Ideal and representation theory of the $L^1$-algebra of a group with polynomial growth

Bruce A. Barnes

Displaying similar documents to “Ideal and representation theory of the $L^{1}$ -algebra of a group with polynomial growth”

An alternative polynomial Daugavet property

Elisa R. Santos (2014)

Studia Mathematica

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We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $m a x_{ω \in} | | I d + ω P | | = 1 + | | P | |$ . We study the stability of the APDP by c₀-, $ℓ_{\infty}$ - and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty} (μ, X)$ and C(K,X), where X has the APDP.

Exponent of growth of polynomial mappings of $C^{2}$ into $C^{2}$

Jacek Chadzyński, Tadeusz Krasiński (1988)

Banach Center Publications

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Root location for the characteristic polynomial of a Fibonacci type sequence

Zhibin Du, Carlos Martins da Fonseca (2023)

Czechoslovak Mathematical Journal

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We analyse the roots of the polynomial $x^{n} - p x^{n - 1} - q x - 1$ for $p ⩾ q ⩾ 1$ . This is the characteristic polynomial of the recurrence relation $F_{k, p, q} (n) = p F_{k, p, q} (n - 1) + q F_{k, p, q} (n - k + 1) + F_{k, p, q} (n - k)$ for $n ⩾ k$ , which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.

Volume of spheres in doubling metric measured spaces and in groups of polynomial growth

Romain Tessera (2007)

Bulletin de la Société Mathématique de France

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Let $G$ be a compactly generated locally compact group and let $U$ be a compact generating set. We prove that if $G$ has polynomial growth, then ${(U^{n})}_{n \in ℕ}$ is a Følner sequence and we give a polynomial estimate of the rate of decay of $\frac{μ (U^{n + 1} ∖ U^{n})}{μ (U^{n})} .$ Our proof uses only two ingredients: the doubling property and a weak geodesic property that we call Property (M). As a matter of fact, the result remains true in a wide class of doubling metric measured spaces including manifolds and graphs. As an application, we obtain...

On the distribution of the roots of polynomial $z^{k} - z^{k - 1} - \dots - z - 1$

Carlos A. Gómez, Florian Luca (2021)

Commentationes Mathematicae Universitatis Carolinae

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We consider the polynomial $f_{k} (z) = z^{k} - z^{k - 1} - \dots - z - 1$ for $k \geq 2$ which arises as the characteristic polynomial of the $k$ -generalized Fibonacci sequence. In this short paper, we give estimates for the absolute values of the roots of $f_{k} (z)$ which lie inside the unit disk.

A set on which the Łojasiewicz exponent at infinity is attained

Jacek Chądzyński, Tadeusz Krasiński (1997)

Annales Polonici Mathematici

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We show that for a polynomial mapping $F = (f ₁, . . ., f ₘ) : ℂ^{n} \to ℂ^{m}$ the Łojasiewicz exponent $_{\infty} (F)$ of F is attained on the set $z \in ℂ^{n} : f ₁ (z) \cdot . . . \cdot f ₘ (z) = 0$ .

Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials

Didier D&#039;Acunto, Krzysztof Kurdyka (2005)

Annales Polonici Mathematici

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Let f: ℝⁿ → ℝ be a polynomial function of degree d with f(0) = 0 and ∇f(0) = 0. Łojasiewicz’s gradient inequality states that there exist C > 0 and ϱ ∈ (0,1) such that ${| \nabla f | \geq C | f |}^{ϱ}$ in a neighbourhood of the origin. We prove that the smallest such exponent ϱ is not greater than $1 - R {(n, d)}^{- 1}$ with $R (n, d) = d {(3 d - 3)}^{n - 1}$ .

Twinning in minerals and metals: remarks on the comparison of a thermoelastic theory with some experimental results. Mechanical twinning and growth twinning. Nota II

Giovanni Zanzotto (1988)

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

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In this Note II we continue the analysis of the phenomenon of mechanical twinning that we began in a preceding Note I $(^{1})$ . Furthermore, we point out some fundamental properties useful in the study of growth twins, for which a fully comprehensive thermoelastic theory is not yet available.

Weak polynomial identities and their applications

Vesselin Drensky (2021)

Communications in Mathematics

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Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$ . The polynomial $f (x_{1}, ..., x_{n})$ of the free associative algebra $K 〈 x_{1}, x_{2}, ... 〉$ is a weak polynomial identity for the pair $(R, V)$ if it vanishes in $R$ when evaluated on $V$ . We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of...

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

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Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality $∥ p^{'} ∥_{[- 1, 1]} \leq \frac{1}{2} {∥ p ∥}_{[- 1, 1]}$ for a constrained polynomial $p$ of degree at most $n$ , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval $(- 1, 1)$ and establish a new asymptotically sharp inequality. ...

Heights of squares of Littlewood polynomials and infinite series

Artūras Dubickas (2012)

Annales Polonici Mathematici

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Let P be a unimodular polynomial of degree d-1. Then the height H(P²) of its square is at least √(d/2) and the product L(P²)H(P²), where L denotes the length of a polynomial, is at least d². We show that for any ε > 0 and any d ≥ d(ε) there exists a polynomial P with ±1 coefficients of degree d-1 such that H(P²) < (2+ε)√(dlogd) and L(P²)H(P²)< (16/3+ε)d²log d. A similar result is obtained for the series with ±1 coefficients. Let $A_{m}$ be the mth coefficient of the square f(x)² of...

The algebra of polynomials on the space of ultradifferentiable functions

Katarzyna Grasela (2010)

Banach Center Publications

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We consider the space $^{ℳ}$ of ultradifferentiable functions with compact supports and the space of polynomials on $^{ℳ}$ . A description of the space $(^{ℳ})$ of polynomial ultradistributions as a locally convex direct sum is given.

Connections partially adapted to a metric $(J^{4} = 1)$ -structure

E. Reyes, A. Montesinos, P. M. Gadea (1987)

Colloquium Mathematicae

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Polynomial flows on $R^{n}$

Gary Hosler-Meisters (1989)

Banach Center Publications

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Varieties of Algebras of Polynomial Growth

Daniela La Mattina (2008)

Bollettino dell'Unione Matematica Italiana

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Let $\mathcal{V}$ be a proper variety of associative algebras over a field $F$ of characteristic zero. It is well-known that $\mathcal{V}$ can have polynomial or exponential growth and here we present some classification results of varieties of polynomial growth. In particular we classify all subvarieties of the varieties of almost polynomial growth, i.e., the subvarieties of $\operatorname{\textbf{var}}(G)$ and $\operatorname{\textbf{var}}(UT_{2})$ , where $G$ is the Grassmann algebra and $UT_{2}$ is the algebra of $2\times 2$ upper triangular matrices.

Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, Niels Besseling (2011)

Studia Mathematica

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We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), $ẋ (t) = A^{- 1} x (t)$ , and the difference equation $x_{d} (n + 1) = (A + I) {(A - I)}^{- 1} x_{d} (n)$ are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ${(e^{A^{- 1} t})}_{t \geq 0}$ is O(∜t), and for $((A + I) {(A - I)}^{- 1}) ⁿ$ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore,...

Sum of squares and the Łojasiewicz exponent at infinity

Krzysztof Kurdyka, Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław Spodzieja (2014)

Annales Polonici Mathematici

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Let V ⊂ ℝⁿ, n ≥ 2, be an unbounded algebraic set defined by a system of polynomial equations $h ₁ (x) = \dots = h_{r} (x) = 0$ and let f: ℝⁿ→ ℝ be a polynomial. It is known that if f is positive on V then ${f |}_{V}$ extends to a positive polynomial on the ambient space ℝⁿ, provided V is a variety. We give a constructive proof of this fact for an arbitrary algebraic set V. Precisely, if f is positive on V then there exists a polynomial $h (x) = \sum_{i = 1}^{r} h ²_{i} (x) σ_{i} (x)$ , where $σ_{i}$ are sums of squares of polynomials of degree at most p, such that f(x) + h(x) >...

Entire functions of exponential type not vanishing in the half-plane $ℑ z > k$ , where $k > 0$

Mohamed Amine Hachani (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $P (z)$ be a polynomial of degree $n$ having no zeros in $| z | < k$ , $k \leq 1$ , and let $Q (z) : = z^{n} \bar{P (1 / \bar{z})}$ . It was shown by Govil that if ${max}_{| z | = 1} | P^{'} (z) |$ and ${max}_{| z | = 1} | Q^{'} (z) |$ are attained at the same point of the unit circle $| z | = 1$ , then $max_{| z | = 1} | P^{'} (z) | \leq \frac{n}{1 + k^{n}} max_{| z | = 1} | P (z) | .$ The main result of the present article is a generalization of Govil’s polynomial inequality to a class of entire functions of exponential type.

Hodge type decomposition

Wojciech Kozłowski (2007)

Annales Polonici Mathematici

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In the space $Λ^{p}$ of polynomial p-forms in ℝⁿ we introduce some special inner product. Let $H^{p}$ be the space of polynomial p-forms which are both closed and co-closed. We prove in a purely algebraic way that $Λ^{p}$ splits as the direct sum $d * (Λ^{p + 1}) \oplus δ * (Λ^{p - 1}) \oplus H^{p}$ , where d* (resp. δ*) denotes the adjoint operator to d (resp. δ) with respect to that inner product.

Extending piecewise polynomial functions in two variables

Andreas Fischer, Murray Marshall (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We study the extensibility of piecewise polynomial functions defined on closed subsets of $ℝ^{2}$ to all of $ℝ^{2}$ . The compact subsets of $ℝ^{2}$ on which every piecewise polynomial function is extensible to $ℝ^{2}$ can be characterized in terms of local quasi-convexity if they are definable in an o-minimal expansion of $ℝ$ . Even the noncompact closed definable subsets can be characterized if semialgebraic function germs at infinity are dense in the Hardy field of definable germs. We also present a piecewise...

The space $S_{α, β}$ and σ-core

Bruno de Malafosse (2006)

Studia Mathematica

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We give some new properties of the space $S_{α, β}$ and we apply them to the σ-core theory. These results generalize those by Choudhary and Yardimci.

Polynomials with values which are powers of integers

Rachid Boumahdi, Jesse Larone (2018)

Archivum Mathematicum

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Let $P$ be a polynomial with integral coefficients. Shapiro showed that if the values of $P$ at infinitely many blocks of consecutive integers are of the form $Q (m)$ , where $Q$ is a polynomial with integral coefficients, then $P (x) = Q (R (x))$ for some polynomial $R$ . In this paper, we show that if the values of $P$ at finitely many blocks of consecutive integers, each greater than a provided bound, are of the form $m^{q}$ where $q$ is an integer greater than 1, then $P (x) = {(R (x))}^{q}$ for some polynomial $R (x)$ .

Polynomial quotients: Interpolation, value sets and Waring's problem

Zhixiong Chen, Arne Winterhof (2015)

Acta Arithmetica

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For an odd prime p and an integer w ≥ 1, polynomial quotients $q_{p, w} (u)$ are defined by $q_{p, w} (u) \equiv (u^{w} - u^{w p}) / p m o d p$ with $0 \leq q_{p, w} (u) \leq p - 1$ , u ≥ 0, which are generalizations of Fermat quotients $q_{p, p - 1} (u)$ . First, we estimate the number of elements $1 \leq u < N \leq p$ for which $f (u) \equiv q_{p, w} (u) m o d p$ for a given polynomial f(x) over the finite field $_{p}$ . In particular, for the case f(x)=x we get bounds on the number of fixed points of polynomial quotients. Second, before we study the problem of estimating the smallest number (called the Waring number) of summands needed to express each...

Duality of measures of non-𝒜-compactness

Juan Manuel Delgado, Cándido Piñeiro (2015)

Studia Mathematica

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Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map $χ_{}$ (respectively, $n_{}$ ) acting on the operators of the surjective (respectively, injective) hull of such that $χ_{} (T) = 0$ (respectively, $n_{} (T) = 0$ ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...

Displaying similar documents to “Ideal and representation theory of the L 1 -algebra of a group with polynomial growth”

Displaying similar documents to “Ideal and representation theory of the $L^{1}$ -algebra of a group with polynomial growth”