Journal of Algebra, ISSN 0021-8693, 11/2015, Volume 441, pp. 475 - 551

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the...

Krull monoids | Maximal orders | Distances | Non-unique factorization | Noncommutative rings | Noncommutative semigroups | CATENARY | UNIQUE FACTORIZATION | NUMBER-THEORY | MATHEMATICS | DECOMPOSITIONS | QUATERNION ORDERS | DIRECTED-GRAPHS | DIVISION RINGS | ALGEBRAS | WEDDERBURN POLYNOMIALS | Computer science | Analysis | Algebra | Mathematics - Rings and Algebras

Krull monoids | Maximal orders | Distances | Non-unique factorization | Noncommutative rings | Noncommutative semigroups | CATENARY | UNIQUE FACTORIZATION | NUMBER-THEORY | MATHEMATICS | DECOMPOSITIONS | QUATERNION ORDERS | DIRECTED-GRAPHS | DIVISION RINGS | ALGEBRAS | WEDDERBURN POLYNOMIALS | Computer science | Analysis | Algebra | Mathematics - Rings and Algebras

Journal Article

Annals of Statistics, ISSN 0090-5364, 2018, Volume 46, Issue 6B, pp. 3308 - 3333

Recommender systems have been widely adopted by electronic commerce and entertainment industries for individualized prediction and recommendation, which...

Maximum block improvement | Tensor completion | Context-aware recommender system | Nonconvex optimization | Cold-start problem | LOW-RANK APPROXIMATION | NUMBER | STATISTICS & PROBABILITY | nonconvex optimization | tensor completion | CONTEXT | EFFECTS MODELS | maximum block improvement | DECOMPOSITIONS | context-aware recommender system | CLUSTERS | SELECTION

Maximum block improvement | Tensor completion | Context-aware recommender system | Nonconvex optimization | Cold-start problem | LOW-RANK APPROXIMATION | NUMBER | STATISTICS & PROBABILITY | nonconvex optimization | tensor completion | CONTEXT | EFFECTS MODELS | maximum block improvement | DECOMPOSITIONS | context-aware recommender system | CLUSTERS | SELECTION

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 2012, Volume 436, Issue 2, pp. 421 - 435

An algorithm is described for the nonnegative rank factorization (NRF) of some completely positive (CP) matrices whose rank is equal to their CP-rank. The...

Nonnegative rank factorization | Isometry | Nonnegative matrix factorization | Rank reduction | Arrowhead matrix | Extreme ray | Maximum independent set | Symmetric positive semidefinite | Principal submatrix | Symmetric | Pivoted Cholesky factorization | Rotation | MATHEMATICS, APPLIED | MODEL | MATHEMATICS | MATRIX FACTORIZATION | Analysis | Algorithms

Nonnegative rank factorization | Isometry | Nonnegative matrix factorization | Rank reduction | Arrowhead matrix | Extreme ray | Maximum independent set | Symmetric positive semidefinite | Principal submatrix | Symmetric | Pivoted Cholesky factorization | Rotation | MATHEMATICS, APPLIED | MODEL | MATHEMATICS | MATRIX FACTORIZATION | Analysis | Algorithms

Journal Article

Integral Equations and Operator Theory, ISSN 0378-620X, 2/2018, Volume 90, Issue 1, pp. 1 - 8

Let A be a commutative unital complex Banach algebra and let $$GL_n(A)$$ GLn(A) be the group of invertible $$n\times n$$ n×n matrices with entries in A. In...

Maximal ideal space | Secondary 20H25 | Unitriangular matrix | Commutative unital Banach algebra | Analysis | Mathematics | Dimension | Primary 47B48 | Functional calculus | MATHEMATICS | NUMBER | MAPPINGS | Algebra | Glutamine

Maximal ideal space | Secondary 20H25 | Unitriangular matrix | Commutative unital Banach algebra | Analysis | Mathematics | Dimension | Primary 47B48 | Functional calculus | MATHEMATICS | NUMBER | MAPPINGS | Algebra | Glutamine

Journal Article

Journal of Number Theory, ISSN 0022-314X, 09/2013, Volume 133, Issue 9, pp. 3033 - 3056

In this paper, we study a conjecture of Gao and Wang concerning a proposed formula K1⁎(G) for the maximal cross number K1(G) taken over all unique...

Unique factorization indexed sequences | Cross number | MATHEMATICS | FINITE ABELIAN-GROUPS | ZERO-SUM PROBLEMS

Unique factorization indexed sequences | Cross number | MATHEMATICS | FINITE ABELIAN-GROUPS | ZERO-SUM PROBLEMS

Journal Article

IEEE Transactions on Geoscience and Remote Sensing, ISSN 0196-2892, 11/2012, Volume 50, Issue 11, pp. 4420 - 4440

In the linear unmixing of hyperspectral images, the observation pixels form a simplex whose vertices correspond to the endmembers, hence finding the endmembers...

Algorithm design and analysis | Symmetric matrices | triangular factorization (TF) | abundance nonnegative constraint (ANC) | Real-time systems | Abundance estimation | hyperspectral unmixing | abundance sum-to-one constraint (ASC) | Matrix decomposition | endmember extraction | Hyperspectral imaging | simplex volume analysis | TRANSFORMATION | NUMBER | end-member extraction | QUANTIFICATION | DECOMPOSITION | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | INDEPENDENT COMPONENT ANALYSIS | ENGINEERING, ELECTRICAL & ELECTRONIC | GEOCHEMISTRY & GEOPHYSICS | ENDMEMBER EXTRACTION ALGORITHMS | REMOTE SENSING | N-FINDR | NONNEGATIVE MATRIX FACTORIZATION | Measurement | Technology application | Usage | Image processing | Maximum likelihood estimates (Statistics) | Analysis | Innovations | Linear models (Statistics) | Linear regression models | Mathematical optimization | Pixels

Algorithm design and analysis | Symmetric matrices | triangular factorization (TF) | abundance nonnegative constraint (ANC) | Real-time systems | Abundance estimation | hyperspectral unmixing | abundance sum-to-one constraint (ASC) | Matrix decomposition | endmember extraction | Hyperspectral imaging | simplex volume analysis | TRANSFORMATION | NUMBER | end-member extraction | QUANTIFICATION | DECOMPOSITION | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | INDEPENDENT COMPONENT ANALYSIS | ENGINEERING, ELECTRICAL & ELECTRONIC | GEOCHEMISTRY & GEOPHYSICS | ENDMEMBER EXTRACTION ALGORITHMS | REMOTE SENSING | N-FINDR | NONNEGATIVE MATRIX FACTORIZATION | Measurement | Technology application | Usage | Image processing | Maximum likelihood estimates (Statistics) | Analysis | Innovations | Linear models (Statistics) | Linear regression models | Mathematical optimization | Pixels

Journal Article

Journal of Algebra, ISSN 0021-8693, 2006, Volume 304, Issue 1, pp. 311 - 323

We determine all factorisations G = A B where the socle of G is a sporadic simple group.

Group factorisations | Sporadic simple groups | MATHEMATICS | sporadic simple groups | group factorisations | LATTICE | MAXIMAL-SUBGROUPS

Group factorisations | Sporadic simple groups | MATHEMATICS | sporadic simple groups | group factorisations | LATTICE | MAXIMAL-SUBGROUPS

Journal Article

Communications in Algebra, ISSN 0092-7872, 02/2019, Volume 47, Issue 2, pp. 541 - 552

In this paper we study probabilistic aspects such as the (cyclic) subgroup commutativity degree and different types of factorization numbers of ZM-groups. We...

Cyclic factorization number | 20P05 | subgroup commutativity degree | Secondary 20D30 | number of maximal factorizations | Primary 20D60 | 20E28 | 20F16 | cyclic subgroup commutativity degree | factorization number | MATHEMATICS | ELEMENTS | FACTORIZATION NUMBERS | COMMUTATIVITY DEGREES | FINITE | Commutativity | Subgroups

Cyclic factorization number | 20P05 | subgroup commutativity degree | Secondary 20D30 | number of maximal factorizations | Primary 20D60 | 20E28 | 20F16 | cyclic subgroup commutativity degree | factorization number | MATHEMATICS | ELEMENTS | FACTORIZATION NUMBERS | COMMUTATIVITY DEGREES | FINITE | Commutativity | Subgroups

Journal Article

Integers, ISSN 1867-0652, 08/2012, Volume 12, Issue 4, pp. 677 - 687

Let be a sequence of elements from an additive finite abelian group , and let denote the cross number of . A zero-sum sequence of nonzero elements from is...

Cross Number | Unique Factorization Sequence | Zero-Sum Sequence

Cross Number | Unique Factorization Sequence | Zero-Sum Sequence

Journal Article

Journal of Algebra, ISSN 0021-8693, 09/2013, Volume 390, pp. 1 - 43

Let O be a holomorphy ring in a global field K, and R a classical maximal O-order in a central simple algebra over K. We study sets of lengths of...

Divisorial ideals | Sets of lengths | Maximal orders | Krull monoids | Brandt groupoid | Global fields | FACTORIZATION | MULTIPLICATIVE IDEAL THEORY | RINGS | MONOIDS | MATHEMATICS | QUATERNION ORDERS | MODULES | NONCOMMUTATIVE KRULL PAIRS | DOMAINS | Algebra | Mathematics - Rings and Algebras | 16U30 | 20M13 | 20M12 | 11R54 | 16H10

Divisorial ideals | Sets of lengths | Maximal orders | Krull monoids | Brandt groupoid | Global fields | FACTORIZATION | MULTIPLICATIVE IDEAL THEORY | RINGS | MONOIDS | MATHEMATICS | QUATERNION ORDERS | MODULES | NONCOMMUTATIVE KRULL PAIRS | DOMAINS | Algebra | Mathematics - Rings and Algebras | 16U30 | 20M13 | 20M12 | 11R54 | 16H10

Journal Article

1994, 2nd ed., Progress in mathematics, ISBN 0817637435, Volume 126, xvi, 464

From the original hard cover edition: In the modern age of almost universal computer usage, practically every individual in a technologically developed society...

Numbers, Prime | Data processing | Factorization (Mathematics) | Mathematics

Numbers, Prime | Data processing | Factorization (Mathematics) | Mathematics

Book

1985, Progress in mathematics, ISBN 0817632913, Volume 57., xvi, 463

Book

Proceedings of the Edinburgh Mathematical Society, ISSN 0013-0915, 2018, Volume 62, Issue 2, pp. 1 - 48

If H is a monoid and a = u(1) ... u(k) is an element of H with atoms (irreducible elements) u(1), ... , uk, then k is a length of a, the set of lengths of a is...

Dedekind prime rings | Hereditary Noetherian prime rings | Krull monoids | catenary degrees | monoids of zero-sum sequences | transfer homomorphisms | ideal class groups | factorization theory | sets of lengths | UNIQUE FACTORIZATION | NUMBER-THEORY | MATHEMATICS | MODULES | Domains | Homomorphisms | Monoids

Dedekind prime rings | Hereditary Noetherian prime rings | Krull monoids | catenary degrees | monoids of zero-sum sequences | transfer homomorphisms | ideal class groups | factorization theory | sets of lengths | UNIQUE FACTORIZATION | NUMBER-THEORY | MATHEMATICS | MODULES | Domains | Homomorphisms | Monoids

Journal Article

Journal of Applied Physics, ISSN 0021-8979, 09/2016, Volume 120, Issue 12

Determining the prime factors of a given number N is a problem that requires super-polynomial time for conventional digital computers. A polynomial-time...

Computers | Digital computers | Interference | Entanglement | Factorization | Quantum computers | Numbers | Algorithms | Logic circuits | Computation | Mathematical analysis | Quantum mechanics | Polynomials

Computers | Digital computers | Interference | Entanglement | Factorization | Quantum computers | Numbers | Algorithms | Logic circuits | Computation | Mathematical analysis | Quantum mechanics | Polynomials

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2016, Volume 508, pp. 62 - 80

Understanding the boundary of the set of matrices of nonnegative rank at most r is important for applications in nonconvex optimization. The Zariski closure of...

Nonnegative rank | Mixture model | Stabilization | Equivariant Gröbner basis | Algebra and Number Theory | Numerical Analysis | Geometry and Topology | Discrete Mathematics and Combinatorics | MATHEMATICS | MATHEMATICS, APPLIED | MAXIMUM-LIKELIHOOD | FACTORIZATIONS | Equivariant Grobner basis | Rankings | Electric generators

Nonnegative rank | Mixture model | Stabilization | Equivariant Gröbner basis | Algebra and Number Theory | Numerical Analysis | Geometry and Topology | Discrete Mathematics and Combinatorics | MATHEMATICS | MATHEMATICS, APPLIED | MAXIMUM-LIKELIHOOD | FACTORIZATIONS | Equivariant Grobner basis | Rankings | Electric generators

Journal Article

Acta Arithmetica, ISSN 0065-1036, 2016, Volume 173, Issue 2, pp. 97 - 120

Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot...

HALF-FACTORIAL SETS | MAXIMAL CARDINALITY | SUBSETS | nonunique factorizations | MATHEMATICS | zero sum sequences | DECOMPOSITIONS | MODULES | Krull monoids | sets of distances | CYCLIC GROUPS | DOMAINS | cross numbers

HALF-FACTORIAL SETS | MAXIMAL CARDINALITY | SUBSETS | nonunique factorizations | MATHEMATICS | zero sum sequences | DECOMPOSITIONS | MODULES | Krull monoids | sets of distances | CYCLIC GROUPS | DOMAINS | cross numbers

Journal Article

17.
Full Text
STRUCTURAL PROPERTIES OF SUBADDITIVE FAMILIES WITH APPLICATIONS TO FACTORIZATION THEORY

ISRAEL JOURNAL OF MATHEMATICS, ISSN 0021-2172, 10/2019, Volume 234, Issue 1, pp. 1 - 35

Let H be a multiplicatively written monoid. Given k epsilon N+, we denote by U-k the set of all l epsilon N+ such that a(1) ... a(k) = b(1) ... b(l) for some...

MATHEMATICS | LENGTHS | MONOIDS | PRODUCTS | SETS

MATHEMATICS | LENGTHS | MONOIDS | PRODUCTS | SETS

Journal Article

IEEE Transactions on Signal Processing, ISSN 1053-587X, 10/2012, Volume 60, Issue 10, pp. 5241 - 5253

Recently, the Liu-Zhang-Wong's Toeplitz space-time block code and the Shang-Xia's Alamouti code-based overlapped space-time block code have been developed to...

GLRT receiver | Phase shift keying | Transmitting antennas | noncoherent space-time block codes | Toeplitz codes | Receiving antennas | Alamouti codes | Vectors | Block codes | unique factorization | full diversity | unique identification | Full diversity | Unique factorization | Unique identification | Noncoherent space-time block codes | DESIGN | COHERENT | CONSTELLATIONS | ENGINEERING, ELECTRICAL & ELECTRONIC | BLIND | MAXIMUM-LIKELIHOOD DETECTION | CHANNELS | COMMUNICATION | MODULATION | RECEIVER | Signal processing | Wireless sensor networks | Usage | Numerical analysis | Likelihood functions | Innovations | Numbers | Codes | Coherence | Receivers | Factorization | Channels | Antennas | Signal to noise ratio

GLRT receiver | Phase shift keying | Transmitting antennas | noncoherent space-time block codes | Toeplitz codes | Receiving antennas | Alamouti codes | Vectors | Block codes | unique factorization | full diversity | unique identification | Full diversity | Unique factorization | Unique identification | Noncoherent space-time block codes | DESIGN | COHERENT | CONSTELLATIONS | ENGINEERING, ELECTRICAL & ELECTRONIC | BLIND | MAXIMUM-LIKELIHOOD DETECTION | CHANNELS | COMMUNICATION | MODULATION | RECEIVER | Signal processing | Wireless sensor networks | Usage | Numerical analysis | Likelihood functions | Innovations | Numbers | Codes | Coherence | Receivers | Factorization | Channels | Antennas | Signal to noise ratio

Journal Article

Journal of Algebra, ISSN 0021-8693, 10/2018, Volume 512, pp. 252 - 294

We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the...

Irreducible sets | Sets of lengths | Set of distances | Catenary degree | Non-unique factorization | Atoms | Power monoids | Transfer techniques | Equimorphisms | Monoids | Sumsets | CATENARY | NUMBER | LARGE SETS | DISTANCES | RINGS | SUBSETS | MATHEMATICS | DECOMPOSITIONS | KRULL MONOIDS

Irreducible sets | Sets of lengths | Set of distances | Catenary degree | Non-unique factorization | Atoms | Power monoids | Transfer techniques | Equimorphisms | Monoids | Sumsets | CATENARY | NUMBER | LARGE SETS | DISTANCES | RINGS | SUBSETS | MATHEMATICS | DECOMPOSITIONS | KRULL MONOIDS

Journal Article

20.
Full Text
A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

Communications in Algebra, ISSN 0092-7872, 09/2018, Volume 46, Issue 9, pp. 4021 - 4041

Let H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular...

13F05 | Davenport constant | 20M13 | 13A05 | zero-sum sequences | class groups | maximal orders | seminormal orders | Krull monoids | Arithmetical characterizations | 11B30 | sets of lengths | 11R27 | MATHEMATICS | DISTANCES | CROSS NUMBER | DOMAINS | Integers | Mathematical analysis | Factorization | Monoids | Rings (mathematics)

13F05 | Davenport constant | 20M13 | 13A05 | zero-sum sequences | class groups | maximal orders | seminormal orders | Krull monoids | Arithmetical characterizations | 11B30 | sets of lengths | 11R27 | MATHEMATICS | DISTANCES | CROSS NUMBER | DOMAINS | Integers | Mathematical analysis | Factorization | Monoids | Rings (mathematics)

Journal Article

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