I wish to understand the following integral, $$E(u,\Sigma)=\int_{\Sigma}det(D^2u|_{\Sigma})de_1\wedge de_2$$ Where $u\in C^2(\Omega)$ is a solution of some elliptic equation, so we have $D^2u$ is semi-positive, $\Omega\subset R^n$ is a open set, $\Sigma$ is a arbitrary smooth surface equipped with the induce metric from $R^n$, and $\Sigma\subset \Omega$, $\{e_1,e_2\}$ is a pair of orthogonal biases of $\Sigma$.

I wish to understand if there is a similar result like Newton-Leibniz formula in the one dimensional case, which is the following: $$\int_{\gamma}\partial_{e_1e_1}ude_1=\int_{\partial \gamma}u=\partial_{e_1}u(\gamma(1))-\partial_{e_1}u(\gamma(0))$$ Where $\gamma: [0,1]\to \Omega$ is a $C^2$ curve and $e_1$ is the gradient direction along the curve.

So my problem is following:

**problem**
Can we get some integral expression for $E(u,\Sigma)=\int_{\Sigma}det(D^2u|_{\Sigma})de_1\wedge de_2$ use just $u_1=\partial_{e_1}u,u_{2}=\partial_{e_2}u $, and the integral domain of the expression is $\partial \Sigma$? more precisely I wish we could find a functional $\hat E(u,u_1,u_2)$ such that,
$$E(u,\Sigma)=\int_{\Sigma}det(D^2u|_{\Sigma})de_1\wedge de_2=\int_{\partial\Sigma}\hat E(u,u_1,u_2)?$$
$\hat E$ is a functional only relate to $u,\nabla u$. Because of $\hat E$ only related with $u,\nabla u$, I would like to say it is the lift of $E(u,\Sigma)$. And may be this is only true for some special domain $\Sigma$, for example $\Sigma$ is a Ball, this is exactly what I excepted.

**Motivation**

Now I need to explain my motivation why I except this is true and why I need this or it variation is true.
The motivation is come from the 1 dimensional version is true, and which is crucial to establish **the mean value principle** for Laplace equation,

$$\frac{1}{\mu(B)}\int_{\partial B(r,x_0)}u(x)dx=u(x_0)$$

I wish to generated the mean value principle to some special nonlinear elliptic equation by this way, although this mean value principle may be not exist, I still wish to explain why the mean value property failed by this way.

**Attempt**

I have four ways to attempt this problem, but there always emerge difficulties I could not settle.

- Use the identity $$exp(tr(A))=det(exp(A))$$ We try to solve the equation $D^2 u=exp^{tr (A)} ...(*)$, we pretend it could be solved then we have: $A=log(D^2(u))=\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{k}{D^2(u)}^k$, so we have, $$E(u,\Sigma)=\int_{\Sigma}e^{tr(\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{k}{D^2(u)}^k)}de_1\wedge de_2$$ It seems much easy to fine $\hat E(u)$ use stokes theorem with $E(u)$ under this form. But there exists two problem, one is that the solvable of $(*)$ and there exists infinity many different solution of $(*)$ if my insight is right and I do not know how to proof the identity we could proved in this way is independent with the choice.

2.

We discretization the problem and consider it in $\mathbb Z^2$ which is a two dimensional affine subspace of $\mathbb Z^n$.

The advantage of discretization is that we can explicate calculate $u_{11},u_{12},u_{21},u_{22}$ now, in fact,

$$u_{11}(x,y) = h^2(u(x+2h,y)+u(x,y)-u(x+h,y)-u(x+h,y)) $$ $$u_{12}(x,y) = h^2(u(x+h,y+h)+u(x,y)-u(x+h,y)-u(x,y+h)) $$ $$u_{21}(x,y) = h^2(u(x+h,y+h)+u(x,y)-u(x+h,y)-u(x,y+h)) $$ $$u_{22}(x,y) = h^2(u(x,y+2h)+u(x,y)-u(x,y+h)-u(x,y+h)) $$

and we could use this to calculate $det(D^2u)$, but after calculate I do not find general principle and what shape should $\Sigma$ be to make the identity, $$\int_{\Sigma}det(D^2u|_{\Sigma})de_1\wedge de_2=\int_{\partial\Sigma}\hat E(u,u_1,u_2)$$ make sense in this way.

3. Investigate the Frobenius integrable condition, which is just mean:

$$L_iL_ju(x)-L_jL_iu(x)=\sum_k c_{ij}^k(x)L_ku(x)$$ should be true, I tried to split $E(u,\Sigma)$ into several parts, and every part of it satisfied the Frobenius integrable condition, i.e.

$$E(u,\Sigma)=\sum_{i=1}^k E_i(u,\Sigma)$$

and $E_i$ satisfied the Frobenius integrable condition. And we investigate each $E_i$ first and combine the result we got together to establish a result for $E(u,\Sigma)$.

But the difficulties comes from that I do not know how to decompose $E(u)$ at all!

4. The last strategy could only get a part of result(instead of identity, we could only get a inequality). thanks to the elliptic condition we know $D^2u$ is semi-positive and the Principal minors of $D^2u$ is also semi-positive, so $D^2u|_{\Sigma}$ is semi-positive. we could consider the function $f(x)=Det(D^2(u)|_{\Sigma})^{1/2}$, which is a concave function so use Jensen inequality we could get following result:

$$\frac1{{\rm vol}\Sigma}\int_{\Sigma}(\det D^2u|_{\Sigma})^{1/2}\le\det\left(\frac1{{\rm vol}\Sigma}\int_{\Sigma} D^2u|_{\Sigma}(x)\right)^{1/2}.$$

May be according this could gain a mean-value inequality but I am not very sure.

May be all of these approaches are useless. In any case, I wish some result could be establish, whatever positive answer or negative answer. I will appreciate to any valuable advice or new idea, thank you very much!

The injective hull for a module always exists, however over certain rings modules may not have *projective covers*. I have a question.

If $A$ is an Artinian module on a Noetherian local ring $R$ then $A$ has projective cover? If not, please give a counter example.

Let $A$, $B$, and $C$ be positive, invertible $4 \times 4$ complex matrices. So we have three nondegenerate "sesquilinear quadratic" forms $\langle Av,w\rangle$, $\langle Bv,w\rangle$, and $\langle Cv,w\rangle$. Say that $v,w \in \mathbb{C}^4$ are *good* for $A$ if they are orthogonal and have the same norm relative to the quadratic form given by $A$, i.e., $\langle Av,w\rangle = 0$ and $\langle Av,v\rangle = \langle Aw,w\rangle$.

Can we always find two nonzero vectors which are simultaneously good for $A$, $B$, and $C$?

I can do this if $A$, $B$, and $C$ commute. Here is the argument, in case it helps. First find an orthonormal basis $\{e_1, e_2, e_3, e_4\}$ of $\mathbb{C}^4$ consisting of simultaneous eigenvectors for $A$, $B$, and $C$. Say $Ae_i = a_ie_i$, etc. Then find a nonzero $\vec{d} = (d_1, d_2, d_3, d_4) \in \mathbb{R}^4$ satisfying $\langle \vec{d}, \vec{a}\rangle = \langle \vec{d},\vec{b}\rangle = \langle \vec{d}, \vec{c}\rangle = 0$. Since the $a_i$ are positive, $\vec{d}$ has at least one strictly positive component and at least one strictly negative component. Then $v = \sum \sqrt{d_i}e_i$, taking the sum over $i$ with $d_i > 0$, and $w = \sum \sqrt{-d_i}e_i$, taking the sum over $i$ with $d_i < 0$, have the desired properties.

This is a followup to the question here: How to show that the following function isn't a polynomial over Q?.

As before, let $b_1,b_2,\dots$ be an enumeration of $\mathbb Q$. The question might be sensitive to the enumeration (but probably not).

**Question 1:** Suppose I define $$f_3(x) = (x-b_1)^3 + (x-b_1)^3(x-b_2)^3 + \dots,$$

how do I show that this function is not a polynomial? I do not believe the answers to the old question extend to this case since they all seem to use the positivity of squares.

Can we show that $f(x)$ is not a polynomial?

There is a even more general version (and the one I was interested in from the beginning) that I was interested in, however I forgot to omit the trivial cases/there was some ambiguity in the second question, so I asked it again here:

Let $X$ be a smooth variety over a perfect field $k$ with $X(k) \neq \emptyset$. Then is the natural map \begin{equation} \mathrm{Pic}(X) \to (\mathrm{Pic}(X_{\bar{k}}))^{\mathrm{Gal}(\bar{k}/k)} \qquad (1) \end{equation} surjective?

Remarks:

- If $X$ is a proper, then the map (1) is in fact an isomorphism. This is usually proved using the Hochshild--Serre spectral sequence.
- The map (1) need not be injective in general. Take $X$ to be the complement of a closed point of degree two in $\mathbb{P}^1_k$. Then $\mathrm{Pic}(X) = \mathbb{Z}/2\mathbb{Z}$ but $\mathrm{Pic}(X_{\bar{k}}) = 0$.

From this answer I learned that Grothendieck proved the following result.

**Theorem.** Every formally smooth morphism between locally noetherian schemes is flat.

The book *Smoothness, Regularity, and Complete Intersection* by Majadas and Rodicio cites the following result.

**Theorem.** Let $(A,\mathfrak m,K)\to (B,\mathfrak n, L)$ be a local homomorphism of noetherian local rings. Then TFAE.

- $B$ is a formally smooth $A$ algebra for the $\mathfrak n$-adic topology;
- $B$ is a flat $A$-module and the $K$ algebra $B\otimes_AK$ is geometrically regular.

The authors then write:

This result is due to Grothendieck [EGA 0$_{\rm{IV}}$ , (19.7.1)]. His proof is long, though it provides a lot of additional information. He uses this result in proving Cohen’s theorems on the structure of complete noetherian local rings. An alternative proof of (I) was given by M. André [An1], based on André –Quillen homology theory; it thus uses simplicial methods, that are not necessarily familiar to all commutative algebraists. A third proof was given by N. Radu [Ra2], making use of Cohen’s theorems on complete noetherian local rings.

**Questions:**

- Are there any English references for the proof of Grothendieck or of André?
- What is the
*conceptual*outline of Grothendieck's proof?

André's *Homologie des Algèbres Commutatives* does not look very geometric to a novice like me and I was hoping perhaps Grothendieck's path was more geometric. I would also like to at least glimpse the big picture of the proof.

Let $p$ be an odd prime. Let $K$ be a finite extension of $\mathbb{Q}_p$ and $\mathcal{O}$ its ring of integers. Let $k$ be the residue field of $\mathcal{O}$.

Let $A$ be an abelian variety over $K$ which has semistable reduction. Let $A_k$ be the special fiber of the Néron model of $A$. Then, there is an exact sequence $$ 0 \to A^0_{k} \to A_{k} \overset{\pi}{\to} \Phi \to 0 $$ where $A^0_k$ is the connected component of the identity and $\Phi$ is the component group.

Here is my question: For simplicity, the component group is cyclic of order $pt$, for some $t$ not divisible by $p$. Let $x \in A_k$ so that $\pi(x)$ generates the $p$-torsion of $\Phi$, which we denote by $\Phi[p]$. Suppose that $x$ is of exact order $p^2$, i.e., $p^2 x=0$ but $ax \neq 0$ for any $1 \leq a < p^2$. Then, is it possible that the map $$A_k [p] \to \Phi[p]$$ induced by $\pi$ surjective? (If it is necessary, we may restrict the case with $e(K/\mathbb{Q}_p)<p-1$.)

For more information, this assumption can be satisfied if $A=J_0(N)$ the Jacobian variety of the modular curve $X_0(N)$ when $N=Mp$ with $(M, p)=1$. If $p \equiv 1 \pmod {12}$, then the component group of $A_p$ is cyclic. The cuspidal divisor $x=0-\infty$ is a rational torsion point and hence maps to $A_p$. Moreover it generates the component group by the map $\pi$. If $M=1$ then the order of $x$ and the order of the component group is same by Mazur but if $M=qr$ with two primes $q, r$ such that $q\equiv -r \equiv 1 \pmod p$, then the above situation can occur.

This question arises from what I find interesting in the recently asked question What is a chess piece mathematically?

My answer to that question was that mathematically, game pieces are in general epi-phenomenal to the main game-theoretic consideration, the underlying game tree. In this sense, strategic decisions do not involve directly game pieces but only choices in the game tree.

But nevertheless, I think a question remains. Namely, how can we recognize from a game tree that there is an underlying description of the game using pieces moving according to certain rules on a game-playing board? In other words, when is a game tree the game tree of a board game? The game trees arising from board games are a special subclass of the class of all game trees.

For this question, let us consider a *game tree* to be a finite or at least a clopen tree, all of whose terminal nodes are labeled as a win for one of the players (a slightly more general situation would be to allow some of the terminal nodes to be labeled with a draw). Play proceeds in the game by each player successively choosing a child node from the current node in the game tree, and the game ends when a terminal node is reached. (Of course a more general question would result from allowing infinite plays and using the usual Gale-Stewart conception of games.)

Of course, we can imagine playing a game by moving a piece on the game tree itself, in a way such that the resulting game tree is the same as the given game tree. But in this case, the board of the game would have the same size as the game tree. But in the case of our familiar board games, such as chess and Go, the boards are considerably smaller than the game trees to which they give rise. So what we really want is a board game whose board and number of pieces is considerably smaller than the game tree itself.

In the general case, the size of the game board and the number of game pieces would be connected in certain ways with the branching degree of the corresponding game tree. For example, in chess there are twenty first moves (each pawn has two moves, and each knight has two moves) and twenty second moves, and in general the number of moves does not greatly exceed this, which is on the order of magnitude of the board size, although the size of the game tree itself is considerably larger than this. In general, for our familiar games, the size of the game tree is much larger than its branching degree.

As a test question, if a game tree has a comparatively low branching degree compared with its size, is there always a way to realize it as a board game?

I'm not sure what counts as a *board game*, but the idea should be that there is a board and pieces that move about on the board according to certain rules, perhaps capturing other pieces, and certain configurations counting as a win, such as capturing a certain *king* piece or whatever. So part of the question is to provide a mathematical definition of what counts as a board game. What is a board game? Are there comparatively simple necessary and sufficient conditions on a game tree that it be realized as the game tree of a board game?

I would be interested also to learn of general sufficient conditions. How shall we think about this?

01/05/2018 2pm US EDST:

Again at Todd Trimble's suggestion, I have removed the original preface which I added to the question (concerning my interaction with Drs. Coxeter and Weiss regarding a somewhat-related question involving a particular interdimensional projection)

01/05/2018: 1pm US EDST:

In addition to the "PS" which I added last night (at Todd Trimble's suggestion.) I have now added a second "PS" (also at Todd Trimble's suggestion - see his last comment in this thread.) This second "PS" addresses the two concerns which Todd Trimble mentioned in his comment: 1) with respect to points in the $E_6$ lattice and points in the $E_8$ lattice, my first "PS" made reference to "patterns of spatial relationships" between these points, and this reference was too vague by usual and customary MO standards; 2) I didn't explain WHY my team was interested in "patterns of spatial relationships" between these two sets of points.

01/04/2018 10pm US EDST:

In addition to the "preface" which I've already added to this question in an attempt to improve it, I have now added a "PS" at the end of the question, in accordance with Todd Trimble's suggestion to narrow the question by stating the particular property which I would like to be invariant under the projections in question.

01/04/2018 - updated 1/5/2018 2pm - the "preface" has been removed at the suggestion of Todd Trimble.

I'm adding this "preface" to the original question in an attempt to address the concern of some voting MO members that the question is not about research level mathematics.

Original question (without preface):

In Regular Polytopes, Coxeter shows that the vertices of every n-dimensional cross-polytope (hyperoctahedron) project onto the vertices of an n-gonal (anti-)prism.

Question 1:

Has this projection ever been used to visualize properties of $E_8$ in 3-space via the octagonal prism (i.e. by expressing roots in terms of the basis defined by the vectors from the center of the prism to its vertices)

Question 2:

Has this projection ever been used to visualize properties of $E_6$ in 3-space via:

i) the nonagonal antiprism (when the roots of $E_6$ are coordinatized in 9-space)

ii) the octagonal prism (when the roots of $E_6$ are coordinatized in 8-space as a subset of the roots of $E_8$.)

Question 3:

Have these projections ever been used to visualize relatonships between $E_6$ amd $E_8$ in 3-space?

Or are there important properties of $E_6$ and $E_8$ that would not be preserved by such projections?

Please note that this question is related to a comment by Tobias Kildetoft in this question

regarding limitations on his computer-graphic capabilities.

Thanks as always for any time anyone can afford to spend considering this matter.

01/04/2018 10pm US EDST: "PS" added at the suggestion of Todd Trimble.

Since $E_6$ is a subgroup of $E_8$ (with roots occurring as a subset of the roots of $E_8$), there will, in general, be patterns of spatial relationships between the points of the $E_6$ lattice and the points of the $E_8$ lattice.

My team is very interested in the nature of these spatial relationships (for reasons which I won't go into here), but it is difficult for us to visualize these relationships as they truly exist in n > 3 -spaces.

So my question was actually posted in order to find out whether the projections mentioned in the above question would faithfully preserve the spatial relationships in question, because if so, then the projected lattices (or portions thereof) would be very helpful to us.

01/05/2018: 1pm US EDST:

This second "PS" addresses the two concerns which Todd Trimble mentioned in his last comment:

1) with respect to points in the $E_6$ lattice and points in the $E_8$ lattice, my first "PS" made reference to "patterns of spatial relationships" between these points, and this reference was too vague by usual and customary MO standards;

2) I didn't explain WHY my team was interested in "patterns of spatial relationships" between these two sets of points.

I. What "patterns of spatial relationships" in particular (between points in the $E_6$ lattice and points in the $E_8$ lattice) ?

My team is interested in whether any points in the $E_8$ lattice tend to "cluster" around any points in the $E_6$ lattice and if so, where, how, and why.

II. Why is my team interested in the question of whether such "clustering" exists?

I think I can best answer this question as follows - hope this answer is satisfactory.

My team is working at two biomolecular levels simultaneously:

1) the level of DNA and RNA polynucleotides and their associated energetics

2) the level of protein polypeptides (amino acid chains) and two of their associated properties (amino acid hydroaffinity and associated tRNA synthetase class)

In addition, because these two levels are interrelated by what is commonly called the "genetic code", my team is working at the junction of these two levels, i.e. the interface at which DNA genes are transcribed into RNA messages which are then translated into the polypeptide chains of protein "primary structures."

At the polynucleotide level, we have empirically determined a set of 240 special nonanucleotides ("tricodons") over the DNA alphabet {t,c,a,g} (or equivalent RNA alphabet {u,c,a,g}, and we have several good reasons to suspect that these 240 special nonanucleotides are an instantiation of the roots of $E_8$.

At the polypeptide level, these 240 special nonanucleotides translate (via the "genetic code" into a set of 72 tripeptides (ordered 3-tuples of amino acids) and again, we have several good reasons to suspect that these 72 tripeptides are an instantiation of the roots of $E_6$.

And what we suspect is that:

1) the "full-precision" genetic code as we know it TODAY (in all its minor variations across the different kingdoms of species or organisms) may have originally arisen as a set of less precise relationships between nonanuclotides and tripeptides;

2) it MAY be possible to characterize this early set of less precise relationships in terms of the way points of the $E_8$ lattice cluster around points of the $E_6$ lattice, IF such clustering does in fact exist.

Notes:

1) the above is somewhat of an over-simplification, but I think it will suffice to convey the general idea;

2) by "full-precision" genetic code, I simply mean that the present-day genetic code is constructed such that every codon encodes exactly one amino acid - though the reverse is of course not true, inasmuch as the "standard" genetic code has 61 "non-STOP" codons encoding only 20 amino acids.

Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$.

Does anybody know of any nice examples of general relationships between the images of the maps $g_*\colon H_*(M)\mapsto H_*(X)$ and $h_*\colon H_*(N)\mapsto H_*(X)$ induced by the restrictions $g=f\mathrel{|M}$ and $h=f\mathrel{|N}$, based on some given data about about these manifolds?

For Example, if $M=N$ and $W=[0,1]\times M$, then $f$ is a homotopy between $g$ and $h$, so $g_*=h_*$. Also, if our manifolds are oriented and compact, $X$ is a connected $n$-manifold, and $f$ is smooth, then the degree of $f\mathrel{\mathrel{|}\partial W}=f\mathrel{\mathrel{|}M\coprod N}$ is zero, so the degrees of $g$ and $h$ are equal up to sign, as are the images of $g_*$ and $h_*$ in degree $n$ homology.

let $\pi:\omega\to\omega$ be permutation and $\mathcal{F}$ is Ramsey ultrafilter on $\omega$. There are uncountable many growing subsequences of $\pi$. Can one proof that one of them has domain in $\mathcal{F}$ ?

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)