New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q-positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.Comment: 18 page

An additive subfamily of enlargements of a maximally monotone operator

We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical $\epsilon$-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the $\epsilon$-subdifferential enlargement

Closedness type regularity conditions for surjectivity results involving the sum of two maximal monotone operators

In this note we provide regularity conditions of closedness type which guarantee some surjectivity results concerning the sum of two maximal monotone operators by using representative functions. The first regularity condition we give guarantees the surjectivity of the monotone operator $S(\cdot + p)+T(\cdot)$, where $p\in X$ and $S$ and $T$ are maximal monotone operators on the reflexive Banach space $X$. Then, this is used to obtain sufficient conditions for the surjectivity of $S+T$ and for the situation when $0$ belongs to the range of $S+T$. Several special cases are discussed, some of them delivering interesting byproducts.Comment: 11 pages, no figure

A convergent algorithm for the hybrid problem of reconstructing conductivity from minimal interior data

We consider the hybrid problem of reconstructing the isotropic electric conductivity of a body $\Omega$ from interior Current Density Imaging data obtainable using MRI measurements. We only require knowledge of the magnitude $|J|$ of one current generated by a given voltage $f$ on the boundary $\partial\Omega$. As previously shown, the corresponding voltage potential u in $\Omega$ is a minimizer of the weighted least gradient problem $$u=\hbox{argmin} \{\int_{\Omega}a(x)|\nabla u|: u \in H^{1}(\Omega), \ \ u|_{\partial \Omega}=f\},$$ with $a(x)= |J(x)|$. In this paper we present an alternating split Bregman algorithm for treating such least gradient problems, for $a\in L^2(\Omega)$ non-negative and $f\in H^{1/2}(\partial \Omega)$. We give a detailed convergence proof by focusing to a large extent on the dual problem. This leads naturally to the alternating split Bregman algorithm. The dual problem also turns out to yield a novel method to recover the full vector field $J$ from knowledge of its magnitude, and of the voltage $f$ on the boundary. We then present several numerical experiments that illustrate the convergence behavior of the proposed algorithm

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

The membrane-spanning 4-domains, subfamily A (MS4A) gene cluster contains a common variant associated with Alzheimer's disease

Background\ud In order to identify novel loci associated with Alzheimer's disease (AD), we conducted a genome-wide association study (GWAS) in the Spanish population.\ud \ud Methods\ud We genotyped 1,128 individuals using the Affymetrix Nsp I 250K chip. A sample of 327 sporadic AD patients and 801 controls with unknown cognitive status from the Spanish general population were included in our initial study. To increase the power of the study, we combined our results with those of four other public GWAS datasets by applying identical quality control filters and the same imputation methods, which were then analyzed with a global meta-GWAS. A replication sample with 2,200 sporadic AD patients and 2,301 controls was genotyped to confirm our GWAS findings.\ud \ud Results\ud Meta-analysis of our data and independent replication datasets allowed us to confirm a novel genome-wide significant association of AD with the membrane-spanning 4-domains subfamily A (MS4A) gene cluster (rs1562990, P = 4.40E-11, odds ratio = 0.88, 95% confidence interval 0.85 to 0.91, n = 10,181 cases and 14,341 controls).\ud \ud Conclusions\ud Our results underscore the importance of international efforts combining GWAS datasets to isolate genetic loci for complex diseases

Genome-wide association analysis of dementia and its clinical endophenotypes reveal novel loci associated with Alzheimer's disease and three causality networks : The GR@ACE project

Introduction: Large variability among Alzheimer's disease (AD) cases might impact genetic discoveries and complicate dissection of underlying biological pathways. Methods: Genome Research at Fundacio ACE (GR@ACE) is a genome-wide study of dementia and its clinical endophenotypes, defined based on AD's clinical certainty and vascular burden. We assessed the impact of known AD loci across endophenotypes to generate loci categories. We incorporated gene coexpression data and conducted pathway analysis per category. Finally, to evaluate the effect of heterogeneity in genetic studies, GR@ACE series were meta-analyzed with additional genome-wide association study data sets. Results: We classified known AD loci into three categories, which might reflect the disease clinical heterogeneity. Vascular processes were only detected as a causal mechanism in probable AD. The meta-analysis strategy revealed the ANKRD31-rs4704171 and NDUFAF6-rs10098778 and confirmed SCIMP-rs7225151 and CD33-rs3865444. Discussion: The regulation of vasculature is a prominent causal component of probable AD. GR@ACE meta-analysis revealed novel AD genetic signals, strongly driven by the presence of clinical heterogeneity in the AD series

New insights into the genetic etiology of Alzheimer's disease and related dementias

Characterization of the genetic landscape of Alzheimer's disease (AD) and related dementias (ADD) provides a unique opportunity for a better understanding of the associated pathophysiological processes. We performed a two-stage genome-wide association study totaling 111,326 clinically diagnosed/'proxy' AD cases and 677,663 controls. We found 75 risk loci, of which 42 were new at the time of analysis. Pathway enrichment analyses confirmed the involvement of amyloid/tau pathways and highlighted microglia implication. Gene prioritization in the new loci identified 31 genes that were suggestive of new genetically associated processes, including the tumor necrosis factor alpha pathway through the linear ubiquitin chain assembly complex. We also built a new genetic risk score associated with the risk of future AD/dementia or progression from mild cognitive impairment to AD/dementia. The improvement in prediction led to a 1.6- to 1.9-fold increase in AD risk from the lowest to the highest decile, in addition to effects of age and the APOE ε4 allele

Multiancestry analysis of the HLA locus in Alzheimer’s and Parkinson’s diseases uncovers a shared adaptive immune response mediated by HLA-DRB1*04 subtypes

Across multiancestry groups, we analyzed Human Leukocyte Antigen (HLA) associations in over 176,000 individuals with Parkinson’s disease (PD) and Alzheimer’s disease (AD) versus controls. We demonstrate that the two diseases share the same protective association at the HLA locus. HLA-specific fine-mapping showed that hierarchical protective effects of HLA-DRB1*04 subtypes best accounted for the association, strongest with HLA-DRB1*04:04 and HLA-DRB1*04:07, and intermediary with HLA-DRB1*04:01 and HLA-DRB1*04:03. The same signal was associated with decreased neurofibrillary tangles in postmortem brains and was associated with reduced tau levels in cerebrospinal fluid and to a lower extent with increased Aβ42. Protective HLA-DRB1*04 subtypes strongly bound the aggregation-prone tau PHF6 sequence, however only when acetylated at a lysine (K311), a common posttranslational modification central to tau aggregation. An HLA-DRB1*04-mediated adaptive immune response decreases PD and AD risks, potentially by acting against tau, offering the possibility of therapeutic avenues