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Is $x$ rational or irrational when $2^{x}+3^{x}=6$. How to show that?
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3 outcome answered question
Hi all,
I've been having trouble finding the Kelly Criterion bet size for an each-way bet.
The above link shows the solution to a problem with 3 distinct and mutually exclusive outcomes. In an each-way bet there are three distinct outcomes (Lose all bets, horse places but doesn't win, horse ... |
Is there way to show $$\int_{1}^{0}\frac{\ln(1-x)}{x}dx=\frac{\pi ^{2}}{6}$$ without using the Riemann zeta function?
|
Given a complete graph of $n$ nodes and a time interval $T = \{t_1, \ldots, t_k\}$. Suppose each edge has a small probability $p$ of being removed from the graph at each time step and that this $p$ is the same for each edge and for each time step $t_i$. I am interested in the number of disconnected nodes in the graph o... |
A series is formed in the following manner:
$A(1) = 1; $
$A(n) = f(m)$ numbers of $f(m)$ followed by $f(m)$ numbers of $0$;
$m$ is the number of digits in $A(n-1).$
Find $A(30)$.
Here $f(m)$ is the remainder when $m$ is divided by $9$.
I have no idea about this problem. Any hint will be helpful.
Note: This ... |
I'm working on a project that requires text analyzing. I'm currently trying to extract keywords from a single document. I referred to this thesis(Keyword Extraction from a Single Document using Word Co-occurrence Statistical Information by Matsuo) and tried to implement its algorithm myself. In the algorithm there's a ... |
So, I have a topology problem here. It goes like this. We have X, Y conected topological spaces and A, B proper subspaces of X and Y respectively. I have to show that $X \times Y - A \times B$ is connected.
I have considered using subspaces like $\{x\} \times Y \cup X \times \{y\}$ where $(x,y) \in X \times Y$, but I d... |
I have a range of percentage like 10% to 80%. Now, I want to divide this range non-linearly in 6 parts so the 2nd part will be greater than 1st.
So minimum value is 10% and it will scale up to 80% and the increase will be greater each time.
Please help me how I can solve this problem?
|
Let $M$ be a W*-algebra with unit $1$.
Q1) Does there exist a normal positive linear functional $\phi$ on $M$ whose support is 1?
Q2) We know that there is a unique central projection $z$ in $M^{**}$ by which every bounded linear functional is decomposed to a normal and singular part:
$$M^*=
\underbrace{zM^*}_{=M_*}\o... |
I have the parametric parabola: $$
y=f(x)=C(x-4)(x-5)+D
$$
where $D$ is fixed.
I want to find for which value of $C$ the distance from the parabola to the point $(4,0)$ is exactly $\frac{1}{3}$ and the point $(4,0)$ lies at the left of the curve.
The staightforward way is to set up this equation: $$
\sqrt{(x-4)^2+(f(x)... |
First of all, a wedge in a vector space is a convex set which is invariant under multiplication by positive real numbers.
Suppose $A$ and $B$ are disjoint, nonempty, convex sets in a locally convex topological vector space $X$ such that $A$ is compact and $B$ is closed, then part (b) of Theorem 3.4 in Rudin's Functiona... |
For $x \in [-\pi,\pi]$ define $$f_{k}(x) = a_{k}\cdot \left(\frac{1+\cos(x)}{2}\right)^{k}$$ where $a_{k}$'s are chosen such that $$\frac{1}{2\pi} \cdot \int_{-\pi}^{\pi} f_{k}(x) \ dx =1 $$ Then show that $2a_{k} \leq \pi \cdot (k+1)$.
All I can see is that as $k \to \infty$, $f_{k}(x) \to 0$ for $x \neq 0$, but I am... |
I have got stuck trying to solve the following problem. Let $X=-zx \frac{\partial}{\partial x} -zy \frac{\partial}{\partial y} + (1-z^2) \frac{\partial}{\partial z}$ be a vector field in $\mathbb{R}^3$. I am asked to determine the expression of the restriction of $X$ to the sphere by means of the following chart $\phi:... |
sincerely, I'm stuck. Then, I have two questions:
if we take $V=\{v\in H^1(0,1) ; v(0)=0\}$ and $Q=\{ w_1,w_2\}$ is a lineary independent set where $w_1 = \frac{*}{\Vert *\Vert_{V\cap H^2(0,1)}}$ and $w_2 = \frac{**}{\Vert **\Vert_{H^1(0,1)}}$
My first question is:
Can we constract a basis in $V\cap H^2(0,1)$ and $H^1... |
Hi i have problem i hope that someone can make this for me more clear:
So i have matrix $A = \begin{bmatrix} -2 & 1 \\ -2 & 0 \\ \end{bmatrix}$
I have to calculate eigenvector as matrix $P$ from it. So form matrix i have:
$ \lambda^2 + 2 \lambda + 2 $ so there will be complex eigenvalues
$\lambda_{1} = -1 - i$
$\lambd... |
A bakery sells three kinds of pastries -pineapple,
choclate and black forest. On a particular day,
the bakery owner sold the following number of
pastries : $90$ pineapple , $120$ chocolate and
$150$ black forest. If none of the customers bought
more than $2$ pastries of each type, what is the
minimum nu... |
Prove that, if $X$ is a linear space and $C \subset X$ is a convex cone, then:$$Aff(C)=C-C =lin(C)$$.
For the first relation, I tried to prove by duble inclusion;let $x \in Aff(C)$; then $x=\sum\lambda_{i}c_{i}$, with $i\in \{1,..,n\}$, $\lambda_{i}\in \mathbb{R}$, with sum $1$; then, I dont know how to find $c_{1}$ an... |
I'm reading Conway's complex analysis book and I'm trying to solve the exercise 5 from page 74.
In this exercise the author asks for the radius of convergence and power series expansion of $\log z$ about $i$.
Can I use the proposition 2.5 on page 35?
When I apply this theorem I get $a_0=\frac{\pi}{2}i, a_1=\frac{1}{i}... |
I know the following:
$$\lim_{x\to 0^-} \frac{e^{1/x}}{x^2} =0$$
I cannot, however, see why. Is there a method that makes this result intuitively clear?
|
Say A is the set of all Real numbers on $[0,1]$ whose decimal expansion contains an infinite number of 8s. I am trying to prove the measurability of this set.
I realize that this is the set of numbers for which 8 is in the $\limsup$ of their decimal expansion. I also know that $\limsup$ of any set is measurable, as i... |
I hope someone can clarify my ideas about the constant implied volatility in the classical Black Scholes framework.
As well known, market practitioners quote the prices of vanilla call and put options in terms of implied volatilities. For inputs $K$, $S$, $r$, $T$ and the price of the option $V$, one can determine the ... |
Consider a function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ differentiable at $0$. Now, consider the sequence
$$
\sqrt{n}(\mu(\frac{h}{\sqrt{n}})-\mu(0))
$$
for $n \in \mathbb{N}$, $h \in \mathbb{R}$. Could you help me to show that
$$
\lim_{n\rightarrow \infty}\sqrt{n}(\mu(\frac{h}{\sqrt{n}})-\mu(0))=h\mu'(0)
$$
where $... |
I was playing around with the Golden Ratio $\Phi = \frac{1 + \sqrt 5}{2}$ on Wolfram Alpha and I noticed that if $F_n$ denotes the $n{th}$ Fibonacci number, then the polynomial $P_n(x) = x^n - F_n x - F_{n-1}$ seems always have a root at $\Phi$ (and hence is divisible by the minimal polynomial $f(x) = x^2 - x - 1$ of $... |
Thanks for reading. Would be lovely if somebody could help me out on this (but not just post the answer) but also how you got there. I'm a programmer and I've ran across this problem which I can't solve. Nobody taught me mathematical probability so I really only know the self-taught basics, please keep that in mind.
Wh... |
So, I have two continuous functions $f(x)$ and $g(x)$. $f,g : \mathbb{R} \longrightarrow \mathbb{R}$ and $f(x) < g(x)$ for all $x$ real.
I have to show that $\{(x,y)\in \mathbb{R} | f(x) \leq y \leq g(x)\} \cong \mathbb{R} \times [0,1]$.
I imagine I have to figure out a bijective, continuous map with continuous invers... |
I know that this seems a childish question but i have not been able to find a proper convention that cover all math operations. Obviously the basic is that exponential takes precedence over multiplication (and division) that takes precedence over sum (and difference). A particular case can be the sign "$-$" that indica... |
I don't know if that's the most accurate title.
I'm trying to prove that one property of trees implies another without using any of the other properties.
This is for homework. But I'm really just curious if the proof was this easy or if someone more clever can find glaring holes. I will cite any help I receive. I'm a C... |
So the question asks:
find the kernel of the linear transformation $T : \mathbb{R}^4 \to \mathbb{R}^3$ defined by $T(x) = Ax$ where $A$ is the matrix:
$$\begin{bmatrix}1 & 0 &1 & 0\\0 & 1 & 1 & 1\\1 & 1 & 1 & 1\end{bmatrix}$$
My solution is:
$$\left[\begin{array}{cccc|c}1 & 0 &1 & 0 &0\\0 & 1 & 1 & 1 &0\\1 & 1 & 1 & ... |
Could someone, please, help me with a inclusion?I am trying to prove that:
$$Aff(A+B)=Aff(A)+Aff(B)$$, where $A,B \subset X$, $X$ a linear space;
I proved that $Aff(A+B)\subset Aff(A)+Aff(B)$,but I am having troubles in proving the reverse one;
I took $x \in Aff(A)+Aff(B)$; so, $x=a+b$, with $a \in Aff(A)$, $b \in Aff(... |
I'm studying O'Neill's "Semi-Riemannian Geometry with applications to Relativity".
I know that the following theorems are related to the Big Bang, but I don't understand how.
Let $M$ be a semi-Riemannian manifold. Suppose $Ric(v,v)>0$ for every timelike $v$ tangent to $M$. Let $S$ be a spacelike Cauchy hypersurface wi... |
What sum would amount to 31,104 in three years at 20% p.a Compound interest, interest being compounded annually ? (Ans: 18,900)
Solution:
\begin{align*}
C I &= A – P \\[1ex]
P &= 31104 – P\left( 1 + \frac{20}{100}\right)^3 \\[1ex]
P &= 31104 – P\left(1 + \frac{1}{5}\right)^3 \\[1ex]
P + P\left(1 + \frac{1}{5}\right... |
I would like to find the form of solutions of $p*k-q*j=r,$ for $(p,q)=1$ for any fixed $r < pq$ and $k,j \in \mathbb{N}$. I tried to look at the divisibility of $p=cq+b.$ But I didn't have any success.
|
Looking at the proof of Bolzano–Weierstrass theorem, it found an interesting lemma (called the peak lemma here) :
Every sequence $(x_n)_{n\in \mathbb{N}}\in \mathbb{R}^\mathbb{N}$ has a monotone subsequence.
Looking at the French Wikipedia article about subsequence, it says that the proof of the peak lemma doesn't gi... |
Any hints on this problem?
I want to find a function $f(n)$ which is:
NOT $f(n) = O(log n)$
NOT $f(n) = \Omega(n)$
So it must hold that: $c_1 * log n < f(n) < c_2 * n$ and $c_1, c_2$ are some constants...
So I am looking for something which is smaller than $n$...can I say that $f(n) = 2 * n$?
|
Can an open set in $\mathbb{R}^n$ and a not open set in $\mathbb{R}^n$ be homeomorphic ?
I guess the answer is no, but I can't prove it.
|
$$\exists p\neq 0 : x\in D (f(x+p)=f(x))$$
where $D$ is the domain of $f(x)$.
$$f(x+p)=\frac{\sin (x+p)}{2+\cos (x+p)}=\frac{\sin p\cos x+\sin x\cos p}{2+\cos p\cos x-\sin x\sin p}$$
$$\frac{\sin x}{2+\cos x}=\frac{\sin p\cos x+\sin x\cos p}{2+\cos p\cos x-\sin x\sin p} /(2+\cos x)(2+\cos p\cos x-\sin x\sin p)$$
$$\sin... |
I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow
To be more specific, it's the diferentiation of (6.9) right at the top of page 220
The way I see it, shouldn't it have been
$$\frac{\partial f}{\partial (y+ \eta \alpha)} \frac{\partial (y+... |
For each $n\in\mathbb{N}$ let
$$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$
Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of continuous functions on $[0,1]$ equipped with the metric
$$d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|.$$
EDITTED:
My working:
For any g... |
Consider exercise 1.3.4 of Norris' Markov Chains. The question is as follows:
Let $\{X_n\}_{n\geq 0}$ be a Markov Chain with state space $S=\{0,1,2,\dots\}$. Suppose the transition probabilities are given by $$p_{01}=1,\,\,\,p_{i,i+1}+p_{i,i-1}=1, \,\,\,p_{i,i+1}=\dfrac{(i+1)^2}{i^2}p_{i,i-1}$$ for $i\geq1$. Show that... |
I have to solve following problem:
$$ u_t(t,x) = \Delta u(t,x) $$
$$ u(0,x) = f(x) $$
I've started:
$$ \frac{\delta}{\delta t} F(u(t,\xi))=F(u_t(t,\xi))=F(\Delta u(t,x))$$
and here I've stoped, because what I also know is:
$$ \frac{\delta^2u}{\delta x_k^2}=(i\xi_k)^2F(u) $$
Unfortunatelly here I've got $$ \Delta u(t,x)... |
I have the function $f:\mathbb{R}\rightarrow \mathbb{R}\:\:f\left(x\right)=x^2-3x$ and it asks me to prove continuity in point $\:x_o=0$ using the epsilon-delta definition.
I know that in order to do so I must find that for any $\epsilon >0$, there exists a $\delta >0$ so that for any $x\in \mathbb{R}$ with $\left|x-x_... |
Given $2$ points in 2-dimensional space $(x_s,y_s)$ and $(x_d,y_d)$, our task is to find whether $(x_d,y_d)$ can be reached from $(x_s,y_s)$ by making a sequence of zero or more operations.
From a given point $(x, y)$, the operations possible are:
a) Move to point (y, x)
b) Move to point (x, -y)
c) Move to point (x+y, ... |
I have a relation on the set $A = \{a,b,c,d\}$:
$$R_1=\{(d,c),(c,a),(b,d),(d,a),(a,a),(b,c),(b,a)\}$$
I need to determine whether this relation has the trichotomy property or not?
If by any chance you do not know about the trichotomy property then here is the definition:
Let $R$ be a relation on a set $A$. Then $R$ has... |
I have to evaluate the following two integrals that I would like to solve,
Sin[θ] 2 π BesselJ[0, k Sin[θ] (μ[j] - μ[i])] Exp[2 k^2 σ^2 Cos[θ]^2],
and
Sin[\[Theta]]^2 2 I \[Pi] BesselJ[1, k Sin[\[Theta]] (\[Mu][j] - \[Mu][i])] Exp[2 k^2 \[Sigma]^2 Cos[\[Theta]]^2]
w.r.t. $\theta$ between the interval [0, $\pi$]. The... |
Intuitively this proposition seems true, but I've been told that is not a trivial thing to prove.
Is there any simple proof (or counter-example) for the proposition:
Consider the closed ball of radius 1: $B_{1}[0] \subset \mathbb{R}^{n+1}$
Given a homeomorfism $h: B_{1}[0] \rightarrow B_{1}[0]$, we have that $h(S^n) ... |
Let $X$ be the set $(\mathbb{R}\backslash \mathbb{N}) \cup \{1\}$. Define a function $f:\mathbb{R} \rightarrow X$ by
$$
f(x) = \left\{ \begin{array}{ll}
x & \mbox{if $x \in \mathbb{R}\backslash \mathbb{N}$};\\
1 & \mbox{if $x \in \mathbb{N}$}.\end{array} \right.
$$
Further, define a topology $\tau$ o... |
Suppose we have the normal subgroups $H,J\subset G $ with the property $|G|=|H|\cdot |J|$ and $H\cap J={e}. $ $ $Prove that $H\times J\cong G$.
I don't really know how to approach this one. I thought about that $HJ=G$ and hence $HJ\cong H\times J$. But how do I construct the isomorphism?
|
Solve simultaneously:
$$ 12x^2-4xy+11y^2=64 $$
$$ 16x^2-9xy+11y^2=78$$
I understand that it can be solved using the quadratic formula by rearranging the equation in $ax^2+bx+c=0 $ form
$$ (12)x^2-(4y)x+(11y^2-64)=0$$
Can this be solved using any other method (substitution or elimination?).
|
I'm getting used to chordal graphs (and some subclasses of chordal graphs). As we know, the concept of a minimal separator is central to chordal graphs. So recall a set $S \subseteq V(G)$ disconnects $a$ from $b$ if every path of $G$ between $a$ and $b$ contains a vertex from $S$. A non-empty set $S \subseteq V(G)$ is ... |
It is given that a monkey types on a 26-letter keyboard with all the keys as lowercase English alphabets. Each letter is chosen independently and uniformly at random. If the monkey types 1,000,000 letters, what is the expected number of times the sequence "proof" appears?
Here is the suggested solution:
Let the random ... |
I have a doubt about weak topology in a Banach space.
Let $\mathcal{B}$ a infinite dimensional Banach space, I understood that the weak topology in $\mathcal{B}$, is the topology generated by $\Sigma = \lbrace \varphi^{-1}(U); \varphi \in \mathcal{B}^{*} \rbrace $, where $U \subset \mathbb{F} $ is open, it is, the co... |
True or False
Every finite group of odd order is isomorphic to a subgroup of $An$, the group of all even permutations.
The question was in entrance exam. I think there is counter example to this statement but i am not reaching that example. Can some one help?
|
How many DFA's exist with two states over the input alphabet $\{0,1\}$?
My attempt :
Input set is given. So, we have 3 parts of DFA which we can change:
Start state
Transition Function
Final state
Start state can be chosen as any one among 2 in 2 ways.
Transition function is from $Q \times Z$ to $Q$, where $Q$ is ... |
We know that $D_{rms}=\sqrt N$ where $N$ is the number of steps taken by the random walker.
Now,consider a situation where a random walker walks $2$ steps in positive direction in the first two steps.
So,the $D_{rms}=\sqrt2\approx1.41$.
But the actual distance is $2$ units.
So why are the two values different? I am re... |
Let $(\mathbb G,*)$ be a Carnot group. Thus, by definition, $\mathbb G$ is a connected and simply connected Lie group whose Lie algebra $\mathfrak g$ admits a stratification, that is
$$\mathfrak g=V_1\oplus\dots\oplus V_r$$
such that
$$[V_1,V_{i}]=V_{i+1}$$
for $1\leq i\leq r-1$ and every other commutator of lenght >... |
I have a set $A = \{a,b,c,d\}$ on which two relations are $R_1=\{(a,b),(a,d),(b,c),(c,a),(c,d),(d,b)\}$ and $R_2=\{(a,b),(b,c),(d,c),(a,d),(a,c)\}$.
What will $R_1\circ R_2$ be? $\circ$ is a the compose after mapping. I don't understand.
|
I believe what I did in this exercise is correct, but I'm wondering if there is a faster way to do this kind of computation. I'm practicing for an exam that requires me to be really fast solving problems.
Exercise.Consider the following inner product in $\mathbb{R}^{4}$
\begin{equation}
\langle(x_{1},x_{2},x_{3},x_{4})... |
Let $V$ be finite-dimensional vector space and let $U$, $W$ be subspaces of $V$.
Suppose that $V=U+W$.
Is there a standard argument for showing that an element belongs to $U$?
For example, if the sum was direct, if would suffice to show that a vector is not $0$ and is not in $W$ to conclude such a thing.
This may be a... |
Problem
Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$?
I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all $a\in\mathbb{R}$ we can say that the set $(a,a)$ is an open interval in $\mathbb{R}$.
But I can't devise a proof of this. Can a... |
As in the question I have to show that
$$\sum_{j=0}^n z^j=\frac{1-z^{n+1}}{1-z}$$
So if we suppose that the above is true then clearly
\begin{align}(1-z)(1+z+z^2+\dots+z^n)&=(1-z)+(z-z^2)+\dots+(z^{n-1}-z^n)+(z^n-z^{n+1})\\&=1-z^{n+1}\end{align}
as required. But what if I didn't know the right hand side of this equatio... |
WolframAlpha tells me that $I_0(x)K_1(x)+I_1(x)K_0(x)=\frac{1}{x}$ for any $x$, but I have no idea how to prove it.
Should I use the recurrence relations for Bessel functions? Or should I use the series expansion around 0?
Is this identity widely known? If so, I would like a reference. It greatly simplifies the expres... |
A Hopf algebra $H$ (over field $k$) is a bialgebra $(H,m,u,\Delta,\epsilon)$ (H, product, unit, coproduct, counit), with an antipode map, $S:H\to H$ such that $$\sum (Sh_{(1)})h_{(2)} = \epsilon(h) = \sum h_{(1)}(Sh_{(2)}), \forall h\in H$$
This antipode map doesn't make sense to me: $\epsilon:H\to k$, and $S:H\to H$, ... |
Applying the following expansions:
$\sqrt[]{\cos x}=1-\frac{x^2}{4}+O(x^4)$
$\sqrt[3]{\cos x}=1-\frac{x^2}{6}+O(x^4)$
$\sin^2 x=x^2+O(x^4)$
gives the correct result on the limit $L=\frac{-1}{12}$.
But when applying more orders of each function:
$\sqrt[]{\cos x}=1-\frac{x^2}{4}-\frac{x^4}{96}+O(x^5)$
$\sqrt[3]{\cos x}=1... |
Is $(-1)^{a+b} = (-1)^{a-b}$ true $\forall a,b \in \mathbb{Z}$ ?
My argument:
$$
(-1)^{a-b} = \frac{(-1)^a}{(-1)^b}
= \frac{(-1)^a}{(-1)^b}.\frac{(-1)^b}{(-1)^b}
= \frac{(-1)^{a+b}}{(-1)^{2b}}
= \frac{(-1)^{a+b}}{1}
= (-1)^{a+b}
$$
Any flaw?
|
If one assumes that the continuum hypothesis is false then there are subsets of the reals of intermediate cardinality, uncountable but smaller than the continuum. What can be said about the necessary
properties of such a set e.g. topological properties other than that it cannot be closed?
|
I would like to give other representation for zeta function using
fundemental arithmitic I have got this: $\zeta(s-1)=\sum_{n=1}^{\infty}\frac{\gcd(n,n)}{{\operatorname{lcm}(n,n)}^{s}}$ where $\gcd(n,n)$ is the greatest common divisor between $(n,n)$ and $\operatorname{lcm}$ is the least common multiple of $(n,n)$.
My... |
$ABCD$ is a quadrilateral. $P$, $Q$ and $R$ are the midpoints of $AB$, $BC$ and $CD$ respectively. If $PQ = 3$, $QR = 4$ and $PR = 5$; find the area of $ABCD$.
Since, $5^2 = 3^2+4^2$, So, $\angle PQR = 90^o$
I can't Find a way to solve this.
Note: This is a problem from BDMO $2010$ National.
|
Let $f: I \to \mathbb{R}$ continuous and $a\in \operatorname{int}(I)$. Suppose that there is $L\in\mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{ y_n-x_n}=L$$ for all sequences $(x_n)$ and $(y_n)$ in $I$ with $x_n < a < y_n$ and $\lim x_n =\lim y_n = a$. Prove that $f$ is differentiable in $a$ and $f'(a)=L$.
I have ... |
How to solve analytically the following system?
$$
\begin{align}
x_1 - x_2 &= 1\\
-1/2x_1 + x_2 -1/2x_3 &= 1\\
-1/2x_1 + x_3 -1/2x_4 &= 1\\
\vdots\\
-1/2x_1 + x_i -1/2x_{i+1} &= 1\\
\vdots\\
-1/2x_1 + x_{n-1} &= 1\\
x_n &= 0
\end{align}
$$
Is there a strategy for solving system with recursive nature like the one above?... |
Inverse Fourier Transform of:
$$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sinx}{x}} \right \} $$
by using convolution theorem.
Since Fourier Transform convolution turns into multiplication - same property holds also for inverse. So, I am thinking that:
$$\mathfrak{F}^{-1} \left \{ e^{-\frac{x^2}{2}}{\frac{sin... |
I don't know the mathematical term used to define a surface whose points can all be connected by staright lines contained in the surface vs cannot all be connected by straight lines contained in the surface.
My field of expertise is Computational Aerodynamics, and I'm trying to prove that a wing with a delta wing needs... |
I've been trying to evaluate$$\lim_{x\to1}{\frac{\sqrt{x^2+3}-2}{\sqrt{x^2+8}-3}}$$
I tried:
(a) Rationizing the numerator -> Error
(b) Rationizing the denominator -> Error
(c) Factoring out $x$ -> Error
Finally, I used L'Hospital's Rule and got the answer $3/2$, but that is not what I am supposed to do, for not ... |
I need help for the following proof of continuity:
Let $E=L_2([t_0,t_1],\mathbb R)$ be a Hilbert space of
square-integrable real-valued functions on $[t_0,t_1]$. Let
$K(t,\tau):[t_0,t_1]\times[t_0,t_1]\to\mathbb R$ be such that:
$$ \int_{t_0}^{t_1}\int_{t_0}^{t_1} |K(t,\tau)|^2dtd\tau<\infty $$
Define $A:E\to E$ b... |
I have just started learning about predicate logic and am having some trouble in figuring out how to actually read the formula as as a sentence.
¬(marriedTo(X , Y ) ⇒ marriedTo(Y , X ))
Would I read the above formula as;
if X is not married to Y, the Y is married to X
?
|
$$\lim _{n\to \infty }\left(\frac{2^{\left(4n+1\right)} \ln\left(2n^3+1\right)+n^5\cdot 10^n}{15^n+4^{\left(2n-1\right)} \ln\left(5n\right)}\right)$$
I tried like that:
$$\approx\lim _{x\to \infty }\left(\frac{2^{\left(4x+1\right)}}{4^{\left(2x-1\right)}}\right)=\lim _{x\to \infty }\left(\frac{16^x2}{\frac{16^x}{4}}\ri... |
I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: C_{i}(\mathbb R P^n) \to C_{i-1}(\mathbb R P^n)$ has degree $1+(-1)^i$. I looked this up in Hatcher but our lecture followe... |
Can anyone help me prove why
$$1-\left(\frac{x-10}{y}\right) \gt 1-\left(\frac{x}{y+10}\right)$$
when $y < x$.
I have no idea how to show this algebraically so i'd really like some guidance.
|
Let $D$ be the region delimited by
$$\partial D:
\begin{cases}
C_1: x^2 + y^2 = 5^2\\
C_2:(x-2)^2+y^2= 1\\
C_3:(x+2)^2+y^2 = 1\\
C_4: x^2+(y-2)^2= 1\\
C_5: x^2+(y+2)^2= 1
\end{cases}
$$
I've sketched this and the region looks like this (the inside of the big circle, intersection the outside of the small circles should... |
The inequality is
$$|z-1|+|z+1|\leq2$$
I used a triangle inequality to show that
Since triangle inequality states:
$$|z+w|\leq|z|+|w|$$
Then
$$|z-1+z+1|\leq|z-1|+|z+1|\leq2$$
So $$|2z|\leq2$$
From this point I expanded out
$$\sqrt{(2x)^2+(2y)^2}\leq2$$
Or $$4x^2+4y^2\leq4$$
So I end up with a disk centred at origin, ... |
How to prove that: Let $x$ and $y$ be real numbers such that $x \leq y + c$ for every $c > 0$. Prove that $x \leq y$. (Hint: Use the contrapositive.)
I am doing homework on Real Analysis and really stuck with this exercise. I hope you guys have some ideas?
|
I have to show the following theorem:
$p\vee \neg p \equiv ((p \vee q)\wedge \neg (\neg p \wedge (\neg q \vee \neg r)))\vee (\neg p \wedge \neg q) \vee (\neg p \wedge\neg r)$
I have proved
$((p \vee q)\wedge \neg (\neg p \wedge (\neg q \vee \neg r)))\vee (\neg p \wedge \neg q) \vee (\neg p \wedge\neg r) \implies p\vee ... |
Let's define the vectors $\mathbf{v}_1,\dots,\mathbf{v}_m \in \mathbb{R}^n$, with $m\leq n$, to be mutually conjugate with respect to matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ if
$$\mathbf{v}_{i}^T\mathbf{A}\mathbf{v}_j = 0, \qquad 1 \leq i,j \leq m, \qquad i\not=j.$$ Assume that $\mathbf{A}$ is nonsingular, and ... |
I have to show that sequences $\left(\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(0,1)$ and $\left(-\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(-1,0)$ decrease monotonically and converge to $0$.
I know that if $x\in(0,1)$ then the sequnce $(x^n)$ converges to $0$ but in this case I have the $(2k+1)!$
|
In the application of Laplace method (or steepest descent) it is often assumed that the dependence on the factor N, on which we are expanding the integral, is only in the argument of the exponential. What if we have an expression to expand for large $N$ of the type:
$\int_{-\infty}^{+\infty} f(N,t) e^{Ng(t) dt}$
where... |
Let $C(\mathbb{R})$ be a Banach space of continuous real-valued functions defined on $\mathbb{R}$, with supremum norm, and let $C_0(\mathbb R)$ be the subspace of functions vanishing at infinity. Is $C_0(\mathbb{R})$ a Banach space?
I try to see it using:
$f\in C_0(\mathbb{R})$ iff for any $\epsilon>0$ there exists $K... |
Spreading $15$ white balls and $12$ black balls into $3$ cells, each of which can contain any number of balls.
$(A.)$ Find the probability that in each cell there will be exactly $5$ white balls.
$(B.)$ Find the probability that in each cell there will be exactly $5$ white balls and exactly $4$ black balls.
$(C.)$ Find... |
How could I show if or not the following piece-wise defined function is continuous at the point $z=-i$?
$$f(z)=\left\{ \begin{matrix} \frac{z^2+2iz-1}{2z^2+iz+1}, & z \neq -i \\ 0, & z=-i \end{matrix} \right. $$
|
I know Pascal's Identity is ${n \choose k}={n-1 \choose k-1}{n-1 \choose k}$, but I am not sure how to set up and use the proof to show that $\sum_{k=0}^{r}C(n+k,k) = C(n+r+1,r)$.
Can anyone help me with this problem?
|
I got stuck in the following problem. Let $M$ be the manifold defined by the equation $x^2+y^2+z^4=1$ and $F: M \to S^2$ defined as $F(x,y,z)=(x,y,z^2)$. I have to calculate the integral $\int_M F^* \omega$ for an arbitrary $\omega$ of degree two in $S^2$.
I believe that the result must be zero. What leaded me to think... |
I came across this question from an older textbook with no answers and I'm a bit stuck. Currently, I have done the following;
let $z=r(\cos x+i \sin x)$
($z \in \mathbb{C}$)
therefore, we now have
$r^n(\cos nx + i \sin nx) = (r\cos x + i(r \sin x+1))^n$
From here its seems that I should equate the real and imaginary pa... |
The integral is defined, for all $x\in\mathbb{R}$ as follows:
$$I= \int_{0}^{1}\dfrac{\left(t-1\right)}{\ln t} t^x\mathrm{d}t.$$
When $I$ converges?
Let $t-1=u$, we have: $u\to 0$ when $t\to 1$.
$$\dfrac{\left(t-1\right)}{\ln t}=\dfrac{u}{\ln\left(u+1\right)}\sim_{0}1.$$
Then,
$$\dfrac{\left(t-1\right)}{\ln t} t^x=\dfr... |
Let $A$ be a local noetherian ring, $B$ and $C$ are finitely generated $A$-algebras and $M$ is a finitely generated $B$-module. Is the natural morphism $\mathrm{Hom}_B(M,B) \otimes_A C \to \mathrm{Hom}_B(M,B \otimes_A C)$ injective? If necessary, assume that $B$ is flat over $A$.
EDIT: Please note that the homomorphism... |
If you roll a die two times, what is the probability the sum of the upturned faces equals $7$?
I can answer this question if I consider the order of the rolled numbers relevant. However, when I disregard the order I get a wrong answer.
My reasoning: we roll the die two times and each roll there are $6$ possible outcome... |
Is there a name for categories with the following property:
The category has zero morphisms, coproducts, and for each family $(X_i)_{i \in I}$ of objects the natural map
$$\hom\bigl(Y,\coprod_{i \in I} X_i\bigr) \to \prod_{i \in I} \hom(Y,X_i),\quad f \mapsto (p_i \circ f)_{i \in I}$$
induced by the projections $p_i : ... |
I want to compute this primitive $$I=\int \frac{3x+2}{x^2+x+1}dx.$$
I split this integral into two part:
$$\int \frac{3x+2}{x^2+x+1}dx=\int \frac{2x+1}{x^2+x+1}dx+\int \frac{x+1}{x^2+x+1}dx,$$
For the first part:
$$\int \frac{2x+1}{x^2+x+1}dx= \ln(|x^2+x+1|)+c_1$$
For the second part: due a change of variable $u=x+1$, ... |
I am completely lost on how to achieve this. I have no idea where to start, nor do I know what to use to find to prove this problem. Can someone help me with this?
|
This question was asked in CSIR NET December 2015. I could not solve this question.Although I know the answer that CSIR posted in their answer key, which is 2. But I cannot understand how 2 is the correct option.
Please help to solve this question.
The statement: “The father of my son is the only child of your parents”... |
In sample variance we divide by n-1 and not n. I know a couple of arguments for this - one is that this is sort of a normalization to ensure that the expected value of sample variance is equal to population variance (I calculated this mathematically and it is sort of a good argument) and the other argument is an intuit... |
In the proof of $p$ | $\binom{p}{k}$ (p divides $\binom{p}{k}$) where $p$ is prime,
what is the simplest way to show that $${(p-1)! \over (k)!(p-k)!}$$ is an integer?
|
This is a question from an exercise in model theory.
Let T be a complete theory, $ \mathfrak{C} $ monster model of T (a $ \kappa $ saturated model of cardinality $ \kappa $ for some large $ \kappa $) and $ A\subseteq|\mathfrak{C}| $. The algebraic closure of A is the union of all finite definable (over A) subsets of $ ... |
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