Combinatorial topology of nonclosed sets I. The first duality law; spectral duality

1960 ◽  
pp. 245-295 ◽  
Author(s):  
K. A. Sitnikov
Keyword(s):  
Combinatorial Topology

Combinatorial Topology of Polyhedra

1940 ◽  
Vol 19 (1-4) ◽  
pp. 260-288
Author(s):  
S. L. Greitzer
Keyword(s):  
Combinatorial Topology

scholarly journals A No-Go Theorem for One-Layer Feedforward Networks

2014 ◽  
Vol 26 (11) ◽  
pp. 2527-2540 ◽  
Author(s):  
Chad Giusti ◽  
Vladimir Itskov
Keyword(s):  
Activity Patterns ◽  
Maximal Activity ◽  
Neural Code ◽  
Surprising Result ◽  
Response Patterns ◽  
Neural Codes ◽  
Combinatorial Topology ◽  
Feedforward Networks ◽  
Combinatorial Codes ◽  
Recurrent Connections

It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This implies the existence of neural codes that cannot arise solely from feedforward processing. We set out to find such codes in the context of one-layer feedforward networks and identified a large class of combinatorial codes that indeed cannot be shaped by the feedforward architecture alone. However, these codes are difficult to distinguish from codes that share the same sets of maximal activity patterns in the presence of subtractive noise. When we coarsened the notion of combinatorial neural code to keep track of only maximal patterns, we found the surprising result that all such codes can in fact be realized by one-layer feedforward networks. This suggests that recurrent or many-layer feedforward architectures are not necessary for shaping the (coarse) combinatorial features of neural codes. In particular, it is not possible to infer a computational role for recurrent connections from the combinatorics of neural response patterns alone. Our proofs use mathematical tools from classical combinatorial topology, such as the nerve lemma and the existence of an inverse nerve. An unexpected corollary of our main result is that any prescribed (finite) homotopy type can be realized by a subset of the form [Formula: see text], where [Formula: see text] is a polyhedron.

Combinatorial Topology, Volume 3.

1961 ◽  
Vol 68 (4) ◽  
pp. 390
Author(s):  
S. S. Cairns ◽  
P. S. Aleksandrov
Keyword(s):  
Combinatorial Topology

scholarly journals Asynchronous Computability Theorem in Arbitrary Solo Models

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 757
Author(s):  
Yunguang Yue ◽  
Fengchun Lei ◽  
Xingwu Liu ◽  
Jie Wu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Computational Power ◽  
Nest Structure ◽  
Combinatorial Topology ◽  
Necessary And Sufficient ◽  
As If

In this paper, we establish the asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which we state a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether their output complexes hold a d-nest structure. As a byproduct, we find an alternative way to distinguish the computational power of d-solo objects for different d.

Combinatorial Topology. Vol. 1

1958 ◽  
Vol 42 (340) ◽  
pp. 155
Author(s):  
M. H. A. Newman ◽  
P. S. Aleksandrov
Keyword(s):  
Combinatorial Topology
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