Results 1  10
of
28
Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals
 Higher Order Logic Theorem Proving and its Applications (HUG `93
, 1993
"... In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from ..."
Abstract

Cited by 33 (4 self)
 Add to MetaCart
In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and cryptosystems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...
On the correctness of linear boundary value problems for systems of ordinary differential equations
 Russian) Bull. Acad. Sci. Georgian SSR
"... Abstract. The sufficient conditions are established for the correctness of the linear boundary value problem dx(t) = dA(t) · x(t) + df(t), l(x) = c0, where A: [a, b] → R n×n and f: [a, b] → R n are matrix and vectorfunctions of bounded variation, c0 ∈ R n, and l is a linear continuous operator ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Abstract. The sufficient conditions are established for the correctness of the linear boundary value problem dx(t) = dA(t) · x(t) + df(t), l(x) = c0, where A: [a, b] → R n×n and f: [a, b] → R n are matrix and vectorfunctions of bounded variation, c0 ∈ R n, and l is a linear continuous operator from the space of ndimentional vectorfunctions of bounded variation into R n. Let the matrix and vectorfunctions, A: [a, b] → R n×n and f: [a, b] → R n, respectively, be of bounded variation, c0 ∈ R n, and let l: BVn(a, b) → R n be a linear continuous operator such that the boundary value problem dx(t) = dA(t) · x(t) + df(t), (1) l(x) = c0 (2) has the unique solution x0. Consider the sequences of matrix and vectorfunctions of bounded variation Ak: [a, b] → R n×n (k = 1, 2,...) and fk: [a, b] → R n (k = 1, 2,...), respectively, the sequence of constant vectors ck ∈ R n (k = 1, 2,...) and the sequence of linear continuous operators lk: BVn(a, b) → R n (k = 1, 2,...). In this paper the sufficient conditions are given for the problem dx(t) = dAk(t) · x(t) + dfk(t), (3) lk(x) = ck (4) to have a unique solution xk for any sufficiently large k and lim k→+ ∞ xk(t) = x0(t) uniformly on [a, b]. (5)
Reasoning About the Reals: the marriage of HOL and Maple
, 1993
"... . Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
. Computer algebra systems are extremely powerful and flexible, but often give results which require careful interpretation or are downright incorrect. By contrast, theorem provers are very reliable but lack the powerful specialized decision procedures and heuristics of computer algebra systems. In this paper we try to get the best of both worlds by careful exploitation of a link between a theorem prover and a computer algebra system. 1 Motivation In the HOL theorem prover[5], a theory of real numbers has been developed, using a rigorous definition in terms of Dedekind cuts [8]. It is therefore possible to apply HOL to areas traditionally within the purview of Computer Algebra Systems (CASs). This offers two main benefits. Firstly, theorem provers are designed to manipulate proofs and theorems in a coherent and structured way, with all concepts clearly defined. By contrast, most CASs have no concept of `logic' as such  they usually take an algebraic expression and return another pur...
Linear integral equations in the space of regulated functions
, 1997
"... In this paper, we investigate the existence of solutions to a wide class of systems of linear integral equations with solutions which can have in the closed interval [0,1] only discontinuities of the rst kind and are leftcontinuous on the corresponding open interval (0,1). The results cover, e.g., ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
In this paper, we investigate the existence of solutions to a wide class of systems of linear integral equations with solutions which can have in the closed interval [0,1] only discontinuities of the rst kind and are leftcontinuous on the corresponding open interval (0,1). The results cover, e.g., the results known for systems of linear generalized differential equations as well as systems of Stieltjes Integral equations. Some possible applications to functional differential equations are discussed as well.
Continuous dependence of solutions of generalized linear differential equations on a parameter
 FUNCT. DIFFER. EQU
, 2009
"... This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bounded varia ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bounded variation on [a, b]. The integrals are understood in the KurzweilStieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations on parameters and initial data. In particular, we generalize in several aspects the known result by Ashordia. Our main goal consists in a more general notion of a solution to the given system. In particular, neither g nor x need not be of bounded variation on [a, b] and, in general, they can be regulated functions.
Approximated Solutions of Generalized Linear Differential Equations
 Institute of Mathematics, Acad. Sci. Czech Rep., Preprint 141/2008 [available as http://www.math.cas.cz/preprint/pre141.pdf
"... Abstract. This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bound ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. This contribution deals with systems of linear generalized linear differential equations of the form ∫ t x(t) = ˜x + d[A(s)] x(s) + g(t) − g(a), t ∈ [a, b], a where − ∞ < a < b < ∞, A is an n × ncomplex matrix valued function, g is an ncomplex vector valued function, A and g have bounded variation on [a, b]. The integrals are understood in the KurzweilStieltjes sense. Our aim is to present some new results on continuous dependence of solutions to linear generalized differential equations on parameters and initial data. In particular, we generalize in several aspects the known result by Ashordia. Our main goal consists in a more general notion of a solution to the given system. In particular, neither g nor x need not be of bounded variation on [a, b] and, in general, they can be regulated functions. AMS Subject Classification. 34A37, 45A05, 34A30. 1
QUASISTATIC EVOLUTION FOR CAMCLAY PLASTICITY: EXAMPLES OF SPATIALLY HOMOGENEOUS SOLUTIONS
"... Abstract. We study a quasistatic evolution problem for CamClay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities. The behavior of the solutions at the jump ti ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We study a quasistatic evolution problem for CamClay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities. The behavior of the solutions at the jump times is studied by a viscous approximation. Keywords: CamClay plasticity, softening behaviour, pressuresensitive yield criteria, nonassociative plasticity, quasistatic evolution, rate independent processes, viscosity approximation.
BOUNDARY VALUE PROBLEMS FOR LINEAR GENERALIZED DIFFERENTIAL EQUATIONS AND THEIR ADJOINTS
, 1971
"... Boundary value problems for linear generalized differential equations and their adjoints Czechoslovak Mathematical Journal, Vol. 23 (1973), No. 2, 183217 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Boundary value problems for linear generalized differential equations and their adjoints Czechoslovak Mathematical Journal, Vol. 23 (1973), No. 2, 183217
ASYMPTOTIC SOLUTIONS OF FORCED NONLINEAR SECOND ORDER DIFFERENTIAL EQUATIONS AND THEIR EXTENSIONS
, 2007
"... Abstract. Using a modified version of Schauder’s fixed point theorem, measures of noncompactness and classical techniques, we provide new general results on the asymptotic behavior and the nonoscillation of second order scalar nonlinear differential equations on a halfaxis. In addition, we extend ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Using a modified version of Schauder’s fixed point theorem, measures of noncompactness and classical techniques, we provide new general results on the asymptotic behavior and the nonoscillation of second order scalar nonlinear differential equations on a halfaxis. In addition, we extend the methods and present new similar results for integral equations and VolterraStieltjes integral equations, a framework whose benefits include the unification of second order difference and differential equations. In so doing, we enlarge the class of nonlinearities and in some cases remove the distinction between superlinear, sublinear, and linear differential equations that is normally found in the literature. An update of papers, past and present, in the theory of VolterraStieltjes integral equations is also presented. 1.