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Review of Deterministic and Random System and Signal Concepts
Published in Ralph D. Hippenstiel, Detection Theory, 2017
In addition, the vector space must have a distance (norm) and an angle (inner product) measure. So if we take a complete inner product space (i.e., a vector space), all of these properties hold. Since the majority of problems and applications deal with discrete-time problems, more attention is brought to these types of problems. Where appropriate, the continuous time type problems arc addressed. The hierarchy for the vector space is as follows: group → ring → field → vector space → algebra → complete normed space (i.e., Banach space) → complete inner product space (real or complex vector space with inner product, i.e., Hilbert space).
Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
An isomorphism from group (ring) S1 to group (ring) S2 is a group (ring) homomorphism ϕ : S1 → S2 that is 1‐1 and onto S2. If an isomorphism exists, then S1 is said to be isomorphic to S2. Write S1 ≅ S2. (See the table on page 119 for numbers of non‐isomorphic groups and the table on page 121 for examples of groups of orders less than 16.)
A Comparative Analysis of Classical Cryptography versus Quantum Cryptography for Web of Medical Things (WoMT)
Published in Aboul Ella Hassanien, Nilanjan Dey, Surekha Borra, Medical Big Data and Internet of Medical Things, 2018
K. Anitha Kumari, G. Sudha Sadasivam
Modern cryptographic algorithms treat the information as numbers/elements in a finite space to obey closure property for applying in encryption and decryption procedures [6]. As basic arithmetic operations are not satisfying the closure property, algebraic structures like group, ring and field are used. The characteristics of an algebraic structure ‘A’ is denoted by char(A). It is said to possess a least one positive integer ‘n’ such that, na = 0 for every a ∈ A.
Synthesis, characterization and antioxidant activity of EGCG complexes with copper and zinc ions
Published in Journal of Coordination Chemistry, 2019
Maysoon Alhafez, Fadi Kheder, Malak Aljoubbeh
Overall, the results suggest the formation of EGCG complexes with copper and zinc. It was proposed that 3-hydroxy group in some polyphenols has a more acidic proton making it along with 4-oxo group a more preferable site for binding the metal ion [19]. Since EGCG lacks both groups the metal ion has two possible binding sites only; pyrogallol group (ring B) or galloyl group (ring D). It was reported that the metal ions preferentially bind to pyrogallol group of EGCG since the calculated unpaired electron density was higher at C3′ position [15]. It was also proposed that EGCG would act as a stable bidentate chelating agent between the pyrogallol groups and Cu(II) [20]. Furthermore, Wu and his colleagues indicated that in neutral to alkaline mediums the C3′, C4′, C4′′-hydroxyl groups are transformed to an anion [21], so the ability to react with cations is favorable through pyrogallol group.
Chalconoid metal chelates: spectral, biological and catalytic applications
Published in Journal of Coordination Chemistry, 2019
A majority of chalcone ligands contain coordination sites preferably on the phenyl ring adjacent to the carbonyl group (ring A). Ligand 5-(4-(dimethylamino)phenyl)-1-(thiophen-2-yl)penta-2,4-dien-1-one (see 16, Scheme 2) is derived from 2-acetylthiophene (ring A) and 4-(dimethylamino)cinnamaldehyde [49]. Thiophene sulphur and (C=O) are involved in metal coordination [50]. Gaber synthesized a chalcone ligand (see 17, Scheme 2) in which a six-membered pyridine ring (ring A) was introduced and observed (N,O) coordination towards metal ions [51].
Redox-responsive hollow mesoporous silica nanoparticles constructed via host–guest interactions for controllable drug release
Published in Journal of Biomaterials Science, Polymer Edition, 2020
Jiangtao Liu, Yan Li, Min Zhao, Zhongli Lei, Hui Guo, Yuping Tang, Hao Yan
In Figure 4(e), the disappearance of the typical absorption peaks of alkyne moiety and azide signals, while an intense broad band appeared at 1626 cm−1 were associated with triazolyl group ring vibrations, indicating the click chemistry was successful. Moreover, the strong absorption band centered at 3376 cm−1 due to O–H vibration of β-CD, together with the formation of intense bands at 1445 and 1335 cm−1 (δ O–H modes), confirmed the presence of β-CD on the HMS surface.