Narrow Results

The pentaquark width is calculated in QCD sum rules. The higher dimension operators contribution is accounted. It is shown, that Γ_Θ should be very small, less than 1 MeV.

We study (1+ε)-factor approximation algorithms for several well-known optimization problems on a given n-point set: (a) diameter, (b) width, (c) smallest enclosing cylinder, and (d) minimum-width annulus. Among our results are new simple algorithms for (a) and (c) with an improved dependence of the running time on ε, as well as the first linear-time approximation algorithm for (d) in any fixed dimension. All four problems can be solved within a time bound of the form O(n+ε^-c) or O(nlog(1/ε)+ε^-c).

Given a finite set of points in ℝ^d, the diameter of is defined as the maximum distance between two points of . We propose a very simple algorithm to compute the diameter of a finite set of points. Although the algorithm is not worst-case optimal, an extensive experimental study has shown that it is extremely fast for a large variety of point distributions. In addition, we propose a comparison with the recent approach of Har-Peled⁵ and derive hybrid algorithms to combine advantages of both approaches.

We study the problem of maintaining a (1 + ∊)-factor approximation of the diameter of a stream of points under the sliding window model. In one dimension, we give a simple algorithm that only needs to store points at any time, where the parameter R denotes the "spread" of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang's recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.

Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, S. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α>1 and β≥1 be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(α log² n) time, and reports in O(α log² n) time an approximation, Ŵ, to the width such that . The algorithm for the diameter problem uses O(βn) space, supports updates in O(β log n) time, and reports in O(β) time an approximation, , to the diameter such that . Thus, for instance, even for α as small as 11, Ŵ/W≤1.01, and for β as small as 9, . All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.

Let S be a set of points in the plane. The width (resp. roundness) of S is defined as the minimum width of any slab (resp. annulus) that contains all points of S. We give a new characterization of the width of a point set. Also, we give a rigorous proof of the fact that either the roundness of S is equal to the width of S, or the center of the minimum-width annulus is a vertex of the closest-point Voronoi diagram of S, the furthest-point Voronoi diagram of S, or an intersection point of these two diagrams. This proof corrects the characterization of roundness used extensively in the literature.

It has been conjectured that for knots K and K′ in S³, w(K # K′) = w(K) + w(K′) - 2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots for which they conjectured that where Bⁿ is a bridge number n knot. We show that for n > 2 none of the knots in produces such counterexamples.

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S³ branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

In this paper, we prove that $w(K)=4w(J)$, where $w(.)$ is the width of a knot and $K$ is the Whitehead double of a nontrivial knot $J$.

We extend the classical definition of width to higher dimensional, smooth codimension two knots and show in each dimension there are knots of arbitrarily large width.

Nuclear radial distance is a prerequisite for generating any alpha-decay half-life formula by taking a suitable effective potential. We study the emission process of alpha particles from an isolated quasi-bound state generated by an effective potential to a scattering state. The effective potential is expressed in terms of Frahn form of potential which is exactly solvable and an analytical expression for half-life is obtained in terms of Coulomb function, wave function and the potential. We then derive a closed-form expression for the decay half-life in terms of the parameters of the potential, Q-value of the system, mass and proton numbers of the nuclei valid for alpha-decay as well as proton-decay. From the nature of variations of half-life as a function of radial distance, we trace the radial independence region where decay time is almost constant. Finally, by overviewing our results and picking that particular radial distance, we predict the half-lives of a series of nuclei by using the closed-form expression. We also predict the half-lives of isotopes of nuclei with $Z=119$ and 120.

We investigate width and Krull–Gabriel dimension over commutative Noetherian rings which are "tame" according to the Klingler–Levy analysis in [4–6], in particular over Dedekind-like rings and their homomorphic images. We show that both are undefined in most cases.

We prove that if $𝔽$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $\phi$ of ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ such that $R(\phi )=1$. This fact implies that ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ do not possess the $R_{\infty }$-property. However, if the transcendece degree of $𝔽$ over $\mathbb{Q}$ is finite, then ${\mathrm{\text{SL}}}_n(𝔽)$ and ${\mathrm{\text{GL}}}_n(𝔽)$ are known to possess the $R_{\infty }$-property [13].

This paper introduces the notion of depth with respect to ideals for unbounded DG-modules, and gives a reduction formula and the local nature of this depth. As applications, we provide several bounds of the depth in special cases, and recover and generalize the known results about the depth of complexes. In addition, the width with respect to ideals for unbounded DG-modules is investigated and the depth and width formulas for DG-modules are generalized.

Given a 2-dimensional surface M and a constant C we construct a Riemannian metric g, so that diameter diam(M, g) = 1 and every 1-cycle dividing M into two regions of equal area has length > C. It follows that there exists no universal inequality bounding 1-width of M in terms of its diameter. This answers a question of Stéphane Sabourau.

We show that for every complete Riemannian surface M diffeomorphic to a sphere with k ≥ 0 holes, there exists a Morse function f : M → ℝ, which is constant on each connected component of the boundary of M and has fibers of length no more than . We also show that on every 2-sphere there exists a simple closed curve of length subdividing the sphere into two discs of area .

Volume and width are two of the most studied ways to measure a body. With the concept of width, natural questions arise on lattice polytopes.

A particular question someone can ask is the maximum width that a lattice polytope can have for arbitrary dimension or which is the finiteness threshold of a hollow lattice polytope. These questions are the main topic of this survey.

Focusing in these concepts have helped to bring new upper bounds in the volume of hollow polytopes with lower bounded width, which lead into new advances in their classification. Some examples are maximal 3-hollow polytopes or empty 4-simplices.

In this survey, we recap all background knowledge regarding this concept and get together all new results that have been published during the recent years. These advances in lattice polytopes may lead to new results in big questions of convex geometry as it is finding new examples or bounds in relation with the Flatness Theorem for convex bodies and polytopes.

The LEP collider has accumulated around 40k W boson pair events in the period 1996-2000. The excellent performance of the four detectors has allowed a precise measurement of the properties of the W boson. In this contribution the final results on the mass and the width of the W boson from the four individual experiments are collected for the first time. In the combination of the W boson mass results systematic uncertainties are important. Because LEP is still measuring these the combination remain preliminary. Nevertheless the mass of the W boson has been combined to a value of m_W = 80.376 ± 0.033 GeV and the width measurement resulted in Γ_W = 2.196 ± 0.083 GeV. The branching ratios of the W boson decay have been measured and preliminary combined into BR(W → lν) = 10.84 ± 0.09 and BR(W → hadrons) = 67.48 ± 0.28.

The virtual-height (virtual-width) of a straight line embedding of a plane graph in the xy-plane is the number of vertices with pairwise distinct y-coordinates (x-coordinates). We show that any plane graph of order n has a straight line embedding of virtual-height at most and virtual-width at most and present a polynomial time algorithm which obtains such an embedding.

The research presented here developed from rather mysterious observations, originally made by the authors independently and in different circumstances, that Lebesgue null sets may have uniquely defined tangent directions that are still seen even if the set is much enlarged (but still kept Lebesgue null). This phenomenon appeared, for example, in the rank-one property of derivatives of BV functions and, perhaps in its most striking form, in attempts to decide whether Rademacher's theorem on differentiability of Lipschitz functions may be strengthened or not.

We describe the non-differentiability sets of Lipschitz functions on ℝⁿ and use this description to explain the development of the ideas and various approaches to the definition of the tangent fields to null sets. We also indicate connections to other current results, including results related to the study of structure of sets of small measure, and present some of the main remaining open problems.