We construct comparison morphisms between two well-known projective resolutions of a monomial algebra $A$: the bar resolution $\operatorname{\mathbb{Bar}} A$ and Bardzell's resolution $\operatorname{\mathbb{Ap}} A$; the first one is used to define the cup product and the Lie bracket on the Hochschild cohomology $\operatorname{HH} ^*(A)$ and the second one has been shown to be an efficient tool for computation of these cohomology groups. The constructed comparison morphisms allow us to show that the cup product restricted to even degrees of the Hochschild cohomology has a very simple description. Moreover, for $A= \mathbb{k} Q/I$ a monomial algebra such that $\dim_ \mathbb{k} e_i A e_j = 1$ whenever there exists an arrow $\alpha: i \to j \in Q_1$, we describe the Lie action of the Lie algebra $\operatorname{HH} ^1(A)$ on $\operatorname{HH} ^ {\ast} (A)$.

Let $X$ be a smooth complex projective irreducible curve of genus $g \geq 3$. Let $G$ be the simple complex exceptional Lie group $F_4$ or $E_6$ and let $M(G)$ be the moduli space of principal $G$-bundles. In this work we describe the group of automorphisms of $M(G)$. In particular, we prove that the only automorphisms of $M(F_4)$ are those induced by the automorphisms of the base curve $X$ by pull-back and that the automorphisms of $M(E_6)$ are combinations of the action of the automorphisms of $X$ by pull-back, the action of the only nontrivial outer involution of $E_6$ on $M(E_6)$ by taking the dual and the action of the third torsion of the Picard group of $X$ by tensor product. We also prove a Torelli type theorem for the moduli spaces of principal $F_4$ and $E_6$-bundles, which we use as an auxiliary result in the proof of the main theorems, but which is interesting in itself. We finally draw some conclusions about the way we can see the natural map $M(F_4) \rightarrow M(E_6)$ induced by the inclusion of groups $F_4 \hookrightarrow E_6$.

We give a characterization of dyadic BMO spaces in terms of Haar wavelet coefficients in spaces of homogeneous type.

Let $\Omega$ be a nonempty open proper and connected subset of $ \mathbb R^ {n} $, $n \geq 3$. Consider the elliptic Schrödinger type operator $L_ {E} u= A_ {E} u+Vu= - \Sigma_{ij} a_ {ij} (x) u_ {x_i x_j} +Vu$ in $ \Omega$, and the linear parabolic operator $L_ {P} u=A_ {P} u+Vu=$ $u_ {t} - \Sigma a_ {ij} (x,t)u_ {x_{i}x_{j}} +Vu$ in $ \Omega_{T} = \Omega \times (0,T)$, where the coefficients $a_ {ij} \in \mathrm{VMO} $ and the potential $V$ satisfies a reverse Hölder condition. The aim of this paper is to obtain a priori estimates for the operators $L_ {E} $ and $L_ {P} $ in weighted Sobolev spaces involving the distance to the boundary and weights in a local $A_ {p} $ class.

We introduce a natural extension of the Riesz potentials to quasi-metric measure spaces with an upper doubling measure. In particular, these operators are defined when the underlying space has components of differing dimensions. We study the behavior of the potential on classical and variable exponent Lebesgue spaces, obtaining necessary and sufficient conditions for its boundedness. The technique we use relies on a geometric property of the measure of the balls which holds both in the doubling and non-doubling situations, and allows us to present our results in a unified way.

We study hom-Nijenhuis operators, $T^ \ast$-extensions and abelian extensions of hom-Lie color algebras. We show that the infinitesimal deformation generated by a hom-Nijenhuis operator is trivial. Many properties of a hom-Lie color algebra can be lifted to its $T^ \ast$-extensions such as nilpotency, solvability and decomposition. It is proved that every finite-dimensional nilpotent quadratic hom-Lie color algebra over an algebraically closed field of characteristic not 2 is isometric to a $T^ \ast$-extension of a nilpotent Lie color algebra. Moreover, we introduce abelian extensions of hom-Lie color algebras and show that there is a representation and a 2-cocycle, associated to any abelian extension.

Pappus' Involution Theorem is useful for proving incidence relations in the hyperbolic and elliptic planes. This fact is exemplified with the proof of a theorem about a family of 4-gons in the hyperbolic and elliptic planes. This non-Euclidean theorem is also re-interpreted in multiple ways, providing some other theorems for different figures in the hyperbolic plane.

In this paper we are interested in studying the properties of Armendariz, Baer, quasi-Baer, p.p. and p.q.-Baer over skew PBW extensions. Using a notion of compatibility, we generalize several propositions established for Ore extensions and present new results for several noncommutative rings which can not be expressed as Ore extensions (universal enveloping algebras, diffusion algebras, and others).

Properties of metallic conjugate connections are stated by pointing out their relation to product conjugate connections. We define the analogous in metallic geometry of the structural and the virtual tensors from the almost product geometry and express the metallic conjugate connections in terms of these tensors. From an applied point of view we consider invariant distributions with respect to the metallic structure and for a natural pair of complementary distributions, the above structural and virtual tensors are expressed in terms of O'Neill–Gray tensor fields.

We study orders in proper $*$-rings that are derived from the core-nilpotent decomposition. The notion of the C-N-star partial order and the S-star partial order is extended from $M_ {n} ( \mathbb{C)} $, the set of all $n \times n$ complex matrices, to the set of all Drazin invertible elements in proper $*$-rings with identity. Properties of these orders are investigated and their characterizations are presented. For a proper $*$-ring $ \mathcal{A} $ with identity, it is shown that on the set of all Drazin invertible elements $a \in \mathcal{A} $ where the core part of $a$ is an EP element, the C-N-star partial order implies the star partial order.

In this paper we classify the irreducible quasifinite highest weight modules over the orthogonal and symplectic types Lie subalgebras of the Lie algebra of the matrix quantum pseudo-differential operators. We also realize them in terms of the irreducible quasifinite highest weight modules of the Lie algebras of infinite matrices with finitely many nonzero diagonals and its classical Lie subalgebras of types B, C and D.

The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order $p$ and degree $k$. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on $p$-forms for all $p$, but not for all natural operators. Here we explain, simplify, and extend their investigations.

We give a gradient estimate of the positive solution to the equation \[ \Delta u=- \lambda^2u, \quad \lambda \geq 0 \] on a complete noncompact Finsler manifold. Then we obtain the corresponding Liouville-type theorem and Harnack inequality for the solution. Moreover, on a complete noncompact Finsler manifold we also prove a Liouville-type theorem for a $C^2$-nonnegative function $f$ satisfying \[ \Delta f \geq cf^d, \quad c>0, \; d>1, \] which improves a result obtained by Yin and He.

We develop a topological duality for the category of mildly distributive meet-semilattices with a top element and certain morphisms between them. Then, we use this duality to characterize topologically the lattices of Frink ideals and filters, and we also obtain a topological representation for some congruences on mildly distributive meet-semilattices.

We use shuffle and stuffle relations to give a simple proof of a reduction formula for length-two multiple polylogarithms evaluated in complex parameters of absolute value 1 in terms of a finite sum of products of length-one polylogarithms. This result was originally due to Nakamura and recently reproved by Panzer by different methods. This generalises results of Borwein and Girgensohn for alternating Euler sums and for multiple zeta values twisted by fourth roots of unity by the first author. We also explore implications for other colored multiple zeta values and present some applications to Mahler measure and Feynman diagrams.

We present a review of results about the shape derivatives of scalar- and vector-valued shape functions, and extend the results from Doğan and Nochetto [ESAIM Math. Model. Numer. Anal. 46 (2012), no. 1, 59-79] to more general surface energies. In that article, Doğan and Nochetto consider surface energies defined as integrals over surfaces of functions that can depend on the position, the unit normal and the mean curvature of the surface. In this work we present a systematic way to derive formulas for the shape derivative of more general geometric quantities, including the Gauss curvature (a new result not available in the literature) and other geometric invariants (eigenvalues of the second fundamental form). This is done for hyper-surfaces in the Euclidean space of any finite dimension. As an application of the results, with relevance for numerical methods in applied problems, we derive a Newton-type method to approximate a minimizer of a shape functional. We finally find the particular formulas for the first and second order shape derivatives of the area and the Willmore functional, which are necessary for the aforementioned Newton-type method.

This paper constructs a Hardy–Littlewood type maximal operator adapted to the Schr ö dinger operator $ \mathcal{L} := - \Delta + |x|^2$ acting on $L^ {2} ( \mathbb{R} ^ {d} )$. It achieves this through the use of the Gaussian grid $ \Delta^{\gamma} _ {0} $, constructed by Maas, van Neerven, and Portal [Ark. Mat. 50 (2012), no. 2, 379-395] with the Ornstein-Uhlenbeck operator in mind. At the scale of this grid, this maximal operator will resemble the classical Hardy–Littlewood operator. At a larger scale, the cubes of the maximal function are decomposed into cubes from $ \Delta^{\gamma} _ {0} $ and weighted appropriately. Through this maximal function, a new class of weights is defined, $A_ {p} ^ {+} $, with the property that for any $w \in A_{p} ^ {+} $ the heat maximal operator associated with $ \mathcal{L} $ is bounded from $L^ {p} (w)$ to itself. This class contains any other known class that possesses this property. In particular, it is strictly larger than $A_ {p} $.

We will prove the existence of a class of transitive maps on the real line $ \mathbb{R} $, with a discontinuity and horizontal asymptotes, whose set of periodic orbits is dense in $ \mathbb{R} $; that is, a class of chaotic families. In addition, we will show a rare phenomenon: the existence of periodic orbits of period three prevents the existence of transitivity.

If $f_n$ is a free parameter, we give a combinatorial closed form solution of the recursion \[ (n+1)^2 u_ {n+1} -f_n u_n-n^2 u_ {n-1} =0, \quad n \geq 1, \] and a related generating function. This is used to give a solution to the Apéry type sequence \[ r_n n^3+r_ {n-1} \left \{ \alpha n^3- \frac{3\alpha}{2} n^2+ \left \{ \frac{\alpha}{2} +2 \theta \right \} n- \theta \right \} +r_ {n-2} (n-1)^3=0, \quad n \geq 2, \] for certain parameters $ \alpha, \theta$. We show from another viewpoint two independent solutions of the last recursion related to certain modular forms associated with a problem of conformal mapping: Let $f( \tau)$ be a conformal map of a zero-angle hyperbolic quadrangle to an open half plane with values $0$, $ \rho$, $1$, $ \infty$ ($0< \rho<1$) at the cusps and define $t=t( \tau): = \frac{1}{\rho} f( \tau) \frac{f(\tau)-\rho}{f(\tau)-1} $. Then the function \[ E( \tau)= \frac{1}{2\pi i} \frac{f'(\tau)}{f(\tau)} \frac{1}{1-\frac{f(\tau)}{\rho}} \] is a solution, as a generating function in the variable $t$, of the above recurrence. In other words, $E( \tau)=r_0+r_1t+r_2 t^2+ \dots$, where $r_0=1$, $r_1=- \theta$, $ \alpha=2- \frac{4}{\rho} $.

Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.

We introduce two generalizations of hyperbolic area for connected, closed, orientable surfaces: the complexity and the simple complexity of a surface. These concepts are defined in terms of collections of branched coverings $M \to \mathbb{P} ^1$, where $M$ is a Riemann surface homeomorphic to $S$ and $ \mathbb{P} ^1$ is the Riemann sphere. We prove that if $S$ is a surface of positive genus, then both the topological complexity and the simple topological complexity of $S$ are linear functions of its genus.

Let $ \mathbb{k} $ be an algebraically closed field of characteristic zero. We construct several families of finite-dimensional Hopf algebras over $ \mathbb{k} $ without the dual Chevalley property via the generalized lifting method. In particular, we obtain 14 families of new Hopf algebras of dimension 128 with non-pointed duals which cover the eight families obtained in our unpublished version, arXiv:1701.01991 [math.QA].

In this paper, the concepts of (generalized) $( \theta, \varphi)$-prederivations and (generalized) Jordan $( \theta, \varphi)$-prederivations on a Lie color algebra are introduced. It is proved that Jordan $( \theta, \varphi)$-prederivations (resp. generalized Jordan $( \theta, \varphi)$-prederivations) are $( \theta, \varphi)$-prederivations (resp. generalized $( \theta, \varphi)$-prederivations) on a Lie color algebra under some conditions. In particular, Jordan $ \theta$-prederivations are $ \theta$-prederivations on a Lie color algebra.