## algorithms – Reductions versus generalizations

In the questions above, we are saying that a special case of any of the problems above are NP-complete complete. This seems like a misnomer though. Shouldn’t this be a specialization rather than a generalization?

No. The problem you are proving NP-complete is a generalization of a known NP-complete problem. E.g. the MAX-SAT example contains Boolean satisfiability; just set $$g$$ large enough to demand satisfaction of all the clauses.

Another part of my confusion is how these so-called generalizations compare or contrast to reductions. Are we saying that some (special) version of a problem is NP-complete and therefore all the other instances are NP-complete? I thought that we needed to show that all instances of the problem are the NP-complete problem.

No, we’re saying that if an NP-complete problem can be specified using the rules that describe a different problem without changing those rules, then that different problem is NP-hard, and is also NP-complete if that different problem is in NP.

Regarding the direction of reductions: we reduce from a NP-complete problem (the hard problem) to the problem that we have at hand. For example, in the reduction from 3SAT to independent set we reduce from 3SAT that we know is NP-complete to what we have at hand, the independent set – did I misunderstand? Are these reductions generalizations? I thought that they were for all instances of the independent set.

No, you should just think of reductions as reductions. A generalization of an NP-complete problem contains that problem as a special case within the generalized problem. A reduction provides a way to go from an instance of one problem to an instance of another without regard to whether one problem is a special case of another.

Are generalizations reductions?

No, they are an expansion of the original problem that still contains the original problem. E.g. Boolean satisfiability is a generalization of 3-satisfiability because 3SAT is contained within the expanded problem set.

## Generalizations of Artin–Verdier duality?

Constructible étale abelian sheafs on $$Spec O_mathbb K$$, for number fields $$mathbb K$$, satisfy Artin-Verdier duality. Are there known any algebraic schemes or algebraic stacks, other than $$Spec O_mathbb K$$ and their open subsets, for which (a version of) A-V duality holds?

The motivation for my question is that Artin-Verdier duality is reminiscent of Poincare duality for $$3$$-dimensional manifolds. Since 3-manifolds can be easily glued of pieces, it is natural to ask about an analog of that construction in which schemes $$Spec O_mathbb K$$ are spliced together.

In particular, one can construct 3-manifolds is by taking branched covers of other 3-manifolds. Obviously, every field extension $$mathbb Ksubset mathbb L$$ defines a branched cover $$Spec O_mathbb Lto Spec O_mathbb K,$$ but perhaps there is a reasonable notion of branched cover $$Yto Spec O_mathbb K$$ in which $$Y$$ satisfies a version of A-V duality, even though it is not $$Spec O_mathbb L$$?

## cv.complex variables – Two generalizations of the Verblunsky Theorem

I learned from this paper about the Verblunsky theorem.

My question is that: What kind of generalizations of this theorem is availlable?

In particular I am interested in the following two possible generalization:

1]Does every arbitrary sequence $${alpha_n} in mathbb{D}_4$$ determine a unique probability measure on $$S^3=partial mathbb{D}_4$$?Does a queternion calculse help for such a generalization?

2]For which kind of sequence $${alpha_n} in mathbb{C}^2simeq mathbb{C}P^2 setminus mathbb{C}P^1$$ we would obtain a unique probability measure on $$S^2=mathbb{C}P^1=partial mathbb{C}^2$$?

To what extent the method of consideration of isometric group of the hyperbolic disck in the above paper be generalized in order to answer to each of the above two questions?

## generalizations of semi-continuity theorems

I would like to know some references (if any) about the generalization of the semicontinuity theorem and "Cohomology and base change" theorem (theorems 12.8 and 12.11 in the book "Algebraic geometry" of Hartshorne). Generalizations to non projective (maybe proper) morphisms. And also generalizations of "Grauert theorem" (corollary 12.9) to a non integral base $$Y$$.

## ag.algebraic geometry – is there a classification of superordinate generalizations of confocal conic sections?

The 1-parameter families of ellipses and hyperbolas with a certain point pair in the plane as focal points result in “orthogonal double sheets” of the plane. That is, once the focal points are established, each point in the plane lies on both a unique ellipse and a unique hyperbola that are orthogonal at the intersection. For a given selection of focal points, the family is described by a single polynomial equation in three variables:

$$P (x, y, a) = 0$$

Where $$a$$ is the semi-major axis of the cone. When $$a$$ is larger than half the distance between the focal points, the curve is an ellipse, and if it is smaller, the curve is a hyperbola.

In addition to the limit case of families of parabolas with a certain focus and a certain axis of symmetry, there do not seem to be any other 1-parameter families of degree 2 curves with the same properties:

• They are defined by a Single polynomial equation in the coordinates and a single parameter.
• You film the plane twice, where the two leaves come from different domains for the parameter.
• The two types of curves are perpendicular at the intersection.

But there must surely be families of such higher degree curves.

I would be grateful for the following:

• Specific examples of higher grade families.
• Any classification of higher grade families.
• Relevant terminology that could be helpful when searching for literature.

## Computational Geometry – Generalizations of the polygon "Curious Tiger"

I do not really know if the polygon I'm describing already has a name, but let me explain the problem solved by the polygon, with a little story:

Imagine a flat terrain with bamboo bushes (represented by the red dots), a monk who has selected two of these shrubs as endpoints of his meditation path (represented by the bold orange line), and on to a tiger that is curious. What's up ,

The tiger wants to circle the monk, running from bush to bush to become invisible there, and of course the tiger wants to clear the entire meditation path from any bush he lies in front of.
For this purpose, the tiger may either contain the bushes of the meditation path or not.

The polygon with the vertices of the meditation path

The polygon with the meditation path vertices is excluded

The solution to the above problem is the polygon formed by the corners of a maximal set of empty triangles with the "meditation path" as the common side $$c$$,

When the endpoints of the meditation path are $$A$$ and $$B$$with associated coordinates $$(0,0)$$ and $$(1,0)$$then the set $$lbrace C_i rbrace$$ third corners of the maximum set of empty triangles with common side $$c$$ is sorted by the side angle $$b_i: = C_i-A$$ with side $$c: = B-A$$ (or according to the angles of the sides $$a_i$$ With $$c$$)

So, this question is not about how to solve the problem of the tiger, nor how to calculate the polygon.

• Is there already a name for one of the variants of the "Curious Tiger" polygon?

• which (if any) generalizations of the 2D problem are higher, $$d$$Dimensional Euclidean spaces have as solution a maximum set of points from which an unobstructed view of the entire low-dimensional "meditation simplex" is possible.
such that this maximum set of points defines the vertices of a $$d$$-dimensional polytope that is topologically equivalent $$(d-1)$$Ball and
that every point of this polytope has an unobstructed view of the entire "meditation simplex"?

The reason why I am interested in this kind of polygons. Polytope is that they would allow new algorithms for two known problems in computer geometry: shape hulls and Tour expansion heuristics for the plane Euclidean TSP can be improved by searching for large "Curious Tiger" polygons (CTPs).

If no holes are allowed, the following heuristic can be applied to both problems:

• Take the edges of the convex hull as a meditation path
• Perform a greedy relaxation on the points contained in multiple CTPs until no pair of these polygons share a common edge.

If holes are allowed, inner edges can also serve as meditation paths:

• Repeat the above steps as long as necessary

In extreme cases, each edge would generate an initial CTP and the relaxation would be done, ensuring that topological constraints are not violated.

An illustrative example showing how the CTP is superior to the distance-based Shapehull algorithms and the Euclidean TSP heuristics for the smallest strain of unilateral vertex insertion.