scholarly journals Some Krein-Milman theorems for order-convexity

1974 ◽  
Vol 18 (3) ◽  
pp. 257-261
Author(s):  
Andrew Wirth
Keyword(s):  
Large Class ◽  
Open Interval ◽  
Theoretic Result ◽  
Interval Topology ◽  
Original Order ◽  
Definition Of

Analogues of the Krein-Milman theorem for order-convexity have been studied by several authors. Franklin [2] has proved a set-theoretic result, while Baker [1] has proved the theorem for posets with the Frink interval topology. We prove two Krein-Milman results on a large class of posets, with the open-interval topology, one for the original order and one for the associated preorder. This class of posets includes all pogroups. Cellular-internity defined in Rn by Miller [3] leads to another notion of convexity, cell-convexity. We generalize the definition of cell-convexity to abelian l-groups and prove a Krein-Milman theorem in terms of it for divisible abelian l-groups.

scholarly journals Cauchy completion of partially ordered groups

1974 ◽  
Vol 18 (2) ◽  
pp. 222-229 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Partial Order ◽  
Open Interval ◽  
Macneille Completion ◽  
Partially Ordered ◽  
Interval Topology ◽  
Original Order ◽  
Ordered Groups ◽  
Cauchy Completion ◽  
Original Group ◽  
Topological Completion

A number of completions have been applied to p.o.-groups — the Dede kind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology — the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this topology; Kowaisky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order.

scholarly journals Pseudoinversion of degenerate metrics

2003 ◽  
Vol 2003 (55) ◽  
pp. 3479-3501 ◽  
Author(s):  
C. Atindogbe ◽  
J.-P. Ezin ◽  
Joël Tossa
Keyword(s):  
Differential Geometry ◽  
Large Class ◽  
Smooth Manifold ◽  
Differential Operators ◽  
Type Operator ◽  
Lorentzian Space ◽  
Lightlike Hypersurface ◽  
Definition Of ◽  
Lightlike Hypersurfaces ◽  
Theoretical Results

Let(M,g)be a smooth manifoldMendowed with a metricg. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metricg∗on the dual bundleTM∗of 1-forms onM. If the metricgis (semi)-Riemannian, the metricg∗is just the inverse ofg. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for whichg∗is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian spaceℝ1n+2.

Quotient groups and realization of tight Riesz groups

1973 ◽  
Vol 16 (4) ◽  
pp. 416-430 ◽  
Author(s):  
John Boris Miller
Keyword(s):  
Normal Subgroup ◽  
Open Interval ◽  
Canonical Homomorphism ◽  
Interval Topology ◽  
Essential Step ◽  
Image Position ◽  
Quotient Groups

Let (G, ≼) be an l-group having a compatible tight Riesz order ≦ with open-interval topology U, and H a normal subgroup. The first part of the paper concerns the question: Under what conditions on H is the structure of (G, ≼, ∧, ∨, ≦, U) carried over satisfactorily to by the canonical homomorphism; and its answer (Theorem 8°): H should be an l-ideal of (G, ≼) closed and not open in (G, U). Such a normal subgroup is here called a tangent. An essential step is to show that ≼′ is the associated order of ≦′.

scholarly journals The SL(2, C) Casson Invariant for Knots and the Â-polynomial

2016 ◽  
Vol 68 (1) ◽  
pp. 3-23 ◽  
Author(s):  
Hans U. Boden ◽  
Cynthia L. Curtis
Keyword(s):  
Large Class ◽  
Product Formula ◽  
Integral Homology ◽  
Connected Sum ◽  
Connected Sums ◽  
Knot Invariant ◽  
Definition Of ◽  
Arbitrary Knots

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

scholarly journals Tight Riesz groups

1972 ◽  
Vol 13 (2) ◽  
pp. 224-240 ◽  
Author(s):  
R. J. Loy ◽  
J. B. Miller
Keyword(s):  
General Theory ◽  
Open Interval ◽  
Discrete Topology ◽  
Topological Groups ◽  
Ordered Group ◽  
Partially Ordered ◽  
Interval Topology ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Positive Elements

The theory of partially ordered topological groups has received little attention in the literature, despite the accessibility and importance in analysis of the group Rm. One obstacle in the way of a general theory seems to be, that a convenient association between the ordering and the topology suggests that the cone of all strictly positive elements be open, i.e. that the topology be at least as strong as the open-interval topology U; but if the ordering is a lattice ordering and not a full ordering then U itself is already discrete. So to obtain in this context something more interesting topologically than the discrete topology and orderwise than the full order, one must forego orderings which make lattice-ordered groups: in fact, the partially ordered group must be an antilattice, that is, must admit no nontrivial meets or joins (see § 2, 10°).

scholarly journals A THEORY OF ALGEBRAIC INTEGRATION

2000 ◽  
Vol 14 (22n23) ◽  
pp. 2293-2297
Author(s):  
R. CASALBUONI
Keyword(s):  
Large Class ◽  
Configuration Space ◽  
Path Integral ◽  
Physical Principle ◽  
Integral Approach ◽  
Path Integral Approach ◽  
The One ◽  
Definition Of

In this paper we study the problem of quantizing theories defined over a nonclassical configuration space. If one follows the path-integral approach, the first problem one is faced with is the one of definition of the integral over such spaces. We consider this problem and we show how to define an integration which respects the physical principle of composition of the probability amplitudes for a very large class of algebras.

scholarly journals Clustering of attribute and/or relational data

2009 ◽  
Vol 6 (2) ◽  
Author(s):  
Anuška Ferligoj ◽  
Luka Kronegger
Keyword(s):  
Large Class ◽  
Scientific Collaboration ◽  
Relational Data ◽  
Criterion Function ◽  
New Developments ◽  
Clustering Problem ◽  
Constraint Problem ◽  
Clustering Approach ◽  
Definition Of ◽  
Clustering Problems

A large class of clustering problems can be formulated as an optimizational problem in which the best clustering is searched for among all feasible clustering according to a selected criterion function. This clustering approach can be applied to a variety of very interesting clustering problems, as it is possible to adapt it to a concrete clustering problem by an appropriate specification of the criterion function and/or by the definition of the set of feasible clusterings. Both, the blockmodeling problem (clustering of the relational data) and the clustering with relational constraint problem (clustering of the attribute and relational data) can be very successfully treated by this approach. It also opens many new developments in these areas. The paired clustering approaches are applied to the Slovenian scientific collaboration data.

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

Generalization of Hölder's Theorem to Ordered Modules

1969 ◽  
Vol 21 ◽  
pp. 149-157 ◽  
Author(s):  
T. M. Viswanathan
Keyword(s):  
Sufficient Conditions ◽  
Open Interval ◽  
Order Structure ◽  
Real Numbers ◽  
Ordered Group ◽  
Quotient Field ◽  
Interval Topology ◽  
Ordered Field ◽  
Topological Completion ◽  
Necessary And Sufficient

Hölder's theorem on archimedean groups states:An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

Program completion in the input language of GRINGO

2017 ◽  
Vol 17 (5-6) ◽  
pp. 855-871 ◽  
Author(s):  
AMELIA HARRISON ◽  
VLADIMIR LIFSCHITZ ◽  
DHANANJAY RAJU
Keyword(s):  
Reverse Engineering ◽  
Large Class ◽  
Logic Program ◽  
Answer Set Programming ◽  
Program Completion ◽  
Engineering Process ◽  
Input Language ◽  
Definition Of ◽  
Answer Set

AbstractWe argue that turning a logic program into a set of completed definitions can be sometimes thought of as the “reverse engineering” process of generating a set of conditions that could serve as a specification for it. Accordingly, it may be useful to define completion for a large class of Answer Set Programming (ASP) programs and to automate the process of generating and simplifying completion formulas. Examining the output produced by this kind of software may help programmers to see more clearly what their program does, and to what degree its behavior conforms with their expectations. As a step toward this goal, we propose here a definition of program completion for a large class of programs in the input language of the ASP grounder gringo, and study its properties.

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