Search

The question of when a vertex algebra is a quantization of the arc space of its associated scheme has recently received a lot of attention in both the mathematics and physics literature. This property was first studied by Tomoyuki Arakawa and Anne Moreau (see their paper in the references), and was given the name \lq\lq classical freeness" by Jethro van Ekeren and Reimundo Heluani [Comm. Math. Phys. 386 (2021), no. 1, pp. 495-550] in their work on chiral homology. Later, it was extended to vertex superalgebras by Hao Li [Eur. J. Math. 7 (2021), pp. 1689–1728]. In this note, we prove the classical freeness of the simple affine vertex superalgebra L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{m|2r}) for all positive integers m,n,r satisfying -\frac {m}{2} + r +n+1 > 0. In particular, it holds for the rational vertex superalgebras L_n(\mathfrak {o}\mathfrak {s}\mathfrak {p}_{1|2r}) for all positive integers r,n. [ABSTRACT FROM AUTHOR]

Let p \geq 7 be a prime number. Let S(3) denote the third Morava stabilizer algebra. In recent years, Kato-Shimomura and Gu-Wang-Wu found several families of nontrivial products in the stable homotopy ring of spheres \pi _* (S) using H^{*,*} (S(3)). In this paper, we determine all nontrivial products in \pi _* (S) of the Greek letter family elements \alpha _s, \beta _s, \gamma _s and Cohen's elements \zeta _n which are detectable by H^{*,*} (S(3)). In particular, we show \beta _1 \gamma _s \zeta _n \neq 0 \in \pi _*(S), if n \equiv 2 mod 3, s \not \equiv 0, \pm 1 mod p. [ABSTRACT FROM AUTHOR]

Equivariant monoids are very important objects in many branches of mathematics: they combine the notion of multiplication and the concept of a group action. In this paper we will construct categories which encode the structure borne by monoids with a group action by combining the theory of product and permutation categories (PROPs) and product and braid categories (PROBs) with the theory of crossed simplicial groups. PROPs and PROBs are categories used to encode structures borne by objects in symmetric and braided monoidal categories respectively, whilst crossed simplicial groups are categories which encode a unital, associative multiplication and a compatible group action. We will produce PROPs and PROBs whose categories of algebras are equivalent to the categories of monoids, comonoids and bimonoids with group action using extensions of the symmetric and braid crossed simplicial groups. We will extend this theory to balanced braided monoidal categories using the ribbon braid crossed simplicial group. Finally, we will use the hyperoctahedral crossed simplicial group to encode the structure of an involutive monoid with a compatible group action. [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

In this short paper, we derive a new result on Umehara algebra. As a consequence, we prove that an indefinite complex hyperbolic space and an indefinite complex projective space do not share a common complex submanifold with induced metrics, answering a question raised in Cheng et al. [ABSTRACT FROM AUTHOR]

In this paper, we prove a result similar to results of Itoh [J. Algebra 150 (1992), pp. 101–117] and Hong-Ulrich [J. Lond. Math. Soc. (2) 90 (2014), pp. 861–878], proving that integral closure of an ideal is compatible with specialization by a general element of that ideal for ideals of height at least two in a large class of rings. Moreover, we show integral closure of sufficiently large powers of the ideal is compatible with specialization by a general element of the original ideal. In a polynomial ring over an infinite field, we give a class of squarefree monomial ideals for which the integral closure is compatible with specialization by a general linear form. [ABSTRACT FROM AUTHOR]

Let (A,\mathfrak {m}) be an excellent normal domain of dimension two containing a field k \cong A/\mathfrak {m}. An \mathfrak {m}-primary ideal I is a p_g-ideal if the Rees algebra A[It] is a Cohen-Macaulay normal domain. If k is algebraically closed then Okuma, Watanabe and Yoshida proved that A has p_g-ideals and furthermore product of two p_g-ideals is a p_g ideal. Previously we showed that if k has characteristic zero then A has p_g-ideals. In this paper we prove that if k is perfect field of positive characteristic then also A has p_g ideals. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40]. [ABSTRACT FROM AUTHOR]

We consider associative algebras with an action by derivations by some finite dimensional and semisimple Lie algebra. We prove that if a differential variety has almost polynomial growth, then it is generated by one of the algebras UT_2(W_\lambda) or End(W_\mu) for some integral dominant weight \lambda,\mu with \mu \neq 0. In the special case L=\mathfrak {sl}_2 we prove that this is a sufficient condition too. [ABSTRACT FROM AUTHOR]

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras. [ABSTRACT FROM AUTHOR]

We prove the strong Lefschetz property for Artinian Gorenstein algebras generated by the relative invariants of prehomogeneous vector spaces of commutative parabolic type. [ABSTRACT FROM AUTHOR]

Let V be a vertex algebra of countable dimension, G a subgroup of AutV of finite order, V^{G} the fixed point subalgebra of V under the action of G, and \mathscr {S} a finite G-stable set of inequivalent irreducible twisted weak V-modules associated with possibly different automorphisms in G. We show a Schur–Weyl type duality for the actions of \mathscr {A}_{\alpha }(G,\mathscr {S}) and V^G on the direct sum of twisted weak V-modules in \mathscr {S} where \mathscr {A}_{\alpha }(G,\mathscr {S}) is a finite dimensional semisimple associative algebra associated with G,\mathscr {S}, and a 2-cocycle \alpha naturally determined by the G-action on \mathscr {S}. It follows as a natural consequence of the result that for any g\in G every irreducible g-twisted weak V-module is a completely reducible weak V^G-module. [ABSTRACT FROM AUTHOR]

Spin^h manifolds are the quaternionic analogue to \text {spin}^c manifolds. We compute the \text {spin}^h bordism groups at the prime 2 by proving a structure theorem for the cohomology of the \text {spin}^h bordism spectrum \mathrm {MSpin^h} as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of \mathrm {MSpin^h} as a wedge sum of familiar spectra. We also compute the decomposition of H^*(\mathrm {MSpin^h};\mathbb {Z}/2\mathbb {Z}) explicitly in degrees up through 30 via a counting process. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

Let \Lambda (f) = K[x][y; f\frac {d}{dx} ] be an Ore extension of a polynomial algebra K[x] over a field K of characteristic zero where f\in K[x]. For a given polynomial f, the automorphism group of the algebra \Lambda (f) is explicitly described. The polynomial case \Lambda (0) = K[x,y] and the case of the Weyl algebra A_1= K[x][y; \frac {d}{dx} ] were done by Jung [J. Reine Angew. Math. 184 (1942), pp. 161–174] and van der Kulk [Nieuw Arch. Wisk. (3) 1 (1953), pp. 33–41], and Dixmier [Bul. Soc. Math. France 96 (1968), pp. 209–242], respectively. Alev and Dumas [Comm. Algebra 25 (1997), pp. 1655–1672] proved that the algebras \Lambda (f) and \Lambda (g) are isomorphic iff g(x) = \lambda f(\alpha x+\beta) for some \lambda, \alpha \in K\backslash \{ 0\} and \beta \in K. Benkart, Lopes and Ondrus [Trans. Amer. Math. Soc. 367 (2015), pp. 1993–2021] gave a complete description of the set of automorphism groups of algebras \Lambda (f). In this paper we complete the picture, i.e. given the polynomial f we have the explicit description of the automorphism group of \Lambda (f). The key concepts in finding the automorphism groups are the eigenform, the eigenroot and the eigengroup of a polynomial (introduced in the paper; they are of independent interest). [ABSTRACT FROM AUTHOR]

Let \mathcal {B}(H) be the algebra of all bounded operators on a Hilbert space H. Let T=V|T| be the polar decomposition of an operator T\in \mathcal {B}(H). The mean transform of T is defined by M(T)=\frac {T+|T|V}{2}. In this paper, we discuss several properties related to the spectrum, the kernel, the image, and the polar decomposition of mean transform. Moreover, we investigate the image and preimage by the mean transform of some class of operators such as positive, normal, unitary, hyponormal, and co-hyponormal operators. [ABSTRACT FROM AUTHOR]

The heat semigroup on discrete hypercubes is well-known to be contractive over L_p-spaces for 1

We prove that for any fixed unitary matrix U, any abelian self-adjoint algebra of matrices that is invariant under conjugation by U can be embedded into a maximal abelian self-adjoint algebra that is still invariant under conjugation by U. We use this result to analyse the structure of matrices A for which A^*A commutes with AA^*, and to characterize matrices that are unitarily equivalent to weighted permutations. [ABSTRACT FROM AUTHOR]

We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let \mathcal {G} be a topological group, and \mathcal {A} a unital symmetric C^*-subalgebra of \mathrm {UC}(\mathcal {G}), the algebra of bounded uniformly continuous functions on \mathcal {G}. Generalizing the notion of a stable metric, we study \mathcal {A}-metrics \delta, i.e., the function \delta (e, \cdot) belongs to \mathcal {A}; the case \mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G}), the algebra of weakly almost periodic functions on \mathcal {G}, recovers stability. If the topology of G is induced by a left invariant metric d, we prove that \mathcal {A} determines the topology of \mathcal {G} if and only if d is uniformly equivalent to a left invariant \mathcal {A}-metric. As an application, we show that the additive group of C[0,1] is not reflexively representable; this is a new proof of Megrelishvili [ Topological transformation groups: selected topics , Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now \mathcal {G} be a metric group, and assume \mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G}), the algebra of bounded left uniformly continuous functions on \mathcal {G}, is a unital C^*-algebra which is the uniform closure of coefficients of representations of \mathcal {G} on members of \mathscr {F}, where \mathscr {F} is a class of Banach spaces closed under \ell _2-direct sums. We prove that \mathcal {A} determines the topology of \mathcal {G} if and only if \mathcal {G} embeds into the isometry group of a member of \mathscr {F}, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability. [ABSTRACT FROM AUTHOR]

We study the maximal abelian von Neumann algebra corresponding to L^\infty (\mathbb {R}) via the Bargmann transform. It is naturally an algebra of operators on the Fock space F^2, but it can also be realized as a function algebra contained in F^2. This provides an interesting example of a C^* algebra whose elements are analytic functions. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the fundamental theorem of algebra for quaternions to polynomials with two monomials in the leading form, while showing that it fails for three. [ABSTRACT FROM AUTHOR]

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gröbner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gröbner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is uncountable, as well as a subalgebra with finitely many distinct initial algebras. [ABSTRACT FROM AUTHOR]

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of q-difference operators. The q-difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of q-difference operators is investigated. [ABSTRACT FROM AUTHOR]

Lei and Lin [Comm. Pure Appl. Math. 64 (2011), pp. 1297–1304] have recently given a proof of a global mild solution of the three-dimensional Navier-Stokes equations in function spaces based on the Wiener algebra. An alternative proof of existence of these solutions was then developed by Bae [Proc. Amer. Math. Soc. 143 (2015), pp. 2887–2892], and this new proof allowed for an estimate of the radius of analyticity of the solutions at positive times. We adapt the Bae proof to prove existence of the Lei-Lin solution in the spatially periodic setting, finding an improved bound for the radius of analyticity in this case. [ABSTRACT FROM AUTHOR]

Ritt's theorem of zeroes and Siedenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs. [ABSTRACT FROM AUTHOR]

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the length of all such modules is bounded. [ABSTRACT FROM AUTHOR]

Examples of Fell algebras with compact spectrum and trivial Dixmier-Douady invariant are constructed to illustrate differences with the case of continuous-trace C^*-algebras. At the level of the spectrum, this translates to only assuming the spectrum is locally Hausdorff (rather than Hausdorff). The existence of (full) projections is the fundamental question considered. The class of Fell algebras studied here arises naturally in the study of Wieler solenoids and applications to dynamical systems will be discussed in a separate paper. [ABSTRACT FROM AUTHOR]

Let R be a finitely generated \mathbb N-graded algebra domain over a Noetherian ring and let I be a homogeneous ideal of R. Given P\in Ass(R/I) one defines the v-invariant v_P(I) of I at P as the least c\in \mathbb N such that P=I:f for some f\in R_c. A classical result of Brodmann [Proc. Amer. Math. Soc. 74 (1979), pp. 16–18] asserts that Ass(R/I^n) is constant for large n. So it makes sense to consider a prime ideal P\in Ass(R/I^n) for all the large n and investigate how v_P(I^n) depends on n. We prove that v_P(I^n) is eventually a linear function of n. When R is the polynomial ring over a field this statement has been proved independently also by Ficarra and Sgroi in their recent preprint [ Asymptotic behaviour of the \text {v}-number of homogeneous ideals , https://arxiv.org/abs/2306.14243, 2023]. [ABSTRACT FROM AUTHOR]

Let \mathcal {A} denote a central hyperplane arrangement of rank n in affine space \mathbb {K}^n over a field \mathbb {K} of characteristic zero and let l_1,\ldots, l_m\in R≔\mathbb {K}[x_1,\ldots,x_n] denote the linear forms defining the corresponding hyperplanes, along with the corresponding defining polynomial f≔l_1\cdots l_m\in R. The focus of the paper is on the ideal J_f\subset R generated by the partial derivatives of f. We conjecture that J_f is a minimal reduction of the ideal \mathbb {I}\subset R generated by the (m-1)-fold products of distinct forms among l_1,\ldots, l_m. We prove this conjecture for an almost generic \mathcal {A} (i.e., any n-1 among the defining linear forms are linearly independent). In this case we obtain a stronger version of a result by Dimca and Papadima, and we confirm the conjecture unconditionally for n=3. We also conjecture that J_f is an ideal of linear type (i.e., the respective symmetric and Rees algebras coincide). We prove this conjecture for n=3. In the sequel we explain the tight relationship between the two ideals J_f, \mathbb {I}\subset R; in particular, we show that in the generic case (J_f)^{\text {sat}}=\mathbb I. As a consequence, we can provide a simpler proof of a conjectured result of Yuzvinsky, proved by Rose and Terao, on the vanishing of the depth of R/J_f. [ABSTRACT FROM AUTHOR]

We define Chen's iterated integrals of forms with values in an Azumaya algebra and show that in the flat case, these iterated integrals provide chain maps, and more generally, the curved structure is preserved. The appropriate trace maps are then defined and their corresponding geometric meanings are studied. We also define and study the wedge product of forms and the concatenation product of paths for these iterated integrals. [ABSTRACT FROM AUTHOR]

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z (the integers). Consequently, results of Eilers, Restorff, Ruiz and Sørensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups). [ABSTRACT FROM AUTHOR]

We call an algebra A commutator-simple if [A,A] does not contain nonzero ideals of A. After providing several examples, we show that in these algebras derivations are determined by a condition that is applicable to the study of local derivations. This enables us to prove that every continuous local derivation D\colon L^1(G)\to L^1(G), where G is a unimodular locally compact group, is a derivation. We also give some remarks on homomorphism-like maps in commutator-simple algebras. [ABSTRACT FROM AUTHOR]

In this paper we show that evolution algebras over any given field \Bbbk are universally finite. In other words, given any finite group G, there exist infinitely many regular evolution algebras X such that Aut(X)\cong G. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme \mathbf {G} over \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra. [ABSTRACT FROM AUTHOR]

We prove a version of a theorem of Auslander for finite group actions or coactions on noetherian polynomial identity Artin-Schelter regular algebra. [ABSTRACT FROM AUTHOR]

We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras. [ABSTRACT FROM AUTHOR]

We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free. [ABSTRACT FROM AUTHOR]

We study the extent of countable saturation for coronas of abelian \mathrm {C}^{*}-algebras. In particular, we show that the corona algebra of C_0(\mathbb {R}^n) is countably saturated if and only if n=1. [ABSTRACT FROM AUTHOR]

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory. [ABSTRACT FROM AUTHOR]

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general "minimal test for injectivity" for a left Artinian ring A: we show that if every injective morphism \Sigma _k\rightarrow A from the k'th socle of A extends to a morphism A\rightarrow A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N\rightarrow M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N\rightarrow M extend (quasi-injectivity) and obtain further applications. [ABSTRACT FROM AUTHOR]

We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory. [ABSTRACT FROM AUTHOR]

Let G be a discrete group, VN(G) the von Neumann algebra generated by the left translation operators and A(G) the Fourier algebra of G. Then VN(G) can be considered as a subspace of \ell ^2(G) and A(G) = \ell ^2(G)*\ell ^2(G). The Banach space dual of A(G) is VN(G). The duality between \varphi \in VN(G) and u \in A(G) is \langle \varphi,u \rangle = \sum _{x \in G}{\varphi (x)u(x)}, if \varphi \in \ell ^{1}(G) or u \in \ell ^2(G); in these two cases, \varphi u \in \ell ^{1}(G). We will show that if G is infinite then there exist \varphi \in VN(G) and u \in A(G) such that \varphi u \notin \ell ^{1}(G). When G is countably infinite, this implies that there exist \varphi \in VN(G), u \in A(G) and an increasing sequence of finite subsets F_n, G = \bigcup F_n such that the sequence \sum _{x \in F_n}{\varphi (x)u(x)} is divergent and therefore \|{\chi _{F}}_n \|_{MA(G)} is unbounded. This leads us to give a preliminary study of the growth of \|\chi _{F}\|_{MA(G)} as |F|, the size of the finite subset F of G, is increasing. Recall that when G = \mathbb {Z}, the additive group of integers, \|\chi _{F}\|_{MA(\mathbb {Z})} = \frac {1}{2\pi }\int _0^{2\pi } \big |\sum _{k \in F} {e^{ikx}}\big | \, d{x}. [ABSTRACT FROM AUTHOR]

We compute the C_p-equivariant dual Steenrod algebras associated to the constant Mackey functors \underline {\mathbb {F}}_p and \underline {\mathbb {Z}}_{(p)}, as \mathrm {H}\underline {\mathbb {Z}}_{(p)}-modules. The C_p-spectrum \mathrm {H}\underline {\mathbb {F}}_p \wedge \mathrm {H}\underline {\mathbb {F}}_p is not a direct sum of RO(C_p)-graded suspensions of \mathrm {H}\underline {\mathbb {F}}_p when p is odd, in contrast with the classical and C_2-equivariant dual Steenrod algebras. [ABSTRACT FROM AUTHOR]

A group G is Jónsson if |H| < |G| whenever H is a proper subgroup of G. Using an embedding theorem of Obraztsov it is shown that there exists a Jónsson group G of infinite cardinality \kappa if and only if there exists a Jónsson algebra of cardinality \kappa. Thus the question as to which cardinals admit a Jónsson group is wholly reduced to the well-studied question of which cardinals are not Jónsson. As a consequence there exist Jónsson groups of arbitrarily large cardinality. Another consequence is that the infinitary edge-orbit conjecture of Babai is true. [ABSTRACT FROM AUTHOR]

In this work, we study the bigraded regularities of the symbolic Rees algebras R_s(J(G)), R_s(I(G)), of the vertex cover ideal J(G) and the edge ideal I(G), of a graph G respectively. We give combinatorial upper bounds for the (1,0)-regularities of R_s(J(G)) and R_s(I(G)). By using this upper bounds, we give general linear upper bounds for reg(J(G)^{(k)}), reg(I(G)^{(k)}) for any k\geq 1. Let G be a graph on n vertices and \deg (J(G)) be the maximum degree of minimal generators of J(G). We show that if G is a non-bipartite graph, then \begin{equation*} k \deg (J(G)) \!\leq \! reg(J(G)^{(k)})\!\leq \! k \deg (J(G))+ \alpha _0(G)-1+|A_0\cup \{x_{i_1}, \ldots, x_{i_r}\}|-r, \end{equation*} for all k \geq 1, where \alpha _0(G) denotes the vertex cover number of G, A_0 is a maximal independent set in G of maximal cardinality, and r is the number of 0-covers that are present in an irreducible representation of the affine cone associated with the irreducible covers of G. Also if G is a non-bipartite perfect graph, then \begin{equation*} 2k \leq reg(I(G)^{(k)})\leq 2k+n-r+1, \end{equation*} for all k \geq 1, where r is the number of 0-covers of \Gamma (G) that are present in an irreducible representation of the affine cone associated with the irreducible covers of \Gamma (G). [ABSTRACT FROM AUTHOR]

Given a commutative local ring (R,\mathfrak m) and an ideal I of R, a family of quotients of the Rees algebra R[It] has been recently studied as a unified approach to the Nagata's idealization and the amalgamated duplication and as a way to construct interesting examples, especially integral domains. When R is noetherian of prime characteristic, we compute the Hilbert-Kunz function of the members of this family and, provided that either I is \mathfrak {m}-primary or R is regular and F-finite, we also find their Hilbert-Kunz multiplicity. Some consequences and examples are explored. [ABSTRACT FROM AUTHOR]

We define the notion of an almost polynomial identity of an associative algebra R, and show that its existence implies the existence of an actual polynomial identity of R. A similar result is also obtained for Lie algebras and Jordan algebras. We also prove related quantitative results for simple and semisimple algebras. [ABSTRACT FROM AUTHOR]

Let X be a finite connected poset and K a field. We give a full description of the Lie automorphisms of the incidence algebra I(X,K). In particular, we show that they are in general not proper. [ABSTRACT FROM AUTHOR]

We show that several well known higher dimensional methods for finding roots of univariate polynomials have infinite orbits that diverge to infinity: in particular, the "Weierstrass" (Durand–Kerner) and the "Ehrlich–Aberth" methods. This is possible for the Jacobi update scheme (all coordinates are updated in parallel) as well as Gauss–Seidel (any coordinate update is used for all subsequent coordinates). These root finding methods are in active use in practice, but very little is known about their global dynamical properties, and diverging orbits were discovered only recently for one of them. Our results are established by a combination of methods from dynamical systems and computer algebra. [ABSTRACT FROM AUTHOR]