9,262 research outputs found
Subalgebras of Matrix Algebras Generated by Companion Matrices
Let be monic polynomials of degree and let
be the corresponding companion matrices. We find necessary and sufficient
conditions for the subalgebra to be a sublattice of finite index in
the full integral lattice , in which case we compute the exact value of
this index in terms of the resultant of and . If is a commutative
ring with identity we determine when , in which case a
presentation for in terms of and is given
Characterizing rings in terms of the extent of injectivity and projectivity of their modules
Given a ring R, we define its right i-profile (resp. right p-profile) to be
the collection of injectivity domains (resp. projectivity domains) of its right
R-modules. We study the lattice theoretic properties of these profiles and
consider ways in which properties of the profiles may determine the structure
of rings and viceversa. We show that the i-profile is isomorphic to an interval
of the lattice of linear filters of right ideals of R, and is therefore modular
and coatomic. In particular, we give a practical characterization of the
i-profile of a right artinian ring. We show through an example that the
p-profile is not necessarily a set, and also characterize the right p-profile
of a right perfect ring. The study of rings in terms of their (i- or p-)profile
was inspired by the study of rings with no (i- or p-) middle class, initiated
in recent papers by Er, L\'opez-Permouth and S\"okmez, and by Holston,
L\'opez-Permouth and Orhan-Ertas. In this paper, we obtain further results
about these rings and we also use our results to provide a characterization of
a special class of QF-rings in which the injectivity and projectivity domains
of any module coincide.Comment: 19 pages, examples and propositions added. Title change
On the Topological Origin of Entanglement in Ising Spin Glasses
The origin of thermal and quantum entanglement in a class of
three-dimensional spin models, at low momenta, is traced to purely topological
reasons. The establishment of the result is facilitated by the gauge principle
which, when used in conjunction with the duality mapping of the spin models,
enables us to recast them as lattice Chern-Simons gauge theories. The thermal
and quantum entanglement measures are expressed in terms of the expectation
values of Wilson lines, loops, and their generalisations. For continuous spins,
these are known to yield the topological invariants of knots and links. For
Ising-like models, they are expressible in terms of the topological invariants
of three-manifolds obtained from finite group cohomology -- the so-called
Dijkgraaf-Witten invariants.Comment: RevTex4, 6 page
Quantum scale biomimicry of low dimensional growth: An unusual complex amorphous precursor route to TiO2 band confinement by shape adaptive biopolymer-like flexibility for energy applications
Crystallization via an amorphous pathway is often preferred by biologically driven processes enabling living species to better regulate activation energies to crystal formation that are intrinsically linked to shape and size of dynamically evolving morphologies. Templated ordering of 3-dimensional space around amorphous embedded non-equilibrium phases at heterogeneous polymer-metal interfaces signify important routes for the genesis of low-dimensional materials under stress-induced polymer confinement. We report the surface induced catalytic loss of P=O ligands to bond activated aromatization of C-C C=C and Ti=N resulting in confinement of porphyrin-TiO(2 )within polymer nanocages via particle attachment. Restricted growth nucleation of TiO2 to the quantum scale (˂= 2 nm) is synthetically assisted by nitrogen, phosphine and hydrocarbon polymer chemistry via self-assembly. Here, the amorphous arrest phase of TiO, is reminiscent of biogenic amorphous crystal growth patterns and polymer coordination has both a chemical and biomimetic significance arising from quantum scale confinement which is atomically challenging. The relative ease in adaptability of non-equilibrium phases renders host structures more shape compliant to congruent guests increasing the possibility of geometrical confinement. Here, we provide evidence for synthetic biomimicry akin to bio-polymerization mechanisms to steer disorder-to-order transitions via solvent plasticization-like behaviour. This challenges the rationale of quantum driven confinement processes by conventional processes. Further, we show the change in optoelectronic properties under quantum confinement is intrinsically related to size that affects their optical absorption band energy range in DSSC.This work was supported by the National Research Foundation of Korea (NRF) grant funded by Korea government (MEST) NRF-2012R1A1A2008196, NRF 2012R1A2A2A01047189, NRF 2017R1A2B4008801, 2016R1D1A1A02936936, (NRF-2018R1A4A1059976, NRF-2018R1A2A1A13078704) and NRF Basic Research Programme in Science and Engineering by the Ministry of Education (No. 2017R1D1A1B03036226) and by the INDO-KOREA JNC program of the National Research Foundation of Korea Grant No. 2017K1A3A1A68. We thank BMSI (A*STAR) and NSCC for support. SJF is funded by grant IAF25 PPH17/01/a0/009 funded by A* STAR/NRF/EDB. CSV is the founder of a spinoff biotech Sinopsee Therapeutics. The current work has no conflicting interests with the company. We would like to express our very great appreciation to Ms. Hyoseon Kim for her technical expertise during HRTEM imaging
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