On upper semicontinuity of the Allen–Cahn twisted eigenvalues

We give an asymptotic upper bound for the kth twisted eigenvalue of the linearized Allen–Cahn operator in terms of the kth eigenvalue of the Jacobi operator, taken with respect to the minimal surface arising as the asymptotic limit of the zero sets of the Allen–Cahn critical points. We use an argument based on the notion of second inner variation developed in Le (On the second inner variations of Allen–Cahn type energies and applications to local minimizers. J. Math. Pures Appl. (9) 103 (2015) 1317–1345).

Guaranteed Robust Tensor Completion via ∗L-SVD with Applications to Remote Sensing Data

This paper conducts a rigorous analysis for the problem of robust tensor completion, which aims at recovering an unknown three-way tensor from incomplete observations corrupted by gross sparse outliers and small dense noises simultaneously due to various reasons such as sensor dead pixels, communication loss, electromagnetic interferences, cloud shadows, etc. To estimate the underlying tensor, a new penalized least squares estimator is first formulated by exploiting the low rankness of the signal tensor within the framework of tensor ∗L-Singular Value Decomposition (∗L-SVD) and leveraging the sparse structure of the outlier tensor. Then, an algorithm based on the Alternating Direction Method of Multipliers (ADMM) is designed to compute the estimator in an efficient way. Statistically, the non-asymptotic upper bound on the estimation error is established and further proved to be optimal (up to a log factor) in a minimax sense. Simulation studies on synthetic data demonstrate that the proposed error bound can predict the scaling behavior of the estimation error with problem parameters (i.e., tubal rank of the underlying tensor, sparsity of the outliers, and the number of uncorrupted observations). Both the effectiveness and efficiency of the proposed algorithm are evaluated through experiments for robust completion on seven different types of remote sensing data.

Cops and robbers on graphs and hypergraphs

Cops and Robbers is a vertex-pursuit game played on a graph where a set of cops attempts to capture a robber. Meyniel's Conjecture gives as asymptotic upper bound on the cop number, the number of cops required to win on a connected graph. The incidence graphs of affine planes meet this bound from below, they are called Meyniel extremal. The new parameters mқ and Mқ describe the minimum orders of k-cop-win graphs. The relation of these parameters to Meyniel's Conjecture is discussed. Further, the cop number for all connected graphs of order 10 or less is given. Finally, it is shown that cop win hypergraphs, a generalization of graphs, cannot be characterized in terms of retractions in the same manner as cop win graphs. This thesis presents some small steps towards a solution to Meyniel's Conjecture.

Cops and robbers on graphs and hypergraphs

Cops and Robbers is a vertex-pursuit game played on a graph where a set of cops attempts to capture a robber. Meyniel's Conjecture gives as asymptotic upper bound on the cop number, the number of cops required to win on a connected graph. The incidence graphs of affine planes meet this bound from below, they are called Meyniel extremal. The new parameters mқ and Mқ describe the minimum orders of k-cop-win graphs. The relation of these parameters to Meyniel's Conjecture is discussed. Further, the cop number for all connected graphs of order 10 or less is given. Finally, it is shown that cop win hypergraphs, a generalization of graphs, cannot be characterized in terms of retractions in the same manner as cop win graphs. This thesis presents some small steps towards a solution to Meyniel's Conjecture.

On the number of distinct k-decks: Enumeration and bounds

<p style='text-indent:20px;'>The <i><inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-deck</i> of a sequence is defined as the multiset of all its subsequences of length <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>. Let <inline-formula><tex-math id="M4">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> denote the number of distinct <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula>-decks for binary sequences of length <inline-formula><tex-math id="M6">\begin{document}$ n $\end{document}</tex-math></inline-formula>. For binary alphabet, we determine the exact value of <inline-formula><tex-math id="M7">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for small values of <inline-formula><tex-math id="M8">\begin{document}$ k $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ n $\end{document}</tex-math></inline-formula>, and provide asymptotic estimates of <inline-formula><tex-math id="M10">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M11">\begin{document}$ k $\end{document}</tex-math></inline-formula> is fixed.</p><p style='text-indent:20px;'>Specifically, for fixed <inline-formula><tex-math id="M12">\begin{document}$ k $\end{document}</tex-math></inline-formula>, we introduce a trellis-based method to compute <inline-formula><tex-math id="M13">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> in time polynomial in <inline-formula><tex-math id="M14">\begin{document}$ n $\end{document}</tex-math></inline-formula>. We then compute <inline-formula><tex-math id="M15">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M16">\begin{document}$ k \in \{3,4,5,6\} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ k \leqslant n \leqslant 30 $\end{document}</tex-math></inline-formula>. We also improve the asymptotic upper bound on <inline-formula><tex-math id="M18">\begin{document}$ D_k(n) $\end{document}</tex-math></inline-formula>, and provide a lower bound thereupon. In particular, for binary alphabet, we show that <inline-formula><tex-math id="M19">\begin{document}$ D_k(n) = O\bigl(n^{(k-1)2^{k-1}+1}\bigr) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M20">\begin{document}$ D_k(n) = \Omega(n^k) $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M21">\begin{document}$ k = 3 $\end{document}</tex-math></inline-formula>, we moreover show that <inline-formula><tex-math id="M22">\begin{document}$ D_3(n) = \Omega(n^6) $\end{document}</tex-math></inline-formula> while the upper bound on <inline-formula><tex-math id="M23">\begin{document}$ D_3(n) $\end{document}</tex-math></inline-formula> is <inline-formula><tex-math id="M24">\begin{document}$ O(n^9) $\end{document}</tex-math></inline-formula>.</p>

Functional ARCH and GARCH Models: A Yule-Walker Approach

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

Estimates for the Finite-Time Ruin Probability of a Time-Dependent Risk Model with a Brownian Perturbation

In this paper, we consider a time-dependent risk model with a Brownian perturbation. In this model, there is a dependence structure between the claim sizes and their corresponding interarrival times. Assuming the claim sizes have subexponential distributions, we obtain the asymptotic lower bound of the finite-time ruin probability. When the claim sizes have distributions from the class L∩D, the asymptotic upper bound of the finite-time ruin probability has been presented. These results confirm that when the claim sizes are heavy-tailed, the asymptotics of the finite-time ruin probability of this time-dependent model are insensitive to the Brownian perturbation.

Functional ARCH and GARCH Models: A Yule-Walker Approach

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound of the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.

Functional ARCH and GARCH Models: A Yule-Walker Approach

Conditional heteroskedastic financial time series are commonly modelled by ARCH and GARCH. ARCH(1) and GARCH processes were recently extended to the function spaces C[0,1] and L2[0,1], their probabilistic features were studied and their parameters were estimated. The projections of the operators on finite-dimensional subspace were estimated, as were the complete operators in GARCH(1,1). An explicit asymptotic upper bound for the estimation errors was stated in ARCH(1). This article provides sufficient conditions for the existence of strictly stationary solutions, weak dependence and finite moments of ARCH and GARCH processes in various Lp[0,1] spaces, C[0,1] and other spaces. In L2[0,1] we deduce explicit asymptotic upper bounds of the estimation errors for the shift term and the complete operators in ARCH and GARCH and for the projections of the operators on a finite-dimensional subspace in ARCH. The operator estimaton is based on Yule-Walker equations. The estimation of the GARCH operators also involves a result concerning the estimation of the operators in invertible, linear processes which is valid beyond the scope of ARCH and GARCH. Through minor modifications, all results in this article regarding functional ARCH and GARCH can be transferred to functional ARMA.