Crystallographic number systems

We introduce the notion of crystallographic number systems, generalizing matrix number systems. Let Γ be a group of isometries of ${\mathbb{R}^d,g}$ an expanding affine mapping of ${\mathbb{R}^d}$ with ${g\circ\Gamma\circ g^{-1}\subset\Gamma}$ and ${\mathcal{D}\subset\Gamma}$ . We say that ${(\Gamma,g,\mathcal{D})}$ is a Γ-number system if every isometry ${\gamma\in \Gamma}$ has a unique expansion $$\gamma=g^n\delta_n g^{-n}\,g^{n-1}\delta_{n-1} g^{-(n-1)}\dots g\delta_{1} g^{-1}\,\delta_0,$$ for some ${n\in \mathbb{N}}$ and ${\delta_0,\ldots,\delta_n\in \mathcal{D}}$ . A tile can be attached to a Γ-number system. We show fundamental topological properties of this tile: they admit the fixed point of g as interior point and tesselate the space by the whole group Γ. Moreover, we give several examples, among them a class of p2-number systems, where p2 is the crystallographic group generated by the π-rotation and two independent translations.     

Schauder estimates for sub-elliptic equations

We establish Hölder estimates of second derivatives for a class of sub-elliptic partial differential operators in ${\mathbb{R}^{N}}$ of the kind $$\mathcal L=\sum_{i,j=1}^{m}a_{ij}(x)X_{i}X_{j}+X_{0},$$ where the X _j ’s are smooth vector fields in ${\mathbb{R}^{N}}$ , and a _ij is a uniformly elliptic matrix. It is assumed that the X _j ’s satisfy homogeneity conditions with respect to a group of dilations δ _r which yield the existence of a composition law ${\circ}$ in ${\mathbb{R}^{N}}$ making the triplet ${\mathbb G=(\mathbb{R}^{N},\circ,\delta_{r})}$ an homogeneous Lie group on which the X _j ’s are left translation invariant. The Hölder norms are defined in terms of this composition law. The main tools used are the Taylor formula for smooth functions on ${\mathbb{G}}$ , some properties of the corresponding Taylor polynomials, and an orthogonality theorem that extends to homogeneous Lie groups a classical theorem of Calderón and Zygmund in the Euclidean setting.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Relations between the K-loop and the defect of an absolute plane

Let ${\cal P}$ be the point set of an absolute plane, let ${\cal {\tilde P}}$ be the set of all point reflections, let ?, resp. ?⁺, be the group of all, resp. of all proper, motions and let $$^\sim:{\cal P\times P\rightarrow \tilde P};\ \ \ (a,\ b)\mapsto\ \widetilde {a,\ b}$$ be the map where ${\widetilde {a,\ b}}$ denotes the uniquely determined point-reflection interchanging a and b. Then $$\delta\:\ {\cal P}^{3}\rightarrow {\cal M}^{+};\ \ \ (a,b,c)\mapsto \delta_{a;b,c}\:=\ {\tilde a}\ {\rm o}\ \widetilde {a,\ b}\ {\rm o}\ \widetilde {b,\ c}\ {\rm o}\ \widetilde {c,\ a}$$ is called the defect function, or shortly the defect. We show that δ_a;b,c is a rotation around the point a where the angle of δ_a;b,c is exactly the angle defect of the triangle (a, b, c) (cfr. 3.5). After fixing a point $o\ \in {\cal P}$ and setting $a+b\:=\widetilde {o,\ a}\ {\rm o}\ {\tilde o}\ (b),\ ({\cal P},+)$ becomes a K-loop and the so called precession function $$\delta_{a,b}\:=\ \big((a+b)^{+}\big)^{-1}\ {\rm o}\ a^{+}\ {\rm o}\ b^+$$ of the loop ( ${\cal P}, +)$ coincides with the defect of the triangle (o, a, ?b) (cfr. (4.4.1)), hence δ_a,b = δ_o;a,?b for all $a, b \in {\cal P}$ . With the order relation of the absolute plane we associate an orientation function $$\Omega\:\ \Delta\ \times \Delta \rightarrow \lbrace -1,+1\rbrace$$ defined on the pairs of triangles (cfr. (2.8)). If (a, b, c) ∈ Δ is a triangle and d a point of the line $\overline {b,\ c}$ ¹, then (cfr. (3.9.2)): $$\delta_{a;b,c}\ {\rm o}\ \delta_{a;c,d}=\delta_{a;b,d}$$ and moreover, if d is even a point of the open segment ]b, c[ then (cfr. (2.8.5)): $$\Omega(a,\ b,\ c;\ a,\ b,\ d)=\Omega(a,\ b,\ d;\ a,\ d,\ c)=+1.$$ Thus the angle defect of the triangle (a, b, c) is the sum of the angle defects of the triangles (a, b, d) and (a, d, c).     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .     

Axiomatizability by {{\forall}{\exists}!}-sentences

A ${\forall\exists!}$ -sentence is a sentence of the form ${\forall x_{1}\cdots x_{n}\exists!y_{1}\cdots y_{m}O(\overline{x},\overline{y})}$ , where O is a quantifier-free formula, and ${\exists!}$ stands for ??there exist unique??. We prove that if ${\mathcal{C}}$ is (up to isomorphism) a finite class of finite models then ${\mathcal{C}}$ is axiomatizable by a set of ${\forall\exists!}$ -sentences if and only if ${\mathcal{C}}$ is closed under isomorphic images, ${\mathcal{C}}$ has the intersection property, and ${\mathcal{C}}$ is closed under fixed-point submodels. This result is employed to characterize the subclasses of finitely generated discriminator varieties axiomatizable by sentences of the form ${\forall\exists!\bigwedge p=q}$ .     

Biharmonic Hypersurfaces in a Conformally Flat Space

Biharmonic hypersurfaces in a generic conformally flat space are studied in this paper. The equation of such hypersurfaces is derived and used to determine the conformally flat metric ${f^{-2}\delta_{ij}}$ on the Euclidean space ${\mathbb{R}^{m+1}}$ so that a minimal hypersurface ${M^{m}\longrightarrow (\mathbb{R}^{m+1}, \delta_{ij})}$ in a Euclidean space becomes a biharmonic hypersurface ${M^m\longrightarrow (\mathbb{R}^{m+1}, f^{-2}\delta_{ij})}$ in the conformally flat space. Our examples include all biharmonic hypersurfaces found in Ou (Pac J Math 248(1):217–232, 2010) and Ou and Tang (Mich Math J 61:531–542, 2012) as special cases.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

On p-tuples of the Grassmann manifolds

We provide a matrix invariant for isometry classes of p-tuples of points in the Grassmann manifold ${G_{n}\left(\mathbb{K}^{d}\right) }$ ( ${\mathbb{K=\mathbb{R}}}$ or ${\mathbb{C}}$ ). This invariant fully characterizes the p-tuple. We use it to classify the regular p-tuples of ${G_{2}\left(\mathbb{R}^{d}\right) }$ , ${G_{3}\left( \mathbb{R}^{d}\right) }$ and ${G_{2}\left( \mathbb{C}^{d}\right) }$ .     

Graded Antisimple Primitive Radical

We introduce the graded version of the antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}} $ , the graded antisimple primitive radical $ {\user1{\mathcal{S}\mathcal{J}}}_{G} $ . We show that $ {\user1{\mathcal{S}\mathcal{J}}}_{G} = {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} = {\user1{\mathcal{S}\mathcal{J}}}^{G} $ when |G| < ∞, where $ {\user1{\mathcal{S}\mathcal{J}}}_{{{\text{ref}}}} $ denotes the reflected antisimple primitive radical and $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ denotes the restricted antisimple primitive radical. Furthermore, we discuss the graded supplementing radical of $ {\user1{\mathcal{S}\mathcal{J}}}^{G} $ .     

Varieties generated by modes of submodes

In a natural way, we can ??lift?? any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra ( ${A, \Omega}$ ) its power algebra of subsets. G. Gr?tzer and H. Lakser proved that for a variety ${\mathcal{V}}$ , the variety ${\mathcal{V}\Sigma}$ generated by power algebras of algebras in ${\mathcal{V}}$ satisfies precisely the consequences of the linear identities true in ${\mathcal{V}}$ . For certain types of algebras, the sets of their subalgebras form subalgebras of their power algebras. They are called the algebras of subalgebras. In this paper, we partially solve a long-standing problem concerning identities satisfied by the variety ${\mathcal{VS}}$ generated by algebras of subalgebras of algebras in a given variety ${\mathcal{V}}$ . We prove that if a variety ${\mathcal{V}}$ is idempotent and entropic and the variety ${\mathcal{V}\Sigma}$ is locally finite, then the variety ${\mathcal{VS}}$ is defined by the idempotent and linear identities true in ${\mathcal{V}}$ .     

Valuation Extensions of Algebras Defined by Monic Gr?bner Bases

Let K be a field, $\mathcal {O}_v$ a valuation ring of K associated to a valuation v: K → Γ?∪?{?∞?}, and m _v the unique maximal ideal of $\mathcal {O}_v$ . Consider an ideal $\mathcal {I}$ of the free K-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on X ₁,...,X _n . If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr?bner basis of ${\cal I}$ in $K\langle X\rangle$ , where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$ -algebra on X ₁,...,X _n , then the valuation v induces naturally an exhaustive and separated Γ-filtration F ^v A for the K-algebra $A=K\langle X\rangle /\mathcal {I}$ , and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$ ; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$ , $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free k-algebra on X ₁,...,X _n , and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$ , then F ^v A determines a valuation function A → Γ?∪?{?∞?}, and thereby v extends naturally to a valuation function on the (skew-)field Δ of fractions of A provided Δ exists.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

The Ricci tensor and structure Jacobi operator of real hypersurfaces in a complex projective space

Let M be a real hypersurface with almost contact metric structure ${(\phi, \xi, \eta, g)}$ in a complex projective space ${P_{n}\mathbb{C}}$ . A Real hypersurface M is said to be a Hopf hypersurface if ξ is principal. In this paper we investigate real hypersurfaces of ${P_{n}\mathbb{C}}$ whose Ricci tensors S satisfy ${\nabla_{\phi\nabla_{\xi}\xi}S = 0}$ . Under some further conditions we characterize Hopf hypersurfaces of ${P_{n}\mathbb{C}}$ .     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Nikodym Maximal Functions Associated with Variable Planes in {\mathbb{R}^{3}}

Given a vector field ${\mathfrak{a}}$ on ${\mathbb{R}^3}$ , we consider a mapping ${x\mapsto \Pi_{\mathfrak{a}}(x)}$ that assigns to each ${x\in\mathbb{R}^3}$ , a plane ${\Pi_{\mathfrak{a}}(x)}$ containing x, whose normal vector is ${\mathfrak{a}(x)}$ . Associated with this mapping, we define a maximal operator ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^1_{loc}(\mathbb{R}^3)}$ for each ${N\gg 1}$ by $$\mathcal{M}^{\mathfrak{a}}_Nf(x)=\sup_{x\in\tau} \frac{1}{|\tau|} \int_{\tau}|f(y)|\,dy$$ where the supremum is taken over all 1/N ×? 1/N?× 1 tubes τ whose axis is embedded in the plane ${\Pi_\mathfrak{a}(x)}$ . We study the behavior of ${\mathcal{M}^{\mathfrak{a}}_N}$ according to various vector fields ${\mathfrak{a}}$ . In particular, we classify the operator norms of ${\mathcal{M}^{\mathfrak{a}}_N}$ on ${L^2(\mathbb{R}^3)}$ when ${\mathfrak{a}(x)}$ is the linear function of the form (a ₁₁ x ₁?+?a ₂₁ x ₂, a ₁₂ x ₁?+?a ₂₂ x ₂, 1). The operator norm of ${\mathcal{M}^\mathfrak{a}_N}$ on ${L^2(\mathbb{R}^3)}$ is related with the number given by $$D=(a_{12}+a_{21})^2-4a_{11}a_{22}.$$