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Leveraging biased click data for optimizing learning to rank systems has been a popular approach in information retrieval. Because click data is often noisy and biased, a variety of methods have been proposed to construct unbiased learning to rank (ULTR) algorithms for the learning of unbiased ranking models. Among them, automatic unbiased learning to rank (AutoULTR) algorithms that jointly learn user bias models (i.e., propensity models) with unbiased rankers have received a lot of attention due to their superior performance and low deployment cost in practice. Despite their differences in theories and algorithm design, existing studies on ULTR usually use uni-variate ranking functions to score each document or result independently. On the other hand, recent advances in context-aware learning-to-rank models have shown that multivariate scoring functions, which read multiple documents together and predict their ranking scores jointly, are more powerful than uni-variate ranking functions in ranking tasks with human-annotated relevance labels. Whether such superior performance would hold in ULTR with noisy data, however, is mostly unknown. In this paper, we investigate existing multivariate scoring functions and AutoULTR algorithms in theory and prove that permutation invariance is a crucial factor that determines whether a context-aware learning-to-rank model could be applied to existing AutoULTR framework. Our experiments with synthetic clicks on two large-scale benchmark datasets show that AutoULTR models with permutation-invariant multivariate scoring functions significantly outperform those with uni-variate scoring functions and permutation-variant multivariate scoring functions., Comment: 4 pages, 2 figures. It has already been accepted and will show in Proceedings of the 29th ACM International Conference on Information and Knowledge Management (CIKM '20), October 19--23, 2020

This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full model \begin{eqnarray*} \left\{ \begin{array}{lcl} \, \, \partial_t n_{\epsilon} + u_{\epsilon} \cdot \nabla n_{\epsilon} &=& \Delta n_{\epsilon} - \nabla \cdot \Big( n_{\epsilon} S(x, n_{\epsilon}, c_{\epsilon})\cdot\nabla c_{\epsilon}\Big) + f(x, n_{\epsilon}, c_{\epsilon}), \\[1mm] \epsilon \partial_t c_{\epsilon} + u_{\epsilon}\cdot\nabla c_{\epsilon} &=& \Delta c_{\epsilon} - c_{\epsilon} + n_{\epsilon} , \\[1mm] \,\,\partial_t u_{\epsilon} + \kappa (u_{\epsilon}\cdot\nabla) u_{\epsilon} &=& \Delta u_{\epsilon} + \nabla P_{\epsilon} + n_{\epsilon} \nabla\phi, \qquad \nabla\cdot u_{\epsilon}=0 \end{array} \right. \end{eqnarray*} to solutions of the parabolic-elliptic counterpart formally obtained on taking $\epsilon\searrow 0$. In smoothly bounded physical domains $\Omega\subset {\mathbb R}^{N}$ with $N\ge 1$, and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on $\nabla c_{\epsilon}$ and $u_{\epsilon}$ are bounded in $L^\lambda((0,T);L^q(\Omega))$ and in $L^\infty((0,T);L^r(\Omega))$, respectively, for some $\lambda\in (2,\infty]$, $q>N$ and $r>\max\{2,N\}$ such that $\frac{1}{\lambda}+\frac{N}{2q}<\frac{1}{2}$. To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system., Comment: 40 pages

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{array} \end{eqnarray*} in $N$-dimensional bounded smooth domains for suitably regular positive initial data. We shall establish the existence of a global bounded classical solution for suitably large $\mu$ and prove that for any $\mu>0$ there exists a weak solution. Moreover, in the case of $\kappa>0$ convergence to the constant equilibrium $(\frac{\kappa}{\mu},0)$ is shown.

In [7] we obtained a formula for the Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring by relating it to the Hilbert depth of powers of the irrelevant maximal ideal. In this paper, we prove that these two Hilbert depth formulas are equivalent to each other. Our result reveals that there is a strong connection between these two classes of seemingly unrelated monomial ideals. We conjecture that their Stanley depths are equivalent as well., Comment: 7 pages

It is well known that Thom-Boardman symbols are realized by non-increasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any non-increasing sequence of nonnegative integers, there is a map-germ with the prescribed sequence as its Thom-Boardman symbol., Comment: 5 pages, no figures

In this paper, we obtain explicit formulas for the Hilbert series and Hilbert depth of squarefree Veronese ideals in a standard graded polynomial ring., Comment: 7 pages, a gap in the previous version is fixed