In this work we show that if the frame property of a Gabor frame with window in Feichtinger's algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger's algebra and any separable lattice of density , , cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

We provide several new -congruences for truncated basic hypergeometric series with the base being an even power of . Our results mainly concern congruences modulo the square or the cube of a cyclotomic polynomial and complement corresponding ones of an earlier paper containing -congruences for truncated basic hypergeometric series with the base being an odd power of . We also give a number of related conjectures including -congruences modulo the fifth power of a cyclotomic polynomial and a congruence for a truncated ordinary hypergeometric series modulo the seventh power of a prime greater than 3.

We establish a new family of -supercongruences modulo the fourth power of a cyclotomic polynomial, and give several related results. Our main ingredients are -microscoping and the Chinese remainder theorem for polynomials.

We study the evolution equations for a regularized version of Dirac-harmonic maps from closed Riemannian surfaces. We establish the existence of a global weak solution for the regularized problem, which is smooth away from finitely many singularities. Moreover, we discuss the convergence of the evolution equations and address the question if we can remove the regularization in the end.

We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.

Given a pair of real functions (, ), we study the conditions they must satisfy for to be the curvature in the arc-length of a closed planar curve for all real . Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

In this paper, a couple of -supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new -analogue of the (E.2) supercongruence by Van Hamme, another one is a new -analogue of a supercongruence by Swisher, while the other results are closely related -supercongruences. The proofs make use of special cases of a very-well-poised summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.

We construct optimal flat functions in Carleman-Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn'kin. Finally, we discuss some examples (including the well-known -Gevrey case) where such optimal flat functions can be obtained in a more explicit way.

Cone-nets are conjugate nets on a surface such that along each individual curve of one family of parameter curves there is a cone in tangential contact with the surface. The corresponding conjugate curve network is projectively invariant and is characterized by the existence of particular transformations. We study properties of that transformation theory and illustrate how several known surface classes appear within our framework. We present cone-nets in the classical smooth setting of differential geometry as well as in the context of a consistent discretization with counterparts to all relevant statements and notions of the smooth setting. We direct special emphasis towards smooth and discrete tractrix surfaces which are characterized as principal cone-nets with constant geodesic curvature along one family of parameter curves.

We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold , we associate an integer-valued function, called , measuring the extent to which fails to be cylindrical. In particular, we show that if the degree is constant and equal to , then the singularities of can only occur along an -dimensional "striction" submanifold. This result allows us to extend the standard classification of developable surfaces in to the whole family of flat and ruled submanifolds without planar points, also known as : an open and dense subset of every rank-one submanifold is the union of , , and regions.

We study polynomial identities satisfied by the mutation product on the underlying vector space of an associative algebra , where ,  are fixed elements of . We simplify known results for identities in degree 4, proving that only two identities are necessary and sufficient to generate them all; in degree 5, we show that adding one new identity suffices; in degree 6, we demonstrate the existence of a significant number of new identities, which induce us to conjecture that the variety generated by mutation algebras of associative algebras is not finitely based.

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for -minimal hypersurfaces and the spectrum of the Hodge Laplacian.

We study the existence of continuous (linear) operators from the Banach spaces of Lipschitz functions on infinite metric spaces vanishing at a distinguished point and from their predual spaces onto certain Banach spaces, including ()-spaces and the spaces and . For pairs of spaces and () we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space contains a bilipschitz copy of the unit sphere of the space , then admits a continuous operator onto and hence onto . Using this, we provide several conditions for a space implying that is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space has the Schur property if and only if for every complete discrete metric space with cardinality () the spaces and are weakly sequentially homeomorphic.