Narrow Results

Rota–Baxter algebras are important in probability, combinatorics, associative Yang–Baxter equation and splitting of algebras. This paper studies the formal deformations of Rota–Baxter algebra morphisms. As a consequence, we develop a cohomology theory of Rota–Baxter algebra morphisms to interpret the lower degree cohomology groups as formal deformations. Finally, we prove the cohomology comparison theorem of Rota–Baxter algebra morphisms, i.e. the cohomology of a morphism of Rota–Baxter algebras is isomorphic to the cohomology of an auxiliary Rota–Baxter algebra.

The concept of Rota–Baxter family algebra is a generalization of Rota–Baxter algebra. It appears naturally in the algebraic aspects of renormalizations in quantum field theory. Rota–Baxter family algebras are closely related to dendriform family algebras. In this paper, we first construct an $L_{\infty }$-algebra whose Maurer–Cartan elements correspond to Rota–Baxter family algebra structures. Using this characterization, we define the cohomology of a given Rota–Baxter family algebra. As an application of our cohomology, we study formal and infinitesimal deformations of a given Rota–Baxter family algebra. Next, we define the notion of a homotopy Rota–Baxter family algebra structure on a given $A_{\infty }$-algebra. We end this paper by considering the homotopy version of dendriform family algebras and their relations with homotopy Rota–Baxter family algebras.

We introduce a new family of deformations of the Witt algebra , F varying in the space of all polynomials with vanishing constant terms, and show the existence of an isomorphism of its formal and analytic completions with those of the Witt algebra. Central extensions of this algebra are considered and the existence of an isomorphism between their formal and analytic completions with those of the Virasoro algebra is proved.

We construct new examples of Kobayashi hyperbolic hypersurfaces in ℙ⁴. They are generic projections of the triple symmetric product V = C (3) of a generic genus g ≥ 6 curve C, smoothly embedded in ℙ⁷.

We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.

We prove that the gonality among the smooth curves in a complete linear system on a K3 surface is constant except for the Donagi–Morrison example. This was proved by Ciliberto and Pareschi under the additional condition that the linear system is ample. The constancy was originally conjectured by Harris and Mumford.

As a consequence we prove that exceptional curves on K3 surfaces satisfy the Eisenbud–Lange–Martens–Schreyer conjecture and explicitly describe such curves. They turn out to be natural extensions of the Eisenbud–Lange–Martens–Schreyer examples of exceptional curves on K3 surfaces.

A characteristic class for deformations of foliations called the Fuks–Lodder–Kotschick class (FLK class) is studied. It seems unknown if there is a real foliation with nontrivial FLK class. In this paper, we show some conditions to assure the triviality of the FLK class. On the other hand, we show that the FLK class is easily to be nontrivial for transversely holomorphic foliations. We present an example and give a construction which generalizes it.

An approach is proposed enabling to effectively describe the behavior of a bosonic system. The approach uses the quantum group GL_p,q(2) formalism. Indeed, considering a bosonic Hamiltonian in terms of the GL_p,q(2) generators, it is shown that its thermodynamic properties are connected to deformation parameters p and q. For instance, the average number of particles and the pressure have been computed. If p is fixed to be the same value for q, our approach coincides perfectly with some results developed recently in this subject. The ordinary results, of the present system, can be found when we take the limit p = q = 1.

The ground-state properties of Zr and Sn isotopes are studied within the relativistic mean field theory. Zr and Sn isotopes have received tremendous attention due to various reasons, including the predicted giant halos in the neutron-rich Zr isotopes, the unique feature of being robustly spherical in the region of ¹⁰⁰Sn~¹³²Sn and the particular interest of Sn isotopes to nuclear astrophysics. Furthermore, four (semi-) magic neutron numbers, 40, 50, 82 and 126, make these two isotopic chains particularly important to test the pairing correlations and the deformations in a microscopic model. In the present work, we carry out a systematic study of Zr and Sn isotopes from the proton drip line to the neutron drip line with deformation effects, pairing correlations and blocking effects for nuclei with odd number of neutrons properly treated. A constrained calculation with quadrupole deformations is performed to find the absolute minimum for each nucleus on the deformation surface. All ground-state properties, including the separation energies, the odd–even staggerings, the nuclear radii, the deformations and the single-particle spectra are analyzed and discussed in detail.

Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor λ_a and Ramond Grassmannian vector in the spinor realization⁴² are proposed. The map of supergravity background into composite supercoordinates: valid up to the second order corrections in deformation parameter h and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector and to vanish for the θ components with the same chiralities. The deformations in the (super)twistor picture are discussed.

The ground state energies, sizes and deformations of 1897 even–even nuclei with 10≤Z ≤110 have been carried out by using the Relativistic Mean Field (RMF) model. In the present calculations, the nonlinear RMF force NL3* recent refitted version of the NL3 force has been used. The BCS (Bardeen–Cooper–Schrieffer) formalism with constant gap approximation has been taken into account for pairing correlations. The predictions of RMF model for the ground state properties of some nuclei have been discussed in detail.

The Muon Spectrometer of the ATLAS experiment is a large and complex system relying on several detector technologies. The simulation and reconstruction of muon events require a very careful description of these detectors, some of which provide a muon trigger while the others make precision measurements of tracks. A precise description of the passive materials, toroidal magnet systems and shield is also needed to account for Coulomb scattering and energy losses. Finally, the spectrometer performance is enhanced by the alignment of its precision chambers, so the detector description model must fully implement misalignments and deformations. We present the Detector Description chain employed in the Muon system and its integration in the ATLAS software framework. It relies on a database technology and a standard set of geometrical primitives common to all ATLAS subsystems. The Muon Detector Description has been used successfully in the context of the ATLAS Data Challenges, where it provides a unique and coherent geometry source for the simulation and reconstruction algorithms. It has also been validated in the context of the experimental program of the ATLAS testbeams, where analyses of the treatment of chamber alignment in track reconstruction relies crucially upon the detector description model.

The Lie algebra of the group SU₂ is constructed from two deformed oscillator algebras for which the deformation parameter is a root of unity. This leads to an unusual quantization scheme, the {J², U_r} scheme, an alternative to the familiar {J², J_z} quantization scheme corresponding to common eigenvectors of the Casimir operator J² and the Cartan operator J_z. A connection is established between the eigenvectors of the complete set of commuting operators {J², U_r} and mutually unbiased bases in spaces of constant angular momentum.

A simple recipe for generating a complete set of mutually unbiased bases in dimension 2j + 1, with 2j integer and 2j + 1 prime, is developed from a single matrix V_a acting on a space of constant angular momentum j and defined in terms of the irreducible characters of the cyclic group C_{2j + 1}. This recipe yields an (apparently new) compact formula for the vectors spanning the various mutually unbiased bases. In dimension (2j + 1)^e, with 2j integer, 2j + 1 prime and e positive integer, the use of direct products of matrices of type V_a makes it possible to generate mutually unbiased bases. As two pending results, the matrix V_a is used in the derivation of a polar decomposition of SU(2) and of a FFZ algebra.

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

In this note, we consider low-dimensional metric Leibniz algebras with an invariant inner product over the complex numbers up to dimension 5. We study their deformations, and give explicit formulas for the cocycles and deformations. We identify among those the metric deformations.

We deal with local density approximations for the kinetic and exchange energy term, ℰ_kin(ρ) and ℰ_ex(ρ), of a periodic Coulomb model. We study asymptotic approximations of the energy when the number of particles goes to infinity and for densities close to the constant averaged density. For the kinetic energy, we recover the usual combination of the von-Weizsäcker term and the Thomas–Fermi term. Furthermore, we justify the inclusion of the Dirac term for the exchange energy and the Slater term for the local exchange potential.

Using HF+BCS method with Skyrme forces we analyze the neutron drip line. It is shown that the drip line may form stability peninsulas. These peninsulas locate along fixed neutron numbers on the nuclear chart, which correspond to magic and new magic numbers, and are the same for all Skyrme forces. It is found that the size of the peninsulas is sensitive to the choice of Skyrme forces and the most extended peninsulas appear with the SkI2 set.

Quantitative studies of shapes in the real world are often usefully enhanced by applications of morphometrics, a branch of multivariate statistics organized to exploit the origin of such data in real Euclidean geometry. This paper reviews one morphometric tool that has recently been designed for analysis of corresponding biological outlines of similar shape, such as arise from organ or tissue boundaries in medical images. The analysis is a hybrid of two techniques, Procrustes analysis and the thin-plate spline, the properties of which are already well-understood separately. The thin-plate spline optimizes one shape metric, the Procrustes methods, another. Combining them leads to an algorithm for averaging groups of outlines, informative displays of sample variation around these averages in full detail, and rigorous multivariate statistical detection and testing of scientific hypotheses linking those patterns to their putative causes or effects. The new tool is demonstrated using outlines of the corpus callosum (connection of the cerebral hemispheres) in midsagittal images of 12 normal human brains and 13 brains of schizophrenics.

We define two families of deformations of probability measures depending on the second free cumulants and the corresponding new associative convolutions arising from the conditionally free convolution. These deformations do not commute with dilation of measures, which means that the limit theorems cannot be obtained as a direct application of the theorems for the conditionally free case. We calculate the general form of the central and Poisson limit theorems. We also find the explicit form for three important examples.