Combinatorial Identities Associated with a Multidimensional Polynomial Sequence

In this paper we combine the knowledge of different structures of a special Appell multidimensional polynomial sequence with the problem of establishing combinatorial identities. The elements of this special polynomial sequence have values in a Clifford algebra, are homogeneous hypercomplex differentiable functions of different degrees and their coefficients properties can be used to stress interesting matrix and combinatorial relations

Non-symmetric Number Triangles Arising from Hypercomplex Function Theory in IR^{n+1}

The paper is focused on intrinsic properties of a one parameter family of non-symmetric number triangles T(n), n ≥ 2, which arises in the construction of hyperholomorphic Appell polynomials

TOTALLY REGULAR VARIABLES AND APPELL SEQUENCES IN HYPERCOMPLEX FUNCTION THEORY

The aim of our contribution is to clarify the relation between totally regular variables and Appell sequences of hypercomplex holomorphic polynomials (sometimes simply called monogenic power-like functions) in Hypercomplex Function Theory. After their introduction in 2006 by two of the authors of this note on the occasion of the 17th IKM, the latter have been subject of investigations by different authors with different methods and in various contexts. The former concept, introduced by R. Delanghe in 1970 and later also studied by K. Gürlebeck in 1982 for the case of quaternions, has some obvious relationship with the latter, since it describes a set of linear hypercomplex holomorphic functions all power of which are also hypercomplex holomorphic. Due to the non-commutative nature of the underlying Clifford algebra, being totally regular variables or Appell sequences are not trivial properties as it is for the integer powers of the complex variable z=x+ iy. Simple examples show also, that not every totally regular variable and its powers form an Appell sequence and vice versa. Under some very natural normalization condition the set of all para-vector valued totally regular variables which are also Appell sequences will completely be characterized. In some sense the result can also be considered as an answer to a remark of K. Habetha in chapter 16: Function theory in algebras of the collection Complex analysis. Methods, trends, and applications, Akademie-Verlag Berlin, (Eds. E. Lanckau and W. Tutschke) 225-237 (1983) on the use of exact copies of several complex variables for the power series representation of any hypercomplex holomorphic function

3D-MAPPINGS AND THEIR APPROXIMATION BY SERIES OF POWERS OF A SMALL PARAMETER

In classical complex function theory the geometric mapping property of conformality is closely linked with complex differentiability. In contrast to the planar case, in higher dimensions the set of conformal mappings is only the set of Möbius transformations. Unfortunately, the theory of generalized holomorphic functions (by historical reasons they are called monogenic functions) developed on the basis of Clifford algebras does not cover the set of Möbius transformations in higher dimensions, since Möbius transformations are not monogenic. But on the other side, monogenic functions are hypercomplex differentiable functions and the question arises if from this point of view they can still play a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. On the occasion of the 16th IKM 3D-mapping methods based on the application of Bergman's reproducing kernel approach (BKM) have been discussed. Almost all authors working before that with BKM in the Clifford setting were only concerned with the general algebraic and functional analytic background which allows the explicit determination of the kernel in special situations. The main goal of the abovementioned contribution was the numerical experiment by using a Maple software specially developed for that purpose. Since BKM is only one of a great variety of concrete numerical methods developed for mapping problems, our goal is to present a complete different from BKM approach to 3D-mappings. In fact, it is an extension of ideas of L. V. Kantorovich to the 3-dimensional case by using reduced quaternions and some suitable series of powers of a small parameter. Whereas until now in the Clifford case of BKM the recovering of the mapping function itself and its relation to the monogenic kernel function is still an open problem, this approach avoids such difficulties and leads to an approximation by monogenic polynomials depending on that small parameter

Microbial environment shapes immune function and cloacal microbiota dynamics in zebra finches <i>Taeniopygia guttata</i>

BACKGROUND: The relevance of the host microbiota to host ecology and evolution is well acknowledged. However, the effect of the microbial environment on host immune function and host microbiota dynamics is understudied in terrestrial vertebrates. Using a novel experimental approach centered on the manipulation of the microbial environment of zebra finches Taeniopygia guttata, we carried out a study to investigate effects of the host's microbial environment on: 1) constitutive immune function, 2) the resilience of the host cloacal microbiota; and 3) the degree to which immune function and host microbiota covary in microbial environments that differ in diversity. RESULTS: We explored immune indices (hemagglutination, hemolysis, IgY levels and haptoglobin concentration) and host-associated microbiota (diversity and composition) in birds exposed to two experimental microbial environments differing in microbial diversity. According to our expectations, exposure to experimental microbial environments led to differences related to specific antibodies: IgY levels were elevated in the high diversity treatment, whereas we found no effects for the other immune indices. Furthermore, according to predictions, we found significantly increased richness of dominant OTUs for cloacal microbiota of birds of the high diversity compared with the low diversity group. In addition, cloacal microbiota of individual females approached their baseline state sooner in the low diversity environment than females in the high diversity environment. This result supported a direct phenotypically plastic response of host microbiota, and suggests that its resilience depends on environmental microbial diversity. Finally, immune indices and cloacal microbiota composition tend to covary within treatment groups, while at the same time, individuals exhibited consistent differences of immune indices and microbiota characteristics. CONCLUSION: We show that microbes in the surroundings of terrestrial vertebrates can influence immune function and host-associated microbiota dynamics over relatively short time scales. We suggest that covariation between immune indices and cloacal microbiota, in addition to large and consistent differences among individuals, provides potential for evolutionary adaptation. Ultimately, our study highlights that linking environmental and host microbiotas may help unravelling immunological variation within and potentially among species, and together these efforts will advance the integration of microbial ecology and ecological immunology

Pascal trapezoids emerging from hypercomplex polynomial sequences

The construction of two di erent representations of special Appell polynomials in (n+1) real variables with values in a Clifford algebra suggested to explore the relation between the respective coe cients. Properties of sequences resulting from such relation and an interesting trapezoidal array of their elements are pointed out.publishe

The number of zeros of unilateral polynomials over coquaternions revisited

The literature on quaternionic polynomials and, in particular, on methods for finding and classifying their zero sets, is fast developing and reveals a growing interest in this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovska and Opfer [Electron Trans Numer Anal. 2017;46:55-70], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, zeros. In this paper we present a full proof of this result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.Research at CMAT was financed by Portuguese Funds through FCT -Fundacao para a Ciencia e a Tecnologia, within the [project number UID/MAT/00013/2013]. Research at NIPE was carried out within the funding with COMPETE reference number POCI-01-0145-FEDER-006683 [project number UID/ECO/03182/2013], with the FCT/MEC's (Fundacao para a Ciencia e a Tecnologia, I.P.) financial support through national funding and by the ERDF through the Operational Programme on 'Competitiveness and Internationalization - COMPETE 2020' under the PT2020 Partnership Agreement

The microbial environment modulates non-genetic maternal effects on egg immunity

BACKGROUND: In a diverse microbial world immune function of animals is essential. Diverse microbial environments may contribute to extensive variation in immunological phenotypes of vertebrates, among and within species and individuals. As maternal effects benefit offspring development and survival, whether females use cues about their microbial environment to prime offspring immune function is unclear. To provide microbial environmental context to maternal effects, we asked if the bacterial diversity of the living environment of female zebra finches Taeniopygia guttata shapes maternal effects on egg immune function. We manipulated environmental bacterial diversity of birds and tested if females increased immunological investment in eggs in an environment with high bacterial diversity (untreated soil) versus low (gamma-sterilized soil). We quantified lysozyme and ovotransferrin in egg albumen and IgY in egg yolk and in female blood, and we used 16S rRNA gene sequencing to profile maternal cloacal and eggshell microbiotas. RESULTS: We found a maternal effect on egg IgY concentration that reflected environmental microbial diversity: females who experienced high diversity deposited more IgY in their eggs, but only if maternal plasma IgY levels were relatively high. We found no effects on lysozyme and ovotransferrin concentrations in albumen. Moreover, we uncovered that variation in egg immune traits could be significantly attributed to differences among females: for IgY concentration in yolk repeatability R = 0.80; for lysozyme concentration in albumen R = 0.27. Furthermore, a partial least squares path model (PLS-PM) linking immune parameters of females and eggs, which included maternal and eggshell microbiota structures and female body condition, recapitulated the treatment-dependent yolk IgY response. The PLS-PM additionally suggested that the microbiota and physical condition of females contributed to shaping maternal effects on egg immune function, and that (non-specific) innate egg immunity was prioritized in the environment with low bacterial diversity. CONCLUSIONS: The microbial environment of birds can shape maternal effects on egg immune function. Since immunological priming of eggs benefits offspring, we highlight that non-genetic maternal effects on yolk IgY levels based on cues from the parental microbial environment may prove important for offspring to thrive in the microbial environment that they are expected to face

Hypercomplex polynomials, vietoris’ rational numbers and a related integer numbers sequence

This paper aims to give new insights into homogeneous hypercomplex Appell polynomials through the study of some interesting arithmetical properties of their coefficients. Here Appell polynomials are introduced as constituting a hypercomplex generalized geometric series whose fundamental role sometimes seems to have been neglected. Surprisingly, in the simplest non-commutative case their rational coefficient sequence reduces to a coefficient sequence S used in a celebrated theorem on positive trigonometric sums by Vietoris (Sitzungsber Österr Akad Wiss 167:125–135, 1958). For S a generating function is obtained which allows to derive an interesting relation to a result deduced by Askey and Steinig (Trans AMS 187(1):295–307, 1974) about some trigonometric series. The further study of S is concerned with a sequence of integers leading to its irreducible representation and its relation to central binomial coefficients.The work of the first and third authors was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013. The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio