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In my posterior probability computation, I got the following integration and I could not figure it out.
$$\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$$ where $a$ is a positive constant
Any help is appreciated.
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I want to be sure that I understand how to show that a structure satisfies a sentence under a variable assignment, and suspect that I'm handling the computation of multiple quantifiers incorrectly. Consider the structure,
(1) $M = \langle \{a\}, \emptyset \rangle$ (i.e., the structure whose domain contains only $a$ a... |
In my textbook, I came across this interesting question which I am currently struggling to solve:
If $\log_6 2 = a$ and $\log_5 3 = b$, express $\log_5 2$ in terms of a and b
The solution given is $\frac{ab}{1-a}$ but I do not know the working behind this. What is it?
|
Currently in my third week of my first ODEs class and I've already encountered something I'm struggling with. My second homework assignment requires me to classify and solve some ODEs. He gave us four classifications: separable, linear, homogeneous, and Bernoulli.
However, when reading through my text, I noticed there ... |
For some arbitrary function: $f(x)$ within the interval $a<x<b$, should I just calculate the roots for $f'(x)$, and the points $f(a)$ & $f(b)$, and make deductions based around what is going on at those points, or is there some more rigorous or arithmetic method?
Thank you
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Let $\Omega S_1$ be the space of loops of $S_1$ based at $x_0 \in S_1$ with the topology of uniform convergence. We know that the path connected components of this space are in one to one correspondence with $\pi_1(S_1)$. Concretely this is because the a path in $\Omega S_1$ between two maps $\phi_a: S_1 \to S_1$ and... |
Let $T:V \rightarrow V$ be a linear operator on a finite dimensional vector space over $F$. Let $W \subset V$ be a subspace which is $T$-invariant. Show that there exists an ordered basis $\mathcal{B}$ for $V$ such that $$[T]_{\mathcal{B}}=\begin{pmatrix}A & B \\ 0 & C \end{pmatrix}$$
where $A$ is a matrix represent... |
Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite matrix and $\mathbf{b}\in\mathbb{R}^n$. How to prove that $A\mathbf{u}=\mathbf{b}$ if and only if $\mathbf{u}$ minimizes the so-called energy function $$F(\mathbf{v})=\frac{1}{2}\mathbf{v}^\mathsf{T}A\mathbf{v}-\mathbf{b}^\mathsf{T}\mathbf{v}.$$
What I try... |
In a linear algebra text, the following is the definition of the inverse of a matrix
An $n\times n$ matrix $A$ is invertible when there exists an $n \times n$ matrix $B$ such that $$AB = BA = I_n$$
And likewise in an abstract algebra textbook, the definition of the inverse of a group is
Given that $G$ is a group wit... |
I am reading a book in discrete mathematics and it assumes that a multiplication of two integers yields an integer.
Although that this book's saying is justifiable since the book is making an assumption, I found that that this is "completely wrong but it is still a good estimation".
let $n \ge 1$ be an integer. Then,... |
Let $\{E_{j}\}$, $j = 1, 2, ..., \infty$, be measurable subsets of $[0,1]$. Also,
$\displaystyle\sum_{j=1}^{\infty} |E_j| = M < \infty$
Let $S_n$ be the set of points in $[0,1]$ contained in at least $n$ of the $E_j$'s. How can we show that that $S_n$ is measurable, with $|S_n| \leq \frac{M}{n}$?
I know that the $E_{... |
Find the area of the region inside
$r=7\sin\theta$ but outside $r=1$.
I have tried finding the area of both using $A=\frac{1}{2}\int_\alpha^\beta r^2 d\theta$, tried arc length...but I can't find the answer.
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A manufacturer has six distinct motors in stock, two of which came
from a particular supplier. The motors must be divided among two
production lines, with three motors going to each line. If the
assignment of motors to lines is random, find the probability that
both motors from the particular supplier are assi... |
Is it solvable to ask to find a recurrence relation for the following sequence 4,7,2,20,31,73,155,332,715,... without any other information? If not, what would be the very least amount of information for me to give someone for them to arrive at $f_n=f_{n-1}+2f_{n-2}+f_{n-3}$?
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Is it possible to find a $3 \times 3$ matrix $A$ such that it's adjoint is $0$?
|
I am confused about how to perform a discreet fourier transform in excel and what the output means. There are many math.stackexchange questions about this topic (see for example Fourier transform for dummies). but I am specifically interested in how to do it in Excel
Here are the first 20 rows of my spreadsheet:
Here ... |
Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$
What would be the process satisfying the following process of $y_t$?
$$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$
I'm not quite sure about differentiating $y_t$. The following is what I did
$$\frac{\partial y_t}{\partial S_t} =dt +dW_t $$
and
$$\frac{\part... |
I need some help figuring out how to work through this problem.
Prove that $ u \cdot v = 1/4 ||u + v||^2 - 1/4||u - v||^2$ for all vectors $u$ and $v$ in $\mathbb{R}^n$.
Sorry, forgot to include my work so far:
I decided to ignore the 1/4 and deal with it later once I had a better understanding of the question.
$= ... |
Show that if $\prod_\alpha X_\alpha$ is normal then so is $X_\alpha$.
This a question of proof-verification.So please suggest the required edits and fault in the logic but please don't give a different proof all together.
Proof:
Let $A_\alpha,B_\alpha$ be disjoint closed subsets in $X_\alpha$.
Consider the projectio... |
Let $G$ be an undirected graph and define
$$P=\{x \in R^{V}: x(u)+x(v) \leq 1 \:\:\text{for all edges}\:\: e=uv,\:\: x \geq 0\}$$
Show that any vertex $v$ of $P$ is half integral.
|
One of the questions in Stewart's "Galois Theory" says:
Let $p(t) \in \mathbb{Q}[t]$. Show that $p(t)$ has a unique expression of the form:
$p(t)=(t-\alpha_1)\cdot \cdot \cdot (t- \alpha_r)q(t)$
(except for reordering the $\alpha_j$) where $\alpha_j \in \mathbb{Q}$ for $1 \leq j \leq r$ and $q(t)$ has no zeros in $\ma... |
In software engineering, there is a coverage metric for testing called modified condition/decision coverage, or MC/DC for short. This metric is well-known in the avionics industry due to showing up in the DO-178B and DO-178C standards as part of the testing requirements for Level A airborne software.
One way of thinkin... |
What does it mean if someone says the "span of vectors" $\{(x,y,z),(a,b,c)\}$ contains the vector $(d,e,f)$? I am making up numbers here. Let me know if I need to insert actual numbers here to clarify.
|
I have been trying to disprove that if I have a sequence of random variables $X_n$, that $X_n \to a$, where $a$ is a constant, in distribution doesn't imply $X_n \to a$ almost surely. One example I came up with was where $X_n \sim N(0, n)$. $X_n$ converges in distribution to $1/2$, but I am not sure how to show almost ... |
How do I prove the following using Natural Deduction?
$$((P \land Q) \rightarrow R) \vdash (P \rightarrow R) \lor (Q\rightarrow R)$$
My current approach:
So instead of proving $(P \rightarrow R) \lor (Q\rightarrow R)$, I figure I just need to prove $(P \rightarrow R)$ and use disjunction introduction.
The problem is m... |
In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result:
If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P (|Z_{v}| \leq \epsilon) \leq \epsilon$
I have computed :
$\mathbb P (|Z_{v}| \leq \epsilon) = 1 - \sqrt\frac{2}{\pi}\int_{\epsilon}^... |
I conjecture: $$\lim_{x\to\infty} \frac{256}{163x}\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$
Is this provable? If this is false, then can I have a function $f$ such that:
$$\lim_{x\to\infty} f(x)\sum_{n=1}^{\lfloor x\rceil} |\sin n|=1$$
|
I am stuck with Step 2 of the Strichartz's estimate. My question is actually a continuation of a topic which has been raised some time ago and it could be seen here Technical question about Strichartz estimate's proof..
I would really appreciate if someone could help me. I actually struggle with the third equality, na... |
Can you give me a sketch of how to Construct all (up to isomorphism) indecomposable representations of a quiver of type $E_{6}$ and $D_{6}$. Perhaps using Gabriel theorem.
Thank you
|
I am trying to solve a problem like
$$\min_{u,v} \Vert X-uv^T \Vert_F^2 + \beta \Vert v \Vert_1 \mbox{ s.t. } \Vert u \Vert_2 \le 1$$
where $X \in \mathbb{R}^{N\times N}$, $u, v \in \mathbb{R}^{N}$,
which is basically like a rank-1 (truncated) SVD decomposition with sparsity constraints. I understand that this object... |
I'm trying to solve this system
$$
M\ddot{X}(t) = KX(t)
$$
where M is a known diagonal matrix and K is a symmetrical known matrix. I'm asked to do the ansatz $Y(t) = M^{1/2}X(t)$ where $M^{1/2} = diag(\sqrt{m_1},...,\sqrt{m_N})$ so that it takes the new form $\ddot{Y}(t)=AY(t)$, can someone show me what they mean by th... |
Given a ring $R$ and $R$-modules $A_R$ and $_{R}B$, we define the tensor product $A \otimes_R B$ as the free abelian group on $A \times B$ modded out by the subgroup generated by the elements of the following form:
$$
(a+a',b)\!-\!(a,b)\!-\!(a',b) \quad\quad
(a,b+b')\!-\!(a,b)\!-\!(a,b') \quad\quad
(ar,b)\!-\!... |
What are good examples of fields of characteristic $2$, starting from the simplest one to more interesting examples?
|
Let $G$ be an undirected simple graph and let $A$ be its adjacency matrix. It is easy to see that $A$ is neither positive semidefinite nor negative semidefinite.
I would like to know if there are interesting graphs $G$ for which smallest eigenvalue of $A$ is at least $-1$. Clearly, cliques and disjoint union of clique... |
I was talking with a friend of mine and we wonder how many sequences of rational numbers on $[0,1]$ there exists. My first attempt was to consider that every sequence like that must be a subset of $\mathbb{Q} \times \mathbb{N}$ and because this is an infinite countable set, then all their subsets (functions) must be an... |
The question is to solve the differential equation :
$$0.5 \frac{dy}{dx}=4.9-0.1y^2$$
What I have attempted:
$$0.5 \frac{dy}{dx}=4.9-0.1y^2$$
$$ \frac{dy}{dx} = \frac{4.9-0.1y^2}{0.5} $$
$$ \frac{dy}{dx} = 9.8 - 0.2y^2 $$
$$ \int \frac{1}{9.8 - 0.2y^2} dy = \int 1 dx $$
$$ \int \frac{10}{98 - 2y^2} dy = \int 1 dx $$... |
$Γ(x)$ = $\int_0^{∞} e^{-t}t^{x-1}dt$
Plugging $(x-1)$ into this equation, I get
$Γ(x-1)$ = $\int_0^{∞} e^{-t}t^{x-2}dt$
Integrating by parts, I eventually end up with $-e^{-t}t^{x-1}]_0^∞$ + $(x-1)$$\int_0^{∞} e^{-t}t^{x-2}dt$
I can turn the right part into $(x-1)Γ(x-1)$, but that still leaves me with $-e^{-t}t^{x-1}]... |
Question
If:
$$f(a) + f(b) = f(ab)$$
$$ f(1) = 0 $$
$$ a<b \implies f(a) < f(b) \forall a,b \in N $$
where $N$ is the set of natural numbers.
Prove or disprove $f$ must be the $\log$ function.
Background
I was recently wondering about the uniqueness of a function given:
$$f(a) + f(b) = f(ab)$$
$$ f(1) = 0 $$
$$ a<b ... |
I ran into some trouble proving the following:
$(R \rightarrow (P \rightarrow Q))\vdash (Q\rightarrow P) \lor (P \rightarrow Q)$
My approach thus far:
Honestly I'm really stuck. So basically my hypothesis requires this $R$, hence the first thing that came to my mind is the Law of Excluded Middle. Using $R$ is trivial. ... |
How does one prove that:
There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, 7,8,9,10)(11,12,13,14,15).$$
|
I am given the definition that
$$ q\longleftrightarrow \left[
\begin{array}{cc}
z&w\\
-\overline{w}&\overline{z}
\end{array}
\right], q = z + j \overline{w}, z = x_0 + i x_1, w = x_2+ix_3 $$
If I want to find the reciprocal of $q$, I know it is $\overline{q} = \overline{z + j \overline{w} } = \over... |
I am trying to prove the following statement regarding nowhere dense sets:
"In a metric space X, the frontier of an open set is the set of accumulation points of a discrete set."
As far as my attempts go, I have gone back and looked at an earlier proof of Baire's Category Theorem and worked out the details (as to my un... |
I am trying to understand the proof of a Lemma in the book 'Nonlinear Systems' by Hasaan Khalil (3rd edition). In the Proof of Lemma 3.1, about Lipschitz continuity of vector valued functions, I am unable to follow the evaluation of a derivative using chain rule. My specific question is as follows.
$f:[a,b] \times D \r... |
I was surprised to encounter a claim made on the internet that the following number is prime:
999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999... |
I need to calculate the number of such trajectories of length $n$ (started at $0$ and end at $a$), that for giving $k$ substitute following rule: the maximum isn't greater than $a$, and for every interval of length $k$ exists such a point, at which the value of the trajectory is equal or greater than at the beginning o... |
If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits? Intuitively this is not the case, but I am not sure how to formally show this, would anyone have any hints? thanks.
|
I am currently studying a basic course in abstract harmonic analysis. I do have a good background in abstract algebra and functional analysis but I have not done a course in Fourier analysis. Is it compulsory to have a background of Fourier analysis to grasp the concepts of abstract harmonic analysis or is a good back... |
Say, I have a function $f(\vec{x}) = \cfrac{1}{2}\vec{x}^{T}Q\vec{x} - \vec{b}^T\vec{x}$, where $Q$ is Symmetric Positive Definite $\in R^{nxn}$.
I want to find $\nabla f(\vec{x} - \nabla f(\vec{x}))$.
So, am I solving this correctly like the following?
I know that,
\begin{equation}\nabla f(\vec{x}) = Q\vec{x}-\vec{b}... |
Consider the following question taken from this link, question number $25$:
We have four boxes. Box $1$ contains $2000$ components of which $5$ percent
are defective. Box $2$ contains $500$ components of which $40$ percent are
defective. Boxes $3$ and $4$ contain $1000$ each with $10$ percent defective.
We select at r... |
If $V_1$ and $V_2$ are vector spaces over the field of rational numbers and $f \colon V_1 \to V_2$ is a map such that $f(x+y)=f(x)+f(y)$ for all $x, y \in V_1$, show that $f$ is a linear transformation.
It is easy to show that $f(0)=0$ and $f(-x)=-x$.
Here scalars are from rationals. If scalar is an integer then clea... |
I am looking for a combinatorial proof for it. I know how to prove it mathematically. Expanding $(1+x)^n$ and replacing $x$ with $1$ will give me the result but I am not able to explain it combinatorially.
Note: This is not a homework question. I am just curious.
|
The question:
a) Differentiate both sides of the geometric series with respect to $r$: $$~~\displaystyle\sum_{i=0}^nr^i=\frac{1-r^{n+1}}{1-r}$$
b) Use the result in part (a) to show that (Assume that $0 < r < 1$). $$\displaystyle\sum_{i=1}^nir^i < \frac{r}{(1-r)^2}$$ for all $n\ge1$.
My attempts:
a) $$\sum_{i=0}^nir... |
I'm struggling with following step in an excercises about Mayer-Vietoris sequences:
In one step the solution says this map is injective since $A \cap B$ is path-connected:
$$ H_0(A \cap B) \stackrel{i_*^{A} \oplus i_*^{B}} \longrightarrow H_0(A) \oplus H_0(B)$$
What I've been thinking is that as $A \cap B$ is path-conn... |
When
m = 2, series is 1,2,3,4,5..
m = 3 : 1,3,6,10,15..
m = 4 : 1,4,10,20,35..
m = 5 : 1,5,15,35,70..
m = 6 : 1,6,21,56,126..
So given value of m, can I find the entire series ?
Thank you
|
A function $f:R→R$ is given. Whenever we choose real numbers $a < b$
and set {$f(x) : a < x < b$} has a biggest element, we call this
element local maximum of function $f$. Prove that a set of all local
maximums of function $f$ is countable.
I think it's impossible to prove, because local maximum is defined as ... |
Problem: Prove that for real $x, y, \alpha, \beta$,
$(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$.
I am looking for an elegant (non-bashy) solution.
It closely resembles Cauchy inequality but $\alpha y +\beta y$ part is creating a problem.
I also tried to define ... |
I have ternary number. Its size is $N$.
Monotonic sequence means every digit smaller or equal to the next digits in the number.
$00122$ - legal monotonic sequence
$1022$ - illegal
What is the possible number of monotonic sequences?
|
Suppose $p = z + j w $ where $z = x_0 + i x_1$ and $w = x_2+ix^3$. Let $q = \alpha + j \beta $ where $\alpha = y_1 + i y_2$ and $\beta = y_2 + i y_3$. How can we multiply $p$ and $q$. Is is just like complex numbers?
$$ pq = (z\alpha - w \beta) + j(\beta w + z \alpha) $$
|
I must calculate the inverse z-transform of $\frac{10}{z-5}-\frac{2}{z-1}$. I decided to use the known formula $H(n-1)a^n\rightarrow \frac{a}{z-a}$, where $H(n)$ is the heaviside signal. I finally get this result: $2H(n-1)5^n-2H(n-1)$, but my professor's result would be $2H(n)(5^{n}-1)$. Can you explain me where's the ... |
I've moved this over to HSM.
It seems odd that something that looks so much like a join [see below] would get given "the wrong symbol". It's even worse when you dualise it and get something called (quite reasonably) the "meet product", but write that as $A\vee B$.
I'm aware the answer might be "it just turned out tha... |
Find all $x$ such that $8^x(3x+1)=4$,and prove that you have found all values of $x$ that satisfy this equation.
My effort
Rewriting the equation I have
\begin{array}
22^{3x}(3x+1)&=2^2 \\
\log_2{2^{3x}(3x+1)} &=\log_2{2^2} \\
3x +\log_2{(3x+1)} &=2 \\
\end{array}
I don't know how to simplify from there,looking it u... |
I'm trying to show that if $h$ is analytic in $D=\{z\in\mathbb{C}: |z-1|<1\},$ $h'(z)=z^{-1},$ and $h(1)=0,$ then $h$ is Log $z$ in $D$. I know Log' $z=z^{-1}$ and Log $1=0.$ However, I don't know how to show $h$ must be Log $z$. Can anyone help me with this?
|
This is what I know:
A sequence $(a_{n})$ is eventually in a set $A \subseteq \mathbb{R} $ if there exists an $N \in \mathbb{N}$ such that $a_{n} \in A $ $\forall n \geq N$
A sequence $(a_{n})$ is frequently in a set $A \subseteq \mathbb{R} $ if for every $N \in \mathbb{N}$ there exists $n \geq N $ such that $a_{n} \i... |
What is the probability of observing three or fewer 6s when rolling a fair die twenty times?
I am trying to figure this out..please be detailed on explaining. Thank you!
|
I would say it has something to do with the numbers that can be expressed as a factor of different prime numbers, but when I get to $8$, that can be changed to $2^3$, which goes against this. Is there some kind of pattern I'm missing?
|
Let $f_*:C_*\to D_*$ be a chain map. I'm stuck in the proof of the following statement:
$f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$, cone($f_*$), is contractible.
(For information about the mapping cone of a chain map see for example here https://ncatlab.org/nlab/show/mapping+cone , definiti... |
Suppose that $f$ is a differentiable function in an open set $E\subset \mathbb{R}^n$, and that $f$ has a local maximum at a point $\mathbf{x}\in E$. Prove that $f'(\mathbf{x})=0.$
I am sorry if this topic is repeated but I solved this myself and
I would like to know is it correct?
Proof: Since function $f$ is differen... |
There is a green rectangular grid of size $n \times m$ with $k$ cows in it.
The objective of the cow is to escape the field. If any one of the cows escape, all cows are set free.
There is a farmer who tries to restrict the field by putting up fences along the border. Initially the grid is not fenced.
Cows coordinate a... |
I'm having a little doubts on the definition of the fundamental class for the Eilenberg-MacLane space $K(A,0)$.
Recall that a fundamental class $\imath_{A,n}$ for a polarized $K(A,n)$ is the element of $H^n(K(A,n),A)$ representing $\phi_n\circ H^{-1}$ via the UCT isomorphism. we use it to prove the brown representabil... |
In a bag with an infinite number of marbles, where a third are red, a third are green and a third are blue. Given that you pick $10$ marbles, of which $3$ are blue, what are the chances of picking $4$ red ones?
I've tried looking for an answer in the form of $\overline3^3\times\overline3^6\times{a\choose b}$, but I'm n... |
There is a coin with a probability $p$ of heads, and $1-p$ of tails. Tosses are independent of each other. When you bet an amount of money $x$, you receive $2x$ if it lands heads, and you lose what you bet if it lands tails.
We are going to use the following strategy:
You start betting an initial amount $x_0$. Every ti... |
Is the following true ?
The inverse of a positive definite matrix is also positive definite and since symmetric then we could write the following:
$A=PP^T, \space B = A^{-1} = P\Lambda P^T$ since $AA^{-1}=I$ we could also write
$AA^{-1}=(PP^T)(P\Lambda P^T) = P(P^TP)\Lambda P^T = PA^T\Lambda P^T = I$ and since $P$ i... |
I was trying to solve the following exercise. Let $K$ a closed subset of $\mathbb{R}$.
$$X:=\left\{f\in L^2[0,1]:f(x)\in K \mbox{ a.e. }\:x\in [0,1] \right\}$$
Then:
1) $X$ is closed under strong convergence in $L^2$.
2) $X$ is in general not closed under weak convergence (find a counterexample).
I proved 1) and also t... |
Let $f: \Bbb R \to \Bbb R$ with
$$f(x)= \begin{cases} {\sin(\sqrt{x})\over\sqrt{x}},& \text{for } x>0\\
1,& \text{for } x=0\\
{\sinh(\sqrt{-x})\over\sqrt{-x}},& \text{for } x<0.\end{cases}$$
(i) Is $f$ continuous on $\Bbb R$ ?
(ii) Is $f$ differentiable on ... |
I'm going mad calculating the sum $\sum_{n=0}^{\infty}n6^{n}z^{n-1}$. I proceeded in this way: $\frac{1}{z}\sum_{n=0}^{\infty}n(6z)^n$ and I'd like to figure out a geometric serie, but how can I take that $n$ factor to the exponent? Thankyou!
|
Q: Let $$A = \begin{pmatrix}
3 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -1 & 3
\end{pmatrix}$$
Find the quadratic form of $q: \mathbb{R}^3 \to \mathbb{R}^3$ represented by A.
and find an orthonormal basis of $\mathbb{R}^3$ which q has a diagonal form.
-
So far I managed to find the quadratic form and used lagrange to get the fo... |
Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I can't prove the monotony given there are also exponential in the derivative. Any hints to that?
|
Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer.
Show that $$T_2 (x) = 2x^2 – 1.$$
My attempt:
$T_2 (x) = \cos (2 \arccos x)$
Because an identity for $\cos(ab)$ doesn't exist, I do not how to simplify further to get to the solution.
|
I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing.
I been reading quit a lot about this topic and I found many solutions
Gaussian elimination: This perhaps the most common approach for this problem. It's running time is $O(N^3)$ above GF (2)
Method of Four Russians: I... |
If I want to solve
$$\frac{d}{d\frac{1}{x}} x$$
is my approach correct?
As
$$\begin{align*} \frac{d}{d\frac{1}{x}}x&=\\
\text{with }\frac{1}{x}&=y\\
\frac{d}{dy}\frac{1}{y}&=-\frac{1}{y^2}\\
&=-\frac{1}{\left(\frac{1}{x}\right)^2}\\
&=-x^2
\end{align*}$$
Is this approach correct, or did I miss something?
|
Although I have a brief understanding of solving linear ODE systems, I got stuck with this non-linear system:
\begin{align*}
\left(x'(t) \right)^2 + k_1 x + k_1 y = 0\\
\left(y'(t) \right)^2 + k_2 x + k_2 y = 0
\end{align*}
Where $x(0)=x_0$ and $y(0)=y_0$.
It is possible to rewrite this problem to a single equation by ... |
What is the generalization of the Pauli matrices and Dirac matrices in higher dimensions? I am actually looking for $\sqrt{\mathbb{I}}$ but I can't use the principal root which is just $\mathbb{I}$. For $n=3$, I found that the Gell-Mann matrices don't work. I'm thinking that these actually are the $SU(n)$ generators. A... |
I'm wondering about this problem on bilinear forms :
We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$
I've proved $\phi$ is a bilinear form and symetric, but how could we do to prove it is a non generate form ?
Besides if we have $\mathbb{S_{n}(R)}$ the subspa... |
I was trying to solve this exercise.
Let $f\in C^2(\mathbb{R})$ a strictly positive function such that $f''$ is bounded. Then prove that $\sqrt{f}$ is Lipschitz.
A first idea was to prove that it's first derivative is bounded, but i find the expression $\dfrac{f'}{\sqrt{f}}$ which gave me no informations, also derivin... |
I was thinking about Jordan curve with infinite length and Koch snowflake seems to be a valid answer intutively. Can anyone give mathematical proof for this?
|
Let $G$ be a group and $N$ a normal subgroup of $G$. Let $H=\{g\in G\mid gn=ng\space \forall n\in N\}$. Prove that $H$ is a normal subgroup of $G$.
I've tried seeing if we can write $H$ as the intersection or product of normal subgroups, but haven't succeeded so far. I'd be grateful if someone could provide a hint in ... |
So I've tried using the definition: $f$ is differentiable at $(x_0,y_0)$ iff
$$
f(x,y)-f(x_0,y_0)=\frac{\partial f}{\partial x}(x_0)\cdot x+\frac{\partial f}{\partial y}(y_0)\cdot y+o(\sqrt {(x-x_0)^2+(y-y_0)^2})
$$
So, trying to prove this relation (substituted $f$ for $f^2$, and performing the chain rule), I got thi... |
This is a question from my exam in Calculus 1.
Problem 6
Let $f: [0,\infty[ \to \mathbb{R}$ be continuously differentiable and $\lim_{x \to \infty} f'(x) = 0$.
a) Show that for $n \in \mathbb{N}: \lim_{n \to \infty} \left( f(n+1) - f(n)\right) = 0$.
b) Is $f$ necessarily bounded? Justify your answer.
I started (a) w... |
Suppose $f: \mathbb{C}\rightarrow \mathbb{R}$
$f$ is continuous, bounded, $f(z)\geq 0$.
Can we claim that the following integration
$$\int_{C_R}f(z)dz$$
is equal to zero? ($C_R$ is a circle centered on 0, with radius R )
and what if $R\rightarrow \infty$?
|
I am trying to re-learn some basic math and I realize I have forgotten most of it.
Evaluate $$\sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1}$$
Call the terms $S_n$ and the total sum $S$.
$$S_n < \frac{1}{n^3} \Rightarrow \sum_{n=1}^{\infty} \frac{n}{n^4+n^2+1} = S < \infty$$
$$S_n = \frac{n}{n^4+n^2+1} = \frac{n}{(n^2+1)^2... |
How to show that any path $\gamma:[a, b]\rightarrow\mathbb C$ is rectifiable and that $L(\gamma)=\int_{a}^{b}|\gamma'(t)|dt$.
Definition:
Consider a partition $P,a=t_0\lt t_1\lt \ldots\lt t_n=b\; of\; [a, b]$ and define
$L(P)=\sum_{i=0}^{n-1}|\gamma(t_{i+1}-\gamma(t_i)|$
The length of the curve , $L(\gamma)= sup\{L(P):... |
Assume that we have a rational polynomial of the form:
$$\chi\left(\omega\right)=\frac{\sum_{n=0}\left(c_n+ic_n^{\dagger}\right)\omega^{n}}{\sum_{n=0}\left(d_n+id_n^{\dagger}\right)\omega^{n}}$$
where $\omega$, $c_n$, $c_n^{\dagger}$, $d_n$, $d_n^{\dagger}$ are all real and $i$ is imagiary i.
Let us further assume that... |
The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum number of outcomes in favour of event $B$ can be?
|
Assume we have a lottery with payouts $(2,3,5)$. So if you buy a ticket you can win a pot which will payout your ticket price multiplied by one of those numbers.
The organizer expects a margin profit of $4\%$ from all tickets. So if the player plays with $1\$$ the mathematical expectation of outcomes will be $0.96$.
In... |
First of all, i don't know if the correct word is normalise or not, but I'll try to explain my issue.
I have a relationship between an object A and an object B equals 0.5 (max is 1)
you can consider this relationship as a vector that its length is 0.5
I have another vector between A and C and its value is 0.3
I have an... |
I have been wondering whether $U(1)$, defined as the group of complex phases (edit for clarity: I mean complex numbers of unit absolute value, such as $e^{i\alpha}$ with $\alpha \in \mathbb{R}$) with multiplication as group operation, is a normal subgroup of $U(2)$, defined as the group of $2\times 2$ unitary matrices.... |
L.S.,
I am reading a paper of D. Blasius, where he states that the numbers Tr$(X^n) $ determine the conjugacy class of a semisimple element $X \in GL_n(\mathbb{C})$. I am having trouble proving this. I understand that for conjugate elements $A,B$ we have Tr$(A^n) =$ Tr$(B^n)$, it follows because $A = GBG^{-1}$ for some... |
I would like to prove the sharp estimate for the $L_p$ norm to power $p$ with $1\leq p <\infty$. What is the constant $C$ here:
$$\left\|\sum_{j=1}^Jf_j\right\|^p_p\leq C\sum_{j=1}^J\|f_j\|_p^p$$
By triangle and Jensen inequalities, we have
$$\left\|\frac{1}{J}\sum_{j=1}^Jf_j\right\|^p_p\leq \left(\frac{1}{J}\sum_{j=1}... |
Let $H$ be a normal subgroup of $K$, which is a subgroup of $G$.
By just sketching some computations it seems that $H$ is not necessarily a normal subgroup of $G$ even if $K$ is a normal subgroup of $G$. I haven't proved it since I didn't really know how to structure the proof.
My motivation for this is understanding ... |
Let $N\in \text{Mat}(10 \times 10,\mathbb{C})$ be nilpotent. Furthermore let $\text{dim} \ker N =3 $, $\text{dim} \ker N^2=6$ and $\text{dim} \ker N^3=7$. What is the Jordan Normal Form?
The only thing I know is that there have to be three blocks, since $\text{dim} \ker N = 3$.
Thank you very much in advance for your... |
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