Cluster Algebras of Type A _{\bf{2}}^{\bf {(1)}}

In this paper we study cluster algebras $\mathcal{A}$ of type $A_2^{(1)}$ . We solve the recurrence relations among the cluster variables (which form a T-system of type $A_2^{(1)}$ ). We solve the recurrence relations among the coefficients of $\mathcal{A}$ (which form a Y-system of type $A_2^{(1)}$ ). In $\mathcal{A}$ there is a natural notion of positivity. We find linear bases B of $\mathcal{A}$ such that positive linear combinations of elements of B coincide with the cone of positive elements. We call these bases atomic bases of $\mathcal{A}$ . These are the analogue of the “canonical bases” found by Sherman and Zelevinsky in type $A_{1}^{(1)}$ . Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of B are parameterized by ?³ via their g-vectors in every cluster. We prove that the denominator vector map in every acyclic seed of $\mathcal{A}$ restricts to a bijection between B and ?³. We find explicit recurrence relations to express every element of $\mathcal{A}$ as linear combinations of elements of B.     

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.     

On the complexity of the closed fragment of Japaridze’s provability logic

We consider the well-known provability logic GLP. We prove that the GLP-provability problem for polymodal formulas without variables is PSPACE-complete. For a number n, let \({L^{n}_0}\) denote the class of all polymodal variable-free formulas without modalities \({\langle n \rangle,\langle n+1\rangle,...}\) . We show that, for every number n, the GLP-provability problem for formulas from \({L^{n}_0}\) is in PTIME.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

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.     

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.     

{\boldsymbol(}\boldsymbol{\mathcal{Z}}_{\bf 1}, \boldsymbol{\mathcal{Z}}_{\bf 2}{\boldsymbol)}-Complete Partially Ordered Sets and Their Representations by \boldsymbol{\mathcal{Q}}-Spaces

The present paper proposes a general theory for $\left( \mathcal{Z}_{1}, \mathcal{Z}_{2}\right) $ -complete partially ordered sets (alias $\mathcal{Z} _{1}$ -join complete and $\mathcal{Z}_{2}$ -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections $\mathcal{Z}_{i}$ (i?=?1,...,4) and $\mathcal{Q} =\left( \mathcal{Z}_{1},\mathcal{Z}_{2},\mathcal{Z}_{3},\mathcal{Z} _{4}\right) $ , the category $\mathcal{Q}$ P of $\left( \mathcal{Z}_{1},\mathcal{Z}_{2}\right) $ -complete partially ordered sets and $\left( \mathcal{Z}_{3},\mathcal{Z}_{4}\right) $ -continuous (alias $\mathcal{ Z}_{3}$ -join preserving and $\mathcal{Z}_{4}$ -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category $\mathcal{Q}$ S of $\mathcal{Q}$ -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory $ \mathcal{Q}$ P _s of $\mathcal{Q}$ P of all $\mathcal{Q}$ -spatial objects and the full subcategory $\mathcal{Q}$ S _s of $\mathcal{Q}$ S of all $\mathcal{Q}$ -sober objects. Here $\mathcal{Q}$ -spatiality and $\mathcal{Q}$ -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to $\mathcal{Z}$ -compact generation and $\mathcal{Z}$ -sobriety have also been pointed out in this paper.     

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.     

On linear Hodge-Newton decomposition for reductive monoids

Let $\bar{\mathbf{G}}$ be an irreducible linear reductive monoid over a characteristic zero field F of fractions of a complete discrete valuation ring $\mathfrak{o}$ , such that its group G of units is split over $\mathfrak{o}$ . This paper concerns a relation between the Hodge point and the Newton point associated to an element $\gamma\in\bar{\mathbf{G}}(F)$ , proved by Kottwitz and Viehmann when $\bar{\mathbf{G}}$ is either a connected reductive $\mathfrak{o}$ -split linear algebraic group over F or the monoid of n×n matrices over F. On the way to proving this relation, we apply the Putcha-Renner theory of linear algebraic monoids over algebraically closed fields to study $\bar{\mathbf{G}}(F)$ by generalizing various results for linear algebraic groups over F such as the Iwasawa, Cartan and affine Bruhat decompositions.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

A Jost–Pais-Type Reduction of Fredholm Determinants and Some Applications

We study the analog of semi-separable integral kernels in \({\mathcal {H}}\) of the type $$ K(x, x') = \left\{\begin{array}{ll} F_1(x) G_1(x'), \quad& a < x' < x < b,\\ F_2 (x)G_2(x'), \quad& a < x < x' < b,\end{array}\right.$$ where \({-\infty \leqslant a < b \leqslant \infty}\) , and for a.e. \({x \in (a, b)}\) , \({F_j (x) \in \mathcal{B}_2(\mathcal{H}_j, \mathcal{H})}\) and \({G_j(x) \in \mathcal {B}_2(\mathcal {H},\mathcal {H}_j)}\) such that F _j (·) and G _j (·) are uniformly measurable, and $$\begin{array}{ll} || F_j ( \cdot) ||_{\mathcal {B}_2(\mathcal {H}_j,\mathcal {H})} \in L^2((a, b)), ||G_j (\cdot)||_{\mathcal {B}_2(\mathcal {H},\mathcal {H}_j)} \in L^2((a, b)), \quad j=1,2, \end{array}$$ with \({\mathcal {H}}\) and \({\mathcal {H}_j}\) , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in \({L^2((a, b);\mathcal {H})}\) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant \({{\rm det}_{L^2((a,b);\mathcal{H})}}\) (I ? α K), \({\alpha \in \mathbb{C}}\) , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \({\mathcal{H}}\) (and \({\mathcal{H}_1 \oplus \mathcal{H}_2}\) ). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.     

Shintani lifting and real-valued characters

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over ${\mathbb{F}_q}$ , and ψ is an irreducible character of G( ${\mathbb{F}_{q^m}}$ ) which is the lift of an irreducible character χ of G( ${\mathbb{F}_q}$ ), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operator of G( ${\mathbb{F}_{q^2}}$ ), and is a real-valued irreducible character in the image of lifting from G( ${\mathbb{F}_q}$ ), then χ must be an orthogonal character. We also study properties of the Frobenius–Schur indicator under Shintani lifting of regular, semisimple, and irreducible Deligne–Lusztig characters of finite reductive groups.     

The size of sums of sets,II

We prove inequalities which give lower bounds for the Lebesgue measures of setsE +K whereK is a certain kind of Cantor set. For example, ifC is the Cantor middle-thirds subset of the circle groupT, then $$m(E)^{1 - log2/log3} \leqq m(E + C)$$ for every BorelE ?T.     

Chebyshev's inequalities for functions monotone in the mean

LetR be the reals ≥ 0. LetF be the set of mapsf: {1, 2, ?,n} →R. Choosew ∈ F withw _i = w(i) > 0. PutW _i = w₁ + ? + w_i. Givenf ∈ F, define \(\bar f\) ∈F by $$\bar f\left( i \right) = \frac{{\left\{ {w_i f\left( 1 \right) + \ldots + w_i f\left( i \right)} \right\}}}{{W_i }}.$$ Callf mean increasing if \(\bar f\) is increasing. Letf ₁, ?, f_t be mean decreasing andf _t+1,?: f_t+u be mean increasing. Put $$k = W_n^u \min \left\{ {w_i^{u - 1} W_i^{t - u} } \right\}.$$ Then $$k\mathop \sum \limits_{i = 1}^n w_i f_1 \left( i \right) \ldots f_{t + u} \left( i \right) \leqslant \mathop \prod \limits_{j = 1}^{t + u} (\mathop \sum \limits_{i = 1}^n w_i f_1 (i)).$$      

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.     

Canonical Extensions and Profinite Completions of Semilattices and Lattices

Canonical extensions of (bounded) lattices have been extensively studied, and the basic existence and uniqueness theorems for these have been extended to general posets. This paper focuses on the intermediate class \({{\boldsymbol{\mathcal{S}}}}_{\wedge}\) of (unital) meet semilattices. Any \({\mathbf S}\in {{\boldsymbol{\mathcal{S}}}}_{\wedge}\) embeds into the algebraic closure system Filt(Filt(S)). This iterated filter completion, denoted Filt²(S), is a compact and \({\textstyle{\bigvee}\,}{\textstyle{\bigwedge}\,}\) -dense extension of S. The complete meet-subsemilattice S ^δ of Filt²(S) consisting of those elements which satisfy the condition of \({\textstyle{\bigwedge}\,}{\textstyle{\bigvee}\,}\) -density is shown to provide a realisation of the canonical extension of S. The easy validation of the construction is independent of the theory of Galois connections. Canonical extensions of bounded lattices are brought within this framework by considering semilattice reducts. Any S in \({{\boldsymbol{\mathcal{S}}}}_{\wedge}\) has a profinite completion, \({\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})\) . Via the duality theory available for semilattices, \({\rm Pro}_{{{\boldsymbol{\mathcal{S}}}}_{\wedge}}({\mathbf S})\) can be identified with Filt²(S), or, if an abstract approach is adopted, with \({\mathbb F_{\sqcup}}({\mathbb F_{\sqcap}}({\mathbf S}))\) , the free join completion of the free meet completion of S. Lifting of semilattice morphisms can be considered in any of these settings. This leads, inter alia, to a very transparent proof that a homomorphism between bounded lattices lifts to a complete lattice homomorphism between the canonical extensions. Finally, we demonstrate, with examples, that the profinite completion of S, for \({\mathbf S} \in {{\boldsymbol{\mathcal{S}}}}_{\wedge}\) , need not be a canonical extension. This contrasts with the situation for the variety of bounded distributive lattices, within which profinite completion and canonical extension coincide.