scholarly journals SCROLLS AND HYPERBOLICITY

2013 ◽  
Vol 24 (04) ◽  
pp. 1350026 ◽  
Author(s):  
C. CILIBERTO ◽  
M. ZAIDENBERG
Keyword(s):  
Lower Bounds ◽  
Hyperbolic Surfaces ◽  
Easy Proof ◽  
New Construction

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in ℙ3of degree d ≥ 5. We also show that there exist Kobayashi hyperbolic surfaces in ℙ3of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Our method yields also some lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d ≥ 15 in ℙ4.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

The Travelling Salesman Problem in symmetric circulant matrices with two stripes

2008 ◽  
Vol 18 (1) ◽  
pp. 165-175 ◽  
Author(s):  
IVAN GERACE ◽  
FEDERICO GRECO
Keyword(s):  
Computational Complexity ◽  
Lower Bound ◽  
Lower Bounds ◽  
Travelling Salesman Problem ◽  
Minimum Cost ◽  
Circulant Matrix ◽  
Upper And Lower Bounds ◽  
Circulant Matrices ◽  
Travelling Salesman ◽  
New Construction

The Symmetric Circulant Travelling Salesman Problem asks for the minimum cost tour in a symmetric circulant matrix. The computational complexity of this problem is not known – only upper and lower bounds have been determined. This paper provides a characterisation of the two-stripe case. Instances where the minimum cost of a tour is equal to either the upper or lower bound are recognised. A new construction providing a tour is proposed for the remaining instances, and this leads to a new upper bound that is closer than the previous one.

TWO CLASSES OF HYPERBOLIC SURFACES IN ℙ3

2000 ◽  
Vol 11 (01) ◽  
pp. 65-101 ◽  
Author(s):  
BERNARD SHIFFMAN ◽  
MIKHAIL ZAIDENBERG
Keyword(s):  
Lower Bounds ◽  
Minimal Degree ◽  
Hyperbolic Surfaces ◽  
Symmetric Square ◽  
Generic Projections

We construct two classes of singular Kobayashi hyperbolic surfaces in ℙ3. The first consists of generic projections of the Cartesian square V = C × C of a generic genus g ≥ 2 curve C smoothly embedded in ℙ5. These surfaces have C-hyperbolic normalizations; we give some lower bounds for their degrees and provide an example of degree 32. The second class of examples of hyperbolic surfaces in ℙ3 is provided by generic projections of the symmetric square V′ = C2 of a generic genus g ≥ 3 curve C. The minimal degree of these surfaces is 16, but this time the normalizations are not C-hyperbolic.

A new construction of mutually unbiased maximally entangled bases in ℂq⊗ℂq

2021 ◽  
pp. 2150014
Author(s):  
Dengming Xu ◽  
Feiya Li ◽  
Wei Hu
Keyword(s):  
Lower Bounds ◽  
Prime Power ◽  
Maximal Size ◽  
New Construction

This paper is devoted to constructing mutually unbiased maximally entangled bases (MUMEBs) in [Formula: see text], where [Formula: see text] is a prime power. We prove that [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd, where [Formula: see text] is the maximal size of the sets of MUMEBs in [Formula: see text]. This highly raises the lower bounds of [Formula: see text] given in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175. It should de noted that the method used in the paper is completely different from that in D. Xu, Quant. Inf. Process. 18(7) (2019) 213; D. Xu, Quant. Inf. Process. 19(6) (2020) 175.

New construction of mutually unbiased bases in square dimensions

2005 ◽  
Vol 5 (2) ◽  
pp. 93-101
Author(s):  
P. Wocjan ◽  
T. Beth
Keyword(s):  
Lower Bounds ◽  
Prime Power ◽  
Hadamard Matrices ◽  
Latin Squares ◽  
Mutually Unbiased Bases ◽  
Asymptotic Growth ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
New Construction

We show that k=w+2 mutually unbiased bases can be constructed in any square dimension d=s^2 provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (s,k)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bounds on the asymptotic growth of the number of mutually orthogonal Latin squares (based on number theoretic sieving techniques), we obtain that the number of mutually unbiased bases in dimensions d=s^2 is greater than s^{1/14.8} for all s but finitely many exceptions. Furthermore, our construction gives more mutually unbiased bases in many non-prime-power dimensions than the construction that reduces the problem to prime power dimensions.

Main Provisions Of The Concept Of Technology Of Diamond Cutting And Drilling Of Building Structures

2019 ◽  
pp. 59-63
Keyword(s):  
Combustion Engine ◽  
Front Surface ◽  
The Body ◽  
Building Structures ◽  
Diamond Cutting ◽  
Variable Diameter ◽  
Diamond Drilling ◽  
Single Structure ◽  
New Construction ◽  
Non Destructive

The main provisions of the concept of technology of diamond cutting and drilling of building structures are considered. The innovativeness of the technology, its main possibilities and advantages are presented. Carrying out works with the help of this technology in underwater conditions expands its use when constructing and reconstructing hydraulic structure. The use of diamond drilling equipment with motors equipped with an internal combustion engine is considered. Drilling holes with a variable diameter during the reconstruction of the runways of airfields makes it possible to combine the landing mats into a single structure. The ability to cut inside the concrete mass, parallel to the front surface, has no analogues among the methods of concrete treatment. The use of this technology for producing blind openings in the body of concrete without weakening the structure is also unique. Work with precision quality in cutting and diamond drilling of concrete and reinforced concrete was noted by architects and began to be implemented in the manufacture of inter-room and inter-floor openings. Non-destructive approach to the fragmentation of building structures allows them to be reused. The technology of diamond cutting and drilling is located at the junction of new construction, repair, reconstruction of buildings and structures, and dismantling of structures. Attention is paid to the complexity and combinatorial application of diamond technology. Economic efficiency and ecological safety of diamond technology are presented. The main directions of further research for the development of technology are indicated.

Sign in / Sign up

Export Citation Format

Share Document