scholarly journals A standard form in (some) free fields: How to construct minimal linear representations

2020 ◽  
Vol 18 (1) ◽  
pp. 1365-1386
Author(s):  
Konrad Schrempf
Keyword(s):  
Normal Form ◽  
Linear Algebra ◽  
Free Field ◽  
Standard Form ◽  
Associative Algebras ◽  
Special Version ◽  
Linear Representations ◽  
Free Fields

Abstract We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

scholarly journals Linearizing the word problem in (some) free fields

2018 ◽  
Vol 28 (07) ◽  
pp. 1209-1230 ◽  
Author(s):  
Konrad Schrempf
Keyword(s):  
Normal Form ◽  
Linear Algebra ◽  
Word Problem ◽  
Commutative Field ◽  
Linear Representations ◽  
Free Fields

We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (positively) test rational identities. Moreover, we provide a construction of minimal linear representations for the inverse of nonzero elements.

A Normal Form in Free Fields

1994 ◽  
Vol 46 (3) ◽  
pp. 517-531 ◽  
Author(s):  
Paul M. Cohn ◽  
Christophe Reutenauer
Keyword(s):  
Normal Form ◽  
Free Field ◽  
Rational Series ◽  
Free Fields

AbstractWe give a normal form for the elements in the free field, following the lines of the minimization theory of noncommutative rational series.

The word problem for free fields

1973 ◽  
Vol 38 (2) ◽  
pp. 309-314 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Normal Form ◽  
Word Problem ◽  
Free Field ◽  
Free Associative Algebra ◽  
Commutative Field ◽  
Skew Field ◽  
Solvable Word Problem ◽  
Skew Fields ◽  
Free Fields ◽  
Solvable Word

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ring R with a field of fractions inverting all full matrices, that if the set of full matrices over R is recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices over R inverted over the field is recursive, but it seems difficult to exploit this more general statement.

scholarly journals A Note on Decomposable and Reducible Integer Matrices

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1125
Author(s):  
Carlos Marijuán ◽  
Ignacio Ojeda ◽  
Alberto Vigneron-Tenorio
Keyword(s):  
Normal Form ◽  
Linear Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integer Matrix ◽  
Hermite Normal Form ◽  
Necessary And Sufficient

We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.

scholarly journals GENERALIZED W3-STRINGS FROM FREE FIELDS

1995 ◽  
Vol 10 (06) ◽  
pp. 515-524 ◽  
Author(s):  
J. M. FIGUEROA-O'FARRILL ◽  
C. M. HULL ◽  
L. PALACIOS ◽  
E. RAMOS
Keyword(s):  
Bosonic Strings ◽  
Free Field ◽  
Space Time ◽  
General Construction ◽  
Special Cases ◽  
Free Fields ◽  
Rule Out ◽  
General Conditions ◽  
String Theories

The conventional quantization of w3-strings gives theories which are equivalent to special cases of bosonic strings. We explore whether a more general quantization can lead to new generalized W3-string theories by seeking to construct quantum BRST charges directly without requiring the existence of a quantum W3-algebra. We study W3-like strings with a direct space-time interpretation — that is, with matter given by explicit free field realizations. Special emphasis is placed on the attempt to construct a quantum W-string associated with the magic realizations of the classical w3-algebra. We give the general conditions for the existence of W3-like strings, and comment on how the known results fit into our general construction. Our results are negative: we find no new consistent string theories, and in particular rule out the possibility of critical strings based on the magic realizations.

scholarly journals GAMBARAN MALOKLUSI DENGAN MENGGUNAKAN HMAR PADA PASIEN DI RUMAH SAKIT GIGI DAN MULUT UNIVERSITAS SAM RATULANGI MANADO

e-GIGI ◽  
2014 ◽  
Vol 2 (2) ◽  
Author(s):  
Vigni Astria Laguhi ◽  
P. S. Anindita ◽  
Paulina N. Gunawan
Keyword(s):  
Normal Form ◽  
Tooth Loss ◽  
Standard Form ◽  
Descriptive Study ◽  
Posterior Region ◽  
Research Subjects ◽  
Patient Study ◽  
Teeth Abnormalities

Abstract: Malocclusions is a form of occlusions that deviates from the standard form which is accepted as a normal form and in Indonesia, the prevelence is still high enough. One of the ways to identify and assess the severity of malocclusions is using the Handicapping Malocclusion Assessment Record Index (HMAR). This study aims to describe the malocclusions patients of  RSGM Unsrat using the HMAR index. This is a descriptive study conducted at the RSGM Unsrat Manado. Research subjects which totaled 34 patient study models. Malocclusions assessment obtained by examination of the study sample according to HMAR index include the tooth defect in one jaw, the second jaw teeth abnormalities relationship in a state of occlusions, and dentofasial defect. Result of research on the teeth in one jaw irregularities showed the highest percentage of tooth loss. Abnormal jaw relations in a state of second gear in the region of the anterior occlusion showed the highest percentage in the form of excessive biting distance and in the posterior region of highest form of canines more distally. Dentofacial abnormalities showed the highest percentage in the form of palatal bite. Research malocclusions severity based on HMAR index showed the highest percentage of severe malocclusions are in need of care.Keywords: Malocclusions, Handicapping Malocclusion Assessment Record Index  Abstrak: Maloklusi adalah suatu bentuk oklusi yang menyimpang dari bentuk standar yang diterima sebagai bentuk normal dan di Indonesia prevalensinya masih cukup tinggi. Salah satu cara mengidentifikasi maloklusi dan menilai keparahan maloklusi  tersebut menggunakan Indeks Handiccaping Assessment Record (HMAR). Penelitian ini bertujuan untuk mengetahui gambaran maloklusi pasien RSGM Unsrat menggunakan Indeks HMAR. Penelitian ini bersifat deskriptif dan dilakukan di RSGM Unsrat Manado. Subjek penelitian berjumlah 34 model studi  pasien. Penilaian maloklusi diperoleh dengan pemeriksaan pada sampel penelitian berdasarkan indeks HMAR  meliputi penyimpangan gigi dalam satu rahang, kelainan hubungan gigi kedua rahang dalam keadaan oklusi, dan kelainan dentofasial. Hasil penelitian pada penyimpangan gigi dalam satu rahang menunjukkan persentase tertinggi pada kehilangan gigi. Kelainan hubungan gigi kedua rahang dalam keadaan oklusi menunjukkan di regio anterior persentase tertinggi berupa jarak gigit berlebih  dan diregio posterior tertinggi berupa gigi kaninus lebih ke distal. Kelainan dentofasial menunjukkan persentase tertinggi berupa palatal bite. Hasil penelitian tingkat keparahan maloklusi berdasarkan indeks HMAR menunjukkan persentase tertinggi pada maloklusi berat sangat memerlukan perawatan. Kata kunci: Maloklusi, Indeks Handicapping Malocclusion Assessment Record

Free Quantum Fields

2020 ◽  
pp. 247-260
Author(s):  
Daniel Canarutto
Keyword(s):  
Free Field ◽  
Gauge Fields ◽  
Quantum Field ◽  
Quantum Fields ◽  
Dirac Fields ◽  
Free Fields ◽  
Explicit Expressions

The notion of free quantum field is thoroughly discussed in the linearised setting associated with the choice of a detector. The discussion requires attention to certain details that are often overlooked in the standard literature. Explicit expressions for generic fields, Dirac fields, gauge fields and ghost fields are laid down, as well the ensuing free-field expressions of important functionals. The relations between super-commutators of free fields and propagators, and the canonical super-commutation rules, follow from the above results.

Causal amplitudes and the Yang-Feldman formalism

1957 ◽  
Vol 53 (4) ◽  
pp. 843-847 ◽  
Author(s):  
J. C. Polkinghorne
Keyword(s):  
Field Theory ◽  
Green’S Functions ◽  
Free Field ◽  
Dispersion Relations ◽  
Matrix Elements ◽  
Field Equations ◽  
Commutation Relations ◽  
The Matrix ◽  
Free Fields ◽  
Set Up

ABSTRACTThe Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

scholarly journals A NOTE ON KERR/CFT AND FREE FIELDS

2010 ◽  
Vol 25 (20) ◽  
pp. 3965-3973 ◽  
Author(s):  
JØRGEN RASMUSSEN
Keyword(s):  
Black Hole ◽  
Boundary Conditions ◽  
Isometry Group ◽  
Free Field ◽  
Kerr Black Hole ◽  
Virasoro Algebra ◽  
Free Fields ◽  
Horizon Geometry

The near-horizon geometry of the extremal four-dimensional Kerr black hole and certain generalizations thereof has an SL (2, ℝ) × U (1) isometry group. Excitations around this geometry can be controlled by imposing appropriate boundary conditions. For certain boundary conditions, the U(1) isometry is enhanced to a Virasoro algebra. Here, we propose a free-field construction of this Virasoro algebra.

scholarly journals An inner orthogonality of Hadamard matrices

1971 ◽  
Vol 12 (2) ◽  
pp. 242-248 ◽  
Author(s):  
K. A. Bush
Keyword(s):  
Normal Form ◽  
Standard Form ◽  
Hadamard Matrices ◽  
Inner Product

We say a matrix H of order 4t is Hadamard if each entry is 1 or — 1, and the inner product of any two rows is zero. We shall consider only Hadamard matrices in normal form with the first row consisting solely of 1 while any two of the remaining rows have the property that hik = hjk = 1, hik = –hjk = 1, —hik = hjk = 1, and –hik = –hjk = 1 each occur t times. We can induce a further normalization by choosing the second row of H to have the first 2t entries 1 and the last 2t entries — 1, and this will be the standard form we consider. We now call the submatrix obtained by deleting the first two rows of H and the first It columns. H therefore is of dimension 4t — 2 x 2t. We prove the following theorem:

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