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Instanton configurations of self-dual and anti-self-dual instantons interplay

Sat, 06/23/2018 - 21:41

Yang-Mills gauge theory is given by the action $$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$ whose Euler-Lagrange equations are the classical equations of motion. The classical solutions are stationary points of this functional under variation. The $M$ is the spacetime base manifold. The $\mathrm{Tr}_\mathfrak{g}$ is the trace over the Lie algebra $\mathfrak{g}$ of the principle compact Lie group $G$-bundle. The $F$ is the field strength of the connection $A$.

Equations of motion are: $$ dF=0, \quad d\star F=0 $$

  1. It is known that one way to satisfy the classical equations of motion are equivalent to $$\star F = \pm F,$$ i.e. the classical solutions of instanton equation to the equation of motion.

  2. It is said that the self-dual $F^+$ and an anti-self-dual $F^-$ are orthogonal to each other, say $$\star F_{\pm} = \pm F_{\pm},$$ w.r.t. the inner product defined by $ (f_1,f_2) = \int f_1\wedge\star f_2$.

Altogether this gives $S_\text{YM}[A] = \int \mathrm{Tr}_\mathfrak{g}(F^+\wedge \star F^+) + \int \mathrm{Tr}_\mathfrak{g}(F^-\wedge \star F^-).$ The second Chern class $$C_2(A) :=\frac{1}{8 \pi^2} \int\mathrm{Tr}_\mathfrak{g}(F\wedge F),$$ (I hope I get the normalization correct.) Notice that $$S_\text{YM}[A] \geq 8 \pi^2 \lvert C_2(A)\rvert ,$$ is locally minimized when the equality holds.

The equality holds exactly when either $F^+ = 0$ or $F^- = 0$, i.e. when the full field strength $F$ is itself either self-dual $F^+$ or anti-self-dual $F^-$, or zero field strength.

Question: At the classical level, it looks that the self-dual instanton configuration ($\star F_{+} = + F_+$) and anti-self-dual instanton configuration ($\star F_{-} = - F_-$) are totally decoupled. However, at the quantum level, in terms of the Yang-Mills functional $Z= \int [DA]\exp[- S_\text{YM}[A]] = \int [DA]\exp[- \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)]$, do the self-dual instanton and anti-self-dual instanton configurations interact with each other in some nontrivial way? Namely, are there some configurations of $F_{+}$ and $F_{-}$, that they interplay with each other, at the even classical or only at quantum level?

If so, what are the examples? How do they alter the classical theory or quantum theory of path integral?

An epllipse through 12 points related to Golden ratio

Sat, 06/23/2018 - 18:34

I am looking for a proof of the problem as follows:

Let $ABC$ be a triangle, let points $D$, $E$ be chosen on $BC$, points $F$, $G$ be chosen on $CA$, points $H$, $I$ be chosen on $AB$, such that $IF$, $GD$, $EH$ parallel to $BC$, $CA$, $AB$ respectively. Denote $A'=DG \cap EH$, $B'=FI \cap GD$, $C'=HE \cap IF$, then 12 points:

$D$, $E$, $F$, $G$, $H$, $I$ and midpoints of $AB'$, $AC'$, $BC'$, $BA'$, $CA'$, $CB'$ lie on an ellipse if only if

$$\frac{\overline{BC}}{\overline{BE}}=\frac{\overline{CB}}{\overline{CD}}=\frac{\overline{CA}}{\overline{CG}}=\frac{\overline{AC}}{\overline{AF}}=\frac{\overline{AB}}{\overline{AI}}=\frac{\overline{BA}}{\overline{BH}}=\frac{\sqrt{5}+1}{2}$$

Note: This problem don't appear in AMM, and I don't have a solution for this problem, but there is the same configuration appear in:

Does every $n\times n\times n$ Latin cube contain a Latin transversal?

Sat, 06/23/2018 - 18:22

In 1967 H. J. Ryser conjectured that every Latin square of odd order has a Latin transversal. Similar to Latin squares, we may consider Latin cubes.

QUESTION: Let $n$ be any positive integer. Does every $n\times n\times n$ Latin cube contain a Latin transversal?

Let $N$ be any positive integer. In 2008, I proved that for the $N\times N\times N$ Latin cube over $\mathbb Z/N\mathbb Z$ formed by the Cayley addition table, each $n\times n\times n$ subcube with $n\le N$ contains a Latin transversal (cf. my paper An additive theorem and restricted sumsets). Motivated by this, in the 2008 paper I conjectured that my above question has a positive answer.

Any comments are welcome!

Compactness of operators and norming sets

Sat, 06/23/2018 - 16:12

Originally asked on MSE.

Let $T$ be a linear map from a normed space $E$ into a Banach space $F$.

Let $D\subset \overline{B}_{F^{*}}$ be norming, i.e. there is $r>0$ such that $\sup\limits_{v\in D}|\left<f,v\right>|\ge r\|f\|$, for every $f\in F$.

It is easy to see that a linear map $T:E\to F$ is bounded iff $T^{*}D$ is bounded.

If $T$ is (weakly) compact, then $T^{*}$ is a (weakly) compact, and then $T^{*}D$ is relatively (weakly) compact.

I am wondering about the converse of the last statement.

I can show that compactness of $T$ follows from relative compactness of $T^{*}D$ under the assumption that $E^{*}$ is separable, but no progress whatsoever about the weak compactness.

Dirichlet series as rational zeta expressions

Sat, 06/23/2018 - 15:45

Let $D(f,s):=\sum_{n=1}^\infty \frac{f(n)}{n^s}$, otherwise known as a Dirichlet series. When $f$ is a multiplicative, number theoretic function, $D(f,s)$ tends to be expressed as a rational product of zeta functions. For example, $D(\mu,n) = \zeta^{-1}(s)$, $D(\sigma,n)=\zeta(s)\zeta(s-1)$, $D(\phi,n) = \zeta(s-1)/\zeta(s)$, and many, many more.

Question 1: Is it possible to deduce for which family of $f$, $D(f,s) = \frac{\prod_{a_i}\zeta(s-a_i)}{\prod_{b_i}\zeta(s-b_i)}$?

Question 2: Conversely, given an arbitrary such rational product of zeta functions, does it always correspond to some multiplicitive number-theroteic function? If yes, have any new "interesting" such functions been discovered by such methods?

As I understand, it the answer these questions should likely follow from Perron's formula, but I'm not seeing an obvious way of proceeding. In fact, the questions seem to reduce to basic generating function theory, along with the inclusion-exclusion principle, as many of the above known number-theoretic identities can be derived from it as such.

Metrizable subspaces of separable spaces

Sat, 06/23/2018 - 15:35

Are metrizable subspaces of separable spaces separable?

Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.

An “almost separable” optimization problem on a graph

Sat, 06/23/2018 - 14:47

I am trying to solve an unconstrained optimization problem with the following properties: I am given a graph $G=(V,E)$ with functions $f_{ij}:\mathbb{R}^2\to \mathbb{R}$ for each edge $(i,j)$. I am trying to choose scalar variables $x_1,\dots,x_n$ for each vertex to minimize $$\sum_{(i,j)\in E } f_{ij} (x_i, x_j) $$. This is almost a separable problem except that a variable can appear in multiple functions. Are there any papers describing ways to exploit this special structure? The functions are continuous but not necessarily convex or concave.

Family of rooted trees parameterized by binary sequences

Sat, 06/23/2018 - 14:36

Let $A=\{1,2\}$. For any $d \in A$ and any sequence $a=(a_1,a_2,\dots)\in A^{\mathbb N}$ the associated rooted tree $T(d,a)$ is recursively defined in the following way. The degree of the root of $T(d,a)$ is $d$, and the result of the removal of its root (together with the $d$ edges issued from it) is the tree $T(a_1,Sa)$ if $d=1$ and two trees $T(1,Sa), T(2,Sa)$ if $d=2$ (here $Sa=(a_2,a_3,\dots)$ denotes the shift of the sequence $a$).

I am interested in any references to the appearances of the family of trees $T(d,a)$ (or anything similar).

Change of rank of a matrix under random sampling

Sat, 06/23/2018 - 13:03

Consider a matrix $X \in \mathcal{R}^{n\times n}$ with rank $r$. Assume the matrix is sampled, using some distribution, and the resulting matrix $Y$ is defined as $Y_{i,j} = X_{i,j}$ if $(i,j)$ is sampled and $0$ otherwise. The interest is to determine how the rank of $X$ changes. For instance, if $X$ is invertible, what conditions, say on the sampling, ensure that $Y$ is also invertible? In particular, the interest is to do analysis on the change of eigenvalues under random sampling. I am looking for any references or works that deal with this question. Initial attempts using the Weyl's perturbation theorem was not fruitful.

What are the periodic Dyck paths?

Sat, 06/23/2018 - 12:31

Let $C$ be the Cartan matrix of a finite dimensional algebra $A$ with finite global dimension, then the Coxeter matrix is defined as $M=-C^{-1}C^T$. $A$ is called periodic in case $M^k=id$ for some $k \geq 1$. The smallest such $k$ is called the period of the algebra. See for example https://www.sciencedirect.com/science/article/pii/S0024379505001709 , https://www.sciencedirect.com/science/article/pii/S0024379502004056and related articles on that.

We can associate to every Dyck path from $(0,0)$ to $(2n-2,0)$ the Nakayama algebra with linear quiver and $n$ simple modules which has this Dyck path as the top boundary of its Auslander-Reiten quiver. This gives a natural bijection.

Now we call a Dyck path periodic in case its corresponding algebra is periodic and the periodic of the Dyck path is defined as the periodic of the corresponding algebra.

I am not experienced with this topic but I wondered:

What are the periodic Dyck paths and is there a formula for the period?

It seems to be very complicated, but I noted the following for $n \leq 9$ which I state here as a conjecture:

Mainconjecture: -The number of Dyck paths from $(0,0)$ to $(2n-2,0)$ with period $n+1$ is equal to $2^{n-2}$.

Other conjectures (that would follow from the main conjecture):

-Those with additionally having global dimension (=global dimension of the corresponding algebra) at most 2 are counted by the Fibonacci sequence (starting with 1,2,... for $n \geq 2$.)

-Those with additionally having global dimension at most 3 are counted by the Tribonacci sequence: https://oeis.org/A000073. (starting with 1,2,... for $n \geq 2$.)

-Those with additionally having global dimension at most 4 are counted by the Tetranacci sequence: https://oeis.org/A000078. (starting with 1,2,... for $n \geq 2$.)

-Probably one can guess how that might continue(Yes, global dimension 5 lead to the pentanacci numbers https://oeis.org/A001591). I found suprisingly much more nice sequences when restricting to certain nice homological conditions, but global dimension seems to be the best suited here.

The conjecture is checked for $n \leq 9$ with the computer. In case it is true, can those Dyck paths be nicely described? Here the Dyck paths for $n \leq 7$ with their period (period 0 means it is now periodic):

http://www.findstat.org/StatisticsDatabase/St001218

edit: Thanks to the comment of Michael Albert, I can see now what should be going on with period $n+1$, and it would be enough to prove the main conjecture to obtain the other conjectures as a corollary. I leave the other conjectures here for completeness.

Binomial coefficients in discrete valuation rings

Sat, 06/23/2018 - 10:19

Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer.

For any integer $d,n\ge 0$, define:

$${\pi^d \choose n} := \frac{\pi^d\cdot(\pi^d -1)\cdot\ldots\cdot(\pi^d-n+1)}{n!}.$$

  • Is ${\pi^d\choose n}$ an element of $V$?
  • For exactly what integers $n\ge 0$ is ${\pi^d\choose n}$ a unit?

Example. If $V = \mathbf{Z}_p$, $\pi = p$, then ${p^d\choose n}$ is zero unless $0\le n\le p^d$, and its $p$-adic valuation is $d-v_p(n)\ge 0$ for $1\le n\le p^d$. In all cases, the answer to the first question is yes, and the answer to the second question is: for $n = 0, c\cdot p^d$, with $(c,p) = 1$.

Symplectic forms and sign of eigenvalues

Sat, 06/23/2018 - 09:01

This question has come out while reading J. Moser "New Aspects in the Theory of Stability of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of elliptic fixed points of Hamiltonian dynamical systems, in the time independent case. I start presenting the framework.

Let us consider the Hamiltonian dynamical system $$ \dot{x}_\nu = H_{y_\nu}(x,y), \qquad \dot{y}_\nu=-H_{x_\nu}(x,y) \qquad \qquad (1) $$

where $\nu=1,2,\dots,n$. Hamiltonian $H$ does not depend on time $t$ and it is assumed to be a real analytic function of $x_\nu,\,y_\nu$, with $\nu=1,2,\dots,n$ in the neighboorhood of $x=y=0$, the expansion of which starts with quadratic terms. Then $x=y=0$ is an equilibrium solution.

One can construct a fundamental system of solutions of exponential form $$ w^{(\nu)}=e^{\gamma_\nu t}p^{(\nu)} $$ where $p^{(\nu)}$ are constant vectors or, in the case of multiple eigenvalues $\gamma_{\nu}$, possibly polinomials in $t$. The numbers $\gamma_\nu$ are obtained as the eigenvalues of the matrix determined by the linear terms of the right-hand side of (1). Suppose that all eigenvalues are distinct and purely imaginary, i.e. of the type $\gamma_\nu=i\beta_\nu$, with $\beta_\nu$ real. So the spectrum has the form $$ \pm i \beta_1, \quad \pm i \beta_2, \,\dots,\, \pm i \beta_n. $$ So one obtains a collection of distinct numbers $\beta_\nu$, $-\beta_\nu$ with $\nu=1,2,\dots,n$

So far, so good.

For later purposes, one needs to define the sign of $\beta_\nu$. The Author says that the sign of $\beta_\nu$ is taken in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ Square brackets are defined as Lagrange Brackets (an outdated therminology, nowadays called symplectic form [see comments]). More precisely, given any two $2n$-dimensional vectors, $x$ and $\tilde{x}$, with components $x_\nu$ and $\tilde{x}_\nu$, their Lagrange Bracket (read: symplectic form) is defined as $$ [x,\tilde{x}]=\sum_{\nu=1}^n (x_\nu\tilde{x}_{\nu+n}-x_{\nu+n}\tilde{x}_\nu) $$ In passing, one has to remember that an Hamiltonian sysytem is marked by the fact that, for any two solutions $x$ and $\tilde{x}$, the Lagrange bracket (read: symplectic form) $[x,\tilde{x}]$ is $t$-independent.

Questions: can someone please show

1) how to explicitly compute:

$$ [w^{(\nu)},w^{(\mu)}] = ? $$

$$ [w^{(\nu)},\overline{w^{(\mu)}}] = ? $$

2) how to prove that $$ [w^{(\nu)},\overline{w^{(\nu)}}] = [p^{(\nu)},\overline{p^{(\nu)}}] $$

$$ [p^{(\nu)},\overline{p^{(\nu)}}] \text{ is purely imaginary} $$ 3) How to define the sign $\beta_{\nu}$ in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ I am from the Physics community, so I kindly ask to display all important passages.

Thomason-Trobaugh Theorem

Sat, 06/23/2018 - 08:18

Let $X$ be a scheme and $U$ be an open subscheme. The proof of the Thomason-Trobaugh Theorem implies that under some mild assumptions, for any perfect complex $F$ on $U$, we have that $F\oplus F[1]$ can be extended to a perfect complex on $X$. I'm just wondering whether there exists examples where $F$ is a perfect complex on $U$ but $F$ itself cannot be extended to $X$? I've found an example when $X$ is the cone $xy-z^{2}=0$ and $U$ is the complement of the origin. Is there an example for smooth $X$?

Clopen subsets of a closed subspace of a spectral space

Sat, 06/23/2018 - 02:08

Let $X$ be a topological space. Set $K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral, if it satisfies all of the following conditions:

1) $X$ is quasi-compact and $T_0$. 2) $K(X)$ is a basis of open subsets of $X$. 3) $K(X)$ is closed under finite intersections. 4) $X$ is sober, i.e. every nonempty irreducible closed subset of $X$ has a (necessarily unique) generic point.

Let $C$ be a closed subset of a spectral topological space $X$. I am looking for equivalent conditions on $C$ under which if $A$ is a clopen(=Closed+Open) subset of $C$, then there exists a clopen subset $B$ of $X$ such that $A=C\cap B$?

On the structure of a finite group of order $144$

Sat, 06/23/2018 - 01:00

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.

QUESTION: Prove $G\cong A_4\times A_4$.

By using Magma, we know there is only one group of order $144$ with an irreducible $\mathbb{C}$-character $\theta$ of degree $9$. Now I want to prove this result without using Magma.

Asymptotic behavior of an integral depending on an integer

Fri, 06/22/2018 - 08:57

A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where $$ f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t $$ Numerical computaions suggest that $$ f(n) = 4 \pi n + o(n) $$ How to justify it? Moreover, is it possible to obtain a good rate of convergence?

Why torsion is only defined for linear connection on TM?

Fri, 06/22/2018 - 04:22

The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, or for a connection obtained as the pullback of a connection on a vector bundle $E \to M$ isomorphic to $TM$ via an isomorphism $\theta \colon TM \to E$ equivalent to a solder form.

Why is that so ? If torsion can be interpreted as the twist of a moving frame along a curve, the same phenomena should occur for a connection on any vector bundle.

Is there a way to define a notion of torsion for any vector bundle ?

Can the number of countable models of a complete first-order theory decrease after adding constants?

Thu, 06/21/2018 - 18:43

If $T$ is a countable complete first-order theory with infinite models, the number of countable models it has, $I(T,\omega)$, must be an element of $N=\{1,3,4,5,6,7,\dots,\omega,\omega_1,2^\omega\}$ (although we don't know if $\omega_1$ can happen). For which pairs $n,m\in N$ does there exist a countable complete theory $T$ with $n$ countable models but $m$ countable models after adding finitely many constants to the theory? Countably many new constants? In particular can we have $m<n$? EDIT: By 'adding constants,' I mean adding constants whose type is completely specified, i.e. expanding by constants and then passing to a complete theory in the expanded language.

Let $n\rightarrow m$ denote the statement "There exists a complete countable theory $T$ and a finite tuple of constants $\overline{a}$ such that $I(T,\omega)=n$ and $I(T_\overline{a},\omega)=m$." And let $n\rightarrow_\omega m$ denote the statement "There exists a complete countable theory $T$ and a countable set of distinct constants $A$ such that $I(T,\omega)=n$ and $I(T_A,\omega)=m$." Some easy results and relevant observations:

  • If $n\rightarrow m$ (resp. $n\rightarrow_\omega m$) and $k \rightarrow \ell$ (resp. $k \rightarrow_\omega \ell$), then $nk \rightarrow m\ell$ (resp. $nk \rightarrow_\omega m\ell$). (Take the disjoint union of the relevant theories.)
  • $n \rightarrow n$ for every $n\in N-\{\omega_1\}$. (This is obvious for $n=1$. There are easy examples for $n=\omega,2^\omega$ and the standard examples for $n=3,4,5\dots$ all have constants which do not increase the number of countable models.)
  • $n^2+n\rightarrow (n+1)^2$ for any $1<n<\omega$. (DLO with $n-1$ colors and a countable set of constants of order type $\omega + \omega^\ast$. By itself this theory has $n^2 + 1$ countable models. Adding a constant in between $\omega$ and $\omega^\ast$ makes the theory have $(n+1)^2$ models.)
  • $1\not\rightarrow n$ and $n\not\rightarrow 1$ for any $n\in N - \{1\}$.
  • $1\rightarrow_\omega 2^\omega$ (For example: DLO.)
  • $1\rightarrow_\omega \omega$ (For example: A structureless set.)
  • $n\not\rightarrow_\omega 1$ for any $n\in N$.
  • $1 \rightarrow_\omega n$ for every $2<n<\omega$. (The standard examples of Ehrenfeucht theories are $\omega$-categorical theories with countably many constants added.)
  • If a theory is not small, then it will have $2^\omega$ countable models after adding any countable set of constants.

Covering inequality for sets of intervals

Thu, 06/21/2018 - 07:50

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point from $P$ by $I_P,J_P$. Now suppose that \begin{align}\tag{*}\label{IP JP ineq}\lvert I_P\rvert\le \lvert J_P\rvert+1\end{align} for all finite subsets $P$, and that the inequality \eqref{IP JP ineq} is optimal in the sense that there exists at least one finite $P$ for which equality is achieved. Here $\lvert\cdot\rvert$ is simply the counting measure.

I have the strong suspicion (supported by numerical experimentation with random sets) that there has to exist some specific interval $(a,b)\in I$ such that $(a,b)$ contains at least one point from each set $P$ for which equality is achieved in \eqref{IP JP ineq}.

It is clear that the statement cannot be true for more general subsets than intervals but I couldn't come up with any argument yet. It is also clear that the claim cannot hold true if the $1$ in \eqref{IP JP ineq} is replaced by $0$.

How to extract an indiscernible sequence from a set in a stable theory?

Wed, 06/20/2018 - 18:58

I was reading Chernikov's notes about stable theories, and he mentions the following fact:

If $T$ is stable and $A$ is some set of parameters large enough then there is some indiscernible sequence $I \subseteq A$ such that $|I|=|A|$.

I have been trying to find a reference of the previous fact but I have not been lucky. Any comment, hint or reference is highly appreciated!

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