Hopf bifurcation and multiple periodic solutions in Lotka–Volterra systems with symmetries

The purpose of this paper is to study Hopf bifurcations in a delayed Lotka–Volterra system with dihedral symmetry. By treating the response delay as bifurcation parameter and employing equivariant degree method, we obtain the existence of multiple branches of nonconstant periodic solutions through a local Hopf bifurcation around an equilibrium. We find that competing coefficients and the response delay in the system can affect the spatio-temporal patterns of bifurcating periodic solutions. According to their symmetric properties, a topological classification is given for these periodic solutions. Furthermore, an estimation is presented on minimal number of bifurcating branches. These theoretical results are helpful to better understand the complex dynamics induced by response delays and symmetries in Lotka–Volterra systems.     

The Fermionic Observable in the Ising Model and the Inverse Kac–Ward Operator

We show that the critical Kac–Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac–Ward operator on any planar graph is given by what we call a fermionic generating function. We also present a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac–Ward operators on isoradial graphs.     

Hamiltonian dynamics of a spaceship in Alcubierre and Gödel metrics: Recursion operators and underlying master symmetries

Theoretical and Mathematical Physics - We study the Hamiltonian dynamics of a spaceship in the background of Alcubierre and Gödel metrics. We derive the Hamiltonian vector fields governing the...     

SU(3) Color Gauge Invariance and the Jaffe–Witten Mass Gap in QCD

We discuss the role of group theory, the theory of distributions, and some theorems of the theory of functions of complex variable in connection with the so-called Jaffe–Witten mass gap in QCD, which is responsible for the large-scale structure of the QCD ground state and hence plays a dominant role in QCD as a theory of strong interactions at low energies. We show how the mass gap may appear without violating the SU(3) color gauge invariance of QCD. The theory of generalized functions (distributions) and the Weierstrass–Sochocki–Casorati theorem are used in order to perform the renormalization of the regularized mass gap.     

How to find solutions,Lie symmetries,and conservation laws of forced Korteweg–de Vries equations in optimal way

It is shown that the forced Korteweg–de Vries (KdV) equation studied in the recent papers [A.H. Salas, Computing solutions to a forced KdV equation, Nonlinear Anal. RWA 12 (2011) 1314–1320] and [M.L. Gandarias, M.S. Bruzón, Some conservation laws for a forced KdV equation, Nonlinear Anal. RWA 13 (2012) 2692–2700] is reduced to the classical (constant-coefficient) KdV equation by point transformations for all values of variable coefficients. The equivalence-based approach proposed in [R.O. Popovych, O.O. Vaneeva, More common errors in finding exact solutions of nonlinear differential equations: part I, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 3887–3899] allows one to obtain more results in a much simpler way.     

Deformation Features and Models of [±45]2s Cross-Ply Fiber-Reinforced Plastics in Tension

Mechanics of Composite Materials - Series of experiments on [±45]2s cross-ply carbon-fiber-reinforced plastic specimens were carried out in tension with various loading programs. In analyzing...     

Whitham Hierarchy and Generalized Picard–Fuchs Operators in the N=2 Susy Yang–Mills Theory for Classical Gauge Groups

Theoretical and Mathematical Physics - We derive infinitely many meromorphic differentials based on the fractional powers of the superpotential arising from hyperelliptic curves. We obtain various...     

Local and Global Analytic Solutions for a Class of Characteristic Problems of the Einstein Vacuum Equations in the “Double Null Foliation Gauge”

The main goal of this work consists in showing that the analytic solutions for a class of characteristic problems for the Einstein vacuum equations have an existence region much larger than the one provided by the Cauchy–Kowalevski theorem due to the intrinsic hyperbolicity of the Einstein equations. To prove this result we first describe a geometric way of writing the vacuum Einstein equations for the characteristic problems we are considering, in a gauge characterized by the introduction of a double null cone foliation of the spacetime. Then we prove that the existence region for the analytic solutions can be extended to a larger region which depends only on the validity of the a priori estimates for the Weyl equations, associated with the “Bel-Robinson norms”. In particular, if the initial data are sufficiently small we show that the analytic solution is global. Before showing how to extend the existence region we describe the same result in the case of the Burger equation, which, even if much simpler, nevertheless requires analogous logical steps required for the general proof. Due to length of this work, in this paper we mainly concentrate on the definition of the gauge we use and on writing in a “geometric” way the Einstein equations, then we show how the Cauchy–Kowalevski theorem is adapted to the characteristic problem for the Einstein equations and we describe how the existence region can be extended in the case of the Burger equation. Finally, we describe the structure of the extension proof in the case of the Einstein equations. The technical parts of this last result is the content of a second paper.     

The angular derivative of Fréchet-holomorphic maps in J*-algebras and complex Hilbert spaces

We study the problem of the existence of the angular derivative of Fréchet-holomorphic maps of the generalized right half-planes defined by non-zero partial isometries in infinite dimensional complex Banach spaces called J^*-algebras. These generalized right half-planes and open unit balls (which are bounded symmetric homogeneous domains) are holomorphically equivalent. The results obtained here also hold in C ^*-algebras, JC^*-algebras, B ^*-algebras and ternary algebras, containing non-zero partial isometries, and in complex Hilbert spaces. Some examples are given. The principal tool we use are general results of the Pick-Julia type in J^*-algebras.     

General Relativity in the Making, 1916–1918: Rudolf J. Humm in Göttingen and Berlin

Mathematische Semesterberichte - In 1916, the Swiss student Rudolf J. Humm arrived in Göttingen to study relativity theory under Hilbert. He enrolled in his courses, attended Klein’s...     

A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) ×?U(4) Yang-Mills in 4D

A Clifford Cl(5, C) Unified Gauge Field Theory of Conformal Gravity, Maxwell and U(4) × U(4) Yang-Mills in 4D is rigorously presented extending our results in prior work. The Cl(5, C) = Cl(4, C) ?Cl(4, C){Cl(5, C) = Cl(4, C) \oplus Cl(4, C)} algebraic structure, behind the 4D (complexified) Conformal Gravity-Maxwell and U(4) × U(4) Yang-Mills unification program advanced in this work, is such that it encodes the direct group product U(2, 2) × U(4) × U(4) = [SU(2, 2)] _spacetime × [U(1) × U(4) × U(4)] _internal and which does not violate the Coleman-Mandula theorem because the spacetime symmetries (conformal group SU(2, 2) in the absence of a mass gap, Poincaré group when there is mass gap) do not mix with the internal symmetries. Similar considerations apply to the supersymmetric case when the symmetry group structure is given by the direct product of the superconformal group (in the absence of a mass gap) with an internal symmetry group so that the Haag-Lopuszanski-Sohnius theorem is not violated. A generalization of the de Sitter and Anti de Sitter gravitational theories based on the gauging of the Cl(4, 1, R), Cl(3, 2, R) algebras follows. We conclude with a few remarks about the complex extensions of the Metric Affine theories of Gravity (MAG) based on GL(4, C) × _s C ⁴, the realizations of twistors and the N{\mathcal{N}} = 1 superconformal su(2, 2|1) algebra purely in terms of Clifford algebras and their plausible role in Witten’s formulation of perturbative N{\mathcal{N}} = 4 super Yang- Mills theory in terms of twistor-string variables.