We construct, for all $d\,\ge \,4$, a cellulation of ${{\mathbb{S}}^{d-1}}$. We prove that these cellulations cannot be polytopal with maximal combinatorial symmetry. Such non-realizability phenomenon was first described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and, to the knowledge of the author, until now there have not been any known examples in higher dimensions. As a starting point for the construction, we introduce a new class of (Wythoffian) uniform polytopes, which we call duplexes. In proving our main result, we use some tools that we developed earlier while studying perfect polytopes. In particular, we prove perfectness of the duplexes; furthermore, we prove and make use of the perfectness of another new class of polytopes which we obtain by a variant of the so-called $E$-construction introduced by Eppstein, Kuperberg and Ziegler.

This paper introduces the concepts of fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity in intuitionistic fuzzy topological spaces. After defining the fundamental concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological spaces, we present intuitionistic fuzzy $\gamma$-open sets and intuitionistic fuzzy $\gamma$-continuity and other results related topological concepts.

Let $f$ be a square-free integer and denote by ${{\Gamma }_{0}}{{\left( f \right)}^{+}}$ the normalizer of ${{\Gamma }_{0}}\left( f \right)$ in $\text{SL}\left( 2,\,\mathbb{R} \right)$. We find the analogues of the cusp form $\Delta$ for the groups ${{\Gamma }_{0}}{{\left( f \right)}^{+}}$.

J. Elton used an application of Ramsey theory to show that if $X$ is an infinite dimensional Banach space, then ${{c}_{0}}$ embeds in $X$, ${{\ell }_{1}}$ embeds in $X$, or there is a subspace of $X$ that fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that if ${{c}_{0}}$ embeds in ${{X}^{*}}$ , then ${{\ell }_{\infty }}$ embeds in ${{X}^{*}}.$ Emmanuele and John showed that if ${{c}_{0}}$ embeds in $K\left( X,\,Y \right)$, then $K\left( X,\,Y \right)$ is not complemented in $L\left( X,\,Y \right)$. Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space ${{L}_{{{w}^{*}}}}\left( {{X}^{*}},\,Y \right)$ of ${{w}^{*}}\,-\,w$ continuous operators is also studied.

This paper is concerned with the boundary behavior of solutions of the Helmholtz equation in ${{\mathbb{R}}^{n}}$. In particular, we give a Littlewood-type theorem to show that the approach region introduced by Korányi and Taylor (1983) is best possible.

We prove a new upper bound for the smallest zero $x$ of a quadratic form over a number field with the additional restriction that $x$ does not lie in a finite number of $m$ prescribed hyperplanes. Our bound is polynomial in the height of the quadratic form, with an exponent depending only on the number of variables but not on $m$.

In this note we present inequalities relating the successive minima of an $o$-symmetric convex body and the successive inner and outer radii of the body. These inequalities join known inequalities involving only either the successive minima or the successive radii.

Let $R$ be a real closed field, let $X\,\subset \,{{R}^{n}}$ be an irreducible real algebraic set and let $Z$ be an algebraic subset of $X$ of codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. We improve this dimension theorem as follows. Indicate by $\mu$ the minimum integer such that the ideal of polynomials in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ vanishing on $Z$ can be generated by polynomials of degree $\le \,\mu$. We prove the following two results: (1) There exists a polynomial $P\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\mu +1$ such that $X\cap {{P}^{-1}}\left( 0 \right)$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. (2) Let $F$ be a polynomial in $R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $d$ vanishing on $Z$. Suppose there exists a nonsingular point $x$ of $X$ such that $F\left( x \right)\,=\,0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G\,\in \,R\left[ {{x}_{1}}\,,\,.\,.\,.\,,\,{{x}_{n}} \right]$ of degree $\le \,\max \{d,\,\mu \,+\,1\}$ such that, for each $t\,\in \,\left( -1,\,1 \right)\,\backslash \,\{0\}$, the set $\{x\in X|F\left( x \right)+tG\left( x \right)=0\}$ is an irreducible algebraic subset of $X$ of codimension 1 containing $Z$. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

A ring $R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to ${{C}^{*}}$-algebras. Various examples to illustrate and delimit our results are provided.

Let $K$ denote an oval, a centrally symmetric compact convex domain with non-empty interior. A family of translates of $K$ is said to have property $T\left( k \right)$ if for every subset of at most $k$ translates there exists a common line transversal intersecting all of them. The integer $k$ is the stabbing level of the family. Two translates ${{K}_{i}}\,=\,K\,+\,{{c}_{i}}$ and ${{K}_{j}}\,=\,K\,+\,{{c}_{j}}$ are said to be $\sigma$-disjoint if $\sigma K\,+\,{{c}_{i}}$ and $\sigma K\,+\,{{c}_{j}}$ are disjoint. A recent Helly-type result claims that for every $\sigma \,>\,0$ there exists an integer $k\left( \sigma \right)$ such that if a family of $\sigma$-disjoint unit diameter discs has property $T\left( k \right)|k\ge k\left( \sigma \right)$, then there exists a straight line meeting all members of the family. In the first part of the paper we give the extension of this theorem to translates of an oval $k$. The asymptotic behavior of $k\left( \sigma \right)$ for $\sigma \,\to \,0$ is considered as well.

Katchalski and Lewis proved the existence of a constant $r$ such that for every pairwise disjoint family of translates of an oval $K$ with property $T\left( 3 \right)$ a straight line can be found meeting all but at most $r$ members of the family. In the second part of the paper $\sigma$-disjoint families of translates of $K$ are considered and the relation of $\sigma$ and the residue $r$ is investigated. The asymptotic behavior of $r\left( \sigma \right)$ for $\sigma \,\to \,0$ is also discussed.

We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.

We show that for compact orientable hyperbolic orbisurfaces, the Laplace spectrum determines the length spectrum as well as the number of singular points of a given order. The converse also holds, giving a full generalization of Huber's theorem to the setting of compact orientable hyperbolic orbisurfaces.

Let $L$ be an $\text{RA}$ loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell \,\mapsto \,{{\ell }^{\theta }}$ denote an involution on $L$ and extend it linearly to the loop ring $RL$. An element $\alpha \,\in \,RL$ is symmetric if ${{\alpha }^{\theta }}\,=\,\alpha$ and skew-symmetric if ${{\alpha }^{\theta }}=-\alpha$ . In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or θ is the canonical involution on $L$, and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or 4.

We shall prove the following shaken Rogers's theorem for homothetic sections: Let $K$ and $L$ be strictly convex bodies and suppose that for every plane $H$ through the origin we can choose continuously sections of $K$ and $L$, parallel to $H$, which are directly homothetic. Then $K$ and $L$ are directly homothetic.

We compute the minimal polynomials of the Ramanujan values ${{t}_{n}}$, where $n\,\equiv \,11\,\bmod \,24$, using the Shimura reciprocity law. These polynomials can be used for defining the Hilbert class field of the imaginary quadratic field $\mathbb{Q}\left( \sqrt{-n} \right)$ and have much smaller coefficients than the Hilbert polynomials.

Let ${{C}_{p}}$ denote the cyclic group of order $p$, where $p\,\ge \,3$ is prime. We denote by ${{V}_{n}}$ the indecomposable $n$ dimensional representation of ${{C}_{p}}$ over a field $\mathbb{F}$ of characteristic $p$. We compute a set of generators, in fact a SAGBI basis, for the ring of invariants $\mathbb{F}{{\left[ {{V}_{2}}\,\oplus \,{{V}_{2}}\,\oplus \,{{V}_{3}} \right]}^{{{C}_{p}}}}$.

The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

If a mapping of several complex variables into projective space is holomorphic in each pair of variables, then it is globally holomorphic.

An irreducible graph manifold $M$ contains an essential torus if it is not a special Seifert manifold. Whether $M$ contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits $M$ into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors.

We say that a numerical semigroup is $d$-squashed if it can be written in the form

$$S\,=\,\frac{1}{N}\langle {{a}_{1}},\,.\,.\,.\,,\,{{a}_{d}}\rangle \,\cap \,\mathbb{Z}$$

for $N$, ${{a}_{1}}\,,\,.\,.\,.\,,\,{{a}_{d}}$ positive integers with $\gcd \left( {{a}_{1}},\,.\,.\,.\,,{{a}_{d}} \right)\,=\,1$. Rosales and Urbano have shown that a numerical semigroup is 2-squashed if and only if it is proportionally modular.

Recent works by Rosales et al. give a concrete example of a numerical semigroup that cannot be written as an intersection of 2-squashed semigroups. We will show the existence of infinitely many numerical semigroups that cannot be written as an intersection of 2-squashed semigroups. We also will prove the same result for 3-squashed semigroups. We conjecture that there are numerical semigroups that cannot be written as the intersection of $d$-squashed semigroups for any fixed $d$, and we prove some partial results towards this conjecture.