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In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2.4}$ (Lai, 2003). The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.

This paper presents a method for generative design of decorative architectural parts such as corbel,moulding and panel, which usually have clear structure and aesthetic details. The method is composed of two components: offline learning and online generation. The offline learning trains a 2D CurveInfoGAN and a 3D VoxelVAE that learn the feature representations of the parts in a dataset. The online generation proceeds with an evolution procedure that evolves to product new generation of part components by selecting, crossing over and mutating features, followed by a feature-driven deformation that synthesizes the 3D mesh representation of new models. Built upon these technical components, a generative design tool is developed, which allows the user to input a decorative architectural model as a reference and then generates a set of new models that are “more of the same” as the reference and meanwhile exhibit some “surprising” elements. The experiments demonstrate the effectiveness of the method and also showcase the use of classic geometric modelling and advanced machine learning techniques in modelling of architectural parts.

Double-blind randomized controlled trials are traditionally seen as the gold standard for causal inferences as the difference-in-means estimator is an unbiased estimator of the average treatment effect in the experiment. The fact that this estimator is unbiased over all possible randomizations does not, however, mean that any given estimate is close to the true treatment effect. Similarly, while predetermined covariates will be balanced between treatment and control groups on average, large imbalances may be observed in a given experiment and the researcher may therefore want to condition on such covariates using linear regression. This article studies the theoretical properties of both the difference-in-means and OLS estimators conditional on observed differences in covariates. By deriving the statistical properties of the conditional estimators, we can establish guidance for how to deal with covariate imbalances.

In this paper, we develop a class of nonlinear compact schemes based on our previous linear central compact schemes with spectral-like resolution [X. Liu, S. Zhang, H. Zhang and C-W. Shu, A new class of central compact schemes with spectral-like resolution I: Linear schemes, Journal of Computational Physics 248 (2013) 235-256]. In our approach, we compute the flux derivatives on the cell-nodes by the physical fluxes on the cell nodes and numerical fluxes on the cell centers. To acquire the numerical fluxes on the cell centers, we perform a weighted hybrid interpolation of an upwind interpolation and a central interpolation. Through systematic analysis and numerical tests, we show that our nonlinear compact scheme has high order, high resolution and low dissipation, and has the same ability to capture strong discontinuities as regular weighted essentially non-oscillatory (WENO) schemes. It is a good choice for the simulation of multiscale problems with shock waves.