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Identities versus bijections [PDF]

, 2004
Consider the infinite product (1+x)(1+x 2)(1+x 3)(1+x 4) · · · and expand it in the usual way into a series \( \sum_{n \geq 0} \) a n x n by grouping together those products that yield the same power x n . By inspection we find for the first terms $$ \label{1} \prod_{k \geq 1} (1+x^{k}) = 1 + x + x^{2} + 2x^{3} + 2x^{4} + 3x^{5} + 4x^{6} + 5x^{7} +
Martin Aigner, Günter M. Ziegler
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Endless Jailbreaks with Bijection Learning

International Conference on Learning Representations
Despite extensive safety measures, LLMs are vulnerable to adversarial inputs, or jailbreaks, which can elicit unsafe behaviors. In this work, we introduce bijection learning, a powerful attack algorithm which automatically fuzzes LLMs for safety ...
Brian R.Y. Huang, Maximilian Li, Leonard Tang   +2 more
semanticscholar   +1 more source

Differences of Bijections

The American Mathematical Monthly, 2019
When is a function from an abelian group to itself expressible as a difference of two bijections? Answering this question for finite cyclic groups solves a problem about juggling.
Daniel J. Velleman, Daniel J. Velleman, Daniel H. Ullman   +2 more
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Bijections for the Schröder Numbers

Mathematics Magazine, 2000
(2000). Bijections for the Schroder Numbers. Mathematics Magazine: Vol. 73, No. 5, pp. 369-376.
Robert A. Sulanke, Louis W. Shapiro
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Bijective A-transducers

20th Annual Symposium on Foundations of Computer Science (sfcs 1979), 1979
In this paper we study bijective a-transducers. We derive necessary and sufficient conditions on pairs of regular sets (R,S) such that a bijective a-transducer, mapping R cnto S exists. The results obtained allow the systematic construction of an a-transducer, mapping a set R onto a set S bijectively for surprisingly "different" regular sets R and S.
Hermann A. Maurer, H. Nivat
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Warnaar’s bijection and colored partition identities, II

The Ramanujan journal, 2012
In our previous paper (J. Comb. Theory Ser. A 120(1):28–38, 2013), we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular ...
Colin Sandon, Fabrizio Zanello
semanticscholar   +1 more source

Efficient bijective parameterizations

ACM Transactions on Graphics, 2020
We propose a novel method to efficiently compute bijective parameterizations with low distortion on disk topology meshes. Our method relies on a second-order solver. To design an efficient solver, we develop two key techniques. First, we propose a coarse shell to substantially reduce the number of collision constraints that are used to ...
Jian-Ping Su, Chunyang Ye, Xiao-Ming Fu, Ligang Liu   +3 more
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KKR type bijection for the exceptional affine algebra E_6^{(1)}

, 2011
For the exceptional affine type E_6^{(1)} we establish a statistic-preserving bijection between the highest weight paths consisting of the simplest Kirillov-Reshetikhin crystal and the rigged configurations.
M. Okado, Nobumasa Sano
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Bijective projection in a shell

ACM Transactions on Graphics, 2020
We introduce an algorithm to convert a self-intersection free, orientable, and manifold triangle mesh T into a generalized prismatic shell equipped with a bijective projection operator to map T to a class of discrete surfaces contained within the shell whose normals ...
Denis Zorin, Zhongshi Jiang, Teseo Schneider, Daniele Panozzo   +3 more
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Identités et bijections

2013
On considere le produit infini (1 + x)(1 + x2)(1 + x3)(1 + x4) ⋯ que l’on developpe selon la methode habituelle sous la forme d’une serie \( \sum {_{n \geqslant 0} a_n x^n } \) en regroupant les termes correspondant a la meme puissance de x.
Martin Aigner, Günter M. Ziegler
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