, $y\in \mathbb{Q}$ and $m$ a square free positive integer, if $2x$, $x^2-my^2 \in \mathbb{Z}$, then for any odd prime $p$ $\mathrm{ord}_p(m)+2\mathrm{ord}_p(y) \geq 0$.
I get that $\mathrm{ord}_p(my^2)=\mathrm{ord}_p(m)+2\mathrm{ord}_p(y)$ and... |
In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections of any function onto the corresponding four eigenspaces may be found through some simple linear algebra.
I would like to ... |
Now, suppose the transformation(in 2d) I am working with has two
separate functions
for $x$ and $y$.
That is, the transformation for $x$ is of the form
$$
x'=\frac{x}{x+y}
$$
and the transformation of $y$ is
$$
y'=\frac{y}{x+y}
$$
Each is an LFT (The schwarzian derivatives are $0$) but is
the transform as a whole still... |
I want to prove that for non-negative random variables with distribution F:
$$E(X^{n}) = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx$$
Is the following proof correct?
$$R.H.S = \int_0^\infty n x^{n-1} P(\{X≥x\}) dx = \int_0^\infty n x^{n-1} (1-F(x)) dx$$
using integration by parts:
$$R.H.S = [x^{n}(1-F(x))]_0^\infty + \int... |
Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$.
I am looking for a general theorem that implies that the Fourier series of $f$ converges to $f$, pointwise, uniformly or absolutely.
|
Here is how I have worked it out so far:
$\log _2(x-4)+\log(x+2)=4$
$\log _2((x-4)(x+2)) = 4$
$(x-4)(x+2)=2^4$
$(x-4)(x+2)=16$
How do I proceed from here?
$x^2+2x-8 = 16$
$x^2+2x = 24$
$x(x+2) = 24$ Which I know is not the right answer
$x^2+2x-24 = 0$ Can't factor this
|
The number of dropped connections per call follows a Poisson distribution. From four calls, the number of dropped connections is $2,\,0,\,3,\,1$.
Obtain the maximum likelihood estimate that the next two calls will be completed without any accidental drops.
I know the maximum likelihood estimate of $\lambda$ is $1.5$.... |
You are dealt five cards from 52 card deck that is thoroughly shuffled. What is the probability that two out of five cards are seven?
Is it:
$${4C2 . 12C3 . (4C1)^3} / 52C5$$
|
I'm out of practice with algebra, and I'm having a mind blank on how to solve for $t$ in the following equation. It's for some collision detection if you're wondering.
$$\Bigl( \bigl(a-y-(qt)\bigr) \bigl(b-x-(rt)\bigr) \Bigr) - \Bigl( \bigl(c-x-(rt)\bigr) \bigl(d-y-(qt)\bigr) \Bigr) = z$$
I'm really not seeing it at t... |
I'm trying to derive the integral form of the Bessel function by finding
the $k$th coefficient of the Laurent series expansion of the function
$f(z) =\exp [\lambda(z-\frac{1}{z})]$.
I managed to get it down to the form
$
J_k(\lambda) =\frac{1}{2\pi}\int_{0}^{2 \pi} e^{i(\lambda \sin\theta - k \theta)} d\theta
=\frac{... |
On p.28 (here) Reid proves that the curve $y^2=x(x-1)(x-\lambda)$ for $\lambda\neq 0,1$ has no rational paramaterisation. At one point (on p.29), he has the equation $$r^2=ap(p-q)(p-\lambda q)$$ where $p$ and $q$ are polynomials over a field $k$, and $a$, $\lambda$ are constants in the field. He then says that "by cons... |
If $A$ is a positive definite matrix, then the solution to the minimization problem $(1/2)x^TAx - b^Tx$ is given by $A^{-1}b$. I'm interested in the generalization of this to a Hilbert space. What conditions on $A$ should be required for this to be true in a Hilbert space? Also is there a reference that includes thi... |
A rectangle can be cut into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different side lengths than the original one. Show that this can be done in countable many ways. Can this be done in uncountable many ways?
|
I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle $\theta$.
My question is: given a pair of spherical caps, how can I determine whether they intersect?
|
Today, I read these two new Logarithmic identity $\displaystyle$ $$a^{\log_a m} = m $$ $$\log_{a^q}{m^p} = \frac{p}{q} \log_a m$$ Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?
|
The rule: "Characteristic of a number($\lt$ 1) with n zeros after decimal and before a significant digit = $\overline{( n + 1 )}$" This rule is mentioned in my module,Goggle lead me to this link.
When I am trying to verify with some random numbers like Log[10, 0.0004356] its giving me $-3.36091$ and Log[10, 0.725] giv... |
Suppose that K is a finite extension over $\mathbb{Q}$. Let $p$ be a prime in $\mathbb{Q}$, let $p\mathcal{O}_K=\mathfrak{P}_1\ldots\mathfrak{P}_n$ be its prime decomposition in $\mathcal{O}_K$, then is it true that any prime ideal in $\mathcal{O}_K$ above $p$ is a factor in the decomposition of $p$?
|
I have $n$ random elements from 0 to $k$. What is the probability that there is some element which appears more than $n/2$ times (or $n/2$ times).
|
Suppose that $q$ is a rational prime, and $K$ is a $p$-extension over $\Bbb{Q}$ where $p$ is prime, then $q$ ramifies in $K$ if and only if $q$ is $p$ or $q$ is congruent to $1 \mod p$. I don't know why this is true, can someone convince me of this?
|
I was given a simple function, very simple. f(x) = bla bla. With a constant and an X. The point was to find for which values of a(the constant) the function is not one-to-one. The domain of the function is limited from zero and upwards. Here is what I put down on the paper:
As long as a is negative, it will not be one... |
Two numbers are equivalent if they have the same digits, for example $2271$ and $1227$ are equivalent. How many equivalent classes are there between $1000$ & $9999$?
I'm not sure I fully understand how this works, is the answer to this question $\dfrac{9\times 10\times 10\times 10}{4!}$? First calculating the total nu... |
This question is about Project Euler 113:
Working from left-to-right if no digit
is exceeded by the digit to its left
it is called an increasing number; for
example, 134468.
Similarly if no digit is exceeded by
the digit to its right it is called a
decreasing number; for example, 66420.
We shall call a posi... |
Original equation:
$g(x) = \frac{(x+1)(x^2+2)(x^3+3)}{\sqrt{x^4+3}}$
If I take ln on both sides, and than differentiate I get this:
$\frac{1}{g(x)} = \frac{1}{x+1}+\frac{1}{x^2+2}+\frac{1}{x^3+3}-\frac{1}{2}* \frac{1}{x^4+3} $
Is this good so far? I don't know what to do about the left side?
|
What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder class, or absolutely continuous.)
|
Under the isomorphism of Hilbert spaces $L^2(S^1)\to\ell^2(\mathbb Z),\quad e^{2\pi i n t}\mapsto e_n$, smooth functions on the circle are mapped to rapidly decaying sequences (see wikipedia).
Is the converse also true? That is, does every rapidly decaying sequence in $\ell^2(\mathbb Z)$ represent a smooth function?
Mo... |
I am facing some trouble in solving this equation:
$$
3\cdot x^{\log_5 2} + 2^{\log_5 x} = 64
$$
Give me some hints to proceed on this.
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I am given $n$ points in 2D.(Each of say approximately equal weight). I want to partition it into $m$ clusters ($m$ can be anything and it is input by the user) in such a way that the center of mass of each cluster is "far" from center of mass of all other clusters. What is a good heuristic approach (it should also be ... |
In "A Short Course on Spectral Theory", page 10, William Arveson asserts that the "$ax+b$ group", ie. the group generated by all dilations and translations of the real line, is isomorpic to the group of all (real) $2\times 2$ matrices of the form
$$ \begin{bmatrix} a & b\\ 0 & \frac{1}{a}\end{bmatrix}, \,\,\,\,\,\, a>... |
If set is closed under some operation,
what does that tell us the about structure
of the groupoid?
It's a question from one of the previous math exams which was shown to me by one of my friends.
Unfortunately, I can't find a good answer to it. As far as I know, the answer is nothing, because of the definition of ... |
Find a general expression for the Laurent Series expansion of
$f(z) = (z-\alpha)^{-n}$
when $|z|>|\alpha|$.
I've tried expanding the function using the binomial theorem, and I've tried
integrating with the general formula for Laurent Series coefficients
using Cauchy's Integral formula, but I can't get the solution,
w... |
This actually triggered me in my mind from here. After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive to see how this holds ?
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I am quite a baffled now, I am not getting by how it can be written that :
\begin{align*}
&\log_{10} \tan 40^\circ \cdot \log_{10} \tan 41^\circ \cdot \log_{10} \tan 42^\circ \cdot\log_{10} \tan 43^\circ \cdots \log_{10} \tan 50^\circ \\
&= \log_{10} \tan 40^\circ + \log_{10} \tan 41^\circ + \log_{10} \tan 42^\circ + ... |
In algebra, we learn that if a function $ f(x) $ has a one-to-one mapping, then we can find the inverse function $ f^{-1}(x) $. The method that I have seen taught is the "horizontal line test": if any horizontal line touches the graph of the function more than once, then it must not be one-to-one. This method is not ex... |
In MATLAB, when you plot something, there's a tool available which is called "curve fitting". And if you have a set of data points and a linear correlation, this tool will easily come up with an equation on the form y = ax + c.
I want to do exactly that, but I need to do it programmatically. I know that I can use poly... |
do you have an example of a non separable locally compact connected metric space?
Thank you
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All, I am having some trouble in checking whether a sequence of functions converges pointwise or uniformly. The problem is, sometimes, my intuition is right and sometimes its wrong. Finding the limit function is the most difficult task I would say.
I am able to understand some simple examples, such as a pointwise conve... |
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