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 EDTSufficient conditions for Strassen’s additivity conjecturehttp://projecteuclid.org/euclid.ijm/1488186021<strong>Zach Teitler</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1071--1085.</p><p><strong>Abstract:</strong><br/>
We give a sufficient condition for the strong symmetric version of Strassen’s additivity conjecture: the Waring rank of a sum of forms in independent variables is the sum of their ranks, and every Waring decomposition of the sum is a sum of decompositions of the summands. We give additional sufficient criteria for the additivity of Waring ranks and a sufficient criterion for additivity of cactus ranks and decompositions.
</p>projecteuclid.org/euclid.ijm/1488186021_20170227040113Mon, 27 Feb 2017 04:01 ESTLong turns, INP’s and indices for free group automorphismshttp://projecteuclid.org/euclid.ijm/1488186022<strong>Thierry Coulbois</strong>, <strong>Martin Lustig</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1087--1109.</p><p><strong>Abstract:</strong><br/>
The goal of this paper is to introduce a new tool, called long turns , which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs in a train track map and hence the index of the induced automorphism.
</p>projecteuclid.org/euclid.ijm/1488186022_20170227040113Mon, 27 Feb 2017 04:01 ESTIndex realization for automorphisms of free groupshttp://projecteuclid.org/euclid.ijm/1488186023<strong>Thierry Coulbois</strong>, <strong>Martin Lustig</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 59, Number 4, 1111--1128.</p><p><strong>Abstract:</strong><br/>
For any surface $\Sigma$ of genus $g\geq1$ and (essentially) any collection of positive integers $i_{1},i_{2},\ldots,i_{\ell}$ with $i_{1}+\cdots+i_{\ell}=4g-4$ Masur and Smillie ( Comment. Math. Helv. 68 (1993) 289–307) have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma\to\Sigma$ with precisely $\ell$ singularities $S_{1},\ldots,S_{\ell}$ in its stable foliation $\mathcal{L}$, such that $\mathcal{L}$ has precisely $i_{k}+2$ separatrices raying out from each $S_{k}$.
In this paper, we prove the analogue of this result for automorphisms of a free group ${F}_{N}$, where “pseudo-Anosov homeomorphism” is replaced by “fully irreducible automorphism” and the Gauss–Bonnet equality $i_{1}+\cdots+i_{\ell}=4g-4$ is replaced by the index inequality $i_{1}+\cdots+i_{\ell}\leq2N-2$ from ( Duke Math. J. 93 (1998) 425–452).
</p>projecteuclid.org/euclid.ijm/1488186023_20170227040113Mon, 27 Feb 2017 04:01 ESTWolfgang Haken: A biographical sketchhttp://projecteuclid.org/euclid.ijm/1498032019<strong>Ilya Kapovich</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 3--9.</p>projecteuclid.org/euclid.ijm/1498032019_20170621040059Wed, 21 Jun 2017 04:00 EDTOn homotopy 3-sphereshttp://projecteuclid.org/euclid.ijm/1498032020<strong>Wolfgang Haken</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 11--30.</p>projecteuclid.org/euclid.ijm/1498032020_20170621040059Wed, 21 Jun 2017 04:00 EDTFour-dimensional Haken cobordism theoryhttp://projecteuclid.org/euclid.ijm/1498032021<strong>Bell Foozwell</strong>, <strong>Hyam Rubinstein</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 1--17.</p><p><strong>Abstract:</strong><br/>
Cobordism of Haken $n$-manifolds is defined by a Haken $(n+1)$-manifold $W$ whose boundary has two components, each of which is a closed Haken $n$-manifold. In addition, the inclusion map of the fundamental group of each boundary component to $\pi_{1}(W)$ is injective. In this paper, we prove that there are $4$-dimensional Haken cobordisms whose boundary consists of any two closed Haken $3$-manifolds. In particular, each closed Haken $3$-manifold is the $\pi_{1}$-injective boundary of some Haken $4$-manifold.
</p>projecteuclid.org/euclid.ijm/1498032021_20170621040059Wed, 21 Jun 2017 04:00 EDTStrip maps of small surfaces are convexhttp://projecteuclid.org/euclid.ijm/1498032022<strong>François Guéritaud</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 19--37.</p><p><strong>Abstract:</strong><br/>
The strip map is a natural map from the arc complex of a bordered hyperbolic surface $S$ to the vector space of infinitesimal deformations of $S$. We prove that the image of the strip map is a convex hypersurface when $S$ is a surface of small complexity: the punctured torus or thrice punctured sphere.
</p>projecteuclid.org/euclid.ijm/1498032022_20170621040059Wed, 21 Jun 2017 04:00 EDTThin surface subgroups in cocompact lattices in $\operatorname{SL}(3,\mathbf{R})$http://projecteuclid.org/euclid.ijm/1498032023<strong>D. D. Long</strong>, <strong>A. W. Reid</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 39--53.</p><p><strong>Abstract:</strong><br/>
We show certain cocompact lattices in $\operatorname{SL}(3,\mathbf{R})$ contain closed surface groups. With further restrictions, we exhibit such lattices containing infinitely many commensurability classes of closed surface groups.
</p>projecteuclid.org/euclid.ijm/1498032023_20170621040059Wed, 21 Jun 2017 04:00 EDTThe $SL(3,\mathbb{C})$-character variety of the figure eight knothttp://projecteuclid.org/euclid.ijm/1498032024<strong>Michael Heusener</strong>, <strong>Vicente Muñoz</strong>, <strong>Joan Porti</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 55--98.</p><p><strong>Abstract:</strong><br/>
We give explicit equations that describe the character variety of the figure eight knot for the groups $\mathrm{SL}(3,\mathbb{C})$, $\mathrm{GL}(3,\mathbb{C})$ and $\mathrm{PGL}(3,\mathbb{C})$. For any of these $G$, it has five components of dimension $2$, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $\mathrm{SL}(2,\mathbb{C})$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.
</p>projecteuclid.org/euclid.ijm/1498032024_20170621040059Wed, 21 Jun 2017 04:00 EDTEmbedding of groups and quadratic equations over groupshttp://projecteuclid.org/euclid.ijm/1498032025<strong>D. F. Cummins</strong>, <strong>S. V. Ivanov</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 99--115.</p><p><strong>Abstract:</strong><br/>
We prove that, for every integer $n\ge2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, that is, every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.
</p>projecteuclid.org/euclid.ijm/1498032025_20170621040059Wed, 21 Jun 2017 04:00 EDTOne-domination of knotshttp://projecteuclid.org/euclid.ijm/1498032026<strong>M. Boileau</strong>, <strong>S. Boyer</strong>, <strong>D. Rolfsen</strong>, <strong>S. C. Wang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 117--139.</p><p><strong>Abstract:</strong><br/>
We say that a knot $k_{1}$ in the $3$-sphere $1$-dominates another $k_{2}$ if there is a proper degree 1 map $E(k_{1})\to E(k_{2})$ between their exteriors, and write $k_{1}\ge k_{2}$. When $k_{1}\ge k_{2}$ but $k_{1}\ne k_{2}$ we write $k_{1}>k_{2}$. One expects in the latter eventuality that $k_{1}$ is more complicated . In this paper, we produce various sorts of evidence to support this philosophy.
</p>projecteuclid.org/euclid.ijm/1498032026_20170621040059Wed, 21 Jun 2017 04:00 EDTTori and Heegaard splittingshttp://projecteuclid.org/euclid.ijm/1498032027<strong>Abigail Thompson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 141--148.</p><p><strong>Abstract:</strong><br/>
In Studies in modern topology (1968) 39–98 Prentice Hall, Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the “interesting” surface increases from zero, more complicated phenomena occur. Kobayashi ( Osaka J. Math. 24 (1987) 173–215) showed that if a 3-manifold $M^{3}$ contains an essential torus $T$, then it contains one which can be isotoped to intersect a (strongly irreducible) Heegaard splitting surface $F$ in a collection of simple closed curves which are essential in $T$ and in $F$. In general, there is no global bound on the number of curves in this collection. We show that given a 3-manifold $M$, a minimal genus, strongly irreducible Heegaard surface $F$ for $M$, and an essential torus $T$, we can either restrict the number of curves of intersection of $T$ with $F$ (to four), find a different essential surface and minimal genus Heegaard splitting with at most four essential curves of intersection, find a thinner decomposition of $M$, or produce a small Seifert-fibered piece of $M$.
</p>projecteuclid.org/euclid.ijm/1498032027_20170621040059Wed, 21 Jun 2017 04:00 EDTWolfgang Haken and the four-color problemhttp://projecteuclid.org/euclid.ijm/1498032028<strong>Robin Wilson</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 149--178.</p><p><strong>Abstract:</strong><br/>
In 1852, Augustus De Morgan, Professor of Mathematics at University College, London, was asked: Can every map be colored with just four colors in such a way that neighboring countries are colored differently? Over a century later, in a controversial proof that made substantial use of a computer, Wolfgang Haken and Kenneth Appel of the University of Illinois answered the question in the affirmative. But how did Haken come to be involved with the problem, and what was his role in its solution?
</p>projecteuclid.org/euclid.ijm/1498032028_20170621040059Wed, 21 Jun 2017 04:00 EDTA knot without a nonorientable essential spanning surfacehttp://projecteuclid.org/euclid.ijm/1498032029<strong>Nathan M. Dunfield</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 179--184.</p><p><strong>Abstract:</strong><br/>
This note gives the first example of a hyperbolic knot in the 3-sphere that lacks a nonorientable essential spanning surface; this disproves the Strong Neuwirth Conjecture formulated by Ozawa and Rubinstein. Moreover, this knot has no even strict boundary slopes, disproving the Even Boundary Slope Conjecture of the same authors. The proof is a rigorous calculation using Thurston’s spun-normal surfaces in the spirit of Haken’s original normal surface algorithms.
</p>projecteuclid.org/euclid.ijm/1498032029_20170621040059Wed, 21 Jun 2017 04:00 EDTFour color theorem from three points of viewhttp://projecteuclid.org/euclid.ijm/1498032030<strong>Yuri Matiyasevich</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 185--205.</p><p><strong>Abstract:</strong><br/>
The Four Color Conjecture, which in 1977 became the Four Color Theorem of Kenneth Appel and Wolfgang Haken, is famous for the number of its reformulations. Three of them found by the author at different time are discussed in this paper.
</p>projecteuclid.org/euclid.ijm/1498032030_20170621040059Wed, 21 Jun 2017 04:00 EDTProposed Property 2R counterexamples examinedhttp://projecteuclid.org/euclid.ijm/1498032031<strong>Martin Scharlemann</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 207--250.</p><p><strong>Abstract:</strong><br/>
In 1985, Akbulut and Kirby analyzed a homotopy $4$-sphere $\Sigma$ that was first discovered by Cappell and Shaneson, depicting it as a potential counterexample to three important conjectures, all of which remain unresolved. In 1991, Gompf’s further analysi showed that $\Sigma$ was one of an infinite collection of examples, all of which were (sadly) the standard $S^{4}$, but with an unusual handle structure.
Recent work with Gompf and Thompson, showed that the construction gives rise to a family $L_{n}$ of $2$-component links, each of which remains a potential counterexample to the generalized Property R Conjecture. In each $L_{n}$, one component is the simple square knot $Q$, and it was argued that the other component, after handle-slides, could in theory be placed very symmetrically. How to accomplish this was unknown, and that question is resolved here, in part by finding a symmetric construction of the $L_{n}$. In view of the continuing interest and potential importance of the Cappell-Shaneson-Akbulut-Kirby-Gompf examples (e.g., the original $\Sigma$ is known to embed very efficiently in $S^{4}$ and so provides unique insight into proposed approaches to the Schoenflies Conjecture) digressions into various aspects of this view are also included.
</p>projecteuclid.org/euclid.ijm/1498032031_20170621040059Wed, 21 Jun 2017 04:00 EDTA state calculus for graph coloringhttp://projecteuclid.org/euclid.ijm/1498032032<strong>Louis H. Kauffman</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 251--271.</p><p><strong>Abstract:</strong><br/>
This paper discusses reformulations of the problem of coloring plane maps with four colors. We give a number of alternate ways to formulate the coloring problem including a tautological expansion similar to the Penrose Bracket, and we give a simple extension of the Penrose Bracket that counts colorings of arbitrary cubic graphs presented as immersions in the plane.
</p>projecteuclid.org/euclid.ijm/1498032032_20170621040059Wed, 21 Jun 2017 04:00 EDTTutte relations, TQFT, and planarity of cubic graphshttp://projecteuclid.org/euclid.ijm/1498032033<strong>Ian Agol</strong>, <strong>Vyacheslav Krushkal</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 273--288.</p><p><strong>Abstract:</strong><br/>
It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.
</p>projecteuclid.org/euclid.ijm/1498032033_20170621040059Wed, 21 Jun 2017 04:00 EDTThe 3D-index and normal surfaceshttp://projecteuclid.org/euclid.ijm/1498032034<strong>Stavros Garoufalidis</strong>, <strong>Craig D. Hodgson</strong>, <strong>Neil R. Hoffman</strong>, <strong>J. Hyam Rubinstein</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 289--352.</p><p><strong>Abstract:</strong><br/>
Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in $q^{1/2}$ with integer coefficients. Our goal is to explain how the 3D-index is a generating series of normal surfaces associated to the ideal triangulation. This shows a connection of the 3D-index with classical normal surface theory, and fulfills a dream of constructing topological invariants of 3-manifolds using normal surfaces.
</p>projecteuclid.org/euclid.ijm/1498032034_20170621040059Wed, 21 Jun 2017 04:00 EDTBoundaries of Kleinian groupshttp://projecteuclid.org/euclid.ijm/1498032035<strong>Peter Haïssinsky</strong>, <strong>Luisa Paoluzzi</strong>, <strong>Genevieve Walsh</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 1, 353--364.</p><p><strong>Abstract:</strong><br/>
We review the theory of splittings of hyperbolic groups, as determined by the topology of the boundary. We give explicit examples of certain phenomena and then use this to describe limit sets of Kleinian groups up to homeomorphism.
</p>projecteuclid.org/euclid.ijm/1498032035_20170621040059Wed, 21 Jun 2017 04:00 EDTTwisted pseudo-differential operator on type I locally compact groupshttp://projecteuclid.org/euclid.ijm/1499760013<strong>H. Bustos</strong>, <strong>M. Măntoiu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 365--390.</p><p><strong>Abstract:</strong><br/>
Let $\mathsf{G}$ be a locally compact group satisfying some technical requirements and $\widehat{\sf{G}}$ its unitary dual. Using the theory of twisted crossed product $C^{*}$-algebras, we develop a twisted global quantization for symbols defined on $\mathsf{G}\times\widehat{\sf{G}}$ and taking operator values. The emphasis is on the representation-theoretic aspect. For nilpotent Lie groups, the connection is made with a scalar quantization of the cotangent bundle $T^{*}(\mathsf{G})$ and with a Quantum Mechanical theory of observables in the presence of variable magnetic fields.
</p>projecteuclid.org/euclid.ijm/1499760013_20170711040038Tue, 11 Jul 2017 04:00 EDTNew characterizations of Besov and Triebel–Lizorkin spaces via the $T1$ theoremhttp://projecteuclid.org/euclid.ijm/1499760014<strong>Fanghui Liao</strong>, <strong>Yanchang Han</strong>, <strong>Zongguang Liu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 391--412.</p><p><strong>Abstract:</strong><br/>
The main purpose of this paper is to provide new characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with the “reverse” doubling property. To achieve our goal, the key idea is to prove a $T1$ theorem with only half the usual smoothness and cancellation conditions.
</p>projecteuclid.org/euclid.ijm/1499760014_20170711040038Tue, 11 Jul 2017 04:00 EDTFixed-point index, the Incompatibility Theorem, and torus parametrizationhttp://projecteuclid.org/euclid.ijm/1499760015<strong>Andrey M. Mishchenko</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 413--445.</p><p><strong>Abstract:</strong><br/>
The fixed-point index of a homeomorphism of Jordan curves measures the number of fixed-points, with multiplicity, of the extension of the homeomorphism to the full Jordan domains in question. The now-classical Circle Index Lemma says that the fixed-point index of a positive-orientation-preserving homeomorphism of round circles is always non-negative. We begin by proving a generalization of this lemma, to accommodate Jordan curves bounding domains which do not disconnect each other. We then apply this generalization to give a new proof of Schramm’s Incompatibility Theorem, which was used by Schramm to give the first proof of the rigidity of circle packings filling the complex and hyperbolic planes. As an example application, we include outlines of proofs of these circle packing theorems.
We then introduce a new tool, the so-called torus parametrization, for working with fixed-point index, which allows some problems concerning this quantity to be approached combinatorially. We apply torus parametrization to give the first purely topological proof of the following lemma: given two positively oriented Jordan curves, one may essentially prescribe the images of three points of one of the curves in the other, and obtain an orientation-preserving homeomorphism between the curves, having non-negative fixed-point index, which respects this prescription. This lemma is essential to our proof of the Incompatibility Theorem.
</p>projecteuclid.org/euclid.ijm/1499760015_20170711040038Tue, 11 Jul 2017 04:00 EDTGrowth of some transversely homogeneous foliationshttp://projecteuclid.org/euclid.ijm/1499760016<strong>Jesús A. Álvarez López</strong>, <strong>Robert Wolak</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 447--457.</p><p><strong>Abstract:</strong><br/>
For transversely homogeneous foliations on compact manifolds whose global holonomy group has connected closure, it is shown that either all holonomy covers of the leaves have polynomial growth with degree bounded by a common constant, or all holonomy covers of the leaves have exponential growth. This is an extension of a recent answer given by Breuillard and Gelander to a question of Carrière. Examples of transversely projective foliations satisfying the above condition were constructed by Chihi and ben Ramdane.
</p>projecteuclid.org/euclid.ijm/1499760016_20170711040038Tue, 11 Jul 2017 04:00 EDTViscosity solutions, ends and ideal boundarieshttp://projecteuclid.org/euclid.ijm/1499760017<strong>Xiaojun Cui</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 459--480.</p><p><strong>Abstract:</strong><br/>
On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold $(M,g)$, there are three kinds of objects that have been studied extensively:
$\bullet $ Viscosity solutions to the Hamilton–Jacobi equation determined by the Riemannian metric;
$\bullet $ Ends introduced by Freudenthal and more general other remainders from compactification theory;
$\bullet $ Various kinds of ideal boundaries introduced by Gromov.
In this paper, we will present some initial relationship among these three kinds of objects and some related topics are also considered.
</p>projecteuclid.org/euclid.ijm/1499760017_20170711040038Tue, 11 Jul 2017 04:00 EDTProbabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension threehttp://projecteuclid.org/euclid.ijm/1499760018<strong>Chenmin Sun</strong>, <strong>Bo Xia</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 481--503.</p><p><strong>Abstract:</strong><br/>
In this article, by following the strategies in dealing with supercritical cubic and quintic wave equations in ( J. Eur. Math. Soc. (JEMS) 16 (2014) 1–30) and ( J. Math. Pures Appl. (9) 105 (2016) 342–366), we obtain that, the equation \begin{equation*}(\partial^{2}_{t}-\Delta)u+|u|^{p-1}u=0,\quad3<p<5\end{equation*} is almost surely global well-posed with initial data $(u(0),\partial_{t}u(0))\in H^{s}(\mathbb{T}^{3})\times H^{s-1}(\mathbb{T}^{3})$ for any $s\in(\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\frac{2}{p-1}$ for $3<p<5$.
</p>projecteuclid.org/euclid.ijm/1499760018_20170711040038Tue, 11 Jul 2017 04:00 EDTAmenability properties of the central Fourier algebra of a compact grouphttp://projecteuclid.org/euclid.ijm/1499760019<strong>Mahmood Alaghmandan</strong>, <strong>Nico Spronk</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 505--527.</p><p><strong>Abstract:</strong><br/>
We let the central Fourier algebra, $\operatorname{ZA}(G)$, be the subalgebra of functions $u$ in the Fourier algebra $\mathrm{A}(G)$ of a compact group, for which $u(xyx^{-1})=u(y)$ for all $x$, $y$ in $G$. We show that this algebra admits bounded point derivations whenever $G$ contains a non-Abelian closed connected subgroup. Conversely when $G$ is virtually Abelian, then $\operatorname{ZA}(G)$ is amenable. Furthermore, for virtually Abelian $G$, we establish which closed ideals admit bounded approximate identities. We also show that $\operatorname{ZA}(G)$ is weakly amenable, in fact hyper-Tauberian, exactly when $G$ admits no non-Abelian connected subgroup. We also study the amenability constant of $\operatorname{ZA}(G)$ for finite $G$ and exhibit totally disconnected groups $G$ for which $\operatorname{ZA}(G)$ is non-amenable. In passing, we establish some properties related to spectral synthesis of subsets of the spectrum of $\operatorname{ZA}(G)$.
</p>projecteuclid.org/euclid.ijm/1499760019_20170711040038Tue, 11 Jul 2017 04:00 EDTSome results on compact almost Ricci solitons with null Cotton tensorhttp://projecteuclid.org/euclid.ijm/1499760020<strong>A. Barros</strong>, <strong>I. Evangelista</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 529--540.</p><p><strong>Abstract:</strong><br/>
The aim of this paper is to prove that a compact almost Ricci soliton with null Cotton tensor is isometric to a standard sphere provided one of the following conditions associated to the Schouten tensor holds: the second symmetric function is constant and positive; two consecutive symmetric functions are non null multiple or some symmetric function is constant and the quoted tensor is positive.
</p>projecteuclid.org/euclid.ijm/1499760020_20170711040038Tue, 11 Jul 2017 04:00 EDTRational singularities and uniform symbolic topologieshttp://projecteuclid.org/euclid.ijm/1499760021<strong>Robert M. Walker</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 541--550.</p><p><strong>Abstract:</strong><br/>
Take $(R,\mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P\subseteq R$, the symbolic power $P^{(Da)}\subseteq P^{a}$ for all $a>0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^{2}$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.
</p>projecteuclid.org/euclid.ijm/1499760021_20170711040038Tue, 11 Jul 2017 04:00 EDTAlternate characterizations of bounded variation and of general monotonicity for functionshttp://projecteuclid.org/euclid.ijm/1499760022<strong>Barry Booton</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 551--561.</p><p><strong>Abstract:</strong><br/>
We find necessary and sufficient conditions for a function to be equal almost everywhere to a function of bounded variation. These results can be applied to broaden the class of general monotone functions.
</p>projecteuclid.org/euclid.ijm/1499760022_20170711040038Tue, 11 Jul 2017 04:00 EDTObstructions for compactness of Hankel operators: Compactness multipliershttp://projecteuclid.org/euclid.ijm/1499760023<strong>Mehmet Çelik</strong>, <strong>Yunus E. Zeytuncu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 563--585.</p><p><strong>Abstract:</strong><br/>
We establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators.
</p>projecteuclid.org/euclid.ijm/1499760023_20170711040038Tue, 11 Jul 2017 04:00 EDTExponential convergence for some SPDEs with Lévy noiseshttp://projecteuclid.org/euclid.ijm/1499760024<strong>Yulin Song</strong>, <strong>Tiange Xu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 587--611.</p><p><strong>Abstract:</strong><br/>
In this paper, we generalize the Malliavin calculus for jump processes in the infinite-dimensional setting and obtain an integration by parts formula for jump processes on Hilbert spaces. By using this formula, we investigate derivative formula and exponential convergence for SPDEs driven by purely jump processes.
</p>projecteuclid.org/euclid.ijm/1499760024_20170711040038Tue, 11 Jul 2017 04:00 EDTGeneralization of the Wiener–Ikehara theoremhttp://projecteuclid.org/euclid.ijm/1499760025<strong>Gregory Debruyne</strong>, <strong>Jasson Vindas</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 2, 613--624.</p><p><strong>Abstract:</strong><br/>
We study the Wiener–Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener–Ikehara theorem, that is, in “if and only if” versions of this theorem.
</p>projecteuclid.org/euclid.ijm/1499760025_20170711040038Tue, 11 Jul 2017 04:00 EDTCommon hypercyclic vectors for certain families of differential operatorshttps://projecteuclid.org/euclid.ijm/1506067283<strong>N. Tsirivas</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 625--640.</p><p><strong>Abstract:</strong><br/>
Let $(k_{n})$ be a strictly increasing sequence of positive integers. If $\sum_{n=1}^{+\infty}\frac{1}{k_{n}}$ $=+\infty$, we establish the existence of an entire function $f$ such that for every $\lambda\in(0,+\infty)$ the set $\{\lambda^{k_{n}}f^{(k_{n})}(\lambda z):n=1,2,\ldots\}$ is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane. This provides the best possible strengthened version of a corresponding result due to Costakis and Sambarino ( Adv. Math. 182 (2004) 278–306). From this, and using a non-trivial result of Weyl which concerns the uniform distribution modulo $1$ of certain sequences, we also derive an entire function $g$ such that for every $\lambda\in J$ the set $\{\lambda^{k_{n}}g^{(k_{n})}(\lambda z):n=1,2,\ldots\}$ is dense in the space of entire functions, where $J$ is “almost” equal to the set of non-zero complex numbers. On the other hand, if $\sum_{n=1}^{+\infty}\frac{1}{k_{n}}<+\infty$ we show that the conclusions in the above results fail to hold.
</p>projecteuclid.org/euclid.ijm/1506067283_20170922040145Fri, 22 Sep 2017 04:01 EDTLocal-to-global rigidity of Bruhat–Tits buildingshttps://projecteuclid.org/euclid.ijm/1506067284<strong>Mikael De La Salle</strong>, <strong>Romain Tessera</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 641--654.</p><p><strong>Abstract:</strong><br/>
A vertex-transitive graph $X$ is called local-to-global rigid if there exists $R$ such that every other graph whose balls of radius $R$ are isometric to the balls of radius $R$ in $X$ is covered by $X$. Let $d\geq4$. We show that the $1$-skeleton of an affine Bruhat–Tits building of type $\widetilde{A}_{d-1}$ is local-to-global rigid if and only if the underlying field has characteristic $0$. For example, the Bruhat–Tits building of $\mathrm{SL}(d,\mathbf{F}_{p}(\!(t)\!))$ is not local-to-global rigid, while the Bruhat–Tits building of $\mathrm{SL}(d,\mathbf{Q}_{p})$ is local-to-global rigid.
</p>projecteuclid.org/euclid.ijm/1506067284_20170922040145Fri, 22 Sep 2017 04:01 EDTConstructions of exotic group $C$∗-algebrashttps://projecteuclid.org/euclid.ijm/1506067285<strong>Matthew Wiersma</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 655--667.</p><p><strong>Abstract:</strong><br/>
Let $\Gamma$ be a discrete group. When $\Gamma$ is nonamenable, the reduced and full group $C$∗-algebras differ and it is generally believed that there should be many intermediate $C$∗-algebras, however few examples are known. In this paper, we give new constructions and compare existing constructions of intermediate group $C$∗-algebras for both generic and specific groups $\Gamma$.
</p>projecteuclid.org/euclid.ijm/1506067285_20170922040145Fri, 22 Sep 2017 04:01 EDTOn the behavior of singularities at the $F$-pure thresholdhttps://projecteuclid.org/euclid.ijm/1506067286<strong>Eric Canton</strong>, <strong>Daniel J. Hernández</strong>, <strong>Karl Schwede</strong>, <strong>Emily E. Witt</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 669--685.</p><p><strong>Abstract:</strong><br/>
We provide a family of examples for which the $F$-pure threshold and the log canonical threshold of a polynomial are different, but such that the characteristic $p$ does not divide the denominator of the $F$-pure threshold (compare with an example of Mustaţă–Takagi–Watanabe). We then study the $F$-signature function in the case that either the $F$-pure threshold and log canonical threshold coincide, or that $p$ does not divide the denominator of the $F$-pure threshold. We show that the $F$-signature function behaves similarly in those two cases. Finally, we include an appendix that shows that the test ideal can still behave in surprising ways even when the $F$-pure threshold and log canonical threshold coincide.
</p>projecteuclid.org/euclid.ijm/1506067286_20170922040145Fri, 22 Sep 2017 04:01 EDTWeighted local Hardy spaces associated to Schrödinger operatorshttps://projecteuclid.org/euclid.ijm/1506067287<strong>Hua Zhu</strong>, <strong>Lin Tang</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 687--738.</p><p><strong>Abstract:</strong><br/>
In this paper, we characterize the weighted local Hardy spaces $h^{p}_{\rho}(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt’s weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space $h^{1}_{\rho}(\omega)$ and the weighted Hardy space $H^{1}_{\mathcal{L}}(\omega)$ associated to Schrödinger operators $\mathcal{L}$ with $\omega\in A_{1}^{\rho,\infty}(\mathbb{R}^{n})$. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^{p}_{\rho}(\omega)$ of quasi-Banach-valued sublinear operators.
</p>projecteuclid.org/euclid.ijm/1506067287_20170922040145Fri, 22 Sep 2017 04:01 EDTPurely infinite totally disconnected topological graph algebrashttps://projecteuclid.org/euclid.ijm/1506067288<strong>Hui Li</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 739--750.</p><p><strong>Abstract:</strong><br/>
We give a sufficient condition on totally disconnected topological graphs such that their associated topological graph algebras are purely infinite.
</p>projecteuclid.org/euclid.ijm/1506067288_20170922040145Fri, 22 Sep 2017 04:01 EDTStrong measure zero sets in Polish groupshttps://projecteuclid.org/euclid.ijm/1506067289<strong>Michael Hrušák</strong>, <strong>Jindřich Zapletal</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 751--760.</p><p><strong>Abstract:</strong><br/>
In the context of arbitrary Polish groups, we investigate the Galvin–Mycielski–Solovay characterization of strong measure zero sets as those sets for which a meager collection of right translates cannot cover the whole group.
</p>projecteuclid.org/euclid.ijm/1506067289_20170922040145Fri, 22 Sep 2017 04:01 EDTFatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open setshttps://projecteuclid.org/euclid.ijm/1506067290<strong>Hyunchul Park</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 761--790.</p><p><strong>Abstract:</strong><br/>
We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.
</p>projecteuclid.org/euclid.ijm/1506067290_20170922040145Fri, 22 Sep 2017 04:01 EDTNon-compact subsets of the Zariski space of an integral domainhttps://projecteuclid.org/euclid.ijm/1506067291<strong>Dario Spirito</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 60, Number 3-4, 791--809.</p><p><strong>Abstract:</strong><br/>
Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\operatorname{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\operatorname{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\operatorname{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.
</p>projecteuclid.org/euclid.ijm/1506067291_20170922040145Fri, 22 Sep 2017 04:01 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 EST