Understanding Dilated Mathematical Relationship between Image Features and the Convolutional Neural Network’s Learnt Parameters

Deep learning, in general, was built on input data transformation and presentation, model training with parameter tuning, and recognition of new observations using the trained model. However, this came with a high computation cost due to the extensive input database and the length of time required in training. Despite the model learning its parameters from the transformed input data, no direct research has been conducted to investigate the mathematical relationship between the transformed information (i.e., features, excitation) and the model’s learnt parameters (i.e., weights). This research aims to explore a mathematical relationship between the input excitations and the weights of a trained convolutional neural network. The objective is to investigate three aspects of this assumed feature-weight relationship: (1) the mathematical relationship between the training input images’ features and the model’s learnt parameters, (2) the mathematical relationship between the images’ features of a separate test dataset and a trained model’s learnt parameters, and (3) the mathematical relationship between the difference of training and testing images’ features and the model’s learnt parameters with a separate test dataset. The paper empirically demonstrated the existence of this mathematical relationship between the test image features and the model’s learnt weights by the ANOVA analysis.

NITRATE REMOVAL IN WOODCHIP DENITRIFICATION BIOREACTOR – AN APPROACH COMBINING MATHEMATICAL MODELLING AND PI CONTROL

A mathematical model of nitrate removal in woodchip denitrification bioreactor based on field experiment measurements was developed in this study. The approach of solving inverse problem for nonlinear system of differential convection-reaction equations was applied to optimize the efficiency of nitrate removal depending on bioreactor’s length and flow rate. The approach was realized through the developed algorithm containing a nonlocal condition with an incorporated PI controller. This allowed to adjust flow rate for varying inflow nitrate concentrations by using PI controller. The proposed model can serve as a useful tool for bioreactor design. The main outcome of the model is a mathematical relationship intended for bioreactor length selection when nitrate concentration at the inlet and the flow rate are known. Custom software was developed to solve the system of differential equations aiming to ensure the required nitrate removal efficiency.

Prediction of Structural and Magnetic Properties of Nanocrystalline Mechanically Alloyed Fe-Ni Powders Using Multi-Gene Genetic Programming Approach

In this paper, a theoretical model based on multi-gene genetic programming (MGGP) approach has been applied to predict the structural and magnetic properties in nanocrystalline Fe–Ni powders prepared by mechanical alloying (MA) using a planetary ball mill. The MGGP model was used to correlate the input parameters (milling speed, chemical composition, and milling time), to output parameters (crystallite size and coercivity) of nanocrystalline Fe–Ni powders. The model obtained was tested with additional data to demonstrate its performance and prediction ability. The MGGP model is a robust and efficient method to find an accurate mathematical relationship between input and output data. A sensitivity analysis study was applied to determine the most influential milling parameters on the crystallite size and coercivity.

Mathematical simulation analysis of optimal testing of shot puter's throwing path

Objective To study the mathematical simulation analysis of shot-putter throwing optimal path. Methods Shot put was simplified as a parabolic motion of a particle, the corresponding mathematical model was established, and the mathematical relationship between the throwing distance and the initial velocity of shot put, the shooting Angle and the shooting height was defined. Results The fitting formula between shooting speed and shooting Angle was obtained by using the fitting method, and the quantitative relationship between them and the ideal shooting Angle was identified. Conclusion The mathematical principle of shot put is revealed through the process of building a model from simple to complex. However, there are still many problems to be solved, among which the height problem is a complex one. At the present level, it is not possible to find a reasonable height, because it involves many factors. However, the development of grey mathematics will provide a beneficial attempt for it to establish a reasonable and scientific model.

A Hypothetical Approach to Cosmic Inflation Incorporating Binary Entropy

The current paper presents an alternative hypothesis for the termination of cosmic inflation based on Huang’s model of spacetime involving the movement of a superfluid through a random resistor network. Using this model, we previously derived a mathematical relationship between the velocity of a reference frame and the probability that a random bond is intact. As an extension of this finding, the permutations of open and closed bonds are now shown to represent potential microstates, thus providing a means of relating motion within the network to binary entropy. Applying this concept to cosmic inflation, termination of this process is an expected consequence of the changes in binary entropy associated with the increasing velocity of expansion.

Investigation of Magneto-/Radio-Metric Behavior in Order to Identify an Estimator Model Using K-Means Clustering and Artificial Neural Network (ANN) (Iron Ore Deposit, Yazd, IRAN)

The study area is located near Toot village in the Yazd province of Iran, which is considered in terms of its iron mineralization potential. In this area, due to radioactivity, radiometric surveys were performed in a part of the area where magnetometric studies have also been performed. According to geological studies, the presence of magnetic anomalies can have a complex relationship with the intensity of radioactivity of radioactive elements. Using the K-means clustering method, the centers of the clusters were calculated with and without considering the coordinates of radiometric points. Finally, the behavior of the two variables of magnetic field strength and radioactivity of radioactive elements relative to each other was studied, and a mathematical relationship was presented to analyze the behavior of these two variables relative to each other. On the other hand, the increasing and then decreasing behavior of the intensity of the Earth’s magnetic field relative to the intensity of radioactivity of radioactive elements shows that it is possible to generalize the results of magnetometric surveys to radiometry without radiometric re-sampling in this region and neighboring areas. For this purpose, using the general regression neural network and backpropagation neural network (BPNN) methods, radiometric data were estimated with very good accuracy. The general regression neural network (GRNN) method, with more precision in estimation, was used as a model for estimating the radiation intensity of radioactive elements in other neighboring areas.

Biased statistical learning of closed-class items

In natural languages, closed-class items predict open-class items but not the other way around. For example, in English, if there is a determiner there will be a noun, but nouns can occur with or without determiners. Here we asked whether statistical learning of closed-class items is also asymmetrical. In three experiments we exposed adults to a miniature language with the one-way dependency “if X then Y”: if X was present, Y was also present, but Y could occur without X. We created different versions of the language in order to ask whether learning depended on which category (X or Y) was an open or closed class. In one condition, X had the main properties of a closed class and Y had the main properties of an open class; in a contrasting condition, X had properties of an open class and Y had properties of a closed class. Learners’ exposure in these two conditions was otherwise identical. Learning was significantly better with closed-class X. Additional experiments demonstrated that it is the perceptual distinctiveness of closed-class items that drives learners to analyze them differently, and that the mathematical relationship between closed- and open-class items influences learning more strongly than their linear order. These results suggest that statistical learning is biased: learners privilege computations in which closed-class items are predictive of, rather than predicted by, open-class items. We suggest that the distributional asymmetries of closed-class items in natural languages—and perhaps the asymmetrical structure of linguistic representations—may arise in part from this learning bias.

Consilience in the Peripheral Sensory Adaptation Response

Measurements of the peripheral sensory adaptation response were compared to a simple mathematical relationship involving the spontaneous, peak, and steady-state activities. This relationship is based on the geometric mean and is found to be obeyed to good approximation in peripheral sensory units showing a sustained response to prolonged stimulation. From an extensive review of past studies, the geometric mean relationship is shown to be independent of modality and is satisfied in a wide range of animal species. The consilience of evidence, from nearly 100 years of experiments beginning with the work of Edgar Adrian, suggests that this is a fundamental result of neurophysiology.

Distribution across Age of Raw and Scaled Intelligence Scores - the DARSIS model

The mean raw score on intelligence tests rises steeply in childhood before stabilizing for adolescents and adults, Little is known however about how the different percentiles, let alone the entire raw score distribution, changes from early childhood to adulthood. This study will show that there is a regular, mathematical relationship between age, raw scores and scaled scores on intelligence test subtests with a high g-loading. Using the norm tables from 60 subtests from 19 different intelligence tests normed in 5 different countries between 1984-2020, a relatively simple model with three coefficients and a variable ’difficulty’ parameter is shown to explain almost all of the variance in the norm tables. Smaller errors are found for the mean of norm tables, and for higher performance, but not for greater age. The year in which a test was normed also did not appear to influence the fit of the model. Possible applications of the DARSIS model, such as the creation of norms for above-level testing, creating extended norms, a reduced parameter space when norming new intelligence tests and calculating reference ages are discussed.

Instructional Plans: Grades K–2

Chapter 5 presents three instructional plans that are geared toward grades K–2. Instructional plans consist of planning necessities, standard alignment, alignment to philosophies approached in earlier chapters, as well as instructional procedures and assessments. Adaptations for other grade-level bands as well as potential extensions are available for each plan. This chapter includes the following instructional plans: Shapes of Electric Guitars, Sound Amplification and Speaker Building, and Measuring Length and Pitch. In Shapes of Electric Guitars, students will design guitar bodies and perform on them using available technology. In Sound Amplification, students will analyze and experiment with sound waves, eventually building their own small speaker. In Measuring Length and Pitch, students will measure pitched tubes to determine the mathematical relationship between pitches.