398,942 research outputs found

### Coupled oscillators and Feynman's three papers

According to Richard Feynman, the adventure of our science of physics is a
perpetual attempt to recognize that the different aspects of nature are really
different aspects of the same thing. It is therefore interesting to combine
some, if not all, of Feynman's papers into one. The first of his three papers
is on the ``rest of the universe'' contained in his 1972 book on statistical
mechanics. The second idea is Feynman's parton picture which he presented in
1969 at the Stony Brook conference on high-energy physics. The third idea is
contained in the 1971 paper he published with his students, where they show
that the hadronic spectra on Regge trajectories are manifestations of
harmonic-oscillator degeneracies. In this report, we formulate these three
ideas using the mathematics of two coupled oscillators. It is shown that the
idea of entanglement is contained in his rest of the universe, and can be
extended to a space-time entanglement. It is shown also that his parton model
and the static quark model can be combined into one Lorentz-covariant entity.
Furthermore, Einstein's special relativity, based on the Lorentz group, can
also be formulated within the mathematical framework of two coupled
oscillators.Comment: 31 pages, 6 figures, based on the concluding talk at the 3rd Feynman
Festival (Collage Park, Maryland, U.S.A., August 2006), minor correction

### Competitively tight graphs

The competition graph of a digraph $D$ is a (simple undirected) graph which
has the same vertex set as $D$ and has an edge between two distinct vertices
$x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x,v)$
and $(y,v)$ are arcs of $D$. For any graph $G$, $G$ together with sufficiently
many isolated vertices is the competition graph of some acyclic digraph. The
competition number $k(G)$ of a graph $G$ is the smallest number of such
isolated vertices. Computing the competition number of a graph is an NP-hard
problem in general and has been one of the important research problems in the
study of competition graphs. Opsut [1982] showed that the competition number of
a graph $G$ is related to the edge clique cover number $\theta_E(G)$ of the
graph $G$ via $\theta_E(G)-|V(G)|+2 \leq k(G) \leq \theta_E(G)$. We first show
that for any positive integer $m$ satisfying $2 \leq m \leq |V(G)|$, there
exists a graph $G$ with $k(G)=\theta_E(G)-|V(G)|+m$ and characterize a graph
$G$ satisfying $k(G)=\theta_E(G)$. We then focus on what we call
\emph{competitively tight graphs} $G$ which satisfy the lower bound, i.e.,
$k(G)=\theta_E(G)-|V(G)|+2$. We completely characterize the competitively tight
graphs having at most two triangles. In addition, we provide a new upper bound
for the competition number of a graph from which we derive a sufficient
condition and a necessary condition for a graph to be competitively tight.Comment: 10 pages, 2 figure

### Lens optics as an optical computer for group contractions

It is shown that the one-lens system in para-axial optics can serve as an
optical computer for contraction of Wigner's little groups and an analogue
computer which transforms analytically computations on a spherical surface to
those on a hyperbolic surface. It is shown possible to construct a set of
Lorentz transformations which leads to a two-by-two matrix whose expression is
the same as those in the para-axial lens optics. It is shown that the lens
focal condition corresponds to the contraction of the O(3)-like little group
for a massive particle to the E(2)-like little group for a massless particle,
and also to the contraction of the O(2,1)-like little group for a space-like
particle to the same E(2)-like little group. The lens-focusing transformations
presented in this paper allow us to continue analytically the spherical O(3)
world to the hyperbolic O(2,1) world, and vice versa.Comment: 14 pages, RevTeX, no figure

### Norm Estimates for the Difference Between Bochner's Integral and the Convex Combination of Function's Values

Norm estimates are developed between the Bochner integral of a vector-valued
function in Banach spaces having the Radon-Nikodym property and the convex
combination of function values taken on a division of the interval [a,b]

### A note on the values of the weighted q-Bernstein polynomials and modified q-Genocchi numbers with weight alpha and beta via the p-adic q-integral on Zp

The rapid development of q-calculus has led to the discovery of new
generalizations of Bernstein polynomials and Genocchi polynomials involving
q-integers. The present paper deals with weighted q-Bernstein polynomials and
q-Genocchi numbers with weight alpha and beta. We apply the method of
generating function and p-adic q-integral representation on Zp, which are
exploited to derive further classes of Bernstein polynomials and q-Genocchi
numbers and polynomials. To be more precise we summarize our results as
follows, we obtain some combinatorial relations between q-Genocchi numbers and
polynomials with weight alpha and beta. Furthermore, we derive an integral
representation of weighted q-Bernstein polynomials of degree n on Zp. Also we
deduce a fermionic p-adic q-integral representation of product weighted
q-Bernstein polynomials of different degrees n1,n2,...on Zp and show that it
can be written with q-Genocchi numbers with weight alpha and beta which yields
a deeper insight into the effectiveness of this type of generalizations. Our
new generating function possess a number of interesting properties which we
state in this paper.Comment: 10 page

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