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In this paper, we establish the inductive Alperin weight condition for the finite simple groups of Lie type A, contributing to the program to prove the Alperin weight conjecture by checking the inductive condition for all finite simple groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set E of finite perimeter, there is an asymptotic limit set (E)∞ corresponding to the asymptotic expansion of E and that every such asymptotic limit (E)∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of (E)∞ is Ahlfors co-dimension 1 regular. [ABSTRACT FROM AUTHOR]

Let V be a vertex operator superalgebra with the natural order 2 automorphism σ. Under suitable conditions on V, the σ-fixed subspace V0 is a vertex operator algebra and the V0-module category CV0 is a modular tensor category. In this paper, we prove that CV0 is a fermionic modular tensor category and the Müger centralizer CV00 of the fermion in CV0 is generated by the irreducible V0-submodules of the V-modules. In particular, CV00 is a super-modular tensor category and CV0 is a minimal modular extension of CV00. We provide a construction of a vertex operator superalgebra Vl for each positive integer l such that C(Vl)0 is a minimal modular extension of CV00. We prove that these modular tensor categories C(Vl)0 are uniquely determined, up to equivalence, by the congruence class of l modulo 16. [ABSTRACT FROM AUTHOR]

Given bivariate means m and M, we can form sequences an, bn defined recursively by an+1 = m(an, bn), bn+1 = M(an,bn) with a0, b0 > 0. These converge (under mild conditions) to a new mean, M(a0, b0), called a compound mean. For m and M homogeneous, M is also homogeneous and satisfies a functional equation. In this paper we study the asymptotic behaviour of M(1,x) as x → ∞ given that of m and M, obtaining the main term up to a possible oscillatory function. We investigate when this oscillatory behaviour is in fact present, in particular for m and M coming from some well-known classes of means. We also present some numerics, which indicate the presence of oscillation is generic. [ABSTRACT FROM AUTHOR]

In 2013, B. Farhi conjectured that for each m ≥ 3, every natural number n can be represented as ⌊x2/m⌋ + ⌊y2/m⌋ + ⌊z2/m⌋ with x,y,z ∈ Z, where ⌊⋅⌋ denotes the floor function. Moreover, in 2015, Z.-W. Sun conjectured that every natural number n can be written as ⌊x2/a⌋ + ⌊y2/b⌋ + ⌊z2/c⌋ with x,y,z ∈ Z, where a,b,c are positive integers and (a,b,c) ≠ (1,1,1),(2,2,2). In this paper, with the help of congruence theta functions, we prove that for each m ≥ 3, B. Farhi's conjecture is true for every sufficiently large integer n. And for a,b,c ≥ 5 with a,b,c pairwise coprime, we also confirm Z.-W. Sun's conjecture for every sufficiently large integer n. [ABSTRACT FROM AUTHOR]

It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number ε >0. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive reflection boundary condition from a local Maxwellian with its fluid quantities given by the solutions to the corresponding compressible Navier-Stokes equations with consistent non-slip boundary conditions whenever ε > 0 is small enough. Specifically, it is shown that for well chosen initial data around constant equilibrium states, the deviation weighted by a velocity function is O(ε1/2) in L∞x,v and O(ε3/2) in L2x,v globally in time. The proof is based on the uniform estimates for the remainder in different functional spaces without any spatial regularity. One key step is to obtain the global-in-time existence as well as uniform-in-ε estimates for regular solutions to the full compressible Navier-Stokes equations in bounded domains when the parameter ε > 0 is involved in the analysis. [ABSTRACT FROM AUTHOR]

In this short note we correct the statement of the main result of [Trans. Amer. Math. Soc. 373 (2020), no. 3, 1909-1940]. That paper presented the rational cohomology ring of a toric arrangement by generators and relations. One of the series of relations given in the paper is indexed over the set circuits in the arrangement's arithmetic matroid. That series of relations should however be indexed over all sets X with |X| = rk(X)+1. Below we give the complete and correct presentation of the rational cohomology ring. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study and classify singular solutions of the Poisson problem { Lsμu = ƒ in Ω\ {0}, u =0 in RN \ Ω for the fractional Hardy operator Lsμu= (−Δ)su + μ/|x|2su in a bounded domain Ω ⊂ RN (N ≥ 2) containing the origin. Here (−Δ)s, s ∈ (0,1), is the fractional Laplacian of order 2s, and μ ≥ μ0, where μ0 = −22s Γ2(N+2s/4)/Γ2(N−2s/4) < 0 is the best constant in the fractional Hardy inequality. The analysis requires a thorough study of fundamental solutions and associated distributional identities. Special attention will be given to the critical case μ = μ0 which requires more subtle estimates than the case μ > μ0. [ABSTRACT FROM AUTHOR]

The aim of this study is to understand to what extent a 1-convex domain with Levi-flat boundary is capable of holomorphic functions with slow growth. This paper discusses a typical example of such domain, the space of all the geodesic segments on a hyperbolic compact Riemann surface. Our main finding is an integral formula that produces holomorphic functions on the domain from holomorphic differentials on the Riemann surface. This construction can be seen as a non-trivial example of L2 jet extension of holomorphic functions with optimal constant. As its corollary, it is shown that the weighted Bergman spaces of the domain is infinite dimensional for any weight order greater than -1 in spite of the fact that the domain does not admit any non-constant bounded holomorphic functions. [ABSTRACT FROM AUTHOR]

Let 1 < p < q ≤ 2. The paper contains the identification of the best constant Cp,q such that the following holds. If X, Y are Hilbert-space valued martingales such that Y is differentially subordinate to X, then we have ||Y||p,∞ ≤ Cp,q ||X||p,q The proof rests on the careful combination of Burkholder's method and optimization arguments. As an application, related sharp Lorentz-norm inequalities for a wide class of Fourier multipliers are obtained. [ABSTRACT FROM AUTHOR]

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples. [ABSTRACT FROM AUTHOR]

This is the last paper in a series of three in which we have studied the Peierls substitution in the case of a weak magnetic field. Here we deal with two 2d Bloch eigenvalues which have a conical crossing. It turns out that in the presence of an almost constant weak magnetic field, the spectrum near the crossing develops gaps which remind of the Landau levels of an effective mass-less magnetic Dirac operator. [ABSTRACT FROM AUTHOR]

Let G be a finite simple connected graph on [n] and [ R = K[x1, . . . , xn] the polynomial ring in n variables over a field K. The edge ideal of G is the ideal I(G) of R which is generated by those monomials xixj for which {i, j} is an edge of G. In the present paper, the possible tuples (n, depth(R/I(G)), reg(R/I(G)), dim R/I(G), deg h(R/I(G))), where deg h(R/I(G)) is the degree of the h-polynomial of R/I(G), arising from Cameron–Walker graphs on [n] will be completely determined. [ABSTRACT FROM AUTHOR]

Let X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T on X such that AT ≔ {x ∈ X : ||Tnx|| → ∞} is not dense and has nonempty interior, whenever X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X for which the set AT and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators. [ABSTRACT FROM AUTHOR]

In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times. As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold. [ABSTRACT FROM AUTHOR]

Given a real number x > 0, we determine qs(x) ≔ inf U(x), where U(x) is the set of all bases q∈ (1,2] for which x has a unique expansion of 0's and 1's. We give an explicit description of qs(x) for several regions of x-values. For others, we present an efficient algorithm to determine qs(x) and the lexicographically smallest unique expansion of x. We show that the infimum is attained for almost all x, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function qs is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of qs. A large part of the paper is devoted to the level sets L(q) ≔ x > 0 : qs(x) = q. We show that L(q) is finite for almost every q, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant qKL = min U(1) ≅ 1.787 we prove that L(qKL) has both infinitely many left- and infinitely many right accumulation points. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Kawamata-Viehweg type vanishing theorem for smooth Fano threefolds, canonical del Pezzo surfaces, and del Pezzo fibrations in positive characteristic. [ABSTRACT FROM AUTHOR]

In this paper we generalize the known De Smet, Dillen, Verstraelen and Vrancken (DDVV)-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices to arbitrary real, complex and quaternionic matrices. Inspired by the Erdős-Mordell inequality, we establish the DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the Böttcher-Wenzel inequality to quaternionic matrices. [ABSTRACT FROM AUTHOR]

Zeta values in Tate algebras were introduced by Pellarin in 2012. They are generalizations of Carlitz zeta values and play an increasingly important role in function field arithmetic. In this paper we prove a conjecture of Pellarin on identities for these zeta values. The proof is based on arithmetic properties of Carlitz zeta values and an explicit formula for Bernoulli-type polynomials attached to zeta values in Tate algebras. [ABSTRACT FROM AUTHOR]

We provide some minor corrections to our paper "Statistics for Iwasawa invariants of elliptic curves". [ABSTRACT FROM AUTHOR]

Let M(2,1)(N) be the infimum of the largest sum-free subset of any set of N positive integers. An old conjecture in additive combinatorics asserts that there is a constant c = c(2,1) and a function ω (N) → ∞ as N → ∞, such that cN + ω (N) < M(2,1)(N) < (c + o(1))N. The constant c(2,1) is determined by Eberhard, Green, and Manners, while the existence of ω (N) is still wide open. In this paper, we study the analogous conjecture on (k,ℓ)-sum-free sets and restricted (k,ℓ)-sum-free sets. We determine the constant c(k,ℓ) for every (k,ℓ)-sum-free sets, and confirm the conjecture for infinitely many (k,ℓ). [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following equations {(−Δ)m+α/2u(x) = ƒ(x,u,Du,⋅⋅⋅), x ∈ Rn, u ∈ C2m+[α],{α}+εloc∩ Lα(Rn), u(x) ≥ 0, x ∈ Rn involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to the above equations. Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to the above fractional higher-order equations with general nonlinearities ƒ(x,u,Du,⋅⋅⋅) including conformally invariant and odd order cases. In particular, we classify nonnegative classical solutions to all odd order conformally invariant equations. Our results completely improve the classification results for third order conformally invariant equations in Dai and Qin (Adv. Math., 328 (2018), 822-857) by removing the assumptions on integrability. We also give a crucial characterization for \alpha-harmonic functions via outer-spherical averages in the appendix. [ABSTRACT FROM AUTHOR]

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph G is uniformly hyperfinite if for any ε > 0 there exists K ≥ 1 such that not only G, but all of its subgraphs of positive measure are (ε,K)-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lovász. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness. [ABSTRACT FROM AUTHOR]

In 2011 Lutwak, Yang and Zhang extended the definition of the Lp-Minkowski convex combination (p ≥ 1) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in Rn, and as a consequence, extended the Lp-Brunn-Minkowski inequality (Lp-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their Lp-Minkowski convex combination—the Lp,s–supremal convolution and prove the Lp-Borell-Brascamp-Lieb type (Lp-BBL) inequalities. Based on the Lp-BBL type inequalities for functions, we extend the Lp-BMI for measurable sets to the class of Borel measures on Rn having (1/s)-concave densities, with s ≥ 0; that is, we show that, for any pair of Borel sets A,B ⊂ Rn, any t ∈ [0,1] and p ≥ 1, one has μ((1-t)⋅p A +p t ⋅p B)p/n+s ≥ (1-t) μ (A)p/n+s + t μ (B)p/n+s, where μ is a measure on Rn having a (1/s)-concave density for 0 ≤ s < ∞. Additionally, with the new defined Lp,s–supremal convolution for functions, we prove Lp-BMI for product measures with quasi-concave densities and for log-concave densities, Lp-Prékopa-Leindler type inequality (Lp-PLI) for product measures with quasi-concave densities, Lp-Minkowski's first inequality (Lp-MFI) and Lp isoperimetric inequalities (Lp-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the p-generalization. [ABSTRACT FROM AUTHOR]

Let R be the power series ring or the polynomial ring over a field k and let I be an ideal of R. Macaulay proved that the Artinian Gorenstein k-algebras R/I are in one-to-one correspondence with the cyclic R-submodules of the divided power series ring \Gamma. The result is effective in the sense that any polynomial of degree s produces an Artinian Gorenstein k-algebra of socle degree s. In a recent paper, the authors extended Macaulay's correspondence characterizing the R-submodules of Γ in one-to-one correspondence with Gorenstein d-dimensional k-algebras. However, these submodules in positive dimension are not finitely generated. Our goal is to give constructive and finite procedures for the construction of Gorenstein k-algebras of dimension one and any codimension. This has been achieved through a deep analysis of the G-admissible submodules of Γ. Applications to the Gorenstein linkage of zero-dimensional schemes and to Gorenstein affine semigroup rings are discussed. [ABSTRACT FROM AUTHOR]

Zarhin has extensively studied restrictions placed on the endomorphism algebras of Jacobians J for which the Galois group associated to their 2-torsion is insoluble and "large" (relative to the dimension of J). In this paper we examine what happens when this Galois group merely contains an element of "large" prime order. [ABSTRACT FROM AUTHOR]

In this paper, we develop an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems. As an application of the theorem, we study the quintic nonlinear Schrödinger equation on the two-dimensional torus iut-Δ u + |u|4u = 0, x ∈ T2 := R2/(2π Z)2, t ∈ R. We obtain a Whitney smooth family of small-amplitude quasi-periodic solutions for the equation. The overall strategy in the proof of the KAM theorem is a normal form techniques sparsing angle-dependent terms, which can be achieved by choosing the appropriate tangential sites. The idea in our proof comes from Geng, Xu, and You [Adv. Math. 226 (2011), pp. 5361–5402], which however has to be substantially developed to deal with the equation above. [ABSTRACT FROM AUTHOR]

In this paper we prove several results regarding decidability of the membership problem for certain submonoids in amalgamated free products and HNN extensions of groups. These general results are then applied to solve the prefix membership problem for a number of classes of one-relator groups which are low in the Magnus-Moldavanskiĭ hierarchy. Since the prefix membership problem for one-relator groups is intimately related to the word problem for one-relator special inverse monoids in the E-unitary case (as discovered in 2001 by Ivanov, Margolis and Meakin), these results yield solutions of the word problem for several new classes of one-relator special inverse monoids. In establishing these results, we introduce a new theory of conservative factorisations of words which provides a link between the prefix membership problem of a one-relator group and the group of units of the corresponding one-relator special inverse monoid. Finally, we exhibit the first example of a one-relator group, defined by a reduced relator word, that has an undecidable prefix membership problem. [ABSTRACT FROM AUTHOR]

Let X be a Kähler manifold which is fibered over a complex manifold Y such that every fiber is a Calabi-Yau manifold. Let ω be a fixed Kähler form on X. By Yau's theorem, there exists a unique Ricci-flat Kähler form ωKE,y on each fiber Xy for y ∈ Y which is cohomologous to ω|Xy. This family of Ricci-flat Kähler forms ωKE,y induces a smooth (1,1)-form ρ on X under a normalization condition. In this paper, we prove that the direct image of ρn+1 is positive on the base Y. We also discuss several byproducts including the local triviality of families of Calabi-Yau manifolds. [ABSTRACT FROM AUTHOR]

Let F be a free pro- p non-abelian group, and let Δ be a commutative Noetherian complete local ring with a maximal ideal I such that char(Δ/I) = p > 0. Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64-69] showed that when p\neq 2, the pro- p congruence subgroup GL21(Δ) = ker (GL2(Δ) Δ → Δ/I → GL2(Δ/I)) admits a pro-p identity, i.e., there exists an element 1 ≠ w ∈ F that vanishes under any continuous homomorphism F → GL21(Δ). In this paper we investigate the case p = 2. The main result is that when char(Δ) = 2, the pro- 2 group GL21(Δ) admits a pro-2 identity. This result was obtained by the use of trace identities that originate in PI-theory. [ABSTRACT FROM AUTHOR]

The topic of this paper is a semi-linear, energy subcritical, defocusing wave equation ∂t2 u − Δ u = − |u|p−1 u in the 3-dimensional space (3 ≤ p < 5) whose initial data are radial and come with a finite energy. We split the energy into inward and outward energies, then apply the energy flux formula to obtain the following asymptotic distribution of energy: Unless the solution scatters, its energy can be divided into two parts: ''scattering energy'', which concentrates around the light cone |x| = |t| and moves to infinity at the light speed, and ''retarded energy'', which is at a distance of at least |t|β behind when |t| is large. Here β is an arbitrary constant smaller than β0(p) = 2(p-2)/p+1. A combination of this property with a more detailed version of the classic Morawetz estimate gives a scattering result under a weaker assumption on initial data (u0,u1) than previously known results. More precisely, we assume ∫R3 (|x|κ + 1) (½|∇u0|2 + ½|u1|2 + 1/p+1 |u|p+1) dx < + ∞. Here β > β0(p) = 1−β0(p) = 5−p/p+1 is a constant. This condition is so weak that the initial data may be outside the critical Sobolev space of this equation. This phenomenon is not covered by previously known scattering theory, as far as the author knows. [ABSTRACT FROM AUTHOR]

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar. In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near ℓ1. For our negative results we generalize an argument of Zienkiewicz. [ABSTRACT FROM AUTHOR]

The paper considers compactness of Sobolev embeddings of non-compact manifolds, restricted to subsets (typically subspaces) defined either by conditions of symmetry (or quasisymmetry) relative to actions of compact groups, or by restriction in the number of variables, i.e. consisting of functions of the form ƒ° φ with a fixed φ. The manifolds are assumed to satisfy general common conditions under which Sobolev embeddings exist. We provide sufficient conditions for compactness of the embeddings, which in many situations are also necessary. [ABSTRACT FROM AUTHOR]

In this paper, we develop an elementary and unified treatment, in the spirit of Rivière and Struwe (Comm. Pure. Appl. Math. 2008), to explore regularity of weak solutions of higher order geometric elliptic systems in critical dimensions without using conservation law. As a result, we obtain an interior Hölder continuity for solutions of the higher order elliptic system of de Longueville and Gastel in critical dimensions Δku = ∑i=0k−1Δi⟨Vi,du⟩ + ∑i=0 k−2Δiδ (widu) quad in B2k, under critical regularity assumptions on the coefficient functions. This verifies an expectation of Rivière, and provides an affirmative answer to an open question of Struwe in dimension four when k = 2. The Hölder continuity is also an improvement of the continuity result of Lamm and Rivière and de Longueville and Gastel. [ABSTRACT FROM AUTHOR]

In this paper, we prove a generalized Donaldson-Uhlenbeck-Yau theorem on Higgs bundles over a class of non-compact Gauduchon manifolds. [ABSTRACT FROM AUTHOR]

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity. [ABSTRACT FROM AUTHOR]

In this paper, we study some algebraic topology aspects of Stringc structures, more precisely, from the perspective of Whitehead tower and the perspective of the loop group of Spinc(n). We also extend the generalized Witten genera constructed for the first time by Chen et al. [J. Differential Geom. 88 (2011), pp. 1-40] to correspond to Stringc structures of various levels and give vanishing results for them. [ABSTRACT FROM AUTHOR]

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2-punctured elliptic curve C. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P1 × P1. We also showcase a special curve Γ isomorphic to C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P1 via a modular map which turns out to be the 2:1 cover ramified in Γ. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles. [ABSTRACT FROM AUTHOR]

We study derived categories arising from quivers with potential associated to a decorated marked surface SΔ, in the sense taken in a paper by Qiu. We prove two conjectures from Qiu's paper in which, under a bijection between certain objects in these categories and certain arcs in SΔ, the dimensions of morphisms between these objects equal the intersection numbers between the corresponding arcs. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the asymptotic behavior of automorphism groups of function fields when genus tends to infinity. Motivated by applications in coding and cryptography, we consider the maximum size of abelian subgroups of the automorphism group Aut(F/Fq) in terms of genus gF for a function field F over a finite field Fq. Although the whole group Aut(F/Fq) could have size Ω(gF4), the maximum size mF of abelian subgroups of the automorphism group Aut(F/Fq) is upper bounded by 4gF + 4 for gF ≥ 2. In the present paper, we study the asymptotic behavior of mF by defining Mq = lim supgF→∞ mF·logq mF/gF, where F runs through all function fields over Fq. We show that Mq lies between 2 and 3 (resp., 4) for odd characteristic (resp., even characteristic). This means that mF grows much more slowly than genus does asymptotically. The second part of this paper is to study the maximum size bF of subgroups of Aut(F/Fq) whose order is coprime to q. The Hurwitz bound gives an upper bound bF ≤ 84(gF - 1) for every function field F/Fq of genus gF ≥ 2. We investigate the asymptotic behavior of bF by defining Bq = lim supgF →∞ bF/gF, where F runs through all function fields over Fq. Although the Hurwitz bound shows Bq ≤ 84, there are no lower bounds on Bq in the literature. One does not even know whether Bq = 0. For the first time, we show that Bq ≥ 2/3 by explicitly constructing some towers of function fields in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we consider the final state problem for the nonlinear Schrödinger equation with a homogeneous nonlinearity of the critical order which is not necessarily a polynomial. In [SIAM J. Math. Anal. 50 (2018), pp. 3251–3270], the first and second authors consider one- and twodimensional cases and give a sufficient condition on the nonlinearity so that the corresponding equation admits a solution that behaves like a free solution with or without a logarithmic phase correction. The present paper is devoted to the study of the three-dimensional case, in which it is required that a solution converge to a given asymptotic profile in a faster rate than in the lower dimensional cases. To obtain the necessary convergence rate, we employ the end-point Strichartz estimate and modify a time-dependent regularizing operator, introduced in the aforementioned article. Moreover, we present a candidate for the second asymptotic profile of the solution. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of two dedicated to the study of period relations of the type L(1/2 + k,Π) ∈ (2πi)d·kΩ(-1)kQ(Π), 1/2 + k critical, for certain automorphic representations Π of a reductive group G. In this paper we discuss the case G = GL(n+1)×GL(n). The case G = GL(2n) is discussed in part two. Our method is representation theoretic and relies on the author's recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation Π under consideration. The new period relations we prove are in accordance with Deligne's Conjecture on special values of L-functions, and the author expects this method to apply to other cases as well. [ABSTRACT FROM AUTHOR]

We consider a nonlinear heat equation with a double source: |u|p-1u and |∇u|q. This equation has a double interest: in ecology, it was used by Souplet (1996) as a population dynamics model; in mathematics, it was introduced by Chipot and Weissler (1989) as an intermediate equation between the semilinear heat equation and the Hamilton-Jacobi equation. Further interest in this equation comes from its lack of variational structure. In this paper, we intend to see whether the standard blow-up dynamics known for the standard semilinear heat equation (with |u|p-1u as the only source) can be modified by the addition of the second source (|∇u|q). Here arises a nice critical phenomenon at blow-up: - when q < 2p/(p+1), the second source is subcritical in size with respect to the first, and we recover the classicial blow-up profile known for the standard semilinear case; - when q = 2p/(p + 1), both terms have the same size, and only partial blow-up descriptions are available. In this paper, we focus on this case, and start from scratch to: - first, formally justify the occurrence of a new blow-up profile, which is different from the standard semilinear case; - second, to rigorously justify the existence of a solution obeying that profile, thanks to the constructive method introduced by Bricmont and Kupiainen together with Merle and Zaag. Note that our method yields the stability of the constructed solution. Moreover, our method is far from being a straightforward adaptation of earlier literature and should be considered as a source of novel ideas whose application goes beyond the particular equation we are considering, as we explain in the introduction. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let (E,F) be a regular and local Dirichlet form on L²(I,m), where I is an interval and m is a fully supported Radon measure on I. We shall first present a complete representation for (E,F), which shows that (E,F) lives on at most countable disjoint "effective" intervals with an "adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C8 c (I) being a special standard core of (E,F) and shall identify the closure of C8 c (I) in (E,F) when C8 c (I) is contained but not necessarily dense in F relative to the E1/2 1 -norm. This paper is partly motivated by a result of Hamza's that was stated in a theorem of Fukushima, Oshima, and Takeda's and that provides a different point of view to this theorem. To illustrate our results, many examples are provided. [ABSTRACT FROM AUTHOR]

This paper forms the second part of a series of three papers by the authors concerning the structure of nilspaces of Antolín Camarena and Szegedy. A nilspace is a compact space X together with closed collections of cubes Cn(X) ⊆ X2n, n = 1, 2, … satisfying some natural axioms. From these axioms it follows that a nilspace can be built as a finite tower of extensions where each of the successive fibers is a compact abelian group. Our main result is a new proof of a result due to Antolín Camarena and Szegedy [Nilspaces, nilmanifolds and their morphisms, arXiv:1009.3825v3 (2012)] stating that if each of these groups is a torus, then X is isomorphic (in a strong sense) to a nilmanifold G/Г. We also extend the theorem to a setting where the nilspace arises from a dynamical system (X, T). These theorems are a key stepping stone towards the general structure theorem in [The structure theory of nilspaces III: Inverse limit representations and topological dynamics, arXiv:1605.08950v1 [math.DS] (2016)] (which again closely resembles the main theorem of Antolín Camarena and Szegedy). The main technical tool, enabling us to deduce algebraic information from topological data, consists of existence and uniqueness results for solutions of certain natural functional equations, again modelled on the theory in Antolín Camarena and Szegedy's paper. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. The first purpose is to establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into an arbitrary projective variety intersecting a family of hypersurfaces in subgeneral position. In our result, the truncation level of the counting functions is estimated explicitly. Our result is an extension of the classical second main theorem of H. Cartan and is also a generalization of the recent second main theorem of M. Ru and improves some recent results. The second purpose of this paper is to give another proof for the second main theorem for the special case where the projective variety is a projective space, by which the truncation level of the counting functions is estimated better than that of the general case. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the k-homogeneous permutation groups (those which act transitively on the subsets of size k of their domain X) where |X| = n and k < n/2. In the process we obtain, for k-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on k-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture: If a transformation semigroup S contains singular maps and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group. For the special case that S contains all constant maps, this conjecture was proved correct more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups, and computational algebra are proposed at the end of the paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define conformal operators that arise from an extension problem of codimension two. To this end we interpret and extend results of representation theory from a purely analytic point of view. The first part of the paper is an interpretation of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces. In the flat case, these results are well known from the representation theory perspective but have been much less explored in the context of non-local operators in partial differential equations. This analytic approach will be needed in order to consider the curved case. In the second part of the paper we construct new boundary operators with good conformal properties that generalize the fractional Laplacian in Rn using an extension problem in which the boundary is of codimension two. Then we extend these results to more general manifolds that are not necessarily symmetric spaces. [ABSTRACT FROM AUTHOR]

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups. [ABSTRACT FROM AUTHOR]

In this paper, we develop a lower bound for the double slice genus of a knot using Casson-Gordon invariants. As an application, we show that the double slice genus can be arbitrarily larger than twice the slice genus. As an analogue to the double slice genus, we also define the superslice genus of a knot, and give both an upper bound and a lower bound in the topological category. [ABSTRACT FROM AUTHOR]