Influence of optical property contrast on the critical distribution of electrostatic torques in double-beam torsional Casimir actuators: Non-linear actuation toward chaotic motion.

Here, we discuss how to achieve the stable actuation of a double beam torsional micro-actuator over the largest possible displacement of the moving component under the influence of Casimir and electrostatic torques, when the rotating component is constructed from different materials. The main part of this study is devoted to finding the optimal distribution of the electrostatic torque between the left and right sides of the micro-actuator to reach the maximum stable operation of the device. The latter is manifested by switching from homoclinic to heteroclinic orbits in the phase portraits. Indeed, the bifurcation curves and the phase portraits have been employed to show the sensitivity of the critical distribution of the electrostatic torque, beyond which the device does show stable performance, on the contrast of the optical properties of the moving component and the applied voltage in a conservative autonomous system. Moreover, for driven systems, the Melnikov function approach and the Poincaré portraits are used to study the presence of chaotic motion, which eventually leads to stiction. It is shown that the application of the optimal distribution of the electrostatic torque can significantly decrease the possibility of chaotic motion, and at this optimal level, the threshold curves reveal less difference between systems with different optical contrast.

Nonlinear actuation of micromechanical Casimir oscillators with topological insulator materials toward chaotic motion: Sensitivity on magnetization and dielectric properties.

We have investigated the dynamical actuation of micro-electromechanical systems under the influence of attractive and repulsive Casimir forces between topological insulator plates as a function of their dielectric function and coating magnetization. The analysis of the Casimir force in the limit of strong and weak magnetization shows that the attractive force, which is produced for plate magnetizations in the same direction, is greater than the repulsive force that is produced for opposite magnetizations. However, both forces remain comparable for intermediate magnetizations. Moreover, for weak magnetization, the attractive force becomes stronger for an increasing dielectric function, while the opposite occurs for the repulsive force. On the other hand, increasing magnetization decreases the influence of the dielectric function on both the repulsive and attractive forces. Furthermore, for conservative systems, bifurcation and phase portrait analysis revealed that increasing magnetization decreases the regime of stable operation for devices with attractive forces, while their operation remains always stable under the presence of repulsive forces. Finally, for non-conservative periodically driven systems, the Melnikov function and Poincaré portrait analysis show that for magnetizations in the same direction leading to strong attractive Casimir forces, chaotic motion toward stiction is highly likely to occur preventing the long-term prediction of actuating dynamics. A remedy for this situation is obtained by the application of any magnetization in opposite directions between the interacting surfaces since the repulsive force makes it possible to prevent stiction.

New insights into genetics underlying of plumage color.

Plumage color can be considered as a social signal in chickens and a breeding identification tool among breeders. The relationship between plumage color and trait groups of immunity, growth and fertility is still a controversial issue. This research aimed to determine the genome-wide additive and epistatic variants affecting plumage color variation in chickens using the chicken Illumina 60k high-density SNP array. Two scenarios of genome-wide additive association studies using all SNPs and independent SNPs were carried out. To perform epistatic association analysis, the LD pruning approach was used to reduce the complexity of the analysis. We detected seven novel significant loci using all of the SNPs in the model and 14 SNPs using the LD pruning approach associated with plumage color. Moreover, 89 significantly associated SNP-SNP interactions (P-value <10-6 ) distributed in 25 chromosomes were identified, indicating that all of the signals together putatively influence the quantitative variation of plumage color. By annotating genes relevant to top SNPs, we have distinguished 18 potential candidate genes comprising HNF4beta, CKMT1B, TBC1D22A, RPL8, CACNA2D1, FZD4, SGMS1, IRF8, OPTN, LOC420362, TRABD, OvoDA1, DAD1, USP6, RBM12B, MIR1772, MIR1709 and MIR6696 and also 89 putative gene-gene combinations responsible for plumage color variation in chickens. Furthermore, several KEGG pathways including metabolic pathway, cytokine-cytokine receptor interaction, focal adhesion, melanogenesis, glycosaminoglycan biosynthesis-keratan sulfate and sphingolipid metabolism were enriched in the gene-set analysis. The results indicated that plumage color is a highly polygenic trait which, in turn, can be affected by multiple coding genes, regulatory genes and gene-gene epistasis interactions. In addition to genes with additive effects, epistatic genes with tiny individual effect sizes but significant effects in a pair have the potential to control plumage coloration in chickens.

Chaotic motion due to lateral Casimir forces during nonlinear actuation dynamics.

We investigated here the influence of the lateral Casimir force on the dynamical actuation of devices with interacting materials covering a broad range of optical properties ranging from poor to good conductors, such as, for example, nitrogen doped SiC and Au, respectively. The conservative actuating system shows a central heteroclinic orbit surrounded by a finite number of homoclinic orbits, because at higher periods, an increased lateral Casimir force will be necessary to counterbalance the restoring force. As a result, the conservative system reaches stable operation sooner for the higher conductivity materials (Au-Au), indicating the significant impact of the material optical properties on the lateral Casimir force. Furthermore, for the non-conservative driven systems, the decrement of the Melnikov parameter α leads to a faster disappearance of the satellite homoclinic orbits in the Poincaré portraits, followed by a strong shrinkage of the central heteroclinic orbit toward unstable chaotic motion. The latter is more pronounced for the lower conductivity materials since comparison shows the Au-Au system to be significantly more stable than the SiC-SiC system. Therefore, in actuating systems where the lateral Casimir force could play a significant role, the higher conductivity materials appear to be a better choice to ensure stable operation against a chaotic motion.

Genome-wide association study of shank length and diameter at different developmental stages in chicken F2 resource population.

In order to find SNPs and genes affecting shank traits, we performed a GWAS in a chicken F2 population of eight half-sib families from five hatches derived from reciprocal crosses between an Arian fast-growing line and an Urmia indigenous slow-growing chicken. A total of 308 birds were genotyped using a 60K chicken SNP chip. Shank traits including shank length and diameter were measured weekly from birth to 12 weeks of age. A generalized linear model and a compressed mixed linear model (CMLM) were applied to achieve the significant regions. The value of the average genomic inflation factor (λ statistic) of the CMLM model (0.99) indicated that the CMLM was more effective than the generalized linear model in controlling the population structure. The genes surrounding significant SNPs and their biological functions were identified from NCBI, Ensembl and UniProt databases. The results indicated that 12 SNPs at 12 different ages passed the LD-adjusted 5% Bonferroni significant threshold. Two SNPs were significant for shank length and nine SNPs were significant for shank diameter. The significant SNPs were located near to or inside 11 candidate genes. The results showed that a number of significant SNPs in the middle ages were higher than the rest. The MXRA8 gene was related to the significant SNP at week 1 that promotes proliferation of growth plate chondrocytes. A unique SNP of Gga_rs16689511 located on chicken Z chromosome within the LOC101747628 gene was related to shank length at three different ages of birds (weeks 8, 9 and 11). The significant SNPs for shank diameter were found at weeks 4 and 7 (four and five SNPs respectively). The identifications of SNPs and genes here could contribute to a better understanding of the genetic control of shank traits in chicken.

Electron scattering in a superlattice of line defects on the surface of topological insulators.

The electron scattering from periodic line defects on the surface of topological insulators with hexagonal warping effect is investigated theoretically by means of a transfer matrix method. The influence of surface line defects, acting as structural ripples on propagation of electrons are studied in two perpendicular directions due to the asymmetry of warped energy contour under momentum exchange. The transmission profiles and the details of resonant peaks which vary with the number of defects and the strength of their potentials are strongly dependent on the direction in which the line defects extend. At low energies, the quantum interference between the incident and reflected propagating electrons has the dominant contribution in transmission resonances, while at high energies the multiple scattering processes on the constant-energy contour also appear because of the strong warping effect. By increasing the spatial separation between the line defects, the minimum value of electrical conductance remains significantly high at low incident energies, while the minimum value may approach zero at high energies as the number of defects is increased. Our findings suggest that the potential ripples on the surface of topological insulators can be utilized to control the local electronic properties of these materials.

Sensitivity of nonequilibrium Casimir forces on low frequency optical properties toward chaotic motion of microsystems: Drude vs plasma model.

Here, we investigate the sensitivity of nonequilibrium Casimir forces to optical properties at low frequencies via the Drude and plasma models and the associated effects on the actuation of microelectromechanical systems. The stability and chaotic motion for both autonomous conservative and nonconservative driven systems were explored assuming good, e.g., Au, and poor, e.g., doped SiC, interacting conductors having large static conductivity differences. For both material systems, we used the Drude and plasma methods to model the optical properties at low frequencies, where measurements are not feasible. In fact, for the conservative actuating system, bifurcation and phase space analysis show that the system motion is strongly influenced by the thermal nonequilibrium effects depending on the modeling of the optical properties at low frequencies, where also the presence of residual electrostatic forces can also drastically alter the actuating state of the system, depending strongly on the material conductivity. For nonconservative systems, the Melnikov function approach is used to explore the presence of chaotic motion rendering predictions of stable actuation or malfunction due to stiction on a long-term time scale rather impossible. In fact, the thermal effects produce the opposite effect for the emerging chaotic behavior for the Au-Au and SiC-SiC systems if the Drude model is used to model the low optical frequencies. However, using the plasma model, only for the poor conducting SiC-SiC system, the chance of chaotic motion is enhanced, while for the good conducting Au-Au system, the chaotic behavior will remain unaffected at relatively short separations (<2 µm).

Dependence of non-equilibrium Casimir forces on material optical properties toward chaotic motion during device actuation.

The sensitivity of nonequilibrium Casimir forces on material optical properties can have strong impact on the actuation of devices. For this purpose, we considered nonequilibrium Casimir interactions between good and poor conductors, for example, gold (Au) and highly doped silicon carbide (SiC), respectively. Indeed, for autonomous conservative systems, the bifurcation and phase portrait analysis have shown that the nonequilibrium Casimir forces can have significant impact on the stable and unstable operating regimes depending on the material optical properties. At a few micrometer separations, for systems with high conductivity materials, an increasing temperature difference between the actuating components can enhance the stable operation range due to the reduction of the Casimir force, while for the poor conductive materials, the opposite takes place. For periodically driven dissipative systems, the Melnikov function and Poincare portrait analysis have shown that for poor conductive systems, the nonequilibrium Casimir forces lead to an increased possibility for chaotic behavior and stiction with an increasing temperature difference between the actuating components. However, for good conducting systems, the thermal contribution to Casimir forces reduces the possibility for chaotic behavior with increasing temperature, as comparison with systems without thermal fluctuations shows. Nevertheless, the positive benefit of good conductors toward increased actuation stability and reduced the chaotic behavior under nonequilibrium conditions can be easily compromised by any voltage application. Therefore, thermal, nonequilibrium Casimir forces can influence the actuation of devices toward unstable and chaotic behavior in strong correlation with their optical properties, and associated conduction state, as well as applied electrostatic potentials.

Sensitivity of chaotic behavior to low optical frequencies of a double-beam torsional actuator.

We investigate here how the optical properties at low frequencies affect the actuation dynamics and emerging chaotic behavior in a double-beam torsion actuator at nanoscale separations (<200nm), where the Casimir forces and torques play a major role. In fact, we take into account differences of the Casimir force due to alternative modeling of optical properties at low frequencies, where measurements are not feasible, via the Drude and plasma models, and repercussions by different material preparation conditions. For conservative autonomous actuation, bifurcation and phase portrait analysis indicate that both factors affect the stability of an actuating device in such a way that stronger Casimir forces and torques will favor increased unstable behavior. The latter will be enhanced by unbalanced application of electrostatic voltages in double-beam actuating systems. For the case of a time-periodic driving force, we use a Melnikov function and a phase plane analysis to study the emerging chaotic behavior with respect to the Drude and plasma modeling and material preparation conditions. We find indications that any factor that leads to stronger Casimir interactions will aid chaotic behavior and prevent long term prediction of the actuating dynamics. Moreover, in a double-beam actuator chaoticity will be amplified by the application of unbalanced electrostatic voltages. Therefore, the details of modeling of optical properties and the material preparations conditions must be carefully considered in the design of actuating devices at nanoscale because here Casimir forces are omnipresent and broadband type interactions.

Random deposition with a power-law noise model: Multiaffine analysis.

We study the random deposition model with power-law distributed noise and rare-event dominated fluctuation. In this model instead of particles with unit sizes, rods with variable lengths are deposited onto the substrate. The length of each rod is chosen from a power-law distribution P(l)∼l^{-(µ+1)}, and the site at which each rod is deposited is chosen randomly. The results show that for µ<µ_{c}=3 the log-log diagram of roughness, W(t), versus deposition time, t, increases as a step function, where the roughness in each interval acts as W_{loc}(t)≈t^{ß_{loc}}. The local growth exponent, ß_{loc}, is zero for µ=1. By increasing the µ exponent, the value of ß_{loc} is increased. It tends to the growth exponent of the random distribution model with Gaussian noise, ß=1/2, at µ_{c}=3. The fractal analysis of the height fluctuations for this model was performed by multifractal detrended fluctuation analysis algorithm. The results show multiaffinity behavior for the height fluctuations at µ<µ_{c} and the multiaffinity strength is greater for smaller values of the µ exponent.

Dependence of chaotic behavior on optical properties and electrostatic effects in double-beam torsional Casimir actuation.

We investigate the influence of Casimir and electrostatic torques on double-beam torsional microelectromechanical systems with materials covering a broad range of conductivities of more than three orders of magnitude. For the frictionless autonomous systems, bifurcation and phase space analysis shows a significant difference between stable and unstable operating regimes for equal and unequal applied voltages on both sides of the double torsional system giving rise to heteroclinic and homoclinic orbits, respectively. For equal applied voltages, only the position of a symmetric unstable saddle equilibrium point is dependent on the material optical properties and electrostatic effects, while in any other case stable and unstable equilibrium points are dependent on both factors. For the periodically driven system, a Melnikov function approach is used to show the presence of chaotic motion rendering predictions of whether stiction or stable actuation will take place over long times impossible. Chaotic behavior introduces significant risk for stiction, and it is more likely to occur for the more conductive systems that experience stronger Casimir forces and torques. Indeed, when unequal voltages are applied, the sensitive dependence of chaotic motion on electrostatics is more pronounced for the highest conductivity systems.

Identification of candidate genes for blood metabolites in Iranian chickens using a genome-wide association study.

1. In order to identify loci associated with metabolic traits, a genome-wide association study was carried out in a chicken F2 population derived from a reciprocal cross between Iranian Urmia indigenous chickens and Arian broiler line using Illumina 60K Chicken single nucleotide polymorphism (SNP) BeadChip. 2. Six traits including plasma level of triglycerides (TGs), cholesterol (Chol), glucose (Glu), total protein, albumin (Alb) and globulin (Glo) were recorded. The association between the identified SNPs and metabolic traits was estimated by general linear model (GLM) and compressed mixed linear model (CMLM). 3. A total of 38 SNPs were identified at the genome-wide significant and suggestive levels, of which 5 SNPs reached a 5% Bonferroni genome-wide significance (P < 2.58E-6) for TG, Alb and Glo through CMLM, and 21 SNPs were significantly associated with TG, Chol, Glu, Alb and Glo through GLM. 4. Gene ontology showed that these SNPs were located within or near the candidate genes responsible for metabolic traits. 5. In conclusion, the identified candidate genes provided novel information for molecular mechanisms underlying metabolic traits. These findings are important in marker-assisted selection in the chicken breeding scheme.

Genomic dissection and prediction of feed intake and residual feed intake traits using a longitudinal model in F2 chickens.

Feed efficiency traits (FETs) are important economic indicators in poultry production. Because feed intake (FI) is a time-dependent variable, longitudinal models can provide insights into the genetic basis of FET variation over time. It is expected that the application of longitudinal models as part of genome-wide association (GWA) and genomic selection (i.e. genome-wide selection (GS)) studies will lead to an increase in accuracy of selection. Thus, the objectives of this study were to evaluate the accuracy of estimated breeding values (EBVs) based on pedigree as well as high-density single nucleotide polymorphism (SNP) genotypes, and to conduct a GWA study on longitudinal FI and residual feed intake (RFI) in a total of 312 chickens with phenotype and genotype in the F2 population. The GWA and GS studies reported in this paper were conducted using ß-spline random regression models for FI and RFI traits in a chicken F2 population, with FI and BW recorded for each bird weekly between 2 and 10 weeks of age. A single SNP regression approach was used on spline coefficients for weekly FI and RFI traits, with results showing that two significant SNPs for FI occur in the synuclein (SNCAIP) gene. Results also show that these regions are significantly associated with the spline coefficients (q 2) for 5- and 6-week-old birds, while GWA study results showed no SNP association with RFI in F2 chickens. Estimated breeding value predictions obtained using a pedigree-based best linear unbiased prediction (ABLUP) model were then compared with predictions based on genomic best linear unbiased prediction (GBLUP). The accuracy was measured as correlation between genomic EBV and EBV with the phenotypic value corrected for fixed effects divided by the square root of heritability. The regression of observed on predicted values was used to estimate bias of methods. Results show that prediction accuracies using GBLUP and ABLUP for the FI measured from 2nd to 10th week were between 0.06 and 0.46 and 0.03 and 0.37, respectively. These results demonstrate that genomic methods are able to increase the accuracy of predicted breeding values at later ages on the basis of both traits, and indicate that use of a longitudinal model can improve selection accuracy for the trajectory of traits in F2 chickens when compared with conventional methods.

Genome-wide association study of cell-mediated immune response in chicken.

Cell-mediated immunity (CMI) causes the intracellular destruction of the antigen or elimination of the host cell to make animals resistant against exogenous antigens and cancers. In this study, a genome-wide association study (GWAS) was carried out to identify genomic regions associated with CMI in chicken using chicken 60k high-density single nucleotide polymorphism (SNP) array. Genomic relationships were taken into account to adjust for population structure. In order to account for multiple testing, chromosome-wise false discovery rate was controlled at 5% and 10% levels. Moreover, a comparison of the power of fixed and mixed linear models based on genomic inflation factor was carried out. Mixed linear model (MLM) had better inflation rate, and therefore the results from MLM were used for subsequent analysis. Three significantly associated SNPs (FDR < 0.05) on chromosome 24 and linkage group E22C19W28_E50C23, and three suggestively associated SNPs (FDR < 0.1) on chromosome 1, 5 and 16 were identified. Pathway analysis showed that two biological pathways, which are related to immune response, were strongly associated with the candidate genes surrounding identified SNPs, and their influences were mostly on antigen processing and presentation, and cellular structure.

Association of mitochondrial function and sperm progressivity in slow- and fast-growing roosters.

The objectives of the current study were to investigate the activity of the electron transport chain (ETC) complexes, adenosine triphosphate (ATP), and reactive oxygen species (ROS) in the sperm mitochondria of 2 Iranian slow- and fast-growing chickens, namely native and Aryan strains. In this study, semen of 133 roosters was analyzed by computer-assisted sperm analysis (CASA). The roosters were categorized by sperm progressivity, as a low or high sperm motility phenotype. Mitochondrial complex I (NADH ubiquinone oxidoreductase), II (succinate dehydrogenase), III (ubiquinol cytochrome C reductase), IV (cytochrome C oxidase) activity, ATP, and ROS production in sperm were assayed. As a result, Aryan roosters with high progressive motility levels (HPL) had the greatest progressivity. Progressive motility was significant in in both strains (Aryan and native; P = 0.020) and motility levels (high and low; P = 0.007). The highest activity of complexes I was observed in Aryan with HPL roosters (P = 0.004). Native roosters with HPLs demonstrated higher complex activities I compared with Aryan and native roosters with low progressive motility levels (LPL) (P = 0.004). Significant differences were observed not only in the mitochondrial amounts of ATP of the strains (P = 0.000) but also between HPL and LPL (P = 0.003). The highest mitochondrial amounts of ATP was found in Aryan roosters with HPL (P = 0.021). Native roosters with LPL had the highest concentration of mitochondrial ROS (P = 0.033). The Aryan roosters with HPL, on the other hand, indicated less concentration of the mitochondrial ROS compared with Aryan having LPL and native roosters with HPL and LPL (P = 0.033). A significant difference was observed in the mitochondrial ROS between the strains (P = 0.004) and between HPL and LPL (P = 0.000). There was positive relationship between progressivity and each of mitochondrial complexes and ATP (r = 0.71, 0.62, 0.90 and 0.65 respectively). Based on our results, the sperm progressivity largely depends on the whole energy production originating in the mitochondrial compartment. Therefore, the deficiency in the function of mitochondria in the sperm and energy production could be responsible for low progressive motility.

Changes in the Expression of the Prolactin Receptor (PRLR) Gene in Different Physiological Stages in the Mammary Gland of the Iranian Adani Goat.

The actions of prolactin hormone are mediated by prolactin receptor (PRLR), and proliferation and differentiation of secretory mammary epithelium are dependent on the presence of its receptors. To understand the PRLR expression pattern in mammary gland of dairy goat during different lactation stages, in this study, we first estimated the milk yield breeding value by multitrait random regression model and then compared the expression of the gene in different physiological stage of mammary gland between high- and low-breeding value groups. We assayed the transcription level of the gene by quantitative real-time PCR method, and its outcomes were analysed by a statistical model containing breeding value groups, sampling times and their interactions as fixed effects. The results indicated that the expression levels of PRLR gene were significantly upregulated in the drying stage (p < 0.01). The transcription pattern of the gene was significantly different between the two breeding value groups (p < 0.01), so that the amount of PRLR mRNA was significantly higher in the low-breeding value groups of animals in the lactation stage (p < 0.01). Based on the results of this study, it could be suggested that the abundance of PRLR transcripts in mammary gland of goat might be changed by some physiological, environmental and genetic factors. Nucleotide variations in the promoter region might be resulted in various transcription activities of the gene which should be studied in a complementary research.

Mitochondrial DNA D-loop sequence variation in maternal lineages of Iranian native horses.

To understand the origin and genetic diversity of Iranian native horses, mitochondrial DNA (mtDNA) D-loop sequences were generated for 95 horses from five breeds sampled in eight geographical locations in Iran. Sequence analysis of a 247-bp segment revealed a total of 27 haplotypes with 38 polymorphic sites. Twelve of 19 mtDNA haplogroups were identified in the samples. The most common haplotypes were found within haplogroup X2. Within-population haplotype and nucleotide diversities of the five breeds ranged from 0.838 ± 0.056 to 0.974 ± 0.022 and 0.011 ± 0.002 to 0.021 ± 0.001 respectively, indicating a relatively high genetic diversity in Iranian horses. The identification of several ancient sequences common between the breeds suggests that the lineage of the majority of Iranian horse breeds is old and obviously originated from a vast number of mares. We found in all native Iranian horse breeds lineages of the haplogroups D and K, which is concordant with the previous findings of Asian origins of these haplogroups. The presence of haplotypes E and K in our study also is consistent with a geographical west-east direction of increasing frequency of these haplotypes and a genetic fusion in Iranian horse breeds.

Taylor's frozen-flow hypothesis in Burgers turbulence.

By detailed analytical treatment of the shock dynamics in the Burgers turbulence with large scale forcing we calculate the velocity structure functions between pairs of points displaced both in time and space. Our analytical treatment verifies the so-called Taylor's frozen-flow hypothesis without relying on any closure and under very general assumptions. We discuss the limitation of the hypothesis and show that it is valid up to time scales smaller than the correlation time scale of temporal velocity correlation function. We support the analytical calculation by performing numerical simulation of the periodically kicked Burgers equation.

Linkage mapping of the locus responsible for forelimb-girdle muscular anomaly of Japanese black cattle on bovine chromosome 26.

Forelimb-girdle muscular anomaly is an autosomal recessive disorder of Japanese black cattle characterized by tremor, astasia and abnormal shape of the shoulders. Pathological examination of affected animals reveals hypoplasia of forelimb-girdle muscles with reduced diameter of muscle fibres. To identify the gene responsible for this disorder, we performed linkage mapping of the disorder locus using an inbred pedigree including a great-grand sire, a grand sire, a sire and 26 affected calves obtained from a herd of Japanese black cattle. Two hundred and fifty-eight microsatellite markers distributed across the genome were genotyped across the pedigree. Four markers on the middle region of bovine chromosome 26 showed significant linkage with the disorder locus. Haplotype analysis using additional markers in this region refined the critical region of the disorder locus to a 3.5-Mb interval on BTA26 between BM4505 and MOK2602. Comparative mapping data revealed several potential candidate genes for the disorder, including NRAP, PDZD8 and HSPA12A, which are associated with muscular function.

Intermittency of height fluctuations in stationary state of the Kardar-Parisi-Zhang equation with infinitesimal surface tension in 1+1 dimensions.

The Kardar-Parisi-Zhang (KPZ) equation with infinitesimal surface tension, dynamically develops sharply connected valley structures within which the height derivative is not continuous. We discuss the intermittency issue in the problem of stationary state forced KPZ equation in 1+1 dimensions. It is proved that the moments of height increments C(a) = <|h(x(1)) - h(x(2))|(a)> behave as |x(1) - x(2)|(xi(a)) with xi(a) = a for length scales |x(1) - x(2)|<