Newest Questions
167,248 questions
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Classification of simple closed curves on the punctured torus
I am looking for a reference for the following fact:
Consider the fundamental group of the once punctured torus $\pi_1(T^\ast)\cong F_2$ generated by elements $a$ and $b$. An oriented curve $[\alpha]$...
- 705
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Is partitioning a graph into matchings of size $m$ polynomially solvable?
Holyer (1981) proved that for each fixed $m\ge 3$, it is NP-complete to determine whether an arbitrary graph can be edge-partitioned into subgraphs isomorphic to the complete graph $k_m$. Dor and ...
- 413
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A relational-power hierarchy and its relation to Jensen's hierarchy and $L$
Fix a coding of ordered pairs, say Quine--Rosser ordered pairs, and write the ordered pair of $a$ and $b$ as
$$
\langle a,b\rangle.
$$
For a set $X$, I want to define a "relational power set"...
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On the definition of elementary embeddings [closed]
Given an elementary embedding $j: M \to N$, the concept of the definition states that whenever $M$ satisfies a formula $\phi$, then through $j$, $N$ also satisfies $\phi$. The question is does it ...
- 109
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Are equivariant coherent sheaf cohomologies on a G-scheme semi-simple?
(I am working over $\mathbb{C}$). Consider a reductive algebraic group $G$ and a $G$-scheme $X$. Are the cohomologies of a $G$-equivariant coherent sheaf on $X$, seen as $G$-representations, semi-...
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60
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Is every cocongruence in $\mathbf{FinGrp}$ effective?
Question. Is every cocongruence in $\mathbf{FinGrp}$ effective?
Here, a cocongruence on an object $X$ in a category is a jointly epimorphic pair $(p,q): X \rightrightarrows Y$ which induces for every ...
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389
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What is the unit distance exponent?
Given $n$ distinct points in the Euclidean plane, what is the greatest number of pairs of points that can be unit distance apart? Paul Erdős conjectured that the answer was $n^{1+o(1)}$. Recently, ...
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Ratio limit Theorem for a centered random walk on $\mathbb{Z}$ under time shifts
Let $(X_i)$ be i.i.d. random variables on $\mathbb{Z}$ with $\mathbb{E}[X_1] = 0$ and define $W_n = \sum_{i=1}^n X_i$. Assume the markov chain is aperiodic. Is it possible to show the following:
If $...
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Is the product-preserving property of a comonad necessary for the automatic Beck–Chevalley condition in an orthogonal site construction?
Consider the following setup. Let C and D be two sites, and let
$\mathcal{E} = \mathrm{Sh}_\infty(C \times D)$
be the $\infty-topos$ of sheaves on their product. Suppose we have:
A cohesive modality (...
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Generalized Mayer‑Vietoris for colimits over finite posets of CW‑complexes
I am trying to find a way to compute the cohomology ring of a space of the form $X:=\mathrm{colim}_{I}X_i$.
Here are the properties for this diagram of spaces:
The index set $I$ is a finite poset.
...
- 171
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172
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How should I seek expert feedback on a possible proof outside my advisor’s area?
I am currently a phd candidate in pure mathematics. Very recently, I think I have found a proof of a statement in geometric measure theory that seems nontrivial but not in my advisor's area (My ...
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1
answer
30
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Characterization of $R=\frac{\|\Delta\|_{1,1}}{\|\Delta\|_{F}}$
Let $P, Q$ be two $M\times M$ projection matrices, both of rank $K$. I want to find an upper bound on the ratio $R=\frac{\|\Delta\|_{1,1}}{\|\Delta\|_{F}}$, where $\Delta = P-Q$. Here $\|\Delta\|_{1,1}...
3
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220
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On Erdős Problem #598
$[\lambda]^\kappa$ denotes $\{A\subseteq\lambda:|A|=\kappa\}$. For (not necesarily infinite) cardinals $\kappa,\lambda,\tau,\mu$, let $\lambda\rightarrow[\kappa]^\tau_\mu$ denote the statement that ...
- 699
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Inequality involving the principal invariants of the sum of two positive definite matrices
$\newcommand{\be}{\boldsymbol{e}}
\newcommand{\bff}{\boldsymbol{f}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bB}{\boldsymbol{B}}
\DeclareMathOperator\tr{tr}
\DeclareMathOperator\trace{tr}
\...
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A question regarding the proof of Lemma 2.3 in Lin's Paper "A New Proof of the Caffarelli-Kohn-Nirenberg Theorem"
Recently, I am reading the paper of Fanghua Lin named "A New Proof of the Caffarelli-Kohn-Nirenberg Theorem".
I have a question regarding the proof of Lemma 2.3, the statement of the lemma ...
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