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Let $V$ be a full flag, $\lambda$ a partition. Consider $$\sigma_\lambda(V) = \{ \Lambda \in G(k,n): \Lambda \cap V_{n-k+i-\lambda_i} \geq i \}.$$ If you have another full flag $V'$, are $\sigma_\lambda(V)$ and $\sigma_\lambda(V')$ isomorphic to each other? It seems that in intersection theory, they only care about the... |
I am finding a Cauchy point to minimize a quadratic problem using gradient projection method. And I get a trouble reading this equation.
$x(t) = x(t_{j−1}) + ∆t*p_{j−1}$ where $∆t=t_{j}-t_{j-1}$. For example let us say that $t_{j-1}=0.0024$. Then what does $x(t_{j−1})=x(0.0024)$ mean and how do I compute it? |
I am finding a Cauchy point to minimize a quadratic problem using gradient projection method. And I get a trouble reading this equation.
$x(t) = x(t_{j−1}) + ∆t*p_{j−1}$ where $∆t=t_{j}-t_{j-1}$.
For example let us say that $t_{j-1}=0.0024$. Then what does $x(t_{j−1})=x(0.0024)$ mean and how do I compute it? |
I want to know whether one can compute $H_k(\Sigma_{g,n})$ with elementary methods like singular chain complex? By $\Sigma_{g,n}$ I mean a genus $g$ surface with $n$ circular holes. If not, I'm not familiar with the standard method, which is Mayer-Vietoris sequence. Could you please introduce me to a lecture or note th... |
First of all let's we remember the following result.
>**Theorem**
>
>Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ in $\alpha$ implies that for $\alpha$.
Let be now $f$ a function from an order $(X,\mathcal U)$ into an o... |
In a town N, every person is either a truth-teller, who always tells the truth, or a liar, who always lies. Every person in town N took part in a survey. "Is winter your favorite season?" was answered "yes" by 40% of respondents. A similar question about spring had 30% of affirmative answers, about summer had 50%, and ... |
> (From Wikipedia) Let $V$ be any vector space over some field $K$ of scalars, and let $T$ be a linear transformation mapping $V$ into $V$, $T:V\to V$. We say that a nonzero vector $v\in V$ is an eigenvector of $T$ if and only if there exists a scalar $\lambda\in K$ such that $T(v)=\lambda v$. The scalar $\lambda$ is t... |
What is the Correct Definition of Eigenvalue? |
**TL/DR: Skip to the end for examples!**
Given that we are allowed $N>0$ turns total (in our case $N=100$), we want to choose "place" for $(N-n)$ turns, followed by choosing "take" for the remaining $n$ turns. (We should prove that, whenever we choose "take" on $n$ turns, it is best if those $n$ turns are at the en... |
I'm exploring the likelihood ratio principle in hypothesis testing, specifically within the context of normal distributions, and I've encountered a challenge in deriving a specific likelihood ratio. The principle is typically used to select a suitable statistic for testing hypotheses, where we compare the likelihood ... |
First of all let's we remember the following result.
>**Theorem**
>
>Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ in $\alpha$ implies that for $\alpha$.
Let be now $f$ a function from an order $(X,\mathcal U)$ into an o... |
> (From Wikipedia) Let $V$ be any vector space over some field $K$ of scalars, and let $T$ be a linear transformation mapping $V$ into $V$, $T:V\to V$. We say that a nonzero vector $v\in V$ is an eigenvector of $T$ if and only if there exists a scalar $\lambda\in K$ such that $T(v)=\lambda v$. The scalar $\lambda$ is t... |
First of all let's we remember the following result.
>**Theorem**
>
>Let be $\lambda$ and ordinal: a predicate $\mathbf P$ is true for any $\alpha$ in $\lambda$ when the truth of $\mathbf P$ for any $\beta$ in $\alpha$ implies that for $\alpha$.
Let be now $f$ a function from an order $(X,\mathcal U)$ into an o... |
Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point.
**Question 1:** Does there exist a supporting hyperplane of $X$ containing $k$?
I _think_ the answer is “yes” via some Hahn-Banach argument, although I’m a little confused about this at the moment. But... |
This questions isn't completely about mathematics, it is also part of computer science. I hope here is the correct place for it.
I studied about the gaussian elimination algorithm (I want to implement it in Octave). The system of linear equations is Ax=b. At each step p, I multiply A to the left with a matrix $T_p =... |
Let $K \subseteq X$ be a compact convex subset of a locally convex space $X$. Let $k \in K$ be an extreme point.
**Question 1:** Does there exist a supporting hyperplane of $X$ containing $k$?
I _think_ the answer is “yes” via some Hahn-Banach argument, although I’m a little confused about this at the moment. But... |
Recently I was studying my notes on Lie algebra, and while I was studying Dynkin diagrams, I came with the following question:
>If we have $D_{1}$ and $D_{2}$ two Dynkin diagrams such that $D_{1}$ is a subdiagram of $D_{2}$, and $R_{1}$ and $R_{2}$ are the root systems corresponding to $D_{1}$ and $D_{2}$, respective... |
The [character][1] of the group elements
>the trace of the matrix representing that group element in the corresponding irreducible representation.
One usage was
>With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class-functions, and this yields the orthog... |
What does the character table of the group tell algebraically? |
Suppose $W_t$ is Brownian motion and consider the following two stopping times:
$$\tau_a \equiv \inf \{t \ge 0 : W_t + at \ge b(t) \} \wedge T$$
and
$$\tau_{-a} \equiv \inf\{t \ge 0: W_t - at \ge b(t)\} \wedge T$$
for some $T > 0$ and an arbiitrary (strictly positive) boundary $b(\cdot)$. Obviously $\tau_a \leq \ta... |
Let $U \subset \mathbb{C}^n$ be a neighborhood of $0=(0,\dots,0)$.
If $z=(z_1,\dots,z_n) \in U$ then there exists $r>0$ such that $|z-0|<r$.
Fix $b_1,\dots,br \in \mathbb{Z}^{n}_{+}$, where $b_i=(b_{i}^{1},\dots,b_{i}^{n})$.
We have to $z^{b_i}=z_1^{b_{i}^{1}}\cdots z_n^{b_{i}^{n}}$.
**Question:** can I say tha... |
Cut-elimination theorem states that any sequent calculus derivation that uses the cut rule also has a derivation that does not make use of the cut rule. I cannot find any explicit examples of such derivations. Are there any?
|
Examples of sequent derivations that uses cut rule that can be modified to not to use cut rule? |
If i am solving for the convolution $f \star (g \cdot h)$, can it be written in some way in terms of the convolutions of $f \star g$ and $f \star h$? |
Theorem for the convolution of a product? |
I have this series
$$\sum_{n=-\infty}^\infty \frac{x+n}{|x+n|^3}$$
Which seemed quite similar to that of (csc(x))^2:
$$\sum_{n=-\infty}^\infty \frac{1}{(x+πn)^2}$$
The only difference is the change of sign throughout the sum, does this series also converge to some function? |
Let $E$ be a Banach space and $B(E)$ (resp.~$S(E)$) be the closed unit ball (resp.~the unit sphere) of the Banach space $E$. $E$ has strictly convex norm if for each pair of elements $x, y \in S(E)$ with $x\neq y$, it is true that $\| \frac12 (x+y)\|<1$.
>If given norm is strictly convex, then each convex subset of... |
Andre's solution was discussed in the link below:
https://math.stackexchange.com/questions/279449/about-andr%C3%A9s-original-solution-to-the-ballot-problem
The solution from Marc Van Leeuwen includes the following simplification, can someone please explain how this was calculated?
$$\binom{p+q}p-2\binom{p+q-... |
I would like to know if there is a video, course or book(the option I would like the least) for most important topics in math for CS, I tried MIT's course is that just the best option? or is there a easier place to learn? |
I'm trying to solve the following problem:
Given real $2$-forms $\omega_1, \omega_2, \omega_3\in\Lambda^{1,1}\mathbb C^2$ satisfying
1. $\omega_1\wedge\omega_1=\omega_2\wedge\omega_2=\omega_3\wedge\omega_3=0$
2. The forms $\omega_1\wedge\omega_2$ and $\omega_2\wedge\omega_3$ are proportional to the standard vo... |
Let $E$ be a Banach space and $B(E)$ (resp.~$S(E)$) be the closed unit ball (resp.~the unit sphere) of the Banach space $E$. $E$ has strictly convex norm if for each pair of elements $x, y \in S(E)$ with $x\neq y$, it is true that $\| \frac12 (x+y)\|<1$.
>If given norm is strictly convex, then each convex subset of... |
For the given functions $f(x) = e^x - \ln(x)$ and its derivative $f'(x) = e^x - \frac 1x$. Finding the local minimum wouldn't be possible because of the presence of both an exponential and a polynomial term. But we can find an approximation of the coordinates.
Since we are focused on the coordinates of the minimum, ... |
What does a function $f(x_0)$ represent if we substitute the x value from $f'(x) = 0$ into $f(x)$? |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) ➕ (γ, δ) = (α + γ, β + δ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ + βγ)... |
why is this associative? |
I've been messing around with the floor function, and I am seeking to find a closed form of:
$$
\sum^n_{i=1}\left\lfloor\frac{n}{i}\right\rfloor
$$
, or is there even one?
For reference, $n$ is an integer. I remember reading something about how this relates to the factors of $n$. I also reckon I'd need the gcd, b... |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
I'm trying to solve the following problem:
Given real $2$-forms $\omega_1, \omega_2, \omega_3\in\Lambda^{1,1}\mathbb C^2$ satisfying
1. $\omega_1\wedge\omega_1=\omega_2\wedge\omega_2=\omega_3\wedge\omega_3=0$
2. The forms $\omega_1\wedge\omega_2$ and $\omega_2\wedge\omega_3$ are proportional to the standard vo... |
> Proposition: If $x,y\in\mathbb{N}$ then for any $\varepsilon>0,$ there
> are infinitely many pairs of positive integers $(n,m)$ such that $$\frac{\left\lvert
y^m-x^n \right\rvert}{y^m} < \varepsilon,$$
>
> i.e. $\displaystyle\large{\frac{x^n}{y^m}} \to 1\ $ as these pairs
> $(n,m) \to (\infty,\infty).$
I th... |
Let (X,\mathcal{A},\mu) be an atomless probability space and let $A\sim B$ whenever $\mu(A\vartriangle B)=0$ for each $A$ and $B$ in $\mathcal{A}$. This way, if $\mathbb{A}$ is the set of $\sim$-equivalences classes in $\mathcal{A}$, then $\mathbb{A}$ inherits $\cap,\cup,\cdot^c$ and $\mu$ from $(X,\mathcal{A},\mu)$, b... |
Does every $\sigma$-homomorphism of a probability algebra come from a measure preserving transformation on the probability space? |
If $x,y\in\mathbb{N},\varepsilon>0$ then are there infinitely many positive integer pairs $(n,m)$ s.t. $\vert\frac{x^n}{y^m}- 1\vert < \varepsilon?$ |
Let $U \subset \mathbb{C}^n$ be a neighborhood of $0=(0,\dots,0)$.
If $z=(z_1,\dots,z_n) \in U$ then there exists $r>0$ such that $|z-0|<r$.
Fix $b_1,\dots,br \in \mathbb{Z}^{n}_{+}$, where $b_i=(b_{i}^{1},\dots,b_{i}^{n})$.
We have to $z^{b_i}=z_1^{b_{i}^{1}}\cdots z_n^{b_{i}^{n}}$.
**Question:** can I say tha... |
For $F$ a familiy of disjoint sets, the set of all prefixes of tuples in $\prod_{S\in F} S$ is
$$ \prod_{S \in F} S \dot{\cup} \{\bot\} \;,$$
which is just a choice of either some element of $S$ or "nothing" (corresponding to $\bot$) for every set $S \in F$ in the family.
I want a bottom-up (finite support) ca... |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
Let $U \subset \mathbb{C}^n$ be a neighborhood of $0=(0,\dots,0)$.
If $z=(z_1,\dots,z_n) \in U$ then there exists $r>0$ such that $|z-0|<r$.
Fix $b_1,\dots,b_m \in \mathbb{Z}^{n}_{+}$, where $b_i=(b_{i}^{1},\dots,b_{i}^{n})$.
We have to $z^{b_i}=z_1^{b_{i}^{1}}\cdots z_n^{b_{i}^{n}}$.
**Question:** can I say th... |
I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for $f,g\in\mathcal S$
$$\langle f(ax),g(x)\rangle=\int_{-\infty}^{\infty}{f(ax)\ g(x)\ dx}$$
$$=\int_{-\infty}^{\inf... |
Can be use $u$-substitution for calculating the adjoint of an operator in Schwartz space? |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for $f,g\in\mathcal S$
$$\langle f(ax),g(x)\rangle=\int_{-\infty}^{\infty}{f(ax)\ g(x)\ dx}$$
$$=\int_{-\infty}^{\inf... |
> Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)) \to L$ as $t \to 1$, then $\lim_{r \to 1^-} f(r) = L. \ [\ldots]$
Proof: We can asume $L = 0$ and $|f| < 1$. Let $K... |
Let $U \subset \mathbb{C}^n$ be a neighborhood of $0=(0,\dots,0)$.
If $z=(z_1,\dots,z_n) \in U$ then there exists $r>0$ such that $|z-0|<r$.
Fix $b_1,\dots,br \in \mathbb{Z}^{n}_{+}$, where $b_i=(b_{i}^{1},\dots,b_{i}^{n})$.
We have to $z^{b_i}=z_1^{b_{i}^{1}}\cdots z_n^{b_{i}^{n}}$.
**Question:** can I say tha... |
**Question**
I want to prove that $V$ is naturally isomorphic to $(V^\ast)^\ast$ without making reference to a basis.
**Attempt**
I have already shown that the map between $V$ and $(V^\ast)^\ast$ is linear and injective. What is left is to show that it is surjective.
We define $\Phi :V\rightarrow (V^\ast)^... |
Give an interpretation where
$$∃x(\neg P(x) ∨ Q(x)) \to \neg(∃xP(x) ∧ ∀x\neg Q(x))$$
is false.
How does someone even begin with questions like this? I have interpreted it in my head and I kind of get it in a sense. But seems like the only thing I know is that since it is an implication, the only way it will be fal... |
**Question**
I want to prove that $V$ is naturally isomorphic to $(V^\ast)^\ast$ without making reference to a basis.
**Note:**
$V$ is finite dimensional space.
**Attempt**
I have already shown that the map between $V$ and $(V^\ast)^\ast$ is linear and injective. What is left is to show that it is surjecti... |
# Question Summary
I have a 2D parametric curve defined by two functions, $f_x(t)$ and $f_y(t)$, and by several parameters that adjust the overall shape of the curve. As-is, if $t$ varies at a constant speed, the velocity of the point $(f_x(t), f_y(t))$ follows a sine curve. I want to adjust the parametric functions... |
Can I reparameterize this parametric curve to have constant velocity with closed-form expressions? |
I'm trying to solve the following problem:
Given real $2$-forms $\omega_1, \omega_2, \omega_3\in\Lambda^{1,1}\mathbb C^2$ satisfying
1. $\omega_1\wedge\omega_1=\omega_2\wedge\omega_2=\omega_3\wedge\omega_3=0$
2. The forms $\omega_1\wedge\omega_2$ and $\omega_2\wedge\omega_3$ are proportional to the standard vo... |
*This is the excercise:*
**Let $B_t$ be 1d Brownian motion with $B_0=0$. Define**
$$
X_t=X_t^x=x\cdot\exp\left(ct+\alpha B_t\right)
$$
**where $c,\alpha$ are constants and $x$ is non-random. Prove directly from the definition that $X_t$ is a Markov process, i.e., that**
$$
\mathbb{E}\left[f\left(X_{t+h}\... |
The constraints are $f(0,0)=1, f(0,k)=0\space \forall k \neq 0, f(N,k)=0 \space \forall k>N\space0\leq p\leq 1$ .
When working on a probability problem, I came across this recursion when working with random walks. For symmetric random walks, with $p=\frac{1}{2}$, I found a closed form solution of $2^{-N}*$ ${N}\choo... |
> Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)) \to L$ as $t \to 1$, then $\lim_{r \to 1^-} f(r) = L. \ [\ldots]$
Proof: We can asume $L = 0$ and $|f| < 1$. Let $K... |
What to do if you don't understand a part of a solution? Beside asking. What are the common strategies? |
What to do if you can't understand a part of a solution to math problem? |
> Supose that $\gamma : [0,1] \to \overline{\mathbb{D}}$ is continuous, $\gamma(t) \in \mathbb{D}$ for $0 \le t < 1$ and $\gamma(1) = 1$. Suppose that $f \in H(\mathbb{D})$ is bounded. If $f(\gamma(t)) \to L$ as $t \to 1$, then $\lim_{r \to 1^-} f(r) = L. \ [\ldots]$
<hr>
Proof: We can asume $L = 0$ and $|f| < 1$. Le... |
I'm currently reading a textbook about abstract algebra. There is a proof that every subgroup of a cyclic group is cyclic. This proof is using the fact as every proof I have found on the Internet that all cyclic groups have the form
$$ <a>={a^n}$$
With n element of the integers.
But I dont think that that i... |
Subgroup of a cyclic group is cyclic ,not complete proof? |
I'm trying to understand the first part of the proof of Lemma 2 in https://www.emis.de/journals/BAG/vol.46/no.2/b46h2hei.pdf.
Given $r > 0$, they claim that for an elliptic curve $E$ there exists another elliptic curve $\widetilde{E}$ with $\pi_r: \widetilde{E} \rightarrow E$ such that $E = \widetilde{E}/G$ where $G... |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
"Let $C\subset \mathbb{R}^2$ be a regular Jordan curve oriented so that the normal unit vector $N(p)$ points towards $I(C)$ (the interior of $C$) for every $p\in C$. Denoting with $L=Lung(C)$ the length of $C$ and with $A=AreaI(C)$ the area of $I(C)$, show that for every $\varepsilon >0$ sufficiently small, $C_{\vareps... |
I'm currently reading a textbook about abstract algebra. There is a proof that every subgroup of a cyclic group is cyclic. This proof is using the fact as every proof I have found on the Internet that all cyclic groups have the form $ \langle a\rangle=\{a^n\}$, where $n \in \mathbb{Z}$. But I don't think that this is ... |
I'm trying to solve the following problem:
Given real $2$-forms $\omega_1, \omega_2, \omega_3\in\Lambda^{1,1}\mathbb C^2$ satisfying
1. $\omega_1\wedge\omega_1=\omega_2\wedge\omega_2=\omega_3\wedge\omega_3=0$
2. The forms $\omega_1\wedge\omega_2$ and $\omega_2\wedge\omega_3$ are proportional to the standard vo... |
I'm dealing with Paul Halmos' Linear Algebra Problem Book and I've found a problem already
The fourth exercise asks me to determine whether the following operation is compliant with the associative principle:
$$(α, β) · (γ, δ) = (αγ − βδ, αδ + βγ)$$
The answer says that it is, because:
$$(αγ − βδ)ε − (αδ +... |
<b>What is Hilbert's reference of and what is a generic function in Hilbert's theorem below?</b>
Khovanskii, A.: Topological approach to 13th Hilbert problem. Department of Mathematics, University of Toronto, February 19, 2019, page 4:
"Theorem 3 (D.Hilbert)
<i>Generic analytic function of $n$ variables can no... |
What is the reference and what is a generic function in Hilbert's theorem about representability by compositions of fewer variables? |
I'm trying to understand the first part of the proof of Lemma 2 in https://www.emis.de/journals/BAG/vol.46/no.2/b46h2hei.pdf.
Given $r > 0$, they claim that for an elliptic curve $E$ there exists another elliptic curve $\widetilde{E}$ with $\pi_r: \widetilde{E} \rightarrow E$ such that $E = \widetilde{E}/G$ where $G... |
I know that if $A$ is semisimple, then every $A$-module is semisimple. But I don't know how to prove then that a short exact sequence
$$0 \to L \to M \to N \to 0$$
splits for every $A$-modules $L,M,N$. Can someone help me please ? Thank you |
The complex solutions always diverge, from what was tried, using
[Lagrange reversion](https://en.m.wikipedia.org/wiki/Lagrange_reversion_theorem), but the real solutions are:
$$\begin{align}\sin(x)^{\cos(x)}=2\mathop \iff^{x=\cos^{-1}(-\sqrt y)}y=1-4^{-\frac1{\sqrt y}}\implies x_k=\left(2k+\frac12\right)\pi+\sin^{-1... |
How to find function $G(v_i)$, such that $\mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}$. Is this a fixed-point problem? |
> Proposition: If $x,y\in\mathbb{N}_{\geq2}$ then for any $\varepsilon>0,$ there
> are infinitely many pairs of positive integers $(n,m)$ such that $$\frac{\left\lvert
y^m-x^n \right\rvert}{y^m} < \varepsilon,$$
>
> i.e. $\displaystyle\large{\frac{x^n}{y^m}} \to 1\ $ as these pairs
> $(n,m) \to (\infty,\infty).$... |
Let $(X,\mathcal{A},\mu)$ be an atomless probability space and let $A\sim B$ whenever $\mu(A\vartriangle B)=0$ for each $A$ and $B$ in $\mathcal{A}$. This way, if $\mathbb{A}$ is the set of $\sim$-equivalences classes in $\mathcal{A}$, then $\mathbb{A}$ inherits $\cap,\cup,\cdot^c$ and $\mu$ from $(X,\mathcal{A},\mu)$,... |
<b>What is Hilbert's reference, a generic function and a germ of an analytic function in Hilbert's theorems below?</b>
Khovanskii, A.: Topological approach to 13th Hilbert problem. Department of Mathematics, University of Toronto, February 19, 2019, page 4:
"Theorem 3 (D.Hilbert)
<i>Generic analytic function o... |
What's the reference, generic function, germ of an analytic function in Hilbert's theorem about representability by composition of fewer variables? |
>[Is there any valid complex or just real solution to $\sin(x)^{\cos(x)} = 2$?](https://math.stackexchange.com/questions/4692173/is-there-any-valid-complex-or-just-real-solution-to-sinx-cosx-2/4693242#4693242)
has a series solution for $$\sin(x)^{\cos(x)}=2\mathop\iff^{x=\cos^{-1}(-\sqrt y)}y=1-4^{-\frac1{\sqrt y}}$... |
I'm currently reading a textbook about abstract algebra.
There is a proof that every subgroup of a cyclic group is cyclic.
This proof is using the fact as every proof I have found on the Internet that all cyclic groups have the form $ \langle a\rangle=\{a^n\}$, where $n \in \mathbb{Z}$. But I don't think that ... |
But what is with the other cyclic groups? Doesn't one also have to consider them? |
For the given functions $f(x) = e^x - \ln(x)$ and its derivative $f'(x) = e^x - \frac 1x$. Finding the local minimum wouldn't be possible because of the presence of both an exponential and a polynomial term. But we can find an approximation of the coordinates.
Since we are focused on the coordinates of the minimum, ... |
How do you go by solving this kind of linear differential equation? $\frac{df(y)}{dy}\frac{dy}{dx} + P(x)f(y) = Q(x)$ |
First of all: I know basically nothing about Euler-Lagrange equations, I'm a first year civil engineering student simply doing this to learn some new stuff. If there are better ways to do this, feel free to tell me.
I have a two-dimensional grid of lines (4x6 intersections, all evenly spaced), six green objects are ... |
(How) Can I use the Euler-Lagrange equation to find the fastest path between certain points? |
A hint to an exercise in my textbook says "Since $B/F$ is finite, it is algebraic, and there are elements $α_1,...,α_n$ with $B=F(α_1,...,α_n)$."
I understand that if $B/F$ is a finite extension then $B$ is algebraic. But why does $B$ being algebraic imply that $B=F(α_1,...,α_n)$?
Thanks in advance! |
If a function has a jump discontinuity on x0, is its antiderivative undefined for x0 or does it exist? |
I saw an integral like the following in a book: For $p, q$ nonnegative integers, $$\int_{e}^{\infty}\frac{r^{2p-2q-1}}{(\log r)^{2q+2}}dr$$ is finite IF and ONLY IF $q \geq p$. I am not sure how to prove it.
If I make the simple substitution $\log r=x$ then the integral above changes to $$\int_{1}^{\infty} e^{{(2p-... |
When is this (gamma integral like) integral finite? |
Do I understand derivatives correctly? As I understand it, derivatives are something like that in the example: f(x) = x^2:
x1=1, so 1
x2=4, so 3
x3=9, so 5
And the derivative is 2x+C, where C = -1. Is this some kind of overly successful example, or did I just misunderstand everything? This does not work wit... |
In the context of proving the [Hoeffding Lemma ][1] I came across a slightly weaker statement in the form of an exercise:
"If $X$ is a real valued random variable and $|X| \leq 1$ a.s. then there exists a random variable $Y$ with values in $\{ -1, +1 \}$ such that
\begin{equation}
E[Y|X] = X \qquad a.s.
\end{e... |
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