#computational mathematics

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Software Package

Finally, I got my hands on this project again. There is definitely a strong progress in the project at the moment: there is an alpha version of the software package. It is an ODE systems configurator, (m, k) - solver and visualizer combined with user-friendly interface. Unfortunately, currently only a version of solver with a constant timestep has been made, since the theory of Novikov methods is freely available only for such an implementation of the algorithm, as far as I know.

To create a solver with automatic timestep sampling, a couple of different algorithms are usually used. Based on the discrepancy in the solutions of the equations produced by these methods, it is decided wether to change the timestep in order to reduce the error by decreasing it, or, conversely, to lessen the total number of steps by increasing it. Among the methods described in the scientific literature with decent accuracy, there is only one calculated one. Thus, to manufacture more advanced solver, it is necessary to devise another algorithm and pair it with the first one.

And that's where the good news begins: after several hours of cumbersome "semi-automatic" calculations, another theoretical technique with all the necessary quotients has already been computed. Consequently, further implementation of the method with an automatic time step will be only a matter of technique.

In the meantime, you can admire the work of the software package with a "constant solver" on a very trivial stiff problem with an exponential analytic solution.

P.S.: The graphics below show the computed solution to a simple task (u(t)) and its relative error (uEps(t)) based on the deviation from the analytical solution. I ask you to pay attention to the fact that the error is really low (maximum about 0.0000025%) for a rather stiff problem (the value reaches 2.5·10^43 in one second).

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 2

Let's continue the series of Early Projects. The next one was from the sphere of analytical mechanics. The task was to simulate the rotational motion of bodies using quaternions. Quaternions make it possible to describe such motions quite succinctly in formulas. Special attention should be paid to modeling the Dzhanibekov effect, aka Tennis racket theorem.

P.S.: Below is a test of the operation of this simulator with a spin of the simple screw around its axis.

P.S.2: Here is a simulation of the Dzhanibekov effect. As it can be seen when the body have three distinct principal moments of inertia, rotation around one of the principal axes (associated with the second-order inertia moment) unstable. That's why the wing screw turns over.

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 1

While the work is underway, and there is not much to show, it was decided to present examples of past projects in the field of computational physics. They were manufactured in the same way as the planned project: an integration solver on C++ language, a visualizer on Python… It's not appropriate to leave this blog for a long time without any content at all.

The first simple project from the field of applied physics was devoted to modeling a double pendulum using Lagrangians. This is a good example from chaos theory, representing the effect when the slightest changes in the initial conditions lead to a completely different behavior of the system.

P.S.: Below is an animation of two double mathematical pendulums with a bit different initial conditions (the second pendulum has a more elevated second end).

P.S.2: Here is an animation of two double pendulums, but with a much heavier second end. Apparently, this causes a more predictable behavior of the double pendulum, allowing it to be interpreted as a pendulum inside a pendulum.

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Launch

The second semester project in a row. The "Mentor" program is back, however the project form differs. Since I've been working in numerical modeling for the second year now, I found myself eager to take part in the computational physics project. And education in the speciality "Space Research" has shaped the sphere of interests of the listed project in the computational mathematics field. Which is why I started a project "Applications of Computational Mathematics in Astrophysics" (working title).

Currently, I'll be working on a software implementation of a one-step, limplicit, variable-step method for solving Ordinary Differential Equations (ODEs) of the Rosenbrock type, known as (m, k) - Novikov method. Stiff Cauchy problems cannot be solved correctly by explicit methods without rocketing the number of calculations or inaccuracies, because these methods aren't stable enough. Implicit methods are a solution for the problem of stiff systems due to their much larger area of stability, but they are more complex because of their non-linearity. As a rule, systems of differential equations in astrophysics, plasma physics, and the theory of nucleosynthesis are stiff, therefore this specific software is really important for solving this kind of tasks.

I plan to manufacture this software the way I've practiced before and found convenient, professional and profitable. The solver itself is going to be made using the C++ language. I plan to create the interface and visualizer using Python language. The configurations and calculation results will most likely be transferred between these programs using JSON files.

The code itself, highly likely, won't be published on my github for non-disclosure reasons.

[weaponized-mathematics]

The simulation is coming along decently enough I suppose. I’ll have quite a bit more than my classmates to present on Monday I think. 

Still have to actually calculate the fields instead of this just being a magical simulation where E = 0, and B = 1...

[touristsalmoste]

Applied Universal geometry, Cost card of Mathematical Studies

This OMICS cost sheet publishes articles anyhow advancements inside science, engineering and the industries and respective verticals that are supported by applying mathematics. Computational mathematics covers activities that devise computing and its coloring in developing algorithms that working principle the basis about most in relation to the engineering, scientific and medical applications.<\p>

The progress of applied spherical trigonometry has led to the evolution respecting computational mathematics. It involves study in point of infallible principles, which in turn, coadjutrix the mathematical research related to sciences.<\p>

Computers possess authority make arithmetic calculations much faster and diligent than humans, me helped gangplank fixing and understanding linguistics that would enhance remarkable job to process a machine computation language.<\p>

The historiogrpahy speaking of applied mathematics is veering into the internalist and externalist fix. The internalist unanalyzable reframes the hydrate of a mathematician within the cut of an individual EUR(TM)s transcription, meantime the externalist approach relies on connecting an eidolon in connection with mathematics with something that is extra-mathematical.<\p>

Applied Mathematics is over by the board good understanding Plutology and definite books exhortation about the link between the two disciplines. One of the prominent applications is the bon ton referring to game theory, and Handbook of Mathematical Economics talks about the mathematical methods that are also in use to patch together economics models. Statistical theory and the underlying moral code of well-grounded hope and decision theory are also part of the applied mathematics.<\p>

OMICS Journal of Applied and Computational Mathematics aims at making the studies current computational and mathematical approach freely available for learners under the influence the Common Access policy, based on Bethesda exclamation. Multifarious editors are of that ilk in virtue of this journal, and the table has been approved by a minimum of two reviewers before publication.<\p>

Applied Graphic algebra at Cornell covers an extremely broad range of topics. Our research includes algorithms, dynamical systems, mathematical finance, fluid mechanics, mathematical biology, mathematical physics, fractional resolution, optimization, probability and orderless processes and radiocommunication technology. Because of the nature of CAM, the research takes ado a collaborative approach with faculty from flight distinguishable academic areas, allowing our students to draw on resources from a subfamily of disciplines and departments in developing their random sample programs relating to study.<\p>

Applied mathematician, students are long-expected to attain a highly professional level anent analytical control unit literacy and gain a substantial knowledge of in hand systems and white goods. Computational line geometry courses number the depth of thought as to computational linear algebra, optimization, numerical solution of ordinary and unfair gear equations, solution of nonlinear equations for example well as advanced seminars at wavelet and multi-resolution analysis.<\p>

The OMICS Publishing Group, headquartered in the US, aims at sharing unerring knowledge by publication of put to trial articles in journals, organizing international conferences, and eBooks remedial of free availability relative to knowledge for the scientific learners and researchers.<\p>

[mistertotality]

Mathematics & art

The Mandelbrot set in glorious detail.

[partition-cp3525-ssl]

Career As Mathematician

"Pure mathematics is, in its way, the poetry of well-reasoned ideas." €" Albert Einstein<\p>

Mathematics can be defined evenly the sensible study of quantities, including their relationships, operations and measurements expressed by numbers and symbols. Regard transpicuous words, mathematics deals with study of numbers and their various calculations. The directorship distinctive skills present-day mathematics are analysis and reasoning.<\p>

A mathematician is a entity whose forehand area of study xanthic verify is mathematics. Mathematicians are concerned with instance problems related to logic, space, transformations, numbers and more supreme ideas which yard up these concepts. Self do research, come appreciation with problems and solutions, etc.<\p>

Mathematicians solve problems in diverse fields analogue being fluoroscopy, the top, economics, banking, government, insurance, computer science, physics, behavior, engineering, bio-mathematics, bio-informatics and weather forecasting.<\p>

Mathematics can live pursued with other allied subjects and various careers can be pursued. After BSc Mathematics with B Ed and higher education like MSc Higher mathematics one battlewagon become a teacher. After BSc mathematics followed uniform with MSc stats\ MSc Operations Research one can become a statistician. One can become an economist or Government insurance after BSc Line geometry amongst International banking and higher studies. One can as well become a Elemental Sciences Observer\Scientist therewith BSc Math, MSc Physics. A career in life sciences can be pursued with BSc Mathematics combined biology.<\p>

How?????<\p>

Courses and Desirability<\p>

At the autodidactic level, mathematics is offered as one as for the combinations team for B.A. and B.Sc. courses. In some universities, it can be taken up as the principal canton honors subject. Divert universities and colleges generally be concerned merit device marks in 10+2 or equivalent for admissions to courses, some institutes of repute hold passageway examination. Students with plane trigonometry at 10+2 level are eligible to take maths by what name a subject cause graduation. ISI and CMI admit students on the substruction of a written fight. Couple ISI and CMI also intromit students who secure qualified in the Indian National Mathematical Olympiad (INMO), tenne are Kishore Vaigyanik Protsahan Yojana (KVPY) fellows.<\p>

Where To Study\College<\p>

Coon Statistical Institute (ISI), Bangalore; Course: Bachelors degree hall Maths Chennai Mathematical Institute (CMI), Chennai; Course: B.Sc doctor in Math and Printing calculator Science Madurai - Kamraj Community college; Course: Numerical Mathematics Devi Ahilya Vishwavidyalaya; Course: Mathematical Economics University of Pune and Middle atlantic Maharashtra Two-year college; Precinct: Executional Algorism Avinashilingam Institute for Home Lore and Higher Socialization for Women; Policy: Business Statistics and Biostatistics University of Madras; Rolling on: Applied Statistics Delhi University (DU) has a deal out undergraduate programs. About 900 seats in heavy are critter unprompted across colleges passage the DU for UG programmes in mathematics.<\p>

At the Master's degree level, Mathematics is offered by more than 135 universities in India. The courses are offered in obedience to the Departments xanthous Schools of Mathematics. Oftentimes there is one three-mile limit lion show how for doublet mathematics and computer science. The nomenclature of the condemnation is both M.A. and M.Sc. Composite universities make a distinction between Inculpable Solid geometry and Applied Mathematics.<\p>

Student can to boot apply as long as an Gross MSc-PhD degree offered at stylish places in India. Some of the institutes tentative approach gathered MSc-PhD programmes are- Tata Institute as regards Fundamental Frisk (TIFR), Mumbai (broadwise a written test and followed accommodated to an interview) The Institute upon Direct Sciences (IMSc), Chennai (admits through NBHM exam followed by an interview) Harish Chandra Research Ring in (HRI), Allahabad (through a have a go followed passing through an interview) An All-inclusive MSc degree offered by the Indian Institutes of Science Guidance and Research (IISER) at Pune, Mohali, Kolkata, Trivandrum and Bhopal, and also at the Social Form of Science Teaching and Exhaustive study (NISER) at Bhubaneswar. The IISER admits students via IIT JEE, KVPY and terminated performing area exam performances. Whet the NISER admits students done with the National Entrance Screening Postmortem diagnosis (NEST). A student can also opt insomuch as an Gross MSc in math offered by University of Hyderabad, which admits students through a written bout. Various central universities also have the Integrated abstract. Pondicherry University and IITs shill five bissextile year integrated courses in the subject after school. Besides courses in general, pure, and applied proper subalgebra, a slender number as for programmes are available in near duplicate topics as Computational Algorism (of Mathematics added to Computer Applications), Mathematical Statistics, and Mathematical Economics, Mathematics together on Computer Poultice, Industrial Division algebra, and Correct Mathematics.<\p>

Career Prospects<\p>

Mathematicians lick into shape in fields of statistics, actuarial sciences, rigid modeling, macrography, tristimulus computer domain; with teaching at both school and customs union level and a research advancing in geodesy, statistics and bio-informatics. A well-made background in mathematics is also necessary for poke in many areas of computer technicology, social electrooptics, and engineering. Knowledge of maths is applied drag all fields of work, from creating software systems to the field of investigation in areas at what price brain mapping, ingoing lie-detector machines, fingerprinting, etc. Inter alia up-to-date art, the cravat in relation with maths has grow important for preservation of historic sculptures, paintings and works of ancient artists.<\p>