Displaying similar documents to “On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters”

On the application of control theory to certain problems for Lagrangian systems, and hyper-impulsive motion for these. I. Some general mathematical considerations on controllizable parameters

Aldo Bressan (1988)

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

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In applying control (or feedback) theory to (mechanic) Lagrangian systems, so far forces have been generally used as values of the control u ( ) . However these values are those of a Lagrangian co-ordinate in various interesting problems with a scalar control u = u ( ) , where this control is carried out physically by adding some frictionless constraints. This pushed the author to consider a typical Lagrangian system Σ , referred to a system χ of Lagrangian co-ordinates, and to try and write some handy...

On Lagrangian systems with some coordinates as controls

Franco Rampazzo (1988)

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

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Let Σ be a constrained mechanical system locally referred to state coordinates ( q 1 , , q N , γ 1 , , γ M ) . Let ( γ ~ 1 γ ~ M ) ( ) be an assigned trajectory for the coordinates γ α and let u ( ) be a scalar function of the time, to be thought as a control. In [4] one considers the control system Σ γ ^ , which is parametrized by the coordinates ( q 1 , , q N ) and is obtained from Σ by adding the time-dependent, holonomic constraints γ α = γ ^ α ( t ) := γ ~ α ( u ( t ) ) . More generally, one can consider a vector-valued control u ( ) = ( u 1 , , u M ) ( ) which is directly identified with γ ^ ( ) = ( γ ^ 1 , , γ ^ M ) ( ) . If one denotes the momenta...

On control theory and its applications to certain problems for Lagrangian systems. On hyper-impulsive motions for these. III. Strengthening of the characterizations performed in parts I and II, for Lagrangian systems. An invariance property.

Aldo Bressan (1988)

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

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In [1] I and II various equivalence theorems are proved; e.g. an ODE ( ) z ˙ = F ( t , z , u , u ˙ ) ( m ) with a scalar control u = u ( ) is linear w.r.t. u ˙ iff ( α ) its solution z ( u , ) with given initial conditions (chosen arbitrarily) is continuous w.r.t. u in a certain sense, or iff ( β ) z ( u , ) satisfies certain conditions by which 1 s t -order discontinuities of u and u ˙ can be treated satisfactorily. In the case when, for z = ( q , p ) equation ( ) is a semi-Hamiltonian system, equivalent to a system of Lagrangian equations of a general type,...

On control theory and its applications to certain problems for Lagrangian systems. On hyperimpulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Applications to Lagrangian systems

Aldo Bressan (1988)

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

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See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case Σ , χ and M satisfy conditions (11.7) when 𝒬 is a polynomial in γ ˙ , conditions (C)-i.e. (11.8) and (11.7) with 𝒬 0 -are proved to be necessary for treating satisfactorily Σ 's hyper-impulsive motions (in which positions can suffer first order discontinuities)....

A remark on complex powers of analytic functions

Giuseppe Zampieri (1985)

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

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Sia K n un compatto, f 0 una funzione analitica all'intorno di K , ed m la massima molteplicità in K degli zeri di f ; si prova che la potenza f λ ( λ , R e λ > 1 m ) è integrabile in K . L'estensione meromorfa dell'applicazione λ f λ da R e λ > 0 a tutto (con valori in 𝒟 ( K ) anziché in L 1 ( K ) ) era già stata provata in [1] e [2].

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

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We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower C 1 -functions, or of two strongly paraconvex functions) coincide with semismooth functions on (i.e. those locally Lipschitz functions on for which f + ' ( x ) = lim t x + f + ' ( t ) and f - ' ( x ) = lim t x - f - ' ( t ) for each x ). Further, for each modulus ω , we characterize the class D S C ω of functions on which can be written as f = g - h , where g and h are semiconvex with modulus C ω (for some C > 0 ) using a new...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions. We use the result to prove that, if ε 0 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any...

Ramsey numbers for trees II

Zhi-Hong Sun (2021)

Czechoslovak Mathematical Journal

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Let r ( G 1 , G 2 ) be the Ramsey number of the two graphs G 1 and G 2 . For n 1 n 2 1 let S ( n 1 , n 2 ) be the double star given by V ( S ( n 1 , n 2 ) ) = { v 0 , v 1 , ... , v n 1 , w 0 , w 1 , ... , w n 2 } and E ( S ( n 1 , n 2 ) ) = { v 0 v 1 , ... , v 0 v n 1 , v 0 w 0 , w 0 w 1 , ... , w 0 w n 2 } . We determine r ( K 1 , m - 1 , S ( n 1 , n 2 ) ) under certain conditions. For n 6 let T n 3 = S ( n - 5 , 3 ) , T n ' ' = ( V , E 2 ) and T n ' ' ' = ( V , E 3 ) , where V = { v 0 , v 1 , ... , v n - 1 } , E 2 = { v 0 v 1 , ... , v 0 v n - 4 , v 1 v n - 3 , v 1 v n - 2 , v 2 v n - 1 } and E 3 = { v 0 v 1 , ... , v 0 v n - 4 , v 1 v n - 3 , v 2 v n - 2 , v 3 v n - 1 } . We also obtain explicit formulas for r ( K 1 , m - 1 , T n ) , r ( T m ' , T n ) ( n m + 3 ) , r ( T n , T n ) , r ( T n ' , T n ) and r ( P n , T n ) , where T n { T n ' ' , T n ' ' ' , T n 3 } , P n is the path on n vertices and T n ' is the unique tree with n vertices and maximal degree n - 2 .

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

On control theory and its applications to certain problems for Lagrangian systems. On hyperimpulsive motions for these. II. Some purely mathematical considerations for hyper-impulsive motions. Applications to Lagrangian systems

Aldo Bressan (1988)

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

Similarity:

See Summary in Note I. First, on the basis of some results in [2] or [5]-such as Lemmas 8.1 and 10.1-the general (mathematical) theorems on controllizability proved in Note I are quickly applied to (mechanic) Lagrangian systems. Second, in case Σ , χ and M satisfy conditions (11.7) when 𝒬 is a polynomial in γ ˙ , conditions (C)-i.e. (11.8) and (11.7) with 𝒬 0 -are proved to be necessary for treating satisfactorily Σ 's hyper-impulsive motions (in which positions can suffer first order discontinuities)....

A remark on complex powers of analytic functions

Giuseppe Zampieri (1985)

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

Similarity:

Sia K n un compatto, f 0 una funzione analitica all'intorno di K , ed m la massima molteplicità in K degli zeri di f ; si prova che la potenza f λ ( λ , R e λ > 1 m ) è integrabile in K . L'estensione meromorfa dell'applicazione λ f λ da R e λ > 0 a tutto (con valori in 𝒟 ( K ) anziché in L 1 ( K ) ) era già stata provata in [1] e [2].

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

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The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt A p -weighted L p -spaces

Ryôhei Kakizawa (2018)

Czechoslovak Mathematical Journal

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We discuss the validity of the Helmholtz decomposition of the Muckenhoupt A p -weighted L p -space ( L w p ( Ω ) ) n for any domain Ω in n , n , n 2 , 1 < p < and Muckenhoupt A p -weight w A p . Set p ' : = p / ( p - 1 ) and w ' : = w - 1 / ( p - 1 ) . Then the Helmholtz decomposition of ( L w p ( Ω ) ) n and ( L w ' p ' ( Ω ) ) n and the variational estimate of L w , π p ( Ω ) and L w ' , π p ' ( Ω ) are equivalent. Furthermore, we can replace L w , π p ( Ω ) and L w ' , π p ' ( Ω ) by L w , σ p ( Ω ) and L w ' , σ p ' ( Ω ) , respectively. The proof is based on the reflexivity and orthogonality of L w , π p ( Ω ) and L w , σ p ( Ω ) and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation...

On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group. The prime graph of G is a simple graph Γ ( G ) whose vertex set is π ( G ) and two distinct vertices p and q are joined by an edge if and only if G has an element of order p q . A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition Γ ( G ) = Γ ( H ) . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that PGL ( 2 , p α ) is recognizable, if p is an odd prime and α > 1 is odd. But for even α , only...

Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

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Let be the set of integers, 0 the set of nonnegative integers and F ( x 1 , x 2 ) = u 1 x 1 + u 2 x 2 be a binary linear form whose coefficients u 1 , u 2 are nonzero, relatively prime integers such that u 1 u 2 ± 1 and u 1 u 2 - 2 . Let f : 0 { } be any function such that the set f - 1 ( 0 ) has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set A of integers such that r A , F ( n ) = f ( n ) for all integers n , where r A , F ( n ) = | { ( a , a ' ) : n = u 1 a + u 2 a ' : a , a ' A } | . We add the structure of difference for the binary linear form F ( x 1 , x 2 ) .

Nodal curves in 3 ( )

Edoardo Ballico, Paolo Oliverio (1984)

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

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Siano d , g , t interi con 0 t g ; se esiste in 3 ( ) una curva connessa, non singolare di grado d e genere g , allora esiste in 3 ( ) una curva irriducibile di grado d , genere aritmetico g e t nodi.