Cantor bouquets, explosions, and Knaster continua : dynamics of complex exponentials

We describe some of the interesting dynamical and topological properties of the complex exponential family [lambda]ez and its associated Julia sets

Checkerboard Julia Sets for Rational Maps

In this paper, we consider the family of rational maps \F(z) = z^n + \frac{\la}{z^d}, where $n \geq 2$, $d\geq 1$, and\la \in \bbC. We consider the case where \la lies in the main cardioid of one of the $n-1$ principal Mandelbrot sets in these families. We show that the Julia sets of these maps are always homeomorphic. However, two such maps \F and $F_\mu$ are conjugate on these Julia sets only if the parameters at the centers of the given cardioids satisfy \mu = \nu^{j(d+1)}\la or \mu = \nu^{j(d+1)}\bar{\la} where j \in \bbZ and $\nu$ is an $n-1^{\rm st}$ root of unity. We define a dynamical invariant, which we call the minimal rotation number. It determines which of these maps are are conjugate on their Julia sets, and we obtain an exact count of the number of distinct conjugacy classes of maps drawn from these main cardioids.Comment: 25 pages, 14 figures; Changes since March 19 version: added nine figures, fixed one proof, added a section on a group actio

A Pseudo Random Numbers Generator Based on Chaotic Iterations. Application to Watermarking

In this paper, a new chaotic pseudo-random number generator (PRNG) is proposed. It combines the well-known ISAAC and XORshift generators with chaotic iterations. This PRNG possesses important properties of topological chaos and can successfully pass NIST and TestU01 batteries of tests. This makes our generator suitable for information security applications like cryptography. As an illustrative example, an application in the field of watermarking is presented.Comment: 11 pages, 7 figures, In WISM 2010, Int. Conf. on Web Information Systems and Mining, volume 6318 of LNCS, Sanya, China, pages 202--211, October 201

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve

Computational Complexity of Iterated Maps on the Interval (Extended Abstract)

The exact computation of orbits of discrete dynamical systems on the interval is considered. Therefore, a multiple-precision floating point approach based on error analysis is chosen and a general algorithm is presented. The correctness of the algorithm is shown and the computational complexity is analyzed. As a main result, the computational complexity measure considered here is related to the Ljapunow exponent of the dynamical system under consideration

PPAR alpha L162V underlies variation in serum triglycerides and subcutaneous fat volume in young males

Background: Of the five sub-phenotypes defining metabolic syndrome, all are known to have strong genetic components ( typically 50 - 80% of population variation). Studies defining genetic predispositions have typically focused on older populations with metabolic syndrome and/or type 2 diabetes. We hypothesized that the study of younger populations would mitigate many confounding variables, and allow us to better define genetic predisposition loci for metabolic syndrome. Methods: We studied 610 young adult volunteers ( average age 24 yrs) for metabolic syndrome markers, and volumetric MRI of upper arm muscle, bone, and fat pre- and post-unilateral resistance training. Results: We found the PPARa L162V polymorphism to be a strong determinant of serum triglyceride levels in young White males, where carriers of the V allele showed 78% increase in triglycerides relative to L homozygotes ( LL = 116 +/- 11 mg/ dL, LV = 208 +/- 30 mg/ dL; p = 0.004). Men with the V allele showed lower HDL ( LL = 42 +/- 1 mg/ dL, LV = 34 +/- 2 mg/ dL; p = 0.001), but women did not. Subcutaneous fat volume was higher in males carrying the V allele, however, exercise training increased fat volume of the untrained arm in V carriers, while LL genotypes significantly decreased in fat volume ( LL = - 1,707 +/- 21 mm(3), LV = 17,617 +/- 58 mm(3); p = 0.002), indicating a systemic effect of the V allele on adiposity after unilateral training. Our study suggests that the primary effect of PPARa L162V is on serum triglycerides, with downstream effects on adiposity and response to training. Conclusion: Our results on association of PPARa and triglycerides in males showed a much larger effect of the V allele than previously reported in older and less healthy populations. Specifically, we showed the V allele to increase triglycerides by 78% ( p = 0.004), and this single polymorphism accounted for 3.8% of all variation in serum triglycerides in males ( p = 0.0037)

AKT1 polymorphisms are associated with risk for metabolic syndrome

Converging lines of evidence suggest that AKT1 is a major mediator of the responses to insulin, insulin-like growth factor 1 (IGF1), and glucose. AKT1 also plays a key role in the regulation of both muscle cell hypertrophy and atrophy. We hypothesized that AKT1 variants may play a role in the endophenotypes that make up metabolic syndrome. We studied a 12-kb region including the first exon of the AKT1 gene for association with metabolic syndrome-related phenotypes in four study populations [FAMUSS cohort (n = 574; age 23.7 ± 5.7 years), Strong Heart Study (SHS) (n = 2,134; age 55.5 ± 7.9 years), Dynamics of Health, Aging and Body Composition (Health ABC) (n = 3,075; age 73.6 ± 2.9 years), and Studies of a Targeted Risk Reduction Intervention through Defined Exercise (STRRIDE) (n = 175; age 40–65 years)]. We identified a three SNP haplotype that we call H1, which represents the ancestral alleles at the three loci and H2, which represents the derived alleles at the three loci. In young adult European Americans (FAMUSS), H1 was associated with higher fasting glucose levels in females. In middle age Native Americans (SHS), H1 carriers showed higher fasting insulin and HOMA in males, and higher BMI in females. In older African-American and European American subjects (Health ABC) H1 carriers showed a higher incidence of metabolic syndrome. Homozygotes for the H1 haplotype showed about twice the risk of metabolic syndrome in both males and females (p < 0.001). In middle-aged European Americans with insulin resistance (STRRIDE) studied by intravenous glucose tolerance test (IVGTT), H1 carriers showed increased insulin resistance due to the Sg component (p = 0.021). The 12-kb haplotype is a risk factor for metabolic syndrome and insulin resistance that needs to be explored in further populations

PPARα L162V underlies variation in serum triglycerides and subcutaneous fat volume in young males

<p>Abstract</p> <p>Background</p> <p>Of the five sub-phenotypes defining metabolic syndrome, all are known to have strong genetic components (typically 50–80% of population variation). Studies defining genetic predispositions have typically focused on older populations with metabolic syndrome and/or type 2 diabetes. We hypothesized that the study of younger populations would mitigate many confounding variables, and allow us to better define genetic predisposition loci for metabolic syndrome.</p> <p>Methods</p> <p>We studied 610 young adult volunteers (average age 24 yrs) for metabolic syndrome markers, and volumetric MRI of upper arm muscle, bone, and fat pre- and post-unilateral resistance training.</p> <p>Results</p> <p>We found the PPARα L162V polymorphism to be a strong determinant of serum triglyceride levels in young White males, where carriers of the V allele showed 78% increase in triglycerides relative to L homozygotes (LL = 116 ± 11 mg/dL, LV = 208 ± 30 mg/dL; p = 0.004). Men with the V allele showed lower HDL (LL = 42 ± 1 mg/dL, LV = 34 ± 2 mg/dL; p = 0.001), but women did not. Subcutaneous fat volume was higher in males carrying the V allele, however, exercise training increased fat volume of the untrained arm in V carriers, while LL genotypes significantly decreased in fat volume (LL = -1,707 ± 21 mm³, LV = 17,617 ± 58 mm³; p = 0.002), indicating a systemic effect of the V allele on adiposity after unilateral training. Our study suggests that the primary effect of PPARα L162V is on serum triglycerides, with downstream effects on adiposity and response to training.</p> <p>Conclusion</p> <p>Our results on association of PPARα and triglycerides in males showed a much larger effect of the V allele than previously reported in older and less healthy populations. Specifically, we showed the V allele to increase triglycerides by 78% (p = 0.004), and this single polymorphism accounted for 3.8% of all variation in serum triglycerides in males (p = 0.0037).</p

INSIG2 gene polymorphism is associated with increased subcutaneous fat in women and poor response to resistance training in men

Background A common SNP upstream of the INSIG2 gene, rs7566605 (g.-10,1025G\u3eC, Chr2:118,552,255, NT_022135.15), was reported to be associated with obesity (Body Mass Index, [BMI]) in a genome-wide association scan using the Framingham Heart Study but has not been reproduced in other cohorts. As BMI is a relatively insensitive measure of adiposity that is subject to many confounding variables, we sought to determine the relationship between the INSIG2 SNP and subcutaneous fat volumes measured by MRI in a young adult population. Methods We genotyped the INSIG2 SNP rs7566605 in college-aged population enrolled in a controlled resistance-training program, (the Functional Polymorphism Associated with Human Muscle Size and Strength, FAMuSS cohort, n = 752 volunteers 18–40 yrs). In this longitudinal study, we examined the effect of the INSIG2 polymorphism on subcutaneous fat and muscle volumes of the upper arm measured by magnetic resonance imaging (MRI) before and after 12 wks of resistance training. Gene/phenotype associations were tested using an analysis of covariance model with age and weight as covariates. Further, the % variation in each phenotype attributable to genotype was determined using hierarchical models and tested with a likelihood ratio test. Results Women with a copy of the C allele had higher levels of baseline subcutaneous fat (GG: n = 139; 243473 ± 5713 mm3 vs. GC/CC: n = 181; 268521 ± 5003 mm3; p = 0.0011); but men did not show any such association. Men homozygous for the G ancestral allele showed a loss of subcutaneous fat, while those with one or two copies of the C allele gained a greater percentage of subcutaneous fat with resistance training (GG: n = 103; 1.02% ± 1.74% vs. GC/CC: n = 93; 6.39% ± 1.82%; p = 0.035). Conclusion Our results show that the INSIG2 rs7566605 polymorphism underlies variation in subcutaneous adiposity in young adult women and suppresses the positive effects of resistance training on men. This supports and extends the original finding that there is an association between measures of obesity and INSIG2 rs7566605 and further implicates this polymorphism in fat regulation