Narrow Results

We generalize results on the connection between existence and uniqueness of integrals and representation theoretic properties for Hopf algebras and compact groups. For this, given a coalgebra C, we study analogues of the existence and uniqueness properties for the integral functor Hom^C(C, -), which generalizes the notion of integral in a Hopf algebra. We show that the coalgebra C is co-Frobenius if and only if dim(Hom^C(C, M)) = dim(M) for all finite dimensional right (left) comodules M. As applications, we give a few new categorical characterizations of co-Frobenius, quasi-co-Frobenius (QcF) coalgebras and semiperfect coalgebras, and re-derive classical results of Lin, Larson, Sweedler and Sullivan on Hopf algebras. We show that a coalgebra is QcF if and only if the category of left (right) comodules is Frobenius, generalizing results from finite dimensional algebras, and we show that a one-sided QcF coalgebra is two-sided semiperfect. We also construct a class of examples derived from quiver coalgebras to show that the results of the paper are the best possible. Finally, we examine the case of compact groups, and note that algebraic integrals can be interpreted as certain skew-invariant measure theoretic integrals on the group.

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson–Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

In this paper, we develop a theory of integration on algebraic quantum groupoids in the form of regular multiplier Hopf algebroids, and establish the main properties of integrals obtained by Van Daele for algebraic quantum groups before — faithfulness, uniqueness up to scaling, existence of a modular element and existence of a modular automorphism — for algebraic quantum groupoids under reasonable assumptions. The approach to integration developed in this paper forms the basis for the extension of Pontrjagin duality to algebraic quantum groupoids, and for the passage from algebraic quantum groupoids to operator-algebraic completions, which both will be studied in separate papers.

In this paper, we introduce a method which can generate a family of growing symmetrical tree networks. The networks are constructed by replacing each edge with a reduced-scale of the initial graph. Repeating this procedure, we obtain the fractal networks. In this paper, we define the average geodesic distance of fractal tree in terms of some integral, and calculate its accurate value. We find that the limit of the average geodesic distance of the finite networks tends to the average geodesic distance of the fractal tree. This result generalizes the paper [Z. Zhang, S. Zhou, L. Chen, M. Yin and J. Guan, Exact solution of mean geodesic distance for Vicsek fractals, J. Phys. A: Math. Gen.41(48) (2008) 7199–7200] for which the mean geodesic distance of Vicsek fractals was considered.

In this paper, we present the definition of integral of fuzzy mappings from the fuzzy number space E into E, and discuss its properties and calculation. And as an application of the integral, we provide a method of robustness estimates for the traffic volumes passing a road each day (24 hours). In addition, we also give another example of the integral characterizing the distance travelled in an imprecise or uncertain environment of time and speed.

We present a quasi-analytic perturbation expansion for multivariate N-dimensional Gaussian integrals. The perturbation expansion is an infinite series of lower-dimensional integrals (one-dimensional in the simplest approximation). This perturbative idea can also be applied to multivariate Student-t integrals. We evaluate the perturbation expansion explicitly through 2nd order, and discuss the convergence, including enhancement using Padé approximants. Brief comments on potential applications in finance are given, including options, models for credit risk and derivatives, and correlation sensitivities.

Let R ⊆ T be a (unital) extension of (commutative) rings, such that the total quotient ring of R is a von Neumann regular ring and T is torsion-free as an R-module. Let T ⊆ B be a ring extension such that B is a reduced ring that is torsion-free as a T-module. Let R* (respectively, A) be the integral closure of R in T (respectively, in B). Then (R*, T) is a normal pair (i.e. S is integrally closed in T for each ring S such that R* ⊆ S ⊆ T) if and only if (A, AT) is a normal pair. This generalizes results of Prüfer and Heinzer on Prüfer domains to normal pairs of complemented rings.

We give necessary and sufficient conditions for the functor F from the category of partial entwined modules to the category of right A-modules to be separable. This leads to a generalized notion of integrals of partial entwining structures. As an application, we prove a Maschke-type theorem for partial entwined modules.

Let R ⊆ S be a unital extension of commutative rings, with the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many -subalgebras of S. Each chain of rings from R to S is finite if dim(R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim(R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

Given any minimal ring extension $k\subset L$ of finite fields, several families of examples are constructed of a finite local (commutative unital) ring $A$ which is not a field, with a (necessarily finite) inert (minimal ring) extension $A\subset B$ (so that $B$ is a separable $A$-algebra), such that $A\subset B$ is not a Galois extension and the residue field of $A$ (respectively, $B$) is $k$ (respectively, $L$). These results refute an assertion of G. Ganske and McDonald stating that if $R\subseteq S$ are finite local rings such that $S$ is a separable $R$-algebra, then $R\subseteq S$ is a Galois ring extension. We identify the homological error in the published proof of that assertion. Let $(A,M)$ be a finite special principal ideal ring (SPIR), but not a field, such that $M$ has index of nilpotency $\alpha$ ($\ge 2$). Impose the uniform distribution on the (finite) set of ($A$-algebra) isomorphism classes of the minimal ring extensions of $A$. If $2\in M$ (for instance, if $A\cong \mathbb{Z}/2^{\alpha }\mathbb{Z}$), the probability that a random isomorphism class consists of ramified extensions of $A$ is at least $2/3$; if $2\notin M$ (for instance, if $A\cong \mathbb{Z}/p^{\alpha }\mathbb{Z}$ for some odd prime $p$), the corresponding probability is at least $3/4$. Additional applications, examples and historical remarks are given.

Let $p$ be an odd prime number and $A:=\mathbb{Z}/p^2\mathbb{Z}$. Assume that either $($i$)$$p=3$ or $($ii$)$$p\ge 5$ and $p$ is congruent to either $5$ or $11$ modulo $12$$($respectively, assume that $p\ge 5$ and $p$ is congruent to either $1$ or $7$ modulo $12)$. Then there exist exactly five $($respectively, exactly seven$)$ isomorphism classes of rings $R$ for which there exists a tower $A\subset {ℬ}\subset R$ of ramified $($integral minimal$)$ ring extensions such that ${ℬ}$ is the only ring properly contained between $A$ and $R$. We produce a set of isomorphism class representatives of such $R$ and, for each such $R$, we show that, up to isomorphism, the corresponding ring ${ℬ}$ is the idealization $A(+){𝔽}_p$. One consequence, for each integer $n>1$ whose prime-power factorization $n=\prod p_i^{e_i}$ (with pairwise distinct prime numbers $p_i$ and positive integers $e_i$) satisfies $e_i\le 4$ for all $i$, is a classification, up to isomorphism, of the rings that have cardinality $n$ and exactly two proper (unital) subrings.

Let $S$ be an additively idempotent semiring and $M_n(S)$ be the semiring of all $n\times n$ matrices over $S$. We characterize the conditions of when the semiring $M_n(S)$ is congruence-simple provided that the semiring $S$ is either commutative or finite. We also give a characterization of when the semiring $M_n(S)$ is subdirectly irreducible for $S$ being almost integral (i.e. $xy+yx+x=x$ for all $x,y\in S$). In particular, we provide this characterization for the semirings $S$ derived from the pseudo MV-algebras.

In this paper we give necessary and sufficient conditions for a comodule magma over a weak Hopf quasigroup to have a total integral, thus extending the theories developed in the Hopf algebra, weak Hopf algebra and non-associative Hopf algebra contexts. From this result we also deduce a version of Maschke’s theorems for right (H, B)-Hopf triples associated to a weak Hopf quasigroup H and a right H-comodule magma B.

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal theory of PDEs.

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we give a new proof by relating it to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed form, and discuss nonlinear Laplace transformations and symmetric PDE models.

We investigate the size of L^p-integrals for exponential sums over k-free numbers and prove essentially tight bounds.