G-Compactness and Groups

Lascar described E_KP as a composition of E_L and the topological closure of EL. We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M_0 consisting of M and X as two sorts, where X is an affine copy of G and in M_0 we have the structure of M and the action of G on X. We prove that the Lascar group of M_0 is a semi-direct product of the Lascar group of M and G/G_L. We discuss the relationship between G-compactness of M and M_0. This example may yield new examples of non-G-compact theories.Comment: 18 page

When a totally bounded group topology is the Bohr Topology of a LCA group

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and sufficient conditions are established in terms of the inner properties of $w$. As an application, an example of a MAP group $(G,t)$ is given such that every closed, metrizable subgroup $N$ of $bG$ with $N \cap G = \{0\}$ preserves compactness but $(G,t)$ does not strongly respects compactness. Thereby, we respond to Questions 4.1 and 4.3 in [comftrigwu]

Group-valued continuous functions with the topology of pointwise convergence

We denote by C_p(X,G) the group of all continuous functions from a space X to a topological group G endowed with the topology of pointwise convergence. We say that spaces X and Y are G-equivalent provided that the topological groups C_p(X,G) and C_p(Y,G) are topologically isomorphic. We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of C_p(X,G). Since R-equivalence coincides with l-equivalence, this line of research "includes" major topics of the classical C_p-theory of Arhangel'skii as a particular case (when G = R). We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if C_p(X,G) is TAP, and (ii) for a metrizable NSS group G, a G^*-regular space X is compact if and only if C_p(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if C_p(X,R) is a TAP group (of countable tightness). We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, sigma-compactness, the property of being a Lindelof Sigma-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed.Comment: Two references were added and one reference was updated. Question 11.1 was resolved in arXiv:0909.2381 [math.GN]. The bibliographic information related to Theorem 10.2 was corrected. Minor typos were corrected as well

The Effect of Gasification Conditions on the Surface Properties of Biochar Produced in a Top-Lit Updraft Gasifier

The effect of airflow rate, biomass moisture content, particle size, and compactness on the surface properties of biochar produced in a top-lit updraft gasifier was investigated. Pine woodchips were studied as the feedstock. The carbonization airflow rates from 8 to 20 L/min were found to produce basic biochars (pH > 7.0) that contained basic functional groups. No acid functional groups were presented when the airflow increased. The surface charge of biochar at varying airflow rates showed that the cation exchange capacity increased with airflow. The increase in biomass moisture content from 10 to 14% caused decrease in the pH from 12 to 7.43, but the smallest or largest particle sizes resulted in low pH; therefore, the carboxylic functional groups increased. Similarly, the biomass compactness exhibited a negative correlation with the pH that reduced with increasing compactness level. Thus, the carboxylic acid functional groups of biochar increased from 0 to 0.016 mmol g−1, and the basic functional group decreased from 0.115 to 0.073 mmol g−1 when biomass compactness force increased from 0 to 3 kg. BET (Brunauer-Emmett-Teller) surface area of biochar was greater at higher airflow and smaller particle size, lower moisture content, and less compactness of the biomassThe effect of airflow rate, biomass moisture content, particle size, and compactness on the surface properties of biochar produced in a top-lit updraft gasifier was investigated. Pine woodchips were studied as the feedstock. The carbonization airflow rates from 8 to 20 L/min were found to produce basic biochars (pH > 7.0) that contained basic functional groups. No acid functional groups were presented when the airflow increased. The surface charge of biochar at varying airflow rates showed that the cation exchange capacity increased with airflow. The increase in biomass moisture content from 10 to 14% caused decrease in the pH from 12 to 7.43, but the smallest or largest particle sizes resulted in low pH; therefore, the carboxylic functional groups increased. Similarly, the biomass compactness exhibited a negative correlation with the pH that reduced with increasing compactness level. Thus, the carboxylic acid functional groups of biochar increased from 0 to 0.016 mmol g−1, and the basic functional group decreased from 0.115 to 0.073 mmol g−1 when biomass compactness force increased from 0 to 3 kg. BET (Brunauer-Emmett-Teller) surface area of biochar was greater at higher airflow and smaller particle size, lower moisture content, and less compactness of the biomas

Spaces of invariant circular orders of groups

Motivated by well known results in low-dimensional topology, we introduce and study a topology on the set CO(G) of all left-invariant circular orders on a fixed countable and discrete group G. CO(G) contains as a closed subspace LO(G), the space of all left-invariant linear orders of G, as first topologized by Sikora. We use the compactness of these spaces to show the sets of non-linearly and non-circularly orderable finitely presented groups are recursively enumerable. We describe the action of Aut(G) on CO(G) and relate it to results of Koberda regarding the action on LO(G). We then study two families of circularly orderable groups: finitely generated abelian groups, and free products of circularly orderable groups. For finitely generated abelian groups A, we use a classification of elements of CO(A) to describe the homeomorphism type of the space CO(A), and to show that Aut(A) acts faithfully on the subspace of circular orders which are not linear. We define and characterize Archimedean circular orders, in analogy with linear Archimedean orders. We describe explicit examples of circular orders on free products of circularly orderable groups, and prove a result about the abundance of orders on free products. Whenever possible, we prove and interpret our results from a dynamical perspective.Comment: Minor errors corrected and exposition improved throughout. Provides a more careful analysis of cases in the proof of Theorem 4.3. Fixed the proof that Archimedean implies fre

On G-Sequential Continuity

Let $X$ be a first countable Hausdorff topological group. The limit of a sequence in $X$ defines a function denoted by $lim$ from the set of all convergence sequences to $X$. This definition was modified by Connor and Grosse-Erdmann for real functions by replacing $lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. \c{C}akall{\i} extended the concept to topological group setting and introduced the concept of $G$-sequential compactness and investigated $G$-sequential continuity and $G$-sequential compactness in topological groups. In this paper we give a further investigation of $G$-sequential continuity in topological groups most of which are also new for the real case.Comment: 15 pages, Research Paper, arXiv admin note: substantial text overlap with arXiv:1006.4706 and arXiv:1105.2203 by other autho

A dichotomy property for locally compact groups

We extend to metrizable locally compact groups Rosenthal's theorem describing those Banach spaces containing no copy of $l_1$. For that purpose, we transfer to general locally compact groups the notion of interpolation ($I_0$) set, which was defined by Hartman and Ryll-Nardzewsky [25] for locally compact abelian groups. Thus we prove that for every sequence $\lbrace g_n \rbrace_{n<\omega}$ in a locally compact group $G$, then either $\lbrace g_n \rbrace_{n<\omega}$ has a weak Cauchy subsequence or contains a subsequence that is an $I_0$ set. This result is subsequently applied to obtain sufficient conditions for the existence of Sidon sets in a locally compact group $G$, an old question that remains open since 1974 (see [32] and [20]). Finally, we show that every locally compact group strongly respects compactness extending thereby a result by Comfort, Trigos-Arrieta, and Wu [13], who established this property for abelian locally compact groups.Comment: To appear in J. of Functional Analysi

Interpolation sets in spaces of continuous metric-valued functions

Let $X$ and $M$ be a topological space and metric space, respectively. If $C(X,M)$ denotes the set of all continuous functions from X to M, we say that a subset $Y$ of $X$ is an \emph{$M$-interpolation set} if given any function $g\in M^Y$ with relatively compact range in $M$, there exists a map $f\in C(X,M)$ such that $f_{|Y}=g$. In this paper, motivated by a result of Bourgain in \cite{Bourgain1977}, we introduce a property, stronger than the mere \emph{non equicontinuity} of a family of continuous functions, that isolates a crucial fact for the existence of interpolation sets in fairly general settings. As a consequence, we establish the existence of $I_0$ sets in every nonprecompact subset of a abelian locally $k_{\omega}$-groups. This implies that abelian locally $k_{\omega}$-groups strongly respects compactness