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1. CJM Online first

Borwein, Peter; Choi, Stephen; Ferguson, Ron; Jankauskas, Jonas
On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk
We investigate the numbers of complex zeros of Littlewood polynomials $p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$ of these types. We show that, if $n+1$ is a prime number, then for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore, we describe some cases when the ratios $N(p)/n$ and $U(p)/n$ have limits as $n \to \infty$ and find the corresponding limit values.

Keywords:Littlewood polynomials, zeros, complex roots
Categories:11R06, 11R09, 11C08

2. CJM Online first

Levandovskyy, Viktor; Shepler, Anne V.
Corrigendum to "Quantum Drinfeld Hecke Algebras"
The last example of the article contains an error which we correct. We also indicate some indices in Theorem 11.1 that were accidently transposed.

Keywords:quantum/skew polynomial rings, noncommutative Groebner bases
Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40

3. CJM 2013 (vol 66 pp. 525)

Berg, Chris; Bergeron, Nantel; Saliola, Franco; Serrano, Luis; Zabrocki, Mike
A Lift of the Schur and Hall-Littlewood Bases to Non-commutative Symmetric Functions
We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.

Keywords:Hall-Littlewood polynomial, symmetric function, quasisymmetric function, tableau
Category:05E05

4. CJM Online first

Levandovskyy, Viktor; Shepler, Anne V.
Quantum Drinfeld Hecke Algebras
We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré-Birkhoff-Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the PBW conditions.

Keywords:skew polynomial rings, noncommutative Gröbner bases, graded Hecke algebras, symplectic reflection algebras, Hochschild cohomology
Categories:16S36, 16S35, 16S80, 16W20, 16Z05, 16E40

5. CJM 2013 (vol 66 pp. 625)

Giambruno, Antonio; Mattina, Daniela La; Zaicev, Mikhail
Classifying the Minimal Varieties of Polynomial Growth
Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with $1$ over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t\lt k$). These varieties are the building blocks of general varieties of polynomial growth. It turns out that for $k\le 4$ there are only a finite number of varieties of polynomial growth $n^k$, but for each $k \gt 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them.

Keywords:T-ideal, polynomial identity, codimension, polynomial growth,
Categories:16R10, 16P90

6. CJM 2012 (vol 65 pp. 863)

Josuat-Vergès, Matthieu
Cumulants of the $q$-semicircular Law, Tutte Polynomials, and Heaps
The $q$-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings where $q$ follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called crossing graph. The case $q=0$ of these cumulants was studied by Lassalle using symmetric functions and hypergeometric series. We show that the underlying combinatorics is explained through the theory of heaps, which is Viennot's geometric interpretation of the Cartier-Foata monoid. This method also gives a general formula for the cumulants in terms of free cumulants.

Keywords:moments, cumulants, matchings, Tutte polynomials, heaps
Categories:05A18, 05C31, 46L54

7. CJM 2012 (vol 65 pp. 600)

Kroó, A.; Lubinsky, D. S.
Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.

Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20

8. CJM 2012 (vol 65 pp. 171)

Lyall, Neil; Magyar, Ákos
Optimal Polynomial Recurrence
Let $P\in\mathbb Z[n]$ with $P(0)=0$ and $\varepsilon\gt 0$. We show, using Fourier analytic techniques, that if $N\geq \exp\exp(C\varepsilon^{-1}\log\varepsilon^{-1})$ and $A\subseteq\{1,\dots,N\}$, then there must exist $n\in\mathbb N$ such that \[\frac{|A\cap (A+P(n))|}{N}\gt \left(\frac{|A|}{N}\right)^2-\varepsilon.\] In addition to this we also show, using the same Fourier analytic methods, that if $A\subseteq\mathbb N$, then the set of $\varepsilon$-optimal return times \[R(A,P,\varepsilon)=\left\{n\in \mathbb N \,:\,\delta(A\cap(A+P(n)))\gt \delta(A)^2-\varepsilon\right\}\] is syndetic for every $\varepsilon\gt 0$. Moreover, we show that $R(A,P,\varepsilon)$ is dense in every sufficiently long interval, in particular we show that there exists an $L=L(\varepsilon,P,A)$ such that \[\left|R(A,P,\varepsilon)\cap I\right| \geq c(\varepsilon,P)|I|\] for all intervals $I$ of natural numbers with $|I|\geq L$ and $c(\varepsilon,P)=\exp\exp(-C\,\varepsilon^{-1}\log\varepsilon^{-1})$.

Keywords:Sarkozy, syndetic, polynomial return times
Category:11B30

9. CJM 2012 (vol 64 pp. 318)

Ingram, Patrick
Cubic Polynomials with Periodic Cycles of a Specified Multiplier
We consider cubic polynomials $f(z)=z^3+az+b$ defined over $\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier $\lambda$. In the case $N=1$, there are infinitely many such objects, and in the case $N\geq 3$, only finitely many (subject to a mild assumption). The case $N=2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.

Keywords:cubic polynomials, periodic points, holomorphic dynamics
Category:37P35

10. CJM 2011 (vol 64 pp. 1036)

Koh, Doowon; Shen, Chun-Yen
Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields
In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

Keywords:extension problems, averaging operator, finite fields, Erdős-Falconer distance problems, homogeneous polynomial
Categories:42B05, 11T24, 52C17

11. CJM 2011 (vol 64 pp. 481)

Chamorro, Diego
Some Functional Inequalities on Polynomial Volume Growth Lie Groups
In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition.

Keywords:Sobolev inequalities, polynomial volume growth Lie groups
Category:22E30

12. CJM 2010 (vol 63 pp. 413)

Konvalinka, Matjaž; Skandera, Mark
Generating Functions for Hecke Algebra Characters
Certain polynomials in $n^2$ variables that serve as generating functions for symmetric group characters are sometimes called ($S_n$) character immanants. We point out a close connection between the identities of Littlewood--Merris--Watkins and Goulden--Jackson, which relate $S_n$ character immanants to the determinant, the permanent and MacMahon's Master Theorem. From these results we obtain a generalization of Muir's identity. Working with the quantum polynomial ring and the Hecke algebra $H_n(q)$, we define quantum immanants that are generating functions for Hecke algebra characters. We then prove quantum analogs of the Littlewood--Merris--Watkins identities and selected Goulden--Jackson identities that relate $H_n(q)$ character immanants to the quantum determinant, quantum permanent, and quantum Master Theorem of Garoufalidis--L\^e--Zeilberger. We also obtain a generalization of Zhang's quantization of Muir's identity.

Keywords:determinant, permanent, immanant, Hecke algebra character, quantum polynomial ring
Categories:15A15, 20C08, 81R50

13. CJM 2010 (vol 63 pp. 200)

Rahman, Mizan
An Explicit Polynomial Expression for a $q$-Analogue of the 9-$j$ Symbols
Using standard transformation and summation formulas for basic hypergeometric series we obtain an explicit polynomial form of the $q$-analogue of the 9-$j$ symbols, introduced by the author in a recent publication. We also consider a limiting case in which the 9-$j$ symbol factors into two Hahn polynomials. The same factorization occurs in another limit case of the corresponding $q$-analogue.

Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensions
Categories:33D45, 33D50

14. CJM 2010 (vol 63 pp. 181)

Ismail, Mourad E. H.; Obermaier, Josef
Characterizations of Continuous and Discrete $q$-Ultraspherical Polynomials
We characterize the continuous $q$-ultraspherical polynomials in terms of the special form of the coefficients in the expansion $\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$ being the Askey--Wilson divided difference operator. The polynomials are assumed to be symmetric, and the connection coefficients are multiples of the reciprocal of the square of the $L^2$ norm of the polynomials. A similar characterization is given for the discrete $q$-ultraspherical polynomials. A new proof of the evaluation of the connection coefficients for big $q$-Jacobi polynomials is given.

Keywords:continuous $q$-ultraspherical polynomials, big $q$-Jacobi polynomials, discrete $q$-ultra\-spherical polynomials, Askey--Wilson operator, $q$-difference operator, recursion coefficients
Categories:33D45, 42C05

15. CJM 2010 (vol 62 pp. 1058)

Chen, Yichao; Liu, Yanpei
On a Conjecture of S. Stahl
S. Stahl conjectured that the zeros of genus polynomials are real. In this note, we disprove this conjecture.

Keywords:genus polynomial, zeros, real
Category:05C10

16. CJM 2010 (vol 62 pp. 261)

Chiang, Yik-Man; Ismail, Mourad E. H.
Erratum to: On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
No abstract.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Problem
Categories:34M10, 33C15, 33C47

17. CJM 2009 (vol 61 pp. 351)

Graham, William; Hunziker, Markus
Multiplication of Polynomials on Hermitian Symmetric spaces and Littlewood--Richardson Coefficients
Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda$ denotes the isomorphism class of an irreducible representation of $K$, let $\rho_\lambda\from K \rightarrow GL(V_{\lambda})$ denote the corresponding irreducible representation and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write $S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of $S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible constituent of $V_\lambda\otimes V_\mu$, is it true that $S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood--Richardson rule.

Keywords:Hermitian symmetric spaces, multiplicity free actions, Littlewood--Richardson coefficients, Jack polynomials
Categories:14L30, 22E46

18. CJM 2008 (vol 60 pp. 1240)

Beliakova, Anna; Wehrli, Stephan
Categorification of the Colored Jones Polynomial and Rasmussen Invariant of Links
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce Khovanov and Lee homology theories categorifying the colored Jones polynomial. Further, we study conditions under which framed colored link cobordisms induce chain transformations between our formal brackets. We conjecture that for special choice of parameters, Khovanov and Lee homology theories of colored links are functorial (up to sign). Finally, we extend the Rasmussen invariant to links and give examples where this invariant is a stronger obstruction to sliceness than the multivariable Levine--Tristram signature.

Keywords:Khovanov homology, colored Jones polynomial, slice genus, movie moves, framed cobordism
Categories:57M25, 57M27, 18G60

19. CJM 2008 (vol 60 pp. 958)

Chen, Yichao
A Note on a Conjecture of S. Stahl
S. Stahl (Canad. J. Math. \textbf{49}(1997), no. 3, 617--640) conjectured that the zeros of genus polynomial are real. L. Liu and Y. Wang disproved this conjecture on the basis of Example 6.7. In this note, it is pointed out that there is an error in this example and a new generating matrix and initial vector are provided.

Keywords:genus polynomial, zeros, real
Categories:05C10, 05A15, 30C15, 26C10

20. CJM 2008 (vol 60 pp. 685)

Savu, Anamaria
Closed and Exact Functions in the Context of Ginzburg--Landau Models
For a general vector field we exhibit two Hilbert spaces, namely the space of so called \emph{closed functions} and the space of \emph{exact functions} and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field.

Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries
Categories:42B05, 81Q50, 42A16

21. CJM 2007 (vol 59 pp. 1223)

Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.
Calderón--Zygmund Operators Associated to Ultraspherical Expansions
We define the higher order Riesz transforms and the Littlewood--Paley $g$-function associated to the differential operator $L_\l f(\theta)=-f''(\theta)-2\l\cot\theta f'(\theta)+\l^2f(\theta)$. We prove that these operators are Calder\'{o}n--Zygmund operators in the homogeneous type space $((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted, $H^1-L^1$ and $L^\infty-BMO$ inequalities are obtained.

Keywords:ultraspherical polynomials, Calderón--Zygmund operators
Categories:42C05, 42C15frcs

22. CJM 2007 (vol 59 pp. 186)

Okoh, F.; Zorzitto, F.
Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields
Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$, arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer, $h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and $\alpha$ is a $K$-linear functional on the space $\krx$ of rational functions in one variable $X$. Every pair $(h,\alpha)$ comes with a polynomial $f$ in $K(X)[Y]$ called the regulator. When the module ${\mathcal M}$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$-algebra $\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For some height functions $h$ of infinite support $I$, the search for a functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes down to having functions $\eta\colon I\ar K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a purely-simple Kronecker module ${\mathcal M}$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\ktil$ and $h$ is finite-valued at least twice on $\ktil$. Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $\krx$ if and only if $r$ is a perfect square in the completions of $\krx$ with respect to all of its valuations.

Keywords:Purely simple Kronecker module, regulating polynomial, Laurent expansions, endomorphism algebra
Categories:16S50, 15A27

23. CJM 2006 (vol 58 pp. 726)

Chiang, Yik-Man; Ismail, Mourad E. H.
On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials
We show that the value distribution (complex oscillation) of solutions of certain periodic second order ordinary differential equations studied by Bank, Laine and Langley is closely related to confluent hypergeometric functions, Bessel functions and Bessel polynomials. As a result, we give a complete characterization of the zero-distribution in the sense of Nevanlinna theory of the solutions for two classes of the ODEs. Our approach uses special functions and their asymptotics. New results concerning finiteness of the number of zeros (finite-zeros) problem of Bessel and Coulomb wave functions with respect to the parameters are also obtained as a consequence. We demonstrate that the problem for the remaining class of ODEs not covered by the above ``special function approach" can be described by a classical Heine problem for differential equations that admit polynomial solutions.

Keywords:Complex Oscillation theory, Exponent of convergence of zeros, zero distribution of Bessel and Confluent hypergeometric functions, Lommel transform, Bessel polynomials, Heine Proble
Categories:34M10, 33C15, 33C47

24. CJM 2006 (vol 58 pp. 401)

Kolountzakis, Mihail N.; Révész, Szilárd Gy.
On Pointwise Estimates of Positive Definite Functions With Given Support
The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition.

Keywords:Fourier transform, positive definite functions and measures, Turán's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems
Categories:42B10, 26D15, 42A82, 42A05

25. CJM 2006 (vol 58 pp. 3)

Ben Saïd, Salem
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras
This paper is devoted to the study of certain zeta distributions associated with simple non-Euclidean Jordan algebras. An explicit form of the corresponding functional equation and Bernstein-type identities is obtained.

Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras
Categories:11M41, 17C20, 11S90
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