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To counteract the contribution of CO₂emissions by cement production and utilization, biochar is being harnessed as a carbon-negative additive in concrete. Increasing the cement replacement and biochar dosage will increase the carbon offset, but there is large variability in methods being used and many researchers report strength decreases at cement replacements beyond 5%. This work presents a reliable method to replace 10% of the cement mass with a vast selection of biochars without decreasing ultimate compressive strength, and in many cases significantly improving it. By carefully quantifying the physical and chemical properties of each biochar used, machine learning algorithms were used to elucidate the three most influential biochar characteristics that control mortar strength: initial saturation percentage, oxygen-to-carbon ratio, and soluble silicon. These results provide additional research avenues for utilizing several potential biomass waste streams to increase the biochar dosage in cement mixes without decreasing mechanical properties.

Graphical Abstract

 

Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of$t$-hooks in the partitions of$n$. We prove that the limiting distribution is normal with mean\[$\mu _t(n)\sim \frac {\sqrt {6n}}{\pi }-\frac {t}{2}$\]and variance\[$\sigma _t^2(n)\sim \frac {(\pi ^2-6)\sqrt {6n}}{2\pi ^3}.$\]Furthermore, we prove that the distribution of the number of hook lengths that are multiples of a fixed$t\geq 4$in partitions of$n$converge to a shifted Gamma distribution with parameter$k=(t-1)/2$and scale$\theta =\sqrt {2/(t-1)}$.

 

Ramanujan’s partition congruences modulo$$\ell \in \{5, 7, 11\}$$$\ell \in \{5,7,11\}$assert that$$\begin{aligned} p(\ell n+\delta _{\ell })\equiv 0\pmod {\ell }, \end{aligned}$$$\begin{array}{c}p(\ell n+{\delta }_{\ell })\equiv 0(mod\ell ),\end{array}$where$$0<\delta _{\ell }<\ell $$$0<{\delta }_{\ell }<\ell$satisfies$$24\delta _{\ell }\equiv 1\pmod {\ell }.$$$24{\delta }_{\ell }\equiv 1(mod\ell ).$By proving Subbarao’s conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes$$\ell \ge 5,$$$\ell \ge 5,$we give a new proof of the conclusion that there are infinitely manymfor which$$p(\ell m+\delta _{\ell })$$$p(\ell m+{\delta }_{\ell })$is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest suchmsatisfies$$m<(\ell ^2-1)/24,$$$m<({\ell }^2-1)/24,$representing a significant improvement to the previous bound.

 

Abstract Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb {C}^{2})^{[n]}$ on $n$ points, as $n\rightarrow +\infty ,$ is a Gumbel distribution . In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_{k}(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$ , then we obtain the asymptotic $$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$ a result which is of independent interest. 

We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)={0.5053\dots}$. As a corollary, we find that $L_{\mathrm{Tam}}(-1)={1.8193\dots}$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}$. We give an application of these results to canonical and Weil heights.