Lower bound for sum of binomial coefficients? Hi! I'm new here. It would be awesome if someone knows a good answer.
Is there a good lower bound for the tail of sums of binomial coefficients? I'm particularly interested in the simplest case $\sum_{i=0}^k {n \choose i}$. It would be extra good if the bound is general enough to apply to $\sum_{i=0}^k {n \choose i}(1-\epsilon)^{n-i}\epsilon^i$.
For the more commonly used upper bound, variants of Chernoff, Hoeffding, or the more general Bernstein inequalities are used. But for the lower bound, what can we do?
One could use Stirling to compute $n!$ and then ${n \choose k}$ and then take the sum:
${n \choose k} = \frac{n!}{k!}{(n-k)!}$, and Stirling's formula (a version due to Robbins) gives
$$n! = \sqrt{2\pi}n^{-1/2}e^{n-r(n)}$$
 with remainder $r(n)$ satisfying $\frac{1}{12n} \leq r(n) \leq \frac{1}{12n+1}$.
For the next step, it's easy to apply Stirling thrice. But, even better, I noticed that Stanica 2001 has a slight improvement to the lower bound that also is simpler to state (but more difficult to prove):
$${n \choose k} \geq \frac{1}{\sqrt{2\pi}}2^{nH(k/n)}n^{1/2}k^{-1/2}(n-k)^{-1/2}e^{-\frac{1}{8n}}$$
for $H(\delta) = -\delta \log \delta -(1-\delta)\log(1-\delta)$ being the entropy of a coin of probability $\delta$.
Now for step 3. If $k$ is small, it's reasonable to approximate the sum by its largest term,
which should be the ${n \choose k}$ term unless $\epsilon$ is even smaller than $k/n$. So that's great, we're done!
But wait. This bound is off by a factor of at most $\sqrt{n}$. It would be better to
be off by at most $1 + O(n^{-1})$, like we could get if we have the appropriate Taylor series. Is there a nice way to do the sum? Should I compute
$\int_{0}^{k/n} 2^{nH(x)}\frac{1}{\sqrt{2\pi}}x^{-1/2}(1-x)^{-1/2}n^{1/2}e^{-1/8n} dx$
and compare that to the discrete sum, and try to bound the difference? (This technique has worked for Stirling-type bounds.) (The terms not dependent on $k$ or $x$ can be moved out of the integral.)
Another approach would be to start from Chernoff rather than Stirling (i.e. "How tight is Chernoff guaranteed to be, as a function of n and k/n?")
Any ideas or references? Thanks!


when is an eigenvalue differentiable with respect to a parameter? Let say we have a symmetric matrix $A(\omega)$ depending smoothly on some variables $\omega \in \Omega$ with $\Omega \subset \mathbb{R}^d$ a $d$-dimensional parameterspace (this means the eigenvalues are real). For calculating the eigenvalues we can make use of the characteristic polynomial $\rho(A( \omega))$ and by searching for the roots of this polynomial, we find the eigenvalues
I am wondering if these eigenvalues are always differentiable with respect to the variables $\omega$. If the eigenvalue has multiplicity > 1, then we can’t be sure. For example, if we take the matrix
$$A(\omega) = \begin{bmatrix} \omega_1+1 & \omega_2 \\ \omega_2 & -\omega_1+1 \end{bmatrix}$$
Then the characteristic polynomial $\rho(A(\omega)) = (\lambda-1)^2 -  \omega_1^2 -  \omega_2^2$. If we set this equal to 0, we get that the eigenvalues fulfil $ (\lambda-1)^2 = \omega_1^2 +  \omega_2^2$ so
$\lambda_1(\omega)= \sqrt{ \omega_1^2 + \omega_2^2} + 1$ and $\lambda_2(\omega) = -\sqrt{ \omega_1^2 + \omega_2^2} + 1$.
This is not differentiable in $\omega_1 = \omega_2 = 0$ for both derivatives $\dfrac{\partial \lambda_i(\omega)}{\partial \omega_j} , i,j = 1,2$ and has there the same eigenvalues $\lambda_1 = \lambda_2 = 1$.  In the case of simple eigenvalues (= multiplicity = 1), do we always have that the eigenvalues are differentiable?


$2$-norm of idempotent matrix Suppose $n > 1$ is an integer. Let $P \in \mathbb C^{n \times n}$ be a matrix such that $P^2=P$ and $1\leqslant {\rm rank}(P)<n$. Prove that $\Vert P \Vert_2 = \Vert I - P \Vert_2$.

I have been working on the problem for hours. Please let me know if any can help. Thanks!


Computing fundamental groups and singular cohomology of projective varieties Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations defining the variety?
I seem to recall that, if the variety is smooth, we can compute the H^{p,q}'s by computer -- and thus the H^n's by Hodge decomposition -- is this correct? However this won't work if the variety is not smooth -- are there any techniques that work even for non-smooth things?
Also I seem to recall some argument that, at least if we restrict our attention to smooth things only, all varieties defined by polynomials of the same degrees will be homotopy equivalent. The homotopy should be gotten by slowly changing the coefficients of the polynomials. Is something like this true? Does some kind of argument like this work?


What is $K_1(\mathrm{Var}_\Bbbk)$? Ok, this is a very naive question, and not seriously motivated. But I was just wondering: did anybody define any (interesting) higher K-theory Grothendieck group of varieties $K_n(\mathrm{Var}_{\Bbbk})$ generalizing the usual Grothendieck ring $K_0(\mathrm{Var}_\Bbbk)$?


Conformally flat manifold with zero scalar I would like to ask the following : Is there any example of a compact conformally flat Riemannian manifold $(M^n,g)$ with $n\geq 4$ which is not flat and has zero scalar curvature?


Largest family of subsets ordered by strict inclusion Given a set S, with cardinality $\kappa$, let F be a family of subsets of S, such that if A and B are any two members of F then either A $\subset$ B or B $\subset$ A.
Question: Can you always find F, such that card(F) = $2^\kappa$ ?
Motivation: I recently proved this is true if S = $\mathbb{N}$.
However, my proof is dependent on very specific properties of $\mathbb{R}$ and $\mathbb{Q}$, so I was wondering if the result could be generalized.
Proof: Let f be a bijection from $\mathbb{Q} \rightarrow \mathbb{N}$ be a bijection.
For every real number r, define Q(r) to be the set of rationals < r.
Now define corresponding subsets of $\mathbb{N}$, N(r) = f(Q(r)).
Clearly, the F = {N(r)} satisfies the criterion and card(F) = $2^{\aleph_0}$
QED


Intuition behind the diagonal lemma while proving Tarski's theorem about truth Let $F$ be a first order logic theory with a set of axioms that are primitively recursive and are supposed to capture arithmetic (in particular, we can define a Gödel number. Let $\mathbb N$ be the standard model for the natural numbers. Tarski proves that it is impossible to find a statement with one free variable $T(x)$ in the language such that for every natural number $n$, $T(n)$ holds if and only if $n$ is the Gödel number of a true statement (in $\mathbb N$).
One possible proof goes through a "diagonal lemma" like so:
(The following is edited in response to the (valid) criticism below.)
Diagonal Lemma: For any statement with one free variable $A(x)$, there exists a term $t$ such that the Gödel number of the sentence $A(t)$ is $n$ and $t$ evaluates to $n$ in the standard model.
Proof: As part of the Gödel numbering, there is a function $sub(x,y)$ that does the following: If $m$ is the Gödel number of a statemnt $\phi(x)$, then $sub(m,n)$ is term that in the standard model corresponds to the Gödel number of the statement $\phi(n)$.
Then, consider the formula $C(x) = A(sub(x,x))$. Let the Gödel number of this be $c$ and consider $B = A(sub(\overline c,\overline c))$ where $\overline c$ is a term that evaluates to $c$. Then, I claim that the Gödel number of $B$ is equal to $sub(c,c)$. Indeed, $sub(\overline c,\overline c)$ is by definition a term corresponding to the the Gödel number of $C(\overline c) = A(sub(\overline c,\overline c)) = B$. This completes the proof.
Question: The above proof is of course extremely simple but also to me, extremely mysterious. Why would anyone want to consider $C(x)$ in the first place?
My first attempt was to treat the Diagonal Lemma as saying that the function $n$ to Gödel number of $A(n)$ has a fixed point in $\mathbb N$. The naive way to proceed might be try some sort of iteration of this function or some modification of this. Can the diagonal lemma be seen as a sophisticated way of making this naive idea work?
Question 2: This is really a bonus question, feel free to ignore everything after this point. I find that I can prove Tarski's theorem directly in the following way:
Let us enumerate in a computable way all the statements with one free variable by $A_n(x)$. Suppose there was a statement $T(x)$ representing truth in the language. Then we can define $B(n) = \neg T(A_n(n))$. That is, $B(n)$ "says" that the statement $A_n(n)$ is false. However, since our enumeration was computable, $B(x)$ is itself a statement and occurs among the $A_n(x)$ at say $n = n_0$. However, then $B(n_0)$ effectively says "This statement is false" and immediately leads to a contradiction.
First: Is the above proof correct? Second: Can this approach be adapted to prove the diagonal lemma itself? Perhaps by enumerating all formulas with two variables or something clever along these lines?


A proof required for this identity Experiments support the below identity.

Question. Is this true? Combinatorial proof preferred if possible.
  $$\sum_{m=0}^n\binom{n-\frac13}m\binom{n+\frac13}{n-m}(1+6m-3n)^{2n+1}
=\left(\frac43\right)^n\frac{(3n+1)!}{n!}.$$

In View of MTyson's suggestion (see below), a generalized question can be asked:

Question. Is this true? Combinatorial proof preferred if possible.
  $$\sum_{m=0}^n\binom{n-y}m\binom{n+y}{n-m}(y+2m-n)^{2n+1}
=y\prod_{k=1}^n4(k^2-y^2).$$


Picard group of a cusp $\DeclareMathOperator\Pic{Pic}$For a commutative ring with unity $R$, if $R_{\mathrm{red}}$ is semi-normal then we have an equivalent criterion which states that $\Pic(R) \cong \Pic(R[t])$. Here $\Pic(R)$ is the Picard group of $R$.
So naturally it is well understood that $\Pic(R) \ncong \Pic(R[s])$ when $R = k[t^2,t^3]$ a cusp, which is a domain that is not semi-normal.
With the help of conductor ideals I was able to show that $\Pic(R) \cong k$ but I am unable to explicitly compute $\Pic(R[s])$ will the same approach via conductor ideals work, for any suggestion or guidance I am thankful.


CW-structure induced by Morse function on Riemannian manifold I have heard a statement in the following direction. Given a compact Riemannian manifold $M$ with a Morse function on it possibly satisfying some extra assumptions. Then this data induces a CW-structure on $M$: to each critical point corresponds a cell - its unstable submanifold. The main difficulty is to construct a compactification of these cells.

I am looking for a reference to a precise statement of the result, possibly with some details of the construction.

At the moment I am less interested in a detailed proof.


When does Pontryagin duality generalize? Let $T$ be a locally compact abelian (LCA) group. For any other LCA group $G$, let
$\hom(G,T)$ be the set of continuous homomorphisms $G\to T$. With the compact-open
topology, $\hom(G,T)$ is certainly a topological group, but is not in general locally
compact, even if $T$ is compact. In any case there is an obvious homomorphism
$$
  \alpha_G : G \to \hom(\hom(G,T),T)
$$
sending $x\in G$ to the functional $\chi\mapsto \chi(x)$.
I have three questions:

For what LCA groups $T$ is $\hom(G,T)$ locally compact for all locally compact
      $G$?
For what groups $T$ is $\alpha_G$ always a (topological) isomorphism?
  More generally, is there any full subcategory $\mathcal L$ of the category of LCA
      groups for which $\hom(G,H)$ is in $\mathcal L$ whenever $G$ and $H$ are in
      $\mathcal L$?

(Edit: I'm most interested in 1 and 2. If there is some $\mathcal L$ satisfying 3
where the objects in $\mathcal L$ can be characterized by some interesting topological
criterion, that would be fantastic. My question is not whether there are "ad-hoc" ways of constructing $\mathcal L$.)


Kernel of modular representation of a finite group Let $G$ be a finite group; let $F$ be a field of characteristic $p > 0$.
If I have an irreducible modular representation $\rho: G \to GL_n(F)$, does $\ker  \rho$ contain all the normal $p$-subgroups of $G$? If so, how does one show this?


Why is the Hodge Conjecture so important? The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm able to understand what it says, and at first glance I do find it a very nice assertion, I cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in algebraic geometry. Which are its implications?
Going a bit further, what about the Tate conjecture?


Multivariate vandermonde matrix I am dealing with a generalized vandermonde matrix, and i wandered if there were simple results availiable on it's invertibility.
Fix a dimension $d$, and let $\mathbf x_1,...\mathbf x_n$ be points in $[0,1]^d$. Consider the vandermonde matrix
$$M = \left\{\mathbf x_i^{\mathbf p}\right\}_{\substack{\lvert \mathbf p \rvert \le m\\ i \in 1,...n}}.$$
Where $\mathbf x^{\mathbf p} = \prod\limits_{i=1}^d x_i^{p_i}$, the $p_i$ being integers.
Fix $n,m$ so that the matrix is square, that is $n = \sum\limits_{i=0}^m \binom{i+d-1}{d-1}$.
Q : Are there conditions so that this matrix is invertible ? Maybe a ref ?
Q2 : If I simulate $\mathbf x_1,...\mathbf x_n$ uniformely from $[0,1]^d$, indepandantly, will the matrix be (almost surely) invertible ?


Is the category of pure Hodge structures abelian semi-simple? Apologies if the question in the title is too elementary.
A reference would be helpful.


Exotic smooth structures on Lie groups? If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $G$, do we know if the underlying topological manifold supports any other exotic smooth structures (necessarily not a Lie group)?
Even a more specific example: Up to diffeomorphism, we have $SO(8)=SO(7)\times S^7$. If we replace the smooth structure on $S^7$ by an exotic one, do we get an exotic smooth structure on $SO(8)$?
Thank you!


Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions? Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense the complexity or power of a theory.
Does anyone know what is the proof theoretic ordinal of $ZFC$ or any non-trivial $ZFC$ extensions? Wikipedia says this is unknown for $ZFC$ as of 2008, but maybe there has been some recent progress? Thank you.


Homomorphisms from powers of Z to Z I believe it is known that if I is a set of non-measurable cardinality, then any homomorphism $Z^I\to Z$ factors through a finite power.  Here $Z$ is the group of integers.  Can anyone give a reference for this?


Major mathematical advances past age fifty From A Mathematician’s Apology, G. H. Hardy, 1940:
"I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. ... I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself."
Have matters improved for the elderly mathematician? Please answer with major discoveries made by mathematicians past 50.


Units in the (stable) center of a Frobenius algebra Let $A$ be a Frobenius algebra with center $Z(A)$ and $I\subset Z(A)$ the ideal of elements in the image of some $A$-bimodule map $A\rightarrow A\otimes A\rightarrow A$, where the second map is multiplication. The stable center is $\underline{Z}(A)=Z(A)/I$. Equivalently, $\underline{Z}(A)=\underline{Hom}_{A^{op}\otimes A}(A,A)$ in the stable category of $A$-bimodules while ${Z}(A)={Hom}_{A^{op}\otimes A}(A,A)$.
I wanted to show that the natural projection $Z(A)\twoheadrightarrow \underline{Z}(A)$ induces a surjection between groups of units $Z(A)^\times\twoheadrightarrow \underline{Z}(A)^\times$. I can show it when $A$ is indecomposable as a bimodule over itself since in that case, if $A$ is not separable then $I$ is nilpotent, and if $A$ is separable then the stable center is trivial.
Any clue about the general case?


Does cutting off the taylor expansion of e^x always give an irreducible polynomial? I am talking of the polynomials:
$P_n(x)$ = $1+x..+x^n/n!$
I've tested this for the first 10 values and it seems so. I know this might be random but I've got a hunch that there's something deeper here.
For prime $n$ this follows from Eisenstein's principle, but I don't have much progress on other values.


Concrete formula for Shapiro's Lemma I wonder if there is a concrete formula to express the isomorphism in the well known Shapiro's Lemma that $H^i(G, \text{CoInd}_{H}^{G}(M)) \simeq H^i(H, M)$, where $H \subset G$ is a subgroup of $G$, $M$ is a $\mathbb{Z}[H]$-module, and $\text{CoInd}_{H}^{G}(M)$ is the co-induced $\mathbb{Z}[G]$-module. Here by 'concrete' I mean, for instance, given an $i$-cocycle $\phi \in Z^i(H, M)$ considered as a map $\phi: H \times \cdots \times H \longrightarrow M$, I wish to find a corresponding cocycle in $Z^i(G, \text{CoInd}_{H}^{G}(M))$, or vice versa.


Homeomorphisms of $S^n\times S^1$ Is every homeomorphism $h$ of $S^n\times S^1$ with identity action in $\pi_1$ pseudo isotopic to a homeomorphism $g$ of $S^n\times S^1$ such that $g(S^n\times x)=S^n\times x$ for each $x\in S^1$? I would be satisfied with an answer for $n>3$.


Given a formal power series ,decide whether there exists a polynomial the series satisfies and if it exists,how to write it down? Given a formal power series $$y(x)=\sum_{i=0}^{\infty} a_i x^i$$ Is there an algorithm that decides whether there exists a polynomial$$ P(x,y)=p_n(x)y^n+p_{n-1}(x)y^{n-1}+\cdots+p_0(x)=0,p_j(x)\in F[x]$$the series satisfies  and if it exists,how to write it down?


$P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \in \mathbb{Z}[X]$ Let $P \in \mathbb{Q}[X]$, is it true that $P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \in \mathbb{Z}[X]$ implies $P \in \mathbb{Z}[X]$ ?
The two conditions are necessary because, if $P=2X^2-\frac{1}{2}$, we have $P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \notin \mathbb{Z}[X]$, but $P \notin \mathbb{Z}[X]$.
And if $P=9X^3+\frac{1}{3}$, we have $P \circ P \circ P \in \mathbb{Z}[X]$ and $P \circ P \notin \mathbb{Z}[X]$, but $P \notin \mathbb{Z}[X]$.


Looking for bound in integral involving Legendre polynomial I'm looking for an upper bound to the following integral or equivalent when $n$ leads to $ +\infty $  to the following expression
$$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) p_n(y)}{(1-xy)} dx dy \right| $$
with
$$ p_n(t):=\frac{1}{n!}(t^n(1-t)^n)^{(n)}.$$
This integral is similar to Beukers integral; after integrating $n$ times to $y$ ,I obtain
$$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) x^n y^n(1-y)^n}{(1-xy)^{n+1}} dx dy \right| ,$$
I don't know what to do after this since I can't have interesting expression inside the integral when I derive n times  $x$ .


Fibers of fibrations of a 3-manifold over $S^1$ Given a fiber bundle $S\hookrightarrow M \rightarrow S^1$  with $M$ (suppose compact closed connected and oriented) 3-manifold and $S$ a compact connected surface, it follows form the exact homotopy sequence that $\pi_1(S)\hookrightarrow \pi_1(M)$. Does this imply that the "fiber" of a 3-manifold which fibers over $S^1$ is well defined?
The answer should be NO, so I am asking: are there simple examples of 3-manifolds which are the total space of two fiber bundles over $S^1$ with fibers two non homeomorphic surfaces?
EDIT: the answer is NO (see Autumn Kent's answer). I'm just seeking for a "practical" example to visualize how this phoenomenon can happen.


Geodesic preserving diffeomorphisms of constant curvature spaces Let $X$ be either Euclidean space $\mathbb{R}^n$,  the sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$.

I would like to have a classification of all diffeomorphisms $X\to X$ which map every geodesic line to a geodesic line.

In the first two cases, the group of all such transformations is strictly larger than the group of isometries, but for the hyperbolic space I am not sure.


If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{  , } \forall x \in \Bbb R^n$ . Then prove that $$f(x) = Ox +b \text{ , for some fixed } O \in O(n,\Bbb R) \text{ ,} \forall x \in \Bbb R^n$$
I have no idea about this problem, so couldn't make any attempt.


Number of divergence free symmetric two tensor in dimension 4 In a  $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any counting formula for such types of tensors?
The divergence-freeness would imply that number of such tensors would be way less than $\frac{ n(n+1)}{2}.$ If the number is $1,$ Einstein's Field equations $G= \kappa T$ become trivial, and so it should not be the case. In fact, the Bach tensor (B) is another traceless symmetric and divergence-free tensor. Can you at least name a couple more of these types of tensors? Please don't tell me to take positive linear combinations of $G, T, B.$ I am looking for some counting formula like the formula that gives a total number of "independent" Riemann curvature tensor components. I am sorry if this question is too vague and naive. Thanks so much


Name of the sum of the $k$-wise product of distinct integers from $1$ to $n$ Let $n$ be a positive integer. For each $k = 1, \ldots, n$, let
$$
S_k(n) := \sum_{1 \le i_1 < \cdots < i_k \le n} i_1 \cdots i_k
$$
be the sum of the $k$-wise product of distinct integers from $1$ to $n$. Does this sum $S_k(n)$ have a name and formula?
For example, $$S_1(n) = 1 + \cdots + n = \binom{n + 1}{2}$$ and $$S_2(n) = [(1 + \cdots + n)^2 - (1^2 + \cdots + n^2)]/2 = n(n+1)(3n^2 - n - 2)/24.$$


Faithful representations of non-amenable Lie groups Can a non-amenable connected Lie group have a faithful finite-dimensional unitary representation?


Geometrical proof of Noether Theorem I am reading a very nice Physics book "The standard model in a nutshell" by D.Goldberg and just read there a mention to Noether Theorem. Of course I knew this outstanding theorem very well from previous physics courses : "A symmetry in the system (Lagrangian) implies a conserved physical quantity". The proof is quite short it follows from the Euler Lagrange equation and a few lines of calculus. But now that I think about it, I am not that happy with it. I fell like such a important result should have a more abstract and geometrical understanding.
Do you know another proof of Noether Theorem written in a differential geometry setting and that can be generalized in a more abstract case?


What is the longest recorded gap between "proof" of a "theorem" and discovery that the result is false I hope this question is not a duplicate. I am motivated by wondering when widely accepted results may be considered have a secure place in the mathematical literature. The question is intended to refer to temporal gap. I am particularly interested in cases where a result generally recognised to be of interest has later turned out to be definitely false (eg by explicit demonstration of a counterexample). However, cases where a widely accepted "proof" has later been shown to be incorrect, yet the result has later been correctly proved are also of some interest (such as the 19th century example of a "proof" of the four colour theorem whose incorrectness went undetected for 11 years, although the theorem is now known to be true). The nature of this question may change over time as formal proof checking becomes more advanced.


Definition of $(\infty,1)$-category in HoTT Reading Lurie's Higher topos theory got me thinking, if we can think of $(\infty,1)-$categories as (1-)categories enriched over $\infty$-groupoid, then wouldn't the internal definition of an $(\infty,1)$-category in Homotopy type theory be the usual univalent category but with homs as usual types instead of sets?
This seems to correspond to the notion of category in the HoTT library although technically the HoTT book defines a category to have hom-sets instead of types.
The first higher category I can think of is $\infty$-groupoid where the type of objects is simply $\mathcal U_0$, and then morphisms are the types $X \to Y$ which I am assuming isn't truncated or anything. Then the circle $S^1$ defined as a higher inductive type is in this category. And the hom-type $S^1\to S^1$ is some sort of elimator of the HIT for $S^1$ which is itself a HIT.
This (I think) obviously only works if our ambient category is $\infty$-groupoid itself (which is some form of HoTT). I am not sure this applies to all $\infty$-toposes.


least square optimization under positive semidefinite constraint i have the following optimization problem:
$min_X \|aX-b\|$
$s.t. \quad X\geq 0$
where $a$, $b$ are vectors and $X$ is a matrix. Is there any possibility of obtain the closed form of the optimized X? Thanks.


Is there a method to find the prime factorization of a number(integer) if the prime factorization of the previous number(integer) is known? There are methods to find out prime factorization of a given number but Is there a method to find the prime factorization of a number(integer) if the prime factorization of the previous number(integer)?. (Using only the information of previous number.)


Square root of normal distribution Let $X$ and $Y$ be independent random variates with the same probability distribution, $P(x)$. Assuming that the product $Z=XY$ is a random variate with normal distribution, say $$f_Z(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2} x^2}$$ what is the form of $P(x)$? Does it exist in closed form?
EDIT: I just realized that a product distribution is necessarily divergent at zero, which means that the normal distribution cannot be a product distribution (at least when $P(x)$ is interpreted as a normal function; maybe there is a generalized function that solves the problem?). Do you agree?


Bott Periodicity: Morse and K-Theory Is there an easy way to see the equivalence of the two statements of Bott periodicity via K-Theory and the original Morse Theory proof?
So, $$BU \times \mathbb{Z} \simeq \Omega^2BU$$ and
$$K(X)\otimes K(S^2) \cong K(X\times S^2)$$
I know there is the Bott paper that explicitly comments on this, but it is written in French. Any help will be greatly appreciated!


Fourier coefficients of real analytic functions on an n-dimension torus Let $(\mathbf{R}^n,\langle\;,\; \rangle)$ be the n-dimensional euclidean space endowed with the standard inner product. For a lattice $L\subseteq \mathbf{R}^n$ we let $cov(L)$ denote the covolume of $L$ with respect to $\langle\;,\; \rangle$. We let $L^*$ denote the dual lattice of $L$ with respect to $\langle\;,\rangle$. Let $T:=\mathbf{R}^n/L$ be the corresponding real torus endowed with the Lebesgue measure dx. For every integrable function $f:T\rightarrow \mathbf{C}$ and each $\xi\in L^*$ we let
$$
a_{\xi}(f):=\frac{1}{cov(L)}\int_T f(x)e^{-2\pi i\langle x,\xi\rangle}dx
$$
denote the $\xi$ Fourier coefficient of $f$. Assume now that $f$ is a real analytic function on $T$. Does this imply that there exists positive constants $c,C\in\mathbf{R}_{>0}$ such that
$$
|a_{\xi}(f)|\leq Ce^{-c||\xi||}\;\;
$$
for all $\xi\in L^*$ ?
Here $||\xi||$ corresponds to the norm of the vector $\xi\in L^*$. If such a result is true I would like to have a reference. I seem to remember that this is true when $n=1$. I would also be interested to know if one can go in the other direction namely if $f$ is a smooth function (infinitely many times differentiable) such that its Fourier coefficients satisfy an inequality as above then $f$ is real analytic.


Elementary reference for algebraic groups I'm looking for a reference on algebraic groups which requires only knowledge of basic material on the theory of varieties which you could find in, for example, Basic Algebraic Geometry 1 by Shafarevich.
If possible, it would be nice to find such a book which also discusses representation theory, but that's not necessary.


Find conditions over $U$ such that optimization over a convex cone generated by $U$ is equal to optimization over $U$ Reading several pappers to prepare my thesis I found the following problem:
We considerer the following optimization problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in\mathcal{C}} & f(x) \\[2pt]
\text{s.t.} & \mathcal{A}x-b \in K \end{array}   \right. \tag{1}
$$
where $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ and $\mathcal{A}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{m}$ are linear functions non-zero, $b\in\mathbb{R}^{m}$, $\mathcal{C}$ is a convex cone in $\mathbb{R}^{n}$ and $K$ is a closed convex cone in $\mathbb{R}^{m}$.
Question: I need to find conditions over a set $U\subset\mathbb{R}^{n}$ such that if $\mathcal{C}$ is the convex cone generated by $U$, then  problem $(1)$ is equivalent to the following problem
$$
\left\{\begin{array}{cl} \max\limits_{x\in U} & f(x) \\[2pt]
\text{s.t.} & \mathcal{A}x-b \in K \end{array}   \right. \tag{2}
?$$


When is a symplectomorphism on a cotangent bundle the lift of a diffeomorphism on the base manifold Let $X$ be a manifold and $T^\ast X$ the cotangent bundle. Let $\alpha$ denote the tautological $1$-form on $T^\ast X$ so that $(T^\ast X, \omega=-d\alpha)$ is a symplectic manifold. I want to know when a symplectomorphism $g:T^\ast X\to T^\ast X$ is the lift of a function $f:X\to X$ (here the lift of $f$ is $f^\sharp:T^\ast X\to T^\ast X \ ,\  (p,\xi_p)\mapsto (f(p),(f^\ast)^{-1}(\xi_p))$. An exercise in Da Silva's "Lectures on Symplectic Geometry" says this happens if and only if $g^\ast\alpha=\alpha$. I can show that the lift of a diffeomorphism preserves $\alpha$, but I am struggling with the converse. Given $g:T^\ast X\to T^\ast X$ such that $g^\ast\alpha=\alpha$, I so far have shown the following:
If $V$ is the symplectic dual of $\alpha$ (i.e. $\omega(V,\cdot)=\alpha$) then $g$ commutes with the flow of $V$ (i.e. $g\circ\theta_t((p,\xi_p))=\theta_t\circ g((p,\xi_p)))$. I can show that the flow of $V$ is $\theta^{(p,\xi_p)}(t)=(p,e^t\xi_p)$ implying for any $\lambda>0$ if $g(p,\sigma_p)=(q,\tau_q)$ then $g(p,\lambda\sigma_p)=(q,\lambda\tau_q)$ and the continuity of $g$ gives this also holds for $\lambda=0$. The hint in Da Silva's book says that this holds for all $\lambda\in\mathbb{R}$, but I don't understand how this follows. After showing this, the hint says to show the existence of a map $f:X\to X$ such that $\pi\circ g=f\circ\pi$ where $\pi:T^\ast X\to X$ is the projection. I thought the obvious map would be $f(p):=\pi\circ g(p,\sigma_p)$ where $\sigma_p\in T^\ast_p X$ is arbitrary. However, even if the above holds for all $\lambda\in\mathbb{R}$ I don't see why this map is well defined.
With such an $f$, I can show that $f^\sharp=g$. Any help would be greatly appreciated.


Subgroups of a topological Group such that quotient space is totally disconnected If $G$  is a topological group and $G_{{e}}$ is the identity component, the it is well known that $G_{{e}}$ is a normal subgroup of $G$ and the quotient group $G/G_{{e}}$ is totally disconnected. What can we say about the subgroup $H$ (need not be normal) of $G$ such that coset space is totally disconnected? Has it been studied for some topological groups such that the coset space is totally disconnected?


If L is a field extension of K, how big is L*/K*? Let $K$ be a field and $L$ an extension of $K$. I wonder how much larger the multiplicative group $L^\times$ of $L$ is than the multiplicative group $K^\times$ of $K$.
I know that if $L=K(t)$ and $t$ is transcendental over $K$, then $L^\times/K^\times$ is isomorphic to the direct product of an infinite number of copies of the integers: Indeed, any (monic) rational function can be uniquely written as the product of a finite number of powers of irreducible polynomials.
In particular, $L^\times/K^\times$ is not finitely generated.
What happens when $L=K(t)$ and $t$ is algebraic over $K$? I can handle the case when $K$ is finite, but what happens in the infinite case?
Even for $L=Q(\sqrt{2})$ it's not immediate to me if $L^\times/K^\times$ is finitely generated or not.


Stable torus that is not a torus Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus.
Is it true that $M$ is homeomorphic to a torus?


Continuity of the product map Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with respect to the minimal/maximal tensor product of $C^*$-algebras?


Examples of algebraic closures of finite index So there are easy examples for algebraic closures that have index two and infinite index: $\mathbb{C}$ over $\mathbb{R}$ and the algebraic numbers over $\mathbb{Q}$. What about the other indices?
EDIT: Of course $\overline{\mathbb{Q}} \neq \mathbb{C}$. I don't know what I was thinking.


Sum of odd reciprocals Can (n-1)/n be expressed as sum of random "odd" distinct reciprocals ever? here 'n' is also an odd.


Bijective map of smooth varieties that is not an isomorphism Let $k$ be an algebraically closed field of characteristic $0$ and let $f:X \rightarrow Y$ be a map between smooth varieties over $k$ that is bijective on closed points.  Musr $f$ be an isomorphism?
The restriction to fields of characteristic $0$ is necessary to rule out the Frobenius map, and the restriction to smooth varieties is necessary  to rule out resolutions of singularities of curves.  My guess is that there will be counterexamples with $X$ and $Y$ algebraic surfaces, but I do not know a rich enough store of examples of surfaces to find one.
Edit: I assert nothing about the maps on tangent spaces (unlike the question that someone has claimed is a duplicate of this one), and would be happy for a counterexample where that is the issue (but as I said, I insist that my varieties be smooth, which rules out the counterexamples in the purported duplicate question).


Relationship between P-noncomputable and P-random sets $P$ means polynomial complexity.
$S_p$ is the class of all $P$_random set, and $S_{pc}$ is the class of all $P$ noncomputable sets, is $S_p \bigcap S_{pc}$ empty? If not empty, any example?
what is the result, if we replace $P$ complexity with $NP$?
Moreover, $S$ is the class of all random sets, and $S_c$ is the class of all incomputable sets,  $S_{c}\bigcap S $ is not empty, what is a set in $S_{c}\bigcap S $,an immune set, productive set, or set of any other kind.
Is the difference class empty?
$P$ means polynomial complexity.
$S_p$ is class of all $P$_random sets, and $S_{pc}$ is class of all  $P$ incomputable sets, is $S_{pc} \setminus S_p$ empty? If not empty, any example?
what is the result, if we replace $P$ complexity with $NP$?
Moreover, $S$ is class of all random sets, and $S_c$ is class of all incomputable sets,  $S_{c}\setminus S $ is not empty, what is a set in the class $S_{c}\setminus S $,an immune set, productive set, or set of any other kind.
I have restored the original post with clarification, and ask the question  I intend  in the final part.


Dual of Banach-valued $L^p$ Let $X$ be an infinite-dimensional  Banach space and let $p\in(1,+\infty)$. We may define $L^p(\mathbb R;X)$. Is it always true that the topological dual of $L^p(\mathbb R;X)$ is $L^{p'}(\mathbb R;X^*)$? Maybe some reflexivity or separability is needed for the Radon-Nikodym argument to work.
Is there a simple realization of  the dual of $L^1(\mathbb R;X)$?


Does Zorn's Lemma imply a physical prediction? A friend of mine joked that Zorn's lemma must be true because it's used in functional analysis, which gives results about PDEs that are then used to make planes, and the planes fly. I'm not super convinced. Is there a direct line of reasoning from Zorn's Lemma to a physical prediction? I'm thinking something like "Zorn's Lemma implies a theorem that says a certain differential equation has a certain property, and that equation models a phenomena that indeed has that property".


Dense property of intersection of Sobolev space I'm using Muscalu and Schlag's textbook (online notes) to study Littlewood-Paley theory in harmonic analysis, where I encounter the following claim:
Pick an arbitrary real number $s$, we have that the intersection of $\cap_{t \in \mathbb{R}}H^{t}$ is dense in $H^{s}$.
The author mentioned that it suffices to prove the following equivalence of norms:
$$c^{-1}\|f-P_{\leq k}f\|_{H^{s}}^2 \leq \sum_{l \geq k}2^{2ls}\|P_{l}f\|_{L^2}^{2} \leq c\|f-P_{\leq k}f\|_{H^{s}}^2 \quad (\forall \ f \in H^s)$$
where $c = c(s) > 0$ above is some constant dependent on $s$ and $P_{\leq k} = \sum_{j \leq k}P_{j}$ is the sum of a subset of Littlewood-Paley projection operators.
I can see how the dense property can be derived from the equivalence of norms exhibited above. However, I have tried using several properties of the Littlewood-Paley projection operators, but none of them helps me gain any progress in showing the inequality above...any hint/idea?


Module of Kahler differentials for manifolds Let $A$ be a $k$-algebra and let $\mathcal{M}_A$ be the set of all $A$-modules. In $\mathcal{M}_A$, there exists a universal object $\Omega_{A/k}$,  called the module of Kahler differentials, and a $k$-derivation $d: A \to \Omega_{A/k}$ such that for any $k$-derivation $D: A \to M$ there exists a map $f: \Omega_{A/k} \to M$ such that $f\circ d = D$.
I'm trying to understand what this object has to be when I look at manifolds that are also smooth affine varities. I think that the answer should be the space of sections of the cotangent bundle over the ring of $C^\infty$ functions on the manifold. I'm unable to figure out how to show this.
Any help would be appreciated.


Transcendence of some number Everybody knows that $\sum_{k=0}^\infty{\frac{1}{2^{2^k}}}$ is transcendental.
Is number $\sum_{k=0}^\infty{\frac{1}{2^{k^2}}}$ algebraic or not?


This is not a dyadic cosine-product The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\sin z}z$, because then it's well-known that $\int_0^{\infty}\frac{\sin z}zdz=\frac{\pi}2$.
I've found the following variant intriguing and curious.

Question. Is this valid? If not, what is the value of the integral?
  $$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{k}=\frac{\pi}4.$$

In case such is known, what is a reference?


Are there any techniques for solving a differential equation of the form $f ' (x) = f( f( x ) )$? I am trying to solve the following differential equation
$$f ' (x) = f( f( x ) ),$$
but I have no idea how. I don't think the chain rule is useful for this.
Although I don't think this differential equation is solvable, I'd like to know if there is any interesting approach to solve a differential equation of this kind, or, at least, a non-trivial solution of the equation.


Surface area of convex hull Let Q be the convex hull of a non-convex polyhedron P. Is it true that the surface area of Q is not greater than the surface area of P?


Request:  intermediate-level proof: every 2-homology class of a 4-manifold is generated by a surface. Hi, everyone:
For the sake of context, I am a graduate student, and I have taken classes in
algebraic topology and differential geometry. Still, the 2 proofs I have found
are a little too terse for me; they are both around 10 lines long, and each line
seems to pack around 10 pages of results. Of course, I am considering cases for
"reasonable" spaces, being the beginner I am at this point.
It would also be great if someone knew of similar results for H_1 (equiv. H_3).
Thanks in Advance.


On starting graduate school and common pitfalls... Hi,
I'll be starting graduate school soon, and when I look back at my college career, there are certain things I wish I could have done differently. In hindsight, I wished I wasn't in such a rush to study [insert your favorite hot topic here] as opposed to pinning down the fundamentals of the course materials I was studying. Probably the best class I took was a seminar where the prof had us read and discuss from classic texts in differential geometry and pdes.  Anyways, now that I'm starting graduate school, I'd like to avoid other common pitfalls that graduate students make.  So my question is:

What are common pitfalls, mistakes or
  misconceptions that you wished
  somebody had told you were wrong?  I'm
  interested in pretty much anything
  from how to conduct research, to what
  courses to take, or anything else.


Weak convergence can imply strong convergence In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?


The distribution of fractional parts $\Big\{ \frac{N}{n} \Big\}$ Let $N$ be a large integer. What is known about distribution of fractional parts $\Big\{ \frac{N}{n} \Big\} \in [0,1)$ after division of $N$ by all odd numbers $n$ in the range $3 \leq n < \sqrt{N}$?


What are some bounds on $\displaystyle \sum_{n\leq N} \mu(n)e(\alpha n)$? As usual, let $e(x):=2\pi i x$. I already know of a bound on $\sum_{n\leq N}\mu(n)$: by the prime number theorem, it is $\ll x\exp(-c(\log x)^{\frac{1}{2}})$ for some $c>0$.

However, what happens when we consider the sum $\displaystyle \sum_{n\leq N} \mu(n)e(\alpha n)$?

I could imagine using partial summation to obtain a leading term which is $\ll N\exp(-c\sqrt{\log N})$using the prime number theorem, but I don't know how good a bound optimising $O(|\alpha|\int_1^N t\exp(-c\sqrt{\log t})\mathrm{d}t)$ would give me. Surely, there are some already available bounds? Thanks!
P.S. I am looking for bounds without assuming GRH.
Follow-up question: Is the circle-method somehow useful in this context?


Gram determinant in dual basis Assume that $V$ is a $n$ -dimensional inner product space.
Basis $(e_1,...,e_n)$, $(f_1,...,f_n)$ are said to be dual if
$\langle e_i, f_j\rangle=0$ for  $i\neq j$, $i,j=1,...,n$ and $\langle e_i, f_i\rangle=1$ for $i=1,...,n$.
The Gram determinant of vectors $x_1,...,x_r\in V$ is the determinant
$G(x_1,...,x_r)=\det[\langle x_i,x_j\rangle]$.
For dual basis $(e_1,...,e_n)$, $(f_1,...,f_n)$ we have:
$f_i=\sum_{j=1}^n  g^{i j}e_j$ for $i=1,...,n$, where $g^{ij}$ are elements of the inverse matrix to  $[g_{ij}]$ with $g_{ij}=\langle e_i,e_j\rangle$.
Is it true for fixed $p\in \{1,...,n-1 \}$ the formula
$$
\frac{G(e_1,..,e_p)}{G(f_{p+1},...,f_{n})}=G(e_1,...,e_n)
$$
and how to prove it.