Let \mathcal C be a concrete category. We prove that if \mathcal {C} admits a universally free object \mathsf F, then there is a projectively universal morphism u\colon \mathsf F\to \mathsf F, i.e., ...a morphism u such that for any B\in \mathcal {C} and \tau \in \operatorname {Mor}(B) there exists an epimorphism \pi \in \operatorname {Mor}(\mathsf F, B) such that \pi \tau = u \pi. This builds upon and extends various ideas by Darji and Matheron Proc. Amer. Math. Soc. 145 (2017), pp. 251–265 who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc.
The accepted methods for testing concrete are not favorable for determining its heterogeneity. The interpretation of the compressive strength result as a product of destructive force and ...cross-section area is burdened with significant understatements. It is assumed erroneously that this is the lowest value of strength at the height of the tested sample. The top layer of concrete floors often crumble, and the strength tested using sclerometric methods does not confirm the concrete class determined using control samples. That is why it is important to test the distribution of compressive strength in a cross-section of concrete industrial floors with special attention to surface top layers. In this study, we present strength tests of borehole material taken from industrial floors using the ultrasonic method with exponential spot heads with a contact surface area of 0.8 mm
and a frequency of 40 kHz. The presented research project anticipated the determination of strength for samples in various cross-sections at the height of elements and destructive strength in the strength testing machine. It was confirmed that for standard and big borehole samples, it is not possible to test the strength of concrete in the top layer of the floor by destructive methods. This can be done using the ultrasonic method. After the analysis, certain types of distributions of strength across concrete floor thickness were chosen from the completed research program. The gradient and anti-gradient of strength were proposed as the new parameters for the evaluation of floor concrete quality.
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Sulphate corrosion of concrete is a complex chemical and physical process that leads to the destruction of construction elements. Degradation of concrete results from the transportation of sulphate ...compounds through the pores of exposed elements and their chemical reactions with cementitious material. Sulphate corrosion can develop in all kind of structures exposed to the corrosive environment. The mechanism of the chemical reactions of sulphate ions with concrete compounds is well known and described. Furthermore, the dependence of the compressive strength of standard cubic samples on the duration of their exposure in the sulphate corrosion environment has been described. However, strength tests on standard samples presented in the scientific literature do not provide an answer to the question regarding the measurement methodology and actual distribution of compressive strength in cross-section of reinforced concrete structures exposed to sulphate ions. Since it is difficult to find any description of this type of test in the literature, the authors undertook to conduct them. The ultrasonic method using exponential heads with spot surface of contact with the material was chosen for the measurements of concrete strength in close cross-sections parallel to the corroded surface. The test was performed on samples taken from compartments of a reinforced concrete tank after five years of operation in a corrosive environment. Test measurements showed heterogeneity of strength across the entire thickness of the tested elements. It was determined that the strength of the elements in internal cross-sections of the structure was up to 80% higher than the initial strength. A drop in the mechanical properties of concrete was observed only in the close zone near the exposed surface.
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Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach ...algebra that cannot be realised as a Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form C(X) for a compact space X, \ell _1(G) for some torsion-free Abelian group, and a simple, unital AF C^*-algebra. Extensions to higher densities are also presented.
In 1973, Diestel published his seminal paper
Grothendieck spaces and vector measures
that drew a connection between Grothendieck spaces (Banach spaces for which weak- and weak*-sequential ...convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr.
Vector measures
. Additionally, Diestel’s paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day.
The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel’s problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.
It is well known that there is no basis for the field of real numbers regarded as a vector space over any proper subfield that is closed under multiplication. Mabry 3 has extended this result to ...bases of arbitrary proper field extensions. The aim of this short communication is to notice that the proof of the result concerning the reals may be adjusted to a larger class of algebras (including full matrix algebras); thereby, we subsume Mabry’s result.
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Using the fact that the normalized matrix trace is the unique linear functional f on the algebra of n × n matrices which satisfies f(I) = 1 and f(AB) = f(BA) for all n × n matrices A and B, we derive ...a well-known formula expressing the normalized trace of a complex matrix A as the expectation of the numerical values of A; that is the function ⟨Ax, x⟩, where x ranges the unit sphere of ℂ
n
.
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Separated sets and Auerbach systems in Banach spaces Hájek, Petr; Kania, Tomasz; Russo, Tommaso
Transactions of the American Mathematical Society,
October 1, 2020, 2020-10-00, Volume:
373, Issue:
10
Journal Article
Peer reviewed
Open access
The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a ...large enough space X, the unit sphere S_X always contains an uncountable (1+)-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of c_0(\omega _1). Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically (1+\varepsilon )-separated subset of any regular cardinality not exceeding the density of X; should the space X be super-reflexive, the unit sphere of X contains such a subset of cardinality equal to the density of X. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.
The present paper contributes to the ongoing programme of quantification of isomorphic Banach space theory focusing on the Hagler–Stegall characterisation of dual spaces containing complemented ...copies of
L
1
. As a corollary, we obtain the following quantitative version of the Hagler–Stegall theorem asserting that for a Banach space
X
, the following statements are equivalent:
X
contains almost isometric contains almost isometric copies of
(
⨁
n
=
1
∞
ℓ
∞
n
)
ℓ
1
;
for all
ε
>
0
,
X
∗
contains a
(
1
+
ε
)
-complemented,
(
1
+
ε
)
-isomorphic copy of
L
1
;
for all
ε
>
0
,
X
∗
contains a
(
1
+
ε
)
-complemented,
(
1
+
ε
)
-isomorphic copy of
C
0
,
1
∗
.
Moreover, if
X
is separable, one may add the following assertion:
for all
ε
>
0
, there exists a
(
1
+
ε
)
-quotient map
T
:
X
→
C
(
Δ
)
so that
T
∗
C
(
Δ
)
∗
is
(
1
+
ε
)
-complemented in
X
∗
, where
Δ
is the Cantor set
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Cracking in non-load-bearing internal partition walls is a serious problem that frequently occurs in new buildings within the short term after putting them into service or even before completion of ...construction. Sometimes, it is so considerable that it cannot be accepted by the occupiers. The article presents tests of cracking in ceramic walls with a door opening connected in a rigid and flexible way along vertical edges. The first analyzes were conducted using the finite element method (FEM), and afterward, the measurements of deformations and stresses in walls on deflecting floors were performed on a full scale in the actual building structure. The measurements enabled to determine floor deformations leading to cracking of walls and to establish a dependency between the values of tensile stresses within the area of the door opening corners and their location along the length of walls and type of vertical connection with the structure.
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