Continuous Isomorphism Research Articles

Crystal-chemical and morphological interpretation of the biocompatibility of compounds in a Ca-Na-Bi-fluorapatite system.

This work is devoted to the study of the features of isomorphism in compounds of a Ca-Na-Bi-P-O-F system with a crystalline structure of the mineral apatite, as well as its effect on the biocompatibility of substances in relation to human cells in an in vitro model. A Ca10-2xBixNax(PO4)6F2 system (x = 0, 1, 2, 3, 4, and 5) is characterized by continuous isomorphism, which follows from the minimum deviations of the unit cell parameters from the Vegard and Rötgers rules. The refinement of the crystal structure showed that the cations are unevenly distributed between the 4f and 6h positions of the crystal structure of apatite: the bismuth ions are predominantly localized in the 6h position, while the sodium ions are concentrated in the 4f position. A standard MTT test of the biocompatibility of compounds with x = 1, 2, 3, and 4, and at x = 1 showed an anomaly in the form of an increased relative cell growth rate. This paper discusses the possible crystal-chemical and morphological reasons for this phenomenon.

On the Continuous Cancellative Semigroups on a Real Interval and on a Circle and Some Symmetry Issues

Let S denote the unit circle on the complex plane and ★:S2→S be a continuous binary, associative and cancellative operation. From some already known results, it can be deduced that the semigroup (S,★) is isomorphic to the group (S,·); thus, it is a group, where · is the usual multiplication of complex numbers. However, an elementary construction of such isomorphism has not been published so far. We present an elementary construction of all such continuous isomorphisms F from (S,·) into (S,★) and obtain, in this way, the following description of operation ★: x★y=F(F−1(x)·F−1(y)) for x,y∈S. We also provide some applications of that result and underline some symmetry issues, which arise between the consequences of it and of the analogous outcome for the real interval and which concern functional equations. In particular, we show how to use the result in the descriptions of the continuous flows and minimal homeomorphisms on S.

Mg-rich thomsonite-Ca from chromitites of the Lekkhoyla occurrence, Voykar-Syn’ya massif, Polar Urals

The paper describes a previously unknown isomorphic series between thomsonite-Ca and a hypothetical Mg-rich zeolite. The Mg-rich thomsonite-Ca was found for the frst time in hydrothermal veins of the Lekkhoyla chromite occurrence (Polar Urals, Russia). The mineral contains up to 6.5 wt. % MgO (up to 1.35 f.u. Mg, O = 20) and exhibits a continuous isomorphism between Mg2+ and Na+ and Ca2+. The formula of the Mg-richest thomsonite-Ca is Na0.37Mg1.35Ca1.59Si5.18 Al4.35O20 ? 6(?)H2O (O = 20). The Raman spectra of thomsonite-Ca and Mg-rich thomsonite-Ca are similar both showing no bands typical of other minerals. Figures 7. Tables 2. References 12.

Topological groups all continuous automorphisms of which are open

A topological space is reversible if each continuous bijection of it onto itself is open. We introduce an analogue of this notion in the category of topological groups: A topological group G is g-reversible if every continuous automorphism of G (=continuous isomorphism of G onto itself) is open. The class of g-reversible groups contains Polish groups, locally compact σ-compact groups, minimal groups, abelian groups with the Bohr topology, and reversible topological groups. We prove that subgroups of Rn are g-reversible, for every positive integer n. An example of a compact (so reversible) metric abelian group having a countable dense non-g-reversible subgroup is given. We also highlight the differences between reversible spaces and g-reversible topological groups. Many open problems are scattered throughout the paper.

On hyperbolic cobordisms and Hurwitz classes of holomorphic coverings

In this article we show that for every collection C \mathcal {C} of an even number of polynomials, all of the same degree d > 2 d>2 and in general position, there exist two hyperbolic 3 3 -orbifolds M 1 M_1 and M 2 M_2 with a Möbius morphism α : M 1 → M 2 \alpha :M_1\rightarrow M_2 such that the restriction of α \alpha to the boundaries ∂ M 1 \partial M_1 and ∂ M 2 \partial M_2 forms a collection of maps Q Q in the same conformal Hurwitz class of the initial collection C \mathcal {C} . Also, we discuss the relationship between conformal Hurwitz classes of rational maps and classes of continuous isomorphisms of sandwich products on the set of rational maps.

Coexisting bournonite-seligmanite and tennantite-tetrahedrite solid solutions in Darasun gold deposit (Eastern Transbaikalia, Russia): Estimation of the mineral formation temperature

Mineral associations, intergrowth features, chemical composition and Sb and As distribution in coexisting fahlore and bournonite-seligmanite solid solutions in the Darasun gold deposit were studied. For the first time the almost complete solid solution was found between bournonite and seligmanite with a continuous isomorphism between Sb and As in the range of Sb/(Sb + As) ratios from 0.21 to 1.00 in the Darasun deposit by EMPA. The composition of the fahlore coexisting with bournonite varies widely: Sb/(Sb + As) ratios 0.03-0.96 and Fe/(Fe + Zn) ratios 0.36-0.87. The coupled compositional evolution from early antimony to late arsenic in coexisting solid solutions has been established. Based on the investigation of the Sb and As distribution between coexisting fahlore and bournonite-seligmanite, the temperatures of their joint deposition and, as a result, of the productive stage, in which they are closely associated with native gold and tellurides (90–335 °С), are estimated.

Coexisting Bournonite–Seligmannite and Tennantite–Tetrahedrite Solid Solutions of the Darasun Gold Deposit, Eastern Transbaikalia, Russia: Estimation of the Mineral Formation Temperature

Mineral associations, intergrowth features, chemical composition and Sb and As distribution in coexisting fahlore and bournonite-seligmanite solid solutions in the Darasun gold deposit were studied. For the first time the almost complete solid solution was found between bournonite and seligmanite with a continuous isomorphism between Sb and As in the range of Sb/(Sb + As) ratios from 0.21 to 1.00 in the Darasun deposit by EMPA. The composition of the fahlore coexisting with bournonite varies widely: Sb/(Sb + As) ratios 0.03-0.96 and Fe/(Fe + Zn) ratios 0.36-0.87. The coupled compositional evolution from early antimony to late arsenic in coexisting solid solutions has been established. Based on the investigation of the Sb and As distribution between coexisting fahlore and bournonite-seligmanite, the temperatures of their joint deposition and, as a result, of the productive stage, in which they are closely associated with native gold and tellurides (90–335 °С), are estimated.

Quotients of locally minimal groups

A topological group G is called locally minimal if there exists a neighbourhood V of 1 in G such that if H is a Hausdorff group and f:G→H is a continuous isomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. This paper is focused on the study of quotients of locally minimal groups.A topological group G is called locally q-minimal if there exists a neighbourhood V of the identity of G such that whenever H is a Hausdorff group and f:G→H is a continuous surjective homomorphism such that f(V) is a neighbourhood of 1 in H, then f is open. We find a close connection between locally q-minimality and divisibility, by showing that a dense subgroup of Rn is locally q-minimal if and only if it is divisible. A description of the locally q-minimal subgroups of the n-dimensional torus Tn is also given.Two weaker versions of local q-minimality are proposed – a topological group G is:(a)locally t-minimal, if all Hausdorff quotients of G are locally minimal;(b)locallyq⁎-minimal, if there exists a neighbourhood V of 1 in G such that whenever H is a Hausdorff group and f:G→H is a continuous surjective homomorphism such that f(V) is a neighbourhood of 1 in H and ker⁡f⊆V, then f is open.While local q-minimality and local t-minimality are preserved by taking quotients, local q⁎-minimality fails to have this property (so does not coincide with local t-minimality), yet it has a relevant advantage. By means of an appropriate notion of local t-density, one can show that a dense subgroup H of a Hausdorff group G is locally q⁎-minimal if and only if G is locally q⁎-minimal and H is locally t-dense in G. Similar criterion for local q-minimality is not available (examples are given to show that a topological group with a dense locally q-minimal subgroup need not be locally q-minimal). The interrelations of these three versions of local minimality, as well as their prominence properties are studied in detail.

On continuous causal isomorphisms on two-dimensional Minkowski spacetimes

It is shown that continuous causal isomorphisms on two-dimensional Minkowski spacetime can be characterized by the invariance of wave equations.

Condensations of paratopological groups

We prove that every Hausdorff Lindelöf paratopological group with countable pseudocharacter admits a condensation onto a separable metrizable space. This result resolves a problem of M. Tkachenko. Also, we show that every regular (Hausdorff) ω-narrow semitopological (paratopological) group with countable Hausdorff number and countable pseudocharacter condenses onto a second countable Urysohn space.We show that each regular precompact paratopological group of countable pseudocharacter admits a continuous isomorphism onto a metrizable separable topological group. Also, we construct a Hausdorff precompact paratopological group with countable pseudocharacter which cannot be condensed onto a Hausdorff first countable space.

Subgroups of products of paratopological groups

We give a characterization of the paratopological groups that can be topologically embedded as subgroups into a product of first-countable (second-countable) Ti paratopological groups for i=0,1. We show that a T1 paratopological group G admits a topological embedding as a subgroup into a topological product of first-countable (second-countable) T1 paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the symmetry number of G is countable, i.e., for every neighborhood U of the identity e in G we can find a countable family γ of neighborhoods of e satisfying ⋂V∈γV−1⊆U. We show that every 2-pseudocompact T1 paratopological group with a countable symmetry number is a topological group.We answer in the negative some questions posed by Manuel Sanchis and Mikhail Tkachenko by constructing an example of a commutative functionally Hausdorff totally ω-narrow paratopological group of countable pseudocharacter H such that there is no continuous isomorphism from H onto a Hausdorff first-countable paratopological group. The group H is not topologically isomorphic to a subgroup of a product of Hausdorff second-countable paratopological groups.

Continuous isomorphisms onto separable groups

A condensation is a one-to-one continuous function onto. We give sufficient conditions for a Tychonoff space to admit a condensation onto a separable dense subspace of the Tychonoff cube Ic and discuss the differences that arise when we deal with topological groups, where condensation is understood as a continuous isomorphism. We also show that every Abelian group G with |G| 2c admits a separable, precompact, Hausdorff group topology, where c = 2!.

The dual of a pseudocompact group refining the topology of the circle group

We present an example of a reflexive, pseudocompact, non-compact, monothetic group H such that the Pontryagin dual H∧ of H is precompact, connected, Baire, and admits a continuous isomorphism onto the circle group. In addition, the countably compact subsets of both groups H and H∧ are finite.

Continuous isomorphisms fromRonto a complete abelian group

AbstractThis paper provides a Bishop-style constructive analysis of the contrapositive of the statement that a continuous homomorphism ofRonto a compact abelian group is periodic. It is shown that, subject to a weak locatedness hypothesis, ifGis a complete (metric) abelian group that is the range of a continuous isomorphism fromR, thenGis noncompact. A special case occurs whenGsatisfies a certain local path-connectedness condition at 0. A number of results about one-one and injective mappings are proved en route to the main theorem. A Brouwerian example shows that some of our results are the best possible in a constructive framework.

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence { σ n : n ∈ N } of cardinals such that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ⩽ | G | ⩽ 2 w ( G ) , where w ( G ) is the weight of G. If G is an infinite minimal abelian group, then either | G | = 2 σ for some cardinal σ, or w ( G ) = min { σ : | G | ⩽ 2 σ } ; moreover, the equality | G | = 2 w ( G ) holds whenever cf ( w ( G ) ) > ω . For a cardinal κ, we denote by F κ the free abelian group with κ many generators. If F κ admits a pseudocompact group topology, then κ ⩾ c , where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F c is equivalent to the Lusin's Hypothesis 2 ω 1 = c . For κ > c , we prove that F κ admits a (zero-dimensional) minimal pseudocompact group topology if and only if F κ has both a minimal group topology and a pseudocompact group topology. If κ > c , then F κ admits a connected minimal pseudocompact group topology of weight σ if and only if κ = 2 σ . Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.

Marked length rigidity for symmetric spaces

We give conditions under which a homomorphism between two Zariski dense subgroups of connected semisimple Lie groups G and G' without compact factors and with trivial center can be extended to a continuous isomorphism between G and G'. In particular we prove the marked length rigidity and the marked translation vector rigidity. This last result was motivated by a Margulis‚s question.

ON THE GALOIS COHOMOLOGICAL DIMENSIONS OF STABLE FIELDS WITH HENSELIAN VALUATIONS

ABSTRACT We present an essentially complete description of the spectra of cohomological p-dimensions of the absolute Galois groups of basic types of Henselian valued stable fields with totally indivisible value groups when p ranges over the set of prime numbers. In the stable closed and formally real case, this is supplemented by a classification of the absolute Galois groups of the corresponding residue fields up-to continuous isomorphisms.

The Wiener-Hopf integral equation for fractional Riesz-Bessel motion

AbstractThis paper gives an approximate solution to the Wiener-Hopf integral equation for filtering fractional Riesz-Bessel motion. This is obtained by showing that the corresponding covariance operator of the integral equation is a continuous isomorphism between appropriate fractional Sobolev spaces. The proof relies on properties of the Riesz and Bessel potentials and the theory of fractional Sobolev spaces.

Countably compact groups satisfying the open mapping theorem

We discuss the structure and Cartesian products of the countably compact groups G that satisfy the following forms of the open mapping theorem: (a) every continuous surjective homomorphism G→H is open, (b) every continuous isomorphism G→H is open (in both cases H is a Hausdorff topological group).

Fourier transforms of measures from the classes [formula omitted]β′ −2 < β ≤ −1

Subclasses U β(E), −2 < β ≤ −1, of the Lévy class L of self-decomposable measures on a Banach space E are examined. They are closed convolution subsemigroups of the semigroup of infinitely divisible probability distributions on E, defined as limit distributions of some prescribed schemes of summation. Each element of U β(E) belongs to the domain of normal attraction of a stable measure with exponent −β. Their Fourier transforms are characterized. The symmetric measures in U β(E) are shown to decompose uniquely into the convolution product of a symmetric stable measure with exponent −β and the probability distribution of a random integral of the form ∫ (0,1) t dY( t β ), where Y is a Lévy process with paths in the Skorohod space D E [0, ∞) and Y(1) has finite (−β)-moment. Topological and algebraic properties of the random-integral mapping S β: L ( Y(1)) → L [∫ (0,1) t dY( t β)] are investigated when E is a Hilbert space. As an application of the fact that S β is a continuous isomorphism, generators for U β are found as the images of compound Poisson distributions. Finally, the connection between the distributions U β and thermodynamic limits in the Ising model with zero external field is pointed out.