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Is $\exists x [(P(x) \vee Q(x))\rightarrow R(x)]$ logically equivalent to $\exists x [(P(x) \rightarrow R(x)) \vee (Q(x)\rightarrow R(x))]$?
What about if I replace $\exists$ with $\forall$? |
Is $\exists x [(P(x) \vee Q(x))\rightarrow R(x)]$ logically equivalent to $\exists x [(P(x) \rightarrow R(x)) \vee (Q(x)\rightarrow R(x))]$? |
What is the meaning of "on" in the expression "operation **on** a set $A$"?
Does it mean that:
- at least one of the operands is in $A$;
- all the operands are in $A$; or
- all the operands are in $A$ and the codomain is $A$?
(There are some similar questions, but they are specific to *binary* operations. A... |
Following this related [Example of tensor (0,2)
acting on two vectors][1] (Nick's answer). I want to ask why in the definition we consider the map $(f\otimes g)\colon V\otimes W \to X\otimes Y$ and not $(f\otimes g)\colon V\times W \to X\otimes Y$
as $(f\otimes g)(a,b)=f(a)\otimes g(b)$ ?
Why we need $V\otimes ... |
From the note: https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/comment-page-1/#comment-682897.
In the coupon collector problem in which one considers an infinite sequence of “coupons” that are iid and uniformly distributed from the finite set ${\{1,\dots,N\}}$, let ${T... |
I used boolean algebra to simplify an expression with $3$ variables. After simplifying, it reduces to $2$ variables. The first truth table has $8$ rows and the second one has $4$. How to prove that they're logically equivalent?
Considering $+$ as or, . as and $\sim$ as not.
The original expression is:
$$F = \s... |
I'm rendering rhombus textures in a $3D$ world:
[![Rhombus textures][1]][1]
Here are the rhombus textures rendered in a grid in perspective projection with $FOV = 45^\circ$:
[![Rhombus textures rendered as a grid in perspective projection][2]][2]
Here is the same grid scene, but in dimetric orthographic project... |
Let $\text{circ}(x, y; R, c)$ be the characteristic function on the plane of a disk of radius $R$ and centre $c=(c_1,c_2)$:
$$\text{circ}\left(\frac{\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}}{R}\right)=\begin{cases}1 & \text{if }\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}< R, \\ \frac 12 & \text{if }\sqrt{(x_1-c_1)^2+(x_2-c_2)^2}=1, \\ 0 &... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Let $f:X\mapsto Y$ be a function between two sets. I am wondering if there is a case where $f^{-1}: \mathscr P(Y) \mapsto \mathscr P(X)$ can not be injective ?
I feel like the fact that $f$ can only give one image to an element of $X$ makes it compulsory for $f^{-1}$ to be injective.
Many thanks |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
I've got a problem (da Silva homework 3, question 3) that I have rewritten as the following:
Let $g:M\to M$ be a symplectomorphism such that it takes $(q,p)\to (q',p')= (f(q,p), h(q,p))$.
I want to show that $\partial f/\partial p=0$ and $\partial^2h/\partial p^2=0$.
I have by the assumption that $g^{*}\alpha... |
Let $f:X\to Y$ be a function between two sets. I am wondering if there is a case where $f^{-1}: \mathscr P(Y) \to \mathscr P(X)$ can not be injective ?
I feel like the fact that $f$ can only give one image to an element of $X$ makes it compulsory for $f^{-1}$ to be injective.
Many thanks |
Consider a nonempty subset $A$ of a smooth manifold $M$ and suppose that every smooth function on $A$ can be extended to a smooth function on $M$. We want to show that $A$ is closed. Is the following proof correct?
Since $M$ is a manifold, $A$ is Hausdorff. Thus $A$ is closed iff it is compact. To construct a finite... |
Let $|\psi_1 \rangle, \dots, |\psi_n \rangle, |\phi_1 \rangle, \dots, |\phi_n\rangle \in \mathbb{C}^m$. Show that if $\langle \psi_i | \psi_j \rangle = \langle \phi_i | \phi_j \rangle$, then there always exists a unitary $U$ such that $U|\psi_i\rangle = |\phi_i \rangle$.
If $n = m$ and $|\psi_i\rangle$ and $|\phi_i\... |
From the note: https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/comment-page-1/#comment-682897.
In the coupon collector problem in which one considers an infinite sequence of “coupons” that are iid and uniformly distributed from the finite set ${\{1,\dots,N\}}$, let ${T... |
Find the domain of:
$f(x) = \frac{1}{(g+1)x^2 + 2(g-1) + g-3}$
for the various values of g$\epsilon R$
I am trying to solve this. First of all i take the g value equal to -1 so the domain is R.<br>
<br>
The next thing that i need to do is to take the g value not equal to -1. <br>
<br>
The discriminant mus... |
Recall the distribution property:
$$a(b+c) = ab + ac$$
In this case, we can distribute the factor of $2k+1$:
$$(2k+1)^2 = (2k+1)(2k+1) = (2k+1)2k + (2k+1)1$$
You can already at this point see why we would have $1$ as your constant: the first term will have $k$'s in it because of the $2k$ factor, while the l... |
The following is a follow-up to this [previous question](https://math.stackexchange.com/questions/4861221/show-that-partial-mu-phi-ast-a-mu-phi-a-mu-phi-ast-partial-mu-phi-a-mu).
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\parti... |
Consider the following function:
$$ f(x,y) = \frac{1}{2}x^2 + \frac{1}{2}y^2 + \sum_{i=1}^{n}\ln\big(\cosh(q_{i})\big) $$
where $q_{i} = \sqrt{(x-x_{i})^2 + (y-y_{i})^2} - L, \ L>0 \ (\text{constant})$. Suppose $x_{i}, y_{i}$ are known points in the 2D space. I would like to understand what does the function $f(x... |
What does a specific function represent in the 2D space? |
This question is a bit related to the question https://math.stackexchange.com/questions/307522/to-what-extent-is-a-scheme-morphism-determined-by-its-topological-map.
Let $X, Y$ be a separated geometrically reduced scheme over a field $k$. Whenever I have a morphism $X \to Y$ I get a morphism $X(k^{\text{alg}}) \to Y... |
If $a_n$ is a decreasing sequence of real numbers and $a_0=1$. How to prove the minimum value of $\sum a_n^2/a_{n+1}$ is 4 without using Cauchy Schwarz inequality? Here's what I got:
If $a_n=1/2^n$, then $a_n^2/a_{n+1}>1/2^n$. I try to prove the minimum is attained when sequence $a_n=1/2^n$ using induction but I fai... |
Find the domain of:
$$f(x) = \frac{1}{(g+1)x^2 + 2(g-1) + g-3}$$
for the various values of g $\epsilon R$
I am trying to solve this. First of all, I take the $g$ value equal to $-1$ so the domain is $R$.<br>
<br>
The next thing that I need to do is to take the g value not equal to $-1$. <br>
<br>
The disc... |
Let $G$ be a locally compact Hausdorff topological group with Haar measure $\mu$. Let $\varphi \colon G \to GL(V)$ be a continuous representation of $G$, where $V$ is a Hilbert space over $\mathbb C$. Let $C_c(G)$ be the vector space of compactly supported functions $G \to \mathbb C$. Then I've seen the definition
$$
... |
While working on a project, I encountered [a paper that includes this sum][1]:
$$
r(ξ, \bar{ξ}) = \frac{1}{\#} \sum_{g ∈ } r_0(g(ξ), g(\bar{ξ}))
$$
where
- $$ is "the tetrahedral group", "a discrete subgroup of isometries $ ⊂ O(3)$";
- $r_0(R) = -\frac{a + b R^2 + (3 - b) R^4 + (1 - a) R^6}{(1 + R^2)^3} + c... |
This question is a bit related to the question https://math.stackexchange.com/questions/307522/to-what-extent-is-a-scheme-morphism-determined-by-its-topological-map.
Let $X, Y$ be separated geometrically reduced schemes over a field $k$. Whenever I have a morphism $X \to Y$, I get a morphism $X(k^{\text{alg}}) \to Y... |
Find the domain of:
$$f(x) = \frac{1}{(g+1)x^2 + 2(g-1)x + g-3}$$
for the various values of $g\in \mathbb{R}$
I am trying to solve this. First of all, I take the $g$ value equal to $-1$ so the domain is $\mathbb{R}$.<br>
<br>
The next thing that I need to do is to take the g value not equal to $-1$. <br>
<... |
Is there a monotone function that is not differentiable everywhere? |
I've recently encountered the following puzzle that seems a bit too "mathy" for Puzzling Stack Exchange so I decided to ask here. This is not a homework.
The numbers from $1$ to $15$ must be put in a sequence with the two numbers fixed:
* The $4$th number is $15$.
* The $8$th number is $8$.
($1$-based counting)... |
If
>We interpret the question as asking for the probability that **precisely two** people share a birthday.
>Imagine lining up our $n\ge 3$ people, and writing down their birthdays. Under the usual not quite accurate assumptions, there are $365^n$ equally likely strings.
>We now count the number of... |
Let $f$ be:
\begin{cases}
0 \space\space if\space\space x \in \mathbb Q \\
1\space\space if \space\space x \notin \mathbb Q \\
\end{cases}
The question wants me to find any values $a$ such that $f$ is continuous at $x = a$. This was my proof:
Suppose $f$ is continuous, so $\lim\limits_{x\to ... |
Let $G$ be a locally compact Hausdorff topological group with Haar measure $\mu$. Let $\varphi \colon G \to GL(V)$ be a continuous representation of $G$, where $V$ is a Hilbert space over $\mathbb C$. Let $C_c(G)$ be the vector space of compactly supported functions $G \to \mathbb C$. Then I've seen the definition
$$
... |
The following is a follow-up to this [previous question](https://math.stackexchange.com/questions/4861221/show-that-partial-mu-phi-ast-a-mu-phi-a-mu-phi-ast-partial-mu-phi-a-mu).
>Consider the following Lagrangian:
>$$\mathcal{L}=\frac14\left(\partial_\mu A_\nu-\partial_\nu A_\mu\right)\left(\partial^\mu A^\nu-\parti... |
According to everything you did up until the last sentence, the answer should be
$${4 \choose 2} \cdot 365 \cdot 364 \cdot 363$$
What makes you think otherwise?
If you used $364 \choose 2$ to represent the number of ways in which you can pick $2$ birthdays for the $2$ people with distinct birthdays (out of th... |
I have the following system of equations:
$$ 0 = \left[ (p_{31} - p_{11})^2 - \omega_{33} (p_{21} - p_{11})^2 \right] x^2_1 + \left[(p_{32} -p_{12} )^2 - \omega_{33} (p_{22} - p_{12})^2 \right] x_2^2+ \left[(p_{33} - p_{13})^2 - \omega_{33} (p_{23} - p_{13})^2 \right]x_3^2$$
$$0 = \left[ (p_{41} - p_{11})^2 - \om... |
In naive set theory, *everything* may be member of a set. $\{\{,\}\}$ heavily abuses this by using things that are meant only for syntax (the comma, the curly brackets) as elements, thus making the whole thing ambiguous (it could denote the set of the set with the comma or the set containing a left and a right curly br... |
By using $(71)$ and $(70)$ identities of Mathworld's [hypergeometric function][1] page,
$$_2F_1(1,1;\tfrac12;\tfrac12)\stackrel{(71)}{=}2\,_2F_1(1,-\tfrac12;\tfrac12;-1)\stackrel{(70)}{=}2(\tfrac{\sqrt\pi}2\tfrac{\Gamma(\tfrac12)}{\Gamma(1)}+1)=\pi+2.$$
But, WolframAlpha calculates it as $\tfrac{\pi}2+2.$ Where is m... |
I have the following system of equations:
$$ 0 = \left[ (p_{31} - p_{11})^2 - \omega_{11} (p_{21} - p_{11})^2 \right] x^2_1 + \left[(p_{32} -p_{12} )^2 - \omega_{11} (p_{22} - p_{12})^2 \right] x_2^2+ \left[(p_{33} - p_{13})^2 - \omega_{11} (p_{23} - p_{13})^2 \right]x_3^2$$
$$0 = \left[ (p_{41} - p_{11})^2 - \om... |
I have the following system of equations:
$$ 0 = \left[ (p_{31} - p_{11})^2 - \omega_{22} (p_{21} - p_{11})^2 \right] x^2_1 + \left[(p_{32} -p_{12} )^2 - \omega_{22} (p_{22} - p_{12})^2 \right] x_2^2+ \left[(p_{33} - p_{13})^2 - \omega_{22} (p_{23} - p_{13})^2 \right]x_3^2$$
$$0 = \left[ (p_{41} - p_{11})^2 - \om... |
In section 3(Page 50) of this [paper][1], it is mentioned that the Kummer map, $\hat{E}(\mathfrak{m}_n)\rightarrow H^{1}(k_n,T)$ along with the Weil pairing induces a cup product of Galois cohomology groups. How is the Kummer map defined? I know that $\hat{E}(\mathfrak{m}_n)$ injects into $E(k_n)$ and then somehow the ... |
I read in a resolution of an exercise that the splitting field of the polynomial $ f(x)=x^{4}+2x^{3}-1 $ over $ \Bbb Z_{3}[x]$ is obtained expanding $\Bbb Z_{3}$ with any root of the polynomial $f(x)$
could someone give me an explanation of this? |
This is a question in a quant interview. Given a bag of $n$ marbles each of distinct weights, perform the following process. Pick two marbles at random, keep the heavier one and return the lighter one. Then, repeat the following process. Pick one marble at random from the rest, and compare it to the marble at hand. Kee... |
Let $\Sigma$ be an arbitrary alphabet, $\mathcal{P}$ denote the set of all prime numbers, and $\omega := \mathbb{N} \cup \{0\}$ Take the following set.
$$
L = \left\{ (x, \alpha, \beta) \in \omega \times \Sigma^{*2} : \left( \exists t \in \mathcal{P} \right) ~ \alpha^{(x-2)(|\alpha| - 1)} = \beta^t \right\}
$$
... |
Say there are 2 players. Each is given one card, and the person with the higher number wins. The first player goes and can either bet 1 dollar or fold (giving the pot to the second player). If the first player bets 1 dollar, the second player can either call by putting 1 dollar in the pot, or fold giving the pot to the... |
If
>We interpret the question as asking for the probability that **precisely two** people share a birthday.
>Imagine lining up our $n\ge 3$ people, and writing down their birthdays. Under the usual not quite accurate assumptions, there are $365^n$ equally likely strings.
>We now count the number of... |
When people talk about Chern classes of $U(1)$ principal bundles, do they really mean the Chern class of the associated vector bundle? As far as I can understand, Chern classes exist only for complex vector bundles, which $U(1)$ is not, so I don't understand why it makes sense to talk about Chern classes/numbers for a ... |
If Chern classes are only defined for vector bundles, why can a $U(1)$ principal bundle have associated Chern classes? |
This is a question in a quant interview. Given a bag of $n$ marbles each of distinct weights, perform the following process. Pick two marbles at random, keep the heavier one and return the lighter one. Then, repeat the following process. Pick one marble at random from the rest, and compare it to the marble at hand. Kee... |
In the Q. 4265006 (https://math.stackexchange.com/questions/4265006/l2-convergence-for-a-simple-function-approximation) at his question related to solve "Problem 2.19." (probable copied from a manual/book)
received the following solution:
Hint: If $g \in L^{2}(0,1)$ and $g$ is orthogonal to $I_{j,n}$ for all... |
Is the set of linear combinations of the functions $I_{j,n}$ n \in N, dense in $L^{2}$? |
As you asked, I will convert my comments in an answer.
I guess you can prove the theorem showing directly that $\beta$
is bijective but what's the point of that? I mean, you have defined tensor product using an extremely powerful universal property, why you don't want to use it? If you think about the contruction ... |
A presentation of a affine complex variety consists of finitely many polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_n]$. A presentation of a projective complex variety consists of finitely many homogeneous polynomials $f_1,...,f_m$ in $\mathbb{C}[x_1,...,x_{n}]$.
Is there something similar for smooth manifolds ... |
Can someone tell me how the pdf of noise (w) is equivalent to the conditional pdf of observations (x) given A, assuming noise is independent of A for the equation x[n]=A+w[n] where A is the mean (and a random variable)?
This is done in order to determine the posterior pdf using Bayes' rule.[![From the book "Fundame... |
Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in \mathbb{R}^n$ and $\mathrm{g}$ is differentiable at some point $\mathrm{f(a)} \in \mathbb{R}^n$. Let us denote the Jaco... |
Let, in the projective plane, $A,B,C,D,E$ and $F$ be the intersection points of the sides of a quadrilateral (4 lines, no 3 of which are concurrent with the same point) so that the sides are $ABF$, $ADE$, $DCF$ and $BCE$ and similarly $A',B',C',D',E'$ and $F'$ be the intersection points of the sides of a second quadri... |
Let $(X,\mathcal B(X),\lambda)$ be some Euclidean space $X\subset \mathbb R^n$ equipped with the Lebesgue measure $\lambda$. Suppose we have a Fredholm integral operator $$T:L^1(X)\to L^1(X):f(\cdot)\mapsto\int_XK(\cdot,y)f(y)\ \mathrm dy,$$ where $K$ is some uniformly bounded kernel, *that is not necessarily symmetric... |
So i want to prove that the wave operator $P=\partial_t - \Delta_x$ isn't hypoelliptic in $R \times R^n$.
My attempt is to use the fact that an operator with constant coefficients is hypoelliptic iff (given $E$ fundamental solution of $P$) $\text{sing supp}(E)=\{0\}$
If we call the family of distributions $E_t(\va... |
I want to prove that the wave operator $P=\partial_t - \Delta_x$ isn't hypoelliptic in $\Bbb R \times\Bbb R^n$.
My attempt is to use the fact that an operator with constant coefficients is hypoelliptic iff (given $E$ fundamental solution of $P$) $\text{sing supp}(E)=\{0\}$
If we call the family of distributions $E... |
In a 2015 paper named "Lecture on the abc conjecture and some of its consequences" by Michel Waldschmidt it is stated that:
It can be easily seen that the abc Conjecture is equivalent to the following statement:
For each $\epsilon > 0$, there exists $\kappa(\epsilon)$ such that, for any abc triple $(a, b, c)$,
... |
I was readying the construction of the Tensor Product that was made on the book **Introduction to Smooth Manifolds**, from John M. Lee. In the **proposition 12.7** he proved that his construction satisfied the universal property of the Tensor Product. His proof follows:
[![enter image description here][1]][1]
... |
I've encoutered an interesting question from an old exam with no solution and i was wondering: how do you build a dfa(deterministic finite automata) from given equivalence classes?
this is the question:
L is a language over {0,1}, for which the equivalence classes of $R_L$ are:
$\left\{w|\#\:_0\left(w\right)\:... |
I've encoutered an interesting question from an old exam with no solution and I was wondering: how do you build a dfa (deterministic finite automata) from given equivalence classes?
this is the question:
L is a language over {0,1}, for which the equivalence classes of $R_L$ are:
$\left\{w|\#\:_0\left(w\right)\... |
Consider a nonempty subset $A$ of a smooth manifold $M$ and suppose that every smooth function on $A$ can be extended to a smooth function on $M$. We want to show that $A$ is closed. Is the following proof correct?
Since $M$ is a manifold, $A$ is Hausdorff. Thus if $A$ is closed if it is compact. To construct a fini... |
This is a question in a quant interview. Given a bag of $n$ marbles each of distinct weights, perform the following process. Pick two marbles at random, keep the heavier one and return the lighter one. Then, repeat the following process. Pick one marble at random from the rest, and compare it to the marble at hand. Kee... |
Consider a nonempty subset $A$ of a smooth manifold $M$ and suppose that every smooth function on $A$ can be extended to a smooth function on $M$. We want to show that $A$ is closed. How might we proceed with a proof?
Since $M$ is a manifold, $A$ is Hausdorff. Thus if $A$ is closed if it is compact. To construct a f... |
I won't be adding anything else to the previous answers but a piece of imagination and loose interpretations to bring a physical intuition about.
**A moving point running along the envelope**
If you look carefully the animation of an sliding bar on the [envelope page](https://en.wikipedia.org/wiki/Envelope_(mathe... |
Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in \mathbb{R}^n$ and $\mathrm{g}$ is differentiable at some point $\mathrm{f(a)} \in \mathbb{R}^n$. Let us denote the Jaco... |
Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in \mathbb{R}^n$ and $\mathrm{g}$ is differentiable at some point $\mathrm{f(a)} \in \mathbb{R}^n$. Let us denote the Jaco... |
Imagine a function $f:X\to X$ and $x\in X$ (keeping $f$ and $X$ vague on purpose) and let's define
$u_1 = f (x)$
$u_2=f^2(x)=f(f(x))$
$u_n=f^n(x)$
Let's also assume that $\forall n, u_{n-1} \neq u_n $ (otherwise its boring) and that $lim_{n\to\infty} u_n$ exists.
Let $u_{\omega} = lim_{n\to\infty} u_n$,... |
Let us have functions $\mathrm{f}: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $\mathrm{g}: \mathbb{R}^m \rightarrow \mathbb{R}^k$ such that $\mathrm{f}$ is differentiable at some point $\mathrm{a} \in \mathbb{R}^n$ and $\mathrm{g}$ is differentiable at some point $\mathrm{f(a)} \in \mathbb{R}^n$. Let us denote the Jaco... |
It is a renamed theorem the existence of a nowhere vanishing vector fields on odd dimensional spheres. Now, in his textbook, L. Tu proposes the following exercise:
>Let $x^1,y^1,\dots,x^n,y^n$ be the standard coordinates on $\mathbb{R}^{2n}$ and consider the unit sphere $S^{2n-1}$ (defined by the equation $\sum\limi... |
I checked my work on an online calculator and it's wrong. Can someone explain why I can't solve it this way and how do I solve it?
$$\int{\frac{1}{sin(x)+cos(x)}dx}$$
$${\rightarrow}\space \int{(sin(x)+cos(x))^{-1}dx} $$
$$ u = sin(x) + cos(x) $$
$$du = (cosx - sinx) \space dx$$
$${\rightarrow}\space... |
Why can't I use u-substitution to solve for ∫1/(sinx+cosx) dx? |
Let $I$ be a proper ideal in $R=\mathbb{C}[x,y]$, generated by two elements
$f_1,f_2$, $I=\langle f_1,f_2 \rangle$.
I wonder if the following claim is true or false:
> **Claim:** If for every $c_1,c_2,d,e \in \mathbb{C}$, each of the ideals
$\langle f_1-c_1,f_2-c_2,x-d \rangle$ and $\langle f_1-c_1,f_2-c_2,y-e ... |
Does a closed form expression exist for
$\\\sum_{k=1}^{n}\left(\sqrt{k+\frac{1}{k}}-\sqrt{k-\frac{1}{k}}\right)$
I obtained
$0.97423066ln(n)+1.2019463$
using log regression and it works very well for my (physics) problems but I was wondering if a better solution exists. Is the Euler-Maclaurin method so... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Let $n > 1$ be an integer not a multiple of $10$.
Is there a short proof that $n = 5$ is the only solution to $n^n \equiv n^{n^n} \pmod {10^{n-1}}$, given the fact that $5^{5^5} \equiv 3125 \pmod{10^4}$ (and even $3125 \equiv 5^{5^5} \pmod{10^5}$)? |
> I think $\exp(t)$ converges for every possible input of $t$, but I haven’t found a proof yet.
For all real and complex inputs of $t$, yes. This is not something trivial to show. You need proper background in set theory, how naturals, integers, reals and complex numbers work. You need to understand not only what i... |
Suppose I have a sequence of random variables $X_{n,i}$ with $i=1,...,n$. Each of these random variables is strictly bounded between $-1$ and $1$. I know that each of these random variables is getting closer and closer to $1$ as $n$ grows.
Specifically, I know that
$$\lim_{n \to \infty} \mathbb{E}[\prod_{i=1}^n ... |
Consider the following exercise
> If $X_1,\dots,X_n$ are independent with same distribution and $S_n=X_1+\cdots+X_n$, prove that
> $$\mathbb{E}[X_i \mid S_n]=\frac{S_n}{n}$$
Proof: Since the variables have the same distribution we have that:
$$S_n=\mathbb{E}[S_n \mid S_n]=\mathbb{E}\left[\sum_{i=0}^n X_i \mid S... |
I was reading a proof of
[![enter image description here][1]][1]
It goes as follows
[![enter image description here][2]][2]
where
[![enter image description here][3]][3]
What I am struggling with is the application of corollary 8.3. In particular, I do not understand why $a$ not being in this set implies th... |
Imagine a function $f:X\to X$ and $x\in X$ (keeping $f$ and $X$ vague on purpose) and let's define
$u_1 = f (x)$
$u_2=f^2(x)=f(f(x))$
$u_n=f^n(x)$
Let's also assume that $\forall n, u_{n-1} \neq u_n $ (otherwise its boring) and that $lim_{n\to\infty} u_n$ exists.
Let $u_{\omega} = lim_{n\to\infty} u_n$,... |
I noticed in Linear Control Theory, we may multiply matrices for controllability $\mathcal{C} \in \mathbb{R}^{n \times np}$ and observability $\mathcal{O} \in \mathbb{R}^{nq \times n}$ as follows $\mathcal{O} \mathcal{C} \in \mathbb{R}^{np \times nq}$. I do know, the independent columns of both matrices span respectiv... |
This is Exercise 2.3.3 in Vershynin's book.
>Let $X\sim \operatorname{Pois}(\lambda)$. Show that for any $t > \lambda$, we have
$$P(X\ge t) \le e^{-\lambda}\left(\frac{e\lambda}{t}\right)^t.$$
Hint: Combine Chernoff’s inequality with the Poisson limit theorem.
**My work:**
$$P(X\ge t) = P(e^{sX}\ge e^{st}) \le ... |
The statement in the title of this question is wrong. Consider $A=M=\mathbb{R}$. Then, clearly any smooth function on $A$ can be extended to one on $M$. But $A$ is not compact.
Of course, $A$ is still closed. Hence, this example does not disprove the statement from your post. It does, however, show that your approac... |
I've been studying about Simpson's paradox, and encountered this question:
[![Question][1]][1]
Why must the positive value be less than 2.7%?
According to the explanation, people can move between groups, which generated a total increase of 0.8% across those years.
Imagine however one person in every group, ex... |
Hello I am working on the following question:
For $c = \frac{2-i}{8}\in\mathbb{C}$, define $z_0 = 0$ and $z_{n+1} = z_n^2 + c$. We want to determine if this converges. I am trying to do it this way:
Suppose the limit exists, call it $L$, then $L = L^2 + c$, solving it there are two possible roots $w_1 = \frac{3}{... |
Below, I will provide three explicit infinite sums with irrational summands that converge to rational numbers. They are examples of <a href="https://en.wikipedia.org/wiki/Rational_zeta_series">rational zeta series</a>, and they all satisfy the additional requirements as stated by the OP.
Let $\zeta(\cdot)$ be the <... |
So I am trying to prove the expression $A^{-1}+B^{-1}$ is non-singular by showing that $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, given $A$, $B$, and $A+B$ are all non-singular matrices. I encountered a step where I had to rearrange the multiplication of an expression. Specifically, I was supposed to go from $(A^{-1}(A+B))^{... |
This paragraph is at the end of Section 6.3 in the book *Testing Statistical Hypotheses* by Lehmann and Romano:
>In most applications, $M(x)$ is a measurable function taking on values in a
Euclidean space and it is convenient to take $\mathcal B$ as the class of Borel sets. If $\phi(x) = \psi[M(x)]$ is then an arbi... |
Suppose $X_i$ is a random variable from $(\Omega, \mathcal{A},P)$ to $(E_i,\mathcal{E}_i)$.
**Why we can check $X_1, X_2, \cdots, X_n$ are independent random variables in the following way?**
The definition for $X_1, X_2, \cdots, X_n$ are independent random variables is the sigma algebra generated by them (i.e. ... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
Hello I am working on the following question:
For $c = \frac{2-i}{8}\in\mathbb{C}$, define $z_0 = 0$ and $z_{n+1} = z_n^2 + c$. We want to determine if this converges. I am trying to do it this way:
Suppose the limit exists, call it $L$, then $L = L^2 + c$, solving it there are two possible roots $w_1 = \frac{3}{... |
Does a closed form expression exist for
$\\\sum_{k=1}^{n}\left(\sqrt{1+\frac{1}{k}}-\sqrt{1
-\frac{1}{k}}\right)$
I obtained
$0.97423066ln(n)+1.2019463$
using log regression and it works very well for my (physics) problems but I was wondering if a better solution exists. Is the Euler-Maclaurin method ... |
So I am trying to prove the expression $A^{-1}+B^{-1}$ is non-singular by showing that $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, given $A$, $B$, and $A+B$ are all non-singular matrices. I encountered a step where I had to rearrange the multiplication of an expression. Specifically, I was supposed to go from $(A^{-1}(A+B))^{... |
So I am trying to prove the expression $A^{-1}+B^{-1}$ is non-singular by showing that $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$, given $A$, $B$, and $A+B$ are all non-singular matrices. I encountered a step where I had to rearrange the multiplication of an expression. Specifically, I was supposed to go from $(A^{-1}(A+B))^{... |
I checked my work on an online calculator and it's wrong. Can someone explain why I can't solve it this way and how do I solve it?
$$\int{\frac{1}{\sin x+ \cos x}dx}$$
$$\rightarrow\space \int (\sin x+ \cos x)^{-1}~dx $$
$$ u = \sin x + \cos x $$
$$du = (\cos x - \sin x) \space dx$$
$$\rightarrow\spa... |
Why can't I use u-substitution to solve for $\int 1/(\sin x+\cos x)~ dx$? |
I am working through the following exercise:
> Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ be a Riemann integrable function on every $[a,b],$ and such that $\int_{-\infty}^\infty |f(x)|\mathrm{d}x < \infty.$ Show that the Fourier sine and cosine transforms exist. That is, for every $\omega \geqslant 0,$ the fol... |
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