Approximating Threshold Circuits by Rational Functions

AbstractMotivated by the problem of understanding the limitations of threshold networks for representing boolean functions, we consider size-depth trade-offs for threshold circuits that compute the parity function. Using a fundamental result in the theory of rational approximation, we show how to approximate small threshold circuits by rational functions of low degree. We apply this result to establish an almost optimal lower bound of Ω(n2/ln2n) on the number of edges of any depth-2 threshold circuit with polynomially bounded weights that computes the parity function. We also prove that any depth-3 threshold circuit with polynomially bounded weights requires Ω(n1.2/ln5/3n) edges to compute parity. On the other hand, we give a construction of a depth d threshold circuit that computes parity with n1+1/Θ(φd) edges where φ = (1 + √5)/2 is the golden ratio. We conjecture that there are no linear size bounded depth threshold circuits for computing parity

Unbounded-Error Classical and Quantum Communication Complexity

Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, \cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum} communication complexity in the {\em one-way communication} model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical one-way communication complexity. In this paper, we extend the arrangement argument to the {\em two-way} and {\em simultaneous message passing} (SMP) models. As a result, we show similarly tight bounds of the unbounded-error two-way/one-way/SMP quantum/classical communication complexities for {\em any} partial/total Boolean function, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is also used to show that the gap between {\em weakly} unbounded-error quantum and classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200

Partition-function zeros of spherical spin glasses and their relevance to chaos

We investigate partition-function zeros of the many-body interacting spherical spin glass, the so-called $p$-spin spherical model, with respect to the complex temperature in the thermodynamic limit. We use the replica method and extend the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex-temperature plane and calculate the density of zeros in each phase. Near the imaginary axis away from the origin, there is a replica symmetric phase having a large density. On the other hand, we observe no density in the spin-glass phases, irrespective of the replica symmetry breaking. We speculate that this suggests the absence of the temperature chaos. To confirm this, we investigate the multiple many-body interacting case which is known to exhibit the chaos effect. The result shows that the density of zeros actually takes finite values in the spin-glass phase, even on the real axis. These observations indicate that the density of zeros is more closely connected to the chaos effect than the replica symmetry breaking.Comment: 22 pages, 8 figure

Unbounded-error One-way Classical and Quantum Communication Complexity

This paper studies the gap between quantum one-way communication complexity $Q(f)$ and its classical counterpart $C(f)$, under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for {\em any} (total or partial) Boolean function $f$, $Q(f)=\lceil C(f)/2 \rceil$, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of $(m,n,p)$-QRAC which is the $n$-qubit random access coding that can recover any one of $m$ original bits with success probability $\geq p$. We can prove that $(m,n,>1/2)$-QRAC exists if and only if $m\leq 2^{2n}-1$. Previously, only the construction of QRAC using one qubit, the existence of $(O(n),n,>1/2)$-RAC, and the non-existence of $(2^{2n},n,>1/2)$-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200

Fluprostenol-Induced MAPK Signaling is Independent of Aging in Fischer 344/NNiaHSd x Brown Norway/BiNia Rat Aorta

The factors that regulate vascular mechanotransduction and how this process may be altered with aging are poorly understood and have not been widely studied. Recent data suggest that increased tissue loading can result in the release of prostaglandin F2 alpha (PGF2α) and other reports indicate that aging diminishes the ability of the aged aorta to activate mitogen activated protein kinase (MAPK) signaling in response to increased loading. Using ex vivo incubations, here we investigate whether aging affects the ability of the aorta to induce phosphorylation of extracellular signal-regulated kinase 1/2 (ERK½-MAPK), p38-MAPK, and Jun N-terminal kinase (JNK-MAPK) activation following stimulation with a PGF2α analog, fluprostenol. Compared to aortas from 6-mo animals, the amounts of ERK½- and p38-MAPK remained unchanged with aging, while the level of JNK-MAPK protein increased by 135% and 100% at 30- and 36-mo, respectively. Aging increased the basal phosphorylation of ERK½ (115% and 47%) and JNK (29% and 69%) (p \u3c0.05) in 30- and 36-mo aortas, while p38 phosphorylation levels remained unaltered. Compared to age-matched controls, fluprostenol induced phosphorylation of ERK½ (310%, 286%, and 554%), p38-MAPK (unchanged, 48%, and 148%), and JNK (78%, 88%, and 95%) in 6-, 30- and 36-mo aortas, respectively. These findings suggest that aging does not affect the ability of the rat aorta to activate ERK½-, p38-MAPK, and JNK-MAPK phosphorylation in response to PGF2α stimulation

The interpretation of the field angle dependence of the critical current in defect-engineered superconductors

We apply the vortex path model of critical currents to a comprehensive analysis of contemporary data on defect-engineered superconductors, showing that it provides a consistent and detailed interpretation of the experimental data for a diverse range of materials. We address the question of whether electron mass anisotropy plays a role of any consequence in determining the form of this data and conclude that it does not. By abandoning this false interpretation of the data, we are able to make significant progress in understanding the real origin of the observed behavior. In particular, we are able to explain a number of common features in the data including shoulders at intermediate angles, a uniform response over a wide angular range and the greater discrimination between individual defect populations at higher fields. We also correct several misconceptions including the idea that a peak in the angular dependence of the critical current is a necessary signature of strong correlated pinning, and conversely that the existence of such a peak implies the existence of correlated pinning aligned to the particular direction. The consistency of the vortex path model with the principle of maximum entropy is introduced.Comment: 14 pages, 7 figure

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Altered Regulation of Contraction-Induced Akt/mTOR/p70S6k Pathway Signaling in Skeletal Muscle of the Obese Zucker Rat

Increased muscle loading results in the phosphorylation of the 70 kDa ribosomal S6 kinase (p70S6k), and this event is strongly correlated with the degree of muscle adaptation following resistance exercise. Whether insulin resistance or the comorbidities associated with this disorder may affect the ability of skeletal muscle to activate p70S6k signaling following an exercise stimulus remains unclear. Here, we compare the contraction-induced activation of p70S6k signaling in the plantaris muscles of lean and insulin resistant obese Zucker rats following a single bout of increased contractile loading. Compared to lean animals, the basal phosphorylation of p70S6k (Thr389; 37.2% and Thr421/Ser424; 101.4%), Akt (Thr308; 25.1%), and mTOR (Ser2448; 63.0%) was higher in obese animals. Contraction increased the phosphorylation of p70S6k (Thr389), Akt (Ser473), and mTOR (Ser2448) in both models however the magnitude and kinetics of activation differed between models. These results suggest that contraction-induced activation of p70S6k signaling is altered in the muscle of the insulin resistant obese Zucker rat

Hydrological, Sedimentological, and Meteorological Observations and Analysis on the Sagavanirktok River

The Dalton Highway near Deadhorse was closed twice during late March and early April 2015 because of extensive overflow from the Sagavanirktok River that flowed over the highway. That spring, researchers from the Water and Environmental Research Center at the University of Alaska Fairbanks (UAF) monitored the river conditions during breakup, which was characterized by unprecedented flooding that overtopped and consequently destroyed several sections of the Dalton Highway near Deadhorse. The UAF research team has monitored breakup conditions at the Sagavanirktok River since that time. Given the magnitude of the 2015 flooding, the Alyeska Pipeline Service Company started a long-term monitoring program within the river basin. In addition, the Alaska Department of Transportation and Public Facilities (ADOT&PF) funded a multiyear project related to sediment transport conditions along the Sagavanirktok River. The general objectives of these projects include determining ice elevations, identifying possible water sources, establishing surface hydro-meteorological conditions prior to breakup, measuring hydro-sedimentological conditions during breakup and summer, and reviewing historical imagery of the aufeis extent. In the present report, we focus on new data and analyze it in the context of previous data. We calculated and compared ice thickness near Franklin Bluffs for 2015, 2016, and 2017, and found that, in general, ice thickness during both 2015 and 2016 was greater than in 2017 across most of the study area. Results from a stable isotope analysis indicate that winter overflow, which forms the aufeis in the river area near Franklin Bluffs, has similar isotopic characteristics to water flowing from mountain springs. End-of-winter snow surveys (in 2016/2017) within the watershed indicate that the average snow water equivalent was similar to what we observed in winter 2015/2016. Air temperatures in May 2017 were low on the Alaska North Slope, which caused a long and gradual breakup, with peak flows occurring in early June, compared with mid-May in both 2015 and 2016. Maximum discharge measured at the East Bank station, near Franklin Bluffs was 750 m3/s (26,485 ft3/s) on May 30, 2017, while the maximum measured flow was 1560 m3/s (55,090 ft3/s) at the same station on May 20, 2015. Available cumulative rainfall data indicate that 2016 was wetter than 2017. ii In September 2015, seven dry and wet pits were dug near the hydro-sedimentological monitoring stations along the Sagavanirktok River study reach. The average grain-size of the sediment of exposed gravel bars at sites located upstream of the Ivishak-Sagavanirktok confluence show relatively constant values. Grain size becomes finer downstream of the confluence. We conducted monthly topo-bathymetric surveys during the summer months of 2016 and 2017 in each pit. Sediment deposition and erosion was observed in each of the pits. Calculated sedimentation volumes in each pit show the influence of the Ivishak River in the bed sedimenttransport capacity of the Sagavanirktok River. In addition, comparison between dry and wet pit sedimentation volumes in some of the stations proves the complexity of a braided river, which is characterized by frequent channel shifting A two-dimensional hydraulic model is being implemented for a material site. The model will be used to estimate the required sediment refill time based on different river conditions.ABSTRACT ..................................................................................................................................... i LIST OF FIGURES ......................................................................................................................... i LIST OF TABLES ....................................................................................................................... xiv ACKNOWLEDGMENTS AND DISCLAIMER ........................................................................ xvi CONVERSION FACTORS, UNITS, WATER QUALITY UNITS, VERTICAL AND HORIZONTAL DATUM, ABBREVIATIONS, AND SYMBOLS .......................................... xvii ABBREVIATIONS, ACRONYMS, AND SYMBOLS .............................................................. xix 1 INTRODUCTION ................................................................................................................... 1 2 STUDY AREA ........................................................................................................................ 2 2.1 Sagavanirktok River near MP318 Site 066 (DSS4) ......................................................... 7 2.2 Sagavanirktok River at Happy Valley Site 005 (DSS3) .................................................. 7 2.3 Sagavanirktok River below the Confluence with the Ivishak River (DSS2) ................... 9 2.4 Sagavanirktok River near MP405 Site 042 (DSS1) ....................................................... 10 3 METHODOLOGY AND EQUIPMENT .............................................................................. 13 3.1 Pits .................................................................................................................................. 13 3.1.1 Excavation............................................................................................................... 13 3.1.2 Surveying ................................................................................................................ 14 3.2 Surface Meteorology ...................................................................................................... 15 3.3 Aufeis Extent .................................................................................................................. 17 3.3.1 Field Methods ......................................................................................................... 18 3.3.2 Imagery ................................................................................................................... 18 3.4 Water Level Measurements ............................................................................................ 19 3.5 Runoff............................................................................................................................. 20 3.6 Suspended Sediment ...................................................................................................... 21 3.7 Turbidity ......................................................................................................................... 22 3.8 Stable Isotopes................................................................................................................ 22 4 RESULTS .............................................................................................................................. 23 4.1 Meteorology ................................................................................................................... 23 4.1.1 Air Temperature ...................................................................................................... 23 4.1.2 Precipitation ............................................................................................................ 31 4.1.2.1 Cold Season Precipitation ................................................................................ 31 4.1.2.2 Warm Season Precipitation ............................................................................. 36 4.1.3 Wind Speed and Direction ...................................................................................... 39 iv 4.2 Aufeis Extent .................................................................................................................. 40 4.2.1 Historical Aufeis at Franklin Bluffs ........................................................................ 41 4.2.2 Delineating Ice Surface Elevation with GPS and Aerial Imagery .......................... 45 4.3 Surface Water Hydrology ............................................................................................... 52 4.3.1 Sagavanirktok River at MP318 (DSS4) .................................................................. 58 4.3.2 Sagavanirktok River at Happy Valley (DSS3) ....................................................... 61 4.3.3 Sagavanirktok River near MP347 (ASS1) .............................................................. 65 4.3.4 Sagavanirktok River below the Ivishak River (DSS2) ........................................... 66 4.3.5 Sagavanirktok River at East Bank (DSS5) near Franklin Bluffs ............................ 70 4.3.6 Sagavanirktok River at MP405 (DSS1) West Channel .......................................... 78 4.3.7 Additional Field Observations ................................................................................ 82 4.3.8 Preliminary Rating Curves and Estimated Discharge ............................................. 85 4.4 Stable Isotopes................................................................................................................ 86 4.5 Sediment Grain Size Distribution .................................................................................. 90 4.5.1 Streambed Sediment Grain Size Distribution ......................................................... 90 4.5.2 Suspended Sediment Grain Size Distribution ......................................................... 94 4.6 Suspended Sediment Concentration ............................................................................... 95 4.6.1 Sagavanirktok River near MP318 (DSS4) .............................................................. 95 4.6.2 Sagavanirktok River at Happy Valley (DSS3) ..................................................... 100 4.6.3 Sagavanirktok River below the Ivishak River (DSS2) ......................................... 105 4.6.4 Sagavanirktok River near MP405 (DSS1) ............................................................ 111 4.6.5 Discussion ............................................................................................................. 114 4.7 Turbidity ....................................................................................................................... 116 4.7.1 Sagavanirktok River near MP318 (DSS4) ............................................................ 116 4.7.2 Sagavanirktok River at Happy Valley (DSS3) ..................................................... 119 4.7.3 Sagavanirktok River below the Ivishak (DSS2) ................................................... 124 4.7.4 Sagavanirktok River near MP405 (DSS1) ............................................................ 126 4.7.5 Discussion ............................................................................................................. 130 4.8 Analysis of Pits............................................................................................................. 130 4.8.1 Photographs of Pits ............................................................................................... 130 4.8.2 GIS Analysis of Pit Bathymetry ........................................................................... 141 4.8.3 Pit Sedimentation .................................................................................................. 142 4.8.4 Erosion Surveys .................................................................................................... 149 4.8.5 Patterns of Sediment Transport Along the River .................................................. 156 v 4.9 Hydraulic Modeling ..................................................................................................... 158 4.9.1 Model Development .............................................................................................. 160 4.9.2 Results of Simulation ............................................................................................ 165 5 CONCLUSIONS ................................................................................................................. 171 6 REFERENCES .................................................................................................................... 174 7 APPENDICES ..................................................................................................................... 18