Narrow Results

In this study, we examine the Umbrella matrix group in Galilean space, which possesses the stochastic matrix property (i.e. leaves the $1={\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill ...\hfill & \hfill 1\hfill \end{array}\right]}^T\in {ℝ}_1^n$ axis fixed) within matrix groups. In our examination, we first obtain the matrix Lie group and Lie algebra in $G^3$ space and then generalize this result. Furthermore, we present the Cayley formula, which gives the transition between the $SO(n)$ Lie group and Lie algebra, for the first time between the Galilean Umbrella matrix Lie group and Lie algebra. Then, we define a case of shear motion along the $1$ axis with the help of a special Galilean transformation and generate rotated surfaces in Galilean space using certain curves.

Given fusion categories of loop group representations at level $l$, we construct subfactors $N\subset M$ of depth $d$ which satisfy the following conditions :

• ⁽¹⁾$d=l$ for type $A_n,C_n,B_2$ and $\beta \cdot l\le d\le l$ for the other types with $\beta$ uniquely determined by the type.
• ⁽²⁾The fusion ring of the associated bimodules is isomorphic to the one of loop group at level $l$.

Actions of Lie groups on presymplectic manifolds are analyzed, introducing the suitable comomentum and momentum maps. The subsequent theory of reduction of presymplectic dynamical systems with symmetry is studied. In this way, we give a method of reduction which enables us to remove gauge symmetries as well as non-gauge "rigid" symmetries at once. This method is compared with other step-by-step reduction procedures. As particular examples in this framework, we discuss the reduction of time-dependent dynamical systems with symmetry, the reduction of a mechanical model of field theories with gauge and non-gauge symmetries, and the gauge reduction of the system made of a conformal particle.

In this paper, we study the restrictions of the minimal representation in the analytic continuation of the scalar holomorphic discrete series from Sp(n, ℝ) to GL⁺(n, ℝ), and from SU(n, n) to GL(n, ℂ) respectively. We work with the realizations of the representation spaces as L²-spaces on the boundary orbits of rank one of the corresponding cones, and give explicit integral operators that play the role of the intertwining operators for the decomposition. We prove inversion formulas for dense subspaces and use them to prove the Plancherel theorem for the respective decomposition. The Plancherel measure turns out to be absolutely continuous with respect to the Lebesgue measure in both cases.

We study the existence problem of proper actions of $SL(2,ℝ)$ on homogeneous spaces $G/H$ of reductive type. Based on Kobayashi’s properness criterion [T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann.285 (1989) 249–263.], we show that $G/H$ admits a proper $SL(2,ℝ)$-action via $G$ if a maximally split abelian subspace of Lie $H$ is included in the wall defined by a restricted root of Lie $G$. We also give a number of examples of such $G/H$.

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.

Let $G$ be a complex affine algebraic group, and let ${\sigma }_1$ and ${\sigma }_2$ be commuting anti-holomorphic involutions of $G$. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms $G^{{\sigma }_1}$ and $G^{{\sigma }_2}$.

Lack of any baryon number in the eightfold way model, and its intrinsic presence in the SU(3)-flavor model, has been a puzzle since the genesis of these models in 1961–1964. First we show that the conventional popular understanding of this puzzle is actually fundamentally wrong, and hence the problem being so old, begs urgently for resolution. In this paper we show that the issue is linked to the way that the adjoint representation is defined mathematically for a Lie algebra, and how it manifests itself as a physical representation. This forces us to distinguish between the global and the local charges and between the microscopic and the macroscopic models. As a bonus, a consistent understanding of the hitherto mysterious medium–strong interaction is achieved. We also gain a new perspective on how confinement arises in quantum chromodynamics.

It is well known that chiral symmetry breaking (χSB) in QCD with N_f = 2 light quark flavors can be described by orthogonal groups as O(4) → O(3), due to local isomorphisms. Here we discuss the question how specific this property is. We consider generalized forms of χSB involving an arbitrary number of light flavors of continuum or lattice fermions, in various representations. We search systematically for isomorphic descriptions by nonunitary, compact Lie groups. It turns out that there are a few alternative options in terms of orthogonal groups, while we did not find any description entirely based on symplectic or exceptional Lie groups. If we adapt such an alternative as the symmetry breaking pattern for a generalized Higgs mechanism, we may consider a Higgs particle composed of bound fermions and trace back the mass generation to χSB. In fact, some of the patterns that we encounter appear in technicolor models. In particular if one observes a Higgs mechanism that can be expressed in terms of orthogonal groups, we specify in which cases it could also represent some kind of χSB of techniquarks.

Number of zero Lyapunov exponents of a system is directly related to the dimension of the manifold of the system attractor. Moreover, this attractor dimension is governed by the algebraic structure of the manifold it lives on. In this work we try to establish a basis for the description of this manifold, which we aim to use in determining zero Lyapunov exponents of a continuous time dynamical system.

The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically, solving the fractional Schrödinger equation based on the Riemann and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

We study the classical flat full causal bulk viscous Friedmann–Robertson–Walker (FRW) cosmological model through the factorization method. The method shows that there exists a relationship between the viscosity parameter s and the parameter γ entering the equations of state of the model. Also, the factorization method allows to find some new exact parametric solutions for different values of the viscous parameter s. Special attention is given to the well-known case s = 1/2, for which the cosmological model admits scaling symmetries. Furthermore, some exact parametric solutions for s = 1/2 are obtained through the Lie group method.

We find fundamental solutions in closed form for a family of parabolic equations with two spatial variables, whose symmetry groups had been determined in an earlier paper by Finkel [12]. We show how these results can be applied in finance to yield closed form solutions for special affine and quadratic two factor term structure models as well as a new class of models with inverse square behavior. The latter can be considered a partial extension to two factors of pricing models related to the Bessel process devised by Albanese and Campolieti [3] and Albanese et al. [2]. A by-product of our results is that Lie's reduction method in this setting leads only to fundamental solutions that can be factorized as products of functions that depend jointly on time and on one spatial coordinate. Thus all the results in this paper extend immediately to n factor models.

We propose an extension of the transform approach to option pricing introduced in Duffie, Pan and Singleton (Econometrica68(6) (2000) 1343–1376) and in Carr and Madan (Journal of Computational Finance2(4) (1999) 61–73). We term this extension the "coherent state transform" approach, it applies when the Markov generator of the factor process can be decomposed as a linear combination of generators of a Lie symmetry group. Then the family of group invariant coherent states determine the transform to price derivatives. We exemplify this procedure deriving a coherent state transform for affine jump-diffusion processes with positive state space. It improves the traditional FFT because inversion of the latter requires integration over an unbounded domain, while inversion of the coherent state transform requires integration over unit ball. We explicitly perform the pricing exercise for some contracts like the plain vanilla options on (credit) risky bonds and on the spread option.

Let G be a Lie group equipped with a set of left invariant vector fields. These vector fields generate a function ξ on Wiener space into G via the stochastic version of Cartan's rolling map. It is shown here that, for any smooth function f with compact support, f(ξ) is Malliavin differentiable to all orders and these derivatives belong to L^p(μ) for all p > 1, where μ is Wiener measure.

We present the subalgebra structure of 𝔰𝔩(3, 𝕆), a particular real form of 𝔢₆ chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group 𝔰𝔩(3, 𝕆) to construct the multiplication table of the corresponding Lie algebra 𝔰𝔩(3, 𝕆). Both the multiplication table and the group are then utilized to find various nested chains of subalgebras of 𝔰𝔩(3, 𝕆), in which the corresponding Cartan subalgebras are also nested where possible. Because our construction involves the Lie group, we simultaneously obtain an explicit representation of the corresponding nested chains of subgroups of SL(3, 𝕆).

The purpose of this paper is to study certain geometric properties of generalized quadrangle $(\mathrm{\Omega },𝔏)$ of type $O_6^{-}(2)$. We define certain root elements which generate a Lie algebra of type $E_6(K)$ for fields $K$ of characteristic two. The construction will be mainly based on the geometric properties of the generalized quadrangle $(\mathrm{\Omega },𝔏)$. In fact, we will explicity construct a Chevalley base of this Lie algebra.

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this paper we discuss the construction of frames for L²(ℝⁿ) using the action of closed reductive subgroups H ⊆ GL(n,ℝ) such that H has an open orbit in ℝⁿ under the action (h,ω) ↦ (h^-1)^T(ω). In particular, we show that if is an affine symmetric space, then there is a canonical way to construct frames.

In this paper we are concerned with the continuous shearlet transform in arbitrary space dimensions where the shear operation is of Toeplitz type. In particular, we focus on the construction of associated shearlet coorbit spaces and on atomic decompositions and Banach frames for these spaces.

Unitary matrices are the quantum gates in quantum computing. We study the question under which conditions is the commutator of two unitary matrices again a unitary matrix. We consider the general case and a complete solution for the 2 × 2 unitary matrices. We also consider Kronecker products and direct sums of unitary matrices.