instruction stringlengths 22 27.2k |
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I haven't been able to find a reference that tells what word (if a word) the L is short for.
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I need to evaluate the following function and then check my answer by taking the derivative:
$$\int e^{ax}\cos(bx)\,dx$$
where $a$ is any real number and $b$ is any positive real number.
I know that you set $u=\cos(bx)$ and $dv=e^{ax} dx$,
and the second time you need to integrate again you set $u=\sin(bx)$ and $dv=e... |
I've got:
$$\frac{e^{ax}(b^2\cos(bx)+a^2\cos(bx))}{a^2+b^2}.$$
Could someone show me how it simplifies to:
$e^{ax} \cos(bx)$?
It looks like the denominator is canceled by the terms that are being added, but then how do I get rid of one of the cosines?
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I have to prove that:
$$x \sec x - \ln |\sec x + \tan x| + C$$
is the indefinite integral of:
$$x \sec x \tan x $$
by taking the derivative.
I've got far enough to get:
$$x\sec x\tan x + \sec x -\dfrac{|\sec x+\tan x|(\sec^2 x + \sec x \tan x)}{|\sec x + \tan x|}.$$
Kind of stuck here. Am I able to cancel out the ... |
In the following integral, p(x) and q(x) are probability distributions. Can you help me to determine in what situation this integral is equal to infinity. For example, I think that such a situations is when only p(x) has an infinite peak.
$$\int_{-\infty}^{\infty}\{\log\frac{p(x)}{q(x)}\}p(x)dx$$
Thank you very much!
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The following equation arises in the water-filling problem:
$\sum_{i=1}^{n} max \left\{ 0, \frac{1}{\nu} - \alpha_i \right\} = 1$
Assuming that all $\alpha_i$s are known, how does one solve for $\nu$?
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Could someone take me through the steps of decomposing
$$\frac{2x^2+11x}{x^2+11x+30}$$
into partial fractions?
More generally, how does one use partial fractions to compute integrals
$$\int\frac{P(x)}{Q(x)}\,dx$$
of rational functions ($P(x)$ and $Q(x)$ are polynomials) ?
This question is asked in an effort to cut d... |
Given a a polynomial with coefficients in $GF(p)$ and degree $d$, will that polynomial always have $d$ roots in $GF(p^d)$?
The text I am reading seems to be implying that this is true but I can't see why.
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How can I prove from first principles that $0!$ is equal to $1$?
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Find the values of $x \in \mathbb{Z}$ such that there is no prime number between $x$ and $x^2$. Is there any such number?
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I'm trying to learn about the "functor of points" approach to algebraic geometry, and I get everything here except for why a $k[\epsilon]/(\epsilon^2)$-point of a scheme corresponds to choosing a point and a tangent vector. Am I correct that the $\kappa_i$ must satisfy the $f_j$'s because we can always mod out by $\eps... |
In a book there is a derivation for y' that comes from
$$\frac{y'}{[1+(y')^2]^{1/2}} = c,$$
where $c$ is a constant. The result they had was
$$y' = \sqrt{\frac{c^2}{1-c^2}}.$$
How did they get this? I tried expanding the square, and other tricks, I cant seem to get their result.
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how to draw a graph with 5 vertices which is:
a)Eulerian cycle but not Hamiltonnian cycle.
b)Hamiltonnian cycle but not Eulerian cycle.
c)Eulerian cycle and Hamiltonnian cycle.
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It it an easy exercise to show that if $X$ is first-countable then for every point $x$ and every subset $A$ we have $x \in \text{cl}A$ iff there exists a sequence $(x_n)_n$ that converges to $x$.
Well, this uses the axiom of choice to create the sequence (I think). What would happen if we don't have that? (I know that ... |
I know that:
1) A function $f:\mathbb{R}^2\to \mathbb{R}^2$, when differentiable at a point, has a $2\times 2$ matrix as a derivative, which is a linear transformation from $\mathbb{R}^2\to \mathbb{R}^2$ best approximating the function linearly in some neighbourhood.
2) There is a ring homomorphism $\mathbb{C} \to Mat_... |
Find four weights such that given four weights and weighing pan (balance scale) you can measure all weights between $1$ to $80$.
I found this one here.Any idea how to solve?
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I was just having a look at Gödels incompletness theorem as found in:
https://www.ii.uni.wroc.pl/~kosciels/pi/canon00-goedel.pdf
I noticed that Gödel used a higher order logic. At least such a logic
was introduced via the Axioms IV and V in the above paper:
Axiom IV (looks to me like comprehension, but is called reduci... |
With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover are (1) linear equation in two variables, (2) quadratic equations in two variables, (3) polynomials in one variable, (4)... |
If I have two lognormal processes ($X$ and $Y$) with mean and volatiilty for each and also correlation between the two, what is the characteristic formula of $X + Y$ (i.e. what is the new mean and new volatility of $X + Y$)?
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e.g. given two numbers $5$ and $6$ the maximum number is $19$, as after $19$ each number can be formed using equation $5n+6m$ by putting different (non-negative) values for $n$ and $m$.
Such number is not possible for $4$ & $6$.
I am not sure how to go-forward. The only thing I have till now is that the number is... |
Need guidance on this problem. Given the following ordinals, determine the order relation (and equalities) - ($\Omega = 2^{\aleph_0}$)
$\omega_1$, $\Omega \cdot 3$, $\omega \cdot \omega_1$, $3^{\Omega}$, $\Omega^{\omega_1}$
My solution (partial):
$\Omega\cdot 3 \gt \Omega\cdot 2 \gt \Omega$, $3^{\Omega} = \lim_{n \lt ... |
Imagine a rectangle (x1 by y1) always has to be drawn with horizontal and vertical lines (so it can't have lines at 45 degrees). If the rectangle is rotated by angle θ, it needs to have a rectangle drawn inside it so that this new rectangle still follows the rules. In the diagram, the black rectangle is the original, t... |
Let $R$ be a ring, and let $I_1,\ldots,I_n$ be ideals in $R$ (or submodules of some $R$-module). Consider the sequence
$$
\bigoplus_{1\leq j < k\leq n} I_j\cap I_k\quad\xrightarrow{f}\quad\bigoplus_{l=1}^n I_l\quad\xrightarrow{g}\quad\sum_{k=1}^n I_k,
$$
where $g$ is given by addition, and $f$ maps $x\in I_j\cap I_k$... |
So it is clear than the nondeterministic versions of computational models such as the Turing Machine is equivalent in "power" to the deterministic model.
Other than showing this fact, what would be the "purpose" in learning nondeterminism?
My instictive answer to this is that the nondeterministic version of a TM is eas... |
I'm trying to show that complements of hyperplanes in $P^{n}$ are affine varieties on my way to a bigger result but I've gotten stuck. Am I missing something- or will the complement be a copy of affine space? I'm not seeing it.
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If $f(x)\nearrow\infty$ in $[a, \infty)$ and $f'(x)$ is continuous in $[a, \infty)$ then $\int_{a}^{\infty}\frac{f'(x)}{f(x)}\sin(f(x))$ converges
I'm not exactly sure what to try here. I thought of Dirichlets test but I can't seem to identify the right functions to make it work.
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There are five real numbers $a,b,c,d,e$ such that
$$a + b + c + d + e = 7$$$$a^2+b^2+c^2+d^2+e^2 = 10$$
How can we find the maximum and minimum possible values of any one of the numbers ?
Source
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For a polynomial of order n with unknown coefficients, what are the ways to find the coefficients from n+1 points on its plot?
I remember one way is to construct a fractional for each point, and the polynomial is the sum of the fractionals, s.t. for each fractional, its value is the desired value for its corresponding... |
The question contains 2 stages:
Prove that ${\aleph_n} ^ {\aleph_1} = {2}^{\aleph_1}\cdot\aleph_n$
This one is pretty clear by induction and by applying Hausdorff's formula.
Prove ${\aleph_\omega} ^ {\aleph_1} = {2}^{\aleph_1}\cdot{\aleph_\omega}^{\aleph_0}$
Is there an easy proof for the second one?
Thanks in ... |
Most math texts claim that $\pi$ is an irrational number. However, I'm having a little bit of trouble understanding that.
Since nobody has calculated all of the digits of $\pi$, how can we know that either:
one of the digits repeats (as in $\frac{10}{3}$)
the number eventually terminates
Note: Please be very descrip... |
I've two questions
(1) How do I determine the distribution of the first order and the highest order statistic for sample of random size N is taken from the continuous uniform(0, $\theta$) and
\begin{equation} \rm{P} (N = n) = \frac{1}{n! (e − 1)} \text{for n = 1, 2, 3, . . . .} \end{equation}
(2) In general without a... |
I am trying to factorize $-6x^5+15x^4-30x^2+30x-13$ for hours:( Could someone help me? I tried making a system of equations from $(Ax^3 + Bx^2 + Cx + D) (Ex^2 + Fx + G)$ but it is a nightmare:(
In case you are interested, the system is:
$AE = -6$
$AF + BE = 15$
$AG + BF + CE = 0$
$BG + CF + DE = -30$
$CG + DF = 30$
$DG... |
Hope I'm not spamming too much by asking questions on separate threads.
I have 2 more questions, not connected one to another, in any way:
1. Show that every stationary set in $\aleph_1$, contains, for every $\alpha<\omega_1$, a closed set of otp $\alpha$.
2. Show without using AC, that there exist a surjective fun... |
The sum $$ I + A + A^2 + A^3 + \cdots $$ equals $$(I-A)^{-1}$$ under the assumption $\rho(A)<1$, which is necessary to make the sum converge. My question: what does the sum
$$ I + A^T A + (A^2)^T A^2 + (A^3)^T A^3 + \cdots$$ equal under the same assumption? Is there a similarly neat expression?
It is true that this la... |
I am reading a cryptography paper, and the authors introduce a function $f(x) = x^{-\omega(1)}$ and call it a negligible function in $x$. What is the possible meaning of this?
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Does anyone know if there exists any small-scale game-theoretical models of the game of contract bridge, similar to the many simplified versions of poker that have been analysed?
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Assuming I have a always positive random variable $X$, $X \in \mathbb{R}$, $X > 0$. Then I am now interested in the difference between the following two expectation values:
$E \left[ \ln X \right]$
$\ln E \left[ X \right]$
Is one maybe always a lower/upper bound of the other?
Many thanks in advance...
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What does the notation $x\equiv y$ mod $2\mathbb{Z}[\zeta_3]$ mean?
|
Everyone likes to prove that Dirichlet convolution is a group operation on the multiplicative arithmetic functions, but what consequence does this have?
Does any important theorem use this fact?
Can general group theory lead to results about these functions (or even better, about numbers) from the this theorem?
Furthe... |
Let $f:X \rightarrow Y$ be a continuous map of topological spaces and let $\cal{F}$ be a sheaf on $X$. Is there an obvious map $H^\ast(Y,f_\ast \mathcal{F} ) \rightarrow H^\ast (X,\cal{F})$?
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I have two points. I need to draw an arc ($<180$°) between them, and I know how long it should be, but nothing else about it.
Knowing either the radius length or the coordinates of the center point of the circle should be enough to draw it directly. I could also calculate the radius if I knew the subtended angle θ. Bu... |
Assuming I have a Gamma distributed random Variable $x \sim Gamma( \alpha, \beta )$. Now I like to have the following two expectation values (integrals):
$E \left[ x \ln x \right]$
$E \left[ \ln \Gamma \left( x \right) \right]$
with $\Gamma \left( x \right)$ being the Gamma function
Many thanks in advance
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Possible Duplicate:
Rational Numbers
Baby Rudin has a very nice construction showing that, given a positive rational number whose square is less than (greater than) two, one can always find a larger (smaller) rational whose square is also less than (greater than) two. Basically, if $p$ is a rational whose square is... |
I'm having a hard time understanding how to find the don't-care values in a Karnaugh map. What does it even mean? If I have a boolean function, say $f(a,b,c,d)=a'bc+abc'+bc'd+a'bc'd$, how would I determine don't-care values? What would I be looking for?
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I'm looking at some class examples on basic postulates and I can't figure out how the 2nd part is reduced (see below). Could someone explain it to me?
eq:
X='a'c+ac+'ac+a'c
this is the part I do not understand, and could use some explanation using the formulas below.
= 'a('c+c) + a('c+c)
The example uses theses formu... |
Say I have a flat rectangular bar of aluminum with a known volume. Sitting directly on top of that piece of aluminum is a rectangular circuit board made of FR4 (standard circuit board material). I can raise and lower the temperature of the aluminum very rapidly and I can monitor the temperature of the aluminum instant... |
Does there exist $K \subseteq \mathbb{R} \backslash \mathbb{Q}$ such that $K$ is compact, and has Lebesgue measure greater than $0$? As I have been trying to think of examples, I suspect that any subset of $\mathbb{R} \backslash \mathbb{Q}$ that is closed can be at most countable, since the closure of an uncountable su... |
Prompted by this question: of course Godel's constructibility axiom implies that $P(S)$ is minimal for any set $S$ and so handily answers the question of the size of the power set of the continuum in $L$. Martin's Maximum is the only other principle I know of that has implications for the size of the continuum (caveat... |
Prove that an odd integer $n>1$ is prime if and only if it is not expressible as a sum of three or more consecutive integers.
I can see how this works with various examples of the sum of three or more consecutive integers being prime, but I can't seem to prove it for all odd integers $n>1$. Any help would be great.
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What is the importance of the rank of a matrix? I know that the rank of a matrix is the number of linearly independent rows or columns (whichever is smaller).
Why is it a problem if a matrix is rank deficient?
Also, why is the smaller value between row and column the rank?
An intuitive or descriptive answer (also in ... |
Find
$$\lim\limits_{x \to \infty}\frac{2^x}{3^{x^2}}$$
I can only reason with this intuitively. since $3^{x^2}$ grows much faster than $2^x$ the limit as $x \to \infty$ of $f$ must be 0.
Is there a more rigorous way to show this?
|
Consider the ternary Cantor set $C$, the associated Cantor function $f$, and the associated Lebesgue-Stieltjes measure $u$.
What is the integral of $f$ over all of $\mathbb{R}$ with respect to $u$?
My attempt:
Under the Lebesgue measure, the integral of the Cantor function is 1/2 using a symmetry argument, but under th... |
I'm trying to follow/understand a research paper that I have, and well, it's been a while since I've done this kind of math. At this point I have an nxn matrix H and from that construct an (n-1)xn matrix H' = $[\textbf{h}_1-\textbf{h}_n$, ..., $\textbf{h}_{1-n}-\textbf{h}_n]^T$. Now "using orthogonal decomposition" I'm... |
If $f$ is a real-valued function defined in a convex open set $E$ of $\mathbb{R}^n$, such that $\frac{\partial f}{\partial x_1}$ is $0$ for every $x\in E$, prove that $f(x)$ does not depend on the coordinate $x_1$. Why do we need convexity of $E$ in order to show this?
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Every month, a girl gets An allowance. Assume last year she had no money, and kept all the money she has earned up to now. Then she spends $\frac{1}{2}$ of her money on clothes, then $\frac{1}{3}$ of the remaining money on games, and then $\frac{1}{4}$ of the remaining money on toys. After she bought all of that, she h... |
I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts about groups and modular arithmetic. Is it too early to start learning category theory? should I wait to take a course on... |
Let $f(x):\mathbb R^n \to \mathbb R$ be a smooth function, and let $f(0)=0$.
I alway see that someone rewrite the function in the form $f(x)=x_1b_1(x_1)+x_2b_2(\overline {x_2})+...+x_nb_n(\overline x)$,
where the $\overline {x_i}=[x_1\ \ x_2\ \ ...\ \ x_i]^\mathrm{T}$
Can someone provide a proof?
|
What is an intuitive meaning of the null space of a matrix? Why is it useful?
I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it.
E.g.: I think of the rank $r$ of a matrix as the minimum number of dimensions that a linear combination of its columns would have; it te... |
The title is self-explanatory . :-)
thanks,
Actually my description is not comprehensive : I want to draw the graphical solution of an operation research problem
maximize $z = 7x_1 + 9x_2$.
under $x_1 + x_2 \leq 8$.
$x_2\leq 4$.
$2x_1 + 3x_2 \leq 19$.
$x_1\geq 0$.
$x_2\geq 0$.
|
I have a polyhedron $P$, in 3D, which consists of $f$ faces, each face consists of $V$ vertexes.
My question is, how to choose a consistent normal orientation for all the faces? Consistent here means that if I unfold all the faces of that polyhedron onto a common plane all the normals will agree.
For the sake of argum... |
In a group of permutations of $n$ elements, there are two permutations $P_1$ and $P_2$ such that $P_2=P_1^e$. $P_1$ and $P_2$ have the same order $o$: $P_1^o = P_2^o$. How can I find $e$?
$n$, $P_1, P_2$ and $o$ are known, $e$ is not known. Problem is to find $e$.
|
I got this from Thomson et al.'s freely available "Elementary Real Analysis" p.356.
They introduce Baire's category theorem through a game where, given two players (A) and (B)
Player (A) is given a subset $A$ of $\mathbb{R}$, and player (B) is given the complementary set $B = \mathbb{R} \backslash A$. Player (A) first... |
Can someone explain intuitively what the Fundamental Theorem of Linear Algebra states? and why specifically it is called the above? Specifically, what makes it 'Fundamental' in the broad scope of the theory.
|
How can I prove that $$n^4 + 4$$ is composite for all $n > 5$?
This problem looked very simple, but I took 6 hours and ended up with nothing :(. I broke it into cases base on quotient remainder theorem, but it did not give any useful information.
Plus, I try to factor it out:
$$n^4 - 16 + 20 = ( n^2 - 4 )( n^2 + 4 ) -... |
Prove that $$(2^k + 1)|( 2^{ k( 2n + 1 ) } + 1 )$$
How can I approach this problem? Using algebra manipulation with the number $$2^{ k( 2n + 1 ) } + 1 $$ (this is what I tried, but failed :( ), or there are some other properties are more useful. Any hint?
Thanks,
Chan
|
Possible Duplicate:
How to factor quadratic $ax^2+bx+c$?
If $x^2 + 2x - 35 = 0$,
then $x = $?
|
Suppose I have a deterministic sequence $\{t_n\}$ that is uniformly distributed on $[0,1]$ (for example $t_n = \{ \pi n \}$, i.e the fractional part of $\pi n$) and a decreasing function $f : \mathbb{R} \rightarrow [0,1]$.
I think it is reasonable to expect that
$\#\{t_k < f(k) : k \leq n \} \approx \sum_{k = 1}^n f(k)... |
Suppose $X$ is a normed space. Prove that $X^\ast$ is rotund iff $X/M$ smooth whenever $M$ is a closed subspace of $X$ such that $X/M$ is two-dimensional.
I know that "a normed space is smooth iff each of its two-dimensional subspaces is smooth." Maybe that will be used.
|
In the Bishop Book on Machine Learning, when discussinf Variational Inference, he gives an example of the univariate Gaussian.
He defines in 10.1.3 the mean prior as:
$p(\mu|\tau)=N(\mu|\mu_0,(\lambda_0*\tau)^{-1})$
What is the meaning of the conditional mean over an initial $\mu_0$ and what would be the analytical for... |
I have just started learning Riemann surfaces and I am using the book by Rick Miranda: Algebraic curves and Riemann Surfaces. #F in section 1.3 asks to determine the genus of the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2 + x_1^2 +x_2^2 +x_3^2 = 0$. #G also has a similar question ... |
Taken from
Cambridge University Press
0521830842 - Foundations of Cryptography: Basic Applications, Volume 2
CHAPTER FIVE - Encryption Schemes
which can be downloaded here
Notation. In the rest of this text, we
write $E_e (\alpha)$ instead of $E(e, \alpha)$ and
$D_d (\beta)$ instead of $D(d, \beta)$. Sometimes,
... |
I have a polygon and I only know the length of its sides, how would I go about figuring out what it's points are?
Assume that the first side starts from the origin $(0,0)$ and extends along the x axis to (length of side $(a, 0)$ ).
I realize that this is trivial for, say a triangle, but is there a rule that can be used... |
As a follow-up to this question, I'd like to ask:
What are examples of rings $R$ with the property that for all finite sets of ideals $I_1,\ldots,I_n$ in $R$ the sequence
$$
\bigoplus_{1\leq j < k\leq n}^n I_j\cap I_k\quad\xrightarrow{f}\quad\bigoplus_{l=1}^n I_l\quad\xrightarrow{g}\quad\sum_{k=1}^n I_k
$$
is exact in ... |
I have figured out that:
Given $N = 10^K$, $D = K + 1$.
What is the formula for a natural number $D = {???}(N)$
How to solve this in Mathematica/WolframAlpha ?
|
$$\sum_{n=2}^{\infty} \frac{ (-1)^n} { \ln(n) +\cos(n)}$$
|
In my programming project I need to evenly distribute various sized circles in ellipse shaped area.
The ellipse area is defined by major and minor axis (4 points).
There is finite number of various sized circles that will all fit into the ellipse.
The order of the circles does not matter.
What is the criterion that... |
What is the simplification of $$\frac{\sin^2 x}{(1+ \sin^2 x +\sin^4 x +\sin^6 x + \cdots)} \space \text{?}$$
|
Why is it that no one seems to factor quadratics into just one bracket Eg:
$$2x^2+8x+6$$
into
$$2x\left(x+4+3\cdot\frac{1}{x}\right)\quad\text{or}\quad 2x\left(x+4+\frac{3}{x}\right)\quad ?$$
|
As I understand the KL Divergence, it measures how different two probability distributions $P$ and $Q$ are.
However, say the two distributions are:
P = [0.2 0.2 0.2 0.4];
Q = [0 0.5 0.1 0.4];
Then the KL-divergence is infinity. What is the justification for these distributions being infinitely different?
|
I was just reading about the e-function in the complex plane and was trying to understand the differences between the real and the complex case. Part of the problem is that the mapping of a 2-D plane to another 2-D plane is hard to visualize.
My question
What are the properties of the complex exponential function in te... |
Suppose $\bar{X}_n$ is the mean of a random sample of size ${n}$ from an exponential distribution with $\lambda$ > 0. Then what does the following statement about convergence mean (how does this converge)?
$$\text{exp} \left(-\frac{1}{\bar{X}_n} \right) \xrightarrow{\rm{P}} \text{exp}(-\lambda) $$
More specifically wha... |
While working with elliptic curves for cryptography reasons, I found the notion of a primitive point, but no definition.
For example, $P(0,6)$ is a primitive point on the elliptic curve $y^2\equiv x^3+2x+2 \mod 17$.
What does that mean? How can I tell if a point is primitive or not?
|
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are able to plug in values for $\textrm{d}x$ and calculate a... |
How does one show this?
Show that the set of all continuous functions $f:[0,1] \rightarrow R$ is a closed subset of set of all functions $g:[0,1] \rightarrow R$, when endowed with the box topology.
What does it mean if a set is endowed with a box topology? What function will fit this? What happens if I change the box t... |
For the past couple of days I have been encountering the word "Torus" quite often.
I would like to know what special properties does the Torus possess that it is studied very much in Mathematics.
A recent article on Toral Automorphisms was given out to students by one of our Professors, which i have posted here.
h... |
Please tell me if what I did is correct or if there's any faster alternatives.
I set $x$ and $y$ axes on the center of the circle with radius $r$, therefore this can be seen as an area described by $x^2+y^2=r$ revolving around $x=-R-r$
$dV$ can be written as
$$dV = 2\pi(R+x)\cdot 2y \cdot dx =2\pi(R+x)\cdot 2 \sqrt{r-... |
I've split the integral into:
$$\int_0^1\frac{1-x^\alpha}{1-x}\mathrm dx=\int_0^{1/2}\frac{1-x^\alpha}{1-x}\mathrm dx+\int_{1/2}^1\frac{1-x^\alpha}{1-x}\mathrm dx$$
I'm trying to find a suitable function in the form of $g(x)=\frac1{(x-1)^\text{?}}$ so I can use the LCT when $x\to 1$ but I can't figure out what needs to... |
I've found the following derivative in my Calculus book and I can't get my my head around the algebra involved. Can anybody help me?
Thanks.
|
As I was a student, I got the following problem:
Let $f$ be a mapping from $[a,b]$ to $\mathbb{R}$ satisfying
$$(1)\quad f((1-\lambda)x+\lambda y)\leq (1-\lambda )f(x)+\lambda f(y)$$
for all $x,y\in [a,b]$ and $\lambda\in [0,1]$. Suppose $f$ is differentiable on $]a,b[$. Show that if $f$ is continuous and $f'$ increasi... |
I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius automorphism at a place.
Thank you
|
I have a question about the relationship between parameters of weakly regular graphs (if it exists)
Consider two weakly regular graphs with parameters $(n,k,\lambda,\beta)$ where n is number of vertices, k is the degree, $\lambda$ is the number of common neighbors of adjacent vertices, and $\beta$ is the number of comm... |
A function $f$ is called fully time-constructible if there exists a Turing machine $M$ which, given a string $1^n$ consisting of $n$ ones, stops after exactly $f(n)$ steps.
Analogously, we can call a function $f$ fully space-constructible if there exists a Turing machine $M$ which, given a string $1^n$ consisting of $n... |
The problem i am supposed to solve for $x$ by completing the square:
$3x^2+9x+5 = 0$
step 1. move constant to right: $3x^2+9x\quad\quad+5 = 0$
step 2. divide by $3$: $x^2+3\quad\quad+\frac{5}{3}$
step 3. $(\frac{1}{2}b)^2$: $(\frac{1}{2}\cdot 3)^2 = \frac{9}{4}$
step 4. add and subtract result from step 3 into equati... |
Let $G=<a,b | a^2b=ba^2, b^2a=ab^2, a^2=b^4>$. Consider a homomorphism $\phi$ from $G$ to $<c| >$ where $\phi(a)=c^2$ and $\phi(b)=c$. Can someone help me decide what group the kernel of this map is? Thanks!
|
In these Karen Smith's notes on representation of finite groups, on page 50 the irreducible representations of $S_3$ are found.
If $\sigma=(1 2 3)$ and $V$ is a complex representation of $S_3$, I don't understand what "the action of $\sigma$ on $V$" means. Is it this:
$\langle\sigma\rangle \times V \to V$
$\sigma \cdot... |
$$\int{96\cos^4(16x)} \ dx$$
Setting $u=16x$, $du=16dx$,
$${\frac{96}{16}}\int{\cos^4(u)} \ du$$
Kinda stuck here, I checked Wolfram Alpha but it suggests using some reduction formula that we haven't learned in my class yet.
@Myself: Using what you posted, I end up getting
$${\frac{3}{2}}\left(\frac{\sin4u}{4}+\frac{3... |
I know that this question is some sort of bridge between Informatics and Mathematics, not knowing the best place where to post this question, I opted for this place because of the type of answer I want (which concerns Math more than anything).
Consider to have a graph represented by a collection of nodes and connection... |
Everyone:
Please forgive that I do not yet know LaTex, bro, and my English ( I am from UCV in Venezuela).
I think I understand concept of bundles almost well, and that, once a vector bundle with a fiber is known/given, that we can define a new fiber pointwise, in manipulating each of the fibers, e.g., we may change ... |
If $X$ has the density function
$$ f_\vartheta (x) = \Big \{ \begin{array}{cc}
(\vartheta - 1)x^{-\vartheta} & x \geq 1\\
0 & otherwise
\end{array}$$
How can I see that $\log X \sim Exp(\vartheta - 1)$?
I had the idea to look at $f(\log x)$ but I think that's not right.
Many thanks for... |
Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$.
(a) Prove that $R$ is an equivalence relation.
Here is what I have so far. Is this correct?
Reflexive:
$a \sim a$
$\implies$ $a+b=a+b$; $(a,b) R (c,d)$
Symm... |
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