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

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

On intersections of compacta in Euclidean space : the metastable case

We prove the following theorem: Let f: X→Rn and g:Y→Rn be any maps of copmpacta X and Y into the Euclidean n-space Rn, n≧5. Suppose that dim(X×)0 there exist maps f\u27:X→Rn and g\u27:Y→Rn sucha that d(f,f\u27)<ε, d(g,g\u27)<ε and f\u27(X)∩g\u27(Y)=0

Constant sign and nodal solutions for parametric anisotropic (p,2)-equations

We consider an anisotropic (Formula presented.) -equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques, and critical groups

The Covering Homotopy Extension Problem for Compact Transformation Groups

It is shown that the orbit space of universal (in the sense of Palais) G-spaces classifies G-spaces. Theorems on the extension of covering homotopy for G-spaces and on a homotopy representation of the isovariant category ISOV are proved

Constructing near-embeddings of codimension one manifolds with countable dense singular sets

The purpose of this paper is to present, for all $n\ge 3$, very simple examples of continuous maps $f:M^{n-1} \to M^{n}$ from closed $(n-1)$-manifolds $M^{n-1}$ into closed $n$-manifold $M^n$ such that even though the singular set $S(f)$ of $f$ is countable and dense, the map $f$ can nevertheless be approximated by an embedding, i.e. $f$ is a {\sl near-embedding}

Association of Thalamic Dysconnectivity and Conversion to Psychosis in Youth and Young Adults at Elevated Clinical Risk

IMPORTANCE: Severe neuropsychiatric conditions, such as schizophrenia, affect distributed neural computations. One candidate system profoundly altered in chronic schizophrenia involves the thalamocortical networks. It is widely acknowledged that schizophrenia is a neurodevelopmental disorder that likely affects the brain before onset of clinical symptoms. However, no investigation has tested whether thalamocortical connectivity is altered in individuals at risk for psychosis or whether this pattern is more severe in individuals who later develop full-blown illness. OBJECTIVES: To determine whether baseline thalamocortical connectivity differs between individuals at clinical high risk for psychosis and healthy controls, whether this pattern is more severe in those who later convert to full-blown illness, and whether magnitude of thalamocortical dysconnectivity is associated with baseline prodromal symptom severity. DESIGN, SETTING, AND PARTICIPANTS: In this multicenter, 2-year follow-up, case-control study, we examined 397 participants aged 12-35 years of age (243 individuals at clinical high risk of psychosis, of whom 21 converted to full-blown illness, and 154 healthy controls). The baseline scan dates were January 15, 2010, to April 30, 2012. MAIN OUTCOMES AND MEASURES: Whole-brain thalamic functional connectivity maps were generated using individuals\u27 anatomically defined thalamic seeds, measured using resting-state functional connectivity magnetic resonance imaging. RESULTS: Using baseline magnetic resonance images, we identified thalamocortical dysconnectivity in the 243 individuals at clinical high risk for psychosis, which was particularly pronounced in the 21 participants who converted to full-blown illness. The pattern involved widespread hypoconnectivity between the thalamus and prefrontal and cerebellar areas, which was more prominent in those who converted to full-blown illness (t(173) = 3.77, P \u3c .001, Hedge g = 0.88). Conversely, there was marked thalamic hyperconnectivity with sensory motor areas, again most pronounced in those who converted to full-blown illness (t(173) = 2.85, P \u3c .001, Hedge g = 0.66). Both patterns were significantly correlated with concurrent prodromal symptom severity (r = 0.27, P \u3c 3.6 x 10(-8), Spearman rho = 0.27, P \u3c 4.75 x 10(-5), 2-tailed). CONCLUSIONS AND RELEVANCE: Thalamic dysconnectivity, resembling that seen in schizophrenia, was evident in individuals at clinical high risk for psychosis and more prominently in those who later converted to psychosis. Dysconnectivity correlated with symptom severity, supporting the idea that thalamic connectivity may have prognostic implications for risk of conversion to full-blown illness

Changes in global and thalamic brain connectivity in LSD-induced altered states of consciousness are attributable to the 5-HT2A receptor

Background:Lysergic acid diethylamide (LSD) has agonist activity at various serotonin (5-HT) and dopamine receptors. Despite the therapeutic and scientific interest in LSD, specific receptor contributions to its neurobiological effects remain unknown. Methods: We therefore conducted a double-blind, randomized, counterbalanced, cross-over study (ClinicalTrials.gov, NCT02451072) during which 24 healthy human participants received either (i) placebo+placebo, (ii) placebo+LSD (100 µg po), or (iii) Ketanserin, a selective 5-HT receptor antagonist,+LSD. We quantified resting-state functional connectivity via a data-driven global brain connectivity method and compared it to cortical gene expression maps. Findings: LSD reduced associative, but concurrently increased sensory-somatomotor brain-wide and thalamic connectivity. Ketanserin fully blocked the subjective and neural LSD effects. Whole-brain spatial patterns of LSD effects matched 5-HT receptor cortical gene expression in humans. Conclusion: Together, these results strongly implicate the 5-HT receptor in LSD's neuropharmacology. This study therefore pinpoints the critical role of 5-HT in LSD's mechanism, which informs its neurobiology and guides rational development of psychedelic-based therapeutics. Funding: Swiss National Science Foundation (SNSF, P2ZHP1_161626, KHP), the Swiss Neuromatrix Foundation (2015 - 0103, FXV), the Usona Institute (2015 - 2056, FXV), the NIH (R01MH112746, JDM; DP5OD012109, AA; R01MH108590, AA), the NIAA ( P50AA012870-16, AA & JHK), the NARSAD Independent Investigator Grant (AA), the Yale CTSA grant (UL1TR000142 Pilot Award, AA), and the Slovenian Research Agency (ARRS J7-6829 & ARRS J7-8275, GR)

Computational modelling of EEG and fMRI paradigms indicates a consistent loss of pyramidal cell synaptic gain in schizophrenia

Background Diminished synaptic gain – the sensitivity of postsynaptic responses to neural inputs – may be a fundamental synaptic pathology in schizophrenia. Evidence for this is indirect, however. Furthermore, it is unclear whether pyramidal cells or interneurons (or both) are affected, or how these deficits relate to symptoms. Methods Participants with schizophrenia diagnoses (PScz, n=108), their relatives (n=57), and controls (n=107) underwent three electroencephalography (EEG) paradigms – resting, mismatch negativity, and 40 Hz auditory steady-state response – and resting functional magnetic resonance imaging. Dynamic causal modelling was used to quantify synaptic connectivity in cortical microcircuits. Results Classic group differences in EEG features between PScz and controls were replicated, including increased theta and other spectral changes (resting EEG), reduced mismatch negativity, and reduced 40 Hz power. Across all four paradigms, characteristic PScz data features were all best explained by models with greater self-inhibition (decreased synaptic gain), in pyramidal cells. Furthermore, disinhibition in auditory areas predicted abnormal auditory perception (and positive symptoms) in PScz, in three paradigms. Conclusions First, characteristic EEG changes in PScz in three classic paradigms are all attributable to the same underlying parameter change: greater self-inhibition in pyramidal cells. Second, psychotic symptoms in PScz relate to disinhibition in neural circuits. These findings are more commensurate with the hypothesis that in PScz, a primary loss of synaptic gain on pyramidal cells is then compensated by interneuron downregulation (rather than the converse). They further suggest that psychotic symptoms relate to this secondary downregulation