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I started by letting $u(x,y) = e^{-x}$ and $v(x,y) = e^{-iy}$ . I then tried to use the cauchy reiman equations : $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$. But my answer did not seem right, did I assign my values of $u$ and $v$ ... |
Find the remainder when ${45^{17^{17}}}$ is divided by 204
I tried using congruence modulo. But I am not able to express it in the form of $a\equiv b\pmod{204}$.
$204=2^2\cdot 3\cdot 17$
|
Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function.
To be precise: Let's define a semi-euclidean domain as a domain $R$ together with a norm $\delta : R \rightarrow \alpha$ for an ordinal $\alpha$ (or even the class of all ordinals) such that for $f,g \in R\ ... |
Given that $(z_n)$ is a sequence of distinct points in $D(0,1)=\{z \in \Bbb C : |z| \lt 1\}$ with $\lim_{n \to \infty} z_n=0$, Can we find an analytic function $f$ such that
$f(z_n)= \begin{cases} 0, & \text{if n is even} \\ 1, & \text{if n is odd} \end{cases}$
?
I know the Uniqueness theorem and Identity theorem. How ... |
If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are the lengths of $\tau_1, \dots, \tau_r$ (supposing we ordered them in such a way that $l_1 \ge l_2 \ge \dots \ge l_r$).
Th... |
"A positive integer is a prime only if it has no divisors other than one and itself." The inverse of this conditional statement is : " A positive integer is not prime if it has divisors other than one and itself."
In logic the inverse isn't equivalent to the original statement. Can someone please clarify it for me??
|
I have to find the branch points of $f(z)=\left( z(z+1)\right )^{1/3}$.
It is clear that $0$ and $-1$ are branch points, but I am not sure about infinity. Making the substituition $x=\frac{1}{z}$ and examining $f(x)=\left(\frac{1}{x}\left(\frac{1}{x}+1\right)\right )^{1/3}$, I do not really see what happens as x tends ... |
We have $n$-persons ($n\ge 3$) cooperative game. And we know that player $1$ and $2$ are symmetric. So for each element $(x_1,x_2,...,x_n)$ from the core we have $x_1=x_2$ ? Is that true ? Never seen example when this is not true, but how do it come from ?
|
Here's Euclid's proof of Infinitude of Primes in Rosen's Discrete Mathematics:
We will prove this theorem using a proof by contradiction. We assume that there are only finitely many primes, $p_1, p_2,..., p_n$. Let
$$Q = p_1p_2...p_n +1$$
By the fundamental theorem of arithmetic, $Q$ is prime or else it can be written... |
A garrison has sufficient food for $75$ soldiers for a period of $90$ days.
After $10$ days, one third of the soldiers leave. After another $10$
days, $5$ soldiers return, From this day on, how many days will the
food last ?
I did
$F \propto s\times d \\
F_{1} =k\times75\times 90 \\
F_{2} =k\times50\times 80 ... |
A square matrix $A$ is called nilpotent if $A^m = 0$ for some positive integer $m$. Show that
A $n \times n$ matrix $A = (a_{i,j})$ in which $a_{i,j} = 0$ if $i ≥ j$ is nilpotent.
|
Does there exist a pathological function which is differentiable any finite number of times as one wishes, but is not differentiable an infinite amount of times?
Is it reasonable for such function to exist?
|
Here are basically my two problems, which I have the answer from WolframAlpha:
$$
\lim_{n\to\infty}(1-\sqrt 2-\sqrt{n+1}+\sqrt{n+2})=1-\sqrt 2
$$
$$
\lim_{n\to\infty}(\sqrt n-2\sqrt{n+1}+\sqrt{n+2})=0
$$
I have no idea how to actually solve them on my own though. At this moment it's gonna be inf - inf (and inf - 2 inf ... |
series of n numbers is given
A operation is performed by:
for i = 2 to N:
Ai = Ai + A(i-1)
our task is to find the xth element of the array (i.e., Ax) after performing the above operation M times.
How could we generalized to find the ith term.
My approcah is
ith(mth opeartion)= ith term(m-1th opeartion) +... |
I got this problem to prove, and I assume I need to use the existence and uniqueness theorem for non-linear ODE's, so I set $y' = f(x,y)$ and differentiating in respect to $y$ gives: $f_y(x,y)$.
And both of them are continuous at $y=\Large {\pi \over 8}$.
But from this point on I don't really know what I'm doing, how ... |
Let $A$ be a square matrix of order $3$. Prove that
$$
\operatorname{adj}(A)
= \tfrac{1}{2} \bigl[
(\operatorname{tr} A)^2 - \operatorname{tr}(A^2) \bigr] I_3
- [\operatorname{tr} A] A + A^2
$$
where $\operatorname{tr}A$ is the trace of $A$.
To begin, I'm not quite sure how to start proving this. I've consid... |
Let $f(x,y) = (e^x \cos y, e^x \sin y)$. What is $f(\Bbb R^2)$?
I know I should take $u = e^x \sin y$ and $v =e^x \cos y$ and try to find a relation but I can't find something
|
What is the value of the following sum?
$$\sum_{i=1}^{2000}\gcd(i,2000)\cos\left(\frac{2\pi\ i}{2000}\right)$$
where
$\gcd$ is the greatest common divisor.
|
Evaluate the integral:$\int_{1}^{2} \frac{\tan^{-1} x}{\tan^{-1} \frac {1}{x^2-3x+3}} dx$
On applying the property $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ I dont seem to reach any where
|
I am reading this part of a research paper
where the author states that the left hand side of equations (12) and (13) must be equal to a constant. However I could not understand the explanation he gives.
Does anyone know what the author means for why the equations are equal to a constant?
Thanks.
|
I need some easy to implement and fast numerical method that finds the root of a nonlinear function (a polynomial in my case) closest to my initial guess. If I know that there is one root $x^\star_k\in I$ and I start in the middle of $I$, is there a numerical method that guarantees convergence to exactly this root?
New... |
I am not sure where to start on this one. I know that $z^n$ can be written as $\sum_{n=0}^{\infty} \frac{1}{1-z}$. But I do not know how to proceed.
|
What are the first few values of derivatives of Riemann xi function at zero? Is there any general formula for calculating the nth derivative of the riemann zeta function at zero? What happens to the value of the nth derivative of xi function as "n" tends to infinity? what is the Taylor series expansion of xi function a... |
If $0 < a \le b \le c \le d$ and $abcd = 1$ prove that $$a+b^2+c^3+d^4 \ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$$
I first thought of multiplying both sides with $\big(\frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}\big)$
Then by applying C-S inequality in the LHS the inequality would be $$16 \g... |
Consider the irreducible rational function in $\mathbb{R}^2$.
$$y=\frac{A(x)}{B(x)}$$
where at least one term is quadratic and the other term has degree either 0, 1 or 2.
The classic way of finding the range of these rational functions is to re-arrange the equation as a quadratic in $x$, and then let the discriminant... |
If $\tan A=\dfrac{1-\cos B}{\sin B}$, prove that $\tan 2A=\tan B$.
My effort:
Here
$$\tan A=\frac{1-\cos B}{\sin B}$$
Now
$$\begin{align}\text{L.H.S.} &=\tan 2A \\[4pt]
&=\frac{2\tan A}{1-\tan ^2A} \\[6pt]
&=\frac{(2-2\cos B)\over\sin B}{1-\frac{(1-\cos B)^2}{\sin^2 B}}
\end{align}$$
On simplification from here, I c... |
How is the principal axis form (quadratic) of a symmetric orthogonally diagonalized matrix
e.g.
$$y^2+6yx+x^2=1$$
transformed into the equation of a hyperbola (or ellipse as well)?
Completing the square?
|
Let $g(x) = f(x)/(x+1)$, where $f(x)$ is differentiable on $x\in[0,5]$, such that $f(0)=4$ and $f(5)=-1$. What is the range of values $g'(c)$ for a $c$ belonging to $[0,5]$?
Considering values of $f(x_i)$, $f(x)$ must decrease at least once from $0$ to $5$. But that is all the information I can use here. Is there anyth... |
Given this chain of equalities:
$$\frac{\sqrt{1-x} - \sqrt{1+x}}{x}=\frac{(\sqrt{1-x} - \sqrt{1+x})(\sqrt{1-x} + \sqrt{1+x})}{x(\sqrt{1-x} + \sqrt{1+x})} = \frac{(\sqrt{1-x})^2 - (\sqrt{1+x})^2}{x(\sqrt{1-x} + \sqrt{1+x})} = \frac{(1-x) - (1+x)}{x(\sqrt{1-x} + \sqrt{1+x})} = \frac{-2x}{x(\sqrt{1-x} + \sqrt{1+x})} = \fr... |
I'm solving a couple of integration problems using the method of changing variables, and would like assistance with two particular problems that I can't seem to solve. I completed rest of the problems in this problem set without much effort, but these two seem impossible.
I've tried changing a few different variables i... |
Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely.
Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, and $A_6$ be the events that the last value is a $1$,$2$,$3$,$4$,$5$, and $6$ respectively.
Let $B$ be the event that the first value is less or equal to the second val... |
I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but
If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. $f(r,t)=g(r)=constant$?
|
$G$ is a finite group of order $n$, then if $a,b\in G : ab^{-1}=ba^{-1}$ and $n$ is odd, then $a=b$.
multiply both sides by $ab^{-1}$ we get $(ab^{-1})^2 = ab^{-1}ab^{-1}=ba^{-1}ab^{-1}=1$ so $ab^{-1}$ is of order $2$ or its $1\in G$ now we need to show that if $n$ odd then there's a single element of order $2$? I do... |
I'm struggling with the proof of the following, well-known result for the Levi-Civita connection of a Lie Group with biinvariant metric, i.e. satisfying
\begin{equation}
g(D_bL_a X_b, D_b L_a Y_b)= g(X_a, Y_a)
\end{equation}
with $L_a$ the left (or right) translation by $a$.
The claim is
\begin{equation}
\nabla_X Y= \f... |
I need to solve $$\frac{\partial u^2}{\partial x\partial y}=0$$ with the boundary conditions: $u(x,y=x^3)=\sin(x^6)$ and $\frac{\partial u}{\partial x}(x,y=x^3)=0$.
I got a particular solution, I thing which is $u_p=A\sin(y^2)$, where $A\in\mathbb{R}$, that satisfies the two boundary conditions, but it is rather a gues... |
This is the question:
I first found $\frac{dv}{ds}=6s^2+5$, then I tried to find $\frac{ds}{dt}$ by messing about a little with implicit differentiation, but I had no luck and I therefore couldn't apply the chain rule (i.e. $a=\frac{dv}{dt}=\frac{ds}{dt}\frac{dv}{ds}$) to find acceleration. The back of my book tells m... |
in this picture a length of square edge is 8 cm. I want to calculate the radius of circle. i try to calculate it, but i don't know how.
I calculate this:
|
What is the difference when someone says that
an expression exclusively depends on $x$
and
an expression only depends $x$?
|
If $m\tan (\theta-30°)=n\tan (\theta+120°)$ then prove that :
$$\cos 2\theta=\frac{m+n}{2(m-n)}$$
My attempt\
Here,
$$m\tan (\theta-30°)=n\tan (\theta+120)$$
$$\frac{\tan (\theta-30°)}{\tan (\theta+120°)}=\frac{n}{m}$$.
Now, what should I do next?
|
As the title says, I can't prove that, no matter what I try. What I've tried so far:
induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates mentioned below and using identities of the binomial coefficient didn't get me anywhere
managed to show $${n \choose k} \le... |
For $\mathbb{R}^2$ we have the euclidian norm
$$(x_1,x_2)\mapsto\sqrt{x_1^2+x_2^2},$$
and the Manhattan norm
$$(x_1,x_2)\mapsto|x_1|+|x_2|.$$
Let $d_E$ and $d_M$ be the metrics defined by these norms, respectively. Then we define $B_\epsilon^E(x)=\{y\in\mathbb{R}^2|d_E(x,y)<\epsilon\}$ and $B_\epsilon^M(x)=\{y\in\mathb... |
Main Question
Can someone give a "constructive" proof of the fact that,
Let $(X,d)$ be a metric space and $x,y\in X$. Let $B_d(x,r_x)$ and $B_d(y,r_y)$ be two open balls centered respectively at $x$ and $y$ for $r_x,r_y\in \mathbb{R}^+$. Suppose that $B_d(x,r_x)\cap B_d(y,r_y)\ne \emptyset$ and $z\in B_d(x,r_x)\cap B_... |
The question is given in the title. I am unable to come up with a counterexample and I'm thinking this could apply to cyclic groups but not necessarily to general groups. Does anyone have any ideas or something to point me in the right direction?
Help would be much appreciated, thank you!
|
In order to prove the normalisation property of a Lorentzian function,
$L = \dfrac{1}{\pi}\displaystyle \int_{-\infty}^\infty \dfrac{b}{(z-a)^2+b^2} dz = 1$
we take a closed contour on the upper-half complex plane. This means we only consider the $z=a+ib$ pole when finding residues. I know this has to do with the windi... |
The rule of Mersenne Prime says that $2^p - 1$ is prime if $p$ is prime.
$2^{11} - 1 = 2047$ satisfies the condition, but it's not a prime as it can be divided by two prime numbers $23$ and $89$. Then, why do we use Mersenne Prime thing at all?
Last I checked, the biggest known prime today is a Mersenne Prime. Can't it... |
If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual space of $V$ is the vector space $V^*=\{F:V\to K:F\ \text{is linear}\}$ for all $i$; $x^{i^*}$ is defined as $x^{i^*}(x^... |
Input parameters:
space dividor (number), vector (vec2)
Desired result:
Divide space in $X$ sectors, then move all vectors to one sector. (Angle of any vector wont be larger then $360/X$.)
Example of behavior that I made but seems to long:
First, calculate $360$/space dividor which results in angle that we use as modu... |
This is the context of the question:
I'm assuming by the space diagonal (although I'm not sure) to be the area of the right-angled traingle created by the diagonal. Let this space be $V$, then we will have $V=\frac{1}{2}\sqrt{2}x^2$, where $x$ is the lenght of the edges at time $t$. Now, $\frac{dV}{dx}=\sqrt{2}x$, and... |
Given $A_{n\times n},B_{n\times n} \in \mathbb C$ then:
if $A$ is unitary and the characteristic polynomial $f_A(x)=f_B(x)$ then $B$ is also unitary.
if $A$ is normal and $f_A(x)=f_B(x)$ then $B$ is also normal.
if $A$ is unitary and $f_A(x)=f_B(x)$ and $m_A(x)=m_B(x)$ then $A$ is similar to $B$.
(1) if the matr... |
I have a question which asks to design a circuit to convert from binary to gray code, using a boolean expression.
Now I understand you have to use XOR to achieve this. And I understand that XOR means that only one or the other can be true, if both are true thats just nomral OR.
I also understanding how to convert betw... |
If $$\frac{m+1}{m-1}=\frac{\cos(\alpha-\beta)}{\sin(\alpha+\beta)}$$, prove that :
$$m=\tan(π/4 +\alpha).\tan(π/4 +\beta)$$.
My attempts/
Here ..
$$\frac{m+1}{m-1}=\frac{\cos(\alpha-\beta)}{\sin(\alpha+\beta)}$$
Applying componendo and dividendo, I got;
$$m=\frac{\cos(\alpha-\beta)+\sin(\alpha+\beta)}{\cos(\alpha-\bet... |
Let $X_1, X_2$ be i.i.d. $\mathcal U(0, 1)$ (continuous) r.v.'s, and let $0 \le R \le 1$ be some number. What is $\mathbb E[\min(X_1, X_2) \mid \min(X_1, X_2) \ge R]$?
My attempt: Let $Y = \min(X_1, X_2)$. Let $F$ be the CDF, and $f$ the pmf of $Y$. We can see that
$$
\begin{align*}
F(y) &= 1 - (1 - y)^2 \\
f(y) ... |
Use Pascal’s formula repeatedly to derive a formula for $\dbinom{n+3}{r}$ in terms of values of $\dbinom{n}{k}$ with $k \leq r.$ (Assume $n$ and $r$ are integers with $n\geq r \geq 3).$
I have a idea about this question but I don't know how to convert $r$ to $k$.
|
If the Difference of radii of larget and smallest Circle passing through the focus of Parabola $$Y^2=4x$$ and toughing parabola in at least one point is
My Approach
Let Circle be $$C: (x-a)^2+y^2=r^2$$
Since it passes through focus of parabola:(1,0)
=> $$r^2=(1-a)^2$$
$$C: (x-a)^2+y^2=(1-a)^2$$
Solving for Parabola an... |
I am going through Linear Algebra right now, we are using the book Elementary Linear Algebra by Andrilli. In one of the theorems he uses this notation without really introducing it. Here is the theorem:
Theorem 2.3 Let AX=B be a system of linear equations. If [C|D] is row equivalent to [A|B], then the system CX=D is eq... |
Suppose $f$ is an analytic function on a domain $D$. Then I need to show that $\overline {f(\bar z)}$ is also analytic.
Here is what I did -
Suppose $f(z) = u(x,y) + iv(x,y)$ where $u$ and $v$ are real functions of $x$ and $y$ and $z = x + iy$. Now $f(\bar z) = u(x,-y) + iv(x,-y) $ and then $\overline {f(\bar z)} = ... |
Suppose you pick a number between $1$ and $30$ uniformly at random. Let $A$ be the event that
the number is even. Let $B$ be the event that the number is divisible by $3$. Let $C$ be the event that the number
is divisible by $5$. Using the above formula, what is the probability that the number is divisible be at least ... |
Let $\mathbb{N} = \{1,2,3,4,\dots \}$ and define the sets $A_k \subset \mathbb{N}$ by
$$
A_k = \{k,2k,3k,\dots \}
$$
for $k = 1,2,\dots$. We denote by $\mathcal{H}$ the collection $\{A_1, A_2, \dots \} \cup \emptyset$.
Show that $\sigma(\mathcal{H})$ (the sigma-algebra generated by $\mathcal{H}$) is equal to $\ma... |
In my problem set there was an exercise involving optimal stopping theory. Here is the problem: There is an exam, a list of $n$ questions and $n$ students. Student $A$ knows answers to $k$ of them. He sees all the people who took the exam before him and knows whether or not they got one of those $k$ questions he knows.... |
How can one possibly find the general solution to the following nonlinear ODE?
$$\frac{dy(x)}{dx}=e^{y(x)/2}$$
I tried Mathematica, which gives the solution
$$y(x)=-2\ln \left[ \frac{1}{2 (-x - c)} \right]$$
However I think that this is not the only one solution. Any ideas to treat the
problem analytically?
What happ... |
If all men were good, there would be no wars.
Some men are not good. Therefore, there will be wars.
I am confused on what sign to use on the some part.
|
Please help me to study the following simple cases:
Let $P$ be a binary predicate symbol. I am trying to find out, if there exists a satisfiable $T$ having infinite models only, for the following cases:
1) T is under {P} language with "="
2) T is under {P} language with "=" and T is finite
3) T is under {P} language wi... |
I was trying to figure out when a Riemann or Lebsgue Stieltjes integral is of bounded variation. For simplicity let $f$ be a increasing RCLL function; when is that
$$\int_0^t g(x) df(x)$$ is of bounded variation?
Clearly if $f \in C^1$ this holds, because then the integral is absolutely continuos and therefore of boun... |
I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me?
Thanks.
|
What is the minimum distance between the curves $f(x) =e^x$ and $g(x) = \ln x$?
I didn't understand how to solve the problem. Please help me.
|
Equation:
$$\log(x^2+2ax)=\log(4x-4a-13)$$
It has only one solution; then exhaustive set of values of $a$ is ??
I don't even know where to begin
The answer is :
$$(-13/4,-13/12) \cup [-1]$$
|
is there any partial sums of harmonic series that add up to an integer? partial sums not as trivial as the first term only i.e. 1, or the powers of 2 i.e the infinite geometric series for 2.
This might be trivially obvious for a subset of permutations of reciprocal of some integers.
PS: there is already another solved... |
Let $p = \sqrt{1+\sqrt{1+\sqrt{1 + \cdots}}}$
The sum $$\sum_{k=2}^{\infty}{\dfrac{\lfloor p^k \rceil}{2^k}}$$
Can be expressed as $\frac{a}{b}$. Where $\lfloor \cdot \rceil$ denotes the nearest integer function. Find $a+b$.
My Work
$p^2 = 1+p \implies p = \dfrac{1+\sqrt{5}}{2} = \phi$
And also $\phi^n = \dfrac... |
Given an arbitrary field $K$, may I define an integer in $K$? I have found how to define an algebraic number in $K$ and how to define an integer algebraic number in $K$.
For instance, let $K=\Bbb{Z}_p$, for $p$ prime (i.e., the quotient of the ring of integers by the principal ideal generated by the prime $p$). Are the... |
In the definition of metric space, only one thing strikes me as unnatural: the requirement that $d(x,y) = 0$ implies $x = y$. As a programmer, I don't find it uncommon to deal with equivalence relations (or, more generally, preorders) whose equivalence classes don't have computable canonical representatives. If I wante... |
The set of values of $p$ such that both the roots of the equation
$$f(x)=(p−5)x^2−2px+(p−4)=0$$ are positive and one of
the roots is less than $2$ and the other root lies between $2$ & $3$
is ?
Are these conditions sufficient?Please check.
$f(2).f(3)<0$
$(p-5)f(2)<0$
$(p-5)f(0)>0$
$\frac{p}{p-5}<0$
P.S:Also t... |
Think of a set with $m+n$ elements as composed of two parts, one with $m$ elements and the other with $n$ elements. Give a combinatorial argument to show that
$\dbinom{m+n}{r}$ = $\dbinom{m}{0}$$\dbinom{n}{r}$ + $\dbinom{m}{1}$$\dbinom{n}{r-1}$ + $...$ + $\dbinom{m}{r}$$\dbinom{n}{0}$
where $n$ and $m$ are positive in... |
In numeric analysis class, we are supposed to show that
$$P\Bigl(\bigcup_{n\in\mathbb N}A_n\Bigr)\le\sum_{n\in\mathbb N}P(A_n).$$
This is easy to show using induction for a union of finitely many $A_n$, but I'm not sure how to prove this for the required union of countably infinite $A_n$. Indeed, it seems that the stat... |
Find the minimum integer $n>1$ for which $2^n > n^{1000}$.
I have taken the $log$ on both sides, but not reached any result. I would appreciate if anybody will solve it accurately. Thanks in advance.
|
An incompressible inviscid fluid, under the influence of gravity, has the velocity field
$$\textbf u = (− \cos(x)\sin(y), \, \, \sin(x)\cos(y), \, \, 0)$$
with the $z$-axis vertically upwards, where $g$ is the acceleration due to gravity.
Show that the Bernoulli function $H = p/ρ + \textbf{u}^2/2 + gz$ is constant on the ... |
Is the following assertion true or false?
There exists a nonzero function $f \in C([0,1])$ such that $$\int_0^1f(x)x^ndx=0 (\forall n \in \mathbb N)$$ holds.
(Hint: use the weierstrass approximation theorem)
|
Let $\mu$ be a complex borel measure on $\Omega$, $|\mu|$ its total variation and $A \subseteq \Omega$ a closed set s.t. for each closed set $B\subseteq A$ we have $\mu(A)=0$. Now does it hold that $|\mu|(A)=0$? I think yes, as it would trivially hold if $\mu(C)=0$ for all measurable subsets of $A$ and the closed sets ... |
Let $X\to Y\to Z$ be three random variables.
The data processing inequality states $I(X;Y)\geq I(X;Z)$.
Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function.
What more can we say about how $I(Y;Z)=I(f(X);Z)$ relates to $I(X;Y)=I(X;f(X))$ and $I(X;Z)$?
I.e. can one somehow add the mutual ... |
I have several questions regarding these methods.
Primal Simplex Method
Does the pivot element always have to be a positive entry in the table?
Does the RHS always have to be positive in the pivot row?
What if I have some negative RHS values?
How does the procedure differ if we are looking for MAX or MIN problem?
Dua... |
Please, can someone help me? I have the following problem:
Let $X$ be a normed space, $Y \subset X$ a linear subspace of $X$ and the function $$d_{Y}(x)=inf\{||x-y||:y \in Y\}$$;I have to prove that $N:X/Y \to \mathbb{R}$, $N(\overline{a})= d_{Y}(a)$ is well defined. I picked $\overline{a}, \overline{b}$ from $X/Y$, wi... |
Let $A \in SO(3,\mathbb R)\setminus\{I\}$ , then is it true that there exist exactly two points in $$S^2:=\{(x,y,z)\in \mathbb R^3:x^2+y^2+z^2=1\}$$ which are fixed by $A$?
Or equivalently we have to show that $1$ is an eigen-value of $A$ and there are exactly two distinct eigenvectors corresponding to it. Now I can ... |
Let $K$ be a compact subset of $\Omega \subset \mathbb{R^m}$, $\Omega$ is open and nonempty and let $f \in C(\Omega)$ have support contained in $K$. For $\epsilon \gt 0$, let
$$f_{\epsilon}(x)=\frac{1}{C(\epsilon)}\int_{K} \phi_{y,\epsilon}(x) f(y) dy$$
where $$C(\epsilon)=\int_{\mathbb{R^m}}\phi_{y,\epsilon}(x) dy$$, ... |
From A First Course in Probability (9th Edition):
3.5 An urn contains 6 white and 9 black balls. If 4 balls are to be
randomly selected without replacement, what is the probability that
the first 2 selected are white and the last 2 black?
This method is straightforward and results in the correct answer (according... |
$x \dfrac{dy}{dx} = y , y (0) = 0, x \geq 0 .$
My Approach :
$\dfrac{dy}{y} = \dfrac{dx}{x},$ by variable separable method, we get
$lny = ln x +c $ and then raising e to both sides will get $ y=x+c.$ Then substituting the intial condtions, we get $c=-1$
so $y = x -1 $is the required solution.
But it's given that it h... |
I'm studying a paper on (biological) Neural Networks, and the paper studies some stability properties of an $N$-sized network, and then, as $N$ tends to infinity, it is proven that a "mean-field result holds".
Now, I've already read the Wikipedia article about "mean-field Theory" and besides the concept of "averaging" ... |
I am given a pretty lengthy Boolean expression:
$$(¬A ∧ ¬B ∧ ¬C ∧ ¬D) ∨ (¬A ∧ B ∧ ¬C ∧ D) ∨ (A ∧ ¬B ∧ C ∧ ¬D) ∨ (A ∧ B ∧ C ∧ D)$$
which I am asked to simplify. The solution should be:
$$((¬D ∨ B) ∧ (D ∨ ¬B)) ∧ ((¬A ∨ C) ∧ (A ∨ ¬C))$$
But I'm not really sure how this reduction can be achieved. Here's my attempts:
Attemp... |
My question is a follow-up question to this one: How to show that a function is computable?
The original question was:
Is the following function $$g(x) = \begin{cases} 1 & \mbox{if } \phi_x(x) \downarrow \mbox{or } x \geq 1 \\ 0 & \mbox{otherwise } \end{cases}$$ computable?
The accepted answer was that this function ... |
Let $\sigma(x)$ be the sum of the divisors of $x$. A number $X$ is called perfect if $\sigma(X) = 2X$. Denote the abundancy index $\sigma(X)/X$ by $I(X)$.
If $N$ is odd and perfect, then $N$ can be written in the Eulerian form $N = {q^k}{n^2}$, where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1... |
How to prove that elementary matrices actually perform their intended row operations: multiplying by a constant, adding a multiple of one row to another, and switching two rows?
I've seen examples of their use, but I haven't seen a proof for an $n$ by $n$ matrix.
|
While looking at Arveson's "An invitation to C* algebras", at the moment of defining type I representations (p. 47), he says that a (non degenerate) representation is type I if every central subrepresentation contains a multiplicity free representation. One line further, he says this is equivalent to the fact that ever... |
For a velocity field
$$ \textbf G(x, y) = (3x^2 − 6y^2 + 1)\textbf i + (x + 4y − 12xy)\textbf j
$$ show that
the work done in moving a particle on the unit circle centred at (1, 0) taking an
anticlockwise direction, starting and ending at the rightmost point of the circle,
can be expressed as
$$W=\oint_{c} dx +\oint_{... |
I am trying to formally prove that limit of $2^n\sin(π/2^n)$ as $n$ approaches infinity is $π$. Generally I can tell limit of each term of product of $∞$ and $0$ respectively, but am little confused how to make use of these facts. Any suggestions?
|
What is the name of the inequality $|x|+|y| \geq |x+y|$?
I remember seeing this inequality and thinking it was the triangle inequality, but that only holds if $x,y,z$ are the side lengths of a triangle and this holds for all real numbers? If so is there a name for this inequality?
|
Let G be a graph of order n. Prove that if deg u + deg v ≥ n - 2 for every pair u, v of nonadjacent vertices of G, then G has at most two components.
|
Prove that for each $n \in N$,
$$f(n) =\frac{(n+1)^n}{n^{n+1}}$$
is monotonic.
First, I can tell that the function is decreasing. If I take $\frac{1}{n}$, the function looks like $\frac{1}{n}(1+\frac{1}{n})^n$. Can this help?
|
I am doing a self-study of algebraic topology, and am having some difficulties comprehending the idea of a non-abelian fundamental group on a path connected space. (See for example Hatcher Exercise 1.1.3 page 37 which implies that such groups do exist.)
If anyone could supply some intuition for such spaces by examples,... |
I came across some series many years ago for calculating PI.
I found that the first member of that series has been known for a long time
in the math world.
It is the set of series defined by:
$$ \pi^k=T\sum_{n=1}^\infty\frac1{n^k}$$
where $[T,k] = [6,2]$, for the well known one.
I have some value pairs for $k$ and $T$ ... |
I am having a little confusion with finding $E[X^{2}]$ that perhaps can be cleared up relatively easily. Here, $A, B, C$ are Poisson random variables with parameters $2.6, 3,$ and $3.4$, respectively. Let $X$ be the number of errors typed in a manuscript. Each of three typists $A,B,C$ are equally likely to type the man... |
In a nXn box(n>3),a number is written on every cell such that the sums along all rows and columns are the same.Not all numbers are the same.What is the biggest possible number of equal values in the box?
I found the correct answer,which is
n(n-1) by imagining a 4x4 box and in the box,all cells except the diagonal ones... |
Disclaimer: I'm an student in engineering, so please forgive me asking this stupid question.
Consider the following tensor coordinates given: The covariant coordinates of the metric tensor $\boldsymbol{g}$ and the contra variant coordinates of a 2-nd order tensor $\boldsymbol{A}$.
$
[g_{ij}]=\begin{bmatrix}
2 &... |
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