fluctuations around classical solutions for gauge theories in lagrangian and hamiltonian approach we analyse the dynamics of gauge theories and constrained systems in general under small perturbations around a classical solution in both lagrangian and hamiltonian formalisms we prove that a fluctuations theory described by a quadratic lagrangian has the same constraint structure and number of physical degrees of freedom as the original non perturbed theory assuming the non degenerate solution has been chosen we show that the number of noether gauge symmetries is the same in both theories but that the gauge algebra in the fluctuations theory becomes abelianized we also show that the fluctuations theory inherits all functionally independent rigid symmetries from the original theory and that these symmetries are generated by linear or quadratic generators according to whether the original symmetry is preserved by the background or is broken by it we illustrate these results with examples

on encoding and enumerating threshold functions in this paper we deal with encoding and enumerating threshold functions defined on n dimensional binary inputs the paper specifies situations in which the unique characterization of functions from a given class is preserved by usage of an appropriate set of discrete moments moreover sometimes such a characterization coding is optimal with respect to the number of necessary bit rate per coded function by estimating the number of possible values of the discrete moments used several upper bounds for different classes of threshold functions are derived some of which are better than those previously known

capacity of multilevel threshold devices the principal purpose of this research is to discover the underlying properties of linear multilevel threshold devices and networks of linear multilevel threshold devices the main theoretical developments of this investigation entail generalizing the function counting theorem associated with linear bi level threshold devices to that of linear multilevel threshold devices the investigation reveals surprising connections that underlie linear inequalities linear separability linear ordering and region counting the results have implications in the field of geometric probability computations based on theoretical analysis support brown s 1964 conjecture and suggest that k k 1 is a natural definition for the information storage capacity of a k level threshold device a lower bound on the number of weights required to implement a universal network of k level threshold devices is derived finally it is shown that the vapnik chervonenki 1971 s dimension vc dimension for the class of multilevel threshold function is d 1 for pattern vectors in spl rscr sup d this vc dimension is also linked to an error rate bound for multilevel threshold functions within the framework of uniform learnability

lagrangian and hamiltonian constraint structure coefficients we introduce the notion of lagrangian and hamiltonian constraint structure coefficients these coefficients appear in the process of investigating the consistency condition on constraints in both formulations then we show that for a first class hamiltonian constrained system one can find all the lagrangian coefficients in terms of hamiltonian ones

new relations between hamiltonian and lagrangian constraints different procedures for obtaining all the lagrangian constraints from the hamiltonian constraints are studied the connection of arbitrary functions in both hamiltonian and lagrangian formalisms is also studied as a consequence the authors give a way to pass directly from the final hamilton dirac equations to the euler lagrange ones

symmetries and constants of motion for constrained lagrangian systems a presymplectic version of the noether theorem the concept of dynamical symmetry for constrained lagrangian systems is analysed and generalized both the direct and converse noether theorems are proved for singular lagrangians

the capacity of multilevel threshold functions lower and upper bounds for the capacity of multilevel threshold elements are estimated using two essentially different enumeration techniques it is demonstrated that the exact number of multilevel threshold functions depends strongly on the relative topology of the input set the results correct a previously published estimate and indicate that adding threshold levels enhances the capacity more than adding variables

a hamiltonian approach to lagrangian noether transformations it is proven that each lagrangian noether symmetry rigid or gauge can be easily obtained from a kind of hamiltonian generator which is a conserved quantity satisfying a simple condition this yields a procedure to construct lagrangian gauge transformations it is also shown that some regularity conditions are needed in order to assure the existence of hamiltonian gauge generators the authors present an example which has no such generators though noether gauge transformations can be constructed for it they apply this method to obtain the covariant gauge transformations of the bosonic string from its hamiltonian constraints

lagrangian approach to the physical degree of freedom count in this paper we present a lagrangian method that allows the physical degree of freedom count for any lagrangian system without having to perform neither dirac nor covariant canonical analyses the essence of our method is to establish a map between the relevant lagrangian parameters of the current approach and the hamiltonian parameters that enter in the formula for the counting of the physical degrees of freedom as is given in dirac s method once the map is obtained the usual hamiltonian formula for the counting can be expressed in terms of lagrangian parameters only and therefore we can remain in the lagrangian side without having to go to the hamiltonian one using the map it is also possible to count the number of first and second class constraints within the lagrangian formalism only for the sake of completeness the geometric structure underlying the current approach developed for systems with a finite number of degrees of freedom is uncovered with the help of the covariant canonical formalism finally the method is illustrated in several examples including the relativistic free particle

noether s theorem and gauge transformations application to the bosonic string and cp n 12 model new results on the theory of constrained systems are applied to characterize the generators of noether s symmetry transformations as a byproduct an algorithm to construct gauge transformations in hamiltonian formalism is derived this is illustrated with two relevant examples

the stability of certain partial difference equations abstract this paper is concerned with the linear delay partial difference equation where and are positive integers a i j b i j p i j are real sequences defined on i j sufficient conditions for this equation to be stable are derived stability of certain nonlinear partial difference equations is studied also

on the singular fornasini marchesini model of 2 d discrete systems the singular fornasini marchesini model of 2 d discrete systems cannot be handled by commonly used z transform techniques a recursive form of the basic equation is found and consequently existence and uniqueness theorems can be formulated the recursive form of the singular fornasini marchesini model also yields a sufficient stability condition for this model a detailed example illustrates the results

bounds for solutions of a six point partial difference scheme abstract this paper is concerned with a class of partial difference equations which arise from discretising the one dimensional heat and the first order wave equations explicit bounds are found for their solutions and subexponential solutions are considered using the green s function we derive a stability criteria for arbitrary solutions of this equation

vector analysis of threshold functions abstract viewing n variable boolean functions as vectors in r 2n we invoke basic tools from linear algebra and linear programming to derive new results on the realizability of boolean functions using threshold gates using this approach we obtain 1 a lower bound on the number of input functions required by a threshold gate implementing a given function 2 a lower bound on the error incurred when a boolean function is approximated by a linear combination of a set of functions 3 a limit on the effectiveness of a well known lower hound technique based on computing correlations among boolean functions for the depth of threshold circuits implementing boolean functions 4 a construction showing that every boolean function of n input variables is a threshold function of polynomially many input functions none of which is significantly correlated with 5 generalizations of some known results on threshold circuit complexity particularly those that are based on spectral analysis

equivalence between the lagrangian and hamiltonian formalism for constrained systems the equivalence between the lagrangian and hamiltonian formalism is studied for constraint systems a procedure to construct the lagrangian constraints from the hamiltonian constraints is given those hamiltonian constraints that are first class with respect to the hamiltonian constraints produce lagrangian constraints that are fl projectable

exponential asymptotic stability of delay partial difference equations this paper is concerned with the linear delay partial difference equation u i j 1 a i j u i 1 j b i j u i j p i j u i j where and are two nonnegative integers a i j b i j p i j are three real sequences defined on i 0 j 0 some sufficient conditions for this equation to be exponentially asymptotically stable are derived

dirac and reduced quantization a lagrangian approach and application to coset spaces a lagrangian treatment of the quantization of first class hamiltonian systems with constraints and hamiltonian linear and quadratic in the momenta respectively is performed the first reduce and then quantize and the first quantize and then reduce dirac s methods are compared a source of ambiguities in this latter approach is pointed out and its relevance on issues concerning self consistency and equivalence with the first reduce method is emphasized one of the main results is the relation between the propagator obtained a la dirac and the propagator in the full space as an application of the formalism developed quantization on coset spaces of compact lie groups is presented in this case it is shown that a natural selection of a dirac quantization allows for full self consistency and equivalence finally the specific case of the propagator on a two dimensional sphere s2 viewed as the coset space su 2 u 1 is worked out

general solutions of a three level partial difference equation abstract this paper is concerned with a linear partial difference equation which includes the well known dufort frankel multilevel difference scheme for the heat equation by introducing green s functions for this equation we obtain via a novel formal approach an explicit formula for all its solutions given exponential initial conditions separable solutions are also found as applications we derive several stability criteria for the solutions of this equation

reduced phase space quotienting procedure for gauge theories we present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration velocity space local expressions for a basis of this kernel are obtained using phase space procedures the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented we show that this reduction procedure is equivalent to the standard dirac method as long as the dirac conjecture holds that the dirac hamiltonian containing the primary first class constraints with their lagrange multipliers can be enlarged to an extended dirac hamiltonian which includes all first class constraints without any change of the dynamics the quotienting procedure is always equivalent to the extended dirac theory even when it differs from the standard dirac theory the differences occur when there are ineffective constraints and in these situations we conclude that the standard dirac method is preferable at least for classical theories an example is given to illustrate these features as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom

higher order conditions for singular lagrangian dynamics for a kth order singular lagrangian the presymplectic equation plus the different mth order differential equation conditions that can be considered in the lagrangian space yield different dynamics which are studied this leads to a new classification of the constraints this study is also performed in the intermediate formalisms which can be defined between the lagrangian and the hamiltonian ones the corresponding classification schemes are then related

higher order lagrangian systems geometric structures dynamics and constraints in order to study the connections between lagrangian and hamiltonian formalisms constructed from a perhaps singular higher order lagrangian some geometric structures are constructed intermediate spaces between those of lagrangian and hamiltonian formalisms partial ostrogradski s transformations and unambiguous evolution operators connecting these spaces are intrinsically defined and some of their properties studied equations of motion constraints and arbitrary functions of lagrangian and hamiltonian formalisms are thoroughly studied in particular all the lagrangian constraints are obtained from the hamiltonian ones once the gauge transformations are taken into account the true number of degrees of freedom is obtained both in the lagrangian and hamiltonian formalisms and also in all the intermediate formalisms herein defined

stability criteria for a class of linear delay partial difference equations this paper is concerned with the linear delay partial difference equation a i j u i 1 j u i j 1 b i j u i j p i j u i j 0 where a i j b i j p i j i j n0 are real sequences sufficient conditions for this equation to be stable are derived some conditions for this equation to be unstable are obtained also

oscillation criteria of a class of partial difference equations with delays this paper is concerned with the linear delay partial difference equation aa m 1 n 1 ba m 1 n ca n m n l da m n p m na m s n t 0 where s and t are two nonnegative integers a b c and d are positive constants and p i j i j e n 0 is a double real sequence sufficient conditions for this equation to be oscillatory are derived

stability criteria for two partial difference equations abstract this paper is concerned with two classes of linear partial difference equations with constant coefficients explicit conditions are derived which are sufficient and or necessary for these equations to have stable solutions

singular lagrangians some geometric structures along the legendre map new geometric structures that relate the lagrangian and hamiltonian formalisms defined upon a singular lagrangian are presented several vector fields are constructed in velocity space that give new and precise answers to several topics such as the projectability of a vector field to a hamiltonian vector field the computation of the kernel of the presymplectic form of a lagrangian formalism the construction of the lagrangian dynamical vector fields and the characterization of dynamical symmetries