We characterize the asymptotic behavior of time-homogeneous doubly stochastic Markov chains. Our investigation revolves around understanding the dynamics of products of doubly stochastic matrices, which in turn allows us to fully characterize three distinct behaviors: cyclicity, convergence towards a special equilibrium matrix, and divergence. Notably, we introduce a novel and comprehensive sufficient condition for the convergence of an infinite product of doubly stochastic matrices.
In this paper we describe how to compute a Saito basis of a cusp, a plane curve with only one Puiseux pair. Moreover, the 1-forms of the Saito basis that we obtain are characterized in terms of their divisorial orders associated to the "cuspidal" divisor of the minimal reduction of singularities of the cusp. We also introduce a new family of analytic invariants for plane curves computed in terms of Saito bases.
In this paper we prove a reverse Hölder inequality for the variable exponent Muckenhoupt weights $\mathcal{A}_{p(\cdot)}$, introduced by the first author, Fiorenza, and Neugeabauer. All of our estimates are quantitative, showing the dependence of the exponent function on the $\mathcal{A}_{p(\cdot)}$ characteristic. As an application, we use the reverse Hölder inequality to prove that the matrix $\mathcal{A}_{p(\cdot)}$ weights, introduced in our previous paper, have both a right and left-openness property. This result is new even in the scalar case.
We show that every Lie ideal in a unital, properly infinite C*-algebra is commutator equivalent to a unique two-sided ideal. It follows that the Lie ideal structure of such a C*-algebra is concisely encoded by its lattice of two-sided ideals. This answers a question of Robert in this setting. We obtain similar structure results for Lie ideals in unital, real rank zero C*-algebras without characters. As an application, we show that every Lie ideal in a von Neumann algebra is related to a unique two-sided ideal, which solves a problem of Brešar, Kissin, and Shulman.
In order to circumvent a fundamental issue when studying densely defined traces on C & lowast;-al-gebras-which we refer to as the Trace Question-we initiate a systematic study of the set TR(A) of self-adjoint traces on the Pedersen ideal of A. The set TR(A) is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for TR(A) and compute TR(A) for principal twisted etale groupoid C & lowast;-algebras. We also answer the Trace Question positively for a large class of C & lowast;-algebras.
We prove a Hankel-variant commutant lifting theorem. This also uncovers the complete structure of the Beurling-type reducing and invariant subspaces of Hankel operators. Kernel spaces of Hankel operators play a key role in the analysis.
Precipitating a notion emerging from recent research, we formalise the study of a special class of compact quantum metric spaces. Abstractly, the additional requirement we impose on the underlying order unit spaces is the Riesz interpolation property. In practice, this means that a 'quantum metric Choquet simplex' arises as a unital C^*-algebra A whose trace space is equipped with a metric inducing the w^*-topology, such that tracially Lipschitz elements are dense in A. This added structure is designed for measuring distances in and around the category of stably finite classifiable C^*-algebras, and in particular for witnessing metric and statistical properties of the space of approximate unitary equivalence classes of unital embeddings of A into a stably finite classifiable C^*-algebra B. As for examples, we recall the construction of classifiable C^*-algebraic quantum metric Bauer simplices that function as noncommutative spaces of observables of compact connected metric spaces (X,ρ). We also explain how to build non-Bauer examples by forming 'tracial quantum crossed products' associated with topological dynamical systems on (X,ρ), and we use classification to show that continuous fields of quantum spaces are obtained by continuously varying either the dynamics or the metric. In the case of deformed isometric actions, we show that equivariant Gromov-Hausdorff continuity implies fibrewise continuity of the quantum structures with respect to Rieffel's quantum Gromov-Hausdorff distance. As an example, we present a field of deformed tracial rotation algebras whose fibres are continuous with respect to a quasimetric that we call the quantum intertwining gap.
Our initial aim was to answer the question: does the Frobenius (symmetric) property transfers from a strongly graded algebra to its homogeneous component of trivial degree? Related to it, we investigate invertible bimodules and the Picard group of a finite dimensional quasi-Frobenius algebra $R$. We compute the Picard group, the automorphism group and the group of outer automorphisms of a $9$-dimensional quasi-Frobenius algebra which is not Frobenius, constructed by Nakayama. Using these results and a semitrivial extension construction, we give an example of a symmetric strongly graded algebra whose trivial homogeneous component is not even Frobenius. We investigate associativity of isomorphisms $R^*\ot_RR^*\simeq R$ for quasi-Frobenius algebras $R$, and we determine the order of the class of the invertible bimodule $H^*$ in the Picard group of a finite dimensional Hopf algebra $H$. As an application, we construct new examples of symmetric algebras.
We show that various methods for explicitly building resolutions of unbounded complexes in fact fail when applied to a rather simple and explicit complex. We show that one way to rescue these methods is to assume Roos (Ab.4^*)-k axiom, which we adapt to encompass also resolutions in the framework of relative homological algebra. In the end we discuss the existence of model structures for relative homological algebra for unbounded complex under the relative (Ab.4^*)-k condition, and present a variety of examples where our results apply.
For every prime number p and integer n>1, a simple, involutive, non-degenerate set-theoretic solution (X,r) of the Yang-Baxter equation of cardinality |X| = p^n is constructed. Furthermore, for every non-(square-free) positive integer m which is not the square of a prime number, a non-simple, indecomposable, irretractable, involutive, non-degenerate set-theoretic solution (X,r) of the Yang-Baxter equation of cardinality |X| = m is constructed. A recent question of Castelli on the existence of singular solutions of certain type is also answered affirmatively.
In this paper, we provide a complete classification of the positive minimal monads whose cohomology is a stable rank 2 bundle on ℙ^3 with Chern classes c_1=-1, c_2=10 and we prove the existence of a new irreducible component of the moduli space ℬ(-1,10) of a rank 2 stable bundles with the given Chern classes. We also show that Hartshorne's conditions on a sequence 𝒳 of 10 integers are sufficient and necessary for the existence of a stable rank 2 bundle with odd determinant and spectrum 𝒳. Furthermore, we prove that the sequence of integers {-2^n-1,-1,0,1^n-1} for n≥4 is realized as the spectrum of a stable rank 2 bundle of odd determinant by computing the minimal generators of its Rao module.
We determine the structure of the singular locus of generic codimension-$q$ logarithmic foliations and its relation with the unfoldings of said foliations. In the case where the ambient variety is the projective space $\mathbb{P}^n$ we calculate the graded ideal defining the scheme of persistent singularities.
We revisit certain localised variants of the Bennett-Carbery-Tao multilinear restriction theorem, recently proved by Bejenaru. We give a new proof of Bejenaru's theorem, relating the estimates to the theory of Kakeya-Brascamp-Lieb inequalities. Moreover, the new proof allows for a substantial generalisation, exploiting the full power of the Kakeya-Brascamp-Lieb theory.
Any power series with nonnegative coefficients has an associated family of probability distributions supported on the nonnegative integers. There is a close connection between the function theoretic properties of the power series and the moments of the family of distributions. In this paper, we describe that interplay, provide simpler proofs of some known results by emphasizing the probabilistic perspective, and present some new theorems.
Let p be a prime. In this paper we provide a lower bound for the number of almost p-rational characters of degree coprime to p in the principal p-block of a finite group of order divisible by p. We further describe the p-local structure of the groups for which the above-mentioned bound is sharp.
We construct Nakayama functors on proper abelian subcategories of triangulated categories with a Serre functor using approximation theory. This, in turn, allows for the construction of Auslander-Reiten translates. As a result, we prove that suitable proper abelian subcategories are dualising k-varieties and have enough projectives if and only if they have enough injectives. As an application, we provide a new proof of the existence of Auslander-Reiten sequences in the category of finite dimensional modules over a finite dimensional algebra.
Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.
A b-contact structure on a b-manifold (M,Z) is a singular Jacobi structure on M satisfying a transversality condition along the hypersurface Z. We show that, in three dimensions, b-contact structures with overtwisted three-dimensional leaves satisfy an existence h-principle that allows prescribing the induced singular foliation. We give a method to classify b-contact structures on a given b-manifold and use it to give a classification on S^3 with either a two-sphere or an unknotted torus as the critical surface. We also discuss generalizations to higher dimensions.
