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How many subgraphs of $K_{9,9}$ are isomorphs of $C_6$? |
Is this particular function a positive semidefinite kernel function? |
I am trying to calculate the residue of $f(z) = \frac{1}{z^2}\sin\left(\frac{1}{z^2}\right)$ at $z=0$. Looking at the Taylor expansion of $f$ around $0$, as $c_{-n}$ (the coefficients of the powers of $z$) is non-zero for infinitely many $n$, $z=0$ must be a essential singularity of the function. However, as the expans... |
>Let $f:\mathbb{C} \to \mathbb{C} \\ \,\ \ \ \ \ z \to (|z|^2-2)\overline{z}$
>Determine the points where f is differentiable, presenting the due expression $f'(z)$.
$u(x,y)=Re((x^2+y^2-2)(x-yi))=x^3+xy^2-2x; \\ v(x,y)=Im((x^2+y^2-2)(x-yi))=-yx^2-y^3+2y $
$u_x=3x^2+y^2-2; \,u_y=2xy \\
v_x= \ \ -2xy;\qquad \... |
I have this recurrence relation
$$h_{n} = h_{n-1} + 4n(n-1) \ \text{for} \ n \geq 1 \ \text{with} \ h_{0} = 0, h_{1} = 0$$
I am asked to find $h_{9}$.
I have attempted to solve this by generating functions and ended up with a closed form solution as:\
$g(x) = \frac{4x}{(1-x)^3} - \frac{4x}{(1-x)^2}$\
I feel... |
>Let $f:\mathbb{C} \to \mathbb{C} \\ \,\ \ \ \ \ z \to (|z|^2-2)\overline{z}$
>Determine the points where f is differentiable, presenting the due expression $f'(z)$.
$u(x,y)=Re((x^2+y^2-2)(x-yi))=x^3+xy^2-2x; \\ v(x,y)=Im((x^2+y^2-2)(x-yi))=-yx^2-y^3+2y $
$u_x=3x^2+y^2-2; \,u_y=2xy \\
v_x= \ \ -2xy;\qquad \... |
[Resolution][1]
"The question is a problem from Leithold's calculus book. I didn't understand the ($x = 5 \cos(t)$). Shouldn't it be ($x = 2 \cos(t)$)?
[Resolution][2]
The theorem in the book is this. From what I understand, you calculate both sides of the equality in the theorem and obtain the same value.Ro... |
Does this series violate the integral test for convergence/divergence? |
Suppose $R$ is a ring and $P$ is a projective generator of right $R$-modules. If $M$ is a left $A$-module, show that the natural map $M \to \text{Hom}_R(P, P \otimes_R M)$ that sends $m \mapsto (p \mapsto p \otimes m)$ is an isomorphism.
Above is a question that arose while I was studying Morita theory. Morita theor... |
**An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?**
**********************
*Intro__________*
On a [previous question](https://math.stackexchange.com/q/4838311/909869) I found that the following approximation could be used:
$$\dfrac{\ln\left(1+x^{\ln(2)}\right)}{\... |
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers? |
First of all, in your reasoning, there has to be one more $7$. So the product looks like $18\cdot9 \cdot8 \cdot8 \cdot7 \cdot7$. And only then an edge goes to the very first vertex.
Second, you overcount each cycle $12$ times in this calculation. Imagine you chose the vertices 1-2-3-4-5-6 and made a cycle in that o... |
Question：
A, B, and C students will participate in a badminton competition, and the agreed format is as follows:
Those who have accumulated two losses are eliminated;
Draw lots before the competition to determine the first two contestants, with the other person taking turns;
The winner of each game will pla... |
Is there a better way to solve this problem of winning and losing probability in sports? |
Suppose $R$ is a ring and $P$ is a projective generator of right $R$-modules. If $M$ is a left $A$-module, show that the natural map $M \to \text{Hom}_R(P, P \otimes_R M)$ that sends $m \mapsto (p \mapsto p \otimes m)$ is an isomorphism.
Above is a question that arose while I was studying Morita theory. Morita theor... |
I am trying to prove the following:
>[![enter image description here][1]][1]
Using only modus ponens and the following axioms:
>[![enter image description here][2]][2]
I'm stuck for some days on it, [I've even made a small program in Mathematica to help me with the substitutions](https://i.imgur.com/uiVpx... |
Proving $A\to (¬A \to B)$ with Łukasiewicz's axioms and modus ponens? |
Does this series violate the integral test condition for convergence/divergence? |
Question：
A, B, and C students will participate in a badminton competition, and the agreed format is as follows:
Those who have accumulated two losses are eliminated;
Draw lots before the competition to determine the first two contestants, with the other person taking turns;
The winner of each game will pla... |
I am trying to answer the following exercise from [Hao's Fundamentals of Logic and Computation: With Practical Automated Reasoning and Verification](https://link.springer.com/book/10.1007/978-3-030-87882-5).
>[![enter image description here][1]][1]
Using only modus ponens and the following axioms:
>[![enter im... |
Question：
A, B, and C students will participate in a badminton competition, and the agreed format is as follows:
Those who have accumulated two losses are eliminated;
Draw lots before the competition to determine the first two contestants, with the other person taking turns;
The winner of each game will pla... |
Suppose we have a sequence of functions $f_n : \mathbb{N} \to [0,1]$ such that
$$ |f_n|_{1} := \sum_{i=1}^{\infty} |f_n(i)| = A $$ for each $n$, where $A > 0$ is given. Suppose that for each $i$ we have $f_n(i) \to g(i)$ as $n\to\infty$, where $g : \mathbb{N} \to [0,1]$ and $|g|_1 = A$.
How do I prove that $|g - f... |
Let $X$ be a quasicompact quasiseparated scheme and $Z$ a closed subset of $X$ with quasicompact complement. Then there exists a perfect complex $F \in D^{\mathrm{perf}}(X)$ with $\operatorname{supp} F = Z$. The proof of this in a paper I'm reading uses the induction principle for qcqs schemes and the following lemma: ... |
I was checking the subadditivity of the function $f\colon x\mapsto \min(1, x)$, *i.e.*, whether $f(x + y)\le f(x) + f(y)$ and was expecting that the entire first quadrant would be colored. However, the following was outputted:
[![enter image description here][1]][1]
But clearly, for nonnegative $x, y$ satisfying ... |
Let four random variables form the Markov chain
$${\displaystyle W\rightarrow X\leftarrow Y\rightarrow Z},$$
such that the conditional distribution of $X$ depends only on $W$ and $Y$ and is conditionally independent of $Z$ given $Y$. What is the relationship between the mutual informations I(W;X|Y) and I(W;X|Z) ... |
Suppose we have a sequence of functions $f_n : \mathbb{N} \to [0,1]$ such that
$$ |f_n|_{1} := \sum_{i=1}^{\infty} |f_n(i)| = A $$ for each $n$, where $A > 0$ is given. Suppose that for each $i$ we have $f_n(i) \to g(i)$ as $n\to\infty$, where $g : \mathbb{N} \to [0,1]$ and $|g|_1 = A$.
How do I prove that $|g - f... |
Consider the following problem:
Alice has decided to participate in a jury with three members, with the verdict decided by majority. To express her disinterest in the case, she decides to vote by flipping a fair coin. The other two members make the correct decision with probability $p \in (0, 1)$. How does this arra... |
Suppose we have a sequence of functions $f_n : \mathbb{N} \to [0,1]$ such that
$$ |f_n|_{1} := \sum_{i=1}^{\infty} |f_n(i)| = A $$ for each $n$, where $A > 0$ is given. Suppose that for each $i$ we have $f_n(i) \to g(i)$ as $n\to\infty$, where $g : \mathbb{N} \to [0,1]$ and $|g|_1 = A$.
How do I prove that $|g - f... |
Suppose we have a sequence of functions $f_n : \mathbb{N} \to [0,1]$ such that
$$ |f_n|_{1} := \sum_{i=1}^{\infty} |f_n(i)| = A $$ for each $n$, where $A > 0$ is given. Suppose that for each $i$ we have $f_n(i) \to g(i)$ as $n\to\infty$, where $g : \mathbb{N} \to [0,1]$ and $|g|_1 = A$.
How do I prove that $|g - f... |
Is a totally ordered set consisting of discrete point sets containing point p a well-ordered set? |
I don't believe there are any general conditions to tell if the sum of $n$ non-convex functions is convex outside of using the definitions of convexity to show it directly. It's very dependent on your functions.
Fortunately, you're just trying to show that the sum of $f$ isn't necessarily always convex (assuming... |
First, let me say that asking for "the subset" is usually a wrong question to ask, because sets usually have more than one subset.
A set that includes only one element is a **singleton**. Singletons always have 2 subsets -- the empty set $\emptyset$ and the entire set itself.
For example the set $\{a\}$ has two ... |
At the end of section 2 of [this paper][1], the authors mention the integral:
$$\frac{\alpha}{Z}\int N(x_i | \mu, \sigma I) N(\mu | 0, \rho I) d \mu$$
(Note: $x_i$ and $\mu$ are $d$-dimensional.)
In the first part of section 3.1, they claim that "by a straightforward calculation," that integral evaluates to
... |
If $m\mid n$ can we find a group $G$ of order $n$ with a subgroup $H$ of order $m$? |
The Axiom of Regularity tells us the following is disallowed:
$$ A = \{ A, B \} $$
where A and B are either sets, or fundamental objects ("lowest hierarchy" as described by Terence Tao's Analysis I). Clearly, here $A$ is a set.
---
Now, as I was doing an exercise I came to a point in the proof resulting in ... |
We know that $e = \lim_{n\to \infty}$$ (1 + {1\over n})^n $. Now, If we expand $\left(1+\frac1n\right)^n$, we get
$$\tag1 \left(1+\frac1n\right)^n = \sum_{k=0}^n{n\choose k}\frac1{n^k}=\sum_{k=0}^n\frac{n!}{(n-k)!n^k}\cdot\frac1{k!} = \frac{1}{n^0} + \frac{n}{n}+ \sum_{k=2}^n \frac1{k!}\prod_{t= 1}^{k-1} \Big(1 - \fra... |
Is my proof that the Euler number is equal to the sum of infinite series correct and rigorous? |
Let $\mathbb{N} = \{1,2,\dots\}$ be the set of natural numbers, and let $C_0$ the the collection of functions on $\mathbb{N}$ that vanishes at infinity. Let ${\cal C}\subset C_0$ be the collection of functions that is finite linear combinations of $f_s(n) = s^n$ where $0\le s<1$. More explicitly,
$$
{\cal C} = \lef... |
In the Hartshorne proof of Bertini's theorem (CH-2 Theorem 8.18), I am having difficulty in the first part of the proof which tells us that the scheme $H \cap X$ is regular at every point.
I could follow till the statement that $x \in H \cap X \iff \varphi_x(f) \in \mathfrak{m}_x$.
(Q-1) I just want to confirm he... |
I'm considering functions on the interval $[0,1]$ which are continuously differentiable, so in the space $C^{1}([0,1])$. On this space I want to show whether or not the two norms $\|f\|_{A} = \|f\|_{\infty} + \|f\|_{1}$ and $\|f\|_{D} = |f(0)| + \|f'
\|_{1}$ are equivalent.
Note that $\|f\|_{\infty}$ is defined as... |
Are these two norms on $C^{1}([0,1])$ equivalent? |
Suppose V is a finite-dimensional complex vector space and S,T ∈ L(V) commute. Prove that there exist a, b ∈ C such that range(S − a) + range(T − b) ≠ V. How would I write out the solution to this question? I can't seem to find a contradiction if I assume the negation of the statement.I don't know how to next proceed o... |
I'm reading van Bentham's *Modal Logic for Open Minds* (section 5.7), where he proves the (weak) completeness of K. I'm trying to work out each step, but I seem to need to use an intuitionistically invalid inference, which I'm not sure if necessary for this proof. I'm also not sure how to extend the same proof to prove... |
I have discovered a pretty weird sequence
$$a_0=1$$
$$a_1= 2$$
$$a_n=\frac{a_{n-1}+a_{n-2}}{a_{n-1}-a_{n-2}}$$
The first few terms are:
$$1,\ 2,\ 3,\ 5,\,\ 4,\ -9,\ \frac{5}{13},\ -\frac{53}{61},\ \frac{423}{1033},...$$
The sequence is very unpredictable and looks very random. I wonder if there is a closed-fo... |
At the end of section 2 of [this paper][1], the authors mention the integral:
$$\frac{\alpha}{Z}\int N(x_i | \mu, \sigma I) N(\mu | 0, \rho I) d \mu$$
(Note: $x_i$ and $\mu$ are $d$-dimensional.)
In the first part of section 3.1, they claim that "by a straightforward calculation," that integral evaluates to
... |
*With* the Deduction Theorem, you can do this:
$1. \alpha \to \beta \ (A.)$
$2. \neg \neg \alpha \ (A.)$
$3. \neg \neg \alpha \to \alpha \ (L5)$
$4. \alpha \ (MP \ 2,3)$
$5. \beta \ (MP 1,4)$
$6. \beta \to \neg \neg \beta \ (L6)$
$7. \neg \neg \beta \ (MP \ 5,6)$
So now we have shown $\alpha \to... |
Here is a hint: As you said, $S$ and $T$ are simultaneously diagonalizable, so there is some basis $\vec{v}_1,\dots,\vec{v}_n$ of $\mathbb{C}$ such that all $\vec{v}_i$ are eigenvectors of both $S$ and $T$. Let $a$ and $b$ be the corresponding eigenvalue to $\vec{v}_1$ in $S$ and $T$, respectively.
Next, consider $S... |
I have a sequence of 54 events (Likert-scale survey items). The outcomes of these events (item responses) correspond to integers.
I need to find the probability (assuming all outcomes for each event are equally likely) of ANY pattern of outcomes repeating 3 or more times consecutively OR of a single outcome repeati... |
I'm having an issue at work where several of us are seemingly arguing over a naming convention in piece of software we're writing.
Basically, suppose we have 3 variables $x, y, z$. $z$ is usually some nice even number like 1000. $x$ and $y$ are integers.
We want to compute a rounded version of $x + y$ (call it $... |
We know that $e = \lim_{n\to \infty}$$ (1 + {1\over n})^n $. Now, If we expand $\left(1+\frac1n\right)^n$, we get
$$\tag1 \left(1+\frac1n\right)^n = \sum_{k=0}^n{n\choose k}\frac1{n^k}=\sum_{k=0}^n\frac{n!}{(n-k)!n^k}\cdot\frac1{k!} = \frac{1}{n^0} + \frac{n}{n}+ \sum_{k=2}^n \frac1{k!}\prod_{t= 1}^{k-1} \Big(1 - \fra... |
We know that $e = \lim_{n\to \infty}$$ (1 + {1\over n})^n $. Now, If we expand $\left(1+\frac1n\right)^n$, we get
$$\tag1 \left(1+\frac1n\right)^n = \sum_{k=0}^n{n\choose k}\frac1{n^k}=\sum_{k=0}^n\frac{n!}{(n-k)!n^k}\cdot\frac1{k!} = \frac{1}{n^0} + \frac{n}{n}+ \sum_{k=2}^n \frac1{k!}\prod_{t= 1}^{k-1} \Big(1 - \fra... |
**Theorem.** Given that W is finite-dimensional and T∈L(V,W). then T′=0 if and only if T=0.
I’m having trouble proving T=0 if T ′ = 0. I see a counter example to this. From hypothesis T′=0 so equivalently given any linear functional ψ∈W′, T′(ψ)=ψ∘T=0. Then T can be non zero when ψ is a zero map i.e. all values in W ... |
We know that $e = \lim_{n\to \infty}$$ (1 + {1\over n})^n $. Now, If we expand $\left(1+\frac1n\right)^n$, we get
$$\tag1 \left(1+\frac1n\right)^n = \sum_{k=0}^n{n\choose k}\frac1{n^k}=\sum_{k=0}^n\frac{n!}{(n-k)!n^k}\cdot\frac1{k!} = \frac{1}{n^0} + \frac{n}{n}+ \sum_{k=2}^n \frac1{k!}\prod_{t= 1}^{k-1} \Big(1 - \fra... |
We know that $e = \lim_{n\to \infty} (1 + {1\over n})^n $. Now, If we expand $\left(1+\frac1n\right)^n$, we get
$$ \left(1+\frac1n\right)^n = \sum_{k=0}^n{n\choose k}\frac1{n^k}=\sum_{k=0}^n\frac{n!}{(n-k)!n^k}\cdot\frac1{k!} = \frac{1}{n^0} + \frac{n}{n}+ \sum_{k=2}^n \frac1{k!}\prod_{t= 1}^{k-1} \Big(1 - \frac{t}{n... |
Here is a hint: Since $S$ and $T$ are commuting matrices, $S$ and $T$ are simultaneously triagonalizable, so there is some basis $\vec{v}_1,\dots,\vec{v}_n$ of $\mathbb{C}$ such for all $1\leq i\leq n$, both $S\vec{v}_i$ and $T\vec{v}_i$ are in $\text{span}(\vec{v}_1,\dots,\vec{v}_i)$. In particular, $\vec{v}_1$ is a c... |
In a cards game, there are ten cards numbered 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/6cH6b.png
The difference between the corresponding cards is calculated such t... |
In a cards game, there are ten cards numbered 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/6cH6b.png
The difference between the corresponding cards is calculated such t... |
Does Von Neumann's Minimax Theorem concluding that it doesn't matter which player moves first (when moving means submitting a probability distribution whose realizations take place after both moves are submitted) apply to zero-sum games only if they consist of a single move from each player, or does it also apply to ze... |
**Theorem.** Given that $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. then $T'=0$ if and only if $T=0$.
I’m having trouble proving $T=0$ if $T' = 0$. I see a counter example to this. From hypothesis $T'=0$ so equivalently given any linear functional $\psi \in W', T'(\psi)= \psi \circ T=0$. Then $T$ can be... |
In a cards game, there are ten cards numbered 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/6cH6b.png
The difference between the corresponding cards is calculated such t... |
Let $(X,\sigma)$ and $(Y,\nu)$ are measurable space and $f:\ X\longrightarrow Y$ is a map that takes each measurable set of $\sigma$ to a measurable set of $\nu$, then is $f$ needed to be measurable, i.e preimage of measurable set in $Y$ is a measurable set in $X$ ?
In general topology, an open map - a map $f:\ X\lo... |
If $f$ maps any measurable set to a measurable set, then is $f$ a measurable map? |
I am wondering if there is symmetry among the single partial derivative of a multivariate diffeomorphic function.
Let $f: \mathcal{X} \rightarrow \mathcal{X}$ be a diffeomorphism defined over $\mathcal{X} \subset \mathbb{R}^d$ (it is a d-dimensional multivariate function). Let $\mathbf{V} \in \mathcal{X}$ be some d-... |
In a cards game, there are ten cards numbered 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/6cH6b.png
The difference between the corresponding cards is calculated such t... |
We use Theorem 5.80 of [Linear Algebra Done Right](https://link.springer.com/content/pdf/10.1007/978-3-031-41026-0.pdf):
> Suppose $V$ is a fnite-dimensional complex vector space and $S, T$ are commuting operators on $V$. Then there is a basis of $V$ with respect to which both $S$ and $T$ have upper-triangular matr... |
For a filtered bialgebra $(A,m,u,\delta,\epsilon)$, with $A_j,j\in\mathbb Z$ filtration, so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$, the assotiated graded space $\text gr A=\bigoplus_{j}A_{j}/A_{j-1}$ has a multiplication and a comultiplication induced from $A$, but the... |
I have that $A$ is a positive definite matrix and $b$ is some fixed vector. Define $r=b-Ax$ to be the residual vector and $e=A^{-1}b-x$ to be the error vector for all vectors $x$. I need to show that $\langle e,r\rangle$ is positive unless $Ax=b$. What I have done so far:
\begin{align*}
\langle e,r\ran... |
How would you solve this?
[![enter image description here][1]][1]
I did the standard way of solving questions like these - I took the Laplace of both sides and used the convolution identity. But the solution I got, $f(t)=3+t^2/2$, does not satisfy the integral equation, nor does it satisfy the initial conditions.... |
Hint: prove that no matter how you arrange the cards, the cards $1,2,3,4,5$ are opposite some higher-value card, while the cards $6,7,8,9,10$ are opposite some lower-value card. (Think about it one card at a time: what would it take for $5$ to end up opposite one of $1,2,3,4$, for example?)
If you prove this, then i... |
Suppose $p$ is an odd prime. I wish to show that the number of solutions to $y^2 \equiv x^2 + 1 \pmod{p}$ is $$p + \sum_{k=0}^{p-1}\left(\frac{k^2 +1}{p}\right).$$
I know that, for any $a$, $y^2 \equiv a \pmod{p}$ admits 1 + $\left(\frac{a}{p}\right)$ solutions (mod $p$). It is thus 'intuitively' true that $y^2 \equi... |
Suppose i have a strategy that is not buy-and-hold type of strategy. It can have unique entry timing and unique exit timing for a single asset and both long, and short positions will be allowed, and sometimes, there may exist some time period where the strategy doesn't have any positions and holds all of it in cash.
... |
Given a surface bundle over surface $S_g\to M\to S_h$ (where $g,h>1$, there is an induced surface group representation
$$\pi_1(S_h)\to \text{SP}(2g,\mathbb R).$$
I was told that there is an associated equivariant holomorphic map
$$\tilde{S_h}=\mathbb H^2\to \text{SP}(2g,\mathbb R)/U(g).$$
I wonder why there is ... |
In a cards game, there are ten cards, 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/3sRqU.png
The difference between the corresponding cards is calculated such that the ... |
The unit sphere has periodic geodesic flow, of period $2 \pi$. By the Killing Hopf Theorem, every manifold with constant positive sectional curvature must be a quotient of the sphere by a group acting freely and properly discontinuously. Does it follow from this that all manifolds with constant positive sectional curva... |
I came across this sum $ \sum_{p \leq N \ and \ p \ prime} \frac 1 p $ in page 22 of Concrete Mathematics by Graham, Knuth and Patashnik. The authors denote that:
> Incidentally, this sum gives the approximate average number of distinct prime
factors of a random integer near N, **since about $1/... |
Why $ \sum_{p \leq N \ and \ p \ prime} \frac 1 p $ counts prime factors of $N$? |
Given a surface bundle over surface $S_g\to X\to S_h$, then there are two associated vector bundles $H^1(S_g,\mathbb C)\to E\to S_h$ and $H^{1,0}(S_g,\mathbb C)\to E_1\to S_h$, s.t. the Toledo invariant of $E$ is $\deg{E_1}$.
I think I know why we have the vector bundle $H^{1,0}(S_g,\mathbb C)\to E_1\to S_h$, but I ... |
Given a closed manifold $M$ of dimension $3D$, we can construct an alternating trilinear map $\alpha(a, b, c) = \int_M a \cup b \cup c$ where $a, b, c \in H^D(M)$.
The question is the realization question: Given an arbitrary alternating trilinear map, is there a closed manifold $M$ that generates such a trilinear ma... |
In the Hartshorne proof of Bertini's theorem (CH-2 Theorem 8.18), I am having difficulty in the first part of the proof which tells us that the scheme $H \cap X$ is regular at every point.
I could follow till the statement that $x \in H \cap X \iff \varphi_x(f) \in \mathfrak{m}_x$.
(Q-1) I just want to confirm he... |
Question：
A, B, and C students will participate in a badminton competition, and the agreed format is as follows:
Those who have accumulated two losses are eliminated;
Draw lots before the competition to determine the first two contestants, with the other person taking turns;
The winner of each game will pla... |
Suppose $J \subseteq \mathbb{C}[x_1,\ldots,x_n]$ is an ideal, and $I = \sqrt{J}$ is its radical. How large can the smallest integer $e$ for which $I^e \subseteq J$ be, specifically in terms of the maximal degree(s) of the Groebner basis of $J$ and $I$, and the number of variables $n$?
Is anything known in this regar... |
Suppose $P < Q$ and $PQ=N$, where both $P$ and $Q$ are prime (including $2$). What's the maximum distance $P$ can ever be from $\sqrt{N}$, if we hold $Q$ constant?
|
Consider the set of compactly supported smooth functions $C^\infty_c(\mathbb{R})$. The dual of this space, $(C^\infty_c(\mathbb{R}))'$ is often referred to as a very large space. Since every $f \in C^\infty_c(\mathbb{R})$ can be mapped to an element in the dual space by integration we know that $C^\infty_c(\mathbb{R}) ... |
How do we know the space of distributions is "big"? |
I'm aware that _by definition_ the Mutual Information (MI) should be non-negative, and there are two related questions here: [(1)][1] and [(2)][2].
However, I can think of an example in Physics where it is (or at least it seems to be) negative. So, please, point to where my reasoning is wrong.
First, write the $M... |
Negative mutual information example. What's wrong about it? |
I'm aware that _by definition_ the Mutual Information (MI) should be non-negative, and there are two related questions here: [(1)][1] and [(2)][2].
However, I can think of an example in Physics where it is (or at least it seems to be) negative. So, please, point to where my reasoning is wrong.
First, write the $M... |
I'm not sure I understand your argument... what does it mean for "the claim to be true for sentences of the form..."? "The claim" is not a statement about all sentences, it is a statement about an equivalence of theories. You seem to be showing each sentence in $T$ is equivalent under $T$ to a quantifier-free sentence,... |
If $p(x) = a_3 + a_2x + a_1x^2 + a_0x^3$ is irreducible in $\mathbb{Q}[x]$, then $q(x) = a_0 + a_1x + a_2x^2 + a_3x^3$ is also irreducible? |
About five years ago, I discovered a property of a parabola, a circle that touches it at two points, and two tangents to it that are parallel to the index of the parabola.
[![enter image description here][1]][1]
In the previous picture $AC⊥BD$
After some trying, I was able to prove this with analytical geometry
... |
$we \ suppose \ f:[0,a]\to \mathbb{R},a>0,f(x) \ is \ monotonically \ increasing \ on \ [0,a] \ and \ f(x) \geq 0 ,\ \forall x\in [0,a].$ $$Calculate: \lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{sin(px)}{x}dx$$
My idea is this:$$let \ t=px \ we \ get \ \int_{0}^{a}f(x)\frac{sin(px)}{x}dx= \int_{0}^{ap}f(\frac{t}{p})... |
I'm not sure I understand your argument... what does it mean for "the claim to be true for sentences of the form..."? "The claim" is not a statement about all sentences, it is a statement about an equivalence of theories. You seem to be showing each sentence in $T$ is equivalent under $T$ to a quantifier-free sentence,... |
For a filtered bialgebra $(A,m,u,\delta,\epsilon)$, with $A_j,j\in\mathbb Z$ filtration, so that $\delta(A_n)\subset\sum_k A_k\otimes A_{n-k}$ and $m(A_k\otimes A_{n-k})\subset A_n$, the assotiated graded space $\text {gr}A=\bigoplus_{j}A_{j}/A_{j-1}$ has a multiplication and a comultiplication induced from $A$, but th... |
In a cards game, there are ten cards, 1 to 10. Two players, seated facing each other, randomly choose 5 cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/3sRqU.png
The difference between the corresponding cards is calculated such that the ... |
How can commutativity of operators be assumed in this example? |
$we \ suppose \ f:[0,a]\to \mathbb{R},a>0,f(x) \ is \ monotonically \ increasing \ on \ [0,a] \ and \ f(x) \geq 0 ,\ \forall x\in [0,a].$ $$Calculate: \lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{sin(px)}{x}dx$$
My idea is this:$$let \ t=px \ we \ get \ \int_{0}^{a}f(x)\frac{sin(px)}{x}dx= \int_{0}^{ap}f(\frac{t}{p})... |
$We \ suppose \ f:[0,a]\to \mathbb{R},a>0,f(x) \ is \ monotonically \ increasing \ on \ [0,a] \ and \ f(x) \geq 0 ,\ \forall x\in [0,a].$ $$Calculate: \lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{\sin(px)}{x}dx$$
My idea is this:$$let \ t=px \ we \ get \ \int_{0}^{a}f(x)\frac{\sin(px)}{x}dx= \int_{0}^{ap}f(\frac{t}{p... |
Suppose $f\colon[0,a]\to \mathbb{R}$ with $a>0$, is monotonically increasing on $[0,a]$ and $f(x) \geq 0$ for all $x\in [0,a]$.Calculate
$$\lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx$$
My idea is as follows. Let $t=px$, then we get $$\int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx= \int_{0}^{... |
Suppose $f\colon[0,a]\to \mathbb{R}$ with $a>0$, is monotonically increasing on $[0,a]$ and $f(x) \geq 0$ for all $x\in [0,a]$. Calculate
$$\lim_{p\to +\infty} \int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx.$$
My idea is as follows. Let $t=px$, then we get $$\int_{0}^{a}f(x)\frac{\sin(px)}{x}\,\mathrm dx= \int_{0}... |
I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after applying the global sections functor. As we have a quasi-isomorphism between sheaf complexes if the homology groups (which are ... |
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