590 research outputs found
Understanding the calculus
A number of significant changes have have occurred recently that give us a golden opportunity to review the teaching of calculus. The most obvious is the arrival of the microcomputer in the mathematics classroom, allowing graphic demonstrations and individual investigations into the mathematical ideas. But equally potent are new
insights into mathematics and mathematics education that suggest new ways of approaching the subject.
In this article I shall consider some of the difficulties encountered studying the calculus and outline a viable alternative approach suitable for specialist and non-specialist mathematics students alike
Why High and Low Performers Leave and What They Find Elsewhere: Job Performance Effects on Employment Transitions
Little is known about how high and low performers differ in terms of why they leave their jobs, and no work examines whether pre-quit job performance matters for post-quit new-job outcomes. Working with a sample of approximately 2,500 former employees of an organization in the leisure and hospitality industry, we find that the reported importance of a variety of quit reasons differs both across and within performance levels. Additionally, we use an ease-of-movement perspective to predict how pre-quit performance relates to post-quit employment, new-job pay, and new-job advancement opportunity. Job type, tenure, and race interacted with performance in predicting new-job outcomes, suggesting explanations grounded in motivation, signaling, and discrimination in the external job market
Consistent discretization and canonical classical and quantum Regge calculus
We apply the ``consistent discretization'' technique to the Regge action for
(Euclidean and Lorentzian) general relativity in arbitrary number of
dimensions. The result is a well defined canonical theory that is free of
constraints and where the dynamics is implemented as a canonical
transformation. This provides a framework for the discussion of topology change
in canonical quantum gravity. In the Lorentzian case, the framework appears to
be naturally free of the ``spikes'' that plague traditional formulations. It
also provides a well defined recipe for determining the measure of the path
integral.Comment: 8 pages, Dedicated to Rafael Sorkin on his 60th birthday, to appear
in Proceedings of the Puri Conference, special issue of IJMP
Cosmological Inflation and the Quantum Measurement Problem
According to cosmological inflation, the inhomogeneities in our universe are
of quantum mechanical origin. This scenario is phenomenologically very
appealing as it solves the puzzles of the standard hot big bang model and
naturally explains why the spectrum of cosmological perturbations is almost
scale invariant. It is also an ideal playground to discuss deep questions among
which is the quantum measurement problem in a cosmological context. Although
the large squeezing of the quantum state of the perturbations and the
phenomenon of decoherence explain many aspects of the quantum to classical
transition, it remains to understand how a specific outcome can be produced in
the early universe, in the absence of any observer. The Continuous Spontaneous
Localization (CSL) approach to quantum mechanics attempts to solve the quantum
measurement question in a general context. In this framework, the wavefunction
collapse is caused by adding new non linear and stochastic terms to the
Schroedinger equation. In this paper, we apply this theory to inflation, which
amounts to solving the CSL parametric oscillator case. We choose the
wavefunction collapse to occur on an eigenstate of the Mukhanov-Sasaki variable
and discuss the corresponding modified Schroedinger equation. Then, we compute
the power spectrum of the perturbations and show that it acquires a universal
shape with two branches, one which remains scale invariant and one with nS=4, a
spectral index in obvious contradiction with the Cosmic Microwave Background
(CMB) anisotropy observations. The requirement that the non-scale invariant
part be outside the observational window puts stringent constraints on the
parameter controlling the deviations from ordinary quantum mechanics...
(Abridged).Comment: References added, minor corrections, conclusions unchange
Categorifying Hecke algebras at prime roots of unity, part I
We equip the type diagrammatic Hecke category with a special derivation,
so that after specialization to characteristic it becomes a -dg
category. We prove that the defining relations of the Hecke algebra are
satisfied in the -dg Grothendieck group. We conjecture that the -dg
Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it
with a basis which may differ from both the Kazhdan-Lusztig basis and the
-canonical basis. More precise conjectures will be found in the sequel.
Here are some other results contained in this paper. We provide an incomplete
proof of the classification of all degree derivations on the diagrammatic
Hecke category, and a complete proof of the classification of those derivations
for which the defining relations of the Hecke algebra are satisfied in the
-dg Grothendieck group. In particular, our special derivation is unique up
to duality and equivalence. We prove that no such derivation exists in
simply-laced types outside of finite and affine type . We also examine a
particular Bott-Samelson bimodule in type , which is indecomposable in
characteristic but decomposable in all other characteristics. We prove that
this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any
characteristic, which is the analogue in the -dg setting of being
indecomposable.Comment: 44 pages, many figures, color viewing essential. V2 contains
corrections from referee reports. To appear in Transactions of the AM
Advanced Methods in Black-Hole Perturbation Theory
Black-hole perturbation theory is a useful tool to investigate issues in
astrophysics, high-energy physics, and fundamental problems in gravity. It is
often complementary to fully-fledged nonlinear evolutions and instrumental to
interpret some results of numerical simulations. Several modern applications
require advanced tools to investigate the linear dynamics of generic small
perturbations around stationary black holes. Here, we present an overview of
these applications and introduce extensions of the standard semianalytical
methods to construct and solve the linearized field equations in curved
spacetime. Current state-of-the-art techniques are pedagogically explained and
exciting open problems are presented.Comment: Lecture notes from the NRHEP spring school held at IST-Lisbon, March
2013. Extra material and notebooks available online at
http://blackholes.ist.utl.pt/nrhep2/. To be published by IJMPA (V. Cardoso,
L. Gualtieri, C. Herdeiro and U. Sperhake, Eds., 2013); v2: references
updated, published versio
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