## Glossary

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$$\displaystyle f(x) \sim \frac{a_0}{2} + \sum_{k\ge 1} \left( a_k \cos kx + b_k \sin kx \right) .$$ The convergence of this series to f(x) at a fixed point x depends only upon the behavior of f(x) in an arbitrary small neighborhood of x.
See Part V, xi
 Abel, Niels Niels Henrik Abel (1802--1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. Abel's formula Abel's formula or Abel's identity is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in terms of a coefficient of the original differential equation. Abscissa Abscissa is the first coordinate (usually horizontal) of a point in a coordinate system. Abscissa of convergence Abscissa is the first coordinate (usually horizontal) of a point in a coordinate system. Adiabatic invariant When the parameters of a physical system vary slowly under the effect of an external perturbation, some quantities are constant to any order of the variable describing the slow rate of change. Such a quantity is called an adiabatic invariant. This does not mean that these quantities are exactly constant but rather that their variation goes to zero faster than any power of the small parameter. Adjoint Suppose that A is a linear operator from one vector space with inner product < , > into another vector space with inner product. The adjoint operator to A is the linear operator A* such that $$\left\langle A\,f, g \right\rangle = \left\langle f, A^{\ast} g \right\rangle$$ for any elements f and g. For example, if $$A = a_2 (x)\, \texttt{D}^2 + a_1 (x)\, \texttt{D} + a_0 (x) ,$$ where $$\texttt{D}$$ = d/dx is the derivative operator, then its adjoint operator acts on a function u as $$A^{\ast}\, u = \texttt{D}^2 \left( a_2 \, u \right) - \texttt{D} \left( a_1 \, u \right) + a_0 (x)\, u(x) .$$ Analytic function A function is analytic at a point if the function has a power series expansion valid in some neighborhood of that point. It may consist of many holomorphic functions, called branches of the analytic function. Arakelian set A closed set E ⊂ ℂ, without holes, is an Arakelian set if, for every closed disc D ⊂ ℂ, the union of all holes of E ∪ D is a bounded set. Arakelian's theorem Arakelyan's theorem states that for every f continuous in E and holomorphic in the interior of E and for every ε > 0 there exists g holomorphic in Ω such that |g − f| < ε on E if and only if Ω* \ E is connected and locally connected. Asymptotic expansion Given a function f(x) and an asymptotic series { gk(x) } at x0, the formal series $$\sum_{k=0}^{\infty}a_k\,g_k(x),$$ where the { ak } are given constants, is said to be an asymptotic expansion of f(x) if $$f(x) - \sum_{k=0}^{n}a_k \,g_k(x)=o(g_n(x))$$ as x → x0 for every n; this is expressed as $$f(x) \sim \sum_{k=0}^{\infty}a_k\, g_k(x) .$$ Basin of attraction Bendixson, I.O. Bernoulli, Daniel Bessel, F.W. Friedrich Wilhelm Bessel (1784--1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method of parallax. See Part VII: section iii (Bessel functions) Bessel equation Bessel equation, first defined by the mathematician Daniel Bernoulli (1700-1782) and then generalized by Friedrich Bessel (1784--1846), is the second order differential equation with regular singular point at the origin and irregula singular point at infinity: $x^2 y'' + x\,y' + \left( x^2 - \nu^2 \right) y = 0 .$ See Part VII: section iii (Bessel functions) Bessel functions Bessel functions, first defined by the mathematician Daniel Bernoulli (1700-1782) and then generalized by Friedrich Bessel (1784--1846), are canonical solutions y(x) of Bessel's differential equation $x^2 y'' + x\,y' + \left( x^2 - \nu^2 \right) y = 0 .$ See Part VII: section iii (Bessel functions) Bessel inequality Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence { e1, e2, …}. The inequality was derived by F.W. Bessel in 1828: $\sum_{n\ge 1} \left\vert \langle {\bf x} , e_k \rangle \right\vert^2 \le \| {\bf x} \|^2 ,$ where ⟨ · , · ⟩ denotes the inner product in the Hilbert space. Bessel series Beta function The beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients $B(x,y) = \int_0^1 t^{x-1} \left( 1- t \right)^{y-1} {\text d} t = \frac{\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)} .$ Bifurcation A bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Blasius problem The Blasius boundary value problem on semi-infinite interval 0 ≤ x < ∞ is $2\,f_{xxx} + f\,f_{xx} =0, \qquad f(0) = f_x (0) =0, \quad f_x (\infty ) = 1,$ where fx denotes the derivative of f(x) with respect to x. It is named after its inventor in 1911 Paul Richard Heinrich Blasius (1883--1970). Blasius constant The constant fxx(x=0) ≈ 0.33205733621... is called the Blasius constant. Boundary data Given a differential equation, the value of the dependent variable on the boundary may be given in many different ways. Boussinesq equation The Boussinesq equation (1872): $\frac{\partial^2 \eta}{\partial t^2} = gh\,\frac{\partial^2 \eta}{\partial x^2} + gh \frac{\partial^2}{\partial x^2} \left( \frac{3}{2}\,\frac{\eta^2}{h} + \frac{1}{3}\,h^2 \frac{\partial^2 \eta}{\partial x^2} \right) .$ Camassa–Holm equation The Camassa–Holm equation (1993) was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water: $u_t + 2 \kappa\,u_x - u_{xxt} + 3u\,u_x = 2u_x u_{xx} + u\,u_{xxx} .$ Chebyshev, P Pafnuty Chebyshev (1821--1894) is known for his fundamental contributions to the fields of probability, statistics, mechanics, and number theory. A number of important mathematical concepts are named after him, including the Chebyshev inequality (which can be used to prove the weak law of large numbers), the Bertrand–Chebyshev theorem, Chebyshev polynomials, and Chebyshev bias. See Part VII: Section IV (Chebyshev Functions) Chebyshev Polynomials There four kinds of Chebyshev Polynomials, usually denoted by Tn(x, Un(x, Vn(x, and Wn(x. They all are eigenfunctions of second order differential equations that form a complete set of orthogonal functions. See Part VII: Section IV (Chebyshev Functions) Clenshaw algorithm also called Clenshaw summation, is a recursive method to evaluate a linear combination of Chebyshev polynomials. It is a generalization of Horner's method for evaluating a linear combination of monomials. See Part V, Section XV (Chebyshev Expansions) Companion matrix companion matrix of the monic polynomial Conjugate harmonic functions A pair of real harmonic functions u and v which are the real and imaginary parts of some analytic function $$f = u + {\bf j}\,v$$ of a complex variable. Crocco's equation The Crocco's equation $\phi\,\frac{\partial^2 \phi}{\partial h^2} + \frac{1}{2}\,f(h) =0$ where f is a given positive function, is usually considered in the unit interval 0 ≤ h ≤ 1 subject to some (mixed) boundary conditions. d'Alembert, J. Jean-Baptiste le Rond d'Alembert (1717--1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Determinant The determinant of an n×n matrix A is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det(A), det A, or |A| and is equal to (-1)n times the constant term in the characteristic polynomial of A. Dini U. An Italian mathematician Ulisse Dini (1845--1918). See Part V, xi Dini criterion A Dini criterion is a sufficient criterion for convergence of Fourier series. See Part V, xi Dirichlet boundary conditions The dependent variable is prescribed on the boundary. This is also called a boundary condition of the first kind. See PartVI E, ii Dirichlet conditions The following (sufficient) conditions that guarantee convergence of the Fourier series were first discovered in 1829 by the German mathematician Peter Gustav Lejeune Dirichlet (1805--1859). f must be absolutely integrable over a period. f must be of bounded variation in any given bounded interval. f must have a finite number of discontinuities in any given bounded interval, and the discontinuities cannot be infinite. See Part V, xi Dirichlet kernel The Dirichlet kernel named after the German mathematician Peter Gustav Lejeune Dirichlet, is the sum of trigomometric functions: $D_n (x) = \frac{1}{2} + \sum_{k=1}^n \cos (kx) = \frac{1}{2}\sum_{k=-n}^n e^{{\bf j}kx} = \frac{\sin\left( n + \frac{1}{2} \right) x}{2\,\sin \frac{x}{2}} .$ See Part V, xi Dispersion relation In applied mathematics, when a solution of the partial differential equation can be represented as the Ehrenpreis integral over some contour L, $u(x,t) = \frac{1}{2\pi} \,\int_{L} {\text d}k\, e^{- \omega (k)\, t - {\bf j}kx} \rho (k) , \qquad {\bf j}^2 = -1,$ then the function ω(k) is referred to as the dispersion relation for the given differential equation. The concept of dispersion relations entered physics with the work of Kronig and Kramers in optics (known as the Kramers–Kronig relations). Dym equation The Dym equation (HD) is the third-order partial differential equation $u_t = u^3 u_{xxx} .$ The Dym equation first appeared in the paper by Martin Kruskal and is attributed to an unpublished paper by Harry Dym (born 1938). Ehrenpreis Principle The Ehrenpreis Fundamental Principle was established by Ehrenpreis and Palamodov in 1970. It states that for the evolution partial differential equation $$u_t + \omega \left( -{\bf j}\partial_x \right) u = 0 ,$$ where ω(ν) is a polynomial, there exists a measure μ(ν) with support L such that $$u(x,t) = \int_L e^{{\bf j}\nu x - \omega (\nu )t} \,{\text d}\mu (\nu ) ,$$ however, the measure μ is not constructed explicitly. Euler's reflection formula $\Gamma \left( 1-z \right) \Gamma (z) = \frac{\pi}{\sin (\pi z)} , \qquad z \notin \mathbb{Z} .$ Fejér, L. Lipót Fejér (1880--1959) was a Hungarian mathematician of Jewish heritage. Fejér was born as Leopold Weisz (his last name means "white"), and changed to the Hungarian name Fejér (which also means "white") around 1900. During the period (1911--1959) he was the chair at Budapest University and led a highly successful Hungarian school of analysis. He was the thesis advisor of mathematicians such as John von Neumann, Paul Erdős, George Pólya, Pál Turán, and many others. Fejér theorem Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f: ℝ → ℂ is a continuous function with period 2π, then the sequence (σn) of Cesàro means of the sequence (sn) of partial sums of the Fourier series of f converges uniformly to f on [-π,π]. Fixed point A fixed point, also known as an invariant point of a function is an element of the function's domain that is mapped to itself by the function. Fokas method The Fokas method (or unified transform method) was originally introduced by A.S. Fokas in 1990s. The method allows to construct solutions to evolution partial differential equations (that admit Lax pairs) in the explicit form that are always uniformly convergent at the boundaries. Fourier, J. Jean-Baptiste Joseph Fourier (1768--1830) was a French mathematician, physicis, and polytician who used Fourier series to solve heat transfer problems. Fourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, and was appointed secretary of the Institut d'Égypte. Fourier transform There are several common conventions for defining the Fourier transform of an integrable complex-valued function f : ℝ → ℂ. We use the following notation for the Fourier transformation and its inverse. $\hat{f} (\xi ) =ℱ\left[ f(x) \right] (\xi ) = f^F (\xi ) = \int_{-\infty}^{\infty} f(t)\,e^{{\bf j} \xi\cdot t} \,{\text d}t$ with the inverse (that is valid for functions satisfying the Dirichlet conditions) $f(t) = ℱ^{-1} \left( \hat{f} \right) = \text{V.P.} \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(\xi )\,e^{-{\bf j} \xi\cdot t} \,{\text d}\xi = \lim_{N\to \infty} \frac{1}{2\pi} \int_{-N}^N \hat{f}(\xi )\,e^{-{\bf j} \xi\cdot t} \,{\text d}\xi = \frac{f(t+0) + f(t-0)}{2} .$ Gamma function The gamma function was introduced by Leonhard Euler, who suggested to use Γ, (the capital letter gamma from the Greek alphabet) for its notation $\Gamma \left( z \right) = \int_0^{\infty} x^{z-1} e^{-x} {\text d}x , \qquad \Re (z) > 0,$ Gauss, C.F. Johann Carl Friedrich Gauss (1777--1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and sciences. Sometimes referred to as the Princeps mathematicorum (Latin for "the foremost of mathematicians"). Glukhovsky-- Dolzhanksy system is a system of the form $$\begin{split} \dot{x} = -\sigma \left( x - y \right) -a yz , \\ \dot{y} = rx - y -xz , \\ \dot{z} = - bz + xy , \end{split}$$ where σ, a, r, b are physical parameters. Green, G George Green (1793--1841) was a British mathematical physicist who introduced several important concepts, among them a theorem similar to the modern Green's theorem. Green function A Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions. Green theorem Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C: $$\int_C P\,{\text d}x + Q\,{\text d}y = \iint_{D} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) {\text d}A .$$ Horner rule also known as Horner's method or Horner's scheme etc, is referred to a polynomial evaluation method named after the British mathematician William George Horner (1786--1837) expressed by $$p(x) = a_0 + a_1 x + \cdots + a_n x^n = a_0 + x \left( a_1 + x \left( a_2 + x \left( a_3 + \cdots + x \left( a_{n-1} + x\,a_n \right) \right) \right) \right)$$ Holomorphic function A function is analytic at a point if the function has a power series expansion valid in some neighborhood of that point. Hypergeometric function The Gaussian or ordinary hypergeometric function 2F1(a,b;c;x) is a special function represented by the hypergeometric series, $$_2F_1 (a,b,c;x) = \sum_{n\ge 0} \frac{a^{\overline{n}} b^{\overline{n}}}{c^{\overline{n}}} \, \frac{x^n}{n!} ,$$ that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE), called hypergeometric differential equation: $$x \left( 1- x \right) \frac{{\text d}^2 y}{{\text d}x^2} + \left[ c - (a+b+1)\,x \right] \frac{{\text d} y}{{\text d}x} -ab\,y =0 .$$ Here $$a^{\overline{n}} = a \left( a+1 \right) \cdots \left( a+n-1 \right)$$ is the $$a$$ rising factorial (sometimes called the Pochhammer function). Inhomogeneous equation An ordinary or partial differential equation is called inhomogeneous (or nonhomogeneous) if it contains an input (driven) function. Integrable equation An evolution partial differential equation is called integrable if it admits a Lax pair. Jefery--Hamel The Jeffery--Hamel flow is a flow created by a converging or diverging channel with a source or sink of fluid volume at the point of intersection of the two plane walls. In dimensioneless variables, it can be modeled by the boundary value problem for the third order differential equation: $$F''' + 2\,R_e\,\alpha \, F\,F' + 4\alpha^2 F' =0 , \qquad F(0) = 1, \quad F' (0) = 0 , \quad F(1) =0 .$$ Here α is the channel half-angle and Re is the Reynolds number of flow. Kadomtsev–Petviashvili equation The Kadomtsev–Petviashvili equation – or KP equation, named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a partial differential equation to describe nonlinear wave motion: $$\partial_x \left( \partial_t u + u\,\partial_x u + \epsilon^2 \partial_{xxx} u \right) + \lambda \,\partial_{yy} u =0 ,$$ where λ = ±1 Kaup–Kupershmidt equation The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation $u_t = u_{xxxxx} + 10\,u_{xxx}u + 25\,u_{xx} u_x + 20\,u^2 u_x = \frac{1}{6} \left( 6\,u_{xxxx} + 60\,u\,u_{xx} + 45\,u_x^2 + 40\,u^3 \right)_x .$ Korteweg–de Vries (KdV) equation The Korteweg–de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. The KdV equation is a nonlinear, dispersive partial differential equation for a function ϕ of two real variables, space x and time t: $$\partial_t \phi + \partial_x^3 \phi -6\,\phi\,\partial_x \phi =0$$ with ∂x and ∂t denoting partial derivatives with respect to x and t. The constant 6 in front of the last term is conventional but of no great significance. The Linearized KdV Equation: $$u_t + u_x + u_{xxx} =0 .$$ Lane--Emden equation The Lane–Emden equation is a dimensionless form of Poisson's equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid. $\frac{1}{\xi^2}\, \frac{\text d}{{\text d}\xi} \left( \xi^2 \frac{{\text d}\theta}{{\text d}\xi} \right) + \theta^n =0 .$ It is named after astrophysicists Jonathan Homer Lane and Robert Emden. Lax pair A Lax pair is a pair of matrices or operators L(t), P(t) dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation: $$\partial_t L = \left[ P, L \right] ,$$ where [P,L] = PL - LP is the commutator. In other words, a partial differential equation (PDE) in two independent variables for function u(x,t) has a Lax pair formulation if the PDE can be written as $A_t - B_x + \left[ A, B \right] = 0 ,$ where both A and B are matrx functions. Morse potential The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. Its Hamiltonian is $H(p,q) = \frac{p^2}{2} + D\left( 1 - e^{-rq} \right)^2 ,$ where q stands for the bond length, D for the dissociation energy, and r for the anharmonic parameter. The exact solution is $q(t) = - \frac{1}{r} \,\ln \frac{1 - (E/D)^{1/2} \cos (\omega t + \varphi_0 )}{1 - E/D} ,$ where E is the total energy, ω is the anharmonic frequency of the oscillator given by $$\omega = \left( 2D - 2E \right)^{1/2} ,$$ and φ0 the initial phase. Neighborhood A neighborhood is any set of points containing the point or subset of interest inside some open set. For example, a neighborhood containing the origin in one dimension could be [-0.1,1], as it contains the point 0 inside the open symmetric interval (-0.1, 0.1). But [0, 1] is not a neighborhood of the origin as it does not contain any open interval centered at zero. In a two-dimensional space, a neighborhood of the origi could be any set containing an open circle with radius epsilon (x² + y² < ε²), which is centered about the origin. See: Part II, iv. Nonlinear Schrödinger equation Nonlinear Schrödinger equation ${\bf j}\,\psi_t = -\frac{1}{2}\,\psi_{xx} + \kappa \left\vert \psi \right\vert^2 \psi ,$ where j is the unit vector in positive vertical direction on the complex plane ℂ. Ordinate Ordinate is the second coordinate (usually vertical) of a point in a coordinate system. Poincaré Map In dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré (1854--1912), is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensional subspace, called the Poincaré section, transversal to the flow of the system. More precisely, one considers a periodic orbit with initial conditions within a section of the space, which leaves that section afterwards, and observes the point at which this orbit first returns to the section. One then creates a map to send the first point to the second, hence the name first recurrence map. The transversality of the Poincaré section means that periodic orbits starting on the subspace flow through it and not parallel to it. See: Part III, Chaos. Poisson kernel The function P(r, x) is called the Poisson kernel: $P(r, x) = \frac{1}{2} + \sum_{n\ge 1} r^n \cos (nx) = \mbox{P.V.} \sum_{n = -\infty}^{\infty} r^{|n|} e^{{\bf j}nx} = \frac{1}{2} \cdot \frac{1 + r^2}{1- 2r\,\cos x + r^2} ,$ See: Part V, Examples. Radiation condition The radiation condition states that a wave equation has no waves incoming from an infinite distance, only outgoing waves. For example, the equation $$u_{t t}=\Delta\, u$$ might have the radiation condition $$u(x,t)\simeq A_{-}\exp(ik(t-x))$$ as $$x\to -\infty$$ and $$u(x,t)\simeq A_{+}\exp(ik(t+x))$$ as $$x\to +\infty .$$ This is also called the Sommerfeld radiation condition. Regular function A function is regular or holomorphic a point if the function has a power series expansion valid in some neighborhood of that point. Resolvent The resolvent of a linear operator A is $$R_{\lambda} = \left( \lambda\,I - A \right)^{-1} ,$$ where I is the identical operator. Resolvent method The resolvent method was developed by Vladimir Dobrushkin in 1980s. The method reduces a boundary value problem to an integral equation of the second order on the boundary, so it reduces a n-dimensional problem to n-1 dimensional one. Rodrigues Formulas Odile Rodrigues (1795--1851) showed that a large class of second-order Sturm--Liouville ordinary differential equations (ODEs) had polynomial solutions that could be put in a compact and useful form now generally called a Rodrigues Formula. See Part VII: Sections iv (Chebyshev Functions), v (Legendre polynomials), vi (Hermite polynomials), and vii (Laguerre polynomials). Riemann-Lebesgue lemma Riemann-Lebesgue lemma: Let f(x) ∈ 𝔏[𝑎, b] be absolutely integrable function on the interval [𝑎. b]. Then $\lim_{n\to\infty} \int_a^b f(x)\,\cos nx\,{\text d}x = \lim_{n\to\infty} \int_a^b f(x)\,\sin nx\,{\text d}x = 0.$ See Part V, xi Schwarzian derivative If y = y(x), then the Schwarzian derivative of y with respect to x is defined to be $\displaystyle \{y,x\} \equiv \frac{-1}{(y')^2} \left[ \frac{y'''}{y''} - \frac{3}{2} \left( \frac{y''}{y'} \right)^2 \right] .$ Shock A shock is a narrow region in which the dependent variable undergoes a large change. Also called a layer'' or a propagating discontinuity.'' Sine-Gordon equation There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted (x, t), the equation reads: $\varphi_{tt} - \varphi_{xx} + \sin\varphi =0 ,$ where partial derivatives are denoted by subscripts. Passing to the light cone coordinates (u, v), akin to asymptotic coordinates where $$u = \frac{x+t}{2} , \quad v = \frac{x-t}{2} ,$$ the equation takes the form: $$\varphi_{uv} = \sin\varphi .$$ Sobolev S.L. Sergei Lvovich Sobolev (1908--1989) was a Russian mathematician who first introduced generalized functions that later were called distributions. He was the first director of the Institute of Mathematics at Akademgorodok near Novosibirsk (Siberia). Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sturm J. Jacques Charles François Sturm (1803--1855) was a French mathematician. Sturm chain A Sturm chain or Sturm sequence is a finite sequence of polynomials p0, p1, ... , pm, of decreasing degree with the following properties: p0 is square-free: i.e. it has no factors of the form $$q^2 (x)$$ for any polynomial q(x). If a is a root of p(x), then the sign of p1(a) is the same as the sign of the derivative p'(a), and in particular is nonzero. If a is a root of pi(x), for some i with 0 < i < m, then both pi-1(a) and pi+1(a) are nonzero. Moreover, the sign of pi-1(a) is the opposite of the sign ofpi+1(a). pm(x)'s sign is constant and nonzero for all x. Sturm Sturm Sturm--Liouville theory A classical Sturm--Liouville theory, named after French mathematicians Jacques Charles François Sturm (1803--1855) and Joseph Liouville (1809--1882), is a generalization of eigenvalue problem for unbounded operators; namely, it is the theory of a real second-order linear differential equation of the form $$\displaystyle \frac{\text d}{{\text d}x} \left[ p(x)\, \frac{{\text d}y}{{\text d}x} \right] + q(x)\, y + \lambda\,w(x)\, y =0 ,$$ where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset. In the simplest of cases all coefficients are continuous on the finite closed interval [a,b], and p has continuous derivative. Singular point Zeeman model In 1972 (Zeeman, E.C.: Differential Equations and Nerve Impulse. Towards a Theoretical Biology, 4, pp. 8-67), Zeeman presented an important set of nonlinear dynamical equations for heartbeat modelling, based on the Van der Pol-Lienard equation. $\begin{split} \varepsilon\,\frac{{\text d}x}{{\text d}t} &= T\, x - x^3 - y, \quad T > 0, \\ \frac{{\text d}y}{{\text d}t} &= x - x_d . \end{split}$ Here variable x represents the length of a muscle fiber in the heart and the variable y is an electrical control variable that triggers the electro-chemical wave leading to the heart contraction. The positive constant T represents a tension of muscle and is related to blood pressure. The constant ε characterizes the heart. The initial conditions are usualy taken as x(0) = 1 and y(0) = 0.