Let $P\in \mathbb{F}_{2}[z]$ be such that $P(0)=1$ and degree $(P)\geq 1$. Nicolas et al. [‘On the parity of additive representation functions’, J. Number Theory73 (1998), 292–317] proved that there exists a unique subset ${\mathcal{A}}={\mathcal{A}}(P)$ of $\mathbb{N}$ such that $\sum _{n\geq 0}p({\mathcal{A}},n)z^{n}\equiv P(z)~\text{mod}\,2$, where $p({\mathcal{A}},n)$ is the number of partitions of $n$ with parts in ${\mathcal{A}}$. Let $m$ be an odd positive integer and let ${\it\chi}({\mathcal{A}},.)$ be the characteristic function of the set ${\mathcal{A}}$. Finding the elements of the set ${\mathcal{A}}$ of the form $2^{k}m$, $k\geq 0$, is closely related to the $2$-adic integer $S({\mathcal{A}},m)={\it\chi}({\mathcal{A}},m)+2{\it\chi}({\mathcal{A}},2m)+4{\it\chi}({\mathcal{A}},4m)+\cdots =\sum _{k=0}^{\infty }2^{k}{\it\chi}({\mathcal{A}},2^{k}m)$, which has been shown to be an algebraic number. Let $G_{m}$ be the minimal polynomial of $S({\mathcal{A}},m)$. In precedent works there were treated the case $P$ irreducible of odd prime order $p$. In this setting, taking $p=1+ef$, where $f$ is the order of $2$ modulo $p$, explicit determinations of the coefficients of $G_{m}$ have been made for $e=2$ and 3. In this paper, we treat the case $e=4$ and use the cyclotomic numbers to make explicit $G_{m}$.

The Banach $^{\ast }$-operator algebras, exhibiting the second-order noncommutative differential structure and the noncommutative Lipschitz structure, that are determined by the unbounded derivation and induced by a closed symmetric operator in a Hilbert space, are explored.

For any real number ${\it\beta}$ with ${\it\beta}>1$, let ${\mathcal{M}}(\,{\it\beta})$ (${\mathcal{N}}(\,{\it\beta})$ respectively) denote the class of analytic functions $f$ in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+\sum _{n=2}^{\infty }a_{n}z^{n}$ and satisfying $\text{Re}\,P_{f}<{\it\beta}$ ($\text{Re}\,Q_{f}<{\it\beta}$ respectively) in $\mathbb{D}$, where $P_{f}=zf^{\prime }(z)/f(z)$ and $Q_{f}=1+zf^{\prime \prime }(z)/f^{\prime }(z)$. Also, for ${\it\beta}>1$, let ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ (${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ respectively) denote the class of analytic functions $g$ of the form $g(z)=z(1+\sum _{n=1}^{\infty }b_{n}z^{-n})$ and satisfying $\text{Re}\,P_{g}<{\it\beta}$ ($\text{Re}\,Q_{g}<{\it\beta}$ respectively) for $z\in {\rm\Delta}=\{z\in \mathbb{C}:1<|z|<\infty \}$. In this paper, we shall determine the coefficient bounds, inverse coefficient bounds, the growth and distortion theorem and the upper bounds for the Fekete–Szegő functional ${\rm\Lambda}_{{\it\lambda}}(f)=a_{3}-{\it\lambda}a_{2}^{2}$ for functions $f$ in the classes ${\mathcal{M}}(\,{\it\beta})$ and ${\mathcal{N}}(\,{\it\beta})$. Further, we shall solve the maximal area problem for functions of the type $z/f(z)$ when $f\in {\mathcal{M}}(\,{\it\beta})$, which is Yamashita’s conjecture for the class ${\mathcal{M}}(\,{\it\beta})$. We shall obtain the radius of convexity for the class ${\mathcal{N}}(\,{\it\beta})$. We shall also determine the coefficient bounds for functions $g$ in the classes ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$ and ${\mathcal{N}}{\rm\Sigma}(\,{\it\beta})$ and the inverse coefficient bounds for functions $g$ in the class ${\mathcal{M}}{\rm\Sigma}(\,{\it\beta})$. All the results are sharp.

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’, Comm. Inst. Sci. Kharkov13 (1936), 9–95] bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of $n$ variables’, Math. USSR Izv.37 (1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’, Israel. J. Math.186 (2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’, J. Approx. Theory162 (2010), 72–93]). Still, given a subset $Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here ${\mathcal{P}}_{d}(\mathbb{R}^{n})$ is the space of real polynomials of degree at most $d$ on $\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality: $\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$ for all $P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$ with $C$ independent of $P$. (Although ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree $d$ in $\mathbb{R}^{n}$, there are many apparently unrelated reasons for $Z\subset [-1,1]^{n}$ to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces $V$ of continuous functions on $[-1,1]^{n}$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for $Z$ to be $V$-norming, partly known, partly new, restricting ourselves to the simplest nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of the set $Z$, showing that it is Lipschitz in the Hausdorff metric.

For an odd prime $p$, a $p$-transposition group is a group generated by a set of involutions such that the product of any two has order 2 or $p$. We first classify a family of $(G,2)$-geodesic transitive Cayley graphs ${\rm\Gamma}:=\text{Cay}(T,S)$ where $S$ is a set of involutions and $T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$. In this case, $T$ is either an elementary abelian 2-group or a $p$-transposition group. Then under the further assumption that $G$ acts quasiprimitively on the vertex set of ${\rm\Gamma}$, we prove that: (1) if ${\rm\Gamma}$ is not $(G,2)$-arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if $T$ is a $p$-transposition group and $S$ is a conjugacy class, then $p=3$ and ${\rm\Gamma}$ is $(G,2)$-arc transitive.

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

It is well known that every elliptic curve over the rationals admits a parametrization by means of modular functions. In this short note, we show that only finitely many elliptic curves over $\mathbf{Q}$ can be parametrized by modular units. This answers a question raised by W. Zudilin in a recent work on Mahler measures. Further, we give the list of all elliptic curves $E$ of conductor up to 1000 parametrized by modular units supported in the rational torsion subgroup of $E$. Finally, we raise several open questions.

We prove some Banach–Stone type theorems for linear isometries of vector-valued continuous function spaces, by making use of the extreme point method.

We extend to soluble $\text{FC}^{\ast }$-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].

A conjecture of Scharaschkin and Skorobogatov states that there is a Brauer–Manin obstruction to the existence of rational points on a smooth geometrically irreducible curve over a number field. In this paper, we verify the Scharaschkin–Skorobogatov conjecture for explicit families of generalized Mordell curves. Our approach uses standard techniques from the Brauer–Manin obstruction and the arithmetic of certain threefolds.

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

This paper is devoted to dualization of paracompactness to the coarse category via the concept of $R$-disjointness. Property A of Yu can be seen as a coarse variant of amenability via partitions of unity and leads to a dualization of paracompactness via partitions of unity. On the other hand, finite decomposition complexity of Guentner, Tessera, and Yu and straight finite decomposition complexity of Dranishnikov and Zarichnyi employ $R$-disjointness as the main concept. We generalize both concepts to that of countable asymptotic dimension and our main result shows that it is a subclass of spaces with Property A. In addition, it gives a necessary and sufficient condition for spaces of countable asymptotic dimension to be of finite asymptotic dimension.

In this paper, we investigate strongly regular congruences on $E$-inversive semigroups $S$. We describe the complete lattice homomorphism of strongly regular congruences, which is a generalization of an open problem of Pastijn and Petrich for regular semigroups. An abstract characterization of left and right traces for strongly regular congruences is given. The strongly regular (sr) congruences on $E$-inversive semigroups $S$ are described by means of certain strongly regular congruence triples $({\it\gamma},K,{\it\delta})$ consisting of certain sr-normal equivalences ${\it\gamma}$ and ${\it\delta}$ on $E(S)$ and a certain sr-normal subset $K$ of $S$. Further, we prove that each strongly regular congruence on $E$-inversive semigroups $S$ is uniquely determined by its associated strongly regular congruence triple.

We raise a question of whether the Riesz transform on $\mathbb{T}^{n}$ or $\mathbb{Z}^{n}$ is characterized by the ‘maximal semigroup symmetry’ that the transform satisfies. We prove that this is the case if and only if the dimension is one, two or a multiple of four. This generalizes a theorem of Edwards and Gaudry for the Hilbert transform on $\mathbb{T}$ and $\mathbb{Z}$ in the one-dimensional case, and extends a theorem of Stein for the Riesz transform on $\mathbb{R}^{n}$. Unlike the $\mathbb{R}^{n}$ case, we show that there exist infinitely many linearly independent multiplier operators that enjoy the same maximal semigroup symmetry as the Riesz transforms on $\mathbb{T}^{n}$ and $\mathbb{Z}^{n}$ if the dimension $n$ is greater than or equal to three and is not a multiple of four.

Let ${\mathcal{A}}$ be a unital ring with involution. Assume that ${\mathcal{A}}$ contains a nontrivial symmetric idempotent and ${\it\phi}:{\mathcal{A}}\rightarrow {\mathcal{A}}$ is a nonlinear surjective map. We prove that if ${\it\phi}$ preserves strong skew commutativity, then ${\it\phi}(A)=ZA+f(A)$ for all $A\in {\mathcal{A}}$, where $Z\in {\mathcal{Z}}_{s}({\mathcal{A}})$ satisfies $Z^{2}=I$ and $f$ is a map from ${\mathcal{A}}$ into ${\mathcal{Z}}_{s}({\mathcal{A}})$. Related results concerning nonlinear strong skew commutativity preserving maps on von Neumann algebras are given.

It is proved that a quantale is projective if and only if it is isomorphic to a derived tensor quantale over a completely distributive sup-lattice. Furthermore, an intrinsic characterization of projectivity is given in terms of inertial sup-lattices and derivations of quantales.

In light of the recent work by Maynard and Tao on the Dickson $k$-tuples conjecture, we show that with a small improvement in the known bounds for this conjecture, we would be able to prove that for some fixed $R$, there are infinitely many Carmichael numbers with exactly $R$ factors for some fixed $R$. In fact, we show that there are infinitely many such $R$.

Motivated by a problem of characterising a family of Cayley graphs, we study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\mathsf{Aut}(G)$. It is shown that such groups correspond to complete multipartite graphs which are normal edge-transitive Cayley graphs.