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1 vote
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Lower bound on the number of balanced graphs

The bound of Ruciński and Vince is for strongly balanced, which is a more strict condition. If only balanced is required, the example of connected regular graphs provides a bound much greater than $n^{...
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6 votes
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Asymptotic behavior of a double oscillatory integral

$\newcommand{\R}{\mathbb R}\newcommand\sgn{\operatorname{sgn}}\newcommand{\vpi}{\varphi}$Obviously, for $a:=\sqrt{2\pi}\,\psi(0)$ we have \begin{equation*} \psi(0)=f(0), \end{equation*} where \...
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1 vote

Calculate asymptotic value of an integral of exponential function

The strategy is as follows: put everything in the integrand in the exponent $e^{f(z)}$, with $f(z)=\zeta z-z^p/p$; calculate the saddle point, the (possibly complex) number $z_0$ where $f'(z)=0$; ...
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$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

The answer by Carlo Beenakker is correct that any contribution to $D_n A_n \sim n^p$ with $p\ge 0$ does not generate any logarithmic asymptotic terms. Instead of numerical checks, one can also examine ...
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$\log(t)$ term in the small time expansion of $\mathrm{Tr}( A e^{-tB} )$

I don't think there will be a logarithmic term. For definiteness, let me take $B_n=n^2$, $D_nA_n=n^p$, and approximate the sum over $n$ by an integral, $$I_p(t)=\int_0^\infty n^p e^{-tn^2}=\tfrac{1}{2}...
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