Search

In this paper, we give closed-form expressions of the p-Frobenius number for the triple of the generalized star numbers an(n−1)+1 for an integer a≥4. When a=6, it is reduced to the famous star number. For the set of given positive integers {a1,a2,…,ak}, the p-Frobenius number is the largest integer N whose number of non-negative integer representations N=a1x1+a2x2+⋯+akxk is at most p. When p=0, the 0-Frobenius number is the classical Frobenius number, which is the central topic of the famous linear Diophantine problem of Frobenius.

We give an explicit formula for the p-Frobenius number of triples associated with Diophantine Equations x2−y2=zr (r≥2), that is, the largest positive integer that can only be represented in p ways by combining the three integers of the solutions of Diophantine equations x2−y2=zr. This result is also a generalization because if r=2 and p=0, the (0-)Frobenius number of the Pythagorean triple has already been given. To find p-Frobenius numbers, we use geometrically easy to understand figures of the elements of the p-Apéry set, which exhibits symmetric appearances.

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let a1,a2,…,al be positive integers such that their greatest common divisor is one. For a nonnegative integer p, denote the p-Frobenius number by gp(a1,a2,…,al), which is the largest integer that can be represented at most p ways by a linear combination with nonnegative integer coefficients of a1,a2,…,al. When p=0, the 0-Frobenius number is the classical Frobenius number. When l=2, the p-Frobenius number is explicitly given. However, when l=3 and even larger, even in special cases, it is not easy to give the Frobenius number explicitly. It is even more difficult when p>0, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers [1] or of repunits [2] for the case where l=3. In this paper, we show the explicit formula for the Fibonacci triple when p>0. In addition, we give an explicit formula for the p-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most p ways. Furthermore, explicit formulas are shown concerning the Lucas triple.

For given positive integers a1,a2,⋯,ak with gcd(a1,a2,⋯,ak)=1, the denumerant d(n)=d(n;a1,a2,⋯,ak) is the number of nonnegative solutions (x1,x2,⋯,xk) of the linear equation a1x1+a2x2+⋯+akxk=n for a positive integer n. For a given nonnegative integer p, let Sp=Sp(a1,a2,⋯,ak) be the set of all nonnegative integer n’s such that d(n)>p. In this paper, by introducing the p-numerical semigroup, where the set N0\Sp is finite, we give explicit formulas of the p-Frobenius number, which is the maximum of the set N0\Sp, and related values for the triple of arithmetic progressions. The main aim is to determine the elements of the p-Apéry set.

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and the p-genus of generalized Fibonacci numerical semigroups. Here, the p-numerical semigroup Sp is defined as the set of integers whose nonnegative integral linear combinations of given positive integers a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with the 0-Frobenius number and the 0-genus is the original numerical semigroup with the Frobenius number and the genus. In this paper, we consider the p-numerical semigroup involving Jacobsthal polynomials, which include Fibonacci numbers as special cases. We can also deal with the Jacobsthal–Lucas polynomials, including Lucas numbers accordingly. An application on the p-Hilbert series is also provided. There are some interesting connections between Frobenius numbers and geometric and algebraic structures that exhibit symmetry properties.

In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy, and Euler numbers. Cauchy numbers can be generalized to the hypergeometric Cauchy numbers. Recently, Barman et al. study more general numbers in terms of determinants, which involve Bernoulli, Euler and Lehmer's generalized Euler numbers. However, Cauchy numbers and their generalizations are not involved in these generalized numbers. In this paper, we study more general numbers in terms of determinants, which involve Cauchy numbers. The motivations and backgrounds of the definition are in an operator related to graph theory. We also give several expressions and identities by Trudi's and inversion formulae.

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants.

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p.

The truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers are defined as analogues of hypergeometric Bernoulli numbers and hypergeometric Cauchy numbers, and as extensions of Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of incomplete Stirling-Carlitz numbers. In this paper, we give several expressions of truncated Bernoulli-Carlitz numbers and truncated Cauchy-Carlitz numbers as natural extensions. One kind of expressions is in continued fractions. Another is in determinants originated in Glaisher, giving several interesting determinant expressions of numbers, including Bernoulli and Cauchy numbers.

In this paper, we introduce the hypergeometric Euler number as an analogue of the hypergeometric Bernoulli number and the hypergeometric Cauchy number. We study several expressions and sums of products of hypergeometric Euler numbers. We also introduce complementary hypergeometric Euler numbers and give some characteristic properties. There are strong reasons why these hypergeometric numbers are important. The hypergeometric numbers have one of the advantages that yield the natural extensions of determinant expressions of the numbers, though many kinds of generalizations of the Euler numbers have been considered by many authors.

Abstract In this paper, we give several characteristics of hypergeometric Bernoulli numbers, including several identities for hypergeometric Bernoulli numbers which the convergents of the continued fraction expansion of the generating function of the hypergeometric Bernoulli numbers entail. We show an analog of Kummer’s congruences in the classical Bernoulli numbers. We also give some determinant expressions of hypergeometric Bernoulli numbers and some relations between the hypergeometric and the classical Bernoulli numbers. By applying Trudi’s formula, we have some different expressions and inversion relations.

There are many kinds of generalizations of Cauchy numbers and polynomials. Recently, a parametric type of the Bernoulli numbers with level 3 was introduced and studied as a kind of generalization of Bernoulli polynomials. A parametric type of Cauchy numbers with level 3 is its analogue. In this paper, as an analogue of a parametric type of Bernoulli polynomials with level 3 and its extension, we introduce a parametric type of Cauchy polynomials with a higher level. We present their characteristic and combinatorial properties. By using recursions, we show some determinant expressions.

The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ-function using the regular representation for the group and the Schur-polynomials. The υ-HF SCF theory on an infinite-dimensional Fock space F∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.

The generalized Frobenius number is the largest integer represented in at most p ways by a linear combination of nonnegative integers of given positive integers a1, a2, ..., ak. When p = 0, it reduces to the classical Frobenius number. In this paper, we give the generalized Frobenius number when aj = (bn+j-1-1)/(b - 1) (b ≥ 2) as a generalization of the result of p = 0 in [16]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a parametric type of Cauchy polynomials and study their characteristic and combinatorial properties. In particular, we show some determinant expressions. [ABSTRACT FROM AUTHOR]

Poly-Cauchy numbers with level 2 are defined by inverse sine hyperbolic functions with the inverse relation from sine hyperbolic functions. In this paper, we introduce the Stirling numbers of the first kind with level 2 in order to establish some relations with poly-Cauchy numbers with level 2. Then, we show several convolution identities of poly-Cauchy numbers with level 2. In particular, that of three poly-Cauchy numbers with level 2 can be expressed as a simple form. [ABSTRACT FROM AUTHOR]

Tibetan weedy barleys reside at the edges of qingke (hulless barley) fields in Tibet (Xizang). The spikes of these weedy barleys contain or lack a brittle rachis, with either two- or six-rowed spikes and either hulled or hulless grains at maturity. Although the brittle rachis trait of Tibetan weedy barleys is similar to that of wild barley (Hordeum vulgare ssp. spontaneum Thell.), these plants share genetic similarity with domesticated barley. The origin of Tibetan weedy barleys continues to be debated. Here, we show that most Tibetan weedy barleys originated from cross-pollinated hybridization of domesticated barleys, followed by hybrid self-pollination and recombination between Non-brittle rachis 1 (btr1) and 2 (btr2). We discovered the specific genetic ancestry of these weedy barleys in South Asian accessions. Tibetan weedy barleys exhibit lower genetic diversity than wild and Chinese landraces/cultivars and share a close relationship with qingke, genetically differing from typical eastern and western barley populations. We classified Tibetan weedy barleys into two groups, brittle rachis (BR) and non-brittle rachis (NBR); these traits align with the haplotypes of the btr1 and btr2 genes. Whereas wild barleys carry haplotype combinations of Btr1 and Btr2, each showing lower proportions in a population, the recombinant haplotype BTR2H8+BTR1H24 is predominant in the BR group. Haplotype block analysis based on whole-genome sequencing revealed two recombination breakpoints, which are present in 80.6% and 16.8% of BR accessions according to marker-assisted analysis. Hybridization events between wild and domesticated barley were rarely detected. These findings support the notion that Tibetan weedy barleys originated via recombination between Btr1 and Btr2 in domesticated barley.

In early spring 2018, significant mosaic disease symptoms were observed for the first time on barley leaves (Hordeum vulgare L., cv. New Sachiho Golden) in Takanezawa, Tochigi Prefecture, Japan. This cultivar carries the resistance gene rym3 (rym; resistance to yellow mosaic). Through RNA-seq analysis, Barley yellow mosaic virus (BaYMV-Takanezawa) was identified in the roots of all five plants (T01–T05) in the field. Phylogenetic analysis of RNA1, encompassing known BaYMV pathotypes I through V, revealed that it shares the same origin as isolate pathotype IV (BaYMV-Ohtawara pathotype). However, RNA2 analysis of isolates revealed the simultaneous presence of two distinct BaYMV isolates, BaYMV-Takanezawa-T01 (DRR552862, closely related to pathotype IV) and BaYMV-Takanezawa-T02 (DRR552863, closely related to pathotype III). The amino acid sequences of the BaYMV-Takanezawa isolates displayed variations, particularly in the VPg and N-terminal region of CP, containing mutations not found in other domains of the virus genome. Changes in the CI (RNA1 amino acid residue 459) and CP (RNA1 amino acid residue 2138) proteins correlated with pathogenicity. These findings underscore the importance of monitoring and understanding the genetic diversity of BaYMV for effective disease management strategies in crop breeding.

Let a and b be relatively prime positive integers. In this paper the weighted sum ∑nεNR(a,b) λn−1nm is given explicitly or in terms of the Apostol- Bernoulli numbers, where m is a nonnegative integer, and NR(a, b) denotes the set of positive integers nonrepresentable in terms of a and b. [ABSTRACT FROM AUTHOR]

In this paper, by using Abel's Lemma on summation by parts, we mainly show that several types of infinite sums involving the generalized q-harmonic numbers can be expressed in terms of linear combinations of the generalized Hurwitz q-zeta values, which are natural q-generalizations of Sofo's results. [ABSTRACT FROM AUTHOR]

We show explicit expressions for an inverse power series over the gaps values of numerical semigroups generated by two and three integers. As an application, a set of identities of the Hurwitz zeta functions is derived. [ABSTRACT FROM AUTHOR]

For a positive integer N, hypergeometric Cauchy numbers cN,n are defined by ... where 2F1(a,b;c;z) is the Gauss hypergeometric function. When N = 1, cn = c1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers (see (4), (6) and (3) in the text). Hypergeometric numbers can be recognized as one of the most natural extensions of the classical Cauchy numbers in terms of determinants (see Section 2), though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In addition, there are some relations between the hypergeometric Cauchy numbers and the classical Cauchy numbers. In this paper, we give the determinant expressions of hypergeometric Cauchy numbers and their generalizations, and show some interesting expressions of hypergeometric Cauchy numbers. As applications, we can get the inversion relations such that hypergeometric Cauchy numbers as cN,n/n! and the numbers N/(N + n) are interchanged in terms of determinants of the so-called Hassenberg matrices. [ABSTRACT FROM AUTHOR]

Poly-Cauchy numbers c(k)n (n ≥ 0, k ≥ 1) have explicit expressions in terms of the Stirling numbers of the first kind. When the index is negative, there exists a different expression. However, the sequence {c(-k)n}n ≥ 0 seem quite irregular for a fixed integer k ≥ 2. In this paper we establish a certain kind of recurrence relations among the sequence {c(-k)n}n ≥ 0, analyzing the structure of poly-Cauchy numbers. We also study those of poly-Cauchy numbers of the second kind, poly-Euler numbers, and poly-Euler numbers of the second kind. Some different proofs are given. As applications, some leaping relations are shown. [ABSTRACT FROM AUTHOR]

Abstract Objective This study aimed to identify the recurrence rate of genetic generalized epilepsy (GGE) and risk factors for recurrence after antiseizure medication (ASM) withdrawal in adolescent patients. Methods We retrospectively reviewed medical records of patients with GGE who were included in the registry at the Department of Child Neurology, National Hospital Organization Nishiniigata Chuo Hospital from 2000 through 2020. The eligibility criteria were as follows: onset of epileptic seizures at

Infection by the Japanese soil-borne wheat mosaic virus (JSBWMV) can lead to substantial losses in the grain yield of barley and wheat crops. While genetically based resistance to this virus has been documented, its mechanistic basis remains obscure. In this study, the deployment of a quantitative PCR assay showed that the resistance acts directly against the virus rather than by inhibiting the colonization of the roots by the virus’ fungal vector Polymyxa graminis. In the susceptible barley cultivar (cv.) Tochinoibuki, the JSBWMV titre was maintained at a high level in the roots during the period December–April, and the virus was translocated from the root to the leaf from January onwards. In contrast, in the roots of both cv. Sukai Golden and cv. Haruna Nijo, the titre was retained at a low level, and translocation of the virus to the shoot was strongly suppressed throughout the host’s entire life cycle. The roots of wild barley (Hordeum vulgare ssp. spontaneum) accession H602 responded in the early stages of infection similarly to those of the resistant cultivated forms, but the host was unable to suppress the translocation of the virus to the shoot from March onwards. The virus titre in the root was presumed to have been restricted by the action of the gene product of Jmv1 (on chromosome 2H), while the stochastic nature of the infection was suppressed by the action of that of Jmv2 (on chromosome 3H), a gene harbored by cv. Sukai Golden but not by either cv. Haruna Nijo or accession H602.

Powdery mildew is a fungal disease devastating to wheat, causing significant quality and yield loss. Flavonoids are important secondary plant metabolites that confer resistance to biotic and abiotic stress. However, whether they play a role in powdery mildew resistance in wheat has yet to be explored. In the present study, we combined transcriptome and metabolome analyses to compare differentially expressed genes (DEGs) and differentially accumulated flavonoids identified in plants with and without powdery mildew inoculation. Transcriptome analysis identified 4,329 DEGs in susceptible wheat cv. Jimai229, and 8,493 in resistant cv. HHG46. The DEGs were functionally enriched using Gene Ontology and Kyoto Encyclopedia of Genes and Genomes, revealing the flavonoid synthesis pathway as the most significant in both cultivars. This was consistent with the upregulation of flavonoid synthesis pathway genes observed by quantitative PCR. Metabolome analysis indicated flavone and flavonol biosynthesis pathways as the most significantly enriched following powdery mildew inoculation. An accumulation of total flavonoids content was also found to be induced by powdery mildew infection. Exogenous flavonoids treatment of inoculated plants led to less severe infection, with fewer and smaller powdery mildew spots on the wheat leaves. This reduction is speculated to be regulated through malondialdehyde content and the activities of peroxidase and catalase. Our study provides a fundamental theory for further exploration of the potential of flavonoids as biological prevention and control agents against powdery mildew in wheat.

Tribonacci numbers have been widely studied in relation with Fibonacci numbers and their generalizations. Tribonacci-type numbers ... are defined by the recurrence relation ... (n ≥ 3) with given initial values .... When T0 = 0 and T1 = T2 = 1, Tn = Tn(0,1,1) are ordinary Tribonacci numbers, which sequence is given by {Tn}n≥0 = 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, .... In this paper, we give some convolution identities for Tribonacci-type numbers with binomial (multinomial) coefficients. [ABSTRACT FROM AUTHOR]

It has been known that some numbers, including Bernoulli, Cauchy, and Euler numbers, have such corresponding numbers in terms of determinants of Hessenberg matrices. There exist inversion relations between the original numbers and the corresponding numbers. In this paper, we introduce the numbers related to harmonic numbers in determinants. We also give several of their arithmetical and/or combinatorial properties and applications. These concepts can be generalized in the case of hyperharmonic numbers. [ABSTRACT FROM AUTHOR]

We define a truncated Euler polynomial Em,n(x) as a generalization of the classical Euler polynomial En(x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial. [ABSTRACT FROM AUTHOR]

The objective of this article is to derive some higher-order identities concerning a general second-order recurrence sequence. [ABSTRACT FROM AUTHOR]

Sucrose nonfermenting 2 (Snf2) family proteins, as the catalytic core of ATP-dependent chromatin remodeling complexes, play important roles in nuclear processes as diverse as DNA replication, transcriptional regulation, and DNA repair and recombination. The Snf2 gene family has been characterized in several plant species; some of its members regulate flower development in Arabidopsis. However, little is known about the members of the family in barley (Hordeum vulgare). Here, 38 Snf2 genes unevenly distributed among seven chromosomes were identified from the barley (cv. Morex) genome. Phylogenetic analysis categorized them into 18 subfamilies. They contained combinations of 21 domains and consisted of 3 to 34 exons. Evolution analysis revealed that segmental duplication contributed predominantly to the expansion of the family in barley, and the duplicated gene pairs have undergone purifying selection. About eight hundred Snf2 family genes were identified from 20 barley accessions, ranging from 38 to 41 genes in each. Most of these genes were subjected to purification selection during barley domestication. Most were expressed abundantly during spike development. This study provides a comprehensive characterization of barley Snf2 family members, which should help to improve our understanding of their potential regulatory roles in barley spike development.

Freezing stress is a major factor limiting production and geographical distribution of temperate crops. Elongator is a six subunit complex with histone acetyl-transferase activity and is involved in plant development and defense responses in Arabidopsis thaliana. However, it is unknown whether and how an elongator responds to freezing stress in plants. In this study, we found that wheat elongator subunit 4 (TaELP4) negatively regulates freezing tolerance through ethylene signaling. TaELP4 promoter contained cold response elements and was up-regulated in freezing stress. Subcellular localization showed that TaELP4 and AtELP4 localized in the cytoplasm and nucleus. Silencing of TaELP4 in wheat with BSMV-mediated VIGS approach significantly elevated tiller survival rate compared to control under freezing stress, but ectopic expression of TaELP4 in Arabidopsis increased leaf damage and survival rate compared with Col-0. Further results showed that TaELP4 positively regulated ACS2 and ACS6 transcripts, two main limiting enzymes in ethylene biosynthesis. The determination of ethylene content showed that TaELP4 overexpression resulted in more ethylene accumulated than Col-0 under freezing stress. Epigenetic research showed that histone H3K9/14ac levels significantly increased in coding/promoter regions of AtACS2 and AtACS6 in Arabidopsis. RT-qPCR assays showed that the EIN2/EIN3/EIL1-CBFs-COR pathway was regulated by TaELP4 under freezing stress. Taken together, our results suggest that TaELP4 negatively regulated plant responses to freezing stress via heightening histone acetylation levels of ACS2 and ACS6 and increasing their transcription and ethylene accumulation.

The objective of this article is to derive some higher-order identities concerning a general second-order recurrence sequence [ABSTRACT FROM AUTHOR]

We prove several explicit formulae for the n-th Bernoulli polynomial Bn(x), in which Bn (x) is equal to an affine combination of the polynomials (x - 1)n, (x - 2)n,...,(x - k - 1)n, where k is any fixed positive integer greater then or equal to n. [ABSTRACT FROM AUTHOR]

Gene targeting (GT) enables precise genome modification—e.g., the introduction of base substitutions—using donor DNA as a template. Combined with clean excision of the selection marker used to select GT cells, GT is expected to become a standard, generally applicable, base editing system. Previously, we demonstrated marker excision via a piggyBac transposon from GT-modified loci in rice. However, piggyBac-mediated marker excision has the limitation that it recognizes only the sequence TTAA. Recently, we proposed a novel and universal precise genome editing system consisting of GT with subsequent single-strand annealing (SSA)-mediated marker excision, which has, in principle, no limitation of target sequences. In this study, we introduced base substitutions into the microRNA miR172 target site of the OsCly1 gene—an ortholog of the barley Cleistogamy1 gene involved in cleistogamous flowering. To ensure efficient SSA, the GT vector harbors 1.2-kb overlapped sequences at both ends of a selection marker. The frequency of positive–negative selection-mediated GT using the vector with overlapped sequences was comparable with that achieved using vectors for piggyBac-mediated marker excision without overlapped sequences, with the frequency of SSA-mediated marker excision calculated as ~40% in the T0 generation. This frequency is thought to be adequate to produce marker-free cells, although it is lower than that achieved with piggyBac-mediated marker excision, which approaches 100%. To date, introduction of precise substitutions in discontinuous multiple bases of a targeted gene using base editors and the prime editing system based on CRISPR/Cas9 has been quite difficult. Here, using GT and our SSA-mediated marker excision system, we succeeded in the precise base substitution not only of single bases but also of artificial discontinuous multiple bases in the miR172 target site of the OsCly1 gene. Precise base substitution of miRNA target sites in target genes using this precise genome editing system will be a powerful tool in the production of valuable crops with improved traits.

As generalizations of the poly-Cauchy numbers of the first kind cn(k) and of the second kind ..., we introduce the concept about q-analogues or extensions of the poly-Cauchy numbers of the first kind..., and of the second kind cn,q(k) and investigate their properties. We also study q-analogues or extensions of the poly-Cauchy polynomials of the first kind ...(k) (z) and of the second kind ...(z). [ABSTRACT FROM AUTHOR]

In this paper, we find certain family of 4k-square (k = 1, 2, . . . ) band matrices whose band width is 2k + 1, and the entries in their arbitrary powers for matrices are expressed by the generalized Fibonacci numbers only. [ABSTRACT FROM AUTHOR]

Seed dispersal among wild species belonging to the tribe Triticeae is typically achieved by the formation of a brittle rachis. The trait relies on the development of a disarticulation layer, most frequently above the rachis node (resulting in wedge type dispersal units), but in some species below the rachis node (resulting in barrel type dispersal units). The genes responsible for the former type are the complementary pair Btr1 and Btr2, while the genetic basis of the latter type has yet to be determined. Aegilops tauschii forms barrel type dispersal units and previous study showed this species lacked an intact copy of Btr1. Here it has been demonstrated that Ae. tauschii carries two of Btr2; and that Btr2 transcript is present in a region below the rachis node where the abscission zone forms. The implication is that in this species, the Btr2 product is involved in the formation of barrel type dispersal units.

In many non-cultivated angiosperm species, seed dispersal is facilitated by the shattering of the seed head at maturity; in the Triticeae tribe, to which several of the world's most important cereals belong, shattering takes the form of a disarticulation of the rachis. The products of the genes Btr1 and Btr2 are both required for disarticulation to occur above the rachis nodes within the genera Hordeum (barley) and Triticum/Aegilops (wheat). Here, it has been shown that both Btr1 and Btr2 are specific to the Triticeae tribe, although likely paralogs (Btr1-like and Btr2-like) are carried by the family Poaceae including Triticeae. Aegilops tauschii (the donor of the bread wheat D genome) lacks a copy of Btr1 and disarticulation in this species occurs below, rather than above the rachis node; thus, the product of Btr1 appears to be required for disarticulation to occur above the rachis node.

Abstract  Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions 0 F1(;c;z). In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

Abstract Background Pre-harvest sprouting frequently occurs in Triticum aestivum (wheat) and Hordeum vulgare (barley) at the end of the maturity period due to high rainfall, particularly in Asian monsoon areas. Seed dormancy is a major mechanism preventing pre-harvest sprouting in these crops. Results We identified orthologous sequences of the major Hordeum vulgare (barley) seed dormancy gene Qsd1 in hexaploid wheat cv. Chinese Spring by performing genomic clone sequencing, followed by transcript sequencing. We detected 13 non-synonymous amino acid substitutions among the three sub-genomes of wheat and found that the Qsd1 sequence in the B sub-genome is most similar to that in barley. The Qsd1 sequence in A genome diploid wheat is highly similar to that in the hexaploid A sub-genome. Wheat orthologs of Qsd1 showed closer similarities to barley Qsd1 than did those of other accessions in the DNA database. Like barley Qsd1, all three wheat Qsd1s showed embryo-specific gene expression patterns, indicating that barley and wheat Qsd1 share an orthologous origin. The alignment of four hexaploid wheat cultivars indicated that the amino acid sequences of three spring cultivars, Chinese Spring, Haruyo Koi, and Fielder, are exactly the same in each sub-genome. Only Kitahonami has three amino acid substitutions at the B sub-genome. Conclusions Kitahonami has a longer seed dormancy period than does Chinese Spring. Sequence polymorphisms between Chiniese Spring and Kitahonami in the B sub-genome may underlie the phenotypic differences in seed dormancy between these hexaploid wheat cultivars.

Triticum turgidum and T. timopheevii are two tetraploid wheat species sharing T. urartu as a common ancestor, and domesticated accessions from both of these allopolyploids exhibit nonbrittle rachis (i.e., nonshattering spikes). We previously described the loss-of-function mutations in the Brittle Rachis 1 genes BTR1-A and BTR1-B in the A and B subgenomes, respectively, that are responsible for this most visible domestication trait in T. turgidum. Resequencing of a large panel of wild and domesticated T. turgidum accessions subsequently led to the identification of the two progenitor haplotypes of the btr1-A and btr1-B domesticated alleles. Here, we extended the haplotype analysis to other T. turgidum subspecies and to the BTR1 homologues in the related T. timopheevii species. Our results showed that all the domesticated wheat subspecies within T. turgidum share common BTR1-A and BTR1-B haplotypes, confirming their common origin. In T. timopheevii, however, we identified a novel loss-of-function btr1-A allele underlying a partially brittle spike phenotype. This novel recessive allele appeared fixed within the pool of domesticated Timopheev’s wheat but was also carried by one wild timopheevii accession exhibiting partial brittleness. The promoter region for BTR1-B could not be amplified in any T. timopheevii accessions with any T. turgidum primer combination, exemplifying the gene-level distance between the two species. Altogether, our results support the concept of independent domestication processes for the two polyploid, wheat-related species.

Seed dormancy allows wild barley grains to survive dry summers in the Near East but has been selected against for industrial applications such as beer and whisky production that require quicker germination. Here Sato et al. show that Qsd1is a major seed dormancy gene in barley and encodes an alanine aminotransferase.

Einkorn and emmer wheat together with barley were among the first cereals domesticated by humans more than 10,000 years ago, long before durum or bread wheat originated. Domesticated einkorn wheat differs from its wild progenitor in basic morphological characters such as the grain dispersal system. This study identified the Non-brittle rachis 1 (btr1) and Non-brittle rachis 2 (btr2) in einkorn as homologous to barley. Re-sequencing of the Btr1 and Btr2 in a collection of 53 lines showed that a single non-synonymous amino acid substitution (alanine to threonine) at position 119 at btr1, is responsible for the non-brittle rachis trait in domesticated einkorn. Tracing this haplotype variation back to wild einkorn samples provides further evidence that the einkorn progenitor came from the Northern Levant. We show that the geographical origin of domesticated haplotype coincides with the non-brittle domesticated barley haplotypes, which suggest the non-brittle rachis phenotypes of einkorn and barley were fixed in same geographic area in today’s South-east Turkey.

Abstract The hydrophobic cuticle covers the surface of the most aerial organs of land plants. The barley mutant eceriferum‐zv (cer‐zv), which is hypersensitive to drought, is unable to accumulate a sufficient quantity of cutin in its leaf cuticle. The mutated locus has been mapped to a 0.02 cM segment in the pericentromeric region of chromosome 4H. As a map‐based cloning approach to isolate the gene was therefore considered unlikely to be feasible, a comparison was instead made between the transcriptomes of the mutant and the wild type. In conjunction with extant genomic information, on the basis of predicted functionality, only two genes were considered likely to encode a product associated with cutin formation. When eight independent cer‐zv mutant alleles were resequenced with respect to the two candidate genes, it was confirmed that the gene underlying the mutation in each allele encodes a Gly‐Asp‐Ser‐Leu (GDSL)‐motif esterase/acyltransferase/lipase. The gene was transcribed in the epidermis, and its product was exclusively deposited in cell wall at the boundary of the cuticle in the leaf elongation zone, coinciding with the major site of cutin deposition. CER‐ZV is speculated to function in the deposition of cutin polymer. Its homologs were found in green algae, moss, and euphyllophytes, indicating that it is highly conserved in plant kingdom.

Wild barley (Hordeum vulgare ssp. spontaneum) has strong grain dormancy, a trait that may enhance its survival in non-cultivated environments; by contrast, cultivated barley (Hordeum vulgare ssp. vulgare) has weaker dormancy, allowing uniform germination in cultivation. Malting barley cultivars have been bred for especially weak dormancy to optimize their use in malt production. Here, we analyzed the genetic mechanism of this difference in seed dormancy, using recombinant inbred lines (RILs) derived from a cross between the wild barley accession ‘H602’ and the malting barley cultivar ‘Kanto Nakate Gold (KNG)’. Grains of H602 and KNG harvested at physiological maturity and dried at 30°C for 7 days had germination of approximately 0 and 100%, respectively. Analysis of quantitative trait loci (QTL) affecting grain dormancy identified the well-known major dormancy QTL SD1 and SD2 (located near the centromeric region and at the distal end of the long arm of chromosome 5H, respectively), and QTL at the end of the long arm of chromosome 4H and in the middle of the long arm of chromosome 5H. We designated these four QTL Qsd1-OK, Qsd2-OK, Qsdw-4H, and Qsdw-5H, and they explained approximately 6, 38, 3, and 13% of the total phenotypic variation, respectively. RILs carrying H602 alleles showed increased dormancy levels for all QTL. The QTL acted additively and did not show epistasis or QTL–environment interactions. Comparison of QTL locations indicated that all QTL except Qsdw-5H are likely the same as the QTL previously detected in the doubled haploid population from a cross between the malting cultivar ‘Haruna Nijo’ and ‘H602.’ We further examined Qsd2-OK and Qsdw-5H by analyzing the segregation of phenotypes and genotypes of F2 progenies derived from crosses between RILs carrying specific segments of chromosome 5H from H602 in the KNG background. This analysis confirmed that the two genomic regions corresponding to these QTL are involved in the regulation of grain dormancy. Germination tests of F1 grains derived from reciprocal crosses between H602 and KNG revealed that the H602 strong dormancy phenotype shows maternal inheritance with incomplete dominance. These results provide new insight into the mechanisms regulating grain dormancy in barley.

The msm 1 and msm 2 cytoplasms cause male sterility in barley. In order to clarify whether these cytoplasms can be used for hybrid seed production in Japan, we investigated the fertility restoration under environmental conditions in Japan using alloplasmic lines and an isogenic line. Fertility of barley lines containing msm1 and msm2 was completely restored by the restorer gene Rfm 1a. The effect of cytoplasm on malting quality was also investigated, but no significant difference in malting quality was observed between lines possessing msm1, msm2, and original cytoplasms. Malting quality also did not differ between Rfm1a/Rfm1a and Rfm1a/rfm1a genotypes, suggesting that Rfm1a and its flanking genes are not associated with heterosis for malting quality. Possibilities of hybrid barley breeding using these male sterile cytoplasms are discussed.

We show that gcd(2!S(2!+1,2), . . . ,(2n+1)!S(2n + 1,2n+ 1)= denominator of B2n, where S(n,k) is the Stirling number of the second kind and Bn is the Bernoulli number. [ABSTRACT FROM AUTHOR]