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H: What Topology does a Straight Line in the Plane Inherit as a Subspace of $\mathbb{R_l} \times \mathbb{R}$ and of $\mathbb{R_l} \times \mathbb{R_l}$
Given a straight line in the plane, what topology does this straight line inherit as a subspace of $\mathbb{R_l} \times \mathbb{R}$ and as a subspace of $\mathbb{R_l} \... |
H: Convergent operatorial series
An exercise I was doing asks (among other things) for the values of $z\in\mathbb{C}$ for which the following (operatorial) series converges absolutely:
$$\sum_{n=0}^{\infty}z^nA^n$$
where $A$ is an operator in the Hilbert space $L^2(0,2\pi)$ such that
$$(Af)(x)=\frac{1}{\pi}\int_0^{2\p... |
H: Two sums for $\pi$
I was looking for a simple way to evaluate the integral $\int_0^\infty \frac{\sin x}{x}dx$ ( a belated look at this question). There are symmetries to be exploited, for one thing. So I had an idea, but the idea depends, optimistically, on two expressions for $\pi$ that I cannot prove, and my que... |
H: Simplifying a fraction
I'm in doubt on how to simplify $ \left (-\dfrac{1}{243} \right )^{-\frac{2}{3}}$.
I've started with $\left (-\dfrac{1}{9\sqrt{3}} \right )^{-\frac{2}{3}}$ but now I'm stuck because of this minus signal in the main fraction ?
Thanks in advance.
AI: Take it one step at a time, first getting ri... |
H: Linear algebra: projection
Consider $P_2$ together with the inner product $(p(x), q(x)) = p(0)q(0) + p(1)q(1) + p(2)q(2)$. Find the projection of $p(x)=x$ onto the subspace $W=\operatorname{span}\{-x+1,x^{2}+2\}$.
How do you solve this question? I don't get what it means to find the projection of $p(x)=x$ onto ... |
H: Proof of relative compactness
Is the set of functions $f$ in $C^1([0,1])$ equipped with $\lVert f \lVert = \lVert f\lVert_\infty+\lVert f'\lVert_\infty$ (Case 1) or $\lVert f\lVert=\lVert f \lVert_\infty$ (Case 2) satisfying $\lvert f(0.5) \lvert \leq 1$ and $\int_0^1 \lvert f'(x)\lvert^2 dx \leq1$ relatively compa... |
H: Precompactness of a set vs. compactness of some operator
I found a remark in Adams' book Sobolev spaces (Acad. Press) which I cannot understand completely. It is on page 34 and says:
We remark that Theorem 2.22 is just a setting, suitable for our purposes, of a well-known theorem stating that the operator-norm lim... |
H: Cartesian equation of a parametric curve
I am trying to find the cartesian equation of the parametric curve
$$x = 1-t^2, \qquad y = t-2, \qquad -2 \leq t \leq 4$$
I am not sure how to proceed but I think a good direction would be to get everything in terms of $x$, eliminating $t$. I get $t = \sqrt{x-1}$ so I put t... |
H: Inconjugate maximal subgroups of a soluble group
I am looking for a proof of the following proposition:
If $M_1,M_2$ are inconjugate maximal subgroups of the finite and soluble group $G$, then $M_1\cap M_2$ is maximal in at least one of $M_1$ or $M_2$.
I know there is a proof of this fact in Doerk and Hawkes' "Fi... |
H: Unique decimal representation using only 0s and 1s (base 10)
Say $x \in [0, 1)$ and $\displaystyle x = \sum_{i = 1}^\infty \frac{a_i}{10^i}$ for $a_i \in \{0, 1\}$. Is it true if $\displaystyle x = \sum_{i = 1}^\infty \frac{b_i}{10^i}$ for $b_i \in \{0, 1\}$ that $a_i = b_i$ for all $i$?
It seems to me that yes thi... |
H: On compact space with order topology
We are familar with that for the first uncountable cardinality $\omega_1$, the topological space $[0,\omega_1]$ is compact. I find the proof for the $\omega_1$, is also for every regular cardinality. So is there a result for every regular cardinality, i.e.,
For every regular car... |
H: Describing multivariable functions
So I am presented with the following question:
Describe and sketch the largest region in the $xy$-plane that corresponds to the domain of the function:
$$g(x,y) = \sqrt{4 - x^2 - y^2} \ln(x-y).$$
Now to be I can find different restrictions like $4 - x^2 - y^2 \geq 0$... but I'm ho... |
H: Parametric equations and line segments
I am not sure how to do this one at all, I can't even start it.
I am suppose to show that $$x = x_1+(x_2-x_1)t \\ y= y_1+(y_2-y_1)t$$where t is between zero and one describes the line segment that joins $P_1(x_1, y_1)$ and $P_2(x_2,y_2)$
I really have no clue what to do and ho... |
H: When is tangent line horizontal?
When is the tangent line of the following function horizontal?
$$y =\frac{\sin x}{e^x}$$
What steps or how can I draw a graph to figure this out or prove this?
AI: Apply the quotient rule to find
$$ y' = \frac{e^x \cos x - e^x \sin x}{e^{2x}} $$
We want $y'=0$, which means we want t... |
H: Computing $\lim_{h\rightarrow 0}\frac{\tan(-\frac{\pi}{4})+1}{h}$
How do I compute a tan limit with a fraction?
$$\lim_{h\rightarrow 0}\frac{\tan(-\frac{\pi}{4})+1}{h}$$
AI: Since $ \tan(- \pi /4 ) + 1 = 0$, we have
$$ \lim_{h \rightarrow 0} \frac{\tan \left ( \frac{- \pi}{4} \right ) + 1}{h} = \lim_{h \righta... |
H: Piece-Wise Discontinuity & Continuity
When is the following function continuous? How would i go about listing the removable
discontinuities and then redefine the function so that it is now continuous
in those places?
$$f(x)= \begin {cases} \sqrt{2x}+7&\text{if }x <2\\
0\
&\text{if }x = 2\\
3^x&\text{if }x > 2 \en... |
H: Topology - The arbitrary union axiom
So, the common answer to why we need the concept of topology is that we need it to talk about things like limits of infinite sequences and continuity. But, when we define the axioms of topology, we have an axiom which says that an arbitrary(countable and uncountable) union of op... |
H: Lebesgue integral (existence and finiteness) of $\sin(1/x^2)$
I think I am getting a little better at these MCT, DCT-type exercises. The issue is to show/prove the existence and finiteness (if they apply) to the following function:
$$f(x)=\sin \left(\dfrac1{x^2} \right)$$
Where applicable I want to show as rigorous... |
H: Derivative of matrix exponential away from the identity
Suppose $X$ and $Y$ are $n\times n$ matrices (assume they are symmetric if you want), is there a reasonable formula for
$$
Z=\frac{d}{dt}|_{t=0} e^{X+tY}?
$$
Obviously if we assume that $[X,Y]=0$ then there is a simple answer.
I assume if there is an reasonabl... |
H: Rate and Distance Question Calc
How do I find the rate at which the distance from the plane to the station is increasing when it is 4 mi away from the station.
A plane flying horizontally at an altitude of 3 mi and a speed of 460 mi/h passes directly over a radar station
AI: This is a related rates question. The ge... |
H: Solving Linear Systems with Singular Matrices
Good morning! For (say, homogenous) linear systems of the form
$$x_{n+1} = A x_n,$$
where $A$ is a nonsingular matrix, each initial value problem can be solved by the method of finding a general solution by means of eigenvalues of $A$. However, for singular matrices, th... |
H: How to determine if 2 points are on opposite sides of a line
How can I determine whether the 2 points $(a_x, a_y)$ and $(b_x, b_y)$ are on opposite sides of the line $(x_1,y_1)\to(x_2,y_2)$?
AI: Explicitly, they are on opposite sides iff
$$((y_1-y_2)(a_x-x_1)+(x_2-x_1)(a_y-y_1))((y_1-y_2)(b_x-x_1)+(x_2-x_1)(b_y-y_1... |
H: A fifth degree polynomial $P(x)$ with integral coefficients.
A fifth degree polynomial $P(x)$ with integral coefficients takes on values $0,1,2,3,4$ at $x=0,1,2,3,4$, respectively.
Which of the following is a possible value for $P(5)$?
A) $5$
B) $24$
C) $125$
D)None of the above
AI: Any polynomial of degree at m... |
H: If $f \circ g$ is invertible, is $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$?
If $f \circ g$ is invertible, is $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$?
If not can someone give me a counterexample?
AI: It is true if $f$ and $g$ are invertible. You can check this directly by using the definition of inverse function.
Bu... |
H: Solve functional equation $f(x_1 x_2) = g_1(x_1) g_2(x_2)$
Let $x_1$ and $x_2$ be real positive numbers. The problem is to find all possible triples of $f$,$g_1$,$g_2$ such that $f(x_1 x_2) = g_1(x_1) g_2(x_2)$.
I suspect that the only one solution is $f(x_1 x_2) = (x_1 x_2)^n$, $g_1(x_1) = x_1^n$, $g_1(x_2) = x_2^... |
H: Entropy expression optimization with Langrange multipliers
I have recently encountered variants of the following expression:
\begin{equation}
S = H(a,b,c,d)-H(a+b,c+d)
\end{equation}
where $H$ is the Shannon entropy function, that is $H(X)=\sum_{x\in X}-x\log x$. And restrictions:
\begin{eqnarray}
a + b &=& t\\
a +... |
H: Cardinality of Mappings on a 2-Sphere
I am wondering about the number of mappings from a point on a sphere to a neighboring point, and a not so neighboring point.
If I take a 2-sphere, and place it on some $x,y,z$-axis and fix those so that the center is at the origin, then I draw a point on the surface close, but ... |
H: Existence of $n^{th}$ root
Disclaimer : This is homework (Tagged)
I am trying to prove that Real numbers have nth roots. Most references do the following :
Let $a \in \mathbb{R}, n \in \mathbb{N}, n > 0$ Then we consider a set X = {$t | t^n< a$}. Then we prove that X is nonempty (since it contains 0) and that it i... |
H: Sum of Absolute Values
I was going over a question and I wanted your opinion(s) on it:
The product of two numbers is 6 and one of the numbers is 5 less than the other. What is the absolute value of the sum of the two numbers?
The numbers I got are 1 and -6 because $b(b+5)=6$.
Now the sum of absolute values woul... |
H: $\frac{(a^2+b^2)}{(1+ab)}$ must be a perfect square if it is an integer
Possible Duplicate:
Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer
I came across this problem, but couldn't solve it.
Let $a,b>0$ be two integers such that $(1+ab)\mid (a^2+b^2)$. Show that the integer $\frac{(a... |
H: Whats the sum of the length of all the sides of a triangle?
You are given triangles with integer sides and one angle fixed at 120 degrees. If the length of the longest side is 28 and product of the remaining to sides is 240, what is the sum of all sides of the triangle?
I have tried to solve it using the formula gi... |
H: Integration Example
How can i find the integration of this example
$$\int \frac{\sin x}{\sin x - \cos x } dx$$
I tried first add cos and then substracting cos but then what about $$\int \frac{\cos x}{\sin x - \cos x } dx\ ?$$
AI: Multiply and divide Numerator by 2 to get $2\sin(x)$ and write it as $(\sin(x)+\cos(x)... |
H: Example of limit
How can I find the limit of this example
$$ \lim_{x\to 0}\frac{\sin^{-1}x-2x}{\sin^{-1}x+2\sin(\frac{1}{2}\sin^{-1}x)[3-4\sin^2(\frac{1}{2}\sin^{-1}x)] } $$
I tried using L'hospital but it was very long. Is there any easy trick? please give me some hint.
AI: what I usually do in such cases: Since ... |
H: Must a function that 'preserves r.e.-ness' be computable itself?
Does there exist a non-recursive function (say, from naturals to naturals) such that the inverse of every r.e. set is r.e.?
If yes, how to construct one?
If no, how to prove that?
Any References?
AI: For any sequence $\mathcal{B} = \langle B_i : i \in... |
H: the meaning of "finite" in the finite covering theorem
A textbook I am using to learn analysis states (in reference to just the real line):
Every system of open intervals covering a closed interval contains a finite subsystem that covers the closed interval.
(the textbook is "Mathematical Analysis I" by V.A. Zori... |
H: Calc Rate of Change Question I think
I have an interesting calc question here but im not sure how to solve it. Can someone perhaps give me a helping hand or guide me through steps?
A balloon that takes images of the earth is shot up in the sky with
rockets from 0 ft off the ground is given by the height of the f... |
H: A number has 101 composite factors.
A number has 101 composite factors. How many prime factors at max A number could have ?
AI: Suppose $m = p_1^{a_1} ... p_n^{a_n}$ has evactly $101$ composite factors.
Then $101 + (1+n) = (a_1 + 1)(a_2+1) ... (a_n+1)$.
But the RHS is at least $2^n$ and it is easily checked that th... |
H: What is the shortest sequence that contains every permutation of $1..n$?
Possible Duplicate:
What is the shortest string that contains all permutations of an alphabet?
How can one create a list of numbers so that by taking $n$ consecutive elements from that list, it is possible get every permutation of numbers f... |
H: Fourier transformation of a test function
Let $f \in C_c^\infty(\mathbb{R}^n)$. Then, $$\hat{f}(\xi)= (2\pi)^{\frac{-n}{2}}\int_{\mathbb{R^n}}\exp(-ix\xi)f(x){d}x$$
can be (i) analytically continued to an entire function and (ii) for $r>0$ there holds: $\hat{f}(\cdot + ib)$ is uniformly rapidly decreasing $\forall\... |
H: $|G|=2^{k}m$ has a normal subgroup of index $2^k$
I am being involved with the following problem:
A group $G$ of order $2^{k}m$ wherein $m$ is odd has a cyclic subgroup of order $2^k$. Prove that $G$ has a normal subgroup of index $2^k$.
Honestly, I have one solution of this problem which is based on induction on... |
H: Stumped: Trapezoid problem with law of sin/cos ...
This one stumped me.
Thought you might enjoy..
What are the lengths of the diagonals?
It should be related to law of sine/cosine.
(The answers are in fraction)
AI: Hint: If you cut off a parallelogram from the left, you get a triangle with a base of 3 and a top an... |
H: Proof that $\mathbb N $ is finite
Obviously this is a false proof. It relies on Berry's paradox.
Assume that $\mathbb{N}$ is infinite. Since there are only finitely many words in the English language, there are only finitely many numbers which can be described unambiguously in less than 15 words. Let $n$ be the ... |
H: Reproducing Kernel of subspace of $L^2(0,1)$
Definition of the problem
Let $\mathcal{H}$ be a Hilbert space which consists of functions, defined on a set $S$. Let $k:S\times S \rightarrow \mathbb{K}$ is our reproducing kernel for $\mathcal{H}$.
Now, let $\mathcal{H}$ be the two-dimensional subspace of $L^2(0,1)$, c... |
H: Induction for sum of Poisson distributed random variables
Given the identically distributed and independant random variables $X_1,X_2,\ldots\sim\operatorname{Po}(\lambda)$ and $S_n=X_1+\ldots+X_n$ show with induction that
$$\Pr[S_n=k]=\frac{(n\lambda)^k}{k!}e^{-n\lambda}.$$
So far for
$n=1$ via the definition o... |
H: Biased alternating random walk on a lattice in 1D
Let's consider a random walk on a fixed lattice with step size 1 in 1 dimensions. In variation to the broadly discussed basic case, with a probability p the next step will be in the opposite direction of the previous step. The direction of the first step is also cho... |
H: Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?
As in the title, I am asking if there is a difference between allowing set-theoretic operations over arbitrarily many sets, and restricting to only countably many sets.
For... |
H: Direct limit of localizations of a ring at elements not in a prime ideal
For a prime ideal $P$ of a commutative ring $A$, consider the direct limit of the family of localizations $A_f$ indexed by the set $A \setminus P$ with partial order $\le$ such that $f \le g$ iff $V(f) \subseteq V(g)$. (We have for such $f \l... |
H: Harmonic function with bounded preimage
I recently saw a question here about bounded/unbounded preimages of a set under a harmonic function. The question asked did not seem to make sense as it was talking about harmonic functions on $\mathbb{R}$, but got me thinking about a corresponding problem for a harmonic func... |
H: How to get a part of a quaternion? e.g. get half of the rotation of a quaternion?
if I have a quaternion which describes an arbitrary rotation, how can I get for example only the half rotation or something like 30% of this rotation?
Thanks in advance!
AI: I believe what you're looking for are exponent and logarithm... |
H: Positivity of a functional.
While going through the Riez Representation theorem i am stuck with the use of positivity of linear functional.
My question is If $\tau$ is a linear functional from $C(X)\to \mathbb C$ , $f\in C(X)$ . Then i didn't understand how positivity is used to write the following :
$\tau(|f|^2)... |
H: Number of times two rescaled, 'fully' monotonic functions can cross
Consider two functions $f: [0,1) \rightarrow \mathbb{R}$ and $g: [0,1) \rightarrow \mathbb{R}$. Suppose $f(x) > g(x)$ for all $x \in [0,1)$. Suppose further that $f$ and $g$ are infinitely differentiable and all derivatives of both $f$ and $g$ are ... |
H: Finding a polynomial for this trigonometric equality
For all natural numbers, find the polynomials such that $$\cos(n\theta)=p_n(\tan(\theta))\cos^n(\theta)$$
It was suggested that taking $p_n(x)=\frac{1}{2}\{(1+ix)^n+(1-ix)^n\}$, but I don't know how?
AI: Let $q_n(x) = (1+ix)^n$.
Then $$q_n(\tan(\theta))\cos^n(\th... |
H: Exposition on hyperelliptic curves
Is there any literature that introduces hyperelliptic curves without the view towards cryptography? Even better is if there are any books that talk about them with (about) the same amount of detail as Silverman does for elliptic curves.
AI: For Genus $2$ there is a very nice book ... |
H: A system of equations with 5 variables: $a+b+c+d+e=0$, $a^3+b^3+c^3+d^3+e^3=0$, $a^5+b^5+c^5+d^5+e^5=10$
Find the real numbers $a, b, c, d, e$ in $[-2, 2]$ that simultaneously satisfy the following relations:
$$a+b+c+d+e=0$$ $$a^3+b^3+c^3+d^3+e^3=0$$ $$a^5+b^5+c^5+d^5+e^5=10$$
I suppose that the key is related to a... |
H: Differentiating Under the Integral Proof
There are many variations of "differentiating under the integral sign" theorem; here is one:
If $U$ is an open subset of $\mathbb{R}^n$ and $f:U \times [a,b] \rightarrow \mathbb{R}$ is continuous with continuous partial derivatives $\partial_1 f, \dots \partial_n f$ then the... |
H: Confusion regarding ML estimate
I was going through this article and they have this log likelihood given by
$$ LL = \sum_{i=1}^n A_i\log p_i + \sum_{i=1}^n A'_i\log(1-p_i).$$
Basically this is the loglikelihood of a logistic regression where pi is the output from the sigmoid function and Ai is the number of entries... |
H: Finding solutions to $(4x^2+1)(4y^2+1) = (4z^2+1)$
Consider the following equation with integral, nonzero $x,y,z$
$$(4x^2+1)(4y^2+1) = (4z^2+1)$$
What are some general strategies to find solutions to this Diophantine?
If it helps, this can also be rewritten as $z^2 = x^2(4y^2+1) + y^2$
I've already looked at On the... |
H: Finding $\frac{d^2 y}{dx^2}$
I am not sure how to do this but I need to find $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$
For $x = t^2 + 1, y= t^2+t$
And then show what t values gives a concave upward.
I know the simple formula to find $\frac{dy}{dx}$
I get $$\frac{dy}{dx} = \frac{y'}{x'}$$
$$\frac{dy}{dx} = \frac{2t+1... |
H: Submodule of free module over a p.i.d. is free even when the module is not finitely generated?
I have heard that any submodule of a free module over a p.i.d. is free.
I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so it does not apply to ... |
H: Convergence of $E(X_n)$ where $X_n=\frac nY 1_{\{Y>n\}}$ for any non-negative rv $Y$
Here is another self-study exercise that I am struggling mightily with:
$X_n=\frac nY 1_{\{Y>n\}}$ for any $Y$ such that $P(0\le Y<\infty)=1$
I am told that $X_n\to X$ a.s for some $X$, and am to show whether $E(X_n)\to E(X)$ as $... |
H: Ring automorphisms of $\mathbb{Z}$ quotients and their products
How many ring automorphisms are there of $\mathbb{Z}/n\mathbb{Z}$? Calling such rings $A_n$, how many ring automorphisms are there of $\prod_1^n A_{n_j}$?
AI: For $\mathbb{Z}/n\mathbb{Z}$, the only ring automorphism is the identity. For finite products... |
H: The adjugate of the adjugate
For any $n > 2$ and any $(n \times n)$-matrix $A$ over an arbitrary field, the adjugate of the adjugate of $A$ equals $\det(A)^{n - 2} A$.
Is there a unified way, without dividing into two cases – $A$ invertible and $A$ non-invertible – to prove this result ?
AI: Exactly the same univer... |
H: Question about infinite intersection
Is $\bigcap\limits_{n=1}^\infty (0, 1 + \frac{1}{n})$ equal to $(0, 1)$ or $(0,1]$? Help is appreciated.
AI: For all $n$, $1\in\left(0,1+\frac{1}{n}\right)$ and $\displaystyle\bigcap_{n=1}^{\infty}\left(0,1+\frac{1}{n}\right)=\left\{x,\,\forall n,\,x\in\left(0,1+\frac{1}{n}\ri... |
H: Show these simple inequalities and $(\log(1+x))^2\le x$ and that $(\log(1+x))^2\le x^2$
Show that $(\log(1+x))^2\le x$ and that $(\log(1+x))^2\le x^2$ for all $x\ge0$.
Both of these inequalities seem to be true, judging from plotting these functions with a grapher.
Can you help me get started? Can this be done by... |
H: If $a^m=b^m$ and $a^n=b^n$ for $(m,n)=1$, does $a=b$?
Possible Duplicate:
Prove that $a=b$, where $a$ and $b$ are elements of the integral domain $D$
Something I'm curious about, suppose $a,b$ are elements of an integral domain, such that $a^m=b^m$ and $a^n=b^n$ for $m$ and $n$ coprime positive integers. Does th... |
H: Convergence in the absence of Dominated Convergence Theorem, and uniform integrability
This question is extended from Resnick's exercise 5.13 in his book A Probability Path.
Let the probability space be the Lebesgue interval, that is, $(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and define $X_n:=\frac{n}{\log n}1_{(... |
H: Proving an implication by proving its dual
My textbook "Discrete and Combinatorial Mathematics, an Applied Introduction" by Ralph P. Grimaldi contains the following definition:
Let $s$ be a statement. If $s$ contains no logical connectives other than $\wedge$ and $\vee$, then the dual of $s$, denoted $s^d$, is the... |
H: Laplace transform
I want to find the Laplace transform of the following signal but I don't know what to do with the absolute value.
$$x(t)=e^{-|t|}\; u(t+1)$$
The first thing it came to my mind is to split in negative and positive sides and then find each one and add them. The problem is that I checked back to the ... |
H: Error Term in Passing from Summation to Integral
I encountered the following in a paper and do not understand how the error term is being bounded. In what follows, $n$ and $k$ are large integer constants.
$$ \sum_{i=0}^{n-1} \ln\left(1 - \frac{i}{k}\right) = \int_0^n
\ln\left(1 - \frac{x}{k}\right) dx \pm e(n,k) ... |
H: Prove the following $\tan(nA)$ expansion
I've figured out the approach. Writing the expansion of $(1 + x)^n$, then replacing $x$ with $i \tan (A)$.
Then separating real and imaginary part and $\tan(nA)$ will be equal to Im/Real.
But, after reaching $(1 + i \tan(A))^n$, I'm unable to convert it into De-Moivre's form... |
H: length of paths between two nodes in a directed acyclic graph
What might be a good way to calculate length of all paths between two nodes in a directed acyclic graph? I don't need the actual paths, just the length. Is there a combinatorial formula for that?
AI: Let $A$ be the adjacency matrix for your graph - so $A... |
H: An example of a $2\times2$ matrix $A$ without zero entries and with eigenvalues $\lambda_{1}=3,\lambda_{2}=-4$
I am trying to do an exercise that asks :
Find an example of a $2\times2$ matrix $A$ without zero entries and with eigenvalues $\lambda_{1}=3,\lambda_{2}=-4$
I am having trouble thinking of a way to fin... |
H: How to average ellipses?
I would like to find the average among a number of ellipses.
The ellipses all have the same center, and the same major axis length, but they have different eccentricities and different orientations, which is what makes the problem challenging.
Averaging two ellipses with the same eccentric... |
H: A non-arithmetical set?
A set is called arithmetical if it can be defined by a first-order formula in Peano arithmetic. I first encountered these sets when exploring the arithmetical hierarchy in the context of computability theory. However, I have not encountered any examples of sets that are not arithmetical.
I... |
H: Finding the smallest possible angles in a triangle
I am having difficulty solving this problem:
In the given figure (x+y) is an integer greater than 110. What is the smallest possible values of (w+z) (ans is 111)?
Any suggestion on how 111 is the answer?
AI: If you draw a line across the big triangle like the ... |
H: How much Set Theory before Topology?
I was reading Baby Rudin for Real Analysis and wanted to explore Topology a little deeper. I bought George Simmons' Introduction to Topology and Modern Analysis and found myself liking it. I am having some problems every once in a while with prerequisites.
How much Set theory do... |
H: Prove that $ \sqrt{\sum_{j=1}^{n}a_j^2} \le \sum_{j=1}^{n}|a_j|$, for all $a_i \in \mathbb{R}, i=\{1,2,...,n\}$
As the title says, prove that $ \sqrt{\sum_{j=1}^{n}a_j^2} \le \sum_{j=1}^{n}|a_j|$, for all $a_i \in \mathbb{R}, i=\{1,2,...,n\}$. I can solve for the case of n=2, but kind of stuck while proving the n-t... |
H: An explicit example of an invariant halfspace of the unilateral shift?
In a recent talk, A. Popov stated the following fact
The unilateral shift on $\ell^2$ has invariant halfspaces.
Halfspaces are closed subspaces whose dimension and codimension are both infinite.
He did not prove it. I know that unilateral shif... |
H: On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?
I don't know exactly how to describe it, but in a programmatic way I want to spiral outward from coordinates 0,0 to infinity (for practical purposes, though really I only need to go to about +/-100,000:+/-100,000)
So... |
H: Understanding Riemann sums
I am trying to understand Riemann sums. As far as I can understand, we have $\Delta x$ which is $\frac{b-a}{n}$ and $n$ is the number of subintervals I want to divide my function between $a$ and $b$. But I still don't understand how to calculate $C_i$.
For example I have a function in my ... |
H: Equalizing percentages
I am having difficulty with the following problem:
Mark was formerly paid a salary of $400 per week and 8% commission on his total sales.Later Mark was given a 20% salary cut and his commission rate was increased to 10%. What is the smallest amount of sales Mark has to make to earn as much ... |
H: Restricting Line Bundles to Hypersurfaces?
The Adjunction Formula is given to be
$K_V = (K_X \otimes [V])_V$
Where $K_V$ is the canonical class on $V\subset X$ and $K_X$ of $X$. And $[V]$ denotes the line bundle associated to $V$.
Now say, Instead of the canonical bundle on $V$, that I'm interested in some general... |
H: Find the modular inverse of $19\pmod{141}$
I'm doing a question that states to find the inverse of $19 \pmod {141}$.
So far this is what I have:
Since $\gcd(19,141) = 1$, an inverse exists to we can use the Euclidean algorithm to solve for it.
$$
141 = 19\cdot 7 + 8
$$
$$
19 = 8\cdot 2 + 3
$$
$$
8 = 3\cdot 2 + 2
$$... |
H: First order ordinary differential equations involving powers of the slope
Are there any general approaches to differential equations like
$$x-x\ y(x)+y'(x)\ (y'(x)+x\ y(x))=0,$$
or that equation specifically?
The problem seems to be the term $y'(x)^2$. Solving the equation for $y'(x)$ like a qudratic equation giv... |
H: Is it true that $\max\limits_D |f(x)|=\max\limits\{|\max\limits_D f(x)|, |\min\limits_D f(x)|\}$?
I came across an equality, which states that
If $D\subset\mathbb{R}^n, n\geq 2$ is compact, for each $ f\in C(D)$, we have the following equality
$$\max\limits_D |f(x)|=\max\limits\{|\max\limits_D f(x)|, |\min\limi... |
H: Flows of left-invariant vector fields on a Lie Group
Let $G$ be a connected Lie group with identity $e$ and let $g\in G$. Must there exist a left-invariant vector field $X_g = dL_g(v)$ (for some $v \in \mathfrak g$) such that the flow $\phi_t(e)$ of $X$ through $e$ passes through $g$? I believe this is equivalent t... |
H: Find ring morphism from ring R to ring S
I am trying to find a method to find ring morphism between two rings. I know the definition and all but I am unsure how to go about finding the ring morphisms between $\mathbb{Z}$ and $\mathbb{Z}_6$
If anyone could help me figure that out it would be great.
Thanks in advance... |
H: Counting the ordered pairs $(A, B)$, where $A$ and $B$ are subsets of $S$ and $A$ is a proper subset of $B$:
Let $S$ be a set of $n$ consecutive natural numbers. How to find the number of ordered pairs $(A, B)$, where $A$ and $B$ are subsets of $S$ and $A$ is a proper subset of $B$.
The answer in my book $ 3^n - 2^... |
H: How do I factor trinomials of the format $ax^2 + bxy + cy^2$?
Take, for example, the polynomial $15x^2 + 5xy - 12y^2$.
AI: What you have is a homogeneous polynomial of degree $2$ in $x$ and $y$ (homogeneous means that the total degree is the same in each monomial).
Divide through by $y^2$, and set $Z=\frac{x}{y}$. ... |
H: The most efficient way to tile a rectangle with squares?
I was writing up a description of the Euclidean algorithm for a set of course notes and was playing around with one geometric intuition for the algorithm that involves tiling an $m \times n$ rectangle with progressively smaller squares, as seen in this animat... |
H: Round a value in a set of number
Given an order set of some number, $S=\{1.3, 1.7, 1.9, 2.8\}$, I would like to know how can I mathematically define a function that round a value to the nearest number in the set $S$.
For example, if I give the value 1.72, I'll receive back the number 1.7.
AI: One possible candidate... |
H: How to convert a series to an equation?
Possible Duplicate:
Value of $\sum\limits_n x^n$
I don't know the technical language for what I'm asking, so the title might be a little misleading, but hopefully I can convey my purpose to you just as well without.
Essentially I'm thinking of this: the series $4^n + 4^{n-... |
H: Find Min and Abs Max with $\ln x$
I know how to do a quadratic version but how do I find the absolute minimum and absolute maximum values of f on the given interval.
$f(x) = x − \ln 2x$, $[\frac{1}{2}, 2]$
I keep getting the wrong answers
AI: The derivative is $1-\frac{2}{2x}$. It is hard to go wrong after that.... |
H: How to find the sum of this infinite series.
How to find the sum of the following series ?
Kindly guide me about the general term, then I can give a shot at summing it up.
$$1 - \frac{1}{4} + \frac{1}{6} -\frac{1}{9} +\frac{1}{11} - \frac{1}{14} + \cdots$$
AI: Your sum is $$S = \dfrac11 - \dfrac14 + \dfrac16 - \dfr... |
H: How do you do linearization Questions?
This is new to me, im trying to work on a linearization problem. I found an online problem
Find the linearization L(x) of the function at a.
$f(x) = x^4 + 2x^2,$ a = −1
What steps do i take to approach this solution?
AI: The linearization is $px+q$, where $y=px+q$ is the equat... |
H: Simplest method to find $5^{20}$ modulo $61$
What is the simplest method to go about finding the remainder of $5^{20}$ divided by $61$?
AI: $5^3 = 125 \equiv 3 \mod 61$, so $5^5 \equiv 25 \times 3 = 75 \equiv 14 \mod 61$,
$5^{10} \equiv 14^2 = 196 \equiv 13 \mod 61$, $5^{20} \equiv 13^2 = 169 \equiv 47 \mod 61$. |
H: Counting descending sequences of positive integers
The complete question I would like to answer is:
Given positive integers $k,n$, how many descending lists of non-negative integers $(x_1~x_2\ldots x_k)$ are there such that $\sum_{i=1}^k x_i = n$?
As a quick first insight, it's clear that the general problem will... |
H: To find the square root of a polynomial
My question is:
Find the value of $k$ such that $$4x^6 - 24x^5 + 20x^4 + 68x^3 -44x^2 - 40x + k$$ is a perfect square.
hey all i have made an edit. Sorry for the inconvenience.
Any help to solve this question would be greatly appreciated.
AI: If the goal is to find $k$ so th... |
H: How do I solve an overdetermined linear system of partial differential equations?
I have two partial differential equations that I want to solve (for $\ \sigma $) by finite differences:
$\ -\frac{\partial \sigma}{\partial x}(x,y,t) -p(x,y,t)\frac{\partial \sigma}{\partial t}(x,y,t) = p(x,y,t)$
$\ -\frac{\partial \s... |
H: weak convergence in $L^p$ plus convergence of norm implies strong convergence
Having trouble with this problem. Any ideas?
Let $\Omega$ be a measure space. Let $f_n$ be a sequence in $L^p(\Omega)$ with $1<p<\infty$ and let $f \in L^p(\Omega)$. Suppose that
$$f_n \rightharpoonup f \text{ weakly in } \sigma(L^p,L^{p'... |
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