Okay this is about complex numbers, and I am using (tg angle = y / x) so when I have only x (real value) in a complex number and y = 0, I figured out that if x is negative then angle is PI, but if x is positive, how can I determine if angle is 0 or 2PI ?

Let $\Gamma$ be a nontrivial group of isometries of $\mathbb{S}^n$, $n \geq 2$, acting properly discontinuously. For $p \in \mathbb{S}^n$, define

$$r(p) = \min_{g \in \Gamma \setminus\{e\} } d(p, g(p)), $$

that is, $r(p)$ is the minimum distance to $p$ of a point in its orbit. As the sphere is a homogeneous manifold, I expect $r$ to be a constant function, although I couldn't prove it. Any thoughts on it?

Suppose, we have a random variable $Y \sim \mathrm{N}\left( Ax, \, \Sigma \right)$ and realisations $y$. I would like to estimate $x$, the parameter of the expected value. The loglikelihood function is $$L(x) =- \frac{1}{2} \left[ \ln \left( \left| \Sigma \right| \right) + \left( y - Ax \right)^T \left( \Sigma \right)^{-1} \left( y - Ax \right) + k \ln(2 \pi) \right],$$ thus $$D_L(x) = \left( y - Ax \right)^T \left( \Sigma \right)^{-1} A,$$ which leads to $$\hat{x} = \left[ A^T \left( \Sigma \right)^{-1} A \right]^{-1} A^T \left( \Sigma \right)^{-1}y,$$ since the covariance matrix is self-adjoint.

My question is now: What happens if a small error matrix $E$ is added to $\Sigma$? Is there anything I can say about the sensitivity apart from giving the formula $$\hat{x}_E = \left[ A^T \left(\Sigma +E\right)^{-1} A \right]^{-1} A^T \left( \Sigma+E \right)^{-1}y?$$ Probably, this was already done but I did not find any sources dealing with this problem in particular. I was thinking about the analyzing the eigenvalues. Is there any other way? Are there any sources you recommend?

In this paper Higher direct images of log canonical divisors and positivity theorems, Fujino generalized the semi-positivity of direct image of relative canonical sheaf with two stronger conditions.

Let $S\longrightarrow C$ be a fibration of smooth projective varieties over complex number $\mathbb{C}$. In my application, assume $(S,D)$ is a log smooth pair, i.e. $D$ is a effective and reduced divisor on $S$ with simple normal crossing, and $C$ is a curve, $S$ has dimension 2. Let $f_0: S_0\longrightarrow C_0$ be the smooth morphism by deleting the singular fibre of $f$, and $\Sigma=C/C_0$ is the ramified divisor of $f$, $S_0=f^{-1}(C_0), D_0:= S_0\cap D$. If the following conditions are satisfies,

- The horizontal part of $D$ is strongly horizontal, i.e. any intersection of its irreducible components is dominant to $C$;
- All the local monodromies on the local system $R^1f_{0*}\mathbb{C}_{S_0/D_0}$ around every component of $\Sigma$ are unipotent.

Then $f_*\omega_{S/C}(D)$ is locally free and semi-positive.

My question is that in the case of surface to curve fibration, can we drop these two conditions.

For the condition 1, take $\sigma:X'\longrightarrow X$ blow up along the intersection of horizontal component of $D$, since $\mathrm{Sing}(D^h)$ are only nodal and we have $K_S'+D'=\sigma^*(K_S+D)$, then $D'$ is an effective and reduced divisor and its horizontal part is strongly horizontal. And this modification dosn't break the monodromy condition which is the condition 2, since they are isomorphic outside $D$. If the condition 2 hold for $f$, then so does $f'$, then we have $f'_*\omega_{S'/C}(D')$ is semi-positive. On the other hand, $f'_*\omega_{S'}(D')\cong f_*\sigma_*\omega_{S'}(D')\cong f_*\omega_{S}(D)$, tensor with $\omega^{*}_C$, we have $f'_*\omega_{S'/C}(D')\cong f_*\omega_{S/C}(D)$, so $f_*\omega_{S/C}(D)$ is semipositive.

For the monodromy condition, by Kawamata's cover strict, we can reduced to the unipotent case, we have the following commutative diagram $f':S'\longrightarrow C'$, $f:S\longrightarrow C$, $\pi:C'\longrightarrow C$, $\rho:X'\longrightarrow X$(I can't insert the commutative diagram here, it is just a square.) where $\pi$ is a finite morphism and $S'$ a resolution of $S\times_{S}S'$. Define $D'=(\rho^*D)_{\mathrm{red}}$ such that $f'$ satisfy the condition 2. A nature ideal is comparing the $\pi^*f_*\omega_{S/C}(D)$ and $f'_*\omega_{S'/C'}(D')$, since this is not base change, we don't have isomorphism, but it seem to have chance to show that when $f'_*\omega_{S'/C'}(D')$ is semi-positive, then so does $f_*\omega_{S/C}(D)$.

If $~~~S = A\ast B \cdot C+ C\cdot C+ A\ast D\ast B \cdot C+E\cdot C $

where $A, B, C, D, E$ are random vectors, whose elements are assumed to be random sample from a normal distribution with mean 0 and variance $\sigma^{2}$, and $\ast$ denotes convolution and $\cdot$ denotes dot product.

Can I say items ($(A\ast B \cdot C)$, $ (C\cdot C)$, etc.) in $S$ are independent of each other? and variance of $S$ is the sum of variances of each item?

Do locally cartesian closed $\infty$-categories form a presentable $\infty$-category? It seems like they should, and that the inclusion $\text{LCC}\rightarrow\text{Cat}$ preserves colimits.

Here is a possible strategy for proof. There is a functor $\text{Cat}\rightarrow\text{Fun}(\Delta^1,\text{Cat})_\text{cart}$ taking an $\infty$-category $\mathcal{C}$ to the cartesian fibration $\text{Fun}(\Delta^1,\mathcal{C})\rightarrow\mathcal{C}$; this fibration classifies the functor $\mathcal{C}\rightarrow\text{Cat}$ which sends $X$ to $\mathcal{C}_{/X}$. The condition that $\mathcal{C}$ is locally cartesian closed is precisely that for all $X\rightarrow Y$, $\mathcal{C}_{/X}\rightarrow\mathcal{C}_{/Y}$ has a chain of two right adjoints (its right adjoint has a right adjoint). This should correspond to some condition on the cartesian fibration, so that $\text{LCC}$ is a pullback of $\text{Cat}\rightarrow\text{Cat}^\text{cart}$ along some subcategory of $\text{Cat}^\text{cart}$. A pullback of presentable categories should be presentable.

But I'm not sure how to handle the "chain of two right adjoints", and I wonder if I am missing an easier argument.

The Euler Lagrange equation states that the time flow is given by a vector field such that when the vector field is contracted with the sympletic form gives dL, where L is the Lagrangian function on the tangent bundle.

Suppose $n_2$ denotes the binary representation of the integer number $n$. Let $X_2(n)=[1_22_2\ldots n_2]$,$n\geq2$, be a binary vector which is obtained by concatenating of binary representation of the numbers from $1$ to $n$. Also, let $X_2^m(n)$,$0\leq m\leq n-1$,denotes the cyclically $m$ shift of the entries of the vector $X_2(n)$. For example, we have $$X_2^0(n)=X_2(n)$$ $$X_2^1(n)=[n_21_22_2\ldots(n-1)_2]$$ $$X_2^2(n)=[(n-1)_2n_21_22_2\ldots(n-2)_2]$$ and so on.

For two binary vectors $X$ and $Y$ (with same length), suppose $|X\cap Y|$ denotes the number of ones common to both $X$ and $Y$.

The conjecture is:

for all $m$ and $k$, we have $|X_2^m(n)\cap X_2^k(n)|\cong 0 \mod 2$ if and only if $n=2^s-1$, for some integer number $s$.

Note: I use $\lfloor \log_2n\rfloor +1$ bits for binary representation of each integer number from $1$ to $n$. So, fedja's example is as follows:

$X_2(3)=[011011]$, $X_2^1(3)=[110110]$ and $X_2^2(3)=[101101]$. We can see the claim is true.

The conjecture is tested for many integer numbers. I appreciate any helpful comments and answers.

Does there exist a method to compute the K-theory $$K(A \rtimes G)$$ for a discrete, countable groupoid $G$ and $G$-$C^*$-algebra $A$? In good cases, say $G$ is ameanable.

Say, via Baum--Connes and a Chern-character? But which Chern character?

(This question has partial overlapp with K-theory of topological groupoids)

I have two curves both of which start at $(0,0)$. They satisfy the following equations:

$ \begin{align*} \frac{dx}{d\theta} &= -\frac{\sin \theta}{\kappa(\theta)} \\ \frac{dy}{d\theta} &= \text{ }\frac{\cos \theta}{\kappa(\theta)}\\ \kappa(\theta) &= \cos^2 \theta \left[(2-x) \cos \theta - (1+y)\sin \theta\right] \end{align*} $

Angle $\theta$ is the angle of the outward normal to the curve. The curve is parametrized by $\theta$ which ranges from $-\pi/4 < \theta_0 < 0$ to $0$.

While the two curves, $\mathcal{C}_1$ and $\mathcal{C}_2$, satisfy the same DEs and start at the origin, their initial data differs in their angle $\theta_0$. Say, $\theta_0^{\mathcal{C}_1} < \theta_0^{\mathcal{C}_2}$.

The picture below is perhaps more helpful to understand what $\theta$ is and also how the curves look.

I am interested in understanding how far the curves are when $\theta = 0$. Obviously, this is very easy to do numerically. But, are there any analytical methods that I can exploit to understand the difference between their $x$-coordinate as well as their $y$-coordinate?

Thanks.

Why does Fermat's last theorem imply immediately the non-existence of many alleged real algebraic numbers in mathematics and science of the following form? $$\sqrt[p]{q}$$ where $(p > 2, q)$ are prime numbers

Long time back, I was shocked by discovering this fact, and strangely the rigorous proof is so elementary to the limit that makes one may be wondering about this, or wither he had truly missed something that led to this rare beast conclusion, so I thought there must be some historical proofs about their true existence that I may be not aware of

Thanking you for help

Cluster-tilted algebras are 1-Gorenstein. Is it known which of those algebras are representation-finite and which are CM-finite (that is have only finitely many Gorenstein projective modules)? More generally, for which 1-Gorenstein algebras is it known that they are CM-finite? Is a classification possible?

Consider $f(x)$, a rapidly decreasing function, such that $\int_0^{\infty} f(x)=0$ and for $x$ near zero: $f(x)=O(x^a)$ (wit $a>0$). Then I calculated the integral of the following sum (which appears in the Summation Poisson Formula) and I found (noting $F(x)=\int_0^{x} f(t) dt$): $$\int_0^{\infty}\sum_{n=1}^{\infty} f(nx) dx= -\frac{1}{2}\int_0^{\infty} \frac{1}{x}F(x) dx = -\frac{1}{2}\int_0^{\infty} \ln(x) f(x) dx$$

(See my previous post to see that the integral on the left is well defined and how above result is obtained using the Poisson summation formula: Interchange of sum and integral (on a "Poisson summation") )

Is there any known reference in literature with similar result (using Poisson Summation formula or not) ?

How to evaluate this integral: $$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=?$$ I'm making use of the integral identity: $\int_{0}^{+\infty }e^{-t(x_{1}+x_{2}\cdots +x_{n})}dt=\frac{1}{x_{1}+x_{2}\cdots +x_{n}}$and then reversing the order of integration with respect to time and space variables. But for $n=1$,then such that,$\int_{0}^{\infty }dt\int_{0}^{1}x^{2}e^{-tx}dx=\int_{0}^{\infty }\frac{2 - e^{-t}(2 + 2t+t^2)}{t^3}dt=\int_{0}^{1}xdx=\frac{1}{2}$,and,$\int_0^1 \int_0^1 \cdots \int_0^1\frac{x_{1}^2+x_{2}^2+\cdots+x_{n}^2}{x_{1}+x_{2}+\cdots+x_{n}}dx_{1}\, dx_{2}\cdots \, dx_{n}=n\int_{0}^{+\infty }\frac{2 - e^{-t}\left ( 2 + 2t+t^2 \right )}{t^3}\left ( \frac{1-e^{-t}}{t} \right )^{n-1}dt$

Is there an existence theorem for linear time-dependent differential operator equations that reduces in case of constant coefficients to the Hille-Yosida theorem?

Show that for any $x, y \in \mathbb R$ with $x + y \neq 0,xy\neq 0$

$$p(x,y) := x^6-2 x^5 y+2 x^5-x^4 y^2-2 x^4 y+x^4+4 x^3 y^3+2 x^3 y-x^2 y^4-4 x^2 y^3-4 x^2 y^2+2 x^2 y-2 x y^5+6 x y^4+2 x y^3+y^6-2 y^5-y^4-2 y^3+y^2 \neq 0$$

I'm sorry,I forget $xy\neq 0$,Now I think it's hold?

Let $k<N$ be natural numbers. In this question we consider graphs whose vertices are size-$k$ subsets of a size-$N$ universe. Consider the following random walk in the graph:

Starting from a set $R$ pick $t$ elements in $R$ uniformly at random and pick a uniformly random set $S$ that contains those $t$ elements ($t$ is a parameter; note that the size of the intersection of $S$ and $R$ may be larger than $t$).

This model is studied in the association schemes literature and has an elegant spectral analysis (see, e.g., https://www.math.uwaterloo.ca/~cgodsil/pdfs/assoc2.pdf).

My question is whether one can prove a hypercontractive inequality in this model (equivalently, a Log Sobolev constant), similarly to what's proven for closely related models in https://projecteuclid.org/download/pdf_1/euclid.aop/1022855885.

I have asked this question on math.stackexchange.com but received no response; hoping someone on here can help.

Suppose a function $f$, representing what I call a "dynamic transposition cipher" taking one string of text $str$ as input, is defined so that it outputs another string of text $res$ which is the transposed characters of $str$ according to the following algorithm:

- Let $i_1 = 1$, $i_2 = $ the number of characters in $str$, and $res = ""$ (empty string)
While $i_2 - i_1 > 1$:

a. Append the character $str[i_1]$ to $res$

b. Append the character $str[i_2]$ to $res$

c. Increment $i_1$ by $1$

d. Decrement $i_2$ by $1$

If $i_1 = i_2$, then append the character $str[i_1]$ to $res$; otherwise, append the characters $str[i_1]$ followed by $str[i_2]$ to $res$.

- Return $res$

Assume that string indices begin at one, i.e. "ABC"$[1] = $ "A" and that each character of $str$ is unique.

For example, $f($"ABCDEF"$)$ would equal "AFBECD", and $f($"AFBECD"$)$ would equal "ADFCBE". We can keep iterating $f \circ f \circ f(x)$, yielding "AEDBFC", "ACEFDB", and finally "ABCDEF", which was our original input. All in all, for any string $str$ of 5 characters, $f(str)$ = 4, since it takes four iterations including the original string.

Let $g(n)$ represent the number of iterations required to transpose a string of $n$ characters back into itself according to $f$.

Following this formula for strings of 1 through 10 characters, there is unusual variation in the number of iterations required: $g(2) = 2$, $g(3) = 3$, $g(4) = 4$, $g(5) = 4$, $g(6) = 6$, $g(7) = 7$, $g(8) = 5$, $g(9) = 5$, and $g(10) = 10$. I have calculated these values up to $g(30)$: $g(28) = 21$, $g(29) = 10$, and $g(30) = 30$.

Plotting these discrete points, we can find several linear functions which collectively fit some of the points: $y = x$ fits some, $y = 0.5x + 1.5$ fits others, while $y = 0.25x + 2.5$ fits still others, while there are many more data pairs not accounted for. I am not noticing much of a pattern that holds consistently besides $y=x$.

What would be a definition of $g(n) \space | \space n \in \Bbb I, n > 0$?

So I am inspired by unitary matrices which preserve the L2-norm of all vectors, so in particular the unit norm vectors. But then I saw that the L1-norm of probability vectors is preserved by matrices whose columns are probability vectors. And this got me thinking: But what are the matrices preserving the L1-norm of arbitrary real unit L1-norm vectors? So basically we extend a probability vector to also allow a sign, but ignoring the signs, this should still be a probability vector; and then we ask for the corresponding structure-preserving matrices.

It is already clear that the columns of such a matrix should be this 'extended' kind of probability vector, because we can multiply the matrix with a standard basis vector which has L1-norm 1. But not all of such matrices preserve this, take for example

$$ M = \frac{1}{2} \left(\begin{matrix} 1 & 1\\ 1 & -1 \end{matrix}\right) $$

and

$$ x = \left( \begin{matrix} 0.3 \\ -0.7 \end{matrix} \right) $$

Then we have

$$ Mx = \left(\begin{matrix} -0.2 \\ 0.5 \end{matrix}\right) $$

which fails the test.

In Bardakov, Algebra and Logic, Vol. 39, No. 4, 2000 I have found the following (page 225, see https://link.springer.com/article/10.1007/BF02681648)

We pronounce tile validity of the following:

**Conjecture.** For every element *z* in the derived subgroup of a free non-Abelian group *F* and for any natural *m*,
$$
\mathrm{cl}(z^m) \geq (m+1/2)\mathrm{cl}(z)
$$

Where *cl* denotes the commutator length of an element (ie. the minimal number to express it as a product of commutators).

This inequality is not true, and $$z = [a, b]$$ may be a counterexample. However, I belive that there may be a typo, so it should rather be $$ \mathrm{cl}(z^m) \geq (m+1)/2 \cdot \mathrm{cl}(z) $$

Unvortunatelly, I could not find it in any other paper/book (including Calegari's "scl"). And the proof in Bardakov is unclear to me.

Do you know any paper, with a proof of the above inequality? Or maybe some counterexample? Or maybe has anybody have any clue why Bardakov did not prove this inequality?