Solution of Belousov's problem

Maks A. Akivis; Vladislav V. Goldberg

Displaying similar documents to “Solution of Belousov's problem”

Quasigroup covers of division groupoids

Jaroslav J. Ježek, Tomáš Kepka, Petr Němec (2023)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a division groupoid that is not a quasigroup. For each regular cardinal $α > | G |$ we construct a quasigroup $Q$ on $G \times α$ that is a quasigroup cover of $G$ (i.e., $G$ is a homomorphic image of $Q$ and $G$ is not an image of any quasigroup that is a proper factor of $Q$ ). We also show how to easily obtain quasigroup covers from free quasigroups.

Medial quasigroups of prime square order

David Stanovský (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove that, for any prime $p$ , there are precisely $2 p^{4} - p^{3} - p^{2} - 3 p - 1$ medial quasigroups of order $p^{2}$ , up to isomorphism.

Linear operator identities in quasigroups

Reza Akhtar (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study identities of the form $L_{x_{0}} ϕ_{1} \dots ϕ_{n} R_{x_{n + 1}} = R_{x_{n + 1}} ϕ_{σ (1)} \dots ϕ_{σ (n)} L_{x_{0}}$ in quasigroups, where $n \geq 1$ , $σ$ is a permutation of ${1, ..., n}$ , and for each $i$ , $ϕ_{i}$ is either $L_{x_{i}}$ or $R_{x_{i}}$ . We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $σ$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n \to \infty$ .

Antiflexible Latin directed triple systems

Andrew R. Kozlik (2015)

Commentationes Mathematicae Universitatis Carolinae

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It is well known that given a Steiner triple system one can define a quasigroup operation $\cdot$ upon its base set by assigning $x \cdot x = x$ for all $x$ and $x \cdot y = z$ , where $z$ is the third point in the block containing the pair ${x, y}$ . The same can be done for Mendelsohn triple systems, where $(x, y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed...

On dicyclic groups as inner mapping groups of finite loops

Emma Leppälä, Markku Niemenmaa (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group with a dicyclic subgroup $H$ . We show that if there exist $H$ -connected transversals in $G$ , then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I (Q)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I (Q)$ is a certain type of a direct product.

Moufang loops of order coprime to three that cyclically extend groups of dihedral type

Aleš Drápal (2016)

Commentationes Mathematicae Universitatis Carolinae

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This paper completely solves the isomorphism problem for Moufang loops $Q = G C$ where $G ⊴ Q$ is a noncommutative group with cyclic subgroup of index two and $| Z (G) | \leq 2$ , $C$ is cyclic, $G \cap C = 1$ , and $Q$ is finite of order coprime to three.

Construction of Mendelsohn designs by using quasigroups of $(2, q)$ -varieties

Lidija Goračinova-Ilieva, Smile Markovski (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $q$ be a positive integer. An algebra is said to have the property $(2, q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2, q)$ if every member of $𝒱$ has the property $(2, q)$ . Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2, q)$ , $2 < q$ , blocks of Steiner system of type $(2, q)$ are obtained. The stated correspondence...

A class of irreducible polynomials

Joshua Harrington, Lenny Jones (2013)

Colloquium Mathematicae

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Let $f (x) = x ⁿ + k_{n - 1} x^{n - 1} + k_{n - 2} x^{n - 2} + \dots + k ₁ x + k ₀ \in ℤ [x]$ , where $3 \leq k_{n - 1} \leq k_{n - 2} \leq \dots \leq k ₁ \leq k ₀ \leq 2 k_{n - 1} - 3$ . We show that f(x) and f(x²) are irreducible over ℚ. Moreover, the upper bound of $2 k_{n - 1} - 3$ on the coefficients of f(x) is the best possible in this situation.

A new characterization for the simple group $PSL (2, p^{2})$ by order and some character degrees

Behrooz Khosravi, Behnam Khosravi, Bahman Khosravi, Zahra Momen (2015)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $| PSL (2, p^{2}) |$ such that $G$ has an irreducible character of degree $p^{2}$ and we know that $G$ has no irreducible character $θ$ such that $2 p ∣ θ (1)$ , then $G$ is isomorphic to $PSL (2, p^{2})$ . As a consequence of our result we prove that $PSL (2, p^{2})$ is uniquely determined by the structure of its complex group algebra.

Asymptotic values of modular multiplicities for ${GL}_{2}$

Sandra Rozensztajn (2014)

Journal de Théorie des Nombres de Bordeaux

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We study the irreducible constituents of the reduction modulo $p$ of irreducible algebraic representations $V$ of the group ${Res}_{K / ℚ_{p}} {GL}_{2}$ for $K$ a finite extension of $ℚ_{p}$ . We show that asymptotically, the multiplicity of each constituent depends only on the dimension of $V$ and the central character of its reduction modulo $p$ . As an application, we compute the asymptotic value of multiplicities that are the object of the Breuil-Mézard conjecture.

A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

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Let $(𝒮, 0)$ be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor $E$ and its irreducible components $E_{i}$ , $i \in I$ . The Nash map associates to each irreducible component $C_{k}$ of the space of arcs through $0$ on $𝒮$ the unique component of $E$ cut by the strict transform of the generic arc in $C_{k}$ . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if $E \cdot E_{i} < 0$ for any $i \in I$ . ...

Monodromy of a family of hypersurfaces

Vincenzo Di Gennaro, Davide Franco (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $Y$ be an $(m + 1)$ -dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$ , and $δ$ be a positive integer such that $ℐ_{Z, Y} (δ)$ is generated by global sections. Fix an integer $d \geq δ + 1$ , and assume the general divisor $X \in | H^{0} (Y, ℐ_{Z, Y} (d)) |$ is smooth. Denote by $H^{m} {(X; ℚ)}_{⊥ Z}^{van}$ the quotient of $H^{m} (X; ℚ)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$ . In the present paper we prove that the monodromy representation on $H^{m} {(X; ℚ)}_{⊥ Z}^{van}$ for the family...

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

Cardinal invariants for κ-box products: weight, density character and Suslin number

W. W. Comfort, Ivan S. Gotchev

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The symbol ${(X_{I})}_{κ}$ (with κ ≥ ω) denotes the space $X_{I} : = \prod_{i \in I} X_{i}$ with the κ-box topology; this has as base all sets of the form $U = \prod_{i \in I} U_{i}$ with $U_{i}$ open in $X_{i}$ and with $| i \in I : U_{i} \neq X_{i} | < κ$ . The symbols w, d and S denote respectively the weight, density character and Suslin number. Generalizing familiar classical results, the authors show inter alia: Theorem 3.1.10(b). If κ ≤ α⁺, |I| = α and each $X_{i}$ contains the discrete space 0,1 and satisfies $w (X_{i}) \leq α$ , then $w (X_{κ}) = α^{< κ}$ . Theorem 4.3.2. If $ω \leq κ \leq | I | \leq 2^{α}$ and $X = {(D (α))}^{I}$ with D(α) discrete, |D(α)| = α, then $d ({(X_{I})}_{κ}) = α^{< κ}$ . Corollaries 5.2.32(a)...

Variations on a question concerning the degrees of divisors of $x^{n} - 1$

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial $x^{n} - 1$ : “How often does $x^{n} - 1$ have a divisor of every degree between $1$ and $n$ ?” In a previous paper, we considered the situation when $x^{n} - 1$ is factored in $ℤ [x]$ . In this paper, we replace $ℤ [x]$ with $𝔽_{p} [x]$ , where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$ .

A generalized exponential map for an affinely homogeneous cone

Umberto Sampieri (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Dato un cono $V$ aperto non vuoto, convesso, regolare e affinemente omogeneo in uno spazio vettoriale reale $W$ di dimensione finita si prova che per ogni $v$ appartenente a $V$ esiste un diffeomorfismo $E_{v}:W\to V$ che soddisfa le condizioni seguenti E1) $E_{v}(0)=v$ ; E2) $\det(dE_{v}(y))=\Phi_{V}(E_{v}(y))^{-1}$ per ogni $y$ appartenente a $W$ ove $\Phi_{V}:V\to\mathbf{R}^{+}$ è la funzione caratteristica di $V$ .

A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${(α_{j})}_{j = 1}^{\infty}$ of scalars, there exists a subsequence ${(α_{k_{j}})}_{j = 1}^{\infty}$ such that either every subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series, or no subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.