Narrow Results

We show that the orbit space of Euclidean space, under the action of a closed and connected Lie group of isometries is homeomorphic to a plane or closed half-plane, if the action is of cohomogeneity two.

In this paper, we investigate analytic symbols $\psi$ and $\phi$ when the weighted composition operator $C_{\psi ,\phi }$ is complex symmetric on general function space $H_s$. As applications, we characterize completely the compactness, normality and isometry of complex symmetric weighted composition operators. Especially, we show that the equivalence of compactness and Hilbert–Schmidtness, and the existence of non-normal complex symmetric operators for such operators, which answers one open problem raised by Noor in [On an example of a complex symmetric composition operators on $H^2(𝔻)$, J. Funct. Anal.269 (2015) 1899–1901] for higher dimensional case.

We study the large and the small isometry groups of Kellendonk–Savinien spectral triples associated to a choice function for a self-similar compact ultrametric Cantor set; in particular, we show that under reasonable assumptions they coincide. To characterize these isometry groups, we use the Michon rooted weighted tree associated to an ultrametric Cantor set: the small and large isometry groups turn out to be equal to the subgroup of the automorphism group of the tree consisting of elements that commute with the choice function in a suitable sense. When the rooted tree is binary, these isometry groups are equal to ${\mathbb{Z}}_2$. We also examine examples of Connes metrics associated to Pearson–Bellissard spectral triples, presenting a gamut of cases in where it is infinite.

In this paper, we study CR maps between hyperquadrics and Winkelmann hypersurfaces. Based on a previous study on the CR Ahlfors derivative of Lamel–Son and a recent result of Huang–Lu–Tang–Xiao on CR maps between hyperquadrics, we prove that a transversal CR map from a hyperquadric into a hyperquadric or a Winkelmann hypersurface extends to a local holomorphic isometric embedding with respect to certain Kähler metrics if and only if the Hermitian part of its CR Ahlfors derivative vanishes on an open set of the source. Our proof is based on relating the geometric rank of a CR map into a hyperquadric and its CR Ahlfors derivative.

The two-dimensional minimal supersymmetric sigma models with homogeneous target spaces $G/H$ and chiral fermions of the same chirality are revisited. In particular, we look into the isometry anomalies in $\mathrm{\text{O}}(N)$ and $\mathrm{\text{CP}}(N-1)$ models. These anomalies are generated by fermion loop diagrams which we explicitly calculate. In the case of $\mathrm{\text{O}}(N)$ sigma models the first Pontryagin class vanishes, so there is no global obstruction for the minimal ${𝒩}=(0,1)$ supersymmetrization of these models. We show that at the local level isometries in these models can be made anomaly free by specifying the counterterms explicitly. Thus, there are no obstructions to quantizing the minimal ${𝒩}=(0,1)$ models with the $S^{N-1}=\mathrm{\text{SO}}(N)/\mathrm{\text{SO}}(N-1)$ target space while preserving the isometries. This also includes $\mathrm{\text{CP}}(1)$ (equivalent to $S^2$) which is an exceptional case from the $\mathrm{\text{CP}}(N-1)$ series. For other $\mathrm{\text{CP}}(N-1)$ models, the isometry anomalies cannot be rescued even locally, this leads us to a discussion on the relation between the geometric and gauged formulations of the $\mathrm{\text{CP}}(N-1)$ models to compare the original of different anomalies. A dual formalism of $\mathrm{\text{O}}(N)$ model is also given, in order to show the consistency of our isometry anomaly analysis in different formalisms. The concrete counterterms to be added, however, will be formalism dependent.

We report a holographic study of a two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the cases of nonvanishing and vanishing cosmological constants. Our result shows that the boundary theory of the two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the case of nonvanishing cosmological constants is the Schwarzian term coupled to a dilaton field, while for the case of vanishing cosmological constant, a theory does not have a kinetic term. We also include the higher derivative term $R^2$, where $R$ is the scalar curvature that is coupled to a dilaton field. We find that the form of the boundary theory is not modified perturbatively. Finally, we show that a lattice holographic picture is realized up to the second-order perturbation of boundary cutoff ${𝜖}^2$ under a constant boundary dilaton field and the nonvanishing cosmological constant by identifying the lattice spacing $a$ of a lattice Schwarzian theory with the boundary cutoff $𝜖$ of the two-dimensional dilaton gravity theory.

We use the $\pi$-orbital axis vector (POAV) analysis to deal with large curvature effect of graphene in the tight-binding model. To test the validities of pseudo-magnetic fields (PMFs) derived from the tight-binding model and the model with Dirac equation coupled to a curved surface, we propose two types of spatially constant-field topographies for strongly-curved graphene nanobubbles, which correspond to these two models, respectively. It is shown from the latter model that the PMF induced by any spherical graphene nanobubble is always equivalent to the magnetic field caused by one magnetic monopole charge distributed on a complete spherical surface with the same radius. Such a PMF might be attributed to the isometry breaking of a graphene layer attached conformably to a spherical substrate with adhesion.

Let K be a two-bridge knot in S³. Then K is also denoted as the four-plat, b(p,q) to indicate its association with some rational number p/q.

The lens space L = L(p,q) admits an isometry, τ, of order 2, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K.

In this paper, the isometries of these orbifolds are classified; this is equivalent to computing the isometries of (S³,K).

When $C\subseteq {𝔽}^n$ is a linear code over a finite field $𝔽$, every linear Hamming isometry of $C$ to itself is the restriction of a linear Hamming isometry of ${𝔽}^n$ to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping $C$ to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.

The monomial transformations mapping $C$ to itself form a group $\mathrm{\text{rMon}}(C)$, and the additive Hamming isometries form a larger group $\mathrm{\text{Isom}}(C)$: $\mathrm{\text{rMon}}(C)\subseteq \mathrm{\text{Isom}}(C)$. The main result says that these two subgroups can be as different as possible: for any two subgroups $H_1\subseteq H_2$, subject to some natural necessary conditions, there exists an additive code $C$ such that $\mathrm{\text{rMon}}(C)=H_1$ and $\mathrm{\text{Isom}}(C)=H_2$.

We construct a family of compact hyperbolic 3-manifolds with totally geodesic boundary, depending on three integer parameters. Then we determine geometric presentations of the fundamental groups of these manifolds and prove that they are cyclic coverings of the 3-ball branched along a specified tangle with two components. Finally, we give a classification of these manifolds up to homeomorphism (resp., isometry), and determine their isometry groups.

It is shown that the connected stable rank of a unital Banach _*-algebra containing two isometries with orthogonal ranges is infinity. Several consequences as its applications and variations are also obtained.

We introduce the class of $n$-quasi-$m$-isometric operators on Hilbert space. This generalizes the class of $m$-isometric operators on Hilbert space introduced by Agler and Stankus. An operator $T\in B(H)$ is said to be $n$-quasi-$m$-isometric if

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on ${{\mathscr{H}}}^2(𝔻)$ and ${{𝒜}}_{\alpha }^2(𝔻),$ the composition operator on ${{𝒮}}^2(𝔻)$ induced by an analytic self-map on $𝔻$ with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on $H^p$, thesis, University of Toledo (1969)] well-known result on ${{𝒜}}_{\alpha }^2(𝔻)$ which characterizes the almost multiplicative operator on ${{\mathscr{H}}}^2(𝔻).$

We discuss the construction of quantum stochastic integrals over the positive plane. Non-commutative representations for the quasi-free CAR and CCR settings are presented extending results obtained previously for martingales over the two parameter plane.

We offer a short invitation to elementary hyperbolic plane geometry. We first examine the contents of Book I of Euclid's Elements and obtain a hyperbolic plane from the Euclidean one by negating Euclid's parallel postulate and keeping all of his other axioms. Then we explore the fundamentals of hyperbolic plane geometry, and study the structure of its isometries. Finally, we obtain certain identities involving the isometries and evaluate them in the upper half-plane model to derive some trigonometric laws for hyperbolic triangles.