Related papers: Generalizations of the Cauchy Integral Theorems

A noncommutative-geometric generalization of classical Weil theory of characteristic classes is presented, in the conceptual framework of quantum principal bundles. A particular care is given to the case when the bundle does not admit…

Mechanics is developed over a differentiable manifold as space of possible positions. Time is considered to fill a one--dimensional Riemannian manifold, so having the metric as lapse. Then the system is quantized with covariant instead of…

We generalize Iskovskih's theorem about surfaces without irregularity and bigenus from the smooth case to regular surfaces over arbitrary fields, with special focus on the case of imperfect fields. This includes surfaces that are…

The Hamilton-Jacobi theory of Classical Mechanics can be extended in a novel manner to systems which are fuzzy in the sense that they can be represented by wave functions. A constructive interference of the phases of the wave functions then…

The circulation around any closed loop is a Lagrangian invariant for classical, smooth solutions of the incompressible Euler equations in any number of space dimensions. However, singular solutions relevant to turbulent flows need not…

We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are…

In this paper we present a detailed study of the quantum conservation laws for Toda field theories defined on the half plane in the presence of a boundary perturbation. We show that total derivative terms added to the currents, while…

The equivalence problem of curves with values in a Riemannian manifold, is solved. The domain of validity of Frenet's theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold new invariants must thus be…

Fundamental domains are found for functions defined by general Dirichlet series and using basic properties of conformal mappings the Great Riemann Hypothesis is studied.

We prove new theorems which are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from $\mathbb{C}$ to affine curves. These include results on…

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function…

For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also…

We explore the mathematical consequences of the assumption of a discrete space-time. The fundamental laws of physics have to be translated into the language of discrete mathematics. We find integral transformations that leave the lattice of…

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…

By reformulating Wang tiles with tensors, we propose a natural generalization to the probabilistic and quantum setting. In this new framework, we introduce notions of tilings and periodicity directly extending their classical counterparts.…

Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…

It is well known that a k-dimensional smooth surface in a Euclidean space cannot be tangent to a non-involutive distribution of k-dimensional planes. In this paper we discuss the extension of this statement to weaker notions of surfaces,…

Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrodinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical…

Quantum mechanics and classical mechanics are two very different theories, but the correspondence principle states that quantum particles behave classically in the limit of high quantum number. In recent years much research has been done on…

The different roles and natures of spacetime appearing in a quantum field theory and in classical physics are analyzed implying that a quantum theory of gravitation is not necessarily a quantum theory of curved spacetime. Developing an…