Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries:

If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).

The of a continuous function over an interval is itself an interval.

Motivation This captures an intuitive property of continuous functions over the real numbers: given f continuous on [1,2] with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn

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Evidence that Pacific tuna mercury levels are driven by marine methylmercury production and anthropogenic inputs

Hélène Paule Angot

Pacific Ocean tuna is among the most-consumed seafood products but contains relatively high levels of the neurotoxin methylmercury. Limited observations suggest tuna mercury levels vary in space and time, yet the drivers are not well understood. Here, we map mercury concentrations in skipjack tuna across the Pacific Ocean and build generalized additive models to quantify the anthropogenic, ecological, and biogeochemical drivers. Skipjack mercury levels display a fivefold spatial gradient, with maximum concentrations in the northwest near Asia, intermediate values in the east, and the lowest levels in the west, southwest, and central Pacific. Large spatial differences can be explained by the depth of the seawater methylmercury peak near low-oxygen zones, leading to enhanced tuna mercury concentrations in regions where oxygen depletion is shallow. Despite this natural biogeochemical control, the mercury hotspot in tuna caught near Asia is explained by elevated atmospheric mercury concentrations and/or mercury river inputs to the coastal shelf. While we cannot ignore the legacy mercury contribution from other regions to the Pacific Ocean (e.g., North America and Europe), our results suggest that recent anthropogenic mercury release, which is currently largest in Asia, contributes directly to present-day human mercury exposure.

National Academy of Sciences2022

Thermodynamic properties of the Shastry-Sutherland model from quantum Monte Carlo simulations

Frédéric Mila, Bruce Normand

We investigate the minus-sign problem that afflicts quantum Monte Carlo (QMC) simulations of frustrated quantum spin systems, focusing on spin S = 1/2, two spatial dimensions, and the extended Shastry-Sutherland model. We show that formulating the Hamiltonian in the diagonal dimer basis leads to a sign problem that becomes negligible at low temperatures for small and intermediate values of the ratio of the inter-and intradimer couplings. This is a consequence of the fact that the product state of dimer singlets is the exact ground state both of the extended Shastry-Sutherland model and of a corresponding "sign-problem-free" model, obtained by changing the signs of all positive off-diagonal matrix elements in the dimer basis. By exploiting this insight, we map the sign problem throughout the extended parameter space from the Shastry-Sutherland to the fully frustrated bilayer model and compare it with the phase diagram computed by tensor-network methods. We use QMC to compute with high accuracy the temperature dependence of the magnetic specific heat and susceptibility of the Shastry-Sutherland model for large systems up to a coupling ratio of 0.526(1) and down to zero temperature. For larger coupling ratios, our QMC results assist us in benchmarking the evolution of the thermodynamic properties by systematic comparison with exact diagonalization calculations and interpolated high-temperature series expansions.

2018

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and po

Infinitesimal

In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinag

MATH-101(g): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-101(a): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-100(b): Advanced analysis I

Dans ce cours, nous étudierons les notions fondamentales de l'analyse réelle, ainsi que le calcul différentiel et intégral pour les fonctions réelles d'une variable réelle.