Illinois Journal of Mathematics Articles (Project Euclid)
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The latest articles from Illinois Journal of Mathematics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 09 Mar 2011 09:09 ESThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Hilbertian matrix cross normed spaces arising from normed ideals
http://projecteuclid.org/euclid.ijm/1264170836
<strong>Takahiro Ohta</strong><p><strong>Source: </strong>Illinois J. Math., Volume 53, Number 1, 1--24.</p><p><strong>Abstract:</strong><br/>
Generalizing Pisier’s idea, we introduce a Hilbertian matrix cross normed space associated with a pair of symmetric normed ideals. When the two ideals coincide, we show that our construction gives an operator space if and only if the ideal is the Schatten class. In general, a pair of symmetric normed ideals that are not necessarily the Schatten class may give rise to an operator space. We study the space of completely bounded mappings between the matrix cross normed spaces obtained in this way and show that the multiplicator norm naturally appears as the completely bounded norm.
</p>projecteuclid.org/euclid.ijm/1264170836_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTA note on nonexistence of multiple black holes in static vacuum Einstein space–timeshttps://projecteuclid.org/euclid.ijm/1506067292<strong>H. Baltazar</strong>, <strong>B. Leandro</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 811--818.</p><p><strong>Abstract:</strong><br/>
The purpose of this note is to study the static vacuum Einstein space–time with half harmonic Weyl tensor, that is, $\delta W^{+}=0$. We prove that there are no multiple black holes on a four-dimensional static vacuum Einstein space–time with half harmonic Weyl tensor.
</p>projecteuclid.org/euclid.ijm/1506067292_20170922040145Fri, 22 Sep 2017 04:01 EDTOn the injective dimension of $\mathscr{F}$-finite modules and holonomic $\mathscr{D}$-moduleshttps://projecteuclid.org/euclid.ijm/1506067293<strong>Mehdi Dorreh</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 819--831.</p><p><strong>Abstract:</strong><br/>
Let $R$ be a regular local ring containing a field $k$ of characteristic $p$ and $M$ be an $\mathscr{F}$-finite module. In this paper, we study the injective dimension of $M$. We prove that $\operatorname{dim}_{R}(M)-1\leq\operatorname{inj.dim}_{R}(M)$. If $R=k[[x_{1},\ldots,x_{n}]]$ where $k$ is a field of characteristic $0$ we prove the analogous result for a class of holonomic $\mathscr{D}$-modules which contains local cohomology modules.
</p>projecteuclid.org/euclid.ijm/1506067293_20170922040145Fri, 22 Sep 2017 04:01 EDTKoszul factorization and the Cohen–Gabber theoremhttps://projecteuclid.org/euclid.ijm/1506067294<strong>C. Skalit</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 833--844.</p><p><strong>Abstract:</strong><br/>
We present a sharpened version of the Cohen–Gabber theorem for equicharacteristic, complete local domains $(A,\mathfrak{m},k)$ with algebraically closed residue field and dimension $d>0$. Namely, we show that for any prime number $p$, $\operatorname{Spec}A$ admits a dominant, finite map to $\operatorname{Spec}k[[X_{1},\ldots,X_{d}]]$ with generic degree relatively prime to $p$. Our result follows from Gabber’s original theorem, elementary Hilbert–Samuel multiplicity theory, and a “factorization” of the map induced on the Grothendieck group $\mathbf{G}_{0}(A)$ by the Koszul complex.
</p>projecteuclid.org/euclid.ijm/1506067294_20170922040145Fri, 22 Sep 2017 04:01 EDTAsymptotic stabilization of Betti diagrams of generic initial systemshttps://projecteuclid.org/euclid.ijm/1506067295<strong>Sarah Mayes-Tang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 845--858.</p><p><strong>Abstract:</strong><br/>
Several authors investigating the asymptotic behaviour of the Betti diagrams of the graded system $\{I^{k}\}$ independently showed that the shape of the nonzero entries in the diagrams stabilizes when $I$ is a homogeneous ideal with generators of the same degree. In this paper, we study the Betti diagrams of graded systems of ideals built by taking the initial ideals or generic initial ideals of powers, and discuss the stabilization of additional collections of Betti diagrams. Our main result shows that when $I$ has generators of the same degree, the entries in the Betti diagrams of the reverse lexicographic generic initial system $\{\operatorname{gin}(I^{k})\}$ are given asymptotically by polynomials and that the shape of the diagrams stabilizes.
</p>projecteuclid.org/euclid.ijm/1506067295_20170922040145Fri, 22 Sep 2017 04:01 EDTNewton’s lemma for differential equationshttps://projecteuclid.org/euclid.ijm/1506067296<strong>Fuensanta Aroca</strong>, <strong>Giovanna Ilardi</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 859--867.</p><p><strong>Abstract:</strong><br/>
The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon.
Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano’s Groebner fan to differential ideals.
</p>projecteuclid.org/euclid.ijm/1506067296_20170922040145Fri, 22 Sep 2017 04:01 EDTOn the classification of rational sphere mapshttps://projecteuclid.org/euclid.ijm/1506067297<strong>John P. D’Angelo</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 869--890.</p><p><strong>Abstract:</strong><br/>
We prove a new classification result for (CR) rational maps from the unit sphere in some $\mathbb{C}^{n}$ to the unit sphere in $\mathbb{C}^{N}$. To do so, we work at the level of Hermitian forms, and we introduce ancestors and descendants.
</p>projecteuclid.org/euclid.ijm/1506067297_20170922040145Fri, 22 Sep 2017 04:01 EDTOn the Krein–Milman–Ky Fan theorem for convex compact metrizable setshttps://projecteuclid.org/euclid.ijm/1520046206<strong>Mohammed Bachir</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 1--24.</p><p><strong>Abstract:</strong><br/>
We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi $-extreme points of a $\Phi $-convex compact metrizable space are replaced by the $\Phi $-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.
</p>projecteuclid.org/euclid.ijm/1520046206_20180302220336Fri, 02 Mar 2018 22:03 ESTThe Hörmander multiplier theorem, I: The linear case revisitedhttps://projecteuclid.org/euclid.ijm/1520046207<strong>Loukas Grafakos</strong>, <strong>Danqing He</strong>, <strong>Petr Honzik</strong>, <strong>Hanh Van Nguyen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 25--35.</p><p><strong>Abstract:</strong><br/>
We discuss $L^{p}(\mathbb{R}^{n})$ boundedness for Fourier multiplier operators that satisfy the hypotheses of the Hörmander multiplier theorem in terms of an optimal condition that relates the distance $\vert \frac{1}{p}-\frac{1}{2}\vert $ to the smoothness $s$ of the associated multiplier measured in some Sobolev norm. We provide new counterexamples to justify the optimality of the condition $\vert \frac{1}{p}-\frac{1}{2}\vert <\frac{s}{n}$ and we discuss the endpoint case $\vert \frac{1}{p}-\frac{1}{2}\vert =\frac{s}{n}$.
</p>projecteuclid.org/euclid.ijm/1520046207_20180302220336Fri, 02 Mar 2018 22:03 ESTAlmost conformally flat hypersurfaceshttps://projecteuclid.org/euclid.ijm/1520046208<strong>Christos-Raent Onti</strong>, <strong>Theodoros Vlachos</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 37--51.</p><p><strong>Abstract:</strong><br/>
We prove a universal lower bound for the $L^{n/2}$-norm of the Weyl tensor in terms of the Betti numbers for compact $n$-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the round sphere and extend a result due to Shiohama and Xu ( J. Geom. Anal. 7 (1997) 377–386) for compact hypersurfaces in any space form.
</p>projecteuclid.org/euclid.ijm/1520046208_20180302220336Fri, 02 Mar 2018 22:03 ESTBi-parameter Littlewood–Paley operators with upper doubling measureshttps://projecteuclid.org/euclid.ijm/1520046209<strong>Mingming Cao</strong>, <strong>Qingying Xue</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 53--79.</p><p><strong>Abstract:</strong><br/>
Let $\mu=\mu_{n_{1}}\times\mu_{n_{2}}$, where $\mu_{n_{1}}$ and $\mu_{n_{2}}$ are upper doubling measures on $\mathbb{R}^{n_{1}}$ and $\mathbb{R}^{n_{2}}$, respectively. Let the pseudo-accretive function $b=b_{1}\otimes b_{2}$ satisfy a bi-parameter Carleson condition. In this paper, we established the $L^{2}(\mu)$ boundedness of non-homogeneous Littlewood–Paley $g_{\lambda}^{*}$-function with non-convolution type kernels on product spaces. This was mainly done by means of dyadic analysis and non-homogenous methods. The result is new even in the setting of Lebesgue measures.
</p>projecteuclid.org/euclid.ijm/1520046209_20180302220336Fri, 02 Mar 2018 22:03 ESTBounds on the norm of the backward shift and related operators in Hardy and Bergman spaceshttps://projecteuclid.org/euclid.ijm/1520046210<strong>Timothy Ferguson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 81--96.</p><p><strong>Abstract:</strong><br/>
We study bounds for the backward shift operator $f\mapsto(f(z)-f(0))/z$ and the related operator $f\mapsto f-f(0)$ on Hardy and Bergman spaces of analytic and harmonic functions. If $u$ is a real valued harmonic function, we also find a sharp bound on $M_{1}(r,u-u(0))$ in terms of $\|u\|_{h^{1}}$, where $M_{1}$ is the integral mean with $p=1$.
</p>projecteuclid.org/euclid.ijm/1520046210_20180302220336Fri, 02 Mar 2018 22:03 ESTOn high-frequency limits of $U$-statistics in Besov spaces over compact manifoldshttps://projecteuclid.org/euclid.ijm/1520046211<strong>Solesne Bourguin</strong>, <strong>Claudio Durastanti</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 97--125.</p><p><strong>Abstract:</strong><br/>
In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered here are the so-called needlets, characterized by strong concentration properties and by an exact reconstruction formula. Furthermore, we consider Poisson point processes over the manifold such that the density function associated to its control measure lives in a Besov space. The main findings of this paper include new rates of convergence that depend strongly on the degree of regularity of the control measure of the underlying Poisson point process, providing a refined understanding of the connection between regularity and speed of convergence in this framework.
</p>projecteuclid.org/euclid.ijm/1520046211_20180302220336Fri, 02 Mar 2018 22:03 ESTStructure of porous sets in Carnot groupshttps://projecteuclid.org/euclid.ijm/1520046212<strong>Andrea Pinamonti</strong>, <strong>Gareth Speight</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 127--150.</p><p><strong>Abstract:</strong><br/>
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\sigma $-porous with respect to the Carnot–Carathéodory (CC) distance. In the first Heisenberg group, we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups, we then construct a Lipschitz function which is Pansu differentiable at no point of a given $\sigma $-porous set and show preimages of open sets under the horizontal gradient are far from being porous.
</p>projecteuclid.org/euclid.ijm/1520046212_20180302220336Fri, 02 Mar 2018 22:03 ESTMaximal torus theory for compact quantum groupshttps://projecteuclid.org/euclid.ijm/1520046213<strong>Teodor Banica</strong>, <strong>Issan Patri</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 151--170.</p><p><strong>Abstract:</strong><br/>
Associated to any compact quantum group $G\subset U_{N}^{+}$ is a canonical family of group dual subgroups $\widehat{\Gamma }_{Q}\subset G$, parametrized by unitaries $Q\in U_{N}$, playing the role of “maximal tori” for $G$. We present here a series of conjectures, relating the various algebraic and analytic properties of $G$ to those of the family $\{\widehat{\Gamma }_{Q}|Q\in U_{N}\}$.
</p>projecteuclid.org/euclid.ijm/1520046213_20180302220336Fri, 02 Mar 2018 22:03 ESTEvaluation of Tornheim’s type of double serieshttps://projecteuclid.org/euclid.ijm/1520046214<strong>Shin-ya Kadota</strong>, <strong>Takuya Okamoto</strong>, <strong>Koji Tasaka</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 171--186.</p><p><strong>Abstract:</strong><br/>
We examine values of certain Tornheim’s type of double series with odd weight. As a result, an affirmative answer to a conjecture about the parity theorem for the zeta function of the root system of the exceptional Lie algebra $G_{2}$, proposed by Komori, Matsumoto and Tsumura, is given.
</p>projecteuclid.org/euclid.ijm/1520046214_20180302220336Fri, 02 Mar 2018 22:03 ESTOn representations of error terms related to the derivatives for some Dirichlet serieshttps://projecteuclid.org/euclid.ijm/1520046215<strong>Jun Furuya</strong>, <strong>T. Makoto Minamide</strong>, <strong>Yoshio Tanigawa</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 187--209.</p><p><strong>Abstract:</strong><br/>
In previous papers, we examined several properties of an error term in a certain divisor problem related to the derivatives of the Riemann zeta-function. In this paper, we obtain representations of error terms related to the derivatives of some Dirichlet series, which can be regarded as generalized versions of a Dirichlet divisor problem and a Gauss circle problem. We also give the upper bounds of the error terms in terms of exponent pairs.
</p>projecteuclid.org/euclid.ijm/1520046215_20180302220336Fri, 02 Mar 2018 22:03 ESTThe expected number of complex zeros of complex random polynomialshttps://projecteuclid.org/euclid.ijm/1520046216<strong>Katrina Ferrier</strong>, <strong>Micah Jackson</strong>, <strong>Andrew Ledoan</strong>, <strong>Dhir Patel</strong>, <strong>Huong Tran</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 211--224.</p><p><strong>Abstract:</strong><br/>
By using the technique introduced in 1995 by Shepp and Vanderbei, we derive an exact formula for the expected number of complex zeros of a complex random polynomial due to Kac. The explicit evaluation of the average intensity function is obtained in closed form in the case of standard normal coefficients. In addition, we provide the limiting expressions for the intensity function and the expected number of zeros in open circular disks in the complex plane.
</p>projecteuclid.org/euclid.ijm/1520046216_20180302220336Fri, 02 Mar 2018 22:03 ESTSum of Toeplitz products on the Hardy space over the polydiskhttps://projecteuclid.org/euclid.ijm/1520046217<strong>Tao Yu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 225--241.</p><p><strong>Abstract:</strong><br/>
In this paper, we obtain several sufficient and necessary conditions for a finite sum of Toeplitz products with form $\sum_{m=1}^{M}T_{f_{m}}T_{g_{m}}$ on the Hardy space over the polydisk to be zero. The methods used in this note are Berezin transform and the essential fiber dimension.
</p>projecteuclid.org/euclid.ijm/1520046217_20180302220336Fri, 02 Mar 2018 22:03 ESTSome combinatorial number theory problems over finite valuation ringshttps://projecteuclid.org/euclid.ijm/1520046218<strong>Thang Pham</strong>, <strong>Le Anh Vinh</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 1-2, 243--257.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{R}$ be a finite valuation ring of order $q^{r}$. In this paper, we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_{q}$ and finite cyclic rings $\mathbb{Z}/p^{r}\mathbb{Z}$, in the setting of finite valuation rings.
</p>projecteuclid.org/euclid.ijm/1520046218_20180302220336Fri, 02 Mar 2018 22:03 ESTCohomology of ideals in elliptic surface singularitieshttps://projecteuclid.org/euclid.ijm/1534924827<strong>Tomohiro Okuma</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 259--273.</p><p><strong>Abstract:</strong><br/>
We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_{g}$-ideal if and only if it is a maximally elliptic singularity of degree $1$.
</p>projecteuclid.org/euclid.ijm/1534924827_20180822040103Wed, 22 Aug 2018 04:01 EDTUltraproducts of crossed product von Neumann algebrashttps://projecteuclid.org/euclid.ijm/1534924828<strong>Reiji Tomatsu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 275--286.</p><p><strong>Abstract:</strong><br/>
We study a relationship between the ultraproduct of a crossed product von Neumann algebra and the crossed product of an ultraproduct von Neumann algebra. As an application, the continuous core of an ultraproduct von Neumann algebra is described.
</p>projecteuclid.org/euclid.ijm/1534924828_20180822040103Wed, 22 Aug 2018 04:01 EDTThe module theory of divided power algebrashttps://projecteuclid.org/euclid.ijm/1534924829<strong>Rohit Nagpal</strong>, <strong>Andrew Snowden</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 287--353.</p><p><strong>Abstract:</strong><br/>
We study modules for the divided power algebra $\mathbf{D}$ in a single variable over a commutative Noetherian ring $\mathbf{K}$. Our first result states that $\mathbf{D}$ is a coherent ring. In fact, we show that there is a theory of Gröbner bases for finitely generated ideals, and so computations with finitely presented $\mathbf{D}$-modules are in principle algorithmic. We go on to determine much about the structure of finitely presented $\mathbf{D}$-modules, such as: existence of certain nice resolutions, computation of the Grothendieck group, results about injective dimension, and how they interact with torsion modules. Our results apply not just to the classical divided power algebra, but to its $q$-variant as well, and even to a much broader class of algebras we introduce called “generalized divided power algebras.” On the other hand, we show that the divided power algebra in two variables over $\mathbf{Z}_{p}$ is not coherent.
</p>projecteuclid.org/euclid.ijm/1534924829_20180822040103Wed, 22 Aug 2018 04:01 EDTA new direct proof of the central limit theoremhttps://projecteuclid.org/euclid.ijm/1534924830<strong>Vladimir Dobrić</strong>, <strong>Patricia Garmirian</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 355--370.</p><p><strong>Abstract:</strong><br/>
We prove the central limit theorem from the definition of weak convergence using the Haar basis, calculus, and elementary probability, and we estimate the rate of convergence off the tails. The use of the Haar basis pinpoints the role of $L^{2}([0,1])$ in the CLT as well as the assumption of finite variance.
</p>projecteuclid.org/euclid.ijm/1534924830_20180822040103Wed, 22 Aug 2018 04:01 EDTA characterization of the Macaulay dual generators for quadratic complete intersectionshttps://projecteuclid.org/euclid.ijm/1534924831<strong>Tadahito Harima</strong>, <strong>Akihito Wachi</strong>, <strong>Junzo Watanabe</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 371--383.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a homogeneous polynomial in $n$ variables of degree $d$ over a field $K$. Let $A(F)$ be the associated Artinian graded $K$-algebra. If $B\subset A(F)$ is a subalgebra of $A(F)$ which is Gorenstein with the same socle degree as $A(F)$, we describe the Macaulay dual generator for $B$ in terms of $F$. Furthermore when $n=d$, we give necessary and sufficient conditions on the polynomial $F$ for $A(F)$ to be a complete intersection.
</p>projecteuclid.org/euclid.ijm/1534924831_20180822040103Wed, 22 Aug 2018 04:01 EDTConvex subquivers and the finitistic dimensionhttps://projecteuclid.org/euclid.ijm/1534924832<strong>Edward L. Green</strong>, <strong>Eduardo N. Marcos</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 385--397.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{Q}$ be a quiver and $K$ a field. We study the interrelationship of homological properties of algebras associated to convex subquivers of $\mathcal{Q}$ and quotients of the path algebra $K\mathcal{Q}$. We introduce the homological heart of $\mathcal{Q}$ which is a particularly nice convex subquiver of $\mathcal{Q}$. For any algebra of the form $K\mathcal{Q}/I$, the algebra associated to $K\mathcal{Q}/I$ and the homological heart have similar homological properties. We give an application showing that the finitistic dimension conjecture need only be proved for algebras with path connected quivers.
</p>projecteuclid.org/euclid.ijm/1534924832_20180822040103Wed, 22 Aug 2018 04:01 EDTSome uniqueness results for Ricci solitonshttps://projecteuclid.org/euclid.ijm/1534924833<strong>J. F. Silva Filho</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 399--413.</p><p><strong>Abstract:</strong><br/>
We investigate the relationship between Ricci soliton structures and homothetic vector fields, especially Killing vector fields. More precisely, we present some characterizations for Ricci solitons endowed with Killing vector fields of constant norm as well as homothetic vector fields. In particular, we relate different Ricci soliton structures on a Riemannian manifolds in order to deduce some uniqueness results.
</p>projecteuclid.org/euclid.ijm/1534924833_20180822040103Wed, 22 Aug 2018 04:01 EDTDistinguishing $\Bbbk$-configurationshttps://projecteuclid.org/euclid.ijm/1534924834<strong>Federico Galetto</strong>, <strong>Yong-Su Shin</strong>, <strong>Adam Van Tuyl</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 415--441.</p><p><strong>Abstract:</strong><br/>
A $\Bbbk$-configuration is a set of points $\mathbb{X}$ in $\mathbb{P}^{2}$ that satisfies a number of geometric conditions. Associated to a $\Bbbk$-configuration is a sequence $(d_{1},\ldots,d_{s})$ of positive integers, called its type, which encodes many of its homological invariants. We distinguish $\Bbbk$-configurations by counting the number of lines that contain $d_{s}$ points of $\mathbb{X}$. In particular, we show that for all integers $m\gg0$, the number of such lines is precisely the value of $\Delta\mathbf{H}_{m\mathbb{X}}(md_{s}-1)$. Here, $\Delta\mathbf{H}_{m\mathbb{X}}(-)$ is the first difference of the Hilbert function of the fat points of multiplicity $m$ supported on $\mathbb{X}$.
</p>projecteuclid.org/euclid.ijm/1534924834_20180822040103Wed, 22 Aug 2018 04:01 EDTOn strict Whitney arcs and $t$-quasi self-similar arcshttps://projecteuclid.org/euclid.ijm/1534924835<strong>Daowei Ma</strong>, <strong>Xin Wei</strong>, <strong>Zhi-Ying Wen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 443--477.</p><p><strong>Abstract:</strong><br/>
A connected compact subset $E$ of $\mathbb{R}^{N}$ is said to be a strict Whitney set if there exists a real-valued $C^{1}$ function $f$ on $\mathbb{R}^{N}$ with $\nabla f|_{E}\equiv 0$ such that $f$ is constant on no non-empty relatively open subsets of $E$. We prove that each self-similar arc of Hausdorff dimension $s>1$ in $\mathbb{R}^{N}$ is a strict Whitney set with criticality $s$. We also study a special kind of self-similar arcs, which we call “regular” self-similar arcs. We obtain necessary and sufficient conditions for a regular self-similar arc $\Lambda $ to be a $t$-quasi-arc, and for the Hausdorff measure function on $\Lambda $ to be a strict Whitney function. We prove that if a regular self-similar arc has “minimal corner angle” $\theta_{\min }>0$, then it is a 1-quasi-arc and hence its Hausdorff measure function is a strict Whitney function. We provide an example of a one-parameter family of regular self-similar arcs with various features. For some values of the parameter $\tau $, the Hausdorff measure function of the self-similar arc is a strict Whitney function on the arc, and hence the self-similar arc is an $s$-quasi-arc, where $s$ is the Hausdorff dimension of the arc. For each $t_{0}\ge 1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t$-quasi-arc for each $t>t_{0}$, but it is not a $t_{0}$-quasi-arc. For each $t_{0}>1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t_{0}$-quasi-arc, but it is a $t$-quasi-arc for no $t\in [1,t_{0})$.
</p>projecteuclid.org/euclid.ijm/1534924835_20180822040103Wed, 22 Aug 2018 04:01 EDTNaimark’s problem for graph $C^{*}$-algebrashttps://projecteuclid.org/euclid.ijm/1534924836<strong>Nishant Suri</strong>, <strong>Mark Tomforde</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 479--495.</p><p><strong>Abstract:</strong><br/>
Naimark’s problem asks whether a $C^{*}$-algebra that has only one irreducible $*$-representation up to unitary equivalence is isomorphic to the $C^{*}$-algebra of compact operators on some (not necessarily separable) Hilbert space. This problem has been solved in special cases, including separable $C^{*}$-algebras and Type I $C^{*}$-algebras. However, in 2004 Akemann and Weaver used the diamond principle to construct a $C^{*}$-algebra with $\aleph_{1}$ generators that is a counterexample to Naimark’s Problem. More precisely, they showed that the statement “There exists a counterexample to Naimark’s Problem that is generated by $\aleph_{1}$ elements.” is independent of the axioms of ZFC. Whether Naimark’s problem itself is independent of ZFC remains unknown. In this paper, we examine Naimark’s problem in the setting of graph $C^{*}$-algebras, and show that it has an affirmative answer for (not necessarily separable) AF graph $C^{*}$-algebras as well as for $C^{*}$-algebras of graphs in which each vertex emits a countable number of edges.
</p>projecteuclid.org/euclid.ijm/1534924836_20180822040103Wed, 22 Aug 2018 04:01 EDTSimplifying branched covering surface-knots by chart moves involving black verticeshttps://projecteuclid.org/euclid.ijm/1534924837<strong>Inasa Nakamura</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 497--515.</p><p><strong>Abstract:</strong><br/>
A branched covering surface-knot is a surface-knot in the form of a branched covering over an oriented surface-knot $F$, where we include the case when the covering has no branch points. A branched covering surface-knot is presented by a graph called a chart on a surface diagram of $F$. We can simplify a branched covering surface-knot by an addition of 1-handles with chart loops to a form such that its chart is the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications for the case when branched covering surface-knots have a non-zero number of branch points, using chart moves involving black vertices.
</p>projecteuclid.org/euclid.ijm/1534924837_20180822040103Wed, 22 Aug 2018 04:01 EDTA note on the simultaneous Waring rank of monomialshttps://projecteuclid.org/euclid.ijm/1534924838<strong>Enrico Carlini</strong>, <strong>Emanuele Ventura</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 517--530.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the complex simultaneous Waring rank for collections of monomials. For general collections, we provide a lower bound, whereas for special collections we provide a formula for the simultaneous Waring rank. Our approach is algebraic and combinatorial. We give an application to ranks of binomials and maximal simultaneous ranks. Moreover, we include an appendix of scripts written in the algebra software Macaulay2 to experiment with simultaneous ranks.
</p>projecteuclid.org/euclid.ijm/1534924838_20180822040103Wed, 22 Aug 2018 04:01 EDTExamples of non-autonomous basins of attractionhttps://projecteuclid.org/euclid.ijm/1534924839<strong>Sayani Bera</strong>, <strong>Ratna Pal</strong>, <strong>Kaushal Verma</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 61, Number 3-4, 531--567.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to present several examples of non-autonomous basins of attraction that arise from sequences of automorphisms of $\mathbb{C}^{k}$. In the first part, we prove that the non-autonomous basin of attraction arising from a pair of automorphisms of $\mathbb{C}^{2}$ of a prescribed form is biholomorphic to $\mathbb{C}^{2}$. This, in particular, provides a partial answer to a question raised in (A survey on non-autonomous basins in several complex variables (2013) Preprint) in connection with Bedford’s Conjecture about uniformizing stable manifolds. In the second part, we describe three examples of Short $\mathbb{C}^{k}$’s with specified properties. First, we show that for $k\geq3$, there exist $(k-1)$ mutually disjoint Short $\mathbb{C}^{k}$’s in $\mathbb{C}^{k}$. Second, we construct a Short $\mathbb{C}^{k}$, large enough to accommodate a Fatou–Bieberbach domain, that avoids a given algebraic variety of codimension $2$. Lastly, we discuss examples of Short $\mathbb{C}^{k}$’s with (piece-wise) smooth boundaries.
</p>projecteuclid.org/euclid.ijm/1534924839_20180822040103Wed, 22 Aug 2018 04:01 EDTOn the Krein–Milman–Ky Fan theorem for convex compact metrizable setshttps://projecteuclid.org/euclid.ijm/1552442654<strong>Mohammed Bachir</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 1--24.</p><p><strong>Abstract:</strong><br/>
We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi$-extreme points of a $\Phi$-convex compact metrizable space are replaced by the $\Phi$-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.
</p>projecteuclid.org/euclid.ijm/1552442654_20190312220428Tue, 12 Mar 2019 22:04 EDTOn the curvature of Einstein–Hermitian surfaceshttps://projecteuclid.org/euclid.ijm/1552442655<strong>Mustafa Kalafat</strong>, <strong>Caner Koca</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 25--39.</p><p><strong>Abstract:</strong><br/>
We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.
</p>projecteuclid.org/euclid.ijm/1552442655_20190312220428Tue, 12 Mar 2019 22:04 EDTQuantum semigroups generated by locally compact semigroupshttps://projecteuclid.org/euclid.ijm/1552442656<strong>M. A. Aukhadiev</strong>, <strong>Y. N. Kuznetsova</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 41--60.</p><p><strong>Abstract:</strong><br/>
Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^{*}$-algebra $C^{*}_{\delta }(S)$ generated by the operators of translation by all elements of $S$ in $L^{2}(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. The same is proved for the von Neumann algebra $\operatorname{VN}(S)$ generated by $C^{*}_{\delta }(S)$.
</p>projecteuclid.org/euclid.ijm/1552442656_20190312220428Tue, 12 Mar 2019 22:04 EDTDonaldson–Thomas invariants of Calabi–Yau orbifolds under flopshttps://projecteuclid.org/euclid.ijm/1552442657<strong>Yunfeng Jiang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 61--97.</p><p><strong>Abstract:</strong><br/>
We study the Donaldson–Thomas type invariants for the Calabi–Yau threefold Deligne–Mumford stacks under flops. A crepant birational morphism between two smooth Calabi–Yau threefold Deligne–Mumford stacks is called an orbifold flop if the flopping locus is the quotient of weighted projective lines by a cyclic group action. We prove that the Donaldson–Thomas invariants are preserved under orbifold flops.
</p>projecteuclid.org/euclid.ijm/1552442657_20190312220428Tue, 12 Mar 2019 22:04 EDTEvery lens space contains a genus one homologically fibered knothttps://projecteuclid.org/euclid.ijm/1552442658<strong>Yuta Nozaki</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 99--111.</p><p><strong>Abstract:</strong><br/>
We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the Chebotarev density theorem and binary quadratic forms in number theory play a key role. We also discuss the Alexander polynomial of homologically fibered knots.
</p>projecteuclid.org/euclid.ijm/1552442658_20190312220428Tue, 12 Mar 2019 22:04 EDTActions of measured quantum groupoids on a finite basishttps://projecteuclid.org/euclid.ijm/1552442659<strong>Jonathan Crespo</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 113--214.</p><p><strong>Abstract:</strong><br/>
In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning actions of locally compact quantum groups on C$^{*}$-algebras ( Comm. Math. Phys. 235 (2003) 139–167). Let $\mathcal{G}$ be a measured quantum groupoid on a finite basis. We prove that if $\mathcal{G}$ is regular, then any weakly continuous action of $\mathcal{G}$ on a C$^{*}$-algebra is necessarily strongly continuous. Following ( K-Theory 2 (1989) 683–721), we introduce and investigate a notion of $\mathcal{G}$-equivariant Hilbert C$^{*}$-modules. By applying the previous results and a version of the Takesaki–Takai duality theorem obtained in ( Bull. Soc. Math. France 145 (2017) 711–802) for actions of $\mathcal{G}$, we obtain a canonical equivariant Morita equivalence between a given $\mathcal{G}$-C$^{*}$-algebra $A$ and the double crossed product $(A\rtimes\mathcal{G})\rtimes\widehat{\mathcal{G}}$.
</p>projecteuclid.org/euclid.ijm/1552442659_20190312220428Tue, 12 Mar 2019 22:04 EDTOn logarithmic differential operators and equations in the planehttps://projecteuclid.org/euclid.ijm/1552442660<strong>Julien Sebag</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 215--224.</p><p><strong>Abstract:</strong><br/>
Let $k$ be a field of characteristic zero. Let $f\in k[x_{0},y_{0}]$ be an irreducible polynomial. In this article, we study the space of polynomial partial differential equations of order one in the plane, which admit $f$ as a solution. We provide algebraic characterizations of the associated graded $k[x_{0},y_{0}]$-module (by degree) of this space. In particular, we show that it defines the general component of the tangent space of the curve $\{f=0\}$ and connect it to the $V$-filtration of the logarithmic differential operators of the plane along $\{f=0\}$.
</p>projecteuclid.org/euclid.ijm/1552442660_20190312220428Tue, 12 Mar 2019 22:04 EDTOn amicable tupleshttps://projecteuclid.org/euclid.ijm/1552442661<strong>Yuta Suzuki</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 225--252.</p><p><strong>Abstract:</strong><br/>
For an integer $k\ge 2$, a tuple of $k$ positive integers $(M_{i})_{i=1}^{k}$ is called an amicable $k$-tuple if the equation \begin{equation*}\sigma (M_{1})=\cdots =\sigma (M_{k})=M_{1}+\cdots +M_{k}\end{equation*} holds. This is a generalization of amicable pairs. An amicable pair is a pair of distinct positive integers each of which is the sum of the proper divisors of the other. Gmelin (Über vollkommene und befreundete Zahlen (1917) Heidelberg University) conjectured that there is no relatively prime amicable pairs and Artjuhov ( Acta Arith. 27 (1975) 281–291) and Borho ( Math. Ann. 209 (1974) 183–193) proved that for any fixed positive integer $K$, there are only finitely many relatively prime amicable pairs $(M,N)$ with $\omega (MN)=K$. Recently, Pollack ( Mosc. J. Comb. Number Theory 5 (2015), 36–51) obtained an upper bound \begin{equation*}MN<(2K)^{2^{K^{2}}}\end{equation*} for such amicable pairs. In this paper, we improve this upper bound to \begin{equation*}MN<\frac{\pi^{2}}{6}2^{4^{K}-2\cdot 2^{K}}\end{equation*} and generalize this bound to some class of general amicable tuples.
</p>projecteuclid.org/euclid.ijm/1552442661_20190312220428Tue, 12 Mar 2019 22:04 EDTAbstract key polynomials and comparison theorems with the key polynomials of Mac Lane–Vaquiéhttps://projecteuclid.org/euclid.ijm/1552442662<strong>J. Decaup</strong>, <strong>W. Mahboub</strong>, <strong>M. Spivakovsky</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 253--270.</p><p><strong>Abstract:</strong><br/>
Let $(K,\nu)$ be a valued field and $K(x)$ a simple purely transcendental extension of $K$. In the nineteen thirties, in order to study the possible extensions of $\nu $ to $K(x)$, S. Mac Lane considered the special case when $\nu $ is discrete of rank $1$, and introduced the notion of key polynomials. M. Vaquié extended this definition to the case of arbitrary valuations.
In this paper we give a new definition of key polynomials (which we call abstract key polynomials ) and study the relationship between them and key polynomials of Mac Lane–Vaquié.
</p>projecteuclid.org/euclid.ijm/1552442662_20190312220428Tue, 12 Mar 2019 22:04 EDTSingular string polytopes and functorial resolutions from Newton–Okounkov bodieshttps://projecteuclid.org/euclid.ijm/1552442663<strong>Megumi Harada</strong>, <strong>Jihyeon Jessie Yang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 271--292.</p><p><strong>Abstract:</strong><br/>
The main result of this paper is that the toric degenerations of flag and Schubert varieties associated to string polytopes and certain Bott–Samelson resolutions of flag and Schubert varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton–Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton–Okounkov bodies of Bott–Samelson varieties with respect to a certain valuation $\nu_{\mathrm{max}}$ coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton–Okounkov bodies of Bott–Samelson varieties with respect to a different valuation $\nu_{\mathrm{min}}$ in terms of Grossberg–Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton–Okounkov bodies coincide.
</p>projecteuclid.org/euclid.ijm/1552442663_20190312220428Tue, 12 Mar 2019 22:04 EDTThe profile decomposition for the hyperbolic Schrödinger equationhttps://projecteuclid.org/euclid.ijm/1552442664<strong>Benjamin Dodson</strong>, <strong>Jeremy L. Marzuola</strong>, <strong>Benoit Pausader</strong>, <strong>Daniel P. Spirn</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 293--320.</p><p><strong>Abstract:</strong><br/>
In this note, we prove the profile decomposition for hyperbolic Schrödinger (or mixed signature) equations on $\mathbb{R}^{2}$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot{H}}^{\frac{1}{2}}$ critical problem. Then, we give the derivation of the profile decomposition in the mass-critical case based on an estimate of Rogers-Vargas ( J. Functional Anal. 241 (2) (2006), 212–231).
</p>projecteuclid.org/euclid.ijm/1552442664_20190312220428Tue, 12 Mar 2019 22:04 EDTInvariant CR mappings between hyperquadricshttps://projecteuclid.org/euclid.ijm/1552442665<strong>Dusty Grundmeier</strong>, <strong>Kemen Linsuain</strong>, <strong>Brendan Whitaker</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 321--340.</p><p><strong>Abstract:</strong><br/>
We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to D’Angelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric for all finite subgroups of $SU(1,1)$. We also extend combinatorial results proven by Loehr, Warrington, and Wilf on determinants of sparse circulant determinants. We apply these results to study CR mappings invariant under finite subgroups of $U(1,1)$.
</p>projecteuclid.org/euclid.ijm/1552442665_20190312220428Tue, 12 Mar 2019 22:04 EDTMultiplicative structure in stable expansions of the group of integershttps://projecteuclid.org/euclid.ijm/1552442666<strong>Gabriel Conant</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 341--364.</p><p><strong>Abstract:</strong><br/>
We define two families of expansions of $(\mathbb{Z},+)$ by unary predicates, and prove that their theories are superstable of $U$-rank $\omega $. The first family consists of expansions $(\mathbb{Z},+,A)$, where $A$ is an infinite subset of a finitely generated multiplicative submonoid of $\mathbb{Z}^{+}$. Using this result, we also prove stability for the expansion of $(\mathbb{Z},+)$ by all unary predicates of the form $\{q^{n}:n\in \mathbb{N}\}$ for some $q\in \mathbb{N}_{\geq 2}$. The second family consists of sets $A\subseteq \mathbb{N}$ which grow asymptotically close to a $\mathbb{Q}$-linearly independent increasing sequence $(\lambda_{n})_{n=0}^{\infty }\subseteq\mathbb{R}^{+}$ such that $\{\frac{\lambda_{n}}{\lambda_{m}}:m\leq n\}$ is closed and discrete.
</p>projecteuclid.org/euclid.ijm/1552442666_20190312220428Tue, 12 Mar 2019 22:04 EDTThe rate of convergence on Schrödinger operatorhttps://projecteuclid.org/euclid.ijm/1552442667<strong>Zhenbin Cao</strong>, <strong>Dashan Fan</strong>, <strong>Meng Wang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 365--380.</p><p><strong>Abstract:</strong><br/>
Recently, Du, Guth and Li showed that the Schrödinger operator $e^{it\Delta }$ satisfies $\lim_{t\rightarrow 0}e^{it\Delta }f=f$ almost everywhere for all $f\in H^{s}(\mathbb{R}^{2})$, provided that $s>1/3$. In this paper, we discuss the rate of convergence on $e^{it\Delta }(f)$ by assuming more regularity on $f$. At $n=2$, our result can be viewed as an application of the Du–Guth–Li theorem. We also address the same issue on the cases $n=1$ and $n>2$.
</p>projecteuclid.org/euclid.ijm/1552442667_20190312220428Tue, 12 Mar 2019 22:04 EDTConcerning $q$-summable Szlenk indexhttps://projecteuclid.org/euclid.ijm/1552442668<strong>Ryan M. Causey</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 381--426.</p><p><strong>Abstract:</strong><br/>
For each ordinal $\xi$ and each $1\leqslant q<\infty$, we define the notion of $\xi$-$q$-summable Szlenk index. When $\xi=0$ and $q=1$, this recovers the usual notion of summable Szlenk index. We define for an arbitrary weak$^{*}$-compact set a transfinite, asymptotic analogue $\alpha_{\xi,p}$ of the martingale type norm of an operator. We prove that this quantity is determined by norming sets and determines $\xi$-Szlenk power type and $\xi$-$q$-summability of Szlenk index. This fact allows us to prove that the behavior of operators under the $\alpha_{\xi,p}$ seminorms passes in the strongest way to injective tensor products of Banach spaces. Furthermore, we combine this fact with a result of Schlumprecht to prove that a separable Banach space with good behavior with respect to the $\alpha_{\xi,p}$ seminorm can be embedded into a Banach space with a shrinking basis and the same behavior under $\alpha_{\xi,p}$, and in particular it can be embedded into a Banach space with a shrinking basis and the same $\xi$-Szlenk power type. Finally, we completely elucidate the behavior of the $\alpha_{\xi,p}$ seminorms under $\ell_{r}$ direct sums. This allows us to give an alternative proof of a result of Brooker regarding Szlenk indices of $\ell_{p}$ and $c_{0}$ direct sums of operators.
</p>projecteuclid.org/euclid.ijm/1552442668_20190312220428Tue, 12 Mar 2019 22:04 EDTExplicit bounds for primes in arithmetic progressionshttps://projecteuclid.org/euclid.ijm/1552442669<strong>Michael A. Bennett</strong>, <strong>Greg Martin</strong>, <strong>Kevin O’Bryant</strong>, <strong>Andrew Rechnitzer</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 62, Number 1-4, 427--532.</p><p><strong>Abstract:</strong><br/>
We derive explicit upper bounds for various counting functions for primes in arithmetic progressions. By way of example, if $q$ and $a$ are integers with $\mathop{\mathrm{gcd}}\nolimits (a,q)=1$ and $3\leq q\leq10^{5}$, and $\theta(x;q,a)$ denotes the sum of the logarithms of the primes $p\equiv a\ (\operatorname{mod}q)$ with $p\leq x$, we show that
\[\vert \theta(x;q,a)-{x}/{\varphi(q)}\vert <\frac{1}{160}\frac{x}{\log x}\] for all $x\geq8\cdot10^{9}$, with significantly sharper constants obtained for individual moduli $q$. We establish inequalities of the same shape for the other standard prime-counting functions $\pi(x;q,a)$ and $\psi(x;q,a)$, as well as inequalities for the $n$th prime congruent to $a\ (\operatorname{mod}q)$ when $q\le1200$. For moduli $q>10^{5}$, we find even stronger explicit inequalities, but only for much larger values of $x$. Along the way, we also derive an improved explicit lower bound for $L(1,\chi)$ for quadratic characters $\chi$, and an improved explicit upper bound for exceptional zeros.
</p>projecteuclid.org/euclid.ijm/1552442669_20190312220428Tue, 12 Mar 2019 22:04 EDTExplicit versions of the local duality theorem in ${\mathbb{C}}^{n}$https://projecteuclid.org/euclid.ijm/1559116821<strong>Richard Lärkäng</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 1--45.</p><p><strong>Abstract:</strong><br/>
We consider versions of the local duality theorem in ${\mathbb{C}}^{n}$ . We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff–Herrera and Andersson–Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by Passare, Lejeune-Jalabert and Lundqvist.
</p>projecteuclid.org/euclid.ijm/1559116821_20190529040039Wed, 29 May 2019 04:00 EDTExtreme points and saturated polynomialshttps://projecteuclid.org/euclid.ijm/1559116822<strong>Greg Knese</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 47--74.</p><p><strong>Abstract:</strong><br/>
We consider the problem of characterizing the extreme points of the set of analytic functions $f$ on the bidisk with positive real part and $f(0)=1$ . If one restricts to those $f$ whose Cayley transform is a rational inner function, one gets a more tractable problem. We construct families of such $f$ that are extreme points and conjecture that these are all such extreme points. These extreme points are constructed from polynomials dubbed $\mathbb{T}^{2}$ -saturated, which roughly speaking means they have no zeros in the bidisk and as many zeros as possible on the boundary without having infinitely many zeros.
</p>projecteuclid.org/euclid.ijm/1559116822_20190529040039Wed, 29 May 2019 04:00 EDTExponential mixing for SPDEs driven by highly degenerate Lévy noiseshttps://projecteuclid.org/euclid.ijm/1559116823<strong>Xiaobin Sun</strong>, <strong>Yingchao Xie</strong>, <strong>Lihu Xu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 75--102.</p><p><strong>Abstract:</strong><br/>
By modifying a coupling method developed by the third author with much more delicate analysis, we prove that a family of stochastic partial differential equations (SPDEs) driven by highly degenerate pure jump Lévy noises are exponential mixing. These pure jump Lévy noises include a finite dimensional $\alpha $ -stable process with $\alpha \in (0,2)$ .
</p>projecteuclid.org/euclid.ijm/1559116823_20190529040039Wed, 29 May 2019 04:00 EDTA cancellation theorem for generalized Swan moduleshttps://projecteuclid.org/euclid.ijm/1559116824<strong>F. E. A. Johnson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 103--125.</p><p><strong>Abstract:</strong><br/>
The module cancellation problem asks whether, given modules $X$ , $X^{\prime}$ and $Y$ over a ring $\Lambda$ , the existence of an isomorphism $X\oplus Y\cong X^{\prime}\oplus Y$ implies that $X\cong X^{\prime}$ . When $\Lambda$ is the integral group ring of a metacyclic group $G(p,q)$ , results of Klingler show that the answer to this question is generally negative. By contrast, in this case we show that cancellation holds when $Y=\Lambda$ and $X$ is a generalized Swan module.
</p>projecteuclid.org/euclid.ijm/1559116824_20190529040039Wed, 29 May 2019 04:00 EDTIntersection homology: General perversities and topological invariancehttps://projecteuclid.org/euclid.ijm/1559116825<strong>David Chataur</strong>, <strong>Martintxo Saralegi-Aranguren</strong>, <strong>Daniel Tanré</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 127--163.</p><p><strong>Abstract:</strong><br/>
Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent perversities with some growth conditions, verifying $\overline{p}(1)=\overline{p}(2)=0$ . King reproves this invariance by associating an intrinsic pseudomanifold $X^{*}$ to any pseudomanifold $X$ . His proof consists of an isomorphism between the associated intersection homologies $H^{\overline{p}}_{*}(X)\cong H^{\overline{p}}_{*}(X^{*})$ for any perversity $\overline{p}$ with the same growth conditions verifying $\overline{p}(1)\geq 0$ .
In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, $\overline{p}$ , which corresponds to the classical topological invariance if $\overline{p}$ is a GM-perversity. We also extend it to the tame intersection homology, a variation of the intersection homology, particularly suited for “large” perversities, if there is no singular strata on $X$ becoming regular in $X^{*}$ . In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification.
</p>projecteuclid.org/euclid.ijm/1559116825_20190529040039Wed, 29 May 2019 04:00 EDTOn semidualizing modules of ladder determinantal ringshttps://projecteuclid.org/euclid.ijm/1559116826<strong>Sean Sather-Wagstaff</strong>, <strong>Tony Se</strong>, <strong>Sandra Spiroff</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 1, 165--191.</p><p><strong>Abstract:</strong><br/>
We identify all semidualizing modules over certain classes of ladder determinantal rings over a field $\mathsf{k}$ . Specifically, given a ladder of variables $Y$ , we show that the ring $\mathsf{k}[Y]/I_{t}(Y)$ has only trivial semidualizing modules up to isomorphism in the following cases: (1) $Y$ is a one-sided ladder, and (2) $Y$ is a two-sided ladder with $t=2$ and no coincidental inside corners.
</p>projecteuclid.org/euclid.ijm/1559116826_20190529040039Wed, 29 May 2019 04:00 EDT