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It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the q-Morris constant term identity[J. Combin. Theory Ser. A 81 (1998), pp. 69–87]. This conjecture was proved and extended by Károlyi, Nagy, Petrov, and Volkov (KNPV) in 2015 [Adv. Math. 277 (2015), pp. 252–282]. In this paper, we obtain two symmetric function generalizations of the q-Baker–Forrester ex-conjecture. These include: (i) a q-Baker–Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's result. [ABSTRACT FROM AUTHOR]

In this paper, we study the Minkowski-type inequality for asymptotically flat static manifolds (M^{n},g) with boundary and with dimension n<8 that was established by McCormick [Proc. Amer. Math. Soc. 146 (2018), pp. 4039–4046]. First, we show that any asymptotically flat static (M^{n},g) which achieves the equality and has CMC or equipotential boundary is isometric to a rotationally symmetric region of the Schwarzschild manifold. Then, we apply conformal techniques to derive a new Minkowski-type inequality for the level sets of bounded static potentials. Taken together, these provide a robust approach to detecting rotational symmetry of asymptotically flat static systems. As an application, we prove global uniqueness of static metric extensions for the Bartnik data induced by both Schwarzschild coordinate spheres and Euclidean coordinate spheres in dimension n < 8 under the natural condition of Schwarzschild stability. This generalizes an earlier result of Miao [Classical Quantum Gravity 22 (2005), pp. L53–L59]. We also establish uniqueness for equipotential photon surfaces with small Einstein-Hilbert energy. This is interesting to compare with other recent uniqueness results for static photon surfaces and black holes, e.g. see V. Agostiniani and L. Mazzieri [Comm. Math. Phys. 355 (2017), pp. 261–301], C. Cederbaum and G. J. Galloway [J. Math. Phys. 62 (2021), p. 22], and S. Raulot [Classical Quantum Gravity 38 (2021), p. 22]. [ABSTRACT FROM AUTHOR]

In this paper, we study 4-dimensional complete hypersurfaces with w-constant mean curvature in the unit sphere. We give a lower bound of the scalar curvature for 4-dimensional complete hypersurfaces with w-constant mean curvature. As a by-product, we give a new proof of the result of Deng-Gu-Wei [Adv. Math. 314 (2017), pp. 278–305] under the weaker topological condition. [ABSTRACT FROM AUTHOR]

Alfred Tarski [J. Symbolic Logic 13 (1948), pp. 107–111] defined \mathbf {D}_{pm} to be the set of all sets of type p, type-theoretically definable by parameterfree formulas of type {\le m}, and asked whether it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m} for m\ge 1. Tarski noted that the negative solution is consistent because the axiom of constructibility \mathbf {V}=\mathbf {L} implies \mathbf {D}_{1m}\notin \mathbf {D}_{2m} for all m\ge 1, and he left the consistency of the positive solution as a major open problem. This was solved in our recent paper [Mathematics 8 (2020), pp. 1–36], where it is established that for any m\ge 1 there is a generic extension of \mathbf {L}, the constructible universe, in which it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m}. In continuation of this research, we prove here that Tarski's sentences \mathbf {D}_{1m}\in \mathbf {D}_{2m} are not only consistent, but also independent of each other, in the sense that for any set Y\subseteq \omega \smallsetminus \{0\} in \mathbf {L} there is a generic extension of \mathbf {L} in which it is true that \mathbf {D}_{1m}\in \mathbf {D}_{2m} holds for all m\in Y but fails for all m\ge 1, m\notin Y. This gives a full and conclusive solution of the Tarski problem. The other main result of this paper is the consistency of \mathbf {D}_{1}\in \mathbf {D}_{2} via another generic extension of \mathbf {L}, where \mathbf {D}_{p}=\bigcup _m\mathbf {D}_{pm}, the set of all sets of type p, type-theoretically definable by formulas of any type. Our methods are based on almost-disjoint forcing of Jensen and Solovay [Some applications of almost disjoint sets, North-Holland, Amsterdam, 1970, pp. 84–104]. [ABSTRACT FROM AUTHOR]

In this paper, we are going to compute the average size of 2-Selmer groups of families of even hyperelliptic curves over function fields. The result will be obtained by a geometric method which is based on a Vinberg's representation of the group G=\text {PSO}(2n+2) and a Hitchin fibration. Consistent with the result over \mathbb {Q} of Arul Shankar and Xiaoheng Wang [Compos. Math. 154 (2018), pp. 188–222], we provide an upper bound and a lower bound of the average. However, if we restrict to the family of transversal hyperelliptic curves, we obtain precisely average number 6. [ABSTRACT FROM AUTHOR]

In [Math. Ann. 367 (2017), pp. 1517–1586] Bar-Natan and the first author show that solutions to the Kashiwara–Vergne equations are in bijection with certain knot invariants: homomorphic expansions of welded foams. Welded foams are a class of knotted tubes in \mathbb {R}^4, which can be finitely presented algebraically as a circuit algebra , or equivalently, a wheeled prop. In this paper we describe the Kashiwara-Vergne groups \mathsf {KV} and \mathsf {KRV}—the symmetry groups of Kashiwara-Vergne solutions—as automorphisms of the completed circuit algebras of welded foams, and their associated graded circuit algebras of arrow diagrams, respectively. Finally, we provide a description of the graded Grothendieck-Teichmüller group \mathsf {GRT}_1 as automorphisms of arrow diagrams. [ABSTRACT FROM AUTHOR]

In this paper, we first study a mapping problem between indefinite hyperbolic spaces by employing the work established earlier by the authors. In particular, we generalize certain theorems proved by Baouendi-Ebenfelt-Huang [Amer. J. Math. 133 (2011), pp. 1633–1661] and Ng [Michigan Math. J. 62 (2013), pp. 769–777; Int. Math. Res. Not. IMRN 2 (2015), pp. 291–324]. Then we use these results to prove a rigidity result for proper holomorphic mappings between type I classical domains, which confirms a conjecture formulated by Chan [Int. Math. Res. Not., doi.org/10.1093/imrn/rnaa373] after the work of Zaitsev-Kim [Math. Ann. 362 (2015), pp. 639-677], Kim [ Proper holomorphic maps between bounded symmetric domains , Springer, Tokyo, 2015, pp. 207–219] and himself. [ABSTRACT FROM AUTHOR]

The theory of Moufang sets essentially deals with groups having a split BN-pair of rank one. Every quadratic Jordan division algebra gives rise to a Moufang set such that its root groups are abelian and a certain condition called special is satisfied. It is a major open question if also the converse is true, i.e. if every special Moufang set with abelian root groups comes from a quadratic Jordan division algebra. De Medts and Segev [Amer. Math. Soc. 360 (2008), pp. 5831–5852] proved in Theorem 5.11 that this is the case for special Moufang set satisfying two conditions. In this paper we prove that these conditions are in fact equivalent and hence either of them suffices. Even more, we can replace them by weaker conditions. [ABSTRACT FROM AUTHOR]

The category of framed correspondences Fr*(k) and framed sheaves were invented by Voevodsky in his unpublished notes [ Notes on framed correspondences , 2001, https://www.math.ias.edu/vladimir/publications]. Based on the theory, framed motives are introduced and studied in Garkusha and Panin [J. Amer. Math. Soc. 34 (2021), pp. 261–313]. These are Nisnivich sheaves of S1-spectra and the major computational tool of Garkusha and Panin. The aim of this paper is to show the following result which is essential in proving the main theorem of Garkusha and Panin: given an infinite perfect base field k, any k-smooth scheme X and any n ≥ 1, the map of simplicial pointed Nisnevich sheaves (−,A1//Gm)∧n+ → Tn induces a Nisnevich local level weak equivalence of S1-spectra Mfr(X × (A1//Gm)∧n) → Mfr(X × Tn). Moreover, it is proven that the sequence of S1-spectra Mfr(X ×Tn × Gm) → Mfr(X × Tn × A1) → Mfr(X × Tn+1) is locally a homotopy cofiber sequence in the Nisnevich topology. Another important result of this paper shows that homology of framed motives is computed as linear framed motives in the sense of Garkusha and Panin. This computation is crucial for the whole machinery of framed motives. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring R of prime characteristic. The methods used employ Lyubeznik's F-modules over R, particularly his F-finite F-modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over R have natural structures as F-modules; these lead to F-module structures on certain local cohomology modules over R, which are exploited, in conjunction with F-module structures on injective R-modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2. [ABSTRACT FROM AUTHOR]

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R+ or R with values in a general Riemannian manifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper of Röckner and the second, third, and fourth authors [SIAM J. Math. Anal. 52 (2020), pp. 2237-2274] on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution of the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative. [ABSTRACT FROM AUTHOR]

We correct a mistake that appeared in the first section of the original article, which appeared in Tran. Amer. Math. Soc. 370 (2018), 99-130. Namely, Corollary 1.16 was false as stated and was subsequently used in later proofs in the paper. In this note it is argued that all the relevant statements after Corollary 1.16 can be saved with at most minor modifications. In particular, all the main results of the original paper remain valid as stated, but some intermediate claims are slightly modified or proved more directly without Corollary 1.16. [ABSTRACT FROM AUTHOR]

In this paper we introduce a certain class of families of Hessenberg varieties arising from Springer theory for symmetric spaces. We study the geometry of those Hessenberg varieties and investigate their monodromy representations in detail using the geometry of complete intersections of quadrics. We obtain decompositions of these monodromy representations into irreducibles and compute the Fourier transforms of the IC complexes associated to these irreducible representations. The results of the paper refine (part of) the Springer correspondece for the split symmetric pair (SL(N),SO(N)) in [Compos. Math. 154 (2018), pp. 2403-2425]. [ABSTRACT FROM AUTHOR]

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler–Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene. [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]

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]

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]

Let R be a standard graded Gorenstein algebra over a field presented by quadrics. In [Compositio Math. 129 (2001), no. 1, 95-121], Conca-Rossi-Valla show that such a ring is Koszul if reg R ≤ 2 or if reg R = 3 and c = codim R ≤ 4, and they ask whether this is true for reg R = 3 in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring R that guarantee the Nagata idealization ~ R = R × ωR(−a−1) is a non-Koszul quadratic Gorenstein ring. We prove there exist rings of regularity 3 satisfying our conditions for all c ≥ 9; this yields a negative answer to the question from the above-mentioned paper. [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field of characteristic different from 2, let g be a positive integer, let ƒ(x) \in K[x] be a degree 2g+1 monic polynomial without multiple roots, let Cƒ: y2 = ƒ(x) be the corresponding genus g hyperelliptic curve over K, and let J be the Jacobian of Cƒ. We identify Cƒ with the image of its canonical embedding into J (the infinite point of Cƒ goes to the zero of the group law on J). It is known [Izv. Math. 83 (2019), pp. 501-520] that if g ≥ 2, then Cƒ(K) contains no points of orders lying between 3 and 2g. In this paper we study torsion points of order 2g + 1 on Cƒ(K). Despite the striking difference between the cases of g = 1 and g ≥ 2, some of our results may be viewed as a generalization of well-known results about points of order 3 on elliptic curves. E.g., if p = 2g + 1 is a prime that coincides with char(K), then every odd degree genus g hyperelliptic curve contains at most two points of order p. If g is odd and ƒ(x) has real coefficients, then there are at most two real points of order 2g + 1 on Cƒ. If ƒ(x) has rational coefficients and g ≤ 51, then there are at most two rational points of order 2g+1 on Cƒ. (However, there exist odd degree genus 52 hyperelliptic curves over Q that have at least four rational points of order 105.) [ABSTRACT FROM AUTHOR]

Let g be the derived subalgebra of a Kac-Moody Lie algebra of finite-type or affine-type, let μ be a diagram automorphism of g, and let L(g,μ) be the loop algebra of g associated to μ. In this paper, by using the vertex algebra technique, we provide a general construction of current-type presentations for the universal central extension g[μ] of L(g,μ). The construction contains the classical limit of Drinfeld's new realization for (twisted and untwisted) quantum affine algebras [Soviet Math. Dokl. 36 (1988), pp. 212-216] and the Moody-Rao-Yokonuma presentation for toroidal Lie algebras [Geom. Dedicata 35 (1990), pp. 283-307] as special examples. As an application, when g is of simply-laced-type, we prove that the classical limit of the μ-twisted quantum affinization of the quantum Kac-Moody algebra associated to g introduced in [J. Math. Phys. 59 (2018), 081701] is the universal enveloping algebra of g[μ]. [ABSTRACT FROM AUTHOR]

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191-7223, is that the structure algebra K[M(X,r)], over a field K, of a finite bijective left non-degenerate solution (X,r) of the Yang-Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M(X,r) is central-by-finite. This however is not true, even in case (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M(X,r) is abelian-by-finite and thus the conclusion that K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution (X,r) is square-free is omitted. [ABSTRACT FROM AUTHOR]

Consider the Hamiltonian abcd system in one dimension, with data posed in the energy space H1 × H1. This model, introduced by Bona, Chen, and Saut, is a well-known physical generalization of the classical Boussinesq equations. The Hamiltonian case corresponds to the regime where a,c < 0 and b = d > 0. Under this regime, small solutions in the energy space are globally defined. A first proof of decay for this 2 × 2 system was given in [J. Math. Pure Appl. (9) 127 (2019), 121-159] in a strongly dispersive regime, i.e., under essentially the conditions b = d > 2/9, a,c < −1/18. Additionally, decay was obtained inside a proper subset of the light cone (−|t|,|t|). In this paper, we improve [J. Math. Pure Appl. (9) 127 (2019), 121-159] in three directions. First, we enlarge the set of parameters (a,b,c,d) for which decay to zero is the only available option, considering now the so-called weakly dispersive regime a,c ~0: we prove decay if now b = d > 3/16, a,c < −1/48. This result is sharp in the case where a = c, since for a,c bigger, some abcd linear waves of nonzero frequency do have zero group velocity. Second, we sharply enlarge the interval of decay to consider the whole light cone, that is to say, any interval of the form |x| ~ |v| t for any |v| < 1. This result rules out, among other things, the existence of nonzero speed solitary waves in the regime where decay is present. Finally, we prove decay to zero of small abcd solutions in exterior regions |x| >> |t|, also discarding super-luminical small solitary waves. These three results are obtained by performing new improved virial estimates for which better decay properties are deduced. [ABSTRACT FROM AUTHOR]

In [J. Inst. Math. Jussieu 14 (2015), pp. 149-184] and [Int. Math. Res. Not. imrn 7 (2017), pp. 2014-2099], the twisted standard L-function L(s, π, χ, st) of a cuspidal representation π of the exceptional group of type G2 was shown to be represented by a family of new-way Rankin-Selberg integrals. These integrals connect the analytic behaviour of L(s, π, χ, st) with that of a family of degenerate Eisenstein series εE(χ, fs, s, g) on quasi-split forms HE of Spin8, induced from Heisenberg parabolic subgroups. The analytic behaviour of the series εE(χ, fs, s, g) in the right half-plane Re(s) > 0 was studied in [Tran. Amer. Math. Soc. 370 (2018), pp. 5983-6039]. In this paper we study the residual representations associated with EE(χ, fs, s, g). [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of the absolute Galois group Gal(Q/Q) attached to modular forms for noncongruence subgroups of SL2(Z). When the underlying modular curves have a model over Q, these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49-77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially GL2-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions. [ABSTRACT FROM AUTHOR]

In this paper we investigate nilpotenct and probabilistically nilpotent Hopf algebras. We define nilpotency via a descending chain of commutators and give a criterion for nilpotency via a family of central invertible elements. These elements can be obtained from a commutator matrix A which depends only on the Grothendieck ring of H. When H is almost cocommutative we introduce a probabilistic method. We prove that every semisimple quasitriangular Hopf algebra is probabilistically nilpotent. In a sense we thereby answer the title of our paper Are we counting or measuring anything? by Yes, we are. [ABSTRACT FROM AUTHOR]

This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, the "splitting" properties of affine supergroups are discussed, i.e., special kinds of factorizations they may admit -- either globally, or pointwise. Almost everything should be more or less known, but seems to be not as clear in the literature (to the author's knowledge) as it ought to. Second, a new contribution to the study of affine supergroups by means of super Harish-Chandra pairs is presented (a method already introduced by Koszul, and later extended by other authors). Namely, a new functorial construction ψ is provided which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) -- thus setting a link with the first part of the paper. One knows that there exists a natural functor Φ from affine supergroups to super Harish-Chandra pairs. Then we show that the new functor ψ -- which goes the other way round -- is indeed a quasi-inverse to Φ, provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and ψ are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, via different approaches. Nevertheless, the novelty in the present paper lies in the construction of a different functor ψ and thus extends the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free and is presented here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex analytic supergeometry. The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

Let A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A** with the (first) Arens multiplication. To each idempotent element u of A** we associate the closed ideal Iu = {a ∊ A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A**. The main results are: a) A closed ideal I of A has a BAI iff there is an idempotent u ∊ A** such that I = Iu and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAI, the quotient algebra A/I is isomorphic to a subalgebra of A**. We also show that a weak* closed invariant subspace X of the group von Neumann algebra VN(G) of an amenable group G is naturally complemented in VN(G) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [ABSTRACT FROM AUTHOR]

Eigenvalues and eigenfunctions of two- and three-dimensional double layer potentials are considered. Let Ω be a C2 bounded region in Rn (n = 2, 3). The double layer potential K : L2(∂Ω) → L2(∂Ω) is defined by (Kψ)(x) ≡ ∫ ∂ Ω ψ(y)·vyE(x, y) dsy, where E(x, y) = ∫1/2π log1/∣x-y∣ , if n = 2, 1/π log1/∣x-y∣ , if n = 3, dsy is the line or surface element and vy is the outer normal derivative on ∂Ω. It is known that K is a compact operator on L2(∂Ω) and consists of at most a countable number of eigenvalues, with 0 as the only possible limit point. This paper aims to establish some relationships among the eigenvalues, the eigenfunctions, and the geometry of ∂Ω. [ABSTRACT FROM AUTHOR]

We investigate the Dirichlet problem for the parablic equation [ u_t = \Delta u^m, m > 0, \] in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [{\em U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403}] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains. [ABSTRACT FROM AUTHOR]

In this paper we study the following question related to Diophantine approximations and geometric measure theory: for a given set Ω find α such that α - θ has bad Diophantine properties simultaneously for all θ ∊ Ω. How do the arising Diophantine inequalities depend on the geometry of the set Ω? We provide several methods which yield different answers in terms of the metric entropy of Ω and consider various examples. Furthermore, we apply these results to explore the asymptotic behavior of the directional discrepancy, i.e., the discrepancy with respect to rectangles rotated in certain sets of directions. It is well known that the extremal cases of this problem (fixed direction vs. all possible rotations) yield completely different bounds. We use rotated lattices to obtain directional discrepancy estimates for general rotation sets and investigate the sharpness of these methods. [ABSTRACT FROM AUTHOR]

The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

In this paper, we show that the chordal Loewner differential equation with C° driving function generates a C°1/2 slit for 1/2 < ° ≼ 2, except when ° = 3/2 the slit is only proved to be weakly C1,1. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit combinatorial realization of the crystal B(λ) for an irreducible highest weight Uq(q(n))-module V (λ) in terms of semistandard decomposition tableaux. We present an insertion scheme for semistandard decomposition tableaux and give algorithms for decomposing the tensor product of q(n)-crystals. Consequently, we obtain explicit combinatorial descriptions of the shifted Littlewood-Richardson coefficients. [ABSTRACT FROM AUTHOR]

The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of $ *$ [ABSTRACT FROM AUTHOR]

For an $n$-component link, Milnor's isotopy invariants are defined for each multi-index $I=i_1i_2...i_m~(i_jin {1,...,n})$. Here $m$ is called the length. Let $r(I)$ denote the maximum number of times that any index appears in $I$. It is known that Milnor invariants with $r=1$, i.e., Milnor invariants for all multi-indices $I$ with $r(I)=1$, are link-homotopy invariant. N. Habegger and X. S. Lin showed that two string links are link-homotopic if and only if their Milnor invariants with $r=1$ coincide. This gives us that a link in $S^3$ is link-homotopic to a trivial link if and only if all Milnor invariants of the link with $r=1$ vanish. Although Milnor invariants with $r=2$ are not link-homotopy invariants, T. Fleming and the author showed that Milnor invariants with $rleq 2$ are self $Delta $-equivalence invariants. In this paper, we give a self $Delta $-equivalence classification of the set of $n$-component links in $S^3$ whose Milnor invariants with length $leq 2n-1$ and $rleq 2$ vanish. As a corollary, we have that a link is self $Delta $-equivalent to a trivial link if and only if all Milnor invariants of the link with $rleq 2$ vanish. This is a geometric characterization for links whose Milnor invariants with $rleq 2$ vanish. The chief ingredient in our proof is Habiro's clasper theory. We also give an alternate proof of a link-homotopy classification of string links by using clasper theory. [ABSTRACT FROM AUTHOR]

Let $G$ be a finite group, $S in mathit {Syl}_p(G)$, and $mathcal S$ be the set subgroups containing $S$. For $M in mathcal S$ and $V = Omega _1Z(O_p(M))$, the paper discusses the action of $M$ on $V$. Apart from other results, it is shown that for groups of parabolic characteristic $p$ either $S$ is contained in a unique maximal $p$-local subgroup, or there exists a maximal $p$-local subgroup in $M in mathcal S$ such that $V$ is a nearly quadratic 2F-module for $M$. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Murre's conjectures on the structure of rational Chow groups and exhibit explicit Chow--Künneth projectors for some examples. More precisely, the examples we study are the varieties which have a nef tangent bundle. For surfaces and threefolds which have a nef tangent bundle, explicit Chow--Künneth projectors are obtained which satisfy Murre's conjectures, and the motivic Hard Lefschetz theorem is verified. [ABSTRACT FROM AUTHOR]

In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type. [ABSTRACT FROM AUTHOR]

We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$ which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category. [ABSTRACT FROM AUTHOR]

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus. [ABSTRACT FROM AUTHOR]

In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of $\mathbb{R}^3$. [ABSTRACT FROM AUTHOR]

A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})

The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold $X$, a rank-two vector bundle $\mathcal{E}$ on $X$, and integers $k\geq 0$, $\delta >0 $, denote by ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ the subscheme of ${\mathbb{P}}(H^0({\mathcal{E}}(k)))$ parametrizing global sections of ${\mathcal{E}}(k)$ whose zero-loci are irreducible $\delta$-nodal curves on $X$. We present a new cohomological description of the tangent space $T_{[s]}({\mathcal{V}}_{\delta} ({\mathcal{E}} (k)))$ at a point $[s]\in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$. This description enables us to determine effective and uniform upper bounds for $\delta$, which are linear polynomials in $k$, such that the family ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ is smooth and of the expected dimension ({\em regular}, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when $X$ is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point $[s] \in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ related to the postulation of the nodes of its zero-locus $C = V(s) \subset X$. Roughly speaking, when the nodes of $C$ are assumed to be in general position either on $X$, or on an irreducible divisor of $X$ having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in $X$, we find upper bounds on $\delta$ which are, respectively, cubic, quadratic and linear polynomials in $k$ ensuring the regularity of ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ at $[s]$. Finally, when $X= \mathbb{P}^3$, we also discuss some interesting geometric properties of the curves given by sections parametrized by ${\mathcal{V}}_{\delta} ({\mathcal{E}} \otimes \mathcal{O}_X(k))$. [ABSTRACT FROM AUTHOR]

In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi $ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds. [ABSTRACT FROM AUTHOR]

In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight $2k+1$ for $\Gamma_{0}(q)$. Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke $L$-functions. [ABSTRACT FROM AUTHOR]

In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space. [ABSTRACT FROM AUTHOR]

This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space. [ABSTRACT FROM AUTHOR]

Let $T(x)=\sum_{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min_{x\in[0,2\pi)}|T(x)|0$ and large $n$ most choices of sign satisfy $\min_{x\in[0,2\pi)}|T(x)|> \eps n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials. [ABSTRACT FROM AUTHOR]

This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta\ln(1+u)$ in $\Omega \times (0, +\infty)$, \[\frac{\partial\ln(1+u)}{\partial n}=\sqrt{1+u}(\ln(1+u))^{\alpha} \quad\text{on} \partial \Omega \times (0, +\infty),] and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha\geq 0$ is a parameter, $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain. After pointing out the mistakes in {\em Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary}, SIAM J. Math. Anal. {\bf 24} (1993), 317--326, by N. Wolanski, which claims that, for $\Omega=B_R$ the ball of $\mathbb{R}^N$, the positive solution exists globally if and only if $\alpha\leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha\leq {1/2}$. [ABSTRACT FROM AUTHOR]

Let $\Omega$ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega$. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that \begin{equation*} \frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx \leq c_{o} , \end{equation*} where $|\Omega|$ is the Lebesgue measure of $\Omega$, and $\omega_{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the \begin{equation*} \sup \{\frac{1}{|\Omega|} \int_{\Omega} e^{n \omega_{n-1}^{\frac{1}{n-1}} u^{\frac{n}{n-1}}} dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 } \end{equation*} is attained. Earlier results include Carleson-Chang (1986, $\Omega$ is a ball in any dimension) and Flucher (1992, $\Omega$ is any domain in 2-dimensions). [ABSTRACT FROM AUTHOR]