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pro vyhledávání: '"Morrill, Thomas"'
Autor:
Morrill, Thomas
A modified form of Euclid's algorithm has gained popularity among musical composers following Toussaint's 2005 survey of so-called Euclidean rhythms in world music. We offer a method to easily calculate Euclid's algorithm by hand as a modification of
Externí odkaz:
http://arxiv.org/abs/2206.12421
Autor:
Morrill, Thomas, Simonič, Aleksander
We establish quasimodularity for a family of residual crank generating functions defined on overpartitions. We also show that the second moments of these $k$th residual cranks admit a combinatoric interpretation as weighted overpartition counts.
Externí odkaz:
http://arxiv.org/abs/2005.01919
Autor:
Morrill, Thomas
Publikováno v:
Bull. Aust. Math. Soc. 103 (2021) 210-217
We examine a recursive sequence in which $s_n$ is a literal description of what the binary expansion of the previous term $s_{n-1}$ is not. By adapting a technique of Conway, we determine limiting behaviour of $\{s_n\}$ and dynamics of a related self
Externí odkaz:
http://arxiv.org/abs/2004.06414
Two analogues of the crank function are defined for overpartitions -- the first residual crank and the second residual crank. This suggests an exploration of crank functions defined for overpartitions whose parts are divisible by an arbitrary $d$. We
Externí odkaz:
http://arxiv.org/abs/1912.05722
Let $V(T)$ denote the number of sign changes in $\psi(x) - x$ for $x\in[1, T]$. We show that $\liminf_{\;T\rightarrow\infty} V(T)/\log T \geq \gamma_{1}/\pi + 1.867\cdot 10^{-30}$, where $\gamma_{1} = 14.13\ldots$ denotes the ordinate of the lowest-l
Externí odkaz:
http://arxiv.org/abs/1910.14203
Autor:
Morrill, Thomas, Trudgian, Tim
We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$ with $1-
Externí odkaz:
http://arxiv.org/abs/1811.12521
Autor:
Morrill, Thomas, Platt, David
In 1984, Robin showed that the Riemann Hypothesis for $\zeta$ is equivalent to demonstrating $\sigma(n) < e^\gamma n \log \log n$ for all $n > 5040$. Robin's inequality has since been proven for various infinite families of power-free integers: $5$-f
Externí odkaz:
http://arxiv.org/abs/1809.10813
Autor:
Morrill, Thomas
In order to explore tonality outside of the `Pythagorean' paradigm of integer ratios, Robert Schneider introduced a musical scale based on the logarithm function. We seek to refine Schneider's scale so that the difference tones generated by different
Externí odkaz:
http://arxiv.org/abs/1804.08067
Autor:
Morrill, Thomas
We generalize the generating series of the Dyson ranks and $M_2$-ranks of overpartitions to obtain $k$-fold variants, and give a combinatorial interpretation of each. The $k$-fold generating series correspond to the full ranks of two families of buff
Externí odkaz:
http://arxiv.org/abs/1702.03558
Autor:
Morrill, Thomas, Trudgian, Tim
Publikováno v:
In Journal of Number Theory July 2020 212:448-457
