Vašek Chvátal, Noé de Rancourt, Guillermo Gamboa Quintero
et al.
Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In a metric space on $n$ points, fewer than $7n^2/6$ suitably placed degenerate triangles suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how cleverly placed, never suffice.
In this simple paper, a small refinement to the Prime Number Theorem (PNT) is proposed, which allows us to limit the error with which said theorem predicts the value of the Prime-counting function π(x); and, in this way, endorse the veracity of the Riemann Hypothesis.Many people know that the Riemann Hypothesis is a difficult mathematical problem - even to understand - without a certain background in mathematics. Many techniques have been used, for more than 150 years, to try to solve it. Among them is the one that establishes that, if the Riemann hypothesis is true, then the error term that appears in the prime number theorem can be bounded in the best possible way.
Gaute Lyngstad, Per Skjelbred, David Michael Swanson
et al.
AbstractPurpose Combining analgesics with different mechanisms of action may increase the analgesic efficacy. The multidimensional pharmacodynamic profiles of ibuprofen 400 mg/paracetamol 1000 mg, ibuprofen 400 mg/paracetamol 1000 mg/codeine 60 mg, and paracetamol 1000 mg/codeine 60 mg and placebo were compared. Methods A randomized, double-blind, placebo-controlled, parallel-group, single-centre, outpatient, and single-dose study used 200 patients of both sexes and homogenous ethnicity after third molar surgery (mean age 24 years, range 19–30 years). Primary outcome was sum pain intensity over 6 h (SPI). Secondary outcomes were time to analgesic onset, duration of analgesia, time to rescue drug intake, number of patients taking rescue drug, sum pain intensity difference (SPID), maximum pain intensity difference, time to maximum pain intensity difference, number needed to treat, prevent remedication and harm values, adverse effects, and patient-reported outcome measure (PROM). Results Analgesia following ibuprofen and paracetamol combination with or without codeine was comparable. Both were better than paracetamol combined with codeine. Secondary variables supported this finding. Post hoc analysis of SPI and SPID revealed a sex/drug interaction trend in the codeine-containing groups where females experienced less analgesia. PROM showed a significant sex/drug interaction in the paracetamol and codeine group, but not in the other codeine-containing group. Especially females reported known and mild side effects in the codeine containing groups. Conclusion Codeine added to ibuprofen/paracetamol does not seem to add analgesia in a sex-mixed study population. Sex may be a confounding factor when testing weak opioid analgesics such as codeine. PROM seems to be more sensitive than traditional outcome measures. ClinicalTrials.gov June 2009 NCT00921700
If $C_n(\mathbb{R}^d)$ denotes the configuration space of $n$ distinct points in $\mathbb{R}^d$, we construct a sequence of maps $(f_m),$ $m \geq 1$, where \[f_m: C_n(\mathbb{R}^d) \times \mathbb{R}^d \to \mathbb{R}^d\] is real analytic, and has the property that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$ and any $m \geq 1$, the map $f_m(\mathbf{x},-): \mathbb{R}^d \to \mathbb{R}^d$ is a rational map whose image lies in the convex hull of $\mathbf{x}$. Our Approximation Conjecture is that for any $\mathbf{x} \in C_n(\mathbb{R}^d)$, the image of the sphere $S^{d-1}$ under our map $f_m(\mathbf{x},-)$ is an approximation of the boundary of the convex hull of $\mathbf{x}$. More precisely, we conjecture that \[ \operatorname{lim}_{m \to \infty} d_H\left(f_m(\mathbf{x},-)(S^{d-1}), \,\partial \operatorname{Conv}(\mathbf{x}) \right) = 0, \] where $d_H(-,-)$ is the Hausdorff distance, $\operatorname{Conv}(\mathbf{x})$ is the convex hull of $\mathbf{x}$ and $\partial$ is the boundary operator. Computer generated plots will be presented in this work.
We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, in the continuous as well as in the smooth case. This answers a question of Makeev in the special case of convex curves. The difficulty of this problem comes from the fact that standard topological arguments to prove the existence of solutions do not apply here due to the lack of sufficient symmetry. Instead, the proof makes use of an area argument of Karasev and Tao, which we furthermore simplify and elaborate on. The continuous case requires an additional analysis of the singular points, and a small miracle, which then extends to show that the problems of inscribing isosceles trapezoids in smooth curves and in piecewise $C^1$ curves are equivalent.
We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. It turns out that for any of these model sets $\varLambda$ there exists a `magic number' $m_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-directions. In particular, for pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible numbers are in that very order 11, 9, 11 and 13.
Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[ I(μ) = \int_X \int_X d(x,y) dμ(x) dμ(y), \] and set $M(X) = \sup I(μ)$, where $μ$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant $M(X)$ and its relationship to the metric properties of $X$ and the functional-analytic properties of a certain subspace of $\mathcal{M}(X)$ when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining $M(X)$, sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of $M(X)$.
Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(μ) = \int_X \int_X d(x,y) dμ(x) dμ(y),\] and set $M(X) = \sup I(μ)$, where $μ$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $α_1, ..., α_n \in \R$ satisfying $\sum_{i=1}^n α_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n α_i α_j d(x_i, x_j) \leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\mathcal{M}_0(X)$, the subspace of $\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\mathcal{M}_0(X)$; and the functional-analytic properties of $\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.
In 2000, A. Koldobsky asked whether two types of generalizations of the notion of an intersection-body, are in fact equivalent. The structures of these two types of generalized intersection-bodies have been studied by the author in [http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence for a positive answer to this question. The purpose of this note is to construct a counter-example, which provides a surprising negative answer to this question in a strong sense. This implies the existence of non-trivial non-negative functions in the range of the spherical Radon transform, and the existence of non-trivial spaces which embed in L_p for certain negative values of p.