91,855 research outputs found
Representation and regularity for the Dirichlet problem for real and complex degenerate Hessian equations
We consider the Dirichlet problem for positively homogeneous, degenerate
elliptic, concave (or convex) Hessian equations. Under natural and necessary
conditions on the geometry of the domain, with the boundary data, we
establish the interior -regularity of the unique (admissible)
solution, which is optimal even if the boundary data is smooth. Both real and
complex cases are studied by the unified (Bellman equation) approach.Comment: 42 pages. Comments are welcom
Finite groups with small number of cyclic subgroups
In this note, we study the finite groups with the number of cylic subgroups
no greater than 6.Comment: 4 page
Interior regularity of fully nonlinear degenerate elliptic equations, I: Bellman equations with constant coefficients
This is the first of a series of papers on the interior regularity of fully
nonlinear degenerate elliptic equations. We consider a stochastic optimal
control problem in which the diffusion coefficients, drift coefficients and
discount factor are independent of the spacial variables. Under suitable
assumptions, for , when the terminal and running payoffs are globally
, we obtain the -smoothness of the value function, which
yields the existence and uniqueness of the solution to the associated Dirichlet
problem for the degenerate Bellman equation.Comment: Assumption 2.2 was corrected and then weakened. The original
Assumption 2.2 after correction is now Remark 2.1. Minor revision was made
accordingly on pages 28 and 29. A few typos were corrected als
On representation and regularity of viscosity solutions to degenerate Isaacs equations and certain nonconvex Hessian equations
We study the smoothness of the upper and lower value functions of stochastic
differential games in the framework of time-homogeneous (possibly degenerate)
diffusion processes in a domain, under the assumption that the diffusion, drift
and discount coefficients are all independent of the spatial variables. Under
suitable conditions (see Assumptions 2.1 and 2.2), we obtain the optimal local
Lipschitz continuity of the value functions, provided that the running and
terminal payoffs are globally Lipschitz. As applications, we obtain the
stochastic representation and optimal interior -regularity of the
unique viscosity solution to the Dirichlet problem for certain degenerate
elliptic, nonconvex Hessian equations in suitable domains, with Lipschitz
boundary data.Comment: 30 pages. Comments are welcom
Universal price impact functions of individual trades in an order-driven market
The trade size has direct impact on the price formation of the stock
traded. Econophysical analyses of transaction data for the US and Australian
stock markets have uncovered market-specific scaling laws, where a master curve
of price impact can be obtained in each market when stock capitalization is
included as an argument in the scaling relation. However, the rationale of
introducing stock capitalization in the scaling is unclear and the anomalous
negative correlation between price change and trade size for small
trades is unexplained. Here we show that these issues can be addressed by
taking into account the aggressiveness of orders that result in trades together
with a proper normalization technique. Using order book data from the Chinese
market, we show that trades from filled and partially filled limit orders have
very different price impact. The price impact of trades from partially filled
orders is constant when the volume is not too large, while that of filled
orders shows power-law behavior with .
When returns and volumes are normalized by stock-dependent averages,
capitalization-independent scaling laws emerge for both types of trades.
However, no scaling relation in terms of stock capitalization can be
constructed. In addition, the relation is
verified, where and are the tail exponents of trade
sizes and returns. These observations also enable us to explain the anomalous
negative correlation between and for small-size trades. We
anticipate that these regularities may hold in other order-driven markets.Comment: 17 pages + supplementary figures. The paper has been significantly
expanded and more Supplementary Information is adde
Quadboost: A Scalable Concurrent Quadtree
Building concurrent spatial trees is more complicated than binary search
trees since a space hierarchy should be preserved during modifications. We
present a non-blocking quadtree-quadboost-that supports concurrent insert,
remove, move, and contain operations. To increase its concurrency, we propose a
decoupling approach that separates physical adjustment from logical removal
within the remove operation. In addition, we design a continuous find mechanism
to reduce its search cost. The move operation combines the searches for
different keys together and modifies different positions with atomicity. The
experimental results show that quadboost scales well on a multi-core system
with 32 hardware threads. More than that, it outperforms existing concurrent
trees in retrieving two-dimensional keys with up to 109% improvement when the
number of threads is large. The move operation proved to perform better than
the best-known algorithm, with up to 47%
Uplink Multicell Processing with Limited Backhaul via Per-Base-Station Successive Interference Cancellation
This paper studies an uplink multicell joint processing model in which the
base-stations are connected to a centralized processing server via rate-limited
digital backhaul links. Unlike previous studies where the centralized processor
jointly decodes all the source messages from all base-stations, this paper
proposes a suboptimal achievability scheme in which the Wyner-Ziv
compress-and-forward relaying technique is employed on a per-base-station
basis, but successive interference cancellation (SIC) is used at the central
processor to mitigate multicell interference. This results in an achievable
rate region that is easily computable, in contrast to the joint processing
schemes in which the rate regions can only be characterized by exponential
number of rate constraints. Under the per-base-station SIC framework, this
paper further studies the impact of the limited-capacity backhaul links on the
achievable rates and establishes that in order to achieve to within constant
number of bits to the maximal SIC rate with infinite-capacity backhaul, the
backhaul capacity must scale logarithmically with the
signal-to-interference-and-noise ratio (SINR) at each base-station. Finally,
this paper studies the optimal backhaul rate allocation problem for an uplink
multicell joint processing model with a total backhaul capacity constraint. The
analysis reveals that the optimal strategy that maximizes the overall sum rate
should also scale with the log of the SINR at each base-station.Comment: JSAC Oct 2013, special issue on VMIM
Capacity of the Gaussian Relay Channel with Correlated Noises to Within a Constant Gap
This paper studies the relaying strategies and the approximate capacity of
the classic three-node Gaussian relay channel, but where the noises at the
relay and at the destination are correlated. It is shown that the capacity of
such a relay channel can be achieved to within a constant gap of \hf \log_2 3
=0.7925 bits using a modified version of the noisy network coding strategy,
where the quantization level at the relay is set in a correlation dependent
way. As a corollary, this result establishes that the conventional
compress-and-forward scheme also achieves to within a constant gap to the
capacity. In contrast, the decode-and-forward and the single-tap
amplify-and-forward relaying strategies can have an infinite gap to capacity in
the regime where the noises at the relay and at the destination are highly
correlated, and the gain of the relay-to-destination link goes to infinity.Comment: accepted to communications letter
Normal property, Jamenson property, CHIP and linear regularity for an infinite system of convex sets in Banach spaces
In this paper, we study different kinds of normal properties for infinite
system of arbitrarily many convex sets in a Banach space and provide the dual
characterization for the normal property in terms of the extended Jamenson
property for arbitrarily many weak*-closed convex cones in the dual space.
Then, we use the normal property and the extended Jamenson property to study
CHIP, strong CHIP and linear regularity for the infinite case of arbitrarily
many convex sets and establish equivalent relationship among these properties.
In particular, we extend main results in [3] on normal property, Jamenson
property, CHIP and linear regularity for finite system of convex sets in a
Hilbert space to the infinite case of arbitrarily many convex sets in Banach
space setting
Object Detection based on LIDAR Temporal Pulses using Spiking Neural Networks
Neural networks has been successfully used in the processing of Lidar data,
especially in the scenario of autonomous driving. However, existing methods
heavily rely on pre-processing of the pulse signals derived from Lidar sensors
and therefore result in high computational overhead and considerable latency.
In this paper, we proposed an approach utilizing Spiking Neural Network (SNN)
to address the object recognition problem directly with raw temporal pulses. To
help with the evaluation and benchmarking, a comprehensive temporal pulses
data-set was created to simulate Lidar reflection in different road scenarios.
Being tested with regard to recognition accuracy and time efficiency under
different noise conditions, our proposed method shows remarkable performance
with the inference accuracy up to 99.83% (with 10% noise) and the average
recognition delay as low as 265 ns. It highlights the potential of SNN in
autonomous driving and some related applications. In particular, to our best
knowledge, this is the first attempt to use SNN to directly perform object
recognition on raw Lidar temporal pulses
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