Supercyclic abelian semigroups of matrices on \mathbb {C}^{n}

We give a complete characterization of a supercyclic abelian semigroup of matrices on \(\mathbb {C}^{n}\) . For finitely generated semigroups, this characterization is explicit and it is used to determine the minimal number of matrices in normal form over \(\mathbb {C}\) that form a supercyclic abelian semigroup on \({\mathbb {C}}^{n}\) . In particular, no abelian semigroup generated by \(n-1\) matrices on \(\mathbb {C}^{n}\) can be supercyclic.     

Structure of \mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k}) for some fields \mathbb {k}=\mathbb {Q}(\sqrt{2p_1p_2},i) with \mathbf {C}l_2(\mathbb {k})\simeq (2, 2, 2)

Let \(p_1 \equiv p_2 \equiv 5\pmod 8\) be different primes. Put \(i=\sqrt{-1}\) and \(d=2p_1p_2\) , then the bicyclic biquadratic field \(\mathbb {k}=\mathbb {Q}(\sqrt{d},i)\) has an elementary abelian 2-class group of rank \(3\) . In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group \(\mathrm {Gal}(\mathbb {k}_2^{(2)}/\mathbb {k})\) of the second Hilbert 2-class field \(\mathbb {k}_2^{(2)}\) of \(\mathbb {k}\) . We study the capitulation problem of the 2-classes of \(\mathbb {k}\) in its seven unramified quadratic extensions \(\mathbb {K}_i\) and in its seven unramified bicyclic biquadratic extensions \(\mathbb {L}_i\) .     

Histograms for stationary linear random fields

Denote the integer lattice points in the \(N\) -dimensional Euclidean space by \(\mathbb {Z}^N\) and assume that \(X_\mathbf{n}\) , \(\mathbf{n} \in \mathbb {Z}^N\) is a linear random field. Sharp rates of convergence of histogram estimates of the marginal density of \(X_\mathbf{n}\) are obtained. Histograms can achieve optimal rates of convergence \(({\hat{\mathbf{n}}}^{-1} \log {\hat{\mathbf{n}}})^{1/3}\) where \({\hat{\mathbf{n}}}=n_1 \times \cdots \times n_N\) . The assumptions involved can easily be checked. Histograms appear to be very simple and good estimators from the point of view of uniform convergence.     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.     

Fixed points and homology of superelliptic Jacobians

Let \(\eta : C_{f,N}\rightarrow \mathbb {P}^1\) be a cyclic cover of \(\mathbb {P}^1\) of degree \(N\) which is totally and tamely ramified for all the ramification points. We determine the group of fixed points of the cyclic covering group \({{\mathrm{Aut}}}(\eta )\simeq \mathbb {Z}/ N \mathbb {Z}\) acting on the Jacobian \(J_N:={{\mathrm{Jac}}}(C_{f,N})\) . For each prime \(\ell \) distinct from the characteristic of the base field, the Tate module \(T_\ell J_N\) is shown to be a free module over the ring \(\mathbb {Z}_\ell [T]/(\sum _{i=0}^{N-1}T^i)\) . We also study the subvarieties of \(J_N\) and calculate the degree of the induced polarization on the new part \(J_N^\mathrm {new}\) of the Jacobian.     

Renewal reward process for T-related fuzzy random variables on (\mathbb {R}^{p}, \mathbb {R}^{q})

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under \(t\) -norm-based fuzzy operations on \(\mathbb {R}^{p}\) and \(\mathbb {R}^{q}\,\,p,\,q \ge 1,\) respectively. Many versions of \(T\) -related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on \(\mathbb {R}^{p}\) . An application example is provided to illustrate the utility of the results.     

Simply transitive geodesic ball packings to \mathbf{S}^2\!\times \!\mathbf{R} space groups generated by glide reflections

The \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry can be derived by the direct product of the spherical plane \(\mathbf{S}^2\) and the real line \(\mathbf{R}\) . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . The \(\mathbf{S}^2\!\times \!\mathbf{R}\) manifolds were classified by Molnár and Farkas in [2] by similarity and diffeomorphism. In Szirmai (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 52(2):413–430, 2011), we have studied the geodesic balls and their volumes in \(\mathbf{S}^2\!\times \!\mathbf{R}\) space; moreover, we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . In this paper, we study the locally optimal ball packings to the \(\mathbf{S}^2\!\times \!\mathbf{R}\) space groups having Coxeter point groups, and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The density of the densest packing is \(\approx 0.80407553\) , may be surprising enough in comparison with the Euclidean result \(\frac{\pi }{\sqrt{18}}\approx 0.74048\) . Molnár has shown in (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 38(2):261–288, 1997) that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere \(\mathcal PS ^3(\mathbf{V}^4,\varvec{V}_4,\mathbb R )\) . In our work, we shall use this projective model of \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry.     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

Logarithmic Mean Oscillation on the Polydisc,Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\) -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\) , \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\) . We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\) .     

Regularity and uniqueness of a class of biharmonic map heat flows

We consider a class of weak solutions of the heat flow of biharmonic maps from \(\Omega \subset \mathbb{R }^n\) to the unit sphere \(\mathbb{S }^L\subset \mathbb{R }^{L+1}\) , that have small renormalized total energies locally at each interior point. For any such a weak solution, we prove the interior smoothness, and the properties of uniqueness, convexity of hessian energy, and unique limit at \(t=\infty \) . We verify that any weak solution \(u\) to the heat flow of biharmonic maps from \(\Omega \) to a compact Riemannian manifold \(N\) without boundary, with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12), has small renormalized total energy locally and hence enjoys both the interior smoothness and uniqueness property. Finally, if an initial data \(u_0\in W^{2,r}(\mathbb{R }^n, N)\) for some \(r>\frac{n}{2}\) , then we establish the local existence of heat flow of biharmonic maps \(u\) , with \(\nabla ^2 u\in L^q_tL^p_x\) for some \(p>\frac{n}{2}\) and \(q>2\) satisfying (1.12).     

Classification of finite Morse index solutions for Hénon type elliptic equation -\Delta u = |x|^{\alpha } u_{+}^{p}

In this paper we investigate the non-autonomous elliptic equations \(-\Delta u = |x|^{\alpha } u_{+}^{p}\) in \( \mathbb{R }^{N}\) and in \( \mathbb{R }_+^{N}\) with the Dirichlet boundary condition, with \(N \ge 2\) , \(p>1\) and \(\alpha >-2\) . We consider the weak solutions with finite Morse index and obtain some classification results.     

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Characterization for Fock-Type Space via Higher Order Derivatives and its Application

The space of entire functions \(\mathcal {F}_{\alpha }^{\infty }\) is mentioned in the paper of S. Janson, J. Peetre, and R. Rochberg. In this paper, we establish a characterization for the space \(\mathcal {F}_{\alpha }^{\infty }\) by \(n\) -th derivatives of entire functions. As an application of this result, we study the boundedness of Li-Stevi?’s integral operators and estimate essential norms of these operators acting on \(\mathcal {F}_{\alpha }^{\infty }\) . Furthermore we describe complete characterizations for boundedness and compactness of the Volterra-type integral operators on \(\mathcal {F}_{\alpha }^{\infty }\) .     

Divide and Conquer Roadmap for Algebraic Sets

Let \(\mathrm{R}\) be a real closed field and \(\hbox {D}\subset \mathrm{R}\) an ordered domain. We describe an algorithm that given as input a polynomial \(P \in \hbox {D}[ X_{1} , \ldots ,X_{{ k}} ]\) and a finite set, \(\mathcal {A}= \{ p_{1} , \ldots ,p_{m} \}\) , of points contained in \(V= {\mathrm{{Zer}}} ( P, \mathrm{R}^{{ k}})\) described by real univariate representations, computes a roadmap of \(V\) containing \(\mathcal {A}\) . The complexity of the algorithm, measured by the number of arithmetic operations in \(\hbox {D}\) , is bounded by \(\big ( \sum _{i=1}^{m} D^{O ( \log ^{2} ( k ) )}_{i} +1 \big ) ( k^{\log ( k )} d )^{O ( k\log ^{2} ( k ))}\) , where \(d= \deg ( P )\) and \(D_{i}\) is the degree of the real univariate representation describing the point \(p_{i}\) . The best previous algorithm for this problem had complexity card \(( \mathcal {A} )^{O ( 1 )} d^{O ( k^{3/2} )}\) (Basu et al., ArXiv, 2012), where it is assumed that the degrees of the polynomials appearing in the representations of the points in \(\mathcal {A}\) are bounded by \(d^{O ( k )}\) . As an application of our result we prove that for any real algebraic subset \(V\) of \(\mathbb {R}^{k}\) defined by a polynomial of degree \(d\) , any connected component \(C\) of \(V\) contained in the unit ball, and any two points of \(C\) , there exists a semi-algebraic path connecting them in \(C\) , of length at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) , consisting of at most \(( k ^{\log (k )} d )^{O ( k\log ( k ) )}\) curve segments of degrees bounded by \(( k ^{\log ( k )} d )^{O ( k \log ( k) )}\) . While it was known previously, by a result of D’Acunto and Kurdyka (Bull Lond Math Soc 38(6):951–965, 2006), that there always exists a path of length \(( O ( d ) )^{k-1}\) connecting two such points, there was no upper bound on the complexity of such a path.     

Bent functions on partial spreads

For an arbitrary prime \(p\) we use partial spreads of \(\mathbb{F }_p^{2m}\) to construct two classes of bent functions from \(\mathbb{F }_p^{2m}\) to \(\mathbb{F }_p\) . Our constructions generalize the classes \(PS^{(-)}\) and \(PS^{(+)}\) of binary bent functions which are due to Dillon.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Differentiability in the Sobolev space W^{1,n-1}

Let \(\Omega \subset {\mathbb {R}}^{n}\) be a domain, \(n \ge 2\) . We show that a continuous, open and discrete mapping \(f \in W_{\mathrm{loc }}^{1,n-1}(\Omega , {\mathbb {R}}^{n})\) with integrable inner distortion is differentiable almost everywhere on \(\Omega \) . As a corollary we get that the branch set of such a mapping has measure zero.     

Classical limit of the tensor product of holomorphic discrete series representations

For three coadjoint orbits \(\mathcal {O}_1, \mathcal {O}_2\) and \(\mathcal {O}_3\) in \(\mathfrak {g}^*\) , the Corwin–Greenleaf function \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) is given by the number of \(G\) -orbits in \(\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}\) under the diagonal action. In the case where \(G\) is a simple Lie group of Hermitian type, we give an explicit formula of \(n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)\) for coadjoint orbits \(\mathcal {O}_1\) and \(\mathcal {O}_2\) that meet \(\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }\) , and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.     

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.     

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for \({\boldsymbol {L}{^{2}}(\mathbb {R}^d)}\) are obtained by applying time-frequency shifts from a lattice in \(\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}\) to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some \(g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}\) . There is always a canonical dual frame, generated by the dual Gabor atom \({\widetilde g}\) . The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from \({a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}\) ). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice \(\boldsymbol {\Lambda }\circ\) . The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.