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H: Intersection of $|z_1 - x|=r$ and $|z_2 - y|=r$
Let $x,y \in \mathbb{R}^k$ ($k\geq 3$), $|x-y|=d>0$ and $r>0$.
Prove that if $2r>d$, then there are infinitely many $z\in \mathbb{R}^k$ such that $|z-x|=|z-y|=r$.
Here's what I have proved;
The existence of such $z$, and
$|z-x|=|z-y|=r$ iff $(z-(x+y)/2)\cdot(... |
H: Proving every element in $1+8\cdot \mathbb{Z}_{2}$ is a square
Let $\mathbb{Z}_{p}$ denote the ring of $p$- adic numbers.
How can I prove that every elements of $1+8\cdot \mathbb{Z}_{2}$ is a square.
I am not comfortable in working $\mathbb{Z}_{p}$'s. So a detailed solution would be of great help and I would lea... |
H: Best known bounds for Ramsey numbers
I realize a similar question has been asked before but what I want to know is a little different and is not answered by the link in the answer to that question. I am interested in knowing the best known general upper and lower bounds (non-asymptotic) for an arbitrary Ramsey numb... |
H: How to get upper-left, upper-right, lower-left and lower-right corners XY coordinates from a rectangle shape.
How can I get the get upper-left, upper-right, lower-left and lower-right corners given XY coordinates from a rectangle shape when I have the following data available to me?
positionX
positionY
width
heigh... |
H: Find monic grade 3 polynomial in $\mathbb Z_p[x]$ then factorize
Let $f = 15x^4+22x^3-x=0$ a polynomial in $\mathbb Z_p[x]$, find the first prime $p$ value that will make $f$ result in being grade 3 and monic. Then factorize $f$ in $\mathbb Z_3[x]$ as product of irreducible factors.
Find the $p$ value
In order for ... |
H: Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$
For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of $p$
So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able... |
H: Group homomorphism to the multiplicative subgroup of a field
Let $G$ be a finite group and let $\varphi:G \rightarrow F^{\times}$ be an homomorphism where $F$ is a field. $H$ is a subgroup of $G$ that contains $Ker(\varphi)$.
Prove that $H \lhd G$ and that $G/H$ is cyclic.
I have no idea how to prove this.
Thank yo... |
H: Where's the problem in this equation? Resulting in $4 = 5$
I just saw this equation and I can't find out where's the problem:
$$25-45 = 16-36$$
$$25- 2 \cdot 5 \cdot \frac{9}{2} = 16- 2\cdot4\cdot\frac{9}{2}$$
$$25 - 2\cdot 5\cdot \frac{9}{2} + \frac{81}{4} = 16 - 2\cdot 4 \cdot \frac{9}{2} + \frac{81}{4}$$
$$\left... |
H: Normal to the plane under the condition describes the cone
The plane $lx+my+nz=0$ moves in such a way that its intersection with the planes $ax+by+cz+d=0$ and $a'x + b'y + c'z+d'=0$ are perpendicular. Show that the normal to the plane through the origin describes in general, a cone of the second degree and find its... |
H: Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$.
Find the magnitude of the acute angle between the lines $2y+3x=4$ and $x+y=5$.
I have no idea how to start the above equation.
I try to draw the graph of $2y+3x=4$ and $x+y=5$ in the calculator but nothing show in the calculator.
The ... |
H: Understanding compact subsets of metric spaces
Please help me understand the following definition:
Let $(X,d)$ be a metric space, a subset $S \in X$ is called compact, if any infinite sequence $\{x_{n}\}_{n\in\Bbb N}\in S$ has a sub-sequence with a limit in S.
What does "if any infinite sequence" mean? Maybe: At... |
H: Suppose I have a function $y=x+1$, then is this function the same as $y=\frac{ x^2+x}{x } $?
Suppose I have a function
$y=x+1$
Then is this funcion the same as
$y=\frac{ x^2+x}{x } $ ?
The domain of x in the first function is $R$ and in the second function is $x\neq 0$.
AI: In the strict sense: they are not the sam... |
H: Invariant subspace under orthogonal matrix
Let $V=\mathbb{R}^{n}$ and $T\,:V\to V$ be defined by $Tv=Av$ where
$A\in M_{n}(\mathbb{R})$ is an orthogonal matrix.
My lecture wrote that if $W\subset V$ is a subspace of $V$ then
if $W$ is $A$ invariant then $W^{\perp}$ is also $A$ invariant.
What I do know is that if $... |
H: Seeking clarification of Lebesgue definition given for $\int _{0}^{1}x^{-a}dx$
I came across the example
"Show that $\int _{0}^{1}x^{-a}dx$ exists as a Lebesgue integral, and is equal to $1/(1-a)$, if $0 < a < 1$; but is infinite if $a\geq 1$.
The Lebesgue definition of the integral is $$\lim _{n\rightarrow \inft... |
H: What is the rotation axis and rotation angle of the composition of two rotation matrix in $\mathbb{R}^{3}$
I was told in class that a rotation matrix is defined by a rotation
angle and rotation axis, if we call the rotation axis $v$ and take
a basis of $\mathbb{R}^{3}=\{v\}\bigoplus\{v\}^{\perp}$ then the
matrix is... |
H: Do we have always $f(A \cap B) = f(A) \cap f(B)$?
Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$?
Thanks in advance
AI: Let $y \in f(A\cap B)$. So there is an $x \in A\cap B$, so $f(x) = y \in... |
H: Is there a simple method to prove that the square of the Sorgenfrey line is not normal?
Is there a simple method to prove that the square of the Sorgenfrey line is not normal?
The method in the book is a little complex.
Could someone help me?
AI: I always use Jones' lemma. It's a handy tool to show non-normality of... |
H: About the inverse of the Jacobian matrix
I have a doubt on Jacobian matrices. Consider the non linear transformation
$$
\left[
\begin{array}{c}
x\\
y\\
z
\end{array}\right]
= \mathbf{G}\left(
\left[
\begin{array}{c}
\hat{x}\\
\hat{y}\\
\hat{z}
\end{array}\right]
\right) =
\left[
\begin{array}{c}
\hat{x}g(\hat{z... |
H: Why does "separable" imply the "countable chain condition"?
Why does "separable" imply the "countable chain condition"?
Thanks for any help.
AI: Let $D$ be a dense subset of $X$ and let $\{ U_i : i \in I \}$ be a pairwise disjoint family of non-empty open sets indexed by $I$.
Define a map $f: I \rightarrow D$ by p... |
H: Solution of functional equation $f(x)=-f(x-a)$
I have a problem with finding solution. I suppose it will be something like $f(x) =G(x)\Re(e^{\frac{x\pi}{a}})$, where $\Re$ is real part of a complex number, $G(x)$ periodic function whith period $\frac{a}{n}$ and $n$ is a natural number. Can you help me? Thanks a lot... |
H: Inclusion of $\mathbb{L}^p$ spaces, reloaded
I have a follow-up from this question. It was proved that, if $X$ is a linear subspace of $\mathbb{L}^1 (\mathbb{R})$ such that:
$X$ is closed in $\mathbb{L}^1 (\mathbb{R})$;
$X \subset \bigcup_{p > 1} \mathbb{L}^p (\mathbb{R})$,
then $X \subset \mathbb{L}^p (\mathbb{R... |
H: Prove that there exists a natural number n for which $11\mid (2^{n} - 1)$
I'm thinking putting it into modulo form: there exists a natural number $n$ for which
$$2^{n}\equiv 1 \pmod {11}$$
but I don't know what to do next and I'm still confused how to figure out remainders when doing modulos, like $2^n\equiv \;?? \... |
H: Shortest distance between two shapes
This is the scenario of my problem. I have an image of two objects ( of arbitrary shape, not convex, not touching or crossing each other, kept a few space apart).
And I am supposed to find the shortest distance between these two shapes.
First thing that came to my mind was to us... |
H: Counting zero-digits between 1 and 1 million
I just remembered a problem I read years ago but never found an answer:
Find how many 0-digits exist in natural numbers between 1 and 1 million.
I am a programmer, so a quick brute-force would easily give me the answer, but I am more interested in a pen-and-paper solut... |
H: Does $\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $ converge/diverge?
How would you prove convergence/divergence of the following series?
$$\sum_{n\ge1} \sin (\pi \sqrt{n^2+1}) $$
I'm interested in more ways of proving convergence/divergence for this series. Thanks.
EDIT
I'm going to post the solution I've found here:
$$a... |
H: Contour Integration of Square Root with Laurent Series
Recently I've been working on branch cuts of square root functions, and come to problems like this:
Find a single-valued analytic branch $f$ of $\sqrt{z^2+z}$ on the set $\{ z \in \mathbb{C}: |z| >1 \}$ such that $f(2) = -\sqrt{6}$.
Evaluate the integral of ... |
H: Convention on non-negative singular values?
In the literature I have on disposal it is stated that singular values are non-negative values, and that, for a symmetric matrix $A$, the SVD and EVD coincide. This would mean that singular values of $A$ are the eigenvalues of $A$, but the eigenvalues of $A$ can be negati... |
H: How to get an absolute total from percentage and $\pm$ total?
So, for a site that a friend is developing, we need to work with a legacy plugin that counts votes as either positive or negative totals, but doesn't already provide an absolute total. We can, however, get the percentage out of the plugin. So my questi... |
H: Finite p-group with a cyclic frattini subgroup.
I have a question about the following theorem that I found in some research.
Is it possible that $E$ is the identity?
I just found this elaborated proof that might help.
AI: Yes, $E$ can be the identity, as anon points out. The dihedral group with $8$ elements and th... |
H: Is every algebraic curve birational to a planar curve
Let $X$ be an algebraic curve over an algebraically closed field $k$.
Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$?
I think I can prove this using Noether Normalization Lemma.
Is this correct? If yes, is it... |
H: How to solve the differential equation $dN/dt=aN-\mu t$ in terms of $t$, $a$, $\mu$ and $N(0)$
The number, $N$, of animals of a certain species at time $t$ years increases at a rate of $aN$ per year by births, but decreases at a rate of $\mu t$ per year by deaths, where $a$ and $\mu$ are positive constants.
Modelle... |
H: convergence of alternating series — weakening a hypothesis
A comment below this answer inspires this question.
Suppose $a_n\in\mathbb{R}$ for $n=1,2,3,\ldots$ and $|a_n|\to0$ as $n\to\infty$.
Further suppose the terms alternate in sign.
If moreover the sequence $\{|a_n|\}_{n=1}^\infty$ is decreasing, then $\display... |
H: $B(V,W)$ is complete if $W$ is
Let $B(V,W)$ be the space of bounded linear maps from $V$ to $W$. Then it is complete with respect to the operator norm. Can you tell me if my proof is correct? Thanks.
It's easy to verify that the operator defines a norm. Let $T_n$ be Cauchy in $B(V,W)$ with respect to $\|\cdot\|$. L... |
H: Subtraction by 1 when solving terms of sequences?
If Arithmetic and Geometric Sequences are simply Linear and Exponential functions respectively. Why then do we subtract the n variable by 1 when solving for certain terms in these sequences?
$$t_n=d(n-1)+a$$
$$t_n=a\cdot r^{n-1}$$
I've tried exploring this question ... |
H: How does homeomorphism map sets boundaries?
I'm at the end of my first course on general topology, but this topic was not well developed.
I can tell that an homeomorphism preserves the quality of a point to be a boundary point for a subset of a topological space. In particular, from space X to space Y, one only nee... |
H: Why is the inradius of any triangle at most half its circumradius?
Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer.
I know of two proofs of this fact.
Proof 1:
The radius of the nine-point circle is half the c... |
H: Problems regarding exponents
Write each of the following expressions in the form $ca^pb^q$ where $c$, $p$, $q$ are numbers:
$\dfrac{(2a^2)^3}{b}$ solved
$\sqrt{9ab^3}$ solved
$\dfrac{a(2/b)}{3/a}$ solved
$\dfrac{ab-a}{b^2-b}$ I tried and got to, $(ab-a)(b^2-b)^{-1}$. I know I'm supposed to bring $b^2-b$... |
H: $\kappa$-complete, $\lambda$-saturated ideal properties
Kunen, II.56. Having trouble proving the properties of the following:
The definition: $S(\kappa,\lambda,\mathbb{I})$ is the statement that $\kappa > \omega$ and $\mathbb{I}$ is a $\kappa$-complete ideal on $\kappa$ which contains each singleton and which is $\... |
H: Why operator systems contain an abundant of positive elements?
I am reading Sec33 of Conway's A Course in Operator Theory, according to his definition,
An operator system is a linear manifold $\mathcal{S}$ in a $C^*$-algebra such that $1\in\mathcal{S}$ and $\mathcal{S}=\mathcal{S}^*$.
Then he makes the comment t... |
H: Calculating the points of tangency for two circles given a picture
I have two circles with the same radius and I want to calculate the points of tangency.
For example, in the picture below, I want to calculate $(x_3, y_3)$ and $(x_4,y_4)$. I have the radius and the distance between the two circles as shown below:
... |
H: supremum norm and submultiplicativity
If $f$, $g \in C(S)$ where $S$ is a compact set in $\mathbb{R}^n$ then it is true that $$\lVert fg \rVert \leq \lVert f \rVert \lVert g \rVert$$
where the norm is the usual supremum norm.
Why is this not true if $S$ is not compact? What other conditions can $S$ satisfy so that... |
H: A little integration paradox
The following integral can be obtained using the online Wolfram integrator:
$$\int \frac{dx}{1+\cos^2 x} = \frac{\tan^{-1}(\frac{\tan x}{\sqrt{2}})}{\sqrt{2}}$$
Now assume we are performing this integration between $0$ and $2\pi$. Hence the result of the integration is zero.
On the oth... |
H: Application of Banach Separation theorem
Let $(\mathcal{H},\langle\cdot,\cdot\rangle)$ be a Hilbert Space, $U\subset \mathcal{H},U\not=\mathcal{H}$ be a closed subspace and $x\in\mathcal{H}\setminus U$. Prove that there exists $\phi\in\mathcal{H}^*$, such that\begin{align}\text{Re } \phi(x)<\inf_{u\in U}\text{Re }... |
H: Why doesn't $2\pi\int_{-1}^1\sqrt{1-x^2}dx$ give the surface area of a sphere of radius $1$?
Possible Duplicate:
Areas versus volumes of revolution
For fun I decided to derive the surface area of a sphere of radius $1$ from the formula for the perimeter of a circle. This integral is what I came up with:
$$2\pi\... |
H: A sequence with infinitely many radicals: $a_{n}=\sqrt{1+\sqrt{a+\sqrt{a^2+\cdots+\sqrt{a^n}}}}$
Consider the sequence $\{a_{n}\}$, with $n\ge1$ and $a>0$, defined as:
$$a_{n}=\sqrt{1+\sqrt{a+\sqrt{a^2+\cdots+\sqrt{a^n}}}}$$
I'm trying to prove here 2 things: a). the sequence is convergent; b). the sequence's limit... |
H: Calculate the total error percentage
Below it the table which contains Actual Count and Error Count for each ID.
USER_ID | Actual_Count | Error_Count
-----------+--------------------+---------------------
1345653 5 4
534140349 5 0
682527813 4 ... |
H: The analogous generalization for the commutativity of unions.
Let $\{I_j\}$ be a family of sets indexed by $J$ and let $$K=\bigcup_{j\in J}I_j$$
Then let $\{A_k\}$ be a family of sets indexed by $K$. The generalization of the associative law for unions is that
$$\bigcup_{k\in K}A_k=\bigcup_{j\in J}\left(\bigcup_{i\... |
H: Finite non-abelian $p$-group cannot split over its center
Show that a finite non-abelian $p$-group cannot split over its center.
I'd be happy for some clues.
AI: This definition is not very common, so it may be worth mentioning here:
Definition. Let $G$ be a group. A subgroup $K$ of $G$ is said to be co-central ... |
H: Solving $\cos^2 \theta + \cos \theta = 2$
Solve the following for $\theta$:
$\cos^2 \theta + \cos \theta = 2$ [Hint: There is only one solution.]
I started this out by changing $\cos^2\theta$ to $\dfrac{1+\cos(2\theta)}{2}+\cos\theta=2$
$1+\cos(2\theta)$ turns into $1+\cos^2\theta-\sin^2\theta$ which all beco... |
H: Quantified definition of the derivative
How do you quantify:
A function $f:\mathrm{dom}(f) \longrightarrow \mathrm{codom}(f)$ is differentiable at every $x$ contained in $\mathrm{dom}(f)$ if the limit
$$\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}$$
exists.
I have looked everywhere for a quantified definition of the a... |
H: How to define Mach Subsonic by the Mach Supersonic?
I read the book Mechanic of fluids shames and I find this relationship:
$$\frac{1+kM_1^2}{1+kM_2^2} =\frac{M_1}{M_2} \left ( \frac{1+\dfrac{(k-1)}{2}M_1^2}{1+\dfrac{(k-1)}{2}M_2^2} \right )^{0.5}$$
where $M_1$ is the Mach number of supersonic flow and $M_2$ is the... |
H: Verifying some trigonometric identities: $\frac{\csc\theta}{\cot\theta}-\frac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
Prove the following:
46. $\dfrac{\csc\theta}{\cot\theta}-\dfrac{\cot\theta}{\csc\theta}=\tan\theta\sin\theta$
I got as far as
Right Side: $\tan\theta\sin\theta$ to $\dfrac{\sin\theta}{\cos... |
H: Are the square and the maximum of distribution functions a distribution function?
Let $F$ and $G$ be (one dimensional) distribution functions. Decide which
of the following are distribution functions.
(a) $F^2$,
(b) $H$, where $H(t) = \max \{F(t),G(t)\}$.
Justify your answer.
I know the definition and propertie... |
H: Why use the derivative and not the symmetric derivative?
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric derivative will give the 'expected slope' for removable discont... |
H: Trigonometric Identities To Prove
$\tan\theta+\cot\theta=\dfrac{2}{\sin2\theta}$
Left Side:
$$\begin{align*}
\tan\theta+\cot\theta={\sin\theta\over\cos\theta}+{\cos\theta\over\sin\theta}={\sin^2\theta+\cos^2\theta\over\cos\theta\sin\theta}
= \dfrac{1}{1\sin\theta\cos\theta}
\end{align*}$$
Right Side:
$$\begin{al... |
H: Nontrivial subring with unity different from the whole ring?
Is there an example of a ring $R$ with unity and a nontrivial subring $J$, such that $1_J \ne 1_R$?
AI: If to you, "ring" means "ring with unity", then the definition of "subring" requires that the unity be the same as that of the larger ring, just like "... |
H: Trigonometric Identities: $\frac{\sin^2\theta}{1+\cos\theta}=1-\cos\theta$
$\dfrac{\sin^2\theta}{1+\cos\theta}=1-\cos\theta$
Right Side:
$1-\cos\theta$ either stays the same, or can be $1-\dfrac{1}{\sec\theta}$
Left Side:
$$\begin{align*}
&= \dfrac{\sin^2\theta}{1+\cos\theta}\\
&= \dfrac{1-\cos^2\theta}{1+\... |
H: $(\sin\theta+\cos\theta)^2=1+\sin2\theta$
49) $(\sin\theta+\cos\theta)^2=1+\sin2\theta$
Left Side:
\begin{align*}
(\sin\theta+\cos\theta)^2=\sin^2\theta+2c\cos\theta\sin\theta+cos^2\theta=1+2\cos\theta\sin\theta
\end{align*}
This can either be $1$ or I can power reduce it. I don't know.
Right Side:
\begin{a... |
H: Looking for a 'second course' in logic and set theory (forcing, large cardinals...)
I'm a recent graduate and will likely be out of the maths business for now - but there are a few things that I'd still really like to learn about - forcing and large cardinals being two of them.
My background is what one would proba... |
H: patterns for u-shaped graphs
When the equation is $Ax + By = C$, you know it will be a straight line. Is there a specific pattern to know (without plotting $x$ and $y$ yet) that the graph will be u-shaped? For example, the equation $y = x^2 - 9x – 12$ forms a u-shape. But how would you know that by looking at it? H... |
H: Minimizers of an expression with little O notation
Suppose that $f(x) = o(\sqrt{x})$ as $x\rightarrow\infty$ and let $x^*(a)$ denote the minimizer of $f(x) + a^{3/2}/x$, that is, the value of $x$ that minimizes said expression (assuming such a value exists). As $a\rightarrow\infty$, is it true that $x^*(a) = \omeg... |
H: Write each expression in the form $ca^pb^q$
Write each expression in the form $ca^pb^q$
c) $\dfrac{a\left(\frac{2}{b}\right)}{\frac{3}{a}}$
\begin{align*}
&= \frac{a\left(\frac{2}{b}\right)}{1}*\frac{\left(\frac{a}{3}\right)}{3}=\dfrac{a^2\left(\frac{2}{b}\right)}{3}=\frac{a^2}{1}*\frac{2}{b}*\frac{1}{3}=\frac{2a^2... |
H: Use equalities to derive important trigonometric functions
The trigonometric functions I must know:
(A) $\sin(-x)=-\sin x$
(B) $\cos(-x)=\cos x$
(C) $\cos(x+y)=\cos x\cos y-\sin y\sin x$
(D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$
$\sin^2x+\cos^2x=1$ (Use (C) and $\cos0=1$)
Can anyone help me just understand wha... |
H: A isometric map in metric space is surjective?
Possible Duplicate:
Isometries of $\mathbb{R}^n$
Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
AI: Here is an alternative to the proof linked to in the comments:
Suppose there existed $x \in X\setminus... |
H: Is it valid to consider average rate of change as a 3 variable function?
The average rate of change can be modeled as a function: $f:D^2 \to \mathbb{R}$ where $D$ is the domain of the primary function in consideration. It maps two variables - the ends of the interval- to the average rate of change of that particula... |
H: How many elements of a given order in a finite group
Let $G$ be a finite group and $n_k$ the number of elements of order $k$ in $G$. Show that $n_3$ is even and $o(G) - n_2$ is odd.
By Lagrange's Theorem, if $k$ does not divide $o(G)$, there are no elements of order $k$ in $G$. That implies
$$3\!\not|\;o(G)\Longri... |
H: The Frobenius endomorphism
Let $\mathbf F$ be a field of prime characteristic $p$. It is known that the Frobenius map $c\phi=c^p~~\forall c\in\mathbf F$ is an endomorphism of $\mathbf F$. Moreover, since the only ideals of $\mathbf F$ are $\{0\}$ and $\mathbf F$, we know that $\ker(\phi)=\{0\}$. This implies that $... |
H: Why does $\{1\cdot a\! \! \pmod p, 2\cdot a\! \! \pmod p,\ldots, (p-1)\cdot a\! \! \pmod p\}$ $= \{1, 2,\ldots, p-1\}$ when $a$ and $p$ are coprime?
Why is it that $\{1\cdot a \pmod p, 2\cdot a \pmod p,\ldots, (p-1)\cdot a \pmod p\} = \{1, 2,\ldots, p-1\}$ (albeit in a different order) when a and p are coprimes?
... |
H: Understanding of Derivatives via Limits
The book I'm reading introduces derivatives via limits. It gives the following example:
$f(x) = 12x-3x^3$
$f'(x)=\lim{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$
$=\lim_{\Delta x\rightarrow 0}\frac{12(x+\Delta x)-(x+\Delta x)^3-(12x-x^3)}{\Delta x}$
$=\lim_{\... |
H: Factorize $f$ as product of irreducible factors in $\mathbb Z_5$
Let $f = 3x^3+2x^2+2x+3$, factorize $f$ as product of irreducible factors in $\mathbb Z_5$.
First thing I've used the polynomial reminder theorem so to make the first factorization:
$$\begin{aligned} f = 3x^3+2x^2+2x+3 = (3x^2-x+3)(x+1)\end{aligned}$$... |
H: Computing operator norm exercise
I did the following exercise (given in my notes) can you tell me if my answer is correct? Thanks.
Exercise: Compute the operator norm of the continuous map $f \mapsto f$ when viewed:
(a) as a map $T: C^1([0, 1]) \to C([0, 1])$
(b) as a map $S: C([0, 1]) \to L^1_\mu([0, 1])$, where ... |
H: Find the acute angle of intersection of the curves $y=\cos x$ and $y=e^{-x}$ at the point $(0,1)$.
Find the acute angle of intersection of the curves $y=\cos x$ and $y=e^{-x}$ at the point $(0,1)$.
My method:
$y=\cos(x)$ $(0,1)$
$1=\cos(0)$
$=0$
$\frac{dy}{dx}=-\sin(x)$
$=-\sin(0)$
$=0$
I did the above st... |
H: Average run lengths for large numbers of trials: Intuition and proof
This article states that the formula for the average run lengths for large numbers of trials is:$$\frac{1}{1-Pr(event\ in\ one\ trial)}.$$
My questions
What is the intuition behind this formula?
Do you know an elementary proof for this result?
AI... |
H: If two sets have the same sum and xor are they necessarily the same?
Let $A = \{A_1, A_2, A_3, \cdots, A_n\}$ and $B = \{B_1, B_2, B_3,\cdots, B_n\}$.
where $A_i\in \mathbb{Z}$ and $B_i\in \mathbb{Z}$.
Say,
$$S_{1} = A_1 + A_2 + A_3 + \cdots + A_n = \sum_{i=1}^{n}{A_{i}} \\
S_{2} = B_1 + B_2 + B_3 + \cdots +... |
H: Lie algebra representation induced from homomorphism between spin group and SO(n,n)
Consider the spin group, we know it is a double cover with the map:
$\rho: Spin(n,n)\longrightarrow SO(n,n)$ s.t $\rho(x)(v)= xvx^{-1}$ where $v$ is an element of 2n dimensional vector space V and $x$ is an element of spin group (mu... |
H: Prove if $x^{2}-5xy-3$ is even, then $x+y$ is odd, where $x,y \in\mathbb{Z}$
I know for you this is easy but for me is not. I give my best shot but it's no use so I need someone to teach about all this stuff.
As I try to solve this one, I come up with this answer:
Suppose $x^2-5xy-3$ is even, then $x=2a + 1$ and $... |
H: An homeomorphism between $\mathbb{R}-\mathbb{Q}$ and $(\mathbb{R}-\mathbb{Q})\cap (0,1)$?
Are $\mathbb{R}-\mathbb{Q}$ and $(\mathbb{R}-\mathbb{Q})\cap (0,1)$ homeomorphic? My claim is they are and I'm trying using this function:$$f:(\mathbb{R}-\mathbb{Q})\cap (0,1) \rightarrow (\mathbb{R}-\mathbb{Q})\cap (0,\infty)... |
H: Counting the number of graphs on n vertices
I want to count the number of simple graphs on $n$ vertices where it is given that there is a fixed $K_k$ among those $n$ vertices. The way I am reasoning is this: the edges within the $n-k$ non-$K_k$ vertices can be filled out in $2^{\tbinom{n-k}{2}}$ ways, and for each ... |
H: Prove that: $ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$
I plan to prove the following integral inequality:
$$ \int_{0}^{1} \ln \sqrt{\frac{1+\cos x}{1-\sin x}}\le \ln 2$$
Since we have to deal with a convex function on this interval i thought of considering the area of the trapeze that can be form... |
H: Proof that operator is compact
Prove that the operator $T:\ell^1\rightarrow\ell^1$ which maps $x=(x_1,x_2,\dots)$ to $\left(x_1,\frac{x_2}{2},\frac{x_3}{3},\dots\right)$ is compact.
For an arbitrary sequence $x^{(N)}\in\ell^1$ one would have extract a convergent subsequence of $T x^{(N)}$. Maybe via the diagonal ... |
H: How to prove that $a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\geq6abc$
Help me prove $a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\geq6abc$
AI: $a^2(1+b^2)+b^2(1+c^2)+c^2(1+a^2)\geq6abc$
Since
$(a-bc)^2\geq 0$,
$(b-ac)^2\geq 0$,
$(c-ab)^2\geq 0$,
then
$a^2+b^2c^2\geq 2abc$,
$b^2+a^2c^2\geq 2abc$,
$c^2+a^2b^2\geq 2abc$.
By of collectted ... |
H: minimum requirement to be $f=g$ , $f$, $g$ are holomorphic
Given that $f,g:\mathbb{C}\rightarrow \mathbb{C}$ are holomorphic, $A=\{x\in\mathbb{R}:f(x)=g(x)\}$. The minimum requirement for $f=g$ is
$A$ is uncountable
$A$ has positive lebesgue measure
$A$ contains a nontrivial interval
$A=\mathbb{R}$
By identit... |
H: Automorphism of an Infinite cyclic group
$\newcommand{\Id}{\operatorname{Id}}$
$f$ is an automorphism of an infinite cyclic group $G$ then
1.$f^n\neq \Id_G$
2.$f^2=\Id_G$
3.$f=\Id_G$
if $f^n=\Id_G$ then every element of $G$ will have finite order but in an infinite cyclic group only identity element has finite ord... |
H: lim sup of sequence of continuous function from $[0,1]\rightarrow [0,1]$
$f_n:[0,1]\to [0,1]$ be a continuous function and let $f:[0,1]\to [0,1]$ be defined by $$f(x)=\operatorname{lim\;sup}\limits_{n\rightarrow\infty}\; f_n(x)$$ Then $f$ is
continuous and measurable
continuous, but need not be measurable
measu... |
H: upper bound for a sum of inverse index-distances
Consider the following sum:
$$\sum_{i=1}^{n} \sum_{\substack{j=1\\ j\neq i}}^{n} \frac{1}{\vert i-j \vert ^{1/2}} \leq const \; n^\alpha$$
What is a good (i.e. also easy to achieve) and the best possible estimate (i.e. $\alpha$ being as small as possible) for this su... |
H: Find a Nonsingular matrix in Jordan Form
Let $$ A=
\begin{pmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 & 1 & 1\\
\end {pmatrix}
$$
Find a nonsingular matrix $P$ such that $P^{-1}AP$ is in Jordan form.
The course I am taking uses the textbook "Matrices and Linear Transformation" by Cullen.
The example in the book explains how... |
H: Interpretation of $f(n) \in o(n)$
Suppose that some function $f(n)$ is in $o(n)$. Is it fomally correct to say that there exists an $N$ such that for all $n \ge N$ it holds that
$$f(n) \le \frac{c n}{g(n)}$$
where $c>0$ is a constant and $g(n)$ is a strictly increasing function of $n$ ?
My reasoning is that $f(n)... |
H: $\tbinom{2p}{p}-2$ is divisible by $p^3$
The problem is as follows:
Let $p>3$ be a prime. Show that $\tbinom{2p}{p}-2$ is divisible by $p^3$. The only thing I can think of is that $(2p)!-2(p!)^2$ is divisible by $p^2$ which doesn't help me much. Can someone point me in the right direction? Is there a combinatorial ... |
H: Simple Linear Algebra Problem
I am working on a question from the first chapter of a Linear Algebra textbook I'm reading.
Let $A=(1,1,-1)$, $B=(-3,2,-2)$, and $C=(2,2,-4)$. Prove that $\Delta ABC$ is a right-angled triangle.
I know that the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ must be $90... |
H: Why these conditions make this map open?
If $A \subset \mathbb{R}^n$ is an open set and $g: A \to \mathbb{R}^n$ is an injective continuously differentiable function such that $\forall x \in A, \, \det g'(x) \neq 0$, does $g(U)$ is open for each $U \subset A$ open? Why? This is about p. 67 of Spivak's Calculus on Ma... |
H: Derived Set of a given subset of Real Line.
Let $A = \{a +\pi b : a , b \in \mathbb{Z}\}$ is a subset of $\mathbb{R}$. What will be the derived set of it?
AI: $A$ is an additive subgroup of $\Bbb R$, which is not discrete since $\pi$ is irrational. A known result about additive sub-groups of $\Bbb R$ shows that $A$... |
H: Why does synthetic division work?
Synthethic division is commonly taught, but I have never actually had a proof/explanation shown to me.
Why does it work?
Work So Far
I related the "$x$" to powers to 10, and then proceeded to relate synthetic division to non-polynomial division, but couldn't seem to find the correl... |
H: $\int\frac{\sin\left(x\right)}{\cos\left(x\right)}\,\mathrm{d}x$ by substitution
I'm trying to solve the following integral:
$$\int\frac{\sin\left(x\right)}{\cos\left(x\right)}\,\mathrm{d}x$$
Using the substitution method with the substitution $u = \sin\left(x\right)$.
The exercise has two parts: the first one is u... |
H: Remembering exact sine cosine and tangent values?
There exists a common trick to remember exact sine cosine and tangent values. The trick is relatively long, so instead of reposting it, please refer to my answer on this page.
Although I have used this trick for a while, I've never understood why it works. I underst... |
H: If $f:D\to \mathbb{R}$ is continuous and exists $(x_n)\in D$ such as that $x_n\to a\notin D$ and $f(x_n)\to \ell$ then $\lim_{x\to a}f(x)=\ell$?
Assertion: If $f:X\setminus\left\{a\right\}\to \mathbb{R}$ is continuous and there exists a sequence $(x_n):\mathbb{N}\to X\setminus\left\{a\right\}$ such as that $x_n\to... |
H: Find a nonsingular matrix P given that A is similar to a Jordan matrix
Given ${\bf A}$ is similar to a Jordan matrix find a nonsingular matrix $\bf P$ such that ${\bf P}^{-1}{\bf AP}={\bf J}$
$$
{\bf A}=
\begin{bmatrix}
0 & 0 & 0\\
1 & 0 & 0\\
0 & 1 & 1\\
\end{bmatrix}
$$
I have worked out
$$
{\bf J}=
\begin{bmatr... |
H: Question about proof that multiplication in Banach algebra is continuous
Here's the proof in my notes:
Where does the last inequality come from? If I want to show that it's continuous at $((x,y)$ I can use the inverse triangle inequality to get
$$ (\|x^\prime\| + \|y\|)\varepsilon \leq (\|x\| + \|y \| + \varepsil... |
H: Why is $\log(b,n) = \lfloor \log_b(n) \rfloor$ primitive recursive?
I read in an introduction to primitive recursive function and Wikipedia that
$$\log(b,n) = \lfloor \log_b(n) \rfloor$$ is primitive recursive. But how can that be? Is there any easy proof (and therefore a definition of the function using only cons... |
H: Entropy of Order Statistic
Consider $n$ independent and identically distributed random variables $ \{X_i\}_{i=1,...n} $ with support on some interval $[a,b]$ and its $n$'th order statistic $\max_{i \in \{1,...n\}} X_i$ . The following "entropy-looking" measure of dispersion of the maximum is
$$ - \int_a^b F^n(x) \l... |
H: if $x^2 \bmod p = q$ and I know $p$ and $q$, how to get $x$?
if $x^2 \bmod p = q$ and I know $p$ and $q$, how to get $x$?
I'm aware this has to do with quadratic residues but I do not know how to actually solve it. $p$ is a prime of form $4k+3$
AI: Euler's theorem says that $\left(\frac{q}{p}\right) \equiv q^{\frac... |
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