Monge–Ampère measures on pluripolar sets

In this article we solve the complex Monge–Ampère problem for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko?odziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge–Ampère measure, then it is a complex Monge–Ampère measure.     

Visibility of the higher-dimensional central projection into the projective sphere

We can describe higher-dimensional classical spaces by analytical projective geometry, if we embed the d-dimensional real space onto a d + 1-dimensional real projective metric vector space. This method allows an approach to Euclidean, hyperbolic, spherical and other geometries uniformly [8]. To visualize d-dimensional solids, it is customary to make axonometric projection of them. In our opinion the central projection gives more information about these objects, and it contains the axonometric projection as well, if the central figure is an ideal point or an s-dimensional subspace at infinity. We suggest a general method which can project solids into any picture plane (space) from any central figure, complementary to the projection plane (space). Opposite to most of the other algorithms in the literature, our algorithm projects higher-dimensional solids directly into the two-dimensional picture plane (especially into the computer screen), it does not use the three-dimensional space for intermediate step. Our algorithm provides a general, so-called lexicographic visibility criterion in Definition and Theorem 3.4, so it determines an extended visibility of the d-dimensional solids by describing the edge framework of the two-dimensional surface in front of us. In addition we can move the central figure and the image plane of the projection, so we can simulate the moving position of the observer at fixed objects on the computer screen (see first our figures in reverse order). Supported by DAAD 2008 Multimedia Technology for Mathematics and Computer Science Education.     

Statistical approximation properties of high order operators constructed with the Chan-Chyan-Srivastava polynomials

In this paper, by including high order derivatives of functions being approximated, we introduce a general family of the linear positive operators constructed by means of the Chan-Chyan-Srivastava multivariable polynomials and study a Korovkin-type approximation result with the help of the concept of A-statistical convergence, where A is any non-negative regular summability matrix. We obtain a statistical approximation result for our operators, which is more applicable than the classical case. Furthermore, we study the A-statistical rates of our approximation via the classical modulus of continuity.     

A Quantum Version of the Désarménien Matrix

We use elements in the quantum hyperalgebra to define a quantum version of the Désarménien matrix. We prove that our matrix is upper triangular with ones on the diagonal and that, as in the classical case, it gives a quantum straightening algorithm for quantum bideterminants. We use our matrix to give a new proof of the standard basis theorem for the q-Weyl module. As well, we show that the standard basis for the q-Weyl module and the basis dual to the standard basis for the q-Schur module are related by the quantum Désarménien matrix.     

The Levi Monge-Ampère equation: Smooth regularity of strictly Levi convex solutions

We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in ℂⁿ⁺¹, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface ℂⁿ⁺¹ and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.     

The idealized quantum two-slit gedanken experiment revisited—Criticism and reinterpretation

An idealized two-slit experiment is envisaged in which the hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space–time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Gödel’s undecidability theorem, we proceed to show that space–time is very likely to be akin to a fuzzy Kähler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space–time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space–time from the Riemannian space of general relativity.The paper’s main philosophy is to emphasize that the quintessence of the two-slit experiment as well as Feynman’s path integral could be given a different interpretation by altering our classical concept of space–time geometry and topology. In turn this would be in keeping with the development in theoretical physics since special and subsequently general relativity. In the final analysis it would seem that we have two different yet, from a positivistic philosophy viewpoint, completely equivalent alternatives to view quantum physics. Either we insist on what we see in our daily experiences, namely, a smooth four-dimensional space–time, and then accept, whether we like it or not, things such as probability waves and complex probabilities. Alternatively, we could see behind the façade of classical space–time a far more elaborate and highly complex fuzzy space–time with infinite hierarchical dimensions such as the so-called Fuzzy K3 or E–Infinity space–time and as a reward for this imaginative picture we can return to real probabilities without a phase and an almost classical picture with the concept of a particle’s path restored. We say almost classical because non-linear dynamics and deterministic chaos have long shown the central role of randomness in classical mechanics and this is reinforced once more in our model which is directly related not to Newtonian motion, but rather to a diffusion-like random walk similar to that used with great skill by Einstein and later on by Nagasawa and particularly the English-Canadian physicist Garnet Ord.     

A modified functional delta method and its application to the estimation of risk functionals

The classical functional delta method (FDM) provides a convenient tool for deriving the asymptotic distribution of statistical functionals from the weak convergence of the respective empirical processes. However, for many interesting functionals depending on the tails of the underlying distribution this FDM cannot be applied since the method typically relies on Hadamard differentiability w.r.t. the uniform sup-norm. In this article, we present a version of the FDM which is suitable also for nonuniform sup-norms, with the outcome that the range of application of the FDM enlarges essentially. On one hand, our FDM, which we shall call the modified FDM, works for functionals that are “differentiable” in a weaker sense than Hadamard differentiability. On the other hand, it requires weak convergence of the empirical process w.r.t. a nonuniform sup-norm. The latter is not problematic since there exist strong respective results on weighted empirical processes obtained by Shorack and Wellner (1986) [25], Shao and Yu (1996) [23], Wu (2008) [32], and others. We illustrate the gain of the modified FDM by deriving the asymptotic distribution of plug-in estimates of popular risk measures that cannot be treated with the classical FDM.     

Weighted integral inequalities for conjugate A-harmonic tensors

We first prove a local weighted integral inequality for conjugate A-harmonic tensors. Then, as an application of our local result, we prove a global weighted integral inequality for conjugate A-harmonic tensors in L^s(μ)-averaging domains, which can be considered as a generalization of the classical result. Finally, we give applications of the above results to quasiregular mappings.     

The characterization of certain quantum stochastic stationary process

In this paper we will obtain a Stone type theorem under the frame of Hilbert C＊-module, such that the classical Stone theorem is our special case. Then we use it as a main tool to obtain a spectrum decomposition theorem of certain stationary quantum stochastic process. In the end, we will give it an interpretation in statistical mechanics of multi-linear response.     

Connections Between Classical and Generalised Trajectories of a State/Signal System

In our earlier article “Well-posed state/signal systems in continuous time”, we originally defined the notion of a trajectory of a state/signal system by means of a generating subspace. However, it was left as an open problem whether the generating subspace is uniquely determined by a given family of all generalised trajectories of a well-posed state/signal system. In this article we give a positive answer to this question and show how this insight simplifies some formulations in the theory of well-posed state/signal systems. The main contribution of the article is an explicit convolution scheme for constructing classical trajectories approximating an arbitrary generalised trajectory. We apply this scheme by studying relationships between classical and generalised trajectories of continuous-time state/signal systems under very weak assumptions. Among others, we show that there exists a space of classical trajectories that is invariant under differentiation and dense in the space of generalised trajectories. Some of our results generalise known results for strongly continuous semigroups and input/state/output systems, but we make no use of decompositions of the signal space into an input space and an output space, and in particular, none of our results depend on well-posedness.     

The Conway–Sloane tetralattice pairs are non-isometric

Conway and Sloane constructed a 4-parameter family of pairs of isospectral lattices of rank four. They conjectured that all pairs in their family are non-isometric, whenever the parameters are pairwise different, and verified this for classical integral lattices of determinant up to 10⁴. In this paper, we use our theory of lattice invariants to prove this conjecture.     

Adaptive Galerkin boundary element methods with panel clustering

In this paper, we will propose a boundary element method for solving classical boundary integral equations on complicated surfaces which, possibly, contain a large number of geometric details or even uncertainties in the given data. The (small) size of such details is characterised by a small parameter and the regularity of the solution is expected to be low in such zones on the surface (which we call the wire-basket zones). We will propose the construction of an initial discretisation for such type of problems. Afterwards standard strategies for boundary element discretisations can be applied such as the h, p, and the adaptive hp-version in a straightforward way. For the classical boundary integral equations, we will prove the optimal approximation results of our so-called wire-basket boundary element method and discuss the stability aspects. Then, we construct the panel-clustering and -matrix approximations to the corresponding Galerkin BEM stiffness matrix. The method is shown to have an almost linear complexity with respect to the number of degrees of freedom located on the wire basket.     

On Weak Generalized Stability and (c,d)-Pseudostable Random Variables via Functional Equations

In this paper we give a first attempt to define and study stable distributions with respect to the weak generalized convolution, focusing our attention on the symmetric weakly stable distribution. As in the case of the classical convolution, characterization of distributions stable in the sense of weak generalized convolution depends on solving some functional equations in the class of characteristic functions. This paper was partially written while the second author was a visiting professor of Delft Institute of Applied Mathematics, Delft University of Technology, Holland.     

Categorical Geometry and Integration Without Points

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419–489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical σ-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function μ with values in [0,1] can be extended to a measure on an abstract σ-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.     

The K-Theory of Heegaard-Type Quantum 3-Spheres

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C^*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C^*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. Dedicated to the memory of Olaf Richter An erratum to this article is available at .     

Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup

For holomorphic modular forms on tube domains, there are two types of known Fourier expansions, i.e. the classical Fourier expansion and the Fourier-Jacobi expansion. Either of them is along a maximal parabolic subgroup. In this paper, we discuss Fourier expansion of holomorphic modular forms on tube domains of classical type along the minimal parabolic subgroup. We also relate our Fourier expansion to the two known ones in terms of Fourier coefficients and theta series appearing in these expansions.     

Linear conic formulations for two-party correlations and values of nonlocal games

In this work we study the sets of two-party correlations generated from a Bell scenario involving two spatially separated systems with respect to various physical models. We show that the sets of classical, quantum, no-signaling and unrestricted correlations can be expressed as projections of affine sections of appropriate convex cones. As a by-product, we identify a spectrahedral outer approximation to the set of quantum correlations which is contained in the first level of the Navascués, Pironio and Acín (NPA) hierarchy and also a sufficient condition for the set of quantum correlations to be closed. Furthermore, by our conic formulations, the value of a nonlocal game over the sets of classical, quantum, no-signaling and unrestricted correlations can be cast as a linear conic program. This allows us to show that a semidefinite programming upper bound to the classical value of a nonlocal game introduced by Feige and Lovász is in fact an upper bound to the quantum value of the game and moreover, it is at least as strong as optimizing over the first level of the NPA hierarchy. Lastly, we show that deciding the existence of a perfect quantum (resp. classical) strategy is equivalent to deciding the feasibility of a linear conic program over the cone of completely positive semidefinite matrices (resp. completely positive matrices). By specializing the results to synchronous nonlocal games, we recover the conic formulations for various quantum and classical graph parameters that were recently derived in the literature.     

Quantum families of quantum group homomorphisms

The notion of a quantum family of maps has been introduced in the framework of C*-algebras. As in the classical case, one may consider a quantum family of maps preserving additional structures (e.g. quantum family of maps preserving a state). In this paper, we define a quantum family of homomorphisms of locally compact quantum groups. Roughly speaking, we show that such a family is classical. The purely algebraic counterpart of the discussed notion, i.e. a quantum family of homomorphisms of Hopf algebras, is introduced and the algebraic counterpart of the aforementioned result is proved. Moreover, we show that a quantum family of homomorphisms of Hopf algebras is consistent with the counits and coinverses of the given Hopf algebras. We compare our concept with weak coactions introduced by Andruskiewitsch and we apply it to the analysis of adjoint coaction.     

Entangled Markov chains

Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated and uniquely determined by a corresponding classical Markov chain. We argue that this construction yields as a corollary, a solution to the problem of constructing quantum analogues of classical random walks which are “entangled” in a sense specified in the paper.The formula giving the joint correlations of these quantum chains is obtained from the corresponding classical formula by replacing the usual matrix multiplication by Schur multiplication.The connection between Schur multiplication and entanglement is clarified by showing that these quantum chains are the limits of vector states whose amplitudes, in a given basis (e.g. the computational basis of quantum information), are complex square roots of the joint probabilities of the corresponding classical chains. In particular, when restricted to the projectors on this basis, the quantum chain reduces to the classical one. In this sense we speak of entangled lifting, to the quantum case, of a classical Markov chain. Since random walks are particular Markov chains, our general construction also gives a solution to the problem that motivated our study.In view of possible applications to quantum statistical mechanics too, we prove that the ergodic type of an entangled Markov chain with finite state space (thus excluding random walks) is completely determined by the corresponding ergodic type of the underlying classical chain. Mathematics Subject Classification (2000) Primary 46L53, 60J99; Secondary 46L60, 60G50, 62B10     

Optimal bounds from below of the critical load for elastic solids subject to uniaxial compression

In the recent papers [1,2] we studied a new procedure based on the Korn inequality for determining sufficient conditions for the Hadamard stability, aimed at determining optimal lower bound estimates for the critical load in bifurcation problems. Here, we discuss the effectiveness of our approach for the classical representative problem of uniaxial compression of a Mooney-Rivlin circular cylinder. We find that our lower bound estimate is effective and advantageous for applications, since it is easily implementable in numerical codes. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)