Jul 18Member-onlyWhy even solvable problems are often unsolvableSolvability is a very interesting concept in mathematics by which one can determine, using sophisticated methods of proof and analysis, whether a certain problem can be solved, be it an equation, system of equations, integral, etc. …Mathematics6 min readMathematics6 min read
Mar 29Member-onlySolving an integral equation with a simple trickIn a paper published in 2011 by Waldemar Klobus (Motion on a vertical loop with friction) an interesting integral equation (more precisely, a Volterra integral equation of the 2nd kind) appears: which describes the motion of an object around a circular loop as shown in Fig.1 below:Mathematics4 min readMathematics4 min read
Mar 26Member-onlySolving difficult integrals with substitutionIntegration by substitution is a very powerful tool for evaluating integrals where the answer may not seem as obvious at first, but under appropriate variable transformations, we can obtain simple and elegant solutions. …Mathematics4 min readMathematics4 min read
Mar 23Member-onlyCauchy’s Integral FormulaIn my previous article (Contour integrals — a simple introduction) I discussed how to perform integrals of complex-valued functions in the complex plane, along a defined contour C by using techniques from line integrals on the real plane. …Mathematics6 min readMathematics6 min read
Mar 23Member-onlyContour integrals — a simple introductionWhen dealing with functions of complex variables, such as f(z) where z = x + iy, their integration with respect to z must be done in a slightly different way than integrals of real-valued functions. The techniques are usually a bit difficult for university students to pick up, especially because…Mathematics4 min readMathematics4 min read
Feb 19Member-onlyNumba: Python’s lord and saviorWhen it comes to high performance computing (HPC), people often think of programming languages like C/C++ or Fortran, especially in the scientific community. Some more adventurous people may even try Julia. However, almost no one ever thinks “let me write a HPC application in Python” unless they really know what…Python7 min readPython7 min read
Jan 30Member-onlyDifferentiating inverse functionsFunctions such as arcsin(x), arccos(x), arctan(x), etc. are considered to be inverse functions of sin(x), cos(x) and tan(x) respectively. They have their own properties and as such, they have derivatives which are noticeably different from their ordinary counterparts. In my previous article: https://medium.com/@oscarnieves100/implicit-differentiation-in-a-nutshell-46031531f34b I discussed how to use implicit differentiation…Mathematics3 min readMathematics3 min read
Jan 30Member-onlyImplicit differentiation in a nutshellWhat is implicit differentiation? When we are dealing with derivatives of functions in calculus, we often encounter functions such as y = f(x) where some variable y can be explicitly expressed as a function of an independent variable x, so the differentiation process is straight-forward, the derivative of y is…Mathematics5 min readMathematics5 min read
Jan 19Member-onlyEssential math for quantum mechanicsAll the things you need to know. Quantum mechanics — and by extension quantum field theory — is a subfield of physics concerned with studying the dynamics of single and many-particle systems at the quantum scale, often designated as “quantum systems” which can encompass fundamental particles (e.g. quarks, leptons, bosons)…Science7 min readScience7 min read
Jan 19Member-onlyBivariate nonlinear regression using PythonIn my previous article: https://oscarnieves100.medium.com/univariate-nonlinear-regression-in-python-deb91d1085cd, I discussed how to do nonlinear regression in Python for univariate data using the module scipy.optimize and the function curve_fit. Now, I will extend that to bivariate regression using the same technique. Unbeknownst to some people, Python’s curve_fit function can also be used for bivariate…Data6 min readData6 min read
