Exponential function

Summary

The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to opine that the exponential function is "the most important function in mathematics". The exponential function satisfies the exponentiation identity e^{x+y} = e^x e^y \text{ for all } x,y\in\mat

Official source

https://en.wikipedia.org/wiki/Exponential_function

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Combinatorial algorithms for wireless information flow

Aurore Amaudruz, Christina Fragouli

A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr, Diggavi and Tse proposed a linear binary deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the binary adjacency matrices describing source-destination cuts. However, since there exists an exponential number of cuts, identifying the capacity through exhaustive search becomes infeasible. In this work, we develop a polynomial time algorithm that discovers the relay encoding strategy to achieve the min-cut value in binary linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between edges to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-bit processing at the intermediate nodes, thereby constructively yielding finite length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.

2009

An Optimal Allocation between R&D and Prototype Funding: The Case of Generation IV International Forum’s Nuclear Energy Initiative

National Research Council of the National Academies, Prospective Evaluation of Applied Energy Research and Development at DOE (Phase One): A First Look Forward (2005) proposes a cost-benefit methodology to evaluate U.S. Department of Energy’s Research, Development, and Demonstration (RD&D) programs. This paper develops the methodology for nuclear energy programs. The RD&D process is analyzed in two stages with two success probabilities: (1) that the technology will transition from the R&D Stage to the Prototype Demonstration Stage, and (2) that the technology will be adopted commercially. It models discounted expected total benefits of an RD&D program as a function of the levels of funding, stage durations, the probabilities of success, and spillovers to other technologies. Project duration is an exponential function of funding. Project success is a logistic function of funding and uncertainty. Spillovers are linear functions of funding at each stage. This specification allows calculation of the marginal effects of changes in funding on discounted expected total benefits for a single technology. The paper uses this method to offer an optimal allocation of pre-prototype R&D funding in the development of the Generation IV International Forum’s advanced nuclear energy systems under a specific parameterization and funding constraint.

2005

Let K be a finite extension of Q(p), let L/K be a finite abelian Galois extension of odd degree and let D-L be the valuation ring of L. We define A(L/K) to be the unique fractional D-L-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Q(p) contained in certain cyclotomic extensions, Erez has described integral normal bases for A(L)/Q(p) that are self-dual with respect to the trace form. Assuming K/Q(p) to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. (C) 2009 Elsevier Inc. All rights reserved.

2009