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We propose the definition of an action σ of the additive group scheme G a on an affine variety Y to be geometrically pure, which ensures the existence of a geometric quotient of Y by the G a -action σ if Y is normal. Namely there exists the quotient morphism q : Y → X to a normal affine variety X such that the graph morphism Ψ : G a × Y → Y × X Y is an isomorphism (see Theorem 2.2). Geometric pureness of the given G a -action is the first criterion ever to guarantee the existence of a geometric quotient Y / G a. As a consequence, we obtain an algebraic characterization of the affine 3-space A 3. [ABSTRACT FROM AUTHOR]

In 2013, while studying a relevant class of polyominoes that tile the plane by translation, i.e., double square polyominoes, Blondin Massé et al. found that their boundary words, encoded by the Freeman chain coding on a four letters alphabet, have specific interesting properties that involve notions of combinatorics on words such as palindromicity, periodicity and symmetry. Furthermore, they defined a notion of reducibility on double squares using homologous morphisms, so leading to a set of irreducible tile elements called prime double squares. The authors, by inspecting the boundary words of the smallest prime double squares, conjectured the strong property that no runs of two (or more) consecutive equal letters are present there. In this paper, we prove such a conjecture using combinatorics on words' tools, and setting the path to the definition of a fast generation algorithm and to the possibility of enumerating the elements of this class w.r.t. standard parameters, as perimeter and area. [ABSTRACT FROM AUTHOR]

We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne–Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the effective morphisms, which are a generalization of separated morphisms. [ABSTRACT FROM AUTHOR]

Given an affine algebraic variety X, we prove that if the neutral component \mathrm {Aut}^\circ (X) of the automorphism group consists of algebraic elements, then it is nested, i.e., is a direct limit of algebraic subgroups. This improves our earlier result (see Perepechko and Regeta [Transform. Groups 28 (2023), pp. 401–412]). To prove it, we obtain the following fact. If a connected ind-group G contains a closed connected nested ind-subgroup H\subset G, and for any g\in G some positive power of g belongs to H, then G=H. [ABSTRACT FROM AUTHOR]

The goal of this article is to characterize the lax pullback complement of a given partial morphism along a total morphism, in an -partial morphism category, where is an exponentiable stable system. To achieve this, we first show that if a lax pullback complement of a partial morphism exists in the partial morphism category, then a lax pullback complement of its total part exists in the base category. For the converse, we consider two cases. In the first case the base category is assumed to be adhesive and the morphism along which we take the lax pullback complement is admissible (i.e., its pushout along a monomorphism forms a pullback complement square), while in the second case the base category is -cohesive, however the morphism along which we take the lax pullback complement is an arbitrary total morphism. Finally as a byproduct we provide the connection between exponentials in the partial morphism category and its base category. [ABSTRACT FROM AUTHOR]

The symmetric 2- (v , k , λ) designs with k > λ (λ − 3) / 2 admitting a flag-transitive point-imprimitive automorphism group are completely classified: they are the known 2-designs with parameters (16 , 6 , 2) , (45 , 12 , 3) , (15 , 8 , 4) or (96 , 20 , 4). [ABSTRACT FROM AUTHOR]

In this paper, we present a vertex operator approach to construct and compute all complex irreducible characters of the general linear group GL n (F q). Green's theory of GL n (F q) is recovered and enhanced under the realization of the Grothendieck ring of representations R G = ⨁ n ≥ 0 R (GL n (F q)) as two isomorphic Fock spaces associated to two infinite-dimensional F -equivariant Heisenberg Lie algebras h ˆ F ‾ ˆ q and h ˆ F ‾ q , where F is the Frobenius automorphism of the algebraically closed field F ‾ q. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. One of the features of the current approach is a simpler identification of the Fock space R G as the Hall algebra of symmetric functions via vertex operator calculus, and another is that we are able to compute in general the character table, where Green's degree formula is demonstrated as an example. [ABSTRACT FROM AUTHOR]

We introduce the notion of ideal mutations in a triangulated category, which generalizes the version of Iyama and Yoshino [15] by replacing approximations by objects of a subcategory with approximations by morphisms of an ideal. As applications, for a Hom -finite Krull-Schmidt triangulated category T over an algebraically closed field K. (1) We generalize a theorem of Jorgensen [17, Theorem 3.3] to a more general setting; (2) We provide a method to detect whether T has Auslander-Reiten triangles or not by checking the necessary and sufficient conditions on its Jacobson radical J : (i) J is functorially finite, (ii) Gh J = CoGh J , and (iii) Gh J -source maps coincide with Gh J -sink maps; (3) We generalize the classical Auslander-Reiten theory by using ideal mutations. [ABSTRACT FROM AUTHOR]

Several philosophers have proposed an "algebraic" view of universals according to which there are operations from universals to universals. It is not obvious, nevertheless, how those operations should be interpreted, and what impact they have for the conditions of identity of universals. There are two main interpretations of the algebra of universals. On one hand, it has been interpreted as ways to "construct" complex universals. On the other hand, it has been interpreted as "morphisms" or "mappings", but not as something that "builds" complex universals from others more basic. In this paper, the comparative advantages of both conceptions are assessed and reasons are offered to prefer the second conception. [ABSTRACT FROM AUTHOR]

We describe the algebras of binary formulas for countably categorical weakly circularly minimal theories with 1-transitive nonprimitive automorphism group and trivial definable closure having convexity rank 1. We find some criterion for commutativity of the algebras. [ABSTRACT FROM AUTHOR]

We give an example of an Archimedean Riesz space on which every automorphism is a strictly positive multiple of the identity. [ABSTRACT FROM AUTHOR]

We study ghost distributions on supersymmetric spaces for the case of basic classical Lie superalgebras. We introduce the notion of interlaced pairs, which are those for which both |$({\mathfrak{g}},{\mathfrak{k}})$| and |$({\mathfrak{g}},{\mathfrak{k}}^{\prime})$| admit Iwasawa decompositions. For such pairs, we define a ghost algebra, generalizing the subalgebra of |${\mathcal{U}}{\mathfrak{g}}$| defined by Gorelik. We realize this algebra as an algebra of |$G$| -equivariant operators on the supersymmetric space itself, and for certain pairs, the "special" ones, we realize our operators as twisted-equivariant differential operators on |$G/K$|⁠. We additionally show that the Harish-Chandra morphism is injective, compute its image for all rank one pairs, and provide a conjecture for the image when |$({\mathfrak{g}},{\mathfrak{k}})$| is interlaced. [ABSTRACT FROM AUTHOR]

Up to a finitéetale covering, we classify a smooth projective 3-fold X with κ(X) =-∞admitting a nonisomorphicétale endomorphism.Wemainly study the case where there exist an FESP Y* constructed fromX by a sequence of blowing-downs of an ESP and an extremal rayR* of NE(Y*) such that the contraction morphisms associated to R* give Y* a smooth del Pezzo fiber space over an elliptic curve. [ABSTRACT FROM AUTHOR]

Let f : Y → X be a finite morphism between Fano manifolds Y and X such that the Fano index of X is greater than 1. On the one hand, when both X and Y are fourfolds of Picard number 1, we show that the degree of f is bounded in terms of X and Y unless X ≅ P 4 ; hence, such X does not admit any non-isomorphic surjective endomorphism. On the other hand, when X = Y is either a fourfold or a del Pezzo manifold, we prove that, if f is an int-amplified endomorphism, then X is toric. Moreover, we classify all the singular quadrics admitting non-isomorphic endomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we give a detailed introduction to the theory of (curved) L ∞ -algebras and L ∞ -morphisms, avoiding the concept of operads and providing explicit formulas. In particular, we recall the notion of (curved) Maurer–Cartan elements, their equivalence classes and the twisting procedure. The main focus is then the study of the homotopy theory of L ∞ -algebras and L ∞ -modules. In particular, one can interpret L ∞ -morphisms and morphisms of L ∞ -modules as Maurer–Cartan elements in certain L ∞ -algebras, and we show that twisting the morphisms with equivalent Maurer–Cartan elements yields homotopic morphisms. We hope that these notes provide an accessible entry point to the theory of L ∞ -algebras. [ABSTRACT FROM AUTHOR]

In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra |$B$|⁠ , we construct a polygon |$\mathcal {S}$| with a checkerboard pattern in its interior, which defines a category |$\text {Diag}(\mathcal {S})$|⁠. The indecomposable objects of |$\text {Diag}(\mathcal {S})$| are the 2-diagonals in |$\mathcal {S}$|⁠ , and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category |$\text {Diag}(\mathcal {S})$| is equivalent to the stable syzygy category of the algebra |$B$|⁠. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type |$\mathbb {A}$|⁠. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon |$\mathcal {S}$| is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the one-sided generalized (α, β)-reverse derivation of a ring R. Let R be a semiprime ring, ϱ be a non-zero ideal of R, α be an epimorphism of ϱ, β be a homomorphism of ϱ (α be a homomorphism of ϱ, β be an epimorphism of ϱ) and γ: ϱ → R be a non-zero (α, β)-reverse derivation. We show that there exists F: ϱ → R, an l-generalized (α, β)-reverse derivation (an r-generalized (α, β)-reverse derivation) associated with γ iff F(ϱ), γ(ϱ) ⊂ CR(ϱ) and F is an r-generalized (β, α)-derivation (an l-generalized (β, α)-derivation) associated with (β, α)-derivation γ on ϱ. This theorem generalized the results of A. Aboubakr and S. Gonzalez proved in [1, Theorem 3.1 and Theorem 3.2 ]. [ABSTRACT FROM AUTHOR]

The Hitchin morphism is a map from the moduli space of Higgs bundles |$\mathscr {M}_{X}$| to the Hitchin base |$\mathscr {B}_{X}$|⁠ , where |$X$| is a smooth projective variety. When |$X$| has dimension at least two, this morphism is not surjective in general. Recently, Chen-Ngô introduced a closed subscheme |$\mathscr {A}_{X}$| of |$\mathscr {B}_{X}$|⁠ , which is called the space of spectral data. They proved that the Hitchin morphism factors through |$\mathscr {A}_{X}$| and conjectured that |$\mathscr {A}_{X}$| is the image of the Hitchin morphism. We prove that when |$X$| is a smooth projective surface, this conjecture is true for vector bundles. Moreover, we show that |$\mathscr {A}_{X}$|⁠ , for any dimension, is invariant under any proper birational morphism and apply the result to study |$\mathscr {A}_{X}$| for ruled surfaces. [ABSTRACT FROM AUTHOR]

In this paper, we study some categorical structures of the category CoarsePro, whose objects are coarse proximity spaces and whose morphisms are coarse proximity maps. We investigate the structure of initial, final, embedding and quotient morphisms in the construct CoarsePro. A special attention is paid to investigate quotients by introducing some conditions that they exist. Also, it is shown that bimorphisms are exactly bijective coarse proximity maps, but not isomorphisms. Consequently, CoarsePro is not balanced. [ABSTRACT FROM AUTHOR]

Let (k μ) μ = 1 4 be a quartet of cyclic cubic number fields sharing a common conductor c = p q r divisible by exactly three prime(power)s, p , q , r . For those components of the quartet whose 3-class group Cl 3 (k μ) ≃ (Z / 3 Z) 2 is elementary bicyclic, the automorphism group M = Gal (F 3 2 (k μ) / k μ) of the maximal metabelian unramified 3-extension of k μ is determined by conditions for cubic residue symbols between p , q , r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k * of all k μ . With the aid of the relation rank d 2 (M) , it is decided whether M coincides with the Galois group G = Gal (F 3 ∞ (k μ) / k μ) of the maximal unramified pro-3-extension of k μ . [ABSTRACT FROM AUTHOR]

In quasigroup and loop theory, a pseudo-automorphism (with single companion) is known to generalize automorphism. In this work, the set of crypto-automorphisms (with twin companion) of a quasigroup with right and left identity elements were shown to form a group. For a quasigroup with right and left identity elements, some results on autotopic characterizations of crypto-automorphisms were established and used to deduce some subgroups of the crypto-automorphism group of a middle Bol loop. The crypto-automorphism group and Bryant-Schneider group (this has been used in the study of the isotopy-isomorphy of some varieties of loops e.g. Bol loops, Moufang loops, Osborn loops) of a loop were found to coincide. [ABSTRACT FROM AUTHOR]

This paper provides a complete determination of which of the alternating groups A n and the symmetric groups S n occur as the automorphism group of some regular or chiral map on an orientable surface, and which of them occur as the automorphism group of a regular map on a non-orientable surface. The situation for some given types (m , k) is also considered, where k is the valency and m is the face-size, with special focus on types with m = 3 , and more particularly with (m , k) = (3 , 7) or (3 , 8) , or their duals. Some observations are made also about what happens for regular and orientably-regular maps with given valency, and for regular and chiral polyhedra. Much but certainly not all of what is presented follows from theorems in previous papers by the author and others, and this one brings them and some new observations together into a single reference. [ABSTRACT FROM AUTHOR]

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of \mathrm {Aut}(\mathbf {F}_{p}[x_1, \dots, x_n]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of \mathrm {Aut}(\mathbf {F}_{p}[x_1, \dots, x_n]) on the n-dimensional affine spaces over finite extensions of \mathbf {F}_p. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree 4 for alternating groups of degree p^7-1 for any odd prime p. [ABSTRACT FROM AUTHOR]

Let C be an additive category. The nilpotent category Nil (C) of C , consists of object pairs (X, x) with X ∈ C , x ∈ End C (X) such that xn = 0 for some positive integer n, and a morphism f: (X, x) → (Y, y) is f ∈ Hom C (X , Y) satisfying fx = yf. A general theory of Nil (C) is established and it is abelian in the case that C is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a result in [Comm. Algebra, 2018, 46(7): 3062–3070]. As an application, it is proved that all self-adjoint functors are naturally isomorphic to Hom and Tensor functors over the category V of finite-dimensional vector spaces. Both Hom and Tensor can be naturally generalized to HOM and Tensor functor over Nil (V) . They are still self-adjoint, but intrinsically different. [ABSTRACT FROM AUTHOR]

Almost split sequences lie in the heart of Auslander-Reiten theory. This paper deals with the structure of almost split sequences with certain ending terms in the morphism category of an Artin algebra Λ. Firstly we try to interpret the Auslander-Reiten translates of particular objects in the morphism category in terms of the Auslander-Reiten translations within the category of Λ-modules, and then use them to calculate almost split sequences. In classical representation theory of algebras, it is quite important to recognize the middle term of almost split sequences. As such, another part of the paper is devoted to discuss the middle term of certain almost split sequences in the morphism category of Λ. As an application, we restrict in the last part of the paper to self-injective algebras and present a structural theorem that illuminates a link between representation-finite morphism categories and Dynkin diagrams. [ABSTRACT FROM AUTHOR]

We obtain some sufficient conditions for potency and virtual potency for automorphism groups and the split extensions of some groups. In particular, considering a finitely generated group residually -finite for every prime , we prove that each split extension of by a torsion-free potent group is a potent group, and if the abelianization rank of is at most 2 then the automorphism group of is virtually potent. As a corollary, we derive the necessary and sufficient conditions of virtual potency for certain generalized free products and HNN-extensions. [ABSTRACT FROM AUTHOR]

Birational properties of generically finite morphisms X→Y$X\rightarrow Y$ of algebraic varieties can be understood locally by a valuation of the function field of X. In finite extensions of algebraic local rings in characteristic zero algebraic function fields which are dominated by a valuation, there are nice monomial forms of the mapping after blowing up enough, which reflect classical invariants of the valuation. Further, these forms are stable upon suitable further blowing up. In positive characteristic algebraic function fields, it is not always possible to find a monomial form after blowing up along a valuation, even in dimension two. In dimension two and positive characteristic, after enough blowing up, there are stable forms of the mapping which hold upon suitable sequences of blowing up. We give examples showing that even within these stable forms, the forms can vary dramatically (erratically) upon further blowing up. We construct these examples in defect Artin–Schreier extensions which can have any prescribed distance. [ABSTRACT FROM AUTHOR]

We study which irreducible morphisms in the category of complexes of fixed size C n (proj A) with n ≥ 2 and A an artin algebra, are irreducible in C [ 0 , n ] (proj A) or either in C n + 1 (proj A). Moreover, we determine which of them are also irreducible in the bounded derived category. [ABSTRACT FROM AUTHOR]

We extend the Burger–Mozes theory of closed, nondiscrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger–Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in \mathbb {N}_{\ge 3}$. Three applications are given. First, we characterize the automorphism types that the quasicentre of a nondiscrete subgroup of $\operatorname {\mathrm {Aut}}(T_{d})$ may feature in terms of the group's local action. In doing so, we explicitly construct closed, nondiscrete, compactly generated subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ with nontrivial quasicentre, and see that the Burger–Mozes theory does not extend further to the transitive case. We then characterize the $(P_{k})$ -closures of locally transitive subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks et al. ['Simple groups of automorphisms of trees determined by their actions on finite subtrees', J. Group Theory 18 (2) (2015), 235–261]. Finally, we offer a new view on the Weiss conjecture. [ABSTRACT FROM AUTHOR]

We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, together with an operation ⊞ that leads to a method for obtaining images, kernels and cokernels of tame persistence morphisms. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self-contained presentation of the Mayer–Vietoris spectral sequence. Then we study the Persistent Mayer–Vietoris spectral sequence and present a solution to the extension problem. This solution is given by finding coefficients that indicate gluings between bars on the same dimension. Finally, we review PerMaViss, an algorithm that computes all pages in the spectral sequence and solves the extension problem. This procedure distributes computations on subcomplexes, while focusing on merging homological information. Additionally, some computational bounds are found which confirm the distribution of the method. [ABSTRACT FROM AUTHOR]

The notions of weak GE-morphism, weak GE-endomorphism are introduced and investigated its properties. Conditions for a self-map on a GE-algebra to be an idempotent and to be a weak GE-endomorphism are provided. The notion of qualified GE-algebra is introduced and its properties are investigated. Condition for a qualified GE-algebra to be an idempotent weak GE-endomorphism is given. The notion of qualified self-map is defined and a necessary and condition for a given self-map on a GE-algebra, the product of two GE-algebras under the qualified self-map to be a qualified GE-algebra is given. For a given qualified GE-algebra, using its kernel, an equivalence relation is induced, and then the quotient set is constructed. A binary operation is given on the quotient set to create a quotient qualified GE-algebra. [ABSTRACT FROM AUTHOR]

Let \mathcal C be a concrete category. We prove that if \mathcal {C} admits a universally free object \mathsf F, then there is a projectively universal morphism u\colon \mathsf F\to \mathsf F, i.e., a morphism u such that for any B\in \mathcal {C} and \tau \in \operatorname {Mor}(B) there exists an epimorphism \pi \in \operatorname {Mor}(\mathsf F, B) such that \pi \tau = u \pi. This builds upon and extends various ideas by Darji and Matheron [Proc. Amer. Math. Soc. 145 (2017), pp. 251–265] who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical systems on compact metric spaces. Specialising from our abstract setting, we conclude that the result applies to various categories of Banach spaces/lattices/algebras, C*-algebras, etc. [ABSTRACT FROM AUTHOR]

A reduction formula for the branching coefficients of the restrictions of representations of a semisimple group to a semisimple subgroup is proved in [ 9 , 17 , 33 ]. This formula holds when the highest weights of the representations belong to a codimension |$1$| face of the Horn cone, which by [ 32 ] corresponds to some Schubert coefficient equal to |$1$|⁠. We prove a similar reduction formula when this Schubert coefficient is equal to |$2$| and show some properties of the class of the branch divisor corresponding to a generically finite morphism naturally defined in this context. [ABSTRACT FROM AUTHOR]

In a category C with an (E , M )-factorization structure for morphisms, we prove that any subclass N of M which is closed under pullbacks determines a transitive quasi-uniform structure on C . In addition to providing a categorical characterization of all transitive quasi-uniform structures compatible with a topology, this result also permits us to establish a number of Galois connections related to quasi-uniform structures on C . These Galois connections lead to the description of subcategories of C determined by quasi-uniform structures. Several examples considered at the end of the paper illustrate our results. [ABSTRACT FROM AUTHOR]

All nonhyperbolic automorphisms of the 2-torus are not structurally stable, and it is generally impossible to predict the dynamics of their arbitrarily small perturbations. In this paper, given a representative of each algebraic conjugacy class of nonperiodic nonhyperbolic maps, a one-parameter family of diffeomorphisms is constructed, in which the zero value of the parameter corresponds to the given map and the nonzero values, to Morse–Smale diffeomorphisms. According to results of V. Z. Grines and A. N. Bezdenezhnykh, a Morse–Smale diffeomorphism of a closed orientable surface which induces a nonperiodic action on the fundamental group has nonempty heteroclinic set. It is proved that, in all of the constructed families, the diffeomorphisms corresponding to nonzero parameter values have nonempty orientable heteroclinic sets in which the number of orbits is determined by the automorphism being perturbed. [ABSTRACT FROM AUTHOR]

In this paper we first study some properties of the finite-dimensional simple restricted Lie algebras. In the literature is found a one-to-one correspondence between them and finite-dimensional simple Lie algebras over a field of positive characteristic. Motivated by this fact, we give a one-to-one correspondence between their morphisms, which allow us to conclude that such categories are equivalent. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of natural transformation between morphisms of quasigroupoids and between morphisms of weak Hopf quasigroups. Also, we prove that natural transformations between morphisms of finite quasigroupoids come from natural transformations between morphisms of weak Hopf quasigroups and, on the other hand, we obtain that every natural transformation for morphisms defined between pointed cosemisimple weak Hopf quasigroups comes from a natural transformation between finite quasigroupoids. [ABSTRACT FROM AUTHOR]

We define the notion of a specialization morphism from a locally noetherian analytic adic space to a scheme. This captures the (classical) specialization morphism associated with a formal scheme. There is a well-behaved theory of compactifications and it turns out that the classical specialization morphism is proper in this setup. As an application, we show that the nearby cycles functor commutes with lower shriek in great generality. [ABSTRACT FROM AUTHOR]

A tri-linear rational map in dimension three is a rational map \phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {P}_\mathbb {C}^3 defined by four tri-linear polynomials without a common factor. If \phi admits an inverse rational map \phi ^{-1}, it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set \mathfrak {Bir}_{(1,1,1)} of tri-linear birational maps, up to composition with an automorphism of \mathbb {P}_\mathbb {C}^3, is a locally closed algebraic subset of the Grassmannian of 4-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on \mathfrak {Bir}_{(1,1,1)} given by composition with automorphisms of (\mathbb {P}_\mathbb {C}^1)^3 defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations. [ABSTRACT FROM AUTHOR]

Let K be an uncountable algebraically closed field of characteristic zero. An affine algebraic variety X over K is toral if it is isomorphic to a closed subvariety of a torus (K*)d. We study the group Aut(X) of regular automorphisms of a toral variety X. We prove that if T is a maximal torus in Aut(X), then X is a direct product Y × T, where Y is a toral variety with a trivial maximal torus in the automorphism group. We show that knowing Aut(Y), one can compute Aut(X). In the case when the rank of the group K[Y]*/K* is dim Y + 1, the group Aut(Y) is described explicitly. [ABSTRACT FROM AUTHOR]

We show that for any smooth Calabi–Yau threefold |$X$| of Picard number |$2$| with infinite birational automorphism group, the numerical dimension |$\kappa _\sigma $| of the extremal rays of the movable cone of |$X$| is |$\frac {3}{2}$|⁠. Furthermore, we provide new examples of Calabi–Yau threefolds of Picard number |$2$| with infinite birational automorphism group. [ABSTRACT FROM AUTHOR]

We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces. [ABSTRACT FROM AUTHOR]

Asynchronous, finite dynamical systems are constructed by composing vertex functions f 1 , f 2 through f n using an order π on { 1 , 2 , ... , n } to form maps f π . This article provides a systematic review and new results for comparing the dynamics of maps f π and f π ′ formed using different composition orders π and π ′ . Comparisons are done at the level of (i) entire phase spaces (functional equivalence), (ii) the periodic orbits (cycle equivalence), and (iii) topological conjugation (dynamical equivalence). Conditions and criteria are given in terms of the dependency graph G of the vertex functions along with associated structures such as the acyclic orientations of G and the automorphism group of G. For each type of equivalence, graph-theoretic measures are provided that give an upper bound for the number of distinct structures that can be observed up to the respective notion of equivalence. Several connections to existing mathematical theory (e.g., combinatorics and graphs) are presented along with open questions. [ABSTRACT FROM AUTHOR]

We define certain subvarieties, called -Hecke correspondences, in Cartesian products of diagram automorphism fixed quiver varieties. These give us generators of diagram automorphism fixed Lie algebras. [ABSTRACT FROM AUTHOR]

In the present work we introduce the notion of proper Moore-concrete subcategory of a given concrete category C over a base category X and show that it agrees with that of reflective modification of C. This allows us to associate with any object C of C an object in B that we call the B - closure of C , and with any morphism in C a unique morphism between the corresponding B -closures that we call the B - closure extension. Next, we provide some general properties of proper Moore-concrete subcategories which will be useful to deeply investigate prekernels , precokernels and pretorsion theories in proper Moore-concrete subcategories with respect to some given class of trivial objects. More in detail, we analyze some different conditions to be put on short sequences of morphisms or on short pre-exact sequences in C in order to get informations on the corresponding B -closure extensions and on the B -closure of the various terms of the associated short sequences of B -closure extensions in B. Furthermore, we induce pretorsion theories on proper Moore-concrete subcategories starting from pretorsion theories on the ambient category C and, more in general, we study the interrelations between pretorsion theories on C and proper Moore-concrete subcategories. Finally, we provide a characterization of trivial morphisms and of prekernels of a functor-structured category when we choose Z to be the class of all projective objects. Next, restricting our attention to set-functor structured categories, we obtain a stable pointed category B / R (and a corresponding subcategory (B / R) ⁎ as a quotient of B with respect to a suitable congruence relation R on the hom-sets and use the associated quotient functor to induce a correspondence between prekernels in the subcategory B ⁎ without the objects with empty ground set and kernels in (B / R) ⁎ , and between precokernels in B ⁎ and weak cokernels in (B / R) ⁎. [ABSTRACT FROM AUTHOR]

We construct extended TQFTs associated to Rozansky–Witten models with target manifolds T ∗ C n . The starting point of the construction is the 3-category whose objects are such Rozansky–Witten models, and whose morphisms are defects of all codimensions. By truncation, we obtain a (non-semisimple) 2-category C of bulk theories, surface defects, and isomorphism classes of line defects. Through a systematic application of the cobordism hypothesis we construct a unique extended oriented 2-dimensional TQFT valued in C for every affine Rozansky–Witten model. By evaluating this TQFT on closed surfaces we obtain the infinite-dimensional state spaces (graded by flavour and R-charges) of the initial 3-dimensional theory. Furthermore, we explicitly compute the commutative Frobenius algebras that classify the restrictions of the extended theories to circles and bordisms between them. [ABSTRACT FROM AUTHOR]

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016). [ABSTRACT FROM AUTHOR]

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic Q$\mathcal {Q}$. Hence, as an application of our duality, we develop a topological semantics for Q$\mathcal {Q}$ using orthomodular spaces and prove soundness and completeness. [ABSTRACT FROM AUTHOR]

We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀ x , y ∈ g , [ x , y ] = − [ y , x ] , that is, the operation [ , ] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 6 , or A u t (g) = G 3 G 1 G 2 G 3 G 4 G 7 G 8 G 5 G 3 , where G i is a commutative subgroup of A u t (g) . We give some subgroups of g's automorphism group and systematically studied the properties of these subgroups. [ABSTRACT FROM AUTHOR]

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles. [ABSTRACT FROM AUTHOR]