Mathematical Modelling

Mathematical Modelling

  • Elman Shahverdiev added an answer:
    Why do ordinary differential equation (ODE) models of cancer suggest different behaviors for cancer cells?

    For validation part of my study, I need a comparison between my model of ductal carcinoma in situ (DCIS) and ODE models of this area. But, I’m really confused because ordinary differential equation (ODE) models of cancer have suggested different behaviors for tumor and immune cell populations. For example, the below behaviors are reported by the survey of Eftimie et al. (2011) [1]:

    • Tumor size decreases exponentially through interactions with the immune cells.
    • Tumor size decreases at first. Then, the decay of immune cells leads to an exponentially increase in it again.
    • Tumor size decays in an oscillatory manner.
    • Tumor size grows in an oscillatory manner.

    I don’t understand the reason of the difference! And, I don’t know which behavior is right. Could anyone possibly help me, please?


    [1] A Validated Mathematical Model of Tumor Growth Including Tumor–Host Interaction, Cell-Mediated Immune Response and Chemotherapy

    Elman Shahverdiev

    Dear Negin, 

    I guess your model is non-linear as it should be in such cases. Then  such systems are prone to different behavior, including quasi periodicity, chaotic, etc., depending on the systems' parameters. Besides, may be even some questions arise about the adequacy of the model under consideration. As far as i know in most cases in cancer dynamics it is necessary to  account for some time delays in the system. In other words, may be it is more reasonable to switch to the delay differential equations (DDE).

    Even with more adequate description of the process may it is worthwhile to consider spatial dynamics, i.e. to use partial differential equation (PDE) approach. Although usually DDE can be used to model PDE dynamics too.

    Hopefully this short note could be some help.

  • Hamid Moa added an answer:
    How to model mathematically plate and shell structural damage subject to blast and explosion loading?

    I find some paper for this topic in low-velocity dynamic loading on stiffened plate(like as grounding of ship bottom hull),but can not find in high-velocity range like explosion(internal or external). all of this re souls are in experimental or numerical method. How to model mathematically plate and shell structural damage subject to blast and explosion loading?

    Hamid Moa

    beat regard SAM

  • Rafael Gontijo added an answer:
    How is the evolution of contact between the asperities of two contacting bodies?

    Our team is studying the problem of thermal contact resistance which depends on the contact between asperities. We are trying to capture the transient phase before a steady state is reached. So we are interested in results of analytical or numerical or experimental investigations into the deformation of asperities just after the contact is established and until the final contact patch is formed. Geometry or material of the two bodies is no constraint. Thanks.

    Rafael Gontijo

    In a very interesting problem in the field of fluid mechanics, published in the Journal of Fluid Mechanics studied the problem of two approaching spheres with different asperities. The contact between these rough particles induced a hydrodynamic diffusive coefficient. Maybe you could find some interesting references there. Here is the link of this manuscript:

  • Ghanshyam G Tejani added an answer:
    Can anyone help with a Particle Swarm Optimization algorithm?
    Can anyone help me with a PSO algorithm? Is there any c/c++ programme available?
    Ghanshyam G Tejani

    Go to

  • A. J. Roberts added an answer:
    Which is the most efficient algorithm/package to solve delay differential equations?

    I wish to know the methods of solving a system of equations that require, and those which don't require computer algebra. Is there a method, which is both theoretically and computationally efficient?

    A. J. Roberts

    "to solve" could mean algebraic, numerical, or approximate.  If you want to construct approximate emergent models near bifurcations (pitchfork, Hopf, higher order), you can use my web service that constructs a center manifold model of ODEs or DDEs for systems a user enters.  My web page gives five examples you can try including a Double Hopf bifurcation in a delay DE.

  • Peter T Breuer added an answer:
    What is the best time complexity of this case?

    Assume that the integer numbers s, K, and ui, for i=1,2, ..., m, and ai, for i=1,2, ..., m, are given. I would like to know what is the time complexity of finding out that if the following system has an integer solution, say X=(x1x2, ..., xm), or not. Actually, it does not matter to find all the solutions, and I only want to know whether there is any integer solution for the system or not.

    The system:

    1. x1 + x2 + ... + xm = s
    2. xi <= ui,  for i = 1, 2, ..., m
    3. a1x1 + a2x2 + ... + amxm <= K

    We know that one can find all the solutions of the (1) and (2), and then check each solution for the inequality (3). However, we think this is not that much efficient. I highly appreciate your consideration in advance.

  • Gunabalan Ramachandiran added an answer:
    Does anyone know the mathematical model/transfer function of the 3-phase inverter?
    Even if you know for single phase, its fine.
    I need to develop a controller for three phase inverter. To design controller, first I need to know the mathematical model/transfer function of the 3-phase inverter.
    Gunabalan Ramachandiran

    The transfer function of three phase inverter is available in Electric motor drives by R.Krishnan, Chapter 8, pp.495-496

  • George Stoica added an answer:
    Do we need a new definition of fractals for big data? Or must fractals be based on power laws?

    So far definitions of fractals are mainly from mathematical point of view for the purpose of generating fractal sets or patterns, either strictly or statistically; see illustrations below (Figure 1 for strict fractals, while Figure 2 for statistical fractals; Fig. 4 for fractals emerged from big data):

    Big data are likely to show fractal because of the underlying heterogeneity and diversity. I re-defined fractal as a set or pattern in which the scaling pattern of far more small things than large ones recurs multiple times, at least twice with ht-index being 3. I show below how geographic forms or patterns generated from twitter geolocation data bear the same scaling property as the generative fractal snowflake.

    Jiang B. and Yin J. (2014), Ht-index for quantifying the fractal or scaling structure of geographic features, Annals of the Association of American Geographers, 104(3), 530–541, Preprint:
    Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint:

    The new definition of fractals enables us to see the fractals emerged from big data. The answer to the question seems obvious. Yes, we need the new definition. BUT, some of my colleagues argued that the newly defined fractals are not fractal anymore, because they do not follow power laws.

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    George Stoica

    I understand, so a new definition is in order. The metric aspect is less important. 

  • Giovanni De Gasperis added an answer:
    Why are physicists stuck with Fortran and not willing to move to Python with NumPy and Scipy?

    Nowadays all of the major Fortran related numerical calculus have exactly mapped equivalent libraries in more modern language framework like Numerical Python (NumPy) and Scientific Python (SciPy). 

    What keeps physicists stuck with Fortran?



    Scientific evidence?

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    Giovanni De Gasperis

    Btw, I like the approach of Software Carpentry web site on one of their Python online course for scientists:

    the time to arrive to a solution is given by:  Ts = Td + Te


        Ts : time to the solution

        Td: time to focus on the problem, think of an algorithm, write a program, debug it

        Te: time of execution

    Nowdays Td>>Te, that is why i am definitively stuck with Python/Numpy/Scipy.


  • Ashwin Nanjappa added an answer:
    What is the most reliable package for c++ which calculates eigen vectors?
    I need to calculate the left eigen vector of a Eigen a good package?
    Ashwin Nanjappa

    If you are already using OpenCV in your project, then its PCA class provides Eigenvectors in just a single line of code.

    Here is a simple example:

  • Agnes Schonbrunn added an answer:
    Does anyone have done a Co-IP for galpha protein with a GPCR?

    I cannot detect my galpha protein when I immunoprecipitate it with the receptor.

    Agnes Schonbrunn

    See lysis buffer and IP conditions in:

    Gu YZ, Schonbrunn A 1997 Coupling specificity between somatostatin receptor sst2A and G proteins: isolation of the receptor-G protein complex with a receptor antibody. Mol Endocrinol 11:527-537

  • Nkuba Nyerere added an answer:
    What is the physical interpretation of local and global stability of a disease free or endemic equilibrium in disease modelling?
    How can you explain the local and global stability of DFE or EE to a non-mathematician?
    Nkuba Nyerere

    Thank you Konstantin K. Avilov 

  • Michael Patriksson added an answer:
    Why the dualization algorithm are not much used in traffic assignment problem with the static demand ?

    I want to know why the direct methods are more qualified than the dual methods for solving the traffic assignment problem with the static demand

    I need advice, what direction I should take in my search?

    Michael Patriksson

    Here is the local RG link:

    • Source
      [Show abstract] [Hide abstract]
      ABSTRACT: When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.
      Mathematical Programming 05/2014; 150(2). DOI:10.1007/s10107-014-0772-2
  • Bin Jiang added an answer:
    How to check whether a system is Linear or Nonlinear?
    System is a black box.
    Bin Jiang

    Very stimulating question!

    I would visualize it in order to see whether it is nonlinear or not, e.g., urban growth and evolution of social media.

    Jiang B. (2015), Head/tail breaks for visualization of city structure and dynamics, Cities, 43, 69-77, Preprint:

    Jiang B. and Miao Y. (2014), The evolution of natural cities from the perspective of location-based social media, The Professional Geographer, xx(xx), xx-xx, DOI: 10.1080/00330124.2014.968886,  Preprint:

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  • Javad Khazaei added an answer:
    Does anyone know a good source on scaling-up models/techniques for chemical pilot plants to industrial scale?
    I am trying build a process model for a bioenergy system. In the literature, there are sufficient lab scale and pilot plant models for this process, but much less industrial (commercial) level data. I am looking for a book or a paper discussing what options we have to mathematically scale up a pilot plant model to industrial level. thanks!
    Javad Khazaei

    hi there

    you think you could find some details on the book titled "Marco Zlokamik; Scale Up in Chemical Engineering".  more over there is another good as well as more complete on scale up, titled "Applied Dimensional Analysis and Modeling".  best

  • Fudong Ge added an answer:
    How do I obtain the duality system of a Caputo fractional-order distributed parameter system?

    For integer-order distributed parameter system, we can get the dual system by utilizing the duality theory. However, for frational order distributed parameter system. the results is few

    Fudong Ge

    ok, Thanks very much .  Prof.Grabowski.   I will check them soon----

  • Bajece Balkan Journal of Electrical added an answer:
    How can data envelopment analysis (DEA) theory be modeled in nonlinear instead of linear one?
    Is it possible to develop a model of DEA in nonlinear functions instead of linear?
  • Luca Dimiccoli added an answer:
    Even when using regularization techniques in linear inverse problems is it recommendable to reduce the condition number previously?

    I found empirically that previous reduction of the condition number is useful even when using for instance Tikhonov-Philllips regularization, however I don't have theoretical backup on this. Has anyone a book or paper which can support this? Thank you in advance

    Luca Dimiccoli

    Try the other way around. Regularize first and then reduce the condition number. It makes more sense to me.

  • George Stoica added an answer:
    What are the limits of measurement in science?
    When I was in high school Bohr's atom of shells, s and p orbitals was introduced in chemistry. Realization was automatic that the world was explained according to theory that was verified by experiment. Through college and graduate school, looking for more complete explanation, theory is challanged but it is not brought to question "what is an electron or proton, if they have mass but are visible only in the sense that they emit light energy as photons that also have mass, "spots of light in orbit around nuclei?, the atom a solar system in minature"? Physicists will say this is not the picture they have evolved, but all that remains is the image of equations on a chalkboard, at best 'the image of things of a particle nature in alteration with things of a light nature'. Can a pieced-together stepwise reality of this nature be accepted? In the Feyman quote below pieces are added that can break any of the established laws "they are not directly observeable" or affect "causality". In this same meaning though neither electrons, protons, photons or atoms are observable and their causal effects are but a matter of humanly constructed theory and similarly based experimental apparatus. The possibility exists that theory and theory based apparatus entail one another and all that might be gotten is that the real universe is identical in this respect...i.e. existence entails the experienced universe and visa-verse.
    "You found out in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light! It may surprise you that there is an amplitude for a photon to go at speeds faster or slower than the conventional speed, c." These virtual photons, however, do not violate causality or special relativity, as they are not directly observable and information cannot be transmitted causally in the theory." (from "Varying c in quantum theory"
    George Stoica

    The limits of measurement keep on changing, and sometimes the errors of measurement are not too good. 

  • Bernd Schmeikal added an answer:
    How can I mathematically model the behaviour of the humans?

    Any suggestion/resources are appreciated.

  • Klaus Thoeni added an answer:
    Which quadrature rule can be used in 3D BEM to evaluate hypersingular (1/r^2) Hadamard integrals?

    I am developing a 3D boundary element code for problems of elastostatics. This is not the focus of my research, I just need it as an instrument. I look  for a simple way of numerical evaluation of Hadamard finite part integrals, that occur in 3D fundamental solutions for tractions. These can be either evaluated on a triangular element or boiled down to 1D integrals of a kind: Integral from 0 to 1 f(x) dx / x^2. Papers on the topic that I've found seem to be messy and contradictory, nobody gives Gauss-like quadrature with points and weights. Any help is greatly appreciated. Thanks!  

    Klaus Thoeni

    There is no need to evaluate the hypersingular integral in order to get the stresses at the boundary. You can use a technique called "stress recovery". The method uses the nodal displacement tangential derivatives of a boundary element to determine the tangential strains at the point of interest. Then, by using Hooke’s law and the equilibrium condition at the boundary, the local stresses can be recovered which then have to be transformed back to the global coordinate system.

  • Zol Bahri Razali added an answer:
    Which is the best model to analyse the dynamic behavior of Magnetorhological Elastomer?

    I am in need of Mathematical model of MRE to study its dynamic behavior under varying magnetic field to design MRE based Dynamic vibration absorber

  • Himanshu Kumar added an answer:
    What are the best ways to do the mathematical modelling of Notch and Wnt pathway?

    Need your opinion related to algorithms like Genetic Algorithm and others......

    Himanshu Kumar

    Thank You so much for informative discussions...

  • Mahtab Zadnoor added an answer:
    Is there any software to determine the number of codewords with fixed GC-content in DNA codes?
    If so where can I find it?
    Mahtab Zadnoor

    hi again

    cloud you please help me with writting C-programming in order to compiling number of G and C, as you mentioned to me.

    I myself have problems by C-programming and cannot write a construction algorithm for compute direct content 01.

    best regards


  • Shyamsunder Yadav added an answer:
    Does anyone have REFPROP FORTRAN source code for calculating the properties of Carbon dioxide?

    For the stability analysis of supercritical natural circulation loop i require the REFPROP FORTRAN source code for getting the properties of carbon dioxide. If anyone is having the code pl share it.

    Shyamsunder Yadav

    Check this link

  • Victor Christianto added an answer:
    Can the ebola virus be considered as an epidemic in a scale-free network?

    Some researchers begin to consider epidemics in scale free network. Does the ebola virus belong to this situation? See Abramson (2001)

    And what are its implicatins for ebola spread prediction and modelling?

    Victor Christianto

    @Jimmy: thanks for your answer. Best wishes

  • John Frederick Chionglo added an answer:
    What are the best tools for simulation and modelling?


    I wanted to know what are good tools for simulation and modelling and does it really need a tool? Can't we build our own software for out own system?

    I am new to this field only thing I know is there are two parts is such system 1) mathematical modelling 2) Graphical representation ( Its my perception need not be true).

    So can I implement mathematical modelling  using any object oriented language on my own and for graphical part  can use some tool to reduce efforts and time.

    Also suggest me some material that must be required to enter into field of simulation and modelling.

    Thank you.

    John Frederick Chionglo

    If you are still looking for tools:

    Try JavaScript and the Acrobat/JavaScript API. You can draw your graphics using a vector-based software such as PowerPoint and export the graphics to PDF. If you have some sample problems in mind, post it and I will try to come up with computer programs written in JavaScript and using the Acrobat/JavaScript API as examples for you.

    Is this the "best" or is this a "good tool" for simulation and modelling? I don't know but it is a tool I will use.

  • Joel Ndam added an answer:
    Why does limit of function not exist when right hand limit and left hand limit are different?

    If right hand limit and left hand limit exist, that means the limit 'exists'.

    But, this is wrong.... Why??

    Even in complex case, lim (z->0), (x^2y)/(x^4+y^2) does not exist... coz depending on the path, the values of limit changes.

    Why does that mean the limit does not exist?

    Joel Ndam

    The limit of a function at any given point is unique, hence it does not exist if left and right hand limits are not the same.

  • Carl L. Devito added an answer:
    Is an explicit form of maximal ideals/homomorphism known of C-star algebra with almost periodic functions?

    Due to the Gelfand-Neumark Theorem algebra of almost periodic functions in the sense of Bohr is isometrically isomorphic to the algebra of complex continuous functions on the space of maximal ideals of the first algebra. This space is compact and is known as Bohr's compactification of the real line. I cannot find any explicite form of elements of this space; only abstract description, Hewitt, Ross monograph for example.

    Carl L. Devito

    I don't have an answer to your question. Since you seen interested in almost periodic functions, let me call your attention to how these functions characterize certain ideals in the Banach algebra L1(R) (under convolution). As you know, spectral synthesis fails in this algebra so, given a closed ideal J we say that J can be synthesized if it is the intersection of the regular maximal ideals that contain it. Then: J can be synthesized if, and only if, there is a (Bohr) almost periodic function b  such that 

    J = { f in L1(R) : f convolved with b is equal to zero}.  The details can be found in the paper "Chacterizations of those ideals in L1(R) which can be synthesized"  Math. Ann. 203, 171-173 (1973).

  • Sándor F. Tóth added an answer:
    How can we model disjoint constraints in linear programming?

    We have two constraints: xij- zuv >=0 , zij - xuv >=0  where xij , zuv, zij and  xuv are continuous variables. We would like that one of the two constraints will be satisfied, so how to model this disjoint constraints?

    Sándor F. Tóth

    @Ed Klotz, good to know CPLEX is doing this on the fly to save us sloppy math programmers...

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