We consider the problem of optimal search for locally feasible solutions of a linear function on the permutations where the linear function takes the values from the given interval. We describe a new method of solving...
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We consider the problem of optimal search for locally feasible solutions of a linear function on the permutations where the linear function takes the values from the given interval. We describe a new method of solving such problem by a targeted search of the permutations that provide locally feasible solutions with minimal search.
The Khrapchenko method of finding a lower bound for the complexity of binary formulas is extended to formulas in k-ary bases. The resulting extension makes it possible to evaluate the complexity of linear Boolean func...
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The Khrapchenko method of finding a lower bound for the complexity of binary formulas is extended to formulas in k-ary bases. The resulting extension makes it possible to evaluate the complexity of linear Boolean functions and a majority function of n variables when realized by formulas in the basis of all k-ary monotone functions and negation as Omega(n(g(k))), where g(k) = 1 + Theta(1/ln k). For a linear function, the complexity bound in this form is unimprovable. For k = 3, the sharper lower bound Omega(n(1.53)) is proved.
New calibration method of the capacitance and frequency domain reflectometry (ECH2O EC-5) for soil water monitoring was developed by introducing the slope-intercept relationship derived from the relationship between t...
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New calibration method of the capacitance and frequency domain reflectometry (ECH2O EC-5) for soil water monitoring was developed by introducing the slope-intercept relationship derived from the relationship between the RAW data recorded by the Em50 datalogger and the thermogravimetrically measured volumetric water content of soil. Numerous soil samples taken from paddy-upland rotational fields in western Japan were used for precise soil-specific calibration experiments to determine the variables of slope-intercept relationship. By using these identified variables and an easily available dataset of the measured volumetric water content and the RAW data, the developed linear function predicted volumetric water content of soils. The developed linear function was validated with the reported literature data, the agricultural fields experiment, and the laboratory experiment. From those three validations, the developed linear function was evaluated to be practically available for laboratory and field researches. Also, it is implied that the developed linear function is extendable to dataloggers except for the Em50 and monitoring devices excluding the EC-5.
This study describes a comparison of how worked examples in selected textbooks from England and Shanghai presented possible learning trajectories towards understanding linear function. Six selected English textbooks a...
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This study describes a comparison of how worked examples in selected textbooks from England and Shanghai presented possible learning trajectories towards understanding linear function. Six selected English textbooks and one Shanghai compulsory textbook were analysed with regards to the understanding required for pure mathematics knowledge in linear function. Understanding was defined as being at five levels: Dependent Relationship, Connecting Representations, Local Properties Noticing, Object Analysis and Inventising. These levels were developed by examining the most prominent theories from the existing literature on understanding function. Findings suggested that the English textbooks constrained the structural aspect of understanding linear function due to a point-to-point view of function, while the Shanghai textbook which focussed on a variable view of function overemphasised the algebraic approach. The discussion explored the drawbacks to each approach and what teachers or textbook writers could do to balance these two approaches in order to facilitate students' understanding towards a structural view of linear function.
This study investigates how students in England and Shanghai understand linear function. Understanding is defined theoretically in terms of five hierarchical levels: Dependent Relationship;Connecting Representations;P...
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This study investigates how students in England and Shanghai understand linear function. Understanding is defined theoretically in terms of five hierarchical levels: Dependent Relationship;Connecting Representations;Property Noticing;Object Analysis;and Inventising. A pilot study instrument presented a set of problems to both cohorts, showing that the English cohort generally operated at the levels of Property Noticing and Object Analysis, whereas the Shanghai cohort reached the higher level of Inventising. The main study explored understanding levels and students' errors within each cohort in detail, in order to gain insights into reasons for apparent differences. The instrument used in the main study included two overlapping items, which were the same for both cohorts, while others were pitched at levels of understanding revealed in the pilot. Analysis of students' solutions revealed that the English students' errors were manifested in a lack of basic skills including dealing with negative numbers, while the Shanghai students showed weaknesses in their ability to use graphs. The discussion highlights different views of understanding as a possible background reason for the contrasts observed. Errors and apparent difficulties suggest implications for teaching linear function in each context.
We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linea...
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We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors.
Background: The calibration curve represents the relationship between known CRM and the response of the instrument (e.g absorbance). Most of the time, in analytical chemistry standards, the linear function (of the 1st...
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Background: The calibration curve represents the relationship between known CRM and the response of the instrument (e.g absorbance). Most of the time, in analytical chemistry standards, the linear function (of the 1st degree) is used to characterize the calibration curves. This, at first glance, has some major advantages: simple equation, simple linearity estimation criterion, and easy way of calculation. However, from a quality point of view, the grade 1 calibration curve does not characterize a working domain very well, because it does not take into account the curvature and has larger residuals. Results: Here we show that, most of the time, the non-linear secondary function fits the calibration curves much better (argument valid for UV-Vis spectrophotometry, atomic absorption spectrometry (FAAS and ETAAS), catalytic combustion and ion chromatography (IC)), having lower residuals, but also a higher coefficient of determination, R 2 . The limitation of the secondary non-linear function use is given by the insufficiency of data related to its qualitative characterization. Through this work, we have introduced new characterization criteria for non-linear secondary functions such as curvature angle and curvature index. Although in the specialized literature, the term linearity of the calibration curve has existed for a long time, in reality it has not been fully interpreted. The curvature index developed in this work allows the controlled implementation of non-linear curves, but also with increasing the precision of the results. Significance: The curvature index characterizes the degree of curvature of the 1st or 2nd degree calibration functions. Using the 2nd degree calibration function allows the improvement of the obtained results, and curvature parameters (curvature angle and curvature index) allow qualitative characterization of many data sets used for calibration. The degree of curvature of a calibration curve is a new term not found in the literature, which wil
It is well-known that the Dedekind-MacNeille completion of a poset is its injective hull. We prove that the injective hull of an ordered universal algebra with respect to a specific class of monomorphisms has properti...
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It is well-known that the Dedekind-MacNeille completion of a poset is its injective hull. We prove that the injective hull of an ordered universal algebra with respect to a specific class of monomorphisms has properties that are similar to the properties of the Dedekind-MacNeille completion of a poset. In particular, this injective hull induces a reflector functor from the category of ordered algebras and continuous morphisms to the category of sup-algebras.
Over the past few decades, a lot of new neural network architectures and deep learning (DL)-based models have been developed to tackle problems more efficiently, rapidly, and accurately. For classification problems, i...
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Over the past few decades, a lot of new neural network architectures and deep learning (DL)-based models have been developed to tackle problems more efficiently, rapidly, and accurately. For classification problems, it is typical to utilize fully connected layers as the network head. These dense layers used in such architectures have always remained the same - they use a linear transformation function that is a sum of the product of output vectors with weight vectors, and a trainable linear bias. In this study, we explore a different mechanism for the computation of a neuron's output. By adding a new feature, involving a product of higher order output vectors with their respective weight vectors, we transform the conventional linear function to higher order functions, involving powers over two. We compare and analyze the results obtained from six different transformation functions in terms of training and validation accuracies, on a custom neural network architecture, and with two benchmark datasets for image classification (CIFAR-10 and CIFAR-100). While the dense layers perform better in all epochs with the new functions, the best performance is observed with a quadratic transformation function. Although the final accuracy achieved by the existing and new models remain the same, initial convergence to higher accuracies is always much faster in the proposed approach, thus significantly reducing the computational time and the computational resources required. This model can improve the performance of every DL architecture that uses a dense layer, with remarkably higher improvement in larger architectures that incorporate a very high number of parameters and output classes.
We consider ordered universal algebras and give a construction of a join-completion for them using so-called D-ideals. We show that this construction has a universal property that induces a reflector from a certain ca...
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We consider ordered universal algebras and give a construction of a join-completion for them using so-called D-ideals. We show that this construction has a universal property that induces a reflector from a certain category of ordered algebras to the category of sup-algebras. Our results generalize several earlier known results about different ordered structures.
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