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Functional magnetic resonance imaging (fMRI) functional connectivity between brain regions is often computed using parcellations defined by functional or structural atlases. Typically, some kind of voxel averaging is performed to obtain a single temporal correlation estimate per region pair. However, several estimators can be defined for this task, with various assumptions and degrees of robustness to local noise, global noise, and region size. In this paper, we systematically present and study the properties of 9 different functional connectivity estimators taking into account the spatial structure of fMRI data, based on a simple fMRI data spatial model. These include 3 existing estimators and 6 novel estimators. We demonstrate the empirical properties of the estimators using synthetic, animal, and human data, in terms of graph structure, repeatability and reproducibility, discriminability, dependence on region size, as well as local and global noise robustness. We prove analytically the link between regional intra-correlation and inter-region correlation, and show that the choice of estimator has a strong influence on inter-correlation values. Some estimators, including the commonly used correlation of averages (ca), are positively biased, and have more dependence to region size and intra-correlation than robust alternatives, resulting in spatially-dependent bias. We define the new local correlation of averages estimator with better theoretical guarantees, lower bias, significantly lower dependence on region size (Spearman correlation 0.40 vs 0.55, paired t-test T=27.2, p = 1. 1 e − 47 ), at negligible cost to discriminative power, compared to the ca estimator. The difference in connectivity pattern between the estimators is not distributed uniformly throughout the brain, but rather shows a clear ventral-dorsal gradient, suggesting that region size and intra-correlation plays an important role in shaping functional networks defined using the ca estimator, and leading to non-trivial differences in their connectivity structure. We provide an open source R package and equivalent Python implementation to facilitate the use of the new estimators, together with preprocessed rat time-series. • Most atlas-based functional connectivity estimates use the correlation of averages. • Six novel estimators are derived using a spatio-temporal signal and noise model • Synthetic, rat and Human Connectome Project data are used to validate estimators • Lower dependency on region size, intra-correlation shown analytically and empirically • Improved reproducibility on graph metrics using freely available codes, datasets [ABSTRACT FROM AUTHOR]

Abstract: We study the equality problem for infinite words obtained by iterating morphisms. In particular, we give a practical algorithm to decide whether or not two words generated by primitive morphisms are equal. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption). [Copyright &y& Elsevier]

Abstract: Let be nonempty subsets of the group of integers modulo a prime p. If , then at least different residue classes can be represented as , where , and . This result complements the solution of a problem of Erdős and Heilbronn obtained by Alon, Nathanson, and Ruzsa. [Copyright &y& Elsevier]

Abstract: Let be a union-closed family of subsets of an m-element set A. Let and for let denote the number of sets in that contain a. Frankl''s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element with . Strengthening a result of Gao and Yu [W. Gao, H. Yu, Note on the union-closed sets conjecture, Ars Combin. 49 (1998) 280–288] we verify the conjecture for the particular case when and . Moreover, for these “large” families we prove an even stronger version via averaging. Namely, the sum of the , for all , is shown to be non-positive. Notice that this stronger version does not hold for all union-closed families; however we conjecture that it holds for a much wider class of families than considered here. Although the proof of the result is based on elementary lattice theory, the paper is self-contained and the reader is not assumed to be familiar with lattices. [Copyright &y& Elsevier]

Abstract: [E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the -Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon''s theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result. [Copyright &y& Elsevier]

Abstract: We show that for any metric space X the condition where is the Menger curvature of the triple and is the 1-dimensional Hausdorff measure on X, guarantees that X is rectifiable. [Copyright &y& Elsevier]

Abstract: In this paper the concept of a contraction for multi-valued mappings in a metric space is introduced and the existence theorems for fixed points of such contractions in a complete metric space are proved. Presented results generalize and improve the recent results of Y. Feng, S. Liu [Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103–112], D. Klim, D. Wardowski [D. Klim, D. Wardowski, Fixed point theorems for set-valued contractions in complete metric spaces, J. Math. Anal. Appl. 334 (2007) 132–139] and several others. The method used in the proofs of our results is new and is simpler than methods used in the corresponding papers. Two examples are given to show that our results are genuine generalization of the results of Feng and Liu and Klim and Wardowski. [Copyright &y& Elsevier]

Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic . It is proved in this paper that if the p-envelope of adL in DerL contains a torus of maximal dimension whose centralizer in adL acts nontriangulably on L, then and L is isomorphic to one of the Melikian algebras . In conjunction with [A. Premet, H. Strade, Simple Lie algebras of small characteristic V. The non-Melikian case, J. Algebra 314 (2007) 664–692, Theorem 1.2], this implies that, up to isomorphism, any finite-dimensional simple Lie algebra over an algebraically closed field of characteristic is either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5. This result finally settles the classification problem for finite-dimensional simple Lie algebras over algebraically closed fields of characteristic . [Copyright &y& Elsevier]

Abstract: Polycrystalline strontium–europium hydroxovanadates Sr10− x Eu x (VO4)6(OH)2− x O x were synthesized and studied by X-ray powder diffraction, infrared absorption and diffuse-reflectance spectroscopies and also thermogravimetry. The single-phase region of isomorphous substitution of Eu for Sr in hydroxovanadate has been found to lie in the compositional interval 0⩽x⩽0.18 at 800°C. Refinement of X-ray diffraction (XRD) patterns by the Rietveld method shows that Eu substitutes for Sr preferentially at the Sr(1) sites of the apatite structure. The substitution results in a more uniform distribution of Sr–O bond lengths in coordination polyhedra both around Sr (1) and Sr (2) sites and also in a slight contraction of the vanadate anion VO4 3−. Synthetic Sr hydroxovanadates absorb a considerable amount of water in two forms: capillary condensed and chemically combined. The latter form may be absorbed at temperatures as high as 500°C. [Copyright &y& Elsevier]

Abstract: We prove that the metric characterization of real normed spaces obtained by T. Oikhberg and H. Rosenthal can be obtained without a continuity assumption provided that the space is at least two-dimensional. In order to get this improvement we first need to understand the exceptional one-dimensional case. [Copyright &y& Elsevier]

Abstract: Let C be a smooth projective irreducible curve over an algebraic closed field k of characteristic 0. We consider Brill–Noether loci over the moduli space of morphisms from C to a Grassmannian of m-planes in and the corresponding Quot schemes of quotients of a trivial vector bundle on C compactifying the spaces of morphisms. We study in detail the case in which , . We prove results on the irreducibility and dimension of these Brill–Noether loci and we address explicit formulas for their cohomology classes. We study the existence problem of these spaces which is closely related with the problem of classification of vector bundles over curves. [Copyright &y& Elsevier]

Abstract: Let R be a nonunital ring. A left R-module M is said to be firm if given by is an isomorphism. The category of firm left R-modules generalizes the usual category of unital modules for a unital ring and it has been used to study the Morita Theory for nonunital rings. It is an open problem if the category of firm modules is an abelian category. In this paper, we prove that, in general, this category is not abelian. [Copyright &y& Elsevier]

Abstract: We classify Moufang loops of order 64 and 81 up to isomorphism, using a linear algebraic approach to central loop extensions. In addition to the 267 groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In addition to the 15 groups of order 81, there are 5 nonassociative Moufang loops of order 81, 2 of which are commutative. [Copyright &y& Elsevier]

Abstract: A 2-cell embedding of a graph in an orientable closed surface is called regular if its automorphism group acts regularly on arcs of the embedded graph. The aim of this and of the associated consecutive paper is to give a classification of regular embeddings of complete bipartite graphs , where . The method involves groups which factorize as a product of two cyclic groups of order so that the two cyclic factors are transposed by an involutory automorphism. In particular, we give a classification of such groups . Employing the classification we investigate automorphisms of these groups, resulting in a classification of regular embeddings of based on that for . We prove that given (for ), there are, up to map isomorphism, exactly regular embeddings of . Our analysis splits naturally into two cases depending on whether the group is metacyclic or not. [Copyright &y& Elsevier]

Abstract: In some games, the impact of higher-order uncertainty is very large, implying that present economic theories may rely critically on the strong common knowledge assumptions they make. Focusing on normal-form games in which the players'' action spaces are compact metric spaces, we show that our key condition, called “global stability under uncertainty,” implies that the maximum change in equilibrium actions due to changes in players'' beliefs at orders higher than k is exponentially decreasing in k. Therefore, given any need for precision, we can approximate equilibrium actions by specifying only finitely many orders of beliefs. [Copyright &y& Elsevier]

Abstract: We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov''s criterion for an integral functor to be fully faithful. We also prove that the original condition of characteristic zero cannot be removed by providing a counterexample in positive characteristic. We contemplate a criterion for equivalence as well. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre is also fully faithful. These results imply the invertibility of the usual relative Fourier–Mukai transform for an elliptic fibration as a direct corollary. [Copyright &y& Elsevier]

Abstract: We define a q-chromatic function and q-dichromate on graphs and compare it with existing graph functions. Then we study in more detail the class of general chordal graphs. This is partly motivated by the graph isomorphism problem. Finally we relate the q-chromatic function to the colored Jones function of knots. This leads to a curious expression of the colored Jones function of a knot diagram as a chromatic operator applied to a power series whose coefficients are linear combinations of long chord diagrams. Chromatic operators are directly related to weight systems by the work of Chmutov, Duzhin, Lando and Noble, Welsh. [Copyright &y& Elsevier]

Abstract: We prove that the Kazhdan–Lusztig polynomials are combinatorial invariants for intervals up to length 8 in Coxeter groups of type A and up to length 6 in Coxeter groups of type B and D. As a consequence of our methods, we also obtain a complete classification, up to isomorphism, of Bruhat intervals of length 7 in type A and of length 5 in types B and D, which are not lattices. [Copyright &y& Elsevier]

Abstract: We study the multiple-sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. By casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case. We further derive and analyze a perturbed projection method for the multiple-sets split feasibility problem and, additionally, furnish alternative proofs to two known results. [Copyright &y& Elsevier]

Abstract: The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings, J. Math. Anal. Appl. 192 (1995), 655–666]. [Copyright &y& Elsevier]

Abstract: In this paper, a new method for handling multicriteria fuzzy decision-making problems based on intuitionistic fuzzy sets is presented. The proposed method allows the degrees of satisfiability and non-satisfiability of each alternative with respect to a set of criteria to be represented by intuitionistic fuzzy sets, respectively. Furthermore, the proposed method allows the decision-maker to assign the degrees of membership and non-membership of the criteria to the fuzzy concept “importance.” The method proposed here can provide a useful way to efficiently help the decision-maker to make his decision. [Copyright &y& Elsevier]

Abstract: An algebraic deformation theory of coalgebra morphisms is constructed. [Copyright &y& Elsevier]

Abstract: Valdivia invented a nondistinguished Fréchet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Köthe''s original nondistinguished Fréchet space serves the same purpose. Indeed, a Fréchet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a -space is quasibarrelled if and only if its tightness is countable. This answers a Cascales/Kąkol/Saxon question and leads to a rich supply of -spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces for κ a cardinal of uncountable cofinality. Our level of generality rises above - or even dual metric spaces to Cascales/Orihuela''s class . The small cardinals and invite a novel analysis of the Grothendieck/Köthe example, and are useful throughout. [Copyright &y& Elsevier]

Abstract: This paper presents some fundamental aspects of the design and the implementation of an automated prover for Zermelo–Fraenkel set theory within the Theorema system. The method applies the “Prove–Compute–Solve” paradigm as its major strategy for generating proofs in a natural style for statements involving constructs from set theory. [Copyright &y& Elsevier]

Traumatic events have been proposed to be associated with hypo-activity of the hypothalamic-pituitary-adrenal (HPA) axis, but data in animal models exposed to severe stressors are controversial and have important methodological concerns. Individual differences in resting or stress levels of corticosterone might explain some of the inconsistencies. We then studied this issue in male rats exposed to 2 h immobilization on boards (IMO), a severe stressor. Thirty-six rats were blood sampled under resting conditions four times a day on three non-consecutive days. Then, they were assigned to control (n = 14) or IMO (n = 22) to study the HPA response to IMO, the stressor-induced alterations in the circadian pattern of corticosterone (CPCORT), and the behavioral and HPA responsiveness to an open-field. Individual differences in pre-IMO resting corticosterone were inconsistent, but averaging data markedly improved consistency. The CPCORT was markedly altered on day 1 post-IMO (higher trough and lower peak levels), less altered on day 3 and apparently normal on day 7. Importantly, when rats were classified in low and high resting corticosterone groups (LCORT and HCORT, respectively), on the basis of the area under the curve (AUC) of the averaged pre-IMO data, AUC differences between LCORT and HCORT groups were maintained in controls but disappeared in IMO rats during the post-IMO week. Open-field hypo-activity and corticosterone sensitization were similar in LCORT and HCORT groups nine days after IMO. A single IMO exposure causes long-lasting HPA alterations, some of them dependent on pre-stress resting corticosterone levels, with no evidence for post-IMO resting corticosterone hypo-activity. • Aggregating corticosterone data from several days improves consistency. • Acute immobilization stress alters circadian corticosterone pattern. • The effects of immobilization depend on pre-stress resting levels. • Immobilization caused long-lasting behavioral and endocrine sensitization. [ABSTRACT FROM AUTHOR]