Displaying similar documents to “On motions with bursting characters for Lagrangian mechanical systems with a scalar control. II. A geodesic property of motions with bursting characters for Lagrangian systems”

Norm inequalities for the difference between weighted and integral means of operator differentiable functions

Silvestru Sever Dragomir (2020)

Archivum Mathematicum

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Let f be a continuous function on I and A , B 𝒮𝒜 I H , the convex set of selfadjoint operators with spectra in I . If A B and f , as an operator function, is Gateaux differentiable on [ A , B ] : = ( 1 - t ) A + t B t 0 , 1 , while p : 0 , 1 is Lebesgue integrable, then we have the inequalities 0 1 p τ f 1 - τ A + τ B d τ - 0 1 p τ d τ 0 1 f 1 - τ A + τ B d τ 0 1 τ ( 1 - τ ) | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ 1 4 0 1 | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ , where f is the Gateaux derivative of f .

Inequalities for the arithmetical functions of Euler and Dedekind

Horst Alzer, Man Kam Kwong (2020)

Czechoslovak Mathematical Journal

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For positive integers n , Euler’s phi function and Dedekind’s psi function are given by φ ( n ) = n p n p prime 1 - 1 p and ψ ( n ) = n p n p prime 1 + 1 p , respectively. We prove that for all n 2 we have 1 - 1 n n - 1 1 + 1 n n + 1 φ ( n ) n φ ( n ) ψ ( n ) n ψ ( n ) and φ ( n ) n ψ ( n ) ψ ( n ) n φ ( n ) 1 - 1 n n + 1 1 + 1 n n - 1 . The sign of equality holds if and only if n is a prime. The first inequality refines results due to Atanassov (2011) and Kannan & Srikanth (2013).

On global controllability of linear time dependent control systems

Alberto Tonolo (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Let A , B be a linear time dependent control process, defined on an open interval J = ] a , ω [ with a - and ω ; in this paper we give a description of the function τ : I J , τ ( t ) = inf { t > t : ( A , B ) is t , t -globally controllable from 0 } where I = { t J : t J with A , B t , t -globally controllable from 0 } .

Dynamic behavior of vector solutions of a class of 2-D neutral differential systems

Arun Kumar Tripathy, Shibanee Sahu (2025)

Mathematica Bohemica

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This work deals with the analysis pertaining some dynamic behavior of vector solutions of first order two-dimensional neutral delay differential systems of the form d d t u ( t ) + p u ( t - τ ) v ( t ) + p v ( t - τ ) = a b c d u ( t - α ) v ( t - β ) . The effort has been made to study d d t x ( t ) - p ( t ) h 1 ( x ( t - τ ) ) y ( t ) - p ( t ) h 2 ( y ( t - τ ) ) + a ( t ) b ( t ) c ( t ) d ( t ) f 1 ( x ( t - α ) ) f 2 ( y ( t - β ) ) = 0 , where p , a , b , c , d , h 1 , h 2 , f 1 , f 2 C ( , ) ; α , β , τ + . We verify our results with the examples.

Legendre polynomials and supercongruences

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), P [ p / 6 ] ( t ) - ( 3 / p ) x = 0 p - 1 ( ( x ³ - 3 x + 2 t ) / p ) ( m o d p ) and ( x = 0 p - 1 ( ( x ³ + m x + n ) / p ) ) ² ( ( - 3 m ) / p ) k = 0 [ p / 6 ] 2 k k 3 k k 6 k 3 k ( ( 4 m ³ + 27 n ² ) / ( 12 ³ · 4 m ³ ) ) k ( m o d p ) , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning k = 0 p - 1 2 k k 3 k k 6 k 3 k / m k ( m o d p ² ) , where m is an integer not divisible by p.

Ostrowski’s type inequalities for complex functions defined on unit circle with applications for unitary operators in Hilbert spaces

S.S. Dragomir (2015)

Archivum Mathematicum

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Some Ostrowski’s type inequalities for the Riemann-Stieltjes integral a b f e i t d u t of continuous complex valued integrands f : 𝒞 0 , 1 defined on the complex unit circle 𝒞 0 , 1 and various subclasses of integrators u : a , b 0 , 2 π of bounded variation are given. Natural applications for functions of unitary operators in Hilbert spaces are provided as well.

Global analytic and Gevrey surjectivity of the Mizohata operator D 2 + i x 2 2 k D 1

Lamberto Cattabriga, Luisa Zanghirati (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The surjectivity of the operator D 2 + i x 2 2 k D 1 from the Gevrey space E s R 2 , s 1 , onto itself and its non-surjectivity from E s R 3 to E s R 3 is proved.