on binary equality sets and a solution to the test set conjecture in the binary case abstract k culik ii and j karhumaki rairo inform theor 14 1980 349 369 conjectured that the equality set of two injective morphisms over a binary alphabet is of the form u v for some possibly empty words u and v here we show that such an equality set is always either of the above form or of the form uw v for some words u w and v as an application we give a simple proof for the test set conjecture in the binary case cf k culik ii and a salomaa j comput system sci 20 1980 379 396 in fact we show that a test set can always be chosen to contain no more than three words

test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

semipositone problems with falling zeros on exterior domains abstract we study boundary value problems of the form u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k belongs to a class of c 1 functions such that lim r k r 0 and f belongs to a class of c 1 functions which are negative at the origin and have falling zeros we discuss the existence and uniqueness of nonnegative radial solutions when is large

uniqueness of positive radial solutions for a class of semipositone problems on the exterior of a ball abstract we study positive radial solutions to u k x f u x e where 0 is a parameter e x r n x r 0 r 0 0 n 2 is the laplacian operator k c r 0 0 satisfies k r 1 r n 0 for r 1 and f c 1 0 r is a concave increasing function satisfying lim s f s s 0 and f 0 0 semipositone we are interested in solutions u such that u 0 as x and satisfy the nonlinear boundary condition u c u u 0 if x r 0 where is the outward normal derivative and c c 0 0 is an increasing function we will establish the uniqueness of positive radial solutions for large values of the parameter

uniqueness of nonnegative solutions for semipositone problems on exterior domains abstract we consider the problem u k x f u x u 0 if x r 0 u 0 as x where is a positive parameter u div u is the laplacian of u x r n n 2 x r 0 k c 1 r 0 0 is such that lim r k r 0 and f c 1 0 r is a concave function which is sublinear at and f 0 0 we establish the uniqueness of nonnegative radial solutions when is large

exact multiplicity for degenerate two point boundary value problems with p convex nonlinearity the exact number of positive solutions of a degenerate quasilinear two point boundary value problem is investigated for the generalization of earlier results concerning the nondegenerate case with convex nonlinearity suitably defined p convex nonlinearities are considered strictly p convex c2 functions having a non negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval we have uniqueness when f 0 0 and 0 1 or 2 solutions when f 0 0 similarly to the non degenerate case provided that the number of solutions is finite however now there may also occur a continuum of solutions connected to a dead core type phenomenon the proof of our results relies on the shooting method for the characterization of the shape of the time map in contrast to the non degenerate case the shooting method does not determine directly the number of solutions owing to the lack of uniqueness of the corresponding ivp exact conditions on the uniqueness of the ivp and in the case of non uniqueness the number and types of its local solutions are given based on this all the positive solutions of the bvp can be compiled

systemes entiers d equations sur un alphabet fini et conjecture d ehrenfeucht the following conjecture has been posed by a ehrenfeucht for each language l over a finite alphabet there exists a finite subset of f of l such that for each pair of morphisms of into a free monoid the equality is true for all in l if and only if it is true for all in f such a finite subset f of l is called a test set of l a system s of equations over the alphabet is called entire if there exists a morphism of into a free monoid such that s 1 we have introduced for each morphism of in a free monoid such that is finite and 1 1 1 a generalized automaton m called the shuffle automaton of which recognizes a rational language k over a finite alphabet and a pair of morphisms of into such that 1 k nivat s 1966 characterization of rational relations is a motivation for this construction and implies that all entire systems over a finite alphabet are rational culik ii and salomaa 1978 have proved that every rational language over a finite alphabet admits a test set culik ii and karhumaki 1983 have shown that if s is a system of equations over a finite alphabet and if there exists a language l over a finite alphabet and morphisms and from into such that s l then if l admits a test set f then t f is a finite equivalent subsystem of s this gives an algorithm which determines for each entire system of equations over a finite alphabet a finite subsystem this result can be improved by using the following theorem spehner 1984 each finitely generated submonoid m of a free monoid has a finite m presentation i e there exists a finite alphabet and a finite part of such that m is isomorphic to the quotient monoid of by the smallest malcev congruence containing is a malcev congruence of if the quotient monoid is embeddable in a group the algorithm for determining the shuffle automaton m and the finite equivalent subsystem of s 1 works with a maximum efficiency because m can be embedded in the minimal automaton of the language k

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

on test sets for checking morphism equivalence on languages with fair distribution of letters abstract a test set for a language l is a finite subset t of l with the property that each pair of morphisms that agrees on t also agrees on l some results concerning test sets for languages with fair distribution of letters are presented the first result is that every d0l language with fair distribution of letters has a test set the second result shows that every language l with fair distribution has a test set relative to morphisms g h which have bounded balance on l these results are generalizations of results of culik ii and karhumaki 1983

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a computational approach for corner and vertex detection corners and vertexes are strong and useful features in computer vision for scene analysis stereo matching and motion analysis here we deal with the development of a computational approach to these important features we consider first a corner model and study analytically its behavior once it has been smoothed using the well known gaussian filter this allows us to clarify the behavior of some well knowncornerness measure based approaches used to detect these points of interest most of these classical approaches appear to detect points that do not correspond to the exact position of the corner a new scale space based approach that combines useful properties from the laplacian and beaudet s measure beaudet 1978 is then proposed in order to correct and detect exactly the corner position an extension of this approach is then developed to solve the problem of trihedral vertex characterization and detection in particular it is shown that a trihedral vertex has two elliptic maxima on extremal contrast surfaces if the contrast is sufficient and this allows us to classify trihedral vertexes in 2 classes vertex and vertex as corner the corner detection approach developed is applied to accurately detect trihedral vertexes using an additional test in order to make a distinction between trihedral vertexes and corners many experiments have been carried out using noisy synthetic data and real images containing corners and vertexes most of the promising results obtained are used to illustrate the experimental section of this paper

diffusive logistic equation with constant yield harvesting i steady states we consider a reaction diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth we prove the existence uniqueness and stability of the maximal steady state solutions under certain conditions and we also classify all steady state solutions under more restricted conditions exact global bifurcation diagrams are obtained in the latter case our method is a combination of comparison arguments and bifurcation theory

on the stability properties of nonnegative solutions of semilinear problems with convex or concave nonlinearity 497 abstract we investigate the stability of nontrivial nonnegative stationary solutions of semilinear initial boundary value problems with convex or concave nonlinearity in the convex case f 0 0 implies instability in the concave case f 0 0 implies stability we also discuss the necessity of the sign condition on f 0

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

positive solution curves of semipositone problems with concave nonlinearities we consider the positive solutions to the semilinear equation where denotes a smooth bounded region in n n 1 and 0 here f 0 is assumed to be monotonically increasing concave and such that f 0 f lim t f t 0 we establish the stability and uniqueness of large positive solutions in terms of f t t when is a ball we determine the exact number of positive solutions for each 0 we also obtain the geometry of the branches of positive solutions completely and establish how they evolve this work extends and complements that of 3 7 where f 0

unbounded connected component of the positive solution set of nonlinear operator equation and its applications abstract in this paper by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters as a result we obtain some existence results for positive solutions of the nonlinear operator equation the main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone

binary equality sets are generated by two words we show that the equality set eq g h of two non periodic binary morphisms g h a is generated by at most two words if the rank of eq g h is two then and begin and end with different letters this in particular implies that any binary language has a test set of cardinality at most two

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

affine morphological multiscale analysis of corners andmultiple junctions in this paper we study the application of the affine morphological scale space amss to the analysis of singularities corners or multiple junctions of the shapes present in a 2 d image we introduce a new family of travelling wave solutions of amss which determines the evolution of the initial shapes given by conics we characterize the evolution of corners accross the scales according to their angle we develop a numerical algorithm to compute amss accross the scales and we present some experimental results about corners and multiple junction detection

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

existence of positive radial solutions for a semipositone elliptic equation abstract in this paper we study the existence of positive radial solutions for a semipositone elliptic equation u f u in u 0 on where is a ball or an annulus in r n with n 2 0 is a parameter and f is a continuous function satisfying the semipositone condition f 0 0 we give a weak and general sufficient condition on f for the existence of positive radial solutions for 0 large we show that the linearized operator at each solution has the nonnegative first eigenvalue these solutions are obtained as minimizers of a lagrangian functional

positive solutions for infinite semipositone problems with falling zeros abstract we consider the positive solutions to the singular problem p u a u f u c u in u 0 on where 0 1 a 0 and c 0 are constants is a bounded domain with smooth boundary and f 0 r is a continuous function we assume that there exist m 0 a 0 p 1 such that a u m f u a u p for all u 0 a simple example of f satisfying these assumptions is f u u p for any p 1 we use the method of sub supersolutions to prove the existence of a positive solution of p when a 2 1 1 and c is small here 1 is the first eigenvalue of operator with dirichlet boundary conditions we also extend our result to classes of infinite semipositone systems