2,193 research outputs found
A hierarchy of languages, logics, and mathematical theories
We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal.
We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic
This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence
Thermal degradation of citrus pectin in low-moisture environment - Influence of acidic and alkaline pre-treatment
Pectin powder is degraded during storage and transport by demethoxylation and depolymerisation. The degradation mechanisms and especially the influence of pre-treatments on the degradation reactions are not completely understood. In this study, commercial citrus pectin was modified by either acidic or alkaline demethoxylation. The modified pectins, as well as the commercial pectin, were thermally degraded during four weeks of storage at 60 °C and 80% relative humidity. Demethoxylation and depolymerisation as well as colour alterations were examined during degradation, and the course of the reactions was monitored.
It was found that the type of pre-treatment during modification determined the material properties and, thus, the water uptake of the modified pectin powders. The resulting water availability in the samples was crucial to the extent of demethoxylation and to the type and intensity of depolymerisation since some of these reactions competed for the water in the climate chamber. The pre-treatment also determined the content of neutral sugars and sodium ions of the modified pectins. High contents of these components limited the extent of degradation in different ways.
A previously assumed third depolymerisation mechanism of pectins, beside backbone hydrolysis and β-elimination, was confirmed.DFG, 268547215, Strukturabhängige Abbaureaktionen von Pektinen und deren Auswirkungen auf nicht-enzymatische Bräunung und technologische Funktionalitä
Two-stage Kondo effect in a four-electron artificial atom
An artificial atom with four electrons is driven through a singlet-triplet
transition by varying the confining potential. In the triplet, a Kondo peak
with a narrow dip at drain-source voltage V_ds=0 is observed. The low energy
scale V_ds* characterizing the dip is consistent with predictions for the
two-stage Kondo effect. The phenomenon is studied as a function of temperature
T and magnetic field B, parallel to the two-dimensional electron gas. The low
energy scales T* and B* are extracted from the behavior of the zero-bias
conductance and are compared to the low energy scale V_ds* obtained from the
differential conductance. Good agreement is found between kT* and |g|muB*, but
eV_ds* is larger, perhaps because of nonequilibrium effects.Comment: 7 pages, 7 figures. Added labels on Fig. 3f and one referenc
Measurements of quasi-particle tunneling in the nu = 5/2 fractional quantum Hall state
Some models of the 5/2 fractional quantum Hall state predict that the
quasi-particles, which carry the charge, have non-Abelian statistics: exchange
of two quasi-particles changes the wave function more dramatically than just
the usual change of phase factor. Such non-Abelian statistics would make the
system less sensitive to decoherence, making it a candidate for implementation
of topological quantum computation. We measure quasi-particle tunneling as a
function of temperature and DC bias between counter-propagating edge states.
Fits to theory give e*, the quasi-particle effective charge, close to the
expected value of e/4 and g, the strength of the interaction between
quasi-particles, close to 3/8. Fits corresponding to the various proposed wave
functions, along with qualitative features of the data, strongly favor the
Abelian 331 state
Transport properties of annealed CdSe nanocrystal solids
Transport properties of artificial solids composed of colloidal CdSe
nanocrystals (NCs) are studied from 6 K to 250 K, before and after annealing.
Annealing results in greatly enhanced dark and photocurrent in NC solids, while
transmission electron microscopy (TEM) micrographs show that the inter-dot
separation decreases. The increased current can be attributed to the
enhancement of inter-dot tunneling caused by the decreased separation between
NCs and by chemical changes in their organic cap. In addition, the absorption
spectra of annealed solids are slightly red-shifted and broadened. These
red-shifts may result from the change of the dielectric environment around the
NCs. Our measurements also indicate that Coulomb interactions between charges
on neighboring NCs play an important role in the tunneling current.Comment: 24 pages,4 figures, 1 tabl
Phase Transitions from Saddles of the Potential Energy Landscape
The relation between saddle points of the potential of a classical
many-particle system and the analyticity properties of its thermodynamic
functions is studied. For finite systems, each saddle point is found to cause a
nonanalyticity in the Boltzmann entropy, and the functional form of this
nonanalytic term is derived. For large systems, the order of the nonanalytic
term increases unboundedly, leading to an increasing differentiability of the
entropy. Analyzing the contribution of the saddle points to the density of
states in the thermodynamic limit, our results provide an explanation of how,
and under which circumstances, saddle points of the potential energy landscape
may (or may not) be at the origin of a phase transition in the thermodynamic
limit. As an application, the puzzling observations by Risau-Gusman et al. on
topological signatures of the spherical model are elucidated.Comment: 5 pages, no figure
Phase transitions and configuration space topology
Equilibrium phase transitions may be defined as nonanalytic points of
thermodynamic functions, e.g., of the canonical free energy. Given a certain
physical system, it is of interest to understand which properties of the system
account for the presence of a phase transition, and an understanding of these
properties may lead to a deeper understanding of the physical phenomenon. One
possible approach of this issue, reviewed and discussed in the present paper,
is the study of topology changes in configuration space which, remarkably, are
found to be related to equilibrium phase transitions in classical statistical
mechanical systems. For the study of configuration space topology, one
considers the subsets M_v, consisting of all points from configuration space
with a potential energy per particle equal to or less than a given v. For
finite systems, topology changes of M_v are intimately related to nonanalytic
points of the microcanonical entropy (which, as a surprise to many, do exist).
In the thermodynamic limit, a more complex relation between nonanalytic points
of thermodynamic functions (i.e., phase transitions) and topology changes is
observed. For some class of short-range systems, a topology change of the M_v
at v=v_t was proved to be necessary for a phase transition to take place at a
potential energy v_t. In contrast, phase transitions in systems with long-range
interactions or in systems with non-confining potentials need not be
accompanied by such a topology change. Instead, for such systems the
nonanalytic point in a thermodynamic function is found to have some
maximization procedure at its origin. These results may foster insight into the
mechanisms which lead to the occurrence of a phase transition, and thus may
help to explore the origin of this physical phenomenon.Comment: 22 pages, 6 figure
Singlet-triplet transition in a single-electron transistor at zero magnetic field
We report sharp peaks in the differential conductance of a single-electron
transistor (SET) at low temperature, for gate voltages at which charge
fluctuations are suppressed. For odd numbers of electrons we observe the
expected Kondo peak at zero bias. For even numbers of electrons we generally
observe Kondo-like features corresponding to excited states. For the latter,
the excitation energy often decreases with gate voltage until a new zero-bias
Kondo peak results. We ascribe this behavior to a singlet-triplet transition in
zero magnetic field driven by the change of shape of the potential that
confines the electrons in the SET.Comment: 4 p., 4 fig., 5 new ref. Rewrote 1st paragr. on p. 4. Revised author
list. More detailed fit results on page 3. A plotting error in the horizontal
axis of Fig. 1b and 3 was corrected, and so were the numbers in the text read
from those fig. Fig. 4 was modified with a better temperature calibration
(changes are a few percent). The inset of this fig. was removed as it is
unnecessary here. Added remarks in the conclusion. Typos are correcte
Effect of Quantum Confinement on Electron Tunneling through a Quantum Dot
Employing the Anderson impurity model, we study tunneling properties through
an ideal quantum dot near the conductance minima. Considering the Coulomb
blockade and the quantum confinement on an equal footing, we have obtained
current contributions from various types of tunneling processes; inelastic
cotunneling, elastic cotunneling, and resonant tunneling of thermally activated
electrons. We have found that the inelastic cotunneling is suppressed in the
quantum confinement limit, and thus the conductance near its minima is
determined by the elastic cotunneling at low temperature (,
: dot-reservoir coupling constant), or by the resonant tunneling of
single electrons at high temperature ().Comment: 11 pages Revtex, 2 Postscript figures, To appear in Phys.Rev.
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