Contents

Preface

Urbanization and industrialization are producing large amounts of heavy metals, which are toxic to humans and the environment. These metals are commonly used for applications such as protecting surfaces, building, painting, in the food industry, and more. In addition to natural sources, these anthropogenic activities introduce heavy metals such as titanium (Ti), cadmium (Cd), chromium (Cr), arsenic (As), mercury (Hg), lead (Pd), nickel (Ni), copper (Cu), zinc (Zn), and others into the environment. In addition, rare earth elements such as neodymium (Nd), europium (Eu), terbium (Tb), dysprosium (Dy), and ytterbium (Yb) are being introduced into our ecosystem including the air, water, soil and sediment, and biological species.  Although minute quantities of some metals are required for the physiological processes in the body, excessive amounts can lead to a number of disorders. This book includes fifteen chapters and presents a comprehensive overview of heavy metals including their physiochemical properties, toxicity, transfer in the environment, legislation, environmental impacts, and mitigation measures. Chapters include scientific research and studies. This book provides insight into the available techniques for removing heavy metals, such as adsorption, their qualitative and quantitative analysis, and future prospects for

**Mazen Khaled Nazal**

Research Institute (RI),

Dhahran, Saudi Arabia

Central South University,

Changsha, Hunan, China

**Dr. Hongbo Zhao**

Professor,

Center for Environment and Marine Studies (CEMS),

King Fahd University for Petroleum and Minerals (KFUPM),

Key Lab of Bio-hydrometallurgy of Ministry of Education,

these removal methods.


**1. Introduction**

In the present paper, we study category algebras and states defined on arbitrary small categories to build a new bridge between category theory (see [1–4] and references therein, for example) and noncommutative probability or quantum probability (see [5–7] and references therein, for example), a generalized probability theory which was originated as a mathematical framework for quantum theory.

A category algebra is, in short, a convolution algebra of functions on a category. For example, on certain categories called finely finite category [8], which is a categorical generalization of locally finite poset, the convolution operation can be defined on the set of arbitrary functions and it becomes a unital algebra called incidence algebra. Many authors have studied the notions of Möbius inversion, which has been one of fundamental part of combinatorics since the pioneering work by Rota [9] on posets, in the context of incidence algebras on categories ([8,10–14], for example).

There is another approach to obtain the notion of category algebra. As is well known, a group algebra is defined as a convolution algebra consisting of finite linear combinations of elements. By generalization with replacing "elements" by "arrows", one can obtain another notion of category algebra (see [13], for example), which also includes monoid algebra (in particular polynomial algebras) and groupoid algebras as examples. Please note that for a category with infinite number of objects, the algebra is not unital.

The category algebras we focus on in the present paper are unital algebras defined on arbitrary small categories, which are slightly generalized versions of algebras studied under the name of the ring of an additive category [15]. These category algebras include the ones studied in [13] as subalgebras in general, and they coincide for categories with finite number of objects. Moreover, one of the algebras we study, called "backward finite category algebra", coincides with incidence algebras for combinatorically important cases originally studied in [9].

**Citation:** Saigo, H. Category Algebras and States on Categories. *Symmetry* **2021**, *13*, 1172. https:// doi.org/10.3390/sym13071172

Academic Editors: Motoichi Ohtsu and Alexey Kanel-Belov

Received: 28 May 2021 Accepted: 26 June 2021 Published: 29 June 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The purpose of this paper is to provide a new framework for the interplay between regions of mathematical sciences such as algebra, probability and physics, in terms of states as linear functional defined on category algebras. As is well known, quantum theory can be considered to be a noncommutative generalization of probability theory. At the beginning of quantum theory, matrix algebras played a crucial role (see [16] for example). In the present paper, we clarify that category algebras can be considered to be generalized matrix algebras and that the notions of states on categories as linear functionals defined on category algebras turns out to be a conceptual generalization of probability measures on sets as discrete categories (For the case of states on groupoid algebras over the complex field C it is already studied [17]).

Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction [18,19] (as for the studies in category theoretic context, see [20–22] for example), we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces (semi-Hilbert modules over rigs), which can be considered to be an extension of the result in [17] for groupoid algebras over C. This construction will provide a basis for the interplay between category theory, noncommutative probability and other related regions such as operator algebras or quantum physics.

**Notation 1.** *In the present paper, categories are always supposed to be small (This assumption may be relaxed by applying some appropriate foundational framework). The set of all arrows in a category* C *is also denoted as* C*.* |C| *denotes the set of all objects, which are identified with corresponding identity arrows, in* C*. We also use the following notations:*

$$\prescript{}{}{\mathcal{C}}'\_{\mathbb{C}} := \mathcal{C}(\mathbb{C}, \mathbb{C}'), \; \mathcal{C}\_{\mathbb{C}} := \sqcup\_{\mathbb{C}' \in [\mathcal{C}]} \mathcal{C}(\mathbb{C}, \mathbb{C}'), \; \prescript{}{\mathcal{C}}' \mathcal{C} := \sqcup\_{\mathbb{C} \in [\mathcal{C}]} \mathcal{C}(\mathbb{C}, \mathbb{C}'),$$

*where* C(*C*, *C* ) *denotes the set of all arrows from C to C .*


*Edited by Nizar Souayah*

Understanding the rapid changes in the evaluation and management of peripheral neuropathies, as well as the complexity of their mechanism, is a mandatory requirement for the practitioner to optimize patient's care. The objective of this book is to update health care professionals on recent advances in the pathogenesis, diagnosis and treatment of peripheral neuropathy. This work was written by a group of clinicians and scientists with large expertise in the field.

Peripheral Neuropathy - A New Insight into the Mechanism,

Evaluation and Management of a Complex Disorder

Peripheral Neuropathy

A New Insight into the Mechanism, Evaluation

and Management of a Complex Disorder

*Edited by Nizar Souayah*

Photo by ktsimage / iStock


**4. Conclusions**

In this study, we have investigated whether different types of surface functionalization of SiNP impact immune effects by differential activation, i.e., the maturation, of APCs. In addition to first-line safety assessment, early insight on efficacy (second-line) is crucial for decision making in pharmaceutical development; in the context of immunotherapy, this represents the investigation of the immune activating and modulating capacities of administered substances, including nanomaterial platforms. Both the immune activating as well as modulating potentials involve APCs as a crucial player. As a suitable surrogate for human APCs, monocyte-derived dendritic cells were used here. We observed negligible changes induced by the differently functionalized SiNPs in the viability and maturation state of APCs, which is an indication of their immunological inertness. Nevertheless, a significant impact on the binding of proteins to the differently functionalized SiNPs was evident, and could not be simply explained by considering the isoelectric point of protein the net charge of particles upon functionalization. Enhanced uptake was noted for the particle system used here, irrespective of functionalization, while surface functionalization even optimized their immunological inertness, suggesting that SiNPs may be a suitable vehicle for the delivery of biopharmaceutical products, in particular for allergen-specific immunotherapy. Inert (nano)carriers afford the opportunity to intentionally load additional immune modifiers for fine-tuning of desired immunological outcome (e.g., immunomodulation towards a regulatory T helper cell response in allergy or immunosuppression in autoimmunity). As protein binding can be modulated by various types of chemical functionalization, functionalized SiNPs could be useful to deliver a range of biopharmaceutical drugs and lessen the potential immune response. Nevertheless, in vivo studies about the safety and efficacy profiles are required.

**Supplementary Materials:** The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/pharmaceutics14051103/s1, Reference [56] is cited in the supplementary materials. Figure S1: Schiff base reaction for testing amino functionalization of SiNP; Figure S2. Gating strategy for moDC experiments; Figure S3. Fluorescence minus one (FMO) controls of moDCs; Figure S4. FTIR spectra of differently functionalized SiNP; Figure S5. NTA and DLS size distribution graphs; Figure S6. SDS-PAGE for estimation of protein binding; Figure S7. Suspension stability test; Table S1. Uptake inhibitors; Figure S8. SDS-PAGE analysis with pHrodo-labelled Bet v 1; Figure S9. Viability of moDCs on exposure to labelled allergen; Figure S10. Viability of moDCs on exposure to selected inhibitors.

**Author Contributions:** Conceptualization: L.J. and M.H. Methodology: B.P., L.J., J.H.-H., M.G. and M.H. Investigation: B.P., L.J., M.G. and H.-H.D. Visualization: B.P. and L.J. Writing—original draft: B.P., L.J. and M.H. Writing—review & editing: B.P., L.J., M.G., J.H.-H., A.D. and M.H. Supervision: L.J. and M.H. Funding acquisition: A.D. and M.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the international PhD program "Immunity in Cancer and Allergy" of the Austrian Science Fund (Open Access Funding by the Austrian Science Fund (FWF), grant number W01213), the EU H2020 NanoCommons project (grant number 731032), the SmartCERIALS project of the Austrian Research Promotion Agency (FFG, grant number 890610) and the Allergy Cancer Bio-Nano Research Centre (ACBN) of the University of Salzburg.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** All data generated or analyzed during this study are included in this manuscript and its supplementary information files. Raw data of the cyto-/chemokine secretion multiplex assay are available at https://doi.org/10.5281/zenodo.6473305 (accessed on 20 April 2022).

**Acknowledgments:** The authors thank Robert Mills-Goodlet; Ingrid Hasenkopf; Alexandra Fux; Angela Schmiedlechner; Sabine Hofer; Norbert Hofstätter; and Nanett Mokran for provision of materials for experiments, conduct of supporting experiments and scientific inputs.