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The principal goal of this paper is to extend the classical problem of finding the values of α ∈ ℂ for which either f ˆ α ( z ) = ∫ 0 z ( f ( ζ ) ∕ ζ ) α d ζ or f α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ are univalent, whenever f belongs to some subclasses of univalent mappings in D , to the case of harmonic mappings, by considering the shear construction introduced by Clunie and Sheil-Small in [4] .

Phase-field models have become popular to simulate cohesive failure problems because of their capability of predicting crack initiation and propagation without additional criteria. In this paper, a new phase-field damage model coupled with general softening laws for cohesive fracture is proposed based on the unified phase-field theory. The commonly used quadratic geometric function in the classical phase-field model is implemented in the proposed model. The modified degradation function related to the failure strength and length scale is used to obtain the length scale insensitive model. Based on the analytical solution of a 1-D case, general softening laws in cohesive zone models can be considered. Parameters in the degradation function can be calibrated according to different softening curves and material properties. Numerical examples show that the results obtained by the proposed model have a good agreement with experimental results and the length scale has a negligible influence on the load-displacement curves in most cases, which cannot be observed in classical phase-field model.

Zhang and Zheng (1990) conjectured on the structure of a solution for a two-dimensional Riemann problem for Euler equation. To resolve this illuminating conjecture, many researchers have studied the simplified 2 × 2 systems. In this paper, 3-pieces Riemann problem for two-dimensional 2 × 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave. We classify twelve topologically distinct solutions and construct analytical and numerical solutions. The computed numerical solutions clearly confirm the constructed analytic solutions.

In this paper we study harmonic maps from super Riemann surfaces in complex projective spaces and projective spaces associated with the super skew-field D . In both cases, we develop the theory of Gaus transforms and study the notion of isotropy, in particular its relation to holomorphic differentials on the super Riemann surface. Moreover, we give a definition of finite type harmonic maps for a special class of maps into C P n | n + 1 and thus obtain a classification for certain harmonic super tori. Furthermore, we investigate the equations satisfied by the underlying objects and give an example of a harmonic super torus in D P 2 whose underlying map is not harmonic.

Our aim in this paper is to provide a theory of discrete Riemann surfaces based on quadrilateral cellular decompositions of Riemann surfaces together with their complex structure encoded by complex weights. Previous work, in particular of Mercat, mainly focused on real weights corresponding to quadrilateral cells having orthogonal diagonals. We discuss discrete coverings, discrete exterior calculus, and discrete Abelian integrals. Our presentation includes several new notions and results such as branched coverings of discrete Riemann surfaces, the discrete Riemann-Hurwitz Formula, double poles of discrete one-forms and double values of discrete meromorphic functions that enter the discrete Riemann-Roch Theorem, and a discrete Abel-Jacobi map., Comment: 35 pages, 9 figures. Minor corrections, new Figures 3, 5, 6, and 7

In this study, the Chebyshev collocation spectral method (CCSM) is developed to solve the radiative integro-differential transfer equation (RIDTE) for one-dimensional absorbing, emitting and linearly anisotropic-scattering cylindrical medium. The general form of quadrature formulas for Chebyshev collocation points is deduced. These formulas are proved to have the same accuracy as the Gauss-Legendre quadrature formula (GLQF) for the F -function (geometric function) in the RIDTE. The explicit expressions of the Lagrange basis polynomials and the differentiation matrices for Chebyshev collocation points are also given. These expressions are necessary for solving an integro-differential equation by the CCSM. Since the integrand in the RIDTE is continuous but non-smooth, it is treated by the segments integration method (SIM). The derivative terms in the RIDTE are carried out to improve the accuracy near the origin. In this way, a fourth order accuracy is achieved by the CCSM for the RIDTE, whereas it's only a second order one by the finite difference method (FDM). Several benchmark problems (BPs) with various combinations of optical thickness, medium temperature distribution, degree of anisotropy, and scattering albedo are solved. The results show that present CCSM is efficient to obtain high accurate results, especially for the optically thin medium. The solutions rounded to seven significant digits are given in tabular form, and show excellent agreement with the published data. Finally, the solutions of RIDTE are used as benchmarks for the solution of radiative integral transfer equations (RITEs) presented by Sutton and Chen (JQSRT 84 (2004) 65–103). A non-uniform grid refined near the wall is advised to improve the accuracy of RITEs solutions.

This paper is devoted to the study of some fundamental problems on modulus and extremal length of curve families, capacity, and n-harmonic functions in the Euclidean space R n . One of the main goals is to establish the existence, uniqueness, and boundary behavior of the extremal function for the conformal capacity cap ( A , B ; Ω ) of a capacitor in R n . This generalizes some well known results and has its own interests in geometric function theory and potential theory. It is also used as a major ingredient in this paper to establish a sharp upper bound for the quasiextremal distance (or QED) constant M ( Ω ) of a domain in terms of its local boundary quasiconformal reflection constant H ( Ω ) , a bound conjectured by Shen in the plane. Along the way, several interesting results are established for modulus and extremal length. One of them is a decomposition theorem for the extremal length λ ( A , B ; Ω ) of the curve family joining two disjoint continua A and B in a domain Ω.

Given a function f : E → C ‾ from a closed subset of a Riemann surface R to the Riemann sphere C ‾ , we seek to approximate f in the spherical distance by functions meromorphic on R. As a consequence we generalize a recent extension of Mergelyan's theorem, due to Fragoulopoulou, Nestoridis and Papadoperakis [12] . The problem of approximating by meromorphic functions pole-free on E is equivalent to that of approximating by meromorphic functions zero-free on E, which in turn is related to Voronin's spectacular universality theorem for the Riemann zeta-function.

Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schoenflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous W 1 , 2 -extension but not even a homeomorphic W 1 , 1 -extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents p 2 . John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory.

In this paper, we construct a uniform formula for the Riemann solutions of the simplified Chapman-Jouguet model. Firstly, we define a new functional, and then, we obtain that the Riemann solutions can be expressed by the maximum value point of this functional, while Riemann solutions may contain some of strong detonation waves, Chapman-Jouguet detonation waves and contact discontinuities. Finally, Chapman-Jouguet deflagration waves are also discussed.

Stress intensity factor solutions for spot welds in two kinds of coach-peel specimens are investigated by finite element analyses. Three-dimensional finite element models are developed to obtain accurate solutions. Various ratios of sheet thickness, specimen width, flange length, and effective wall height to nugget radius are considered. The results confirm the functional dependence on the nugget radius and sheet thickness of Zhang’s analytical solution. Two geometric functions in terms of the normalized geometric parameters for Zhang’s solution are obtained. Based on the analytical and computational results, the dimensions of coach-peel specimens and the corresponding approximate stress intensity factor solutions are suggested.

In this article we consider the class A H ( p ) of all sense preserving harmonic functions f in the open unit disc D having a simple pole at z = p ∈ ( 0 , 1 ) with the normalizations f ( 0 ) = f z ( 0 ) − 1 = 0 . We first derive a sufficient condition for univalence of such functions. Next we consider the class S H 0 ( p ) which consists of all functions belonging to A H ( p ) and are univalent with the additional normalization f z ¯ ( 0 ) = 0 . We study the class S H 0 ( p ) in the geometric function theoretic viewpoint. As a byproduct of our investigation, we see that consideration of nonzero pole yields nontrivial lower bounds for the Taylor coefficients of such functions. We also prove a growth result for a non-vanishing subclass associated with S H 0 ( p ) .

In this note we construct the simplest unitary Riemann surface braid group representations geometrically by means of stable holomorphic vector bundles over complex tori and the prime form on Riemann surfaces. Generalised Laughlin wave functions are then introduced. The genus one case is discussed in some detail also with the help of noncommutative geometric tools, and an application of Fourier–Mukai–Nahm techniques is also given, explaining the emergence of an intriguing Riemann surface braid group duality.

In this paper we consider the solid-solid phase transformation in martensitic materials and present two numerical procedures for solving exactly the Riemann problems of a 3 i? 3 system of conservation laws 21]. A particular attention is given to the configurations of the phase boundaries. For a Riemann problem whose initial states are specified in different phases, we first assume that the phase boundary is stationary and then find the solution through an iteration method 24]. Configuration of the transition front is then determined based on this stationary-phase-boundary solution 12]. The solution with dynamic phase change is calculated by listing all the relations in the Riemann problem and solving the resulting nonlinear system. Another approach, which avoids solving this system, is also proposed where the solution is obtained by computing the intersection of two projection curves. A front capturing/tracking method 25] implementing these Riemann solvers is presented to approximate initial value problems with propagating transition fronts. This approach captures the phase boundary sharply without artificial smearing in the physically unstable region.

A new linearized characteristic upwind approximate Riemann solver is proposed based on the blending of the Riemann invariants. The scheme is based on the linearization of the differential relations along the characteristic curves of the one-dimensional hyperbolic conservation laws. It involves construction of the parameters at the cell interface such that they are consistent with the system of characteristic relations. A third order MUSCL reconstruction was used in the present work. Upwinding in the scheme is carried out by upwinding of the Riemann invariants. In this paper the development of the scheme from the compatibility relations for the Euler equations is described. The scheme was then applied to solve various cases of shock tube problems, two-dimensional Riemann problems and finally, as a practical example, it was used to carry out LES of a subsonic jet. The scheme was found to perform remarkably well for all the cases and is extremely computationally efficient, thus making it ideal for practical applications where high accuracy and high speed computation are required.

Spacetime discontinuous Galerkin finite element methods (cf. Abedi et al. (2013), Abedi et al. (2010), Miller and Haber (2008)) rely on ‘target fluxes’ on element boundaries that are computed via local one-dimensional Riemann solutions in the direction normal to the element face. In this work, we provide details of converting a space–time flux expressed in differential forms into a standard one-dimensional Riemann problem on the element interface. We then demonstrate a generalized solution procedure for linearized hyperbolic systems based on diagonalization of the governing system of partial differential equations. The generalized procedure is particularly useful for the implementation aspects of coupled multi-physics applications. We show that source terms do not influence the Riemann solution in the spacetime setting. We provide details for implementation of coordinate transformations and Riemann solutions. Exact Riemann solutions for some linearized systems of equations are provided as examples, including an exact, semi-analytic Riemann solution for generalized thermoelasticity with one relaxation time.

In this paper the Wiener–Hopf (or Riemann–Hilbert) factorization of a class of symbols important in applications is studied. The symbols in this class involve outer functions that appear in applications such as diffraction by strip gratings and infinite-dimensional integrable systems. The method proposed is based on the reduction of a vector Riemann–Hilbert to a scalar problem on an appropriate Riemann surface. Two examples are given leading to the Riemann sphere and to an elliptic curve.

The aim of this work is to further extend the notion of d-summability due to Harrison and Norton in the beginning of the 1990s. Explicit examples are given to illustrate how our notion can be applied to describe the geometry of a simply connected bounded open subset of C in a more delicate manner than the latter one. Applications on the solvability conditions for the Riemann boundary value problems for analytic functions over closed curves merely required to be summable in the generalized sense are described.

In recent years, there has been some interest in applying ideas and methods taken from Physics in order to approach several challenging mathematical problems, particularly the Riemann Hypothesis. Most of these kinds of contributions are suggested by some quantum statistical physics problems or by questions originated in chaos theory. In this article, we show that the real part of the non-trivial zeros of the Riemann zeta function extremizes the grand potential corresponding to a simple model of one-dimensional classical lattice gas, the critical point being located at 1/2 as the Riemann Hypothesis claims., Grupo de Aplicaciones Matemáticas y Estadísticas de la Facultad de Ingeniería

Assuming the Generalized Riemann Hypothesis (GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.

The Riemann problem for one dimensional generalized Chaplygin gas dynamics is considered. Its two characteristic fields are genuinely nonlinear, but the nonclassical solutions appear. The formation of mechanism for δ -shock is analyzed, that is the one-shock curve and the two-shock curve do not intersect each other in the phase plane. The Riemann solutions are constructed, and the generalized Rankine–Hugoniot conditions and the δ -entropy condition are clarified. By the interaction of the delta-shock wave with the elementary waves, the generalized Riemann problem for this system is presented. Furthermore, by studying the limits of the solutions as perturbed parameter e approaches zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data. Some numerical simulations are given to illustrate our analysis.

This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z:|z

The authors show the existence a bi univocal application between Riemann zeta functions and dynamic processes under the control of Non Integer Differential Operator. They show that, in the Fourier space, Riemann zeta function is related to hyperbolic geodesics with angles at infinity determined by the non integer parts of the power laws. The authors assert that Riemann Conjecture can be considered as a geometrical phase transition based upon the cancelation of the geometrical symmetries at infinity. A quasi self similarity of the zeta functions is associated to the self similarity of the dynamics. This characteristic assures the validity of the Riemann conjecture.

Riemann identities on theta functions are derived using properties of eigenvectors corresponding to the discrete Fourier transform Φ ( 2 ) . In particular we get various fourth order identities of classical Jacobi theta functions.

In the present paper, we study the Riemann problem for degenerate equations in which the pressure is constant over some range of stains. By the Liu-entropy condition, the Riemann solutions of twelve regions in the v-u plane are completely constructed. Moreover, we obtain the invariant regions for the Riemann solutions. (C) 2012 Elsevier Inc. All rights reserved.

A tt⁎-bundle is constructed by a harmonic map from a Riemann surface into an n-dimensional sphere. This tt⁎-bundle is a high-dimensional analogue of a quaternionic line bundle with a Willmore connection. For the construction, a flat connection is decomposed into four parts by a fiberwise complex structure.

It is proved that the generalized Riemann problem for the Aw–Rascle vehicular traffic flow with phase transitions admits a unique local solution in the class of piecewise C 1 functions and in a neighborhood of the origin this solution possesses a structure similar to the solution of the corresponding Riemann problem. Especially, when the vacuum state is involved, the Riemann solution is also well-posed, which allows us to fit experimental data well.

In direct marketing, customers are usually asked to take a specific action, and their responses are recorded over time and stored in a database. Based on the response data, we can estimate the number of customers who will ultimately respond, the number of responses anticipated to receive by a certain period of time, and the like. The goal of this article is to derive and propose several estimation methods and compare their performances in a Monte Carlo simulation. The response patterns can be described by a simple geometric function, which relates the number of responses to elapsed time. The “maximum likelihood” estimator appears to be the most effective method of estimating the parameters of this function. As we have more sample observations, the maximum likelihood estimates also converge to the true parameter values rapidly.

Article in press, corrected proof, A new concept of meromorphic $\Sigma$-factorization, for H\"{o}lder continuous functions defined on a contour $\Gamma$ that is the pullback of $\dot{\mathbb{R}}$ (or the unit circle) in a Riemann surface $\Sigma$ of genus 1, is introduced and studied, and its relations with holomorphic $\Sigma$-factorization are discussed. It is applied to study and solve some scalar Riemann-Hilbert problems in $\Sigma$ and vectorial Riemann-Hilbert problems in $\mathbb{C}$, including Wiener-Hopf matrix factorization, as well as to study some properties of a class of Toeplitz operators with $2 \times 2$ matrix symbols., Fundação para a Ciência e a Tecnologia (FCT)

Stress intensity factor solutions for spot welds in cross-tension specimens are investigated by finite element analyses. Three-dimensional finite element models are developed to obtain accurate solutions. Various ratios of sheet thickness, half specimen width and half effective specimen length to nugget radius are considered. The computational results confirm the functional dependence on the nugget radius and sheet thickness of Zhang’s analytical solutions. The results also provide three geometric functions in terms of normalized half specimen width and normalized half effective specimen length to Zhang’s analytical solutions. Based on the analytical and computational results, the dimensions of cross-tension specimens and the corresponding approximate stress intensity factor solutions are suggested.

A compact Riemann surface X of genus g > 1 which can be realized as a p-sheeted covering of the Riemann sphere for some prime p is called p-gonal. If there is an automorphism of order p on X which permutes the sheets, we call X cyclic p-gonal. Here we classify conformal actions on cyclic p-gonal Riemann surfaces of genus g > ( p − 1 ) 2 up to topological conjugacy and determine which of them can be maximal.

By making use of the well-known assertions given in Miller and Mocanu (1978) [13] and Nunokawa (1993) [14], certain theorems concerning p-valently meromorphic (strongly) starlike and (strongly) convex functions obtained in this investigation are firstly proved and then their certain consequences which will be interesting or important for analytic and geometric function theory are pointed out. (C) 2011 Elsevier Inc. All rights reserved.

The generalized Grotzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors. In this paper we study the concavity of the generalized Grotzsch function with respect to Holder means.

For integrity assessment of fuel pins of nuclear reactors using fracture mechanics technique, the solutions for fracture mechanics parameters such as stress intensity factor (SIF), η and γ functions, J – R curve are necessary. These parameters are different for different pin geometries with various types of postulated flaws. These geometries do not conform to that of ASTM standard and hence, the above-mentioned parameters are usually not available. In this work, the authors have devised a pin-loaded tension (PLT) setup to determine the geometric functions for evaluation of stress intensity factors from experimental data using two different analytical methods. A 3D finite element (FE) analysis of the whole PLT setup has been carried out to evaluate the stress intensity factors and hence to validate the analytical solutions. FE analysis is carried out for various pin geometries (with different diameter and thickness values) to evaluate the limit loads and these solutions have been used to evaluate the η and γ functions which are essential for evaluation of the J-integral. Later, a load normalization technique has been used to estimate the J – R curves of a typical fuel pin geometry using the above η and γ functions. The results have been compared with available solutions from the literature.

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.

In this paper, we study the global structure stability of the Riemann solution u = U ( x t ) for general n × n quasilinear hyperbolic systems of conservation laws under a small BV perturbation of the Riemann initial data. We prove the global existence and uniqueness of piecewise C 1 solution containing only n contact discontinuities to a class of the generalized Riemann problem, which can be regarded as a small BV perturbation of the corresponding Riemann problem, for general n × n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution u = U ( x t ) to the corresponding Riemann problem. Our result indicates that this kind of Riemann solution u = U ( x t ) mentioned above for general n × n quasilinear hyperbolic systems of conservation laws possesses a global nonlinear structure stability under a small BV perturbation of the Riemann initial data. Some applications to quasilinear hyperbolic systems of conservation laws arising in physics, particularly to the system describing the motion of the relativistic string in Minkowski space R 1 + n , are also given.

In 1975, S.M. Voronin proved the universality of the Riemann zeta-function ζ ( s ) . This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts ζ ( s + i τ ) . In the paper, we consider the functions F ( ζ ( s ) ) which are universal in the Voronin sense.

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.

We derive a geometric phase using the quantum kinematic approach within the complex quantum Hamilton–Jacobi formalism. The single valuedness of the wave function implies that the geometric phase along an arbitrary path in the complex plane must be equal to an integer multiple of 2π. The nonzero geometric phase indicates that we travel along the path through the branch cut of the phase function from one Riemann sheet to another.

Motivated by the fact that exponential and Laplace distributions have rational characteristic functions and are both geometric infinitely divisible (GID), we investigate the latter property in the context of more general probability distributions on the real line with rational characteristic functions of the form P(t)/Q (t), where P(t) = 1 + a(1)it + a(2)(it)(2) and Q (t) = 1 + b(1)it + b(2)(it)(2). Our results provide a complete characterization of the class of characteristic functions of this form, and include a description of their GID subclass. In particular, we obtain characteristic functions in the class and the subclass that are neither exponential nor Laplace. (C) 2009 Elsevier Inc. All rights reserved.

We present some results on two meromorphic functions from S to the Riemann sphere sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane., 15 pages

We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case. (C) 2009 Elsevier Inc. All rights reserved.

We propose a decomposition principle for the Riemann–Stieltjes integration and study some of its implications which lead to three distinct forms for the Riemann integral for monotonically varying differentiable functions.

In this paper, the stress intensity factor solutions for spot welds in U-shape specimens are investigated by finite element analyses. Three-dimensional finite element models are developed for U-shape specimens to obtain accurate stress intensity factor solutions. In contrast to the existing investigations of the stress intensity factor solutions based on the finite element analyses, various ratios of the sheet thickness, half specimen width, half specimen length, and corner radius to the nugget radius are considered in this investigation. The computational results confirm the functional dependence on the nugget radius and sheet thickness of Zhang’s analytical solutions. The computational results provide a geometric function in terms of the normalized half specimen width, normalized half specimen length, and normalized corner radius to Zhang’s analytical solutions. The computational results also provide a geometric function in terms of the aspect ratio of the specimen to complete Lin and Pan’s analytical solution. Finally, based on the analytical and computational results, the dimensions of U-shape specimens are suggested.

In this paper we find the maximal order of an automorphism of a trigonal Riemann surface of genus g, g5. We find that this order is smaller for generic than for cyclic trigonal Riemann surfaces, showing that generic trigonal surfaces have “less symmetry” than cyclic trigonal surfaces. Finally we prove that the maximal order is attained for infinitely many genera in both the cyclic and the generic case. Original Publication:Antonio F. Costa and Milagros Izquierdo, Maximal order of automorphisms of trigonal Riemann surfaces, 2010, Journal of Algebra, (323), 1, 27-31.http://dx.doi.org/10.1016/j.jalgebra.2009.09.041Copyright: Elsevier Science B.V., Amsterdamhttp://www.elsevier.com/

The weighted arithmetic mean of two copulas is a copula. In some cases, geometric and harmonic means also provide copulas. There are copulas specially appropriate to be combined by using weighted geometric means. With this method of construction we combine Farlie–Gumbel–Morgentern and Ali–Mikhail–Haq copulas to obtain families of copulas which can be expressed in terms of double power series. The Gumbel–Barnett copula is also considered and a new copula is proposed, which arises as the first order approximation of the weighted geometric mean of two copulas. Invariance of two multivariate distributions (Cuadras–Auge and Johnson–Kotz) by weighted geometric and arithmetic means is also studied.

The aim of this Note is to present dispersive estimates for strictly hyperbolic equations with time dependent coefficients that have integrable derivatives. We will relate the time decay rate of L p – L q norms of solutions to certain geometric indices associated to the characteristics of the limiting equation. Results will be applied to the global solvability of Kirchhoff type equations with small data, and to the dispersive estimates for their solutions. To cite this article: T. Matsuyama, M. Ruzhansky, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

We consider the reconstruction of elastic inclusions embedded inside of a planar region, bounded or unbounded, with isotropic inhomogeneous elastic parameters by measuring displacements and tractions at the boundary. We probe the medium with complex geometrical optics solutions having polynomial-type phase functions. Using these solutions we develop an algorithm to reconstruct the exact shape of a large class of inclusions including star-shaped domains and we implement numerically this algorithm for some examples.

If f is continuous on the interval [ a , b ] , g is Riemann integrable (resp. Lebesgue measurable) on the interval [ α , β ] and g ( [ α , β ] ) ⊂ [ a , b ] , then f ○ g is Riemann integrable (resp. measurable) on [ α , β ] . A well-known fact, on the other hand, states that f ○ g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a 2 c -dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f ∈ V ∖ { 0 } and g ∈ W ∖ { 0 } , f ○ g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g ∈ W ∖ { 0 } there exists a c-dimensional space V of measurable functions such that f ○ g is not measurable for all f ∈ V ∖ { 0 } .

In this paper we introduce a slight modification of the definition of migrativity for aggregation functions that allows useful characterization of this property. Among other things, in this context we prove that there are no t-conorms, uninorms or nullnorms that satisfy migrativity (with the product being the only migrative t-norm, as already shown by other authors) and that the only migrative idempotent aggregation function is the geometric mean. The k-Lipschitz migrative aggregation functions are also characterized and the product is shown to be the only 1-Lipschitz migrative aggregation function. Similarly, it is the only associative migrative aggregation function possessing a neutral element. Finally, the associativity and bisymmetry of migrative aggregation functions are discussed.