scholarly journals POVM construction: A simple recipe with applications to symmetric states

2017 ◽  
Vol 15 (06) ◽  
pp. 1750042
Author(s):  
Swarnamala Sirsi ◽  
Karthik Bharath ◽  
S. P. Shilpashree ◽  
H. S. Smitha Rao
Keyword(s):  
Orthonormal Basis ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Irreducible Representations ◽  
Projection Operators ◽  
Simple Method ◽  
Positive Operator Valued Measures ◽  
Higher Dimensional ◽  
Symmetric States

We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.

The Dirac spinor in six dimensions

1968 ◽  
Vol 64 (3) ◽  
pp. 765-778 ◽  
Author(s):  
E. A. Lord
Keyword(s):  
Lorentz Group ◽  
Dimensional Space ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Indefinite Metric ◽  
Dirac Spinor ◽  
Six Dimensions ◽  
Spinor Representations ◽  
Dirac Spinors

AbstractThe spinor representations of the rotation group in a six-dimensional space with indefinite metric are shown to be four-component spinors, which become the usual Dirac spinors when the formalism is restricted to a four-dimensional subspace. Eriksson's work on the five-dimensional Lorentz group is found to result from a restriction of the six-dimensional treatment to a five-dimensional subspace, and the algebraic significance of Eriksson's work is thereby clarified.

Rotations in a Non-Orthogonal Frame

2018 ◽  
Author(s):  
Shengnan Lu ◽  
Xilun Ding ◽  
Gregory S. Chirikjian
Keyword(s):  
Lie Group ◽  
Reference Frames ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Rotation Group ◽  
Dimensional Subspace ◽  
Iwasawa Decomposition ◽  
Discrete Family ◽  
Orthogonal Frame ◽  
Orthogonal Coordinates

This paper is concerned with describing the space of matrices that describe rotations in non-orthogonal coordinates. In scenarios such as in crystallography, conformational analysis of polymers, and in the study of deployable mechanisms and rigid origami, non-orthogonal reference frames are natural. For example, non-orthogonal vectors in the direction of atomic bonds in a molecule, the lattice coordinates of a crystal, or the directions of links in a mechanism are intrinsic. In these cases it is awkward to impose an artificial orthonormal reference frame rather than choosing one that is defined by the geometry of the object being studied. With these applications in mind, we fully characterize the space of all possible non-orthogonal rotations. We find that in the 2D case, this space is a three-dimensional subset of the special linear group, SL(2, R), which is itself a three-dimensional Lie group. In the 3D case we find that the space of nonorthogonal rotations is a seven-dimensional subspace of SL(3, R), which is an eight-dimensional Lie group. In the 2D case we use the Iwasawa decomposition to fully characterize the solution. In the 3D case we parameterize this seven-dimensional space by conjugating elements of the rotation group SO(3) by elements of a discrete family of of four-parameter subgroups of GL(3, R), and using this we derive an inversion formula to extract classical orthogonal rotations from those expressed in non-orthogonal coordinates.

The classification of states of the nuclear f -shell

1952 ◽  
Vol 210 (1103) ◽  
pp. 497-508 ◽  
Keyword(s):  
Group Theory ◽  
Permutation Group ◽  
Short Range ◽  
Dimensional Space ◽  
Rotation Group ◽  
Irreducible Representations ◽  
Closed Shells ◽  
Nuclear Configuration

Following the methods of Racah and of Jahn, the orbital states arising from the nuclear configuration f n in Russell-Saunders coupling are classified by means of group theory. States are classified according to the irreducible representations of the permutation group, and of the rotation group in the seven-dimensional space spanned by the orbital states of a single f -particle. On the assumption of short-range attractive forces, this classification allows predictions regarding the ordering of levels in the f -shell; and, in particular, the lowest states of nuclei which contain only the simple f n configuration (apart from closed shells) may be predicted as a function of the isotopic number. Numerical values of the Slater integrals for the f -shell with a Gauss interaction are given as a function of the range of the interaction.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

Simplifying Schmidt number witnesses via higher-dimensional embeddings

2004 ◽  
Vol 4 (3) ◽  
pp. 207-221
Author(s):  
F. Hulpke ◽  
D. Bruss ◽  
M. Levenstein ◽  
A. Sanpera
Keyword(s):  
Hilbert Space ◽  
Schmidt Number ◽  
Convex Sets ◽  
Entanglement Witness ◽  
Simple Method ◽  
Dimensional Hilbert Space ◽  
Higher Dimensional ◽  
Arbitrary Convex

We apply the generalised concept of witness operators to arbitrary convex sets, and review the criteria for the optimisation of these general witnesses. We then define an embedding of state vectors and operators into a higher-dimensional Hilbert space. This embedding leads to a connection between any Schmidt number witness in the original Hilbert space and a witness for Schmidt number two (i.e. the most general entanglement witness) in the appropriate enlarged Hilbert space. Using this relation we arrive at a conceptually simple method for the construction of Schmidt number witnesses in bipartite systems.

scholarly journals Automatic and Online Detection of Rotor Fault State

2018 ◽  
Vol 7 (1) ◽  
pp. 43 ◽  
Author(s):  
Ali Ouanas ◽  
Ammar Medoued ◽  
Salim Haddad ◽  
Mourad Mordjaoui ◽  
D. Sayad
Keyword(s):  
Induction Motor ◽  
Dimensional Space ◽  
Operating Conditions ◽  
Online Detection ◽  
K Nearest Neighbor ◽  
Simple Method ◽  
Important Place ◽  
Motor Test ◽  
Wide Range ◽  
Fault State

In this work, we propose a new and simple method to insure an online and automatic detection of faults that affect induction motor rotors. Induction motors now occupy an important place in the industrial environment and cover an extremely wide range of applications. They require a system installation that monitors the motor state to suit the operating conditions for a given application. The proposed method is based on the consideration of the spectrum of the single-phase stator current envelope as input of the detection algorithm. The characteristics related to the broken bar fault in the frequency domain extracted from the Hilbert Transform is used to estimate the fault severity for different load levels through classification tools. The frequency analysis of the envelope gives the frequency component and the associated amplitude which define the existence of the fault. The clustering of the indicator is chosen in a two-dimensional space by the fuzzy c mean clustering to find the center of each class. The distance criterion, the K-Nearest Neighbor (KNN) algorithm and the neural networks are used to determine the fault type. This method is validated on a 5.5-kW induction motor test bench.Article History: Received July 16th 2017; Received: October 5th 2017; Accepted: Januari 6th 2018; Available onlineHow to Cite This Article: Ouanas, A., Medoued, A., Haddad, S., Mordjaoui, M., and Sayad, D. (2017) Automatic and online Detection of Rotor Fault State. International Journal of Renewable Energy Development, 7(1), 43-52.http://dx.doi.org/10.14710/ijred.7.1.43-52

scholarly journals Celestial Spheres

2018 ◽  
Vol 2 (2) ◽  
pp. 168-186
Author(s):  
Austin M. Freeman
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Good Evidence ◽  
Medieval Period ◽  
The Bible ◽  
Higher Dimensional ◽  
Spatial Dimensions ◽  
Modern Physics ◽  
The Subject ◽  
Three Dimensional Space

Angels probably have bodies. There is no good evidence (biblical, philosophical, or historical) to argue against their bodiliness; there is an abundance of evidence (biblical, philosophical, historical) that makes the case for angelic bodies. After surveying biblical texts alleged to demonstrate angelic incorporeality, the discussion moves to examine patristic, medieval, and some modern figures on the subject. In short, before the High Medieval period belief in angelic bodies was the norm, and afterwards it is the exception. A brief foray into modern physics and higher spatial dimensions (termed “hyperspace”), coupled with an analogical use of Edwin Abbott’s Flatland, serves to explain the way in which appealing to higher-dimensional angelic bodies matches the record of angelic activity in the Bible remarkably well. This position also cuts through a historical equivocation on the question of angelic embodiment. Angels do have bodies, but they are bodies very unlike our own. They do not have bodies in any three-dimensional space we can observe, but are nevertheless embodied beings.

Sign in / Sign up

Export Citation Format

Share Document