Recent MathOverflow Questions

Long wavelength instability: Linear Vs nonlinear phenomenon

Math Overflow Recent Questions - Mon, 08/14/2017 - 14:48

I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by:

$\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}(f(x,t))$, where $\mathbf{N}$ is a nonlinear second order operator.

Now, I have been able to establish that $f_0(x)$ is exponentially linearly stable for all initial perturbations that have frequency greater than some constant $c$. This has been shown by converting the problem into Fourier domain, in which the **linearized PDE (around $f_0(x)$) decouples into (infinite) system of ODEs, one for each frequency.**

However, the decoupling doesn't hold for the nonlinear PDE. So an initial perturbation which is linearly stable may excite modes of frequency lower than $c$ due to coupling in the nonlinear equation. Hence, we cannot naively claim that "linear stability => nonlinear stability" in this case.

So I am looking for examples where such a situation has been studied. Probably in fluid mechanics or other physical phenomenon ? I am hoping to either prove or disprove the notion of nonlinear stability for the system I am dealing with.

Square lying on moving chord of a simple closed curve

Math Overflow Recent Questions - Mon, 08/14/2017 - 11:25

Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate point $c_1$ on the left and $c_2$ on the right side to chord ab, such that $ac_1bc_2$ is a square. Continuously moving (sliding) a chord ab in such a way that it goes to ba (and $a(t)$ never equals $b(t)$, i.e. $\|a(t)−b(t)\|>0 ~\forall t$), points $c_1$ and $c_2$ will draw lines $L_1$ and $L_2$. These curves will form a closed curve $L = L_1 + L_2$. My hypothesis is that for any such trajectory and for any $C$, $L$ will intersect $C$, i.e. curve $L$ can never be completely inside or outside $C$. Is it true?

P.S. Curve $C$ can be non smooth.

Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions?

Math Overflow Recent Questions - Mon, 08/14/2017 - 09:10

I am interested in asking the following question:

What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation of his question from his slide presentation, "Proof Theory for Set Theory"

I ask this question for the following reasons:

(i). The model of $ZF$ + $V$=$L$ is the minimal submodel of $ZF$ for every transitive model of $ZF$, so it would seem that in every transitive model of $ZF$, constructible sets exist, and in fact, every constructible set exists in a transitive model of $ZF$.

(ii). Since every hereditarily finite set is constructible, it would seem reasonable(?) to infer that $\mathscr P_{Def}$($x$) is the 'correct' power set for $ZF$ (as opposed to the usual impredicative powerset $\mathscr P$($x$)--see Koepke's preprint for his mini-course, "Simplified Constructibility Theory", for the distinction).

Considering these reasons, I can refine the question as follows:

Are the constructible sets the only sets that the axioms of $ZF$ alone can prove to exist?

Extending a module structure in a family

Math Overflow Recent Questions - Mon, 08/14/2017 - 08:28

Let $k$ be a field (you can assume it to be algebraically closed) and let $A$ and $B$ be two non-commutative algebras over $k[t]$; let us assume that both are free and finitely generated as $k[t]$-modules. Let $i:A\hookrightarrow B$ be an embedding of $k[t]$-algebras which becomes an isomorphism if we invert $t$. Let now $M$ be a finitely generated $A$-module, which is free over $k[t]$. Note that $M[t^{-1}]$ is automatically a $B$-module (since $B$ is a subalgebra of $B[t^{-1}]=A[t^{-1}]$.

Let $A_0=A/tA, B_0=B/tB, M_0=M/tM$. We have a homomorphism $i_0:A_0\to B_0$ which might be no longer injective. Assume that the $A_0$-action on $M_0$ extends UNIQUELY to an action of $B_0$.

$\mathbf{Question:}$ Is it true that in this case $M$ (considered as a subspace of $M[t^{-1}]$) is automatically a $B$-module?

The intuition behind the question is this: it seems natural that the uniqueness of the extension at 0 should imply some kind of "continuity" (which will say that $M_0$ considered as a module over $B_0$ fits into "continuous family" of $B$-modules).

Unique words in dihedral groups

Math Overflow Recent Questions - Sun, 08/13/2017 - 21:04

Suppose $x$ is a word over the alphabet $\{0,1\}$. Let $a$, $b$ be elements of the group Dih$_k$ for some $k$.

Let $\varphi=\varphi_{a,b,k}$ be the map from words over $\{0,1\}$ to elements of the dihedral group Dih$_k$ (having $2k$ elements) such that $\varphi(0)=a$, $\varphi(1)=b$, and $\varphi$ takes concatenation to multiplication: $\varphi(xy)=\varphi(x)\varphi(y)$.

Let's say that $x$ is dihedrally simple if there is some $\varphi_{a,b,k}$ such that $\varphi(x)\ne\varphi(y)$ for all $y\ne x$ of the same length as $x$.

Computer experimentation suggests the

Conjecture: The dihedrally simple words form a regular language, namely $S\cup T$ where $$ S=\bigcup_{n=0}^\infty \{0^n,\quad 0^{n-1}1,\quad (01)^{n/2},\quad 01^{n-1},\quad 01^{n-2}0\} $$ where $(01)^{t+\frac12}=(01)^t0$, and $T$ is obtained from $S$ by interchanging 0 and 1.

My question is: Is the Conjecture similar to anything in the literature? Or do you see how to prove it?

Edit: to answer @LucGuyot's question below: this definition arises in my model of quantum security from a recent UCNC'17 paper (see also arXiv version) except that there is an additional constraint there, that we map the locked state $|0\rangle$ to the unlocked state $|1\rangle$.

The relevance of dihedral groups is that Dih$_k$ is representable as a subgroup of the projective unitary group $\mathrm{PU}(2, \mathbb C)$. The interpretation then is that $x$ is a secret code which should be punched into a quantum device with buttons labeled 0 and 1 and which trigger unitary operations $U_A$, $U_B$. Any code of the same length as $x$ but different from $x$ will not have the same effect (say, unlocking the device). Any attempt to inspect the state of the device during entering of the code, as one might do with a simple padlock, will constitute a measurement of the device and therefore reset the quantum superposition.

Traceless nonsymmetric block checkerboard matrix orbits and numerical ranges

Math Overflow Recent Questions - Sun, 08/13/2017 - 17:20

Let $A \in \mathbb{R}^{m \times m}$ be nonsymmetric traceless matrix with a zero/non-zero pattern which is symmetric and persymmetric (i.e. symmetric in the northeast-to-southwest diagonal).

If $A$ has any kind of block checkerboard pattern, or an offset of a block checkerboard pattern that conserves symmetry and persymmetry (of the pattern), then $A^k, k=2n+1, n\in\mathbb{N}_0$ is traceless.

1) Is there a necessary and sufficient condition on the zero/non-zero pattern to get a traceless $A^k, k=2n+1, n\in\mathbb{N}_0$?

If $A^k, k=2n+1, n\in\mathbb{N}_0$ is traceless then the spectrum of $A$ is symmetric with respect to the imaginary axis.

2) Is the numerical range (i.e. field of values, $W(A)=\left\{ \frac{v^* A v}{v^* v}, v \in \mathbb{C}^m, v\ne 0 \right\}$) always symmetric with respect to the imaginary axis?

Examples: \begin{align} \pmatrix{0 &0 &3 &-4\\ 0 &0 &-1 &3 \\ 3 &-1 &0 &0 \\ -4 &3 &0 &0} \end{align}

\begin{align} \pmatrix{0 &-9 &1 &0\\ -9 &0 &0 &-1 \\ -1 &0 &0 &1 \\ 0 &5 &7 &0} \end{align}

Effectiveness of the Intervention Program

Math Overflow Recent Questions - Sun, 08/13/2017 - 17:06

I have a control and treatment data and I would like to compare the effectiveness of the treatment program. What is the best statistical test to analyze the effectiveness of the program. Any input will be very valuable.

Perimeters of nested convex spherical polygons

Math Overflow Recent Questions - Sun, 08/13/2017 - 16:53

I seek a reference—not a proof—that if $P_1$ and $P_2$ are two convex polygons on a sphere composed of geodesic segments, contained in a hemisphere, and $P_1 \subseteq P_2$, then the perimeter of $P_1$ is $\le$ the perimeter of $P_2$. This is a spherical variant of the analogous claim in the plane, which was proved by Noam Elkies in an earlier MO question. I know how to prove the spherical version mimicking Noam's proof; the essence was used here.           I need to use this claim in a paper, and I'd prefer to cite a reference rather than include my own proof.

A query on discrepancy of a recursively defined tensor product sequence

Math Overflow Recent Questions - Sun, 08/13/2017 - 16:06

Given an $\eta\in(0,1)$ and letting $\otimes$ be matrix kronecker/tensor product defined in http://mathworld.wolfram.com/KroneckerProduct.html we define non-negative integer vectors recursively by $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z^2$$ $$v_t=\begin{bmatrix}a_t&b_t\end{bmatrix}\otimes v_{t-1} \in\Bbb Z^{2^t}$$ $$w_t=v_t\boxtimes v_t\in\Bbb Z^{2^{t+1}-1}$$ with $\boxtimes$ being defined as coefficients of product of polynomials whose coefficients are vectors $v_t$ where $$\forall t,t'\in\mathbb N\mbox{ with }t\neq t'\mbox{ }\mathsf{GCD}(a_t,a_{t'})=\mathsf{GCD}(b_t,b_{t'})=1$$ $$\forall t,t'\in\mathbb N,\mbox{ }\mathsf{GCD}(a_t,b_{t'})=1$$$$\forall t,t'\in\mathbb N,\mbox{ }0<a_t,b_{t'}$$ $$\forall t,t'\in\mathbb N, \mbox{ }a_t,b_{t'}\approx n^\eta$$ holds and take non-zero integer vectors $u\in\Bbb Z^{2^{t+1}-1}$ and $u'\in\Bbb Z^{2^{t}}$ such that $\langle u,w_t\rangle=0$ and $\langle u',v_t\rangle=0$ at some $t\in\Bbb N$.

There exists $\beta_t,\gamma_t\in(0,1)$ such that at least one coordinate of $u$ has magnitude bigger than $n^{\beta_t}$ and at least one coordinate of $u'$ has magnitude bigger than $n^{\gamma_t}$.

$\gamma_1=\eta=\beta_1$ is correct.

Discrepancy theory via Erdos-Turan-Koksma should give a value on $\beta_t,\gamma_t\in(0,1)$.

  1. What is a good estimate for $\beta_t,\gamma_t$?

  2. Does $\gamma_t=\frac t{2^t-1}\eta=\beta_t$ at every $t\in\Bbb N$?

I am most interested in $\beta_3,\beta_4,\gamma_3,\gamma_4$.

Studying the anomalies mathematically [on hold]

Math Overflow Recent Questions - Sun, 08/13/2017 - 15:43

The efficient market theory which says that there’s nothing in the data, let’s say price data, which will indicate anything about the future, because the price is sort of always right, the price is always right in some sense. But that’s just not true. According to the famous mathematician, James Simons, there's some mathematical ways to predict prices. There's some anomalies in the data we could study. These are not big anomalies, since otherwise people could see them quickly and predict them. Then these anomalies have to be small and put together can allow to predict well the stock market. I know the mathematics behind is basically statistical, but I do not know where to start.

What types of anomalies can James Simons refer to build his models? How could we study, predict them mathematically?

Stats central limit theorem vs variance property [on hold]

Math Overflow Recent Questions - Sun, 08/13/2017 - 15:08

(Assuming independent events) So there is the CLT which states Sum of (X[i]) from 1 to n has variance of n*var(X[i])

And variance has the property where

  1. Var(aX) = a^2*Var(X)
  2. Var(X+Y) = Var(X) + Var(Y) + (2Cov(X,Y) = 0 since independent)

So my question is when is it alright to use #1 instead of #2 or the CLT?

Is #1 only to be used when multiplying a single instance of a RV and when using multiple instances of the RV we use #2/the CLT where we can just strait up multiply?

Please help I have an exam tomorrow and this is really confusing.

Long identity-free sequences of permutations

Math Overflow Recent Questions - Sun, 08/13/2017 - 15:03

Let $N = [n]$ and for any subset $A \subseteq N$, let $S_A$ denote the subgroup of the symmetric group $S_n$ that fixes all objects outside $A$. Say that a sequence $A_1, \dots, A_k \subseteq N$ is "identity free" if the equivalence $$s_1 s_2 \dots s_k = \mathop{id} \qquad \text{where } s_i \in S_{A_i} \text{ for all } i$$ has no solutions, except for the trivial one where $s_1 = s_2 = \dots = s_k = \mathop{id}$.

Have these been studied under any name that I can search for? While I would be interested in any discussion at all, I am particularly interested in the question of the extremal length $L$ of the longest identity-free sequence when all $A_i$ are constrained to have some fixed size $\alpha$. The only upper bounds on $L$ I have been able to observe so far come from the easy observation that no two $A_i$ may overlap on more than one element (and then apply Cauchy-Schwarz).

The algebraic connectivity of graphs with large isoperimetric number

Math Overflow Recent Questions - Sun, 08/13/2017 - 14:11

Let $G = (V,E)$ be an undirected graph with maximum degree $\Delta$. the isoperimetric number of $G$, denoted $i(G)$, is defined by $$i(G) = \min_{|S| \leq |V|/2} \frac{e(S,\bar{S})}{|S|}$$ where $e(S,\bar{S})$ is the number of edges between $S$ and its complement $\bar{S}$. The Cheeger inequality asserts that: $$\frac{\lambda_2}{2} \leq i(G) \leq \sqrt{\lambda_2(2\Delta-\lambda_2)}$$ where $\lambda_2$ is the second smallest eigenvalue of the Laplacian matrix $L=D-A$ (which also known as the algebraic connectivity of $G$). In general, $i(G)$ can be far from $\frac{\lambda_2(G)}{2}$. For example, if $C_n$ is the cycle of order $n$, then $i(C_n) = \Omega(\frac{1}{n})$ and $\lambda_2(C_n) = O(\frac{1}{n^2})$. Can this be occurred if $i(G)$ be sufficiently large? Specifically, is this true that

if $i(G) \geq 1$, then $\lambda_2(G) \ge 1$?

Is there any known result similar to this statement?

Complexity of induction formulas in proof theoretic ordinals

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:48

According to The Art of Ordinal Analysis, the proof theoretic ordinal of a theory $T$ is the least ordinal $\alpha$ such that:

$${\bf ERA}+TI(\alpha,ECP)\vdash Con(T)$$

In above definition, $ECP$ stands for Elementary computable predicates and $TI(\alpha, A)$ stands for transfinite induction up to $\alpha$ for predicates in complexity class $A$.

Q. Is it possible to reduce the complexity of predicates for transfinite induction in above definition to a smaller complexity class?

For example, Is $Con(T)$ provable from ${\bf ERA}+TI(\beta,P)$ for some ordinal $\beta$ ?($P$ stands for polynomial time predicates.)

McLean theorem for Fano varieties?

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:48

Well-known McLean theorem states that deformations of special Lagrangian $L$ submanifolds in Calabi-Yau manifold are unobstructed and in bijection with harmonic 1-forms on $L$. The proof relies on the holomorphic calibration, which is not available for Fano varieties. My question is: is there any similar result for local deformations of minimal Lagrangians in Fano variety $X$ (by minimality I mean that Riemannian volume is stationary under deformations by arbitrary normal vector fields)?

In 'Minimal Lagrangian tori in Kahler-Einstein manifolds', Prof. Goldstein constructs a bijection between MLAGs in $X$ and conic SLAGs in the total space of the canonical bundle of $X$. Thus, deformation theory of MLAGs can be related to that of SLAGs but one has to preserve the conic condition. On the other hand, not all Fano varieties admit Kahler-Einstein metric.

In 'The Novikov-Veselov hierarchy of equations and integrable deformations of minimal Lagrangian tori in $\mathbb{C}P^2$', Prof. Mironov describes certain deformations of MLAG tori in $\mathbb{C}P^2$ arising from Novikov-Veselov hierarchy. Probably, not every deformation is realized by this construction.

In 'Special Lagrangian cones in $\mathbb{C}^3$ and primitive harmonic maps', Prof. McIntosh has obtained an algebro-geometric description of the moduli space of MLAGs in $\mathbb{C}P^2$ but the spectral curve construction is rather indirect so I am not sure that it leads to usable description of local deformations.

A sum involving Euler totient function

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:47

I wanted to know if we know any asymptotic formula/bound for the following sum

$$ \sum_{1 \leq a < n \\(a,n) = 1 } \phi(a) \ .$$

A trivial upper bound could be $$ \sum_{a=1}^{n} \phi(a) = \frac{3}{\pi^2} n^2 + O(n^{1+\epsilon}), $$ but I wanted to know if we could do better than that. Thanks in advance for the help!

References

https://en.wikipedia.org/wiki/Euler%27s_totient_function#Other_formulae

How can I analytically integrate the exponent of a polynomial?

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:47

I am trying to derive the mean transition path time across a specific one-dimensional potential of mean force,

$U(x) = V_0 (x^2 - a^2)^2 + (1/2)k(x-b)^2$

And I essentially need to evaluate the integral

$$ I = \int_\ e^{- \beta U(x)}dx\ $$ where $\beta$ is inverse Temperature.

I cannot use the Laplace method since, in general, I cannot find a global maximum to apply Taylor's expansion. Particularly, is there any generalized hypergeometric type function that may be a solution to my problem?

Predecessors and Successors of regular silting objects in bounded derived categories of wild hereditary algebras

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:36

Let $\Lambda$ be a wild hereditary algebra and let $T$ be one of its regular silting objects (i.e. all indecomposable direct summands of $T$ are shifts of indecomposable regular modules). What do we know about predecessors and successors of $T$ in the silting quiver of $D^b(\Lambda)$? In particular can we have shifts of $\Lambda$ as predecessors or successors of $T$?

Dimensions of irreducible representations of $GL(n,F_q)$ are polynoms in q having roots ONLY at roots of unity and zero?

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:30

Consider the group $GL(n,F_q)$ for finite field $F_q$, consider its irreducible representations over complex numbers.

Questions Is my understanding correct that the dimensions of all such irreps are polynomials in $q$ with integer coefficients having zeros only at roots of unity and zero ?

I understand that the answer should be contained in Green's 1955 paper THE CHARACTERS OF THE FINITE GENERAL LINEAR GROUPS, but as for me paper is difficult for extracting information.

Questions 2 If anwer is yes - is there any conceptual/nice reason for it ?

Questions 3 If someone can give some nice formula for such dimensions that would be quite helpful.

Questions 4 For other finite groups of Lie type is there any similar phenomena ?

Questions 5 From the perspective of $F_1$ it would be nice to know is there always a manifold such that number of $F_q$ points is given by these polynomials ? (Flags are of that type).

Let me give some examples known to me supporting the positive answer to the question:

  • For $GL(2,F_q)$ dimensions are : $1$ (det-like irreps) , $q+1$ (principal series), $q-1$ (cuspidal), $q$ (Steinberg = irregular principal series). See e.g. MO273764, MO271389.

  • In general "regular princinpal series" - irreps induced from non-trivial characters of the Borel subgroup will have dimension $[n]_q!$. Just because $GL/Borel = Flag$ manifold has such number of points.

  • Cuspidal irreps: the degree of a cuspidal character of $GL(n, q)$ is $(q − 1)(q^2 − 1)· · ·(q^{n-1}-1)$ (see page 135 Corollary 5.4.5. of very nice thesis containing huge amount of concrete information).

  • For the so-called unipotent irreps there is q-analogue of "hook formula". The degrees of the unipotent characters are “polynomials in q”: $ q^{d(λ)} \frac{(q^n − 1)(q^{n−1} − 1)· · ·(q − 1)}{ \prod_{h(λ)}(q^h − 1) }$ with a certain d(λ) ∈ N, and where h(λ) runs through the hook lengths of λ. See nice survey by G. Hiss FINITE GROUPS OF LIE TYPE AND THEIR REPRESENTATIONS (top page 26, section 3.2.6).

  • From above source - section 3.2.7: The degrees of the unipotent characters of $GL(5,q)$ for table $(5)$ dim = $ 1 $, for table $(4, 1)$ dim = $q(q + 1)(q^2 + 1)$ for table $(3, 2)$ dim = $q^2(q^4 + q^3 + q^2 + q + 1)$ for table $(3, 1^2)$ dim = $q^3(q^2 + 1)(q^2 + q + 1)$ for table $ (2^2, 1)$ dim = $q^4(q^4 + q^3 + q^2 + q + 1)$ for table $(2, 1^3)$ dim = $q^6(q + 1)(q^2 + 1)$ for table $(1^5)$ dim = $ q^{10}$

Infinite Point Set triangulations and Spectral Graph Theory

Math Overflow Recent Questions - Sun, 08/13/2017 - 13:08

I'm working with infinite point set triangulations. There is a subset of them which I need to characterize as finite or not. Given that deleting finitely many vertex of an infinite graph doesn't change it's essencial spectrum:

http://www.sciencedirect.com/science/article/pii/0024379582901112

Is Infinite Graph Spectra a useful tool of analysis? Any other approach that would be better?

thanks

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