#algebraic-geometry

[wassupblucoat]

So apparently the thing that you’re supposed to do before you’re allowed into the algebraic geometry club is to do every exercise in Atiyah and MacDonald.  I didn’t want to do my actual homework today so I just worked on that.  At least the book is short so I might actually finish it.

[wjmolina]

Geometry Conundrums

I recently enjoyed this exercise:

Is \(H^1\left(\mathbb C^*,\mathbb Z\right)=\mathbb Z\)? (Čech Cohomology)

Hints:

If \(X\) is a contractible topological space, then \(H^i\left(X,\mathbb Z\right)=0\) for all \(i\in\left\{1,2,\dots\right\}\).

Leray’s Theorem

Here is another fun one:

Show that the Picard group \(\text{Pic}(\Delta)=H^1(\Delta,\mathcal O_\Delta^*)\) of holomorphic line bundles on the unit disk \(\Delta\) is trivial.

Hint:

Behnke, Stein, and Cartan showed that if \(X\) is a connected, non-compact Riemann surface, then \(H^1(X,\mathcal O_X)=H^2(X,\mathcal O_X)=0\).

[arxiv-math]

Authors: Joseph Salmon, Rebecca Willett, Ery Arias-Castro

This paper describes a simple image noise removal method which combines a preprocessing step with the Yaroslavsky filter for strong numerical, visual, and theoretical performance on a broad class of images. The framework developed is a two-stage approach. In the first stage the image is filtered with a classical denoising method (e.g., wavelet or curvelet thresholding). In the second stage a modification of the Yaroslavsky filter is performed on the original noisy image, where the weights of the filters are governed by pixel similarities in the denoised image from the first stage. Similar prefiltering ideas have proved effective previously in the literature, and this paper provides theoretical guarantees and important insight into why prefiltering can be effective. Empirically, this simple approach achieves very good performance for cartoon images, and can be computed much more quickly than current patch-based denoising algorithms.

[arxiv-math]

Authors: Mingyu Guo, Vincent Conitzer

We study a more powerful variant of false-name manipulation in Internet auctions: an agent can submit multiple false-name bids, but then, once the allocation and payments have been decided, withdraw some of her false-name identities (have some of her false-name identities refuse to pay). While these withdrawn identities will not obtain the items they won, their initial presence may have been beneficial to the agent's other identities. We define a mechanism to be false-name-proof with withdrawal (FNPW) if the aforementioned manipulation is never beneficial. FNPW is a stronger condition than false-name-proofness (FNP).

[arxiv-math]

Authors: Amol Sasane

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-\pi,\pi) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(\pi (t -n)))/(\pi (t-n)) (t in R). In this article we prove a generalization of this result under the assumption that F is a compactly supported distribution with its support contained in (-\pi,\pi).

[arxiv-math]

Authors: Pierre-Malo Deni'elou, Nobuko Yoshida, Andi Bejleri, Raymond Hu

For many application-level distributed protocols and parallel algorithms, the set of participants, the number of messages or the interaction structure are only known at run-time. This paper proposes a dependent type theory for multiparty sessions which can statically guarantee type-safe, deadlock-free multiparty interactions among processes whose specifications are parameterised by indices. We use the primitive recursion operator from G"odel's System T to express a wide range of communication patterns while keeping type checking decidable. To type individual distributed processes, a parameterised global type is projected onto a generic generator which represents a class of all possible end-point types. We prove the termination of the type-checking algorithm in the full system with both multiparty session types and recursive types. We illustrate our type theory through non-trivial programming and verification examples taken from parallel algorithms and Web services usecases.

[arxiv-math]

Authors: Lev Kazakovtsev

In this paper, we consider an approach to the parallelizing of the algorithms realizing the modified probability changigng method with adaptation and partial rollback procedure for constrained pseudo-Boolean optimization problems. Existing optimization algorithms are adapted for the shared memory and clusters (PVM library). The parallel efficiency is estimated for the lagre-scale non-linear pseudo-Boolean optimization problems with linear constraints. Initially designed for unconstrained optimization, the probability changing method (MIVER) allows us finding the approximate solution of different linear and non-linear pseudo-Boolean optimization problems with constraints. Although, in case of large-scale problems, the computational demands are also very high and the precision of the result depends on the time spent. In case of the constrained optimization problem, even the search of any permissibly solution can take very large computational resources. The rapid development of the parallel processor systems which are often implemented even in the computer systems designed for home use allows to reduce significantly the time spent to find the acceptable solution with a speed-up close to ideal.

[arxiv-math]

Authors: Zai Yang, Lihua Xie, Cishen Zhang

Sparse Bayesian learning (SBL) is a popular approach to sparse signal recovery in compressed sensing (CS). In SBL, the signal sparsity information is exploited by assuming a sparsity-inducing prior for the signal that is then estimated using Bayesian inference. In this paper, a new sparsity-inducing prior is introduced and efficient algorithms are developed for signal recovery. The main algorithm is shown to produce a sparser solution than existing SBL methods while preserving their desirable properties. Numerical simulations with one-dimensional synthetic signals and two-dimensional images verify our analysis and show that for sparse signals the proposed algorithm outperforms its SBL peers in both the signal recovery accuracy and computational speed. Its improved performance is also demonstrated in comparison with other state-of-the-art methods in CS.