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The purpose of this paper is to investigate the valency of planar harmonic mappings of bounded boundary rotation of the open unit disc D. The paper is motivated by the earlier work of the first two authors [Complex Analysis Oper. Theory 5 (2011), 767-774] and the recent work of T. Hayami [Complex Var. Elliptic Equ. 59 (2014), 1214-1222]. First, the authors give a counterexample showing that both the main result of Hayami, Theorem 2.1, and the related conjecture, Conjecture 4.1, are false. Second, the authors give a valency criterion for planar harmonic mappings of bounded boundary rotation of D, proving an ameliorated statement of Theorem 2.1 and settling a modified version of Conjecture 4.1. [ABSTRACT FROM AUTHOR]

We continue and complete our previous paper "Lifts of projective congruence groups" concerning the question of whether there exist noncongruence subgroups of SL2 (Z) that are projectively equivalent to one of the groups Γ0(N) or Γ1(N). A complete answer to this question is obtained: In case of Γ0(N) such noncongruence subgroups exist precisely if N ∉ {3, 4, 8} and we additionally have either that 4 | N or that N is divisible by an odd prime congruent to 3 modulo 4. In case of Γ1(N) these noncongruence subgroups exist precisely if N > 4. As in our previous paper the main motivation for this question is the fact that the above noncongruence subgroups represent a fairly accessible and explicitly constructible reservoir of examples of noncongruence subgroups of SL2(Z) that can serve as the basis for experimentation with modular forms on noncongruence subgroups. [ABSTRACT FROM AUTHOR]

In this paper, we provide some algebraic structures of convex subrings of *C, a nonstandard extension of the field of complex numbers C. In particular, a detailed description of the prime spectrum of any convex subring of *C is given. To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain. We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum. [ABSTRACT FROM AUTHOR]

In this paper we establish infinite families of prime divisibility properties for the partition function p(n, m,N) enumerating partitions into at most m parts with no part larger than N. These partition numbers are the coefficients of the Gaussian polynomials [N+m m]. The proof of the main theorem connects three fundamental partition functions: the general partition function p(n), the restricted partition function p(n,m) which counts partitions of n into at most m parts, and p(n, m,N). [ABSTRACT FROM AUTHOR]

When Journé established the classical multi-parameter singular integral theory in 1985, it was formulated in the language of vector-valued Calderón-Zygmund theory. In 2011, Pott and Villarroya formulated a new T1 Theorem for product spaces. In their work, they replaced the vectorvalued formulations for several mixed type conditions. Inspired by their work, Martikainen redefined the bi-parameter operators. Originally, it was thought that this new class of operators was larger (and contained the ones defined by Journé). In this paper we prove that both classes are actually equal. [ABSTRACT FROM AUTHOR]

In this paper, we establish the non-positivity of the second eigenvalue of the Schrödinger operator-div(Pr ∇ ·)-Wr² on a closed hypersurface Σn of ℝn+1, where Wr is a power of the (r + 1)-th mean curvature of Σn, which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature. [ABSTRACT FROM AUTHOR]

We give an integrability condition on a function Ψ guaranteeing that for almost all (or almost no) x ∈ R, the system |qx - p| < Ψ(t), |q| < t is solvable in p ∈ Z, q ∈ Z {0} for sufficiently large t. Along the way, we characterize such x in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we show that two balanced triangulations of a closed surface are not necessarily connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are the octahedral sphere. [ABSTRACT FROM AUTHOR]

In a previous paper we provided some update in the treatment of the finiteness theorem for rational maps of finite degree from a fixed variety to varieties of general type. In the present paper we present another improvement, introducing the natural parametrization of maps by means of the space of linear projections in a suitable projective space, and this leads to some new insight into the geometry of the finiteness theorem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d ≤ 5. In analogy with the classical Erdős-Kac theorem, we prove for Sd-extensions that the number of such primes is normally distributed with mean and variance log log X. [ABSTRACT FROM AUTHOR]

This paper considers "geometric" ideal triangulations of cusped hyperbolic 3-manifolds, i.e. decompositions into positive volume ideal hyperbolic tetrahedra. We exhibit infinitely many geometric ideal triangulations of the figure eight knot complement. As far as we know, this is the first construction of infinitely many geometric triangulations of a cusped hyperbolic 3-manifold. In contrast, our approach does not extend to the figure eight sister manifold, and it is unknown if there are infinitely many geometric triangulations for this manifold. [ABSTRACT FROM AUTHOR]

In this paper, we show that the asymptotic expansion of a double Laplace-type integral with a non-stationary minimum point, located on the boundary of the domain of integration, is governed by the order of contact between the boundary curve and the level curve of the phase through the minimum point. This achievement will enable us to construct complete asymptotic expansions in more general settings. Especially, the problem will be completely solved if the phase and the boundary curve of the domain of integration are analytic near the minimum point. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the cycline subalgbra of a k-graph C*-algebra is maximal abelian, and show when it is a Cartan subalgebra (in the sense of Renault). [ABSTRACT FROM AUTHOR]

In this paper, we prove that solutions to the equation (−Δ)su = 0 in RN, for s ∊ (0, 1), are affine. This allows us to prove the uniqueness of the Riesz potential [ABSTRACT FROM AUTHOR]

In 1965, E. C. Zeeman proved that the (±1)-twist spin of any knotted sphere in (n−1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (±1)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (±1)-twist spins are always unknotted. [ABSTRACT FROM AUTHOR]

In this paper, we prove that any pair of q-commuting contractions on a Hilbert space dilates to a pair of q-commuting unitaries, where |q| = 1. We generalize this result to a (G, q)-commuting n-tuple (T1, . . . , Tn) of strict contractions, where G is an acyclic graph with vertex set {1, . . . ,n}. We further generalize it to a family of (G, q)-commuting strict contractions, where G is an acyclic graph on an infinite set of vertices. [ABSTRACT FROM AUTHOR]

This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant bΔ of a simplicial d-manifold Δ taking values in the range 0 ≤ bΔ ≤ d - 1. The main result is that bΔ influences connectivity in the following way: The graph of a d-dimensional simplicial compact manifold Δ is (2d - bΔ)-connected. The parameter bΔ has the property that bΔ = 0 if the complex Δ is flag. Hence, our result interpolates between Barnette's theorem (1982) that all d-manifold graphs are (d + 1)-connected and Athanasiadis' theorem (2011) that flag d-manifold graphs are 2d-connected. The definition of bΔ involves the concept of banner triangulations of manifolds, a generalization of flag triangulations. [ABSTRACT FROM AUTHOR]

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations. [ABSTRACT FROM AUTHOR]

Let A = (A, V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable, a notion due to Stanley. Jambu and Terao showed that every supersolvable arrangement is inductively free, a notion previously due to Terao. So this is a natural subclass of this particular class of free arrangements. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A = (A(W),V) be the associated hyperplane arrangement of W. In a forthcoming paper by the authors, we determine all inductively free reflection arrangements. The aim of this paper is to classify all supersolvable reflection arrangements. Moreover, we characterize the irreducible arrangements in this class by the presence of modular elements of rank 2 in their intersection lattice. [ABSTRACT FROM AUTHOR]

Provan and Billera introduced notions of (weak) decomposability of simplicial complexes as a means of attempting to prove polynomial upper bounds on the diameter of the facet-ridge graph of a simplicial polytope. Recently, De Loera and Klee provided the first examples of simplicial polytopes that are not weakly vertex-decomposable. These polytopes are polar to certain simple transportation polytopes. In this paper, we refine their analysis to prove that these d-dimensional polytopes are not even weakly O( v d)-decomposable. As a consequence, (weak) decomposability cannot be used to prove a polynomial version of the Hirsch Conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a partial solution to the following question of Köthe: For which rings R is it true that every left (or both left and right) R-module is a direct sum of cyclic modules? Let R be a ring in which all idempotents are central. We prove that if R is a left Köthe ring (i.e., every left R-module is a direct sum of cyclic modules), then R is an Artinian principal right ideal ring. Consequently, R is a Köthe ring (i.e., each left and each right R-module is a direct sum of cyclic modules) if and only if R is an Artinian principal ideal ring. This is a generalization of a Köthe-Cohen-Kaplansky theorem. [ABSTRACT FROM AUTHOR]

In this paper, we set up the complete bubble tree theory for approximate harmonic maps from a Riemann surface with tension fields bounded in Zygmund class Lln+ L. Some special cases of this theory have previously been used in a number of papers. On the other hand, one can see that this bubble tree theory is not true for the general target manifold if we only assume that the tension fields are bounded in L¹ uniformly. [ABSTRACT FROM AUTHOR]

In this paper we consider sums of squares of vector fields in R2 satisfying Hörmander's condition and with polynomial, but non-(quasi-)homogeneous, coefficients. We obtain a Gevrey hypoellipticity index which we believe to be sharp. The general operator we consider is P = X² + Y² +LΣj=1 Z²j, with X = Dx, Y = a0(x, y)xq-1Dy, Zj = aj(x, y)xpj-1ykj Dy, with aj(0, 0) ≠ 0, j = 0, 1, . . ., L and q > pj, {kj} arbitrary. The theorem we prove is that P is Gevrey-s hypoelliptic for s ≥ 1/1-T, T = maxjq-pj/qkj. [ABSTRACT FROM AUTHOR]

Let R denote a commutative Noetherian (not necessarily local) ring and I an ideal of R of dimension one. The main purpose of this paper is to generalize, and to provide a short proof of, K. I. Kawasaki's theorem that the category μ(R, I)cof of I-cofinite modules over a commutative Noetherian local ring R forms an Abelian subcategory of the category of all R-modules. Consequently, this assertion answers affirmatively the question raised by R. Hartshorne in his article Affine duality and cofiniteness [Invent. Math. 9 (1970), 145-164] for an ideal of dimension one in a commutative Noetherian ring R. [ABSTRACT FROM AUTHOR]

In a paper by Z. Lu and J. Rowlett, it is shown that the eigenvalues of the weighted Laplacian can be approximated by eigenvalues of a naturally associated family of narrow graphs. In this paper, we generalize this result to the p-Laplacian. Our approach features overcoming the nonlinearity of the p-Laplacian when p ≠ 2, which is different from the Laplacian case. [ABSTRACT FROM AUTHOR]

We study the blowup dynamics of solutions to the L2-supercritical nonlinear SchrØdinger equation and prove that the blowup solution with bounded H sc norm must concentrate at least a fixed amount of the Hsc norm and, also, its Lpc norm must concentrate at least a fixed Lpc norm. We show these properties without any further symmetry assumptions on the solution and partly generalize the results obtained in papers of Holmer and Roudenko and of Zhu, which only deal with the radially symmetric case. Our proof is based on the profile decomposition theorems. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and multiplicity of positive periodic solutions for first-order vector differential equations. By using the Leray-Schauder alternative theorem and the Kransnosel'skii fixed point theorem, we show that the differential equations under the periodic boundary value conditions have at least two positive periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, we complete the classification of quasi-alternating Montesinos links. We show that the quasi-alternating Montesinos links are precisely those identified independently by Qazaqzeh-Chbili-Qublan and Champanerkar-Ording. A consequence of our proof is that a Montesinos link L is quasi-alternating if and only if its double branched cover is an L-space, and bounds both a positive definite and a negative definite 4-manifold with vanishing first homology. [ABSTRACT FROM AUTHOR]

The spectrum of the singular indefinite Sturm-Liouville operator ... with a real potential q ∊ L¹(ℝ) covers the whole real line, and, in addition, non-real eigenvalues may appear if the potential q assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction have been obtained. In this paper the bound ... on the absolute values of the non-real eigenvalues λ of A is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the L¹-norm of the negative part of q. [ABSTRACT FROM AUTHOR]

In this paper we investigate the relation between measure-expansiveness and hyperbolicity. We prove that non-atomic invariant ergodic measures with all of their Lyapunov exponents positive are positively measure-expansive. We also prove that local diffeomorphisms robustly positively measure-expansive are expanding. Finally, we prove that a C¹-volume-preserving diffeomorphism that cannot be accumulated by positively measure-expansive diffeomorphisms has a dominated splitting. [ABSTRACT FROM AUTHOR]

Akemann and Weaver (2014) have shown a remarkable extension of Weaver's KSr Conjecture (2004) in the form of approximate Lyapunov's theorem. This was made possible thanks to the breakthrough solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava (2015). In this paper we show a similar type of Lyapunov's theorem for continuous frames on non-atomic measure spaces. In contrast with discrete frames, the proof of this result does not rely on the recent solution of the Kadison-Singer problem. [ABSTRACT FROM AUTHOR]

In this paper, we construct new nonstandard hulls of topological vector spaces using convex subrings of ∗R (or ∗C) and we show that such spaces are complete. Some examples of locally convex spaces are provided to illustrate our construction. Namely, we show that the new nonstandard hull of the space of polynomials is the algebra of Colombeau’s entire holomorphic generalized functions. The proof is based on the existence of global representatives of entire generalized functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which is related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone nonincreasing under Ricci flow. Moreover, we show that, just the same as Perelman's reduced volume, the weighted reduced volume entropy has the value (4п) n/2 if and only if the Ricci flow is the trivial flow on flat Euclidean space. [ABSTRACT FROM AUTHOR]

We show that the existence of an almost Souslin Kurepa tree is consistent with ZFC. We also prove their existence in L. These results answer two questions from a paper by Zakrzewski. [ABSTRACT FROM AUTHOR]

In this paper, we study the bilinear Littlewood-Paley square function introduced by M. Lacey. We give an easy proof of its boundedness from Lp(…d) × Lq(…d) into Lr(…d), d ≥ 1, for all possible values of exponents p, q, r, i.e. for 2 ≤ p,q ≤∞, 1 ≤ r ≤ ∞ satisfying 1/p + 1/q = 1/r. We also prove analogous results for bilinear square functions on the torus group …d. [ABSTRACT FROM AUTHOR]

In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we prove some rigidity theorems for compact hypersurfaces without the constancy condition on the mean curvature or the scalar curvature in locally symmetric Riemannian manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new expected dimension formula for certain rank two Brill-Noether loci with fixed special determinant. This answers a question asked by Osserman and also leads to a new and much simpler proof of a theorem in his 2015 work. Our result generalizes the well-known result by Bertram, Feinberg and independently Mukai on expected dimension of rank two Brill-Noether loci with canonical determinant and partially verifies a conjecture (in rank two) of Grzegorczyk and Newstead on coherent systems. [ABSTRACT FROM AUTHOR]

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3-manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a (g, b)-bridge surface for a knot K in S3 carries any geometric information related to the knot exterior. In this paper, we show that - unlike in the case of Heegaard splittings - hyperbolic volume and genus g-bridge numbers are completely independent. That is, for any g, we construct explicit sequences of knots with bounded volume and unbounded genus g-bridge number, and explicit sequences of knots with bounded genus g-bridge number and unbounded volume. [ABSTRACT FROM AUTHOR]

In this paper, we answer a question of Li, Ngo, and Rhoades concerning a set of q-series related to the q-hypergeometric series σ from Ramanujan's Lost Notebook. Our results parallel a theorem of Cohen which says that σ, along with its partner function σ*, encode the coefficients of a Maass waveform of eigenvalue 1/4. [ABSTRACT FROM AUTHOR]

We classify all possible h-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension ⩽17 and in socle degree 5 and codimension ⩽25. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein h-vector is 13 for socle degree 4 and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein h-vector is (1, 13, 12, 13, 1), which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a longstanding open question in this area. All of our results are characteristic free. [ABSTRACT FROM AUTHOR]

The celebrated Erd˝sKo-Rado theorem determines the maximum size of a k-uniform intersecting family. The Hilton–Milner theorem determines the maximum size of a k-uniform intersecting family that is not a subfamily of the so-called Erd˝Ko-Rado family. In turn, it is natural to ask what the maximum size of an intersecting k-uniform family that is neither a subfamily of the Erd˝os–Ko–Rado family nor of the Hilton-Milner family is. For k ≥ 4, this was solved (implicitly) in the same paper by Hilton–Milner in 1967. We give a different and simpler proof, based on the shifting method, which allows us to solve all cases k ≥ 3 and characterize all extremal families achieving the extremal value. [ABSTRACT FROM AUTHOR]

In this paper, we consider problems associated with the higher order fractional Laplacian. Through the method of moving planes, we derive rotational symmetry of positive solutions and show their dependence on the xn variable only. We also establish the equivalence between a semilinear higher order fractional Laplacian equation and its corresponding integral equation, so as to further deduce a Liouville type theorem and obtain a priori estimates for positive solutions. [ABSTRACT FROM AUTHOR]

In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus (ℝ/ℤ)³, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domain theorem with equality (Courant- sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains in the torus, deduced, as in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. [ABSTRACT FROM AUTHOR]

By a classical result of Kadec and Pełczyński (1962), every normalized weakly null sequence in Lp, p > 2, contains a subsequence equivalent to the unit vector basis of ℓ² or to the unit vector basis of ℓp. In this paper we investigate the case 1 ≤ p < 2 and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure μ of the sequence satisfies ∫∨x²dμ(x) ∈ Lp/2. [ABSTRACT FROM AUTHOR]

In this paper efficient criteria for uniform asymptotic stability and boundedness of fractional-order functional differential equations are proved. To this end the Lyapunov-like functions and Mittag-Leffler functions are used. [ABSTRACT FROM AUTHOR]

We prove that for certain positive integers k, such as 12, a normal subgroup of a finite group which consists of kth powers is necessarily soluble. This gives rise to new solubility criteria, and solves an open problem from a 2013 paper by the authors. [ABSTRACT FROM AUTHOR]

In this paper we investigate complete critical metrics of the L²-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature, and we characterize critical metrics with nonnegative scalar curvature in dimensions three and four. [ABSTRACT FROM AUTHOR]

Shift invariant subspaces in the vector-valued Hardy space H2(E) play important roles in Nagy-Foias operator model theory. A theorem by Beurling, Lax and Halmos characterizes such invariant subspaces by operatorvalued inner functions ɵ(z). When E = H2(ⅅ), H2(E) is the Hardy space over the bidisk H2(ⅅ2). This paper shows that for some well-known examples of invariant subspaces in H2(ⅅ2), the function ɵ(z) turns out to be strikingly simple. [ABSTRACT FROM AUTHOR]

We settle all problems concerning the additivity of the Gerlits- Nagy property and related additivity numbers posed by Scheepers in his tribute paper to Gerlits. We apply these results to compute the minimal number of concentrated sets of reals (in the sense of Besicovitch) whose union, when multiplied with a Gerlits-Nagy space, need not have Rothberger's property. We apply these methods to construct a large family of spaces whose product with every Hurewicz space has Menger's property. Our applications extend earlier results of Babinkostova and Scheepers. [ABSTRACT FROM AUTHOR]