Integrating factor

Jan Mařík

Displaying similar documents to “Integrating factor”

On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr Mykhaylyuk, Anatolij Plichko (2015)

Colloquium Mathematicae

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We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function $F_{y} (x) : = f_{y}^{'} (x, y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f_{y x}^{''}$ exists. We construct a separately twice differentiable function whose partial derivative $f_{x}^{'}$ is discontinuous with respect to the second variable on a...

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

Differentiation of n-convex functions

H. Fejzić, R. E. Svetic, C. E. Weil (2010)

Fundamenta Mathematicae

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The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n - 1)} = f_{(n - 1)}$ except on a countable set. Moreover $f_{(n - 1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

Generalized α-variation and Lebesgue equivalence to differentiable functions

Jakub Duda (2009)

Fundamenta Mathematicae

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We find conditions on a real function f:[a,b] → ℝ equivalent to being Lebesgue equivalent to an n-times differentiable function (n ≥ 2); a simple solution in the case n = 2 appeared in an earlier paper. For that purpose, we introduce the notions of $C B V G_{1 / n}$ and $S B V G_{1 / n}$ functions, which play analogous rôles for the nth order differentiability to the classical notion of a VBG⁎ function for the first order differentiability, and the classes $C B V_{1 / n}$ and $S B V_{1 / n}$ (introduced by Preiss and Laczkovich) for Cⁿ smoothness....

Vitali Lemma approach to differentiation on a time scale

Chuan Jen Chyan, Andrzej Fryszkowski (2004)

Studia Mathematica

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A new approach to differentiation on a time scale is presented. We give a suitable generalization of the Vitali Lemma and apply it to prove that every increasing function f: → ℝ has a right derivative f₊’(x) for $μ_{Δ}$ -almost all x ∈ . Moreover, $\int_{[a, b)} f ₊^{'} (x) d μ_{Δ} \leq f (b) - f (a)$ .

Boundary value problems with compatible boundary conditions

George L. Karakostas, P. K. Palamides (2005)

Czechoslovak Mathematical Journal

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If $Y$ is a subset of the space $ℝ^{n} \times ℝ^{n}$ , we call a pair of continuous functions $U$ , $V$ $Y$ -compatible, if they map the space $ℝ^{n}$ into itself and satisfy $U x \cdot V y \geq 0$ , for all $(x, y) \in Y$ with $x \cdot y \geq 0$ . (Dot denotes inner product.) In this paper a nonlinear two point boundary value problem for a second order ordinary differential $n$ -dimensional system is investigated, provided the boundary conditions are given via a pair of compatible mappings. By using a truncation of the initial equation and restrictions of its...

Oscillation criteria for fourth order half-linear differential equations

Jaroslav Jaroš, Kusano Takaŝi, Tomoyuki Tanigawa (2020)

Archivum Mathematicum

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Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form $(| y^{''} |^{α} sgn y^{''})^{''} + q (t) {| y |}^{α} sgn y = 0, t \geq a > 0, A$ where $α > 0$ is a constant and $q (t)$ is positive continuous function on $[a, \infty)$ , are given in terms of an increasing continuously differentiable function $ω (t)$ from $[a, \infty)$ to $(0, \infty)$ which satisfies $\int_{a}^{\infty} 1 / (t ω (t)) d t < \infty$ .

Zeros of solutions of certain higher order linear differential equations

Hong-Yan Xu, Cai-Feng Yi (2010)

Annales Polonici Mathematici

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We investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $f^{(k)} + a_{k - 1} (z) f^{(k - 1)} + \dots + a ₁ (z) f^{'} + D (z) f = 0$ , (1) where $D (z) = Q ₁ (z) e^{P ₁ (z)} + Q ₂ (z) e^{P ₂ (z)} + Q ₃ (z) e^{P ₃ (z)}$ , P₁(z),P₂(z),P₃(z) are polynomials of degree n ≥ 1, Q₁(z),Q₂(z),Q₃(z), $a_{j} (z)$ (j=1,..., k-1) are entire functions of order less than n, and k ≥ 2.

Absolute continuity with respect to a subset of an interval

Lucie Loukotová (2017)

Commentationes Mathematicae Universitatis Carolinae

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The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset $M$ of an interval $I$ . This generalization is based on adding more requirements to disjoint systems ${(a_{k}, b_{k})}_{K}$ from the classical definition of absolute continuity – these systems should be not too far from $M$ and should be small relative to some covers of $M$ . We discuss basic properties of relative absolutely continuous functions and compare this class with other...

Criterion of the reality of zeros in a polynomial sequence satisfying a three-term recurrence relation

Innocent Ndikubwayo (2020)

Czechoslovak Mathematical Journal

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This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${P_{i}}_{i = 1}^{\infty}$ generated by a three-term recurrence relation $P_{i} (x) + Q_{1} (x) P_{i - 1} (x) + Q_{2} (x) P_{i - 2} (x) = 0$ with the standard initial conditions $P_{0} (x) = 1, P_{- 1} (x) = 0,$ where $Q_{1} (x)$ and $Q_{2} (x)$ are arbitrary real polynomials.

A new look at an old comparison theorem

Jaroslav Jaroš (2021)

Archivum Mathematicum

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We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval $[a, b)$ when it is known that certain majorant Riccati equation has a global solution on $[a, b)$ .

On boundary value problems for systems of nonlinear generalized ordinary differential equations

Malkhaz Ashordia (2017)

Czechoslovak Mathematical Journal

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $d x = d A (t) \cdot f (t, x), h (x) = 0$ is established, where $f : [a, b] \times ℝ^{n} \to ℝ^{n}$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A : [a, b] \to ℝ^{n \times n}$ with bounded total variation components, and $h : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x (t_{1} (x)) = ℬ (x) \cdot x (t_{2} (x)) + c_{0},$ where $t_{i} : {BV}_{s} ([a, b], ℝ^{n}) \to [a, b]$ $(i = 1, 2)$ and $ℬ : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ are continuous...

On the first sign change in Mertens' theorem

Jan Büthe (2015)

Acta Arithmetica

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The function $\sum_{p \leq x} 1 / p - l o g l o g (x) - M$ is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

-simplicity of interval max-min matrices

Ján Plavka, Štefan Berežný (2018)

Kybernetika

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A matrix $A$ is said to have $𝐗$ -simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗 = {x : \underset{̲}{x} \leq x \leq \bar{x}}$ containing a constant vector is the unique solution of the system $A \otimes y = x$ in $𝐗$ . The main result of this paper is an extension of $𝐗$ -simplicity to interval max-min matrix $𝐀 = {A : \underset{̲}{A} \leq A \leq \bar{A}}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have $𝐗$ -simple image eigenspace. $𝐗$ -simplicity of interval matrices in max-min algebra are studied and equivalent conditions for...