Search

In this short paper we prove that every sufficiently large odd integer can be written as a sum of one prime, two squares of primes and 83 83 powers of 2 2 .

In this paper we present a probabilistic construction of countable abelian p p -groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian p p -groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability 1 1 , produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian p p -groups. Valuated abelian p p -groups are natural generalizations of abelian p p -groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian p p -groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.

In this paper we present two-sided estimates of Chernoff type for the weighted L w 2 L_{w}^{2} -distance of a smooth function to the k k -dimensional space of all polynomials of degree less than k k , whenever the weight function w w solves the Pearson differential equation and generates a finite or infinite sequence of classical orthogonal polynomials. These inequalities are simple corollaries of a unified general theorem, which is the main result of the paper.

In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s s -convexity, and the estimates of the products of two harmonic s s -convex functions are also considered.

In this paper, we extend the work of Cao-Chen [9] on Bach-flat gradient Ricci solitons to classify $n$-dimensional ($n\ge 5$) complete $D$-flat gradient steady Ricci solitons. More precisely, we prove that any $n$-dimensional complete noncompact gradient steady Ricci soliton with vanishing $D$-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well., 10 pages; final version to appear in Proc. Amer. Math. Soc. arXiv admin note: text overlap with arXiv:1105.3163

In this paper we prove the following result: Let A A be a nonunital Banach algebra with a bounded approximate identity. Then A A cannot be both Arens regular and weakly sequentially complete. The paper also contains some applications of this result.

A convex subset B B of a real locally convex space X X is said to have the separation property if it can be separated from every closed convex subset A A of X X , which is disjoint from B B , by a closed hyperplane. The strong separation theorem says that if B B is weakly compact, then it has the separation property. In this paper, we present two versions of the converse and discuss an application of them. For example, we prove that a normed space is reflexive if and only if its closed unit ball has the separation property. Results in this paper can be considered as supplements of the famous theorem of James.

This paper contributes to the general program which aims to eliminate an unbounded search from proofs and procedures in computable structure theory. A countable structure in a finite language is punctual if its domain is ω \omega and its operations and relations are primitive recursive. A function f f is punctual if both f f and f − 1 f^{-1} are primitive recursive. We prove that there exists a countable rigid algebraic structure which has exactly two punctual presentations, up to punctual isomorphism.

In this paper, we give a complete classification of κ \kappa -solutions of Kähler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

The wandering subspace problem for an analytic norm-increasing m m -isometry T T on a Hilbert space H \mathcal {H} asks whether every T T -invariant subspace of H \mathcal {H} can be generated by a wandering subspace. An affirmative solution to this problem for m = 1 m=1 is ascribed to Beurling-Lax-Halmos, while that for m = 2 m=2 is due to Richter. In this paper, we capitalize on the idea of weighted shift on a one-circuit directed graph to construct a family of analytic cyclic 3 3 -isometries which do not admit the wandering subspace property and which are norm-increasing on the orthogonal complement of a one-dimensional space. Further, on this one-dimensional space, their norms can be made arbitrarily close to 1 1 . We also show that if the wandering subspace property fails for an analytic norm-increasing m m -isometry, then it fails miserably in the sense that the smallest T T -invariant subspace generated by the wandering subspace is of infinite codimension.

In this paper we construct an infinite family of paramodular forms of weight 2 2 which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta block conjecture of Gritsenko–Poor–Yuen (2013) related to the most important infinite series of theta blocks of weight 2 2 and q q -order 1 1 . We also consider some applications of this result.

In this paper we derive formulas for the adjoints of a class of composition operators with irrational symbol, in particular, the n n -th root functions. We discuss these formulas on both the Hardy space and the Bergman space.

This paper completely resolves the asymptotic development of order 2 2 by Γ \Gamma -convergence of the mass-constrained Cahn–Hilliard functional. Important new results on the slow motion of interfaces for the mass preserving Allen–Cahn equation and the Cahn–Hilliard equations in higher dimension are obtained as an application.

In this paper, the p p -Minkowski capacity measures of asymmetry in terms of the q q -mixed capacity, which have the well-known Minkowski measure of asymmetry as a special case, are defined, and some properties of these measures are studied. In addition, we extend the p p -Minkowski capacity measures of asymmetry to the corresponding Orlicz version.

Let A A be a semisimple Banach algebra with ‖ ‖ \left \| \right \| , which is a dense subalgebra of a semisimple Banach algebra B B with | | \left | \right | such that ‖ ‖ \left \| \right \| majorizes | | \left | \right | on A A . The purpose of this paper is to investigate the dual property between the algebras A A and B B . Some well-known results follow from this paper.

Dana P. Williams raised in [Proc. Am. Math. Soc., Ser. B, 2016] the following question: Must a second countable, locally compact, transitive groupoid have open range map? This paper gives a negative answer to that question. Although a second countable, locally compact transitive groupoid G may fail to have open range map, we prove that we can replace its topology with a topology which is also second countable, locally compact, and with respect to which G is a topological groupoid whose range map is open. Moreover, the two topologies generate the same Borel structure and coincide on the fibres of G., Comment: 7 pages

In this paper, we prove the following “weak bounded negativity conjecture”, which says that given a complex smooth projective surface X X , for any reduced curve C C in X X and integer g g , assume that the geometric genus of each component of C C is bounded from above by g g ; then the self-intersection number C 2 C^2 is bounded from below.

M. K. Da̧bkowski and J. H. Przytycki introduced the n n th Burnside group of a link, which is an invariant preserved by n n -moves. Using this invariant, for an odd prime p p , they proved that there are links which cannot be reduced to trivial links via p p -moves. It is generally difficult to determine if p p th Burnside groups associated to a link and the corresponding trivial link are isomorphic. In this paper, we give a necessary condition for the existence of such an isomorphism. Using this condition we give a simple proof for their results that concern p p -move reducibility of links.

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, the indefinite nonlinearity, and depend on the real parameter $\lambda$. A special focus is made on the extreme value of Nehari manifold $\lambda^*$, which determines the threshold of applicability of Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where parameter $\lambda$ lies above the threshold $\lambda^*$ is obtained., Comment: 14 pages

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

It has been known since the work of A.A. Kirillov that any smooth Jordan curve in the plane can be represented by its so-called fingerprint, an orientation preserving smooth diffeomorphism of the unit circle onto itself. In this paper, we give a new, simple proof of a theorem of Ebenfelt, Khavinson and Shapiro stating that the fingerprint of a polynomial lemniscate of degree $n$ is given by the $n$-th root of a Blaschke product of degree $n$ and that conversely, any smooth diffeomorphism induced by such a map is the fingerprint of a polynomial lemniscate of the same degree. The proof is easily generalized to the case of rational lemniscates, thus solving a problem raised by the previously mentioned authors., 6 pages

A description of how to assign to a full exceptional collection on a variety X X a DG category C \mathcal {C} such that the bounded derived category of coherent sheaves on X X is equivalent to the bounded derived category of C \mathcal {C} is given in a 1990 work by Bondal and Kapranov, Framed triangulated categories. In this paper we show that the category C \mathcal {C} can be chosen to have finite-dimensional spaces of morphisms. We describe how it behaves under mutations and present an algorithm allowing us to calculate it for full exceptional collections with vanishing Ext k ^k groups for k > 1 k>1 .

The technique of embedding a complex group algebra into the von Neumann regular ring associated with the corresponding W ∗ {W^ * } group algebra is exploited to prove that certain localizations of a group ring K G KG possess the invariant basis property. From this it follows, using a method of S. Rosset, that certain K G KG -modules have zero Euler characteristic. The assumptions are that K K is a commutative integral domain of characteristic zero, and that G G has a nontrivial, torsion-free, abelian normal subgroup A A . The main result of the paper is that the localization of K G KG obtained by inverting all elements of the form α − a \alpha - a , where α \alpha is a nonzero element of K K and a a is a nontrivial element of A A , has the invariant basis property; more generally, this localization and all its matrix rings are directly finite. (M. Smith has extended the methods of this paper to cover the localization of K G KG obtained by inverting all nonzero elements of K A KA .) Given a K G KG -module M M which has a finite free resolution, such that M M is finitely generated over K K , it is proved that the Euler characteristic of M M is zero. This verifies an unpublished result of Rosset.

A recent paper by J. B. Diaz and Dorothy Browne Shaffer extends the Gauss-Lucas Theorem to n n -dimensional Euclidean space. The authors leave open certain natural questions concerning the existence of “zeros of the derivative". This paper answers three such questions, and suggests several other questions for further investigation.

It is the purpose of this paper to show that if a space X X admits a closure-preserving cover of compact, closed sets, then X X is metacompact. This paper also provides a characterization of those closure-preserving covers of compact sets admitted by σ \sigma -compact spaces.

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

In this paper, we link the nature of special values of certain Dirichlet L L -functions to those of multiple gamma values.

Sendov conjecture tells that if $P$ denotes a complex polynomial having all his zeros in the closed unit disk and $a$ denote a zero of $P$, the closed disk of center $a$ and radius 1 contains a zero of the derivative $P'$. The main result of this paper is a proof of Sendov conjecture when the polynomial $P$ has a degree higher than a fixed integer $N$. We will give estimates of its integer $N$ in terms of $|a|$. To obtain this result, we will study the geometry of the zeros and critical points (i.e. zeros of $P'$) of a polynomial which would contradict Sendov conjecture., Comment: 14 pages, 5 figures

Let R R denote a commutative Noetherian (not necessarily local) ring and I I an ideal of R 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 M ( R , I ) c o f \mathscr {M}(R, I)_{cof} of I I -cofinite modules over a commutative Noetherian local ring R R forms an Abelian subcategory of the category of all R 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 R .

For any prime p p , the theory of p p -local compact groups is modelled on the p p -local homotopy theory of classifying spaces of compact Lie groups and p p -compact groups and generalises the earlier concept of p p -local finite groups. These objects have maximal tori and Weyl groups, although the Weyl groups need not be generated by pseudoreflections. In this paper, we study the rational p p -adic cohomology of the classifying space of a p p -local compact group and prove that just as for compact Lie groups, it is isomorphic to the ring of invariants of the Weyl group action on the cohomology of the classifying space of the maximal torus. This is applied to show that unstable Adams operations on p p -local compact groups are determined in the appropriate sense by the map they induce on rational cohomology.

Let R be a commutative noetherian local ring, and M a finitely generated R-module of infinite projective dimension. It is well-known that the depths of the syzygy modules of M eventually stabilize to the depth of R. In this paper, we investigate the conditions under which a similar statement can be made regarding dimension. In particular, we show that if R is equidimensional and the Betti numbers of M are eventually non-decreasing, then the dimension of any sufficiently high syzygy module of M coincides with the dimension of R., Comment: 8 pages; to appear in Proc. Amer. Math. Soc

In a 1998 paper, Kaneko and Zagier explain unpublished work of Atkin which exhibits an infinite sequence of polynomials with the property that when suitable polynomials are reduced mod p p for a prime p p , one gets the locus of supersingular elliptic curves. Here we generalize this phenomenon by considering the continued fraction expansions of modular and quasimodular forms.

In a previous paper, D. W. Hajek showed that if a space X X is a T 3 {T_3} space and A A is a compact subset of W X WX , the Wallman compactification of X X , then X ∩ A X \cap A is a closed subset of X X . This raises the question of whether this “closed intersection” property characterizes the T 3 {T_3} spaces among the Hausdorff spaces. In the present paper, the authors show this conjecture is false by giving an example of a nonregular Hausdorff space whose Wallman compactification is a KC \operatorname {KC} (compact closed)-space, and, hence, trivially satisfies this “closed intersection” property.

In a recent paper Freese and Murphy said a complete, convex, externally convex metric space has the vertical angle property provided for each four of its distinct points p p , q q , r r , s s , if m m is a midpoint of p p and q q and of r r and s s , then p r = q s pr = qs . In this paper we say a line L L is parallel to a line N N in such a space provided L L and N N contain points p p , r r , and q q , s s , respectively, such that the segments S ( p , q ) S\left ( {p,q} \right ) and S ( r , s ) S\left ( {r,s} \right ) have a common midpoint m m . We further assume that if line L L is parallel to line N N and line N N is parallel to line R R , then L L is parallel to R R . The main result of this paper is that such a space is a real strictly convex Banach space. Since real strictly convex Banach spaces have all of the above properties, the characterization is then complete.

In this paper we show that a smooth subvariety Z Z on an odd-dimensional complex projective smooth variety X X is determined by the sufficiently many Hodge conjectures it solves on hypersurfaces Y Y on X X of high degrees containing Z Z .

A Cohen-Kaplansky domain is an atomic domain with only a finite number of irreducibles. In this paper, we show that localizations of certain orders of rings of integers are necessarily CK-domains, and then prove there exists a closed form formula for the number of irreducible elements in several different cases of these types of rings. Modulo a variant of the Goldbach Conjecture, this construction allows us to answer a question posed by Cohen and Kaplansky over 60 years ago regarding the construction of a CK-domain containing n n nonprime irreducible elements for every positive integer n n .

In recent work, Beukers characterized A {A} -hypergeometric systems having a full set of algebraic solutions. He accomplished this by (1) determining which A {A} -hypergeometric systems have a full set of polynomial solutions modulo p p for almost all primes p p and (2) showing that these systems come from geometry. He then applied a fundamental theorem of N. Katz, which says that such systems have a full set of algebraic solutions. In this paper we establish some connections between nonresonant A A -hypergeometric systems and de Rham-type complexes, which leads to a determination of which A A -hypergeometric systems come from geometry. We do not use the fact that the system is irreducible or find integral formulas for its solutions.

In this paper we prove that for every unbounded convex closed set C C in c 0 c_0 there exists a nonexpansive mapping T : C → C T:C\to C which is fixed point free. This result solves in a negative sense a question that has remained open for some time in Metric Fixed Point Theory.

Let f f be a diffeomorphism of a closed n n -dimensional smooth manifold. In this paper, we show that if f f has the C 1 C^1 -stably inverse shadowing property on a transitive set, then it admits a dominated splitting.

This paper is devoted to a study of geodesics of Finsler metrics via Zermelo navigation. We give a geometric description of the geodesics of the Finsler metric produced from any Finsler metric and any homothetic field in terms of navigation representation, generalizing a result previously only known in the case of Randers metrics with constant S S -curvature. As its application, we present explicitly the geodesics of the Funk metric on a strongly convex domain.

This paper gives an asymptotic expansion for the solution of the boundary problem with respect to the Laplace-Beltrami operator. We also consider some examples in the domain whose boundary is real ellipsoid, where the boundary problem does not have a C n C^n solution up to the boundary.

In this paper we study an operator T T on a Banach space E E which is M M -hypercyclic for some subspace M M of E E . We give a sufficient condition for such an operator to be M M -hypercyclic and use it to answer negatively two questions asked by Madore and Martínez-Avendaño. We also give a sufficient condition for T T to be M M -hypercyclic for all finite co-dimensional subspaces M M in E E .

We study the almost Daugavet property, a generalization of the Daugavet property. It is analysed what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is: if $Z$ is a closed subspace of a separable almost Daugavet space $X$ such that the quotient space $X/Z$ contains no copy of $\ell_1$, then $Z$ has the almost Daugavet property, too., 5 pages

Let k k be a field, V V an r r -dimensional k k -vector space, and W W a finite subgroup of A u t k ( V ) \mathrm {Aut}_k(V ) . Let S = S [ V # ] S = S[V^{\#}] be the symmetric algebra on V # V^\# , the k k -dual of V V , and R = S W R = S^W the ring of invariants under the natural action of W W on S S . Define P ∗ P_* to be the quotient algebra S ⊗ R k S\otimes _R k . Steinberg has shown that R R is polynomial if k k is the field of complex numbers and the quotient algebra P ∗ = S ⊗ R k P_* = S\otimes _R k satisfies Poincaré duality. In this paper we use elementary methods to prove Steinberg’s result for fields of characteristic 0 0 or of characteristic prime to the order of W W . This gives a new proof even in the characteristic zero case. Theorem 0.1. If the characteristic of k k is zero or prime to the order of W W and P ∗ P_* satisfies Poincaré duality, then R R is isomorphic to a polynomial algebra on r r generators.

Let $K$ be a fixed number field, and assume that $K$ is Galois over $\qq$. Previously, the author showed that when estimating the number of prime ideals with norm congruent to $a$ modulo $q$ via the Chebotar\"ev Density Theorem, the mean square error in the approximation is small when averaging over all $q\le Q$ and all appropriate $a$. In this article, we replace the upper bound by an asymptotic formula. The result is related to the classical Barban-Davenport-Halberstam Theorem in the case $K=\qq$., Comment: Preprint of an old paper

The topological complexity T C ( X ) \mathsf {TC}(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X X , viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n n distinct points in Euclidean m m -space for all m ≥ 2 m\ge 2 and n ≥ 2 n\ge 2 ; the answer was previously known in the cases m = 2 m=2 and m m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.

Let H H be a complex Hilbert space. The symbol A # B A\# B stands for the geometric mean of the positive bounded linear operators A , B A,B on H H in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation # \# . We prove that if dim ⁡ H ≥ 2 \dim H\geq 2 , any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on H H .

In this paper, we give an estimate from below of the bounding genera for homology 3 3 -spheres defined by Y. Matsumoto in terms of w w -invariants. In particular, combining with Matsumoto’s estimates we determine the values of the bounding genera for several infinite families of Brieskorn homology 3 3 -spheres.

In this paper we identify the “nil-terms” for Waldhausen’s algebraic K K -theory of spaces functor as the reduced K K -theory of a category of equivariant spaces equipped with a homotopically nilpotent endomorphism.

In this paper we consider the Jacobian conjecture for a map f f of complex affine spaces of dimension n n . It is well known that if f f is proper, then the conjecture will hold. Using topological arguments, specifically Smith theory, we show that the conjecture holds if and only if f f is proper onto its image.