A new tautological relation of ${{\overline{M}}_{3,\,1}}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

In his “lost” notebook, Ramanujan stated two results, which are equivalent to the identities

$$\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{n}} \right)}^{5}}}{\left( 1-{{q}^{5n}} \right)}=1-5\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{d} \right)d \right){{q}^{n}}$$

and

$$q\underset{n=1}{\mathop{\overset{\infty }{\mathop{\prod }}\,}}\,\frac{{{\left( 1-{{q}^{5n}} \right)}^{5}}}{\left( 1-{{q}^{n}} \right)}=\underset{n=1}{\mathop{\overset{\infty }{\mathop{\sum }}\,}}\,\left( \underset{d|n}{\mathop{\sum }}\,\left( \frac{5}{n/d} \right)d \right){{q}^{n}}.$$

We give several more identities of this type.

It is with great pleasure that we present this special geometry issue of the Canadian Mathematical Bulletin dedicated to Professor Tibor Bisztriczky on occasion of his sixtieth birthday. All articles were prepared by the contributors especially for this collection. This issue presents new results from current investigations in the fields of the combinatorial theory of polytopes, convexity, the theory of packing and covering, and combinatorial geometry, subjects where Professor Bisztriczky made significant contributions and where he is currently engaged in active research. Here we take the opportunity to thank the contributors for submitting their most valuable work and the referees who devoted their efforts and time to make this special volume possible.

Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than ${{x}^{0.33}}$ Carmichael numbers $n\,\le \,x$ with the property that $\varphi \left( n \right)$ is a perfect square. We also obtain similar results for higher powers.

With an increase in size, configurations of points and lines in the plane usually become complicated and hard to analyze. The “floral” configurations we are introducing here represent a new type that makes accessible and visually intelligible even configurations of considerable size. This is achieved by combining a large degree of symmetry with a hierarchical construction. Depending on the details of the interdependence of these aspects, there are several subtypes that are described and investigated.

In Connections on a parabolic principal bundle over a curve, $I$ we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of ${{\mathbb{Z}}_{4}}$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Let ${{R}_{n}}\left( \alpha \right)$ be the $n!\,\times \,n!$ matrix whose matrix elements ${{\left[ {{R}_{n}}\left( \alpha \right) \right]}_{\sigma p}}$ , with $\sigma$ and $p$ in the symmetric group ${{G}_{n}}$ , are ${{\alpha }^{\ell \left( \sigma {{p}^{-1}} \right)}}$ with $0\,<\,\alpha \,<\,1$, where $\ell \left( \text{ }\!\!\pi\!\!\text{ } \right)$ denotes the number of cycles in $\text{ }\pi \text{ }\in {{G}_{n}}.$ We give the spectrum of ${{R}_{n}}$ and show that the ratio of the largest eigenvalue ${{\text{ }\!\!\lambda\!\!\text{ }}_{0}}$ to the second largest one (in absolute value) increases as a positive power of $n$ as $n\,\to \,\infty$.

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6.

If $K$ is a number field with ${{n}_{k}}\,=\,\left[ k\,:\,\mathbb{Q} \right]$, and ${{\xi }_{k}}$ the symmetrized Dedekind zeta function of the field, the inequality

$$\Re \frac{\xi _{k}^{'}\left( \sigma \,+\,\text{i}t \right)}{{{\xi }_{k}}\left( \sigma \,+\,\text{i}t \right)}\,>\,\frac{\xi _{k}^{'}\left( \sigma \right)}{{{\xi }_{k}}\left( \sigma \right)}$$

for $t\,\ne \,0$ is shown to be true for $\sigma \,\ge \,1\,+\,8/n_{k}^{\frac{1}{3}}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k\,=\,\mathbb{Q}$ the inequality is extended to $\sigma \,\ge \,1$ for all $t$ sufficiently large or small and to the region $\sigma \,\ge \,1\,+\,1/\left( \log \,t\,-\,5 \right)$ for all $t\,\ne \,0$. This answers positively a question posed by Lagarias.

We use some classical estimates for polynomial roots to provide several irreducibility criteria for polynomials with integer coefficients that have one sufficiently large coefficient and take a prime value.

In this paper we study holomorphic maps between almost Hermitian manifolds. We obtain a new criterion for the harmonicity of such holomorphic maps, and we deduce some applications to horizontally conformal holomorphic submersions.

The projection body of order one ${{\Pi }_{1}}K$ of a convex body $K$ in ${{\mathbb{R}}^{n}}$ is the body whose support function is, up to a constant, the average mean width of the orthogonal projections of $K$ onto hyperplanes through the origin.

The paper contains an inequality for the support function of ${{\Pi }_{1}}K$, which implies in particular that such a function is strictly convex, unless $K$ has dimension one or two. Furthermore, an existence problem related to the reconstruction of a convex body is discussed to highlight the different behavior of the area measures of order one and of order $n\,-\,1$.

New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

Let $\tau \left( n \right)$ be the Ramanujan $\text{ }\!\!\tau\!\!\text{ }$-function. We prove that for any integer $N$ with $\left| N \right|\,\ge \,2$ the diophantine equation

$$\underset{i=1}{\overset{148000}{\mathop{\sum }}}\,\,\tau \left( {{n}_{i}} \right)\,=\,N$$

has a solution in positive integers ${{n}_{1}},\,{{n}_{2}},\,.\,.\,.\,,\,{{n}_{148000}}$ satisfying the condition

$$\underset{1\le i\le 148000}{\mathop{\max }}\,{{n}_{i}}\ll |N{{|}^{2/11}}{{e}^{-c\log |N|/\log \log |N|}},$$

for some absolute constant $c\,>\,0$.

In this paper, we give the ${{L}^{p}}$ boundedness for a class of parabolic Littlewood–Paley $g$-function with its kernel function $\Omega$ is in the Hardy space ${{H}^{1}}\left( {{S}^{n-1}} \right)$.

The (classical, small quantum, equivariant) cohomology ring of the grassmannian $G\left( k,\,n \right)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ (Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.

We strengthen certain results concerning actions of $\left( \mathbb{C},\,+ \right)$ on ${{\mathbb{C}}^{3}}$ and embeddings of ${{\mathbb{C}}^{2}}$ in ${{\mathbb{C}}^{3}}$, and show that these results are in fact valid over any field of characteristic zero.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of ${{C}^{*}}$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand–Naimark representation theorem for commutative ${{C}^{*}}$-algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

A classical theorem of Rogers states that for any convex body $K$ in $n$-dimensional Euclidean space there exists a covering of the space by translates of $K$ with density not exceeding $n\,\log \,n\,+\,n\,\log \,\log \,n\,+\,5$. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of $n$ the same bound can be attained by a covering which is the union of $O\left( \log \,n \right)$ translates of a lattice arrangement of $K$.