German history experienced: My studies, my teachers

Conclusion  I wanted to describe my meetings with personalities who have influenced my Me and show how different from today our world half a century ago actually was. I also wanted with gratitude to bring to life the memory of people who were not only important scientists but also—Menschen. This article is based on a talk given at the University of Stuttgart, October 22, 1998. The author and the Editor thank George Seligman for his advice in preparing the present version. Much of the material appeared also in “Menschen, mein Studium, meine Lehrer” inMitteilungen der DMV 4 (1999), 43-52.     

Confusion about Bohm

Conclusion  It is important to distinguish between physics, which is an experimental science, andphysicists, who are people. The latter are most certainlynot objective. Thus, Lipschütz-Yevick’s assertion that Ruskai says that the publication of Bohm’s controversial articles in thePhysical Review is evidence of the objectivity of the establishment towards [Bohm] is not supported by my statement It should be noted that even though studying the foundations of quantum mechanics has long been far from the mainstream, it has never been suppressed. The papers of Bohm, Bell,et al. were published in reputable journals, … Reasonable people may disagree on the significance of a particular theory or individual’s contribution. It is here, rather than in the physicsper se, that questions of social influence are likely to arise. I have commented elsewhere, e.g., [17], on the role that gender sometimes plays. In a subsequent article, I will also discuss the distinction between the effect of the social and political climate on the development of the careers of individuals and the development of physics. The articles by Cronin and Lipschütz-Yevick have stimulated me to think anew about a number of issues related to Bohmian mechanics, for which a full discussion requires clarification of some technical issues regarding the EPR experiment and non-locality. These will be discussed in a forthcoming article.     

Generalizing the Disappearing Act: A Reply to István Aranyosi

In “The Reappearing Act” István Aranyosi postulates a new way of seeing to solve a puzzle posed in “The Disappearing Act;” an object that is exactly shaded can be seen simply by virtue of its contrast with its environment – just like a shadow. This object need not reflect, refract, absorb or block light. To undermine the motive for this heretical innovation, I generalize the puzzle to situations involving inexact shading. Aranyosi cannot extend his solution to these variations because he needs to conserve principles of camouflage. On the bright side, the solution to the puzzle that I propose in my book Seeing Dark Things does extend to these variations.     

Fermats ?ad?quare? – und kein Ende?

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, ?quare and ad?quare, which do not differ semantically but are used by him obviously in different meanings. While ?quabitur is used unambiguously in the sense of “is equal” the meaning of ad?quabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word ad?quare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “?quare” and “ad?quare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that ad?quare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule spécial, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words ?quare and ad?quare. The same distinction is made by Nicolas Bourbaki introducing his ?théorie égalitaire?. Notwithstanding: both verbs stand for a ?relation d’égalité?. On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.     

Large sets of <Emphasis Type="Italic">t</Emphasis>-designs from groups

This paper addresses questions related to the existence and construction of large sets of t-(v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa’s 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of “highlight” topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them. (Communicated by Peter Horák)     

A non-well-founded primitive recursive tree provably well-founded for co-r.e. sets

 We construct by diagonalization a non-well-founded primitive recursive tree, which is well-founded for co-r.e. sets, provable in Σ₁ ⁰. It follows that the supremum of order-types of primitive recursive well-orderings, whose well-foundedness on co-r.e. sets is provable in Σ₁ ⁰, equals the limit of all recursive ordinals ω₁ ^ck . RID="ID=" <E5>Mathematics Subject Classification (2000):&ensp;03B30</E5>, 03F15 RID="ID=" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium f&uuml;r Bildung, Wissenschaft, Forschung und Technologie. RID="ID=" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 &ndash; these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. M&Ouml;LLERFELD for remarks on a previous title. Received: 27 July 2000 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):&ensp;03B30</E5>, 03F15 RID=" ID=" Supported by the Deutschen Akademie der Naturforscher Leopoldina grant #BMBF-LPD 9801-7 with funds from the Bundesministerium f&uuml;r Bildung, Wissenschaft, Forschung und Technologie. RID=" ID=" I would like to thank A. SETZER for his hospitality during my stay in Uppsala in December 1998 &ndash; these investigations are inspired by a discussion with him; S. BUSS for his hospitality during my stay at UCSD and for valuable remarks on a previous version of this paper; and M. M&Ouml;LLERFELD for remarks on a previous title.     

The asymptotic expansion of the solution to the Cauchy problem for a parabolic equation in a perforated space

We consider the Cauchy problem for a second-order parabolic equation with rapidly oscillating coefficients of the forma(ɛ ⁻ x,ɛ ^−2k t) in a perforated space ℝ _ɛ ⁿ . We construct a complete asymptotic expansion for the solution of the said problem and obtain an estimate for the remainder term in this expansion. Bibliography: 8 titles. To my dear Teacher, Olga Arsenievna Oleinik This research was supported in part by Grant MIE000 from the International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 19, pp. 000-000. 0000.     

Singular Integrals and Potential Theory (II)

In this note I give an extension of the T1-Theorem of David and Journé. In this extension the standard estimates on the kernel are replaced by “average standard estimates” but the classical proof is essentially identical. I then use that extension to prove the L^p-boundedness of some natural rough singular integrals. The applications of these singular integrals in Potential Theory has already been described in my previous paper in the area. Received: September 2008     

Recollections of Fred Almgren

While visiting Princeton for the 1997–98 year I have had the privilege of using Fred’s office and helping Jean to go through his papers. It has been moving to see the magnitude of Fred’s helpful correspondence with students and colleagues, some of which Jean and I returned to them with an invitation to contribute short recollections. This article gives some selections from their responses. For more recollections see Mackenzie’s article on “Fred Almgren: Lover of Mathematics, Family, and Life’s Adventures” (Notices AMS 44, 1997, 1102–1106) and the memorial issue ofExperimental Mathematics (Vol. 6, 1997, 1–12).     

Non-constructive properties of cardinal numbers

Let Σ be some standard set theory (Eg. Zermelo Fraenkel or Von Neumann-Bernays-Godel) which does not contain the axiom of choice. Using Σ as the underlying set theory, we shall study operations on infinite cardinals, closely related to exponentiation, and compare the results with known results about exponentiation. I should like to take this opportunity to thank both my husband, Herman and my son, Arthur, for their help in preparing this paper.     

‘Seeing is believing’ and ‘believing is seeing’

The principal concern of my paper is a distinction between two ways of appreciating works of art, characterised here in terms of the phrases ‘seeing is believing’ and ‘believing is seeing’. I examine this distinction in the light of an epistemological requirement at times at least grounded in what David Davies, in his Art as Performance, refers to as the ‘common sense theory of art appreciation’ in order to assess exactly what aspect of the philosophical approach generally known as aesthetic empiricism his account commits him to reject. I argue that the ‘experiential requirement’, if only conceived in a slightly broader way than is usual, might very well have an important role to play not only in the appropriate appreciation of works that do not demonstrate the need for such a requirement (primarily works of late modern and conceptual art), but also in the ontological account Davies himself favours.     

Degree Sequences and the Existence of <Emphasis Type="Italic">k</Emphasis>-Factors

We consider sufficient conditions for a degree sequence π to be forcibly k-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially k-factor graphical. We first give a theorem for π to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most β ≥ 0. These theorems are equal in strength to Chvátal’s well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for π to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from k = 1 to k = 2, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a k-factor will increase superpolynomially in k. This suggests the desirability of finding a theorem for π to be forcibly k-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any k ≥ 2, based on Tutte’s well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness.     

Additive Completion of Lacunary Sequences

To the memory of Pál Erdős Thirty years ago I read the following question of Erdőos [4]: "Does there exist a sequence with so that every sufficiently large number is of the form ? $10" I sent my solution to Erdős in a letter (in Hungarian). He translated my letter into English and sent it to the Canadian Math. Bulletin; this became my first paper to appear. In this paper we will find, among others, the best value of the constant c in the above question, which was also asked by Erdős. Received March 30, 2000 RID="*" ID="*" Supported by Hungarian National Foundation for Scientific Research, Grants No. T 025617 and T 29759.     

Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive δ such that, for every non-trivial measurable set S⊂ℝ, there is a point at which both the lower densities of S and of ℝ∖S are at least δ. The problem of determining the supremum of possible values of this δ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.     

Immersions of spheres and algebraically constructible functions

Let Λ be an algebraic set and let (n is even) be a polynomial mapping such that for each there is r(λ) > 0 such that the mapping g _λ  =  g(· , λ) restricted to the sphere S ⁿ (r) is an immersion for every 0  <  r  <  r (λ), so that the intersection number I(g _λ|S ⁿ (r)) is defined. Then is an algebraically constructible function. I. Karolkiewicz and A. Nowel supported by the grant BW/5100-5-0286-7.     

Infima of superharmonic functions

Let Ω be a Greenian domain in ℝ ^d , d≥2, or—more generally—let Ω be a connected -Brelot space satisfying axiom D, and let u be a numerical function on Ω, , which is locally bounded from below. A short proof yields the following result: The function u is the infimum of its superharmonic majorants if and only if each set {x:u(x)>t}, t∈ℝ, differs from an analytic set only by a polar set and , whenever V is a relatively compact open set in Ω and x∈V.     

Existence of statistical solutions of a stochastic Karman system in a bounded region

The existence of a statistical solution to the stochastic Karman system of equations is proved without imposing constraints on the growth of the initial probability measure. Estimates of the moments of the solution are obtained, and the phase space is defined more precisely. The existence of a solution of the Cauchy problem for the corresponding Kolmogorov equation is given under the assumption that the initial mean energy is finite. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 22–37, July, 1995. I wish to express my gratitude to A. V. Fursikov for his interest in my work.     

Brauer groups,embedding problems,and nilpotent groups as Galois groups

Let ℚ _ab denote the maximal abelian extension of the rationals ℚ, and let ℚ_abnil denote the maximal nilpotent extension of ℚ _ab . We prove that for every primep, the free pro-p group on countably many generators is realizable as the Galois group of a regular extension of ℚ_abnil(t). We also prove that ℚ_abnil is not PAC (pseudo-algebraically closed). This paper was inspired by the author's participation in a special year on the arithmetic of fields at the Institute for Advanced Studies at the Hebrew University of Jerusalem. I would like to express my appreciation to the Institute for its hospitality, and to the organizers, especially Moshe Jarden. Partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund-Japan Technion Society Research Fund.