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Let G be a finite group, p a prime, and k a field of characteristic p. We introduce the notion of an endotrivial chain complex of p -permutation kG -modules, and study the corresponding group E k (G) of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of kG. The elements of E k (G) are determined uniquely by integral invariants arising from the Brauer construction and a degree one character G → k ×. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of E k (G) , and as an application, give complete descriptions of E k (G) for abelian groups and p -groups of normal p -rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism Λ : E k (G) → O (T (k G)) , where O (T (k G)) is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of Λ must satisfy. [ABSTRACT FROM AUTHOR]

Let G be a finite group and X be a subgroup of G. We investigate the topological properties of the poset C X (G) of cosets Hx in G with X ≤ H < G. We show that C X (G) is non-contractible if G is solvable or N G (X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe's work in [8] (2016). We also give some divisibility properties of the Euler characteristic of C X (G) when X is a p -group, which follows K. S. Brown's classical result in [3] (2000). [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the dimension of dual modules of local cohomology of Stanley–Reisner rings to obtain a new vector. This vector contains important information on the Serre's condition (S r) and the CM t property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d − 1) -dimensional simplicial complexes with codimension two which are (S d − 3) but they are not Cohen–Macaulay. By using this characterization, we obtain a condition to equality of projective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h -vectors and give a negative answer to a known question regarding these vectors. [ABSTRACT FROM AUTHOR]

We examine virtual resolutions of Stanley–Reisner ideals for a product of projective spaces by providing sufficient conditions for a simplicial complex to be virtually Cohen–Macaulay (to have a virtual resolution with length equal to its codimension). In particular, we show that all balanced simplicial complexes are virtually Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

Tensors are ubiquitous in mathematics and the sciences, as they allow to store information in a concise way. Decompositions of tensors may give insights into their structure and complexity. In this work, we develop a new framework for decompositions of tensors, taking into account invariance, positivity and a geometric arrangement of their local spaces. We define an invariant decomposition with indices arranged on a simplicial complex which is explicitly invariant under a group action. We give a constructive proof that this decomposition exists for all invariant tensors, after possibly enriching the simplicial complex. We further define several decompositions certifying positivity, and prove similar existence results, as well as inequalities between the corresponding ranks. Our results generalize results from the theory of tensor networks, used in the study of quantum many-body systems, for example. [ABSTRACT FROM AUTHOR]

We put a cochain complex structure CH ⁎ (Z K) on the cohomology of a moment-angle complex Z K and call the resulting cohomology the double cohomology, HH ⁎ (Z K). We give three equivalent definitions for the differential, and compute HH ⁎ (Z K) for a family of simplicial complexes containing clique complexes of chordal graphs. [ABSTRACT FROM AUTHOR]

Let R be a ring such that (GP , GP ⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R -modules. Recently, the first author defined for any complex X the relative cohomology functors Ext GP ⁎ (X , −) as H − ⁎ (Hom (G , −)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which Ext GP i (X , Q) = 0 for all i > n and all R -modules Q ∈ GP ⊥. This result gives a "Gorenstein" version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors Ext R ⁎ (− , −). [ABSTRACT FROM AUTHOR]

We reformulate the problem of bounding the total rank of the homology of perfect chain complexes over the group ring F p [ G ] of an elementary abelian p -group G in terms of commutative algebra. This extends results of Carlsson for p = 2 to all primes. As an intermediate step, we construct an embedding of the derived category of perfect chain complexes over F p [ G ] into the derived category of p -DG modules over a polynomial ring. [ABSTRACT FROM AUTHOR]

We study 2-term tilting complexes of Brauer tree algebras in terms of simplicial complexes. We show the symmetry and convexity of the lattice polytope corresponding to the simplicial complex of 2-term tilting complexes. Via a geometric interpretation of derived equivalences, we show that the f -vector of the simplicial complexes of Brauer tree algebras only depends on the number of the edges of the Brauer trees and hence it is a derived invariant. In particular, this result implies that the number of 2-term tilting complexes, which is in bijection with support τ -tilting modules, is a derived invariant. Moreover, we apply our result to the enumeration problem of Coxeter-biCatalan combinatorics. [ABSTRACT FROM AUTHOR]

The mitochondrial cytochrome c oxidase, the terminal enzyme of the respiratory chain, contains heme and copper centers for electron transfer. The conserved COX2 subunit contains the Cu A site, a binuclear copper center. The copper chaperones SCO1, SCO2, and COA6, are required for Cu A center formation. Loss of function of these chaperones and the concomitant cytochrome c oxidase deficiency cause severe human disorders. Here we analyzed the molecular function of COA6 and the consequences of COA6 deficiency for mitochondria. Our analyses show that loss of COA6 causes combined complex I and complex IV deficiency and impacts membrane potential-driven protein transport across the inner membrane. We demonstrate that COA6 acts as a thiol-reductase to reduce disulfide bridges of critical cysteine residues in SCO1 and SCO2. Cysteines within the CX 3 CX N H domain of SCO2 mediate its interaction with COA6 but are dispensable for SCO2–SCO1 interaction. Our analyses define COA6 as thiol-reductase, which is essential for Cu A biogenesis. Image 1 • Loss of COA6 affects respiratory chain complexes IV and I. • Decreased membrane potential-driven protein import due to loss of COA6. • Cysteine residues in the CX 3 CX N H motif of SCO2 mediate COA6 interaction. • COA6 acts as thiol reductase for copper metallochaperones during Cu A biogenesis. [ABSTRACT FROM AUTHOR]

• Pt-Co hollow nanospheres were synthesized via a sequence reduction. • Co-B-O complex is a key intermediates for forming the hollow-sphered nanostructure. • The optimized Pt-Co hollow nanospheres exhibit efficient activity towards HER. • Hollow nanospheres is a effective way to improve the utilization of precious metals. Improving utilization of precious metals is one of the crucial challenges for electrochemical energy catalysis. Tuning precious metal structure to expose active sites is an effective strategy to reduce mass loading in the practical application. Herein, Pt-Co hollow nanospheres were synthesized via a sequential reduction strategy using amorphous Co-B-O complex as the key spheres-forming intermediates. The modulation mechanism of Pt-Co hollow nanospheres formation was established by investigating critical factors of addition sequence of reducing agent, the type of reducing agents and the ratio of Pt and Co precursors. The optimized Pt-Co hollow nanospheres with the Pt/Co molar ratio of 1:1 by this method is characterized by the average diameter of around 30 nm and the shell thickness of 2.3 nm. They displays superior performance for enhanced hydrogen evolution reaction (HER) in the aspect of activity and stability in 0.5 M H 2 SO 4 , in which the mass activity of the Pt-Co hollow spheres exhibits 2.82 times higher than that of commercial Pt/C and its stability also has a significant enhancement. The outstanding activity of the Pt-Co hollow nanospheres for HER is mainly attributed to its highly accessible electron/mass transport path, expose active sites on the ultra-thin shells and strain effect induced by the hollow sphere structure. This hollow nanospheres with ultra-thin shell provides a promising way to develop highly efficient precious metal based electrocatalysts. [ABSTRACT FROM AUTHOR]

Let R be a Noetherian commutative ring and M a R -module with pd R M ≤ 1 that has rank. Necessary and sufficient conditions were provided in [1] for an exterior power ∧ k M to be torsion free. When M is an ideal of R similar necessary and sufficient conditions were provided in [2] for a symmetric power S k M to be torsion free. We extend these results to a broad class of Schur modules L λ / μ M. En route, for any map of finite free R modules ϕ : F → G we also study the general structure of the Schur complexes L λ / μ ϕ , and provide necessary and sufficient conditions for the acyclicity of any given L λ / μ ϕ by computing explicitly the radicals of the ideals of maximal minors of all its differentials. [ABSTRACT FROM AUTHOR]

Let Δ be a connected, pure 2-dimensional simplicial complex embedded in R 2 and let C r (Δ ˆ) be the homogenized spline module of Δ with smoothness r as in [7]. To study C r (Δ ˆ) , Schenck and Stillman developed in [7] the quotient complex S • / J •. In [8] , Schenck and Stiller conjectured that the regularity of H 1 (S • / J •) is less than 2 r + 1. In this article, we pose a counterexample to this " 2 r + 1 " conjecture. We also propose some modifications to the conjecture. [ABSTRACT FROM AUTHOR]

In a previous paper, we have defined polynomial Witt vectors functor from vector spaces over a perfect field k of positive characteristic p to abelian groups. In this paper, we use polynomial Witt vectors to construct a functorial Hochschild-Witt complex W C H ⁎ (A) for any associative unital k -algebra A , with homology groups W H H ⁎ (A). We prove that the group W H H 0 (A) coincides with the group of non-commutative Witt vectors defined by Hesselholt, while if A is commutative, finitely generated, and smooth, the groups W H H i (A) are naturally identified with the terms W Ω A i of the de Rham-Witt complex of the spectrum of A. [ABSTRACT FROM AUTHOR]

Abstract Let A be a Grothendieck category. We prove that there is an infinite regular cardinal λ such that A has λ -pure injective envelopes. Furthermore, we show that any λ -pure acyclic complex in A is a directed union of contractible subcomplexes. This provides an answer to a question which is conjectured by Simson in [14, Conjecture 3.4]. [ABSTRACT FROM AUTHOR]

Abstract We prove that any derived equivalence between derived-discrete algebras of finite global dimension is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. [ABSTRACT FROM AUTHOR]

Abstract We apply the Auslander–Buchweitz approximation triangles to study (pre)silting subcategories in a triangulated catego- ry T. An Auslander–Reiten type correspondence between the class of silting subcategories of T and that of certain covariantly finite subcategories of T is established. We introduce a relation on presilting subcategories of T , which can be used to obtain another silting subcategory from a given one. We also give a Bazzoni's characterization for two presilting subcategories satisfying such a relation. The results can be used to improve the Auslander–Reiten type correspondence and Bazzoni's characterization for small silting complexes established by Wei. [ABSTRACT FROM AUTHOR]

Abstract We prove that the (n − 2) -dimensional surface area (perimeter) of central hyperplane sections of the n -dimensional unit cube is maximal for the hyperplane perpendicular to the vector (1 , 1 , 0 , ... , 0). This gives a positive answer to a question of Pełczyński who solved the three dimensional case. We study both the real and the complex versions of this problem. We also use our result to show that the answer to an analogue of the Busemann–Petty problem for the surface area is negative in dimensions 14 and higher. [ABSTRACT FROM AUTHOR]

Abstract We prove a Jensen-disc type theorem for polynomials p ∈ R [ z ] having all their zeros in a sector of the complex plane. This result is then used to prove the existence of a collection of linear operators T : R [ z ] → R [ z ] which map polynomials with their zeros in a closed convex sector | arg ⁡ z | ≤ θ < π / 2 to polynomials with zeros in a smaller sector | arg ⁡ z | ≤ γ < θ. We, therefore, provide the first example of a zero-sector reducing operator. [ABSTRACT FROM AUTHOR]

Abstract We study complex interpolation scales obtained by vector valued amalgamation and the derivations they generate. We study their trivial and singular character and obtain examples showing that the hypotheses in the main theorems of Castillo et al. (2017) [9] are not necessary. [ABSTRACT FROM AUTHOR]

Abstract In this paper we give a formula for the homotopy groups of (n − 1) -connected 2 n -manifolds as a direct sum of homotopy groups of spheres in the case the n th Betti number is larger than 1. We demonstrate that when the n th Betti number is 1 the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulas for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore. [ABSTRACT FROM AUTHOR]

We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces/annihilators. We characterize the kernels and images of the related trace operators and discuss duality relationships between trace spaces. We elaborate that many properties of the classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We arrive at abstract trace Hilbert complexes that can be formulated using quotient spaces/annihilators. We show that, if a Hilbert complex admits stable "regular decompositions" with compact lifting operators, then the associated trace Hilbert complex is Fredholm. Incarnations of abstract concepts and results in the concrete case of the de Rham complex in three-dimensional Euclidean space will be discussed throughout. [ABSTRACT FROM AUTHOR]

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the "volume" of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz. [ABSTRACT FROM AUTHOR]

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given. [ABSTRACT FROM AUTHOR]

In this paper, we prove a C 2 , α -estimate for the solution to the complex Monge–Ampère equation det ⁡ ( u i j ¯ ) = f with 0 < f ∈ C α , under the assumption that u ∈ C 1 , β for some β < 1 which depends on n and α . [ABSTRACT FROM AUTHOR]

The theory of k -regular graphs is closely related to group theory. Every k -regular, bipartite graph is a Schreier graph with respect to some group G , a set of generators S (depending only on k ) and a subgroup H . The goal of this paper is to begin to develop such a framework for k -regular simplicial complexes of general dimension d . Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of k -regular simplicial complexes as quotients of one universal object: the k -regular d -dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on d and k . Along the way we answer a question from Parzanchevski and Rosenthal (2016) on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton’s graph covering theorem. This approach also suggests a random model for k -regular d -dimensional multicomplexes. [ABSTRACT FROM AUTHOR]

In his seminal 1983 paper, Jim Lawrence introduced lopsided sets and featured them as asymmetric counterparts of oriented matroids, both sharing the key property of strong elimination. Moreover, symmetry of faces holds in both structures as well as in the so-called affine oriented matroids. These two fundamental properties (formulated for covectors) together lead to the natural notion of “conditional oriented matroid” (abbreviated COM). These novel structures can be characterized in terms of three cocircuits axioms, generalizing the familiar characterization for oriented matroids. We describe a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces. A realizable COM is represented by a hyperplane arrangement restricted to an open convex set. Among these are the examples formed by linear extensions of ordered sets, generalizing the oriented matroids corresponding to the permutohedra. Relaxing realizability to local realizability, we capture a wider class of combinatorial objects: we show that non-positively curved Coxeter zonotopal complexes give rise to locally realizable COMs. [ABSTRACT FROM AUTHOR]

Schoenberg’s theorem for the complex Hilbert sphere proved by Christensen and Ressel in 1982 by Choquet theory is extended to the following result: Let L denote a locally compact group and let D ¯ denote the closed unit disc in the complex plane. Continuous functions f : D ¯ × L → C such that f ( ξ ⋅ η , u − 1 v ) is a positive definite kernel on the product of the unit sphere in ℓ 2 ( C ) and L are characterized as the functions with a uniformly convergent expansion f ( z , u ) = ∑ m , n = 0 ∞ φ m , n ( u ) z m z ¯ n , where φ m , n is a double sequence of continuous positive definite functions on L such that ∑ φ m , n ( e L ) < ∞ ( e L is the neutral element of L ). It is shown how the coefficient functions φ m , n are obtained as limits from expansions for positive definite functions on finite dimensional complex spheres via a Rodrigues formula for disc polynomials. Similar results are obtained for the real Hilbert sphere. [ABSTRACT FROM AUTHOR]

It has long been known that the differential operator D represents a typical example of unbounded operators on many Banach spaces including the classical Fock spaces, the Fock–Sobolev spaces, and the generalized Fock spaces where the weight decays faster than the Gaussian weight. In this note we identify Fock type spaces where the operator admits boundedness, compactness and membership in the Schatten S p class spectral structures. We also showed that its nontrivial spectrum while acting on such spaces is precisely the closed unit disk D ‾ in the complex plane. [ABSTRACT FROM AUTHOR]

Let P 1 and P 2 be two points in C 2 such that the complex line joining them is tangent to the unit sphere ∂ B 2 . We prove that if a real analytic function f on ∂ B 2 extends holomorphically on every complex line through P 1 and P 2 , then f extends holomorphically to the whole ball B 2 . [ABSTRACT FROM AUTHOR]

Consider 2 n points on the unit circle and a reference dissection D ∘ of the convex hull of the odd points. The accordion complex of D ∘ is the simplicial complex of subsets of pairwise noncrossing diagonals with even endpoints that cross a connected set of diagonals of the dissection D ∘ . In particular, this complex is an associahedron when D ∘ is a triangulation, and a Stokes complex when D ∘ is a quadrangulation. We exhibit a bijection between the facets of the accordion complex of D ∘ and some dual objects called the serpent nests of D ∘ . This confirms in particular a prediction of F. Chapoton (2016) in the case of Stokes complexes. [ABSTRACT FROM AUTHOR]

Let f be a polynomial (or polynomial system) with all simple roots. The root separation of f is the minimum of the pair-wise distances between the complex roots. A root separation bound is a lower bound on the root separation. Finding a root separation bound is a fundamental problem, arising in numerous disciplines. We present two new root separation bounds: one univariate bound, and one multivariate bound. The new bounds improve on the old bounds in two ways: (1) The new bounds are usually significantly bigger (hence better) than the previous bounds. (2) The new bounds scale correctly , unlike the previous bounds. Crucially, the new bounds are not harder to compute than the previous bounds. [ABSTRACT FROM AUTHOR]

This paper gives necessary and sufficient conditions for an essentially normal operator T to have its C ⁎ -algebra C ⁎ ( T ) generated by a complex symmetric operator. Also it is completely determined when C ⁎ ( T ) is ⁎-isomorphic to a C ⁎ -algebra singly generated by complex symmetric operators. These both depend only on the singular part of T . [ABSTRACT FROM AUTHOR]

Let k be a field and let Λ be a finite dimensional k -algebra. We prove that every bounded complex V • of finitely generated Λ-modules has a well-defined versal deformation ring R ( Λ , V • ) which is a complete local commutative Noetherian k -algebra with residue field k . We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k -algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra. [ABSTRACT FROM AUTHOR]

For an associative ring R , the projective level of a complex F is the smallest number of mapping cones needed to build F from projective R -modules. We establish lower bounds for the projective level of F in terms of the vanishing of homology of F . We then use these bounds to derive a new version of The New Intersection Theorem for level when R is a commutative Noetherian local ring. [ABSTRACT FROM AUTHOR]

We introduce and study the new concepts of cosilting complexes, cosilting modules and AIR-cotilting modules. We prove that the three concepts AIR-cotilting modules, cosilting modules and quasi-cotilting modules coincide with each other, in contrast with the dual fact that AIR-tilting modules, silting modules and quasi-tilting modules are different. Further, we show that there are bijections between the following four classes (1) equivalence classes of AIR-cotilting (resp., cosilting, quasi-cotilting) modules, (2) equivalence classes of 2-term cosilting complexes, (3) torsion-free cover classes and (4) torsion-free special precover classes. We also extend a classical result of Auslander and Reiten on the correspondence between certain contravariantly finite subcategories and cotilting modules to the case of cosilting complexes. [ABSTRACT FROM AUTHOR]

We prove a special case of a conjecture of Dao–Kurano concerning the vanishing of Hochster's theta pairing. The proof uses Adams operations on both topological K -theory and perfect complexes with support. [ABSTRACT FROM AUTHOR]

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly ( d + 1 ) -colored) triangulation of a combinatorial d -manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d -manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m ≥ d + 2 and any closed combinatorial d -manifold M , two m -colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings. [ABSTRACT FROM AUTHOR]

We consider the 0-order perturbed Lamé operator − Δ ⁎ + V ( x ) . It is well known that if one considers the free case, namely V = 0 , the spectrum of − Δ ⁎ is purely continuous and coincides with the non-negative semi-axis. The first purpose of the paper is to show that, at least in part, this spectral property is preserved in the perturbed setting. Precisely, developing a suitable multipliers technique, we will prove the absence of point spectrum for Lamé operator with potentials which satisfy a variational inequality with suitable small constant. We stress that our result also covers complex-valued perturbation terms. Moreover the techniques used to prove the absence of eigenvalues enable us to provide uniform resolvent estimates for the perturbed operator under the same assumptions about V . [ABSTRACT FROM AUTHOR]

We generalize the problem of univalence of the integral of f ′ ( z ) α when f is univalent to the complex harmonic mappings. To do this, we extend the univalence criterion by Ahlfors in [1] to those mappings. [ABSTRACT FROM AUTHOR]

We study various complexity properties of suffix-free regular languages. A sequence ( L k , L k + 1 , … ) of regular languages in some class, where n is the quotient complexity of L n , is most complex if its languages L n meet the complexity upper bounds for all basic measures. It is known that there exist such most complex sequences in several classes of regular languages. In contrast to this, we prove that there does not exist a most complex sequence in the class of suffix-free regular languages. However, we do exhibit two such sequences that together meet upper bounds for all basic measures. [ABSTRACT FROM AUTHOR]

Let Λ be a derived-discrete algebra. We show that the Krull–Gabriel dimension of the homotopy category of projective Λ-modules, and therefore the Cantor–Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobiński and Krause [8] . We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [17] . Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective Λ-modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple. [ABSTRACT FROM AUTHOR]

The class of objects we consider are algebraic relations between the four kinds of classical Jacobi theta functions θ j ( z | τ ) , j = 1 , … , 4 , and their derivatives. We present an algorithm to prove such relations automatically where the function argument z is zero, but where the parameter τ in the upper half complex plane is arbitrary. [ABSTRACT FROM AUTHOR]

We establish several strengthened versions of Lurie's Tannaka duality theorem for certain classes of spectral algebraic stacks. Our most general version of Tannaka duality identifies maps between stacks with exact symmetric monoidal functors between ∞-categories of quasi-coherent complexes which preserve connective and pseudo-coherent complexes. [ABSTRACT FROM AUTHOR]

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in the case where the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin–Tits group. These lifts turn out to be generalizations of Mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups. [ABSTRACT FROM AUTHOR]

Given an infinite field k and a simplicial complex Δ, a common theme in studying the f - and h -vectors of Δ has been the consideration of the Hilbert series of the Stanley–Reisner ring k [ Δ ] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of h d − 1 a (Δ) , the dimension over k in degree d − 1 of k [ Δ ] / (Θ) , for any complex Δ of dimension d − 1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series. [ABSTRACT FROM AUTHOR]

This paper aims to find the most general combinatorial conditions under which a moment–angle complex ( D 2 , S 1 ) K is a co- H -space, thus splitting unstably in terms of its full subcomplexes. In this way we study to which extent the conjecture holds that a moment–angle complex over a Golod simplicial complex is a co- H -space. Our main tool is a certain generalisation of the theory of labelled configuration spaces. [ABSTRACT FROM AUTHOR]

The Hom complex Hom ( T , G ) of graphs is a CW-complex associated to a pair of graphs T and G , considered in the graph coloring problem. It is known that certain homotopy invariants of Hom ( T , G ) give lower bounds for the chromatic number of G . For a fixed finite graph T , we show that there is no homotopy invariant of Hom ( T , G ) which gives an upper bound for the chromatic number of G . More precisely, for a non-bipartite graph G , we construct a graph H such that Hom ( T , G ) and Hom ( T , H ) are homotopy equivalent but χ ( H ) is much larger than χ ( G ) . The equivariant homotopy type of Hom ( T , G ) is also considered. [ABSTRACT FROM AUTHOR]

We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic. [ABSTRACT FROM AUTHOR]