A deformed Hermitian Yang-Mills Flow

We study a new deformed Hermitian Yang-Mills Flow in the supercritical case. Under the same assumption on the subsolution as Collins-Jacob-Yau \cite{cjy2020cjm}, we show the longtime existence and the solution converges to a solution of the deformed Hermitian Yang-Mills equation which was solved by Collins-Jacob-Yau \cite{cjy2020cjm} by the continuity method.Comment: This version is more readable and some references are adde

The Neumann problem of special Lagrangian type equations

We study the Neumann problem for special Lagrangian type equations with critical and supercritical phases. These equations naturally generalize the special Lagrangian equation and the k-Hessian equation. By establishing uniform a priori estimates up to the second order, we obtain the existence result using the continuity method. The new technical aspect is our direct proof of boundary double normal derivative estimates. In particular, we directly prove the double normal estimates for the 2-Hessian equation in dimension 3. Moreover, we solve the classical Neumann problem by proving the uniform gradient estimate.Comment: 24 page

Evaluation of Pedestrian Level of Service at Signalised Intersections from the Elderly Perspective

The crossing decisions and behaviour of elderly pedestrians are affected by the pedestrian level of service (PLOS). In this paper, an evaluation model was established to analyse the relationship between the traffic environment and the perceived evaluation of elderly pedestrians. Firstly, the characteristic parameters of the selected intersections and the perceived evaluation data of elderly pedestrians at the synchronisation scenery were extracted using manual recording and questionnaire-based truncation methods. The correlation between the perceived evaluation data of elderly pedestrians and the traffic parameters were tested with respect to the dimensions of safety, convenience and efficiency. Then, the significant parameters affecting PLOS were recognised. Based on the traffic characteristic parameters, the PLOS evaluation model from the elderly perspective was established using the fuzzy linear regression method. PLOS classification thresholds were obtained using the fuzzy C-means clustering algorithm. The data from two intersections were used to validate the model. The results show that the difference between the actual and the predicted PLOS values of the two crosswalks were 0.2 and 0.1, respectively. Thus, the proposed PLOS evaluation model in this paper can be used to accurately predict the PLOS from the elderly perspective using the traffic data of signalised intersections

The parabolic quaternionic Monge-Amp\`{e}re type equation on hyperK\"{a}hler manifolds

We prove the long time existence and uniqueness of solution to a parabolic quaternionic Monge-Amp\`{e}re type equation on a compact hyperK\"{a}hler manifold. We also show that after normalization, the solution converges smoothly to the unique solution of the Monge-Amp\`{e}re equation for $(n-1)$-quaternionic psh functions

The Monge-Amp\`{e}re equation for (n−1)(n-1)-quaternionic PSH functions on a hyperK\"{a}hler manifold

We prove the existence of unique smooth solutions to the quaternionic Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic form type equation. We derive $C^0$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without assuming the flatness of underlying hyperK\"{a}hler metric comparing to previous results [14].Comment: 31 page

Targeted Activation Penalties Help CNNs Ignore Spurious Signals

Neural networks (NNs) can learn to rely on spurious signals in the training data, leading to poor generalisation. Recent methods tackle this problem by training NNs with additional ground-truth annotations of such signals. These methods may, however, let spurious signals re-emerge in deep convolutional NNs (CNNs). We propose Targeted Activation Penalty (TAP), a new method tackling the same problem by penalising activations to control the re-emergence of spurious signals in deep CNNs, while also lowering training times and memory usage. In addition, ground-truth annotations can be expensive to obtain. We show that TAP still works well with annotations generated by pre-trained models as effective substitutes of ground-truth annotations. We demonstrate the power of TAP against two state-of-the-art baselines on the MNIST benchmark and on two clinical image datasets, using four different CNN architectures.Comment: 24 pages including appendix; extended version of a paper accepted to AAAI-2024 under the same titl

The Dirichlet problem of the homogeneous kk-Hessian equation in a punctured domain

In this paper, we consider the Dirichlet problem for the homogeneous $k$-Hessian equation with prescribed asymptotic behavior at $0\in\Omega$ where $\Omega$ is a $(k-1)$-convex bounded domain in the Euclidean space. The prescribed asymptotic behavior at $0$ of the solution is zero if $k>\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and $-|x|^{\frac{2k-n}{n}}+O(1)$ if $k<\frac{n}{2}$. To solve this problem, we consider the Dirichlet problem of the approximating $k$-Hessian equation in $\Omega\setminus \overline{B_r(0)}$ with $r$ small. We firstly construct the subsolution of the approximating $k$-Hessian equation. Then we derive the pointwise $C^{2}$-estimates of the approximating equation based on new gradient and second order estimates established previously by the second author and the third author. In addition, we prove a uniform positive lower bound of the gradient if the domain is starshaped with respect to $0$. As an application, we prove an identity along the level set of the approximating solution and obtain a nearly monotonicity formula. In particular, we get a weighted geometric inequality for smoothly and strictly $(k-1)$-convex starshaped closed hypersurface in $\mathbb R^n$ with $\frac{n}{2}\le k<n$.Comment: 33 pages. arXiv admin note: text overlap with arXiv:2207.1350