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H: n-th Power of a self-adjoint linear operator
I am trying to do an exercise that goes like this
Let $H$ be an Hilbert space and $T:H\rightarrow H$ a bounded self-adjoint linear operator and $T\neq 0$ then $T^n\neq 0$.
So my idea was to do this by induction on $n$. Suppose that $T^2=0$ then we will have that $\lang... |
H: $X$ contractible implies reduced homology groups are trivial.
Let $X$ be a contractible space, i.e. the identity map $1_X$ is homotopic to a constant map.
I know the following two theorems in Hatcher:
(1) If $f,g: X \to Y$ are homotopic maps, then $f_* = g_*: H_n(X) \to H_n(Y)$
(2) If $f: X \to Y$ is a homotopy equ... |
H: Viewing a discrete category with a single object as the full subcategory of $\textbf{Set}$ generated by a singleton set.
Let us write $\mathbf{1}$ for the discrete category with a single
object $\star$;
$$1:\mathbf{1}\rightarrow\textbf{Set},\,\star\mapsto\{*\}$$ is the
functor which maps the unique object $\star$ ... |
H: Strategy to beat the casino with unlimited amount of money (Martingales)
Brzezniak and Zastawniak's book on stochastic processes shows that that there is no way to beat the casino by having a finite amount of money available:
Let $(X_1,X_2,\cdots)$ be independent random variables and let $(\alpha_1,\alpha_2, \cdots... |
H: Find points that are on two elliptic curves in $F_p$
I have some basic cryptography question but I don't know if my calculations are not overcomplicated and if there's some simpler solution. Following algorithm is explained here.
Problem
There are two elliptic curves in $F_5$ with equations:
a) $y^2 = x^3 + 2x + 1 ... |
H: Optimally reversing a list by swapping
Let $\sigma_1, \sigma_2, \dots, \sigma_{n-1} \in S_n$ be the "adjacent transpositions", so $\sigma_i = (i, i+1)$ is the permutation which swaps $i$ and $i+1$. Recall that an inversion in a permutation $\pi$ is a pair $(i, j)$ with $i < j$ and $\pi(i) > \pi(j)$. It isn't hard t... |
H: Identifiying a topological space given by a quotient
Let be $\mathcal{D}^{2}=\{z\in\mathbb{C}:|z|\leq 1\}$ and $\mathcal{D}^{2}(\frac{1}{2})=\{z\in\mathbb{C}:|z|\leq \frac{1}{2}\}$. I'm asked to identify who is the collapse $\mathcal{D}^{2}/\mathcal{D}^{2}(\frac{1}{2})$. My intuition is that this quotient space is ... |
H: Complementary of a set
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be function. If there exists only $x_0 \in \mathbb{R}$ such that $f(x_0)=0$ and I consider the set
$$A:=\{ x \in \mathbb{R}\; ; \; x<x_0 \; \text{and} \: f(s)<0,\; \forall \; s \in (x,x_0)\},$$
then, what is the $A$ complementary? That is, what i... |
H: Show that $\mathrm{span}(1 + x, 3 - 2x^2)$ is a subspace of $P_2(\mathbb{R})$.
I know that to show that a subset $S$ is a subspace of $V$:
Show $S \neq 0$,
If $\alpha \in \mathbb{R}$ and $x \in S$, then $\alpha x \in S$
If $x, y \in S$ then $x + y \in S$.
But I don't understand what to do anymore with algebrai... |
H: Models of extensions of ZFC, within $L_{\omega_1^{CK}}$?
Is it the case that every consistent recurively axiomatized extension of $\sf ZFC$ has a model in a level of the constructible universe below $L_{\omega_1^{CK}}$? If not, then what is the least consistency strength of such an extension?
AI: It depends on what... |
H: Differentiability of a function and its roots
Suppose we know that $g$ if thrice differentiable and that $g(x) = 1000$ has exactly 10 distinct solution. Then is it always true that $g'''(x)$ has atleast 7 roots? In general if a function $g$ is $n$ times differentiable, and the function has k distinct solutions, the... |
H: Topological spaces as a group under cartesian product
I was wondering if one can have a group structure on the set of equivalence classes of (homeomorphic) topological spaces, where the law of composition is directly derived from the Cartesian product. That is, $[A] \times [B] = [A \times B]$.
The identity axiom is... |
H: Arrangements of BANANAS where the A's are separated
How many arrangements of the word BANANAS are there where the $3$ A's are separated?
I know that once chosen the places for the three A's, there are $\dfrac{4!}{2!}=12$ possible arrangements for the rest of the letters (we divide by $2!$ because there are $2$ N'... |
H: constant morphisms from projective, integral schemes over k
Let $X$ be a projective, irreducible, reduced scheme over $k$ and $Y$ be an affine $k$-scheme, where $k$ is algebraically closed. Prove that every morphism $f : X → Y$ is constant.
I know that for a general scheme $X$ which is irreducible as well as redu... |
H: Converse to a proposition regarding minimal structures
Suppose $M$ is an infinite structure where the only parameter-free definable subsets is either finite or cofinite. Must $M$ be a minimal structure, that is one where all the definable subsets with parameters are finite or cofinite? And what if we strengthen the... |
H: Baby Rudin Theorem 2.23 — How do we know a limit point exists for the complement of an open set?
In Principles, Rudin states the following in his proof of Theorem 2.23 (that within a metric space, a set $E$ is open iff its complement is closed):
Suppose $E$ is open. Let $x$ be a limit point of $E^c \dots$
How do ... |
H: Differentiating a function that includes variance
I have to differentiate the following function and evaluate it at $x =1$ (so I search for $f'(1)$)
$f(x) = Var[ln(x*A + (1-x) * B)], \space \space \space x \in (0,1)$, A and B are two i.i.d. random variables.
Honestly, I have no idea what to do here exactly.
My firs... |
H: Inflection points vs critical points
I was reading what inflection and critical points are here. If we assume that a function is defined on an open interval. Then does it mean we can say that all inflection points of the function are critical points of the function?
AI: No.
Critical points of a function are where a... |
H: Pseudoinverse and SVD
For the SVD $$\textbf{A=U}\boldsymbol\Sigma \textbf{V}^{*}$$
Where $\textbf{U}$ and $\textbf{V}$ are unitary
By partitioning the matrix $\textbf{A}$, we have the following:
$$\textbf{A} = \left[\begin{array}{c|c}
\textbf{U}_{1} &\textbf{U}_{2}
\end{array}
... |
H: Dimension of union of two varieties
Suppose $X$ and $Y$ are two varieties. By varieties, I mean affine varieties or quasi-affine varieties or projective varieties or quasi-projective varieties. Suppose Krull dimension of $X$ is $n$ and Krull dimension of $Y$ is $m$ and, without lost of generality, assume $n\leq m$.... |
H: Is $C_0$ dense in $l^{\infty}$
Is $C_0$ dense in $l^{p}$ with $1\leq p\leq \infty$ where $C_0=\{ (x_n): x_n\rightarrow 0, x_n\in R\}$. Well I think that if $p<\infty$ is true because by definition if i take $y=(y_n)\in l^p$ then $\sum (y_n)^p <\infty$ so $(y_n)^p \rightarrow 0$ imply $y_n \rightarrow 0$ then I can ... |
H: For what values of $n\in\mathbf{N}$, $n^{4}-7n^{3}-2n^{2}+n+4$ is multiple of 9.
I am given a suggestion which is to use that if $a\equiv b \mod n$ then for all $m\in\mathbf{N}$ we have $a^m\equiv b^m\mod n$. But I don't know how to apply it and where to bigging this problem really. Any suggestions in how to attack... |
H: Question about nonempty assumption for left inverse iff injective and right inverse iff surjective
The following is a theorem from Elements of Set Theory by Enderton:
Assume that $F:A\to B$ and that $A$ is nonempty. Then the following
hold:
(a) There exists a function $G:B\to A$ such that $G\circ F$ is the
identit... |
H: Is $TREE(4)$ bigger than $TREE(3)$
I've recently heard about the enormous number $TREE(3)$ in a youtube video, and I was wondering if $TREE(4)$ would be bigger?
AI: Yes, it is enormously larger. People reference $TREE(3)$ because it is already huge, but the function is monotonically increasing. |
H: Why is regularization used in linear regression?
I already understand that the point of regularization is to penalize (drive down) higher-order parameters for a model thereby increasing its generality. Outside of polynomial regression, I do not understand why regularization would be needed for linear models such as... |
H: Sum of infinite series by considering Maclaurin Series for $e^x$
I want to find:
$ \sum_{n = 0}^{ \infty } \frac{1505n + 1506}{3^n(n+1)!}$
We have:
$e^\frac{x}{3} = \sum_{n = 0}^{ \infty } \frac{x^n}{3^n(n!)}$
which we could integrate on both sides to get:
$ \int e^\frac{x}{3} dx = \sum_{n = 0}^{ \infty } \frac{ x... |
H: Taylor series of $\sin(x^2)$
I am stuck on a problem for my calc 2 course. We are being asked to use Taylor series centered around x=0 (Maclaurin series) to approximate $\sin(x^2)$ and we are being asked to calculate the first five (non-zero) terms in the series and then integrate using our approximation. The issue... |
H: The existence of uncountable regular cardinals in $ZF$
If $ZF$ is consistent, then since there is a model of $ZF+\mathrm{cf}(\omega_{1})=\omega_{0}$, $ZF\not\vdash\mathrm{cf}(\omega_{1})=\omega_{1}$, and since $ZFC$ is consistent, $ZF\not\vdash\mathrm{cf}(\omega_{1})\not=\omega_{1}$.
Can it be proved in $ZF$ that t... |
H: Is a continuous image of a normal space normal?
Problem Let $f:X\rightarrow Y$ be closed continuous surjective map. Show that if $X$ is normal then So is $Y$.
What if we drop the 'closed' condition? I want a counter example. I know the proof of this theorem.
AI: Let $X$ be $\Bbb N$ with the discrete topology, let... |
H: Ice Cream Flavors Probability Problem(Need to check my answer)
A store sells 26 flavors or ice cream(A-Z). We choose six flavors at random(repeats allowed). Find the probability that there are 2 of flavor A and at least 2 of flavor B.
My approach:
Of the six we choose, 2 for sure need to be flavor A. Then of the fo... |
H: Is the interior of the union of $n$ closed balls equal to the union of the interiors of the $n$ closed balls?
I am reading "Calculus on Manifolds" by Michael Spivak.
I am solving problem 1-22 on p.10 now.
If the following equality holds, I can solve the problem.
Let $B_1, \dots, B_n$ be closed balls in $\mathbb{R}^... |
H: $z^{\frac{4}{3}} = -2$ ; How to know which complex roots to keep from this equation
So I recently came upon the following complex algebra problem:
$$
z^{\frac{4}{3}} = -2
$$
So, to solve it I have to find the z values that solve the following:
$$
z = (-2)^{\frac{3}{4}}
$$
To do this I express -2 in exponential for... |
H: Conditional probability given V=3
X and Y are independent exponential random variables with mean 1. If V = X+Y find the conditional density of X given V = 3
Therefore $$f(x) = e^{-x}, x>0$$ and $$f(y) = e^{-y},y>0$$
To get the pdf of V, I'm using the convolution formula $$fv(v) = \int_0^v fx(x)fy(v-x) dx = ve^{-v}$... |
H: Integrate $\int_0^2 \frac{\ln\left(1+x\right)}{x^2-x+1} \mathop{dx}$
Challenge problem $$\int_0^2 \frac{\ln\left(1+x\right)}{x^2-x+1} \mathop{dx}$$
First thought $u=1+x$, $$ \int_1^3 \frac{\ln{(u)}}{u^2-3u+3} \mathop{du}$$
Here complex analysis or what?
Tips please.
AI: $\mathcal{Hint:}$
Substitute $u=3/t$
By the w... |
H: Which of these sets are subspaces of $P_3$?
Which of the following sets is a subspace of $P_{3}$?
a. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = 0 \text{ and } a_2 = a_3 \}$
b. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = 1 \text{ and } a_2 = 2a_3 \}$
c. $\{ a_0 + a_1t + a_2t^2 + a_3t^3 \ | \ a_1 = a_2^2 \}$
d. $\... |
H: Explanation of part of a particular proof by induction that the harmonic series diverges
One proof by induction that a harmonic series diverges begins $\sum_{j=1}^{2^{n+1}}\frac{1}{j}>\frac{(n+1)+1}{2}=\frac{n+2}{2}$ so:
$$\begin{aligned}\sum_{j=1}^{2^{n+1}}\frac{1}{j}&=\sum_{j=1}^{2^{n}}\frac{1}{j}+\sum_{j=2^{n}+1... |
H: What is this event space?
A fair coin is tossed $10$ times (so that heads appears with probability $\frac{1}{2}$ at each toss).
Describe the appropriate probability space in detail for the two cases when
(a) the outcome of every toss is of interest
(b) only the total number of tails is of interest.
In the first cas... |
H: Percentage value higher than 100%
I have two values A= 3.8620E+00 B = 1.4396E+00
According to this post, to calculate how much A is higher than B in percentage we do this:
((A-B)/B)*100 = ((3.8620E+00 - 1.4396E+00)/1.4396E+00)*100 = 168.2690%
Does this mean the value A is 168.2690% higher than B?
Does this calcul... |
H: Why is $\varepsilon_1$ not smaller?
I'm thinking that $\varepsilon_1$ could in theory be $lim[\omega+1,\omega^{\omega+1},\omega^{\omega^{\omega+1}},...]. $
This, in my opinion, would be smaller than the normal definition of $\varepsilon_1$
AI: If $\alpha$ is properly smaller than $\varepsilon_0$, $\omega^\alpha$ is... |
H: Why $L^2\cap L^p $ is not dense in $L^{\infty}$?
Ein Euclidean space $\mathbb{R}^n$.
Why $L^2\cap L^p$ is not dense in $L^{\infty}$? I have that $L^2\cap L^p$ is dense in $L^p$ with $1\leq p<\infty.$
Indeed, for $g\in L^p$ with $1\leq p<\infty$, let $(g_j)_{j}$ with $g_j=\rho_j*g$ where $\rho_j$ is a mollifier fun... |
H: How many basic rotations do I need to make any rotation in $n$ dimensions?
Suppose that I want to make some rotation in $n$-dimensional space $\mathbb{R}^n$. I can construct a basic rotation $R_{ij}$ in some plane spanned by two basis vectors $e_i$ and $e_j$. Then I obviously can construct any rotation by multiplyi... |
H: Describe the triangle whose angles will be in the equivalence class of (60, 60, 60).
The relation here is the largest angle in common. At a first glance I think that it should be a equilateral triangle but now I think it should be as (60, 55,65) what do you guys think can we describe this tringle?
AI: So here we go... |
H: Diophantine equation : $6^m+2^n+2=x^2$
Find $m,n,x\in\mathbb{N}$ such that $6^m+2^n+2=x^2$.
My first approach is to show that for $m,n\geq2$, there exist no solution for $x$ by using modulo $4$.
Case $1$ : $m=1$, $x^2=2^n+8$.
As $n\geq1\implies2\mid RHS\implies2\mid x^2\implies4\mid x^2\implies4\mid LHS\implies ... |
H: A path to combinatorics: proving that there are at least 2 people who have same number of people between them
"An even number of persons are seated around a table. After a break, they are again seated around the same table, not necessarily in the same places. Prove that at least two persons have the same number of ... |
H: Proof for this integral inequality
I am trying to prove that for $p>1$, $f(x)$ be non-negative and non-increasing function, then
$$,
\left( \frac{1}{x}\int_{0}^{x}f(t)F^{p-1}(t)dt\right) -\frac{p-1}{p}\left(
\frac{1}{x}\int_{0}^{x}F^{p}(t)dt\right) \leq \frac{1}{p}F^{p}(x) \tag 1
$$
where $$F(x)=\frac{1}{x}\int_{0... |
H: Grothendieck group of coherent sheaves is not a ring?
My question is motivated by the fact that the Grothendieck group $K^0(X)$ of vector bundles on $X$ can be given a ring structure via the tensor product. But it seems to me that the Grothendieck group of coherent sheaves $K_0(X)$ has no such structure. Why?
Let $... |
H: How many equivalence classes will there be?
Consider the subset $T\subseteq \mathbb{Z}\times \mathbb{Z}\times \mathbb{Z}$ where the three numbers will be the corner angles (in degrees)
of a (real) triangle. For example $(30, 70, 80)\in T$ but
$(10, 30, 50) \not\in T$ (since $10 + 30 + 50 < 180$), and
$(−10, 20, 170... |
H: What is the value of $p^2q + q^2r + r^2p$ for the given cubic equation?
If $p$, $q$, $r$ are the real roots of the equation $x^3-6x^2+3x+1=0$, then find the value of $p^2q + q^2r + r^2p$.
My Attempt:
I tried $(p+q+r)(pq+qr+rp)$ but couldn't really figure out what to do with the extra terms. The roots are also not... |
H: Let $A$ be a torsion abelian group. Then $A$ has no $p$-torsion iff $A \otimes \Bbb Z_{(p)} = 0$.
Let $A$ be a torsion abelian group, $p$ a prime. Then $A$ has no $p$-torsion iff $A \otimes \Bbb Z_{(p)} = 0$.
I could prove one directioin $\Rightarrow$. But how does one prove the converse?
AI: Lets consider the exac... |
H: Find range of the function $f(x)=\sqrt {2\{x\}-\{x\}^2}-\frac 34$
Let $f(x)=y$ and $\{x\}=a$
$$(y+\frac 34)^2 =2a-a^2$$
$$y^2+\frac 94 +\frac{3y}{2} =2a-a^2$$
If I had a singular $\{x\}$ term I could have simply applied the inequality $0\le \{x\}<1$
But instead I have a polynomial. What should I do in this case?
AI... |
H: Suppose $N_1,..,N_r$ are submodules such that $\cap_{i=0}^r N_i=\{0\}$ and $M/N_i$ are semisimple for all $i$. Then M is semisimple.
Suppose $N_1,..,N_r$ are submodules such that $\cap_{i=0}^r N_i=\{0\}$ and $M/N_i$ are semisimple for all $i$. Then M is semisimple.
I am stuck with the above problem. All I can show ... |
H: Evaluate the integral $\int_{\gamma} e^{1/z}dz$
Consider the mapping on the complex plane given by $w = e^{1/z}$.
(a) What is the image of the set $\{z : |z|<1\}$?
(b) Sketch the image of the line $y = x$.
(c) Find a sequence of points in the pre-image of the point $w = i$ which converges to $0$.
(d) Evaluate the i... |
H: Show that $\sum\limits_{j,k=2}^\infty\frac{1}{j^k}$ converges and calculate the limit of the series
Show that $\sum\limits_{j,k=2}^\infty\frac{1}{j^k}$ converges and calculate the limit of the series.
My approach:
We look if one of the iterated series converges absolutly.
$$\sum\limits_{j=2}^\infty\left(\sum\limit... |
H: Prove that the congruence $x^{5} \equiv a \pmod p$ has a solution for every integer $a$
Let $p$ be a prime such that $5 \nmid p-1$. Prove that the polynomial congruence $x^{5} \equiv a \pmod p$ has a solution for every integer $a$.
I struggle to solve the case where $p \nmid a$. I've thought about using the existe... |
H: Can a holomorphic function $f$ can be a product of $z$ and some $g$ that $g$ is holomorphic
Can a holomorphic function $f$ at $D(0,1)$ can we present $f$ as $f=zg(z)$ where $g$ is holomoprhic and why?
AI: Yes, if and only if $f(0)=0$. It is clear that if $f(z)=zg(z)$, then $f(0)=0$. On the other hand, if $f(0)=0$, ... |
H: Exact Value of Infinite Sum $ \sum_{n = 0}^{\infty } \frac{{(-1)}^n}{n!(n+2)}$
Find exact value of the sum:
$ \sum_{n = 0}^{\infty } \frac{{(-1)}^n}{n!(n+2)} $
We could manipulate as follows:
$ \sum_{n = 0}^{\infty } \frac{{(-1)}^n}{n!(n+2)} = \sum_{n = 0}^{\infty } \frac{{(-1)}^n(n+1)}{(n+2)!} = \sum_{n = 0}^{\in... |
H: What is the difference between LMVT integral and differential form?
Differential : $$f'(c)=\frac{f(b)-f(a)}{b-a}$$
Integral : $$\int_a^b f(x) \, dx = f(c)(b - a)$$
I am confused what to apply when, I will appreciate any hints also
AI: The integral form is just a particular case of the first form.
If $f : [a,b] \rig... |
H: Dedekind cut with $-3$
Im studying some set theory, and my book gives me this definition:
"A Dedekind left set is a subset of $r$ of $ℚ$ with the following properties:
$r$ is a proper, non-empty subset of $ℚ$
if $q∈r$ and $p<q$, then $p∈r$
$r$ has no greatest element
A real number is a Dedekind left set and $ℝ$ ... |
H: Given positive definite $X\in\mathbb{R}^{4\times 4}$, find $Y\in\mathbb{R}^{4\times 2}$, such that $YY'\approx X$
Given positive definite $X\in\mathbb{R}^{4\times 4}$, I want to find $Y\in\mathbb{R}^{4\times 2}$, such that $YY'\approx X$.
My attempt:
Using SVD, $X=U\Sigma U^*$. Let $U_i$ be $i'$th column of $U$, ... |
H: What maths rule allows this expression with powers to be rewritten as below?
I have been reading through a programming book and the author asked to calculate the following expression in the program.
$$4*5^3+6*5^2+7*5+8$$
I approached this by expanding the expression like so:
$$(4*5*5*5)+(6*5*5)+(7*5)+8$$
$$500+150+... |
H: Question on when to use polar coordinates to prove existence of limit/ does the method always work?
Show that the following limit exists or does not exist (general example)
$$\lim \limits_{(x,y) \to (0,0)} \dfrac{e^{-x^2-y^2}-1}{x^2+y^2}$$
i) Direct substitution of $x=0$ , $y=0$ leads to indeterminate form of $\... |
H: Line integral over a broken line
I have a question about line integrals,
Q: Evaluate the line integral,
$$\int_C(2ye^{x^2-z}\cos(y^2)-9xy^2)dy+(12z-e^{x^2-z}\sin(y^2))dz+(2xe^{x^2-z}\sin(y^2)-3y^3)dx$$
Where C is the broken line from $A(0, \sqrt{\pi}, 3)$ to $B(0, \sqrt{\frac{\pi}{2}}, -1)$ connecting $(0, \sqrt{\p... |
H: Embedding vs continuous injection?
In Hatcher's algebraic topology, I read the following:
Let $h: D^k \to S^n$ be an embedding. Then $\tilde{H}_i(S^n \setminus
h(D^k)) = 0$ for all $i$.
Here an embedding is a map that is a homeomorphism onto its image.
Would it be correct to replace the word "embedding" by conti... |
H: Computing the $H^s$ norm of a time-dependent integral operator
Let $T>0$ arbitrary but fixed. Consider $f:\mathbb{R}\to\mathbb{R}$ any function in the Schwartz class. Now define the quantity:$$
J(t):=\int_{\vert \tau T\vert>1}(i\tau)^{-1}e^{it\tau}\widehat{f}(\tau)d\tau.
$$
I am reading a book on which the author s... |
H: "Proof" of $0=1$ in set theory
Ok, so here is a proof of "$0 = 1$" I came up with today. You can do in set-theory, where natural numbers are defined in the usual way.
Proof: Let $\mathsf{Succ}$ be the function that takes any natural number and adds one to it. Then we have $\mathsf{Succ}(0) = 1$. The image of $\math... |
H: Show that there are infinitely many such matrices $B$ for which $AB = A$ holds.
Let $$A=\begin{bmatrix}2 & 4\\3 & 6\end{bmatrix}$$ Show that there are infinitely many such matrices $B$ for which
$AB = A$ holds.
The idea that I thought is that in order $AB = A$ holds B must be identity matrix.However the answer is d... |
H: Properties of ${\wr \hspace {-3pt}\wr \hspace {-1pt}A\hspace {-1pt}\wr \hspace {-3pt}\wr } := \inf \{\vert Ax\vert \,\big \vert \, \vert x\vert = 1\}$
For A $\in L(\mathbb {R}^n)$ define ${\wr \hspace {-3pt}\wr \hspace {-1pt}A\hspace {-1pt}\wr \hspace {-3pt}\wr } := \inf \{\vert Ax\vert \,\big \vert \, \vert x\vert... |
H: Excluding a specific index when writing out a sum
Given the sum:
$$
\sum_{i=1}^{m}a_i\prod_{j\neq i}(n_j+x)=0
$$
If I were to write it without the sum symbol how would I go about excluding $j\neq i$ when writing it in the form
$$
a_1\prod_{j=2}^{m}(n_j+x)(where j\neq 1)+a_2\prod_{j=1}^{m}(n_j+x)(where j\neq 2)+\cdo... |
H: $\gamma(\lambda x)\leq \gamma(x)$ for $|\lambda| \leq 1$
Let $E$ be a TVS and let $(V_n)_{n \in \mathbb{N}}$ be a fundamental system of neighborhoods of $0$ in $E$ such that $\bigcap_{n \in \mathbb{N}} V_n=\{0\}$. Let us set $W_1=V_1$ and define by induction the sequence $(W_n)_{n \in \mathbb{N}}$ os balanced neigh... |
H: finding connected components
I want to find the number of connected components of $\mathbb{R}^2\backslash\mathbb{Q}^2$. My approach is since $\mathbb{Q}^2$ is a countable set. Then its compliment that is $\mathbb{R}^2\backslash\mathbb{Q}^2$ is path connected and thus connected. So does that mean it has one connecte... |
H: Why take the maximum eigenvalue in computing the matrix $2$-norm?
We know that the matrix $2$-norm is defined as
$$\|A\|_2 := \sqrt{\lambda_{\max}(A^T A)}$$
Why do we consider the maximum eigenvalue of $A^T A$?
AI: For a self-adjoint matrix $T$ the Courant-minmax-principle says:
$$\lambda_{j+1}=\operatorname{min}_... |
H: How to test data for log-normal distribution?
Can data be log-normally distributed by not normally?
AI: A variable $X$ has a log-normal distribution if and only if $\log(X)$ has a normal distribution, so to test wheter data is log-normally distributed, you can simply test wheter the log-transformed data is normally... |
H: How does topology work when taking charts on a Psuedo-Riemannian manifold?
I'll first explain why I think taking charts is sane when working with Riemannian manifolds, and then show what I believe breaks down in the Pseudo-Riemannian case with a particular choice of a Pseudo Riemannian manifold (Minkowski space). I... |
H: The closure is a subset of a closed set - Metric spaces
I'm trying to show that if $X$ is a metric space and $E, F \subset X$ where $F$ is a closed set and $E \subset F$ then we have $\bar{E} \subset F.$
Where $\bar{E}$ dentotes the closure of the set $E$. In other words $\bar{E} = E \cup E'$.
$E'$ denotes the set ... |
H: Evaluate $\int_{-\pi}^{\pi} \frac{x^2}{1+\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$
I came across this integral:$$\int_{-\pi}^{\pi} \frac{x^2}{1+\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$$
I tried $u=x+\pi$
$$\int_{-\pi}^{\pi} \frac{(x+\pi)^2}{1-\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$$
but had no success.
I also tried... |
H: How to do probability question (v)? Why is my logic wrong?
One plastic toy aeroplane is given away free in each
packet of cornflakes. Equal numbers of red, yellow,
green and blue aeroplanes are put into the packets.
Henry, a quality controller employed by the cornflakes
manufacturer, opens a number of packe... |
H: combinations and proability
There are 120 books on 24 topics with 5 volumes on each topic.
What is the probability of choosing 3 books such that they all belong to different topics?
What is the probability of choosing "r" books such that at least one book is on a repeated topic?
AI: What is the probability of choos... |
H: How would one use Bézout's theorem to prove that if $d = \gcd(a,b)\ \text{then} \ \gcd(\dfrac{a}{d}, \dfrac{b}{d}) = 1$.
Note: I have checked the questions with the same title and I am after something more specific.
I am doing my first course in discrete mathematics, and came across the following proposition that I... |
H: Weak * Convergence
I am trying to do the following exercise :
Let $X$ be a separable Banach space , M a bounded set of $X'$, the dual of $X$, show that for every sequence $f_n$ there is a subsequence that is weak* convergent to some $f \in X'$.
Now my tough process with this was that since $X$ is separable and th... |
H: Divisibility Number Theory problem, explanation needed
I can't understand the solution of the following problem:
$x$,$y$,$z$ are pairwise distinct natural numbers show that $(x-y)^5$ + $(y-z)^5$ + $(z-x)^5$ is divisible by $5(x-y)(y-z)(z-x)$. No need to explain the div. by 5.
The sol. says:
$(x-y)^5$ + $(y-z)^5$ + ... |
H: Understanding defination of Sobolev space
I was reading the definition of Sobolev space form book Evans.
From that, I understand the following,
$W^{k,p}(U)$ is space of locally integrable function whose all derivative up to order k are in $L^p$.
So $W^{k,p}(U)\subset L_{loc}^1(U)\cap L^p(U)$.
I thought locally sum... |
H: Sign Convention in Using Divergence Theorem to calculate Flux with Example
In showing my thought process in solving the following problem using the divergence theorem, I hope someone could point out where the selection of the sign comes into play. While the question below does define the orientation of the normal v... |
H: Calculating the volume of an ellipsoid with triple integrals
I am trying to find the volume of the ellipsoid $\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2=1$ by making the substitution $u=x/a$, $v=y/b$ and $w=z/c$.
With this substitution, the equation becomes $u^2+v^2+w^2=1$.... |
H: Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$
Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$. In my attempt i try to use Weierstrass Test, $\frac{1}{n^s}=\frac{1}{n^{Re(s)+iIm(s)}}=\frac{1}{e^{\log n^{\Re(s)+iIm(s)}}}=\frac{1}{e^{Re(s)\log n}e^{iIm(s)\log n}}$ then $|\frac{1}{n^s}... |
H: Numerable union of open measurable sets
So I've come across this question:
given $A_{1},A_{2},...\ $ Jordan measurable sets in $\mathbb{R}^{n}$, and given that $A=\cup_{n=1}^{\infty}{A_{n}}$ is bounded, is $A$ J-measurable?.
Now, I can think of $\cup_{q\in[0,1]\cap\mathbb{Q}}{\{q\}}$ as a counterexample for this,... |
H: Definition of the operator norm
I know that this definition is correct, is the bottom one also fine?
$A \in L(\mathbb {R}^m,\mathbb {R}^n)$
$\Vert A \Vert _{op}:=\sup \{{\vert Ax\vert } \big \vert \,x \in \mathbb {R}^m,\vert \,\vert x\vert \leq 1\} $
$\vert Ax\vert$ is the Euclidiean norm in $\mathbb {R}^n$ and $\... |
H: A problem on proving inner product formula $\langle x,y\rangle=\sum_{k=1}^\infty\langle x,e_k\rangle\overline{\langle y,e_k\rangle}$
Let $\{e_k\}_{k\in\Bbb N}$ be orthonormal basis of a inner product space over $\Bbb C$.
I want to prove that $\langle x,y\rangle=\sum_{k=1}^\infty\langle x,e_k\rangle\overline{\langle... |
H: Probability that a side length is greater than 2
In triangle $ABC$, we have $\angle B=60^\circ$, $\angle C=90^\circ$, and $AB=2$.
Let $P$ be a point chosen uniformly at random inside $ABC$. Extend ray $BP$ to hit side $AC$ at $D$. What is the probability that $BD<\sqrt 2$?
I drew a picture for this but now I am stu... |
H: Prove the following properties of a matrix
Let $S$ a field and $A$ a matrix, $(A \in M_{m,n}(K)$) so the rank of A is $dim(im(A))$
$ T \in GL_{m}(K), S \in GL_{n}(K)$ where $GL$ is a general linear group
Prove the properties:
$ker(TA)=ker(A)$
$im(AS) = im(A)$
$dim(ker(TAS))=dim(ker(A))$ and $dim(im(TAS))=dim(im(A)... |
H: If $A \in L(\mathbb {R}^n)$ with $\Vert A \Vert _{{op}} \neq 0$, then $A \in GL_n$.
Which of these following statements is true? $n \in \mathbb {N}$
a) It exists $A, B \in L(\mathbb {R}^n)$ with $\Vert A\Vert _{{op}} \neq 0$, $\Vert B\Vert _{{op}} \neq 0$, but $\Vert AB\Vert _{{op}} = 0$.
b) If $A \in L(\mathbb {R}... |
H: Asking about reason of convergence of series on Apostol Mathematical analysis ( Chapter - Lebesgue Integral)
While studying Apostol Mathematical analysis I am unable to find reason of following argument whose picture follows :
In first line of last paragraph can someone please tell how author wrote " the series o... |
H: Determinant of identity plus the sum of two outer products in $\{-1,1\}^d$
Let $a > 0$, $d \in \mathbb{N}$ and define
$$
A= I_d + a(v_1 v_1^T + v_2v_2^T)
$$
where $I_d$ is the $d$-dimensional identity matrix and $v_1, v_2 \in \{-1, 1\}^d$. What is the determinant of $A$? I wanted to use something like the Matrix D... |
H: How do I understand $f \equiv 1$ as the limit of step functions?
I am a beginner in Lebesgue Integral and I am learning Analysis from Tomorrow Apostol's Mathematical Analysis. Apostol gives following definition of step function
A function $s$, defined on compact interval $[a, b]$, is called a step function if th... |
H: Principal ultrafilters and a bijection
In "Compositionality in Montague Grammar" (https://pdfs.semanticscholar.org/0b5d/ab9d1718d6ca0c7211c0d81c9a65e4a03759.pdf), talking about classical higher-order logic, Markus Kracht writes that
Montague assumed that.../John/ no longer denotes the individual John but rather th... |
H: How can I calculate an audio volume in a human like perception?
Let's say I have a value v, such that $x \in [0, 1]$, and v represents digital audio volume. knowing that human perception of loudness is logarithmic, how can I find a function $f(v) = v'$ to interpolate, such that $v'$ respect that rule?
I've tried li... |
H: When does a family of function is uniformly equicontinuous?
The family of functions {f} is called uniformly equicontinuous iff .....
This is the question i have to complete the blanks and then prove the statement.
What i can think of is The family of functions {f} is called uniformly equicontinuous iff each member ... |
H: How can we sum up $\sin^m$ and $\cos^m$ series when the angles are in arithmetic progression?
How can we sum up $\sin^m$ and $\cos^m$ series when the angles are in arithmetic progression?
Does an identity exist similar to 1.1 and 1.2 for $m>1$?, and
Does an approximate or estimate exist?
For example here is the s... |
H: Integral of $(d^2y/dx^2) dy$
I saw this as a step in a calculation and it was confusing. The left cancels down to:
$$\int \frac{d^2y}{dx^2}dy$$
But surely the answer is $\frac{d^2y}{dx^2}y$; instead, $\frac {d^2}{dx^2}$ is being treated as a constant and not $\frac {d^2y}{dx^2} $
https://www.wolframalpha.com/input... |
H: Dynamic Height Resizing of Multiple Elements
Good afternoon,
I'm hoping to get some help/adice or a nudge in the right direction on the following problem:
I'm using a programme where I define a container and inside that container I'm placing three elements. First element is an image that occupies the top of the con... |
H: If $X\sim\mathrm{exp}(1)$ and $Y\sim\mathrm{exp}(1),$ is $X=Y$?
If two random variables have the same distribution it doesnt automatically mean that they are equal.
If $X\sim N(0,1)$ and $X=-Y,$ so is $Y\sim N(0,1).$ Because of the symmetry.
But if $X\sim\mathrm{exp}(1)$ and $Y\sim\mathrm{exp}(1).$ Could $X$ be not... |
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