On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.     

A single instance of the Pythagorean theorem implies the parallel postulate

This note could find use as enrichment material in a course on the classical geometries; its preliminary results could also be used in an advanced calculus course. It is proved that if a, b and c are positive real numbers such that a ₂ + b ² = a ², then cosh (a) cosh (b) > cosh (c). The proof of this result uses an inequality involving combinatorial symbols and properties of absolutely convergent infinite series. As a corollary, it follows that if a neutral geometry G contains at least one right triangle which satisfies the conclusion of the Pythagorean Theorem, then G is (isomorphic to) Euclidean geometry and, hence, satisfies Playfair's form of the parallel postulate.     

Determination of the strong phase in D0-->K+pi- using quantum-correlated measurements

We exploit the quantum coherence between pair-produced D0 and D[over]0 in psi(3770) decays to study charm mixing, which is characterized by the parameters x and y, and to make a first determination of the relative strong phase delta between D0-->K+pi- and D[over]0-->K+pi-. Using 281 pb(-1) of e+e- collision data collected with the CLEO-c detector at Ecm=3.77 GeV, as well as branching fraction input and time-integrated measurements of RM identical with (x2 + y2)/2 and RWS identical with Gamma(D0-->K+pi-)/Gamma(D[over]0-->K+pi-) from other experiments, we find cosdelta=1.03(-0.17)(+0.31)+/-0.06, where the uncertainties are statistical and systematic, respectively. By further including other mixing parameter measurements, we obtain an alternate measurement of cosdelta=1.10+/-0.35+/-0.07, as well as x sindelta=(4.4(-1.8)(+2.7)+/-2.9)x10(-3) and delta=(22(-12-11)(+11+9)) degrees .     

Commutative Algebras In Which Polynomials Have Infinitely Many Roots

Let K be a field. Then there exists a commutative K-algebra A such that each polynomial in K[X] of degree at least 2 has infinitely many roots in A. If B is a finite-dimensional commutative K-algebra and char(K) ≠ 3 (resp., char(K ) = 3), then X ² + X + 1 (resp., X ² + X-1) has only finitely many roots in B. Relevant examples are also given, especially of K-algebras of the form K + N, where N is the nilradical.     

Spherically symmetric inhomogeneous dust collapse in higher dimensional space-time and cosmic censorship hypothesis

We consider a collapsing spherically symmetric inhomogeneous dust cloud in higher dimensional space-time. We show that the central singularity of collapse can be a strong curvature or a weak curvature naked singularity depending on the initial density distribution.     

On comparing two chains of numerical semigroups and detecting arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T):=(N-N) and L(T)= \cup { (hN-hN) \colon h \geq 1 } . If S is a numerical semigroup, define strictly increasing finite sequences { B _i (S) \colon 0 ≤ i ≤β (S) } and { L _i (S) \colon 0 ≤ i ≤λ (S) } of semigroups by B ₀ (S):=S=:L ₀ (S) , B _{β (S)} (S):= \Bbb N =: L _{λ (S)} (S) , B _i+1 (S):=B(B _i (S)) for 0<i< β (S) , L _i+1 (S):=L(L _i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B ₂ (S) need not be a subset of L ₂ (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B ₂ (S) \subseteq L ₂ (S) in each of the following cases: S is symmetric;S has maximal embedding dimension;S has embedding dimension e(S) ≤ 3 . Moreover, if either e(S)=2 or S is pseudo-symmetric of maximal embedding dimension, then B _i (S) \subseteq L _i (S) for each i , 0 ≤ i ≤λ (S) . For each integer n \geq 2 , an example is given of a (necessarily non-Arf) semigroup S such that β (S) = λ (S)=n , B _i (S) = L _i (S) for all 0 ≤ i ≤ n-2 , and B _n-1 (S) \subsetneqq L _n-1 (S) . April 4, 2000