Mathematical Rules Research Articles

The Effect of Accuracy and Uncertainty in Test Performance on Optimal Early Detection Survey Budget Allocations

Early detection of pests at a low population prevalence is a key pillar of surveillance for plant health. A novel pest is expected to undergo exponential growth when it first occurs in a host population. This allows for the use of mathematical rules of thumb for the number of samples required to detect presence before it reaches a target prevalence given the diagnostic performance of the tests used. Previous work assumes that pest presence is diagnosed by applying a single diagnostic test to each sample. However, diagnostic decisions are often made based on outcomes of multiple tests applied to the same sample in a consistent test program; for example, an assessment of symptoms followed by a PCR test applied to samples for which symptoms were observed. Each test can have different testing costs, as well as distinct and independent diagnostic performance and uncertainty values. A framework to optimize early detection surveys by minimizing overall costs for test programs that apply up to three diagnostic tests has been developed. The framework assesses the consequences of the test order and logical rule by which diagnoses are determined on costs across a range of pest prevalence. Explicit definitions of uncertainty in key parameters are incorporated to assess the consequences of uncertainty about their true value. Use of the framework is exemplified with two hypothetical case studies exploring the potential impact of selecting a suboptimal test program (based on Xylella fastidiosa) and investment in test improvements (based on lateral flow devices applied to Phytophthora pluvialis). [Formula: see text] Copyright © 2023 The Author(s). This is an open access article distributed under the CC BY-NC-ND 4.0 International license .

Plastic Surgery: Principles and the Golden Formulas

The entire field of Plastic Reconstructive and Aesthetic surgery is technically based on 3 elements, “Viability” of tissues; “Transfer” of tissues; and the extent of tissue defect or “Tension”. These were expressed in 3 mathematical formulas and rules. An updated “Definition “of the specialty has been suggested.

A Fourier-based explanation of 1D-CNNs for machine condition monitoring applications

Neural networks (NN) have generated extensive interest in the field of machine condition monitoring (MCM). Many applications are however adapting structures and approaches from fields where NNs are highly established (e.g., image classification). This is done despite many neural network layers presenting strong similarities to signal processing steps that are well established in MCM. This paper aims at providing a mathematical analysis of typical NN architectures and layers, with specific attention to similarities to common signal processing steps such as filtering, downsampling, and enveloping. The set of strong analogies between NN and traditional MCM tools enables this work to provide MCM-specific guidelines for the design and tuning of NN architectures for MCM-specific applications. Most MCM applications are often characterised by periodic and cyclostationary signals, a situation not frequently encountered in many other NN applications. Hence, the Fourier-based explanation of the typical NN layers will be specific to the solution of typical MCM problems (e.g., diagnostics of gears and bearings). The study is aligned with the current trends of explainable AI and physics informed neural networks (PINN), in the sense that it aims at producing mathematical rules for a rational design of the network, drawing on the expertise accumulated in decades of signal processing for MCM. This is however done without upsetting traditional NN architectures, but rather trying to provide clear reasons for the setting of network parameters. The analytical work will be verified with numerical tests, which will also enable practical considerations, and by application of the suggested guidelines to the popular Case Western Reserve University (CWRU) experimental dataset.

Adsorption of Thymol Blue dye from Aqueous solution using modified Chitosan

The main objective of this research project is to perform some experimental procedures to study the ability of the novel adsorbent used in this project. The chemical modification of the adsorbent is the chemical oxidation of pure powder chitosan using some specific redox pairs which are potassium dichromate and sodium bisulphite. After preparing the modified chitosan, there are some parameters that must be discussed during the adsorption process. The first parameter is the time parameter, showing experimentally the optimum time for adsorption under certain conditions by collecting values of absorbance and concentration of thymol blue adsorbed after the process from the spectrophotometer showing that the optimum time is 120 minutes. While for the optimum pH experiment, it is shown that optimum pH is 4, 0.5% v/v is optimum concentration and 0.05 grams is the optimum dose for the adsorbent. Then after computing these conditions and values of absorbance, some mathematical calculations and classifications are done to show the reaction kinetics classification of the process. By performing some equations using mathematical rules it is provided that the reaction is not a first order reaction, showing by calculations and graphs that the reaction is a second order reaction.

ANALYSIS ON STUDENTS’ NUMERACY SKILLS IN SOLVING PROPORTION PROBLEMS IN THE CONTEXTS OF CANDI UMBUL TRADITIONAL MARKET

The weaknesses in students' abilities in applying mathematical concepts and rules to non-routine problems caused them difficulties in solving them. This study was aimed at analyzing students' numeracy skills in solving proportion problems with the contexts of Candi Umbul Traditional Market. The study was qualitative descriptive research involving two seventh-grade students in Indonesia as research subjects. The two student subjects were selected from 10 students who completed the numeracy task, one with the most correct answer and one with the most incorrect answer. The ten students who completed the numeracy task were the ones with the highest test scores among 32 students in the topics of proportion and social arithmetics. The research instruments used in the study were students' work sheets (in the form of essay tests) and interviews. The data analysis technique used was descriptive analysis. Findings show that not every student with high math scores had high numeracy skills. This is evident from the fact that students faced problems when completing proportion items in the context of Candi Umbul Traditional Market; they encountered difficulties in solving the problems. The results of this study are expected to serve as a reference for further research on students' numeracy skills.

Theory of short-term and long-term resistance of structures based on the principle of plastic fracture

The authors analyze the theory used in many countries, containing two independent directions: 1) the theory of stability of rod systems, including flat frames; 2) the theory of calculation of structural elements from various materials. The main feature of these theories is the application of the principle of plastic fracture. The assumption about a plastic hinge, due to the inconsistency with the experimental data, is supplemented by the incorrect application of theories of infinite elastic deformations, as well as of infinite creep deformations, which are incompatible with this hinge. Using the rules of mathematics, the principles of mechanics and the results of reliable experiments, it has been revealed that the analyzed theory contains several theories for different applications that reject each other, including the erroneous ones.

Hidden Signs? Postcritique to Solomon Marcus’s Final Article

Solomon Marcus’ final article “Hidden signs” (2016) is no coherent story but gives the sketchy outlines of his lecture notes. Marcus call needs a further explanation to his intellectual background. Marcus advanced mathematical poetics interacting between the “natural” form of literature and the “artificial” scientific fields, to be epitomized by mathematical or statistical rules. Literature and the sciences share the use of linguistic signs and symbols, but French structuralism refused to recognize the transdisciplinary relations of literature with sciences. Thus Marcus’ mathematical poetics can be set against Barthes’ negative contention by agreeing and disagreeing with the “subversive” structuralists (Lévy-Strauss and Calvino). By positioning the idea of mythical and playful interactions of science with literary phenomena, Marcus positioned this tentative hypothesis to accept the revolutionary understanding of his thoughts. The grounds of Marcus’ debate accept the understanding of his revolutionary thoughts, but the postcritical response advances the intellectual conditions and political implications of Marcus’ scholarship, which caused him to abandon the method of structuralism and embark on the semiotic expansion of the dynamism of Peirce and Derrida.

Some Fundamental Research Tools in Mathematical Sciences

Research in mathematical sciences either builds up the insight or breaks the boundary of the literature of the pertaining mathematical research area. A mathematical research technique is an attentive, persistent and systematic approach based on the logical rules of inference and mathematical rules of inference to find something new. Mathematical modeling, construction of theorems with the proofs, design of algorithms, data with simulation could be considered as the fundamental tools in mathematical research. In this paper, we discuss some fundamental research tools which are useful to do research in mathematical sciences.

Evolutionary safety of lethal mutagenesis driven by antiviral treatment.

Nucleoside analogs are a major class of antiviral drugs. Some act by increasing the viral mutation rate causing lethal mutagenesis of the virus. Their mutagenic capacity, however, may lead to an evolutionary safety concern. We define evolutionary safety as a probabilistic assurance that the treatment will not generate an increased number of mutants. We develop a mathematical framework to estimate the total mutant load produced with and without mutagenic treatment. We predict rates of appearance of such virus mutants as a function of the timing of treatment and the immune competence of patients, employing realistic assumptions about the vulnerability of the viral genome and its potential to generate viable mutants. We focus on the case study of Molnupiravir, which is an FDA-approved treatment against Coronavirus Disease-2019 (COVID-19). We estimate that Molnupiravir is narrowly evolutionarily safe, subject to the current estimate of parameters. Evolutionary safety can be improved by restricting treatment with this drug to individuals with a low immunological clearance rate and, in future, by designing treatments that lead to a greater increase in mutation rate. We report a simple mathematical rule to determine the fold increase in mutation rate required to obtain evolutionary safety that is also applicable to other pathogen-treatment combinations.

Localized Modes and Photonic Band Gap Sensitivities with 1D Fibonacci Quasi-Crystals Filled with Sinusoidal Modulated Plasma

Using the transfer matrix method, the electromagnetic responses of 1D deformed and non-deformed quasi-periodic photonic crystals arranged in accordance with the Fibonacci sequence are theoretically studied. The gallium selenide (GeSe) and plasma materials (that is, electron density) are used to construct the multilayer Fibonacci structures. If this study is experimentally validated in the future, we intend to insert two transparent polymer film materials at the top and bottom of the structure, which are intended to protect the plasma material and prevent it from escaping and spreading outside the structure. The effect of the order of the Fibonacci sequence, the deformation of the thickness of the layers using a mathematical rule and the role of the plasma material in the reflectance response are discussed. We notice that the position and the width of photonic band gaps are sensitive to the Fibonacci sequence, the thickness and the density of the plasma material layers. In addition, the intensity of the resonance peaks can be controlled by adjusting the plasma material properties. The width of the photonic band gaps can be notably enlarged by applying a structural deformation along the stacks. The proposed structures have potential applications in tunable filters, micro-cavities for LASER equipment, which allow us to obtain an intense laser, and they are very useful in the communication field.

Avaliação de Políticas Sustentáveis

Sustainability policy evaluation and assessment seeks to answer the key question, sustainability of what and whom? Consequently, sustainability issues are multidimensional in nature and feature a high degree of conflict, uncertainty and complexity. Social multi-criteria evaluation (SMCE) has been explicitly designed for public policies; it builds on formal modelling techniques whose main achievement is the fact that the use of different evaluation criteria translates directly into plurality of values and dimensions underpinning a policy process. SMCE aims at being inter/multi-disciplinary (with respect to the technical team), participatory (with respect to the community) and transparent. SMCE can help deal with three different types of sustainability-related policy issues: 1) epistemological uncertainty (human representation of a given policy problem necessarily reflects perceptions, values and interests of those structuring the problem); 2) complexity (the existence of different levels and scales at which a hierarchical system can be analyzed implies the unavoidable existence of non-equivalent descriptions of it both in space and time); and 3) mathematical manipulation rules of relevant information (compensability versus non-compensability, preference modelling of intensities of preference, mixed information on criterion scores, weights as trade-offs versus weights as importance coefficients, choice of a proper ranking algorithm). This paper focuses on the these three issues and provides an overview of the SMCE approaches to them.

Numeracy of Eighth Grade Students in Solving AKM-Like Problems Based on Mathematical Ability

Abstract: Numeracy is the ability to locate, use, interpret, evaluate, and communicate mathematical information and ideas in the real context. This research purpose to describe the numeracy of eighth grade students in solving AKM-like problems in equations and inequalities subdomain based on high, moderate, and low mathematical abilities. The research subjects were eighth grade students consisting of one student with high mathematical ability, one student with moderate mathematical ability, and one student with low mathematical ability. The research method used in this research is qualitative descriptive research. Data were obtained by numeracy test. Students with high mathematical abilities present the information obtained in the form of equations and inequalities, use mathematical rules and procedures on equations and inequalities, interpret the results in the context of the problem, evaluate the results of problem solving through supposed, and communicate the results of their interpretation to others both orally and writing appropriately. Students with moderate mathematical abilities present the information obtained in the form and use procedures and rules of equations and inequalities in solving problems appropriately. However, students with moderate mathematical abilities interprets the results inaccurately so that in communicating the results of the interpretation is also inaccurate and evaluate the results only by correcting or recalculating. Students with low mathematical abilities do not present information in the form of equations and inequalities, nor do they use procedures and rules of equations and inequalities in solving problems. The interpretation of students with low mathematical abilities is also incorrect so that communicating the results of interpretations is not correct. In addition, students with low mathematical abilities do not evaluate the results of problem solving, either through supposed or correcting and recalculating.

استخدم طريقة السلسلة الامامية لتشخيص مرض السكرى

Diabetes Mellitus (DM) is a disease that affects many people who are overweight or could be a descendant of this disease.There are several classifications of diabetes, namely: Type1, type2 other specific and gestational diabetes.Society needs a technology to diagnose early diabetes from symptoms. In this expert system uses forward chaining method that is built using the PHP programming language and MySQL database. Knowledge Representation is useful in representing phenomena into a mathematical rule that will be stored in a database. The database consists of several tables, including symptoms, symptoms disease, disease, and questions. Knowledge Base materials sourced from internet and medical books on diabetes. The purpose of this system is that people can diagnose the diabetes disease. Keywords: Forward chaining Method, Expert System, Knowledge Base, PHP, Diabetes disease.

Dosimetrically triggered adaptive radiotherapy for head and neck cancer: Considerations for the implementation of clinical protocols.

Defining dosimetric rules to automatically detect patients requiring adaptive radiotherapy (ART) is not straightforward, and most centres perform ad-hoc ART with no specific protocol. This study aims to propose and analyse different steps to design a protocol for dosimetrically triggered ART of head and neck (H&N) cancer. As a proof-of-concept, the designed protocol was applied to patients treated in TomoTherapy units, using their available software for daily MVCT image and dose accumulation. An initial protocol was designed by a multidisciplinary team, with a set of flagging criteria based only on dose-volume metrics, including two action levels: (1) surveillance (orange flag), and (2) immediate verification (red flag). This protocol was adapted to the clinical needs following an iterative process. First, the protocol was applied to 38 H&N patients with daily imaging. Automatic software generated the daily contours, recomputed the daily dose and flagged the dosimetric differences with respect to the planning dose. Second, these results were compared, by a sensitivity/specificity test, to the answers of a physician. Third, the physician, supported by the multidisciplinary team, performed a self-analysis of the provided answers and translated them into mathematical rules in order to upgrade the protocol. The upgraded protocol was applied to different definitions of the target volume (i.e. deformed CTV + 0, 2 and 4mm), in order to quantify how the number of flags decreases when reducing the CTV-to-PTV margin. The sensitivity of the initial protocol was very low, specifically for the orange flags. The best values were 0.84 for red and 0.15 for orange flags. After the review and upgrade process, the sensitivity of the upgraded protocol increased to 0.96 for red and 0.84 for orange flags. The number of patients flagged per week with the final (upgraded) protocol decreased in median by 26% and 18% for red and orange flags, respectively, when reducing the CTV-to-PTV margin from 4 to 2mm. This resulted in only one patient flagged at the last fraction for both red and orange flags. Our results demonstrate the value of iterative protocol design with retrospective data, and shows the feasibility of automatically-triggered ART using simple dosimetric rules to mimic the physician's decisions. Using a proper target volume definition is important and influences the flagging rate, particularly when decreasing the CTV-to-PTV margin.

Discrimination of artificial intelligence in healthcare

Currently, artificial intelligence (AI) plays an important role in various fields as a primary worker or assistant, especially in healthcare. AI has many functions that can be performed even better than humans do due to the high speed of compiling a large amount of data from various sources (Internet and electronic health records), which increases the productivity of doctors. Less discussed, though equally important, is the discrimination of AI in healthcare. Doctors, who have a lot of responsibility, cannot rely on AI due to multiple vulnerabilities. First, AI collects huge amounts of data. However, AI does not guarantee error-free results, despite its non-autonomy and operator control, since human factors can play a role, becoming a source of inaccuracy. Therefore, a high risk of using poor quality data in further decision-making processes exists. Errors, biases (ethnic, gender, age, and social), and data gaps worsen AI outcomes, leading, for example, to discrimination against minorities and inaccurate prescriptions. In addition, AI may act unethically or even violate laws (e.g., Title VI, Section 1557 of the Affordable Care Act, which prohibits discrimination by race, color, national origin, sex, age, or disability in certain healthcare programs and activities).
 Secondly, AI may be called secretive due to the indeterminacy of the algorithms. This means that no one can explain how or why the AI reached this or that decision, as stated by Judea Pearl in The Book of Why: The New Science of Cause and Effect. Therefore, doctors cannot check the facts and make sure that the analysis is done correctly and the conclusion is accurate. Difficulties in solving real-world problems using AI that cannot be solved by formal and mathematical rules of logic as humans do (e.g., natural language and face recognition) must be recognized. AI makes the work of medical professionals easier, however, posing many unresolved problems that mislead doctors and contribute to wrong decisions.

Morphological analysis of historical landscapes based on cultural DNA approach

Recent trends in landscape architecture investigate new approaches, methods, and technologies to understand, monitor, manage, continuity, and sustainably develop heritage landscapes. Cultural DNA (CD) represents designs' transferable geometric, behavioural and functional properties. The morphological structure of Persian historical gardens possesses discoverable hidden patterns, computable through the CD's mathematical and computational models. This study aims to investigate and discover effective parameters in the spatial structure of historical Persian gardens using Space Syntax. Six Persian Gardens were surveyed, and spatial structures were analysed using DepthMap10 software. The results show that these spaces follow meaningful mathematical patterns and rules that can be formulated in the context of generative processes and generalized for the evolutionary continuity of gardens. However, further research is proposed to develop more advanced computational methods, such as artificial intelligence (AI) algorithms and AI-based decision support tools, to help to generate design scenarios and to transmit culture through the design process.

Simple experimental realization of optical Hilbert Hotel using scalar and vector fractional vortex beams

Historically, infinity was long considered a vague concept—boundless, endless, larger than the largest—without any quantifiable mathematical foundation. This view changed in the 1800s through the pioneering work of Georg Cantor, who showed that infinite sets follow their own seemingly paradoxical mathematical rules. In 1924, David Hilbert highlighted the strangeness of infinity through a thought experiment now referred to as the Hilbert Hotel paradox, or simply Hilbert’s Hotel. The paradox describes a “fully” occupied imaginary hotel having an infinite number of single-occupancy rooms. The manager can always find a room for new guests by simply shifting current guests to the next highest room, leaving the first room vacant. The investigation of wavefield singularities has uncovered the existence of a direct optical analogy to Hilbert’s thought experiment. Since then, efforts have been made to investigate the properties of Hilbert’s Hotel by controlling the dynamics of phase singularities in “fractional” order optical vortex beams. Here, we have taken such proposals to the next level and experimentally demonstrated Hilbert’s Hotel using both phase and polarization singularities of optical fields. Using a multi-ramped spiral-phase-plate and a supercontinuum source, we generated and controlled fractional order vortex beams for the practical implementation of Hilbert’s Hotel in scalar and vector vortex beams. Using a multi-ramped spiral-phase-plate, we show the possibility for complicated transitions of the generalized Hilbert’s Hotel. The generic experimental scheme illustrates the usefulness of structured beams in visualizing unusual mathematical concepts and also for fractional vector beams driven by fundamental and applied research.

The Secrets inside the Astonishing Reuleaux Triangle and its Practicable Uses

The idea of the study was mainly from the concerns about whether wheels of different shapes can also fit well in cars. This paper mainly involves the demonstration of some theoretical features of the Reuleaux triangle and some practical situations of using the triangle as wheels. The aim of the study is to find out some basic features and mathematical rules of the Reuleaux triangle, in addition, the practical uses of this shape will also be studied. In order to learn expertise about this triangle, the author went through papers about geometry and some demonstrations by different mathematicians from different times. The author then made a few assumptions and tried to prove himself to find the rules of this shape more deeply. In conclusion, it found out that the Reuleaux triangle is actually a perfect constant width curve and it can be used as wheels under proper conditions. Apart from that, this paper also concluded an exact method of drawing different constant width curves.

Mechanical characterisation of novel aperiodic lattice structures

This paper compares the mechanical properties of a class of lattice metamaterials with aesthetically-pleasing patterns that are governed by the mathematics of aperiodic order. They are built up of ordered planar rod networks and exhibit higher non-crystallographic rotational symmetries. However, they lack the translational symmetry associated with periodic lattice metamaterials. We present schematics illustrating their development based on pattern-unique mathematical substitution rules and exploit a numerical framework from previous work to demonstrate that they exhibit fascinating near-isotropic properties. The lattice structures are compared to the well-known hexagonal lattice with respect to their elastic anisotropy measure and the proximity of their bulk and shear moduli to Hashin–Shtrikman-Walpole limits. The study lends insight into the cost-constrained benefits of introducing additional connectivities between aperiodically-ordered point sets. The results show that aperiodic lattices have the potential to yield superior mechanical properties to periodic ones subject to the mechanical rigidity of the underlying shapes that constitute the pattern. The inherent ‘near-isotropy’ associated with these aperiodic structures, even with uniform strut thicknesses and at low fractional densities, and the ordered and varied orientation of their lattice struts present them as promising mesoscale architecture for solving complex multi-axially loaded structural optimization problems, providing inspiration for this study.

Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.