Quaternionic Cartan Coverings and Applications

We prove that, under some generic non-degeneracy assumptions, real analytic centrally symmetric plane domains are determined by their Dirichlet (resp. Neumann) spectra. We prove that the conditions are open-dense for real analytic strictly convex domains. One step is to use a Maslov index calculation to show that the second derivative of the defining function of a centrally symmetric domain at the endpoints of a bouncing ball orbit is a spectral invariant. This is also true for up-down symmetric domains, removing an assumption from the proof in that case.

The quaternionic functions method is an analytical tool used in elasticity theory. For isotropic elasticity, there are known a few variants of three-dimensional analogues of the Kolosov-Muskhelishvili formulae. In this case, a general solution of the Lamé equation for the spatial theory of elasticity is expressed in terms of two regular quaternionic or monogenic Clifford functions. For the anisotropic elasticity, a close approach exists when equations of equilibrium are factorized by means of matrix algebra. In this report, we will discuss the quaternionic factorization method in the transversely isotropic theory of elasticity. The model of an elastic media with such symmetry is described by five elastic constants, and for example, can be used in the mechanics of rocks in permafrost conditions.

A plane domain G w , symmetric with respect to the origin O, is extended by “sector reflections” to a symmetric domain Ğ w . A “mapping function” [Wbreve](z) of Ğ w (mapping the unit disk G z onto Ğ w ) is constructed in an elementary way in terms of a mapping function w(z) of G w using essentially the reflection principle. In the case of polygonal domains considered here, the results can be interpreted as relations between Schwarz-Christoffel integrals. This method applies to most domains considered in [2]. where only the ratio of conformal radii Ro(Ğ w )/R o (G w ) was determined. The study of the mapping functions themselves was suggested by Max Schiffer.

Geometric function theory over quaternionic slice domains

This article focuses on the integral theory of discrete regular quaternionic functions and the connection between discrete and classical quaternionic analysis. We develop a new approach to the discrete integral theory based on the concepts ‘discrete boundary measure’ and ‘discrete normal vector’. By combining the discrete integral theory and our results on discrete approximations to domains in \({\mathbb {R}}^4\), we reveal a convergence relation between the discrete and continuous regular functions: a quaternionic function is regular if and only if it is the scaling limit of some discrete regular functions.

Solvability of the first Cousin problem and vanishing of higher cohomology groups for domains which are not domains of holomorphy. II

Recently, much attention has been given to the solvability of (otherwise intractable) combinatorial optimization problems when instances are confined to series‐parallel graphs. Substantially less is known, however, regarding those that remain hard on these particularly sparse structures. While a few ad hoc cases have been shown to be difficult, little appears to be known in terms of more generic settings. In this paper, we contend with this state of affairs by establishing that a class of Steiner subgraph‐like problems are NP‐complete on series‐parallel graphs while remaining easy (even trivial) on trees.

In 1961, at A.M.S. Symposium on Convexity, P.C. Hammer proposed the following problem: how many X-ray pictures of a convex planar domain D must be taken to permit its exact reconstruction? Richard Gardner writes in his fundamental 2006 book [4] that X-rays in four different directions would do the job. The present paper points at the possibility that in certain asymptotical sense X-rays in only three different directions can be enough for approximate reconstruction of centrally symmetric convex domains. The accuracy of reconstruction would tend to become perfect in the limit, as the directions of the three X-rays change, all three converging to some given direction. The analysis leading to that conclusion is based on two lemmas of Section 1 and Pleijel type identity for parallel X-rays derived in Sections 2 and 3. These tools together supply a systemof two differential equations with respect to two unknown functions that describe the two branches of the domain boundary D. The system is easily resolved. The solution intended to provide a complete tomography reconstruction of D, happens however to depend on a two dimensional parameter, whose “real value” remains unknown. So tomography reconstruction of D becomes possible if a satisfactory approximation to that unknown “real value” can be found. In the last section a test procedure for the individual candidates for “approximate real value” of the parameter is described. A uniqueness theorem concerning tomography of circular discs is proved.

A relational data base is herein defined as a collection of normalized relations (relations in first normal form) and a collection of domains. A normalized relation may be viewed as a table, wherein each row of the table corresponds to a tuple of the relation, and the entries in a given column belong to the set of values constituting the domain underlying that column. The domains of a data base have an abstract existence apart from the data base relations.The data base also includes various types of semantic integrity rules, which specify additional properties of the data in the data base. One such type of semantic integrity rule is the domain definition. A domain definition includes the precise description of the set of values (objects) constituting the domain. In a normalized data base, all domains are sets of atomic data values. A domain definition also includes a specification of the ordering on the values in a domain, for comparability purposes. In addition, a domain definition contains a specification of the action that is to occur if an attempt is made to violate the restriction that every entry in each column of a relation must be from the underlying domain of that column.A nonprocedural language permitting the high level expression of domain definitions is defined. Language details and examples are presented, and the syntax and informal semantics of the domain definition language are given. This approach to domain definition is analyzed in terms of its impact on other aspects of data base semantic integrity. The relationship with the data base system in general is outlined. An analysis of intradomain and interdomain comparability is included. An introduction to relevant implementation issues is also presented. Emphasis is placed on a general approach to implementation and implementation techniques, rather than on a specific system.

A relational data base is herein defined as a collection of normalized relations (relations in first normal form) and a collection of domains. A normalized relation may be viewed as a table, wherein each row of the table corresponds to a tuple of the relation, and the entries in a given column belong to the set of values constituting the domain underlying that column. The domains of a data base have an abstract existence apart from the data base relations. The data base also includes various types of semantic integrity rules, which specify additional properties of the data in the data base. One such type of semantic integrity rule is the domain definition. A domain definition includes the precise description of the set of values (objects) constituting the domain. In a normalized data base, all domains are sets of atomic data values. A domain definition also includes a specification of the ordering on the values in a domain, for comparability purposes. In addition, a domain definition contains a specification of the action that is to occur if an attempt is made to violate the restriction that every entry in each column of a relation must be from the underlying domain of that column. A nonprocedural language permitting the high level expression of domain definitions is defined. Language details and examples are presented, and the syntax and informal semantics of the domain definition language are given. This approach to domain definition is analyzed in terms of its impact on other aspects of data base semantic integrity. The relationship with the data base system in general is outlined. An analysis of intradomain and interdomain comparability is included. An introduction to relevant implementation issues is also presented. Emphasis is placed on a general approach to implementation and implementation techniques, rather than on a specific system.

A relational data base is herein defined as a collection of normalized relations (relations in first normal form) and a collection of domains. A normalized relation may be viewed as a table, wherein each row of the table corresponds to a tuple of the relation, and the entries in a given column belong to the set of values constituting the domain underlying that column. The domains of a data base have an abstract existence apart from the data base relations. The data base also includes various types of semantic integrity rules, which specify additional properties of the data in the data base. One such type of semantic integrity rule is the domain definition. A domain definition includes the precise description of the set of values (objects) constituting the domain. In a normalized data base, all domains are sets of atomic data values. A domain definition also includes a specification of the ordering on the values in a domain, for comparability purposes. In addition, a domain definition contains a specification of the action that is to occur if an attempt is made to violate the restriction that every entry in each column of a relation must be from the underlying domain of that column. A nonprocedural language permitting the high level expression of domain definitions is defined. Language details and examples are presented, and the syntax and informal semantics of the domain definition language are given. This approach to domain definition is analyzed in terms of its impact on other aspects of data base semantic integrity. The relationship with the data base system in general is outlined. An analysis of intradomain and interdomain comparability is included. An introduction to relevant implementation issues is also presented. Emphasis is placed on a general approach to implementation and implementation techniques, rather than on a specific system.

We have developed a new derivative-free algorithm based on Radial Basis Functions (RBFs). Derivative-free optimization is an active field of research and several algorithms have been proposed recently. Problems of this nature in the industrial setting are quite frequent. The reason is that in a number of applications the optimization process contains simulation packages which are treated as black boxes. The development of our own algorithm was originally motivated by an application in biomedical imaging: the medical image registration problem. The particular characteristics of this problem have incited us to develop a new optimization algorithm based on trust-region methods. However it has been designed to be generic and to be applied to a wide range of problems. The main originality of our approach is the use of RBFs to build the models. In particular we have adapted the existing theory based on quadratic models to our own models and developed new procedures especially designed for models based on RBFs. We have tested our algorithm called BOOSTERS against state-of-the-art methods (UOBYQA, NEWUOA, DFO). On the medical image registration problem, BOOSTERS appears to be the method of choice. The tests on problems from the CUTEr collection show that BOOSTERS is comparable to, but not better than other methods on small problems (size 2-20). It is performing very well for medium size problems (20-80). Moreover, it is able to solve problems of dimension 200, which is considered very large in derivative-free optimization. We have also developed a new class of algorithms combining the robustness of derivative-free algorithms with the faster rate of convergence characterizing Newtonlike-methods. In fact, they define a new class of algorithms lying between derivative-free optimization and quasi-Newton methods. These algorithms are built on the skeleton of our derivative-free algorithm but they can incorporate the gradient when it is available. They can be interpreted as a way of doping derivative-free algorithms with derivatives. If the derivatives are available at each iteration, then our method can be seen as an alternative to quasi-Newton methods. At the opposite, if the derivatives are never evaluated, then the algorithm is totally similar to BOOSTERS. It is a very interesting alternative to existing methods for problems whose objective function is expensive to evaluate and when the derivatives are not available. In this situation, the gradient can be approximated by finite differences and its costs corresponds to n additional function evaluations assuming that Rn is the domain of definition of the objective function. We have compared our method with CFSQP and BTRA, two gradient-based algorithms, and the results show that our doped method performs best. We have also a theoretical analysis of the medical image registration problem based on maximization of mutual information. Most of the current research in this field is concentrated on registration based on nonlinear image transformation. However, little attention has been paid to the theoretical properties of the optimization problem. In our analysis, we focus on the continuity and the differentiability of the objective function. We show in particular that performing a registration without extension of the reference image may lead to discontinuities in the objective function. But we demonstrate that, under some mild assumptions, the function is differentiable almost everywhere. Our analysis is important from an optimization point of view and conditions the choice of a solver. The usual practice is to use generic optimization packages without worrying about the differentiability of the objective function. But the use of gradient-based methods when the objective function is not differentiable may result in poor performance or even in absence of convergence. One of our objectives with this analysis is also that practitioners become aware of these problems and to propose them new algorithms having a potential interest for their applications.

In this chapter we give further means to extend the domain of definition of an analytic function. We shall apply Theorem 1.2 of Chapter III in the following context. Suppose we are given an analytic function f of an open connected set U. Let V be open and connected, and suppose that U ∩ V is not empty, so is open. We ask whether there exists an analytic function g on V such that f = g on U ∩ V, or only such that f(z) = g(z) for all z in some set of points of U ∩ V which is not discrete. The above-mentioned Theorem 1.2 shows that such a function g if it exists is uniquely determined. One calls such a function g a direct analytic continuation of f, and we also say that (g, V) is a direct analytic continuation of (f, U). We use the word “direct” because later we shall deal with analytic continuation along a curve and it is useful to have an adjective to distinguish the two notions. For simplicity, however, one usually omits the word “direct” if no confusion can result from this omission. If a direct analytic continuation exists as above, then there is is unique analytic function h on U ∪ V such that h = f on U and h = g on V.

Correction Note: Correction to "On the Domains of Definition of Analytic Characteristic Functions"

We investigate the possibility of linear elasticity as an infinitesimal theory based on a genuinely linear response function which retains its validity even for finite deformations. Careful consideration of the domain of definition of the stress response function, the definition of linearity and the notion of material frame-indifference leads to our main result that an exact linear constitutive theory for elastic solids is impossible. We then generalize our result to viscoelasticity theory where the stress response is dependent on deformation gradient histories.