A volumetric Penrose inequality for conformally flat manifolds

We consider asymptotically flat Riemannian manifolds with nonnegative scalar curvature that are conformal to $\R^{n}\setminus \Omega, n\ge 3$, and so that their boundary is a minimal hypersurface. (Here, $\Omega\subset \R^{n}$ is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by $(V/\beta_{n})^{(n-2)/n}$, where $V$ is the Euclidean volume of $\Omega$ and $\beta_{n}$ is the volume of the Euclidean unit $n$-ball. This gives a partial proof to a conjecture of Bray and Iga \cite{brayiga}. Surprisingly, we do not require the boundary to be outermost.Comment: 7 page

Weighted Sobolev spaces of radially symmetric functions

We prove dilation invariant inequalities involving radial functions, poliharmonic operators and weights that are powers of the distance from the origin. Then we discuss the existence of extremals and in some cases we compute the best constants.Comment: 38 page

Moment inversion problem for piecewise D-finite functions

We consider the problem of exact reconstruction of univariate functions with jump discontinuities at unknown positions from their moments. These functions are assumed to satisfy an a priori unknown linear homogeneous differential equation with polynomial coefficients on each continuity interval. Therefore, they may be specified by a finite amount of information. This reconstruction problem has practical importance in Signal Processing and other applications. It is somewhat of a ``folklore'' that the sequence of the moments of such ``piecewise D-finite''functions satisfies a linear recurrence relation of bounded order and degree. We derive this recurrence relation explicitly. It turns out that the coefficients of the differential operator which annihilates every piece of the function, as well as the locations of the discontinuities, appear in this recurrence in a precisely controlled manner. This leads to the formulation of a generic algorithm for reconstructing a piecewise D-finite function from its moments. We investigate the conditions for solvability of the resulting linear systems in the general case, as well as analyze a few particular examples. We provide results of numerical simulations for several types of signals, which test the sensitivity of the proposed algorithm to noise

Continuous-Wave Second-Harmonic Generation in Orientation-Patterned Gallium Phosphide Waveguides at Telecom Wavelengths

A new process to produce orientation-patterned gallium phosphide (OP-GaP) on GaAs with almost perfectly parallel domain boundaries is presented. Taking advantage of the chemical selectivity between phosphides and arsenides, OP-GaP is processed into suspended shallow-ridge waveguides. Efficient second-harmonic generation from telecom wavelengths is achieved in both Type-I and Type-II polarisation configurations. The highest observed conversion efficiency is 200% W–1 cm–2, with a bandwidth of 2.67 nm in a 1 mm-long waveguide. The variation of the conversion efficiency with wavelength closely follows a squared cardinal sine function, in excellent agreement with theory, confirming the good uniformity of the poling period over the entire length of the waveguide

The climatic and genetic heritage of Italian goat breeds with genomic SNP data

Local adaptation of animals to the environment can abruptly become a burden when faced with rapid climatic changes such as those foreseen for the Italian peninsula over the next 70 years. Our study investigates the genetic structure of the Italian goat populations and links it with the environment and how genetics might evolve over the next 50 years. We used one of the largest national datasets including > 1000 goats from 33 populations across the Italian peninsula collected by the Italian Goat Consortium and genotyped with over 50 k markers. Our results showed that Italian goats can be discriminated in three groups reflective of the Italian geography and its geo\u2011political situation preceding the country unification around two centuries ago. We leveraged the remarkable genetic and geographical diversity of the Italian goat populations and performed landscape genomics analysis to disentangle the relationship between genotype and environment, finding 64 SNPs intercepting genomic regions linked to growth, circadian rhythm, fertility, and inflammatory response. Lastly, we calculated the hypothetical future genotypic frequencies of the most relevant SNPs identified through landscape genomics to evaluate their long\u2011term effect on the genetic structure of the Italian goat populations. Our results provide an insight into the past and the future of the Italian local goat populations, helping the institutions in defining new conservation strategy plans that could preserve their diversity and their link to local realities challenged by climate change

Runs of homozygosity in the Italian goat breeds: impact of management practices in low-input systems

Background: Climate and farming systems, several of which are considered as low-input agricultural systems, vary between goat populations from Northern and Southern Italy and have led to different management practices. These processes have impacted genome shaping in terms of inbreeding and regions under selection and resulted in differences between the northern and southern populations. Both inbreeding and signatures of selection can be pinpointed by the analysis of runs of homozygosity (ROH), which provides useful information to assist the management of this species in different rural areas. Results: We analyzed the ROH distribution and inbreeding (FROH) in 902 goats from the Italian Goat Consortium2 dataset. We evaluated the differences in individual ROH number and length between goat breeds from Northern (NRD) and Central-southern (CSD) Italy. Then, we identified the signatures of selection that differentiate these two groups using three methods: ROH, ΔROH, and averaged FST. ROH analyses showed that some Italian goat breeds have a lower inbreeding coefficient, which is attributable to their management and history. ROH are longer in breeds that are undergoing non-optimal management or with small population size. In several small breeds, the ROH length classes are balanced, reflecting more accurate mating planning. The differences in climate and management between the NRD and CSD groups have resulted in different ROH lengths and numbers: the NRD populations bred in isolated valleys present more and shorter ROH segments, while the CSD populations have fewer and longer ROH, which is likely due to the fact that they have undergone more admixture events during the horizontal transhumance practice followed by a more recent standardization. We identified four genes within signatures of selection on chromosome 11 related to fertility in the NRD group, and 23 genes on chromosomes 5 and 6 related to growth in the CSD group. Finally, we identified 17 genes on chromosome 12 related to environmental adaptation and body size with high homozygosity in both groups. Conclusions: These results show how different management practices have impacted the level of genomic inbreeding in two Italian goat groups and could be useful to assist management in a low-input system while safeguarding the diversity of small populations

The climatic and genetic heritage of Italian goat breeds with genomic SNP data

GENOTYPE DATABASE ITALIAN GOAT CONSORTIU

Signatures of selection and environmental adaptation across the goat genome post-domestication

Background: Since goat was domesticated 10,000 years ago, many factors have contributed to the differentiation of goat breeds and these are classified mainly into two types: (i) adaptation to different breeding systems and/or purposes and (ii) adaptation to different environments. As a result, approximately 600 goat breeds have developed worldwide; they differ considerably from one another in terms of phenotypic characteristics and are adapted to a wide range of climatic conditions. In this work, we analyzed the AdaptMap goat dataset, which is composed of data from more than 3000 animals collected worldwide and genotyped with the CaprineSNP50 BeadChip. These animals were partitioned into groups based on geographical area, production uses, available records on solid coat color and environmental variables including the sampling geographical coordinates, to investigate the role of natural and/or artificial selection in shaping the genome of goat breeds. Results: Several signatures of selection on different chromosomal regions were detected across the different breeds, sub-geographical clusters, phenotypic and climatic groups. These regions contain genes that are involved in important biological processes, such as milk-, meat- or fiber-related production, coat color, glucose pathway, oxidative stress response, size, and circadian clock differences. Our results confirm previous findings in other species on adaptation to extreme environments and human purposes and provide new genes that could explain some of the differences between goat breeds according to their geographical distribution and adaptation to different environments. Conclusions: These analyses of signatures of selection provide a comprehensive first picture of the global domestication process and adaptation of goat breeds and highlight possible genes that may have contributed to the differentiation of this species worldwide

The eigenvalue problem for the ∞-Bilaplacian

We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ∞:=infu∈W2,∞(Ω)∖{0}∥Δu∥L∞(Ω)∥u∥L∞(Ω), Λ∞:=infu∈W2,∞(Ω)∖{0}‖Δu‖L∞(Ω)‖u‖L∞(Ω), where Ω⊆RnΩ⊆Rn is a bounded C1,1C1,1 domain and W2,∞(Ω)W2,∞(Ω) is a class of weakly twice differentiable functions satisfying either u=0u=0 on ∂Ω∂Ω or u=|Du|=0u=|Du|=0 on ∂Ω∂Ω . Our first main result, obtained through approximation by LpLp -problems as p→∞p→∞ , is the existence of a minimiser u∞∈W2,∞(Ω)u∞∈W2,∞(Ω) satisfying {Δu∞∈Λ∞Sgn(f∞)Δf∞=μ∞ a.e. in Ω, in D′(Ω), {Δu∞∈Λ∞Sgn(f∞) a.e. in Ω,Δf∞=μ∞ in D′(Ω), for some f∞∈L1(Ω)∩BVloc(Ω)f∞∈L1(Ω)∩BVloc(Ω) and a measure μ∞∈M(Ω)μ∞∈M(Ω) , for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue Λ∞Λ∞ on the domain, establishing the validity of a Faber–Krahn type inequality: among all C1,1C1,1 domains with fixed measure, the ball is a strict minimiser of Ω↦Λ∞(Ω)Ω↦Λ∞(Ω) . This result is shown to hold true for either choice of boundary conditions and in every dimension

Genome-wide SNP profiling of worldwide goat populations reveals strong partitioning of diversity and highlights post-domestication migration routes

Background: Goat populations that are characterized within the AdaptMap project cover a large part of the worldwide distribution of this species and provide the opportunity to assess their diversity at a global scale. We analysed genome-wide 50 K single nucleotide polymorphism (SNP) data from 144 populations to describe the global patterns of molecular variation, compare them to those observed in other livestock species, and identify the drivers that led to the current distribution of goats. Results: A high degree of genetic variability exists among the goat populations studied. Our results highlight a strong partitioning of molecular diversity between and within continents. Three major gene pools correspond to goats from Europe, Africa and West Asia. Dissection of sub-structures disclosed regional gene pools, which reflect the main post-domestication migration routes. We also identified several exchanges, mainly in African populations, and which often involve admixed and cosmopolitan breeds. Extensive gene flow has taken place within specific areas (e.g., south Europe, Morocco and Mali-Burkina Faso-Nigeria), whereas elsewhere isolation due to geographical barriers (e.g., seas or mountains) or human management has decreased local gene flows. Conclusions: After domestication in the Fertile Crescent in the early Neolithic era (ca. 12,000 YBP), domestic goats that already carried differentiated gene pools spread to Europe, Africa and Asia. The spread of these populations determined the major genomic background of the continental populations, which currently have a more marked subdivision than that observed in other ruminant livestock species. Subsequently, further diversification occurred at the regional level due to geographical and reproductive isolation, which was accompanied by additional migrations and/or importations, the traces of which are still detectable today. The effects of breed formation were clearly detected, particularly in Central and North Europe. Overall, our results highlight a remarkable diversity that occurs at the global scale and is locally partitioned and often affected by introgression from cosmopolitan breeds. These findings support the importance of long-term preservation of goat diversity, and provide a useful framework for investigating adaptive introgression, directing genetic improvement and choosing breeding targets