Datasets:

MMLA-ORG
/

Cayley-Math

text stringlengths 1 9.22k	batch stringclasses 2 values	packed_row int64 0 103k	label_count int32 1 2.05k	source_group stringlengths 16 36	source_url stringlengths 32 171	license stringclasses 3 values	attribution stringlengths 5 1.37k
# The Alekseev-Meinrenken diffeomorphism arising from the Stokes phenomenon Xiaomeng Xu ###### Abstract The Alekseev-Meinrenken diffeomorphism is a distinguished diffeomorphism from the space of $n\times n$ Hermitian matrices to the space of $n\times n$ positive definite Hermitian matrices. This paper derives the ex...	initial-50m	0	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
Therefore, to relate $\nu(u)$ to $\Gamma_{AM}$, we need to break the symmetry of $\nu(u)$, by choosing specific $u$ such that the particular chain structure (3) can come out. The naive observation is that if $u_{n}\gg u_{n-1}\gg\cdots\gg u_{1}$ in $u={\rm diag}(u_{1},...,u_{n})$, the $n\times n$ linear system (1) seem...	initial-50m	1	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
\frac{u_{k+1}-u_{k}}{u_{k}-u_{k-1}}\rightarrow\infty$ for $k=2,...,n-1$ and $\frac{u_{k+1}-u_{k}}{u_{k}-u_{k-1}}\rightarrow 0$ for $k=2,...,n-1$. See [28] for more details. Another application of Theorem 1.4 is that it enables us to write down an explicit Moser flow of the Hamiltonian vector fields on the coadjoint or...	initial-50m	2	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
$\displaystyle C(\cdot,u)\colon{\rm Herm}(n)\rightarrow{\rm U}(n).$ (5) In a global picture, the connection matrix is related to the Stokes matrices by the following monodromy relation, which follows from the fact that a simple negative loop (i.e., in clockwise direction) around $0$ is a simple positive loop (i.e., ...	initial-50m	3	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
Phi(u)$ of (7) with the prescribed asymptotics (10). Therefore, $\Phi_{0}\in{\rm Herm}(n)$ parameterizes the Hermitian matrix valued solutions of (7) on $U_{\rm id}$. We then denote by $\Phi(u;\Phi_{0})$ the solution of (7) with the prescribed asymptotics $\Phi_{0}$ in the sense of Theorem 2.10. In [28], the Stokes m...	initial-50m	4	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
{\mathfrak{h}_{\rm reg}}(\mathbb{R})$ is a limit point $u={\rm diag}(u_{1},...,u_{n})$, where some $u_{i}$ collapse in a comparable speed. In particular, the limit of $u={\rm diag}(u_{1},...,u_{n})$, as $\frac{u_{k+1}-u_{k}}{u_{k}-u_{k-1}}\rightarrow+\infty$ for all $k=2,...,n-1$ and $u_{2}-u_{1}\rightarrow 0$, is a po...	initial-50m	5	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
\ldots,\lambda_{k}^{(k)}$. The action is well-defined since $U^{-1}tU$ does not depend on the choice of $U$, and preserves the Gelfand- Tsetlin map (15). The actions of the various $T(k)$’s commute, hence they define an action of the Gelfand-Tsetlin torus $T(1)\times\cdots\times T(n-1)\cong{\rm U}(1)^{\frac{(n-1)n}{2}}...	initial-50m	6	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
}s$ are the ordered set of eigenvalues of the upper left $k$-th submatrix $A^{(k)}$ of $A\in{\rm Herm}(n)$. Then $\mu$ is a continuous map from ${\rm Herm}^{+}(n)$ onto the Gelfand-Tsetlin cone $\mathcal{C}(n)$. Thimm torus actions. Let $\mathcal{C}_{0}(n)\subset\mathcal{C}(n)$ denote the subset where all of the eigen...	initial-50m	7	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
n)\rightarrow T^{1}\times\cdot\cdot\cdot\times T^{n-1}=T^{\frac{n(n-1)}{2}}$ be a map from ${\rm Herm}_{0}(n)$ to the Gelfand-Tsetlin tours. Associated to $\theta$, there is a gauge transformation $\mathcal{X}_{\theta}:{\rm Herm}_{0}(n)\rightarrow{\rm Herm}_{0}(n);\ A\mapsto{\theta(A)}\bullet A,$ (29) where $\theta(A...	initial-50m	8	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
+1)}(A)},$ for $1\leq j\leq k+1,1\leq i\leq k$, and $\displaystyle\widetilde{C}_{k+1,j}$ $\displaystyle=$ $\displaystyle\frac{e^{\frac{\lambda^{(k+1)}_{j}-(A)_{k+1,k+1}}{4}}\prod_{v=1}^{k+1}\Gamma\left(1+\frac{\lambda^{(k+1)}_{v}-\lambda^{(k+1)}_{j}}{2\pi{\mathrm{i}}}\right)}{N_{j}^{(k+1)}(A)\cdot\prod_{v=1}^{k}\Gamm...	initial-50m	9	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
which only differs from $\mathcal{T}(A)$ by the angle of the variables $a^{(i)}_{j}$. For the angles $\psi^{(i)}_{j}$ of the variables $a^{(i)}_{j}$, by (31) we have $\psi^{(i)}_{j}(\mathcal{X}_{\theta}(A))=\theta_{j}^{(i)}(A)+\psi^{(i)}_{j}(A)+\pi{\mathrm{i}}$. It implies $\displaystyle D^{(i)}(\mathcal{X}_{\theta}(A...	initial-50m	10	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
$\displaystyle\widetilde{C}\left(E_{k},\delta_{k}(\mathcal{X}_{\theta}(A))\right)_{ii}=1,\ \ \ \ \ \ \text{for}\ k<i\leq n,$ $\displaystyle\widetilde{C}\left(E_{k},\delta_{k}(\mathcal{X}_{\theta}(A))\right)_{ij}=0,\ \ \ \ \ \ \text{otherwise}.$ Here $N_{j}^{(k)}$ and $H^{(k)}_{ij}$ are given by $\displaystyle N_{j}^{...	initial-50m	11	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
\displaystyle g(u;A)Ag(u;A)^{-1}$ $\displaystyle=$ $\displaystyle D(u;A)\cdot\left((u_{2}-u_{1})^{\frac{-\lambda^{(k)}(A)}{2\pi{\mathrm{i}}}}\times\overrightarrow{\underset{k=2,...,n-1}{\prod}}\left(\frac{u_{k+1}-u_{k}}{u_{k}-u_{k-1}}\right)^{\frac{-\lambda^{(k)}(A)}{2\pi{\mathrm{i}}}}\bullet A\right)\cdot D(u;A)^{-1}....	initial-50m	12	2,048	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
endents, Physica 1D (1980), 80-158. * [17] M. Jimbo, T. Miwa and K. Ueno, Monodromy preserving deformations of linear differential equations with rational coefficients I, Physica 2D (1981), 306–352. * [18] M. Loday-Richaud, Divergent series, summability and resurgence II. Simple and multiple summability, vol. 2154 ...	initial-50m	13	757	arxiv:2401.12792	https://arxiv.org/abs/2401.12792	https://creativecommons.org/licenses/by/4.0/	Xiaomeng Xu
# Determination of a pair of newforms from the product of their twisted central values Pramath Anamby Department of Mathematics Indian Institute of Science Education and Research Pune – 411008, India<EMAIL_ADDRESS>and Ritwik Pal Stat-Math Unit Indian Statistical Institute Kolkata – 700108, India<EMAIL_ADDRESS> ###...	initial-50m	14	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
terms. Among the error terms, writing $q_{2}=Q_{1}^{c}$ for $c\geq 203/100$, we note that $Q_{1}^{599/200+\epsilon}q_{2}\begin{cases}\ll Q_{1}^{701/200+\epsilon}q_{2}^{3/4+\epsilon}&\text{ if }203/100\leq c\leq 51/25;\\\ \gg Q_{1}^{701/200+\epsilon}q_{2}^{3/4+\epsilon}&\text{ if }c>51/25.\end{cases}$ Here we want to ...	initial-50m	15	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
mathcal{M}$, which is independent of the prime $p$. ### Acknowledgements. The authors thank NBHM, DAE for the financial support through NBHM postdoctoral fellowship. The first named author thanks IISER, Pune, where he is a postdoctoral fellow. The second named author thanks ISI, Kolkata, where he is a postdoctoral fe...	initial-50m	16	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
}\mathbf{S}_{dn}&=\sum_{q_{1}\in\mathcal{D}}\sum_{m,n}q_{1}\frac{\lambda_{f}(m)\lambda_{g}(n)}{(mn)^{1/2}}\Big{(}\sideset{}{{}^{}}{\sum}_{\chi(q_{1})}\chi(\bar{m}n\bar{p})\Big{)}\Big{(}\sideset{}{{}^{}}{\sum}_{\psi(q_{2})}\psi(n\bar{p})\Big{)}V_{f}\Big{(}\tfrac{m}{Xq_{\chi}^{1/2}}\Big{)}V_{g}\Big{(}\tfrac{nY}{q_{\chi...	initial-50m	17	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
_{1}}$ to get $p+rq_{2}\equiv mp\bmod{q_{1}}$. We observe that for each $m\leq q_{1}^{1+\epsilon}X$, there is a unique $r$ modulo $q_{1}$ that satisfies the above congruence relation as $q_{2}$ is prime. Thus for each $m\leq q_{1}^{1+\epsilon}X$ the numbers of possible $r$ are at most $(q_{1}q_{2})^{\epsilon}/Y$. Hence...	initial-50m	18	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
$\displaystyle=\sum_{q_{1}\in\mathcal{D}}\sum_{m,n}\frac{\phi(q_{1}q_{2})}{q_{2}}\frac{\lambda_{f}(m)\lambda_{g}(n)}{(mn)^{1/2}}S(1,np\bar{q_{1}}^{2};q_{2})S(1,mnp\bar{q_{2}}^{2};q_{1})V_{f}\Big{(}\tfrac{m}{Xq_{\chi}^{1/2}}\Big{)}V_{g}\Big{(}\tfrac{n}{Yq_{\chi\psi}^{1/2}}\Big{)}$ $\displaystyle\quad\quad+O(Q_{1}^{2+\e...	initial-50m	19	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
}_{m,q_{1},q_{1}^{\prime},q_{2}}=\frac{N}{q_{1}q_{1}^{\prime}q_{2}}\sum_{n\in\mathbb{Z}}\mathcal{T}_{1}(n;m,q_{1},q_{1}^{\prime},q_{2})\int_{\mathbb{R}}W(y)e\left(-\frac{nNy}{q_{1}q_{1}^{\prime}q_{2}}\right)dy,$ where $\mathcal{T}(n;m,q_{1},q_{1}^{\prime},q_{2})$ is given by (4.13) $\sum_{\alpha\bmod q_{1}q_{1}^{\pri...	initial-50m	20	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
$ By adding and subtracting the missing $j=p$ term, we get (4.24) $\mathfrak{T}^{2}(n;q_{1},q_{1}^{\prime},q_{2})=q_{2}\sideset{}{{}^{*}}{\sum}_{j\text{ mod }q_{2}}e_{q_{2}}(\bar{n}\bar{q_{1}}\bar{q_{1}^{\prime}}^{3}(p-j)+pq_{1}^{\prime 2}\bar{n}\bar{q_{1}^{\prime}}(p-j)\bar{j})-1.$ The sum over $j$ is now $\ll\|S(-\...	initial-50m	21	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
from the zero-frequency term $np=mq_{2}^{2}$. To see this, since $n\ll(Q_{1}q_{2})^{1+\epsilon}Y$ and $p\ll\log Q_{1}$, $np\ll(Q_{1}q_{2})^{1+\epsilon}Y$. Thus by requiring that $Q_{1}\ll q_{2}$, we see that $np\ll q_{2}^{2+\epsilon}Y<mq_{2}^{2}$, by choosing $Y<q_{2}^{-\delta}$ for some small $\delta>0$. Now, we note...	initial-50m	22	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
2}X^{-1/2}+Q_{1}^{2+\epsilon}q_{2}^{1/2+\epsilon}X^{-1/2}Y^{-1/2}.$ Finally, for $T_{3}$ we estimate (4.41) $T_{3}\ll\sum_{q_{1}\in\mathcal{D}}q_{1}\underset{n\leq(q_{1}q_{2})^{1+\epsilon}/Y}{\sum_{m\leq q_{1}^{1+\epsilon}/X}}\frac{\|\lambda_{f}(m)\lambda_{g}(n)\|}{(mn)^{1/2}}q_{1}^{\frac{1}{2}}\ll Q_{1}^{7/2+\epsilon}...	initial-50m	23	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
ideset{}{{}^{}}{\sum}_{\chi(q_{1})}\sideset{}{{}^{}}{\sum}_{\psi(q_{2})}\overline{\chi}(p)\overline{\psi}(p)\overline{G(\chi)}^{2}L(1/2,f\times\chi)L(1/2,g\times\chi\psi)=\frac{\lambda_{g}(p)}{p^{1/2}}\cdot\pi_{2}(Q_{1})Q_{1}^{\frac{5}{2}}+O(Q_{1}^{\frac{1099}{200}+\epsilon}).$ Now, for $i=1,2$ if we have $f_{i}\in ...	initial-50m	24	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
2}+\epsilon}q_{h}^{\frac{1}{4}+\epsilon}X^{\frac{1}{2}})\\\ &=O(Q^{2+\epsilon}q_{h}^{\frac{1}{4}+\epsilon}X^{\frac{1}{2}}).\end{split}$ Now we put $X=Q^{-1/2}$ in (5.6) and (5.7) respectively and combine them to get (5.4). ∎ By putting $h=g\times\chi$ and taking $p$ to be such that $\lambda_{g}(p)\neq 0$ we easily co...	initial-50m	25	2,048	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
sum_{m\ll M}\left\|\tfrac{\lambda_{f}(m)\lambda_{g}(n)}{(mn)^{1/2}}\right\|^{2}\\\ &\ll(q_{2}+N)(Q_{1}^{2}+M)\log N\log M\\\ &\ll Q_{1}^{2}q_{2}^{1+1/c+\epsilon}\log Q_{1}.\end{split}$ In the last inequality we have used that $q_{2}\asymp Q_{1}^{c}$ and that $q_{2}+N\ll N\ll q_{2}^{1+1/c+\epsilon}$, $Q_{1}^{2}+M\ll Q_{1...	initial-50m	26	1,775	arxiv:2401.12891	https://arxiv.org/abs/2401.12891	https://creativecommons.org/licenses/by/4.0/	Pramath Anamby, Ritwik Pal
# Odd-quadratic Lie superalgebras with a weak filiform module as an odd part Elisabete Barreiro Elisabete Barreiro, University of Coimbra, CMUC, Department of Mathematics, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal E-mail<EMAIL_ADDRESS>, Saïd Benayadi Saïd Benayadi, Laboratoire de Mathématiques IECL UMR...	initial-50m	27	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
verifying then $\mbox{\rm dim }{\mathfrak{g}}_{\bar{0}}-\mbox{\rm dim }[{\mathfrak{g}}_{\bar{0}},{\mathfrak{g}}_{\bar{0}}]\geq 2$, which is a contradiction. In fact, these two structures together have proven to be very restrictive. Therefore, in this paper we present a mild version of this concept, we study odd-quadra...	initial-50m	28	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
displaystyle B(X,Y)=0$ for all $\displaystyle Y\in{\mathfrak{g}}$, then $\displaystyle X=0$. Otherwise, $B$ is called degenerate. ###### Definition 1.3. A Lie superalgebra ${\mathfrak{g}}$ is odd-quadratic if there exists a bilinear form $B:{\mathfrak{g}}\times{\mathfrak{g}}\to{\mathfrak{g}}$ such that $B$ is supersy...	initial-50m	29	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
({\mathfrak{h}}^{*})$ by ${\mathfrak{g}}$ (by means of $\varphi$). ###### Theorem 1.1. [1] Let $({\mathfrak{g}},B)$ be an odd-quadratic Lie superalgebra and ${\mathfrak{h}}$ a Lie superalgebra. Consider $\psi:{\mathfrak{h}}\to Der_{a}({\mathfrak{g}},B)$ a morphism of Lie superalgebras. Define the linear map $\widetil...	initial-50m	30	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
$(V^{})_{i}:=\\{f\in V^{}:f(V_{m-i})=\\{0\\}\\}.$ It is clear that $(V^{})_{i}$ is a vector subspace of $V^{}$ and $\\{e_{m-(i-1)}^{},\dots,e_{m}^{}\\}$ is a basis of $(V^{})_{i}$, for $i\in\\{1,\dots,m\\}$, where $\\{e_{1}^{},\dots,e_{m}^{}\\}$ is a dual basis of $\mathcal{B}$. So $\mbox{\rm dim }(V^{})_{...	initial-50m	31	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
. ∎ ###### Theorem 2.1. There are no odd-quadratic Lie superalgebra $(\mathfrak{g},[\cdot,\cdot],B)$ such that the $\mathfrak{g}_{\bar{0}}$-module $\mathfrak{g}_{\bar{1}}$ is filiform and $\mbox{\rm dim }(\mathfrak{g}_{\bar{0}})\geq 2$. ###### Proof. Suppose that $(\mathfrak{g},[\cdot,\cdot],B)$ is an odd-quadratic...	initial-50m	32	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
}$-module $P({\mathfrak{h}}^{})$ is weak filiform with flag $P({\mathfrak{h}}^{})=V_{m}\supset V_{m-1}\supset\dots\supset V_{2}\supset V_{1}=\\{0\\}$ where each $V_{i}$ has the basis $\\{X_{1}^{},\dots,X_{i}^{}\\}$ for $i\in\\{2,\dots,m\\}$. ###### Example 3.2. Let $m\in\mathbb{N}\setminus\\{0,1,2\\}$. Let us c...	initial-50m	33	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez
{g}})_{\bar{0}}\oplus V_{1}$ (see [3, Lemma 2.1]). ###### Proposition 3.3. Let $(\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}},B)$ be an odd-quadratic non-abelian Lie superalgebra with $\mathfrak{g}_{\bar{0}}\neq\\{0\\}$ and $\mbox{\rm dim }{\mathfrak{g}}_{\bar{1}}=m>0,$ and such that ${\mathfra...	initial-50m	34	2,048	arxiv:2401.13017	https://arxiv.org/abs/2401.13017	https://creativecommons.org/licenses/by/4.0/	Elisabete Barreiro, Sa\"id Benayadi, Rosa M. Navarro, J.M. S\'anchez

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Cayley-Math

Cayley-Math is a collaboratively maintained collection of mathematical text for language-model research. Contributors can publish cleaned collections as separately versioned components. The first component is a frozen mirror of the mathematics subset prepared for Anholon Phase 1.

Collection	Location	Maintenance
Anholon Phase 1 mathematics mirror	Phase 1 bundle	Anholon data maintainers
OpenAI mathematics manuscripts	OpenAI Math component	Source extraction and OMP DeepSeek metadata
Contributed mathematics collections	Component guide	Each component's contributors

Phase 1 Files

Use	Files	Coverage
Train with the frozen release	NumPy archives	250,000,000 training labels across both batches
Browse every packed mathematics window	Data Studio	146,965 training rows plus validation and test
Trace sources and terms	Source catalog · Attribution guide	All included components

Phase 1 Summary

Split	Loss-bearing labels	Packed rows
Train, initial batch	50,000,000	43,833
Train, continuation	200,000,000	103,132
Development	1,024	1
Test	1,024	1

Sources include mathematical arXiv passages cached by Common Pile, excerpts from the Stacks Project, and textbooks by Wen-Wei Li. Each window has up to 2,048 token positions and uses the same 32,768-piece SentencePiece vocabulary as the full corpus.

Repository Structure

phase1/
├── math/                # Mathematics archive and packed manifest
├── attribution/         # Document-level sources, licenses, and notices
├── preview/            # Full text viewer under full/
├── sources.json        # Component counts and provenance references
└── upload-manifest.json # File sizes and hashes by repository path
components/              # Independently contributed collections
tokenizer/               # SentencePiece model and vocabulary for Phase 1
examples/                # Synthetic illustration of a source record

Browse the dataset

Select anholon-phase1 in Data Studio. The train split contains all 43,833 initial mathematics windows and all 103,132 continuation mathematics windows. The original validation and test splits each contain one row. Select openai-math for the independently contributed collection.

Each Parquet row decodes one frozen window and records its batch, row number within that batch and split, supervised label count, source, license and attribution. Padding and control tokens are omitted in the displayed text. Consecutive windows can share a boundary token. The NumPy archives contain the original token IDs and label masks.

The Parquet files and their hashes are in phase1/preview/full. The Stacks Project rows have a compressed editable UTF-8 copy with the source notices and GFDL text.

Data Fields

Each shard contains three int32 arrays with shape [rows, 2048]. The input_ids array uses zero for padding. The labels array stores shifted targets and masks unused positions with -100. The role_ids array is zero throughout plain pretraining text.

Loading the Training Pack

Set HF_DATA_REVISION to a commit SHA for reproducible access. The bundle uses the included Phase 1 tokenizer.

import os
import subprocess
from pathlib import Path

from huggingface_hub import snapshot_download

root = Path(snapshot_download(
    repo_id="MMLA-ORG/Cayley-Math",
    repo_type="dataset",
    revision=os.environ.get("HF_DATA_REVISION", "main"),
    endpoint="https://huggingface.co",
    allow_patterns=["phase1/math/**", "phase1/upload-manifest.json", "tokenizer/**"],
    local_dir="cayley-phase1",
))
math = root / "phase1" / "math"
subprocess.run(["tar", "--zstd", "-xf", str(math / "math.tar.zst"), "-C", str(math)], check=True)

Use math/manifest.json to locate and verify the extracted initial arrays. The continuation bundle has its own archive and manifest. Development and test are small monitoring splits.

Processing and Source Terms

The Phase 1 mirror preserves the original mathematics group assignments, splits and training labels. When building a new training mix from Cayley-Math, record the selected commit, component paths and file digests. Check overlap with any existing Anholon mathematics data before combining the components.

Source terms are recorded per document. Common Pile arXiv text and Wen-Wei Li excerpts retain their applicable Creative Commons terms. The Stacks Project passages retain GFDL 1.2 or later. See the batch notes and attribution and notices in phase1/attribution/math-rights/.

Citation and Revision

For research records, cite MMLA-ORG, Cayley-Math, the component and its Hugging Face revision. Cite the component's upstream works as listed in its attribution. See the MMLA project for the model family.

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