Showing papers in "Glasgow Mathematical Journal in 2021"
TL;DR: In this article, the authors studied the indexing systems that correspond to equivariant Steiner and linear isometries operads, and showed that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits.
Abstract: We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.
10 citations
TL;DR: In this article, practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients with respect to the Fundamental Principle of Ehrenpreis-Palamodov were discussed.
Abstract: We discuss practical methods for computing the space of solutions to an arbitrary homogeneous linear system of partial differential equations with constant coefficients. These rest on the Fundamental Principle of Ehrenpreis-Palamodov from the 1960s. We develop this further using recent advances in computational commutative algebra.
7 citations
TL;DR: For any odd prime p, the authors constructed an infinite family of imaginary quadratic fields whose class numbers are divisible by p, and gave a corollary which settled Iizuka's conjecture for the case n = 1 and p > 2.
Abstract: For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary which settles Iizuka’s conjecture for the case n = 1 and p > 2.
7 citations
TL;DR: In this article, the authors consider the set of primes of K such that the reduction is well defined and has order coprime to m. This set admits a natural density, which they are able to express as a finite sum of products of -adic integrals, where varies in the prime divisors of m. They deduce that the density is a rational number whose denominator is bounded (up to powers of m) in a very strong sense.
Abstract: Let A be the product of an abelian variety and a torus over a number field K, and let be a square-free integer. If is a point of infinite order, we consider the set of primes of K such that the reduction is well defined and has order coprime to m. This set admits a natural density, which we are able to express as a finite sum of products of -adic integrals, where varies in the set of prime divisors of m. We deduce that the density is a rational number, whose denominator is bounded (up to powers of m) in a very strong sense. This extends the results of the paper Reductions of points on algebraic groups by Davide Lombardo and the second author, where the case m prime is established.
7 citations
TL;DR: In this paper, the authors consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers and show that the cluster variables satisfy linear recurrences with periodic coefficients.
Abstract: We consider frieze sequences corresponding to sequences of cluster mutations for affine D- and E-type quivers. We show that the cluster variables satisfy linear recurrences with periodic coefficients, which imply the constant coefficient relations found by Keller and Scherotzke. Viewing the frieze sequence as a discrete dynamical system, we reduce it to a symplectic map on a lower dimensional space and prove Liouville integrability of the latter.
6 citations
TL;DR: By disregarding generators away from homological degree 0, the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of theKhovanov complex are used to give an alternative way to compute Rasmussen s-invariants of knots.
Abstract: We use the divide-and-conquer and scanning algorithms for calculating Khovanov cohomology directly on the Lee- or Bar-Natan deformations of the Khovanov complex to give an alternative way to compute Rasmussen s-invariants of knots. By disregarding generators away from homological degree 0, we can considerably improve the efficiency of the algorithm. With a slight modification, we can also apply it to a refinement of Lipshitz–Sarkar.
6 citations
TL;DR: For an n-tuple of positive invertible operators on a Hilbert space, this paper presented some variants of Ando-Hiai type inequalities for deformed means from an n variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map.
Abstract: For an n-tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n-variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the power mean from the deformed mean in terms of the generalized Kantorovich constants under the operator order. Moreover, we improve the norm inequality for the operator power means related to the Log-Euclidean mean in terms of the Specht ratio.
6 citations
TL;DR: In this article, the authors give necessary and sufficient criteria for skew polynomials of low degree to be irreducible and give examples of new division algebras Sf.
Abstract: Let D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.
6 citations
TL;DR: In this article, it was shown that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid, and that such ideals satisfy the torsion condition of Huneke and Wiegand.
Abstract: We prove that each positive power of the maximal ideal of a commutative Noetherian local ring is Tor-rigid and strongly rigid. This gives new characterizations of regularity and, in particular, shows that such ideals satisfy the torsion condition of a long-standing conjecture of Huneke and Wiegand.
6 citations
TL;DR: In this article, the authors introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Goresstein flat precovers.
Abstract: In this paper, we introduce and study the Gorenstein relative homology theory for unbounded complexes of modules over arbitrary associative rings, which is defined using special Gorenstein flat precovers. We compare the Gorenstein relative homology to the Tate/unbounded homology and get some results that improve the known ones.
5 citations
TL;DR: In this article, it was shown that γn+1(G) has finite (m, n) -bounded order, which generalizes the much-celebrated theorem of B. H. Neumann that the commutator subgroup of a BFC-group is finite.
Abstract: Let γn = [x1,…,xn] be the nth lower central word. Denote by Xn the set of γn -values in a group G and suppose that there is a number m such that for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.
TL;DR: In this paper, it was shown that a four-dimensional compact oriented half-conformally flat Riemannian manifold M 4 is topologically or topologically equivalent to a connected sum of copies of the complex projective plane or the 4-sphere.
Abstract: In this note, we prove that a four-dimensional compact oriented half-conformally flat Riemannian manifold M 4 is topologically or , provided that the sectional curvatures all lie in the interval In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the 4-sphere.
TL;DR: In this paper, the authors define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes, and investigate the dependence on the prime and establish homology bounds in terms of p-adjacency simplicial volumetric volumes.
Abstract: We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume.
TL;DR: In this paper, the relationship of the σ-nilpotent length of a σi-group with its σ -primary norm was studied, and it was shown that the ratio of the length of the shortest normal chain of a group G to the residual of all normal subgroups of G is at most r (r > 1) if and only if lσ (G/Nσ (g)) ≤ r.
Abstract: Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes . Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× … × Gt for some σ-primary groups G1, … , Gt; σ-soluble if every chief factor of G is σ-primary. We use to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, Then the subgroup Nσ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if lσ (G/ Nσ (G)) ≤ r.
TL;DR: In this article, it was shown that a finite index regular inclusion of $II_1$-factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra.
Abstract: We prove that a finite index regular inclusion of $II_1$-factors with
commutative first relative commutant is always a crossed product subfactor with
respect to a minimal action of a biconnected weak Kac algebra. Prior to this,
we prove that every finite index inclusion of $II_1$-factors which is of depth
$2$ and has simple first relative commutant (respectively, is regular and has
commutative or simple first relative commutant) admits a two-sided Pimsner-Popa
basis (respectively, a unitary orthonormal basis)
TL;DR: In this article, the centers of the categories of tilting modules for G = SL2 in prime characteristic, of tilts modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g = 1, 2.
Abstract: In this note we compute the centers of the categories of tilting modules for G=SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g=1,2.
TL;DR: For a wide range of integers s (2 < s < p − 2), it was shown in this article that the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.
Abstract: It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.
TL;DR: In this article, the authors examined various averages of the function r3(n) and found that the understanding of this function is surprisingly poor, and examined various average representations of it.
Abstract: This paper is concerned with the function r3(n), the number of representations of n as the sum of at most three positive cubes, , Our understanding of this function is surprisingly poor, and we examine various averages of it. In particular and
TL;DR: In this article, it was shown that any finite group can not appear as the automorphism group of a finite graph, and a number of negative results were shown for the case of quantum groups.
Abstract: A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.
TL;DR: In this article, the authors construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $BU_\mathbb{R}(-)$.
Abstract: We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $BU_\mathbb{R}(-)$. The calculus produces a Taylor tower, the $n$-th layer of which is classified by a spectrum with an action of $C_2 \ltimes U(n)$.
We further give model categorical considerations, producing a zig-zag of Quillen equivalences between spectra with an action of $C_2 \ltimes U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the $n$-th layer of the Taylor tower.
TL;DR: For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, a characterization of closed ideals of C0(X,A) was discussed in this paper.
Abstract: For a locally compact Hausdorff space X and a C*-algebra A with only finitely many closed ideals, we discuss a characterization of closed ideals of C0(X,A) in terms of closed ideals of A and a class of closed subspaces of X. We further use this result to prove that a closed ideal of C0(X)⊗min A is a finite sum of product ideals. We also establish that for a unital C*-algebra A, C0(X,A) has the centre-quotient property if and only if A has the centre-quotient property. As an application, we characterize the closed Lie ideals of C0(X,A) and identify all the closed Lie ideals of HC0(X)⊗minB(H), H being a separable Hilbert space.
TL;DR: In this paper, it was shown that for a R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff m is D4.
Abstract: A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.
TL;DR: In this article, the authors determined (non-)triviality of Samelson products of inclusions of factors of the mod $p$ decomposition of a Lie group for the special case of exceptional Lie groups.
Abstract: We determine (non-)triviality of Samelson products of inclusions of factors of the mod $p$ decomposition of $G_{(p)}$ for $(G,p)=(E_7,5),(E_7,7),(E_8,7)$. This completes the determination of (non-)triviality of those Samelson products in $p$-localized exceptional Lie groups when $G$ has $p$-torsion free homology.
TL;DR: In this paper, it was shown that there are infinitely many knots of every fixed genus which do not admit surgery to an L-space, despite resembling algebraic knots and L -space knots in general.
Abstract: We show there exist infinitely many knots of every fixed genus $g\geq 2$ which do not admit surgery to an L-space, despite resembling algebraic knots and L-space knots in general: they are algebraically concordant to the torus knot $T(2,2g+1)$ of the same genus and they are fibred and strongly quasipositive.
TL;DR: In this article, it was shown that if there exists a finite nilpotent group with a subgroup whose vertices are proper subgroups of a finite group and in which two vertices $H$ and $K$ are joined by an edge, then the subgroup is supersoluble.
Abstract: Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a finite nilpotent group $X$ with $\Delta(G)\cong \Delta(X),$ then $G$ is supersoluble.
TL;DR: In this article, it was shown that the profinite completion of a projective group is projective, i.e., the group is complete when all members of the group are projective.
Abstract:
We prove that the profinite completion of a profinite projective group is projective.
TL;DR: In this article, the notion of Gorenstein silting complexes was introduced and studied, which is a generalization of the tilting modules in the context of a single-input single-output (SISO) configuration.
Abstract: Abstract We introduce and study the notion of Gorenstein silting complexes, which is a generalization of Gorenstein tilting modules in Gorenstein-derived categories. We give the equivalent characterization of Gorenstein silting complexes. We give a sufficient condition for a partial Gorenstein silting complex to have a complement.
TL;DR: In this article, it was shown that a Markov operator T is uniformly P-ergodic if and only if T − P is characterized by the spectral radius of T − p, where P is a projection.
Abstract: In the present paper, we deal with asymptotical stability of Markov operators acting on abstract state spaces (i.e. an ordered Banach space, where the norm has an additivity property on the cone of positive elements). Basically, we are interested in the rate of convergence when a Markov operator T satisfies the uniform P-ergodicity, i.e. , here P is a projection. We have showed that T is uniformly P-ergodic if and only if , . In this paper, we prove that such a β is characterized by the spectral radius of T − P. Moreover, we give Deoblin’s kind of conditions for the uniform P-ergodicity of Markov operators.
TL;DR: In this article, the authors consider a family of Lehn-Lehn-Sorger-van Straten hyperkahler 8folds with a non-symplectic automorphism of order 3.
Abstract: We consider a $10$-dimensional family of Lehn-Lehn-Sorger-van Straten hyperkahler eightfolds which have a non-symplectic automorphism of order $3$. Using the theory of finite-dimensional motives, we show that the action of this automorphism on the Chow group of $0$-cycles is as predicted by the Bloch-Beilinson conjectures. We prove a similar statement for the anti-symplectic involution on varieties in this family. This has interesting consequences for the intersection product in the Chow ring of these varieties.