Subconvexity for sup-norms of cusp forms on $$\mathrm{PGL}(n)$$

Let F be an \(L^2\)-normalized Hecke Maaß cusp form for \(\Gamma _0(N) \subseteq {\mathrm{SL}}_{n}({\mathbb {Z}})\) with Laplace eigenvalue \(\lambda _F\). If \(\Omega \) is a compact subset of \(\Gamma _0(N)\backslash {\mathrm{PGL}}_n/\mathrm{PO}_{n}\), we show the bound \(\Vert F|_{\Omega }\Vert _{\infty } \ll _{ \Omega } N^{\varepsilon } \lambda _F^{n(n-1)/8 - \delta }\) for some constant \(\delta = \delta _n> 0\) depending only on n.     

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 two-weight $$\mathbb {Z}_{2^k}$$-codes

We determine the possible homogeneous weights of regular projective two-weight codes over \(\mathbb {Z}_{2^k}\) of length \(n>3\), with dual Krotov distance \(d^{\lozenge }\) at least four. The determination of the weights is based on parameter restrictions for strongly regular graphs applied to the coset graph of the dual code. When \(k=2\), we characterize the parameters of such codes as those of the inverse Gray images of \(\mathbb {Z}_4\)-linear Hadamard codes, which have been characterized by their types by several authors.     

There is exactly one $${\mathbb {Z}}_2{\mathbb {Z}}_4$$-cyclic 1-perfect code

Let \(\mathcal{C}\) be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive code of length \(n > 3\). We prove that if the binary Gray image of \(\mathcal{C}\) is a 1-perfect nonlinear code, then \(\mathcal{C}\) cannot be a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic code except for one case of length \(n=15\). Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive 1-perfect code gives a \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-additive extended 1-perfect code. We also prove that such a code cannot be \({\mathbb {Z}}_2{\mathbb {Z}}_4\)-cyclic.     

On Submanifolds with 2-Type Pseudo-Hyperbolic Gauss Map in Pseudo-Hyperbolic Space

In this paper, we study pseudo-Riemannian submanifolds of a pseudo-hyperbolic space \(\mathbb H^{m-1}_s (-1) \subset \mathbb E^m_{s+1}\) with 2-type pseudo-hyperbolic Gauss map. We give a characterization of proper pseudo-Riemannian hypersurfaces in \(\mathbb H^{n+1}_s (-1) \subset \mathbb E^{n+2}_{s+1}\) with non-zero constant mean curvature and 2-type pseudo-hyperbolic Gauss map. For \(n=2\), we prove classification theorems. In addition, we show that the hyperbolic Veronese surface is the only maximal surface fully lying in \(\mathbb H^4_2 (-1) \subset \mathbb H^{m-1}_2 (-1)\) with 2-type pseudo-hyperbolic Gauss map. Moreover, we prove that a flat totally umbilical pseudo-Riemannian hypersurface \(M^n_t\) of the pseudo-hyperbolic space \(\mathbb {H}^{n+1}_t(-1) \subset \mathbb E^{n+2}_{t+1}\) has biharmonic pseudo-hyperbolic Gauss map.     

Some results on linear codes over the ring $$\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4$$

In this paper, we mainly study the theory of linear codes over the ring \(R =\mathbb {Z}_4+u\mathbb {Z}_4+v\mathbb {Z}_4+uv\mathbb {Z}_4\). By using the Chinese Remainder Theorem, we prove that R is isomorphic to a direct sum of four rings. We define a Gray map \(\Phi \) from \(R^{n}\) to \(\mathbb {Z}_4^{4n}\), which is a distance preserving map. The Gray image of a cyclic code over R is a linear code over \(\mathbb {Z}_4\). We also discuss some properties of MDS codes over R. Furthermore, we study the MacWilliams identities of linear codes over R and give the generator polynomials of cyclic codes over R.     

Space–Time Fractional Equations and the Related Stable Processes at Random Time

In this paper, we consider the general space–time fractional equation of the form \(\sum _{j=1}^m \lambda _j \frac{\partial ^{\nu _j}}{\partial t^{\nu _j}} w(x_1, \ldots , x_n ; t) = -c^2 \left( -\varDelta \right) ^\beta w(x_1, \ldots , x_n ; t)\), for \(\nu _j \in \left( 0,1 \right] \) and \(\beta \in \left( 0,1 \right] \) with initial condition \(w(x_1, \ldots , x_n ; 0)= \prod _{j=1}^n \delta (x_j)\). We show that the solution of the Cauchy problem above coincides with the probability density of the n-dimensional vector process \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), where \(\varvec{S}_n^{2\beta }\) is an isotropic stable process independent from \(\mathcal {L}^{\nu _1, \ldots , \nu _m}(t)\), which is the inverse of \(\mathcal {H}^{\nu _1, \ldots , \nu _m} (t) = \sum _{j=1}^m \lambda _j^{1/\nu _j} H^{\nu _j} (t)\), \(t>0\), with \(H^{\nu _j}(t)\) independent, positively skewed stable random variables of order \(\nu _j\). The problem considered includes the fractional telegraph equation as a special case as well as the governing equation of stable processes. The composition \(\varvec{S}_n^{2\beta } \left( c^2 \mathcal {L}^{\nu _1, \ldots , \nu _m} (t) \right) \), \(t>0\), supplies a probabilistic representation for the solutions of the fractional equations above and coincides for \(\beta = 1\) with the n-dimensional Brownian motion at the random time \(\mathcal {L}^{\nu _1, \ldots , \nu _m} (t)\), \(t>0\). The iterated process \(\mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t)\), \(t>0\), inverse to \(\mathfrak {H}^{\nu _1, \ldots , \nu _m}_r (t) =\sum _{j=1}^m \lambda _j^{1/\nu _j} \, _1H^{\nu _j} \left( \, _2H^{\nu _j} \left( \, _3H^{\nu _j} \left( \ldots \, _{r}H^{\nu _j} (t) \ldots \right) \right) \right) \), \(t>0\), permits us to construct the process \(\varvec{S}_n^{2\beta } \left( c^2 \mathfrak {L}^{\nu _1, \ldots , \nu _m}_r (t) \right) \), \(t>0\), the density of which solves a space-fractional equation of the form of the generalized fractional telegraph equation. For \(r \rightarrow \infty \) and \(\beta = 1\), we obtain a probability density, independent from t, which represents the multidimensional generalization of the Gauss–Laplace law and solves the equation \(\sum _{j=1}^m \lambda _j w(x_1, \ldots , x_n) = c^2 \sum _{j=1}^n \frac{\partial ^2}{\partial x_j^2} w(x_1, \ldots , x_n)\). Our analysis represents a general framework of the interplay between fractional differential equations and composition of processes of which the iterated Brownian motion is a very particular case.     

Rotational Surfaces in $$S^{3}$$ with Constant Mean Curvature

There is a two-parametric family of rotational symmetric CMC surfaces; more precisely, for every real number H and every \(C\ge 2(H+\sqrt{1+H^2})\) there is a rotational symmetry surface \(\Sigma _{H,C}\) with mean curvature H. Perdomo (Asian J Math 14:73–108, 2010) showed that for every H between \(\cot \left( \frac{\pi }{m}\right) \) and \(\frac{m^2-2}{2\sqrt{m^2-1}}\) there exists an embedded rotational symmetric example with non-constant principal curvatures that is invariant under the cyclic group \(Z_m\). Recently Andrews and Li (J Differ Geom 99:169–189, 2015) showed that these embedded CMC tori are the only embedded genus 1 surfaces with CMC on the sphere. In this paper we complete the study of this family of CMC surfaces and we show that for every integer \(m>2\), there is a properly immersed example in this family that contains a great circle and is invariant under the cyclic group \(Z_m\). We will say that these examples contain the axis of symmetry. We also show that every non-isoparametric surface \(\Sigma _{H,C}\) is either properly immersed and invariant under the cyclic group \(Z_m\) for some integer \(m>1\) or it is dense in the region bounded by two isoparametric tori if the surface \(\Sigma _{H,C}\) does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface \(\Sigma _{H,C}\) contains the axis of symmetry.     

Estimates for coefficients of certain <Emphasis Type="Italic">L</Emphasis>-functions

Let \(\pi _{\varphi }\) (or \(\pi _{\psi }\)) be an automorphic cuspidal representation of \(\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})\) associated to a primitive Maass cusp form \(\varphi \) (or \(\psi \)), and \(\mathrm{sym}^j \pi _{\varphi }\) be the jth symmetric power lift of \(\pi _{\varphi }\). Let \(a_{\mathrm{sym}^j \pi _{\varphi }}(n)\) denote the nth Dirichlet series coefficient of the principal L-function associated to \(\mathrm{sym}^j \pi _{\varphi }\). In this paper, we study first moments of Dirichlet series coefficients of automorphic representations \(\mathrm{sym}^3 \pi _{\varphi }\) of \(\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})\), and \(\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }\) of \(\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})\). For \(3 \le j \le 8\), estimates for \(|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|\) on average over a short interval have also been established.     

On the Dilation of Truncated Toeplitz Operators

An operator \(S_{\varphi ,\psi }^{u}\in \mathcal {L}(L^2)\) is called the dilation of a truncated Toeplitz operator if for two symbols \(\varphi ,\psi \in L^{\infty }\) and an inner function u,

$$\begin{aligned} S_{\varphi ,\psi }^{u}f=\varphi P_uf+\psi Q_uf \end{aligned}$$

holds for \(f\in {L}^{2}\) where \(P_{u}\) denotes the orthogonal projection of \(L^2\) onto the model space \(\mathcal { K}_{u}^2=H^2{\ominus }{{u}H^2}\) and \(Q_u=I-P_u.\) In this paper, we study properties of the dilation of truncated Toeplitz operators on \(L^{2}\). In particular, we provide conditions for the dilation of truncated Toeplitz operators to be normal. As some applications, we give several examples of such operators.

     

The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$

Let \(\mathbb {F}_{p^m}\) be a finite field of cardinality \(p^m\), where p is a prime, and k, N be any positive integers. We denote \(R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\cdots +u^{k-1}F_{p^m}\) (\(u^k=0\)) and \(\lambda =a_0+a_1u+\cdots +a_{k-1}u^{k-1}\) where \(a_0, a_1,\ldots , a_{k-1}\in F_{p^m}\) satisfying \(a_0\ne 0\) and \(a_1=1\). Let r be a positive integer satisfying \(p^{r-1}+1\le k\le p^r\). First we define a Gray map from \(R_k\) to \(F_{p^m}^{p^r}\), then prove that the Gray image of any linear \(\lambda \)-constacyclic code over \(R_k\) of length N is a distance preserving linear \(a_0^{p^r}\)-constacyclic code over \(F_{p^m}\) of length \(p^rN\). Furthermore, the generator polynomials for each linear \(\lambda \)-constacyclic code over \(R_k\) of length N and its Gray image are given respectively. Finally, some optimal constacyclic codes over \(F_{3}\) and \(F_{5}\) are constructed.     

The shape of $$ \mathbb {Z}/\ell \mathbb {Z}$$-number fields

Let \(\ell \) be a prime and let \(L/ \mathbb {Q}\) be a Galois number field with Galois group isomorphic to \( \mathbb {Z}/\ell \mathbb {Z}\). We show that the shape of L, see Definition 1.2, is either \(\frac{1}{2}\mathbb {A}_{\ell -1}\) or a fixed sub-lattice depending only on \(\ell \); such a dichotomy in the value of the shape only depends on the type of ramification of L. This work is motivated by a result of Bhargava and Shnidman, and a previous work of the first named author, on the shape of \( \mathbb {Z}/3 \mathbb {Z}\) number fields.     

The nonlocal Liouville-type equation in $$\mathbb {}$$ and conformal immersions of the disk with boundary singularities

In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to

$$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$

(1)

with \(K_k\) bounded in \(L^\infty \) and \(e^{u_k}\) bounded in \(L^1\) uniformly with respect to k, we show that up to extracting a subsequence \(u_k\) can blow-up at (at most) finitely many points \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\), or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–Merle and Li–Shafrir on the 2-dimensional Liouville equation, but providing sharp quantization estimates (\(\alpha _j=\pi \) and \(\alpha _j\ge \pi \)) which are not known in dimension 2 under the weak assumption that \((K_k)\) be bounded in \(L^\infty \) and is allowed to change sign.

     

Unitary Equivalence of Composition C$$^*$$-Algebras on the Hardy and Weighted Bergman Spaces

Let \(\varphi \) be an arbitrary linear-fractional self-map of the unit disk \({\mathbb {D}}\) and consider the composition operator \(C_{-1, \varphi }\) and the Toeplitz operator \(T_{-1,z}\) on the Hardy space \(H^2\) and the corresponding operators \(C_{\alpha , \varphi }\) and \(T_{\alpha , z}\) on the weighted Bergman spaces \(A^2_{\alpha }\) for \(\alpha >-1\). We prove that the unital C\(^*\)-algebra \(C^*(T_{\alpha , z}, C_{\alpha , \varphi })\) generated by \(T_{\alpha , z}\) and \(C_{\alpha , \varphi }\) is unitarily equivalent to \(C^*(T_{-1, z}, C_{-1, \varphi }),\) which extends a known result for automorphism-induced composition operators. For maps \(\varphi \) that are not automorphisms of \({\mathbb {D}}\), we show that \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })\) is unitarily equivalent to \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})\), where \({\mathcal {K}}_{\alpha }\) and \({\mathcal {K}}_{-1}\) denote the ideals of compact operators on \(A^2_{\alpha }\) and \(H^2\), respectively, and apply existing structure theorems for \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}\) to describe the structure of \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}\), up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps \(\varphi \) that fix a point on the unit circle.     

Appell polynomial sequences with respect to some differential operators

We present a study of a specific kind of lowering operator, herein called \(\Lambda \), which is defined as a finite sum of lowering operators and might be presented by various configurations. We characterize the polynomial sequences fulfilling an Appell relation with respect to \(\Lambda \), and considering a concrete cubic decomposition of a simple Appell sequence, we prove that the polynomial component sequences are \(\Lambda \)-Appell, with \(\Lambda \) defined as previously, although by a three term sum. Ultimately, we prove the non-existence of orthogonal polynomial sequences which are also \(\Lambda \)-Appell, when \(\Lambda \) is the lowering operator \(\Lambda =a_{0}D+a_{1}DxD+a_{2}\left( Dx\right) ^2D\), where \(a_{0}\), \(a_{1}\) and \(a_{2}\) are constants and \(a_{2} \ne 0\). The case where \(a_{2}=0\) and \(a_{1} \ne 0\) is also naturally recaptured.     

A partition relation for pairs on $$\omega ^{\omega ^\omega }$$

We consider colorings of the pairs of a family \(\mathcal {F}\subseteq {{\mathrm{FIN}}}\) of topological type \(\omega ^{\omega ^k}\), for \(k>1\); and we find a homogeneous family \(\mathcal {G}\subseteq \mathcal {F}\) for each coloring. As a consequence, we complete our study of the partition relation \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,m}}\) identifying \(\omega ^{\omega ^\omega }\) as the smallest ordinal space \(\alpha <\omega _1\) satisfying \({\forall l>1,\, \alpha \rightarrow ({{\mathrm{top}}}\;\omega ^2+1)^2_{l,4}}\).     

On the Chaoticity of Some Tensor Product Weighted Backward Shift Operators Acting on Some Tensor Product Fock–Bargmann Spaces

In Advances in Mathematical Physics (2011) we showed that the weighted shift \(z^{p}\frac{d^{p+1}}{dz^{p+1}} (p=0, 1, 2,\ldots )\) acting on classical Bargmann space \(\mathbb {B}_{p}\) is chaotic operator. In Journal of Mathematical physics (2014), we constructed an chaotic weighted shift \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1} (p=0, 1, 2,\ldots )\) on some lattice Fock–Bargmann \(\mathbb {E}_{p}^{\alpha }\) generated by the orthonormal basis \( {e_{m}^{(\alpha ,p)}(z) = e_{m}^{\alpha } ; m=p, p+1,\ldots }\) where \( {e_{m}^{\alpha }(z) = (\frac{2\nu }{\pi })^{1/4}e^{\frac{\nu }{2}z^{2}}e^{-\frac{\pi ^{2}}{\nu }(m +\alpha )^{2} +2i\pi (m +\alpha )z}; m \in \mathbb {N}}\) with \(\nu , \alpha \) are real numbers; \(\nu > 0\), \(\mathbb {M}\) is an weighted shift and \(\mathbb {M^{*}}\) is the adjoint of the \(\mathbb {M}\). In this paper we study the chaoticity of tensor product \(\mathbb {M}^{*^{p}}\mathbb {M}^{p+1}\otimes z^{p}\frac{d^{p}}{dz^{p+1}} (p=0, 1, 2, \ldots )\) acting on \(\mathbb {E}_{p}^{\alpha }\otimes \mathbb {B}_{p}\).     

Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

Properties of singular integral operators $$\varvec{S}_{\varvec{\alpha ,\beta }}$$

For \(\alpha , \beta \in L^{\infty } (S^1),\) the singular integral operator \(S_{\alpha ,\beta }\) on \(L^2 (S^1)\) is defined by \(S_{\alpha ,\beta }f:= \alpha Pf+\beta Qf\), where P denotes the orthogonal projection of \(L^2(S^1)\) onto the Hardy space \(H^2(S^1),\) and Q denotes the orthogonal projection onto \(H^2(S^1)^{\perp }\). In a recent paper, Nakazi and Yamamoto have studied the normality and self-adjointness of \(S_{\alpha ,\beta }\). This work has shown that \(S_{\alpha ,\beta }\) may have analogous properties to that of the Toeplitz operator. In this paper, we study several other properties of \(S_{\alpha ,\beta }\).     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.