Topological Ring Research Articles

Another generalization of the m-topology

It is well known that the mI-topology is a generalization of the mtopology on C(X), see [1]. Given two subsets A,B ⊆ X such that A ∪ B = X, we are going to define a topology on C(X) namely the m(A,B)-topology, finer than the m-topology and C(X) with this topology becomes a topological ring. Connectedness in this space is studied and it is shown that if A,B are closed realcompact subsets of X, then the component of the zero function in C(X) with m(A,B)-topology is the ideal CK(X). Mathematics Subject Classification: 54C40

Bertotti-Robinson solutions ofD=5Einstein-Maxwell-Chern-Simons-Lambda theory

We present a series of new solutions in five-dimensional Einstein-Maxwell-Chern-Simons theory with an arbitrary Chern-Simons coupling $\gamma$ and a cosmological constant $\Lambda$. For general $\gamma$ and $\Lambda$ we give various generalizations of the Bertotti-Robinson solutions supported by electric and magnetic fluxes, some of which presumably describe the near-horizon regions of black strings or black rings. Among them there is a solution which could apply to the horizon of a topological AdS black ring in gauged minimal supergravity. Others are horizonless and geodesically complete. We also construct extremal asymptotically flat multi-string solutions for $\Lambda=0$ and arbitrary $\gamma$.

Off-Diagonal Bethe Ansatz and Exact Solution of a Topological Spin Ring

A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.

Algebraic theory of Colombeauʼs generalized numbers

Let K¯ denote the commutative ring of Colombeauʼs full generalized numbers. Endowed with Scarpalezosʼ sharp topology it becomes a topological ring. We study the algebraic and topological properties of this topological ring. In particular, we prove that the group of units of K¯ is dense in the sharp topology, determine its boolean algebra, show that it has minimal primes, describe them completely which results in a complete classification of the maximal ideals. From the description of the prime and maximal ideals, it becomes clear that they should be determined by certain ultra-filters.

Simplicial differential calculus, divided differences, and construction of Weil functors

We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points growths linearly with the degree, and not exponentially as in the classical, ``cubic'' approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X^{k+1}).

Notes on topological rings

We prove that every infinite nilpotent ring R admits a ring topology T for which (R, T ) has an open totally bounded countable subring with trivial multiplication. A new example of a compact ring R for which R2 is not closed, is given. We prove that every compact Bezout domain is a principal ideal domain.

Existence of covering topological R-modules

Let R be a topological ring with identity and M a topological (left) R-module such that the underlying topology of M is path connected and has a universal cover. Let 0 ∈ M be the identity element of the additive group structure of M, and N a submodule of the R-module π1(M,0). In this paper we prove that if R is discrete, then there exists a covering morphism p: (e MN, ˜ 0) → (M,0) of topological R-modules with characteristic group N and such that the structure of R-module on M lifts to e MN. In particular, if N is a singleton group, then this cover becomes a universal cover.

Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficients

In this paper we investigate the solvability of inhomogeneous linear systems of partial differential-difference equations with constant coefficients and also the corresponding duality problem in how far the solutions of the corresponding homogeneous systems determine the equations. For ordinary delay-differential (DD) equations these behavioral problems were investigated in a seminal paper by Glüsing-Lürssen (SIAM J Control Optim 35:480–499, 1997) and in later papers by Habets, Glüsing-Lürssen, Vettori and Zampieri. In these papers the delay-differential operators are considered as distributions with compact support which act on smooth functions or on arbitrary distributions via convolution. The entire analytic Laplace transforms of the distributions with compact support play an important part in the quoted papers. In our approach the partial differential-difference operators belong to various topological operator rings $$A$$ of holomorphic functions on subsets of $$\mathbb C ^n$$ and are thus studied in the frequency domain, the arguments of these (operator) functions being interpreted as generalized frequencies. We show that the topological duals $$A^{\prime }$$ of these operator rings with the canonical action of $$A$$ on $$A^{\prime }$$ have strong elimination and duality properties for $$_A A^{\prime }$$ -behaviors and admit concrete representations as spaces of analytic functions of systems theoretic interest. In particular, we study systems with generalized frequencies in the vicinity of suitable compact sets. An application to elimination for systems of periodic signals is given. We also solve an open problem of the quoted authors for DD-equations with incommensurate delays and analytic signals. Module theoretic methods in context with DD-equations have also been used by other authors, for instance by Fliess, Mounier, Rocha and Willems.

The topological jacobson radical of rings. II

In this paper, topologically primitive rings and rings possessing a faithful topologically irreducible module and bounded by this module are considered for the investigation of properties of their topological Jacobson radical. We investigate the topological Jacobson radical in some classes of topological rings such as left topologically Artinian rings, topological rings possessing a basis of neighborhoods of zero consisting of ideals, compact rings, and bounded strictly linearly compact rings.

Weil and Grothendieck approaches to adelic points

In [We, Ch. 1], Weil defines a process of “adelization” of algebraic varieties over global fields. There is an alternative procedure, due to Grothendieck, using adelic points. One aim of this (largely) expository note is to prove that for schemes of finite type over global fields (i.e., without affineness hypotheses), and also for separated algebraic spaces of finite type over such fields, Weil’s adelization process naturally coincides (as a set) with the set of adelic points in the sense of Grothendieck (and that in the affine case the topologies defined by these two viewpoints coincide; Grothendieck’s approach doesn’t provide a topology beyond the affine case). The other aim is to prove in general that topologies obtained by Weil’s method satisfy good functorial properties, including expected behavior with respect to finite flat Weil restriction of scalars. The affine case suffices for most applications, but the non-affine case is useful (e.g., adelic points of G/P for connected reductive groups G and parabolic subgroups P ). We also discuss topologizing X(k) for possibly non-separated algebraic spaces X over locally compact fields k; motivation for this is given in Example 5.5. Although everything we prove (except perhaps for the case of algebraic spaces) is “well known” folklore, and [Oes, I, §3] provides an excellent summary in the affine case, some aspects are not so easy to extract from the available literature. Moreover, (i) some references that discuss the matter in the non-affine case have errors in the description of the topology on adelic points, and (ii) much of what we prove is needed in my paper [Con], or in arithmetic arguments in [CGP]. In effect, these notes can be viewed as an expanded version of [Oes, I, §3], and I hope they will provide a useful general reference on the topic of adelic points of algebro-geometric objects (varieties, schemes, algebraic spaces) over global fields. In §2 we carry out Grothendieck’s method in the affine case over any topological ring R, characterizing the topology on sets of R-points by means of several axioms. The generalization to arbitrary schemes of finite type via a method of Weil is developed in §3. We explore properties of these topologies in §4, especially for adelic points and behavior with respect to Weil restriction of scalars. Finally, in §5 everything is generalized to the case of algebraic spaces. Notation. We write AF to denote the adele ring of a global field F , and likewise A n F denotes Euclidean n-space over AF . There is no risk of confusion with the common use of such notation to denote affine n-space over SpecF since we avoid ever using this latter meaning for the notation.

Ring completion of rig categories

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category ℛ we construct a natural additive group completion ℛ̅ that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category ℛ (also known as a symmetric bimonoidal category), the additive group completion ℛ̅ will be a commutative ring category. In an accompanying paper we show how to use this construction to prove the conjecture that the algebraic K-theory of the connective topological K-theory ring spectrum ku is equivalent to the algebraic K-theory of the rig category 𝒱 of complex vector spaces.

Discrete module categories and operations in K-theory

We study the categories of discrete modules for topological rings arising as the rings of operations in various kinds of topological K-theory. We prove that for these rings the discrete modules coincide with those modules which are locally finitely generated over the ground ring.

Topological Rings of Bounded and Compact Group Homomorphisms on a Topological Ring

Let X be a topological ring. In this paper, we consider the three classes of nr-bounded, br-bounded and cr-continuous group homomorphisms defined on X and denote these classes by Bnr(X);Bbr(X) and Bcr(X), respectively. We show that the operations of addition and product in Bnr(X);Bbr(X) and Bcr(X) are continuous with respect to the topology considered on them. Moreover, we introduce the classes of nr-compact and br-compact group homomorphisms, and we study the situations where a class of bounded group homomorphisms coincides with the corresponding class of compact group homomorphisms.

Bound states and persistent currents in topological insulator rings

We analyze theoretically the bound state spectrum of an Aharonov Bohm (AB) ring in a two-dimensional topological insulator using the four-band model of HgTe-quantum wells as a concrete example. We calculate analytically the circular helical edge states and their spectrum as well as the bound states evolving out of the bulk spectrum as a function of the applied magnetic flux and dimension of the ring. We also analyze the spin-dependent persistent currents, which can be used to measure the spin of single electrons. We further take into account the Rashba spin-orbit interaction which mixes the spin states and derive its effect on the ring spectrum. The flux tunability of the ring states allows for coherent mixing of the edge- and the spin degrees of freedom of bound electrons which could be exploited for quantum information processing in topological insulator rings.

A study of extremally disconnected topological spaces

We show that if an extremally disconnected space X has a homogeneous compactification, then X is finite. It follows that if a totally bounded topological group has a dense extremally disconnected subspace, then it is finite. The techniques developed in this article also imply that if the square of a topological group G has a dense extremally disconnected subspace, then G is discrete. See also Theorem 3.12. We also establish a sufficient condition for an extremally disconnected topological ring to be discrete (Theorem 3.9). A theorem on the structure of an arbitrary homeomorphism of an extremally disconnected topological group onto itself is proved (see Theorem 3.7 and Corollary 3.8).

Topological Ring Currents in the “Empty” Ring of Benzo-Annelated Perylenes

Cyclic conjugation in benzo-annelated perylenes is examined by means of the topological π-electron ring currents calculated for each of their constituent rings, in a study that is an exact analogy of a recent investigation by Gutman et al. based on energy-effect values for the corresponding rings in each of these structures. "Classical" approaches, such as Kekulé structures, Clar "sextet" formulas, and circuits of conjugation, predict that the central ring in perylene is "empty" and thus contributes negligibly to cyclic conjugation. However, conclusions from the present calculations of topological ring currents agree remarkably with those arising from the earlier study involving energy-effect values in that, contrary to what would be predicted from the classical approaches, rings annelated in an angular fashion relative to the central ring of these perylene structures materially increase the extent of that ring's involvement in cyclic conjugation. It is suggested that such close quantitative agreement between the predictions of these two superficially very different indices (energy effect and topological ring current) might be due to the fact that, ultimately, both depend, albeit in ostensibly quite different ways, only on an adjacency matrix that contains information about the carbon-carbon connectivity of the conjugated system in question.

Formal Power Series

We present a formalization of the topological ring of formal power series in Isabelle/HOL. We also formalize formal derivatives, division, radicals, composition and reverses. As an application, we show how formal elementary and hyper-geometric series yield elegant proofs for some combinatorial identities. We easily derive a basic theory of polynomials. Then, using a generic formalization of the fraction field of an integral domain, we obtain formal Laurent series and rational functions for free.

Coordination behavior of 1,4-diallylpiperazinium and 1-allyloxybenzotriazole toward silver(I) in the crystalline π-complexes [Ag2(C4H8N2(C3H5)2(H+)2)(H2O)2(NO3)2](NO3)2 and [Ag(C6H4N3(OC3H5)(NO3))

Reactions of a solution of AgNO3 in aqueous methanol with solutions of 1,4-diallylpiperazine (acidified with HNO3 to pH = 4) and 1-allyloxybenzotriazole in ethanol gave the crystalline silver(I) π-complexes [Ag2(C4H8N2(C3H5)2(H+)2)(H2O)2(NO3)2](NO3)2 (I) and [Ag(C6H4N3(OC3H5)(NO3))] (II). Their crystal structures were determined by X-ray diffraction. Crystals of complexes I and II are monoclinic, space group P21/c; for I: a = 7.053(3)A, b = 9.389(3)A, c = 15.488(4)A, β = 91.60°, V = 1025.3(6)A3, Z = 4; for II: a = 10.650(4)A, b = 15.062(5)A, c = 7.412(4)A, β = 104.20(3)°, V = 1152.6(8)A3, Z = 4. In both structures, the organic components act as bidentate ligands forming with AgNO3 34- and 14-membered topological rings, respectively. In complex I, the nearly tetrahedral environment of the Ag(I) atom is made up of the olefinic C=C bond, the O atoms of the nitrate anions, and the water molecule. 1-Allyloxybenzotriazole in structure II causes the deformation of the coordination polyhedron of Ag into a trigonal pyramid via inclusion of the ligand N atom in its coordination sphere. The topological units of the complexes form infinite polymer layers linked by anionic NO 3 − bridges. In structure I, these layers are united through a system of hydrogen bonds into a three-dimensional framework.

Topological Rings of Endomorphisms

We study the following question: for which Abelian groups do their rings of endomorphisms admit a compact ring topology? Fuchs [4] asked this question for Abelian p-groups. A classification of torsion Abelian groups A of cardinality <2ω and such that 𝔈𝔫𝔡(A) admits a compact ring topology is given. We construct a torsion complete p-group of cardinality 2ω whose ring of endomorphisms admits a compact ring topology. Some facts about non-torsion Abelian groups whose rings of endomorphisms admit a compact ring topology are derived. We also construct an Abelian group A for which the Jacobson radical J(𝔈𝔫𝔡(A)) is not closed with respect to the finite topology. Motivated by this example, we introduce a new radical for topological rings and derive some of its basic properties.

Lattice-Valued Convergence Ring and Its Uniform Convergence Structure

Considering L a frame, we introduce the notion of stratified L-neighborhood topological ring, produce some characterization theorems including its Luniformizability. With the help of the notions of stratified convergence structures attributed to Gunther Jäger [10], we introduce and study various subcategories of stratified L-convergence rings. In doing so, we bring into light, among others, the notions of stratified L-uniform group and stratified L-uniform convergence group in an attempt to show that every stratified L-convergence ring carries in a natural way the stratified L-uniform convergence structure of Jäger and Burton [15], and the category of stratified L-uniform groups and uniformly continuous group homomorphisms, SLUnifGrp is isomorphic to the category of principal stratified L-uniform convergence groups and uniformly continuous group homomorphisms, SL-PUConvGrp. Introducing the notion of stratified L-Cauchy ring, we show that the category of stratified L-Cauchy rings and Cauchy-continuous ring homomorphisms, SL-ChyRng is topological over the category of rings, Rng with respect to the forgetful functor, and that every stratified L-Cauchy ring is a stratified L-convergence ring. We observe that if L is a Boolean algebra, then each stratified L-uniform convergence ring serves as a natural example of a stratified L-Cauchy ring. We give necessary and suficient conditions for a stratified L-convergence structure and a ring structure to be a stratified L-convergence ring.