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1. CJM Online first

Du, Zhuoran; Fang, Yanqin; Gui, Changfeng
A class of degenerate elliptic equations with nonlinear boundary conditions
We consider positive solutions of the problem \begin{equation} (*)\qquad \left\{ \begin{array}{l}-\mbox{div}(x_{n}^{a}\nabla u)=bx_{n}^{a}u^{p}\;\;\;\;\;\mbox{in}\;\;\mathbb{R}_{+}^{n}, \\ \frac{\partial u}{\partial \nu^a}=u^{q} \;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mbox{on}\;\;\partial \mathbb{R}_{+}^{n}, \\ \end{array} \right. \end{equation} where $a\in (-1,0)\cup(0,1)$, $b\geq 0$, $p, q\gt 1$ and $\frac{\partial u}{\partial \nu^a}:=-\lim_{x_{n}\rightarrow 0^+}x_{n}^{a}\frac{\partial u}{\partial x_{n}}$. In special case $b=0$, it is associated to fractional Laplacian equation $(-\Delta)^{s}u=u^{q} $ in entire space $\mathbb{R}^{n-1}$. We obtain the existence of positive axially symmetric solutions to ($*$) for the case $a\in (-1,0)$ in $n\geq3$ for supercritical exponents $p\geq\frac{n+a+2}{n+a-2}, \;\;q\geq\frac{n-a}{n+a-2}$. The nonexistence is obtained for the case $a\in (-1,0)$, $b\geq 0$ and any $p,~q\gt 1$ in $n=2$ as well.

Keywords:existence, non-existence, positive solutions, degenerate elliptic equation, nonlinear boundary conditions, symmetry, monotonicity
Categories:35D30, 35J70, 35J25

2. CJM Online first

Kaftal, Victor; Ng, Ping Wong; Zhang, Shuang
Strict comparison of positive elements in multiplier algebras
Main result: If a C*-algebra $\mathcal{A}$ is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra $\operatorname{\mathcal{M}}(\mathcal{A})$ also has strict comparison of positive elements by traces. The same results holds if ``finitely many extremal traces" is replaced by ``quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If $\mathcal{A}$ is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Keywords:strict comparison, bi-diagonal form, positive combinations
Categories:46L05, 46L35, 46L45, 47C15

3. CJM Online first

Runde, Volker; Viselter, Ami
On positive definiteness over locally compact quantum groups
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

Keywords:bicrossed product, locally compact quantum group, non-commutative $L^p$-space, positive-definite function, positive-definite measure, separation property
Categories:20G42, 22D25, 43A35, 46L51, 46L52, 46L89

4. CJM Online first

Marquis, Timothée; Neeb, Karl-Hermann
Isomorphisms of twisted Hilbert loop algebras
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a ``minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitly as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.

Keywords:locally affine Lie algebra, Hilbert-Lie algebra, positive energy representation
Categories:17B65, 17B70, 17B22, 17B10

5. CJM 2013 (vol 65 pp. 1287)

Reihani, Kamran
$K$-theory of Furstenberg Transformation Group $C^*$-algebras
The paper studies the $K$-theoretic invariants of the crossed product $C^{*}$-algebras associated with an important family of homeomorphisms of the tori $\mathbb{T}^{n}$ called Furstenberg transformations. Using the Pimsner-Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products, whose corresponding $n\times n$ integer matrices are unipotent of maximal degree, always have the same rank $a_{n}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a combinatorial significance. For example, every $a_{2n+1}$ is just the number of ways that $0$ can be represented as a sum of integers between $-n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple, explicit formula for the asymptotic behavior of the sequence $\{a_{n}\}$ is given. Finally, we describe the order structure of the $K_{0}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

Keywords:$K$-theory, transformation group $C^*$-algebra, Furstenberg transformation, Anzai transformation, minimal homeomorphism, positive cone, minimal homeomorphism
Categories:19K14, 19K99, 46L35, 46L80, , 05A15, 05A16, 05A17, 15A36, 17B10, 17B20, 37B05, 54H20

6. CJM 2012 (vol 65 pp. 757)

Delanoë, Philippe; Rouvière, François
Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved
The squared distance curvature is a kind of two-point curvature the sign of which turned out crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, an indirect one via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

Keywords:symmetric spaces, rank one, positive curvature, almost-positive $c$-curvature
Categories:53C35, 53C21, 53C26, 49N60

7. CJM 2007 (vol 59 pp. 673)

Ash, Avner; Friedberg, Solomon
Hecke $L$-Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace
Categories:11M41, 11F30, , 11F55, 11H06, 11R47

8. CJM 2006 (vol 58 pp. 449)

Agarwal, Ravi P.; Cao, Daomin; Lü, Haishen; O'Regan, Donal
Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations
Positive solutions are obtained for the boundary value problem \[\begin{cases} -( | u'| ^{p-2}u')' =\lambda f( t,u),\;t\in ( 0,1) ,p>1\\ u( 0) =u(1) =0. \end{cases} \] Here $f(t,u) \geq -M,$ ($M$ is a positive constant) for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper-lower solution method.

Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solution
Category:34B15

9. 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

10. CJM 2005 (vol 57 pp. 897)

Berezhnoĭ, Evgenii I.; Maligranda, Lech
Representation of Banach Ideal Spaces and Factorization of Operators
Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calder{\'o}n--Lozanovski\u\i\ construction. Factorization theorems for operators in spaces more general than the Lebesgue $L^{p}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de~Francia theorem on factorization of the Muckenhoupt $A_{p}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from $L^{p}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calder{\'o}n--Lozanovski\u\i\ construction are involved in the proofs.

Keywords:Banach ideal spaces, weighted spaces, weight functions,, Calderón--Lozanovski\u\i\ spaces, Orlicz spaces, representation of, spaces, uniqueness problem, positive linear operators, positive sublinear, operators, Schur test, factorization of operators, f
Categories:46E30, 46B42, 46B70

11. CJM 2004 (vol 56 pp. 1259)

Paterson, Alan L. T.
The Fourier Algebra for Locally Compact Groupoids
We introduce and investigate using Hilbert modules the properties of the {\em Fourier algebra} $A(G)$ for a locally compact groupoid $G$. We establish a duality theorem for such groupoids in terms of multiplicative module maps. This includes as a special case the classical duality theorem for locally compact groups proved by P. Eymard.

Keywords:Fourier algebra, locally compact groupoids, Hilbert modules,, positive definite functions, completely bounded maps
Category:43A32

12. CJM 2004 (vol 56 pp. 776)

Lim, Yongdo
Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
We explicitly describe the best approximation in geodesic submanifolds of positive definite matrices obtained from involutive congruence transformations on the Cartan-Hadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of positive definite matrices. An explicit calculation for the minimal distance function from the geodesic submanifold ${\mathrm{Sym}}(p,{\mathbb R})^{++}\times {\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in ${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is given in terms of metric and spectral geometric means, Cayley transform, and Schur complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$

Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, Cartan-Hadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform
Categories:15A48, 49R50, 15A18, 53C3

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