Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA computational framework for connection matrix theory
156
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The connection matrix is a powerful algebraic topological tool from Conley index theory that captures relationships between isolated invariant sets. Conley index theory is a topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, the ideas of Conley have been cast into a purely computational form. In this paper we introduce a computational, categorical framework for the connection matrix theory. This contribution transforms the computational Conley theory into a computational homological theory for dynamical systems. More specifically, within this paper we have two goals: 1) We cast the connection matrix theory into appropriate categorical, homotopy-theoretic language. We identify objects of the appropriate categories which correspond to connection matrices and may be computed within the computational Conley theory paradigm by using the technique of reductions. 2) We describe an algorithm for this computation based on algebraic-discrete Morse theory.
rate research
Read More
For primes $pgeq 5 $, $K(KU_p)$ -- the algebraic $K$-theory spectrum of $(KU)^{wedge}_p$, Morava $K$-theory $K(1)$, and Smith-Toda complex $V(1)$, Ausoni and Rognes conjectured (alongside related conjectures) that $L_{K(1)}S^0 mspace{-1.5mu}xrightarrow{mspace{-2mu}text{unit} , i}~mspace{-7mu}(KU)^{wedge}_p$ induces a map $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to K(KU_p)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1)$ that is an equivalence. Since the definition of this map is not well understood, we consider $K(L_{K(1)}S^0) wedge v_2^{-1}V(1) to (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p}$, which is induced by $i$ and also should be an equivalence. We show that for any closed $G < mathbb{Z}^times_p$, $pi_ast((K(KU_p) wedge v_2^{-1}V(1))^{hG})$ is a direct sum of two pieces given by (co)invariants and a coinduced module, for $K(KU_p)_ast(V(1))[v_2^{-1}]$. When $G = mathbb{Z}^times_p$, the direct sum is, conjecturally, $K(L_{K(1)}S^0)_ast(V(1))[v_2^{-1}]$ and, by using $K(L_p)_ast(V(1))[v_2^{-1}]$, where $L_p = ((KU)^{wedge}_p)^{hmathbb{Z}/((p-1)mathbb{Z})}$, the summands simplify. The Ausoni-Rognes conjecture suggests that in [(-)^{hmathbb{Z}^times_p} wedge v_2^{-1}V(1) simeq (K(KU_p) wedge v_2^{-1}V(1))^{hmathbb{Z}^times_p},] $K(KU_p)$ fills in the blank; we show that for any $G$, the blank can be filled by $(K(KU_p))^mathrm{dis}_mathcal{O}$, a discrete $mathbb{Z}^times_p$-spectrum built out of $K(KU_p)$.
Slang is a common type of informal language, but its flexible nature and paucity of data resources present challenges for existing natural language systems. We take an initial step toward machine generation of slang by developing a framework that models the speakers word choice in slang context. Our framework encodes novel slang meaning by relating the conventional and slang senses of a word while incorporating syntactic and contextual knowledge in slang usage. We construct the framework using a combination of probabilistic inference and neural contrastive learning. We perform rigorous evaluations on three slang dictionaries and show that our approach not only outperforms state-of-the-art language models, but also better predicts the historical emergence of slang word usages from 1960s to 2000s. We interpret the proposed models and find that the contrastively learned semantic space is sensitive to the similarities between slang and conventional senses of words. Our work creates opportunities for the automated generation and interpretation of informal language.
We present a generalization of the induced matching theorem and use it to prove a generalization of the algebraic stability theorem for $mathbb{R}$-indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how the generalized algebraic stability theorem enables the computation of rigorous error bounds in the space of persistence diagrams that go beyond the typical formulation in terms of bottleneck (or log bottleneck) distance.
In recent work, Hess and Shipley defined a theory of topological coHochschild homology (coTHH) for coalgebras. In this paper we develop computational tools to study this new theory. In particular, we prove a Hochschild-Kostant-Rosenberg type theorem in the cofree case for differential graded coalgebras. We also develop a coBokstedt spectral sequence to compute the homology of coTHH for coalgebra spectra. We use a coalgebra structure on this spectral sequence to produce several computations.
Matrix Completion is the problem of recovering an unknown real-valued low-rank matrix from a subsample of its entries. Important recent results show that the problem can be solved efficiently under the assumption that the unknown matrix is incoherent and the subsample is drawn uniformly at random. Are these assumptions necessary? It is well known that Matrix Completion in its full generality is NP-hard. However, little is known if make additional assumptions such as incoherence and permit the algorithm to output a matrix of slightly higher rank. In this paper we prove that Matrix Completion remains computationally intractable even if the unknown matrix has rank $4$ but we are allowed to output any constant rank matrix, and even if additionally we assume that the unknown matrix is incoherent and are shown $90%$ of the entries. This result relies on the conjectured hardness of the $4$-Coloring problem. We also consider the positive semidefinite Matrix Completion problem. Here we show a similar hardness result under the standard assumption that $mathrm{P} e mathrm{NP}.$ Our results greatly narrow the gap between existing feasibility results and computational lower bounds. In particular, we believe that our results give the first complexity-theoretic justification for why distributional assumptions are needed beyond the incoherence assumption in order to obtain positive results. On the technical side, we contribute several new ideas on how to encode hard combinatorial problems in low-rank optimization problems. We hope that these techniques will be helpful in further understanding the computational limits of Matrix Completion and related problems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
