instruction stringlengths 22 27.2k |
|---|
I am trying to upper bound $\sum_{l=1}^\infty (1 - n! 2^{-ln} {2^l \choose n})$.
I tried to use the simple ${2^l \choose n} \ge (\frac{2^l}{n})^n$, but then $l$ disappears and it diverges. Any ideas?
|
Given $c \in \mathbb{R}$ and $m,n \in \mathbb{N}$, is it possible to combine the inequations
$n < m$
$\frac{8}{c} < m$
into one inequation?
I argued that due $n < m$ and $\frac{8}{c} < m$, there has to be one $n$ between $m$ and $\frac{8}{c}$, being $n = \left\lceil \frac{8}{c} \right\rceil$, so that $m > n > \frac... |
Let $a,b,c,x,y,z\in\mathbb{R}$. Prove that $$ \left(\frac{ax+by+cz}{x-y}\right)^2+\left(\frac{ay+bz+cx}{y-z}\right)^2+\left(\frac{az+bx+cy}{z-x}\right)^2\geq(c-a)^2+(c-b)^2$$
|
I've been playing around with this equation:
$\displaystyle\int_{-\pi}^\pi{\displaystyle\frac{1-e^{3it}}{1-e^{it}}dt}$
Now it seems to me that we can (possibly) split the integral into four seperate integrals; one for each quadrant of the complex plane. Doing this, we can determine the complex part if we know the rea... |
From its definition a combination $\binom{n}{k}$, is the number of distinct subsets of size $k$ from a set of $n$ elements.
This is clearly an integer, however I was curious as to why the expression
$$\frac{n!}{k!(n-k)!}$$ always evaluates to an integer.
So far I figured:
$n!$, is clearly divisible by $k!$, and $(n-k)!... |
This is a follow up of Upper bound binomial sum
I was working on the problem in the above thread and noticed an interesting thing. I wanted to try and improve the bound Derek had (which was a quadratic in $n$).
If we reformulate the problem as Derek has (since for this we need $2^k \geq n$, so we forget the original pr... |
I am currently a mathematics graduate student at Western Kentucky University in Bowling Green, KY. I am looking for some kind of summer opportunity to participate in during summer 2011.
Does anyone have any suggestions of good opportunities or a good list of opportunities?
I would really appreciate it!
I am more intere... |
Question: Let $v = (1,1,0,1) \in \mathbb{R}^{4}$ Let $$V:= \{f: \mathbb{R}^{4} \rightarrow \mathbb{R}^{2} \mbox{ linear } | f(v) = 0\}.$$
(a) Show that $V$ forms a vectorspace.
(b) Determine dim$(V)$. (Tip: Describe the elements in $V$ by extending $v$ to a basis of $\mathbb{R}^{4}$).
My attempt: for (a): ok i know th... |
The length of any curve that goes from A to B (other than the line segment AB) is greater than the length of that segment. This statement may be a theorem? Or, necessarily, is an axiom?
It seems to me that any attempt to build a theorem eventually turning into begging.
Thank you very much!
Paulo Argolo
|
I was trying to wrap my brain around this but could not think of anything. I am very poor at maths but trying to learn of it.
Edit:
Exact question is
If $x^2+x-6$ is a composite function $f(g(x))$; then figure out $f(x)$ and $g(x)$ ?
To me information looks incomplete to figure out such details.
|
Prove the following: $\displaystyle a^{n}-b^{n} = (a-b) \sum\limits_{k=0}^{n-1} a^{k}b^{n-1-k}$.
So one could use induction on $n$? Could one also use trichotomy or some type of combinatorial argument?
|
This is a follow up to a question I asked earlier. I had previously asked that if a space was regular if and only if any closed set $Z$ was the intersection of all open sets containing $Z$, but the backwards implication turned out to be untrue. I'm now curious to see if this new bi-implication is fully true.
My defini... |
I'm trying to classify (at least as fully as possible) the values of $t$ in $(0,\pi/2)$ for which the following equation has a solution for some natural number, $n$.
$$\frac1{n}\sin(nt)=\frac1{n+2}\sin((n+2)t)$$
Does anyone know of any results that will help me?
|
Does anyone have any suggestions for abstract algebra books particularly suited to self-study?
Here is some background and motivation, if it's helpful.
I'm currently a junior in high school, but I have some familiarity with groups, rings, polynomials and fields, as I went through Fraleigh's book. I liked its writing ... |
Assume $f$ has a finite derivative and $|f'(x)| \leq y < 1$ for all $x \in (a,b)$
$f$ is continuous and $a \leq f(x) \leq b$ for all $x \in [a,b]$. Prove $f$ has a unique fixed point in $[a,b]$.
So far I have for every c in (a,b) |f'(c)| ≤ y
=> lim x->c |f(x) - f(c)|/|x-c| ≤ y
=> lim x->c |f(x) - f(c)| ≤ y lim x->c |... |
Is there a fundamental relationship between circuits and cycles in an arbitrary graph? Are they mutually independent objects?
Thanks
|
As is well known, if $x,y$ are natural numbers then their GCD $(x,y)$ has a representation $(x,y) = ax + by$ where $a,b$ are natural. Now let's prove that if $(x,y)=1$ then $(xy,x+y) = 1$. Starting from $ax+by=1$, $$1 = (ax+by)^2 = a^2x^2 + b^2y^2 + 2abxy = (a^2x+b^2y)(x+y) - (a-b)^2xy.$$ The reverse implication is tri... |
I was wondering if we have a finite group $G$ with subgroups $H$ and $K$. With $X:=G/H$.
Is there a simple way of finding $X^K$ and when will $X^K=\emptyset$?
Also does this have any baring on when $G/H≅ G/K$?
(This is the first question I've ever asked here so apologies for errors etc.)
|
Quote from Don Zagier (Mathematicians: An Outer View of the Inner World):
" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, imagine you have a series of numbers such that if you add 1 to any number you will get the product of its left and right neigh... |
What would be an example of a matrix that would not have a $LDL^T$ decomposition? This is a trivial case but I was thinking of a zero matrix which would result in L being an identity matrix and D would be a zero matrix. When you compute $LDL^T$ it still gives you the original zero matrix so I don't believe this is corr... |
Let $f(x,y)$, with $0\le x,y\le 1$, be integrable in $y$ for each $x$. Further suppose $\partial f(x,y)/\partial x$ is a bounded function of $(x,y)$. Show that $\partial f(x,y)/\partial x$ is a measurable function of $y$ for each $x$ and
$$\frac{d}{dx}\int_0^1{f(x,y)\,dy} = \int_0^1{\frac{\partial}{\partial x}f(x,y)\,d... |
For
$$f(x,y) = \begin{cases}
\frac{xy}{x^2+y^2} &\text{if }(x,y)\neq (0,0);\\
0 &\text{if }(x,y)=(0,0).\end{cases}$$
I'm trying to prove that $\frac{\partial f}{\partial x}$, the partial derivative with respect to $x$, exists.
Taking the partial derivative freehand I get
$$\frac{y(x^2+y^2)-xy(2x)}{(x^2+y^2)^2}=
\... |
How many elements are in the set:
$\{(x,y,z) \mid x+y+z = k \text{ and } x,y,z \in \{0,\dotsc,n\}\}$
I can see that when $k \le n$, it's $\binom{k+2}{2}$.
And, by symmetry, the value for $k$ is the same as that for $3n-k$.
So, I've figured out the two pyramids, and I'm just missing the octahedron in the middle.
|
How can I prove that $a^x$ < $a^{x + \Delta x}$, where $\Delta x > 0$, $a$ is a constant, and $a > 1$ (in my case, $a=2$)?
I don't want to use a graph, of course.
|
Let $x$ be a positive rational number of the form $\displaystyle x = \sum\limits_{k=1}^{n} \frac{a_k}{k!}$ where each $a_k$ is a nonnegative integer with $a_k \leq k-1$ for $k \geq 2$ and $a_n >0$. Prove that $a_1 = [x]$, $a_k = [k!x]-k[(k-1)!x]$ for $k = 2, \dots, n$, and that $n$ is the smallest integer such that $n... |
As far as I am concerned, probability distribution function is for discrete random variables while probability density function is for continuous random variables. To find the probability value of continuous random variable, we have to take the total area under the function which differ from discrete random variable, w... |
The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me.
Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the distinction, then? Why not just have scales with $12$ notes? (apparen... |
The following result must be well known, but it is instructive to give it here in view of the recent posts $Y= X+N$ what is ${\rm E}(X|Y=y)$ and https://mathoverflow.net/questions/47168/ex-1-x-1-x-2-where-x-i-are-integrable-independent-infinitely-divisib/47204#47204. This is also an interesting exercise in its own righ... |
For Riemann zeta function and $L$-functions of number fields, the width of critical strip is $1$. For $L$-functions of modular forms of weight $k$, the width of the critical strip is $k$.
Why is there a variation in the width of the critical strip for various $L$-functions? Is there a conceptual explanation or an under... |
For proving the quadratic reciprocity, Gauss sums are very useful. However this seems an ad-hoc construction. Is this useful in a wider context? What are some other uses for Gauss sums?
|
The context is the definition of Hecke Größencharakter:
http://en.wikipedia.org/wiki/Hecke_character
This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters are characters of the multiplicative groups of $\mathbb Z/p\mathbb Z$. An appropriate generalization would be instead to co... |
Let X be a topological space and A be an abelian group. Give A the discrete topology. For any open set U of X, Let $\cal A(U)$ be the group of all continuous aps of U into A. Thus with the usual restriction maps we obtain a sheaf $\cal A$.
So, why for every connected open set U, $\cal A(U)\cong A$ for all U?
And, what... |
I need to solve this equation. Please help me!
$$\dfrac{-\mathrm d^2u}{\mathrm dt^2}+\dfrac{u}{LC}=\dfrac{V}{LC},\qquad L,C,V,I=\text{constant}$$
and $u(0)=0$, $\left. \dfrac{\mathrm du}{\mathrm dt}\right\vert_{t=0}=\dfrac{I}{C}$
|
Is there a rational number (with denominator not greater than 200) between 15/106 and 16/113?
|
The Manin-Drinfeld theorem asserts that for a modular curve $X_0(N)$ and Jacobian $J_0(N)$ with the former being embdedded in the latter under the map that takes $i\infty$ to $0$, the cusps are torsion.
The proof Manin-Drinfeld theorem seems to use Hecke operators and thus seems to be valid only for congruence subgroup... |
The Coordinates of two points are A(-2,6) and B(9,3). Find the coordinates of the point C on the x-axis such that AC = BC
|
If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
|
Possible Duplicate:
Value of $\sum x^n$
Proof to the formula
$$1+x+x^2+x^3+\cdots+x^n = \frac{x^{n+1}-1}{x-1}.$$
|
In English (I think this is universal anyway) we use the 1s, 10s, and 100s in a cycle. One, one thousand, one million; twenty two, twenty two thousand, twenty two million; one hundred and forty six, one hundred and forty six thousand, one hundred and forty six million, etc.
Who decided to cycle every (3x+1)th digit?
|
Factoring a rational function as:
$$\frac{(x-a)(x-b)\cdots(x-z)}{(x-A)(x-B)\cdots(x-Z)}$$
Clearly displays the x-intercepts (roots) and vertical asymptotes (poles) of a rational function.
Similarly, dividing the denominator D into the numerator N to get N = DQ+R and writing N/D as Q + R/D clearly displays the end-beha... |
For my understanding of modular forms, I have "properly" read the last chapter of Serre's book on arithmetic. I have not "properly" read the setup of Hecke operators for congruence subgroups.
I was intending to read the paper of Atkin and Lehner for filling this lacuna, as well as for understanding the proof of Atkin-L... |
What is a good reference for the story of congruences such as
$$\displaystyle \tau(n) \equiv \sigma(11)(n) \mod\ 691$$
with a conceptual explanation with connections to étale cohomology, etc?
|
What is the compelling need for introducing a theory of $p$-adic integration?
Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a Lebesgue measure on $p$-adic spaces, and just integrate real or complex valued functions on $p$-adic spaces, or is somethin... |
This is a follow up of Limit of $S(n) = \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$
More details can be found in the above thread.
Let $S(n) = \displaystyle \sum_{k=1}^{\infty} \left(1 - \prod_{j=1}^{n-1}\left(1-\frac{j}{2^k}\right)\right)$
Mike has proved that $S(n)$ in fact div... |
An annuity immediate has quarterly payments of $1000$ for $25$ years at a rate of $6 \%$ converted quarterly. Find its present value.
The effective interest rate $i$ is $0.06/4$. So let $v = \frac{1}{1+i}$. Then wouldn't the present value be $1000 \left(\frac{1-v^{25}}{i} \right)$?
|
I want to find the volume of a torus with a given thickness and a given radius.
Let r be the radius of a circle with its midpoint at $M(0|b)$ ($b \geq r$). Now I want to rotate this circle about the x-axis, that is to say about a circular path which has the length $2 \pi \cdot |b|$. So I thought I'd simply integrate:
$... |
If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ?
My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series (of coefficient) which is nothing but $\frac{2}{3}\ln 2$.But this approach took some time for me (I can't solved it durin... |
How can I prove that dividing an irrational number by an irrational number (besides itself) can result in an integer?
|
The series for $\log (1+x)$ is convergent(by estimating radius of convergence) only with $|x| <1$. Then how is it still true for $x = 1$?
How is it still apparently true when $x$ is a roots of unity other than $-1$? Such as in the answers to this question: Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k... |
From the wikipedia page:
http://en.wikipedia.org/wiki/Motive_(algebraic_geometry)
it appears that the category of Mixed motives $MM(k)$ over a field $k$ is still conjectural; but there is a good derived category $DMM(k)$ already constructed for this.
What is a good reference for this construction, and why this derived... |
It is a bit easier to understand the homology $H_1(X, \mathbb Z)$ for various compact surfaces in analogy with handles and so on. There seems to be a nice intuitive picture with handles, holes, etc to think of the first homology group, and similar heuristics for higher homology groups.
But almost all axiomatic treatmen... |
Sheaf cohomology is a well-studied topic with a lot of references available. For example Hartshorne's book. But for a certain paper I am reading now, I have to understand sheaf homology.
Could someone tell me some references? This is just sheaf homology in the topological sense; not in algebraic geometry.
|
Consider the power series ring in noncommutative associative formal variables $X, Y$ over the field $\mathbb C$. Let $P$ be a power series. Define the exponential $e^P$ and logarithm $\log (1+P)$ using the usual series expansions. The exponential maps from a neighborhood of $0$ to a neighborhood of $1$ and the logarith... |
I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain.
For convolution on Lebesgue-integrable real-valued functions defined on a group equipped with measure, does each of the following properties holds? If not, what extra requir... |
Let $M$ be a smooth connected manifold. Is it always possible to find a connected dense open subset $U$ of $M$ which is diffeomorphic to an open subset of R$^n$?
If we don't require $U$ to be connected, the answer is yes: it is enough to construct a countable collection of disjoint open "affines" whose union is dense,... |
I'm trying to solve a differential equation problem that has to do with barometric pressure at a certain altitude.
The rate at which barometric pressure decreases with altitude is proportional to the pressure at that altitude. If the altitude in feet, then the constant of proportionality is 3.7 x 105. The barometric pr... |
I have a question about the set of functions from a set to another set. I am wondering about the degenerate cases. Suppose $X^Y$ denotes the set of functions from a set $Y$ to a set $X$, why is $|Y^{\emptyset}|=1$ but $|\emptyset^Y|=0$ where $Y\neq \emptyset$?
|
I hope that someone's up for the challenge; I'm attempting to solve this via computer:
\begin{equation}
\int_{-\pi}^\pi{\displaystyle \frac{e^{i\cdot a\cdot t}(e^{i\cdot b\cdot t}-1)(e^{i\cdot c \cdot t}-1)}{(e^{i\cdot t}-1)(e^{i\cdot d \cdot t}+1)(e^{i\cdot f \cdot t}-1)} \dots dt}
\end{equation}
I want to know if it'... |
I have the following problem. There is a binary relation $\succeq$ defined on a separable metric space $X$, and $\succeq$ is total, transitive, and closed (as a subset of $X\times X$, in the product metric). There exists a set $Y\subseteq X$ that bounds the elements of $X$ in order (that is, for each $x \in X$ one can ... |
Let $\{r_1, r_2, r_3,\ldots \}$ be an enumeration of the rationals in $[0,1]$, and let $(a_k)$ be a sequence such that $\sum_{k=1}^{\infty}{a_k}$ converges absolutely. Set $f_k(x) = a_k\,|x-r_k|^{-1/2}$. I'd like to show that $$\sum_{k=1}^{\infty}{\int_{[0,1]}{|f_k(x)|\,dx}}<\infty\,.$$
I've rewritten the sum as $$\su... |
So I am studying for my calculus midterm and I had the following question:
If $f_{x}(a, b)$ and $f_{y}(a,b)$ both exist then $f$ is differentiable at $(a,b)$
I answered true, but the answer is false. Why is that? I haven't done real analysis yet (dropped it this term) so I won't know what your talking about if its too... |
Why is it that cubics are separable over fields that are not of characteristic $2$ or $3$?
This is the starting point for some a discussion of the Galois group of a cubic, but I seem to be stuck right off the bat.
|
Can elliptic arc (defined as part of an ellipse, with extent not greater than $90˚$) be represented by quadratic Bezier curve?
|
I went through this thread on this question asking about a text on number theory, but my question is slightly different.
I was looking for a short and sweet book on elementary number theory. I don't need it to be a definitive treatise on number theory. I just need to have book which covers enough material to advance t... |
I've never had to do this before, so I'm not really sure how to do it. These problems also don't even really relate to what the subject of the book is as well.
Given: $f(px+(1-p)y)\le pf(x)+(1-p)f(y)$
Consider the function
$f(x)\left\{ \begin{array}{cc}
x & x\le1\\
2-x & x>1\end{array}\right\} $
Show that $f(x)$ is... |
I was trying to solve this congruence equation: $2ax+b\equiv 0 \pmod 5$. Here, $b$ is any integer and $a$ is an integer not divisible by $5$. Does a solution always exist? I don't understand why they have set the condition above on $a$. As should be clear from my other question on elementary number theory, I am not ver... |
Question: Let $U$ denote a subset of $\mathbb{R}$. We say that $U$ is an open set when for every $x \in U$ there is an $\epsilon > 0$ for which $U_{\epsilon}(x) = (x - \epsilon, x +\epsilon)$ is contained in $U$. Show:
(i) For $a,b \in \mathbb{R}$ the interval $(a,b)$ is open, but the interval $[a,b]$ is not.
(ii) If $... |
This question arises from a Fourier transform class I took about a year back.
The poisson summation formula is:
$$\displaystyle \sum_{n= - \infty}^{\infty} f(n) = \displaystyle \sum_{k= - \infty}^{\infty} \hat{f}(k)$$
where $\hat{f}$ is the Fourier Transform of $f(x)$.
It is interesting since this is true for all $f(x... |
In the situation of having a high entropy random number generator, that generates numbers in the range of 0 and 2,147,000,000.
If i have a list of 1,000,000 integer values, what are the chances that a random number will already be on my list?
|
Need some help on following induction problem:
$$\dfrac1{1 \cdot 2} + \dfrac1{2 \cdot 3} + \dfrac1{3 \cdot 4} + \cdots + \dfrac1{n \cdot (n+1)} = \dfrac{n}{n+1}$$
|
For a common basis, I'll state basic definitions of a category and the relation type I'm thinking of. They're here for quick clarity, not precision, so feel free to revise for an answer.
Category:
A collection of objects, a collection of arrows each with a source and target among said objects, identity arrows, and a co... |
Is the essential difference between one-variable calculus and multi-variable calculus exterior differential or something else?
|
Lets imagine that we have some experiments. Each experiment may result in one of the outcomes: A, B, C. So we have probability distributions for each experiment $P_A, P_B, P_C$ which is context-dependent, e.g.:
$Context_1 \Rightarrow \{P_A^1, P_B^1, P_C^1\},$ experimental outcome is A
$Context_2 \Rightarrow \{P_A^2, P... |
I am learning about discrete probability distributions and found 2 definitions for Geometric Distributions from wikipedia:
1. The probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ...}
2. The probability distribution of the number Y = X − 1 of failur... |
I am looking for a proof that this inequality:
$$(W_1+W_2+\cdots+W_n)^a \leq W_1^a +\cdots + W_n^a$$
is valid.
I have a power function $f(W)=W^a$ where $a$ is a real number, constant but usually $0\lt a\lt 1$. The $W_i$ are real positive numbers.
Then I want to check if $\left(\sum W_i\right)^a \leq \sum(W_i^a)$
Fo... |
I think the cone (or what is also called the "half cone") is a differential manifold but not a smooth manifold.
Can anyone help me understand this the nuts and bolts way?
How explicitly can I write down the differential structure on the cone?
Also what is the natural metric on it?
(Is that what is called the "Sasakian ... |
I have seen this claim on Wikipedia and other places. Which branch of mathematics does this result come from?
|
Let $A$ be the infinitesimal generator of a $C_0$-semigroup $(S(t))_{t \geq 0}$. Now, for every $x_0 \in X$ the map $t \mapsto S(t) x_0$ is a mild solution of
$$
\dot{x} = Ax, \quad x(0) = x_0.\tag{*}
$$
Now, a continuous function $x: [0, \infty) \to X$ is called a mild solution of $\text{(*)}$ if $\int_0^t x(s) \, ds ... |
Omar Khayyam is known for his significant progress in solving cubic polynomial equations. For example, his biography on www-history.mcs.st-andrews.ac.uk says
(...) This problem in turn led Khayyam to
solve the cubic equation x^3 + 200x =
20x^2 + 2000 and he found a positive
root of this cubic by considering the
inters... |
Given $N$ random iid variables, $X_1, \ldots, X_N$, with a uniform probability distribution on $[0, 1)$ what is the distribution of $\displaystyle \max_{i = 1 \ldots N}(X_i)$?
|
Possible Duplicate:
$a^{1/2}$ is either an integer or an irrational number.
Will every $n^{th}$ root of $2$ be an irrational number? If yes, how can I prove that?
|
What does this expression mean?
$$f\in C^2[a,b]$$
More specifically, I don't know what $C$ means.
|
When we consider the Taylor Series expansion of $f(x)=b^x$ for some $b \in \mathbb{R}$, we see that $$b^x = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}x^n.$$ We can substitute $x$ for $b^x$ to find that $$ b^{b^{x}} = 1 + \sum_{n=1}^{\infty}\frac{(\log(b))^n}{n!}b^{xn}.$$
Now, let's say that we want to find a functi... |
Prove that $\lim\limits_{x \to 2} \frac{x^{2}-2x+9}{x+1}$ using an epsilon delta proof.
So I have most of the work done. I choose $\delta = min{\frac{1}{2}, y}$,
$f(x)$ factors out to $\frac{|x-3||x-2|}{|x+1|}$
But $|x-3| \lt \frac{3}{2}$ for $\delta = \frac{1}{2}$ and also $|x+1| > 5/2$ (I'll spare you the details)... |
Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.
If $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then it follows from implicit function theorem that
$\text{Re}(f(z))^{-1}(c)$ is at least locally a differentiable curve in the plane.
... |
Many years ago a lecturer of mine had a photocopy of a page from a book containing a math problem ( I think it was a simple quadradic equation ) that was stated/solved in Cuneiform, Arabic, Latin scripts and Finally in modern math notation.
I have contacted my lecturer but he has no idea where it was from, nor I have b... |
At university, we are currently introduced in various methods of proving that a series converges. For example the ComparisonTest, the RatioTest or the RootTest. However we aren't told of how to calculate the point of convergence. The only series of which we know the point of convergence is the geometric series.
I beli... |
In a previous post, the following inequality has been proven
$${\left( {\sum\limits_{i = 1}^n {{W_i}} } \right)^a} \le \sum\limits_{i = 1}^n {{W_i}^a}$$
where $W_i\gt 0$, $0\lt a\lt 1$.
I guess it is more correct to say that this is always greater, and it is valid for $a\gt 0$ not just $0\lt a \lt 1$.
I am trying to se... |
I am working through an economics paper and I need to take the derivative of the following function:
$h\left(\overline{\omega}\right) = \int^{\infty}_{\overline{\omega}} \omega \Phi \left(d\omega\right)$
Even though I don't understand it well, I can do the derivative for the case
$g\left(\overline{\omega}\right) = \in... |
Let $$p_n(x)e^{-x^2}$$ be the $n$th derivative of $$e^{-x^2}.$$
Find a formula for $p_n(x)$. We have $p_1(x)=-2x, p_2(x)=4x^2-2$, etc. But what is the general formula for $p_n$?
|
I am reminded of this question that appeared in a regional Physics Olympiad I had appeared. I was wondering if there is a "mathematical" way of doing it.
If you start from a point $A$ at midnight along a specified path down a mountain and reach a specified point $B$ exactly 24 hours later. At this point you reverse you... |
Let $A$ and $B$ be finite-dimensional Hermitian matrices, and let $v$ $:=$ $max${$||A||,||B||$}.
Show that $||(e^{-iAt/m}e^{-iBt/m})^m -e^{-i(A+B)t}||\leq\epsilon$ provided $m = \Omega(v^2t^2/\epsilon)$.
|
Can anyone recommend a good self-contained reference for completion of rationals to get reals using Cauchy sequences?
|
I made the following observation following a suggestion from a friend. This question is about the Microsoft calculator program implemented on Windows. Since I don't have access to any other calculator at the moment, I cannot confirm whether this is a general feature.
If you enter an integer in the calculator program, ... |
Let $f:A \rightarrow B$.
The complements indicated below are taken within $A$ or $B$.
I need to prove that
$f(S)^c \subseteq f(S^c)$, $\forall S \subseteq A$, if and only if $f$ is surjective.
So I need to prove $f(A) = B$ right?
How can I prove this?
|
Let $Q= \{q_1,\ldots,q_n\}$ be a set of permutations of some finite set $X$. Let
$Q' = \{q \in Q^* \mid q=1\in S_X\}$, where $S_X$ is the symmetric group on $X$. Is $Q'$ a regular language?
|
Let's consider a $2\times 2$ linear system:
$$
A\bf{u} = b
$$
The solution will still be the same even after we interchange the rows in $A$ and $B$. I know this to be true because algebraically, we will get the same set of equations before and after the row interchange.
However, the vectors in columns of $A$ and $B$ a... |
I was assisting a TA for an introductory calculus class with the following limit,
$$\lim_{x \rightarrow \infty} \left(\frac{20x}{20x+4}\right)^{8x}$$
and I came to simple solution which involved evaluating the "reciprocal" limit
$$\lim_{z \rightarrow 0} \left(\frac{1}{1+\frac{z}{5}}\right)^{8/z}$$
by using the Taylor... |
Possible Duplicate:
On the sequence $x_{n+1} = \sqrt{c+x_n}$
Where does this sequence converge?
$\sqrt{7},\sqrt{7+\sqrt{7}},\sqrt{7+\sqrt{7+\sqrt{7}}}$,...
|
I have read a result on computing the highest power of a prime that divides $n!$. I was wondering if there are any results on how to compute the highest power of a prime dividing $f(x)$, where $f$ is a function on the real numbers that outputs integer values.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.