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Let a function f belong to the Lebesgue class , , and let be the Fourier transform of f. The classical theorem of E. Titchmarsh states that if the function f belongs to the Lipschitz class , , then...

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb {R}^{d}$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor, which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of $\mathbb {R}^{d}$ . In this paper we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.

The main result is Theorem 1: A flow on a connected open set X ⊂ R d is globally periodic provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32], and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally related periods (Theorem 2).

The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

We obtain improved versions of a classical theorem of Rogosinski concerning the partial sums of a bounded holomorphic function defined on the open unit disk D . Further, we establish refined versions of a generalized Bohr inequality for holomorphic functions mapping D inside a simply connected or convex domain Ω ⊊ C . In addition, we improve on the classical Bohr inequality for the family of holomorphic self mappings of D and for its subfamily consisting of functions that fix the origin.

International audience; The classical theorem of Erdős & Wintner furnishes a criterion for the existence of a limiting distribution for a real, additive arithmetical function. This work is devoted to providing an effective estimate for the remainder term under the assumption that the conditions in the criterion are fulfilled. We also investigate the case of a conditional distribution.

We consider nonlinear problems governed by the fractional p-Laplacian in presence of nonlocal Neumann boundary conditions and we show three different existence results: the first two theorems deal with a p-superlinear term, the last one with a source having p-linear growth. For the p-superlinear case we face two main difficulties. First: the p-superlinear term may not satisfy the Ambrosetti-Rabinowitz condition. Second, and more important: although the topological structure of the underlying functional reminds the one of the linking theorem, the nonlocal nature of the associated eigenfunctions prevents the use of such a classical theorem. For these reasons, we are led to adopt another approach, relying on the notion of linking over cones.

This paper is devoted to the investigation of a kind of generalized Caputo semilinear fractional differential inclusions with deviated-advanced nonlocal conditions. Solvability of the problem is established by means of the Leray-Schauder’s alternative approach with the help of the Lagrange mean-value classical theorem. Finally, some examples are given to delineate the efficient of theoretical results.

The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model., 5 figures

The original version of the article was published in [1]. Unfortunately, the original version of this article contains a mistake: in Theorem 6.2 appears that β(n, Δ) = (n−Δ+5)/4 but the correct statement is β(n, Δ) = (n − Δ + 4)/4. In this erratum we correct the theorem and give the correct proof.

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ 0) be the set of graphs G with n vertices and minimum degree δ 0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ 0) = min{δ(G) | G ∈ H(n, δ 0)}, b(n, δ 0) = max{δ(G) | G ∈ H(n, δ 0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ 0) and we compute the precise value of a(n, δ 0), α(n, Δ) and β(n, Δ) for all values of n, δ 0 and Δ, respectively.

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct application of a classical theorem due to Bobkov and Gotze.

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erdos, Pach, Pollack and Tuza. We use these bounds in order to study hyperbolic graphs (in the Gromov sense). Since computing the hyperbolicity constant is an almost intractable problem, it is natural to try to bound it in terms of some parameters of the graph. Let H ( n , δ 0 ) be the set of graphs G with n vertices and minimum degree δ 0 . We study a ( n , δ 0 ) : = min ⁡ { δ ( G ) | G ∈ H ( n , δ 0 ) } and b ( n , δ 0 ) : = max ⁡ { δ ( G ) | G ∈ H ( n , δ 0 ) } . In particular, we obtain bounds for b ( n , δ 0 ) and we compute the precise value of a ( n , δ 0 ) for all values of n and δ 0 .

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.

This chapter introduces the reader to stochastic homogenization problems describing processes in heterogeneous media whose microstructure is not periodic and, moreover, cannot be described with certainty. As in the previous chapters we stick to the case study conductivity problem but assume no periodicity of the coefficients. Instead we consider the case when the coefficients are rapidly oscillating random fields. We present a detailed proof of the classical theorem on existence of the homogenized limit for problems with stationary and ergodic random coefficients. Before proving this theorem we introduce the reader to stochastic models of heterogeneous media. This is done using basic examples such as the random checkerboard and the Poisson cloud. These examples are easy on the intuitive level but their rigorous mathematical understanding requires significant effort for someone with no experience in stochastic modeling. We conclude the chapter by a brief discussion of recent developments and provide a number of references for further reading.

International audience; Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that |L| is bounded below by c/e. We show here that if c ≤ 3m, then S satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus g(S) = |N \ S| goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.

We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to S 2 are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is complex analytic iff it is complex analytic when restricted to any complex curve.

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible $$\mathfrak{s}\mathfrak{l}$$ (n, $$\mathbb{F}$$ )-modules that appeared in the ℤ2-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. Three infinite subfamilies of these modules have the minimal Gelfand-Kirillov dimension. They contain weight modules with unbounded weight multiplicities and completely pointed modules.

We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph., 6 pages

In this paper, we consider numerical and trigonometric series with a very general monotonicity condition. First, a fundamental decomposition is established from which the sufficient parts of many classical results in Fourier analysis can be derived in this general setting. In the second part of the paper a necessary and sufficient condition for the uniform convergence of sine series is proved generalizing a classical theorem of Chaundy and Jolliffe.

A classical theorem of Paley asserts the existence of an infinite family of quadratic characters whose character sums become exceptionally large. In this paper, we establish an analogous result for characters of any fixed even order. Previously our bounds were only known under the assumption of the Generalized Riemann Hypothesis., 6 pages; minor correction in Proposition 2.4

Using the Dieudonne theory we will study a reduction of an abelian variety with complex multiplication at a prime. Our results may be regarded as generalization of the classical theorem due to Deuring for CM-elliptic curves. We will also discuss a sufficient condition for a prime at which the reduction of a CM-curve is maximal.

Schur’s classical theorem states that for a group \(G\), if \(G/Z(G)\) is finite, then \(G'\) is finite. Baer extended this theorem for the factor group \(G/Z_n(G)\), in which \(Z_n(G)\) is the \(n\)-th term of the upper central series of \(G\). Hekster proved a converse of Baer’s theorem as follows: If \(G\) is a finitely generated group such that \(\gamma _{n+1}(G)\) is finite, then \(G/Z_n(G)\) is finite where \(\gamma _{n+1}(G)\) denotes the \((n+1)\)st term of the lower central series of \(G\). In this paper, we generalize this result by obtaining the same conclusion under the weaker hypothesis that \(G/Z_n(G)\) is finitely generated. Furthermore, we show that the index of the subgroup \(Z_n(G)\) is bounded by a precisely determined function of the order of \(\gamma _{n+1}(G)\). Moreover, we prove that the mentioned theorem of Hekster is also valid under a weaker condition that \(Z_{2n}(G)/Z_{n}(G)\) is finitely generated. Although in this case the bound for the order of \(\gamma _{n+1}(G)\) is not achieved.

Let G be a finite connected graph. We give an asymptotically tight upper bound on the size of G in terms of order, diameter and minimum degree. Our result is a strengthening of an old classical theorem of Ore [Diameters in graphs, J. Combin. Theory, 5 (1968), 75–81] if minimum degree is prescribed and constant.

Luzin’s classical theorem states that any measurable function of one variable is “almost” continuous. This is no longer true for measurable functions of several variables. The search for a correct analogue of Luzin’s theorem leads to the notion of virtually continuous functions of several variables. This, probably new, notion appears implicitly in statements such as embedding theorems and trace theorems for Sobolev spaces. In fact, it reveals their nature of being theorems about virtual continuity. This notion is especially useful for the study and classification of measurable functions, as well as in some questions on dynamical systems, polymorphisms, and bistochastic measures. In this work we recall the necessary definitions and properties of admissible metrics, define virtual continuity, and describe some of its applications. A detailed analysis will be presented elsewhere.

A classical theorem of Bun Wong implies that a strongly pseudoconvex domain with transitive automorphism group must be biholomorphic to the unit ball. This result has been quite influential, and has been extended and modified in a number of fascinating ways. We discuss several variations and implications of this theorem, and present some new results as well.

This is a largely expository account of various aspects of the Borsuk–Ulam theorem, including extensions of the classical theorem to families of maps parametrized by a base space and to multivalued maps. The main technical tool is the Euler class with compact supports.

In this paper, we use a technique introduced in the paper [P. Dankelmann, R.C. Entringer, Average distance, minimum degree and spanning trees, J. Graph Theory 33 (2000), 1-13] to obtain a strengthening of an old classical theorem by Erdos et al. [P. Erdos, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree, J. Combin. Theory B 47 (1989), 73-79] on radius and minimum degree. To be more detailed, we will prove that if G is a connected graph of order n with the minimum degree @d, then the radius G does not [email protected]+1+1,where t is the irregularity index (that is the number of distinct terms of the degree sequence of G) which has been recently defined in the paper [S. Mukwembi, A note on diameter and the degree sequence of a graph, Appl. Math. Lett. 25 (2012), 175-178]. We claim that our result represent the tightest bound that ever been obtained until now.

A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation \(\gamma \) in \(\text {PGL}(2,\mathbb {Z})\) there exist s and t for which the complete quotients \(x_s\) and \(y_t\) coincide. In this paper we give an upper bound in terms of \(\gamma \) for the smallest indices s and t.

We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings., Comment: 40 pages, 5 figures. To appear in Crelle's Journal

Linear upper bounds are provided for the size of the torsion homology of negatively curved manifolds of finite volume in all dimensions $d\ne 3$. This extends a classical theorem by Gromov. In dimension $3$, as opposed to the Betti numbers, the size of torsion homology is unbounded in terms of the volume. Moreover, there is a sequence of $3$-dimensional hyperbolic manifolds that converges to $\mathbb{H}^3$ in the Benjamini--Schramm topology while its normalized torsion in the first homology is dense in $[0,\infty]$. In dimension $d\geq 4$ a somewhat precise estimate is given for the number of negatively curved manifolds of finite volume, up to homotopy, and in dimension $d\ge 5$ up to homeomorphism. These results are based on an effective simplicial thick-thin decomposition which is of independent interest., Comment: final version; to appear in JEMS

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) vd([d − 1]), where vd([n]) is the image of the Veronese map vd acting on [n], the n-dimensional projective space over a suitable field. To understand this we construct dependent sets on the Veronesean and show how to construct small independent sets of size nd + 2 on vd([n]). For d = 2 such sets of 2n + 2 points in [n] have been called “associated” and we observe that they correspond to self-dual codes of length 2n + 2.

A classical theorem of Wiener (Ann Math 33:1-100, 1932 )o n the form of a doubly invariant subspace of the shift operator in L 2 over (-π, π) is generalized in three directions: The interval (-π, π) is replaced by a locally compact abelian group, L 2 is replaced by L α ,α ∈ (0, ∞), and the measure as well as the functions of L α may be

We study the classical theorem of Bun Wong and Rosay about domains with non-compact automorphism group and strongly pseudoconvex orbit accumulation point. We formulate and prove a version of the result in the language of the Kobayashi metric

A classical theorem of Polya states that if f is an entire function taking integer values at the non-negative integers and satisfying the growth condition \(f(z) = o (\mid z \mid^M2^{\mid z \mid}){\rm as}\ z \rightarrow \infty\), for some M > 0, then there exist polynomials P1, P2 with f(z) ≡ P1(z)2z + P2(z). It is shown that the same result holds for functions analytic in a half-plane Re z ≥ A.

The paper proves some general facts on commutators that refer to Fuglede-Putnam classical theorem in the spectral theory of not necessarily selfadjoint operators.

This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.

We propose a heuristic for evaluating model adequacy for the Cox proportional hazard model by comparing the population cumulative hazard with the baseline cumulative hazard. We illustrate how recent results from the theory of competing risk can contribute to analysis of data with the Cox proportional hazard model. A classical theorem on independent competing risks allows us to assess model adequacy under the hypothesis of random right censoring, and a recent result on mixtures of exponentials predicts the patterns of the conditional subsurvival functions of random right censored data if the proportional hazard model holds.

A new proof and a genuine generalization to systems of first order equations is given from Poincare classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.

A classical theorem of L. Carleson states that the injection map from the Hardy space H p into L p ( d μ ) is bounded if and only if the positive measure μ on the unit disc is a bounded Carleson measure. In this paper Forelli–Rudin estimates for arbitrary positive measures on the unit disc are proved, and then these estimates are applied to characterize bounded s-Carleson measures in terms of α-Bloch- and F ( p , q , s ) -functions.

In this paper we generalize a version of the classical Calderón–Zygmund theorem on principle value integrals in generalized Lebesgue spaces Lp(⋅) proved in [J. Reine Angew. Math. 563 (2003) 197–220], to kernels, which do not satisfy standard estimates on Rd+1. This result will be used in part II of this paper to prove the classical theorem on halfspace estimates of Agmon, Douglis, and Nirenberg [Comm. Pure Appl. Math. 12 (1959) 623–727] for generalized Lebesgue spaces Lp(⋅).

In this paper, two theorems for the convergence of Durand-Kerner method in parallel circular iteration are given. The convergent condition of single-step method's circular iteration is relaxed compared with the classical theorem in the same field, while the one of the first proposed double-step method's is obtained accurately as well. An unique phenomenon appears that both of the constants concerned with either condition are [[email protected]]^-^1, where @f=(5+1)/2~1.61803399.

In this paper the classical theorem of Zareckiĭ about regular relations is generalized and an intrinsic characterization of regularity is obtained. Based on the generalized Zareckiĭ theorem and the intrinsic characterization of regularity, the authors give a characterization of monotone normality of ordered spaces. A new proof of the Urysohn-Nachbin lemma is presented which is quite different from the classical one.

We give a quantitative proof to Eremenko’s theorem (2000), which extends Bloch’s classical theorem to the class of n n -dimensional K K -quasiregular mappings.

In this paper various quasiconvexity notions are considered and compared. The main goal is to show that, under the assumption of upper semicontinuity, Jensen-type quasiconvexity properties are equivalent to the corresponding ordinary quasiconvexity property. The results thus obtained are analogous to the classical theorem of Bernstein and Doetsch for convex functions. Finally, the connection between approximate Jensen quasiconvexity and approximate quasiconvexity is investigated. Mathematics subject classification (2000): 26B25, 39B62.

We obtain sufficient conditions for the existence of almost periodic solutions of almost periodic linear differential equations thereby extending Favard’s classical theorem. These results are meant to complement previous results of the authors who have shown by means of a counterexample that the boundedness of all solutions is not, by itself, sufficient to guarantee the existence of an almost periodic solution to a linear almost periodic differential equation.

In this paper, we present a new unified approach and an elementary proof of a very general theorem on the existence of a semicontinuous or continuous utility function representing a preference relation. A simple and interesting new proof of the famous Debreu Gap Lemma is given. In addition, we prove a new Gap Lemma for the rational numbers and derive some consequences. We also prove a theorem which characterizes the existence of upper semicontinuous utility functions on a preordered topological space which need not be second countable. This is a generalization of the classical theorem of Rader which only gives sufficient conditions for the existence of an upper semicontinuous utility function for second countable topological spaces.

We deduce the pointwise one-dimensional multiplicative inequalities \( |f^{(k)}(x)| \le CMf(x)^{1-k/m}Mf^{(m)}(x)^{k/m} \), where Mh is the Hardy-Littlewood maximal function of h from a classical theorem of Kolmogorov. This implies that in the one-dimensional case the Gagliardo-Nirenberg inequalities in weighted L p spaces with Muckenhoupt‚s weights are consequences of Kolmogorov's theorem.

Let R be a ring and S be a nonempty subset of R. A mapping f : R −→ R is said to be centralizing (resp. commuting) on S if (x,f(x)) ∈ Z(R) (resp. (x,f(x)) = 0) for all x ∈ S. The purpose of this paper is to generalize the classical theorem of Posner (7, Theorem 2) and to extend a result of Bell and Martindale (1, Theorem 3) for a generalized derivation of a semiprime ring R which is commuting on a left ideal of R.