Simplifying Bode's maximum available feedback design

Bode's method for obtaining 'maximum obtainable feedback' is a good example of a nontrivial feedback system design technique, but it is largely overlooked. This paper shows how the associated mathematics can be simplified and linear elements used in its implementation, so as to make it accessible for teaching to undergraduates.

Achieving or exceeding maximum available feedback

A feedback system for control or electronics should have high loop gain, so that its output is close to its desired state, and the effects of changes in the system and of disturbances are minimised. Bode proposed a method for single loop feedback systems to obtain the maximum available feedback, defined as the largest possible loop gain over a bandwidth pertinent to the system, with appropriate gain and phase margins. The method uses asymptotic approximations, and this paper describes some novel adjustments to the asymptotes, so that the final system often exceeds the maximum available feedback. The implementation of the method requires the cascading of a series of lead-lag element. This paper describes a new way to determine how many elements should be used.

Maximum induced subgraph of a recursive circulant

The recursive circulant RC(2(n), 4) enjoys several attractive topological properties. Let max_epsilon(G) (m) denote the maximum number of edges in a subgraph of graph G induced by m nodes. In this paper, we show that max_epsilon(RC(2n,4))(m) = Sigma(i)(r)=(0)(p(i)/2 + i)2(Pi), where p(0) > p(1) > ... > p(r) are nonnegative integers defined by m = Sigma(i)(r)=(0)2(Pi). We then apply this formula to find the bisection width of RC(2(n), 4). The conclusion shows that, as n-dimensional cube, RC(2(n), 4) enjoys a linear bisection width. (c) 2005 Elsevier B.V. All rights reserved.

The evolution of maximum body size of terrestrial mammals

The extinction of dinosaurs at the Cretaceous/Paleogene (K/Pg) boundary was the seminal event that opened the door for the subsequent diversification of terrestrial mammals. Our compilation of maximum body size at the ordinal level by sub-epoch shows a near-exponential increase after the K/Pg. On each continent, the maximum size of mammals leveled off after 40 million years ago and thereafter remained approximately constant. There was remarkable congruence in the rate, trajectory, and upper limit across continents, orders, and trophic guilds, despite differences in geological and climatic history, turnover of lineages, and ecological variation. Our analysis suggests that although the primary driver for the evolution of giant mammals was diversification to fill ecological niches, environmental temperature and land area may have ultimately constrained the maximum size achieved.

Vertical and horizontal processes in the global atmosphere and the maximum entropy production conjecture

The objective of this paper is to reconsider the Maximum Entropy Production conjecture (MEP) in the context of a very simple two-dimensional zonal-vertical climate model able to represent the total material entropy production due at the same time to both horizontal and vertical heat fluxes. MEP is applied first to a simple four-box model of climate which accounts for both horizontal and vertical material heat fluxes. It is shown that, under condition of fixed insolation, a MEP solution is found with reasonably realistic temperature and heat fluxes, thus generalising results from independent two-box horizontal or vertical models. It is also shown that the meridional and the vertical entropy production terms are independently involved in the maximisation and thus MEP can be applied to each subsystem with fixed boundary conditions. We then extend the four-box model by increasing its resolution, and compare it with GCM output. A MEP solution is found which is fairly realistic as far as the horizontal large scale organisation of the climate is concerned whereas the vertical structure looks to be unrealistic and presents seriously unstable features. This study suggest that the thermal meridional structure of the atmosphere is predicted fairly well by MEP once the insolation is given but the vertical structure of the atmosphere cannot be predicted satisfactorily by MEP unless constraints are imposed to represent the determination of longwave absorption by water vapour and clouds as a function of the state of the climate. Furthermore an order-of-magnitude estimate of contributions to the material entropy production due to horizontal and vertical processes within the climate system is provided by using two different methods. In both cases we found that approximately 40 mW m−2 K−1 of material entropy production is due to vertical heat transport and 5–7 mW m−2 K−1 to horizontal heat transport

Last Glacial Maximum CO2 and δ13C successfully reconciled

During the Last Glacial Maximum (LGM, ∼21,000 years ago) the cold climate was strongly tied to low atmospheric CO2 concentration (∼190 ppm). Although it is generally assumed that this low CO2 was due to an expansion of the oceanic carbon reservoir, simulating the glacial level has remained a challenge especially with the additional δ13C constraint. Indeed the LGM carbon cycle was also characterized by a modern-like δ13C in the atmosphere and a higher surface to deep Atlantic δ13C gradient indicating probable changes in the thermohaline circulation. Here we show with a model of intermediate complexity, that adding three oceanic mechanisms: brine induced stratification, stratification-dependant diffusion and iron fertilization to the standard glacial simulation (which includes sea level drop, temperature change, carbonate compensation and terrestrial carbon release) decreases CO2 down to the glacial value of ∼190 ppm and simultaneously matches glacial atmospheric and oceanic δ13C inferred from proxy data. LGM CO2 and δ13C can at last be successfully reconciled.

High resolution Northern Hemisphere wintertime mid-latitude dynamics during the Last Glacial Maximum

Hourly winter weather of the Last Glacial Maximum (LGM) is simulated using the Community Climate Model version 3 (CCM3) on a globally resolved T170 (75 km) grid. Results are compared to a longer LGM climatological run with the same boundary conditions and monthly saves. Hourly-scale animations are used to enhance interpretations. The purpose of the study is to explore whether additional insights into ice age conditions can be gleaned by going beyond the standard employment of monthly average model statistics to infer ice age weather and climate. Results for both LGM runs indicate a decrease in North Atlantic and increase in North Pacific cyclogenesis. Storm trajectories react to the mechanical forcing of the Laurentide Ice Sheet, with Pacific storms tracking over middle Alaska and northern Canada, terminating in the Labrador Sea. This result is coincident with other model results in also showing a significant reduction in Greenland wintertime precipitation – a response supported by ice core evidence. Higher-temporal resolution puts in sharper focus the close tracking of Pacific storms along the west coast of North America. This response is consistent with increased poleward heat transport in the LGM climatological run and could help explain “early” glacial warming inferred in this region from proxy climate records. Additional analyses shows a large increase in central Asian surface gustiness that support observational inferences that upper-level winds associated with Asian- Pacific storms transported Asian dust to Greenland during the LGM.

The maximum rate of mammal evolution

How fast can a mammal evolve from the size of a mouse to the size of an elephant? Achieving such a large transformation calls for major biological reorganization. Thus, the speed at which this occurs has important implications for extensive faunal changes, including adaptive radiations and recovery from mass extinctions. To quantify the pace of large-scale evolution we developed a metric, clade maximum rate, which represents the maximum evolutionary rate of a trait within a clade. We applied this metric to body mass evolution in mammals over the last 70 million years, during which multiple large evolutionary transitions occurred in oceans and on continents and islands. Our computations suggest that it took a minimum of 1.6, 5.1, and 10 million generations for terrestrial mammal mass to increase 100-, and 1,000-, and 5,000- fold, respectively. Values for whales were down to half the length (i.e., 1.1, 3, and 5 million generations), perhaps due to the reduced mechanical constraints of living in an aquatic environment. When differences in generation time are considered, we find an exponential increase in maximum mammal body mass during the 35 million years following the Cretaceous–Paleogene (K–Pg) extinction event. Our results also indicate a basic asymmetry in macroevolution: very large decreases (such as extreme insular dwarfism) can happen at more than 10 times the rate of increases. Our findings allow more rigorous comparisons of microevolutionary and macroevolutionary patterns and processes. Keywords: haldanes, biological time, scaling, pedomorphosis