Complex symmetry of Toeplitz operators with continuous symbols

In this note, we consider the question of when a Toeplitz operator on the Hardy–Hilbert space \(H^2\) of the open unit disk \(\mathbb {D}\) is complex symmetric, focusing on symbols \(\phi :\mathbb {T}\rightarrow \mathbb {C}\) that are continuous on the unit circle \(\mathbb {T}=\partial \mathbb {D}\). A closed curve \(\phi \) is called nowhere winding if the winding number of \(\phi \) is 0 about every point not in the range of \(\phi \). It is then shown that if \(T_\phi \) is complex symmetric, then \(\phi \) must be nowhere winding. Hence if \(\phi \) is a simple closed curve, then \(T_\phi \) cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve \(\gamma :[a,b]\rightarrow \mathbb {C}\), it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of \(\gamma \).     

Isometric deformations of surfaces preserving the third fundamental form

We study the following problem: To what extend is a surface in the Euclidean space \(\mathbb{R}^{4}\) determined by the third fundamental form? We prove the existence of families of surfaces in \(\mathbb{R}^{4}\) which allow isometric deformations with isometric but not congruent Gaussian images. In particular, we provide a method which gives locally all surfaces in \(\mathbb{ R}^{4}\) with conformal Gauss map that allow such deformations. As a consequence, we have a way for constructing non-spherical pseudoumbilical surfaces in \(\mathbb{R}^{4}.\)     

Extending Structures for Lie Conformal Algebras

The \(\mathbb {C}[\partial ]\)-split extending structures problem for Lie conformal algebras is studied. In this paper, we introduce the definition of unified product of a given Lie conformal algebra R and a given \(\mathbb {C}[\partial ]\)-module Q. This product includes some other interesting products of Lie conformal algebras such as twisted product, crossed product, and bicrossed product. Using this product, a cohomological type object is constructed to provide a theoretical answer to the \(\mathbb {C}[\partial ]\)-split extending structures problem. Moreover, using this general theory, we investigate crossed product and bicrossed product in detail, which give the answers for the \(\mathbb {C}[\partial ]\)-split extension problem and the \(\mathbb {C}[\partial ]\)-split factorization problem respectively.     

Cohomogeneity one actions on Minkowski spaces

We study isometric cohomogeneity one actions on the \((n+1)\)-dimensional Minkowski space \(\mathbb {L}^{n+1}\) up to orbit-equivalence. We give examples of isometric cohomogeneity one actions on \(\mathbb {L}^{n+1}\) whose orbit spaces are non-Hausdorff. We show that there exist isometric cohomogeneity one actions on \(\mathbb {L}^{n+1}\), \(n \ge 3\), which are orbit-equivalent on the complement of an n-dimensional degenerate subspace \(\mathbb {W}^n\) of \(\mathbb {L}^{n+1}\) and not orbit-equivalent on \(\mathbb {W}^n\). We classify isometric cohomogeneity one actions on \(\mathbb {L}^2\) and \(\mathbb {L}^3\) up to orbit-equivalence.     

Geometric compactification of moduli spaces of half-translation structures on surfaces

In this paper, we give an equivariant compactification of the space \({\mathbb {P}}{\text {Flat}}(\Sigma )\) of homothety classes of half-translation structures on a compact, connected, orientable surface \(\Sigma \). We introduce the space \({\mathbb {P}}{\text {Mix}}(\Sigma )\) of homothety classes of mixed structures on \(\Sigma \), that are \({\text {CAT}}(0)\) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are \({\mathbb {R}}\)-trees and completions of surfaces endowed with half-translation structures. Endowing \({\text {Mix}}(\Sigma )\) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that \({\mathbb {P}}{\text {Mix}}(\Sigma )\) is an equivariant compactification of \({\mathbb {P}}{\text {Flat}}(\Sigma )\), thus allowing us to understand in a geometric way the degenerations of half-translation structures on \(\Sigma \). We finally compare our compactification to the one of Duchin–Leininger–Rafi, based on geodesic currents on \(\Sigma \), by the mean of the translation distances of the elements of the covering group of \(\Sigma \).     

On the admissible sets of type $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ over recursively saturated models

Some effective expression is obtained for the elements of an admissible set \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) as template sets. We prove the Σ-reducibility of \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) to \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\) of a regular theory, give a criterion for uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\), and establish uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{N})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\Re ')\), where \(\mathfrak{N}\) and \(\Re '\) are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each countably saturated model \(\mathfrak{M}\) of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with Σ-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.     

Regularity for Inhomogeneous Dirichlet Problems of Some Schrödinger Equations on Domains

Let \(n\ge 3, \Omega \) be a bounded, simply connected and semiconvex domain in \({\mathbb {R}}^n\) and \(L_{\Omega }:=-\Delta +V\) a Schrödinger operator on \(L^2 (\Omega )\) with the Dirichlet boundary condition, where \(\Delta \) denotes the Laplace operator and the potential \(0\le V\) belongs to the reverse Hölder class \(RH_{q_0}({\mathbb {R}}^n)\) for some \(q_0\in (\max \{n/2,2\},\infty ]\). Assume that the growth function \(\varphi :\,{\mathbb {R}}^n\times [0,\infty ) \rightarrow [0,\infty )\) satisfies that \(\varphi (x,\cdot )\) is an Orlicz function and \(\varphi (\cdot ,t)\in {\mathbb {A}}_{\infty }({\mathbb {R}}^n)\) (the class of uniformly Muckenhoupt weights). Let \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) be the Musielak–Orlicz–Hardy space whose elements are restrictions of elements of the Musielak–Orlicz–Hardy space, associated with \(L_{{\mathbb {R}}^n}:=-\Delta +V\) on \({\mathbb {R}}^n\), to \(\Omega \). In this article, the authors show that the operators \(VL^{-1}_\Omega \) and \(\nabla ^2L^{-1}_\Omega \) are bounded from \(L^1(\Omega )\) to weak-\(L^1(\Omega )\), from \(L^p(\Omega )\) to itself, with \(p\in (1,2]\), and also from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to the Musielak–Orlicz space \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself. As applications, the boundedness of \(\nabla ^2{\mathbb {G}}_D\) on \(L^p(\Omega )\), with \(p\in (1,2]\), and from \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) to \(L^\varphi (\Omega )\) or to \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) itself is obtained, where \({\mathbb {G}}_D\) denotes the Dirichlet Green operator associated with \(L_\Omega \). All these results are new even for the Hardy space \(H^1_{L_{{\mathbb {R}}^n},\,r}(\Omega )\), which is just \(H_{\varphi ,\,L_{{\mathbb {R}}^n},\,r}(\Omega )\) with \(\varphi (x,t):=t\) for all \(x\in {\mathbb {R}}^n\) and \(t\in [0,\infty )\).     

Lifting Maps from the Symmetrized Polydisc in Small Dimensions

The spectral unit ball \(\Omega _n\) is the set of all \(n\times n\) matrices M with spectral radius less than 1. Let \(\pi (M) \in \mathbb {C}^n\) stand for the coefficients of the characteristic polynomial of a matrix M (up to signs), i.e. the elementary symmetric functions of its eigenvalues. The symmetrized polydisc is \({{\mathbb {G}}}_n:=\pi (\Omega _n)\). When investigating Nevanlinna–Pick problems for maps from the disk to the spectral ball, it is often useful to project the map to the symmetrized polydisc (for instance to obtain continuity results for the Lempert function): if \(\Phi \in {\mathrm {Hol}}(\mathbb {D}, \Omega _n)\), then \(\pi \circ \Phi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\). Given a map \(\varphi \in {\mathrm {Hol}}(\mathbb {D}, {{\mathbb {G}}}_n)\), we are looking for necessary and sufficient conditions for this map to “lift through given matrices”, i.e. find \(\Phi \) as above so that \(\pi \circ \Phi = \varphi \) and \(\Phi (\alpha _j) = A_j\), \(1\le j \le N\). A natural necessary condition is \(\varphi (\alpha _j)=\pi (A_j)\), \(1\le j \le N\). When the matrices \(A_j\) are derogatory (i.e. do not admit a cyclic vector) new necessary conditions appear, involving derivatives of \(\varphi \) at the points \(\alpha _j\). We prove that those conditions are necessary and sufficient for a local lifting. We give a formula which performs the global lifting in small dimensions (\(n \le 5\)), and a counter-example to show that the formula fails in dimensions 6 and above.     

On kernels and nuclei of rank metric codes

For each rank metric code \(\mathcal {C}\subseteq \mathbb {K}^{m\times n}\), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When \(\mathcal {C}\) is \(\mathbb {K}\)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When \(\mathbb {K}\) is a finite field \(\mathbb {F}_q\) and \(\mathcal {C}\) is a maximum rank distance code with minimum distance \(d<\min \{m,n\}\) or \(\gcd (m,n)=1\), the kernel of the associated translation structure is proved to be \(\mathbb {F}_q\). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over \(\mathbb {F}_q\) must be a finite field; its right nucleus also has to be a finite field under the condition \(\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1\). Let \(\mathcal {D}\) be the DHO-set associated with a bilinear dimensional dual hyperoval over \(\mathbb {F}_2\). The set \(\mathcal {D}\) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to \(\mathbb {F}_2\). Also, its middle nucleus must be a finite field containing \(\mathbb {F}_q\). Moreover, we also consider the kernel and the nuclei of \(\mathcal {D}^k\) where k is a Knuth operation.     

The finite basis problem for infinite involution semigroups of triangular 2 $$\times $$ 2 matrices

Let \(T_n(\mathbb {F})\) and \(UT_n(\mathbb {F})\) be the semigroups of all upper triangular \(n\times n\) matrices and all upper triangular \(n\times n\) matrices with 0s and/or 1s on the main diagonal over a field \(\mathbb {F}\) with \(\mathsf {char}(\mathbb {F})=0\), respectively. In this paper, we address the finite basis problem for \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) as involution semigroups under the skew transposition. By giving a sufficient condition under which an involution semigroup is nonfinitely based, we show that both \(T_2(\mathbb {F})\) and \(UT_2(\mathbb {F})\) are nonfinitely based, and that there is a continuum of nonfinitely based involution monoid varieties between the involution monoid variety \(\mathsf {var} UT_2(\mathbb {F})\) generated by \(UT_2(\mathbb {F})\) and the involution monoid variety \(\mathsf {var} T_2(\mathbb {F})\) generated by \(T_2(\mathbb {F})\). Moreover, \(\mathsf {var} UT_2(\mathbb {F})\) cannot be defined within \(\mathsf {var} T_2(\mathbb {F})\) by any finite set of identities.     

Paley–Wiener Theorems of Generalized Fourier Transform Associated with a Cherednik Type Operator on the Real Line

We consider a singular differential-difference operator \(\Lambda \) on the real line which generalizes the Cherednik operator associated with the reflection group \(\mathbb {Z}_2\) on \(\mathbb {R}\). We establish the Paley–Wiener theorems for the generalized Fourier transform on \(\mathbb {R}\) tied to \(\Lambda \).     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

A characterization of $$\mathbb {Z}_{2}\mathbb {Z}_{2}[u]$$-linear codes

We prove that the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is exactly the class of \(\mathbb {Z}_2\)-linear codes with automorphism group of even order. Using this characterization, we give examples of known codes, e.g. perfect codes, which have a nontrivial \(\mathbb {Z}_2\mathbb {Z}_2[u]\) structure. Moreover, we exhibit some examples of \(\mathbb {Z}_2\)-linear codes which are not \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear. Also, we state that the duality of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes is the same as the duality of \(\mathbb {Z}_2\)-linear codes. Finally, we prove that the class of \(\mathbb {Z}_2\mathbb {Z}_4\)-linear codes which are also \(\mathbb {Z}_2\)-linear is strictly contained in the class of \(\mathbb {Z}_2\mathbb {Z}_2[u]\)-linear codes.     

Logarithmic and Linear Potentials of Signed Measures and Markov Property of Associated Gaussian Fields

We consider the family of finite signed measures on the complex plane \(\mathbb {C}\) with compact support, of finite logarithmic energy and with zero total mass. We show directly that the logarithmic potential of such a measure sits in the Beppo Levi space, namely, the extended Dirichlet space of the Sobolev space of order 1 over \(\mathbb {C}\), and that the half of its Dirichlet integral equals the logarithmic energy of the measure. We then derive the (local) Markov property of the Gaussian field \(\textbf {G}(\mathbb {C})\) indexed by this family of measures. Exactly analogous considerations will be made for the Beppo Levi space over the upper half plane \(\mathbb {H}\) and the Cameron-Martin space over the real line \(\mathbb {R}\). Some Gaussian fields appearing in recent literatures related to mathematical physics will be interpreted in terms of the present field \(\textbf {G}(\mathbb {C})\).     

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let \({\mathbb {K}(\mathbb {R}^{d})}\) denote the cone of discrete Radon measures on \(\mathbb {R}^{d}\). There is a natural differentiation on \(\mathbb {K}(\mathbb {R}^{d})\): for a differentiable function \(F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}\), one defines its gradient \(\nabla ^{\mathbb {K}}F\) as a vector field which assigns to each \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) an element of a tangent space \(T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))\) to \(\mathbb {K}(\mathbb {R}^{d})\) at point η. Let \(\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}\) be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on \(\mathbb {R}^{d}\). In particular, μ is a probability measure on \(\mathbb {K}(\mathbb {R}^{d})\) such that the set of atoms of a discrete measure \(\eta \in \mathbb {K}(\mathbb {R}^{d})\) is μ-a.s. dense in \(\mathbb {R}^{d}\). We consider the corresponding Dirichlet form

$$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$

Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on \(\mathbb {K}(\mathbb {R}^{d})\) which is properly associated with the Dirichlet form \(\mathcal {E}^{\mathbb {K}}\).

     

On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$, their $$\mathbb {Z}_4$$Z4 images,and new codes

In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).     

On the Connection Between Ritt Characteristic Sets and Buchberger–Gröbner Bases

For any polynomial ideal \(\mathcal {I}\), let the minimal triangular set contained in the reduced Buchberger–Gröbner basis of \(\mathcal {I}\) with respect to the purely lexicographical term order be called the W-characteristic set of \(\mathcal {I}\). In this paper, we establish a strong connection between Ritt’s characteristic sets and Buchberger’s Gröbner bases of polynomial ideals by showing that the W-characteristic set \(\mathbb {C}\) of \(\mathcal {I}\) is a Ritt characteristic set of \(\mathcal {I}\) whenever \(\mathbb {C}\) is an ascending set, and a Ritt characteristic set of \(\mathcal {I}\) can always be computed from \(\mathbb {C}\) with simple pseudo-division when \(\mathbb {C}\) is regular. We also prove that under certain variable ordering, either the W-characteristic set of \(\mathcal {I}\) is normal, or irregularity occurs for the jth, but not the \((j+1)\)th, elimination ideal of \(\mathcal {I}\) for some j. In the latter case, we provide explicit pseudo-divisibility relations, which lead to nontrivial factorizations of certain polynomials in the Buchberger–Gröbner basis and thus reveal the structure of such polynomials. The pseudo-divisibility relations may be used to devise an algorithm to decompose arbitrary polynomial sets into normal triangular sets based on Buchberger–Gröbner bases computation.     

Slope instability of projective spaces blown up along a line

Let \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) be a Kähler manifold obtained by blowing up a complex projective space \(\mathbb {P}^n\) along a line \(\mathbb {P}^1\). We prove that \(\text {Bl}_{\mathbb {P}^1} \mathbb {P}^n\) is slope unstable with respect to any polarisation, and hence, it does not admit constant scalar curvature Kähler metrics in any rational Kähler class.