On the Critical-Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective.

We study the 'critical moments' of subcritical Gaussian multiplicative chaos (GMCs) in dimensions d ≤ 2 . In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments.

A Spin Glass Model for the Loss Surfaces of Generative Adversarial Networks.

We present a novel mathematical model that seeks to capture the key design feature of generative adversarial networks (GANs). Our model consists of two interacting spin glasses, and we conduct an extensive theoretical analysis of the complexity of the model's critical points using techniques from Random Matrix Theory. The result is insights into the loss surfaces of large GANs that build upon prior insights for simpler networks, but also reveal new structure unique to this setting which explains the greater difficulty of training GANs.

Quantification of Spatial and Temporal Trends in Atmospheric Mercury Deposition across Canada over the Past 30 Years.

Mercury (Hg) is a pollutant of concern across Canada and transboundary anthropogenic Hg sources presently account for over 95% of national anthropogenic Hg deposition. This study applies novel statistical analyses of 82 high-resolution dated lake sediment cores collected from 19 regions across Canada, including nearby point sources and in remote regions and spanning a full west-east geographical range of ∼4900 km (south of 60°N and between 132 and 64°W) to quantify the recent (1990-2018) spatial and temporal trends in anthropogenic atmospheric Hg deposition. Temporal trend analysis shows significant synchronous decreasing trends in post-1990 anthropogenic Hg fluxes in western Canada in contrast to increasing trends in the east, with spatial patterns largely driven by longitude and proximity to known point source(s). Recent sediment-derived Hg fluxes agreed well with the available wet deposition monitoring. Sediment-derived atmospheric Hg deposition rates also compared well to the modeled values derived from the Hg model, when lake sites located nearby (<100 km) point sources were omitted due to difficulties in comparison between the sediment-derived and modeled values at deposition "hot spots". This highlights the applicability of multi-core approaches to quantify spatio-temporal changes in Hg deposition over broad geographic ranges and assess the effectiveness of regional and global Hg emission reductions to address global Hg pollution concerns.

Sources of atmospheric metal(loid) pollution recorded in Thompson Manitoba lake sediment cores within the Canadian boreal biome.

Global atmospheric emissions and subsequent deposition of numerous metal(loid)s has increased markedly since the industrial revolution. Due to a paucity of long-term metal(loid) flux measurements, the magnitude and timing of change are largely unknown, resulting in limited ability to predict time-scales of ecosystem recovery in response to emission decreases. In the absence of long-term data, palaeo-reconstructions provide continuous records of atmospheric metal(loid) deposition on an ecosystem, and landscape, scale. Here, we use high-resolution dated lake sediment cores to reconstruct the last c. 100 years of atmospheric anthropogenic deposition of a full suite (40) of metal(loid)s near a large nickel (Ni) and copper (Cu) smelter in an other-wise largely "pristine" region of northern Canada (Thompson, Manitoba). Anthropogenic depositional fluxes were compared to other regions of Canada including Kejimkujik National Park in Nova Scotia, Experimental Lakes Area in Ontario, as well as the Flin Flon, Manitoba Cu and zinc (Zn) smelter, located ~200 km southwest of Thompson. Deposition of 12 metal(loid)s were enriched above baseline (pre-1915) levels: antimony (Sb) > palladium (Pd) > bismuth (Bi) > mercury (Hg) > cadmium (Cd) > Ni > lead (Pb) > arsenic (As) > strontium (Sr) > Cu > platinum (Pt) > Zn. Spatio-temporal patterns in depositional fluxes and inventories demonstrate that 6 of these metal(loid)s were sourced primarily from the smelter, while As, Hg, Pb, Pt, Sb and Zn were sourced primarily from global and/or regional sources. Comparison of anthropogenic fluxes and inventories to available emissions data showed that Cu and Ni deposition has plateaued since the late 1970s despite dramatic smelter emission decreases between 2005 and 2014. We hypothesize that this discrepancy is due to releases of terrestrial metal(loid)s by climate-driven permafrost degradation, which is widespread across the region and will likely continue to drive increased metal(loid) fluxes to northern Canadian lakes for unknown time-scales.

Widespread Atmospheric Tellurium Contamination in Industrial and Remote Regions of Canada.

High tech applications, primarily photovoltaics, have greatly increased demand for the rare and versatile but toxic element tellurium (Te). Here we examine dated lake sediment Te concentration profiles collected near potential point sources (metal smelters, coal mining/combustion facilities, oil sands operations) and from rural regions and remote natural areas of Canada. Te contamination was most prevalent near a Cu/Zn smelter where observed deposition infers 21 g Te released per metric ton (t) of Cu processed. Globally, 9,500 t is predicted to have been atmospherically deposited near Cu smelters post-1900. In a remote area of central Canada (Experimental Lakes Area; ELA), preindustrial Te deposition rates were equivalent to the estimated average global mass flux supplied from natural sources; however more surprisingly, modern Te deposition rates were 6-fold higher and comparable with Te measurements in precipitation. We therefore suggest that sediment cores reliably record atmospheric Te deposition and that anthropogenic activities have significantly augmented atmospheric Te levels, making it an emerging contaminant of potential concern. Lake water residence time was found to influence lake sediment Te inventories among lakes within a region. The apparent settling rate for Te was comparable to macronutrients (C, N, P), likely indicative of significant biological processing of Te.

Anthropogenic mercury deposition in Flin Flon Manitoba and the Experimental Lakes Area Ontario (Canada): A multi-lake sediment core reconstruction.

High-resolution records of anthropogenic mercury (Hg) deposition were constructed from 9 lakes located 5-75km from the Flin Flon, Manitoba smelter (formerly one of North America's largest atmospheric Hg point sources) and 5 lakes in Experimental Lakes Area (ELA), Ontario; a region remote from major Hg point sources. Anthropogenic Hg deposition, as both a flux and inventory, was determined after accounting for lake-specific natural Hg background concentrations, changes in sedimentation and sediment focusing. Results show that records of anthropogenic flux and inventory of Hg were remarkably consistent among the ELA lakes, but varied by 2 orders of magnitude among Flin Flon lakes. The relation between Hg inventories (normalized for prevailing wind direction) and distance from the smelter was used to estimate the total Hg fallout within a 50km radius in 5year time-steps, thus providing a quantitative spatial-temporal Hg depositional history for the Flin Flon region. The same relation solved for 8 cardinal directions weighted by the inverse of the previously applied wind direction normalization generates a map of Hg inventory and deposition on the landscape (Supplementary video). This novel application of sediment core data constructs a landscape model and allows for a visualization of contaminant deposition with respect to a point major source in both space and time. The propensity for Hg to undergo long-range, even global transport explains why Hg deposition within 50km of Flin Flon was ~11% of estimated releases. That is until smelter releases were reduced >10-fold (post-2000), after which observed deposition exceeded smelter releases, suggesting landscape re-emission/remobilization of legacy Hg is a major ongoing regional source of Hg.

Pair correlation and twin primes revisited.

We establish a connection between the conjectural two-over-two ratios formula for the Riemann zeta-function and a conjecture concerning correlations of a certain arithmetic function. Specifically, we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result. This casts a new light on the underpinnings of the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe.

Moments of zeta and correlations of divisor-sums: I.

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously, this approach has proved unsuccessful in computing moments beyond the eighth, even heuristically. A careful analysis of the second and fourth moments illustrates the nature of the problem and enables us to identify the terms that are missed in the standard application of these methods.

Anadromous char as an alternate food choice to marine animals: a synthesis of Hg concentrations, population features and other influencing factors.

This study was conducted to confirm sporadic measurements made over the late 1970s to the early 1990 s which determined that mercury (Hg) concentrations were low in anadromous char across Arctic and subarctic Canada including northern Québec and Labrador. Over 2004-2013, anadromous char populations across northern Canada were investigated at 20 sites for Hg concentrations and life history characteristics. Hg concentrations were extremely low in anadromous char muscle, typically <0.05 µg/g (wet weight) and, at each location, generally increased with fish length, age and nitrogen isotope (δ(15)N) ratio and decreased with condition factor and %lipid; correlations with carbon isotope (δ(13)C) ratio were inconsistent. Location and year were significant variables influencing Hg concentrations over the study area; longitude and latitude also were significant influencing variables. Char length, weight, age, condition factor and lipid content explained additional variance. A tendency towards higher Hg concentrations with increasing latitude may be partially related to decreasing growth of char towards the north. However, Hg concentrations in char were positively correlated with growth rates suggesting that Hg concentrations in char also were higher in the more productive study areas, including to the west where mainland riverine inputs of terrestrial carbon, nutrients, and Hg were greater. The data base for assessing time trends in char was limited by the small number of years investigated at most locations, variable fish size across years, small sample size, etc. Where temporal trends were detected, they were of increase on the long term (1970s, 1980s or early 1990 s to the present) but of decrease on the short term (early 2000s to present) with Nain (Labrador) showing the converse pattern. Higher Hg concentrations were also related to lower condition factor and cooler springs. Hg concentrations in anadromous char are compared with other terrestrial, aquatic and marine vertebrates in traditional diets. The known information on anadromous char is reviewed including population features, habitat, and harvests. Future Hg trend monitoring should focus on specific locations and harvest areas within these areas to better assess trends and influencing factors.

Local structure and speciation of platinum in fresh and road-aged North American sourced vehicle emissions catalysts: an X-ray absorption spectroscopic study.

Given emerging concerns about the bioavailability and toxicity of anthropogenic platinum compounds emitted into the environment from sources including vehicle emission catalysts (VEC), the platinum species present in selected North American sourced fresh and road-aged VEC were determined by Pt and Cl X-ray absorption spectroscopy. Detailed analysis of the Extended X-ray Absorption Fine Structure at the Pt L3 and L2 edges of the solid phase catalysts revealed mainly oxidic species in the fresh catalysts and metallic components dominant in the road-aged catalysts. In addition, some bimetallic components (Pt-Ni, Pt-Pd, Pt-Rh) were observed in the road-aged catalysts from supporting Ni-, Pd-, and Rh-K edge XAS studies. These detailed analyses allow for the significant conclusion that this study did not find any evidence for the presence of chloroplatinate species in the investigated solid phase of a Three Way Catalyst or Diesel Oxidation Catalysts.

Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) and disordered landscapes.

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞. In addition, we argue that it leads to multi-fractal-like behaviour in the total length µN(x) of the intervals in which |pN(θ)|>N(x), x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line s = 1/2 + it of given constant length and present the results of numerical computations of the large values of ζ(1/2 + it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Mercury trends in predatory fish in Great Slave Lake: the influence of temperature and other climate drivers.

Here we report on trends in mercury (Hg) concentrations in lake trout (Salvelinus namaycush), burbot (Lota lota), and northern pike (Esox lucius) from Great Slave Lake, located in the Mackenzie River Basin (MRB) and investigate how climate factors may be influencing these trends. Hg concentrations in lake trout and burbot increased significantly over the early 1990s to 2012 in the two major regions of the lake; no trend was evident for northern pike over 1999-2012. Temporal variations in Hg concentrations in lake trout and burbot were similar with respect to timing of peaks and troughs. Inclusion of climate variables based on annual means, particularly temperature, improved explanatory power for variations in Hg over analyses based only on year and fish length; unexpectedly, the temperature coefficient was negative. Climate analyses based on growing season means (defined as May-September) had less explanatory power suggesting that trends were more strongly associated with colder months within the year. Inclusion of the Pacific/North American index improved explanatory power for the lake trout model suggesting that trends may have been affected by air circulation patterns. Overall, while our study confirmed previously reported trends of Hg increase in burbot in the MRB, we found no evidence that these trends were directly driven by increasing temperatures and productivity.

Elucidation of structure and nature of the PdO-Pd transformation using in situ PDF and XAS techniques.

The PdO-Pd phase transformation in a 4 wt% Pd/Al2O3 catalyst has been investigated using in situ X-ray absorption spectroscopy (XAS) and in situ X-ray total scattering (also known as high-energy X-ray diffraction) techniques. Both the partial and total pair distribution functions (PDF) from these respective techniques have been analysed in depth. New information from PDF analysis of total scattering data has been garnered using the differential PDF (d-PDF) approach where only correlations orginating from PdO and metallic Pd are extracted. This method circumvents problems encountered in characerising the catalytically active components due to the diffuse scattering from the disordered γ-Al2O3 support phase. Quantitative analysis of the palladium components within the catalyst allowed for the phase composition to be established at various temperatures. Above 850 °C it was found that PdO had converted to metallic Pd, however, the extent of reduction was of the order ca. 70% Pd metal and 30% PdO. Complementary in situ XANES and EXAFS were performed, with heating to high temperature and subsequent cooling in air, and the results of the analyses support the observations, that residual PdO is detected at elevated temperatures. Hysteresis in the transformation upon cooling is confirmed from XAS studies where reoxidation occurs below 680 °C.

Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function ζ(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random-matrix theory, and the theory of the Riemann zeta function.

Biomarker responses in nesting, common eiders in the Canadian arctic in relation to tissue cadmium, mercury and selenium concentrations.

Populations of many North American sea ducks are declining. Biomarkers may offer valuable insights regarding the health and fitness of sea ducks in relation to contaminant burdens. In this study we examined body condition, immune function, corticosterone stress response, liver glycogen levels and vitamin A status in relation to tissue concentrations of mercury, selenium and cadmium in female common eiders during the nesting period. The study was conducted in the eastern Canadian arctic during July, 2000. Hepatic mercury, selenium and renal cadmium concentrations ranged 1.5-9.8, 6.5-47.5 and 74-389 microg/g, dry wt, respectively. Mercury concentrations were negatively related to dissection body mass, heart mass and fat mass. Cadmium concentrations were negatively related to mass at capture and dissection mass after controlling for the mercury concentration-dissection mass relationship. Cell-mediated immunity was assessed by the skin swelling reaction to an injection of phytohemagglutinin-P, and was unrelated to metal concentrations. After adjusting the corticosterone concentration to account for the time between capture and sampling, there was a negative relationship between the residual corticosterone concentration and selenium. Liver glycogen concentrations were not significantly related to metal concentrations. Mercury concentrations were positively related to those of hepatic retinol and retinyl palmitate and the ratio of the retinol to retinyl palmitate in liver. They were negatively related to the ratio of plasma to liver retinol. Our findings do not indicate that exposure to metals may have adversely affected the health of these birds. They do, however, suggest that more research is required to elucidate mechanisms by which exposure to these metals could impact body condition.

The semiclassical functional equation.

A semiclassical analog of the functional equation for the Riemann zeta function is considered. In the case of the zeta function itself, this equation forms the basis for a finite approximation to the Dirichlet series, known as the approximate functional equation. In the same way, the semiclassical functional equation can be shown to give rise to a finite approximation to the semiclassical representation of the quantum spectral determinant as a sum over classical pseudo-orbits. This finite approximation has been called the Riemann-Siegel look-alike formula. The formal nature of the derivation of this result is discussed and the fact that it appears to imply a remarkable relationship between long and short pseudo-orbits is shown.