9,028 research outputs found
Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study
A biform theory is a combination of an axiomatic theory and an algorithmic
theory that supports the integration of reasoning and computation. These are
ideal for formalizing algorithms that manipulate mathematical expressions. A
theory graph is a network of theories connected by meaning-preserving theory
morphisms that map the formulas of one theory to the formulas of another
theory. Theory graphs are in turn well suited for formalizing mathematical
knowledge at the most convenient level of abstraction using the most convenient
vocabulary. We are interested in the problem of whether a body of mathematical
knowledge can be effectively formalized as a theory graph of biform theories.
As a test case, we look at the graph of theories encoding natural number
arithmetic. We used two different formalisms to do this, which we describe and
compare. The first is realized in , a version of Church's
type theory with quotation and evaluation, and the second is realized in Agda,
a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds,
Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer
Science, Vol. 10383, pp. 9-24, Springer, 201
Anomalous price impact and the critical nature of liquidity in financial markets
We propose a dynamical theory of market liquidity that predicts that the
average supply/demand profile is V-shaped and {\it vanishes} around the current
price. This result is generic, and only relies on mild assumptions about the
order flow and on the fact that prices are (to a first approximation)
diffusive. This naturally accounts for two striking stylized facts: first,
large metaorders have to be fragmented in order to be digested by the liquidity
funnel, leading to long-memory in the sign of the order flow. Second, the
anomalously small local liquidity induces a breakdown of linear response and a
diverging impact of small orders, explaining the "square-root" impact law, for
which we provide additional empirical support. Finally, we test our arguments
quantitatively using a numerical model of order flow based on the same minimal
ingredients.Comment: 16 pages, 7 figure
Autocorrelation of Random Matrix Polynomials
We calculate the autocorrelation functions (or shifted moments) of the
characteristic polynomials of matrices drawn uniformly with respect to Haar
measure from the groups U(N), O(2N) and USp(2N). In each case the result can be
expressed in three equivalent forms: as a determinant sum (and hence in terms
of symmetric polynomials), as a combinatorial sum, and as a multiple contour
integral. These formulae are analogous to those previously obtained for the
Gaussian ensembles of Random Matrix Theory, but in this case are identities for
any size of matrix, rather than large-matrix asymptotic approximations. They
also mirror exactly autocorrelation formulae conjectured to hold for
L-functions in a companion paper. This then provides further evidence in
support of the connection between Random Matrix Theory and the theory of
L-functions
A Bit-String Model for Biological Aging
We present a simple model for biological aging. We studied it through
computer simulations and we have found this model to reflect some features of
real populations.Comment: LaTeX file, 4 PS figures include
Market impact and trading profile of large trading orders in stock markets
We empirically study the market impact of trading orders. We are specifically
interested in large trading orders that are executed incrementally, which we
call hidden orders. These are reconstructed based on information about market
member codes using data from the Spanish Stock Market and the London Stock
Exchange. We find that market impact is strongly concave, approximately
increasing as the square root of order size. Furthermore, as a given order is
executed, the impact grows in time according to a power-law; after the order is
finished, it reverts to a level of about 0.5-0.7 of its value at its peak. We
observe that hidden orders are executed at a rate that more or less matches
trading in the overall market, except for small deviations at the beginning and
end of the order.Comment: 9 pages, 7 figure
Long-range memory model of trading activity and volatility
Earlier we proposed the stochastic point process model, which reproduces a
variety of self-affine time series exhibiting power spectral density S(f)
scaling as power of the frequency f and derived a stochastic differential
equation with the same long range memory properties. Here we present a
stochastic differential equation as a dynamical model of the observed memory in
the financial time series. The continuous stochastic process reproduces the
statistical properties of the trading activity and serves as a background model
for the modeling waiting time, return and volatility. Empirically observed
statistical properties: exponents of the power-law probability distributions
and power spectral density of the long-range memory financial variables are
reproduced with the same values of few model parameters.Comment: 12 pages, 5 figure
Distribution of parallel vortices studied by spin-polarized neutron reflectivity and magnetization
We present the studies of non-uniformly distributed vortices in Nb/Al
multilayers at applied field near parallel to film surface by using
spin-polarized neutron reflectivity (SPNR) and DC magnetization measurements.
We have observed peaks above the lower critical field, Hc1, in the M-H curves
from the multilayers.
Previous works with a model calculation of minimizing Gibbs free energy have
suggested that the peaks could be ascribed to vortex line transitions for
spatial commensuration in a thin film superconductor. In order to directly
determine the distribution of vortices, we performed SPNR measurements on the
multilayer and found that the distribution and density of vortices are
different at ascending and descending fields. At ascending 2000 Oe which is
just below the first peak in the M-H curve, SPNR shows that vortices are mostly
localized near a middle line of the film meanwhile the vortices are distributed
in broader region at the descending 2000 Oe. That is related to the observation
of more vortices trapped at the descending field. As the applied field is
sightly tilted (< 3.5degree), we observe another peak at a smaller field. The
peak position is consistent with the parallel lower critical field (Hc1||). We
discuss that the vortices run along the applied field below Hc1|| and rotate
parallel to the surface at Hc1||.Comment: 17 pages, 9 figure
Lower order terms in the full moment conjecture for the Riemann zeta function
We describe an algorithm for obtaining explicit expressions for lower terms
for the conjectured full asymptotics of the moments of the Riemann zeta
function, and give two distinct methods for obtaining numerical values of these
coefficients. We also provide some numerical evidence in favour of the
conjecture.Comment: 37 pages, 4 figure
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