Quadratic-monomial generated domains from mixed signed, directed graphs

2019 ◽  
Vol 29 (02) ◽  
pp. 279-308
Author(s):  
Michael A. Burr ◽  
Drew J. Lipman
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Combinatorial Structure ◽  
Signed Directed Graph ◽  
Odd Cycle ◽  
Combinatorial Classification ◽  
Special Case ◽  
Text Condition

Determining whether an arbitrary subring [Formula: see text] of [Formula: see text] is a normal or Cohen-Macaulay domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. We provide a complete characterization of the normality, normalizations, and Serre’s [Formula: see text] condition for quadratic-monomial generated domains. For a quadratic-monomial generated domain [Formula: see text], we develop a combinatorial structure that assigns, to each quadratic monomial of the ring, an edge in a mixed signed, directed graph [Formula: see text], i.e. a graph with signed edges and directed edges. We classify the normality and the normalizations of such rings in terms of a generalization of the combinatorial odd cycle condition on [Formula: see text]. We also generalize and simplify a combinatorial classification of Serre’s [Formula: see text] condition for such rings and construct non-Cohen–Macaulay rings.

scholarly journals The Underlying Order Induced by Orthogonality and the Quantum Speed Limit

2021 ◽  
Vol 3 (3) ◽  
pp. 376-388
Author(s):  
Francisco J. Sevilla ◽  
Andrea Valdés-Hernández ◽  
Alan J. Barrios
Keyword(s):  
Energy Spectrum ◽  
Energy Distribution ◽  
Finite Time ◽  
Complete Characterization ◽  
Comprehensive Analysis ◽  
Orthogonality Condition ◽  
Speed Limit ◽  
Physical Quantities

We perform a comprehensive analysis of the set of parameters {ri} that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time τ, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between τ and the energy spectrum and allowing the classification of {ri} into families organized in a 2-simplex, δ2. Furthermore, the states determined by {ri} are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguishes those ris in δ2 correspondent to states whose orthogonality time is limited by the Mandelstam–Tamm bound from those restricted by the Margolus–Levitin one. Our results offer a complete characterization of the physical quantities that become relevant in both the preparation and study of the dynamics of three-level states evolving towards orthogonality.

Kenoplumbomicrolite, (Pb,□)2Ta2O6[□,(OH),O], a new mineral from Ploskaya, Kola Peninsula, Russia

2018 ◽  
Vol 82 (5) ◽  
pp. 1049-1055 ◽  
Author(s):  
Daniel Atencio ◽  
Marcelo B. Andrade ◽  
Luca Bindi ◽  
Paola Bonazzi ◽  
Matteo Zoppi ◽  
...  
Keyword(s):  
Kola Peninsula ◽  
Northern Region ◽  
Complete Characterization ◽  
New Mineral ◽  
X Ray Diffraction ◽  
International Mineralogical Association ◽  
Cell Parameters ◽  
Mohs Hardness

ABSTRACTThis study presents a complete characterization of kenoplumbomicrolite, (Pb,□)2Ta2O6[□,(OH),O], occurring in an amazonite pegmatite from Ploskaya Mountain, Western Keivy Massif, Kola Peninsula, Murmanskaja Oblast, Northern Region, Russia.Kenoplumbomicrolite occurs in yellowish brown octahedral, cuboctahedral and massive crystals, up to 20 cm, has a white streak, a greasy lustre and is translucent. The Mohs hardness is ~6. Attempts to measure density (7.310–7.832 g/cm3) were affected by the ubiquitous presence of uraninite inclusions. Reflectance values were measured in air and immersed in oil. Kenoplumbocrolite is optically isotropic. The empirical formula is (Pb1.30□0.30Ca0.29Na0.08U0.03)Σ2.00(Ta0.82Nb0.62Si0.23Sn4+0.15Ti0.07Fe3+0.10Al0.01)Σ2.00O6[□0.52(OH)0.25O0.23]Σ1.00 (from the crystal used for the structural study) and (Pb1.33□0.66Mn0.01)Σ2.00(Ta0.87Nb0.72Sn4+0.18Fe3+0.11W0.08Ti0.04)Σ2.00O6[□0.80(OH)0.10O0.10]Σ1.00 (average including additional fragments). The mineral is cubic, space group Fd$\overline 3 $m. The unit-cell parameters refined from powder X-ray diffraction data are a = 10.575(2) Å and V = 1182.6(8) Å3, which are in accord with those obtained previously from a single crystal of a = 10.571(1) Å, V = 1181.3(2) Å3 and Z = 8. The mineral description and its name have been approved by the Commission on New Minerals, Nomenclature and Classification of the International Mineralogical Association (IMA2015-007a).

scholarly journals Complexity and polymorphisms for digraph constraint problems under some basic constructions

2016 ◽  
Vol 26 (07) ◽  
pp. 1395-1433 ◽  
Author(s):  
Marcel Jackson ◽  
Tomasz Kowalski ◽  
Todd Niven
Keyword(s):  
Directed Graphs ◽  
Complete Characterization ◽  
Constraint Satisfaction Problems ◽  
Local Consistency ◽  
First Order ◽  
Graph Theoretic ◽  
The Stability ◽  
Basic Graph

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context, we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the “few subpowers algorithm” if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of “strict width” and solvability by few subpowers are unstable under first-order reductions. The analysis also yields a complete characterization of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.

scholarly journals A CHARACTERIZATION OF SELF-ADJOINT OPERATORS DETERMINED BY THE WEAK FORMULATION OF SECOND-ORDER SINGULAR DIFFERENTIAL EXPRESSIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 385-404 ◽  
Author(s):  
MOHAMED EL-GEBEILY ◽  
DONAL O'REGAN
Keyword(s):  
Complete Characterization ◽  
Second Order ◽  
Inner Product ◽  
Type I ◽  
Weak Formulation ◽  
Friedrichs Extension ◽  
Sesquilinear Form ◽  
Adjoint Operators ◽  
Special Case

AbstractIn this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

The complexity of resolution refinements

2007 ◽  
Vol 72 (4) ◽  
pp. 1336-1352 ◽  
Author(s):  
Joshua Buresh-Oppenheim ◽  
Toniann Pitassi
Keyword(s):  
Theorem Proving ◽  
Complete Characterization ◽  
Exponential Separation ◽  
Semantic Resolution ◽  
Special Case

AbstractResolution is the most widely studied approach to propositional theorem proving. In developing efficient resolution-based algorithms, dozens of variants and refinements of resolution have been studied from both the empirical and analytic sides. The most prominent of these refinements are: DP (ordered), DLL (tree), semantic, negative, linear and regular resolution. In this paper, we characterize and study these six refinements of resolution. We give a nearly complete characterization of the relative complexities of all six refinements. While many of the important separations and simulations were already known, many new ones are presented in this paper; in particular, we give the first separation of semantic resolution from general resolution. As a special case, we obtain the first exponential separation of negative resolution from general resolution. We also attempt to present a unifying framework for studying all of these refinements.

Stationary distributions under mutation-selection balance: structure and properties

1996 ◽  
Vol 28 (01) ◽  
pp. 227-251 ◽  
Author(s):  
Reinhard Bürger ◽  
Immanuel M. Bomze
Keyword(s):  
Existence And Uniqueness ◽  
Probability Distributions ◽  
Semigroup Theory ◽  
Complete Characterization ◽  
Stationary Distributions ◽  
Borel Measures ◽  
Classical Models ◽  
Balance Structure

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

Stationary distributions under mutation-selection balance: structure and properties

1996 ◽  
Vol 28 (1) ◽  
pp. 227-251 ◽  
Author(s):  
Reinhard Bürger ◽  
Immanuel M. Bomze
Keyword(s):  
Existence And Uniqueness ◽  
Probability Distributions ◽  
Semigroup Theory ◽  
Complete Characterization ◽  
Stationary Distributions ◽  
Borel Measures ◽  
Classical Models ◽  
Balance Structure

A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integro-differential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron–Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained.

On the Classification of 3D Periodic Polyhedral Cellular Systems

2008 ◽  
Vol 589 ◽  
pp. 341-348
Author(s):  
Tamás Réti ◽  
Enikő Bitay
Keyword(s):  
Materials Science ◽  
Cellular Systems ◽  
Combinatorial Structure ◽  
Space Filling ◽  
Shape Factors ◽  
3D Space ◽  
Triply Periodic ◽  
Combinatorial Classification ◽  
Periodic Space

In several fields of materials science space-filling polyhedral systems are generally used for modeling and characterizing the microstructure of polycrystalline and cellular materials. In this paper a simple quantitative method designated to classify 3D triply periodic, space-filling, cellular systems is outlined. The concept of the proposed method is based on the known analogy between the combinatorial structure of 3D space-filling polyhedral systems and of 4D polytopes. For classification purposes various topological shape indices are defined and tested. It is demonstrated that using two appropriately selected shape factors (asymmetry and compactness coefficients) a global combinatorial classification of cellular systems can be performed.

scholarly journals A Classification of Ramanujan Unitary Cayley Graphs

10.37236/478 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Andrew Droll
Keyword(s):  
Cayley Graph ◽  
Regular Graph ◽  
Adjacency Matrix ◽  
Complete Characterization ◽  
Absolute Value ◽  
Ramanujan Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

The unitary Cayley graph on $n$ vertices, $X_n$, has vertex set ${\Bbb Z}/{n\Bbb Z}$, and two vertices $a$ and $b$ are connected by an edge if and only if they differ by a multiplicative unit modulo $n$, i.e. ${\rm gcd}(a-b,n) = 1$. A $k$-regular graph $X$ is Ramanujan if and only if $\lambda(X) \leq 2\sqrt{k-1}$ where $\lambda(X)$ is the second largest absolute value of the eigenvalues of the adjacency matrix of $X$. We obtain a complete characterization of the cases in which the unitary Cayley graph $X_n$ is a Ramanujan graph.

scholarly journals Parastrophically equivalent quasigroup equations

2010 ◽  
Vol 87 (101) ◽  
pp. 39-58 ◽  
Author(s):  
Aleksandar Krapez ◽  
Dejan Zivkovic
Keyword(s):  
Functional Equations ◽  
Complete Characterization ◽  
Cubic Graphs ◽  
Quadratic Equations ◽  
Phd Thesis

Fedir M. Sokhats'kyi recently posed four problems concerning parastrophic equivalence between generalized quasigroup functional equations. Sava Krstic in his PhD thesis established a connection between generalized quadratic quasigroup functional equations and connected cubic graphs. We use this connection to solve two of Sokhats'kyi's problems, giving also complete characterization of parastrophic cancellability of quadratic equations and reducing the problem of their classification to the problem of classification of connected cubic graphs. Further, we give formulas for the number of quadratic equations with a given number of variables. Finally, we solve all equations with two variables.

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