On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

Left annihilator of generalized derivations on Lie ideals in prime rings

Let \(R\) be a prime ring, \(L\) a noncentral Lie ideal of \(R\), \(F\) a generalized derivation with associated nonzero derivation \(d\) of \(R\). If \(a\in R\) such that \(a(d(u)^{l_1} F(u)^{l_2} d(u)^{l_3} F(u)^{l_4} \ldots F(u)^{l_k})^{n}=0\) for all \(u\in L\), where \(l_1,l_2,\ldots ,l_k\) are fixed non negative integers not all are zero and \(n\) is a fixed integer, then either \(a=0\) or \(R\) satisfies \(s_4\), the standard identity in four variables.     

Two definite integrals involving products of four Legendre functions

The definite integrals \( \int _{-1}^1x[P_\nu (x)]^4\mathrm{d}x\) and \( \int _{0}^1x[P_\nu (x)]^2\{[P_\nu (x)]^2-[P_\nu (-x)]^2\}\mathrm{d}x\) are evaluated in closed form, where \( P_\nu \) stands for the Legendre function of degree \( \nu \in \mathbb C\). Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green’s functions and Epstein zeta functions.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

Some results on generalized strong external difference families

A generalized strong external difference family (briefly \((v, m; k_1,\dots ,k_m; \lambda _1,\dots ,\lambda _m)\)-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we give some nonexistence results for GSEDFs. In particular, we prove that a \((v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)\)-GSEDF does not exist when \(k_1+k_2+k_3< v\). We also give a first recursive construction for GSEDFs and prove that if there is a \((v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )\)-GSEDF, then there is a \((vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )\)-GSEDF with \(v>1\), \(t>1\) and \(v\equiv t\equiv 1\pmod 2\). Then we use it to obtain some new GSEDFs for \(m=2\). In particular, for any prime power q with \(q\equiv 1\pmod 4\), we show that there exists a \((qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})\)-GSEDF, where \(t=p_1p_2\dots p_n\), \(p_i>1\), \(1\le i\le n\), \(p_1, p_2,\dots ,p_n\) are odd integers.     

Newly appointed editors

Let k be a field and \(k(x_0,\ldots ,x_{p-1})\) be the rational function field of p variables over k where p is a prime number. Suppose that \(G=\langle \sigma \rangle \simeq C_p\) acts on \(k(x_0,\ldots ,x_{p-1})\) by k-automorphisms defined as \(\sigma :x_0\mapsto x_1\mapsto \cdots \mapsto x_{p-1}\mapsto x_0\). Denote by P the set of all prime numbers and define \(P_0=\{p\in P:\mathbb {Q}(\zeta _{p-1})\) is of class number one\(\}\) where \(\zeta _n\) a primitive n-th root of unity in \(\mathbb {C}\) for a positive integer n; \(P_0\) is a finite set by Masley and Montgomery (J Reine Angew Math 286/287:248–256, 1976). Theorem. Let k be an algebraic number field and \(P_k=\{p\in P: p\) is ramified in \(k\}\). Then \(k(x_0,\ldots ,x_{p-1})^G\) is not stably rational over k for all \(p\in P\backslash (P_0\cup P_k)\).     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).     

On limit theorems for Banach-space-valued linear processes

Let \( {\left( {{\epsilon_i}} \right)_{i \in \mathbb{Z}}} \) be i.i.d. random elements in a separable Banach space \( \mathbb{E} \), and let \( \mathop {\left( {{a_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) be continuous linear operators from \( \mathbb{E} \) to a Banach space \( \mathbb{F} \) such that \( \sum\nolimits_{i \in \mathbb{Z}} {\left\| {{a_i}} \right\|} \) is finite. We prove that the linear process \( \mathop {\left( {{X_n}} \right)}\nolimits_{n \in \mathbb{Z}} \) defined by \( {X_n}: = \sum\nolimits_{i \in \mathbb{Z}} {{a_i}} \left( {{\epsilon_{n - i}}} \right) \) inherits from \( \mathop {\left( {{\epsilon_i}} \right)}\nolimits_{i \in \mathbb{Z}} \) the central limit theorem and functional central limit theorems in various Banach spaces of \( \mathbb{F} \)-valued functions, including Hölder spaces.     

ANNIHILATORS OF HIGHEST WEIGHT $$ \mathfrak{s}\mathfrak{l} $$(∞)-MODULES

We give a criterion for the annihilator in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) of a simple highest weight \( \mathfrak{s}\mathfrak{l} \)(∞)-module to be nonzero. As a consequence we show that, in contrast with the case of \( \mathfrak{s}\mathfrak{l} \)(n), the annihilator in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) of any simple highest weight \( \mathfrak{s}\mathfrak{l} \)(∞)-module is integrable, i.e., coincides with the annihilator of an integrable \( \mathfrak{s}\mathfrak{l} \)(∞)-module. Furthermore, we define the class of ideal Borel subalgebras of \( \mathfrak{s}\mathfrak{l} \)(∞), and prove that any prime integrable ideal in U(\( \mathfrak{s}\mathfrak{l} \)(∞)) is the annihilator of a simple \( \mathfrak{b} \) ⁰-highest weight module, where \( \mathfrak{b} \) ⁰ is any fixed ideal Borel subalgebra of \( \mathfrak{s}\mathfrak{l} \)(∞). This latter result is an analogue of the celebrated Duoflo Theorem for primitive ideals.     

On Weighted Average Interpolation with Cardinal Splines

Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\), Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\). In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\), whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\). The case of even degree \(d\) is also studied.     

On the family of elliptic curves $$\varvec{y^2=x^3-m^2x+p^2}$$

In this paper, we study the torsion subgroup and rank of elliptic curves for the subfamilies of \(E_{m,p} : y^2=x^3-m^2x+p^2\), where m is a positive integer and p is a prime. We prove that for any prime p, the torsion subgroup of \(E_{m,p}(\mathbb {Q})\) is trivial for both the cases {\(m\ge 1\), \(m\not \equiv 0\pmod 3\)} and {\(m\ge 1\), \(m \equiv 0 \pmod 3\), with \(gcd(m,p)=1\)}. We also show that given any odd prime p and for any positive integer m with \(m\not \equiv 0\pmod 3\) and \(m\equiv 2\pmod {32}\), the lower bound for the rank of \(E_{m,p}(\mathbb {Q})\) is 2. Finally, we find curves of rank 9 in this family.     

On left-invariant Hörmander operators in $ {\mathbb{R}^N} $. applications to Kolmogorov–Fokker–Planck equations

If \( \mathcal{L} = \sum\limits_{j = 1}^m {X_j^2} + {X_0} \) is a Hörmander partial differential operator in \( {\mathbb{R}^N} \), we give sufficient conditions on the \( {X_{{j^{\text{S}}}}} \) for the existence of a Lie group structure \( \mathbb{G} = \left( {{\mathbb{R}^N},*} \right) \), not necessarily nilpotent, such that \( \mathcal{L} \) is left invariant on \( \mathbb{G} \). We also investigate the existence of a global fundamental solution Γ for \( \mathcal{L} \), providing results that ensure a suitable left-invariance property of Γ. Examples are given for operators \( \mathcal{L} \) to which our results apply: some are new; some have appeared in recent literature, usually quoted as Kolmogorov–Fokker–Planck-type operators. Nontrivial examples of homogeneous groups are also given.     

CCZ equivalence of power functions

Let \(F\simeq {{\mathrm{GF}}}(p^n)\) be a finite field of characteristic p and \(p_k\) and \(p_\ell \) be power functions on F defined by \(p_k(x)=x^k\) and \(p_\ell (x)=x^\ell \) respectively. We show, that \(p_k\) and \(p_\ell \) are CCZ equivalent, if and only if there exists a positive integer \(0\le a< n\), such that \(\ell \equiv p^a k \pmod {p^n-1}\) or \(k\ell \equiv p^a \pmod {p^n-1}\).     

Calabi-Yau manifolds with isolated conical singularities

Let \(X\) be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let \(L\) be an ample line bundle on \(X\). Assume that the pair \((X,L)\) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point \(x \in X\) there exist a Kähler-Einstein Fano manifold \(Z\) and a positive integer \(q\) dividing \(K_{Z}\) such that \(-\frac{1}{q}K_{Z}\) is very ample and such that the germ \((X,x)\) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of \(\frac{1}{q}K_{Z}\). We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing \(2\pi c_{1}(L)\) on \(X\) is asymptotic at a polynomial rate near \(x\) to the natural Ricci-flat Kähler cone metric on \(\frac{1}{q}K_{Z}\) constructed using the Calabi ansatz. In particular, our result applies if \((X, \mathcal{O}(1))\) is a nodal quintic threefold in \(\mathbf {P}^{4}\). This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.     

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 \(t\in [0,\infty )\).     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

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.