instruction stringlengths 12 30k |
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> let $T_1, T_2, T_3$ be Exponential($\lambda$), all independent and let $Y_1 = T_1$, $Y_2 = T_1 + T_2$, $Y_3 = T_1 + T_2 + T_3$
Find pdf of $Y_2$ given $Y_1 = y_1$
First method:
- computed the pdf for $Y_1$, $Y_2$, $Y_3$ using the convolution integral and found that they only depend on the variable of paramet... |
Denesting $\sqrt w = \sqrt{a+b\sqrt{n}}\,$ can be done by a simple formula that I discovered as a teenager.
$ {\bf Simple\ Denesting\ Rule} \ \ \overbrace{\rm \color{#0a0}{subtract\ out}\ \sqrt{norm}^{\phantom .}}^{\textstyle\!\!\! w \to w - \sqrt{ww'} =:\, s\!\!\!\!\!\!}\!\!\!, \ {\rm then}\ \ \overbrace{\color{brow... |
Problem: Find The expected number of coin flips until two heads in a row or three tails in a row occurs.
Answer: $\frac{21}{5}$.
This problem is not hard to just manually solve using states and creating a system of equations—I’ve already done this myself. However, it is a little computation-heavy.
It is well-k... |
**Algorithm pseudocode:**
___
<p>
1 def find max(data): <br>
2 biggest = data[0] # The initial value to beat <br>
3 for val in data: # For each value: <br>
4    if val > biggest # if it is greater than the best so far, <br>
5       biggest = val # we have found a new best (so ... |
Is the Asymptotic complexity of the find max algorithm O(n) or O(n^2)? |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/NYWiQ.png
This proof is from Charles C. Pugh - Real Mathematical Analysis P68. It uses $\lim\limits_{} S$ to denote the set of all limit points of $S$. It shows that $\lim\limits_{} S$ is closed.
It starts by considering $p$ is a limit ... |
Let $(a,b,c)$ be the sides of a triangle inscribed inside a unit circle such that the vertices of the triangle are distributed uniformly on the circumference. Let $x$ be a real and let
$$
f(x) = P\left(\frac{1}{a-b+x} + \frac{1}{b-c+x} \ge \frac{1}{c-a+x}\right)
$$
The plot of $x$ vs $f(x)$ is show below along ... |
How to solve a recurrence relation with full history such as $ T(n) = n + \sum_{i=1}^{n-1} 2T(i)/3i $? |
How to solve a recurrence relation with full history $ T(n) = n + \sum_{i=1}^{n-1} 2T(i)/3i $? |
Prove or reject my solution to "How quickly can you type this unary string?" |
How to solve a recurrence relation with full history $ T(n) = n + \sum_{i=1}^{n-1} \frac{2T(i)}{3i}$? |
Problem: Find The expected number of coin flips until two heads in a row or three tails in a row occurs.
Answer: $\frac{21}{5}$.
This problem is not hard to just manually solve using states and creating a system of equations—I’ve already done this myself. However, it is a little computation-heavy.
It is well-k... |
**Exercise.**
Suppose $T \in \mathcal{L}(V,W)$ and $w_1,\ldots,w_m$ is a basis of $\text{range} \ T$. Hence for each $v \in V$, there exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that
$$
Tv = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m,
$$
thus defining functions $\varphi_1, \ldots, \varphi... |
Why does there exist unique numbers $\varphi_1(v), \ldots, \varphi_m(v)$ such that $ Tv = \varphi_1(v)w_1 + \cdots + \varphi_m(v)w_m$? |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
Cardinality of a set of random variables (measurable functions): $|L_0|=|L_p|=|L_\infty|$ for any $p >0$? |
The linear system: $Ax=b$
I know that if the system is underdetermined, there will be closed-form solutions. But if $b\sim any~distribution$, are there closed-form solutions?
More detailedly, this is what I am trying to solve:
$$
\arg \min_x \lVert x - X\rVert_2, X\sim any~distribution, X, x \in \mathbb{R}^{n\t... |
Assume Line $l$ is the axis of the ball, and the radius of the ball is $R$.
I want to calculate the Moment of Inertia for the ball with cylinder:
I take cylinder in the ball as is shown in the img:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/AeP1e.jpg
Assume the angle noted o... |
How to calculate the Moment of Inertia for a ball(radius=R) with cylinder? |
why is the co-prime part not mentioned in the definition of the rational number? |
Why is this method incorrect? (Combinatorics) |
An automorphism on a finite graph $G$ is defined as an isomorphic map from $G$ to itself. How is automorphism defined on infinite graphs? For example, on the ray graph $G_r: 1 \rightarrow 2 \rightarrow 3 \rightarrow \cdots$, the identity map is obviously an automorphism. But what about the map $f(G_r) = 2 \rightarrow 3... |
> Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$?
My attempt: if $x/a_n$ is rational for any $n$, then $r_n=x/a_n$ is OK$. If $x/a_n$ is irrational, use Dirichlet? $|r_n-x/a_n|<\frac{C}{r_n^2}? Oh...
$|r_na_n-x|<\frac{1}{n}$, $|a_n-\frac{x}{r_n}|<\cdots$...Oh. |
Let $a_n\geq a>0$, for any $x\in\Bbb R$, can we find a rational sequence $r_n$ such that $r_na_n\to x$? |
(Some result from WolframAlpha, acknowledged later in this answer)
I assume that, according to $(1)$ in the question,
$$T(1) = 1+(\text{Empty sum}) = 1.$$
Inspired by the closed form given by WolframAlpha, from the given recurrence:
$$\begin{align*}
T(n+1) &= 1 + T(n) \left(1+\frac{2}{3n}\right)\\
T(n+1) ... |
## What I'm trying to derive
The cartesian position of the lunar ascending node relative to the true equator and equinox of date reference frame. My issue is I'm getting a bit tripped up with reference frames.
## My attempt
Consider the moon at a time $t$. Let $r,v$ be the position and velocity of the moon at ... |
In [this Wikipedia article][1], they derive the variance of a Monte Carlo estimator $Q_N$ for a function $f: \mathbb R^m \rightarrow \mathbb R$, using $N$ samples drawn uniformly over an integration region $\Omega \subset \mathbb R^m$ (which has volume $V$).
They show that the variance of $Q_N \propto V^2 \frac{\sig... |
Does the error of monte carlo integration scale with the number of dimensions or not? |
Assuming $M$ is a compact Riemannian manifold. $\gamma_j:[0,1]\rightarrow M$ are equicontinuous curves, namely, there is constant $L$ such that
$$
d(\gamma_j(t_1),\gamma_j(t_2))\le L(t_2-t_1)
~~~~~~
\forall t_1\le t_2 \in [0,1],~~\forall \gamma_j\in\{\gamma_j\}
$$
And the length of $\gamma_j$ converge, assuming ... |
I am a student of mathematics and have been increasingly feeling doubtful about whether or not I really understand the theory. What I mean by this is highly attributed to the style of textbooks in presenting information.
Authors simply write definitions and make vague connections, but I never get to know why or how ... |
How can I motivate and deal with speculation? |
In groups, if $H,K$ are subgroups of a finite group $G$, then $[HK:K]=[H:H\cap K]$ where this index counts number of cosets.
In field theory, of $K_1,K_2$ are finite extension of a field $F$ contained in a common field $E$, then it may not be the case that $[K_1K_2:K_1]=[K_2:K_1\cap K_2]$.
Under what conditions... |
I've learned about differentiability of several variables.
For $f(x,y)$, if partial derivatives exists and its partial derivatives are continuous, then $f(x,y)$ is differentiable. But the inverse of the theorem is wrong. There is counterexample $xy\sin(1/(x^2+y^2))$,
which is differentiable but its partial derivati... |
> Problem: Six people are to sit at a circular table. Two of the people are not
> to sit immediately beside each other. Find the number of ways that the
> six people can be seated
I'm really stuck on this question. The correct answer is $72$, but my solutions seem to be incorrect (I get $432$ ways as an answer). I... |
(Some result from WolframAlpha, acknowledged later in this answer)
Inspired by the closed form given by WolframAlpha, from the given recurrence and the particular solution $T(n) = 3n$ mentioned by @Sil:
$$\begin{align*}
&&
T(n+1) &= 1 + T(n) \left(1+\frac{2}{3n}\right)\\
&(-)& 3(n+1) &= 1 + 3n \left(1+\frac{2}... |
when H is order r subgroup of G, $gHg^{(-1)}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
H1,H2,H3,H4, where H1={e,f}, H2={e,rf}, H3={e,$r^2f$} ... |
when H is order r subgroup of G, $gHg^{(-1)}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
H1,H2,H3,H4, where H1={e,f}, H2={e,rf}, H3={e,$r^2f$} ... |
How to use the fundamental principle of counting to find out the total number of subsets of a set $A$? $A$ has n distinct elements.
I understand in principle that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elements considered together. However, I want to write t... |
How to use the fundamental principle of counting to find out the total number of subsets of a set $A$? $A$ has $n$ distinct elements.
I understand in principle that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elements considered together. However, I want to write... |
How to use the fundamental principle of counting to find out the total number of subsets of a set $A$? $A$ has $n$ distinct elements.
I understand in principle that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elements considered together. However, I want to write... |
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I understand in principle that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elements considered together. H... |
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I perceive that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elements considered together. However, I want ... |
This is **NOT** a homework question.
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I perceive that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elem... |
I am trying to calculate the mean for $Z = a_1 X_1 + a_2 X_2$. As a first step, I want to find the pdf for $(a_1 X_1, a_2 X_2)$ (I plan to convolve).
$X_1$ and $X_2$ are jointly normal. How can I do this? (specifically, how does the pdf of $(X_1, X_2)$ relate to that of $(a_1 X_1, a_2 X_2)$? |
This is **NOT** a homework question.
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I perceive that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elem... |
How to understand Galois Representations as an undergraduate? |
This is **NOT** a homework question.
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I perceive that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elem... |
This is **NOT** a homework question.
How to use the fundamental principle of counting to derive the result that the total number of subsets of a set $A$ is $2^n$? $A$ has $n$ distinct elements.
I perceive that for building a subset from $A$, I have two choices for each element and $2^n$ choices for all $n$ elem... |
I have a function - $B_{DGC}(x,y) = K_{(x,y)} \times e^{-(\frac{(I_x - I_y)^2}{2\sigma^2})}\times \frac{1}{d(x,y)} \times \frac{1}{\delta(x,y)_{DGC}}$. The components $K_{(x,y)}$ and $\frac{1}{\delta(x,y)_{DGC}}$ are semi metric, $\frac{1}{d(x,y)}$ is metric and $e^{-(\frac{(I_x - I_y)^2}{2\sigma^2})}$ is quasi metric.... |
Is the product of metric and semi-metric functions a semi metric? |
I'm studying this series
\begin{align*}
a_1=\mathrm{i}, \quad a_{n+1}=\mathrm{i} + \frac{\mathrm{i}}{a_n},
\end{align*}
where $\mathrm{i}$ is the imaginary unit, in order for me to see clearly the structure of this sequence, I separate the real part and the imaginary part as a sequence, as follows.
\begin{align*}
... |
I've learned about differentiability of several variables.
For $f(x,y)$, if partial derivatives exists and its partial derivatives are continuous, then $f(x,y)$ is differentiable. But the inverse of the theorem is wrong. There is counterexample $xy\sin(1/(x^2+y^2))$,
which is differentiable but its partial derivati... |
The definition of a rational number is a ratio between two integers. There is no requirement that the integers be coprime (else $\frac 24$ *wouldn't* be rational when it is). Notice that from the definition *alone* there is no statement that all rational numbers *can* be written as a ratio of coprime numbers nor that... |
How to count all possible $2^n$ subsets using the fundamental principle of counting? |
Suppose a finite set of plane curves parameterized by $t$: $p_i(t)=(x_i(t),y_i(t)), i=1...N$, satisfy
- boundedness: $|p_i(t)|\leq 1\ \ \forall i$
- smoothness: $p_i(t)$ differentiable to any order, $\forall i$
- having non-postive curvature: $(\dot{x}_i\ddot{y}_i-\dot{y}_i\ddot{x}_i)\leq 0\ \ \forall i,t$
Q... |
I recently came across the following claim in a paper:
$$\langle X^i - \alpha \rangle = \langle X - \alpha^{i^{-1}} \rangle$$
as the generators have the same roots in (in some extension of the base field $\mathbb{Z}/\mathbb{Z}p$, (prime $p$).
I can see that they would have same roots in an appropriate extensio... |
Two principal ideals are equal if their generator has the same root in some extension? |
Can some give an easy general method to find general term of sequences whose difference is in AP?
Example: 1,4,8,13,19....
The difference is 3,4,5... which is in AP.
Through vigorous testing and analysis you get the general term as $ (n^2 + 3n - 2)/2$
but how to derive it mathematically |
I'm not sure but,
I guess (from my book) an asymptotic approximation of $(1-\frac{1}{\sqrt{n}})^n$ would be
$$(1-\frac{1}{\sqrt{n}})^n \space\space \to \space\space e^{-\sqrt{n}-\frac{1}{2} + O(n^{-1/2})}$$ as n increases.
Can you explain to me why?
(That's just my guessing, so you can correct it)
|
Andrew Pressley's Elementary Differential Geometry textbook claims on page $21$ that "...we can always assume that a closed curve is unit-speed and that its period is equal to its length."
In the previous paragraph, Pressley derives this assuming the curve $\gamma$ is regular. This is obvious since regularity is eq... |
Can closed curves be assumed unit speed? |
This might be anything from trivial to not trivial, but my probability theory is too rusty to get even started with this.
> Supppose I have $N$ real numbers $(x_1, x_2, \ldots, x_N)$, each sampled
> independently from $\mathcal{N}(0,1)$. *Conditional on their sum being
> $0$*, how could I calculate the probability... |
Let $M$ be a type III$_1$ factor and $N$ be a type III$_1$ subfactor of $M$.
If we know that $P$ contains $N$ amd also is a subalgebra of $M$. Can we determine the type of $P$. Is it of type III? |
When I took undergraduate complex analysis, the instructor was trying to prove an inequality, and he used a technique as below:
$$\int \lim_{n\rightarrow \infty} f_n = \lim_{n\rightarrow \infty} \int f_n \cdots (*)$$
And he claimed that we can use this, because all $f_n$s are measurable and $f_n$ converges unifor... |
when H is order r subgroup of G, $gHg^{(-1)}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
H1,H2,H3,H4, where H1={e,f}, H2={e,rf}, H3={e,$r^2f$} ... |
Andrew Pressley's Elementary Differential Geometry textbook claims on page $21$ that "...we can always assume that a closed curve is unit-speed and that its period is equal to its length."
In the previous paragraph, Pressley derives this assuming the curve $\gamma$ is regular. This is obvious since regularity is eq... |
So I want to set up a schedule of double matches: player A and B vs player C and D. I have a few constraints for setting it up:
- Each player plays exactly 4 times
- The scheme should be as fair as possible. With this I mean that I want to achieve that every player sees as many different other players as possible... |
In post-quantum cryptography, there's a suite of algorithms based on "modular lattice". These schemes are defined in terms of vectors and matrices whose elements are polynomials of a fixed degree (e.g. 256 for ML-KEM/[Kyber](https://eprint.iacr.org/2017/634) as well as ML-DSA/[Dilithium](https://eprint.iacr.org/2017/63... |
Let $A$ be a measurable subset of $\mathbb{R}$ such that $0<\mu(A)<\infty$. Define for $x\in\mathbb{R}$, $$f(x):=\int \chi_A(x+t)\chi_A(t)dt.$$
We can know that $f(x)$ is continuous on $\mathbb{R}$. Show that there exists $r>0$ so that $(-r, r)\subset A-A:=\{x-y: x, y\in A\}$.
---
My idea is as follows.
Not... |
How to show that $O=(-r, r)$? |
I'm new to stochastic processes and have problems understanding martingales, conditional probability, $\sigma$-algebras etc.
I have two proofs that I'm now sure how to handle.
**Problem 1**.
Prove that an integrable stochastic process $\{X(t),\mathcal{F}_t, t\in \mathbb{T}\}$ is a martingale if and only if for a... |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
When I took undergraduate complex analysis, the instructor was trying to prove an inequality, and he used a technique as below:
$$\int \lim_{n\rightarrow \infty} f_n dz= \lim_{n\rightarrow \infty} \int f_n dz \cdots (*)$$
And he claimed that we can use this, because all $f_n$s are measurable and $f_n$ converges u... |
I READ A new introduction to modal logic at page 176-177
it said in modal logic, irreflexive frame preserves the one rule, i.e.
Gabb.
[![enter image description here][1]][1]
and proof's content is as follow :
[![enter image description here][2]][2]
[1]: https://i.stack.imgur.com/jxZD0.png
[2]: http... |
why this procedure of proof valid? |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
$$\forall n\in \mathbb{N},\left.\frac{d^{2n+1}}{dx^{2n+1}}(x^{\ln x})\right|_{x=e^{\frac{n}{2}}}=0$$
How to prove this equation?I find that if n=0 or n=1 ,the answer is 0 .But I can't prove that it is true for all the nonegative integer.My idea is , calculate the analytical expression of its n-th derivative and then s... |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
It's my understanding that ZFC is axiomatized in terms of propositional logic. However, propositional logic is just a logic which deals with propositions, and on its own has no inherent axioms. Yet, mathematics is clearly subject to certain rules of inference and tautologies not explicitly stated in ZFC, such as modus ... |
What exactly is the propositional calculus that ZFC is founded on? |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
New to optimization, trying to understand the characteristics of a problem I am in the process of modelling, and looking to solve in Python. Given is a decision variable **k**,the sum of **N** other decision variables **c**, and two constants, **a** and **T**. Both **a** and **T** are positive numbers.
I want to max... |
How to prove this inequality involving a small number? |
I need to random numbers of the form $\frac 1 {(1+x) \log 2}$ over the interval $[0, 1]$. I know that I can generate this using inverse transform method. Are there any other methods beside it? |
$$\forall n\in \mathbb{N},\left.\frac{d^{2n+1}}{dx^{2n+1}}(x^{\ln x})\right|_{x=e^{\frac{n}{2}}}=0$$
How to prove this equation?I find that if $n=0$ or $n=1$ ,the answer is $0$ .But I can't prove that it is true for all the nonegative integer.My idea is , calculate the analytical expression of its n-th derivative and ... |
Given two polynomials $P,Q$ and consider $D=\{\tfrac{P(n)}{Q(n)}:n\in\mathbb{N}\}$, can one find polynomials $P^\prime,Q^\prime$ such that $D-D=\{\tfrac{P(n)}{Q(n)}-\tfrac{P(m)}{Q(m)}:n,m\in\mathbb{N}\}=\{\tfrac{P^\prime(N)}{Q^\prime(N)}:N\in\mathbb{N}\}$?
Clearly, this can happen. For example, one can check that wi... |
When is the image of $\mathbb{N}$ under a bivariate rational function the same as under a univariate rational function? |
when H is order r subgroup of G, $gHg^{-1}$ is yet anther order r subgroup of G.
suppose H is non-normal, mark all subgroups produced by H as Set: {$gHg^{(-1)}$ | for all g in G}
I tried dihedral d_4 with order 2 non-normal subgroup, there are total 4 of them,
H1,H2,H3,H4, where H1={e,f}, H2={e,rf}, H3={e,$r^2f$} et... |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
Suppose that we are given a smooth, closed, connected homology sphere $M^n \subset \mathbb{R}^{n+1}$ with $n \geq 6$. I want to kill some elements in the fundamental group of $M$ by surgeries. As far as I understand it correctly, this is done by the following procedure:
1) Choosing a representative $S^1 \longrightar... |
$$\forall n\in \mathbb{N},\left.\frac{d^{2n+1}}{dx^{2n+1}}(x^{\ln x})\right|_{x=e^{\frac{n}{2}}}=0$$
How to prove this equation?I find that if $n=0$ or $n=1$, the answer is $0$. But I can't prove that it is true for all the nonegative integer. My idea is, calculate the analytical expression of its $n$-th derivative an... |
How can we compare the cardinalities of different sets of random variables on a given probability space $(\Omega, \mathcal F, P)$.
Is $|L_1|>|L_\infty|$, while $L_\infty$ is dense in $L_1$?
When does the following $$|L_p|=|L_\infty|$$ hold for all $p\ge1$?
How large can be the set $L_0$ of all random var... |
I'm referring to [this version][1] of Kripke-Platek set theory minus axiom of infinity. Let $S$ be a Dedekind infinite set, and $f\colon S \to S$ be an injection with $f(S) \neq S$. A proof I know picks $s \in S \setminus f(S)$ and constructs function $g(n) = f^n(s)$ by recursion, and use replacement to obtain $\omega ... |
Suppose symmetric $A$ and $u$ are given, can we exactly identify the smallest eigenvalue of $A+uu^T$ and its corresponding eigenvector (based on $u$ and given eigenpairs of $A$)? It is okay to add any additional meaningful conditions on $A$ in this question.
Even though the method to identify the whole spectrum is ... |
[![enter image description here][1]][1]
How can I tackle this problem with as few abstract ideas as possible when attempting to 'purely' integrate from left to right? I've tried a few different versions of this term, but I'm not sure how to get into an answer. Is there another way to approach this?
[1]: https... |
What is the simplest way to approach this integral problem? To me, this problem seemed to necessitate some kind of gimmick. |
The well-known equation
$$u''(x)= \frac{u(x+h) - 2 u(x) + u(x-h)}{h^2}+\mathcal{O}(h^2)$$
is derived by adding
$$u(x+h)= u(x)+u'(x)h+\frac{1}{2}u''(x)h^2+\mathcal{O}(h^2)$$
$$u(x-h)= u(x)-u'(x)h+\frac{1}{2}u''(x)h^2+\mathcal{O}(h^2)$$
How does it look like if we have different step sizes such as
$$u(x+h_1)= ... |
Suppose diagonal $D\in \mathbb R^{n\times n}$ with $D\succeq 0, v\in \mathbb R,$ and $\alpha>$ are given, can we exactly identify the smallest eigenvalue of $D+\alpha vv^T$ and its corresponding eigenvector (based on $v$ and given eigenpairs of $D$)?
Even though the method to identify the whole spectrum is given in ... |
I have a very specific problem that I'm not even sure how to approach.
I have a function $G(t,s,W,W')$, such that $G(s,t,W,W') = -G(t,s,W,W')$, where $t$ and $s$ are real variables while $W$ is a real function and $W'$ is its derivative. I then need to find conditions on $W$ such that $G$ has the form
$$G(t,s,W,W')... |
Suppose diagonal $D\in \mathbb R^{n\times n}$ with $D\succeq 0, v\in \mathbb R^n,$ and $\alpha>0$ are given. Can we $\textit{exactly}$ identify the smallest eigenvalue of $D+\alpha vv^T$ and its corresponding eigenvector (based on $v$ and given eigenpairs of $D$)?
Even though the method to identify the whole spectru... |
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