Monadic MV-algebras II: Monadic implicational subreducts

In this paper, we study the class of all monadic implicational subreducts, that is, the ${\{\rightarrow, \forall,1\}}$ -subreducts of the class of monadic MV-algebras. We prove that this class is an equational class, which we denote by ${\mathcal{ML}}$ , and we give an equational basis for this variety. An algebra in ${\mathcal{ML}}$ is called a monadic ?ukasiewicz implication algebra. We characterize the subdirectly irreducible members of ${\mathcal{ML}}$ and the congruences of every monadic ?ukasiewicz implication algebra by monadic filters. We prove that ${\mathcal{ML}}$ is generated by its finite members. Finally, we completely describe the lattice of subvarieties, and we give an equational basis for each proper subvariety.     

The subvariety of commutative residuated lattices represented by twist-products

Given an integral commutative residuated lattice L, the product L × L can be endowed with the structure of a commutative residuated lattice with involution that we call a twist-product. In the present paper, we study the subvariety ${\mathbb{K}}$ of commutative residuated lattices that can be represented by twist-products. We give an equational characterization of ${\mathbb{K}}$ , a categorical interpretation of the relation among the algebraic categories of commutative integral residuated lattices and the elements in ${\mathbb{K}}$ , and we analyze the subvariety of representable algebras in ${\mathbb{K}}$ . Finally, we consider some specific class of bounded integral commutative residuated lattices ${\mathbb{G}}$ , and for each fixed element ${{\bf L} \in \mathbb{G}}$ , we characterize the subalgebras of the twist-product whose negative cone is L in terms of some lattice filters of L, generalizing a result by Odintsov for generalized Heyting algebras.     

Classification of Sol lattices

Sol geometry is one of the eight homogeneous Thurston 3-geometries $${\bf E}^{3}, {\bf S}^{3}, {\bf H}^{3}, {\bf S}^{2}\times{\bf R}, {\bf H}^{2}\times{\bf R}, \widetilde{{\bf SL}_{2}{\bf R}}, {\bf Nil}, {\bf Sol}.$$ In [13] the densest lattice-like translation ball packings to a type (type I/1 in this paper) of Sol lattices has been determined. Some basic concept of Sol were defined by Scott in [10], in general. In our present work we shall classify Sol lattices in an algorithmic way into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean 3-lattices, but infinitely many Sol affine equivalence classes, in each type. Then the discrete isometry groups of compact fundamental domain (crystallographic groups) can also be classified into infinitely many classes but finitely many types, left to other publication. To this we shall study relations between Sol lattices and lattices of the pseudoeuclidean (or here rather called Minkowskian) plane [1]. Moreover, we introduce the notion of Sol parallelepiped to every lattice type. From our new results we emphasize Theorems 3?C6. In this paper we shall use the affine model of Sol space through affine-projective homogeneous coordinates [6] which gives a unified way of investigating and visualizing homogeneous spaces, in general.     

Antibasis theorems for {\Pi^0_1} classes and the jump hierarchy

We prove two antibasis theorems for ${\Pi^0_1}$ classes. The first is a jump inversion theorem for ${\Pi^0_1}$ classes with respect to the global structure of the Turing degrees. For any ${P\subseteq 2^\omega}$ , define S(P), the degree spectrum of P, to be the set of all Turing degrees a such that there exists ${A \in P}$ of degree a. For any degree ${{\bf a \geq 0'}}$ , let ${\textrm{Jump}^{-1}({\bf a) = \{b : b' = a \}}}$ . We prove that, for any ${{\bf a \geq 0'}}$ and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}^{-1} ({\bf a}) \subseteq S(P)}$ then P contains a member of every degree. For any degree ${{\bf a \geq 0'}}$ such that a is recursively enumerable (r.e.) in 0', let ${Jump_{\bf \leq 0'} ^{-1}({\bf a)=\{b : b \leq 0' \textrm{and} b' = a \}}}$ . The second theorem concerns the degrees below 0'. We prove that for any ${{\bf a\geq 0'}}$ which is recursively enumerable in 0' and any ${\Pi^0_1}$ class P, if ${\textrm{Jump}_{\bf \leq 0'} ^{-1}({\bf a)} \subseteq S(P)}$ then P contains a member of every degree.     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

Conic volume ratio of the packing cone associate to a convex body

Given a convex body ${K\subset\mathbb{R}^n}$ (n??? 1) which contains o in its interior and ${{\bf u} \in S^{n-1}}$ , we introduce conic volume ratio r(K, u) of K in the direction of u by $$r(K, {\bf u})=\frac{vol(cone(K,{\bf u})\cap B_2^n)}{vol(B_2^n)},$$ where cone(K, u) is the packing cone of K in the direction of u. We prove that if K is an o-symmetric convex body in ${\mathbb{R}^n}$ and r(K, u) is a constant function of u, then K must be a Euclidean ball.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

On the hull-kernel and inverse topologies as frames

We consider some frame-theoretic properties of the hull-kernel and the inverse topologies on the set of minimal prime ideals of an algebraic frame with the finite intersection property on its compact elements. Denote by Alg _do the subcategory of Frm consisting of such frames together with dense onto coherent maps. We construct a functor ${{\sf T} : {\bf Alg}_{\rm do} \rightarrow {\bf Frm}}$ T : Alg do → Frm and a natural transformation ${\tau : {\sf E} \rightarrow {\sf T}}$ τ : E → T , where E is the inclusion functor from Alg _do to Frm.     

On restricted representations of the extended special type Lie superalgebra {\bar{S}(m, n, 1)}

Let F be an algebraically closed field of prime characteristic p > 2, and let ${\mathfrak{g}=\bar{S}(m, n, {\bf 1})}$ be the extended special type Lie superalgebra over F. Simple restricted ${\mathfrak{g}}$ -modules are classified. Moreover, a sufficient and necessary condition is provided for restricted baby Kac modules to be simple.     

Hyperflock determining line sets and totally regular parallelisms of PG {(3,\,{\mathbb{R}})}

By a totally regular parallelism of the real projective 3-space ${\Pi_3:={{\rm PG}}(3, \mathbb {R})}$ we mean a family T of regular spreads such that each line of Π ₃ is contained in exactly one spread of T. For the investigation of totally regular parallelisms the authors mainly employ Klein’s correspondence λ of line geometry and the polarity π ₅ associated with the Klein quadric H ₅ (for details see Chaps. 1 and 3). The λ-image of a totally regular parallelism T is a hyperflock of H ₅, i.e., a family H of elliptic subquadrics of H ₅ such that each point of H ₅ is on exactly one subquadric of H. Moreover, ${\{\pi_5({{\rm span}} \,\lambda(\mathcal {X}))\vert\mathcal {X}\in\bf{T}\}=:\mathcal {H}_{\bf{T}}}$ is a hyperflock determining line set, i.e., a set ${\mathcal {Z}}$ of 0-secants of H ₅ such that each tangential hyperplane of H ₅ contains exactly one line of ${\mathcal {Z}}$ . We say that ${{{\rm dim}}({{\rm span}}\,\mathcal {H}_{\bf{T}})=:d_{\bf{T}}}$ is the dimension of T and that T is a d _T - parallelism. Clifford parallelisms and 2-parallelisms coincide. The examples of non-Clifford parallelisms exhibited in Betten and Riesinger [Result Math 47:226–241, 2004; Adv Geom 8:11–32, 2008; J Geom (to appear)] are totally regular and of dimension 3. If ${\mathcal{G}}$ is a hyperflock determining line set, then ${\{\lambda^{-1}\,{\rm (}\pi_5(X){\,\cap H_5)\,|\, X\in\mathcal{G}\}}}$ is a totally regular parallelism. In the present paper the authors construct examples of topological (see Definition 1.1) 4- and 5-parallelisms via hyperflock determining line sets.     

A new linear quotient of C 4 admitting a symplectic resolution

We show that the quotient C ⁴/G admits a symplectic resolution for ${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$ . Here Q ₈ is the quaternionic group of order eight and D ₈ is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements ?Id of each. It is equipped with the tensor product representation ${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$ . This group is also naturally a subgroup of the wreath product group ${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$ . We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C ⁴/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.     

Characters, fields, Schur indices and divisibility

Let G be a finite nilpotent group. Suppose that G ₀ is a subgroup of G and that ${\psi}$ is an irreducible character of G ₀. Consider the set S whose elements are the natural numbers $${\rm m}_{\bf Q}(\chi)[{\bf Q}(\chi) : {\bf Q}]$$ as ${\chi}$ runs through the irreducible characters of G which contain ${\psi}$ as a summand when restricted to G ₀. Here m _Q (χ) is, as usual, the rational Schur index of ${\chi}$ , and ${[{\bf Q}(\chi) : {\bf Q}]}$ is the degree of the extension of the field of values of the character as an extension of the rationals. We prove that then the minimum element of S divides all the other elements of S. The result is not true when G is an arbitrary finite group. We also consider some variations of this result.     

More covers of the Boolean variety of unital ℓ-groups

Within the lattice of varieties of pseudo MV-algebras, the variety ${\mathcal{B}}$ of Boolean algebras is the least nontrivial variety. Komori identified all varieties of (commutative) MV-algebras that cover ${\mathcal{B}}$ . The authors previously identified all solvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ . We will show the existence of continuum many nonsolvable varieties of pseudo MV-algebras that cover ${\mathcal{B}}$ , show that periodically primitive u?-groups cannot generate Boolean covers, and show that all noncommutative varieties that are Boolean covers must be Top Boolean.     

Igusa’s p-adic local zeta function associated to a polynomial mapping and a polynomial integration measure

For p prime, we give an explicit formula for Igusa’s local zeta function associated to a polynomial mapping ${{\bf f} = (f_1, \ldots, f_t) : {\bf Q}_p^{n} \to {\bf Q}_p^{t}}$ , with ${f_1, \ldots, f_t \in {\bf Z}_p[x_1, \ldots, x_n]}$ , and an integration measure on ${{\bf Z}_p^{n}}$ of the form ${|g(x)||dx|}$ , with g another polynomial in Z _p [x ₁, . . ., x _n ]. We treat the special cases of a single polynomial and a monomial ideal separately. The formula is in terms of Newton polyhedra and will be valid for f and g sufficiently non-degenerated over F _p with respect to their Newton polyhedra. The formula is based on, and is a generalization of results in Denef and Hoornaert (J Number Theory 89(1):31–64, 2001), Howald et?al. (Proc Am Math Soc 135(11):3425–3433, 2007) and Veys and Zú?iga-Galindo (Trans Am Math Soc 360(4):2205–2227, 2008).     

On superdiffusive behavior of a passive tracer in a Poisson shot noise field

In this paper, we investigate the trajectory of the passive tracer model governed by the ordinary differential equation $$ \frac{{\rm d} {\bf x} (t)}{{\rm d}t} = {\bf F} ({\bf x}(t)), \quad {\bf x}(0)= {\bf x}_{0}, $$ where F(x) is a zero mean, homogeneous, isotropic Poisson shot noise random field. We prove the superdiffusive character of the trajectories under certain conditions on the energy spectrum of the velocity field.     

The norm of polynomials in large random and deterministic matrices

Let ${{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}$ be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices ${{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}$ , possibly random but independent of X _N , for which the operator norm of ${P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}$ converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y _N and of the polynomials P, we get for a large class of matrices the ??no eigenvalues outside a neighborhood of the limiting spectrum?? phenomena. We give examples of diagonal matrices Y _N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.     

Regular prism tilings in {{\widetilde{\bf SL_{2}R}}} space

\({{\widetilde{\bf SL_{2}R}}}\) geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from the 3-dimensional Lie group of all 2 × 2 real matrices with determinant one. Our aim is to describe and visualize the regular infinite or bounded p-gonal prism tilings in \({{\widetilde{\bf SL_{2}R}}}\) . For this purpose we introduce the notion of infinite and bounded prisms, prove that there exist infinitely many regular infinite p-gonal face-to-face prism tilings \({\mathcal{T}^i_p(q)}\) and infinitely many regular bounded p-gonal non-face-to-face \({{\widetilde{\bf SL_{2}R}}}\) prism tilings \({\mathcal{T}_p(q)}\) for integer parameters \({p,q; 3 \leq p, \frac{2p}{p-2} < q}\) . Moreover, we describe the symmetry group of \({\mathcal{T}_p(q)}\) via its index 2 rotational subgroup, denoted by pq2 ₁ . Surprisingly this group already occurred in our former work (Molnár et al., J Geometry, 95:91–133, 2009) in another context. We also develop a method to determine the data of the space filling regular infinite and bounded prism tilings. We apply the above procedure to \({\mathcal{T}^i_3(q)}\) and \({\mathcal{T}_3(q)}\) where 6 < q and visualize them and the corresponding tilings. E. Molnár showed, that homogeneous 3-spaces have a unified interpretation in the projective 3-sphere \({\mathcal{PS}^3}\) and 3-space \({\mathcal{P}^3({\bf V}^4,{\bf V}_4, {\bf R})}\) . In our work we will use this projective model of \({{\widetilde{\bf SL_{2}R}}}\) and in this manner the prisms and prism tilings can be visualized on the Euclidean screen of a computer.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

On topological diversity of rectilinear representations of countably infinite graphs

We prove that, for each simple graph G whose set of vertices is countably infinite, there is a family ${\varvec{\mathcal{R}}(\varvec{G})}$ of the cardinality of the continuum of graphs such that (1) each graph ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ is isomorphic to G, all vertices of H are points of the Euclidean space E ³, all edges of H are straight line segments (the ends of each edge are the vertices joined by it), the intersection of any two edges of H is either their common vertex or empty, and any isolated vertex of H does not belong to any edge of H; (2) all sets ${\varvec{\mathcal{B}}(\varvec{H})}$ ( ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ ), where ${\varvec{\mathcal{B}}(\varvec{H})\subset \mathbf{E}^3}$ is the union of all vertices and all edges of H, are pairwise not homeomorphic; moreover, for any graphs ${\varvec{H}_1 \in \varvec{\mathcal{R}}(\varvec{G})}$ and ${\varvec{H}_2 \in \varvec{\mathcal{R}}(\varvec{G})}$ , ${\varvec{H}_1 \ne \varvec{H}_2}$ , and for any finite subsets ${\varvec{S}_i \subset \varvec{\mathcal{B}}(\varvec{H}_i)}$ (i = 1, 2), the sets ${\varvec{\mathcal{B}}(\varvec{H}_1){\setminus} \varvec{S}_1}$ and ${\varvec{\mathcal{B}}(\varvec{H}_2){\setminus} \varvec{S}_2}$ are not homeomorphic.     

Completely representable lattices

It is known that a lattice is representable as a ring of sets iff the lattice is distributive. CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets. jCRL is the class of DLs which have representations preserving arbitrary joins, mCRL is the class of DLs which have representations preserving arbitrary meets, and biCRL is defined to be \({{\bf jCRL} \cap {\bf mCRL}}\) . We prove

${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$

where the marked inclusions are proper.

Let L be a DL. Then \({L \in {\bf mCRL}}\) iff L has a distinguishing set of complete, prime filters. Similarly, \({L \in {\bf jCRL}}\) iff L has a distinguishing set of completely prime filters, and \({L \in {\bf CRL}}\) iff L has a distinguishing set of complete, completely prime filters.Each of the classes above is shown to be pseudo-elementary, hence closed under ultraproducts. The class CRL is not closed under elementary equivalence, hence it is not elementary.