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most recent 30 from mathoverflow.net 2017-12-12T17:39:16Z

Escaping local SOS minima while solving system of nonlinear equations

Sun, 09/03/2017 - 15:26

One of the approaches of solving systems of nonlinear equations ($f_1=f_2=0$, here I take 2D case for simplicity) is minimizing sum of squares $s=f_1^2+f_2^2$. Direction can be chosen as Newton step or gradient of s. The problem I see here is that it is possible to get into local minimum of s, where $s>0$. Such points are also bad, because $\det{J}=0$ and Newton step leads to the infinity. Are there any developed tricks how to escape such local minimums, which are not interesting in root-finding process?

Thanx.

Linear dependence between prime gaps

Sun, 09/03/2017 - 15:24

There is an equal number of twin and cousin primes. The analysis is given in Marek Wolf "On the Twin and Cousin Primes". What is the best formula, algorithm or approach to find other linear dependences between prime gaps?

Provability in $S^1_2$

Sun, 09/03/2017 - 15:17

What are some examples of natural true statements of the form $∀n φ(n)$ ($φ$ is a polynomial time computation/test) that are unprovable in $S^1_2$?

Examples may be unconditional or dependent on reasonable computational complexity conjectures.

While this class of statements naturally corresponds to correctness of polynomial time algorithms (where correctness of the output is coNP), much of mathematics is reflected in these statements, including, for example, Fermat's Last Theorem if it is stated in a way that does not require exponentiation to be total, as well as various numeric tests of open conjectures in number theory.

$S^1_2$ is in some ways the weakest natural base theory for reverse mathematics. It consists of basic arithmetical axioms, closure of unary numbers under multiplication, and polynomial induction on NP predicates: $φ(0) ∧ ∀n (φ(n)→φ(2n)∧φ(2n+1)) ⇒ ∀n φ(n)$ where $φ$ is an NP formula (i.e. φ is $Σ^b_1$; φ may have other free variables; numbers are binary numbers). It is closely connected to polynomial time computation: An $S^1_2$ proof of $∀x ∃y φ(x,y)$ ($φ$ is an NP formula) can be converted into a polynomial time algorithm for finding an example $y$ given $x$ (however, the conversion uses cut-elimination and is not polynomial in proof length). For this class of formulas, $S^1_2$ is conservative over PV$_1$, which is (modulo choice of language and formalization) $S^1_2$ with the polynomial induction restricted to $φ$ that are polynomial time computations.

It remains open whether $S^1_2$ proves P=PSPACE, but assuming plausible computational complexity conjectures, there are sharp limits on provability in $S^1_2$. For example, if factoring is hard, then $S^1_2$ does not prove that every nonprime number -- as tested, for example, by AKS primality test -- has a nontrivial factor (and conversely, the ordinary definition of prime numbers would not provably satisfy many results in number theory). However, these examples can be conceptually grouped with $Π^0_2$ statements in that unprovability depends on number/set existence axioms beyond the power of the base theory. A recurring conjecture is that while the proofs may be hard, the propositions are usually provable if we have have the required existence axioms and basic properties.

A plausible conjecture is that typical (in current mathematical and computer science literature) true statements of the form $∀n φ(n)$ (polynomial time computable $φ$) are already provable in $S^1_2$. The answers may illuminate how accurately the conjecture holds, or show clear limits to this type of polynomial time reasoning.

Cyclic polygons generalized to higher dimensions

Sun, 09/03/2017 - 14:58

Many theorems hold for cyclic polygons—convex polygons inscribed in a circle. Perhaps the most basic is this, from the reference cited below:

Theorem. There exists a cyclic polygon of $n \ge 3$ sides of lengths $\ell_i > 0$ if and only if each $\ell_k$ is less than the sum of the other lengths. And this polygon is unique.

         
          (Wikipedia image from article: Circumscribed circle.)

Kouřimská, Hana, Lara Skuppin, and Boris Springborn. "A variational principle for cyclic polygons with prescribed edge lengths." Advances in Discrete Differential Geometry. Springer Berlin Heidelberg, 2016. 177-195.

My question is:

Q. What is the closest higher-dimensional analog of this theorem? E.g., in $\mathbb{R}^3$ the areas would be prescribed.

I am familiar with Minkowski's theorem on the existence of a polytope realizing given facet areas/volumes and facet normals. What I am wondering is: If one assumes the polytope is inscribed in a sphere, can we reduce the information needed to justify existence/uniqueness? In other words, can Minkowski's theorem be "strengthened" by presuming the inscribed-in-a-sphere condition?

Related: Japanese Theorem” on cyclic polygons: Higher-dimensional generalizations?

Can rank-into-rank embeddings be made to have the branching property in some forcing extension?

Sun, 09/03/2017 - 14:06

Suppose that $X$ is a self-distributive algebra with an element $1$ such that $1*x=x,x*1=1$ for $x\in X$. Then let $\preceq^{X}$ be the smallest pre-ordering on $X\setminus\{1\}$ such that $x\preceq^{X}x*y$ whenever $x,y\in X,x\neq 1,x*y\neq 1$.

Let $\mathcal{E}_{\lambda}^{+}$ denote the set of all non-trivial elementary embeddings $j:V_{\lambda}\rightarrow V_{\lambda}$. Then we say that a collection $(j_{x})_{x\in X}$ of elementary embeddings has the branching property if for all limit ordinals $\gamma<\lambda$, the partial ordering $\preceq^{\langle\{j_{x}|x\in X\}\rangle/\equiv^{\gamma}}$ is a tree whose roots are precisely the elements $[j_{x}]_{\gamma}$ where $\mathrm{crit}(j_{x})<\gamma$.

Suppose that $X$ is an index set and $j_{x}\in\mathcal{E}_{\lambda}^{+}$ for each $x\in X$. Then does there exist a forcing extension $V[G]$ such that in $V[G]$ there are elementary embeddings $\overline{j_{x}}:V[G]_{\lambda}\rightarrow V[G]_{\lambda}$ such that $\overline{j_{x}}$ extends $j_{x}$ for each $x\in X$ and in $V[G]$ the system $(\overline{j_{x}})_{x\in X}$ has the branching property?

This question has obvious several generalizations such as when the elementary embeddings are maps $j:V_{\lambda+1}\rightarrow V_{\lambda+1}$, so I am also looking for answers in these generalized cases.

Basic question related to Stieltjes integral

Sun, 09/03/2017 - 14:01

I am reading this paper. I am stuck on something, which I think is something basic but I haven't been able to figure it out yet, and I was hoping someone could explain it to me.

Let $$ \sigma(u) = \frac{1}{2\pi} \int_{-T}^T F(t) \frac{u^{1/2 + it} - M^{1/2 + it}}{1/2 + it} dt + O(N (\log N)^2/T), $$ where $F$ is a analytic function and $M$, $T$ are fixed positive numbers. Then on page 52 of the paper it is stated $$ \int_M^N e(u \lambda) d\sigma = \frac{1}{2\pi} \int_{-T}^T F(t) \int_M^N e(u \lambda) u^{-1/2 + it} \ du \ dt + E $$ where $$ E \ll (\log N)^2 (1 + |\lambda| N)N/T. $$ I was wondering about how I can show this bound on the error term. I would greatly appreciate any explanation. Thank you very much!

Seminorms on tensor products of affinoid algebras

Sun, 09/03/2017 - 13:41

Let $A \leftarrow C \rightarrow B$ be affinoid $K$-algebras, where $K$ is a non-archimedean field with non-trivial absolute value. Equipping $A$, $B$, $C$ with the supremum seminorms, there is a canonical seminorm $\nu_1$ on $A \otimes_C B$: $$\nu_1(f) = \inf \max_i |a_i|_{\sup} |b_i|_{\sup},$$ the infimum taken over all representations $f = \sum a_i \otimes b_i$. Then $\nu_1$ extends by continuity to a seminorm on the completed tensor product $A \widehat{\otimes}_C B$ (where the latter is endowed with any residue norm). On the other hand, $A \widehat{\otimes}_C B$ is known to be an affinoid algebra in its own right, so has its own supremum seminorm $\nu_2$.

Does $\nu_1 = \nu_2$?

I'm hopeful this is true, but doubt it (although I have no specific counter-example). It's clear that $\nu_2 \le \nu_1$, and the equality holds if and only if $\nu_1$ is power-multiplicative (i.e. $\nu_1(f^n) = \nu_1(f)^n$).

If it helps, we may assume $C$ is a Tate algebra and $\text{Sp} \, A$, $\text{Sp} \, B$ are affinoid subdomains of $\text{Sp} \, C$.

Contour of soap film intersection on a doubly curved surface

Sun, 09/03/2017 - 13:04

When is soap/shampoo film enclosing a small pressure is formed on a flat surface or the inside of a circular cone, it forms a hemispherical bubble or respectively a segment of a sphere.

$$\kappa_1+\kappa_2 =H =p/T$$

( Pressure $p$, surface tension $T$ are constants). The CMC (constant mean curvature) surface integrates to a spherical shape whose curvature increases with $p/T.$

Similarly When a bubble forms enclosing air with pressure on an arbitrary patch of shape $f(x,y,z)=0,$ what would be the shape of the interfacial closed oval closed ring that is seen to definitely form, as a function of $(p/T, f)?$

When soap bubble moves to a new location, the ring changes shape according to $f$ alone.

Whence “homomorphism” and “homomorphic”?

Sun, 09/03/2017 - 11:49

Today homomorphism (resp. isomorphism) means what Jordan (1870) had called isomorphism (resp. holoedric isomorphism). How did the switch happen?

“Homomorphic” (and “homomorphism” as “property of being homomorphic”) are e.g. in de Séguier (1904, pp. 65–66) and the last edition of Weber (1912, p. 195). “Homomorphism” as map of groups is e.g. in Schur (1924, p. 192). But none of these sound like a first.

I asked this on hsm a week ago, but got no answer there.

Profinite extension of a Lie group

Sun, 09/03/2017 - 11:36

Let $H,G,K$ be three topological groups, we say that $G$ is an extension of $K$ by $H$ if the following short sequence $$0\rightarrow H\rightarrow G\rightarrow K\rightarrow 0$$ is exact. (If $H$ is a subgroup of $G$ this is equivalent to $K\cong G/H$)

Now, assume that $H,G,K$ are compact abelian topological groups, therefore, by the structure theorem for compact abelian Lie groups, we have that $K$ is a Lie group if and only if it is isomorphic to $\mathbb{T}^n\times C_k$ where $\mathbb{T}$ is the circle group ($S^1$ or $\mathbb{R}/\mathbb{Z}$), $C_k$ is a finite abelian group and $n\in\mathbb{N}$.

It is well known that every extension of a Lie group by a Lie group is again a Lie group. In particular, every extension of a Lie group by a finite group is a Lie group.

My question is: whether every extension of a compact abelian Lie group by a pro-finite group is a subgroup of $\mathbb{T}^n\times D$ where $D$ is a profinite group for some $n\in\mathbb{N}$.

I am most interested in the case where $H$ is a direct product of finite groups.

Any proof / reference is appreciated.

Edit: I also assume that the groups are Hausdorff. Also every morphism is both topological and algebraic.

Solving ODE via contact geometry

Sun, 09/03/2017 - 09:10

I have been reading H. Geiges' "A Brief History of Contact Geometry and Topology". According to him contact transformations were introduced as a geometric tool to study systems of differential equations. I am aware that he refers to V.I. Arnold's "Geometrical Methods in the Theory of ODEs" for details, but couldn't find a satisfactory example.

I am looking for a concrete example where a contact geometric point of view helps in answering a question about an ODE.

Integrality of iterates of rational functions

Sun, 09/03/2017 - 07:28

Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of rational numbers, except in the rare cases where some iterate is a pole of $f$.

In the special case that $f(x) = \frac{a}{x^m}$ it happens that, when $n \ne 0$ is divisible by $a$ (also a nonzero integer), the iterates of $n$ under $f$ are integers infinitely often and non-integers infinitely often.

Are there any other examples where $n, f(n), f(f(n)), f(f(f(n)), ...$ contains both infinitely many integers and infinitely many non-integers?

Why are simplicial objects monadic over split (contractible) simplicial objects?

Sat, 09/02/2017 - 18:33

Given an augmented simplicial object $d_\bullet:X_\bullet \to \Delta X_{-1}$, suppose there's a simplicial map $s_\bullet :\Delta X_{-1}\to X_\bullet$ making $d_\bullet$ a deformation retract, i.e such that $d_\bullet$ is both a retract and a homotopy-section of $s_\bullet$. This is equivalent to providing the augmented simplicial object with an "extra degeneracy" (just an alternative description of the simplicial homotopy axioms in this case). Such an extra degeneracy of an augmented simplicial object will be called a splitting.

Let $\mathrm{S}$-$s\mathsf C$ denote the category of split simplicial objects (with fixed splitting) and simplicial arrows between them respecting the simplicial homotopies. This category admits a forgetful functor to simplicial objects $s\mathsf C$.

On page 20 of Duskin's Simplicial Methods and the Interpretation of Triple Cohomology (AMS page), the author remarks this forgetful functor is a left adjoint to a shifting functor defined by deleting the top face map, viewing the top degeneracy as an extra one, and shifting to a lower index. (The simplicial homotopy is defined by $h_i=s_0^{n-i}s_{n+1}d_0^{n-i}$.)

Moreover, the author writes this shifting functor $s\mathsf C\to \mathrm{S}$-$s\mathsf C$ is monadic. In other words, simplicial objects are monadic over split simplicial objects.

  1. What's the intuition behind the fact the shifting functor actually takes values in split simplicial objects? This seems strange to me - as if saying a simplicial object becomes contractible if you forget the top face map. How can that be?

  2. What's the intuition behind monadicity? An algebra over a split simplicial object (which is already a structured simplicial object) is an arrow to it, so how can additional structure on an already structured object yield back the original notion of object and arrow?

Added. I am looking for naive geometric intuition for these facts, namely: 1. that a simplicial object becomes contractible upon merely forgetting a face map and reindexing; 2. simplicial objects are monadic over split ones. Ideally, I would like an example of what the contraction deformation retract actually does to the Décalage of a non-contractible simplicial complex, say the boundary of a tetrahedron.

Asymptotic expansion of Riemann's prime counting function

Sat, 09/02/2017 - 16:55

Riemann's prime counting function is given as

$$J(n)=\sum_{k=1}^{\infty}\frac{\mu(k)}{k}\operatorname{li}(n^{1/k})$$

the approximations

\begin{align} \operatorname{li}(n)\sim J(n)\tag{1}\\ \operatorname{li}(n)-\sqrt{n}/\log n\sim J(n)\tag{2}\\ (1-\sqrt{n}/\log n)\operatorname{li}(n+\sqrt{n}\log n)\sim J(n)\tag{3}\\ \end{align}

get increasingly closer. Could these be the first few terms of an asymptotic expansion of $J(n),$ or can $(3)$ not be taken any further?

Almost Complex manifolds of constant curvature

Sat, 09/02/2017 - 07:12

Edited (after R. Bryant comment)

Let $(M,\cal J,g)$ be a almost Hermitian manifold (not necessary integrable). i.e., ${\cal J}^2=-I$ and $g({\cal J} X,{\cal J} Y)=g(X,Y)$. Suppose that $\{X_i,{\cal J}X_i\}$ be any local orthonormal ${\cal J}$-frame and the following relation hold for $i\neq j$ $$g(Q{\cal J}X_i,{\cal J}X_i)=g(QX_j,X_j),\quad K(X_i,X_j)=K(X_i,{\cal J}X_i);$$ where $Q$ and $K$ are Ricci operator and sectional curvature respectively. Then

Can be deduce that $(M,\cal J,g)$ is of constant curvature?

Your advice or suggestions will be much appreciated and welcomed.

Can this inequality be proved?

Sat, 09/02/2017 - 06:51

It appears that for each integer $k\geq2$, there is always one integer $c$ that satisfies the inequalities below. Can this be proved?

$$\frac{3^k-2^k}{2^k-1}<c\leq \frac{3^k-1}{2^k}.$$

Note that for $k\geq2$ the lower bound is always a proper fraction and will never match an integer.

Edit Simultaneous 3-variable equation with: $c>0, 1\leq s<2^k-1, 0\leq t<2^k,$

$$\frac{s}{2^k-1}+\frac{3^k-2^k}{2^k-1}=c=\frac{3^k-1}{2^k}-\frac{t}{2^k}.$$

The exact bounds are designed to show that $$\left\lfloor\left(\frac{3^k-1}{2^k-1}\right)\right\rfloor=\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor.$$

the intersection of maximal ideal

Thu, 08/31/2017 - 20:39

Let R be a ring.both we know the well-known Chinese reminder theorem.and we also know that it is not true for arbitrary index set.

  • My question :if the Krull dimensin of R is 0(That is:every prime ideal is maximal),let $m\in Spec R$ and $\alpha =\cap_{p\in Spec R-m}p$.Can we get $m+\alpha=R$?
  • in some sense,if the Krull dimension of R is 0 and the radical of R is 0,can we get the insection of arbitrary proper subset of $Spec R$ is not 0?

I feel it's right.

Hypersurfaces whose equation is not known

Thu, 08/31/2017 - 20:27

I would like to find some well-known/interesting hypersurfaces which arise as parametrizations where implicitization is computationally too difficult.

I have software which computes the Newton polytope of such hypersurfaces and would like to use it on an interesting example!

What are some great parametrized hypersurfaces that would be of interest? I am currently working on the Luroth invariant, but would love more examples.

For what it's worth, the Luroth invariant is degree 54 in 15 variables and it is proving to be just within reach of the software. Something around this size or slightly smaller would be perfect for me.

What's with equivariant homotopy theory over a compact Lie group?

Thu, 08/31/2017 - 20:12

For some reason -- I'm not quite sure why -- I've developed the impression that I'm supposed to "tiptoe" around equivariant homotopy theory over groups that are not finite. Should I?

Let me explain. Please correct me / tell me what I'm missing.

  1. Elmendorff's theorem holds for an arbitrary compact group. So I'm pretty sure at least that there's no disagreement over what the $G$-equivariant categories -- stable and unstable -- should be for an arbitary compact group. Please correct me if I'm wrong here.

  2. I sometimes get the impression that there are real problems in setting up $G$-equivariant homotopy theory when $G$ is not finite -- for example, all the foundational $\infty$-categorical work of late seems to be done in the finite or profinite case. But I don't know what these problems could possibly be! By (1), it's perfectly clear what the orbit category should be. When stabilizing, the Haar measure should allow the kind of "sums" required to construct transfers. Where's the problem? If it's possible to do motivic equivariant homotopy theory over non-finite algebraic groups, then surely it's possible to work over compact Lie groups!

  3. It's true that non-discrete Lie groups are inconvenient to model as simplicial groups. Is this really a stumbling block?

  4. Perhaps people just focus on the groups they actually intend to work with. I don't know many existing or potential applications of equivariant homotopy theory over infinite groups. This is probably just my ignorance, since for that matter I don't know many existing or potential applications even over finite groups. For example in this overflow question I see Kervaire invariant one, the Segal conjecture, Galois descent for $\mathbb{C}$ over $\mathbb{R}$, and the study of spaces with $G$-action. These are all for $G$ finite, except maybe the last one, but it's also the vaguest. The other application that comes to mind is cyclotomic spectra and THH, where $G = S^1$ (but we use a universe with only the proper closed subgroups), which brings me to another confusion:

  5. THH should admit a genuine $S^1$-equivariant structure, but people tend to just use the cyclotomic part, using only the proper closed subgroups (which are all finite). Is this because (a) we'd like to use the $S^1$ part, but there's no model of it that's reasonable to work with, (b) we wouldn't have any use for the $S^1$ part even if we could get our hands on it (seems unlikely -- shouldn't this correspond to information at the infinite place?), or (c) even if we could get our hands on the $S^1$ part, thinking about it in equivariant terms would be the "wrong" approach, or require additional corrections, or (d) some other reason?

shouldnt this expresions mean the same? 1mer square = to [on hold]

Thu, 08/31/2017 - 18:30

-"1 meter square" phonetic shouldnt expresion mean the same than "one houndred centimetrees"

In other words 1m2 = to said 100cm2

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