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 + j = μ n c n - j z n - j , 1 μ n , having all its zeros on | z | = k , k 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, [ a , b ) f ' ( x ) d μ Δ 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 n × n , we call a pair of continuous functions U , V Y -compatible, if they map the space n into itself and satisfy U x · V y 0 , for all ( x , y ) Y with x · y 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...

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 ) + + a ( z ) f ' + D ( z ) f = 0 , (1) where D ( z ) = Q ( z ) e P ( z ) + Q ( z ) e P ( z ) + Q ( z ) e P ( 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.

On the first sign change in Mertens' theorem

Jan Büthe (2015)

Acta Arithmetica

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The function p 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).

Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

Jean Mawhin, Katarzyna Szymańska-Dębowska (2016)

Mathematica Bohemica

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A couple ( σ , τ ) of lower and upper slopes for the resonant second order boundary value problem x ' ' = f ( t , x , x ' ) , x ( 0 ) = 0 , x ' ( 1 ) = 0 1 x ' ( s ) d g ( s ) , with g increasing on [ 0 , 1 ] such that 0 1 d g = 1 , is a couple of functions σ , τ C 1 ( [ 0 , 1 ] ) such that σ ( t ) τ ( t ) for all t [ 0 , 1 ] , σ ' ( t ) f ( t , x , σ ( t ) ) , σ ( 1 ) 0 1 σ ( s ) d g ( s ) , τ ' ( t ) f ( t , x , τ ( t ) ) , τ ( 1 ) 0 1 τ ( s ) d g ( s ) , in the stripe 0 t σ ( s ) d s x 0 t τ ( s ) d s and t [ 0 , 1 ] . It is proved that the existence of such a couple ( σ , τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

Multiple end solutions to the Allen-Cahn equation in 2

Michał Kowalczyk, Yong Liu, Frank Pacard (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

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An entire solution of the Allen-Cahn equation Δ u = f ( u ) , where f is an odd function and has exactly three zeros at ± 1 and 0 , e.g. f ( u ) = u ( u 2 - 1 ) , is called a 2 k end solution if its nodal set is asymptotic to 2 k half lines, and if along each of these half lines the function u looks (up to a multiplication by - 1 ) like the one dimensional, odd, heteroclinic solution H , of H ' ' = f ( H ) . In this paper we present some recent advances in the theory of the multiple end solutions. We begin with the description of the moduli space...

Approximate and L p Peano derivatives of nonintegral order

J. Marshall Ash, Hajrudin Fejzić (2005)

Studia Mathematica

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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

Entire functions of exponential type not vanishing in the half-plane z > k , where k > 0

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let P ( z ) be a polynomial of degree n having no zeros in | z | < k , k 1 , and let Q ( z ) : = z n P ( 1 / z ¯ ) ¯ . It was shown by Govil that if max | z | = 1 | P ' ( z ) | and max | z | = 1 | Q ' ( z ) | are attained at the same point of the unit circle | z | = 1 , then max | z | = 1 | P ' ( z ) | n 1 + k n max | z | = 1 | P ( z ) | . The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

Real C k Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

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We prove a C k version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

Biholomorphic maps determined on the boundary

Nozomu Mochizuki (1977)

Annales de l'institut Fourier

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Let D be a bounded domain in C n such that the boundary b D is topologically S 2 n - 1 in R 2 n with a regular point; let f : D ˜ C n be a holomorphic map where D ˜ is a neighborhood of D . If f is one-to-one when restricted to b D , then f : D f ( D ) is biholomorphic.