A Short Note on Essentially Σ1 Sentences

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ₁ (i.e., it is Σ₁ under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ₁ formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ ₀  + Supexp and we address a similar problem for IΔ ₀  + Exp.     

On Probability and Moment Inequalities for Supermartingales and Martingales

Burkholder’s type inequality is stated for the special class of martingales, namely the product of independent random variables. The constants in the latter are much less than in the general case which is considered in Nagaev (Acta Appl. Math. 79, 35–46, 2003; Teor. Veroyatn. i Primenen. 51(2), 391–400, 2006). On the other hand, the moment inequality is proved, which extends these by Wittle (Teor. Veroyatn. i Primenen. 5(3), 331–334, 1960) and Dharmadhikari and Jogdeo (Ann. Math. Stat. 40(4), 1506–1508, 1969) to martingales.     

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).     

Tilting Theory and Functor Categories I. Classical Tilting

Tilting theory has been a very important tool in the classification of finite dimensional algebras of finite and tame representation type, as well as, in many other branches of mathematics. Happel (1988) and Cline et al. (J Algebra 304:397–409 1986) proved that generalized tilting induces derived equivalences between module categories, and tilting complexes were used by Rickard (J Lond Math Soc 39:436–456, 1989) to develop a general Morita theory of derived categories. On the other hand, functor categories were introduced in representation theory by Auslander (I Commun Algebra 1(3):177–268, 1974), Auslander (1971) and used in his proof of the first Brauer–Thrall conjecture (Auslander 1978) and later on, used systematically in his joint work with I. Reiten on stable equivalence (Auslander and Reiten, Adv Math 12(3):306–366, 1974), Auslander and Reiten (1973) and many other applications. Recently, functor categories were used in Martínez-Villa and Solberg (J Algebra 323(5):1369–1407, 2010) to study the Auslander–Reiten components of finite dimensional algebras. The aim of this paper is to extend tilting theory to arbitrary functor categories, having in mind applications to the functor category Mod (mod_Λ), with Λ a finite dimensional algebra.     

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.     

A Characterization of the Finite Veronesean by Intersection Properties

A combinatorial characterization of the Veronese variety of all quadrics in PG(n, q) by means of its intersection properties with respect to subspaces is obtained. The result relies on a similar combinatorial result on the Veronesean of all conics in the plane PG(2, q) by Ferri [Atti Accad. Naz. Lincei Rend. 61(6), 603?C610 (1976)], Hirschfeld and Thas [General Galois Geometries. Oxford University Press, New York (1991)], and Thas and Van Maldeghem [European J. Combin. 25(2), 275?C285 (2004)], and a structural characterization of the quadric Veronesean by Thas and Van Maldeghem [Q. J. Math. 55(1), 99?C113 (2004)].     

Families of Finite-Dimensional Hopf Algebras with the Chevalley Property

We introduce and study new families of finite-dimensional Hopf algebras with the Chevalley property that are not pointed nor semisimple arising as twistings of quantum linear spaces. These Hopf algebras generalize the examples introduced in Andruskiewitsch et al. (Mich Math J 49(2):277–298, 2001), Etingof and Gelaki (Int Math Res Not 14:757–768, 2002, Math Res Lett 8:249–255, 2001).     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

Preduals of Quadratic Campanato Spaces Associated to Operators with Heat Kernel Bounds

Let L be a nonnegative, self-adjoint operator on \(L^{2}(\mathbb {R}^{n})\) with the Gaussian upper bound on its heat kernel. As a generalization of the square Campanato space \(\mathcal {L}^{2,\lambda }_{-\Delta }(\mathbb R^{n})\) , in Duong et al. (J. Fourier Anal. Appl. 13:87–111, 2007) the quadratic Campanato space \(\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})\) is defined by a variant of the maximal function associated with the semigroup {e ^{?t L} } _t≥0. On the basis of Dafni and Xiao (J. Funct. Anal. 208:377–422, 2004) and Yang and Yuan (J. Funct. Anal. 255:2760–2809, 2008) this paper addresses the preduality of \(\mathcal {L}_{L}^{2,\lambda }(\mathbb {R}^{n})\) through an induced atom (or molecular) decomposition. Even in the case L = ?Δ the discovered predual result is new and natural.     

Some Integral Representation for Meta-Monogenic Function in Clifford Algebras Depending on Parameters

In recent years, the integral representation problems have been studied in many context and generalities. For example, for the monogenic and meta functions in some Clifford type algebras, see [10, 11]. In this paper we construct a Cauchy-Pompeiu type formula for meta-monogenic operator of order ${n, (D-\lambda)^n, \lambda \in \mathbb{R}}$ , and its conjugate ${(\overline{D} - \lambda)^n}$ in a Clifford algebra depending on parameters ${\mathcal{A}_n(2, \alpha_j, \gamma_{ij})}$ . Using these explicit representation formula of Cauchy-Pompeiu type we will show some applications.     

Whitney algebras and Grassmann��s regressive products

Geometric products on tensor powers ${\Lambda(V)^{\otimes m}}$ of an exterior algebra and on Whitney algebras (Crapo and Schmitt in J. Comb. Theory A 91:215?C263, 2000) provide a rigorous version of Grassmann??s regressive products of 1844 (Grassmann in Die Lineale Ausdehnungslehre, Verlag von Otto Wigand, Leipzig, 1844). We study geometric products and their relations with other classical operators on exterior algebras, such as the Hodge *?operators and the join and meet products in Cayley?CGrassmann algebras (Barnabei et?al. in J. Algebra 96:120?C160, 1985; Stewart in Nature 321:17, 1986). We establish encodings of tensor powers ${\Lambda(V)^{\otimes m}}$ and of Whitney algebras W ^m (M) in terms of letterplace algebras and of their geometric products in terms of divided powers of polarization operators. We use these encodings to provide simple proofs of the Crapo and Schmitt exchange relations in Whitney algebras and of two typical classes of identities in Cayley?CGrassmann algebras.     

Integral Bases of Cluster Algebras and Representations of Tame Quivers

In Caldero and Keller (Invent Math 172:169–211, 2008) and Sherman and Zelevinsky (Mosc Math J 4(4):947–974, 2004), the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$ , respectively. In this paper, we will deduce ?-bases for cluster algebras of affine types.     

The Mixed Curvatures on C N

The aim of the present paper is devoted to the investigation of some geometrical properties on the middle envelope in terms of the invariants of the third quadratic form of the normal line congruence C_N . The mixed middle curvature and mixed curvature on C_N are obtained in tenus of the Mean and Gauss curvatures of the surface of reference. Our study is considered as a continuation to Stephanidis ([1], [2], [3], [4], [5]). The technique adapted here is based on the methods of moving frames and their related exteriour forms [6] and [7].     

Generation and Enumeration of Some Classes of Interval Orders

In this paper we consider the class of interval orders, recently considered by several authors from both an algebraic and an enumerative point of view. According to Fishburn’s Theorem (Fishburn J Math Psychol 7:144–149, 1970), these objects can be characterized as posets avoiding the poset 2?+?2. We provide a recursive method for the unique generation of interval orders of size n?+?1 from those of size n, extending the technique presented by El-Zahar (1989) and then re-obtain the enumeration of this class, as done in Bousquet-Melou et al. (2010). As a consequence we provide a method for the enumeration of several subclasses of interval orders, namely AV(2?+?2, N), AV(2?+?2, 3?+?1), AV(2?+?2, N, 3?+?1). In particular, we prove that the first two classes are enumerated by the sequence of Catalan numbers, and we establish a bijection between the two classes, based on the cardinalities of the principal ideals of the posets.     

Computing Final Polynomials and Final Syzygies Using Buchberger’s Gröbner Bases Method

Final polynomials and final syzygies provide an explicit representation of polynomial identities promised by Hilbert’s Nullstellensatz. Such representations have been studied independently by Bokowski [2,3,4] and Whiteley [23,24] to derive invariant algebraic proofs for statements in geometry. In the present paper we relate these methods to some recent developments in computational algebraic geometry. As the main new result we give an algorithm based on B. Buchberger’s Gröbner bases method for computing final polynomials and final syzygies over the complex numbers. Degree upper bound for final polynomials are derived from theorems of Lazard and Brownawell, and a topological criterion is proved for the existence of final syzygies. The second part of this paper is expository and discusses applications of our algorithm to real projective geometry, invariant theory and matrix theory.     

Filtrations on the knot contact homology of transverse knots

We construct a new invariant of transverse links in the standard contact structure on ${\mathbb R }^3.$ This invariant is a doubly filtered version of the knot contact homology differential graded algebra (DGA) of the link, see (Ekholm et al., Knot contact homology, Arxiv:1109.1542, 2011; Ng, Duke Math J 141(2):365–406, 2008). Here the knot contact homology of a link in ${\mathbb R }^3$ is the Legendrian contact homology DGA of its conormal lift into the unit cotangent bundle $S^*{\mathbb R }^3$ of ${\mathbb R }^3$ , and the filtrations are constructed by counting intersections of the holomorphic disks of the DGA differential with two conormal lifts of the contact structure. We also present a combinatorial formula for the filtered DGA in terms of braid representatives of transverse links and apply it to show that the new invariant is independent of previously known invariants of transverse links.     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

Burgers type equations, Gelfand’s problem and Schumpeterian dynamics

Burgers?? equations have been introduced to study different models of fluids (Bateman, 1915, Burgers, 1939, Hopf, 1950, Cole, 1951, Lighthill andWhitham, 1955, etc.). The difference-differential analogues of these equations have been proposed for Schumpeterian models of economic development (Iwai, 1984, Polterovich and Henkin, 1988, Belenky, 1990, Henkin and Polterovich, 1999, Tashlitskaya and Shananin, 2000, etc.). This paper gives a short survey of the results and conjectures on Burgers type equations, motivated both by fluid mechanics and by Schumpeterian dynamics. Proofs of some new results are given. This paper is an extension and an improvement of (Henkin, 2007, 2011).     

On a variant of Rado’s selection lemma and its equivalence with the Boolean prime ideal theorem

We establish that, in ZF (i.e., Zermelo–Fraenkel set theory minus the Axiom of Choice AC), the statement RLT: Given a set I and a non-empty set \({\mathcal{F}}\) of non-empty elementary closed subsets of 2 ^I satisfying the fip, if \({\mathcal{F}}\) has a choice function, then \({\bigcap\mathcal{F} \ne \emptyset}\) , which was introduced in Morillon (Arch Math Logic 51(7–8):739–749, 2012), is equivalent to the Boolean Prime Ideal Theorem (see Sect. 1 for terminology). The result provides, on one hand, an affirmative answer to Morillon’s corresponding question in Morillon (2012) and, on the other hand, a negative answer—in the setting of ZFA (i.e., ZF with the axiom of extensionality weakened to permit the existence of atoms)—to the question in Morillon (2012) of whether RLT is equivalent to Rado’s selection lemma.     

Growth,entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfies some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod–Shafarevich algebras. This paper provides bounds for the growth function on images of Golod–Shafarevich algebras based upon the number of defining relations. This extends results from Smoktunowicz and Bartholdi (Q J Math. doi:10.1093/qmath/hat005 2013) and Smoktunowicz (J Algebra 381:116–130, 2013). Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky (A private communication, 2013) by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov (2013). Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss (Noncommutative deformations and flops, ArXiv:1309.0698v2 [math.AG]).