On the relational complexity of a finite permutation group

The relational complexity \(\rho (X,G)\) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or (preferably) precise values for \(\rho \) in natural cases; and (b) a rough determination of the primitive permutation groups with \(\rho \) either very small (bounded) or very large (much larger than the logarithm of the degree). The rough version of (a) is relevant to (b). Our main result is an explicit characterization of the binary (\(\rho =2\)) primitive affine permutation groups. We also compute the precise relational complexity of \({{\mathrm{Alt}}}_n\) acting on k-sets, correcting (Cherlin in Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars, 1996–1999, pp. 15–48, Birkhäuser 2000, Example 5).     

Counting nonsingular matrices with primitive row vectors

We give an asymptotic expression for the number of nonsingular integer $n\times n$ -matrices with primitive row vectors, determinant $k$ , and Euclidean matrix norm less than $T$ , as $T\rightarrow \infty $ . We also investigate the density of matrices with primitive rows in the space of matrices with determinant $k$ , and determine its asymptotics for large $k$ .     

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} $ .     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The circle method and bounds for L-functions - I

Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $\chi $ be a primitive character of conductor $M$ . For the twisted $L$ -function $L(s, f\otimes \chi )$ we establish the hybrid subconvex bound $$\begin{aligned} L\left( \frac{1}{2}+it, f\otimes \chi \right) \ll (M(3+|t|))^{\frac{1}{2}-\frac{1}{18}+\varepsilon }, \end{aligned}$$ for $t\in \mathbb{R }$ . The implied constant depends only on the form $f$ and $\varepsilon $ .     

A construction for strength-3 covering arrays from linear feedback shift register sequences

We present a new construction for covering arrays inspired by ideas from Munemasa (Finite Fields Appl 4:252–260, 1998) using linear feedback shift registers (LFSRs). For a primitive polynomial \(f\) of degree \(m\) over \(\mathbb F _q\) , by taking all unique subintervals of length \(\frac{q^m-1}{q-1}\) from the LFSR generated by \(f\) , we derive a general construction for optimal variable strength orthogonal arrays over an infinite family of abstract simplicial complexes. For \(m=3\) , by adding the subintervals of the reversal of the LFSR to the variable strength orthogonal array, we derive a strength-3 covering array over \(q^2+q+1\) factors, each with \(q\) levels that has size only \(2q^3-1\) , i.e. a \(\text {CA}(2q^3-1; 3, q^2+q+1, q)\) whenever \(q\) is a prime power. When \(q\) is not a prime power, we obtain results by using fusion operations on the constructed array for higher prime powers and obtain improved bounds. Colbourn maintains a repository of the best known bounds for covering array sizes for all \(2 \le q \le 25\) . Our construction, with fusing when applicable, currently holds records of the best known upper bounds in this repository for all \(q\) except \(q = 2,3,6\) . By using these covering arrays as ingredients in recursive constructions, we build covering arrays over larger numbers of factors, again providing significant improvements on the previous best upper bounds.     

Ultraproducts of continuous posets

We consider local partial clones defined on an uncountable set E having the form Polp(\({\mathfrak{R}}\)), where \({\mathfrak{R}}\) is a set of relations on E. We investigate the notion of weak extendability of partial clones of the type Polp(\({\mathfrak{R}}\)) (in the case of E countable, this coincides with the notion of extendability previously introduced by the author in 1987) which allows us to expand to uncountable sets results on the characterization of Galois-closed sets of relations as well as model-theoretical properties of a relational structure \({\mathfrak{R}}\). We establish criteria for positive primitive elimination sets (sets of positive primitive formulas over \({\mathfrak{R}}\) through which any positive primitive definable relation over \({\mathfrak{R}}\) can be expressed without existential quantifiers) for finite \({\mathfrak{R}}\) as well as for \({\mathfrak{R}}\) having only finite number of positive primitive definable relations of any arity. Emphasizing the difference between countable and uncountable sets, we show that, unlike in the countable case, the characterization of Galois-closed sets InvPol(\({\mathfrak{R}}\)) (that is, all relations which are invariant under all operations from the clone Pol(\({\mathfrak{R}}\)) defined on an uncountable set) cannot be obtained via the application of finite positive primitive formulas together with infinite intersections and unions of updirected sets of relations from \({\mathfrak{R}}\).     

On the distinctness of modular reductions of primitive sequences over Z/(232−1)

This paper studies the distinctness of modular reductions of primitive sequences over ${\mathbf{Z}/(2^{32}-1)}$ . Let f(x) be a primitive polynomial of degree n over ${\mathbf{Z}/(2^{32}-1)}$ and H a positive integer with a prime factor coprime with 2³²?1. Under the assumption that every element in ${\mathbf{Z}/(2^{32}-1)}$ occurs in a primitive sequence of order n over ${\mathbf{Z}/(2^{32}-1)}$ , it is proved that for two primitive sequences ${\underline{a}=(a(t))_{t\geq 0}}$ and ${\underline{b}=(b(t))_{t\geq 0}}$ generated by f(x) over ${\mathbf{Z}/(2^{32}-1), \underline{a}=\underline{b}}$ if and only if ${a\left( t\right) \equiv b\left( t\right) \bmod{H}}$ for all t ≥ 0. Furthermore, the assumption is known to be valid for n between 7 and 100, 000, the range of which is sufficient for practical applications.     

On affine sub-families of Grain-like structures

Grain is one of eSTREAM hardware-oriented finalists. It contains a cascade connection of an 80-bit primitive linear feedback shift registers (\({{\mathrm{LFSR}}}\)) into an 80-bit nonlinear feedback shift register (\({{\mathrm{NFSR}}}\)). The variant Grain-128 has a cascade connection with both \({{\mathrm{LFSR}}}\) and \({{\mathrm{NFSR}}}\) of order 128. We consider Grain-like structures, i.e., the cascade connection of a primitive \({{\mathrm{LFSR}}}\) into an \({{\mathrm{NFSR}}}\) of the same order. It is easy to know that in such a structure, all the affine sub-families of the \({{\mathrm{NFSR}}}\) are also the affine sub-families of the cascade connection. We prove that if the degree of the characteristic function of the \({{\mathrm{NFSR}}}\) is bigger than 2, then affine sub-families of the cascade connection must also be affine sub-families of the \({{\mathrm{NFSR}}}\). The same result holds if the order of the primitive \({{\mathrm{LFSR}}}\) is bigger than the order of the \({{\mathrm{NFSR}}}\).     

Solvable Permutation Groups and Orbits on Power Sets

We consider the class ? of finitely generated toral relatively hyperbolic groups. We show that groups from ? are commutative transitive and generalize a theorem proved by Benjamin Baumslag in [3 Baumslag, B. (1967). Residually free groups. Prceedings of the London Mathematical Society 17(3):402–418.[Crossref] , [Google Scholar]] to this class. We also discuss two definitions of (fully) residually-𝒞 groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii [18 Ol'shanskii, A. Yu. (1993). On residualing homomorphisms and G-subgroups of hyperbolic groups. International Journal of Algebra Computation 3:365–409.[Crossref] , [Google Scholar]] and Osin [22 Osin, D. V. (2010). Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of mathematics 172:1–39.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the equivalence of the two definitions for 𝒞 = ?. This is a generalization of the similar result obtained by Ol'shanskii for 𝒞 being the class of torsion-free hyperbolic groups. Let Γ ∈ ? be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14 Kharlampovich, O., Myasnikov, A. (2012). Limits of relatively hyperbolic groups and Lyndon's completions. Journal of the European Math. Soc. 14:659–680.[Crossref], [Web of Science ®] , [Google Scholar]] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from ?. On the other hand, we give an example of a finitely generated torsion-free fully residually-? group that does not embed into a group from ?; ? is the class of hyperbolic groups.     

Poisson Hopf algebra related to a twisted quantum group

A Poisson algebra ?[G] considered as a Poisson version of the twisted quantized coordinate ring ?_q,p[G], constructed by Hodges et al. [11 Hodges, T. J., Levasseur, T., Toro, M. (1997). Algebraic structure of multi-parameter quantum groups. Adv. Math. 126:52–92.[Crossref], [Web of Science ®] , [Google Scholar]], is obtained and its Poisson structure is investigated. This establishes that all Poisson prime and primitive ideals of ?[G] are characterized. Further it is shown that ?[G] satisfies the Poisson Dixmier-Moeglin equivalence and that Zariski topology on the space of Poisson primitive ideals of ?[G] agrees with the quotient topology induced by the natural surjection from the maximal ideal space of ?[G] onto the Poisson primitive ideal space.     

Galois groups of some iterated polynomials over cyclotomic extensions

Let \(\varphi _p(z)=(z-1)^p+2-\zeta _p\), where \(\zeta _p\in \bar{\mathbb {Q}}\) is a primitive pth root of unity. Building on previous work, we show that the nth iterate \(\varphi _p^n(z)\) has Galois group \([C_p]^n\), an iterated wreath product of cyclic groups, whenever p is not a Wieferich prime.     

Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect

The purpose of this paper is to prove that a primitive Hilbert cusp form \(\mathbf{g}\) is uniquely determined by the central values of the Rankin–Selberg L-functions \(L(\mathbf{f}\otimes \mathbf{g}, \frac{1}{2})\), where \(\mathbf{f}\) runs through all primitive Hilbert cusp forms of weight \(k\) for infinitely many weight vectors \(k\). This result is a generalization of the work of Ganguly et al. (Math Ann 345:843–857, 2009) to the setting of totally real number fields, and it is a weight aspect analogue of our previous work (Hamieh and Tanabe in Trans Am Math Soc, arXiv:1609.07209, 2016).     

Cell decomposition of some unitary group Rapoport–Zink spaces

In this paper we study the \(p\) -adic analytic geometry of the basic unitary group Rapoport–Zink spaces \(\mathcal {M}_K\) with signature \((1,n-1)\) . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Théorie de la réduction pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special fiber \(\mathcal {M}_{red}\) defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain \(\mathcal {D}_K\) in \(\mathcal {M}_K\) for the action of \(G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)\) , the product of the associated \(p\) -adic reductive groups, and prove that \(\mathcal {M}_K\) admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720, 2013).     

Some modular relations for Ramanujan’s function k(q)=r(q)r^2(q^2)

In both his second and lost notebooks, Ramanujan introduced and studied a function \(k(q)=r(q)r^2(q^2)\) , where \(r(q)\) is the Rogers–Ramanujan continued fraction. Ramanujan also recorded five beautiful relations between the Rogers–Ramanujan continued fraction \(r(q)\) and the five continued fractions \(r(-q)\) , \(r(q^2)\) , \(r(q^3)\) , \(r(q^4)\) , and \(r(q^5)\) . Motivated by those relations, we present some modular relations between \(k(q)\) and \(k(-q)\) , \(k(-q^2)\) , \(k(q^3)\) , and \(k(q^5)\) in this paper.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

Algorithmic aspects of Steiner convexity and enumeration of Steiner trees

For a set \(W\) of vertices of a connected graph \(G=(V(G),E(G))\) , a Steiner W-tree is a connected subgraph \(T\) of \(G\) such that \(W\subseteq V(T)\) and \(|E(T)|\) is minimum. Vertices in \(W\) are called terminals. In this work, we design an algorithm for the enumeration of all Steiner \(W\) -trees for a constant number of terminals, which is the usual scenario in many applications. We discuss algorithmic issues involving space requirements to compactly represent the optimal solutions and the time delay to generate them. After generating the first Steiner \(W\) -tree in polynomial time, our algorithm enumerates the remaining trees with \(O(n)\) delay (where \(n=|V(G)|\) ). An algorithm to enumerate all Steiner trees was already known (Khachiyan et al., SIAM J Discret Math 19:966–984, 2005), but this is the first one achieving polynomial delay. A by-product of our algorithm is a representation of all (possibly exponentially many) optimal solutions using polynomially bounded space. We also deal with the following problem: given \(W\) and a vertex \(x\in V(G)\setminus W\) , is \(x\) in a Steiner \(W'\) -tree for some \(\emptyset \ne W' \subseteq W\) ? This problem is investigated from the complexity point of view. We prove that it is NP-hard when \(W\) has arbitrary size. In addition, we prove that deciding whether \(x\) is in some Steiner \(W\) -tree is NP-hard as well. We discuss how these problems can be used to define a notion of Steiner convexity in graphs.     

Analytic sets and extension of holomorphic maps of positive codimension

Let D, \(D'\) be arbitrary domains in \({\mathbb C}^n\) and \({\mathbb C}^N\) respectively, \(1<n\le N\), both possibly unbounded and \(M \subseteq \partial D\), \(M'\subseteq \partial D'\) be open pieces of the boundaries. Suppose that \(\partial D\) is smooth real-analytic and minimal in an open neighborhood of \({\bar{M}}\) and \(\partial D'\) is smooth real-algebraic and minimal in an open neighborhood of \({\bar{M}'}\). Let \(f: D\rightarrow D'\) be a holomorphic mapping such that the cluster set \(\mathrm{cl}_{f}(M)\) does not intersect \(D'\). It is proved that if the cluster set \(\mathrm{cl}_{f}(p)\) of some point \(p\in M\) contains some point \(q\in M'\) and the graph of f extends as an analytic set to a neighborhood of \((p, q)\in {\mathbb {C}}^n \times {\mathbb C}^N\), then f extends as a holomorphic map to a dense subset of some neighborhood of p. If in addition, \(M =\partial D\), \(M'=\partial D'\) and \(M'\) is compact, then f extends holomorphically across an open dense subset of \(\partial D\).     

On the least primitive root in number fields

Let K be an algebraic number field and $ \mathfrak{O} $ _K its ring of integers. For any prime ideal $ \mathfrak{p} $ , the group $ (\mathfrak{O}_K /\mathfrak{p})* $ of the reduced residue classes of integers is cyclic. We call any element of a generator of the group $ (\mathfrak{O}_K /\mathfrak{p})* $ a primitive root modulo $ \mathfrak{p} $ . Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal $ \mathfrak{p} $ under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.