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For $n \in \mathbb{N}$ a given dimension and $m \geq 1$, consider the lower triangular matrix $A(m)\in \mathbb{R}_+^{n\times n}$ defined by
$$A_{i,j}(m) := \dfrac{1}{m+i}\dfrac{1}{m-1+i} \quad \text{ if } i \leq j $$
$$A_{i,j}(m) := 0 \quad \text{ if } i \geq j+1 $$
I am interested in understanding the behavio... |
Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property $P(1)$
2. Inductive step: Prove that for all $k ≥ 1, P(k) ⇒ P(k + 1)$
(a) Why might the above mechanism not constitute a vali... |
Suppose $X_i$ is independently and identically distributed over $i$, and $E(|X_i|^d)$ with $d>0$ is not finite or undefined:
- I am wondering whether $\sum_{i=1}^N |X_i|^d$ is of any asymptotic order. Take [Cauchy distribution](https://en.wikipedia.org/wiki/Cauchy_distribution#Mean) as an example. If $d=1$, then $N^{... |
First i will say that the exact same question has been uploaded to the site already few years ago but it seems no final answer were given there
x = {a,b,c} and y = {1,2,3,4,5,6,7}
i need to count how many injective functions f: x -> y there are such that
f(a)!= 1,2 f(b)!= 2 f(c) != 3
my instictive approach to... |
First i will say that the exact same question has been uploaded to the site already few years ago but it seems no final answer were given there
x = {a,b,c} and y = {1,2,3,4,5,6,7}
i need to count how many injective functions f: x -> y there are such that
f(a)!= 1,2 f(b)!= 2 f(c) != 3
my instictive approach to... |
First i will say that the exact same question has been uploaded to the site already few years ago but it seems no final answer were given there
x = {a,b,c} and y = {1,2,3,4,5,6,7}
i need to count how many injective functions f: x -> y there are such that
f(a)!= 1,2 f(b)!= 2 f(c) != 3
my instictive approach to... |
I have 10 squads but only space for 4 to be in the office on any given day of the week. How to create the optimal rota? |
Given a semi simple Lie algebra's Dynkin diagram/Cartan matrix one can easily find out the weights of any particular representation. The weights of the defining representation are enough to give the mutually commuting i.e. Cartan generators in Cartan-Weyl basis directly. But how do we derive explicit matrix representat... |
How to derive explicit matrix representations for the non-diagonal generators of a simple Lie algebra? |
Prove that that the function $f:\mathbb r^3\to \mathbb R$ defined by $f(x,y,z)=ye^x+xz^2$ is differentiable at the point $\vec a=\langle 0,-4,2\rangle$ using definition.
My attempt:- Let $\vec{h}=\langle h_1,h_2,h_3\rangle.$ Then we need to prove
$$\lim_{\vec{h}\to \vec 0}\frac{f(\vec a+\vec{h})-f(\vec a)-f_x(\ve... |
Working on a long, tedious proof, and just want to check myself.
Given two square symmetric real matrices $A$ and $B$, the claim is that
$$\lambda_{\min}\left(A + B\right) \le \lambda_{\max}\left(A\right) + \lambda_{\min}\left(B\right),$$ where $\lambda_{\min}$ and $\lambda_{\max}$ are the smallest and the largest... |
Is this upper bound on the smallest eigenvalue of $A+B$ correct? |
I have 2 equal sized groups of people and each group selected A, B, or C for 6 questions. The first group was given no guideline and the second group was given a guideline. I have a hypothesis that the guideline caused people to answer the questions a specific way, and thus the answers in the second group would be more... |
Is there a hypothesis test that two entropies are different? |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
Given the sequence $a_n = n(2+(-1)^n)$.
Find $\mathop{\overline{\lim}}\limits_{n\to\infty} a_n$ and $\mathop{\underline{\lim}}\limits_{n\to\infty} a_n$.
The following is how I approach this problem.
$\mathop{\lim}\limits_{n\to\infty} a_{2n} = \mathop{\lim}\limits_{n\to\infty} n(2+1) = \infty$
and
$\matho... |
What is the conventional name/notation for the function $ N(n) = \begin{cases} 1 & n \le 0 \\ n & n > 0 \end{cases}$? |
Prove that that the function $f:\mathbb R^3\to \mathbb R$ defined by $f(x,y,z)=ye^x+xz^2$ is differentiable at the point $\vec a=\langle 0,-4,2\rangle$ using definition.
My attempt:- Let $\vec{h}=\langle h_1,h_2,h_3\rangle.$ Then we need to prove
$$\lim_{\vec{h}\to \vec 0}\frac{f(\vec a+\vec{h})-f(\vec a)-f_x(\ve... |
What's the problem with the fact that $\infty$ is not a constant? For each $n\in\Bbb N$, $a_n\geqslant n$, and therefore $\lim_{n\to\infty}a_n=\infty$. So, $\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=\infty$.
In general, if $l\in\Bbb R\cup\{\pm\infty\}$, then $\lim_{n\to\infty}a_n=l$ if and only if $\limsup_{n\... |
Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}$ be decreasing subsets such that $\cap B_n= x_0\in\mathcal{X}$.
Let $S = \{i\leq n: X_i\in B_n\}$.
I want to show that
$$
\frac{1}{|S|}\sum_{i\in S}Y_i \overset{P}{\to} \mathbb{... |
I am reading an interesting paper [One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational](https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=427&option_lang=eng) by Zudilin. We fix odd numbers $q$ and $r$, $q\geq r+4$ and a tuple $\eta_0,\eta_1,...,\eta_q$ of positive integer parameters satisfying... |
How would I go about proving this? I tried by considering maps from Y to Z and the space of all such maps would have dimY * dimZ but I have no idea how to extend it to the entire vector space.
My guess is that it would be dimY*dimZ + (dim(V/Y))*dimW but I’m not sure about this. |
Let Y and Z be subspaces of the finite dimensional vector spaces V and W, respectively. Let R={α∈L(V,W):α(Y)≤Z}. What is the dimension of R? |
Spherical coordinates will help you to notice that any term containing $h_1^{p_1}h_2^{p_2}h_3^{p_3}$ with $p_1+p_2+p_3>1$ will tend to $0$ in the limit $\vec h\to\vec 0$. How is this important? It means that you can skip all higher order terms, like $h_i^2$ or $h_ih_j$.
$$f(\vec a+\vec h)-f(\vec a)=(-4+h_2)e^{h_1}+h_1... |
# Aside: Higher Order Terms #
> Why do I not need to consider higher order terms? Looking at Taylor Series would it make it more accurate?
This is a largely-unrelated question that I will not address here; I just want to point out that it applies equally well in a single variable case like $\mathrm dy=\dfrac{\mat... |
Let $\Lambda_{n} = \mathbb{Z}^2 \cap [0,n]^2$. Let $P_n$ be the number of paths which start on the left hand side of this box, $\{0\} \times [0,n]$, finish on the right hand side of this box, $\{n\} \times [0,n]$, and do not visit any vertex twice. The paths can move in any direction.
There are some naive bounds we ... |
I'm interested in the ring $(\mathbb Z/30\mathbb Z,+,\times)$, the elements of which I'll write down in bold. For example $$\textbf{7}=7+30\mathbb Z=\{...-83,-53,-23,7,37,67...\}.$$
I'm interested in particular with $$U:=(\mathbb Z/30\mathbb Z)^\times=\{\textbf{1},\textbf{7},\textbf{11},\textbf{13},\textbf{17},\textbf... |
so I wanted to ask if there is anything that allows for this case:
I have 3 statements A, B, and C and I wish to change them from the form:
(A xor B)
(A xor C)
Into
A xor B xor C
Is this possible and how does one do this? Preferably with references as I wish to increase my knowledge.
A little ... |
Regarding the comments, it seems question is about this formulation:
> Let $i : M \to \mathbb{R}^n$ be the inclusion of a smooth submanifold of $\mathbb{R}^n$, $TM$ be the tangent bundle of $M$ and $\nu(M)$ be its normal bundle defined as:
$$
\nu(M) = \{(p,v)\in M \times \mathbb{R}^n \mid \forall w \in T_pM,~ \lan... |
I know this question has been asked before, and I throughly looked at all the answers, but can't find one that matches my math knowledge, or the contents of the course the exercise is derived from.
In $A=\mathbb{C}[x,y,z]$ I have the ideal $I=(xz-y^2,x^3-yz,z^2-x^2y)$. I need to find the height of this ideal. There ... |
It's not a typo.
What is a "tuple"? Normally we think of a "tuple" with index set $I$ as a function with domain $I$. So, viewed as a set of ordered pairs, a tuple $(x_i)_{i\in I}$ would be the set
$$\{ (i,x_i)\mid i\in I\}.$$
For index sets that are of the form $I=\{1,2,3,\ldots,n\}$, we often denote this as $(a_... |
I'm faced with the problem
$\min x \in R^2: (1/2) x' M x$
subject to $a'x = b, c'x = 1, x \ge 0$
where $c = [1, 1, ..., 1]$ and $a = [a_1, a_2, ..., a_n]'$, where $a_j$ are in $R$, $b$ is in $R$ as well, and $M$ is a positive definite real symmetric matrix.
My first idea was setting up the Lagrangian
$L(... |
I understand that they can both be used to mean "if and only if" or similarly link two statements together, but in the general sense, what is the difference between the two symbols ($\longleftrightarrow$ and $\iff$)?
Thank you! |
What is the technical difference between $\longleftrightarrow$ and $\iff$? |
It's not a typo.
What is a "tuple"? Normally we think of a "tuple" with index set $I$ as a function with domain $I$. So, viewed as a set of ordered pairs, a tuple $(x_i)_{i\in I}$ would be the set
$$\{ (i,x_i)\mid i\in I\}.$$
For index sets that are of the form $I=\{1,2,3,\ldots,n\}$, we often denote this as $(a_... |
In case you did not understand the hint, then recall a linear transformation must preserve zero
(That is $T(\textbf{0})=\textbf{0}$, this is because $T(0+y)=T(0)+T(y)$ for any $y\in\mathbb{R}^2$.)
So does your "linear transformation" preserve zero?
In fact, a linear map should have some 'linear structure', si... |
How would I go about proving this? I tried by considering maps from $Y$ to $Z$ and the space of all such maps would have $\dim Y\times\dim Z$ but I have no idea how to extend it to the entire vector space.
My guess is that it would be $\dim Y\times\dim Z + (\dim(V/Y)\times\dim W$ but I’m not sure about this. |
Let $Y$ and $Z$ be subspaces of the finite dim vector spaces $V$ and $W$, respectively. Let $R=\{α\in L(V,W):α(Y)\subset Z]}$. What is $\dim R$? |
Probably not a great question but I was thinking about how to apply multivariable calculus in real life and came up with a question.
If you have a multivariable function with variables $x,y,z$ and a single output.
Does the order of the partial derivatives in the gradient matter? As in, are $(\partial x, \partial y,... |
Let $Y$ and $Z$ be subspaces of the finite dim vector spaces $V$ and $W$, respectively. Let $R=\{α\in L(V,W):α(Y)\subset Z\}$. What is $\dim R$? |
so I wanted to ask if there is anything that allows for this case:
I have $3$ statements $A, B$, and $C$ and I wish to change them from the form:
$(A \operatorname{xor} B)$
$(A \operatorname{xor} C)$
Into
$A \operatorname{xor} B \operatorname{xor} C$
Is this possible and how does one do this? P... |
Are you trying to integrate over a semi-infinite interval, i.e. you want $\int_0^\infty f(x) dx$? Just making the interval really large (e.g. you mention $10^{10}$) is a probably bad idea, because the quadrature routine will start with too few points over that huge interval and have trouble refine.
Instead, one ap... |
When does a translation of a family of vectors span the same space as the untranslated vectors? |
Suppose $f(x) = g(x) h(x)$ and the number of direction changes in both $g$ and $h$ are known. Is there a simple way to prove that the number of direction changes in $f$ is bounded by the number of direction changes of $g$ and $h$?
As a simple case, consider $h$ as monotonic - it seems intuitive that $f$ cannot have ... |
I'm studying general topology and a question has come to my mind.
We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.
I can see the ambiguity of this d... |
As a continuation of this question relating the [Minimum $k$ for which every positive integer of the interval $(kn, (k+1)n)$ is composite](https://math.stackexchange.com/questions/4877970/minimum-k-for-which-every-positive-integer-of-the-interval-kn-k1n-is-co) and this other one on the [divisibility of numbers in inter... |
I am currently solving the heat equation in cylindrical coordinates. I have found my general solutions using separation of variables. I was wondering, when do I apply the initial and boundary conditions? Is it when each solution is separated or when I bring them together of the form u(rho,z,phi,t). Any help would be gr... |
I'm studying general topology and a question has come to my mind.
We have defined a topological property to be a property which a (viz. any) topological space can satisfy or not satisfy, and such that, if satisfied by a space, is also satisfied by every space homeomorphic to it.
I can see the ambiguity of this d... |
Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property $P(1)$
2. Inductive step: Prove that for all $k ≥ 1, P(k) ⇒ P(k + 1)$
(a) Why might the above mechanism not constitute a vali... |
This question is a response to the answer given in [this SE post][1], concerning the use of the Vitali covering lemma to prove the Hardy-Littlewood maximal inequality in general dimension. For reference, the covering lemma is
> **Lemma (Vitali).** Let $B_1,\ldots,B_n$ be a finite collection of open balls in $\mathbb... |
For a cylinder, the volume formula is given by $V = \pi r^2 H $. To find the differential \( dV \), we differentiate \( V \) with respect to \( r \) and \( H \), yielding $\left( dV = 2 \pi r H dr + \pi r^2 dH \right)$. Upon integrating \( dV \) again, we obtain $( 2 \pi r^2 H )$. However, I'm confused about where the ... |
$\textbf{Hint}$
Following lulu's advice to use friendly nomenclature for symbols, and realising that the red balls are irrelevant,
Let $w$ = # of white balls, $b$ = # of black balls, $n = w+b$, then
P(black) $=p = \dfrac{b}{n},\;\;$ P(white) $=1-p = q$
The earliest the $k_{th}$ black could come would be ... |
For a cylinder, the volume formula is given by $V = \pi r^2 H $. To find the differential \( dV \), we differentiate \( V \) with respect to \( r \) and \( H \), yielding $\left( dV = 2 \pi r H dr + \pi r^2 dH \right)$. Upon integrating \( dV \) again, we obtain $( 2 \pi r^2 H )$. However, I'm confused about where the ... |
Suppose we want to prove that a property $P$ is true for every integer in $ℕ_{odd}$ = $\{1,3,5,7,9,...\}$. Consider the following induction mechanism:
1. Base case: Verify the property $P(1)$
2. Inductive step: Prove that for all $k ≥ 1, P(k) ⇒ P(k + 1)$
(a) Why might the above mechanism not constitute a vali... |
I was reading this [post][1] and in the proof of the fact
>If $C$, $D$ are smooth curves and $g(C) \geq 1$, then any blow-up of $C \times D$ contains at most *finitely many* $(-1)$-curves.
Here we take the projection $p:S\to C$, where $\pi:S\to C\times D$ is a blow-up. Then the proof noted that no rational curv... |
I am interested in finding a closed-form formula to calculate the area of intersection between two circles with known radii, $r_1$ and $r_2$, where $r_1 \geq r_2$, and a known distance $d$ between their centers. Importantly, I need this formula to work even in scenarios where the distance $d$ exceeds the sum of the rad... |
$\textbf{Hint}$
Following lulu's advice to use friendly nomenclature for symbols, and realising that the red balls are irrelevant,
Let $w$ = # of white balls, $b$ = # of black balls, $n = w+b$, then
P(black) $=p = \dfrac{b}{n},\;\;$ P(white) $=1-p = q$
The earliest the $k_{th}$ black could come would be ... |
A right $G$-torsor (or a [principal homogeneous space][1]) is defined to be a set $X$ together with a free and transitive right $G$-action $\lhd: X\times G \to X$. It is often said that "A $G$-torsor is like the group $G$ that forgot its identity element". What, roughly does this last statement mean? I've had a hard ti... |
What does it mean that "A $G$-torsor is like the group $G$ that forgot its identity element"? |
What is the matter in defining the necessity operator internally $\Box: \Omega \to \Omega$? |
I know that the Weil pairing defined on elliptic curves is a bilinear & non-degenerate pairing. However is the pairing perfect? Do we have,
$$E[m]\cong \text{Hom}(E[m],\mu_m)$$
If so, how? |
Suppose we have two IID random variables $G$ and $G'$ drawn from some unknown distribution, and their Monte-Carlo empirical observations $\hat{G}$ and $\hat{G'}$. Is there a way to upper bound the probability $P[\text{sgn}(G-G')=\text{sgn}(\hat{G}-\hat{G'})]$?
The lower bound is easy: just consider the case where th... |
Consider a dataset, $\mathcal{D} = (X,Y)$ consisting of points in input space and ground truth values. The regression model; $y_i = f(x_i) + \varepsilon_i$ where $\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$ is a GP regression model if we endow a $0$-mean $\mathcal{GP}$ as prior distribution over the entire function sp... |
How to incorporate previous mean and covariance when predicting current distribution in Gaussian Process regression? |
I want to calculate the limit $\lim_{z \to 0} \left(\frac{z}{\overline{z}} \right)^2$, if we take $z=re^{i \Theta}$ note that
$$\left(\frac{z}{\overline{z}} \right)^2=\left(\frac{re^{i\Theta}}{re^{-i\Theta}} \right)^2=e^{4i\Theta}$$
This may help clarify things a bit, but I still don't see the value of the limit.... |
I know this question has been asked before, and I throughly looked at all the answers, but can't find one that matches my math knowledge, or the contents of the course the exercise is derived from.
In $A=\mathbb{C}[x,y,z]$ I have the ideal $I=(xz-y^2,x^3-yz,z^2-x^2y)$. I need to find the height of this ideal. There ... |
I'm trying to determine the converge and divergence of the following sum:
$$\sum_{n=1}^\infty \frac{\cos{n\pi}}{n^{3/4}}$$
I thought that I could do a direct comparison test, and could compare it with this series:
$$\sum_{n=1}^\infty \frac{-1}{n^{3/4}}$$
And it is a divergent p-series since $\frac{3}{4... |
Say that an integral domain $\mathcal{X}$ is an *almost-PID* if $\mathcal{X}$ is not a PID but every finitely generated ideal of $\mathcal{X}$ is principal. The question of whether almost-PIDs exist came up in my class recently (the standard proof that elements of PIDs can be factored into irreducibles shows that from ... |
Suppose we have a 1-dimensional space: a number line. We can name each element of it using its module and a "sign"(+/-).
Is there a system which have something like additional signs that are used to name numbers? In other words: Is there any property like *signness* which unite "+" and "-", and **can we create new o... |
Hint 1: $AE:EC=1:2$
> Consider the ratio of area $\Delta ABE:\Delta CBE$. They are triangle with common vertex, so the ratio is equal to the base ratio $AE:EC$. On the other hand we have the area of the triangles are $$\Delta ABE=\dfrac{1}{2}(3)(BE)\sin\angle ABE$$
$$\Delta CBE=\dfrac{1}{2}(6)(BE)\sin\angle CBE$$
... |
We show that if $\sigma(n)$ is exponential then the series tends towards $1/2$ but retains an oscillation about this value and does not completely converge.
Let $\sigma(n)=3^n$ and sum from $n=0$ to $n=\infty$, thus
$f(x)=\lim_{x\to1^-}\sum\limits_{n=0}^\infty (-1)^n x^{3^n}.$
The following graph shows a plot ... |
Hint 1: $AE:EC=1:2$
> Consider the ratio of area $\Delta ABE:\Delta CBE$. They are triangle with common vertex, so the ratio is equal to the base ratio $AE:EC$. On the other hand we have the area of the triangles are $$\Delta ABE=\dfrac{1}{2}(3)(BE)\sin\angle ABE$$
$$\Delta CBE=\dfrac{1}{2}(6)(BE)\sin\angle CBE$$
... |
Given $f:S^1\to S^1: z\mapsto z^n$. I'm trying to show that $Df_p$ is orientation preserving: Let's try to show it at $p= (1,0)$. Tangent space at p is one dimensional and is spanned by basis vector $\partial/\partial y$. I assume $S^1$ to be oriented anticlockwise.
I want to show that $Df_p (\partial/\partial y)$ ... |
Derivative of the map $f:S^1\to S^1: z\mapsto z^n$ is orientation preserving or not? |
In the framework of optimal control we have introduced in class the HJB equation to solve optimal control problem of the form
$$
\inf_{u\in\mathcal{U}}\left[\int_{0}^{T}L(t,x_u(t),u(t))dt + h(x_u(T));\quad x’(t)=f(t,x(t),u(t)), x(0)=x_0\right]
$$
Where $x(t)\in\mathbb{R}^{d}$, $u(t)$ is a control, $L$ a cost f... |
Is this a valid "easy" proof that two free groups are isomorphic if and only if their rank is the same? |
What does a parametric equation mean? |
I've been reading through the Polytope-Wiki entry on [helices][1]. To my understanding, an $n$-gonal helix is and blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$... |
What's the difference between a zig-zag and a helix? |
The Direct Comparison test states that if we have two series $a_n$ and $b_n$ with $b_n ≥ 0$ and $a_n ≤ b_n$ for all n.
If b converges so does a.
if a diverges so does b.
I was given the series $$ \sum_{n=2}^{\infty}\frac{1}{n^n}$$ and I attempted to use the direct comparison test on this but was unable to com... |
I know we don't have an equality $\sup_{y \in \mathbb{R}} \mathbb{E} f(X,y) = \mathbb{E} \sup_{y \in \mathbb{R}}f(X,y)$ in general. But how about a simpler case, namely, is equality
$$\sup_{y \in \mathbb{R}} \mathbb{E} \sum_{k=1}^nX_kf(ky) = \mathbb{E}\sup_{y \in \mathbb{R}} \sum_{k=1}^nX_kf(ky)$$
true, where ... |
Suppose we have a 1-dimensional space: a number line. We can name each element of it using its module and a "sign"(+/-).
Is there a system which have something like additional signs that are used to name numbers? In other words: Is there any property like *signness* which unite "+" and "-", and **can we create new o... |
In his book *Number Systems and the Foundations of Analysis*, Elliot Mendelson first defines a *Peano system* (p. 53). He then goes on to prove that two arbitrary Peano systems are isomorphic — they are essentially the same (p. 80). He then makes a basic existence assumption — there exists a Peano system (p. 83).
In... |
This question is a response to the answer given in [this SE post][1], concerning the use of the Vitali covering lemma to prove the Hardy-Littlewood maximal inequality in general dimension. For reference, the covering lemma is
> **Lemma (Vitali).** Let $B_1,\ldots,B_n$ be a finite collection of open balls in $\mathbb... |
I've been reading through the Polytope-Wiki entry on [helices][1]. To my understanding, an $n$-gonal helix is a blend of a planar $n$-gon $\{n\}$ with the regular linear apeirogon $\{\infty\}$. The blend of the apeirogon with a line segment produces the only two dimensional "helix", the zig-zag. Finally, $\{\infty\}$ i... |
I'm in doubt about my arguments for this problem, in addition to having thought of an alternative argument after writing the solution. Please give me feedback on them, constructive criticism, and of course if they are valid.
Best regards.
**Suppose that f is a continuous function on [-1,1] such that $x^2+(f(x))^2... |
Is the derivative of a real-valued positive definite matrix is also a real-valued positive definite matrix?
If that is not always the case, when it is guaranteed?
|
I am interested in a simple parametric function: $$y(t) = \frac{a}{b} (1-e^{-b t})$$
Where $a$ and $b$ are real parameters greater than zero; $y(0)=0$; and $t$ is time in discrete units, e.g. months or years.
If the value of $a$ can vary arbitrarily by time unit, $a(t)$, is there a solution for $y(t)$?
Note th... |
Is a smooth compactly supported function multiplied by a continuous but not necessarily differentiable function already lipschitz continuous?
right now Im just interested in the answer. If you know something I would appreciate it if you could share it with me.
maybe there is a sufficient condition for uniform con... |
Smooth function times continuous is Lipschitz? |
I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent:
> That is, given $a,b,c \sim \operatorname{N}\left(0,1\right)$, what is ( a numerical approximation to 5 digits of )
$\operatorname{Pr... |
Antie, a smart ant living on a smooth surface $S$ of $\mathbb{R}^3$, would like to evaluate the Gauss curvature $K$ at a certain point $P\in S$. Antie is aware of Gauss Theorema Egregium, according to which Gauss curvature may be evaluated using the first fundamental form of the surface. So, Antie grabs a ruler and a ... |
$\textbf{Hint}$
Following lulu's advice to use friendly nomenclature for symbols, and realising that the red balls are irrelevant,
Let $w$ = # of white balls, $b$ = # of black balls, $n = w+b$, then
P(black) $=p = \dfrac{b}{n},\;\;$ P(white) $=1-p = q$
The earliest the $k_{th}$ black could come would be ... |
I am interested in a simple parametric function: $$y(t) = \frac{a}{b} (1-e^{-b t})$$
Where $a$ and $b$ are real parameters greater than zero; $y(0)=0$; and $t$ is time in discrete units, e.g. months or years.
If the value of $a$ can vary arbitrarily by time unit, $a(t)$, is there a solution for $y(t)$?
Note th... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given that $T(x)\in ... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given $T(x)\in R $ ... |
Does a solution exist for this discrete time function with variable parameter? |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given $T(x)\in R $ ... |
By using
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+x^2}dx=|E_{2n}|\left(\frac{\pi}{2}\right)^{2n+1}\tag{a}$$
and
$$\text{Li}_{a}(-z)+(-1)^a\text{Li}_{a}(-1/z)=-2\sum_{k=0}^{\lfloor{a/2}\rfloor }\frac{\eta(2k)}{(a-2k)!}\ln^{a-2k}(z)\tag{b}$$
I managed to find:
$$\int_0^\infty\frac{\ln^{2n}(x)}{1+yx^2}dx=\frac{... |
let $X \sim Bin(10,p)$
We would like to test the following hypotheses:
$H_{0}: p=0.5 $ vs $H_{1} : p=0.3$
The Statistic $T(x)$ is defined as $T(x)=x$ and the critical (rejection) region is $R=\{x:T(x)\le4\}$
I want to calculate the conditional probability of the null hypothesis given $T(x)\in R $ ... |
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