Extensions of Noether's Second Theorem: from continuous to discrete systems

A simple local proof of Noether's Second Theorem is given. This proof immediately leads to a generalization of the theorem, yielding conservation laws and/or explicit relationships between the Euler--Lagrange equations of any variational problem whose symmetries depend upon a set of free or partly-constrained functions. Our approach extends further to deal with finite difference systems. The results are easy to apply; several well-known continuous and discrete systems are used as illustrations

Rate of lineage origin explains the diversity anomaly in the World’s mangrove vegetation

The contribution of nonecological factors to global patterns in diversity is evident when species richness differs between regions with similar habitats and geographic area. Mangrove environments in the Eastern Hemisphere harbor six times as many species of trees and shrubs as similar environments in the New World. Genetic divergence of mangrove lineages from terrestrial relatives, in combination with fossil evidence, suggests that mangrove diversity is limited by evolutionary transition into the stressful marine environment, the number of mangrove lineages has increased steadily over the Tertiary with little global extinction, and the diversity anomaly in mangrove vegetation reflects regional differences in the rate of origin of new mangrove lineages

A new, flaccid, decurrent leaf variety of Juniperus poblana from Mexico: J. poblana var. decurrens R. P. Adams

Analyses of nrDNA and four cp DNAs (petN-psbM, trnS-trnG, trnD-trnT, trnL-trnF) plus morphology and leaf essential oils revealed that the weeping (flaccid), decurrent leafed junipers near Topia, Durango are closely related to J. poblana (formerly J. flaccida var. poblana) and should be recognized as a new variety, J. poblana var. decurrens R. P. Adams var. nov. The leaf oil of J. p. var. decurrens is dominated by α-pinene (53.2%) with moderate amounts of β-pinene (5.3%), myrcene (4.3%), δ-2-carene (1.2%), δ-3-carene (2.5%), limonene (3.2%), β-phellandrene (3.1%), terpinolene (1.0%), (E)-caryphyllene (1.1%), and germacrene D (1.5%) and shares eleven unique terpenes with J. poblana

Chloroplast capture in Juniperus sabina var. balkanensis R. P. Adams and A. N. Tashev, from the Balkan peninsula: A new variety with a history of hybridization with J. thurifera

example of chloroplast capture has been identified in Juniperus sabina from Bulgaria and Greece in the Balkan peninsula. Nuclear DNA and overall morphology clearly indicate a close relationship to Juniperus sabina, whereas the cpDNA from these populations is very uniform and is nearly identical to that of J. thurifera, an unrelated species currently growing in France, Spain and Morocco. The new taxon is recognized as Juniperus sabina var. balkanensis R. P. Adams and A. Tashev. At present, this new variety is known only from locations in Bulgaria and Greece

Chloroplast capture by a new variety, Juniperus sabina var. balkanensis R. P. Adams and A. N. Tashev, from the Balkan peninsula: A putative stabilized relictual hybrid between J. sabina and ancestral J. thurifera

An example of chloroplast capture has been found in Juniperus sabina from Bulgaria and Greece in the Balkan peninsula. The cpDNA from these populations is very uniform and is nearly identical to that of J. thurifera (currently growing in France, Spain and Morocco). The new taxon is recognized as Juniperus sabina var. balkanensis R. P. Adams and A. Tashev. At present, the variety, with the thurifera cpDNA, is known only from Bulgaria and Greece

Long distance gene flow facilitated by bird-dispersed seeds in wind-pollinated species: A story of hybridization and introgression between Juniperus ashei and J. ovata told by nrDNA and cpDNA

nrDNA and cpDNA were sequenced of J. ashei and J. ovata from populations throughout their ranges. No J. ashei populations were found to be pure in their nrDNA for every tree, however all J. ashei trees in every population contained only the pure J. ashei chloroplast type. Populations of J. ovata in trans-Pecos Texas were almost pure in both nrDNA and cp DNA. Several plants in the J. ashei range contained J. ovata-type nrDNA (White Cliffs, AR, 3/10); Ranger, TX (1/5); Waco, TX (1/12). Every J ashei population contained at least 1 plant with hybrid (heterozygous) nrDNA and 3 J. ovata populations contained putative hybrids (by nrDNA), but one population had only pure J. ovata trees. The presence of ovata germplasm within J. ashei populations seems best explained by long distance bird dispersal of J. ovata seeds (thence seedlings and J. ovata trees and hybrids) in the disjunct J. ashei populations. The reason for the absence of ovata paternal cp, which is distributed by pollen in J. ashei populations is not known. Judged by distribution of cp data, there is very little movement of cp genomes. In contrast, nrDNA polymorphisms indicate there is considerable gene flow between J. ashei and J. ovata, but primarily in the direction of J. ovata to J. ashei which may be explained by a combination of bird migration pattern and recurring and preferential F1-hybrid formation

Chloroplast capture by a new variety, Juniperus sabina var. balkanensis R. P. Adams and A. N. Tashev, from the Balkan peninsula: A putative stabilized relictual hybrid between J. sabina and ancestral J. thurifera

An example of chloroplast capture has been found in Juniperus sabina from Bulgaria and Greece in the Balkan peninsula. The cpDNA from these populations is very uniform and is nearly identical to that of J. thurifera (currently growing in France, Spain and Morocco). The new taxon is recognized as Juniperus sabina var. balkanensis R. P. Adams and A. Tashev. At present, the variety, with the thurifera cpDNA, is known only from Bulgaria and Greece

Non-Commutative Batalin-Vilkovisky Algebras, Homotopy Lie Algebras and the Courant Bracket

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent \Delta operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie brackets with a fixed nilpotent Lie algebra element Q. We find the most general Jacobi-like identity that such a hierarchy satisfies. The numerical coefficients in front of each term in these generalized Jacobi identities are related to the Bernoulli numbers. We suggest that the definition of a homotopy Lie algebra should be enlarged to accommodate this important case. Finally, we consider the Courant bracket as an example of a derived bracket. We extend it to the "big bracket" of exterior forms and multi-vectors, and give closed formulas for the higher Courant brackets.Comment: 42 pages, LaTeX. v2: Added remarks in Section 5. v3: Added further explanation. v4: Minor adjustments. v5: Section 5 completely rewritten to include covariant construction. v6: Minor adjustments. v7: Added references and explanation to Section

Jacobi structures revisited

Jacobi algebroids, that is graded Lie brackets on the Grassmann algebra associated with a vector bundle which satisfy a property similar to that of the Jacobi brackets, are introduced. They turn out to be equivalent to generalized Lie algebroids in the sense of Iglesias and Marrero and can be viewed also as odd Jacobi brackets on the supermanifolds associated with the vector bundles. Jacobi bialgebroids are defined in the same manner. A lifting procedure of elements of this Grassmann algebra to multivector fields on the total space of the vector bundle which preserves the corresponding brackets is developed. This gives the possibility of associating canonically a Lie algebroid with any local Lie algebra in the sense of Kirillov.Comment: 20 page

On classification of discrete, scalar-valued Poisson Brackets

We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the intersection points of certain projective hypersurfaces. In addition, they can be reduced to cubic PB of standard Volterra lattice by discrete Miura-type transformations. Finally, improving a consolidation lattice procedure, we obtain new families of non-degenerate, vector-valued and first order dDGPBs, which can be considered in the framework of admissible Lie-Poisson group theory.Comment: 24 page