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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.

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

In this short paper, we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two squares of primes and a k-th power of prime for any integer $$k \ge 3$$ . Our results improve the recent results due to Brudern (in: Sander, Steuding, Steuding (eds) From arithmetic to zeta-functions, Springer, Cham 2016). The similar method can be also applied to some related questions in this direction, and this can improve the previous results.

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.

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

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

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.

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 .

In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.

Let I be the Stanley–Reisner ideal of a simplicial complex Δ. In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I(r) in the case Δ is a union of disjoint matroids, where I(r) is the rth symbolic power of I. As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127–140, 2014) for the k-Buchsbaumness of S/I(r).

We note that Theorem 5.4 of the paper ``Characteristic cycles of perverse sheaves and Milnor fibers" (by P. Nang and K. Takeuchi, published online in Math. Zeitschrift, October 15, 2004) holds for "any" non-zero complex number $a$., Comment: 1page

We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.

It was shown by F. Wilhelm that Gromov’s filling radius of any positively curved closed Riemannian manifolds are less than that of the round sphere unless they are isometric to each other. In this short paper, we adapt his proof to see that the same is true for any positively curved closed Alexandrov spaces as well.

This short paper is based on the talk on the conference Operator Algebras and Related Topics held on July 23–27, 2010, Beijing. The author surveys recent developments of the noncommutative gravity in joint works with Chaichian, Tureanu, Sun, Wang, Xie and Zhang.

In this short paper, we study the existence of common universal series for uncountable families of specific linear operators. In particular we deal with some derived forms of Seleznev’s theorem and we obtain common universal elements in the space of formal power series in several complex variables.

This paper deals with Schrodinger operators as they were treated by Kato - Ikebe [3]. It is shown that every element of the domain of definition of the adjoint of such an operator has locally square integrable distributional derivatives up to the order 2. For this the potential of the Schrodinger operator must fulfil a local Stummel condition; if the potential is only locally square integrable a somewhat weaker statement is possible for three dimensions (see remark 2 at the end of this paper).

This paper presents a review of the elementary theories on the bending of straight and curved beams, on plates and shells, using asymptotic approximations of the basic linearized equations of elasticity in three dimensions. The maximun norm has been chosen to specify the orders of magnitude of the quantities involved. The expansions are given as usual in terms of the small geometrical parameter characterizing the thinness of the structure. Most of the ideas and results are well known. Nevertheless, in the cases where more than one small parameter may be involved, such as small curvatures (shallow structures) or the small loading parameter used to linearize the equations of elasticity, the discussion on the limits of validity of the different theories lead to some interesting newer aspects. Moreover, the main ideas presented in this paper concerning multiple parameter expansions may be applied to discuss the behaviour of the structures and to obtain valuable analytical results in more complicated situations such as moderate and strong anisotropy, dynamic problems, stability etc.

Supplement to the paper cited in the title. One presents the results of the calculations on an electronic computer, by the method developed in that paper, proving Minkowski's conjecture regarding the critical determinant of the region ∣x∣p+∣y∣p 2.035 and for 1.01⩽p⩽1.99.

In the mentioned paper of Hadwiger an upper bound is given for α (A)·N(A), where α (A) is the density of covering ℝn by lattice translates ofA⊂ℝn andN(A) is the number of neighbours ofA. Using “dual” descriptions of α (A) andN(A), in the paper a sharpening of Hadwiger's inequality as well as a lower bound for α (A)·N(A) are proved.

Why do I think that Z.v.G.II was an epoch-making paper? (1) It was the paradigm for subsequent efforts to classify the possible structures for any mathematical object. Hawkins [15] documents the fact that Killing’s paper was the immediate inspiration for the work of Cartan, Molien, and Maschke on the structure of linearassociative algebras which culminated in Wedderburn’s theorems. Killing’s success was certainly an example which gave Richard Brauer the will to persist in the attempt to classify simple groups. (2) Weyl’s theory of the representation of semi-simple Lie groups would have been impossible without ideas, results, and methods originated by Killing in Z.v.G.II. Weyl’s fusion of global and local analysis laid the basis for the work of Harish-Chandra and the flowering of abstract harmonic analysis. (3) The whole industry of root systems evinced in the writings of I. Macdonald, V. Kac, R. Moody, and others started with Killing. For the latest see [21]. (4) The Weyl group and the Coxeter transformation are in Z.v.G.II. There they are realized not as orthogonal motions of Euclidean space but as permutations of the roots. In my view, this is the proper way to think of them for general Kac-Moody algebras. Further, the conditions for symmetrisability which play a key role in Kac’s book [17] are given on p. 21 of Z.v.G.II. (5) It was Killing who discovered the exceptional Lie algebra E8, which apparently is the main hope for saving Super-String Theory—not that I expect it to be saved! (6) Roughly one third of the extraordinary work of Elie Cartan was based more or less directly on Z.v.G.II.

In this short paper we prove that even though the latest transitive operator has only one negative entry, we cannot get rid of it by simple operations. The research is motivated by, still unsolved, invariant subspace problem for positive operators.

In an unpublished paper, Araque, Hall and Magnanti considered polyhedra associated with the Capacitated Vehicle Routing Problem (CVRP) in the special case of unit demands. Among the valid and facet-inducing inequalities presented in that paper were the so-called multistar and partial multistar inequalities, each of which came in several versions. Some related inequalities for the case of general demands have appeared subsequently and the result is a rather bewildering array of apparently different classes of inequalities. The main goal of the present paper is to present two relatively simple procedures that can be used to show the validity of all known (and some new) multistar and partial multistar inequalities, in both the unit and general demand cases. The procedures provide a unifying explanation of the inequalities and, perhaps more importantly, ideas that can be exploited in a cutting plane algorithm for the CVRP. Computational results show that the new inequalities can be useful as cutting planes for certain CVRP instances.

In a paper which appeared a few years ago the authors investigated the exponential representation of functions analytic in the upper half-plane with positive imaginary part there [1]. We refer to that paper in the sequel as A-D. One of the principal results of A-D, there called Theorem A, can be extended to a considerably more general result, the proof of which is perhaps simpler than that given in A-D. We give the extended version of Theorem A here. We will use the notations and results of A-D without further explanations. Before we present the extension and its proof we would like to add some information that by oversight was omitted from the list of fundamental properties of the functions in the class P given in Section 1 of A-D. In such a comprehensive review one should mention that the classical theorem on representation of a positive harmonic function in a circle by a Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of L. H. Loomis [3] were not given: XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0 common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r. XVIII. I f for two values of 0 in the interval 0 < 0 < ~

In a recent paper M.C. Cheng proposed new criteria for the simplex algorithm which guarantee that (i) a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution, (ii) a basic variable of a basic solution will remain basic in an optimal basic solution. This comment gives (i) a slight generalization of the first result and (ii) a counterexample to the second proposition.