On metric spaces with the properties of de Groot and Nagata in dimension one

A metric space $(X,d)$ has the de Groot property $GP_n$ if for any points $x_0,x_1,...,x_{n+2}\in X$ there are positive indices $i,j,k\le n+2$ such that $i\ne j$ and $d(x_i,x_j)\le d(x_0,x_k)$. If, in addition, $k\in\{i,j\}$ then $X$ is said to have the Nagata property $NP_n$. It is known that a compact metrizable space $X$ has dimension $dim(X)\le n$ iff $X$ has an admissible $GP_n$-metric iff $X$ has an admissible $NP_n$-metric. We prove that an embedding $f:(0,1)\to X$ of the interval $(0,1)$ into a locally connected metric space $X$ with property $GP_1$ (resp. $NP_1$) is open provided $f$ is an isometric embedding (resp. $f$ has distortion Dist(f)=\|f\|_\Lip\cdot\|f^{-1}\|_\Lip<2). This implies that the Euclidean metric cannot be extended from the interval $[-1,1]$ to an admissible $GP_1$-metric on the triode $T=[-1,1]\cup[0,i]$. Another corollary says that a topologically homogeneous $GP_1$-space cannot contain an isometric copy of the interval $(0,1)$ and a topological copy of the triode $T$ simultaneously. Also we prove that a $GP_1$-metric space $X$ containing an isometric copy of each compact $NP_1$-metric space has density not less than continuum.Comment: 10 page

On the second homotopy group of SC(Z)SC(Z)

In our earlier paper (K. Eda, U. Karimov, and D. Repov\v{s}, \emph{A construction of simply connected noncontractible cell-like two-dimensional Peano continua}, Fund. Math. \textbf{195} (2007), 193--203) we introduced a cone-like space $SC(Z)$. In the present note we establish some new algebraic properties of $SC(Z)$

Working memory related brain network connectivity in individuals with schizophrenia and their siblings

A growing number of studies have reported altered functional connectivity in schizophrenia during putatively “task-free” states and during the performance of cognitive tasks. However, there have been few systematic examinations of functional connectivity in schizophrenia across rest and different task states to assess the degree to which altered functional connectivity reflects a stable characteristic or whether connectivity changes vary as a function of task demands. We assessed functional connectivity during rest and during three working memory loads of an N-back task (0-back, 1-back, 2-back) among: (1) individuals with schizophrenia (N = 19); (2) the siblings of individuals with schizophrenia (N = 28); (3) healthy controls (N = 10); and (4) the siblings of healthy controls (N = 17). We examined connectivity within and between four brain networks: (1) frontal–parietal (FP); (2) cingulo-opercular (CO); (3) cerebellar (CER); and (4) default mode (DMN). In terms of within-network connectivity, we found that connectivity within the DMN and FP increased significantly between resting state and 0-back, while connectivity within the CO and CER decreased significantly between resting state and 0-back. Additionally, we found that connectivity within both the DMN and FP was further modulated by memory load. In terms of between network connectivity, we found that the DMN became significantly more “anti-correlated” with the FP, CO, and CER networks during 0-back as compared to rest, and that connectivity between the FP and both CO and CER networks increased with memory load. Individuals with schizophrenia and their siblings showed consistent reductions in connectivity between both the FP and CO networks with the CER network, a finding that was similar in magnitude across rest and all levels of working memory load. These findings are consistent with the hypothesis that altered functional connectivity in schizophrenia reflects a stable characteristic that is present across cognitive states

The behavior of solutions of a parametric weighted (p, q)-laplacian equation

We study the behavior of solutions for the parametric equation (Formula presented), under Dirichlet condition, where (Formula presented) is a bounded domain with a C2-boundary (Formula presented) are weighted versions of p-Laplacian and q-Laplacian. We prove existence and nonexistence of nontrivial solutions, when f (z, x) asymptotically as x → ±∞ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When λ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators