C*-Algebras for Quantum Solids

We prove that in every compact space of Delone sets in ${\mathbb {R}}^d$ , which is minimal with respect to the action by translations, either all Delone sets are uniformly spread or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty–Fell topology, which is the natural topology on the space of closed subsets of ${\mathbb {R}}^d$ . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

On the existence of continuous (approximate) roots of algebraic equations

We show that Kellendonk’s tiling semigroup of a finite local complexity substitution tiling is self-similar, in the sense of Bartholdi, Grigorchuk and Nekrashevych. We extend the notion of the limit space of a self-similar group to the setting of self-similar semigroups, and show that it is homeomorphic to the Anderson–Putnam complex for such substitution tilings, with natural self-map induced by the substitution. Thus, the inverse limit of the limit space, given by the limit solenoid of the self-similar semigroup, is homeomorphic to the translational hull of the tiling.

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

Introduction. When heating with high-frequency currents (HFCs) at high speeds, more significant strengthening effects can be observed compared to using machine generators. Therefore, hardening at high frequencies is more efficient. However, the increase in the generator frequency results in a decrease in the depth of penetration of eddy currents and an increased unevenness of heating across the cross-section. The application of a constant external magnetic field during HFC hardening can increase the depth of eddy current penetration and create more uniform heating. Unfortunately, there is not enough information available on the effect of the external magnetic field on HFC heating processes and phase transformations in steel. Currently, there are no quantitative estimates for the impact of an external magnetic field on changes in the kinetics of electric heating and the penetration depth of eddy currents. In connection with the above, the aim of this paper is to investigate changes in the kinetics of high-frequency heating of iron-carbon alloys when an external constant magnetic field is applied and, and, based on this, to consider the potential for technological applications. Materials and Methods. Theoretical assessment of the influence of an external magnetic field on the change in the kinetics of electric heating and the penetration depth of eddy currents is based on the general theory of induction heating kinetics. An experimental study of the influence of a magnetic field on the kinetics of high-frequency current heating was performed on samples of 45 steel, pearlitic gray (SCh30), and ferritic malleable cast iron (KCh30-6). The temperature distribution over the cross-section of ferromagnetic materials during induction heating with an external magnetic field has been studied using special samples of iron, 45 steel, and SCh30 gray pearlitic cast iron. Electric tempering processes have been investigated on samples of U8A steel using a vacuum tube generator (heating temperature — 450℃, heating rate — 750℃/s). Changes in austenite grain size after high-speed heating with external magnetization have been examined on samples of reduced-hardenability 55PP steel. To study the processes of thermal treatment in a magnetic field during experiments involving heating samples using high-frequency currents, a specially designed electromagnet was created to apply an external constant magnetic field. Results. Theoretical curves were constructed for heating conditions with and without an external constant magnetic field. Experimental data on the effect of an external constant magnetic field on induction heating in the surface layer of various materials were summarized in kinetic diagrams. Evidence that the observed changes were due to increased depth of penetration of eddy currents came from experiments on cylindrical samples of 45 steel with different wall thicknesses. Kinetic curves were provided for estimating the temperature field (at 6 points at different depths) during high-frequency current heating with and without external magnetization. The paper presents experimental data on the micro-hardness distribution across the cross-section of a U8 steel sample after quenching, quenching and electric tempering, quenching and electric tempering with external magnetization, and quenching and bulk tempering. It also includes the results of the study of the austenite grain size of 55PP steel after high-speed heating with external magnetization and conventional (slow) deep heating. Discussion. The application of a high-intensity external constant magnetic field during the first quasi-stationary process resulted in a decrease in the rate of induction heating of the ferromagnetic material and an increase in the depth of its uniform heating. However, above the Curie point, the effect of the magnetic field was negligible due to the low magnetic susceptibility of the material, and the heating rate remained unchanged as if there was no field present. In addition, due to the insignificant difference in the values of magnetic permeability below and above the Curie point during heating in the field, the thermal curve did not exhibit the characteristic inflection typical of kinetic curves observed during the transition of the surface layer to a paramagnetic state. Experiments with electric tempering have demonstrated that by applying an external field, it was possible to temper a material to the desired depth and it could be done on a single high-frequency current setup. The size of the austenite grains after high-speed heating with magnetization was reduced compared to conventional deep heating of steel with low hardenability, eliminating the issue of induction heating for low-hardenability steel. Conclusion. The results of the study demonstrated that the use of an external magnetic field enabled the achievement of strengthening effects during heating at higher frequencies, thereby eliminating the drawbacks of such heating methods.

A basic assumption of tiling theory is that adjacent tiles can meet in only a finite number of ways, up to rigid motions. However, there are many interesting tiling spaces that do not have this property. They have 'fault lines', along which tiles can slide past one another. We investigate the topology of a certain class of tiling spaces of this type. We show that they can be written as inverse limits of CW complexes, and their Čech cohomology is related to properties of the fault lines.

αo̧mpactness in bifuzzy topological spaces

Almost compactness and near compactness in fuzzy topological spaces

Let X be a compact Hausdorff space and C(X) the Banach space of all real-valued continuous functions on X under the sup norm. If H is a linear subspace of C(X) which separates the points of X and contains the constant functions, then it is well known that there exists a smallest closed subset of X, called the Silov boundary of X relative to H and denoted by OHX, with the property that each hCH attains its maximum value on afHX. A point xeX is called an H-extremal point if the only positive linear functional u on C(X) such that u(h) = h(x) for all hCH is the evaluation functional 4. where O.,(f) =f(x) for all fE C(X). The set of H-extremal points of X is called the Choquet boundary of X relative to H and will be denoted by VHX. H. Bauer has shown [1, 2.1 ] that the Choquet boundary of X relative to H is nonempty and the Silov boundary of X relative to H is equal to the closure of the Choquet boundary. Bauer has also introduced (see [1 ]) an abstract Dirichlet problem for the above setting. If S:DHX is closed and xCX, then MS denotes the set of positive linear functionals u on C(S) such that u(h I S) =h(x) for all hCH. MS is always nonempty. The measures in MX'HX are called the H-harmonic measures belonging to x. A function f in C(X) is said to be H-harmonic if for every xEX and every u M we have u(f) =f(x). The set of H-harmonic functions is denoted by H. The Dirichlet problem is said to be solvable for X relative to H if 1^ j aHX = c(aHX). The Dirichlet problem is solvable if and only if to each x in X belongs exactly one H-harmonic measure (see [1, Satz 9]). Let X1, X2 be compact Hausdorff spaces and H1, H2 separating linear subspaces of C(X1), C(X2) respectively, which contain the constant functions. Equipped with the usual product topology, the cartesian product X1 X X2 is a compact Hausdorff space. In this paper we show that a subfamily H1+H2 of C(XIXX2) can be defined, in a natural way, for which VH1+H2Xl X X2 exists and VH1+H2Xl X X2 = VH1X1X VH2X2. We then give two theorems on the relation of the

It is known that a positive, compactly supported function $f \in L^1(\mathbb R)$ can tile by translations only if the translation set is a finite union of periodic sets. We prove that this is not the case if $f$ is allowed to have unbounded support. On the other hand we also show that if the translation set has finite local complexity, then it must be periodic, even if the support of $f$ is unbounded.

Let T be an aperiodic and repetitive tiling of R^d with finite local complexity. Let O be its tiling space with canonical transversal X. The tiling equivalence relation R_X is the set of pairs of tilings in X which are translates of each others, with a certain (etale) topology. In this paper R_X is reconstructed as a generalized "tail equivalence" on a Bratteli diagram, with its standard AF-relation as a subequivalence relation. Using a generalization of the Anderson-Putnam complex, O is identified with the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is built from this sequence, and its set of infinite paths dB is homeomorphic to X. The diagram B is endowed with a horizontal structure: additional edges that encode the adjacencies of patches in T. This allows to define an etale equivalence relation R_B on dB which is homeomorphic to R_X, and contains the AF-relation of "tail equivalence".

We show that for multi-colored Delone point sets with finite local complexity and uniform cluster frequencies the notions of pure point diffraction and pure point dynamical spectrum are equivalent.

For a topological space $X$, let $X_\delta$ be the space $X$ with $G_\delta$-topology of $X$. For an uncountable cardinal $\kappa$, we prove that the following are equivalent: (1) $\kappa$ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelof degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindelof degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindelof and the weak Lindelof degrees of compact Hausdorff spaces with $G_\delta$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_\delta$-topology. For the square of a Lindelof space, using weak $G_\delta$-topology, we prove that the following are consistent: (1) the least $\omega_1$-strongly compact cardinal is the supremum of the (weak) Lindelof degrees of the squares of regular $T_1$ Lindelof spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelof spaces.

Linear extension operators of bounded norms

A new system was developed for measuring AC current distributions in high T/sub c/ superconducting tapes under an external AC magnetic field. In our measurement the current distributions are indirectly found from the results of the self-field distributions measured by an array of pickup coils located around the tape. For improved measurement accuracy under the external magnetic field, a compensation coil was used to cancel the component of the external magnetic field from the signal of pickup coil. In order to confirm the validity of this system, measurement of current distributions in a test conductor composed of insulated copper wires was carried out under an external AC magnetic field. Using this system, we measured the current distributions in multifilamentary Ag alloy sheathed Bi-2223 tape as a function of the angle of the external AC magnetic field at 77 K. Further, we investigated the characteristics of the current distributions for the case of a simultaneous application of AC transport current and transverse AC magnetic field. On the basis of the obtained results the practicality of our measurement system is discussed.