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Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

In this short paper, we show that there does not exist a noncompact Type-I [Formula: see text]-solution of the Ricci flow with positive curvature in dimension 3.

Kudryavtseva and Mazorchuk exhibited Schur–Weyl duality between the rook monoid algebra [Formula: see text] and the subalgebra [Formula: see text] of the partition algebra [Formula: see text] acting on [Formula: see text]. In this paper, we consider a subalgebra [Formula: see text] of [Formula: see text] such that there is Schur–Weyl duality between the actions of [Formula: see text] and [Formula: see text] on [Formula: see text]. This paper studies the representation theory of partition algebras [Formula: see text] and [Formula: see text] for rook monoids inductively by considering the multiplicity free tower [Formula: see text] Furthermore, this inductive approach is established as a spectral approach by describing the Jucys–Murphy elements and their actions on the canonical Gelfand–Tsetlin bases, determined by the aforementioned multiplicity free tower, of irreducible representations of [Formula: see text] and [Formula: see text]. Also, we describe the Jucys–Murphy elements of [Formula: see text] which play a central role in the demonstration of the actions of Jucys–Murphy elements of [Formula: see text] and [Formula: see text].

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

In the proof of Theorem 3.7 of the original paper Internat. J. Math. 30(14) (2019) 1950078, 1–25, two inequalities are used that do not hold in general. In this note, we prove an extra propositions which allows us to give a proof of Theorem 3.7 without using the aforementioned inequalities. Hence, all the results in the original paper are valid. We have also posted a corrected version in the arXiv.

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a [Formula: see text]-valued Brown functor where [Formula: see text] is the category of bicommutative Hopf algebras over a field [Formula: see text] : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary [Formula: see text]-valued homology theory.

In this paper and its successor, we make an application of the hyperboloidal foliation method in [Formula: see text] space-time dimension. After the establishment of some technical tools in this paper, we will prove further the global existence of small regular solution to a class of hyperbolic system composed by a wave equation and a Klein–Gordon equation with null couplings. Our method belongs to vector field method and, more precisely, is a combination of the normal form and the hyperboloidal foliation method.

In the conformal (Möbius) geometry of submanifolds, using algebraic invariants of the shape operator to construct conformal invariants is a frequently used method. But it does not apply to [Formula: see text]-dim Lorentzian hypersurfaces of the last type, the minimal polynomial of whose shape operator has a triple root. In this paper, using the obstruction of some distribution to be integrable, a new method to construct conformal invariants is introduced. Using this method, a complete conformal invariant system is constructed for generic conformally flat Lorentzian hypersurfaces of the last type. We find that such kind of hypersurfaces allows an infinite parameter family of non-equivalent deformations, which implies they are more abundant than the other three types. For non-generic conformally flat Lorentzian hypersurfaces of the last type, the isometric geometry is studied and a fundamental theorem is obtained in this paper.

The paper introduces the [Formula: see text]th-order slant Toeplitz operator on the Lebesgue space of [Formula: see text]-torus, where [Formula: see text] such that [Formula: see text] for all [Formula: see text]. It investigates certain properties of [Formula: see text]th-order slant Toeplitz operators on the Lebesgue space [Formula: see text]. The paper deals with a system of operator equations, characterizing the [Formula: see text]th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

A homogeneous pattern is a linear multivariate polynomial without constant term. Bras-Amorós and García-Sánchez introduced the notion of pattern for numerical semigroups, which generalizes the definition of Arf numerical semigroups. The notion of pattern for numerical semigroups is extended in this paper into a family of homogeneous patterns [Formula: see text]. A numerical semigroup admitting a family of homogeneous patterns [Formula: see text] at [Formula: see text]-level is characterized. We pay our attention in this paper to the families with the action of some permutation groups, especially those consisting of certain partition stabilizers. Stable or sensitive patterns are focused on and we characterize them for several specific permutation groups.

In this paper, we prove the claim given in the title. A group [Formula: see text] is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to [Formula: see text] factors through projection to a canonical free subgroup. Higman, in his seminal 1952 paper [Unrestricted free products and varieties of topological groups, J. London Math. Soc. 27 (1952) 73–81], proved that free groups are noncommutatively slender. Such groups were first defined by Eda in [Free [Formula: see text]-products and noncommutatively slender groups, J. Algebra 148 (1992) 243–263]. Eda has asked which finitely presented groups are noncommutatively slender. This result demonstrates that random finitely presented groups in the few-relator sense are noncommutatively slender.

In this paper, we study exchange rings and clean rings [Formula: see text] with [Formula: see text] (or otherwise). Analogues of a theorem of Camillo and Yu characterizing clean and strongly clean rings with [Formula: see text] are obtained for such rings (as well as for exchange rings) using the viewpoint of exchange equations introduced in a recent paper of the authors. We also study a new class of rings including von Neumann regular rings in which square roots of one (instead of idempotents) can be lifted modulo left ideals, and conjecture that such rings are exchange rings. This conjecture holds for commutative rings, and would hold for all rings if it holds for semiprimitive rings of characteristic [Formula: see text].

In this paper, we make a correction to Theorem 3.3 of the aforementioned paper (Internat. J. Math. 28(13) (2017) 1750099). We also provide a counterexample to the theorem and point out an error in the proof.

In this paper, we investigate the uniqueness problems of meromorphic functions concerning differential polynomials sharing a small function. In particular, the resuts of the paper improve the results due to Waghamore and Anand [11].

Let [Formula: see text] be a commutative ring and [Formula: see text] be an infinite discrete group. The algebraic [Formula: see text]-theory of the group ring [Formula: see text] is an important object of computation in geometric topology and number theory. When the group ring is Noetherian, there is a companion [Formula: see text]-theory of [Formula: see text] which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of [Formula: see text]-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. It has some expected properties such as independence from the choice of a word metric. We prove that, whenever [Formula: see text] is a regular Noetherian ring of finite global homological dimension and [Formula: see text] has finite asymptotic dimension and a finite model for the classifying space [Formula: see text], the natural Cartan map from the [Formula: see text]-theory of [Formula: see text] to [Formula: see text]-theory is an equivalence. On the other hand, our [Formula: see text]-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.

In this paper, we study the notions of [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings, [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system. We produce some properties and characterizations for [Formula: see text]-prime and [Formula: see text]-semiprime ideals of semirings in terms of [Formula: see text]-[Formula: see text]-system and [Formula: see text]-[Formula: see text]-system, respectively. We initiate the notion of prime semiring and semiprime semiring to set up characterizations for these two ideals of a semiring. The concept of power semiring associated to a semigroup is also studied in this paper.

In this paper we study group actions on hyperbolic Λ-metric spaces, where Λ is an ordered abelian group. Λ-metric spaces were first introduced by Morgan and Shalen in their study of hyperbolic structures and then Chiswell, following Gromov's ideas, introduced the notion of hyperbolicity for such spaces. Only the case of 0-hyperbolic Λ-metric spaces (that is, Λ-trees) was systematically studied, while the theory of general hyperbolic Λ-metric spaces was not developed at all. Hence, one of the goals of the present paper was to fill this gap and translate basic notions and results from the theory of group actions on hyperbolic (in the usual sense) spaces to the case of Λ-metric spaces for an arbitrary Λ. The other goal was to show some principal difficulties which arise in this generalization and the ways to deal with them.

A new pair of asymptotic invariants for finitely presented groups, called intrinsic and extrinsic tame filling functions, are introduced. These filling functions are quasi-isometry invariants that strengthen the notions of intrinsic and extrinsic diameter functions for finitely presented groups. We show that the existence of a (finite-valued) tame filling function implies that the group is tame combable. Bounds on both intrinsic and extrinsic tame filling functions are discussed for stackable groups, including groups with a finite complete rewriting system, Thompson's group F, and almost convex groups., This paper was derived from part of the paper at arXiv:1109.6309

There are two different concepts for hypersubstitutions for algebraic systems [K. Denecke and D. Phusanga, Hyperformulas and solid algebraic systems, Studia Logica 90(2) (2008) 263–286; J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. In this paper, we follow the more natural and practicable one given in [J. Koppitz and D. Phusanga, The monoid of hypersubstitutions for algebraic systems, J. Announcements Union Sci. Sliven 33(1) (2018) 120–127]. On the other hand, in [S. Leeratanavalee and K. Denecke, Generalized hypersubstitutions and strongly solid varieties, General Algebra and Applications, Proc. of 59th Workshop on General Algebra; 15th Conf. for Young Algebraists Potsdam 2000 (Shaker Verlag, 2000), pp. 135–145], the concept of the monoid of generalized hypersubstitutions was introduced. Following both ideas, one obtains the concept of a monoid of generalized hypersubstitutions for algebraic systems in a canonical way. The purpose of this paper is the study of the monoid of generalized hypersubstitutions for algebraic systems. We characterize the idempotent as well as regular elements in this monoid.

In this paper, we start the program of the existence of the smooth equivariant geodesics in the equivariant Mabuchi moduli space of Kähler metrics on type II cohomogeneity one compact Kähler manifold. In this paper, we deal with the manifolds [Formula: see text] obtained by blowing up the diagonal of the product of two copies of a [Formula: see text].

In this paper, we first introduce a new class of [Formula: see text]-preinvex functions, which are called [Formula: see text]-preinvex functions, then we give some new estimates of Hermite–Hadamard, Ostrowski and Simpson type inequalities using Riemann–Liouville fractional integral. The obtained results in this paper generalize the well-known results in recent years.

Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface and $G$ be a finite subgroup in $\operatorname{Aut}(X)$. In this paper we study when the quotient surface $X / G$ can be non-rational over $\Bbbk$. Obviously, if there are no smooth $\Bbbk$-points on $X / G$ then it is not $\Bbbk$-rational. Therefore under assumption that the set of smooth $\Bbbk$-points on $X / G$ is not empty we show that there are few possibilities for non-$\Bbbk$-rational quotients. The quotients of del Pezzo surfaces of degree $2$ and greater are considered in the author's previous papers. In this paper we study the quotients of del Pezzo surfaces of degree $1$. We show that they can be non-$\Bbbk$-rational only for the trivial group or cyclic groups of order $2$, $3$ and $6$. For the trivial group and the group of order $2$ we show that both $X$ and $X / G$ are not $\Bbbk$-rational if the $G$-invariant Picard number of $X$ is $1$. For the groups of order $3$ and $6$ we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non-$\Bbbk$-rational del Pezzo surfaces of degree $1$ such that the $G$-invariant Picard number of $X$ is $1$. As a result of complete classification of non-$\Bbbk$-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of $\Bbbk$-rational surfaces, and obtain some corollaries concerning fields of invariants of $\Bbbk(x , y)$., Comment: 35 pages, 4 figures, 1 table. arXiv admin note: text overlap with arXiv:1709.02006

In this paper, we study the uniqueness and local behavior at the origin of positive solutions to [Formula: see text] where [Formula: see text], [Formula: see text]) is a bounded smooth domain and [Formula: see text], and [Formula: see text] is a nonnegative continuous function over [Formula: see text]. In the two cases: (i) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text]; (ii) [Formula: see text], [Formula: see text] and [Formula: see text] on [Formula: see text], we establish the uniqueness of positive solutions and the exact blow-up rate of positive solutions at the origin. This paper seems to be the first one to deal with the latter case.

This is a sequel to our previous paper of oriented bivariant theory [14]. In 2001 M. Levine and F. Morel constructed algebraic cobordism $\Omega_*(X)$ for schemes $X$ over a field $k$ in an abstract way and later M. Levine and R. Pandhairpande reconstructed it more geometrically. In this paper in a similar manner we construct an algebraic cobordism $\Omega^*(X \xrightarrow {\pi_X} S)$ for a scheme $X$ over a fixed scheme $S$ in such a way that if the target scheme $S$ is the point $pt = \operatorname{Spec} k$, then $\Omega^{-i}(X \xrightarrow {\pi_X} pt)$ is isomorphic to Levine-Morel's algebraic cobordism $\Omega_i(X)$, Comment: 35 pages; any comments are welcome

In a joint paper [4] by Otsubo, Terasoma and the first author, we proved that the special value 3F2(a,b,q;a+b,q;1) of the generalized hypergeometric function is a linear combination of log of algebraic numbers if the triplet (a,b,q) of rational numbers satisfies a certain numerical condition. However there remains a question how to obtain explicit descriptions of the values. In this paper, we give a method to do this, which is a further development of the technique in [4]., Comment: 22pages, To appear in Communications in Contemporary Mathematics

This paper introduces the concept of lattice vector space and establishes many important results. Also, this paper deals with linear transformations on lattice vector spaces and discusses their elementary properties. We prove that every finite dimensional lattice vector space is isomorphic to [Formula: see text] and show that the set of all columns (or the set of all rows) of an invertible matrix over [Formula: see text] is a basis for [Formula: see text].

The area of [Formula: see text]-calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of [Formula: see text]-calculus was initiated by Jackson [Jackson, On [Formula: see text]-definite integrals, Quart. J. Pure Appl. Math. 41 (1910) 193–203; On [Formula: see text]-functions and a certain difference operator, Trans. Roy. Soc. Edinburgh 46 (1908) 253–281.], who was the first to develop [Formula: see text]-integral and [Formula: see text]-derivative in a systematic way. In this paper, we make use of the concept of [Formula: see text]-calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of [Formula: see text]-starlike and [Formula: see text]-convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this [Formula: see text]-calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of [Formula: see text] found in [Mishra and Gochhayat, Fekete–Szego problem for [Formula: see text]-uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl. 347(2) (2008) 563–572] for [Formula: see text], [Formula: see text].

Let [Formula: see text], [Formula: see text] be a bounded (semi-)convex domain in [Formula: see text] and the non-negative potential [Formula: see text] belong to the reverse Hölder class [Formula: see text]. Assume that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the Muckenhoupt weight class on [Formula: see text], the boundary of [Formula: see text]. In this paper, the authors show that, for any [Formula: see text], the Neumann problem for the Schrödinger equation [Formula: see text] in [Formula: see text] with boundary data in (weighted) [Formula: see text] is uniquely solvable. The obtained results in this paper essentially improve the known results which are special cases of the results obtained by Shen [Indiana Univ. Math. J. 43 (1994) 143–176] and Tao and Wang [Canad. J. Math. 56 (2004) 655–672], via extending the range [Formula: see text] of [Formula: see text] into [Formula: see text].

In this paper, by using the Faber polynomial expansions we can find the coefficient bounds for [Formula: see text] subclass of meromorphic bi-univalent functions. The results presented in this paper would generalize and improve some recent works.

Let [Formula: see text] be a commutative ring with unity [Formula: see text] and let [Formula: see text] be an ideal expansion. In the first part of this paper, we extend the concept of [Formula: see text]-primary ideals to [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction and [Formula: see text] is an ideal expansion. We introduce some of the ideal expansion [Formula: see text] and define [Formula: see text]-[Formula: see text]-primary ideals, where [Formula: see text] is an ideal reduction. Also, we investigate ideal expansions satisfying some additional conditions and prove more properties of the generalized [Formula: see text]-[Formula: see text]-primary ideals with respect to such an ideal expansion [Formula: see text]. In the second part of this paper we investigate 2-absorbing [Formula: see text]-primary ideals which unify 2-absorbing ideals and 2-absorbing primary ideals, where [Formula: see text] is an ideal expansion. A number of results in the two parts are given.

In this paper, we prove Adams inequality with exact growth condition in the four-dimensional hyperbolic space ℍ4, ∫ℍ4 e32π2u2 − 1 (1 + |u|)2 dvg ≤ C∥u∥L2(ℍ4)2, (0.1) for all u ∈ Cc∞(ℍ4) with ∫ ℍ4(P2u)udvg ≤ 1. We will also establish an Adachi–Tanaka-type inequality in this setting. Another aspect of this paper is the Lions lemma in the hyperbolic space. We prove Lions lemma for the Moser functional and for a few cases of the Adams functional on the whole hyperbolic space.

In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov–Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This paper provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.

This is the first of two articles studying the structure of left adequate and, more generally, of left Ehresmann monoids. Motivated by a careful analysis of normal forms, we introduce here a concept of proper for a left adequate monoid M. In fact, our notion is that of T-proper, where T is a submonoid of M. We show that any left adequate monoid M has an X*-proper cover for some set X, that is, there is a left adequate monoid [Formula: see text] that is X*-proper, and an idempotent separating surjective morphism [Formula: see text] of the appropriate type. Given this result, we may deduce that the free left adequate monoid on any set X is X*-proper. In a subsequent paper, we show how to construct T-proper left adequate monoids from any monoid T acting via order-preserving maps on a semilattice with identity, and prove that the free left adequate monoid is of this form. An alternative description of the free left adequate monoid will appear in a paper of Kambites. We show how to obtain the labeled trees appearing in his result from our structure theorem. Our results apply to the wider class of left Ehresmann monoids, and we give them in full generality. We also indicate how to obtain some of the analogous results in the two-sided case. This paper and its sequel, and the two of Kambites on free (left) adequate semigroups, demonstrate the rich but accessible structure of (left) adequate semigroups and monoids, introduced with startling insight by Fountain some 30 years ago.

Let G = H2n + 1 be the 2n + 1-dimensional Heisenberg group and H be a connected Lie subgroup of G. Given any discontinuous subgroup Γ ⊂ G for G/H, a precise union of open sets of the resulting deformation space [Formula: see text] of the natural action of Γ on G/H is derived since the paper of Kobayshi and Nasrin [Deformation of Properly discontinuous action of ℤk and ℝk+1, Internat. J. Math.17 (2006) 1175–1190]. We determine in this paper when exactly this space is endowed with a smooth manifold structure. Such a result is only known when the Clifford–Klein form Γ\G/H is compact and Γ is abelian. When Γ is not abelian or H meets the center of G, the parameter and deformation spaces are shown to be semi-algebraic and equipped with a smooth manifold structure. In the case where Γ is abelian and H does not meet the center of G, then [Formula: see text] splits into finitely many semi-algebraic smooth manifolds and fails to be a Hausdorff space whenever Γ is not maximal, but admits a manifold structure otherwise. In any case, it is shown that [Formula: see text] admits an open smooth manifold as its dense subset. Furthermore, a sufficient and necessary condition for the global stability of all these deformations to hold is established.

This paper deals with the classification problems of Leibniz central extensions of linear deformations of a Lie algebra. It is known that any n-dimensional filiform Lie algebra can be represented as a linear deformation of n-dimensional filiform Lie algebra μn given by the brackets [ei, e0] = ei+1, i = 0,1,…,n - 2, in a basis {e0, e1,…,en - 1}. In this paper we consider a linear deformation of μn and its Leibniz central extensions. The resulting algebras are Leibniz algebras, this class is denoted here by Ced (μn). We choose an appropriate basis of Ced (μn) and give general isomorphism criteria. By using the isomorphism criteria, one can classify the class Ced (μn) for any fixed n. Two relevant maple programs are provided.

In this paper, the definition of right nil n-injective ring is given, it is a generalization of nil injective rings. The aim of this paper is to investigate properties of nil n-injective rings. Various results are developed, many extending known results.

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the [Formula: see text] knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler–Ricci flow.

Subtle issues arise when extending homotopy invariants to spaces of functions having little regularity, e.g., Sobolev spaces containing discontinuous functions. Sometimes it is not possible to extend the invariant at all, and sometimes, even when the formulas defining the invariants make sense, they may not have expected properties (e.g., there are maps having non-integral degree).In this paper, we define a complete set of homotopy invariants for maps from three-manifolds to the two-sphere and show that these invariants extend to finite Faddeev energy maps and maps in suitable Sobolev spaces. For smooth maps, our description is proved to be equivalent to Pontrjagin's original homotopy classification from the 1930's. We further show that for the finite energy maps the invariants take on exactly the same values as for smooth maps. We include applications to the Faddeev model.The techniques that we use would also apply to many more problems and/or other functionals. We have tried to make the paper accessible to analysts, geometers and mathematical physicists.

An algebra is entropic if its basic operations are homomorphisms. The paper is focused on representations of such algebras. We prove the following theorem: An entropic algebra without constant basic operations which satisfies so called Szendrei identities and such that all its basic operations of arity at least two are surjective is a subreduct of a semimodule over a commutative semiring. Our theorem is a straightforward generalization of Ježek's and Kepka's theorem for groupoids. As a consequence we obtain that a mode (entropic and idempotent algebra) is a subreduct of a semimodule over a commutative semiring if and only if it satisfies Szendrei identities. This provides a complete solution to the problem in mode theory asking for a characterization of modes which are subreducts of semimodules over commutative semirings. In the second part of the paper we use our theorem to show that each entropic cancellative algebra is a subreduct of a module over a commutative ring. It extends a theorem of Romanowska and Smith about modes.

The object of the present paper is to introduce a type of non-flat Riemannian manifold called pseudo cyclic Ricci symmetric manifold and study its geometric properties. Among others it is shown that a pseudo cyclic Ricci symmetric manifold is a special type of quasi-Einstein manifold. In this paper we also study conformally flat pseudo cyclic Ricci symmetric manifolds and prove that such a manifold can be isometrically immersed in a Euclidean manifold as a hypersurface.

This is the final paper in a series of four, concerning the surface 𝕋 × 𝕋 embedded in ℂℙ8, where 𝕋 is the one-dimensional torus. In this paper we compute the fundamental group of the Galois cover of the surface with respect to a generic projection onto ℂℙ2, and show that it is nilpotent of class 3. This is the first time such a group is presented as the fundamental group of a Galois cover of a surface.

In this paper we classify finite special Moufang sets 𝕄(U,τ) of odd characteristic. The characteristic 2 case was handled in another paper by De Medts and the author (see [3]). We prove that U is elementary abelian. Then, we show that 𝕄(U,τ) is the unique Moufang set whose little projective group is PSL2(|U|). The emphasis of this paper is on obtaining elementary proofs. Section 3 deals with root subgroups in any Moufang set and may be of independent interest.

In this paper we give equational and structural characterizations of all algebras representing the sequence (0, 3,1,1,…). In the first part of the paper we prove that an algebra without constants has exactly three distinct essentially unary term operations and exactly one essentially n-ary term operation for every n > 1 if and only if it is clone equivalent to a generic of one of three varieties of algebras defined in Section 2. The second part of the paper is devoted to the representations of algebras from these three varieties by means of semilattice ordered systems of algebras.

This paper is adjacent to [5] and motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The notions of geometrically equivalent algebras and weakly geometrically equivalent algebras (cf. [2]) are intimately related to the above problem (the latter notion being related to automorphisms of the category Θ0 is replaced in the present paper by a more natural one: "automorphic equivalence"). Two algebras are said to have the same algebraic geometry if they are automorphically equivalent. If an automorphic equivalence of algebras is provided by an inner automorphism of Θ0, then this equivalence coincides with the geometric equivalence. However, not all automorphisms of Θ0 are inner (see [5, Theorem 2]). In this paper we study the situation when the automorphic equivalence of algebras is provided by a non-inner automorphism. Assuming Θ to be an IBM variety of one-sorted algebras we prove that two algebras H1 and H2 are automorphically equivalent in Θ if and only if H1 is geometrically equivalent to [Formula: see text], where [Formula: see text] is the algebra with the same domain as H2 and with the new operations induced by some system of words W = {wω(x)|ω ∈ Ω, Ω is the signature of algebras in Θ. Thus, the notion of automorphic equivalence is reduced to a more simple one of geometric equivalence.

This paper is the second in a series of papers concerning Hirzebruch surfaces. In the first paper in this series, the fundamental group of Galois covers of Hirzebruch surfaces Fk(a, b), where a, b are relatively prime, was shown to be trivial. For the general case, the conjecture stated that the fundamental group is [Formula: see text] where c = gcd (a, b) and n = 2ab + kb2. In this paper, we degenerate the Hirzebruch surface F1(2, 2), compute the braid monodromy factorization of the branch curve in ℂ2, and verify that, in this case, the conjecture holds: the fundamental group of the Galois cover of F1(2, 2) with respect to a generic projection is isomorphic to [Formula: see text].

In this paper we start the program of constructing generalized special Lagrangian torus fibrations for Calabi–Yau hypersurfaces in toric varieties near the large complex limit, with respect to the restriction of a toric metric on the toric variety to the Calabi–Yau hypersurface. The construction is based on the deformation of the standard toric generalized special Lagrangian torus fibration of the large complex limit X0. In this paper, we will deal with the region near the smooth top dimensional torus fibers of X0 and its mirror dual situation — the region near the 0-dimensional fibers of X0.

In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalize Artin–Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads us to some interesting observations, in particular, we offer a method of generating a random braid.