DelayAndPeriodicity

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be splitted into two parts: pseudo-continuous and strongly unstable. The pseudo-continuous part of the spectrum mediates destabilization of periodic solutions.Comment: 24 pages, 9 figure

On the stability of periodic orbits in delay equations with large delay

We prove a necessary and sufficient criterion for the exponential stability of periodic solutions of delay differential equations with large delay. We show that for sufficiently large delay the Floquet spectrum near criticality is characterized by a set of curves, which we call asymptotic continuous spectrum, that is independent on the delay.Comment: postprint versio

Optimum ground states for spin-32\frac{3}{2} chains

We present a set of {\em optimum ground states} for a large class of spin-$\frac{3}{2}$ chains. Such global ground states are simultaneously ground states of the local Hamiltonian, i.e. the nearest neighbour interaction in the present case. They are constructed in the form of a matrix product. We find three types of phases, namely a {\em weak antiferromagnet}, a {\em weak ferromagnet}, and a {\em dimerized antiferromagnet}. The main physical properties of these phases are calculated exactly by using a transfer matrix technique, in particular magnetization and two spin correlations. Depending on the model parameters, they show a surprisingly rich structure.Comment: LaTeX, 22 pages, 6 embedded Postscript figure

The EMIF-AD Multimodal Biomarker Discovery study: design, methods and cohort characteristics.

There is an urgent need for novel, noninvasive biomarkers to diagnose Alzheimer's disease (AD) in the predementia stages and to predict the rate of decline. Therefore, we set up the European Medical Information Framework for Alzheimer's Disease Multimodal Biomarker Discovery (EMIF-AD MBD) study. In this report we describe the design of the study, the methods used and the characteristics of the participants. Participants were selected from existing prospective multicenter and single-center European studies. Inclusion criteria were having normal cognition (NC) or a diagnosis of mild cognitive impairment (MCI) or AD-type dementia at baseline, age above 50 years, known amyloid-beta (Aβ) status, availability of cognitive test results and at least two of the following materials: plasma, DNA, magnetic resonance imaging (MRI) or cerebrospinal fluid (CSF). Targeted and untargeted metabolomic and proteomic analyses were performed in plasma, and targeted and untargeted proteomics were performed in CSF. Genome-wide SNP genotyping, next-generation sequencing and methylation profiling were conducted in DNA. Visual rating and volumetric measures were assessed on MRI. Baseline characteristics were analyzed using ANOVA or chi-square, rate of decline analyzed by linear mixed modeling. We included 1221 individuals (NC n = 492, MCI n = 527, AD-type dementia n = 202) with a mean age of 67.9 (SD 8.3) years. The percentage Aβ+ was 26% in the NC, 58% in the MCI, and 87% in the AD-type dementia groups. Plasma samples were available for 1189 (97%) subjects, DNA samples for 929 (76%) subjects, MRI scans for 862 (71%) subjects and CSF samples for 767 (63%) subjects. For 759 (62%) individuals, clinical follow-up data were available. In each diagnostic group, the APOE ε4 allele was more frequent amongst Aβ+ individuals (p < 0.001). Only in MCI was there a difference in baseline Mini Mental State Examination (MMSE) score between the A groups (p < 0.001). Aβ+ had a faster rate of decline on the MMSE during follow-up in the NC (p < 0.001) and MCI (p < 0.001) groups. The characteristics of this large cohort of elderly subjects at various cognitive stages confirm the central roles of Aβ and APOE ε4 in AD pathogenesis. The results of the multimodal analyses will provide new insights into underlying mechanisms and facilitate the discovery of new diagnostic and prognostic AD biomarkers. All researchers can apply for access to the EMIF-AD MBD data by submitting a research proposal via the EMIF-AD Catalog

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays (SD-DDEs), transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).Comment: minor revision, correcting mistakes in formulation of Lemma 2.3 and A.5 (which are also present in the Journal paper): center of neighborhood must be in $C^1$, which is the case for the main theore