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H: Pascal's Triangle Problem?
Guys I have been trying to solve this problem for a long time but cant seem to come up with anything. Problem is essentially a Pascal triangle but I cant figure out how to sum up a column for a specific time, t. Please help. Following is the problem:
Consider a bacterial cell constrained... |
H: In graph theory, what is the difference between a "trail" and a "path"?
I'm reading Combinatorics and Graph Theory, 2nd Ed., and am beginning to think the terms used in the book might be outdated. Check out the following passage:
If the vertices in a walk are distinct, then the walk is called a path. If the edges ... |
H: Probability of getting 499000–501000 heads if a fair coin is flipped $10^6$ times
A fair coin is flipped $10^6$ times.
What's the probability that the number of heads is at least 499000 and at most 501000?
I'm not sure how to even go about starting this. Does it involve the $Q$ function?
AI: By the central limit th... |
H: $\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(x,y)}{\partial y}$ implies $f(x,y)=g(x+y)$
Assume $f(x,y) \in C^{(1)}(\Bbb{R}^2)$,If $$\frac{\partial f(x,y)}{\partial x}=\frac{\partial f(x,y)}{\partial y}$$ for all $(x,y) \in \Bbb{R}^2$. Show that there exists a function $g(t)$,such that $f(x,y)=g(x+y)$.
AI: Y... |
H: Variaton of parameters
Find a particular solution to the differential equation
$-6 y'' + y' + y = t^2 - t - 5 e^{-4 t} $.
This is my work so far.
I know I should be setting things to match up with the right hand side of the differential but I am stuck. I feel that my guess of at^2 + bt + c (e^(-4t)) is inco... |
H: How to show that this process is a Markov chain?
This question is from DEGROOT's "Probability and Statistics".
Question:
Suppose that a coin is tossed repeatedly in such a way
that heads and tails are equally likely to appear on any
given toss and that all tosses are independent, with the
following exception... |
H: Convert Parametric form of plane convert to Cartesian, to determine if it is a subspace
I found a linear algebra problem asking to determine if different structures form subspaces.
In one of the examples, it gives a parametric equation of a plane.
It has one constant vector followed by the two parameters, something... |
H: Proving Multiplication by Positive Element
I need help proving this simple fact.
If $a,b,c \in \mathbb{Z}$, all $a,b,c>0$, and $a>b$, then $ca>b$.
How do I prove this?
AI: Observe $cb \geq b$ iff $cb - b \geq 0$ iff $b(c-1) \geq 0$.
This last inequality holds, since $b$ and $c-1$ are both nonnegative.
Similarly, we... |
H: Expected value calculation.
Let $E(h,k,i,j)$ , $i \leq h$,$j\leq k$ be the expected number of the common numbers of two independently chosen subsets of a set of $h$ elements and $k$ elements respectively, where the first and second subsets respectively contain $i$ and $j$ elements. How to calculate $E(h,k,i,j)$ for... |
H: Find two vector b such that Ab = b
I have a question which I am unable to solve.
I have this matrix:
$
A = \begin{bmatrix}
1 & -1 & 1 & -1\\
1 & 1 & 1 & 1\\
1 & 2 & 4 & 8\\
1 & 3 & 9 & 27
\end{bmatrix}
$
I have to find two vector $\bf{b}$ such that $A\bf{b}=\bf{b}$
I know that the vector $0$ is one of the answer b... |
H: Question about the essential supremum
Say $f: E \to \bar{\mathbb{R}} $ is measurable function. Define the Essential Supremun $esssup f$ as $ \inf \{z : f \leq z \; \text{a.e.} \} $
a.e. means almost everywhere which means everywhere except on sets of measure zero.
MY question is: While studying the concept I have m... |
H: Determining whether a subspace of the metric space of real sequences is separable
Let $$X=\left\{(a_n)_{n \in \mathbb N} \in \mathbb R^N : \exists n_0 \in \mathbb N\, \forall n\ge n_0 \big(a_n\le \sqrt{n}\big)\right\}$$ with the metric $$d\big((a_n)_{n \in \mathbb N},(b_n)_{n \in \mathbb N}\big)=\sup_{n \in \mathbb... |
H: Calculus remark I forgot.
Suppose we have a set $X$. Let $a = \inf X$. then for all $n$, $a + \frac{1}{n} \in X$.
maybe this is wrong anyway I am trying to show it. So, we know $a \leq x $ for all $x \in X$. By archimidean, can find $n$ such that $a + \frac{1}{n} \leq x$. Im stuck here. Any help would be greatly ... |
H: Big O estimate of simple while loop
Give a big-O estimate for the number of operations, where an operation is an addition or a multiplication, used in this segment of an algorithm (ignoring comparisons used to test the conditions in the while loop).
i := 1
t := 0
while i ≤ n
t := t + i
i := 2i
My attempt:
n ... |
H: how to construct a homeomorphism between $T$ and $S$?
$S=\{x^2+y^2\le 1\}$, $T=\{|x|+|y|\le 1\}$, could anyone tell me how to construct a homeomorphism between $T$ and $S$?
AI: Hint: We can consider both $S$ and $T$ in $\mathbb C$ as $S=\{z:|z|\leq 1\}$ and $T=\{z:z=a+ib,|a|+|b|\leq 1\}$, then define $f:T\rightarro... |
H: Is a space where only finite subsets are compact sets always discrete?
If in a topological space only finite subsets are compact sets, is it then the discrete topological space?
Thank you.
AI: A topological space is anticompact if all compact sets are finite. In this answer Stefan H pointed out that an uncountable... |
H: upper bound for a function f
Let $f(z)=z^3e^{z^2}-e^{3z^2}$.
For $R>0$, I need an upper bound for
$$\max_{z\in\bar{B(0,R)}}|f(z)|,$$
which will look nice if I take its natural logarithm, that is
$$\ln|\{\max_{z\in\bar{B(0,R)}}|f(z)|\}|.$$
Hope you can help. Thanks!
AI: Note that if $a$ and $b$ are positive numbers,... |
H: When is $\sum_{k=1}^{\infty}a(k) \sum_{k=1}^{\infty}b(k)\ge \sum _{k=1}^{\infty}a(k)b(k)?$
Given two convergent series $\sum_{k=1}^{\infty}a(k)$ and $\sum_{k=1}^{\infty}b(k)$ under what conditions is valid the inequality:
$$\sum_{k=1}^{\infty}a(k)\sum_{k=1}^{\infty}b(k)\ge\sum_{k=1}^{\infty}a(k)b(k)$$
Thanks.
AI: A... |
H: Integral over X as supremum of integrals over finite subsets of X.
I am trying to prove that
$$\int f d\mu = \sup \left\lbrace \int_E f d \mu, E \in S, \mu(E)< \infty \right\rbrace,$$
given that $\int f d\mu < \infty$. $f$ is a positive measurable function $f:X\rightarrow \mathbb{R}$. $S$ is $X$'s $\sigma$-algebra.... |
H: Trouble finding the Laurent Expansion .
I'm having trouble progressing through (in my experience) the tedious calculations required to obtain a Laurent Expansion of a complex function.
The problem arises in finding the series within the annulus. Thank you in advanced for your help.
AI: Hint for (a): where are the ... |
H: How to calculate total profit in a set amount of days with a 2% daily gain?
I want to know if there is a formula that would give me the total amount at the end of a 365 day period with a 2% daily gain. Adding 365 days up would take for ever.
AI: Yeah, it's called compound interest. Note that if you have some amount... |
H: How is $\frac{\big(\frac{3}{2}\big)^{99}-1}{\big(\frac{3}{2}\big)^{100}-1}\approx\frac{1}{\big(\frac{3}{2}\big)}$
I read somewhere that $$\frac{\big(\frac{3}{2}\big)^{99}-1}{\big(\frac{3}{2}\big)^{100}-1}\approx\frac{1}{\big(\frac{3}{2}\big)}$$I don't know how to have it. Please let me know how this is approximate... |
H: Help finding the interior of a set
Let $$S=\{(x,y,z) \in \mathbb{R}^3|\,\,\, 0<x<1,\,\, y^2+z^2 \le1 \}$$
I know this might seem very simple but I don't understand the geometric representation of it my 3D graphing is really rusty right now. My guess is that the interior would be $$int(S)=\{(x,y,z) \in \mathbb{R}^3|... |
H: Convergence to a delta distribution
Is it Okay to say that when I have a probability density $P(x;\mu,\sigma)$, with $\mu$ the first moment of the probability density and $\sigma$ the square root of the second central moment of the probability density that the following relation holds:
$\lim\limits_{\sigma\rightarr... |
H: Is this an accurate way to represent n! using Π?
I recently learned of the $\Pi$ symbol, and was wondering if the following is an accurate way to represent $n!$:
$\Pi_{i=0}^{n-1} n - i$
AI: Yes, apart from the missing parentheses: make it $$\prod_{i=0}^{n-1}(n-i)\;,$$ and you’ll be fine. As $i$ runs from $0$ up thr... |
H: How to get the transformation?
Let T be the transformation of the xy-plane that reflects each vector through the x-axis and then doubles the vector's length. If A is the 2*2 matrix such that
then A=. I don't know how to get this answer.Please help me, thank you.
AI: You remember the first week of linear algebra wh... |
H: 3 Intersecting lines point, triangulation.
I currently have 3 circles that intersect each other. At these points I create a line, although I am stumped at how I can find the point of which all 3 lines intersect?
I know the coordinate center of each circle as well as its radius and each of the lines start and end c... |
H: Are these two surfaces topological equivalents?
I believe they are not the same since after deforming both, we get a surface with 5 holes in in the first picture, and a picture with 4 holes in the second picture. What do you think?
AI: The first seems to be the connected sum of five tori ($S^1\times S^1$), while t... |
H: Can an infinite permutatation be decomposed into finite number of infinite cycles?
Let $\sigma \in Perm(\mathbb{N})$ the set of permutations on the naturals. Then can $\sigma$ be written as a finite composition of possibly infinite disjoint cycles?
AI: Not necessarily. Consider the permutation given by
$$\sigma(n)... |
H: For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$.
For each of the following functions, what do the first- and the second-order optimality conditions say about wether 0 is a minimum on $\mathbb{R}$.
$f_1(x)=x^2$
$f_2(x)... |
H: Negate the following sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$
Negate the sentence: $\forall x, |x−a| < δ \Rightarrow |f(x)−L| < \varepsilon$
For my negation I got:
$\exists x, |x-a| < δ \Rightarrow |f(x)-L| \geq \varepsilon$
Would that be correct?
AI: You are correct that $\lnot(\forall x... |
H: Filling 4l, 5l bottles from two 10l bottles
There are two bottles of 10litre each filled with water.
Now two persons having empty bottles of 4litre and 5litre want to take 2litres of water each from the previous 10litre bottles..
Now you have to pour the water without wasting or throwing it.
This is an question in... |
H: $K$ is a three digit number such that the ratio of the number to the sum of its digits is least.Find the difference between the hundreds and the tens…
$K$ is a three digit number such that the ratio of the number to the sum of its digits is least. What is the difference between the hundreds and the tens digits of ... |
H: A function satisfy $x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}=0$ in a convex domain implies it is a constant
Assume $D \subset \mathbf{R}^2$ is a convex domain which contain original point. $f \in C^{(1)}(D)$,if $$x\frac{\partial f(x,y)}{\partial x}+y\frac{\partial f(x,y)}{\partial y}... |
H: What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point?
'Im trying to solve the following problem:
What figure does one obtain from a Möbius band if one shrinks the boundary circle to a point?
I don't really quite understand the problem. What does it mean the boundary circle of... |
H: Sinusoid sum of cosine and sine
I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple:
Generally Fourier series works because a sinusoid can be recomposed from a linear combination of a cosine and sine. ... |
H: General solution to a recursive equation
What is the general solution of the following recursive equation?
$$N(t)=(1+f)\cdot\left(N(t-1)-N(t-T)+N(t-T-1)\right)$$
By "general solution" I mean an equation where $N(t)$ stands alone on the left hand-side and is expressed only in terms of f and T only.
This recursive eq... |
H: Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups
Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$.
A. Find integer $n$ such that $U(n)$ contains subgroup isomorphic to $Z_5\oplus Z_5$.
B. Show that ther... |
H: Calculating the instantaneous position of an object constant speed and varying direction
Given an initial position for an object that's moving at constant speed but with a direction that varies in a linear fashion over time, how can I calculate the instantaneous position of the object?
For example, if an object is ... |
H: Taking a power of a polynomial to make it symmetric
Suppose I have a non-symmetric multi-variable polynomial in $n$ variables $P(x_1,x_2,...,x_n)$. For example $P$ might be $x_1^2+x_2$ or $x_1-x_2$
Under what conditions will some power $m$ of $P$ (that is $P(x_1,x_2,...,x_n)^m$) be a symmetric polynomial in $x_1,x_... |
H: Fastest way to check if $x^y > y^x$?
What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that?
The issue is that $x$ and $y$ can be very large.
AI: If both $x$ and $y$ are positive then you can just check: $$ \frac{\log(x)}{x} \gt \frac{\log(y)}{y}$$
so if both $x$ and $y$ ar... |
H: What am I doing incorrectly; logarithms?
We have an increasing number of books on a bookshelf. Every year, 2 books are added and each book is twice as long as the previous book. At the beginning of 1935 the volume was 1 cm thick. We define the 'velocity of the front cover' as the thickness of that volume divided by... |
H: How to prove that $f$ is injective if $\exists h \in H: |f^{-1}(\{h\})| = 1$?
Let $G,H$ be groups and $f: G \to H$ a group homomorphism. I want to show that the following statements are equivalent:
$f$ is injective.
$\ker f = \{e_G\}$.
$\exists h \in H: |f^{-1}(\{h\})|=1$.
The standard solution suggests to show t... |
H: Find an efficient algorithm to calculate $\sin(x) $
Suggest an efficient algorithm to determine the value of the
function $ \sin(x) $ for $ x \in [-4\pi, 4\pi] $.
You can use only Taylor series and $ +, -, *, /$.
I know, that $$\sin(x)=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n + 1)!}$$
but I can't find an effici... |
H: Is $x^8+1$ irreducible in $\mathbb{R}[x]$
Question is to check if :
$x^8+1$ is irreducible over $\mathbb{R}[x]$.
even before this I tried to see $x^4+1$ and $x^2+1$.
for $x^2+1$, it does not have a root in $\mathbb{R}$ So, it is irreducible.
for $x^4+1$, checking for roots does not imply anything.
So I tried to s... |
H: Arithmetic Progressions containing integers close to a power of 2
Consider an arithmetic progression of the form $\{kq: k \in \mathbb{Z}\}$, where $q$ is an odd integer. Do such APs always contain a number of the form $2^n \pm 1$?
I was initially interested in the largest $K(q)$ for which $\{kq: 0 \leq k < K(q)\}$... |
H: If $U$ is open and $B\subset U$ is a closed ball, is there an open set $V\subset U$ such that $V\supset B$ and $V,\partial V$ are Jordan Measurable?
I'm studing Evans PDE book, especifically, the proof of theorem 6 here.
Let $U\subset\mathbb{R}^n$ be an open set and $f:U\to\mathbb{R}$ a locally integrable function.... |
H: Prove the straight line as a tangent to a parabola.
I was going through some past exam papers and I came across this problem and I'm bit puzzled on how to approach this, could someone please help me out?
Equation of the parabola $y^2-7x-8y+14=0$, prove that the straight line given by $7x+6y=13$ is a tangent and fi... |
H: Limit of sequences ($\lim x_n = a > 0 \Rightarrow \lim x_n ^{1/k} = a^{1/k}$)
I need to show that if $(x_n)_{n \in \mathbb{N}}$ is a sequence such that $\lim x_n = a>0$, then $\lim \sqrt[k]{x_n} = \sqrt[k]{a}$.
It was suggest to use the equality $(x-a) = (x^{1/k} - a^{1/k})\left(\sum_{i=0}^{k-1} x^{i/k} a^{-i + 1/... |
H: Writing a permutation as products of transpositions
If a can write a permutation $\sigma$ as a product like $\Delta \alpha \beta$, where $\Delta$ is a product of transpositions (in fact, anything) and $\alpha$ and $\beta$ are two disjoint transpositions, so the symbols moved by $\alpha$ and $\beta$ belong to the su... |
H: Where to take Real Analysis and Linear Algebra?
I am undergraduate in economics. As you may know, most prestigious departments in economics now require their aspirants to have taken Real Analysis (and Linear Algebra, too) before entering their programs.
I have time to take these courses but don't know whether any c... |
H: Formal proof $\binom{n}{k}$ is an integer
In mathematics one defines:
$\left(\begin{array}{c}n\\k\end{array}\right)=\displaystyle\frac{n!}{k!\cdot (n-k)!}$
This is the number of combinations of $k$ elements from a collection of $n$ elements.
I was wondering if it is possible to prove that the result is an integer i... |
H: Is any differentiable function $f : (0,1)\rightarrow [0,1]$ is uniformly continuous
Question is to check if :
any differentiable function $f : (0,1)\rightarrow [0,1]$ is uniformly continuous.
I know that any continuous function on compact subset of $\mathbb{R}$ is uniformly continuous.
As $(0,1)$ is not compact, w... |
H: Are complete intersection prime ideals of regular rings regular ideals?
Let $(R, \mathfrak{m})$ be a regular local ring and let $\mathfrak{p}$ be a prime ideal of $R$ which is a complete intersection, i.e. the minimal number of generators of $\mathfrak{p}$ equals its height $h$. Then by Macaulays theorem there is a... |
H: When f is absolutely integrable and contiunous prove that $\sqrt{f}$ is absolutely integrable.
If $f: (a ,b) \rightarrow [0,\infty)$ is continuous and absolutely integrable on (a,b), then prove that $\sqrt{f}$ is absolutely integrable on (a,b).
I have that $\sqrt{f}$ is locally integrable. I am working on $|\sqrt{f... |
H: Showing the limit of a flow must be an equilibrium point under certain restrictions.
I'm stumped on how to approach this one:
Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$
(at least) continuously differentiable.... |
H: marching band conductor
Let $f(x)$ be the unique polynomial that satisfies:
$f(n)=\sum_{i=1}^{n} i^{101}$, for all positive integers $n$.
The leading coefficient of $f(n)$ can be expressed as $\frac {a}{b}$, where $a$ and $b$ are positive co prime integers. What is the value of $a+b$?
AI: If $f_k(n)$ is the polynom... |
H: Prove the following statement...
So, how can I prove this limit: $$\lim_{x \to x_0} \frac{1}{(x-x_0)^2}=+\infty$$
What I've tried is to multiply both numerator and denominator with (x+x0), but I guess that's wrong, what can I do to solve this? Please help...
AI: What happens to the denominator as $x \to x_0$?
Sinc... |
H: How can these be the weights of the adjoint representation?
This is perhaps a stupid question. We consider $G =\text{SU}(3)$ and $\pi : G \to \textrm{GL}(\mathfrak{g})$ the adjoint representation that sends $g \in G$ to $Ad_g$ that acts on the Lie algebra $\mathfrak{g}$ be the following formula. For $X \in \mathfra... |
H: Matlab - finding gradient
Given the function:
$$v(x,y) = x + e^{-((x-1)^2 + (y-1)^2)}$$
I am supposed to calculate the gradient of this expression in Matlab for x defined in the interval -1:0.1:0.9 and y defined in the interval -2:0.1:1.9. My task is to find the absolute value of the gradient of this function, and... |
H: Question about the complement
If $E=\{(x,y):x\in\mathbb{R},y\in(-\infty,0)\}$, what is the complement of $E$ in $\mathbb{R}^2$?
Is $E^c=\phi\times[0,+\infty)$?Is this an empty set?
AI: I will try to give a bit more detail than DonAntonio's answer contains.
Recall the definition of complement in a given universe of... |
H: comparing bit lengths of binary numbers
Suppose I have two binary numbers x and y that have bit lengths of nx and ny which are unknown. I'm looking for a fast method of comparing their bit lengths without computing their bit lengths; I think I found one using bitwise OR, and a pair of left shifts, but I'm not sure:... |
H: What is the anti derivative of $ f(x) = \int\frac{1}{1+e^{-x}}dx $?
I know u-substitution and integration by parts, but I don't know how to solve this one.
AI: Use
$$\int\frac{f'(x)}{f(x)}dx=\log f(x)+C$$
and
$$\frac1{1+e^{-x}}=\frac{e^x}{e^x+1}$$ |
H: Are all prime numbers up to Mersenne Prime 48 known?
In January this year the biggest prime number so far has been found and it is a Mersenne Prime. $$2^{57885161}-1$$.
My question:
Are all prime numbers from 0 up to $2^{57885161}-1$ found, or not?
If you only look for prime numbers which are Mersenne Primes, you s... |
H: Equivalence of Definitions for $T_1$
A topological space $(X,T)$ is $T_1$ (or Fréchet) if and only if for every $x\in X$, $\{x\}$ is the intersection of all neighborhoods $N\in N_x$
I have to use the following definition:
$(X,T)$ is $T_1$ if for $x,y\in X$ exists a neighborhood $ U_x$ of $x$ not containing $y$ and... |
H: How does $7\log(8x) = 7\ln8x$?
I was working on some math homework with a program called scientific notebook. I was check that I was writing something correctly.
The original equation is $(\log(x^4)+\log(x^5))/\log(8x)=7$
I then converted it to $\log(x^{(4+5)})=7\log(8x)$
I was expecting to get $\log((8x)^7)$ when ... |
H: Recurrence $T(n) = T({2n\over5}) +n$ using Master Theorem
Solve the recurrence
$$T(n) = T\left({2n\over5}\right) +n$$
My attempt:
$a=1$,$\ b=\frac 52$, $f(n)=n$
For the most part I believe that is correct. Now I was wondering if my math is correct in this next step. $n^{\log_b a}$ if $a=1$ and $b=\frac 52$ then:
$... |
H: Superposition of Sine and Cosine functions
I was wondering this:
Let a and b be real numbers. Is it always possible to find real numbers c, d, and e such that $$a\sin(x)+b\cos(x)=c\cos\left(\frac{x+d}e\right)$$
Why is this the case?
Thanks.
AI: $$a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\left(\dfrac{a}{\sqrt{a^2+b^2}} \sin{... |
H: Integrate the following.
Given
$$\dfrac{dN}{N}=4\pi\left(\dfrac{m}{2\pi K_b T}\right)^{3/2}v^2e^{mv^2/2K_bT}dv $$
How to integrate this from $v_0$ to infinity.?
AI: You want
$$\int_{v_0}^{\infty} dv \, v^2 \, e^{- a v^2} $$
where $a = m/(2 K_b T)$. (I know this because I know this is the Maxwell-Boltzmann distrib... |
H: proof that convergence in mean implies convergence in probability
I'm attempting to understand a proof, but I am failing to see how a step is pulled off.
Claim: $\text{if } f_n \longrightarrow_{L_p} f$ then $f_n \longrightarrow_{P} f$
Proof: Let $\epsilon > 0$. Then
$P\left( \left\lbrace \omega: |f_n(\omega) - f(\o... |
H: Solving $\lim_{n \to \infty}{\frac{n^a}{\log\left(\left| \log(n^a)\right|\right)}}$
I haven't practiced limits for years, now I need them to solve an exercise and I don't know whether I have come up with the right solution.
$$\lim_{n \to \infty}{\frac{n^a}{\log\left(\left| \log(n^a)\right|\right)}}$$
where $a$ is a... |
H: Selecting 1 ball of each type from the box
There are n White balls and n Black balls in a box. You repeatedly withdraw 2 balls simultaneously till the box is empty. Find the probability such that every withdrawal consists of 1 White ball and 1 Black ball?
EDIT:-let's just find the probability for the first simulta... |
H: How to determine whether critical points (of the lagrangian function) are minima or maxima?
$f(x,y) = 2x+y$ subject to $g(x,y)=x^2+y^2-1=0$. The Lagrangian function is given by
$$
\mathcal{L}(x,y,\lambda)=2x+y+\lambda(x^2+y^2-1),
$$
with corresponding
$$
\nabla \mathcal{L}(x,y,\lambda)= \begin{bmatrix} 2 + 2\lamb... |
H: Prove that $u \circ f $ is plurisubharmonic on $\Omega_1$
I'm trying to show that the theorem in my book:
Let $f: \Omega_1 \to \Omega_2$ be a holomorphic map between open sub - sets $\Omega_1, \Omega_2$ of $\Bbb C^n$. If $u$ is plurisubharmonic on $\Omega_2$ ($u \in PSH(\Omega_2)$) then $u \circ f $ is plurisubhar... |
H: Inductive step in the induction: $\sum^{n}_{i=0} q^i = \frac {1-q^{n+1}}{1-q}\times2$
I am trying induction for the following formula:
$$\sum^{n}_{i=0} q^i = \frac {1-q^{n+1}}{1-q}\times2$$
I have done the initial step which gives me for $n=1$ for both sites $1+q$
In the inductive step I wrote:
$$(\sum^{n}_{i=0} q^... |
H: Are all values of $\sin(x)$ algebraic.
Can we prove that for all $x$ in $(0,2\pi)$ $\sin(x)$ is an algebraic number?
I have seen people express various values of $\sin(x)$ like $\sin(3)$ and $\sin(30)$ using radicals so I suspect that all values of $\sin(x)$ must be algebraic. Is that correct? Can we prove it?
AI: ... |
H: Prove that $\lim_{n\to\infty} a_k$ is nonnegative for a convergent sequence of nonnegative terms $a_k.$
Suppose we have a convergent sequence $(a_k)$ such that $a_k\ge 0 $ for all $k\ge 1.$
Show that $\lim_{n\to\infty} a_k\ge 0$.
I have to prove by contradiction. This is the first time I've dealt with a quest... |
H: Evaluate the limit of $(\sqrt{5-x}-2)/(\sqrt{2-x}-1)$ as $x\to 1$
Can you help me with it and explain the steps
$$\lim_{x \to 1} \frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}$$
I tried to multiply at conjugate expression but I failed.
AI: Hint: $$\frac{\sqrt{5-x}-2}{\sqrt{2-x}-1}=\frac{(\sqrt{5-x}-2)(\sqrt{5-x}+2)(\sqrt{2-x}+1... |
H: Where does log(x) / x take maximum value?
If the base of the logarithm is e, one can say log(x)/x takes maximum at e.
If the base of the logarithm is 10, one can say log(x)/x takes maximum at 10.
But log10(x)/x is nothing but (loge(e)/loge(10))/x.
The two functions are just a constant multiple (1/loge(10)) of each ... |
H: Check whether W is a T-invariant subspace of V
$V$ = $C([0,1])$
$T(f(t))$ =[ $\int\limits_0^1f(x)dx$]$t$
$W$ = {$f \in V $ : $f(t)$ = $at+b$ $for$ $some$ $a$ $and$ $b$}
I got $T(f(t))$ for $f(t)$ = $at+b$ is,
[ $\int\limits_0^1(ax+b)dx$]$t$ = $(a/2 + b)$$t$ ,which does not belong to $W$.
Hence $W$ is not a $T$-inva... |
H: $\sigma$-algebra generated by open sets coincides with $\sigma$-ring generated by open sets.
Under the topic of Metric spaces in my measure theory book I came across this definition that says:
"Denote by $B$ the $\sigma$-ring generated by the class of all the open sets of X. The sets of $B$ are called Borel-sets."
... |
H: Find $\lim_{x\to 2} \frac{x - \sqrt{3 x - 2}}{(x^2 - 4)}$
Can you help me with this limit?
What do I have to do ?
I tried to multiply by the conjugate expression but it didn't work
$$
\lim_{x\to 2} \frac{x - \sqrt{3 x - 2}}{(x^2 - 4)}.
$$
Thanks
AI: Hint: After multiplying by the conjugate note that $x^2-(3x-2)=x^2... |
H: Column Space of AA' is equal to column of A.
This is my question. How to show that the column space of matrix A
is just equal to the column space of AA'?.. A' represents the transpose of A.
I know that the column space of AA' is a subset of the column space of A
which is just trivial. But the other way around I sti... |
H: relation between Jordan form and Jordan decomposition
How to get the Additive Jordan Decomposition of a matrix from its Jordan Canonical Form?
I tried with few matrices and its Jordan Canonical Forms, but I could not understand. So please explain the relation.
Thanks in advance.
AI: Let $A = S J S^{-1}$ be a Jordan... |
H: show that a measure is complete
If $(X,\mathcal{M},\mu)$ is a measure space and $\mathcal{\overline{M}}:=\{E\cup F:E\in\mathcal{M}\text{ and }F\subset N\text{ for some }N\in \mathcal{N}\}$ is a completion of $\mathcal{M}$ with respect to $\mu$ where $\mathcal{N}:=\{N\in\mathcal{M}:\mu(N)=0\},$ then $\mathcal{\overl... |
H: derivative of an integral which has a bound with multiple variables?
I want to find the derivative with respect to x of:
$$\int_0^{{\frac{x}{\sqrt4t}}} {e}^{-s^2}\,\mathrm{d}s$$
where t and x are both independent variables.
I thought you should use the fundamental theorem of calculus. However, since the upper bound... |
H: Ways to prove the Zsigmondy's theorem
How to prove the Zsigmondy's Theorem with cyclotomic polynomials?
Is there a proof which does not rely on cyclotomic polynomials? How many ways are there?
AI: There are several proofs available for Zsigmondy's theorem: Zsigmondy (1892), Birkhoff and Vandiver (1904), Dickson (19... |
H: How to determine whether a line is at the clockwise side of another line?
This is my first post in math-overflow , I am trying to implement an algorithm where , i am given 2 lines , two lines have one point in common , i need to determine
if one line is at the clockwise side of another line ? My points are in 2D.
L... |
H: Can someone help me solve this limits question?
$$\begin{align}\lim x → ∞\end{align}$$
$$\begin{align}
f(x) = {\frac{2^{x+1}+{3^{x+1}}}{2^x + 3^x}} \\
\end{align}$$
I tried using L Hopitable but that gives the same expression. Also tried using substitution but I didn't get anywhere. Help would be appreciated.
AI: T... |
H: Best way to learn from textbooks that have the Theorem-Proof format?
I've always loved the theorem-proof format in textbooks, e.g. Hardy and Wright's Introduction to the theory of numbers. However, the problem is that I can't remember anything I read in this format because usually books of this kind don't have exer... |
H: Uniqueness of map by dot product
I know that for a map on a complex vector space we have that if $\langle Ax,x \rangle = 0$
then $A = 0$ via the standard polarization trick. But what is the case if we are talking about real vector spaces? Is this then also true?
AI: No, consider the rotation by $90^\circ$ in $\mat... |
H: Laplace Equation -> Boundary Conditions
Given $\nabla^2 \phi=0$,
With B.C.
$\Gamma_1 =-V$ (Left side of rectangle)
$\Gamma_2 = V$ (right side of rectangle)
$\Gamma_0 = 0$ (top and bottom sides of rectangle)
Can Separation of Variables be applied to this problem? Will the superposition principle have to be ... |
H: "$x$ times as many as" versus "$x$ times more than"?
I keep coming across such questions during my GRE preparation:
A is how many times of B versus A is how many times greater than B
A is what percentage of B versus A is what percentage greater than B
Some websites treat both phrasings in each pair as meaning the... |
H: Cauchy-Schwarz inequality in $\mathbb{R}^3$
Use the Cauchy-Schwarz inequality on the euclidean space $ \mathbb{R}^3 $ (usual inner product) to show that, given 3 strictly positive numbers $a_1, a_2, a_3$ we have
$$ (a_1 + a_2 + a_3) \left(\frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}\right) \geq 9 $$
My attempt ... |
H: domain of composition of continuous functions
Let $f$ and $g$ be continuous functions from $R^2$ to $R^2$.
$f$ and $g$ both have open domains. That is, $f$ and $g$ are both defined on open sets of $\mathbf{R}^2$.
Is it true that the domain of $f(g)$ is also open?
By domain of $f(g)$, I mean the set of all $x \in \m... |
H: Common direction between two vectors
I have two vectors with the same origin and I need to find the common direction between them, that is the vector perpendicular to the line that join them. For instance, referring to this image I need the vector that points perpendicularly from $A$ to the line joining $B$ and $D$... |
H: Solving system of equations by using simple iteration method
I have a problem:
$$\begin{cases}
\sin(x) + 2y = 2 \\
\cos(y - 1) + x = 0.7
\end{cases}
$$
with margin of error 0.00001
And I need to solve this by using Fixed-point iteration method.
Can someone help me with that? Algorithm for full solution might be the... |
H: common between family of lines
suppose that there is a question like this :
i just want to make sure that i have understand everything,maybe it is meant by this question that graph structure is common,i mean their graph is straight line right?because let us take several value of $m$
$1.m=1$
$f(x)=1+(x+3)=x+4$
... |
H: Showing a set is open in $\mathbb{R}^2$
Show that $\mathbb{R}^2 \setminus \{(0,0)\}$ is open in $\mathbb{R}^2$.
I'm not sure if this is obvious I can't give enough details. Every point in $U=D(v,||v||)$ has a $\epsilon$-neighbourhood contained in U such that $v=(a,b)\ne (0,0)$.
Can someone please help me on descri... |
H: Complex integral of exponential and power
Let $n$ be an integer. Compute $$\int_{|z|=1}e^zz^{-n}dz.$$
If I parametrize $z(t)=e^{it}$ for $t\in[0,2\pi]$, this becomes $$\int_0^{2\pi}e^{e^{it}}e^{-nit}ie^{it}dt = i\int e^{e^{it}-(n-1)it} dt$$ and this looks too complicated.
Also I thought about using Cauchy's form... |
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