315,097 research outputs found
Prospect and Markowitz Stochastic Dominance
Levy and Levy (2002, 2004) develop the Prospect and Markowitz stochastic dominance theory with S-shaped and reverse S-shaped utility functions for investors. In this paper, we extend Levy and Levy's Prospect Stochastic Dominance theory (PSD) and Markowitz Stochastic Dominance theory (MSD) to the first three orders and link the corresponding S-shaped and reverse S-shaped utility functions to the first three orders. We also provide experiments to illustrate each case of the MSD and PSD to the first three orders and demonstrate that the higher order MSD and PSD cannot be replaced by the lower order MSD and PSD. Prospect theory has been regarded as a challenge to the expected utility paradigm. Levy and Levy (2002) prove that the second order PSD and MSD satisfy the expected utility paradigm. In our paper we take Levy and Levy's results one step further by showing that both PSD and MSD of any order are consistent with the expected utility paradigm. Furthermore, we formulate some other properties for the PSD and MSD including the hierarchy that exists in both PSD and MSD relationships; arbitrage opportunities that exist in the first orders of both PSD and MSD; and that for any two prospects under certain conditions, their third order MSD preference will be ???the opposite??? of or ???the same??? as their counterpart third order PSD preference. By extending Levy and Levy's work, we provide investors with more tools for empirical analysis, with which they can identify the first order PSD and MSD prospects and discern arbitrage opportunities that could increase his/her utility as well as wealth and set up a zero dollar portfolio to make huge profit. Our tools also enable investors to identify the third order PSD and MSD prospects and make better choices.Prospect stochastic dominance, Markowitz stochastic dominance, risk seeking, risk averse, S-shaped utility function, reverse S-shaped utility function
Levy processes: Capacity and Hausdorff dimension
We use the recently-developed multiparameter theory of additive Levy
processes to establish novel connections between an arbitrary Levy process
in , and a new class of energy forms and their corresponding
capacities. We then apply these connections to solve two long-standing problems
in the folklore of the theory of Levy processes. First, we compute the
Hausdorff dimension of the image of a nonrandom linear Borel set
, where is an arbitrary Levy process in
. Our work completes the various earlier efforts of Taylor [Proc.
Cambridge Phil. Soc. 49 (1953) 31-39], McKean [Duke Math. J. 22 (1955)
229-234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370-375, J. Math.
Mech. 10 (1961) 493-516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53-73],
Pruitt [J. Math. Mech. 19 (1969) 371-378], Pruitt and Taylor [Z. Wahrsch. Verw.
Gebiete 12 (1969) 267-289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90-102,
J. London Math. Soc. (2) 17 (1978) 567-576, Probab. Theory Related Fields 112
(1998) 1-11], Hendricks [Ann. Math. Stat. 43 (1972) 690-694, Ann. Probab. 1
(1973) 849-853], Kahane [Publ. Math. Orsay (83-02) (1983) 74-105, Recent
Progress in Fourier Analysis (1985b) 65-121], Becker-Kern, Meerschaert and
Scheffler [Monatsh. Math. 14 (2003) 91-101] and Khoshnevisan, Xiao and Zhong
[Ann. Probab. 31 (2003a) 1097-1141], where is computed under
various conditions on , or both.Comment: Published at http://dx.doi.org/10.1214/009117904000001026 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Curing basis-set convergence of wave-function theory using density-functional theory: a systematically improvable approach
The present work proposes to use density-functional theory (DFT) to correct
for the basis-set error of wave-function theory (WFT). One of the key ideas
developed here is to define a range-separation parameter which automatically
adapts to a given basis set. The derivation of the exact equations are based on
the Levy-Lieb formulation of DFT, which helps us to define a complementary
functional which corrects uniquely for the basis-set error of WFT. The coupling
of DFT and WFT is done through the definition of a real-space representation of
the electron-electron Coulomb operator projected in a one-particle basis set.
Such an effective interaction has the particularity to coincide with the exact
electron-electron interaction in the limit of a complete basis set, and to be
finite at the electron-electron coalescence point when the basis set is
incomplete. The non-diverging character of the effective interaction allows one
to define a mapping with the long-range interaction used in the context of
range-separated DFT and to design practical approximations for the unknown
complementary functional. Here, a local-density approximation is proposed for
both full-configuration-interaction (FCI) and selected
configuration-interaction approaches. Our theory is numerically tested to
compute total energies and ionization potentials for a series of atomic
systems. The results clearly show that the DFT correction drastically improves
the basis-set convergence of both the total energies and the energy
differences. For instance, a sub kcal/mol accuracy is obtained from the
aug-cc-pVTZ basis set with the method proposed here when an aug-cc-pV5Z basis
set barely reaches such a level of accuracy at the near FCI level
Avalanche Dynamics in Evolution, Growth, and Depinning Models
The dynamics of complex systems in nature often occurs in terms of
punctuations, or avalanches, rather than following a smooth, gradual path. A
comprehensive theory of avalanche dynamics in models of growth, interface
depinning, and evolution is presented. Specifically, we include the Bak-Sneppen
evolution model, the Sneppen interface depinning model, the Zaitsev flux creep
model, invasion percolation, and several other depinning models into a unified
treatment encompassing a large class of far from equilibrium processes. The
formation of fractal structures, the appearance of noise, diffusion with
anomalous Hurst exponents, Levy flights, and punctuated equilibria can all be
related to the same underlying avalanche dynamics. This dynamics can be
represented as a fractal in spatial plus one temporal dimension. We develop
a scaling theory that relates many of the critical exponents in this broad
category of extremal models, representing different universality classes, to
two basic exponents characterizing the fractal attractor. The exact equations
and the derived set of scaling relations are consistent with numerical
simulations of the above mentioned models.Comment: 27 pages in revtex, no figures included. Figures or hard copy of the
manuscript supplied on reques
Density Functionals in the Presence of Magnetic Field
In this paper density functionals for Coulomb systems subjected to electric
and magnetic fields are developed. The density functionals depend on the
particle density, , and paramagnetic current density, . This
approach is motivated by an adapted version of the Vignale and Rasolt
formulation of Current Density Functional Theory (CDFT), which establishes a
one-to-one correspondence between the non-degenerate ground-state and the
particle and paramagnetic current density. Definition of -representable
density pairs is given and it is proven that the set of
-representable densities constitutes a proper subset of the set of
-representable densities. For a Levy-Lieb type functional , it
is demonstrated that (i) it is a proper extension of the universal
Hohenberg-Kohn functional, , to -representable densities,
(ii) there exists a wavefunction such that
, where is the Hamiltonian without
external potential terms, and (iii) it is not convex. Furthermore, a convex and
universal functional is studied and proven to be equal the convex
envelope of . For both and , we give upper and lower
bounds.Comment: 26 page
Negative Harmony: Experiments with the Polarity in Music
I set out to experiment and justify the use of a new theory called Negative Harmony in 21st century music. Negative Harmony is a musical avenue from which composers can glean new tones within traditional music theory rules. I took inspiration from the current leading authority on the topic, Jacob Collier, as well as older scholars from the 20th century, such as Ernst Levy and George Rochberg. I conducted research on the theory by finding its relation to major and minor chords, and how these mirrored chords worked from a theory standpoint. I then composed two original works, one piano piece and one piece for SATB choir and piano. I aimed to find the best balance between the unfamiliar negative chords and the familiar positive chords. I then looked to justify the use of this theory through the writings of scholars and modern music listeners and casual music makers
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