The Kruskal-Katona Theorem and a Characterization of System Signatures

2015 ◽  
Vol 52 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Alessandro D'Andrea ◽  
Luca De Sanctis
Keyword(s):  
Finite Number ◽  
Structure Function ◽  
Uniform Distribution ◽  
Simplicial Complexes ◽  
Number Of Components ◽  
Full Characterization ◽  
Signature Vector

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

scholarly journals The Kruskal-Katona Theorem and a Characterization of System Signatures

2015 ◽  
Vol 52 (2) ◽  
pp. 508-518 ◽  
Author(s):  
Alessandro D'Andrea ◽  
Luca De Sanctis
Keyword(s):  
Finite Number ◽  
Structure Function ◽  
Uniform Distribution ◽  
Simplicial Complexes ◽  
Number Of Components ◽  
Full Characterization ◽  
Signature Vector

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

The Kruskal-Katona Theorem and a Characterization of System Signatures

2015 ◽  
Vol 52 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Alessandro D'Andrea ◽  
Luca De Sanctis
Keyword(s):  
Finite Number ◽  
Structure Function ◽  
Uniform Distribution ◽  
Simplicial Complexes ◽  
Number Of Components ◽  
Full Characterization ◽  
Signature Vector

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

A Thorough Theoretical Exploration of Intriguing Characteristics of Cyclo[18]carbon: Geometry, Bonding Nature, Aromaticity, Weak Interaction, Reactivity, Excited States, Vibrations, Molecular Dynamics and Various Molecular Properties

2019 ◽  
Author(s):  
Tian Lu ◽  
Qinxue Chen ◽  
Zeyu Liu
Keyword(s):  
Molecular Dynamics ◽  
Excited States ◽  
Electronic Excitation ◽  
Ring Structure ◽  
Molecular Properties ◽  
Molecular Vibration ◽  
Full Characterization ◽  
Long Time ◽  
Almost All

Although cyclo[18]carbon has been theoretically and experimentally investigated since long time ago, only very recently it was prepared and directly observed by means of STM/AFM in condensed phase (Kaiser et al., <i>Science</i>, <b>365</b>, 1299 (2019)). The unique ring structure and dual 18-center π delocalization feature bring a variety of unusual characteristics and properties to the cyclo[18]carbon, which are quite worth to be explored. In this work, we present an extremely comprehensive and detailed investigation on almost all aspects of the cyclo[18]carbon, including (1) Geometric characteristics (2) Bonding nature (3) Electron delocalization and aromaticity (4) Intermolecular interaction (5) Reactivity (6) Electronic excitation and UV/Vis spectrum (7) Molecular vibration and IR/Raman spectrum (8) Molecular dynamics (9) Response to external field (10) Electron ionization, affinity and accompanied process (11) Various molecular properties. We believe that our full characterization of the cyclo[18]carbon will greatly deepen researchers' understanding of this system, and thereby help them to utilize it in practice and design its various valuable derivatives.

A Thorough Theoretical Exploration of Intriguing Characteristics of Cyclo[18]carbon: Geometry, Bonding Nature, Aromaticity, Weak Interaction, Reactivity, Excited States, Vibrations, Molecular Dynamics and Various Molecular Properties

2019 ◽  
Author(s):  
Tian Lu ◽  
Qinxue Chen ◽  
Zeyu Liu
Keyword(s):  
Molecular Dynamics ◽  
Excited States ◽  
Electronic Excitation ◽  
Ring Structure ◽  
Molecular Properties ◽  
Molecular Vibration ◽  
Full Characterization ◽  
Long Time ◽  
Almost All

Although cyclo[18]carbon has been theoretically and experimentally investigated since long time ago, only very recently it was prepared and directly observed by means of STM/AFM in condensed phase (Kaiser et al., <i>Science</i>, <b>365</b>, 1299 (2019)). The unique ring structure and dual 18-center π delocalization feature bring a variety of unusual characteristics and properties to the cyclo[18]carbon, which are quite worth to be explored. In this work, we present an extremely comprehensive and detailed investigation on almost all aspects of the cyclo[18]carbon, including (1) Geometric characteristics (2) Bonding nature (3) Electron delocalization and aromaticity (4) Intermolecular interaction (5) Reactivity (6) Electronic excitation and UV/Vis spectrum (7) Molecular vibration and IR/Raman spectrum (8) Molecular dynamics (9) Response to external field (10) Electron ionization, affinity and accompanied process (11) Various molecular properties. We believe that our full characterization of the cyclo[18]carbon will greatly deepen researchers' understanding of this system, and thereby help them to utilize it in practice and design its various valuable derivatives.

scholarly journals Full Characterization of Minimal Linear Codes as Cutting Blocking Sets

2021 ◽  
pp. 1-1
Author(s):  
Chunming Tang ◽  
Yan Qiu ◽  
Qunying Liao ◽  
Zhengchun Zhou
Keyword(s):  
Linear Codes ◽  
Blocking Sets ◽  
Full Characterization

scholarly journals Special Issue: Advances in Computational Electromagnetics

2021 ◽  
Vol 7 (6) ◽  
pp. 89
Author(s):  
Valerio De Santis
Keyword(s):  
Rare Earth ◽  
Magnetic Materials ◽  
Permanent Magnets ◽  
Computational Electromagnetics ◽  
Superconducting Materials ◽  
Special Issue ◽  
Full Characterization ◽  
Recent Advances

Recent advances in computational electromagnetics (CEMs) have made the full characterization of complex magnetic materials possible, such as superconducting materials, composite or nanomaterials, rare-earth free permanent magnets, etc [...]

scholarly journals Full characterization of the transmission properties of a multi-plane light converter

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Pauline Boucher ◽  
Arthur Goetschy ◽  
Giacomo Sorelli ◽  
Mattia Walschaers ◽  
Nicolas Treps
Keyword(s):  
Full Characterization ◽  
Transmission Properties ◽  
Plane Light

scholarly journals A New Piano-Stool Ruthenium(II) P-Cymene-Based Complex: Crystallographic, Hirshfeld Surface, DFT, and Luminescent Studies

Crystals ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 13
Author(s):  
Mohd. Muddassir ◽  
Abdullah Alarifi ◽  
Mohd. Afzal
Keyword(s):  
Density Functional ◽  
Weak Interactions ◽  
Energy Levels ◽  
Hirshfeld Surface ◽  
Binding Energies ◽  
Orbital Energy ◽  
Aminosalicylic Acid ◽  
Full Characterization

A new complex (Ru(η6-p-cymene)(5-ASA)Cl2) (1) where 5-ASA is 5-aminosalicylic acid has been prepared by reacting the ruthenium arene precursors ((η6-arene)Ru(μ-Cl)Cl)2, with the 5-ASA ligands in a 1:1 ratio. Full characterization of complex 1 was accomplished by elemental analysis, IR, and TGA following the structure obtained from a single-crystal X-ray pattern. The structural analysis revealed that complex 1 shows a “piano-stool” geometry with Ru-C (2.160(5)- 2.208(5)Å), Ru-N (2.159(4) Å) distances, which is similar to equivalents sister complex. Density functional theory (DFT) was used to calculate the significant molecular orbital energy levels, binding energies, bond angles, bond lengths, and spectral data (FTIR, NMR, and UV–VIS) of complex 1, consistent with the experimental results. The IR and UV–VIS spectra of complex 1 were computed using all of the methods and choose the most appropriate way to discuss. Hirshfeld surface analysis was also executed to understand the role of weak interactions such as H⋯H, C⋯H, C-H⋯π, and vdW interactions, which play a significant role in the crystal environment’s stability. Moreover, the luminescence results at room temperature show that complex 1 gives a more intense emission band positioned at 465 nm upon excitation at 330 nm makes it a suitable candidate for the building of photoluminescent material.

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