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Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor t.....(5)$
$\overline{t}\lor s.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
How to use resolution in this formula? |
Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor t.....(5)$
$\overline{t}\lor s.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
Here is the "halting first" problem: given two programs $P$ and $Q$, return 0 if $P$ stops after less steps than $Q$, return $1$ if $Q$ stops after less steps than $P$, and otherwise output $0$ or $1$.
Is the "halting first" problem equivalent to the halting problem? If we have an oracle for the halting problem, we ... |
Is being able to decide which program stop first equivalent to the halting problem? |
By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$-bimodules and the category of cocontinuous, additive endofunctors on the category of $R$-bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its enveloping ring.
The category of cocontinuous, additive functors carries ... |
I know that the Monty Hall problem is old news. However, I can't get this out of my head and I am looking forward to your thoughts about it.
If the rules of the game are known from the beginning, especially that the game host will guarantee to open a door containing a goat, isn't this information extremely relevant ... |
Monty Hall problem - doesn't it matter that you know from the start that a goat will be revealed with 100% certainty? |
Let $F$ be an irreducible, integral polynomial. Is it true that$$|\{\nu:F(\nu)\equiv 0\mbox{ mod } n,\ 0\le\nu<n\}|\ll n^{\epsilon}$$as $n\rightarrow+\infty$? How can one show it? |
By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$-bimodules and the category of cocontinuous, additive endofunctors on the category of $R$-bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its enveloping ring.
The category of cocontinuous, additive functors carries ... |
To find the square root of $125$ by hand:
$$
\begin{array}{l|l}
& 1,25.00 & 11.1 \\
&1 & \\
\hline
21 & \space \space \space \space 25 & \\
&\space \space \space \space21 & \\
\hline
221& \space \space \space \space \space \space 4.00& \\
&\space \space \space \space \space \space221&
\end{array}
... |
For every line in the 2d plane, we can construct a shape with an "inside and outside" (often a circle) such that the shape is cut by the line into two symmetrical parts.
Does this property of lines extend to all curves?
My idea is if you "zoom in" on a curve, if you get small enough, it might become a line segme... |
I want to calculate the expected rate cut/change given futures contracts on the basic interest rate, knowing when the rate cut occurs (resembling the FED's FOMC). Consider compound rates and a 252 workday-year.
-DI1 is the interest rate's futures contract due to 7 workdays with an annual rate of 11.16% per year;
-D... |
How to estimate interest rate cut from futures contracts? |
Most formulations of the LCP derived from the KKT conditions of a QP tackle problems with non-negativity constraints $x\ge 0$. [Wikipedia][1] presents an alternative without the non-negative constraints but that requires the $Q$ matrix (from the quadratic objective) to be non-singular.
My question is: if one can onl... |
Here is the "halting first" problem: given two programs $P$ and $Q$, return 0 if $P$ stops after less steps than $Q$, return $1$ if $Q$ stops after less steps than $P$, and otherwise output $0$ or $1$. (So if neither problem halts, or if both halts in the same number of steps, the oracle may output either $0$ or $1$ fr... |
To say "R[x] is defined to be the set of all *symbols*..." is not clear. What is a *symbol*?
More formally, a polynomial is a sequence $$(a_0,a_1, a_2,...)$$ where the $a_i \in R$ are almost all zero, which means that there exists an $n$ such that $$\forall m>n, a_m=0$$
The *indeterminate* $X$ is **defined as** $$... |
**I am trying to simplify:**
$(p \lor \neg q) \land (p \lor q) $
One thing, I identify from the table is this:
$(p \lor q) \land (p \lor r) $ is **second distributive law** which becomes $p \lor (q \land r) $
**My steps:**
1. $(p \lor \neg q) \land (p \lor q) $
2. $p \lor (q \lor \neg q) $
3. $p$... |
Why is the distributive law incorrect here? |
Suppose a QCQP problem :
$$ \min_{x\in\mathbb{R}^{n}}f\left(x\right)=\frac{1}{2}x^{T}P_{0}x+q_{0}^{T}x$$
$$ s.t:\begin{cases}
\frac{1}{2}x^{T}P_{i}x+q_{i}^{T}x+r_{i}\le0 & i=1,2\dots,m:m\le n
\newline Ax=b
\end{cases}$$
Where $P_i $ are all positive semi definite.
a SOCP problem :
$$ \min_{y\in\mathbb{R}^{... |
When solving a recurrence relation with multiple repeated roots, would I add the two general solutions I get from each root?
For example this recurrence relation:
$a_{n+5} + 7a_{n+4} + 19a_{n+2} + 16a_{n+1} + 4a_{n} = 0$
Gives me the characteristic roots of $(r+1)^3(r+2)^2$, which is $r = -1$ with a multiplic... |
Hint: If you have a two digit number represented by $10a+b$, b represents the units. What does $a$ represent?
>!(In general, a number can be represented by $10^a x_1 + 10^{a-1} x_2 + 10^{a-2} x_3 \dots 10^{a-n-1} x_{n-1} + 10^{a-n} x_n$, where $x_1, x_2, x_3, \dots x_{n-1}, x_n$ are the digits of that number.) |
How to integrate $\frac{x^N\log(1+x)}{\sqrt{x^2+x_1^2}\sqrt{x^2+x_2^2}}$? |
Can someone please help clarify how to calculate the outer product of two basis vectors, assuming $\mathbb{R}3$? From the definition, I understand that $\textbf{a} \otimes \textbf{b}$ $= a_ib_j = c_{ij}$. Another way to write that is $\textbf{a} \otimes \textbf{b}$ $= a_ib_j\textbf{e}{_i}$ $ \otimes \textbf{e}{_j}$.
... |
Let $ M $ be a smooth manifold, let $ p\in M $ and let $ X\in \mathscr X(M) $ be a fixed vector field on $ M $. Given some other vector field $ Y $ the _Lie derivative_ of $ X $ along $ Y $ evaluated at the point $ p $ is defined as the limit
$$
(L_YX)\rvert_p = \lim_{t\to 0}\frac{\mathrm T_{\Phi_t^Y(p)}(\Phi_{-t}^Y)... |
We have $ X, Y \sim Bin(N,1/2) $ independent variables, and I'm asked to find $ \def\P{\mathbb P}\P(X > Y) $.
Considering the identity $ \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $, we can have a symmetry argument, saying $ \def\P{\mathbb P}\P(X > Y) = \def\P{\mathbb P}\P(Y > X) $, and calculating $ \def\P{\mat... |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function in analytic number theory, and there is a large literature on computing upper and lower bounds on $\Psi(x,y)$.
What are the best explicit upper and lower bounds known ... |
Let function $f:[0,1]^{N+1} \mapsto \mathbb{R}$ be continuously differentiable.
Let $Y_1,\cdots,Y_{N}$ be random variables with support on $[0,1]$ with continuously differentiable joint PDF.
$$g(x):= Pr(f(x,Y_1,\cdots,Y_{N})<0).$$
There is $y^* \in (0,1)$ such that for $Y_1\gtrless y^*$,
$$
f(0,Y_1,\cdots,Y... |
I want to evaluate the following expectation.
$$\mathbb{E}_{U\text{~Haar}} ||I-U||$$
where $||\cdot||$ is the operator norm (i.e., the maximum eigenvalue.). This is equivalent to the integral of
$$
\frac{1}{n!(2\pi)^n}\max_i|1-e^{i\theta_i}|\prod_{j\le k}|e^{i\theta_j}-e^{i\theta_k}|^2
$$
over $(\theta_i,...,\... |
We have $ X, Y \sim Bin(N,1/2) $ independent variables, and I'm asked to find $ \def\P{\mathbb P}\P(X > Y) $.
Considering the identity $ \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $, we can have a symmetry argument, saying $ \def\P{\mathbb P}\P(X > Y) = \def\P{\mathbb P}\P(Y > X) $, and calculating $ \def\P{\mat... |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function in analytic number theory, and there is a large literature on computing upper and lower bounds on $\Psi(x,y)$.
What are the best explicit upper and lower bounds known ... |
Let $P^*(n)$ be the largest prime factor of $n$, and let $\Psi(x,y) = | \{ n \mid n \leq x \wedge P^*(n) \leq y\}|$. This is a well-studied function in analytic number theory, and there is a large literature on computing upper and lower bounds on $\Psi(x,y)$.
What are the best explicit upper and lower bounds known ... |
Let $\mathcal{H}_*$ be a ordinary homology theory, that is, a homology theory that satisfies the [Eilenburg-Steenrod Axioms][1]. I wanted to show that $\mathcal{H}_1(S^1)\cong \mathcal{H}_0(S^1)\cong R$, where $H_0(*)\cong R$, by using the Mayer-Vietoris sequence.
The standard way to do this is to use open cover $U... |
Another way to solve this problem is as follows. Place $A,B,C,D$ on the $xy$ plane. Let $ a = AB , b = BC , c = CD, d = DA $, and let $V$ be the apex with a $z$ coordinate that is positive. Let $ e= VA, f = VB, g = VC, h = VD $.
Now express the coordinates of all the points using three angles. Angles $\theta$ an... |
By the Eilenberg-Watts theorem there is a monoidal equivalence between the category of $R^e$-bimodules and the category of cocontinuous, additive endofunctors on the category of $R$-bimodules. $R$ is a ring and $R^e=R\otimes \overline{R}$ its enveloping ring.
The category of cocontinuous, additive functors carries ... |
>How many composition of $n$ are there in which each part is at least $c$ ?
I don't think my way of doing it is completely correct.
Let $S=\bigcup_{k \geq 0} N_{\geq c}^k$ where $k=0$ is the empty composition of 0 . The generating function for a single part of the composition that is at least $c$ would be $x^c+x^{... |
Define an Interface as $I = <\theta,\phi>$, where $\theta$ defines a set of typenames and $\phi$ consists of function declarations $(f: t_n \times ... t_k \mapsto t)$, that takes arguments of type $t$ and returns a value of type $t \in \theta$. If a function takes 0 arguments, it is said to be a constant. **Let $V$ be ... |
I faced the following problem recently. When I was working on an electricity market clearing problem, it is hard for me to derive the dual problem of it, which is necessary to obtain the optimal price.
The original prime problem:
\begin{align}
&\mathop{Maximize}_{p_g^G, p_d^D, \theta_n}{\quad SW = \sum_d U_d p_d^... |
Consider a coin-tossing event in which at each toss $i$ the probability of having H is $p_i$. What is the probability of having exactly $k$ consecutive H in $n$ toss? Let's say k=2 and n=6; then here we want to know the probability of all the outcomes listed below.
HHTTTT
TTHHTT
TTTTHH
THHTTT
THTHHT
... |
Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor t.....(5)$
$\overline{t}\lor s.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
I faced the following problem recently. When I was working on an electricity market clearing problem, it is hard for me to derive the dual problem of it, which is necessary to obtain the optimal price.
The original prime problem:
\begin{align}
&\mathop{Maximize}_{p_g^G, p_d^D, \theta_n}{\quad SW = \sum_d U_d p_d^... |
Another way to solve this problem is as follows. Place $A,B,C,D$ on the $xy$ plane. Let $ a = AB , b = BC , c = CD, d = DA $, and let $V$ be the apex with a $z$ coordinate that is positive. Let $ e= VA, f = VB, g = VC, h = VD $.
Now express the coordinates of all the points using three angles. Angles $\theta$ an... |
I need to proof if for any n $\geqslant$ 2 ; $a_n = \frac{a_{n-1}+a_{n+1}}{2}$ is correct, then the sequence is arithmetic progression.
I converted that formula to this:
$${a_n-a_{n-1}}=a_{n+1}-a_n$$
And at this point it's pretty obvious, that we could say the diffrence between the element and his neighbors is the... |
There's a lot of salient questions here, but I think that addressing the following might be satisfactory.
> Suppose that we are given a 3-times differentiable function $f:\Bbb R \to \Bbb R^3$. Let $s:\Bbb R \to \Bbb R$ be the path-length function of $f$, so that $f \circ s^{-1}$ is parameterized by path-length.
> ... |
[![enter image description here][1]][1]
We are seeing at representations $sl_3(\mathbb C)$ here from Fulton and Harris. The space $\mathfrak h$ is collection of diagonal matrices and we have written $\mathfrak{sl_3}(\mathbb C) = \mathfrak h \oplus (\oplus \mathfrak g_{\alpha})$ which is obtained by adjoint action of... |
I was reading the definition of CW-complex in terms of pushouts given by Lück's *Algebraische Topologie: Homologie und Mannigfaltigkeiten* (Chapter 3). It is stated (though not proven) that such a topological space is always Hausdorff. I guess this follows readily by induction if we can prove that the adjunction space ... |
Is the adjunction space of two Hausdorff spaces also Hausdorff? |
Is there a notion of an "adjoint Matrix" $M^*$ to some finite-dimensional square Matrix $M$, specifically with the property $$\forall x,y\in \mathbb{C}^n\quad\langle Mx,y\rangle=\langle x,M^*y\rangle$$ I'm aware of the existence of the adjugate Matrix and some of its properties, but don't seem to be able to see if it s... |
I'm reading this [problem][1] and I can understand how they got the output for the first four test cases. But the last one I can't really arrive at it. Is there some mathematical concept that I can apply to get to it instead of just writing a program that counts every permutation by brute force?
Also is there some b... |
So far, I have shown that $S^1 \setminus \{(1, 0)\}$ *is homeomorphic* to $(0, 1) \subset \mathbb{R}$:
Define the function $f: S^1 \setminus \{(1, 0)\} \rightarrow (0, 1)$ by $f\big((\cos(\theta), \sin(\theta))\big) = \theta / (2\pi)$ (since each point $(x, y) \subset S^1 $ may be represented uniquely by $x = \cos(\... |
Suppose I have a window function $h$. If I specify the bandwidth for $h$ to be $c>0$, so $\text{supp}(h) \subset [-c/2,c/2]$, then what and when can I say about the support of the Fourier transform $\widehat{h}$?
Similarly, the other way around, if I have $\text{supp}(\widehat{h}) \subset [-c/2,c/2]$, then what and ... |
There's a lot of salient questions here, but I think that addressing the following might be satisfactory.
> Suppose that we are given a 3-times differentiable function $f:\Bbb R \to \Bbb R^3$. Let $s:\Bbb R \to \Bbb R$ be the path-length function of $f$, so that $f \circ s^{-1}$ is parameterized by path-length.
> ... |
Suppose you choose a rational number at random and reduce it; what is the probability that the denominator is odd?
I looked around and heard you can't define a uniform probability measure on the rationals so the question is useless, but I ran some monte carlo simulations and no matter what bounds I chose, I always ... |
Is it true that if $K$ is a field of characteristic $0$ in which every polynomial of degree 3 is reducible, then $\bar{K}/K$ has degree 1 or 2? I have thought of taking $\alpha$ algebraic over $K$. Then $K\subset K(\alpha)\subset\bar{K}$. From the condition on the degree of irreducible polynomials $[K(\alpha):K]=1,2$ a... |
In problems like this I find it useful to use a homogenized form of polar coordinates: Let $x^4=r\cos\theta$ and $y^2=r\sin\theta$, with $0\le\theta\le\pi/2$. Then
$$\left|\frac{x^3y^3(y^4-x^8)}{(y^4+x^8)^2}\right| = \frac{r^{3/4}r^{3/2}r^2|\cos 2\theta||\cos\theta|^{3/4}|\sin\theta|^{3/2}}{r^4} \le r^{3/4+3/2-2} = r^... |
Suppose there is a dataset $X = [x_1, \cdots, x_n]$ where each element $x_i$ has a set of features (weights) $A_i = [a^1_i, a^2_i, \cdots , a^k_i]$. We then can have a multiple weighted average according to each feature like following:
$\bar{x}^1=\frac{\sum_i a^1_ix_i}{\sum_i a^1_i}$ and $\bar{x}^2=\frac{\sum_i a^2_i... |
Say I am given a function $f : \mathbb{C} \to \mathbb{C}$, where $f(z) = z^n$ for $n \geq 1$ is a positive integer, and given a disk $D(0, \varepsilon) = \{z \in \mathbb{C} : |z| < \varepsilon\}$, and I want to find the image of $f$ under $D(0, \varepsilon)$, i.e. I find $f(D(0, \varepsilon))$. So by definition of the ... |
It is often claimed that the following equation holds for the [modular discriminant](https://en.wikipedia.org/wiki/Weierstrass_elliptic_function#Modular_discriminant) $\Delta=g_2^3-27g_3^2$ of the Weierstrass elliptic funtion and the [Dedekind eta ($\eta$) function](https://en.wikipedia.org/wiki/Dedekind_eta_function) ... |
So I am having problems facing this exercise to find an example of such function. The conditions are: f is R[a, b], g:[a, b]—>R, f(x)=g(x) almost everywhere in [a, b]. Is there any g not R[a, b]? |
Suppose there is a dataset $X = [x_1, \cdots, x_n]$ where each element $x_i$ has a set of features (weights) $A_i = [a^1_i, a^2_i, \cdots , a^k_i]$. We then can have a multiple weighted average according to each feature like following:
$\bar{x}^1=\frac{\sum_i a^1_ix_i}{\sum_i a^1_i}$ and $\bar{x}^2=\frac{\sum_i a^2_i... |
Suppose I have a window function $h$. If I specify the bandwidth for $h$ to be $c>0$, so $\text{supp}(h) \subset [-c/2,c/2]$, then what and when can I say about the support of the Fourier transform $\widehat{h}$?
Similarly, the other way around, if I have $\text{supp}(\widehat{h}) \subset [-c/2,c/2]$, then what and ... |
I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$, where $\kappa:[0,T]\to(0,\infty)$ is linear and $W$ is a cylindrical Wiener process on $H:=L^2([0,1)^2)$. I somehow need to compute/estimate (or at least find a sharp upper bound on) $\operatorname E... |
So I am having problems facing this exercise to find an example of such function.
Let $f\in R([a, b])$. Can we find a function $g\colon [a,b]\to \mathbb R$ such that $f=g$ almost everywhere and $g$ not Riemann integrable? |
Suppose there is a dataset $X = [x_1, \cdots, x_n]$ where each element $x_i$ has a set of features (weights) $A_i = [a^1_i, a^2_i, \cdots , a^k_i]$. We then can have a multiple weighted average according to each feature like following:
$\bar{x}^1=\frac{\sum_i a^1_ix_i}{\sum_i a^1_i}$ and $\bar{x}^2=\frac{\sum_i a^2_i... |
One of the steps of deriving the equations for the parametric curve of a cycloid is the following:
[![enter image description here][1]][1]
Here we say that the green distance travelled by the circle and the length of the circle's green arc are equal, which then (alongside other steps) allows us to derive the para... |
One of the steps of deriving the equations for the parametric curve of a cycloid is the following:
[![enter image description here][1]][1]
Here we establish that the distance PT is equal to the distance OT, which then (alongside other steps) allows us to derive the parametric equation of the cycloid.
Every vid... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
Is the function$$f:\mathbb{R}^{m \times n}\rightarrow R$$ defined by
$$f(X)=\text{rank}(X),$$ a smooth function? |
Is Rank($X$) a smooth function where $X$ is a $m \times n$ matrix? |
Is the function$$f:\mathbb{R}^{m \times n}\rightarrow \mathbb{N}$$ defined by
$$f(X)=\text{rank}(X),$$ a smooth function? |
I have two functions $f_a(x) = y_a$ and $f_t(x) = y_t$, where the dimension of the domain is larger than range.
Finally, I want to compute the linearised mapping from $y_t$ to $y_a$ via ${}^t J_a = \frac{\partial y_a}{\partial y_t}$.
I can not compute this directly but can compute the Jacobians $J_a= \frac{\par... |
My question is the following:
_Assume a point $A$ outside some circle, and a point $X$ on the same circle, such that $AX$ is ***not*** a diameter of the circle. Draw $Y$ as the point of intersection between the circle and the (extesion of the) line $AX$. Draw the tangents to the circle in points $X$ and $Y$, and den... |
How to use resolution in this formula with XOR? |
My question is the following:
_Assume a point $A$ outside some circle, and a point $X$ on the same circle, such that $AX$ is ***not*** a diameter of the circle. Draw $Y$ as the point of intersection between the circle and the (extesion of the) line $AX$. Draw the tangents to the circle in points $X$ and $Y$, and den... |
My question is the following:
_Assume a point $A$ outside some circle, and a point $X$ on the same circle, such that $AX$ is ***not*** a diameter of the circle. Draw $Y$ as the point of intersection between the circle and the (extesion of the) line $AX$. Draw the tangents to the circle in points $X$ and $Y$, and den... |
I'm currently going through an article where I came across these two 1-D coupled differential equations.
$$\frac{dA}{dz} = a_1B(z)e^{-i\beta z} $$
$$\frac{dB}{dz} = a_2A(z)e^{i\beta z} $$
with these three conditions
$$ \frac{d}{dz}(|A(z)|^2 +|B(z)|^2)=0 $$
$$ a_1 =-a_2^* $$
$$ B(0)=B_0, A(0)=0 $$
The article... |
Let $X$ be a random variable with known distribution $N(\mu, \sigma)$. Let $Y$ be a count variable, i.e., it is a discrete random variable with known finite expectation and finite variance, which takes integer values in the range $[0, n]$, for finite $n$. Let their population correlation be $\rho > 0$.
What in gener... |
Under what conditions does a Normal random variable have approximately the same conditional distribution at every value of a discrete variable? |
I'm currently going through an article where I came across these two 1-D coupled differential equations.
$$\frac{dA}{dz} = a_1B(z)e^{-i\beta z} $$
$$\frac{dB}{dz} = a_2A(z)e^{i\beta z} $$
with these three conditions
$$ \frac{d}{dz}(|A(z)|^2 +|B(z)|^2)=0 $$
$$ a_1 =-a_2^* $$
$$ B(0)=B_0, A(0)=0 $$
The article... |
There's a lot of salient questions here, but I think that addressing the following might be satisfactory.
> Suppose that we are given a 3-times differentiable function $f:\Bbb R \to \Bbb R^3$. Let $s:\Bbb R \to \Bbb R$ be the path-length function of $f$, so that $f \circ s^{-1}$ is parameterized by path-length.
> ... |
Can we give bounds or infer any type of information about the spectrum of the point-wise product of two matrices $A$ and $B$ given knowledge about the spectrum of $A$ and $B$ ?
Are there non trivial conditions on $A$ and $B$ that would make reasoning about these easier ?
The inequality on the product of eigenva... |
If $R = GF(3)[x]$, that is the set of all polynomials with coefficients in $\{0,1,2\}$
and $I = R{\langle x^3+1\rangle}$, the ideal of $R$ generated by $x^3+1$, then show that the factor ring $R/I$ is a field.
I have found that $R/I = \{a+bx + cx^2 + I\ | \ a,b,c \in R\}$, how can I show that this is a field?
... |
Trying to find a general solution to a simplified equation of inertial movement in a non-inertial reference frame. For simplicity I'm restricting motion to the plane of rotation and making sure the vector of rotation is constant on the Z-axis. My issue is that I only get one solution, which should be the motion of a st... |
My question is the following:
_Assume a point $A$ outside some circle, and a point $X$ on the same circle, such that $AX$ is ***not*** a diameter of the circle. Draw $Y$ as the point of intersection between the circle and the (extesion of the) line $AX$. Draw the tangents to the circle in points $X$ and $Y$, and den... |
Assume we have two random variables $X$ and $Y$. Then, assume that $X|a < X < b$ has density $f^{a,b}(x)$. Then, assume that $g_{1}(y)<g_{2}(y)$ two functions. Is the following true: $X|g_{1}(Y) < X < g_{2}(Y), Y = y_{0}$ has density $f^{g_{1}(y_{0}),g_{2}(y_{0})}(x)$? What if $X$ and $Y$ are independent? |
**Hint** One can eliminate one of the functions via differentiation and substitution to produce an o.d.e. in, say, $A$ alone. Differentiating the equation for $A'$ gives
$$
A'' = a_1 e^{-i \beta z} (B' - i \beta B).
$$
Substituting for $B'$ using the second equation (in particular using that $a_1 a_2 = a_1(-\overli... |
My question is the following:
_Assume a point $A$ outside some circle, and a point $X$ on the same circle, such that $AX$ is ***not*** a diameter of the circle. Draw $Y$ as the point of intersection between the circle and the (extesion of the) line $AX$. Draw the tangents to the circle in points $X$ and $Y$, and den... |
In a paper I have encountered
> The solid angle is the magnetostatic potential normalised to be $4\pi$ cyclic, i.e it takes the values in $\mathbb{R} / 4\pi \mathbb{Z} ≅ S^1$.
What does this mean? It seems to describe a quotient space related somehow (≅) to the unit circle? |
What does $\mathbb{R} / 4\pi \mathbb{Z} ≅ S^1$ mean? |
I am interested in the asymptotics for large order and large argument for the modified Bessel function of first kind
$$I_{\frac{N-1}{2}} (Nz) $$
where
* $z$ is a fixed positive real number $z>0$ and
* $N$ is natural number.
I want to pass $N$ to $+\infty$.
I checked the NIST collection and the book by Abramo... |
We are to find under which conditions for $u$ and $v$ the rank-one update of the identity $A = I - uv^{T}$, is invertible.
Apparenetly the condition is simply that $1 + v^T u \neq 0$. I can see that we must require that $Ax \neq 0$ for all $x$ which leads me to $x \neq (v^Tx)u$.
This is where I get stuck. The sol... |
I am familiar with the Generalized singular value decomposition (GSVD). In GSVD, given matrices $A_1 \in \mathbb{C}^{m_1 \times n}$ and $A_2 \in \mathbb{C}^{m_2 \times n}$ it is possible to decompose the two matrices such that
\begin{align*}
A_1 &= U_1 \Sigma_1 Q^H, \\
A_2 &= U_2 \Sigma_2 Q^H.
\end{align*}
Is it p... |
Variation of Generalized singular value decomposition with more than two matrices? |
I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$ with Neumann boundary conditions, where $\kappa:[0,T]\to(0,\infty)$ is linear and $W$ is a cylindrical Wiener process on $H:=L^2([0,1)^2)$. I somehow need to compute/estimate (or at least find a sharp... |
Let $f:[0,\infty)->[0,\infty)$ be a concave function with $p \in [0,\frac{1}{2}]$.
a) Show that $f$ is integrable on every interval $[a,b] \subset R$.
b) Show that the function $F:[0,\infty)->[0,\infty)$, $F(x)=\int_{0}^{x^{p}}f(t)dt$ is concave.
I know there is a theoren that says if a function is concav... |
Building off of D S's comments (and the wikipedia article on harmonic quadrilaterals), we can see that $BXCY$ is harmonic.
However, if we let $C'$ be the other (one being $B$) intersection of ray $ZB$ and the circle, then $ZBC'$ is a line and $BXC'Y$ is harmonic. Since for fixed points $B,X,Y$ there is only one poin... |
I'm taking an algebraic geometry course and I'm trying to solve the next problem:
Take the curve $V$ parametrized by $(t,t^3,t^4)$. Show that $V$ is an algebraic set and find $\mathbb{I}(V)$, where $\mathbb{I}(V)$ denotes the ideal of all polynomials that are zero over $V$.
So far, I've shown that $V$ can be seen a... |
I have some difficulties to figure out what the category of IND-schemes and PRO-schemes are...
Are they full subcategory of algebraic stacks? Are they full subcategory of algebraic spaces? |
What are the categories of IND and PRO schemes? |
Consider the following pair of *a priori* independent problems...
> A manufacturer's revenue is $\$100s^{\frac{1}{3}}h^{\frac{2}{3}}$, where $s$ is the number of tons of steel they purchase and $h$ is the number of hours of labor they hire. Steel costs $\$2000$ per ton, labor costs $\$20$ per hour, and their budget... |
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