A ${\mathcal Z}$ -subalgebra $U_{\mathcal Z}^\jmath {(n)}$ ( ${\mathcal Z}=\mathbb Z[\upsilon ,\upsilon ^{-1}]$ ) for the i -quantum group ${\mathbf {U}}^{\jmath }(n)$ over the field $\mathbb Q(\upsilon )$ is constructed by two of the authors [‘A new realisation of the i -quantum group $U^\jmath {(n)}$ ’, J. Pure Appl. Algebra 226 (1) (2022), Paper no. 106793, 27 pages, Theorem 6.5], using a Beilinson–Lusztig–MacPherson (BLM) type realisation. In this paper, we construct bases for $U_{\mathcal Z}^\jmath {(n)}$ , including the monomial basis conjectured in [‘A new realisation of the i -quantum group $U^\jmath {(n)}$ ’, J. Pure Appl. Algebra 226 (1) (2022), Paper no. 106793, 27 pages, Remark 6.6(4)]. This proves that the ${\mathcal Z}$ -algebra $U_{\mathcal Z}^\jmath {(n)}$ is a free ${\mathcal Z}$ -module. Hence, $U_{\mathcal Z}^\jmath {(n)}$ is in fact an integral form of Lusztig type. This construction is further extended to the i -quantum hyperalgebra over a field of any characteristic. By specialising $\upsilon $ to an l th primitive root $\varepsilon $ of $1$ with l odd, a realisation of the quotient of modulo the ideal generated by $d_i^l-1$ , for all $1\leqslant i\leqslant n+1$ , is also given as a by-product.
A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of J-trivial monoids is Cross if and only if it excludes 14 specific almost Cross varieties. Consequently, these 14 varieties exhaust all almost Cross varieties of J-trivial monoids.
In this paper, we show the existence, uniqueness and stability of nontrivial solutions to the followingMinkowski-curvature problems on unbounded domains: (x '/root 1-x '(2))' = f(t,x), t >= t(0), lim(t ->infinity)x(t) = psi(0), lim(t ->infinity)x '(t)e(t )= 0, wheref:[t(0),infinity) & times; R -> Ris continuous, t(0 )> 0 and psi(0 )is an element of Rare some given constants. Moreover, thisunique solution is obtained as the uniform limit of the sequence of successive approximations.
In this paper, we describe & eacute;tale Boolean right restriction monoids in terms of Boolean inverse monoids. We show that the Thompson groups arise naturally in this context.
In this paper, we define and study noncommutative affine pencils of conics, and give a complete classification result. We also fully classify four-dimensional Frobenius algebras. It turns out that the classification of noncommutative affine pencils of conics is the same as the classification of four-dimensional Frobenius algebras.
In this paper, we develop a continuity method for the Dirichlet problem of Hessian equations on Riemannian manifolds. Such equations, introduced by Caffarelli et al. ['The Dirichlet problem for nonlinear second-order elliptic equations III: functions of eigenvalues of the Hessians', Acta Math. 155 (1985), 261-301] are defined in terms of the eigenvalues of the Hessian and a given pair $(f,\Gamma )$ , where f is a symmetric function defined in a symmetric cone $\Gamma \subset \mathbb {R}<^>n$ and $\Gamma $ specifies the set of admissible eigenvalues for the solution. Our method combines techniques from Morse theory with a characterization of the pair $(f,\Gamma )$ . More precisely, in the type-2 case, we first construct admissible functions using Morse theory and then solve the Dirichlet problem without any additional assumptions on the boundary or the subsolution. Building on this characterization of the pair, we can approximate the type-1 equation by a family of type-2 equations.
This paper proves that the integrality of algebraic Witt vectors over imaginary quadratic fields is decidable; based on this, some related problems are also discussed.
We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.
In this paper, we investigate dynamical properties of monoid actions on symbolic systems. First, we establish mixing properties and an entropy formula for such actions. To describe chaotic behavior, we introduce and distinguish several types of Li-Yorke chaos. Our main result shows that positive entropy is equivalent to locally Li-Yorke chaos, a notion that strengthens the classical definition of Li-Yorke chaos.
Forh>1, we consider the reaction-diffusion equation: Delta(h)(infinity)u(x)=f(x,u(x),Du(x)),x is an element of Omega, where Delta(h)(infinity)denotes theh-degree infinity Laplacian,f is an element of C(Omega & times;R & times;Rn)satisfies 0 <= f(x,delta t,p)<=Lambda(x)delta(gamma)f(x,t,p), a positive function Lambda(x)is an element of C(Omega),gamma is an element of[0,h),t>0, and delta>0 is small enough. Such anequation may cause a dead-core region, that is, an unknown region where the nonnegative solutionvanishes completely. We establish a flattening estimate for the viscosity solution and obtain sharpC((h+1)/(h-gamma))-regularity along the free boundary partial derivative{u>0}boolean AND Omega. Using the sharp regularity, we proveLiouville-type theorems for the global solution and give the porosity of the free boundary. In the end,for the limit case gamma=h, we show that if the viscosity solution vanishes at a point, then the dead-coreregion must vanish.
We study the boundedness of the Mordell-Weil rank and the growth of the v-primary part of the Tate-Shafarevich group of p-supersingular abelian varieties of $\mathrm {GL}_2$ -type with real multiplication over $\mathbb Z_p$ -extensions of number fields, where v is a prime lying above p. Building on the work of Iovita and Pollack in the case of elliptic curves, under precise ramification and splitting conditions on p, we construct explicit systems of local points using the theory of Lubin-Tate formal groups. We then define signed Coleman maps, which in turn allow us to formulate and analyse signed Selmer groups. Assuming these Selmer groups are cotorsion, we prove that the Mordell-Weil groups are bounded over any subextensions of the ${\mathbb Z}_p$ -extension and provide an asymptotic formula for the growth of the v-primary part of the Tate-Shafarevich groups. Our results extend those of Kobayashi, Pollack, and Sprung on p-supersingular elliptic curves.
A ${\mathcal Z}$ -subalgebra $U_{\mathcal Z}\jmath {(n)}$ ( ${\mathcal Z}=\mathbb Z[\upsilon ,\upsilon {-1}]$ ) for the i-quantum group ${\mathbf {U}}{\jmath }(n)$ over the field $\mathbb Q(\upsilon )$ is constructed by two of the authors ['A new realisation of the i-quantum group $U\jmath {(n)}$ ', J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Theorem 6.5], using a Beilinson-Lusztig-MacPherson (BLM) type realisation. In this paper, we construct bases for $U_{\mathcal Z}\jmath {(n)}$ , including the monomial basis conjectured in ['A new realisation of the i-quantum group $U\jmath {(n)}$ ', J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Remark 6.6(4)]. This proves that the ${\mathcal Z}$ -algebra $U_{\mathcal Z}\jmath {(n)}$ is a free ${\mathcal Z}$ -module. Hence, $U_{\mathcal Z}\jmath {(n)}$ is in fact an integral form of Lusztig type. This construction is further extended to the i-quantum hyperalgebra over a field of any characteristic. By specialising $\upsilon $ to an l th primitive root $\varepsilon $ of $1$ with l odd, a realisation of the quotient of modulo the ideal generated by $d_il-1$ , for all $1\leqslant i\leqslant n+1$ , is also given as a by-product.
The notion of weighted $\alpha $ -composition was introduced by Ruhan Zhao in the 1990s. In this paper, we study several analytic function spaces that are closely related to weighted $\alpha $ -composition. These include $\alpha $ -Bloch spaces, $F(p,q,s)$ spaces, and Campanato spaces. We obtain derivative-free characterizations for $\alpha $ -Bloch spaces and $F(p,q,s)$ spaces, which improve some previous results in the literature. We also obtain a certain version of Carleson measures for Campanato spaces and $F(p,q,s)$ spaces.
We give a length one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu-Renault groupoids of systems (X,T) where X is compact Hausdorff and totally disconnected and T is a local homeomorphism with domain a clopen subset of X. We also show that algebra of such a partial action groupoid over a field has global dimension at most 2 when the space is second countable.
The primary objective of this paper is to show how the theory of groupoids and their $C^*$ -algebras provide new proofs and extensions of Mallat’s famous theorems on the construction of multiresolution analyses that form the basis of wavelet theory. This work was inspired in large part by the research of Iain Raeburn and co-authors.
In this paper, we establish a lower bound for the maxima of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double-version convolution formula and combined with special GCD sums, our result generalizes the work of Bondarenko et al. ['A dichotomy for extreme values of zeta and Dirichlet L-functions', Bull. Lond. Math. Soc. 55 (2023), 2963-2975]. We also establish a lower bound via the resonance method when the real part is near the critical line; both of the above results refine parts of Yang's ['Extreme values of derivatives of the Riemann zeta function', Mathematika 68 (2022), 486-510] work.
In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here we verify some of these conjectures for fusion systems on an extraspecial group of order p^3, which contain among them the Ruiz-Viruel exotic fusion systems at the prime 7. As a byproduct we verify Robinson's ordinary weight conjecture for principal p-blocks of almost simple groups G realizing such (nonconstrained) fusion systems.
The averaged distance structure of one-dimensional regular model sets is determined via their pair-correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair-correlation functions. We introduce this approach and derive two concrete examples, which display unexpectedly complex and wild behaviour.
Given a permutation group G, the derangement graph of G is defined with vertex set G, where two elements x and y are adjacent if and only if $xy<^>{-1}$ is a derangement. We establish that if G is transitive with degree exceeding 30, then the derangement graph of G contains a complete subgraph with four vertices. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that $|A : G| = 3$ and U is a subgroup of G satisfying $G = \bigcup _{a \in A} U<^>a$ , then $|G : U| \leq 10$ . This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.
We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold X, such a representation involves an assignment of a complex vector bundle on X to each node of the double quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the Atiyah-Drinfel'd-Hitchin-Manin equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperk & auml;hler. Finally, we produce concrete examples that recover known moduli spaces.
