On symmetric differences of NP-hard sets with weakly P-selective sets

AbstractThe symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-⩽Pm-hard set H such that H - AϵPbtt(sparse) and A — HϵPm(sparse) then P = NP. So no NP-⩽Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse)

Can Buyers Reveal for a Better Deal?

We study small-scale market interactions in which buyers are allowed to credibly reveal partial information about their types to the seller. Previous recent work has studied the special case where there is one buyer and one good, showing that such communication can simultaneously improve social welfare and ex ante buyer utility. With multiple buyers, we find that the buyer-optimal signalling schemes from the one-buyer case are actually harmful to buyer welfare. Moreover, we prove several impossibility results showing that, with either multiple i.i.d. buyers or multiple i.i.d. goods, maximizing buyer utility can be at odds with social efficiency, which is a surprising contrast to the one-buyer, one-good case. Finally, we investigate the computational tractability of implementing desirable equilibrium outcomes. We find that, even with one buyer and one good, optimizing buyer utility is generally NP-hard, but tractable in a practical restricted setting

Evolutionary accessibility of modular fitness landscapes

A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension $L$, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken [Proc. Natl. Acad. Sci. USA 92:9657 (1995)] where the genome is decomposed into disjoint sets of loci (`modules') that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman's NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure.Comment: 26 pages, 12 figures; final version with some typos correcte

Coverage, Continuity and Visual Cortical Architecture

The primary visual cortex of many mammals contains a continuous representation of visual space, with a roughly repetitive aperiodic map of orientation preferences superimposed. It was recently found that orientation preference maps (OPMs) obey statistical laws which are apparently invariant among species widely separated in eutherian evolution. Here, we examine whether one of the most prominent models for the optimization of cortical maps, the elastic net (EN) model, can reproduce this common design. The EN model generates representations which optimally trade of stimulus space coverage and map continuity. While this model has been used in numerous studies, no analytical results about the precise layout of the predicted OPMs have been obtained so far. We present a mathematical approach to analytically calculate the cortical representations predicted by the EN model for the joint mapping of stimulus position and orientation. We find that in all previously studied regimes, predicted OPM layouts are perfectly periodic. An unbiased search through the EN parameter space identifies a novel regime of aperiodic OPMs with pinwheel densities lower than found in experiments. In an extreme limit, aperiodic OPMs quantitatively resembling experimental observations emerge. Stabilization of these layouts results from strong nonlocal interactions rather than from a coverage-continuity-compromise. Our results demonstrate that optimization models for stimulus representations dominated by nonlocal suppressive interactions are in principle capable of correctly predicting the common OPM design. They question that visual cortical feature representations can be explained by a coverage-continuity-compromise.Comment: 100 pages, including an Appendix, 21 + 7 figure

Exchange-Repairs: Managing Inconsistency in Data Exchange

In a data exchange setting with target constraints, it is often the case that a given source instance has no solutions. In such cases, the semantics of target queries trivialize. The aim of this paper is to introduce and explore a new framework that gives meaningful semantics in such cases by using the notion of exchange-repairs. Informally, an exchange-repair of a source instance is another source instance that differs minimally from the first, but has a solution. Exchange-repairs give rise to a natural notion of exchange-repair certain answers (XR-certain answers) for target queries. We show that for schema mappings specified by source-to-target GAV dependencies and target equality-generating dependencies (egds), the XR-certain answers of a target conjunctive query can be rewritten as the consistent answers (in the sense of standard database repairs) of a union of conjunctive queries over the source schema with respect to a set of egds over the source schema, making it possible to use a consistent query-answering system to compute XR-certain answers in data exchange. We then examine the general case of schema mappings specified by source-to-target GLAV constraints, a weakly acyclic set of target tgds and a set of target egds. The main result asserts that, for such settings, the XR-certain answers of conjunctive queries can be rewritten as the certain answers of a union of conjunctive queries with respect to the stable models of a disjunctive logic program over a suitable expansion of the source schema.Comment: 29 pages, 13 figures, submitted to the Journal on Data Semantic

On the Feasibility of Multi-Mode Antennas in UWB and IoT Applications below 10 GHz

While on the one hand 5G and B5G networks are challenged by ultra-high data rates in wideband applications like 100+ Gbps wireless Internet access, on the other hand they are expected to support reliable low-latency Internet of Things (IoT) applications with ultra-high connectivity. These conflicting challenges are addressed in a system proposal dealing with both extremes. In contrast to most recent publications, focus is on the frequency domain below 10~GHz. Towards this goal, multi-mode antenna technology is used and different realizations, offering up to eight uncorrelated ports per radiator element, are studied. Possible baseband architectures tailored to multi-mode antennas are discussed, enabling different options regarding precoding and beamforming

Effective microscopic models for sympathetic cooling of atomic gases

Thermalization of a system in the presence of a heat bath has been the subject of many theoretical investigations especially in the framework of solid-state physics. In this setting, the presence of a large bandwidth for the frequency distribution of the harmonic oscillators schematizing the heat bath is crucial, as emphasized in the Caldeira-Leggett model. By contrast, ultracold gases in atomic traps oscillate at well-defined frequencies and therefore seem to lie outside the Caldeira-Leggett paradigm. We introduce interaction Hamiltonians which allow us to adapt the model to an atomic physics framework. The intrinsic nonlinearity of these models differentiates them from the original Caldeira-Leggett model and calls for a nontrivial stability analysis to determine effective ranges for the model parameters. These models allow for molecular dynamics simulations of mixtures of ultracold gases, which is of current relevance for optimizing sympathetic cooling in degenerate Bose-Fermi mixtures.Comment: 14 pages, 7 figure

On Linearly Constrained Minimum Variance Beamforming

Beamforming is a widely used technique for source localization in signal processing and neuroimaging. A number of vector-beamformers have been introduced to localize neuronal activity by using magnetoencephalography (MEG) data in the literature. However, the existing theoretical analyses on these beamformers have been limited to simple cases, where no more than two sources are allowed in the associated model and the theoretical sensor covariance is also assumed known. The information about the effects of the MEG spatial and temporal dimensions on the consistency of vector-beamforming is incomplete. In the present study, we consider a class of vector-beamformers defined by thresholding the sensor covariance matrix, which include the standard vector-beamformer as a special case. A general asymptotic theory is developed for these vector-beamformers, which shows the extent of effects to which the MEG spatial and temporal dimensions on estimating the neuronal activity index. The performances of the proposed beamformers are assessed by simulation studies. Superior performances of the proposed beamformers are obtained when the signal-to-noise ratio is low. We apply the proposed procedure to real MEG datasets derived from five sessions of a human face-perception experiment, finding several highly active areas in the brain. A good agreement between these findings and the known neurophysiology of the MEG response to human face perception is shown