Nilpotence order growth of recursion operators in characteristic p

We prove that the killing rate of certain degree-lowering "recursion operators" on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big mod-p Hecke algebra in the genus-zero case. We sketch the application for p=2 and p=3 in level one. The case p=2 was first established in by Nicolas and Serre in 2012 using different methods

Mod-2 dihedral Galois representations of prime conductor

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida

Mod-2 dihedral Galois representations of prime conductor

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.Comment: 16 pages; v2: final submitted versio

Efficient Graph Laplacian Estimation by Proximal Newton

The Laplacian-constrained Gaussian Markov Random Field (LGMRF) is a common multivariate statistical model for learning a weighted sparse dependency graph from given data. This graph learning problem can be formulated as a maximum likelihood estimation (MLE) of the precision matrix, subject to Laplacian structural constraints, with a sparsity-inducing penalty term. This paper aims to solve this learning problem accurately and efficiently. First, since the commonly used $\ell_1$-norm penalty is inappropriate in this setting and may lead to a complete graph, we employ the nonconvex minimax concave penalty (MCP), which promotes sparse solutions with lower estimation bias. Second, as opposed to existing first-order methods for this problem, we develop a second-order proximal Newton approach to obtain an efficient solver, utilizing several algorithmic features, such as using Conjugate Gradients, preconditioning, and splitting to active/free sets. Numerical experiments demonstrate the advantages of the proposed method in terms of both computational complexity and graph learning accuracy compared to existing methods

Deep congruences + the Brauer-Nesbitt theorem

We prove that mod-$p$ congruences between polynomials in $\mathbb Z_p[X]$ are equivalent to deeper mod-$p^{1+v_p(n)}$ congruences between the $n^{\rm th}$ power-sum functions of their roots. We give two proofs, one combinatorial and one algebraic. This result generalizes to torsion-free $\mathbb Z_{(p)}$-algebras modulo divided-power ideals. As a direct consequence, we obtain a refinement of the Brauer-Nesbitt theorem for finite free $\mathbb Z_p$-modules with an action of a single linear operator, with applications to the study of Hecke modules of mod-$p$ modular forms

An algorithm for orienting graphs based on cause-effect pairs and its applications to orienting protein networks.

Acknowledgments: I would like to thank my thesis advisor, Prof. Roded Sharan, for the initial idea and the excellent guidance throughout the research. I would like to thank Prof. Uri Zwick and Prof. Vineet Bafna for substantial contribution to this work and for co-authoring the paper, upon which this thesis is based. I also thank Andreas Beyer and Silpa Suthram for providing the kinase-substrate data, Oved Ourfali for his help with Integer Programming implementation, and Rani Hod for his help with some theoretical issues. Abstract In recent years we have seen a vast increase in the amount of protein-protein interaction data. Study of the resulting biological networks can provide us a better understanding of the processes taking place within a cell. In this work we consider a graph orientation problem arising in the study of biological networks. Given an undirected graph and a list of ordered source-target pairs, the goal is to orient the graph so that a maximum number of pairs will admit a directed path from the source to the target. We show that the problem is NP-hard and hard to approximate to within a constant ratio. We then study restrictions of the problem to various graph classes, and provide an O(log n) approximation algorithm for the general case. We show that this algorithm achieves very tight approximation ratios in practice and is able to infer edge directions with high accuracy on both simulated and real network data

Big images of two-dimensional pseudorepresentations

Bella\"iche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation $\rho$ of a profinite group on a local pro-$p$ domain $A$ contains a nontrivial congruence subgroup of ${\rm SL}_2(B)$ for a certain subring $B$ of $A$. We enlarge Bella\"iche's ring and give this new $B$ a conceptual interpretation in terms of conjugate self-twists of $\rho$, symmetries that naturally constrain its image. As a corollary, this new $B$ is optimal among congruence subgroups contained in the image. We also interpret the new $B$ vis-a-vis the adjoint trace ring of $\rho$, which we show is a more natural ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for ${\rm GL}_2$ Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and $p$-adic Hida or Coleman families of elliptic and Hilbert modular forms

Dual array EEG-fMRI : An approach for motion artifact suppression in EEG recorded simultaneously with fMRI

Objective: Although simultaneous recording of EEG and MRI has gained increasing popularity in recent years, the extent of its clinical use remains limited by various technical challenges. Motion interference is one of the major challenges in EEG-fMRI. Here we present an approach which reduces its impact with the aid of an MR compatible dual-array EEG (daEEG) in which the EEG itself is used both as a brain signal recorder and a motion sensor. Methods: We implemented two arrays of EEG electrodes organized into two sets of nearly orthogonally intersecting wire bundles. The EEG was recorded using referential amplifiers inside a 3 T MR-scanner. Virtual bipolar measurements were taken both along bundles (creating a small wire loop and therefore minimizing artifact) and across bundles (creating a large wire loop and therefore maximizing artifact). Independent component analysis (ICA) was applied. The resulting ICA components were classified into brain signal and noise using three criteria: 1) degree of two-dimensional spatial correlation between ICA coefficients along bundles and across bundles; 2) amplitude along bundles vs. across bundles; 3) correlation with ECG. The components which passed the criteria set were transformed back to the channel space. Motion artifact suppression and the ability to detect interictal epileptic spikes following daEEG and Optimal Basis Set (OBS) procedures were compared in 10 patients with epilepsy. Results: The SNR achieved by daEEG was 11.05 +/- 3.10 and by OBS was 8.25 +/- 1.01 (p <0.00001). In 9 of 10 patients, more spikes were detected after daEEG than after OBS (p <0.05). Significance: daEEG improves signal quality in EEG-fMRI recordings, expanding its clinical and research potential. (C) 2016 Elsevier Inc. All rights reserved.Peer reviewe