Frobenius polynomials for Calabi-Yau equations

Sei $\pi:X\rightarrow S$ eine \"uber $\Z$ definierte Familie von Calabi-Yau Varietaten der Dimension drei. Es existiere ein unter dem Gauss-Manin Zusammenhang invarianter Untermodul $M\subset H^3_{DR}(X/S)$ von Rang vier, sodass der Picard-Fuchs Operator $P$ auf $M$ ein sogenannter {\em Calabi-Yau } Operator von Ordnung vier ist. Sei $k$ ein endlicher K\"orper der Charaktetristik $p$, und sei $\pi_0:X_0\rightarrow S_0$ die Reduktion von $\pi$ \uber $k$. F\ur die gew\ohnlichen (ordinary) Fasern $X_{t_0}$ der Familie leiten wir eine explizite Formel zur Berechnung des charakteristischen Polynoms des Frobeniusendomorphismus, des {\em Frobeniuspolynoms}, auf dem korrespondierenden Untermodul $M_{cris}\subset H^3_{cris}(X_{t_0})$ her. Sei nun $f_0(z)$ die Potenzreihenl\osung der Differentialgleichung $Pf=0$ in einer Umgebung der Null. Da eine reziproke Nullstelle des Frobeniuspolynoms in einem Teichm\uller-Punkt $t$ durch $f_0(z)/f_0(z^p)|_{z=t}$ gegeben ist, ist ein entscheidender Schritt in der Berechnung des Frobeniuspolynoms die Konstruktion einer $p-$adischen analytischen Fortsetzung des Quotienten $f_0(z)/f_0(z^p)$ auf den Rand des $p-$adischen Einheitskreises. Kann man die Koeffizienten von $f_0$ mithilfe der konstanten Terme in den Potenzen eines Laurent-Polynoms, dessen Newton-Polyeder den Ursprung als einzigen inneren Gitterpunkt enth\alt, ausdr\ucken,so beweisen wir gewisse Kongruenz-Eigenschaften unter den Koeffizienten von $f_0$. Diese sind entscheidend bei der Konstruktion der analytischen Fortsetzung. Enth\alt die Faser $X_{t_0}$ einen gew\ohnlichen Doppelpunkt, so erwarten wir im Grenz\ubergang, dass das Frobeniuspolynom in zwei Faktoren von Grad eins und einen Faktor von Grad zwei zerf\allt. Der Faktor von Grad zwei ist dabei durch einen Koeffizienten $a_p$ eindeutig bestimmt. Durchl\auft nun $p$ die Menge aller Primzahlen, so erwarten wir aufgrund des Modularit\atssatzes, dass es eine Modulform von Gewicht vier gibt, deren Koeffizienten durch die Koeffizienten $a_p$ gegeben sind. Diese Erwartung hat sich durch unsere umfangreichen Rechnungen best\atigt. Dar\uberhinaus leiten wir weitere Formeln zur Bestimmung des Frobeniuspolynoms her, in welchen auch die nicht-holomorphen L\osungen der Gleichung $Pf=0$ in einer Umgebung der Null eine Rolle spielen.

Bottle microresonators for applications in quantum optics and all-optical signal processing

This thesis reports on the experimental realization, characterization and application of a novel microresonator design. The so-called “bottle microresonator” sustains whispering-gallery modes in which light fields are confined near the surface of the micron-sized silica structure by continuous total internal reflection. While whispering-gallery mode resonators in general exhibit outstanding properties in terms of both temporal and spatial confinement of light fields, their monolithic design makes tuning of their resonance frequency difficult. This impedes their use, e.g., in cavity quantum electrodynamics (CQED) experiments, which investigate the interaction of single quantum mechanical emitters of predetermined resonance frequency with a cavity mode. In contrast, the highly prolate shape of the bottle microresonators gives rise to a customizable mode structure, enabling full tunability. The thesis is organized as follows: In chapter I, I give a brief overview of different types of optical microresonators. Important quantities, such as the quality factor Q and the mode volume V, which characterize the temporal and spatial confinement of the light field are introduced. In chapter II, a wave equation calculation of the modes of a bottle microresonator is presented. The intensity distribution of different bottle modes is derived and their mode volume is calculated. A brief description of light propagation in ultra-thin optical fibers, which are used to couple light into and out of bottle modes, is given as well. The chapter concludes with a presentation of the fabrication techniques of both structures. Chapter III presents experimental results on highly efficient, nearly lossless coupling of light into bottle modes as well as their spatial and spectral characterization. Ultra-high intrinsic quality factors exceeding 360 million as well as full tunability are demonstrated. In chapter IV, the bottle microresonator in add-drop configuration, i.e., with two ultra-thin fibers coupled to one bottle mode, is discussed. The highly efficient, nearly lossless coupling characteristics of each fiber combined with the resonator's high intrinsic quality factor, enable resonant power transfers between both fibers with efficiencies exceeding 90%. Moreover, the favorable ratio of absorption and the nonlinear refractive index of silica yields optical Kerr bistability at record low powers on the order of 50 µW. Combined with the add-drop configuration, this allows one to route optical signals between the outputs of both ultra-thin fibers, simply by varying the input power, thereby enabling applications in all-optical signal processing. Finally, in chapter V, I discuss the potential of the bottle microresonator for CQED experiments with single atoms. Its Q/V-ratio, which determines the ratio of the atom-cavity coupling rate to the dissipative rates of the subsystems, aligns with the values obtained for state-of-the-art CQED microresonators. In combination with its full tunability and the possibility of highly efficient light transfer to and from the bottle mode, this makes the bottle microresonator a unique tool for quantum optics applications.

Moduli spaces of (G,h)-constellations

Given a reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G → N_0, we construct the moduli space M_Ө(X) of Ө-stable (G,h)-constellations on X, which is a common generalisation of the invariant Hilbert scheme after Alexeev and Brion and the moduli space of Ө-stable G-constellations for finite groups G introduced by Craw and Ishii. Our construction of a morphism M_Ө(X) → X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G. Furthermore, we determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C²)⁶ as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for the realisation of such calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

The geometry of Lagrangian fibres

If the generic fibre f−1(c) of a Lagrangian fibration f : X → B on a complex Poisson– variety X is smooth, compact, and connected, it is isomorphic to the compactification of a complex abelian Lie–group. For affine Lagrangian fibres it is not clear what the structure of the fibre is. Adler and van Moerbeke developed a strategy to prove that the generic fibre of a Lagrangian fibration is isomorphic to the affine part of an abelian variety.rnWe extend their strategy to verify that the generic fibre of a given Lagrangian fibration is the affine part of a (C∗)r–extension of an abelian variety. This strategy turned out to be successful for all examples we studied. Additionally we studied examples of Lagrangian fibrations that have the affine part of a ramified cyclic cover of an abelian variety as generic fibre. We obtained an embedding in a Lagrangian fibration that has the affine part of a C∗–extension of an abelian variety as generic fibre. This embedding is not an embedding in the category of Lagrangian fibrations. The C∗–quotient of the new Lagrangian fibration defines in a natural way a deformation of the cyclic quotient of the original Lagrangian fibration.

Neuropsychological impairments and the impact of dystrophin mutations on general cognitive functioning of patients with Duchenne muscular dystrophy

Mutations in the dystrophin gene have long been recognised as a cause of mental retardation. However, for reasons that are unclear, some boys with dystrophin mutations do not show general cognitive deficits. To investigate the relationship between dystrophin mutations and cognition, the general intellectual abilities of a group of 25 boys with genetically confirmed Duchenne muscular dystrophy were evaluated. Furthermore, a subgroup underwent additional detailed neuropsychological assessment. The results showed a mean full scale intelligence quotient (IQ) of 88 (standard deviation 24). Patients performed very poorly on various neuropsychological tests, including arithmetics, digit span tests and verbal fluency. No simple relationship between dystrophin mutations and cognitive functioning could be detected. However, our analysis revealed that patients who lack the dystrophin isoform Dp140 have significantly greater cognitive problems.

Comparison of implant stability by means of resonance frequency analysis for flapless and conventionally inserted implants

Background: Resonance frequency analysis (RFA) is a noninvasive technique for the quantitative assessment of implant stability. Information on the implant stability quotient (ISQ) of transmucosally inserted implants is limited. Purpose: The aim of this investigation was to compare the ISQ of conventionally inserted implants by raising a muco-periostal flap with implants inserted using a flapless procedure. Materials and Methods: Forty elderly patients with complete edentulous maxilla were consecutively admitted for treatment with implant-supported prostheses. A computer tomography was obtained for the computer-assisted implant planning. One hundred ten implants were placed conventionally in 23 patients (flap-group) and 85 implants in 17 patients by means of the flapless method (flapless-group) using a stereolithographic template. RFA measurements were performed after implant placement (baseline) and after a healing time of 12 weeks (reentry). Results: All implants exhibited clinically and radiographically successful osseointegration. Bone level did not change significantly neither for genders nor type of surgical protocol. Mean ISQ values of the flapless-group were significantly higher at baseline (p < .001) and at reentry (p < .001) compared with the flap-group. The ISQ values were significantly lower at reentry compared with baseline for the flap-group (p = .028) but not for the flapless-group. This group showed a moderate, but insignificant increase. RFA measurements of males resulted in ISQ values that were thoroughly higher as compared with females at both time-points in both groups. Correlation between RFA and bone level was not found. Conclusions: The flapless procedure showed favorable conditions with regard to implant stability and crestal bone level. Some changes of the ISQ values that represent primary (mechanical) and secondary (bone remodeling) implant stability were observed in slight favor of the flapless method and male patients. In properly planned and well-selected cases, the minimal invasive transmucosal technique using a drill-guide is a safe procedure.

Social Cognition And Functional Brain Imaging: Extending The Broader Autism Phenotype

Evidence suggests that the social cognition deficits prevalent in autism spectrum disorders (ASDs) are widely distributed in first degree and extended relatives. This ¿broader autism phenotype¿ (BAP) can be extended into non-clinical populations and show wide distributions of social behaviors such as empathy and social responsiveness ¿ with ASDs exhibiting these behaviors on the lower ends of the distributions. Little evidence has previously shown relationships between self-report measures of social cognition and more objective tasks such as face perception in functional magnetic resonance imaging (fMRI) and event-related potentials (ERPs). In this study, three specific hypotheses were addressed: a) increased social ability, as measured by an increased Empathy Quotient, decreased Social Responsiveness Scale (SRS-A) score, and increased Social Attribution Task score, will predict increased activation of the fusiform gyrus in response to faces as compared to houses; b) these same measures will predict N170 amplitude and latency showing decreased latency and increased amplitude for faces as compared to houses with increased social ability; c) increased amygdala volume will predict increased fusiform gyrus activation when viewing faces as compared to houses. Findings supported all of the hypotheses. Empathy scores significantly predicted both right FFG activation [F(1,20) = 4.811, p = .041, ß = .450, R2 = 0.20] and left FFG activation [F(1,20) = 7.70, p = .012, ß = .537, R2 = 0.29]. Based on ERP results increased right lateralization face-related N170 was significantly predicted by the EQ [F(1,54) = 6.94, p = .011, ß = .338, R2 = 0.11]. Finally, total amygdala volume significantly predicted right [F(1,20) = 7.217, p = .014, ß = .515, R2 = 0.27] and left [F(1,20) = 36.77, p < .001, ß = .805, R2 = 0.65] FFG activation. Consistent with the a priori hypotheses, traits attributed to the BAP can significantly predict neural responses to faces in a non-clinical population. This is consistent with the face processing deficits seen in ASDs. The findings presented here contribute to the extension of the BAP from unaffected relatives of individuals with ASDs to the general population. These findings also give continued evidence in support of a continuous distribution of traits found in psychiatric illnesses in place of a traditional, dichotomous ¿all-or-nothing¿ diagnostic framework of neurodevelopmental and neuropsychiatric disorders.

Social Cognition and Neural Substrates of Face Perception: Implications for Neurodevelopmental and Neuropsychiatric Disorders

BACKGROUND: Social cognition is an important aspect of social behavior in humans. Social cognitive deficits are associated with neurodevelopmental and neuropsychiatric disorders. In this study we examine the neural substrates of social cognition and face processing in a group of healthy young adults to examine the neural substrates of social cognition. METHODS: Fifty-seven undergraduates completed a battery of social cognition tasks and were assessed with electroencephalography (EEG) during a face-perception task. A subset (N=22) were administered a face-perception task during functional magnetic resonance imaging. RESULTS: Variance in the N170 EEG was predicted by social attribution performance and by a quantitative measure of empathy. Neurally, face processing was more bilateral in females than in males. Variance in fMRI voxel count in the face-sensitive fusiform gyrus was predicted by quantitative measures of social behavior, including the Social Responsiveness Scale (SRS) and the Empathizing Quotient. CONCLUSIONS: When measured as a quantitative trait, social behaviors in typical and pathological populations share common neural pathways. The results highlight the importance of viewing neurodevelopmental and neuropsychiatric disorders as spectrum phenomena that may be informed by studies of the normal distribution of relevant traits in the general population. Copyright 2014 Elsevier B.V. All rights reserved.

A 600 kb deletion syndrome at 16p11.2 leads to energy imbalance and neuropsychiatric disorders

BACKGROUND: The recurrent ~600 kb 16p11.2 BP4-BP5 deletion is among the most frequent known genetic aetiologies of autism spectrum disorder (ASD) and related neurodevelopmental disorders. OBJECTIVE: To define the medical, neuropsychological, and behavioural phenotypes in carriers of this deletion. METHODS: We collected clinical data on 285 deletion carriers and performed detailed evaluations on 72 carriers and 68 intrafamilial non-carrier controls. RESULTS: When compared to intrafamilial controls, full scale intelligence quotient (FSIQ) is two standard deviations lower in carriers, and there is no difference between carriers referred for neurodevelopmental disorders and carriers identified through cascade family testing. Verbal IQ (mean 74) is lower than non-verbal IQ (mean 83) and a majority of carriers require speech therapy. Over 80% of individuals exhibit psychiatric disorders including ASD, which is present in 15% of the paediatric carriers. Increase in head circumference (HC) during infancy is similar to the HC and brain growth patterns observed in idiopathic ASD. Obesity, a major comorbidity present in 50% of the carriers by the age of 7 years, does not correlate with FSIQ or any behavioural trait. Seizures are present in 24% of carriers and occur independently of other symptoms. Malformations are infrequently found, confirming only a few of the previously reported associations. CONCLUSIONS: The 16p11.2 deletion impacts in a quantitative and independent manner FSIQ, behaviour and body mass index, possibly through direct influences on neural circuitry. Although non-specific, these features are clinically significant and reproducible. Lastly, this study demonstrates the necessity of studying large patient cohorts ascertained through multiple methods to characterise the clinical consequences of rare variants involved in common diseases.

A modified calculation of ankle-brachial pressure index is far more sensitive in the detection of peripheral arterial disease

BACKGROUND: Ankle-brachial pressure index (ABI) is a simple, inexpensive, and useful tool in the detection of peripheral arterial occlusive disease (PAD). The current guidelines published by the American Heart Association define ABI as the quotient of the higher of the systolic blood pressures (SBPs) of the two ankle arteries of that limb (either the anterior tibial artery or the posterior tibial artery) and the higher of the two brachial SBPs of the upper limbs. We hypothesized that considering the lower of the two ankle arterial SBPs of a side as the numerator and the higher of the brachial SBPs as the denominator would increase its diagnostic yield. METHODS: The former method of eliciting ABI was termed as high ankle pressure (HAP) and the latter low ankle pressure (LAP). ABI was assessed in 216 subjects and calculated according to the HAP and the LAP method. ABI findings were confirmed by arterial duplex ultrasonography. A significant arterial stenosis was assumed if ABI was <0.9. RESULTS: LAP had a sensitivity of 0.89 and a specificity of 0.93. The HAP method had a sensitivity of 0.68 and a specificity of 0.99. McNemar's test to compare the results of both methods demonstrated a two-tailed P < .0001, indicating a highly significant difference between both measurement methods. CONCLUSIONS: LAP is the superior method of calculating ABI to identify PAD. This result is of great interest for epidemiologic studies applying ABI measurements to detect PAD and assessing patients' cardiovascular risk.

Immediate loading of implants with 3-unit fixed partial dentures: a 12-month clinical study

PURPOSE: The aim of the present clinical trial was to evaluate the 12-month success rate of titanium dental implants placed in the posterior mandible and immediately loaded with 3-unit fixed partial dentures. MATERIALS AND METHODS: Patients with missing mandibular premolars and molars were enrolled in this study. To be included in the study, the implants had to show good primary stability. Implant stability was measured with resonance frequency analysis using the Osstell device (Integration Diagnostics). Implants were included in the study when the stability quotient (ISQ) exceeded 62. Clinical measurements, such as width of keratinized tissue, ISQ, and radiographic assessment of peri-implant bone crest levels, were performed at baseline and at the 12-month follow-up. The comparison between the baseline and the 12-month visits was performed with the Student t test for paired data (statistically significant at a level of alpha = 0.05). RESULTS: Forty implants with a sandblasted, large grit, acid-etched (SLA) surface (Straumann) were placed in 20 patients. At 12 months, only 1 implant had been lost because of an acute infection. The remaining 39 implants were successful, resulting in a 1-year success rate of 97.5%. Neither peri-implant bone levels, measured radiographically, nor implant stability changed significantly from baseline to the 12-month follow-up (P > .05). DISCUSSION: The immediate functional loading of implants placed in this case series study resulted in a satisfactory success rate. CONCLUSION: The findings from this clinical study showed that the placement of SLA transmucosal implants in the mandibular area and their immediate loading with 3-unit fixed partial dentures may be a safe and successful procedure.

Measurement of dental implant stability by resonance frequency analysis and damping capacity assessment: comparison of both techniques in a clinical trial

PURPOSE: Two noninvasive methods to measure dental implant stability are damping capacity assessment (Periotest) and resonance frequency analysis (Osstell). The objective of the present study was to assess the correlation of these 2 techniques in clinical use. MATERIALS AND METHODS: Implant stability of 213 clinically stable loaded and unloaded 1-stage implants in 65 patients was measured in triplicate by means of resonance frequency analysis and Periotest. Descriptive statistics as well as Pearson's, Spearman's, and intraclass correlation coefficients were calculated with SPSS 11.0.2. RESULTS: The mean values were 57.66 +/- 8.19 implant stability quotient for the resonance frequency analysis and -5.08 +/- 2.02 for the Periotest. The correlation of both measuring techniques was -0.64 (Pearson) and -0.65 (Spearman). The single-measure intraclass correlation coefficients for the ISQ and Periotest values were 0.99 and 0.88, respectively (95% CI). No significant correlation of implant length with either resonance frequency analysis or Periotest could be found. However, a significant correlation of implant diameter with both techniques was found (P < .005). The correlation of both measuring systems is moderate to good. It seems that the Periotest is more susceptible to clinical measurement variables than the Osstell device. The intraclass correlation indicated lower measurement precision for the Periotest technique. Additionally, the Periotest values differed more from the normal (Gaussian) curve of distribution than the ISQs. Both measurement techniques show a significant correlation to the implant diameter. CONCLUSION: Resonance frequency analysis appeared to be the more precise technique.

Intellectual outcome, motor skills and BMI of children with congenital hypothyroidism: a population-based study

AIM: To evaluate intellectual outcome, motor skills and anthropometric data of children with congenital hypothyroidism (CH). METHODS: Children with permanent CH who were born in 1999 in Bavaria were eligible for this prospective, population-based study. Cognitive performance was evaluated by the Kaufman Assessment Battery for Children and motor skills were assessed by the motor test, Motoriktest für vier-bis sechsjahrige Kinder (MOT) 4-6. RESULTS: Eighteen of 21 eligible children participated (86%). Median age of the children was 5.5 years (range 4.9-5.8). Treatment with levothyroxine was started after a median of 7.2 days (range 4-15) with a median dose of 12.0 microg/kg (range 7.2-17.0). Mean intelligence quotient (IQ) of the children was 100.4 (standard deviation [SD] 10.1): no children had IQ values below the normal range. Reactivity and speed of movement were significantly reduced in children with CH. Children with an initial thyroid-stimulating hormone (TSH) value of >200 mU/L performed significantly worse than children with TSH value of

Cerebellar disorders in childhood: cognitive problems

Over the last decade, increasing evidence of cognitive functions of the cerebellum during development and learning processes could be ascertained. Posterior fossa malformations such as cerebellar hypoplasia or Joubert syndrome are known to be related to developmental problems in a marked to moderate extent. More detailed analyses reveal special deficits in attention, processing speed, visuospatial functions, and language. A study about Dandy Walker syndrome states a relationship of abnormalities in vermis lobulation with developmental problems. Further lobulation or volume abnormalities of the cerebellum and/or vermis can be detected in disorders as fragile X syndrome, Downs's syndrome, William's syndrome, and autism. Neuropsychological studies reveal a relation of dyslexia and attention deficit disorder with cerebellar functions. These functional studies are supported by structural abnormalities in neuroimaging in these disorders. Acquired cerebellar or vermis atrophy was found in groups of children with developmental problems such as prenatal alcohol exposure or extreme prematurity. Also, focal lesions during childhood or adolescence such as cerebellar tumor or stroke are related with neuropsychological abnormalities, which are most pronounced in visuospatial, language, and memory functions. In addition, cerebellar atrophy was shown to be a bad prognostic factor considering cognitive outcome in children after brain trauma and leukemia. In ataxia teleangiectasia, a neurodegenerative disorder affecting primarily the cerebellar cortex, a reduced verbal intelligence quotient and problems of judgment of duration are a hint of the importance of the cerebellum in cognition. In conclusion, the cerebellum seems to play an important role in many higher cognitive functions, especially in learning. There is a suggestion that the earlier the incorrect influence, the more pronounced the problems.

Steady-state diffusion imaging for MR in-vivo evaluation of reparative cartilage after matrix-associated autologous chondrocyte transplantation at 3 tesla--preliminary results

OBJECTIVES: To demonstrate the feasibility of time-reversed fast imaging with steady-state precession (FISP) called PSIF for diffusion-weighted imaging of cartilage and cartilage transplants in a clinical study. MATERIAL AND METHODS: In a cross-sectional study 15 patients underwent MRI using a 3D partially balanced steady-state gradient echo pulse sequence with and without diffusion weighting at two different time points after matrix-associated autologous cartilage transplantation (MACT). Mean diffusion quotients (signal intensity without diffusion-weighting divided by signal intensity with diffusion weighting) within the cartilage transplants were compared to diffusion quotients found in normal cartilage. RESULTS: The global diffusion quotient found in repair cartilage was significantly higher than diffusion values in normal cartilage (p<0.05). There was a decrease between the earlier and the later time point after surgery. CONCLUSIONS: In-vivo diffusion-weighted imaging based on the PSIF technique is possible. Our preliminary results show follow-up of cartilage transplant maturation in patients may provide additional information to morphological assessment.