Illinois Journal of Mathematics Articles (Project 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 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 EDTCurvature inequalities and extremal operatorshttps://projecteuclid.org/euclid.ijm/1564646431<strong>Gadadhar Misra</strong>, <strong>Md. Ramiz Reza</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 193--217.</p><p><strong>Abstract:</strong><br/>
A curvature inequality is established for contractive commuting tuples of operators $\mathbf{T}$ in the Cowen–Douglas class $B_{n}(\Omega )$ of rank $n$ defined on some bounded domain $\Omega $ in $\mathbb{C}^{m}$ . Properties of the extremal operators (that is, the operators which achieve equality) are investigated. Specifically, a substantial part of a well-known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.
</p>projecteuclid.org/euclid.ijm/1564646431_20190801040055Thu, 01 Aug 2019 04:00 EDTPolynomial time relatively computable triangular arrays for almost sure convergencehttps://projecteuclid.org/euclid.ijm/1564646432<strong>Vladimir Dobrić†</strong>, <strong>Patricia Garmirian</strong>, <strong>Marina Skyers</strong>, <strong>Lee J. Stanley</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 219--257.</p><p><strong>Abstract:</strong><br/>
We start from a discrete random variable, $\mathbf{O}$ , defined on $(0,1)$ and taking on $2^{M+1}$ values with equal probability—any member of a certain family whose simplest member is the Rademacher random variable (with domain $(0,1)$ ), whose constant value on $(0,1/2)$ is $-1$ . We create (via left-shifts) independent copies, $\mathbf{X}_{i}$ , of $\mathbf{O}$ and let $\mathbf{S}_{n}:=\sum _{i=1}^{n}X_{i}$ . We let $\mathbf{S}^{*}_{n}$ be the quantile of $\mathbf{S}_{n}$ . If $\mathbf{O}$ is Rademacher, the sequence $\{\mathbf{S}_{n}\}$ is the equiprobable random walk on $\mathbb{Z}$ with domain $(0,1)$ . In the general case, $\mathbf{S}_{n}$ follows a multinomial distribution and as $\mathbf{O}$ varies over the family, the resulting family of multinomial distributions is sufficiently rich to capture the full generality of situations where the Central Limit Theorem applies.
The $\mathbf{X}_{1},\ldots ,\mathbf{X}_{n}$ provide a representation of $\mathbf{S}_{n}$ that is strong in that their sum is equal to $\mathbf{S}_{n}$ pointwise. They represent $\mathbf{S}^{*}_{n}$ only in distribution. Are there strong representations of $\mathbf{S}^{*}_{n}$ ? We establish the affirmative answer, and our proof gives a canonical bijection between, on the one hand, the set of all strong representations with the additional property of being trim and, on the other hand, the set of permutations, $\pi _{n}$ , of $\{0,\ldots,2^{n(M+1)}-1\}$ , with the property that we call admissibility . Passing to sequences, $\{\pi _{n}\}$ , of admissible permutations, these provide a complete classification of trim, strong triangular array representations of the sequence $\{\mathbf{S}^{*}_{n}\}$ . We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function $\tau ^{\mathbf{O}}_{1}$ which embodies the complexity of $\mathbf{O}$ itself. The trim, strong triangular array representation corresponding to the second of these is as close as possible to the representation of $\{\mathbf{S}_{n}\}$ provided by the $\mathbf{X}_{i}$ .
</p>projecteuclid.org/euclid.ijm/1564646432_20190801040055Thu, 01 Aug 2019 04:00 EDTThe Dirichlet problem for the constant mean curvature equation in $\operatorname{Sol}_{3}$https://projecteuclid.org/euclid.ijm/1564646433<strong>Patricía Klaser</strong>, <strong>Ana Menezes</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 259--297.</p><p><strong>Abstract:</strong><br/>
We prove a version of the Jenkins–Serrin theorem for the existence of constant mean curvature graphs over bounded domains with infinite boundary data in $\operatorname{Sol}_{3}$ . Moreover, we construct examples of admissible domains where the results may be applied.
</p>projecteuclid.org/euclid.ijm/1564646433_20190801040055Thu, 01 Aug 2019 04:00 EDTWhen the Zariski space is a Noetherian spacehttps://projecteuclid.org/euclid.ijm/1564646436<strong>Dario Spirito</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 299--316.</p><p><strong>Abstract:</strong><br/>
We characterize when the Zariski space $\operatorname{Zar}(K|D)$ (where $D$ is an integral domain, $K$ is a field containing $D$ , and $D$ is integrally closed in $K$ ) and the set $\operatorname{Zar}_{\mathrm{min}}(L|D)$ of its minimal elements are Noetherian spaces.
</p>projecteuclid.org/euclid.ijm/1564646436_20190801040055Thu, 01 Aug 2019 04:00 EDTK-theory and K-homology of finite wreath products with free groupshttps://projecteuclid.org/euclid.ijm/1564646437<strong>Sanaz Pooya</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 317--334.</p><p><strong>Abstract:</strong><br/>
This article investigates an explicit description of the Baum–Connes assembly map of the wreath product $\Gamma=F\wr\mathbb{F}_{n}=\bigoplus_{\mathbb{F}_{n}}F\rtimes\mathbb{F}_{n}$ , where $F$ is a finite and $\mathbb{F}_{n}$ is the free group on $n$ generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space $\underline{\mathrm{E}}\Gamma$ . As a result of our computations, we obtain that $\mathrm{K}_{0}(\mathrm{C}^{*}_{\mathrm{r}}(\Gamma))$ is the free abelian group of countable rank with a basis consisting of projections in $\mathrm{C}^{*}_{\mathrm{r}}(\bigoplus_{\mathbb{F}_{n}}F)$ , and $\mathrm{K}_{1}(\mathrm{C}^{*}_{\mathrm{r}}(\Gamma))$ is the free abelian group of rank $n$ with a basis represented by the unitaries coming from the free group.
</p>projecteuclid.org/euclid.ijm/1564646437_20190801040055Thu, 01 Aug 2019 04:00 EDTTwo generalizations of Auslander–Reiten duality and applicationshttps://projecteuclid.org/euclid.ijm/1564646438<strong>Arash Sadeghi</strong>, <strong>Ryo Takahashi</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 2, 335--351.</p><p><strong>Abstract:</strong><br/>
This paper extends Auslander–Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander–Reiten conjecture.
</p>projecteuclid.org/euclid.ijm/1564646438_20190801040055Thu, 01 Aug 2019 04:00 EDTMaximal displacement and population growth for branching Brownian motionshttps://projecteuclid.org/euclid.ijm/1568858864<strong>Yuichi Shiozawa</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 3, 353--402.</p><p><strong>Abstract:</strong><br/>
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the nonextinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
</p>projecteuclid.org/euclid.ijm/1568858864_20190918220801Wed, 18 Sep 2019 22:08 EDTAngular derivatives and semigroups of holomorphic functionshttps://projecteuclid.org/euclid.ijm/1568858865<strong>Nikolaos Karamanlis</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 3, 403--424.</p><p><strong>Abstract:</strong><br/>
A simply connected domain $\Omega \subset \mathbb{C}$ is convex in the positive direction if for every $z\in \Omega $ , the half-line $\{z+t:t\geq 0\}$ is contained in $\Omega$ . We provide necessary and sufficient conditions for the existence of an angular derivative at $\infty $ for domains convex in the positive direction which are contained either in a horizontal half-plane or in a horizontal strip. This class of domains arises naturally in the theory of semigroups of holomorphic functions, and the existence of an angular derivative has interesting consequences for the semigroup.
</p>projecteuclid.org/euclid.ijm/1568858865_20190918220801Wed, 18 Sep 2019 22:08 EDTBorcea–Voisin mirror symmetry for Landau–Ginzburg modelshttps://projecteuclid.org/euclid.ijm/1568858866<strong>Amanda Francis</strong>, <strong>Nathan Priddis</strong>, <strong>Andrew Schaug</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 3, 425--461.</p><p><strong>Abstract:</strong><br/>
Fan–Jarvis–Ruan–Witten theory is a formulation of physical Landau–Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau–Ginzburg/Calabi–Yau correspondence, several birational morphisms of Calabi–Yau orbifolds should correspond to isomorphisms in Fan–Jarvis–Ruan–Witten theory. In this paper, we exhibit some of these isomorphisms that are related to Borcea–Voisin mirror symmetry. In particular, we develop a modified version of Berglund–Hübsch–Krawitz mirror symmetry for certain Landau–Ginzburg models. Using these isomorphisms, we prove several interesting consequences in the corresponding geometries.
</p>projecteuclid.org/euclid.ijm/1568858866_20190918220801Wed, 18 Sep 2019 22:08 EDTCorrection and notes to the paper “A classification of Artin–Schreier defect extensions and characterizations of defectless fields”https://projecteuclid.org/euclid.ijm/1568858867<strong>Franz-Viktor Kuhlmann</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 3, 463--468.</p><p><strong>Abstract:</strong><br/>
We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.
</p>projecteuclid.org/euclid.ijm/1568858867_20190918220801Wed, 18 Sep 2019 22:08 EDTDistance sets over arbitrary finite fieldshttps://projecteuclid.org/euclid.ijm/1568858868<strong>Doowon Koh</strong>, <strong>Sujin Lee</strong>, <strong>Thang Pham</strong>, <strong>Chun-Yen Shen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 3, 469--484.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the Erdős distinct distances problem for Cartesian product sets in the setting of arbitrary finite fields. More precisely, let $\mathbb{F}_{q}$ be an arbitrary finite field and $A$ be a set in $\mathbb{F}_{q}$ . Suppose $|A\cap (aG)|\le |G|^{1/2}$ for any subfield $G$ and $a\in \mathbb{F}_{q}^{*}$ , then \begin{equation*}\vert \Delta _{\mathbb{F}_{q}}(A^{2})\vert =\vert (A-A)^{2}+(A-A)^{2}\vert \gg \vert A\vert ^{1+\frac{1}{21}}.\end{equation*} Using the same method, we also obtain some results on sum–product type problems.
</p>projecteuclid.org/euclid.ijm/1568858868_20190918220801Wed, 18 Sep 2019 22:08 EDTHartogs domains and the Diederich–Fornæss indexhttps://projecteuclid.org/euclid.ijm/1574154081<strong>Muhenned Abdulsahib</strong>, <strong>Phillip S. Harrington</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 485--511.</p><p><strong>Abstract:</strong><br/>
We study a geometric property of the boundary on Hartogs domains which can be used to find upper and lower bounds for the Diederich–Fornæ ss index. Using this property, we are able to show that under some reasonable hypotheses on the set of weakly pseudoconvex points, the Diederich–Fornæss index for a Hartogs domain is equal to one if and only if the domain admits a family of good vector fields in the sense of Boas and Straube. We also study the analogous problem for a Stein neighborhood basis and show that, under the same hypotheses, if the Diederich–Fornæss index for a Hartogs domain is equal to one, then the domain admits a Stein neighborhood basis.
</p>projecteuclid.org/euclid.ijm/1574154081_20191119040141Tue, 19 Nov 2019 04:01 ESTRational growth in virtually abelian groupshttps://projecteuclid.org/euclid.ijm/1574154082<strong>Alex Evetts</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 513--549.</p><p><strong>Abstract:</strong><br/>
We show that any subgroup of a finitely generated virtually abelian group $G$ grows rationally relative to $G$ , that the set of right cosets of any subgroup of $G$ grows rationally, and that the set of conjugacy classes of $G$ grows rationally. These results hold regardless of the choice of finite weighted generating set for $G$ .
</p>projecteuclid.org/euclid.ijm/1574154082_20191119040141Tue, 19 Nov 2019 04:01 ESTHypersurfaces with constant principal curvatures in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times \mathbb{R}$https://projecteuclid.org/euclid.ijm/1574154083<strong>Rosa M. B. Chaves</strong>, <strong>Eliane Santos</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 551--574.</p><p><strong>Abstract:</strong><br/>
In this paper, we classify the hypersurfaces in $\mathbb{S}^{n}\times\mathbb{R}$ and $\mathbb{H}^{n}\times \mathbb{R}$ , $n\neq 3$ , with $g$ distinct constant principal curvatures, $g\in \{1,2,3\}$ , where $\mathbb{S}^{n}$ and $\mathbb{H}^{n}$ denote the sphere and hyperbolic space of dimension $n$ , respectively. We prove that such hypersurfaces are isoparametric in those spaces. Furthermore, we find a necessary and sufficient condition for an isoparametric hypersurface in $\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{H}^{n}\times \mathbb{R}$ with flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\mathbb{R}^{n+2}\supset\mathbb{S}^{n}\times \mathbb{R}$ and $\mathbb{L}^{n+2}\supset\mathbb{H}^{n}\times \mathbb{R}$ , having constant principal curvatures.
</p>projecteuclid.org/euclid.ijm/1574154083_20191119040141Tue, 19 Nov 2019 04:01 ESTCalabi–Yau structures and special Lagrangian submanifolds of complexified symmetric spaceshttps://projecteuclid.org/euclid.ijm/1574154084<strong>Naoyuki Koike</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 575--600.</p><p><strong>Abstract:</strong><br/>
It is known that there exist Calabi–Yau structures on the complexifications of symmetric spaces of compact type. In this paper, we first construct explicit complete Ricci-flat Kaehler metrics (which give Calabi–Yau structures) for complexified symmetric spaces of arbitrary rank in terms of the Schwarz’s theorem. We consider the case where the Calabi–Yau structure arises from the generalized Stenzel metric. In the complexified symmetric spaces equipped with such a Calabi–Yau structure, we give constructions of special Lagrangian submanifolds of any given phase which are invariant under the actions of symmetric subgroups of the isometry group of the original symmetric space of compact type.
</p>projecteuclid.org/euclid.ijm/1574154084_20191119040141Tue, 19 Nov 2019 04:01 ESTThe zero sets of $\ell^{p}_{A}$ are nestedhttps://projecteuclid.org/euclid.ijm/1574154085<strong>Raymond Cheng</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 601--618.</p><p><strong>Abstract:</strong><br/>
It is shown by construction that if $1\lt p_{1}\lt p_{2}\lt \infty$ , then there exists a sequence of points in the open unit disk that is a zero set for $\ell^{p_{2}}_{A}$ but not for $\ell^{p_{1}}_{A}$ . The proof utilizes a zero set criterion for $\ell^{p}_{A}$ based on a notion of $p$ -inner functions.
</p>projecteuclid.org/euclid.ijm/1574154085_20191119040141Tue, 19 Nov 2019 04:01 ESTSimple factor dressings and Bianchi–Bäcklund transformationshttps://projecteuclid.org/euclid.ijm/1574154086<strong>Joseph Cho</strong>, <strong>Yuta Ogata</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 619--631.</p><p><strong>Abstract:</strong><br/>
In this paper, we directly show the known equivalence of simple factor dressings of extended frames and the classical Bianchi–Bäcklund transformations for constant mean curvature surfaces. In doing so, we show how the parameters of classical Bianchi–Bäcklund transformations can be incorporated into the simple factor dressings method.
</p>projecteuclid.org/euclid.ijm/1574154086_20191119040141Tue, 19 Nov 2019 04:01 ESTA remark on thickness of free-by-cyclic groupshttps://projecteuclid.org/euclid.ijm/1574154087<strong>Mark Hagen</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 63, Number 4, 633--643.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a free group of positive, finite rank and let $\Phi \in \operatorname{Aut}(F)$ be a polynomial-growth automorphism. Then $F\rtimes _{\Phi}\mathbb{Z}$ is strongly thick of order $\eta $ , where $\eta $ is the rate of polynomial growth of $\phi$ . This fact is implicit in work of Macura, whose results predate the notion of thickness. Therefore, in this note, we make the relationship between polynomial growth of and thickness explicit. Our result combines with a result independently due to Dahmani–Li, Gautero–Lustig, and Ghosh to show that free-by-cyclic groups admit relatively hyperbolic structures with thick peripheral subgroups.
</p>projecteuclid.org/euclid.ijm/1574154087_20191119040141Tue, 19 Nov 2019 04:01 ESTOn the homogeneous ergodic bilinear averages with Möbius and Liouville weightshttps://projecteuclid.org/euclid.ijm/1583463695<strong>E. H. el Abdalaoui</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 1--19.</p><p><strong>Abstract:</strong><br/>
It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero; that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},\nu)$ , and $a,b\in\mathbb{Z}$ , then for any $f,g\in L^{2}(X)$ , for almost all $x\in X$ ,
\[\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{\nu}(n)f(T^{an}x)g(T^{bn}x)\mathop{\rightarrow}_{N\rightarrow{+\infty}}0,\] where $\boldsymbol{\nu}$ is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan’s estimation. Moreover, we establish that if $T$ is weakly mixing and its restriction to its Pinsker algebra has singular spectrum, then for any integer $k\geq1$ , for any $f_{j}\in L^{\infty}(X)$ , $j=1,\ldots,k$ , for almost all $x\in X$ , we have
\[\frac{1}{N}\sum_{n=1}^{N}\boldsymbol{\nu}(n)\prod _{j=1}^{k}f_{j}(T^{nj}x)\mathop{\rightarrow}_{N\rightarrow{+\infty}}0.\]
</p>projecteuclid.org/euclid.ijm/1583463695_20200305220150Thu, 05 Mar 2020 22:01 ESTScattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equationshttps://projecteuclid.org/euclid.ijm/1583463696<strong>Xing Cheng</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 21--48.</p><p><strong>Abstract:</strong><br/>
We consider the Cauchy problem for the nonlinear Schrödinger (NLS) equation with double nonlinearities with opposite sign, with one term mass-critical and the other term mass-supercritical and energy-subcritical, which includes the well-known two-dimensional cubic-quintic NLS equation arising in the study of the boson gas with 2- and 3-body interactions. We prove global well-posedness and scattering in $H^{1}(\mathbb{R}^{d})$ below the threshold for nonradial data when $1\le d\le 4$ .
</p>projecteuclid.org/euclid.ijm/1583463696_20200305220150Thu, 05 Mar 2020 22:01 ESTExtending Huppert’s conjecture to almost simple groups of Lie typehttps://projecteuclid.org/euclid.ijm/1583463697<strong>Farrokh Shirjian</strong>, <strong>Ali Iranmanesh</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 49--69.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a finite group and $\mathrm{cd}(G)$ be the set of all irreducible complex character degrees of $G$ without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if $H$ is an almost simple group of Lie type with $\mathrm{cd}(G)=\mathrm{cd}(H)$ , then there exists an Abelian normal subgroup $A$ of $G$ such that $G/A\cong H$ . It is furthermore shown that $G$ is not necessarily the direct product of $H$ and $A$ . In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.
</p>projecteuclid.org/euclid.ijm/1583463697_20200305220150Thu, 05 Mar 2020 22:01 ESTClassification of Darboux transformations for operators of the form $\partial_{x}\partial_{y}+a\partial_{x}+b\partial_{y}+c$https://projecteuclid.org/euclid.ijm/1583463698<strong>Ekaterina Shemyakova</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 71--92.</p><p><strong>Abstract:</strong><br/>
Darboux transformations are nongroup-type symmetries of linear differential operators. One can define Darboux transformations algebraically by the intertwining relation $ML=L_{1}M$ or the intertwining relation $ML=L_{1}N$ in the cases when the former is too restrictive.
Here we show that Darboux transformations for operators of the form $L=\partial _{x}\partial _{y}+a\partial _{x}+b\partial _{y}+c$ (sometimes referred to as 2D Schrödinger operators or Laplace operators) are always compositions of atomic Darboux transformations of two different well-studied types of Darboux transformations, provided that the chain of Laplace transformations for the original operator is long enough.
</p>projecteuclid.org/euclid.ijm/1583463698_20200305220150Thu, 05 Mar 2020 22:01 ESTLocally conformally flat metrics on surfaces of general typehttps://projecteuclid.org/euclid.ijm/1583463699<strong>Mustafa Kalafat</strong>, <strong>Özgür Kelekçi</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 93--103.</p><p><strong>Abstract:</strong><br/>
We prove a nonexistence theorem for product-type manifolds. In particular, we show that the 4-manifold $\Sigma _{g}\times \Sigma _{h}$ obtained from the product of closed surfaces does not admit any locally conformally flat metric arising from discrete and faithful representations for genus $g\geq 2$ and $h\geq 1$ .
</p>projecteuclid.org/euclid.ijm/1583463699_20200305220150Thu, 05 Mar 2020 22:01 ESTParabolic vector bundles on Klein surfaceshttps://projecteuclid.org/euclid.ijm/1583463700<strong>Indranil Biswas</strong>, <strong>Florent Schaffhauser</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 105--118.</p><p><strong>Abstract:</strong><br/>
Given a discrete subgroup $\Gamma $ of finite co-volume of $\mathbf{PGL}(2,\mathbb{R})$ , we define and study parabolic vector bundles on the quotient $\Sigma $ of the (extended) hyperbolic plane by $\Gamma $ . If $\Gamma $ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of degree two of $\Sigma $ . We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in a natural bijective correspondence with the equivalence classes of real and quaternionic unitary representations of $\Gamma $ . Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.
</p>projecteuclid.org/euclid.ijm/1583463700_20200305220150Thu, 05 Mar 2020 22:01 ESTMaximum principles for generalized Schrödinger equationshttps://projecteuclid.org/euclid.ijm/1583463701<strong>Masayoshi Takeda</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 1, 119--139.</p><p><strong>Abstract:</strong><br/>
We define three function spaces related to a Schrödinger form and its semigroup: two are spaces of excessive functions defined through the Schrödinger semigroup, and one is the space of weak subsolutions defined through the Schrödinger form. We define the maximum principle for each space and prove the equivalence of three maximum principles. Moreover, we give a necessary and sufficient condition for each maximum principle in terms of the principal eigenvalue of time-changed processes.
</p>projecteuclid.org/euclid.ijm/1583463701_20200305220150Thu, 05 Mar 2020 22:01 ESTCoarse dimension and definable sets in expansions of the ordered real vector spacehttps://projecteuclid.org/euclid.ijm/1588298624<strong>Erik Walsberg</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 141--149.</p><p><strong>Abstract:</strong><br/>
Let $E\subseteq \mathbb{R}$ . Suppose there is an $s\gt 0$ such that \begin{equation*}\bigl\vert \bigl\{k\in \mathbb{Z},-m\leq k\leq m-1:[k,k+1]\cap E\neq\emptyset \bigr\}\bigr\vert \geq m^{s}\end{equation*} for all sufficiently large $m\in\mathbb{N}$ . Then there is an $n\in \mathbb{N}$ and a linear $T:\mathbb{R}^{n}\to \mathbb{R}$ such that $T(E^{n})$ is dense. As a corollary, we show that if $E$ is in addition nowhere dense, then $(\mathbb{R},\lt ,+,0,(x\mapsto \lambda x)_{\lambda \in \mathbb{R}},E)$ defines every bounded Borel subset of every $\mathbb{R}^{n}$ .
</p>projecteuclid.org/euclid.ijm/1588298624_20200430220348Thu, 30 Apr 2020 22:03 EDTBoundary behavior of the Carathéodory and Kobayashi–Eisenman volume elementshttps://projecteuclid.org/euclid.ijm/1588298625<strong>Diganta Borah</strong>, <strong>Debaprasanna Kar</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 151--168.</p><p><strong>Abstract:</strong><br/>
We study the boundary asymptotics of the Carathéodory and Kobayashi–Eisenman volume elements on smoothly bounded convex finite type domains and Levi corank one domains.
</p>projecteuclid.org/euclid.ijm/1588298625_20200430220348Thu, 30 Apr 2020 22:03 EDTReducing invariants and total reflexivityhttps://projecteuclid.org/euclid.ijm/1588298626<strong>Tokuji Araya</strong>, <strong>Olgur Celikbas</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 169--184.</p><p><strong>Abstract:</strong><br/>
Motivated by a recent result of Yoshino and the work of Bergh on reducible complexity, we introduce reducing versions of invariants of finitely generated modules over local rings. Our main result considers modules which have finite reducing Gorenstein dimension and determines a criterion for such modules to be totally reflexive in terms of the vanishing of Ext. Along the way, we give examples and applications, and in particular, prove that a Cohen–Macaulay local ring with canonical module is Gorenstein if and only if the canonical module has finite reducing Gorenstein dimension.
</p>projecteuclid.org/euclid.ijm/1588298626_20200430220348Thu, 30 Apr 2020 22:03 EDTOn the Banach algebra structure for $C^{(\mathbf{n})}$ of the bidisc and related topicshttps://projecteuclid.org/euclid.ijm/1588298627<strong>Ramiz Tapdıgoglu</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 185--197.</p><p><strong>Abstract:</strong><br/>
Let $C^{(\mathbf{n})}=C^{(\mathbf{n})}(\mathbb{D}\times \mathbb{D})$ be a Banach space of complex valued functions $f(x,y)$ that are continuous on the closed bidisc $\overline{\mathbb{D}\times \mathbb{D}}$ , where $\mathbb{D}=\{z\in \mathbb{C}:|z|\lt 1\}$ is the unit disc in the complex plane $\mathbb{C}$ and has $n$ th partial derivatives in $\mathbb{D}\times \mathbb{D}$ which can be extended to functions continuous on $\overline{\mathbb{D}\times \mathbb{D}}$ . The Duhamel product is defined on $C^{(\mathbf{n})}$ by the formula \begin{equation*}(f\circledast g)(z,w)=\frac{\partial ^{2}}{\partial z\partial w}\int _{0}^{z}\int _{0}^{w}f(z-u,w-v)g(u,v)\,dv\,du.\end{equation*} In the present paper we prove that $C^{(\mathbf{n})}(\mathbb{D}\times\mathbb{D})$ is a Banach algebra with respect to the Duhamel product $\circledast$ . This result extends some known results. We also investigate the structure of the set of all extended eigenvalues and extended eigenvectors of some double integration operator $W_{zw}$ . In particular, the commutant of the double integration operator $W_{zw}$ is also described.
</p>projecteuclid.org/euclid.ijm/1588298627_20200430220348Thu, 30 Apr 2020 22:03 EDTOn the converse law of large numbershttps://projecteuclid.org/euclid.ijm/1588298628<strong>H. Jerome Keisler</strong>, <strong>Yeneng Sun</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 199--225.</p><p><strong>Abstract:</strong><br/>
Given a triangular array with $m_{n}$ random variables in the $n$ th row and a growth rate $\{k_{n}\}_{n=1}^{\infty}$ with $\limsup_{n\to\infty}(k_{n}/m_{n})\lt 1$ , if the empirical distributions converge for any subarrays with the same growth rate, then the triangular array is asymptotically independent. In other words, if the empirical distribution of any $k_{n}$ random variables in the $n$ th row of the triangular array is asymptotically close in probability to the law of a randomly selected random variable among these $k_{n}$ random variables, then two randomly selected random variables from the $n$ th row of the triangular array are asymptotically close to being independent. This provides a converse law of large numbers by deriving asymptotic independence from a sample stability condition. It follows that a triangular array of random variables is asymptotically independent if and only if the empirical distributions converge for any subarrays with a given asymptotic density in $(0,1)$ . Our proof is based on nonstandard analysis, a general method arisen from mathematical logic, and Loeb measure spaces in particular.
</p>projecteuclid.org/euclid.ijm/1588298628_20200430220348Thu, 30 Apr 2020 22:03 EDTTwo weighted inequalities for operators associated to a critical radius functionhttps://projecteuclid.org/euclid.ijm/1588298629<strong>B. Bongioanni</strong>, <strong>E. Harboure</strong>, <strong>P. Quijano</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 227--259.</p><p><strong>Abstract:</strong><br/>
In the general framework of $\mathbb{R}^{d}$ equipped with Lebesgue measure and a critical radius function, we introduce several Hardy–Littlewood type maximal operators and related classes of weights. We prove appropriate two weighted inequalities for such operators as well as a version of Lerner’s inequality for a product of weights. With these tools we are able to prove factored weight inequalities for certain operators associated to the critical radius function. As it is known, the harmonic analysis arising from the Schrödinger operator $L=-\Delta+V$ , as introduced by Shen, is based on the use of a related critical radius function. When our previous result is applied to this case, it allows to show some inequalities with factored weights for all first and second order Schrödinger–Riesz transforms.
</p>projecteuclid.org/euclid.ijm/1588298629_20200430220348Thu, 30 Apr 2020 22:03 EDTZeros of derivatives of strictly nonreal meromorphic functionshttps://projecteuclid.org/euclid.ijm/1588298630<strong>J. K. Langley</strong>. <p><strong>Source: </strong>Illinois Journal of Mathematics, Volume 64, Number 2, 261--290.</p><p><strong>Abstract:</strong><br/>
A number of results are proved concerning the existence of nonreal zeros of derivatives of strictly nonreal meromorphic functions in the plane.
</p>projecteuclid.org/euclid.ijm/1588298630_20200430220348Thu, 30 Apr 2020 22:03 EDT