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 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.

ON FINITE TYPE 3-MANIFOLD INVARIANTS I

1996 ◽  
Vol 05 (04) ◽  
pp. 441-461 ◽  
Author(s):  
STAVROS GAROUFALIDIS
Keyword(s):  
Vector Space ◽  
Finite Type ◽  
Commutative Algebra ◽  
Integral Homology ◽  
Knot Invariants ◽  
Finite Type Invariants ◽  
Definition Of ◽  
Type 3

Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.

Characterizations for the n-strong Drazin invertibility in a ring

2020 ◽  
pp. 2150141
Author(s):  
Honglin Zou ◽  
Jianlong Chen ◽  
Huihui Zhu ◽  
Yujie Wei
Keyword(s):  
Positive Integer ◽  
Generalized Inverse ◽  
Product Formula ◽  
Drazin Inverse ◽  
Equivalent Definition ◽  
New Type ◽  
Definition Of ◽  
Existence Criteria

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

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 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.

Connected sum of virtual knots and F-polynomials

2021 ◽  
Author(s):  
Maxim Ivanov
Keyword(s):  
Infinite Family ◽  
Connected Sum ◽  
Connected Sums ◽  
Virtual Knots ◽  
Knot Diagrams

It is known that connected sum of two virtual knots is not uniquely determined and depends on knot diagrams and choosing the points to be connected. But different connected sums of the same virtual knots cannot be distinguished by Kauffman’s affine index polynomial. For any pair of virtual knots [Formula: see text] and [Formula: see text] with [Formula: see text]-dwrithe [Formula: see text] we construct an infinite family of different connected sums of [Formula: see text] and [Formula: see text] which can be distinguished by [Formula: see text]-polynomials.

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.

scholarly journals A Definition of Two-Dimensional Schoenberg Type Operators

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1364
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Second Order ◽  
Order Of Approximation ◽  
Two Dimensional ◽  
Moduli Of Continuity ◽  
Definition Of ◽  
Arbitrary Knots

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity.

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