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In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments cn(?) ? cn = ?10tnd?(t), n = 0, 1, 2, ..., of some singular distributions ? (with support ? 0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).

We point out that the analysis of Piotrowski and Sladkowski of the paradox described by Astumian has a flaw since they consider a different game.

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

This note gives the correct form of a result (Theorem 2, p. 478) published recently in [Math. Slovaca 60 (2010), 471–484].

Let c e IN. For d e ℤ with gcd (d, c) = 1 let δ(d, c) be defined by d · δ(d, c) 1 mod c, 1 ⩽ δ(d, c) ⩽ c. Let s, t e IN with 1 ⩽ s ⩽ c, 1 ⩽ t ⩽ c. The main result is that for arbitrary fixed e > 0, but uniformly over c, s and t Equivalently, the points (d, δ(c,d)) are approximately uniformly distributed in [0, c] × [0, c]: The two-dimensional discrepancy of {(d,δ (c, d)): 1 ⩽ d ⩽ c and (c, d)= 1} in [0,c] × [0,c] is Oe(ce-1/2).

This note is related to a nice short paper of M. Ochiai. We prove in a very fast way that the two-parameter family of Heegaard diagrams, constructed by Ochiai, encodes the genuine three-sphere. The result is obtained, up to isotopy, by using a sequence of only three moves in this order: a Whitehead–Zieschang reduction, a band sum and a cancellation of a handle.

Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

This complement includes stronger versions of the theorems of the paper “A positive answer to the basis problem”, practically with the same proofs.

On construit dans [R] des representants explicites des fibres principaux de fibre S 3 au-dessus de S 7 ; on en deduit des actions libres et explicites de S 3 , dont les quotients sont des spheres exotiques de dimension 7. La description de ces actions est basee sur une formule de classification des fibres. Or cette formule est incorrecte. Le theoreme 1 corrige cette formule ainsi que la formule de classification ; le theoreme 2 exhibe les indices corrects des spheres exotiques de dimension 7 qui apparaissent comme quotient des actions libres de S 3 .

A SIRS epidemiological model with two subpopulations and vital dynamics is analyzed. Both subpopulations sizes are considered constant by assuming that the birth and the death rates are equal. We consider the case where one subpopulation is a core, that is a very infectious small group, responsible for a big fraction of the incidence. For this case thresholds are determined and the main equilibrium points for the four dimensional system are shown to be globally stable by using a known Theorem of Markus on asymptotically autonomous systems. This system models the dynamics of typhoid fever , where the core is the group of food handlers . The results presented in this work are an extension of those presented in [3].

We correct two clerical errors made in the paper ”A Lower Bound for the Length of Closed Geodesics on a Finsler Manifold”.

In a preceding paper of Jin and Wang the authors describe computations concerning differential systems of the form x=y, y=-x+a 1 x 2 +a 2 xy+a 3 y 2 +a 4 x x3 +a 5 x 2 y+a 6 xy 2 +a 7 y 3 . We give proofs of the results suggested by these computations and provide a full description of the bifurcation of small-amplitude limit cycles in a subclass of these systemes

This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].

We generalize and give a simple proof of a Theorem of Blasco. For a topological space X , C(X) denotes the algebra of real valued continuous functions on X under the operation (αf + gh)(x) := αf(x) + h(x)g(x) for each f, g, h ∈ C(X) and α ∈ R. An algebra A on X means a subalgebra of C(X) which contains the constant functions. By a ∗-algebra A on X we mean an algebra on X and all bounded functions in A separate points from the closed sets. Let A be an algebra on X and π : A −→ R be a (algebra) homomorphism and α be a cardinal number. π is said to be α-evaluating if for each subset B of A with cardinality at most α there exits xB ∈ X such that π(f) = f(xB) for each f ∈ B. If α = |N|(=the cardinal number of the set of the natural numbers N), the α-evaluating homomorphism is called countable evaluating. Let α be a cardinal number. Recall that a topological space X is called αLindelof if for each open cover U has a subcover V such that the cardinality of V is at most α. A lindelof space means a |N|-Lindelof space. The following theorem is given in [1] as one of the main results. Theorem 1. Let X be a completely regular Hausdorff space. Then X is Lindelof if and only if each countable evaluating homomorphism π on a ∗-algebra A on X is point evaluating. Received June 9, 2004; accepted June 30, 2004. Communicated by Bor-Luh Lin. 2000 Mathematics Subject Classification: Primary 54C35, 46E25.

Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.

In Sec. 1 a correction is given of the estimate of the Hausdorff dimension and an estimate of the fractal dimension of a bounded subset of a Hilbert space, semiinvariant with respect to a flattening transformation. In Sec. 2 the results, proved by the author for semigroups with a continuous group parameter t∈R+≡[0, ∞), are carried over to the case when t runs through the semigroup ℑ+≡{t∈ℑ∣t⩾0} of some additive group ℑ⊂R=(−∞, ∞).

In Venkaiah [1] an algorithm for solving linear optimization problems based on the idea of the projective algorithm of Karmarkar, is proposed. The essential simplification in the new algorithm is the use of a fixed projection operator. In this way the algorithm requires onlyO(n2) operations to obtain a sufficient exact solution. In this note it is shown that in some special cases the algorithm of Venkaiah yields a feasible solution that is far from the optimal one.

Let H(Dn) be the space of holomorphic functions on the unit polydisk Dn, and let , where p, q> 0, α = (α1,…,αn) with αj > -1, j =1,..., n, be the class of all measurable functions f defined on Dn such thatwhere Mp(f,r) denote the p-integral means of the function f. Denote the weighted Bergman space on . We provide a characterization for a function f being in . Using the characterization we prove the following result: Let p> 1, then the Cesàro operator is bounded on the space .

One result concerning the series representation for the continuous Jacobi transform in Walter and Zayed [1] is improved, the same thought also can be applied to the related results in [1].

SynopsisSome disparate ideas in the literature are drawn together. The work of P. J. Olver and his associates on Lagrangians which vanish for arbitrary variations, the so-called null Lagrangians, is viewed as a parallel development to Witten's study of topological field theories. A theorem of Olver, that all hyperjacobians are expressible as divergences, and are thus candidates for the construction of null Lagrangians, is shown to follow directly from the observation that such entities appear in a power series development of the general associative product, and this technique facilitates the construction of multi-dimensional examples.

In section 8.3 of our paper "Duality and Flat Base Change on Formal Schemes" (http://arXiv.org/abs/alg-geom/9708006) some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations.

In the previous paper [T] we gave a classification of hermitian forms over the real function fieldk=R(t) and its completionsk v with respect to valuationsv trivial onR. Unfortunately in the local case the arguments given for cases A and D, in general, were not correct. Therefore the resulting local and local-global classifications obtained were incorrect. I would like also to thank Dr. D. Hoffmann for pointing out these mistakes and the referee for useful comments. Here we would like to make necessary corrections to [T]. We keep the same notation used there, except that in the first paragraph,J is not the standard involution of a quaternion division algebraD (with basis {1,i,j,ij}). All hermitian forms will be hermitian forms with respect toJ, with values inD.

In the second paragraph on page 30 [ 1 ], we wrote ‘Making a critical point precritical decreases the number of points about which we have to worry’. Indeed, we stopped worrying about these points so much, that we forgot to mention them in several statements. This includes Lemma 3.13, Theorem 3.14 and the Main Theorem. Thus, the possibility of critical points being precritical (not only sinking or recurrent) has to be added to those statements. In particular, the Main Theorem should read as follows.

The date of publication of Helmholtz's first paper on the foundations of geometry is discussed.

The concepts of parallel transport and intrinsic (Gaussian) curvature arising in the differential geometry of surfaces may be pleasantly and concretely investigated using paper models and f...

As early as the 16th century, Simon Jacob, a German reckoning master, noticed that the worst case in computing the greatest common divisor of two numbers by the Euclidean algorithm occurs if these numbers are equimultiples of two consecutive members of the Fibonacci sequence.

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

Let E be a real Banach space. A norm-one element e in E is said to be an isometric reflection vector if there exist a maximal subspace M of E and a linear isometry F : E → E fixing the elements of M and satisfying F (e) = −e. We prove that each of the conditions (i) and (ii) below implies that E is a Hilbert space. (i) There exists a nonrare subset of the unit sphere of E consisting only of isometric reflection vectors, (ii) There is an isometric reflection vector in E, the norm of E is convex transitive, and the identity component of the group of all surjective linear isometries on E relative to the strong operator topology is not reduced to the identity operator on E.

The unpublished scientific papers of Joseph Liouville located in the Institut de France in Paris have been investigated by several scholars in recent years, and it has been pointed out on a number of occasions that the Paris collection is incomplete. The present article analyzes the history of Liouville's unpublished papers and gives an initial survey of additional papers discovered in Bordeaux in 1983–1984. Besides the material originating from Liouville, the Bordeaux collection also contains many manuscripts of Charles-Francois Sturm which had long been searched for and had therefore been assumed by most to have been lost. Furthermore, there are documents connected with the transmission and editing of the unpublished papers of Evariste Galois, which were also once in Liouville's possession.

Three papers from the early history of efficient parametric estimation are reprinted with commentary: (1) Fisher (1912), "On an absolute criterion for fitting frequency curves"; (2) Engledow and Yule (1914), "The determination of the best value of the coupling-ratio from a given set of data"; and (3) Fisher (1922), "The systematic location of genes by means of crossover observations."

We show functoriality of γα in our joint paper with A. Csaszar “Further remarks on δ- and θ-modifications”.

Dans un article anterieur, nous avons etudie les langages m-disjonctifs et s-disjonctifs sur un alphabet X ainsi que les relations d'ordre partiel compatibles qui leur sont associees. Cependant, nous n'avions considere jusqu'a present que des alphabets d'au moins deux lettres. Dans cette note, nous traitons le cas d'un alphabet a une lettre et nous montrons que tout langage disjonctif sur un alphabet a une lettre est s-disjonctif

Professor Zbigniew Lipecki from the University of Wroclaw has told me that the definition of “small ball property” introduced in the paper “Metric spaces with the small ball property” by E. Behrends and V. Kadets (Studia Math. 148, No. 3, 275-287, 2001), has been investigated under different names earlier in the literature and that some of the results in this paper and in my paper quoted in the title are rediscoveries of known facts. In particular this applies to Proposition 2.3. I am grateful to Professor Lipecki for providing the following references to us: – Ch. Bandt, Mathematika 28 (1981), 206-210. – R. Duda and R. Telgarsky, Czechoslovak Math. J. 18(93) (1968), 66-82. – E. Szpilrajn-Marczewski, Fund. Math. 15 (1930), 126-27. – E. Szpilrajn-Marczewski, Fund. Math. 22 (1934), 303-311. – B. Tsaban, Real Analysis Exchange 27 (2001/2002), 757-764.

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation $$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$ wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

We give a simplified proof of an imbedding theorem by C. Fefferman [3].

The deadly coronavirus disease 2019 (COVID-19) has recently affected each corner of the world. Many governments of different countries have imposed strict measures in order to reduce the severity of the infection. In this present paper, we will study a mathematical model describing COVID-19 dynamics taking into account the government action and the individuals reaction. To this end, we will suggest a system of seven fractional deferential equations (FDEs) that describe the interaction between the classical susceptible, exposed, infectious, and removed (SEIR) individuals along with the government action and individual reaction involvement. Both human-to-human and zoonotic transmissions are considered in the model. The well-posedness of the FDEs model is established in terms of existence, positivity, and boundedness. The basic reproduction number (BRN) is found via the new generation matrix method. Different numerical simulations were carried out by taking into account real reported data from Wuhan, China. It was shown that the governmental action and the individuals' risk awareness reduce effectively the infection spread. Moreover, it was established that with the fractional derivative, the infection converges more quickly to its steady state.

In the paper [1], the authors obtain a KAM theorem for nearly integrable hamiltonian systems under the Russmann’s non-degeneracy condition, which is known to be sharpest one for small divisor conditions. However, the Remark 1.3 is wrong because we have ignored the null set − ∗, which may contain zeros of ω of high order such that (1.4) does not hold for all p ∈ . The Remark 1.3 might mislead the readers that the condition (1.5) of Theorem B is equivalent to the Russmann’s non-degeneracy condition. Actually, the Russmann’s non-degeneracy condition is equivalent to the condition (1.4) of Theorem A as proved in [1]. Under the Russmann’s non-degeneracy condition (1.4), as proved in the Remark 3.1 the condition (1.5) holds if we replace n − 1 by a sufficiently large number N depending on h, and then the conclusion of Theorem B remains valid if in the measure estimate n − 1 is replaced by N .

Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid's, which handled some basic concepts awkwardly...