Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. A point of the plane belongs to a curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the points, the inflection points. More advanced questions involve the topology of the curve and relations between the curves given by different equations, Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. In the 20th century, algebraic geometry split into several subareas, the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field.
The study of the points of a variety with coordinates in the field of the rational numbers or in a number field became arithmetic geometry. The study of the points of an algebraic variety is the subject of real algebraic geometry. A large part of singularity theory is devoted to the singularities of algebraic varieties, with the rise of the computers, a computational algebraic geometry area has emerged, which lies at the intersection of algebraic geometry and computer algebra. It consists essentially in developing algorithms and software for studying and finding the properties of explicitly given algebraic varieties and this means that a point of such a scheme may be either a usual point or a subvariety. This approach enables a unification of the language and the tools of algebraic geometry, mainly concerned with complex points. Wiless proof of the longstanding conjecture called Fermats last theorem is an example of the power of this approach. For instance, the sphere in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 −1 =0. A slanted circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 −1 =0, x + y + z =0, first we start with a field k.
In classical algebraic geometry, this field was always the complex numbers C and we consider the affine space of dimension n over k, denoted An. When one fixes a system, one may identify An with kn. The purpose of not working with kn is to emphasize that one forgets the vector space structure that kn carries, the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k
Engineers design materials and systems while considering the limitations imposed by practicality, regulation and cost. The word engineer is derived from the Latin words ingeniare and ingenium, the work of engineers forms the link between scientific discoveries and their subsequent applications to human and business needs and quality of life. His/her work is predominantly intellectual and varied and not of a mental or physical character. It requires the exercise of original thought and judgement and the ability to supervise the technical, he/she is thus placed in a position to make contributions to the development of engineering science or its applications. In due time he/she will be able to give authoritative technical advice, much of an engineers time is spent on researching, locating and transferring information. Indeed, research suggests engineers spend 56% of their time engaged in various information behaviours, Engineers must weigh different design choices on their merits and choose the solution that best matches the requirements.
Their crucial and unique task is to identify, Engineers apply techniques of engineering analysis in testing, production, or maintenance. Analytical engineers may supervise production in factories and elsewhere, determine the causes of a process failure and they estimate the time and cost required to complete projects. Supervisory engineers are responsible for major components or entire projects, Engineering analysis involves the application of scientific analytic principles and processes to reveal the properties and state of the system, device or mechanism under study. Most engineers specialize in one or more engineering disciplines, numerous specialties are recognized by professional societies, and each of the major branches of engineering has numerous subdivisions. Civil engineering, for example, includes structural and transportation engineering and materials engineering include ceramic, mechanical engineering cuts across just about every discipline since its core essence is applied physics.
Engineers may specialize in one industry, such as vehicles, or in one type of technology. Several recent studies have investigated how engineers spend their time, that is, research suggests that there are several key themes present in engineers’ work, technical work, social work, computer-based work, information behaviours. Amongst other more detailed findings, a recent work sampling study found that engineers spend 62. 92% of their time engaged in work,40. 37% in social work. The time engineers spend engaged in activities is reflected in the competencies required in engineering roles. There are many branches of engineering, each of which specializes in specific technologies, typically engineers will have deep knowledge in one area and basic knowledge in related areas. When developing a product, engineers work in interdisciplinary teams. For example, when building robots an engineering team will typically have at least three types of engineers, a mechanical engineer would design the body and actuators
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, biology, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century.
Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work, typically to solve mathematical problems. Mathematics is concerned with numbers, quantity, space, one of the earliest known mathematicians was Thales of Miletus, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, the number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was All is number. It was the Pythagoreans who coined the term mathematics, and with whom the study of mathematics for its own sake begins, the first woman mathematician recorded by history was Hypatia of Alexandria. She succeeded her father as Librarian at the Great Library and wrote works on applied mathematics. Because of a dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked.
Science and mathematics in the Islamic world during the Middle Ages followed various models and it was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences, an example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many working under Muslim rule in medieval times is that they were often polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham, the Renaissance brought an increased emphasis on mathematics and science to Europe. As time passed, many gravitated towards universities. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking. ”Thus, overall, science became the focus of universities in the 19th and 20th centuries.
Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. ”Mathematicians usually cover a breadth of topics within mathematics in their undergraduate education, and proceed to specialize in topics of their own choice at the graduate level. In some universities, a qualifying exam serves to test both the breadth and depth of an understanding of mathematics, the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their knowledge and professional methodology. With professional focus on a variety of problems, theoretical systems
A physicist is a scientist who has specialized knowledge in the field of physics, the exploration of the interactions of matter and energy across the physical universe. A physicist is a scientist who specializes or works in the field of physics, physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists can apply their knowledge towards solving real-world problems or developing new technologies, some physicists specialize in sectors outside the science of physics itself, such as engineering. The study and practice of physics is based on a ladder of discoveries. Many mathematical and physical ideas used today found their earliest expression in ancient Greek culture and Asian culture, the bulk of physics education can be said to flow from the scientific revolution in Europe, starting with the work of Galileo and Kepler in the early 1600s. New knowledge in the early 21st century includes an increase in understanding physical cosmology.
The term physicist was coined by William Whewell in his 1840 book The Philosophy of the Inductive Sciences, many physicist positions require an undergraduate degree in applied physics or a related science or a Masters degree like MSc, MPhil, MPhys or MSci. In a research oriented level, students tend to specialize in a particular field, Physics students need training in mathematics, and in computer science and programming. For being employed as a physicist a doctoral background may be required for certain positions, undergraduate students like BSc Mechanical Engineering, BSc Electrical and Computer Engineering, BSc Applied Physics. etc. With physics orientation are chosen as research assistants with faculty members, the highest honor awarded to physicists is the Nobel Prize in Physics, awarded since 1901 by the Royal Swedish Academy of Sciences. The three major employers of career physicists are academic institutions and private industries, with the largest employer being the last, physicists in academia or government labs tend to have titles such as Assistants, Professors, Sr.
/Jr. As per the American Institute for Physics, some 20% of new physics Ph. D. s holds jobs in engineering development programs, while 14% turn to computer software, a majority of physicists employed apply their skills and training to interdisciplinary sectors. For industry or self-employment. and in science and programming. Hence a majority of Physics bachelors degree holders are employed in the private sector, other fields are academia and military service, nonprofit entities and teaching
A biologist, is a scientist who has specialized knowledge in the field of biology, the scientific study of life. Biologists involved in fundamental research attempt to explore and further explain the mechanisms that govern the functioning of living matter. Biologists involved in applied research attempt to develop or improve more specific processes and understanding, in such as medicine, industry. While biologist can apply to any scientist studying biology, most biologists research, in this way, biologists investigate large-scale organism interactions, whole multicellular organisms, tissues and small-scale cellular and molecular processes. Other biologists study less direct aspects of life, such as phylogeny, Biologists conduct research based on the scientific method, to test the validity of a theory, with hypothesis formation and documentation of methods and data. There are many types of biologists, some work on microorganisms, while others study multicellular organisms. Many jobs in biology as a field require an academic degree, a doctorate or its equivalent is generally required to direct independent research, and involves a specialization in a specific area of biology.
Many biological scientists work in research and development, some conduct fundamental research to advance our knowledge of living organisms, including bacteria and other pathogens. This research enhances understanding and adds to the database of literature. Furthermore, it aids the development of solutions to problems in areas such as human health. These biological scientists work in government and private industry laboratories. Many expand on specialized research that started in post-graduate qualifications. Biological scientists who work in applied research or product development often use knowledge gained by research to further knowledge in particular fields or applications. For example, this research may be used to develop new pharmaceutical drugs and medical diagnostic tests, increase crop yields. These scientists must consider the effects of their work. Some biologists conduct laboratory experiments involving animals, plants or microorganisms, some biological research occurs outside the laboratory and may involve natural observation rather than experimentation.
For example, a botanist may investigate the plant species present in a particular environment, swift advances in knowledge of genetics and organic molecules spurred growth in the field of biotechnology, transforming the industries in which biological scientists work. Biological scientists can now manipulate the genetic material of animals and plants and applied research on biotechnological processes, such as recombining DNA, has led to the production of important substances, including human insulin and growth hormone
A scientist is a person engaging in a systematic activity to acquire knowledge that describes and predicts the natural world. In a more restricted sense, a scientist may refer to an individual who uses the scientific method, the person may be an expert in one or more areas of science. The term scientist was coined by the theologian and historian of science William Whewell and this article focuses on the more restricted use of the word. Scientists perform research toward a comprehensive understanding of nature, including physical and social realms. Philosophers aim to provide an understanding of fundamental aspects of reality and experience, often pursuing inquiries with conceptual, rather than empirical. When science is done with a goal toward practical utility, it is called applied science, an applied scientist may not be designing something in particular, but rather is conducting research with the aim of developing new technologies and practical methods. When science seeks to answer questions about aspects of reality it is sometimes called natural philosophy.
Science and technology have continually modified human existence through the engineering process, as a profession the scientist of today is widely recognized. Jurisprudence and mathematics are often grouped with the sciences, some of the greatest physicists have been creative mathematicians and lawyers. There is a continuum from the most theoretical to the most empirical scientists with no distinct boundaries, in terms of personality, interests and professional activity, there is little difference between applied mathematicians and theoretical physicists. Scientists can be motivated in several ways, many have a desire to understand why the world is as we see it and how it came to be. They exhibit a strong curiosity about reality, other motivations are recognition by their peers and prestige, or the desire to apply scientific knowledge for the benefit of peoples health, the nations, the world, nature or industries. Scientists tend to be motivated by direct financial reward for their work than other careers.
As a result, scientific researchers often accept lower average salaries when compared with other professions which require a similar amount of training. The number of scientists is vastly different from country to country, for instance, there are only 4 full-time scientists per 10,000 workers in India while this number is 79 for the United Kingdom and the United States. According to the US National Science Foundation 4.7 million people with science degrees worked in the United States in 2015, across all disciplines, the figure included twice as many men as women. Of that total, 17% worked in academia, that is, at universities and undergraduate institutions, 5% of scientists worked for the federal government and about 3. 5% were self-employed. Of the latter two groups, two-thirds were men, 59% of US scientists were employed in industry or business, and another 6% worked in non-profit positions
In mathematics and physics, a soliton is a self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium, solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell who observed a solitary wave in the Union Canal in Scotland and he reproduced the phenomenon in a wave tank and named it the Wave of Translation. A single, consensus definition of a soliton is difficult to find, more formal definitions exist, but they require substantial mathematics. Moreover, some use the term soliton for phenomena that do not quite have these three properties. Dispersion and non-linearity can interact to produce permanent and localized wave forms, consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of different frequencies.
Since glass shows dispersion, these different frequencies will travel at different speeds, there is the non-linear Kerr effect, the refractive index of a material at a given frequency depends on the lights amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the effect, and the pulses shape will not change over time. See soliton for a detailed description. The soliton solutions are obtained by means of the inverse scattering transform. The mathematical theory of equations is a broad and very active field of mathematical research. Some types of tidal bore, a phenomenon of a few rivers including the River Severn, are undular. Other solitons occur as the internal waves, initiated by seabed topography. Atmospheric solitons exist, such as the Morning Glory Cloud of the Gulf of Carpentaria, the recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons. A topological soliton, called a defect, is any solution of a set of partial differential equations that is stable against decay to the trivial solution.
Soliton stability is due to constraints, rather than integrability of the field equations. Thus, the differential equation solutions can be classified into homotopy classes, there is no continuous transformation that will map a solution in one homotopy class to another
A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli.
In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The formal sciences are often excluded as they do not depend on empirical observations, disciplines which use science, like engineering and medicine, may be considered to be applied sciences. However, during the Islamic Golden Age foundations for the method were laid by Ibn al-Haytham in his Book of Optics. In the 17th and 18th centuries, scientists increasingly sought to formulate knowledge in terms of physical laws, over the course of the 19th century, the word science became increasingly associated with the scientific method itself as a disciplined way to study the natural world. It was during this time that scientific disciplines such as biology, Science in a broad sense existed before the modern era and in many historical civilizations. Modern science is distinct in its approach and successful in its results, Science in its original sense was a word for a type of knowledge rather than a specialized word for the pursuit of such knowledge.
In particular, it was the type of knowledge which people can communicate to each other, for example, knowledge about the working of natural things was gathered long before recorded history and led to the development of complex abstract thought. This is shown by the construction of calendars, techniques for making poisonous plants edible. For this reason, it is claimed these men were the first philosophers in the strict sense and they were mainly speculators or theorists, particularly interested in astronomy. In contrast, trying to use knowledge of nature to imitate nature was seen by scientists as a more appropriate interest for lower class artisans. A clear-cut distinction between formal and empirical science was made by the pre-Socratic philosopher Parmenides, although his work Peri Physeos is a poem, it may be viewed as an epistemological essay on method in natural science. Parmenides ἐὸν may refer to a system or calculus which can describe nature more precisely than natural languages. Physis may be identical to ἐὸν and he criticized the older type of study of physics as too purely speculative and lacking in self-criticism.
He was particularly concerned that some of the early physicists treated nature as if it could be assumed that it had no intelligent order, explaining things merely in terms of motion and matter. The study of things had been the realm of mythology and tradition, however. Aristotle created a less controversial systematic programme of Socratic philosophy which was teleological and he rejected many of the conclusions of earlier scientists. For example, in his physics, the sun goes around the earth, each thing has a formal cause and final cause and a role in the rational cosmic order. Motion and change is described as the actualization of potentials already in things, while the Socratics insisted that philosophy should be used to consider the practical question of the best way to live for a human being, they did not argue for any other types of applied science
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order, individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, the fields of mathematics and statistics use formal definitions of randomness. In statistics, a variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events, Random variables can appear in random sequences. A random process is a sequence of variables whose outcomes do not follow a deterministic pattern. These and other constructs are extremely useful in probability theory and the applications of randomness.
Randomness is most often used in statistics to signify well-defined statistical properties, Monte Carlo methods, which rely on random input, are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators, Random selection is a method of selecting items from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable and that is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen we can say the selection process is random. In ancient history, the concepts of chance and randomness were intertwined with that of fate, many ancient peoples threw dice to determine fate, and this evolved into games of chance.
Most ancient cultures used various methods of divination to attempt to circumvent randomness, the Chinese of 3000 years ago were perhaps the earliest people to formalize odds and chance. The Greek philosophers discussed randomness at length, but only in non-quantitative forms and it was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus had a impact on the formal study of randomness. The early part of the 20th century saw a growth in the formal analysis of randomness. In the mid- to late-20th century, ideas of information theory introduced new dimensions to the field via the concept of algorithmic randomness