A calculus student’s understanding of graphical approach to the derivative through quantitative reasoning

The concept of derivative is used in many areas including applied problems and requiring mathematical modelling in different disciplines. One of the most important approaches for teaching the derivative is to support students in visualizing the concept. Also, it is necessary to shift researchers and teachers’ focuses to students’ dynamic mental actions while learning derivative in order to conduct effective teaching process. With this necessity, I focused on the perspective of quantitative reasoning related to the graphical approach to the derivative. This study aims to reveal a calculus student’s mental actions related to the graphical approach to the derivative. The data were collected from a first-year calculus student engaged in the task requiring graphical interpretation of the derivative. Results showed that the student’s understanding of the slope shaped her inferences about the tangent line because the quantity of ratio is prior knowledge for learning the instantaneous rate of change. Besides, as the student had the idea of correspondence related to the concept of function, she had difficulties in interpreting the global view of the derivate. This result suggests that having global view of the derivative requires a strong understanding of function and rate.

Electrochemical tomography as a nondestructive technique to study localized corrosion of metals

We present an approach, termed electrochemical tomography (ECT), for the in-situ study of corrosion phenomena in general, and for the quantification of the instantaneous rate of localized corrosion in particular. Traditional electrochemical techniques have limited accuracy in determining the corrosion rate when applied to localized corrosion, especially for metals embedded in opaque, porous media. One major limitation is the generally unknown anodic surface area. ECT overcomes these limitations by combining a numerical forward model, describing the electrical potential field in the porous medium, with electrochemical measurements taken at the surface, and using a stochastic inverse method to determine the corrosion rate, and the location and size of the anodic site. Additionally, ECT yields insight into parameters such as the exchange current densities, and it enables the quantification of the uncertainty of the obtained solution. We illustrate the application of ECT for the example of localized corrosion of steel in concrete.

Hybrid quantum classical graph neural networks for particle track reconstruction

The Large Hadron Collider (LHC) at the European Organisation for Nuclear Research (CERN) will be upgraded to further increase the instantaneous rate of particle collisions (luminosity) and become the High Luminosity LHC (HL-LHC). This increase in luminosity will significantly increase the number of particles interacting with the detector. The interaction of particles with a detector is referred to as “hit”. The HL-LHC will yield many more detector hits, which will pose a combinatorial challenge by using reconstruction algorithms to determine particle trajectories from those hits. This work explores the possibility of converting a novel graph neural network model, that can optimally take into account the sparse nature of the tracking detector data and their complex geometry, to a hybrid quantum-classical graph neural network that benefits from using variational quantum layers. We show that this hybrid model can perform similar to the classical approach. Also, we explore parametrized quantum circuits (PQC) with different expressibility and entangling capacities, and compare their training performance in order to quantify the expected benefits. These results can be used to build a future road map to further develop circuit-based hybrid quantum-classical graph neural networks.

USING A FUNCTION DERIVATIVE IN ECONOMIC TASKS

The article focuses on the application of mathematical methods in economics, in particular discussing economic problems that are easily solved using derivatives. The purpose of the article is to show students the way and opportunity to use mathematical methods to solve economic problems. To this end, the article discusses and analyzes several economic tasks in detail, which will be interesting and easy for students to master. I considered the derivative of a function as the rate of change and introduced the definition: The instantaneous rate of change of the function f with respect to x at a point is called the derivative if it exists. With the help of this definition I have discussed and explained Task 1: Suppose that the increase in production of a certain product over a period of time is described by a function And population growth is described by the following function: Where is the number of years from the initial period, then the production of these products per capita is given by the function: Find the growth rate of product production. By solving this task I came to the conclusion that after a year the production of products per capita increases. In the following tasks I used the method of finding the extremum values of a function using a derivative, namely I equated the first-order derivative to 0, found the critical points, and with the help of the second-order derivative I determined the extremes of the function. I discussed task 2: For the production of X volume of products, the firm plans a cost that is calculated by the formulan . For what volume of production will the average cost be the smallest? Find the numerical value of this small expense. Solving this problem, I came to the conclusion that the given function of the average cost takes on the least value when the output volume is a unit , and this value is equal to: which is the marginal cost when producing the volume output. I discussed Task 3: How many products should be sold in order for a firm to profit maximally if the derivative cost function is known: And return function: Here I came to the conclusion: if 600 units of the product are sold, then the firm's profit will be maximum and it will be numerically equal At the end of the article I discussed such a task 4 of applied optimization. What is the minimum amount of material needed to make a 2 liter cylindrical jar? Where I came to the conclusion: the smallest amount of material will be spent to get a cylindrical vessel if we take the height 2 times the radius of the base.

Drivers of Atmosphere-Ocean CO2 Flux in Northern Norwegian Fjords

High-latitude fjords and continental shelves are shown to be sinks for atmospheric CO2, yet large spatial-temporal variability and poor regional coverage of sea-air CO2 flux data, especially from fjord systems, makes it difficult to scale our knowledge on how they contribute to atmospheric carbon regulation. The magnitude and seasonal variability of atmosphere-sea CO2 flux was investigated in high-latitude northern Norwegian coastal areas over 2018 and 2019, including four fjords and one coastal bay. The aim was to assess the physical and biogeochemical factors controlling CO2 flux and partial pressure of CO2 in surface water via correlation to physical oceanographic and biological measurements. The results show that the study region acts as an overall atmospheric CO2 sink throughout the year, largely due to the strong undersaturation of CO2 relative to atmospheric concentrations. Wind speed exerted the strongest influence on the instantaneous rate of sea-air CO2 exchange, while exhibiting high variability. We concluded that the northernmost fjords (Altafjord and Porsangerfjord) showed stronger potential for instantaneous CO2 uptake due to higher wind speeds. We also found that fixation of CO2 was likely a significant factor controlling ΔpCO2 from April to June, which followed phenology of spring phytoplankton blooms at each location. Decreased ΔpCO2 and the resulting sea-air CO2 flux was observed in autumn due to a combined reduction of the mixed layer with entrain of high CO2 subsurface water, damped biological activity and higher surface water temperatures. This study provides the first measurements of atmospheric CO2 flux in these fjord systems and therefore an important new baseline for gaining a better understanding on how the northern Norwegian coast and characteristic fjord systems participate in atmosphere carbon regulation.

Variability and Randomness of the Instantaneous Firing Rate

The apparent stochastic nature of neuronal activity significantly affects the reliability of neuronal coding. To quantify the encountered fluctuations, both in neural data and simulations, the notions of variability and randomness of inter-spike intervals have been proposed and studied. In this article we focus on the concept of the instantaneous firing rate, which is also based on the spike timing. We use several classical statistical models of neuronal activity and we study the corresponding probability distributions of the instantaneous firing rate. To characterize the firing rate variability and randomness under different spiking regimes, we use different indices of statistical dispersion. We find that the relationship between the variability of interspike intervals and the instantaneous firing rate is not straightforward in general. Counter-intuitively, an increase in the randomness (based on entropy) of spike times may either decrease or increase the randomness of instantaneous firing rate, in dependence on the neuronal firing model. Finally, we apply our methods to experimental data, establishing that instantaneous rate analysis can indeed provide additional information about the spiking activity.

Bearded dragons (Pogona vitticeps) with reduced scalation lose water faster but do not have substantially different thermal preferences

Whether scales reduce cutaneous evaporative water loss in lepidosaur reptiles (Superorder Lepidosauria) such as lizards and snakes has been a contentious issue for nearly half a century. Furthermore, while many studies have looked at whether dehydration affects thermal preference in lepidosaurs, far fewer have examined whether normally hydrated lepidosaurs can assess their instantaneous rate of evaporative water loss and adjust their thermal preference to compensate in an adaptive manner. We tested both of these hypotheses using three captive-bred phenotypes of bearded dragon (Pogona vitticeps) sourced from the pet trade: ‘Wild Types’ with normal scalation, ‘Leatherbacks’ exhibiting scales of reduced prominence, and scaleless bearded dragons referred to as ‘Silkbacks’. Silkbacks on average lost water evaporatively at about twice the rate that Wild Types did. Leatherbacks on average were closer in their rates of evaporative water loss to Silkbacks than they were to Wild Types. Additionally, very small (at most ∼1°C) differences in thermal preference existed between the three phenotypes that were not statistically significant. This suggests a lack of plasticity in thermal preference in response to an increase in rate of evaporative water loss, and may be reflective of a thermal ‘strategy’ as employed by thermoregulating bearded dragons that prioritises immediate thermal benefits over the threat of future dehydration. The results of this study bolster an often-discounted hypothesis regarding the present adaptive function of scales and have implications for the applied fields of animal welfare and conservation.

SIS Epidemic Model Birth-and-Death Markov Chain Approach

We are interested in describing the dynamics of the infected size of the SIS Epidemic model using the Birth-Death Markov process. The Susceptible-Infected-Susceptible (SIS) model is defined within a population of constant size $M$; the size is kept constant by replacing each death with a newborn healthy individual. The life span of each individual in the population is modelled by an exponential distribution with parameter $\alpha$; the disease spreads within the population is modelled by a Poisson process with a rate $\lambda_{I}$. $\lambda_{I}=\beta I(1-\frac{I}{M}) $ is similar to the instantaneous rate in the logistic population growth model. The analysis is focused on the disease outbreak, where the reproduction number $R=\frac{\beta} {\alpha} $ is greater than one. As methodology, we use both numerical and analytical approaches. The numerical approach shows that the infected size dynamics converge to a stationary stochastic process. And the analytical results determine the distribution of the stationary stochastic process as a normal distribution with mean $(1-\frac{1}{R}) M$ and Variance $\frac{M}{R} $ when $M$ becomes larger.

Preliminary observation on the sustainability of white sardine, Escualosa thoracata (Valenciennes, 1847), exploited from the central west coast of India

Background The present study assessed the growth and mortality parameters of the white sardine, Escualosa thoracata which is having high local demand. The white sardine gained importance due to its taste, and high demand in domestic markets as compared to the oil sardine necessitated a study on this resource to know the present status of exploitation level along the central west coast of India. Results A total of 3026 individuals of different size groups of E. thoracata were randomly collected from the Burondi fish landing center of the Ratnagiri district of Maharashtra. The asymptotic length (L∞) and growth coefficient (K) were estimated to be 115 mm and 1.9 year−1, respectively, by ELEFAN-I and 135 mm and 1.2 year−1 by the scattergram. The value of t0 by von Bertalanffy plot was estimated to be −0.000012 year. The fish attained a length of 65 mm, 94 mm, and 114 mm at the end of 0.5, 1, and 1.5 years of its life, respectively. The instantaneous rate of total mortality (Z), natural mortality (M), and fishing mortality (F) were estimated to be 8.07 year−1, 2.55 year−1, and 5.52 year−1, respectively. The exploitation rate (U) was calculated as 0.65, and the exploitation ratio (E) was 0.68. Conclusion The growth, mortality, and other population parameters observed in the present study will help to understand the current stock status, which is pointing toward the over-fishing condition (E ˃ 0.50) of the white sardine in the study area. Therefore, the present investigation suggests reducing the fishing pressure on E. thoracata along the central west coast of India for the sustainability of the resource.