The Hunter-Saxton equation: remarkable structures of symmetries and conserved densities

In this paper, we present extraordinary algebraic and geometrical structures for the Hunter-Saxton equation: infinitely many commuting and non-commuting $x,t$-independent higher order symmetries and conserved densities. Using a recursive relation, we explicitly generate infinitely many higher order conserved densities dependent on arbitrary parameters. We find three Nijenhuis recursion operators resulting from Hamiltonian pairs, of which two are new. They generate three hierarchies of commuting local symmetries. Finally, we give a local recursion operator depending on an arbitrary parameter. As a by-product, we classify all anti-symmetric operators of a definite form that are compatible with the Hamiltonian operator $D_x^{-1}$

First and Second Language Acquisition of Recursive Operations: Two Studies

Linguistic theory has increasingly revolved around the notion of recursion. Most recently, many have advocated a view wherein it forms the essence of the Language Acquisition Device (LAD) purportedly contained in the human mind, while others have argued that it remains a separate and not necessarily related component of language processing. L1 Acquisition theory has suggested that appropriate recursive input is required to activate the LAD’s recursive faculties; nonetheless, L1 recursive structures may resist instruction and cause initial confusion among children. The effect that any of this may have on L2 has only begun to be studied. This dissertation attempts to fill this gap in knowledge by describing two experiments which concentrate on the interpretation of adjacent prepositional phrases (PPs). The first experiment exploits the similarity of PPs in Spanish and English by using identical prompts in both languages and with both L1 and L2 speakers, while the second experiment studies the growth of their recursivity in L1 acquisition in English. Both experiments also study the effect that unique pairs of prepositions have on this as well as the effect created by extending the chain of adjacent PPs beyond only two. The results provide a valuable insight into the interpretation of these structures. Recursive responses suggest an L2 path to acquisition which may result in L1 levels of performance. Yet Spanish and English each display their own behavior patterns, revealing dissimilarities that suggest Spanish possesses a more productive right-recursive rule than does English. Growth in L1 child English is also clearly observed in some scenarios but not in all. The important role individual prepositions clearly play is observed in both experiments, with unique pairs having unique levels of recursion. Increasing complexity of the NP based on number of PPs also entrenches recursion in interpretation. Nonetheless, certain participants resist recursion in multiple scenarios, a fact which may support an argument for targeted recursive input

Recursive Neural Networks Can Learn Logical Semantics

Tree-structured recursive neural networks (TreeRNNs) for sentence meaning have been successful for many applications, but it remains an open question whether the fixed-length representations that they learn can support tasks as demanding as logical deduction. We pursue this question by evaluating whether two such models---plain TreeRNNs and tree-structured neural tensor networks (TreeRNTNs)---can correctly learn to identify logical relationships such as entailment and contradiction using these representations. In our first set of experiments, we generate artificial data from a logical grammar and use it to evaluate the models' ability to learn to handle basic relational reasoning, recursive structures, and quantification. We then evaluate the models on the more natural SICK challenge data. Both models perform competitively on the SICK data and generalize well in all three experiments on simulated data, suggesting that they can learn suitable representations for logical inference in natural language

Quantification of the differences between quenched and annealed averaging for RNA secondary structures

The analytical study of disordered system is usually difficult due to the necessity to perform a quenched average over the disorder. Thus, one may resort to the easier annealed ensemble as an approximation to the quenched system. In the study of RNA secondary structures, we explicitly quantify the deviation of this approximation from the quenched ensemble by looking at the correlations between neighboring bases. This quantified deviation then allows us to propose a constrained annealed ensemble which predicts physical quantities much closer to the results of the quenched ensemble without becoming technically intractable.Comment: 9 pages, 14 figures, submitted to Phys. Rev.

BattRAE: Bidimensional Attention-Based Recursive Autoencoders for Learning Bilingual Phrase Embeddings

In this paper, we propose a bidimensional attention based recursive autoencoder (BattRAE) to integrate clues and sourcetarget interactions at multiple levels of granularity into bilingual phrase representations. We employ recursive autoencoders to generate tree structures of phrases with embeddings at different levels of granularity (e.g., words, sub-phrases and phrases). Over these embeddings on the source and target side, we introduce a bidimensional attention network to learn their interactions encoded in a bidimensional attention matrix, from which we extract two soft attention weight distributions simultaneously. These weight distributions enable BattRAE to generate compositive phrase representations via convolution. Based on the learned phrase representations, we further use a bilinear neural model, trained via a max-margin method, to measure bilingual semantic similarity. To evaluate the effectiveness of BattRAE, we incorporate this semantic similarity as an additional feature into a state-of-the-art SMT system. Extensive experiments on NIST Chinese-English test sets show that our model achieves a substantial improvement of up to 1.63 BLEU points on average over the baseline.Comment: 7 pages, accepted by AAAI 201

All Maximal Independent Sets and Dynamic Dominance for Sparse Graphs

We describe algorithms, based on Avis and Fukuda's reverse search paradigm, for listing all maximal independent sets in a sparse graph in polynomial time and delay per output. For bounded degree graphs, our algorithms take constant time per set generated; for minor-closed graph families, the time is O(n) per set, and for more general sparse graph families we achieve subquadratic time per set. We also describe new data structures for maintaining a dynamic vertex set S in a sparse or minor-closed graph family, and querying the number of vertices not dominated by S; for minor-closed graph families the time per update is constant, while it is sublinear for any sparse graph family. We can also maintain a dynamic vertex set in an arbitrary m-edge graph and test the independence of the maintained set in time O(sqrt m) per update. We use the domination data structures as part of our enumeration algorithms.Comment: 10 page

Layered Quantum Key Distribution

We introduce a family of QKD protocols for distributing shared random keys within a network of $n$ users. The advantage of these protocols is that any possible key structure needed within the network, including broadcast keys shared among subsets of users, can be implemented by using a particular multi-partite high-dimensional quantum state. This approach is more efficient in the number of quantum channel uses than conventional quantum key distribution using bipartite links. Additionally, multi-partite high-dimensional quantum states are becoming readily available in quantum photonic labs, making the proposed protocols implementable using current technology.Comment: 11 pages, 5 figures. In version 2 we extended section 4 about the dimension-rate trade-off and corrected minor error

Analytical description of finite size effects for RNA secondary structures

The ensemble of RNA secondary structures of uniform sequences is studied analytically. We calculate the partition function for very long sequences and discuss how the cross-over length, beyond which asymptotic scaling laws apply, depends on thermodynamic parameters. For realistic choices of parameters this length can be much longer than natural RNA molecules. This has to be taken into account when applying asymptotic theory to interpret experiments or numerical results.Comment: 10 pages, 13 figures, published in Phys. Rev.