Mathematical Laws Research Articles

Growth of a Tessellation: Geometric rules for the Development of Stingray Skeletal Patterns.

The skeletons of sharks and rays, fashioned from cartilage, and armored by a veneer of mineralized tiles (tesserae) present a mathematical challenge: How can the continuous covering be maintained as the skeleton expands? This study, using microCT and custom visual data analyses of growing stingray skeletons, systematically examines tessellation patterns and morphologies of the many thousand interacting tesserae covering the hyomandibula (a skeletal element critical to feeding), over a two-fold developmental change in hyomandibula length. The number of tesserae remains surprisingly constant, even as the hyomandibula expands isometrically, with all hyomandibulae displaying self-similar distributions of tesserae shapes/sizes. Although the distribution of tesserae geometries largely agrees with the rules for polyhedra tiling of complex surfaces-dominated by hexagons and a minor fraction of pentagons and heptagons, but very few other polygons-the agreement with Euler's classic mathematical laws is not perfect. Contrary to the assumed uniform growth rate (which is shown would create geometric incompatibilities), larger tesserae grow faster to accommodate skeletal expansion. It is hypothesized that this local regulation of global system complexity is driven by tension (from cartilaginous core expansion) in the fibers connecting tesserae, with strain-responsive cells orchestrating local mineral apposition.

A New Mathematical Method for Dividing the Area of Land with a Trapezoidal Shape Longitudinally

Land with a trapezoidal shape is often divided by the attempt method. This research attempts to find a new way to divide this type of land by deriving a mathematical law. It can be applied to divide trapezoidal-shaped land into a number of shares of equal area, as well as equal degree of benefit from public services as much as possible.

Two-Dimensional Laplace NMR Reconstruction through Deep Learning Enhancement.

Laplace NMR is a powerful tool for studying molecular dynamics and spin interactions, providing diffusion and relaxation information that complements Fourier NMR used for composition determination and structure elucidation. However, Laplace NMR demands sophisticated signal processing algorithms such as inverse Laplace transform (ILT). Due to the inherently ill-posed nature of ILT problems, it is generally challenging to perform satisfactory Laplace NMR processing and reconstruction, particularly for two-dimensional Laplace NMR. Herein, we propose a proof-of-concept approach that blends a physics-informed strategy with data-driven deep learning for two-dimensional Laplace NMR reconstruction. This approach integrates prior knowledge of mathematical and physical laws governing multidimensional decay signals by constructing a forward process model to simulate relationships among different decay factors. Benefiting from a noniterative neural network algorithm that automatically acquires prior information from synthetic data during training, this approach avoids tedious parameter tuning and enhances user friendliness. Experimental results demonstrate the practical effectiveness of this approach. As an advanced and impactful technique, this approach brings a fresh perspective to multidimensional Laplace NMR inversion.

Research on charging monitoring method for lithium-ion batteries based on magnetic field sensing

With the rapid and continuous development of new energy vehicles, safety issues during charging have become increasingly prominent. Thus, there is an urgent need to strengthen charging safety management. However, current charging detection methods have limitations, such as difficulty in monitoring local battery pack currents, and indirect monitoring through temperature and gas sensors lead to lag. To address these limitations, this article proposes a safety monitoring method based on an improved lithium-ion battery magnetic field sensing model that covers the entire charging process. Specifically, thermal field analysis was introduced to the existing electric magnetic coupling model, resulting in an electric thermal magnetic multi-physical field coupling model. By conducting experiments, the study verified that the accuracy of the improved model has increased by 15.5 %. Based on this model, the study proposed a method for predicting and warning of abnormal currents using BP neural network and mathematical statistical laws. The results showed that the current prediction accuracy of this method reached 96.4 %. Such high accuracy ensures real-time comprehensive monitoring of the current information of lithium-ion battery packs, thereby eliminating the hidden dangers of charging safety accidents caused by the lack of local current detection.

Design and mathematical modeling of an amphibious quadcopter for versatile operations

Amphibious drone is an Unmanned Aerial Vehicle (UAV) capable of performing in both air and under water for application in various fields, such as marine, military, and underwater habitat research, but complexity in the design and provable mathematical model to withstand the arguably swift changes in the forces during the transition phase makes it difficult to build a sustainable UAV that can operate seamlessly in both media. Beginning with the mathematical principles and physical laws, the basic concept of operation is arrived at in the present study for both media (air and water) keeping the prime objective as developing a reliable drone design and mathematical model that will satisfactorily describe the behavior of the amphibious drone with accuracy by defining the coordinate system to describe the amphibious drone’s kinematics. Basic forces and torques are considered to explicitly describe the drone dynamics using Newton–Euler equations, and the final equation derived is the matrix Newton’s second law. Based on the mathematical model, the final design of the drone is arrived at considering the feasibility of withstanding forces, placement of commercial-off-the-shelf components, and the amount of thrust required to carry the seamless operation. A prototype is built with active buoyancy control technique to control the underwater depth of the drone, which clearly satisfies the design and mathematical model developed in this study.

A Theoretical Study of the Separation of Raw Cotton from Air Flow Under the Influence of Centrifugal Force

This article is about the improvement of the separator structure, which separates the cotton from the air stream after the delivery of raw cotton to the technological processes in air ginneries, in which an effective method is proposed compared to previous analogues. This is discusses the process of separating raw cotton from air flow using centrifugal force and its mathematical laws. The trajectories of the movement of cotton in the proposed device were also determined and the corresponding graphs were obtained.

Role of the Cerebellum in the Construction of Functional and Geometrical Spaces.

The perceptual and motor systems appear to have a set of movement primitives that exhibit certain geometric and kinematic invariances. Complex patterns and mental representations can be produced by (re)combining some simple motor elements in various ways using basic operations, transformations, and respecting a set of laws referred to as kinematic laws of motion. For example, point-to-point hand movements are characterized by straight hand paths with single-peaked-bell-shaped velocity profiles, whereas hand speed profiles for curved trajectories are often irregular and more variable, with speed valleys and inflections extrema occurring at the peak curvature. Curvature and speed are generically related by the 2/3 power law. Mathematically, such laws can be deduced from a combination of Euclidean, affine, and equi-affine geometries, whose neural correlates have been partially detected in various brain areas including the cerebellum and the basal ganglia. The cerebellum has been found to play an important role in the control of coordination, balance, posture, and timing over the past years. It is also assumed that the cerebellum computes forward internal models in relationship with specific cortical and subcortical brain regions but its motor relationship with the perceptual space is unclear. A renewed interest in the geometrical and spatial role of the cerebellum may enable a better understanding of its specific contribution to the action-perception loop and behavior's adaptation. In this sense, we complete this overview with an innovative theoretical framework that describes a possible implementation and selection by the cerebellum of geometries adhering to different mathematical laws.

Physics Based Animation Using Computer Graphics

Physics-based animation is a multidisciplinary area that uses ideas from physics, computer science, and mathematics to create realistic and dynamic movements in virtual settings. The basic ideas and methods of physics-based animation are introduced in this research paper, with an emphasis on how mathematical models and physical laws are used to produce realistic motion in computer-generated images. To construct realistic virtual environments, key disciplines covered include fluid simulation, fabric simulation, rigid body dynamics, and soft body dynamics. The report highlights recent developments in real-time physics simulations and addresses other topics such as processing economy, accuracy, and scalability. Applications for physics-based animation can be found in many different fields, such as virtual reality, simulation training, movies, and video games. The pursuit of more precise and effective physics-based animation is essential as technology develops to produce captivating and realistic virtual worlds. Keywords— Physics-based animation(PBA), virtual environment, Physics simulation, Dynamics, Animation software, Real-time physics, Collision detection, Fluid dynamics, Particle systems, Cloth simulation, Smooth particle hydrodynamics, finite element method, Navier-Stokes, Rigid body dynamics, Soft body dynamics, Deformation, Motion capture, Character animation, Kinematics, Rendering, Game development, Virtual reality, Augmented reality, Computer graphics, Simulation accuracy

Internally Disclosed Sets of a Real Space

There is no single set in a real space, for which an exact mathematical definition would not exist by the mathematical symmetry laws. We discuss a theory in which a real number axis is defined at the new level, namely, at the level of real space allowing one to formulate and prove the theorem on the basis of its internal disclosure. This new theory of a set makes it possible to introduce a notion of the full compactness of sets of a real space, confirming their availability in the defined symmetry of elements of a definitely symmetrical line.

Statistical analysis of physico-mechanical parameters of sandstones occurring in orogenic settings

In northeastern Sicily (Italy), sandstone rock masses widely crop out as cover deposits over crystalline terrains belonging to the orogenic belt. Despite being part of the same geological formation, these sandstones are characterized by highly different features in terms of texture and physico-mechanical properties. This poses a scientific question on the possibility of tracing these rocks to a single statistical model, which could be representative of their main engineering geological properties. Therefore, it is worth investigating on the possible reasons of such differences, that should be searched either in the current geographical sandstone distribution or in the rock texture. For this study, sandstone samples were collected from different sites and were analyzed at both the hand and thin section scales. Three sandstone types were recognized, characterized by a different texture. Then, the laboratory characterization allowed estimating their main physico-mechanical and ultrasonic properties, such as porosity, density, mechanical strength, deformability, and ultrasonic velocities. The rock mechanical strength proved linked to the rock compactness and to the presence of lithic fragments, while pores and a pseudo-matrix between grains represent weakening features. Rock data were also statistically analyzed by grouping the specimens according to a geographical criterion, with respect to their sampling area, but no link was found between location and rock properties. Finally, with the aim of achieving mathematical laws that could be used to predict some rock properties from others, useful for practical purposes when dealing with such a high property variability, single and multiple regression analyses were carried out. Results show that the Uniaxial Compressive Strength, porosity, and P-wave velocity are the best predictors for a quick, indirect estimation of the main physico-mechanical parameters. The methodological approach developed for this research can be taken as reference to study other worldwide cases, involving rocks characterized by a wide range of physico-mechanical properties and covering large regional territories.

Mathematical definition and rules of the splitting/merging patterns in bundles of human peripheral nerve segment

Accurately measuring the spatial extension distance of nerve bundles in completing a split/merge is impossible because no clear mathematical definition exists for the starting and ending positions in nerve-bundle splitting/merging. We manually count the number of nerve-bundle splits/merges in long nerve segments, which is labor-intensive, inefficient, and prone to counting errors. Currently, the mathematics are unclear for the nerve-bundle diameter before and after splitting/merging. This paper explores these problems and proposes nerve-bundle splitting/merging rules. Based on the method of defining the beginning and ending positions of nerve-bundle splitting/merging, we explored the mathematical law of equivalent diameter of nerve bundles before and after splitting/merging. The experimental results revealed that the moving average of circularity of nerve bundle accurately defines the beginning and ending positions of nerve-bundle splitting/merging. The diameter of the nerve bundles before and after split/merge approximately conforms to the principles of the Da Vinci formula. The proposed automatic counting algorithm based on centroid offset matching obtains the number of split/merged nerve bundles in the sequence scan images with 100 % accuracy. The mathematical definition of the starting and ending positions of nerve-bundle splitting/merging proposed in this paper is accurate and strict and is the foundation of subsequent research. The proposed automatic counting algorithm based on centroid offset matching (ACA-COM) can accurately and efficiently count the number of times the nerve bundles split and merge in sequential images. The mathematical law satisfied by the diameter of the nerve bundles before and after splitting/merging reflects that the nerve bundles tend to have better capability to resist breaking.

A Random Journey Through the Math of Gambling

The laws of chance are often subtle and deceptive. This is why games of chance work. People are convinced that they obey seemingly intuitive laws, while the underlying mathematical structure reveals a different and more complex reality. This article is a brief and rigorous journey through the implications that the mathematical laws governing stochastic processes have on gambling. It addresses a specific process, the random walk, and analyze some instances of fair and unfair games by highlighting the fallacy of many of our intuitions and beliefs. The paper gradually moves from the analysis of the random walk properties to a comprehensive description of the ruin problem. The introduction of the idea of transient and persistent states concludes the discussion. Much emphasis is placed on concrete examples and on the numerical values, in particular of the involved probabilities, and the interpretation of the results is always more central than the demonstrative technical details, which are nevertheless available to the reader.

A Newly Defined Electromagnetic Dural Armor Functioned as a Brain Protecting Cerebrosphere: A Preliminary Theoretical Analysis

Background: Electric and magnetic field-generating systems must be insulated in order to maintain their balance. It is certain that the brain, which has a very intense electric and magnetic field, is insulated by the dura mater and cerebrospinal fluid (CSF) that surround it. In this article, the electrophysical properties of these structures will be postulated in accordance with the laws of mathematics and physics.
 Material and Methods: In human samples, on the other hand, the morphological features of EEG waves were examined with parameters such as the number of scalp hairs and scalp thickness, conductivity, skull thickness, ratios between cranial and brain volumes, and the thickness of the subarachnoid space where CSF circulates, and ventricular volumes. Since this study is postulative, the data were not detailed by statistical evaluation.
 Results: With the geometric shapes of EEG waves; scalp thickness and number of hairs, skull thickness, depth of subarachnoid space, ventricular volumes, thickness of dura mater. EEG artifacts were excessive in pediatric cases with closed fontanelles or in adults with bone defects. There were statistically varying safety limits between 0.05

Heat and mass transfer characteristics of Al2O3/H2O and (Al2O3+Ag)/H2O nanofluids adjacent to a solid sphere: A theoretical study

A theoretical study is carried out to explore the heat and mass transfer features in the flow of two types nanofluid, i.e., A l 2 O 3 / H 2 O micropolar nanofluid and ( A l 2 O 3 + Ag ) / H 2 O hybrid nanofluid, adjacent to a solid sphere. Furthermore, simultaneous effects of linear thermal radiation and chemical reaction are also explored on the heat and mass transfer characteristics. In mathematical formulation basic conservation laws are utilized along with the Soret and Doufer effects as a coupling agents in the energy and mass concentration equations. The normalized variables are used to transform the governing system of non-linear partial differential equations into the system of dimensionless partial differential equations. Stream function reduces the number of dependent variables and the resulting system is solved numerically by Keller box method. Several plots are presented to explore the impact of involved parameters on the different profiles and engineering interest quantities for both types of fluids. The temperature profile and Nusselt number are large in magnitude for hybrid nanofluid as compared to A l 2 O 3 /water based nanofluid. The heat transfer and mass transfer rates rises with thermal radiation parameter and chemical reaction parameter, respectively.

Optimization of the current spray angle of the nozzles of the adaptive distribution system of a single-support boom sprayer

One of the unsolved problems of plant protection mechanization is the uneven distribution of pesticides due to vertical vibrations of the sprayer boom. At the same time, the variation of the treatment band with each sprayer can reach 38% or more. This issue is even more relevant for single-support wheelbarrow sprayers, which are used in breeding and primary seed production. This significantly affects the quality of the technological operation. A technical solution to this problem could be the use of sprayers with variable spray geometry. Such a solution requires an analytical substantiation of the dependence of the spray pattern angle on the position angle of the sprayer boom. The purpose of the proposed research is to develop a technique for optimizing the spray angle of sprayer nozzles, taking into account the vertical vibrations of the distribution rod. Geometric and mathematical modeling, theoretical analysis of qualitative and agrotechnical parameters of spraying, substantiation of the laws of the technological process of spraying based on the known laws of physics and classical mathematics, as well as a method of nomogramming functions of several variables were used as scientific methods. It has been established that the uniformity of the application of protective equipment is determined not only by the position of the rod in the vertical plane, but also by the value of the root spray angle, the height of the installation of the sprayer and its position relative to the center of oscillation of the rod. An analytical expression is derived that relates the current width of the processing strip from one sprayer to the specified factors. The dependence of the optimal spray angle on the angle of inclination of the sprayer boom is also proposed. A nomogram has been developed that takes into account permissible deviations in the uneven distribution of the working fluid during spraying. The optimization technique is new. It is applicable to the development of software for the sprayer control complex, which is equipped with an adaptive pesticide distribution system in changing conditions.

A Study on Pricing and Replenishment Decision of Vegetables in Fresh Superstores Based on Time Series Analysis

This paper focuses on an in-depth study of the vegetable pricing and replenishment decision problem in fresh produce superstores. Using data preprocessing and multiple mathematical models, the sales and correlation laws between different categories and single products of vegetable products are analyzed. Firstly, the cyclical pattern of sales is revealed through the analysis of historical sales data; secondly, the pricing and replenishment strategies are optimized by using multiple linear regression and time series models; finally, an optimization model is constructed to predict future sales and replenishment, taking into account the spatial limitation of sales.

ОПРЕДЕЛЕНИЕ РАСЧЁТНОЙ ДЛИНЫ ПОЯСА ФЕРМЫ ПРИ УПРУГОМ ПОДКРЕПЛЕНИИ ИЗ ПЛОСКОСТИ.

An algorithm for determining the effective length for a compressed chord of a U-shaped truss using numerical methods is presented, and the results are compared with analytical calculations [6]. The classic bracing solution for bracing compressed truss chords of any spans involves the installation of bracing blocks that prevent the truss nodes from moving out of plane. In engineering practice, there are solutions in which there is no constructive possibility of installing tie blocks along compressed chords of trusses. An example is the U-shaped truss of overhead open pedestrian bridges, in which the running surface is located at the level of the lower chords, and the upper chords of the trusses are located at the level of the handrails of the fence. The design solution of a U-shaped truss is presented in Figure 1. The standards [1] provide formulas for determining the design length of a truss chord (continuous rod) with different compressive forces in sections, but there is no engineering method for calculating the design length of a truss chord depending on elastic lateral torsional. Due to the fact that domestic standards [1], [2] and [4] establish clear mathematical laws for calculating the design lengths of compressed rods, solving the posed engineering problem is possible, for example, using numerical methods.

APPLICATION OF LANCHESTER'S MATHEMATICAL LAWS IN MILITARY STRATEGY

Mathematics, as the science of numbers, structures, and models, plays an important role in many aspects of military operations and strategies. From calculating the probability of success or failure of military operations to determining the best ways to deploy troops and resources, mathematics provides military strategists with powerful tools to make informed and effective decisions. One of the most important aspects of using mathematics in the military is developing strategies for combat operations. From determining the location of troops and the size of forces to the conditions of operations, mathematical methods and models help to optimize decisions. To determine the probability of success of military operations, mathematics uses probability theory and statistics. This allows us to estimate the probability of achieving the goal, taking into account various factors such as military equipment, enemy location, and other external conditions. The analysis of previous military conflicts and data allows us to statistically estimate probability distributions and predict military events. Mathematics also plays an important role in solving logistics and resource allocation problems. Determining the optimal route for the movement of troops and resources can be formulated as a path optimization problem. Mathematical methods are also used in the field of intelligence and the development of new technologies. Cryptography, which protects important information from unauthorized access, is based on complex mathematical algorithms. Mathematical models can also be used to simulate military operations, study the impact of new weapons systems, or analyze missile trajectories. It is worth adding that mathematical laws help to analyze and predict the outcomes of military conflicts, in particular, to determine the impact of the size and effectiveness of the enemy forces on the probability of success. In this article, the main focus is on the laws of Lanchester's mathematical models. The article presents the derivation of Lanchester's linear law and an example of its application. The mathematical principles of the quadratic Lanchester's law are considered on an example. It is indicated and highlighted how these mathematical laws can be applied in the context of the Russian-Ukrainian conflict.

Fully Regular Sets of an Imaginary Space

There is no single set in an imaginary space, for which an exact mathematical definition would not exist by the mathematical symmetry laws. We discuss a theory in which appears an imaginary number axis strictly defifined at the level of imaginary space, allowing one to formulate and prove the theorem on the basis of its internal disclosure. This new theory of a set makes it possible to introduce a notion of the full compactness of sets of an imaginary space, confirming their availability in the defined symmetry of elements of a definitely symmetrical line.

Laws: historical transformation, modern understanding, classification

The article considers the historical transformations of the law, and argues that many contemporary scientists are wrong to ignore it. The paper criticizes the approach to the status of laws, which is determined by the degree of their compliance with the criteria of objectivity and rationality. Instead, a classification is constructed that includes the laws of logic and mathematics, natural sciences, technology, as well as social and legal. To consider the processes of unfolding laws in the evolution of reality, the concept of nomogenesis is used. The unification of laws is achieved due to the fact that they are all determined by the balance of their characteristic tendencies: objective / subjective, rational / extra-rational, determination / will. Special attention is paid to social and legal laws. The case of defining three social laws is offered.