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I'm taking medical physiology and of course, none of the equations are properly explained. The closest I've been able to get is [this page][1]. Even here however, steps from equations 14 to equation 15 is waved away as "careful algebraic manipulations").
What exacly is happening here?
We start with
$$RQ=\fra... |
I am just beginning my study in field theory, and I am doing a question about the trace of $\alpha \in K$ from $K$ to $F$ where $K,F$ are fields:
>Let $K/F$ be a finite field extension and $\alpha ∈ K$. Suppose $L$ is a Galois extension of $F$ containing $K$ and $H$ is the subgroup of $\mathrm{Gal}(L/F)$ correspondi... |
The vertices of a triangle are three unifomly random points on a circle. The side lengths are, in random order, $a,b,c$.
>Show that $P(ab>c)=\frac12$.
The result is strongly suggested by simulations, and by my attempt shown below.
The simplicity of the result suggests that there may be an intuitive explanation... |
$$
\text{Understanding Fixed-Point Iteration: Exploring Convergence and Root Determination}
$$
The equation $$2x^2 - 7x + 6 = 0$$ has two positive roots. We aim to determine these using the following fixed-point iteration:
$$
x_{k+1} = \frac{{8x_k - 2x_k^2}}{{k - 6}}
$$
**a)** It is found that only the large... |
The vertices of a triangle are three unifomly random points on a unit circle. The side lengths are, in random order, $a,b,c$.
>Show that $P(ab>c)=\frac12$.
The result is strongly suggested by simulations, and by my attempt shown below.
The simplicity of the result suggests that there may be an intuitive explan... |
Suppose we are measuring the heights of different people. Our measures will be in $cm$, but the variance will be in $cm^2$. If we were measuring the time to complete a race course, our measures would be in $seconds$ , but the variance would be $seconds^2$
What do those units mean? Why are they there?
I'm not ... |
I am reading Conways A Course in Functional Analysis.
A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\cap_{p \in P} ${x:p(x)=0 }$=(0)$.
I was looking for an example of a topological vector space that is not a locally convex space.
... |
I am trying to solve the following differential equation $$x''+sin(x)=f(t),$$
1-I know the solution of the homogeneous system $x_h(t)$, is it possible to use it to find the general solution of the non-homogeneous system? I am looking for a solution in the following form $$x(t)=\int_{-\infty}^t \xi(t-t') f(t') dt' $$ b... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> **If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integer... |
Yesterday night I watched the [video][1] on youtube and tried my chance :) I was able to rule out some small number of cases.
> **If $b^{2a}-4a^b$ is a perfect square, then any $a>1$ must be even.**
*Proof.* We use the already proven fact that $b^{2a}-4a^b$ is a perfect square iff there are positive integer... |
Let $U \subset \mathbb R^2$ be an open, regular (meaning $U$ is the interior of its closure), bounded and simply connected set. Let $\gamma : [0,1] \rightarrow \partial U$ be a simple curve such that $\gamma(0) \neq \gamma(1)$.
Is $\partial U \setminus \gamma((0,1))$ still connected ?
I think this should be true.... |
I know that $f(x)$ is a parabola, and $f(-2x) = x - x^2$.
Can I assume that $f(x) = -\frac{1}{2}x - \frac{1}{4}x^2$ (real numbers only)?
I got this equation because if I were to plug in $x = -2x$ I would get the original function.
|
If $f(g(x))$ is known, then can we find $f(x)$? |
I have learnt the Cheeger inequality for a graph $G$:
$$\lambda_2/2\le h(G)\le\sqrt{2\lambda_2}.$$
But does the equality hold? For a non-connected graph, it is obvious. But for a connected graph, I have tried some examples and find the first equality may hold, but fail at the second one. Are there any conditions ... |
Considering $k,x\in \mathbb{R}$
$$k^x=1+\sum_{n=1}^{\infty}\frac{x^n(\ln(k))^n}{n!}$$
Does it have an elementary proof?
\
\
$\Large\text{Maclaurin series}$
\
I attempt to solve by Maclaurin Series:
$$f(x)=\sum_{n=1}^{\infty}\frac{x^n f^{(n)}(0)}{n!}$$
It is well known that the $n$th derivatives of the function... |
Let $A=\frac{\mathbb C[X,Y]}{(X^3-Y^2)}$.I am asked to show that $\mathbf m=(\overline X-1,\overline Y-1)$ is a maximal ideal of $A$ which I have shown successfully.Now I am asked to compute the $\mu(\mathbf mA_m)$ which is the unique cardinality of a minimal generating set of $\mathbf mA_m$ which I am unable to do.I k... |
Recently, i've been thinking of the diophantine equation:
$A^{B^C}+D^{E^F} = G^{H^I}$
Where, $A, B, C, D, E, F, G, H, I \in [2, \infty)\cap \mathbb{N}$.
I discovered that one can construct a solution if one finds perfect powers $P, Q$ and a positive integer $n$ that satisfies $nP +1 = Q$. It is not hard to see... |
Assume you have an acute triangle with sides $a>b>c$. Then, after some calculation, we know that the biggest inscribed square that lies on side $a$, i.e. that has two vertices on $a$ and one vertex on each of $b$ and $c$ has side length $ah_a/(a+h_a)=2aA/(2A+a^2)$. Similarly, the biggest square that lies on side $b$ is... |
While analyzing a discrete version if Ito's Lemma, I became interested in a family of functions defined by the following:
$$
f(n, k) = \frac{f(n+1, k+1) + f(n+1, k-1)}{2}
\tag{$1$}
$$
If $f$ satisfies (1), then we call $f$ a backwards harmonic function. For example, $f(k, n) = k^2 - n$ satisfies the above, as... |
Let $G$ be group of $n^{th}$ roots of unity. Assume that $G=<\alpha>$, where $\alpha ^n=1$.
Now we define a group action, $G$ act on $\mathbb{C}$, by $(x,z)\mapsto xz$. The Quotient space $\mathbb{C}/G$ is homeomorphic to a cone, that is $\mathbb{C}/G$ is homeomorphic to cone of $S^1 \times [0,\infty)$. The open sets ... |
Is it possible to give a complex structure on two dimensional cone? |
Let $\mathcal{G}$ be the family of simple connected graphs on $n$ vertices, where each graph has more than $m$ edges, and the maximal degree is less than $\frac{n-1}{2}$. We can assume that m is large, i.e., $m \geq c\frac{n(n-1)}{2}$ for some $0 < c <0.5$.
I don't need a specific formula, but rather a bound on the ... |
I am reading Conways A Course in Functional Analysis.
A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in P} \{x:p(x)=0 \}=(0)$.
I was looking for an example of a topological vector space that is not a locally convex space... |
How to compute a minimal spanning set and the minimal spanning number for the following ring? |
From Legendre's three-square theorem, we know that solutions to
$$
n = x^2 + y^2 + z^2, x,y,z \in \mathbb{Z}
$$
exist for all $n \in \mathbb{N}$ except when $n$ is of the form $4^a(8b + 7)$
for nonnegative integers $a$ and $b$. Legendre's result admits the possibility
that any of $x,y,z$ can be zero. In his bo... |
Let $(X_n)_{n \ge 1}$ be independent with $E(X_k) = \mu$ and $E((X_k)^4) \le C$ for some $C>0$. Prove that if $S_n = X_1 + ... + X_n$ then $\frac{S_n}{n} \to \mu$ almost surely as $n \to \infty$.
The proof begins by assuming $\mu = E(X_k) = 0$, then claims
$$E((S_n)^4) = E \left( \left( \sum_{k=1}^n X_k \right)^... |
#### Understanding Fixed-Point Iteration: Exploring Convergence and Root Determination
The equation $$2x^2 - 7x + 6 = 0$$ has two positive roots. We aim to determine these using the following fixed-point iteration:
$$
x_{k+1} = \frac{{8x_k - 2x_k^2}}{{k - 6}}
$$
**a)** It is found that only the larger of the t... |
Maybe it's a pathetic question to a degree, but I was wondering about this when going through the definitions.
Given a probability space $(\Omega, \mathcal A, P)$, can we use the measure $P$ again to define a (real) random variable $X: \mathcal A \rightarrow \mathbb R, \, A \mapsto P(A)$?
Then the push forward me... |
Using the probability measure as the random variable also. Possible? |
We know that exponential functor $(-)^X$ can be defined as right adjoint to the functor $ -\times X$.
\
Consider two $\frak{C}$-objects $X$ and $Y$ and a functor $F:\frak{C}\to \frak{C}$. Suppose $\frak{C}$$(Z \times X,Y) \cong \frak{C}$$(Z,F(Y))$ naturally in $Z$ (so *almost* an adjunction). Then I was wondering wh... |
Is there some way of understanding more deeply why exactly it is that the probability mass function or probability density functions of various random variables have the shapes that they have when you plot them?
For example, a priori, it is pretty obvious that a continuous, uniformly distributed r.v. will have a PD... |
Is there some way of understanding more deeply why exactly it is that the probability mass function or probability density functions of various random variables have the shapes that they have when you plot them?
For example, a priori, it is pretty obvious that a continuous, uniformly distributed r.v. will have a PD... |
I'm taking medical physiology and of course, none of the equations are properly explained. The closest I've been able to get is [this page][1]. Even here however, steps from equations 14 to equation 15 is waved away as "careful algebraic manipulations").
What exacly is happening here?
We start with
$$RQ=\fra... |
*Remark:* My attemps is WRONG as I fasly assumed that the $Y_n$ were independant. I ve corrected this in my own answer to this post. This answer is correct but not enough well wrotten, read the selected answer for a better quality answer.
## Question:
Let $a > 0$. For $n \in \mathbb{N}$, let the random variable ... |
Problem: Decompose $V(x^2+y^2-1, x^2-Z^2-1)\subset A^3(C)$ into irreducible algebraic components.
Attempt: Let $f[y](x) = x^2 + (y^2-1) \in C[y][x]$, then $f[y](x+1) = x^2 + 2x + y^2$. Let $g[x](y) = y^2 + (x^2+2x)$, then it's irreducible by Eisenstein's Criterion (x is irreducible). Hence $x^2+y^2-1$ is irreducible... |
Problem: Decompose $V(x^2+y^2-1, x^2-Z^2-1)\subset A^3(C)$ into irreducible algebraic components.
Attempt: Let $f[y](x) = x^2 + (y^2-1) \in C[y][x]$, then $f[y](x+1) = x^2 + 2x + y^2$. Let $g[x](y) = y^2 + (x^2+2x)$, then it's irreducible by Eisenstein's Criterion (x is irreducible). Hence $x^2+y^2-1$ is irreducible... |
I am reading a proof of the irrationality of $e^2$ and I am stuck on the following inequality:<br>
Let $S := -a\left(\frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} \mp ...\right)$ (just because of space issues).
$$
-\frac{a}{n} < S < -a\left(\frac{1}{n+1} - \frac{1}{(n+1)^2} - \frac{1}{(n+1)^3} - .... |
In measure theory, an algebra on a set $X$ is a collection $\mathcal A\subset\mathcal P(X)$ with properties
> (1) $X\in\mathcal A$
(2) For any $A\in\mathcal A,\ X\setminus A\in\mathcal A$
(3... |
What is the relationship between concept of algebra in measure theory and algebra over a ring? |
Problem: Decompose $V(x^2+y^2-1, x^2-z^2-1)\subset A^3(C)$ into irreducible algebraic components.
Attempt: Let $f[y](x) = x^2 + (y^2-1) \in C[y][x]$, then $f[y](x+1) = x^2 + 2x + y^2$. Let $g[x](y) = y^2 + (x^2+2x)$, then it's irreducible by Eisenstein's Criterion (x is irreducible). Hence $x^2+y^2-1$ is irreducible... |
I've tried to prove that $x^2+y^2-1$ is irreducible in $C[x,y,z]$.
Attempt: Let $f[y](x) = x^2 + (y^2-1) \in C[y][x]$, then $f[y](x+1) = x^2 + 2x + y^2$. Let $g[x](y) = y^2 + (x^2+2x)$, then it's irreducible by Eisenstein's Criterion (x is irreducible). Hence $x^2+y^2-1$ is irreducible.
Now I know that $x^2+y^2-1... |
Which step of using Eisenstein's Criterion goes wrong? |
Given $a,b \in \mathbb Q$ and $ab \neq 0.\,$ Let us consider trinomials $x^n+ax^k+b,\,$ with integers $n > k$, $n > 8$, and $gcd(n,k) = 1$.
Questions:
1. Do you know examples of *reducible* and solvable other than the following ones?
$$x^9+19x+20 = \\ \left(x+1\right) \left(x^4-2x^3+2x^2-4x+5\right) \left(x^4+x^... |
In solving for isosceles almost and nearly Pythagorean triples, I have observed that the set of triples:
$$ S_n = \{ (1,1,1), (2,2,3) ,(5,5,7), (12, 12, 17), (29, 29, 41) \dots \}$$
are iso-APTs for odd $n$ and iso-NPTs for even $n$. Moreover, I've observed that the next terms are given by: $$(x+y, x+y, x+2y)$$... |
Determine the number of $5$ digit numbers in which no digit occurs more than twice (numbers starting with $0$ are allowed).
I see there $3$ possible groups of scenarios:
1. No digits repeat $ = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6$
2. We have one pair of same digits $ = 10 \cdot \binom{5}{2} \cdot 9 \cdot 8 \cdo... |
Considering $k,x\in \mathbb{R}$
$$k^x=1+\sum_{n=1}^{\infty}\frac{x^n(\ln(k))^n}{n!}$$
Does it have an elementary proof?
\
\
I attempt to solve by Maclaurin Series:
$$f(x)=\sum_{n=1}^{\infty}\frac{x^n f^{(n)}(0)}{n!}$$
It is well known that the $n$th derivatives of the function $k^x$ are $\frac{d}{dx}k^x=k^x\ln (... |
Prove the equation:
$$\sum_{k=1}^n{{k}\choose{j}}k = {{n+1}\choose{j+1}}n - {{n+1}\choose{j+2}}$$
My solution:
We have $n+1$ players numbered from $1$ to $n+1$. We want to play a team game that requires:
- $j$ players in team $A$,
- $n-j$ players in team $B$,
- $1$ special player, who is a player on one ... |
**The equation given is wrong.**
====
To make the equations easier to read, I will use the substitute symbols
$$a=RQ \\ b=F_ACO_2 \\ c=F_IO_2 \\ d=F_AO_2$$
Using which we can write your original equation as
$$a=\frac{\frac{b}{1-c-b}}{\frac{c}{1-c}-\frac{d}{1-d-b}}\tag{1}$$
--
The equation we want to end ... |
I want to define a forgetful functor $\beta:\textbf{Top}^0\to \textbf{Top}$, where $\textbf{Top}^0$ is the category of pointed spaces and pointed continuous maps.
I know that $\beta$ should forget what the base point of any pointed space is. But I am not sure exactly what this means.
Is it that we simply forget... |
Let us consider the vector space of complex $n \times n$ matrices.
Let $\{ V_i \}_{i=1,2,\cdots,n^2}$ be a trace-orthogonal basis of matrices, i.e.,
$$ \forall i,j \in \{1,2,\cdots,n^2\} : \quad \textrm{tr}(V_i^\dagger V_j) = n \; \delta_{ij}. $$
In general, such matrices do not need to be unitary (e.g., the [Ge... |
Is this basis for complex matrices necessarily a unitary basis? |
For $|q| < 1$, $|t|<1$, then
$$1+\sum_{n=1}^{\infty} \frac{(1-a)(\cdots)(1-aq^{n-1})t^n}{(1-q)(\cdots)(1-q^n)}=\prod_{n=0}^{\infty}\frac{(1-atq^n)}{(1-tq^n)}=\sum_{n=0}^{\infty}A_nt^n$$
Fromwhat I am reading they show that $A_n-A_{n-1}=q^nA_n-aq^{n-1}A_{n-1} \implies A_n = \frac{(1-aq^{n-1})}{(1-q^n)}A_{n-1}$
I c... |
**The equation given is wrong.**
====
To make the equations easier to read, I will use the substitute symbols
$$a=RQ \\ b=F_ACO_2 \\ c=F_IO_2 \\ d=F_AO_2$$
Using which we can write your original equation as
$$a=\frac{\frac{b}{1-c-b}}{\frac{c}{1-c}-\frac{d}{1-d-b}}\tag{1}$$
--
The equation we want to end ... |
Study the punctual, absolute and uniform convergence of the following series of functions:
$$\sum_{n=1}^\infty\left[\arcsin\left(\frac{2x^2-1}{x}\right)\right]^n$$
I have done this. Is it correct?
**1. Pointwise Convergence**:
For each fixed $x$, we need to check if the sequence of functions converges. Let's ... |
I wanted to invert the following Laplacian expression which was not possible with simple manipulation.
$$ \frac{s}{s^2 + \omega^2}\exp\bigg(-\sqrt{\frac{s}{\alpha}}x\bigg) $$
Thus, I used the Taylor Series on each part and simplified it as shown:
$$ \implies \frac{1}{s} \bigg( 1 + \frac{\omega^2}{s^2}\bigg)^{... |
We know that:
$$\prod_{n=1}^\infty(e^{-n}+1)=\left(-1,\frac1e\right)_\infty$$
with the [Q Pocchhammer symbol $(a,q)_k$](https://www.wolframalpha.com/input?i=QPochhammer&assumption=%7B%22C%22%2C+%22QPochhammer%22%7D+-%3E+%7B%22Function%22%7D), but what if we made it:
$$P=\prod_{n=1}^\infty(ne^{-n}+1)= 2.273968452... |
Determine the number of $5$ digit numbers in which no digit occurs more than twice (numbers starting with $0$ are allowed).
I see there $3$ possible groups of scenarios:
1. No digits repeat $ = 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6$
2. We have one pair of same digits $ = 10 \cdot \binom{5}{2} \cdot 9 \cdot 8 \cdo... |
For $|q| < 1$, $|t|<1$, then
$$1+\sum_{n=1}^{\infty} \frac{(1-a)(\cdots)(1-aq^{n-1})t^n}{(1-q)(\cdots)(1-q^n)}=\prod_{n=0}^{\infty}\frac{(1-atq^n)}{(1-tq^n)}=\sum_{n=0}^{\infty}A_nt^n$$
Fromwhat I am reading they show that $A_n-A_{n-1}=q^nA_n-aq^{n-1}A_{n-1} \implies A_n = \frac{(1-aq^{n-1})}{(1-q^n)}A_{n-1}$
I c... |
Problem
==============
I encountered this [statement](https://ncatlab.org/nlab/show/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable) on ncatlab, which says that **weakly Lindelöf spaces with a $\sigma$-locally finite basis are second-countable**. The proof given below the statement... |
Is a $\sigma$-locally finite collection of open sets locally countable? |
Does there exist a monoid $(M;*,1)$ and a submagma $S$ of $M$, such that $S$ is not a submonoid of $M$, but $S$ is "monoidable", meaning it contains an identity element, and also, $S$ is not idempotent either? The reason I am asking this question is because I was considering the monoid $(N;*,1)$ of nonnegative integers... |
Existence of monoidable submagma of monoid which is neither a submonoid nor idempotent? |
*I’m still not an expert in integration, and I’m still learning, so sorry if it’s a trivial question:*
Here’s the general integral:
$$\int \frac{a}{x^2 + 1}\text{ } dx, a \in \mathbb R$$
I know that this antiderivative is $a \tan^{-1} (x)$.
Now, I want to compute
$$\int_{-\infty}^{\infty} \frac{a}{x^2 ... |
Is this a formal math symbol? |
Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\Lambda(n)$ represents an arbitrary number theoretic function. So far I've solved it for $\Lambda(n)=2T_n$ for $T_n$ t... |
Is this a formal symbol?
[![enter image description here][1]][1]
I've never seen it before.
[1]: https://i.stack.imgur.com/D6EOn.jpg |
Is "$\prec$" a formal symbol?
[![enter image description here][1]][1]
I've never seen it before.
[1]: https://i.stack.imgur.com/D6EOn.jpg |
Is "$\prec$" a formal math symbol? |
I've been trying to understand the decomposition of the second tangent bundle, $TTQ$, of a smooth Riemannian manifold, $(Q,g)$, into so-called horizontal and vertical bundles, $TTQ = HTQ \oplus VTQ$, using the Levi-Civita connection $\nabla$. I know that such a decomposition can be applied to much more general vector b... |
for $x_{0}\in \mathbb{R}^{n}$ I denote by $G^{k}(x_{0})\: (1\le k\le \infty )$ the $\mathbb{R}$-algebra of germs of real-valued $C^{k}$-functions at $x_{0}$. Recall that a derivation at $x_{0}$ is a linear form $D:G^{k}(x_{0})\to\mathbb{R}$ which satisfies the product rule $D(f\cdot g)=f(x_{0})\cdot Dg+g(x_{0})\cdot Df... |
I am reading Conways A Course in Functional Analysis.
A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in P} \{x:p(x)=0 \}=(0)$.
I was looking for an example of a topological vector space that is not a locally convex space... |
Assume we are given a pair of calibrated cameras with projection matrices $\mathbf \Pi_0$ and $\mathbf \Pi_1$. From Hartley and Zisserman (pg 246), the fundamental matrix is
\begin{align}
\mathbf F
&= [\mathbf e_1]_\times \mathbf\Pi_1 \mathbf\Pi_0^\dagger \\
&= [\mathbf \Pi_1 \mathbf c_0]_\times \mathbf\Pi_1 \mathb... |
Show that the set of permutations $\{\sigma \mid \sigma(i)\le i \text{ for }1\le i\le 6 \}$ is a subgroup of $S_6$.
I am stuck at how I am supposed to reason about this question. I'd assume I'd first check closure, then the group axioms, but how am I to deal with the huge variety of choices of the image? Is there su... |
In brezis 1.8 i need to show that de gauge (or Minkowski funtional) of the set $$ C= \bigg\{ u \in C[0,1]: \int_{0}^{1} \vert {u(t)}^{2}\vert dt < 1 \bigg \} $$
is a norm, where $C[0,1]$ have the norm $\Vert u \Vert=\max \{ u(t) | t \in t\}$ , but I don't know how to do this.
I just prove that $C$ is convex, simet... |
How to prove that the Gauge is a norm? Brezis's hint 1.8 |
Show $\sum_{n=0}^\infty \frac{(cq^n)_{\infty}}{(bq^n)_{\infty}}=\sum_{m=0}^\infty \frac{(c/b)_mb^mq^{nm}}{(q)_m}$
This was taken from the following equation:
$$\frac{(b)_\infty}{(c)_\infty} \sum_{n=0}^\infty \frac{(a)_nt^n}{(q)_n} \cdot \frac{(cq^n)\infty}{(bq^n)\infty}=\frac{(b)_\infty}{(c)_\infty} \sum_{n=0}^\inf... |
In brezis 1.8 i need to show that de gauge (or Minkowski funtional) of the set $$ C= \bigg\{ u \in C[0,1]: \int_{0}^{1} \vert {u(t)}^{2}\vert dt < 1 \bigg \} $$
is a norm, where $C[0,1]$ have the norm $\Vert u \Vert=\max \{ u(t) | t \in t\}$ , but I don't know how to do this.
I just prove that $C$ is convex, simet... |
*I’m still not an expert in integration, and I’m still learning, so sorry if it’s a trivial question:*
Here’s the general integral:
$$\int \frac{a}{x^2 + 1}\text{ } dx, a \in \mathbb R$$
I know that this antiderivative is $a \tan^{-1} (x)$.
Now, I want to compute
$$\int_{-\infty}^{\infty} \frac{a}{x^2 ... |
In Brezis 1.8 I need to show that the gauge (or Minkowski functional) of the set $$ C= \bigg\{ u \in C[0,1]: \int_{0}^{1} \vert {u(t)}^{2}\vert dt < 1 \bigg \} $$
is a norm, where $C[0,1]$ have the norm $\Vert u \Vert=\max \{ u(t) | t \in t\}$, but I don't know how to do this.
I just prove that $C$ is convex, sy... |
I am a high school student and we firstly learn something about abstract algebra this week.I want to learn the idea behing the difference between $a*b$ and $b*a$
It is given that $a*b=a^3 -b^2+4a+3b$ where $*$ is an algebraic operation. It is asked to find $b*a$.
I thought that I should put "a" in place of $b$ i... |
I am reading Conways A Course in Functional Analysis.
A locally convex space is a topological vector space whose topology is defined by a family of seminorms $\mathcal{P}$ such that $\bigcap_{p \in P} \{x:p(x)=0 \}=(0)$.
I was looking for an example of a topological vector space that is not a locally convex space... |
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le3,$ $0\le y\le4,$. Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is?
My approach:
I tried finding the probability of the poi... |
Why doesn't the indefinite integral act as a function even though it is? |
We know that:
$$\prod_{n=1}^\infty(e^{-n}+1)=\left(-1,\frac1e\right)_\infty$$
with the [Q Pocchhammer symbol $(a,q)_k$](https://www.wolframalpha.com/input?i=QPochhammer&assumption=%7B%22C%22%2C+%22QPochhammer%22%7D+-%3E+%7B%22Function%22%7D), but what if we made it:
$$P=\prod_{n=1}^\infty(ne^{-n}+1)= 2.273968452... |
**Background**
The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the limit is taken over all partitions of the interval
$[a,b]$ whose norms approach zero.
We can do something r... |
[According to the wikipedia definition of Jacobian](https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant), if $f:\mathbb{R}^{n}\to \mathbb{R}$, then $Jf=\nabla f^{T}$, where $Jf$ is the jacobian of $f$ and $\nabla f$ is its gradient. Note: I simply took the special case $f:\mathbb{R}^{n}\to \mathbb{R}^{m}, m=1... |
In I.M. James' *Topological and uniform spaces*, the proof for the second half of Proposition 11.5, on p.143, makes me confused. There, he proves a unformizable space is completely regular.
>The outline of his proof is, for a uniform space $X$, a closed subset $H$ and a point $x\in X\setminus H$, he wants to find a... |
Problem
==============
I encountered this [statement](https://ncatlab.org/nlab/show/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable) on ncatlab, which says that **weakly Lindelöf spaces with a $\sigma$-locally finite basis are second-countable**. The proof given below the statement... |
I define the Fourier transform of a function $g$ to be $ \hat{g}(\lambda) = \int_{-\infty}^\infty g(x)e^{-i\lambda x} \, \mathrm{d}x $ with an associated inverse formula
$$ g(x)= \frac{1}{2\pi} \int_{-\infty}^\infty \hat{g}(\lambda) e^{-i\lambda x} \, \mathrm{d}\lambda. $$
Fix $c>0$. I wish to verify this latte... |
I have this assignment on finite automatas and tried to solve it but I have a question on the string s to use for the pumping lemma. I have the intuition that for q = square root of 2, I might want to use a troncature of q and then pump the y part (from the format x.y). In my case can I use for example 1.011 as s and t... |
Consider the equation
$$t\frac{\partial^2}{\partial t^2} \sum_{n=1}^\infty \Phi_n(x,t)=-x\frac{\partial}{\partial x}\sum_{n=1}^\infty \Phi_n(x,t)+\sum_{n=2}^{\infty}\Lambda(n)\Phi_n(x,t)$$
where $\Lambda(n)$ represents an arbitrary number theoretic function, that is a function $\Lambda:\Bbb N \to \Bbb C.$ So far ... |
If $A$ is a $C^\ast$-algebra and $I$ is an ideal of $A$, then $Z(I)=Z(A)\cap I$? where Z(.) is the center.
I have done that $Z(A)\cap I$ trivially contained in $Z(I)$. But I am getting no idea how to do converse containment. Is there any idea or hint how to do it. Is some property of $C^\ast$-algebra will be used to... |
It's a bit ambiguous; have you fixed $X$? If I interpret your statement in the following way:
> Suppose $X\in\mathfrak{C}$ is given and $F^X:\mathfrak{C}\to\mathfrak{C}$ is a functor for which there are isomorphisms $\mathfrak{C}(X\times Z,Y)\cong\mathfrak{C}(Z,F^X(Y))$ which are natural in $Z$, for every $Y$ fixed.... |
I am working with an infinite series of the form $f(x) = \sum_{i=0}^{\infty} a_i x^i$, and I am interested in understanding how to transform this series into a new series where each coefficient $a_i$ is replaced by its reciprocal, yielding a series of the form $\sum_{i=0}^{\infty} \frac{1}{a_i} x^i$.
Given the ini... |
I am a high school student and we firstly learn something about abstract algebra this week.I want to learn the idea behing the difference between $a*b$ and $b*a$
It is given that $a*b=a^3 -b^2+4a+3b$ where $*$ is an algebraic operation. It is asked to find $b*a$.
I thought that I should put "a" in place of $b$ i... |
I am a high school student and we firstly learn something about abstract algebra this week.I want to learn the idea behing the difference between $a*b$ and $b*a$
It is given that $a*b=a^3 -b^2+4a+3b$ where $*$ is an algebraic operation. It is asked to find $b*a$.
I thought that I should put "a" in place of $b$ i... |
If $A$ is a $C^\ast$-algebra and $I$ is an ideal of $A$, then $Z(I)=Z(A)\cap I$? where Z(.) is the center.
I have read in a paper https://doi.org/10.1093/imrn/rnaa133 that it is well known fact that $Z(I)=Z(A)\cap I$ for a $C^\ast$-algebra $A$ and $I$ two sided closed ideal of $A$.
I have done that $Z(A)\cap I$ t... |
It's a bit ambiguous; have you fixed $X$? If I interpret your statement in the following way:
> Suppose $X\in\mathfrak{C}$ is given and $F^X:\mathfrak{C}\to\mathfrak{C}$ is a functor for which there are isomorphisms $\mathfrak{C}(X\times Z,Y)\cong\mathfrak{C}(Z,F^X(Y))$ which are natural in $Z$, for every $Y$ fixed.... |
If $A$ is a $C^\ast$-algebra and $I$ is an ideal of $A$, then $Z(I)=Z(A)\cap I$? where Z(.) is the center.
I have read in a paper https://doi.org/10.1093/imrn/rnaa133 that it is well known fact that $Z(I)=Z(A)\cap I$ for a $C^\ast$-algebra $A$ and $I$ two sided closed ideal of $A$.
I have done that $Z(A)\cap I$ t... |
I define the Fourier transform of a function $g$ to be $ \hat{g}(\lambda) = \int_{-\infty}^\infty g(x)e^{-i\lambda x} \, \mathrm{d}x $ with an associated inverse formula
$$ g(x)= \frac{1}{2\pi} \int_{-\infty}^\infty \hat{g}(\lambda) e^{-i\lambda x} \, \mathrm{d}\lambda. $$
Fix $c>0$. I wish to verify this latte... |
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