In this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the ``finite dimensional'' case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy--Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

We provide concise series representations for various L-series integrals. Different techniques are needed below and above the abscissa of absolute convergence of the underlying L-series.

We classify the left-invariant metrics with nonnegative sectional curvature on $\SO(3)$ and $U(2)$.

Consider a real potential $V$ on $\RR^d$, $d\geq 2$, and the Schr\"odinger equation: \begin{equation} \tag{LS} \label{LS1} i\partial_t u +\Delta u -Vu=0,\quad u_{\restriction t=0}=u_0\in L^2. \end{equation} In this paper, we investigate the minimal local regularity of $V$ needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of $1/2$ derivative) on solutions of \eqref{LS1}. Prior works show some dispersive properties when $V$ (small at infinity) is in $L^{d/2}$ or in spaces just a little larger but with a smallness condition on $V$ (or at least on its negative part). In this work, we prove the critical character of these results by constructing a positive potential $V$ which has compact support, bounded outside $0$ and of the order $(\log|x|)^2/|x|^2$ near $0$. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator $P:=-\Delta+V$. The elementary construction of $V$ consists in sticking together concentrated, truncated potential wells near $0$. This yields a potential oscillating with infinite speed and amplitude at $0$, such that the operator $P$ admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.

In this paper, we study the tensor product $\pi=\sigma^{\min}\otimes \sigma^{\min}$ of the minimal representation $\sigma^{\min}$ of $O(p,q)$ with itself, and decompose $\pi$ into a direct integral of irreducible representations. The decomposition is given in terms of the Plancherel measure on a certain real hyperbolic space.

In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra $\ell^1(S)$ for various semigroups $S$. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than $a$ for some integer $a$. Our results are of two types: in some cases, we show that some cohomology groups are $0$, while in some other cases, we show that some cohomology groups are Banach spaces.

Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$ denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq m$, $(d,m)=1$, satisfy the polynomial $$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1} x^{\phi(m)-1} + \dots + c_{\phi(m)},$$ where the sum and product are taken over a complete system of reduced residues modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely, $$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$ where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$ of reduced residues modulo $m$. Questions concerning the explicit determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their beginning coefficients), their reducibility over the rational field ${\bf Q}$ and duplication among the factors are studied. The treatment is similar to what has been done for period polynomials for finite fields.

Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, i.e., frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames.

We study a real hypersurface $M$ in a complex space form $\mn$, $c \neq 0$, whose shape operator and structure tensor commute each other on the holomorphic distribution of $M$.

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

In this paper, we consider Hermitian harmonic maps from Hermitian manifolds into convex balls. We prove that there exist no non-trivial Hermitian harmonic maps from closed Hermitian manifolds into convex balls, and we use the heat flow method to solve the Dirichlet problem for Hermitian harmonic maps when the domain is a compact Hermitian manifold with non-empty boundary.

We extend results of P.~M. Gauthier, W. Hengartner and A.~A. Nersesyan on simultaneous approximation and interpolation on Arakelian sets.

Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of ``coquecigrues,'' in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction.

We study the structure of the sets of symmetric sequences and spreading models of an Orlicz sequence space equipped with partial order with respect to domination of bases. In the cases that these sets are ``small'', some descriptions of the structure of these posets are obtained.

We show that there exists a non-archimedean Fr\'echet-Montel space $W$ with a basis and with a continuous norm such that any non-archimedean Fr\'echet space of countable type is isomorphic to a quotient of $W$. We also prove that any non-archimedean nuclear Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear Fr\'echet space with a basis and with a continuous norm.

Let $G(X)$ denote the number of positive composite integers $n$ satisfying $\sum_{j=1}^{n-1}j^{n-1}\equiv -1 \tmod{n}$. Then $G(X)\ll X^{1/2}\log X$ for sufficiently large $X$.

It is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products.

We show that there exists a closed infinite dimensional subspace of $H^\infty(B^n)$ such that every function of norm one is universal for some sequence of automorphisms of $B^n$.

A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained.

We prove that every real algebraic integer $\alpha$ is expressible by a difference of two Mahler measures of integer polynomials. Moreover, these polynomials can be chosen in such a way that they both have the same degree as that of $\alpha$, say $d$, one of these two polynomials is irreducible and another has an irreducible factor of degree $d$, so that $\alpha=M(P)-bM(Q)$ with irreducible polynomials $P, Q\in \mathbb Z[X]$ of degree $d$ and a positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.

Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$ defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$ realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$.

Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times \SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional $CW$-complex of the homotopy type of an $n$-sphere. We study the automorphism group $\Aut (G)$ in order to compute the number of distinct homotopy types of spherical space forms with respect to free and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$ is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as well.

We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$.

Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and suppose that $A$ has an AF-skeleton with only finitely many extreme traces. Then the corona algebra ${\mathcal M}(A)/A$ is purely infinite in the sense of Kirchberg and R\o rdam, which implies that $A$ has the corona factorization property.

The original Sato--Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate a modular analogue of the Sato--Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate--Tate measure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform.

Dans le papier Beyond Endoscopy une id\'ee pour entamer la fonctorialit\'e en utilisant la formule des traces a \'et\'e introduite. Maints probl\`emes, l'existence d'une limite convenable de la formule des traces, est eqquiss\'ee dans cette note informelle mais seulement pour $GL(2)$ et les corps des fonctions rationelles sur un corps fini et en ne pas resolvant bon nombre de questions.

Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible $C^*$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a $C^*$-extension which is not even invertible up to homotopy.

In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta functions, because as $q$ radially approaches any point $e^{2\pi ir}$ ($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$. In this paper we establish the relationship between two families of mock theta functions.

We prove Beurling's theorem for rank $1$ Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space.

Let $p$ be a prime greater than or equal to 17 and congruent to 2 modulo 3. We use results of Beukers and Helou on Cauchy--Liouville--Mirimanoff polynomials to show that the intersection of the Fermat curve of degree $p$ with the line $X+Y=Z$ in the projective plane contains no algebraic points of degree $d$ with $3 \leq d \leq 11$. We prove a result on the roots of these polynomials and show that, experimentally, they seem to satisfy the conditions of a mild extension of an irreducibility theorem of P\'{o}lya and Szeg\"{o}. These conditions are conjecturally also necessary for irreducibility.

We obtain Hauptmoduls of genus zero congruence subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is a prime and $w_p$ is the Atkin--Lehner involution. We then use the Hauptmoduls, along with modular functions on $\Gamma_1(p)$ to construct families of cyclic extensions of quadratic number fields. Further examples of cyclic extension of bi-quadratic and tri-quadratic number fields are also given.

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is of Codazzi type.

In this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the $p$-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

Let $M$ be a symplectic $4$-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

We study two sufficient conditions that imply global injectivity for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any point of $\R^2$ is not zero. One is based on the notion of half-Reeb component and the other on the Palais--Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais--Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais--Smale condition it is not a necessary condition for the global injectivity of the map $X$.

We extend the notion of linking number of an ordinary link of two components to that of a singular link with transverse intersections in which case the linking number is a half-integer. We then apply it to simplify the construction of the Seifert matrix, and therefore the Alexander polynomial, in a natural way.

Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely studied in the last forty years. They turned out to be relevant in various domains of mathematics, such as the theory of finite automata, number theory, fractals or dynamical systems. Several results were extended by Dar\'oczy and K\'atai for expansions in complex bases. We introduce an adaptation of the so-called greedy algorithm to the complex case, and we generalize one of their main theorems.

We show that, for most of the elliptic curves $\E$ over a prime finite field $\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number field containing the endomorphism ring of $\E$ over $\F_p$ is sufficiently large. We also obtain an asymptotic formula for the number of distinct quadratic number fields generated by the endomorphism rings of all elliptic curves over $\F_p$.

We will show that any extension of a product of two Cantor minimal $\Z$-systems is affable in the sense of Giordano, Putnam and Skau.

Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

In the present paper, we introduce a modification of the Meyer-K\"{o}nig and Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and $f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r-$th order generalization of our operators and study their approximation properties.

Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of ${\rm Ext}$-groups in the category of smooth representations of $G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$. We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal representation $\pi$ of a $p$-adic group $G$. We also consider an example of identifying the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain representations of $p$-adic ${\rm GL}_{2n}$.

Let $\mathcal{F}$ be a family of vector fields on a manifold or a subcartesian space spanning a distribution $D$. We prove that an orbit $O$ of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$ and it has constant rank on $O$. This result implies Frobenius' theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

We prove that a separable, nuclear, purely infinite, simple $C^*$-algebra satisfying the universal coefficient theorem is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.

This paper presents some results on the simple exceptional Jordan algebra over an algebraically closed field $\Phi$ of characteristic not $2$. Namely an example of simple decomposition of $H(O_3)$ into the sum of two subalgebras of the type $H(Q_3)$ is produced, and it is shown that this decomposition is the only one possible in terms of simple subalgebras.

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.

We prove that for a simply connected closed $4$-dimensional manifold, its embeddings into the sphere of dimension $6$ are all concordant to each other.

We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity.

Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A block design $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda$ elements of $\cA$. More generally, for a fixed simple graph $G$, a graph design $\GD(v,G,\lambda)$ is a collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda$ elements of $\cA$. A famous result of Wilson says that for a fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is resolvable if $\cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

It is shown that Schatten $p$-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative $L_p$-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative $L_p$ spaces. As a consequence, the class of non-commutative $L_p$-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative $L_p$-spaces. For $p=1$ this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

If $A$ is a subset of the set of reflections of a finite Coxeter group $W$, we define a sub-$\ZM$-module $\DC_A(W)$ of the group algebra $\ZM W$. We discuss cases where this submodule is a subalgebra. This family of subalgebras includes strictly the Solomon descent algebra, the group algebra and, if $W$ is of type $B$, the Mantaci--Reutenauer algebra.

The paper offers a study of the inverse Laplace transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{-1}$ where $I_n$ is the modified Bessel function of the first kind and $r$ is a parameter. The present study is a continuation of the author's previous work %[\textit{Canadian Mathematical Bulletin} 45] on the singular behavior of the special case of the functions in question, $r$=1. The general case of $r \in [0,1]$ is addressed, and it is shown that the inverse Laplace transforms for such $r$ exhibit significantly more complex behavior than their predecessors, even though they still only have two different types of points of discontinuity: singularities and finite discontinuities. The functions studied originate from non-stationary fluid-structure interaction, and as such are of interest to researchers working in the area.

In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary of the unit disc $\partial\mathbb D$ we construct a holomorphic function $f\in\mathbb O(\mathbb D)$ such that \[ \int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\] and\[ E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\] In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.

Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$.

Soit $q$ une puissance d'un nombre premier $p$. Dans cette note on \'etablit la g\'en\'eralisation suivante d'un th\'eor\`eme de Wintenberger : tout sous-groupe ab\'elien ferm\'e du groupe des $\mathbb F_q$-auto\-morphismes continus du corps des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration de ramification est un groupe filtr\'e isomorphe au groupe de Galois d'une extension ab\'elienne d'un corps local {\`a} corps r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification de l'extension en num\'erotation inf\'erieure.

Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a non-negative integer. It is known that if the local cohomology module $\H^i_\fa(M)$ is finitely generated for all $i<t$, then $\Hom_R(R/\fa, \H^t_\fa(M))$ is finitely generated. In this paper it is shown that if $\H^i_\fa(M)$ is Artinian for all $i<t$, then $\Hom_R(R/\fa, \H^t_\fa(M))$ need not be Artinian, but it has a finitely generated submodule $N$ such that $\Hom_R(R/\fa,\H^t_\fa(M))/N$ is Artinian.

Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be any algebraic subalgebra reductive in $\mathfrak g$. For any simple finite dimensional $\mathfrak k$-module $V$, we construct simple $(\mathfrak g,\mathfrak k)$-modules $M$ with finite dimensional $\mathfrak k$-isotypic components such that $V$ is a $\mathfrak k$-submodule of $M$ and the Vogan norm of any simple $\k$-submodule $V'\subset M, V'\not\simeq V$, is greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$-modules $M$ are subquotients of the fundamental series of $(\mathfrak g,\mathfrak k)$-modules.

We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is a three space property.

It is shown that if a Banach space is saturated with infinite dimensional subspaces in which all ``special" $n$-tuples of vectors are equivalent with constants independent of $n$-tuples and of $n$, then the space contains asymptotic-$l_p$ subspaces for some $1 \leq p \leq \infty$. This extends a result by Figiel, Frankiewicz, Komorowski and Ryll-Nardzewski.

Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.

No abstract.