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How to determine the solutions of
$(\frac{\pi}{12}+\frac{5}{3}x^2-x(\sqrt{(1-x)})) = \arcsin x$
are (compass and straightedge)constructible numbers?
I got this question from my student ,in my calculus class, to find the implicit derivative of "$(\frac{\pi}{12}+\frac{5}{3}x^2-x(\sqrt{(1-x)})) = \arcsin x$ ... |
How to determine a (compass and straightedge)constructible number from a trigonometric equation? |
I understand the logical statement that: $$\max\{{X_{1},\ldots,X_{n}}\}(\omega) \le y \iff X_{1}(\omega) \le y,\ldots,X_{n}(\omega) \le y$$
But my issue falls with $\max\{\}$ itself. Shouldn't the output from this function be a single random variable?
Why is say $\max\{1,2,3,3\} = 3$, but the $\max\{A, B, C\}(\omega)... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
I am following Glimm & Jaffe's *Quantum Physics* where they define a generating functional
$$S\{f\} = \int e^{i\phi(f)} d\mu.$$
Here $d\mu$ is a Borel probability measure over the space of distributions $\mathcal{D}'(\mathbb{R}^d)$ and $f \in \mathcal{D}(\mathbb{R}^d)$ (the space of smooth functions with compact sup... |
Let $X = (X_1, X_2)$ be a non-negative bivariate random vector. Let $Y_i = \phi _i(X_i), i = 1, 2,$ be one-to-one
transformations with $\phi_i(X_i)$ differentiable functions, and let $Y = (Y_1, Y_2)$ be a non-negative bivariate random
vector connected with $X$ by the relation $Y_i = \phi_i(X_i)$.
I want to tran... |
I don't think this is correct. When the rooms cannot be differentiated, the answer shouldn't simply be $\binom {9}{3} \binom {6}{4} \binom {2}{2}$ divided by $3^9$, because you're assuming that the rooms are distinguishable. For example, you're counting $2, 2, 5$ and $5, 2, 2$ as distinct cases. To clarify, the reasoni... |
Given a Pearson correlation coefficient (PCC) and a list of values (target), can I generate another list of values with roughly the same PCC as the targeting list?
For example, given a list of values x = [1,2,3,4,5] and a PCC=0.8. I want to generate another list y = [a,b,c,d,e], so that x and y will have a PCC aroun... |
Is it possible to reverse the Pearsonr? |
Consider a truncated icosahedron with 12 pentagons and 20 hexagons. Starting from a hexagonal face, we go to any neighboring polygon randomly with equal probability. What is the expected number of steps it takes for us to visit the starting hexagon a second time?
I know that this is easily solvable with a Markov Ch... |
Let $X = (X_1, X_2)$ be a non-negative bivariate random vector. Let $Y_i = \phi _i(X_i), i = 1, 2,$ be one-to-one
transformations with $\phi_i(X_i)$ differentiable functions, and let $Y = (Y_1, Y_2)$ be a non-negative bivariate random
vector connected with $X$ by the relation $Y_i = \phi_i(X_i)$.
I want to tran... |
I want to show that $P(A \cup B^c)=P(A)-P(A \cup B)+1$, so I try to use the identity
$$A \cup B=A \cup (B \cap A^c)$$
But it seems that when I do it for the equality on the left, the sets do not end up being disjoint and therefore I cannot separate the union as a sum.
Maybe as $A \subseteq A \cup B$ then $P(A ... |
Let $X = (X_1, X_2)$ be a non-negative bivariate random vector. Let $Y_i = \phi _i(X_i), i = 1, 2,$ be one-to-one
transformations with $\phi_i(X_i)$ differentiable functions, and let $Y = (Y_1, Y_2)$ be a non-negative bivariate random
vector connected with $X$ by the relation $Y_i = \phi_i(X_i)$.
I want to tran... |
As commented by Blue, the claim is not true in general.
In the following, I'm going to prove the following claim :
**Claim 1** : If $\angle AVC =\angle BVD\color{red}{\not=90^\circ}$, then triangles $VAC$ and $VBD$ have equal areas and equal perimeters.
*Proof* :
Applying the law of cosines to $\triangle{... |
Here is an excerpt from the beginning of Chapter 8 of Fulton's Young Tableaux:
[![First Paragraph of Fulton's Young Tableaux Chapter 8][1]][1].
I do not understand how we can parameterize just be diagrams $\lambda$ with at most $m$ rows? What about the length of the rows? I feel like this should read "diagrams $\la... |
Let us introduce all the points in a clear order, and show the claimed properties one by one.
To have usual, standard notations, i will switch in notation from $\Delta XYZ$ and $\Delta X'Y'Z'$ to $\Delta ABC$ and $\Delta A'B'C'$.
Then instead of the points $A,B,C,D,E,F,G$ i will use (in order to avoid collisions) $... |
I was reading a series of articles on numerical integration of highly oscillatory integrals, e.g.,
1. S. Olver, Numerical approximation of highly oscillatory integrals
2. S. Xiang, H. Wang, Fast integration of highly oscillatory integrals with exotic oscillators
3. A. Iserles, S.P. Nørsett, Efficient quadratu... |
At a high level, a "step" in the gradient descent algorithm always moves perpendicular to the level set of the current iteration. This property is a direct result of the definition of the gradient vector itself (it always points in the direction of steepest ascent). [This article][1] provides a good (and relatively inf... |
I'm new to proof-writing in Real Analysis, and came across the following problem:
**Prove $\lim(a_nb_n)=\lim(a)\lim(b)$**. **Let $\lim(a), \lim(b)$ exist and $\lim(a)=L_a, \lim(b)=L_b$**
The first thing I was tempted to do was the following manipulation...
$$|a_nb_n-L_aL_b|=\left|a_n\left(b_n-\frac{L_aL_b}{a_n... |
I was reading a series of articles on numerical integration of highly oscillatory functions, e.g.,
1. S. Olver, Numerical approximation of highly oscillatory integrals
2. S. Xiang, H. Wang, Fast integration of highly oscillatory integrals with exotic oscillators
3. A. Iserles, S.P. Nørsett, Efficient quadratu... |
In order to show something, we need that something clearly defined in our hands. We need to show that an "expression converges". Well, it is a sequence. Then which are its terms? So as the dots are placed in the expression
"$\displaystyle \sqrt[3]{5\sqrt[3]{3\sqrt[3]{5\sqrt[3]{3 \cdots}}}}$"
it looks like we are sta... |
What is the difference between elements in a set and subsets? Why isn’t a subset also an element of a set? Or is it? I’ve been told that the two are totally different relations, but I struggle to understand the difference. Is it a matter of their logical consequences, the rules by which they’re governed, something else... |
Difference between elements and subsets? |
I've seen related questions, but none quite like this one. For a rational number $r$ let $[r]$ be the largest integer less than or equal to $r$, e.g., $[\frac{1}{2}] = 0, [2] = 2$, and $[3\frac{1}{3}] = 3$. Prove $v_p n! = [n/p] + [n/p^2]+[n/p^3]+\cdots$. My professor encourages us to use this new function: Let $\#_{p,... |
If you differentiate the rooms, the answer to the question in the title is $2^{10}$. If you don’t differentiate the rooms, you should divide by $2$ since every distribution is counted $2$ times now. So the answer is $2^9$.
You can also use the [Stirling numbers of the second kind][1] since they are used to determin... |
It's not true that if $Y$ is a deterministic function of $X$, then $h(Y|X) = 0.$ In fact the differential entropy of a constant is either $-\infty$ or undefined, depending on your tastes. Formulaically this simply suggests that $I(X;Y) = \infty$. This should make sense operationally: if you can transmit a real number $... |
It's not true that if $Y$ is a deterministic function of $X$, then $h(Y|X) = 0.$ In fact the differential entropy of a constant is either $-\infty$ or undefined, depending on your tastes. Formulaically this simply suggests that $I(X;Y) = \infty$. This should make sense operationally: if you can transmit a real number $... |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{+}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is a natural smooth action $G\times \text{Conf}_n(\mathbb R^k)\to ... |
Suppose I have a function $g(t)=\frac{df(t)}{dt}$. Then $f(t) = f(0)+\int_0^tg(\tau)d\tau$.
Now if I apply the Laplace transform to the first equation I get $g(s) = sf(s)$. Hence, $f(s)=\frac{g(s)}{s}$. But then what happened to the $f(0)$? |
Given an $n\times n$ grid partially filled with the numbers $0,\ldots, n-1$, we can play a Sudoku-like game by trying to fill in the rest of the grid so that the end result is the Cayley table for a group.
I am curious about the following value: let $g(n)$ represent the smallest natural number such that if $g(n)$ s... |
If a set is a bowl of cereal, then each of the pieces of cereal is an element, and the subsets are all the possible bowls of cereal you could get by taking out some (or possibly none, or possibly all) of the cereal.
"But", I hear you say, "couldn't I take out all except for a single piece?". And the answer is yes, y... |
Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{\times}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group, then there is a natural smooth action $G\times \text{Conf}_n(\mathbb R^k... |
If a set is a bowl of cereal, then each of the pieces of cereal is an element, and the subsets are all the possible bowls of cereal you could get by taking out some (or possibly none, or possibly all) of the cereal.
"But", I hear you say, "couldn't I take out all except for a single piece?". And the answer is yes, y... |
What is the difference between elements in a set and subsets? Why isn’t a subset also an element of a set? Or is it? I’ve been told that the two are totally different relations, but I struggle to understand the difference. Is it a matter of their logical consequences, the rules by which they’re governed, something else... |
Let $K$ be a finite extension of $\Bbb{Q}_p$.
Let $E/K$ be an elliptic curve over $K$.
Let $E(K)$ be the Mordell-Weil group of $E/K$.
Let $\widehat{E(K)}$ be the profinite completion of $E(K)$, that is, its completion with respect to finite index subgroups.
Is it true that $E(K)=\widehat{E(K)}$ ?
If we a... |
I have some confusion regarding the existence of the follow limit:<br>$$\lim _{a \rightarrow \infty} \frac{\int_0^a \sin ^4 x d x}{a}$$
According to the answer key of the material I found this question in (from a local study institute), this limit does not exist.
I also verified this using Desmos, and it is true th... |
Why does the limit $\lim _{a \rightarrow \infty} \frac{\int_0^a \sin ^4 x d x}{a}$ not exist? |
Let $E=E(k), K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1Kdk=G$ is the Catalan constant. We can also find, by switching integral trick that $$\int_0^1E(k)dk\\
=\int_0^{\frac\pi 2}\int_0^1\sqrt{1-k^2\sin^2\theta}d\theta\\
=\int_0^{\frac\... |
Both methods give the same results. The first method, which I used in https://math.stackexchange.com/questions/4751226/find-the-complete-integral-of-following-partial-differential-equation-z-p2?noredirect=1&lq=1, gives the following result:
$$
z=\frac{1}{2}(x+a)^2+\frac{1}{2}(y+b)^2+ab. \tag{1}
$$
Now, let's use th... |
Let $E=E(k), K=K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1K\,dk=G$ is the Catalan constant. We can also find, by switching integral trick that $$\int_0^1E\,dk\\
=\int_0^{\frac\pi 2}\int_0^1\sqrt{1-k^2\sin^2\theta}d\theta\\
=\int_0^{\fr... |
Are all constants of integration $C$ equal? |
after some helpful feedback on my first question, I will repost my question with some more additional information. <p>
For a programming project, I need to calculate the L-series to a Groessencharacter. In order to use some built-in SAGE functions for certain Zetafunctions, I have the following problem:
Given a field... |
After some helpful feedback on my first question, I will repost my question with some more additional information. <p>
For a programming project, I need to calculate the L-series to a Groessencharacter. In order to use some built-in SAGE functions for certain Zetafunctions, I have the following problem:
Given a field... |
After some helpful feedback on my first question, I will repost my question with some more additional information. <p>
For a programming project, I need to calculate the L-series to a Groessencharacter. In order to use some built-in SAGE functions for certain Zetafunctions, I have the following problem:
Given a field... |
I have the following integral.
$$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$
with
$$ \alpha \; \epsilon \; \mathbb{R}^+, \quad q, b, c \; \epsilon \; \mathbb{R}, \quad j = \sqrt{-1}$$
I have asked this before in different forms, but I have settled down with the following approxi... |
$\Delta_0$-separation admits a rather elegant semantic interpretation. We consider a first-order theory of categories with terminal elements. The sole predicate is equality on morphisms. A $\Delta_0$ formula in this theory is one in which for every quantifier there is an object $X$ for which the variable ranges over th... |
$\Delta_0$-separation admits a rather elegant semantic interpretation. We consider a first-order theory of categories with terminal elements. The sole predicate is equality on morphisms. A $\Delta_0$ formula in this theory is one in which for every quantifier there is an object $X$ for which the variable ranges over th... |
Suppose that we have an arithmetic function $f(x)$ defined as follows:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/B0D3S.png
What are the methods in the literature that will make this function continuous and differentiable? However, it should be noted that this function can be... |
How to make arithmetic function continuous? |
When we want to know why the Author wrote something is a particular way , we have to take the (w)holistic view : What were the Axioms the Author used , What were the Definitions used , How were earlier theorems developed , Where will the current theorem get used ....
**Here there are a couple of reasons why it i... |
---
I was reading Princeton companion and I got the following at page no.160
Take two parallel planes, each punctured at n points.
>"Label the holes 1 to n in each plane, and run a string
from each hole in the first plane to one in the second,
in such a way that no two strings go to the same hole.
The resu... |
Okay, so I was asked this question in an interview on a machine learning expert position. To be honest, the question itself (and the hint by the interviewer) seemed quite ill-phrased, which probably is the reason I ended up failing the interview, and he thought I must be super dumb. Here is the original question.
>... |
What Is the Probability The Second Kid Is a Boy? |
Probability of 9 people going into 3 rooms with 4,3,2 peoplerespectively if the room can and cannot be differentiated
I would like to verify my thinking logic on this question.
For the case where the room can be differentiated, I take the sample space to be $3^{9}$, which is the number of ways the 9 people can go... |
When we want to know why the Author wrote something is a particular way , we have to take the (w)holistic view : What were the Axioms the Author used , What were the Definitions used , How were earlier theorems developed , Where will the current theorem get used ....
**Here there are a couple of reasons why it i... |
How to evaluate the following integral/
$$\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k+x^m}\, dx $$
For the case where $\beta=1$ and $m=2$, we can consider the following integral:
$$
\begin{aligned}
I(\alpha, k) & =\int_0^{\infty} \frac{x^\alpha}{k^2+x^2} d x \\
& =\frac{k^{\alpha-1}}{2} \int_0^{\in... |
Let $E=E(k), K=K(k)$ be the complete elliptic integrals of the second and the first kind respectively. It is well-known that $\frac12\int_0^1K\,dk=G$ is the Catalan constant. We can also find, by switching integral trick that
\begin{align}
\int_0^1E\,dk
&=\int_0^{\frac\pi 2}\int_0^1\sqrt{1-k^2\sin^2\theta}d\theta\\... |
Here are two related theorems. I have trouble proving the second one.
>**Monotonicity theorem**: Let $f$ be a real valued function defined on $\mathbb R$. The function is continuous at a point $a$ and $f$ is monotonically increasing on $(a-r,a]$ and monotonically decreasing on $[a,a+r)$. Then, $f$ has a local maximu... |
First derivative test - how to prove it from the "monotonicity theorem"? |
This is a question from very (very) basic calculus, but it concerns indefinite integrals and the constant $ C $ we always add when we find the antiderivative. This concerns some problems with subtracting two indefinite integrals.
Say we have $ f(x) = \sin(\theta) $
And we have $ h(x) = \cos(\theta) $
(These a... |
This is a question from very (very) basic calculus, but it concerns indefinite integrals and the constant $ C $ we always add when we find the antiderivative. This concerns some problems with subtracting two indefinite integrals.
Say we have $ f(\theta) = \sin(\theta) $
And we have $ h(\theta) = \cos(\theta) $
... |
Given a symplectic matrix $S \in \text{Sp}(2n,\mathbb Z)$ whereby $S^T\Omega S=\Omega$ with
$$\Omega=\left(\begin{matrix}0&I_n\\-I_n&0\end{matrix}\right)$$
what known decompositions exist such that the matrix $S$ is decomposed into $S=ABC$ whereby $A,B,C \in \text{Sp}(2n,\mathbb Z)$?
When $S'\in\text{Sp}(2n,\math... |
This is a question from very (very) basic calculus, but it concerns indefinite integrals and the constant $ C $ we always add when we find the antiderivative. This concerns some problems with subtracting two indefinite integrals.
Say we have $ f(\theta) = \sin(\theta) $
And we have $ h(\theta) = \cos(\theta) $
... |
Is $\mathbb{Q}(\sqrt(2)\sqrt[3](5)) \subset \mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$?
I've tried writing any element $x \in \mathbb{Q}(\sqrt(2)\sqrt[3](5)$ with $x = a + b\sqrt(2)\sqrt[3](5)$ in terms of $a + b(\sqrt{2}+\sqrt[3]{5})$, but I can't find a way. Anyone know how to approach this? |
Is $\mathbb{Q}(\sqrt(2)\sqrt[3](5) \subset \mathbb{Q}(\sqrt{2}+\sqrt[3]{5})$? |
I would like to show that the bilinear form $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), where $G(x)\in\mathbb{R}^{n\times n}$ is positive definite - or find a counterexample if it does not hold.
I tried transforming to the associated quadratic forms $$A = \frac{1}{4}(x+\tanh(x))^T G(x) (x+\tanh(... |
How to formally define this squared matrix? |
How to evaluate the following integral/
$$\int_{0}^{\infty} \, \frac{x^\alpha \log^n{(x^\beta)}}{k^m+x^m}\, dx $$, $\alpha, \beta, m, n$ are real and $k$ is a positive integer.
For the case where $\beta=1$ and $m=2$, we can consider the following integral:
$$
\begin{aligned}
I(\alpha, k) & =\int_0^{\infty} \frac... |
[![Graph that is to be assessed][1]][1]
[1]: https://i.stack.imgur.com/XNEAk.png
Certain necessary conditions for a Hamiltonian circuit such as the graph being 2-connected, having zero pendants are met. Dirac's and Ore's theorem provide sufficient conditions, which are not satisfied by this graph, so it may p... |
Does the graph have a Hamiltonian circuit or a Hamiltonian path? |
I have encountered the following problem studying some practical applications of scheduling systems.
Given a set of symbol types *s1...sn* with associated frequencies of occurrence *f1..fn*, distribute the *f1+...+fn* symbols in a cyclic sequence such that uniformity in the distribution of each symbol type is maximi... |
I want to solve and draw the solution of PDE such that
\begin{align}
\rho_t +2\rho\rho_x=0
\end{align}
and with initial condition
\begin{align}
\rho(x,0)=
\begin{cases}
1 & \text{if $x<0 $}\\
2 & \text{if $0\le x \le1$}\\
1 &\text{if $x >1$}\\
\end{cases}
\end{align}
This is a form of Cauc... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
I have the following integral.
$$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$
with
$$ \alpha \; \epsilon \; \mathbb{R}^+, \quad q, b, c \; \epsilon \; \mathbb{R}, \quad j = \sqrt{-1}$$
I have asked this before in different forms, but I have settled down with the following approxi... |
Why does this sum diverge and does not converge towards 0?
$\sum_{n=0}^{+\infty}(-1)^{n}$
I assume it's because there is no way to define how we count it I guess. However, is there a domain in mathematics where we would accept that this series is equal to zero, e.g.
$\sum_{n=0}^{+\infty}(-1)^{n}=\sum_{n=0}^{+\... |
Where does $\sum_{n=0}^{+\infty}(-1)^{n}$ converge? |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
I'll try to answer your question in case I understood you right.
What about injectivity? If $f \in L^2(\mathbb{R})$ and $T_af = 0$ then for any $x \in \mathbb{R}$ we have $f(x-a)=-f(x+a)$ which immediately yields that $f$ is periodic with period $4a$. Consequently, $f = 0$ because there are no periodic functions in ... |
I have the following integral.
$$ G(q) = \int_{0}^{\infty} y^\alpha \exp(-y) \exp(-j b q\exp(-c y)) dy $$
with
$$ \alpha \; \epsilon \; \mathbb{R}^+, \quad q, b, c \; \epsilon \; \mathbb{R}, \quad j = \sqrt{-1}$$
I have asked this before in different forms, but I have settled down with the following approxi... |
The **least upper bound** property of the real numbers (lub) can be used to prove the **intermediate value theorem (IVT)** as well as the **Bolzano-Weierstrass** theorem (BWT).
I've been trying to prove the IVT without the lub property, using the BWT as an axiom. The sketch of my proof goes as follows:
**Lemma**:... |
I'm sorry if this is a noob question but I got this statement asked on a test:
$$\forall\ x \exists y \,F(x,y) \rightarrow \exists y \forall x \, G(x,y)$$
where
$$M = \lbrace \text{Natural numbers > 0} \rbrace \\
F(x,y): \, x <y \\
G(x,y): \, x > y\\$$
The statement is supposed to be false. I get that the... |
Let two vector spaces $V$,$W$ over the field of real numbers $\mathbb{R}^d$ (or just over $\mathbb{R}$ to make things easier).
Is there a unique isomorphism between $V \otimes W$ and $\mathbb{R}^{d^2}$?
ie, can we interpret $V \otimes W$ as being a vector space on $\mathbb{R}^{d^2}$.
So on the easiest case scena... |
Is the tensor product of two vector space on R^d isomorphic to R^d²? |
If I have a $n\times n$ unimodular matrix $A \in \text{GL}(n,\mathbb Z)$, i.e., with elements $A_{ij} \in \mathbb Z$, is there some way to decompose the matrix into a product of matrices $A=A^{(1)}A^{(2)} \dots A^{(k)}$ such that each $A^{(l)}$ has elements $(A^{(l)}_{ij}-d/2 )\mod d= A^{(l)}_{ij}$? |
In Excercise 10.4 (p.243) in Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, it writes
Let $a,b$ be a regular sequence in domain $R$, and let $S=R[x]$ be the polynomial ring in one variable over $R$.
Show that $ax-b$ is a prime ideal of S (See Excercese 17.2 for a sort of converce . Ge... |
Is the tensor product of two vector space on $R^d$ isomorphic to $R^{d^2}$? |
In Excercise 10.4 (p.243) in Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, it writes
Let $a,b$ be a regular sequence in a domain $R$, and let $S=R[x]$ be the polynomial ring in one variable over $R$.
Show that $ax-b$ is a prime ideal of S (See Excercese 17.2 for a sort of converce . ... |
Let $V$,$W$ be two fine-dimensional vector spaces over the field of real numbers $\mathbb{R}$. Assume the dimension of both spaces is d.
Is there a unique isomorphism between $V \otimes W$ and $\mathbb{R}^{d^2}$?
ie, can we interpret elements of $V \otimes W$ as being vectors in a $\mathbb{R}^{d^2}$ space?
So ... |
Is the tensor product of two vector space on $R$ isomorphic to $R^{d^2}$? |
Since you say you are mostly interested in squaring a polynomial, have you ever tried using the "box method"? It's often used for small polynomials, but can be used for any number of terms (that can be written on a piece of paper). See, for example, [YouTube](https://youtu.be/3OvHmKjVTGU?si=UGHd9qsqrpApTh1C)
On a gr... |
If I have two complex numbers $ a,b \in \mathbb{C} $，and $ |a|^2 + |b|^2 = 1 $，I need to get the max of
\begin{aligned}
(aa^*-bb^*)^2 - [(ab^*)^2 - (a^*b)^2]^2
\end{aligned}
In the answer book, the auther let
\begin{aligned}
a = \cos \frac{\beta}{2}
\\
b = \sin \frac{\beta}{2} e^{i\alpha}
\end{aligned}
the... |
How to get the max of $(aa^*-bb^*)^2 - [(ab^*)^2 - (a^*b)^2]^2$? |
I have been working though CSE 373 Lectures by Skiena and I cannot understand his explanation of why multiplication by a constant does not change the asymptotics:
$O(c*f(n)) \to O(f(n))$
In the lectures, Skiena gives the example with $c = 100$ and $f(n) = n^3$. Next, he proceeds with showing that there is a cons... |
Let $X$ be a positive random variable. I know that $\mathbb{P}[X \leq a] \geq q$. Any hints about how to find an upper bound on $E[X]$ in terms of $q$ and $a$?
Using Markov inequality, I can obtain $E[X] \geq a(1-q)$. But I need an upper bound, not a lower one. Any hints? Thanks! |
Say I have a $\sigma$-algebra $\mathcal{F}$ and its completion $\mathcal{F}^{\ast}$. If a set $E\in \mathcal{F}^{\ast}$ but $E\notin \mathcal{F}$, does that mean that I definitely have sets $A,B\in\mathcal{F}$ s.t. $A\subset E\subset B$ and $\mu\left(A\backslash B\right)=0$?
That is how we defined the $\sigma$-algeb... |
Given your colleague has **two kids**, and **hiddenly** adopting binary biological sex (each kid is either a boy (B) or a girl (G)), the **sample space** is
$$\Omega=\{(G,G), (G,B), (B,G), (B,B) \}$$
with the following probability measure:
$$\mathbb P ((G,G))=p^2, \mathbb P ((G,B))=\mathbb P ((G,B))=p(1-p), \m... |
$\Delta_0$-separation admits a rather elegant semantic interpretation. We consider a first-order theory of categories with terminal elements. The sole predicate is equality on morphisms. A $\Delta_0$ formula in this theory is one in which for every quantifier there is an object $X$ for which the variable ranges over th... |
Since you say you are mostly interested in squaring a polynomial, have you ever tried using the "box method"? It's often used for small polynomials, but can be used for any number of terms (that can be written on a piece of paper). See, for example, [YouTube](https://youtu.be/3OvHmKjVTGU?si=UGHd9qsqrpApTh1C)
On a gr... |
A common schema in intuitionistic, constructive set theories is that of *$\it{\Delta_0}$ separation*:
$$
\forall x\exists y\forall z(z\in y\leftrightarrow z\in x\wedge\phi(x, z))\text{ provided $\phi$ is a $\Delta_0$ formula}
$$
So chosen as $\Delta_0$ sets are clearly predicatively defined, and full separation... |
I have been working though CSE 373 Lectures by Skiena and I cannot understand his explanation of why multiplication by a constant does not change the asymptotics:
$O(c*f(n)) \to O(f(n))$
In the lectures, Skiena gives the example with $c = 100$ and $f(n) = n^3$. Next, he proceeds with showing that there is a cons... |
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