scholarly journals Universal points in the asymptotic spectrum of tensors

2021 ◽  
Author(s):  
Matthias Christandl ◽  
Péter Vrana ◽  
Jeroen Zuiddam
Keyword(s):  
Asymptotic Spectrum

scholarly journals A Generalization of Strassen’s Theorem on Preordered Semirings

Order ◽  
2021 ◽  
Author(s):  
Péter Vrana
Keyword(s):  
Compact Hausdorff Space ◽  
Polynomial Growth ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Equivalent Condition ◽  
Real Numbers ◽  
Asymptotic Spectrum ◽  
Ring Of Continuous Functions ◽  
Archimedean Property ◽  
Locally Compact Hausdorff Space

AbstractGiven a commutative semiring with a compatible preorder satisfying a version of the Archimedean property, the asymptotic spectrum, as introduced by Strassen (J. reine angew. Math. 1988), is an essentially unique compact Hausdorff space together with a map from the semiring to the ring of continuous functions. Strassen’s theorem characterizes an asymptotic relaxation of the preorder that asymptotically compares large powers of the elements up to a subexponential factor as the pointwise partial order of the corresponding functions, realizing the asymptotic spectrum as the space of monotone semiring homomorphisms to the nonnegative real numbers. Such preordered semirings have found applications in complexity theory and information theory. We prove a generalization of this theorem to preordered semirings that satisfy a weaker polynomial growth condition. This weaker hypothesis does not ensure in itself that nonnegative real-valued monotone homomorphisms characterize the (appropriate modification of the) asymptotic preorder. We find a sufficient condition as well as an equivalent condition for this to hold. Under these conditions the asymptotic spectrum is a locally compact Hausdorff space satisfying a similar universal property as in Strassen’s work.

scholarly journals On the Asymptotic Spectrum of Finite Element Matrix Sequences

2007 ◽  
Vol 45 (2) ◽  
pp. 746-769 ◽  
Author(s):  
Bernhard Beckermann ◽  
Stefano Serra‐Capizzano
Keyword(s):  
Finite Element ◽  
Asymptotic Spectrum ◽  
Element Matrix

scholarly journals On the Non-Adaptive Zero-Error Capacity of the Discrete Memoryless Two-Way Channel

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1518
Author(s):  
Yujie Gu ◽  
Ofer Shayevitz
Keyword(s):  
Linear Programming ◽  
Linear Codes ◽  
Single Letter ◽  
Capacity Region ◽  
Probability Of Error ◽  
Asymptotic Spectrum ◽  
Point To Point ◽  
The One ◽  
Outer Bound ◽  
Spectrum Of Graphs

We study the problem of communicating over a discrete memoryless two-way channel using non-adaptive schemes, under a zero probability of error criterion. We derive single-letter inner and outer bounds for the zero-error capacity region, based on random coding, linear programming, linear codes, and the asymptotic spectrum of graphs. Among others, we provide a single-letter outer bound based on a combination of Shannon’s vanishing-error capacity region and a two-way analogue of the linear programming bound for point-to-point channels, which, in contrast to the one-way case, is generally better than both. Moreover, we establish an outer bound for the zero-error capacity region of a two-way channel via the asymptotic spectrum of graphs, and show that this bound can be achieved in certain cases.

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