Recent Questions - MathOverflow
The following are the titles of recent articles syndicated from Recent Questions - MathOverflow
Add this feed to your friends list for news aggregation, or view this feed's syndication information.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.

[ << Previous 20 ]
Sunday, December 8th, 2019
LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
10:15 am
An objective cost function with an equality constraint

I was looking to minimize the following objective function with respect to $x_i$:

$$ \min \space \sum_{i=1}^N h_i \cdot P_i \cdot s_i \cdot \left( 4.85 - 0.3924 \left( \frac{Q_i }{s_i}\right)^{1.3} (1 - x_i)^{1.3} - 5.359\left(\frac{Q_i}{s_i}\right)^{0.135} (1-x_i)^{0.135} \right) $$ where $Q$, $s$, $P$ and $h$ are constants per $i$ and strictly positive.

Given an equality constraint:

$$B = \frac{\displaystyle\sum_{i=1}^N D_i \cdot x_i}{\displaystyle\sum_{i=1}^N D_i}$$ where $0 \le B \le 1$ and $0 \le x_i \le 1$ and $D$ is a constant per $i$ and strictly positive. $B$ is chosen upfront to be a value (e.g. $0.95$).

Let's say, the derivative of the objective function with respect to $x_i$ is called $a_i$. I replace $x_i$ in the derivative with the value of $B$ for my approximation.

Then I tried a simple approximation in the form of:

$$ x_i = 1-(1-B)\cdot\left(\frac{a_i \times \sum_{i=1}^n D_i}{\sum_{i=1}^n D_i \times a_i}\right)$$

I tested this extensively and it seems a very good approximation. My only questions are:

  1. Why does this deliver such a good approximation?

  2. Can we find a better approximation in the same form?

You can see the problem as N-items that need a number between 0 and 1 ($x_i$) and this incurs a cost: this is the objective function and needs to be minimized over the total group. But in the end, multiplying each $x_i$ with $D_i$ has to be equal to $B$.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
10:15 am
Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the subobject represents the possible values of the variable for which the statement is true. By associating logical operators with operations on subobjects, we obtain an internal logic with which to reason about the relationships between subobjects of all objects in the topos.

The internal logic of monoidal categories (optionally symmetric, closed, *-autonomous, etc), as usually described, is different. Now, a proposition of linear logic is associated with an object in the monoidal category, rather than a subobject. The monoidal category is therefore analogous to the poset of subobjects of a single object in a topos, rather than to the whole original topos. It is possible to extend this framework (eg Seely p. 10) using a base category $\bf S$ corresponding to free variable types, and an indexed category containing a monoidal category for each element of $\bf S$. However, this base category is not itself monoidal.

Is there a type of internal logic that lets us reason about the relationships of subobjects of any object in a monoidal base category?

An example application would be in quantum mechanics. A statement about a quantum system with Hilbert space ${\bf H}_A$ corresponds to a subspace $S_A \subseteq {\bf H}_A$. If we a have second system with Hilbert space ${\bf H}_B$ side by side with our first system, then a statement about both of them corresponds to a subspace of ${\bf H}_A \otimes {\bf H}_B$. Given subspaces $S_A \subseteq {\bf H}_A$ and $S_B \subseteq {\bf H}_B$, we can define a subspace $S_A \otimes S_B \subseteq {\bf H}_A \otimes {\bf H}_B$ as the span of $\{ a\otimes b | a \in S_A, b \in S_B\}$. Alternatively, if a single quantum system could be either in a state of type A or type B, then its Hilbert space is ${\bf H}_A \oplus{\bf H}_B$. Again, given two subspaces $S_A \subseteq {\bf H}_A$ and $S_B \subseteq {\bf H}_B$, we can define a subspace $S_A \oplus S_B \subseteq {\bf H}_A \oplus {\bf H}_B$.

I guess a way to ask the question is this. Given a monoidal category $M$ (optionally symmetric, closed, *-autonomous, etc), is there a monoidal functor that maps $M$ to a monoidal category representing the subobjects of all objects of $M$? Thus, given two objects $A$ and $B$ of $M$, and subobjects $S_A \in Sub(A)$, $S_B \in Sub(B)$, we could obtain $S_A \otimes S_B \in Sub(A \otimes B)$, $S_A \oplus S_B \in Sub(A \oplus B)$, $S_A^\bot \in Sub(A^*)$, $S_A \rightarrow S_B \in Sub(A \rightarrow B)$, etc. We could then use the linear logic of the second category to reason about subobjects of all objects in $M$.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
9:45 am
What does a partial map classifier look like as a sheaf?

[Cross-posted from M.SE, where it didn't get an answer]

In constructive logic, it's possible for a set $X$ to satisfy $$\forall x,y \in X.\, x = y$$ while being non-trivial. Such a set is called a subsingleton. Now consider the set of all subsingletons over a set $S$, denoted $S_\perp$. The question is, what are its sections in a sheaf topos? Or rather, how do the sections of $S$ relate to the sections of $S_\perp$?

I have a guess: The sections of an open subset $U$ of $S_\perp$ are pairs $(V, f)$ where $V$ is an open subset of $U$ and $f$ is a section of $V$ in the sheaf $S$.

I guess one could compute what the sections are by applying Kripke-Joyal semantics to the expression $\{X \in \mathcal P(S) \mid \forall x,y \in X.\, x=y\}$. I'm trying to figure out how to do that from Page 22 of this: https://rawgit.com/iblech/internal-methods/master/notes.pdf

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
9:45 am
Majority-driven manipulations of integer vectors

Motivation. Recently I was watching two people play a game that involved arranging sticks in a number of heaps and moving them around in certain allowed ways that I think I was able to infer from observation. (They told me the name of the game in their language, but I can't remember it.)

Problem. Let $\mathbb{N}_0$ denote the set of non-negative integers and let $n\geq 4$ be an integer. For $a,b\in\mathbb{N}_0^n$ say that $b$ arises from a majority move from $a$ if

  1. $\sum a_i =\sum b_i$, and
  2. $|\{i: b_i > a_i\}| > |\{i: b_i < a_i\}|$. (This second condition motivates the "majority" part of nomenclature).

Let $\bf{1}_n$ be the member of $\mathbb{N}_0^n$ in which all entries are $1$. Let $S_n$ be the set of members of $\mathbb{N}_0^n$ that arise from some sequence of majority moves from ${\bf 1}_n$. Let $M_n$ be the maximal value of any entry of any member of $S_n$.

What is the value of $M_n$? Do we have $\lim M_n/n = 1$? (Correct answers to either question will be accepted.)

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
8:33 am
Can the category of linear orders be fully embedded in the category of partial orders?

Q(1): Can the category of partial orders be fully embedded in the category of linear orders?


Vopěnka's principle, or VP, is a very intriguing axiom with many equivalent forms and consequences spanning universal logic (in the Barwise sense), large cardinals, model theory, and category theory.

VP for $C$ for a (large locally presentable) category $C$ means that there is no large discrete full subcategory of $C$. It is widely known that VP for $C$ for most "sufficiently expressive" categories $C$ is simply equivalent to all of VP (see Joel David Hamkin's answer to this question).

The notion of "sufficiently expressive" can be formalized as follows: if $A$ fully embeds into $B$ and VP for $B$ holds, then any large discrete full subcategory of $A$ necessarily produces large discrete full subcategory of $B$. So, perhaps a suitable notion of "sufficiently expressive" is simply a full embedding. For example, the categories of structures of a given language can be fully embedded in the category of graphs, showing that, for example, VP for graphs is the same as VP for structures.


This leads me to the question above; an affirmative answer (which I believe is impossible) would also definitively show that VP for linear orders is indeed equivalent to VP itself. However, I have strong doubts about this. Perhaps it needs to be weakened?

Q(2): Is the category of linear orders a reflective subcategory of the category of partial orders?

It isn't immediately obvious that this even relates to VP anymore. However, as shown in Adamek-Rosicky's Locally Presentable and Accessible Categories the following is equivalent to WVP (Weak Vopenka's principle, weakened form):

For $C$ a locally presentable category, every full subcategory $D↪C$ which is closed under limits is a reflective subcategory.

Now it becomes clear! The category of partial orders is certainly locally presentable, and the category of linear orders is closed under limits. Under the Weak Vopenka's principle therefore, the answer to Q(2) is affirmative! But is using such a powerful axiom unnecessary? The original hypothesis also implies Q(2); if strong large cardinal axioms are indeed necessary for Q(2), that doesn't bode very well for our friend Q(1).

This leads me to Q(3), where we investigate the consistency strengths of Q(1) and Q(2):

Q(3): Is it consistent that Q(2) fails? If so, what is the consistency strength of Q(2)? What is the consistency strength of Q(1)?

And finally, of course, the last question that motivated this whole thing in the first place:

Q(4): Is VP for linear orders equivalent to VP?

Adámek, Jiří; Rosický, Jiří, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series. 189. Cambridge: Cambridge University Press. xiv, 316 p. (1994). ZBL0795.18007.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
6:36 am
Examples of proofs using induction or recursion on a big recursive ordinal

There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?

The original proof of Ramsey theorem and Hales-Jewett theorem use induction on $\omega^2$, but the using is not essential, because Erdos and Shelah have given better bounds by using induction just on $\omega$. And further more $\omega^2$ shouldn't be considered big.

A typical use of big ordinal induction is proving the consistence of axiom systems, for example, using $\varepsilon_0$-induction to prove the consistence of PA. This is one kind of examples.

The existence of Goodstein function uses the induction on $\varepsilon_0$, and I think it's just a directly explaining of how do recursion on ordinal works.

Are there more examples?

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
6:05 am
Has anyone researched this variant of separable rational connectedness?

If I am correct, then one of the definitions of rational chain connectedness is that a variety $ X $ is rationally chain connected if

1) there are schemes $ \mathcal{C} $ and $ T $ with a flat, proper map $ \pi: \mathcal{C} \to T $ such that all fibres $ \pi $ of closed points of $ T $ are connected rational curves,

2) there is a map $ f: \mathcal{C} \to X $ such that $ (f,f): \mathcal{C} \times \mathcal{C} \to X \times X $ is dominant.

Rational chain connectedness of a smooth variety in characteristic zero implies rational connectedness. Generally when it comes to concepts like uniruledness, rational connectedness, and unirationality, in positive characteristic separability is the main obstruction to any of the theorems one might like to see hold working. If one adds the appropriate separability condition then such theorems generally work, e.g., a separably uniruled variety $ X $ has the property that the plurigenera vanish, while a uniruled variety does not necessarily have this property.

Has anyone thought of changing the above definition by adding the requirement that the map $ f $ is Etale and that the map $ \pi $ be smooth of relative dimension $ \dim(\mathcal{C}) -\dim(T) $? If a variety satisfying this condition is ``separably rationally chain connected'' (assuming that this term does not denote something else), then has anyone looked into trying to prove that a smooth variety is separably rationally chain connected if and only if it is separably rationally connected? Thank you.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
6:05 am
Completed tensor product is exact

In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on the category of Banach spaces...". He gave no clue how to prove it, but it is known that the same fact is not true for Archimedean Banach spaces. Is the statement correct, and how can it be proved?

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:34 am
Does anyone know a reference in the literature regrading a proof that every projective hypersurface with vanishing canonical divisor is uniruled

I saw a result in notes on by Olivier Debarre (Rational Curves on Hypersurfaces, Lecture notes for the II Latin American School of Algebraic Geometry and Applications 1-12 of June 2015 in Cabo Frio, Brazil) that if $ Z $ is a hypersurface in $ \mathbb{P}^{n}_{\mathbb{C}} $, of degree less than or equal to $ n $, then $ Z $ is uniruled, even if $ Z $ is not smooth. Does anyone know a reference for this fact. All of the books I have looked in use smoothness. If anyone knows a reference, I would greatly appreciate it. Thank you.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:34 am
Motivation for theta correspondence from classical invariant theory?

In Howe's "Remarks on classical invariant theory", he writes in the introduction

Perhaps not least important, the results on duality make precise in a strong way the striking analogy, remarked on by a number of authors, between the Clifford algebra and the spin representation on the one hand and the Weyl algebra and the oscillator or (Segal-Shale) Weil or metaplectic or... representation on the other (or as physicists would say, the canonical anticommutation relations and canonical commutation relations).

And then he generalizes the Schur-Weyl duality in a general setting, gives reformulation of many theorems in classical invariant theory, and gives many examples in section 4. But in this paper, he doesn't mention the word "theta correspondence". And there is another paper of him named “transcending classical invariant theory”, in the introduction he remarks after Theorem IA.

(b) If G or G' is compact, then Theorems 1 and lA are already known and have been treated by several authors [G, GK, HI, KY, Sa). In fact, they then essentially amount to a version of Classical Invariant Theory [HI, H2, WH). This special case will be a stepping stone to the general result.

But it's still unclear for me how he ends up theta correspondence (in the modern form for general reductive pairs) from classical invariant story? Can anyone tell more about the story and the motivations? Thanks for any illustrating examples.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:34 am
Does anyone know of a uniruled hypersurface over $ \mathbb{C} $, which is not rationally connected?

Does anyone know of any projective hypersurfaces which are uniruled, but not rationally connected. Note that I am not requiring that the hypersurface be smooth or even normal.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:34 am
How to use van der Corput's lemma to get the following estimates?

Recently, I'am reading Tao's article Spherically Averaged Endpoint Strichartz Estimates for the Two-Dimensional Schrodinger equation, in which there are some problems that I can't solve by myself.

We define Bessel function by

\begin{equation} J_n(\lambda)=\frac{1}{2 \pi} \int_{0}^{2\pi} e^{i\lambda \cos\theta} e^{in\theta} d\theta \end{equation}

and decompose $J_n$ smoothly by \begin{equation} J_n(\lambda)=m_0(\lambda)+m_1(\lambda)+\sum_{2^j \gg n} m_j(\lambda), \end{equation}

where $m_0,m_1,m_j$ are supported on $|r| \ll n , |r|\sim n$ and $|r| \sim 2^j \gg n$ respectively.

As for $m_1$ and its derivative, Tao says using van der Corput's lemma, we can get the following estimates:

  • \begin{equation}|m_1(\lambda)|\lesssim n^{-1/3}(1+n^{-1/3}|\lambda-n|)^{-1/4},\end{equation}
  • \begin{equation}|m_1'(\lambda)| \lesssim n^{-1/2}. \end{equation}

Can someone present details of the argument to the two estimates above?

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:05 am
Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod

We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for finitely generated modules.

Now consider the bounded derived category of $\mathbb C[x]/x^2$-modules, how to classify indecomposable objects with finitely generated cohomology in this category? Examples include $\mathbb C[x]/x^2 \overset{x}{\rightarrow} \mathbb C[x]/x^2 \overset{x}{\rightarrow}... \overset{x}{\rightarrow} \mathbb C[x]/x^2 $ and $\mathbb C$.

Note there exists non-formal object.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:05 am
On the Dirichlet series for $1/\zeta(s)$ for real $s$ and the zeros of zeta

For $\Re(s)>1$, it is well known that $$\frac{1}{\zeta(s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$$ where $\mu$ denotes the Mobius function and $\zeta$ is the Riemann zeta function. I have heard that if the series on the right-hand side has an analytic continuation REAL $s_{0} \in (1/2, 1)$, then $\zeta(s)\neq 0$ for every $s$ with$\Re(s)=s_0$. But since all nontrivial zeros of the zeta function are complex, why does the analytic continuation of the said series to real values of $s$ have anything to do with the complex zeros ?

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
5:05 am
Rate of convergence of the prime zeta function P(2)

For an application in statistical group theory, we need explicit upper and lower bounds that an expert in number theory (I am not one) may know how to prove.

Question 1: What are "good" bounds $f_1(x)<\displaystyle\sum_{p>x}\frac{1}{p^2}<f_2(x)$ where $p>x$ is prime?

For our application, sharper bounds than the following $f_1(x),f_2(x)$ are desirable: $$ \frac{1}{12\left(\frac{x}{\log(x)-4}+1\right)^4}<\sum_{p>x}\frac{1}{p^2}<\frac{1}{x-1}.$$ The lower bound can hold for $x>x_1$ and the upper bound for $x>x_2$ provided $x_1,x_2$ are "small".

Question 2: Is there a function $f(x)$ and explicit positive constants $c_1,c_2$ such that $$c_1f(x)<\sum_{p>x}\frac{1}{p^2}<c_2f(x)?$$

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
4:37 am
Nontrivial Compactly Supported Harmonic Functions [closed]

Prove that there is no non-trivial compactly supported harmonic function

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
4:37 am
How to choose a Electric Dirt Bikes for Kids?

If you really want your child to have some fun time with himself, you should be looking for the electric dirt bikes for teenager and pick the best one. Here We provide information about all type of gear information, and review. We try to provide information about all type of gear like Dirt Bike, ATVs, Three Wheeler, Four Wheller, Motorcross, Motorcycle, etc for our visitor.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:45 am
line bundles and jacobians

Let $Y$ be a smooth projective complex curve of genus 2 and $f : X \to Y$ a finite etale cover. Choose two distinct points A and B on $Y$ and let $\Sigma\subset X$ be the set of complex points $P$ such that $f(P)$ is A or B. For any $x\in \Sigma$ choose an element $L(x) \in Pic(Y)[2]$. Does there exist an element $L\in Pic(X\times Y)[2]$ such that, for every $x \in \Sigma$, the restriction of $L$ to $x\times Y = Y$ is equal to $L(x)$?

Angelo says that the restriction of $L$ to $x\times Y = Y$ is constant. Is it so and why?

OK. Thank you for the answer.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:45 am
Multiple tangents on a plane curve corresponds to ordinary $r$-fold points on the dual curve

Let $X$ be a (smooth, irreducible) curve of degree $d$ in $\mathbb{P}^2_k$ where $k$ is an algebraically closed field of characteristic $0.$ We say that a line of $\mathbb{P}^2_k$ is a multiple tangent of $X$ if it is tangent to $X$ at more than one point. If $L$ is a multiple tangent of $X,$ tangent to $X$ at the points $P_1, \ldots, P_r$ and if none of the $P_i$ is an inflection point, show that the corresponding point of the dual curve $X^*$ is an ordinary $r$-fold point, meaning that it is a point of multiplicity $r$ with distinct tangent directions.

This is a question from Hartshorne, and while I believe I know how to prove it (with different methods), I don't think is the kind of solution the exercise warrants. Let me briefly outline the methods I know of how to prove this:

  1. Assume that we know that the Gauss map $X \rightarrow X^*$ is birational. Then it follows that $X$ is the normalization of $X^*$ and we can identify the Gauss map with successive blow-ups at singularities of $X^*.$ Then one checks that essentially by definition that the point $L$ is an ordinary $r$-fold point. To know that the Gauss map is birational, one can argue by the biduality theorem for plane curves in characteristic $0.$
  2. Use the Lefschetz principle to reduce to when $X$ is a curve over $\mathbb{C}$ and argue using the analytic topology / by geometry.
  3. Pass to completions and try to define the Gauss map locally.

All of these methods seems to me to be somewhat against the spirit of the material that Hartshorne has introduced so far in the book. The first of the above items uses the biduality theorem, which, while not extremely hard to prove, still is quite a lot for this The second is not algebraic at all the last seems somewhat of a stretch.

I would be very grateful for a purely algebraic solution of this exercise. Let me explain what I mean by a purely algebraic solution. In some sense, it should only use elementary machinery from scheme theory and not go to the analytic category. If completions are neccessary, they can be used but I would prefer something not using that. Further, I would (if possible) want it to be a proof that would hold true in characteristic $p>0$ as well.

This question has previously been posted to math.stackexchange, but the answers I received was not what I was looking for so I thought I would post it here. I apologize if this is not the right forum for this question, but I am really curious.

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.
3:45 am
divisorial ideals

Let $I$ be an ideal of a domain. Then is there an ideal $J$ properly located between $I$ and $I^{\nu}$? Here $I^{\nu}$ is divisor of $I$.

[ << Previous 20 ]

LJ.Rossia.org makes no claim to the content supplied through this journal account. Articles are retrieved via a public feed supplied by the site for this purpose.