The inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?
    

     

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T, we show that the Schröder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following:

1. 1.
   For any U-rank-1 type q ∈ S(acl ^eq (?)) and any automorphism f of the monster model C, there is some n < ω such that f ⁿ (q) is not almost orthogonal to q ? f(q) ? … ? f ^n?1(q)
    
2. 2.
   T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic.
    

We conclude that for countable, weakly minimal theories, the Schröder-Bernstein property is absolute between transitve models of ZFC.     

Linear Optimization with Box Constraints in Banach Spaces

Let X be a partially ordered real Banach space, let a,b∈X with a≤b. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max 〈φ,x〉, s.t. a≤x≤b, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author’s doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions:

1. (i)
   When is a closed subspace of a BOP space a BOP space?
    
2. (ii)
   When is the range of a bounded linear map a BOP space?
    
3. (iii)
   Is the quotient space of a BOP space a BOP space?
    

     

On a problem of P. Hall for Engel words

Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ ^*(G), respectively. The following problems are generally attributed to P. Hall:

1. (I)
   If π is a set of primes and |G : θ ^*(G)| is a finite π-group, is θ(G) also a finite π-group?
    
2. (II)
   If θ(G) is finite and G satisfies maximal condition on its subgroups, is |G : θ ^*(G)| finite?
    
3. (III)
   If the set \({\{\theta(g_1,\ldots,g_n) \;|\; g_1,\ldots,g_n\in G\}}\) is finite, does it follow that θ(G) is finite?
    

We investigate the case in which θ is the n-Engel word e _n  = [x, _n y] for \({n\in\{2,3,4\}}\) .

     

More on stability of almost surjective <Emphasis Type="Italic">ε</Emphasis>-isometries of Banach spaces

Let X and Y be two Banach spaces, and f: X → Y be a standard ε-isometry for some ε ≥ 0. In this paper, by using a recent theorem established by Cheng et al. (2013–2015), we show a sufficient condition guaranteeing the following sharp stability inequality of f: There is a surjective linear operator T: Y → X of norm one so that

$$\left\| {Tf(x) - x} \right\| \leqslant 2\varepsilon , for all x \in X.$$

As its application, we prove the following statements are equivalent for a standard ε-isometry f: X → Y:

1. (i)
   lim inf _t→∞ dist(ty, f(X))/|t| < 1/2, for all y ∈ S _Y ;
    
2. (ii)
   \(\tau(f)\equiv sup_{y\epsilon S_{Y}}\) lim inf _t→∞dist(ty, f(X))/|t| = 0;
    
3. (iii)
   there is a surjective linear isometry U: X → Y so that

   $$\left\| {f(x) - Ux} \right\| \leqslant 2\varepsilon , for all x \in X.$$
    

This gives an affirmative answer to a question proposed by Vestfrid (2004, 2015).     

Derivations cocentralizing multilinear polynomials on left ideals

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X ₁, . . . , X _t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that

$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$

for all \({x_1,\ldots,x_t\in\lambda}\). Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds:

1. (1)
   f (X₁, . . . , X _t ) is central-valued on eRCe;
    
2. (2)
   λ(d + δ)(λ) = 0 and f (X₁, . . . , X _t )² is central-valued on eRCe;
    
3. (3)
   char R = 2 and eRCe satisfies st ₄(X ₁, X ₂, X ₃, X ₄), the standard polynomial identity of degree 4.
    

     

Non-singular Solutions of Normalized Ricci Flow on Noncompact Manifolds of Finite Volume

The main result of this paper shows that if g(t) is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold M of finite volume, then the Euler characteristic number χ(M)≥0. Moreover, if χ(M)≠0, there exists a sequence of times t _k →∞, a double sequence of points \(\{p_{k,l}\}_{l=1}^{N}\) and domains \(\{U_{k,l}\}_{l=1}^{N}\) with p _k,l∈U _k,l satisfying the following:

1. (i)
   \(\mathrm{dist}_{g(t_{k})}(p_{k,l_{1}},p_{k,l_{2}})\rightarrow\infty\) as k→∞, for any fixed l₁≠l₂;
    
2. (ii)
   for each l, (U _k,l,g(t _k ),p _k,l) converges in the \(C_{\mathrm{loc}}^{\infty}\) sense to a complete negative Einstein manifold (M _∞,l ,g _∞,l ,p _∞,l ) when k→∞;
    
3. (iii)
   \(\operatorname {Vol}_{g(t_{k})}(M\backslash\bigcup_{l=1}^{N}U_{k,l})\rightarrow0\) as k→∞.
    

     

Multi-invariant measures and subsets on nilmanifolds

Given a Z^r-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action

1. (1)
   the factor measure has zero entropy under every element of the action
    
2. (2)
   the factor action is virtually cyclic.
    

We also deduce a rigidity property for invariant closed subsets.

     

On semistable principal bundles over a complex projective manifold,II

Let (X, ω) be a compact connected Kähler manifold of complex dimension d and \({E_G\,\longrightarrow\,X}\) a holomorphic principal G–bundle, where G is a connected reductive linear algebraic group defined over \({\mathbb{C}}\). Let Z(G) denote the center of G. We prove that the following three statements are equivalent:

1. (1)
   There is a parabolic subgroup \({P\,\subset\,G}\) and a holomorphic reduction of structure group \({E_P\,\subset\,E_G}\) to P, such that the corresponding L(P)/Z(G)–bundle

   $E_{L(P)/Z(G)}\,:=\,E_P(L(P)/Z(G))\,\longrightarrow\,X$

   admits a unitary flat connection, where L(P) is the Levi quotient of P.
    
2. (2)
   The adjoint vector bundle ad(E _G ) is numerically flat.
    
3. (3)
   The principal G–bundle E _G is pseudostable, and

   $\int\limits_X c_2({\rm ad}(E_G))\omega^{d-2}\,=\,0.$
    

If X is a complex projective manifold, and ω represents a rational cohomology class, then the third statement is equivalent to the statement that E _G is semistable with c ₂(ad(E _G )) = 0.

     

On the sectionwise connectedness of a contingent

Let X be a real normed space and let f: ? → X be a continuous mapping. Let T _f (t ₀) be the contingent of the graph G(f) at a point (t ₀, f(t ₀)) and let S ⁺ ? (0,∞) × X be the “right” unit hemisphere centered at (0, 0 _X ). We show that

1. 1.
   If dimX < ∞ and the dilation D(f, t ₀) of f at t ₀ is finite then T _f (t ₀) ∩ S ⁺ is compact and connected. The result holds for \(T_f (t_0 ) \cap \overline {S^ + } \) even with infinite dilation in the case f: [0,∞) → X.
    
2. 2.
   If dimX = ∞, then, given any compact set F ? S ⁺, there exists a Lipschitz mapping f: ? → X such that T _f (t ₀) ∩ S ⁺ = F.
    
3. 3.
   But if a closed set F ? S ⁺ has cardinality greater than that of the continuum then the relation T _f (t ₀) ∩ S ⁺ = F does not hold for any Lipschitz f: ? → X.
    

     

Chains of saturated models in AECs

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)

1. (1)
   The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.
    
2. (2)
   There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).
    
3. (3)
   There exists a unique limit model of size \(\lambda \).
    

Our proofs use independence calculus and a generalization of averages to this non first-order context.

     

Flow equivalence of sofic shifts

We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients.

1. (1)
   An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them.
    
2. (2)
   The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs.
    

     

Some topics in the history of harmonic analysis in the twentieth century

In December 2015 I gave a series of six lectures at the Indian Institute of Science in which I sketched the thematic development of some of the main techniques and results of 20th-century harmonic analysis. The subjects of the lectures were, briefly, as follows:

1. 1.
   Fourier series, 1900-1950.
    
2. 2.
   Singular integrals (part I).
    
3. 3.
   H ^p , BMO, and singular integrals (part II).
    
4. 4.
   Littlewood-Paley theory: the history of a technique.
    
5. 5.
   Harmonic analysis on groups.
    
6. 6.
   Wavelets.
    

I emphasized interconnections, both the way in which the material in the first lecture provided the roots out of which most of the developments in the other lectures grew, and the ways in which those developments interacted with each other. I included sketches of as many proofs as the time would permit: some very brief, but some fairly complete, especially those whose methodology is an important part of the subject. Much was omitted, of course, and there was a natural bias toward the areas where I have spent periods of my own mathematical life. Many developments, particularly those of the final quarter-century, received at most a brief mention.This paper is a written account of these lectures with a few more details fleshed out, a few topics reorganized, and a few items added. I hope that others may find it an interesting narrative and a useful reference, and that it may lead some of them to share my enjoyment of exploring the original sources. I have tried to provide the references to those sources wherever possible, and for the more recent developments I also provide references to various expository works as the occasion arises. For the pre-1950 results discussed here and their proofs, however, there is one canonical reference, which I give here once and for all: Antoni Zygmund’s treatise [96]. (The more fundamental ones can also be found in Folland [29].)     

Speed,Gradient and Workrate in Uphill Running

A series of treadmill experiments is described concerned with a runner's speed, heart-rate and the gradient. Together with the results of similar experiments, some of them carried out over 50 years ago, the results suggest that for a given heart-rate, log(speed) is linearly related to gradient, and that for a given gradient, heart-rate is linearly related to speed. The results suggest:

1. 1)
   that athletes who run p% faster on the level will run p% faster up a slope, if they maintain the same heart-rate;
    
2. 2)
   that athletes will use the same number of heart beats running up a hill of uniform slope no matter how fast or slowly they run;
    
3. 3)
   that athletes should run directly up any slope of less than about 20° and try to zigzag up slopes greater than this.
    

     

Mean dimension of $${\mathbb{Z}^k}$$-actions

Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a \({\mathbb{Z}^k}\)-action on a compact metric space X, we study the following three problems closely related to mean dimension.

1. (1)
   When is X isomorphic to the inverse limit of finite entropy systems?
    
2. (2)
   Suppose the topological entropy \({h_{\rm top}(X)}\) is infinite. How much topological entropy can be detected if one considers X only up to a given level of accuracy? How fast does this amount of entropy grow as the level of resolution becomes finer and finer?
    
3. (3)
   When can we embed X into the \({\mathbb{Z}^k}\)-shift on the infinite dimensional cube \({([0,1]^D)^{\mathbb{Z}^k}}\)?
    

These were investigated for \({\mathbb{Z}}\)-actions in Lindenstrauss (Inst Hautes Études Sci Publ Math 89:227–262, 1999), but the generalization to \({\mathbb{Z}^k}\) remained an open problem. When X has the marker property, in particular when X has a completely aperiodic minimal factor, we completely solve (1) and a natural interpretation of (2), and give a reasonably satisfactory answer to (3).A key ingredient is a new method to continuously partition every orbit into good pieces.     

On relaxations of contraction constants and Caristi’s theorem in <Emphasis Type="Italic">b</Emphasis>-metric spaces

In this paper, the following facts are stated in the setting of b-metric spaces.

1. (1)
   The contraction constant in the Banach contraction principle fully extends to [0, 1), but the contraction constants in Reich’s fixed point theorem and many other fixed point theorems do not fully extend to [0, 1), which answers the early stated question on transforming fixed point theorems in metric spaces to fixed point theorems in b-metric spaces.
    
2. (2)
   Caristi’s theorem does not fully extend to b-metric spaces, which is a negative answer to a recent Kirk–Shahzad’s question (Remark 12.6) [Fixed Point Theory in Distance Spaces. Springer, 2014].
    

     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Perturbation of chains of de Branges spaces

We investigate the structure of the set of de Branges spaces of entire functions which are contained in a space L²(μ). Thereby, we follow a perturbation approach. The main result is a growth dependent stability theorem. Namely, assume that measures μ₁ and μ₂ are close to each other in a sense quantified relative to a proximate order. Consider the sections of corresponding chains of de Branges spaces C₁ and C₂ which consist of those spaces whose elements have finite (possibly zero) type with respect to the given proximate order. Then either these sections coincide or one is smaller than the other but its complement consists of only a (finite or infinite) sequence of spaces.

Among other situations, we apply—and refine—this general theorem in two important particular situations

1. (1)
   the measures μ₁ and μ₂ differ in essence only on a compact set; then stability of whole chains rather than sections can be shown
    
2. (2)
   the linear space of all polynomials is dense in L²(μ₂); then conditions for density of polynomials in the space L²(μ₂) are obtained.
    

In the proof of the main result, we employ a method used by P. Yuditskii in the context of density of polynomials. Another vital tool is the notion of the index of a chain, which is a generalisation of the index of determinacy of a measure having all power moments. We undertake a systematic study of this index, which is also of interest on its own right.

     

Semistability and Restrictions of tangent bundle to curves

We consider all complex projective manifolds X that satisfy at least one of the following three conditions:

1. (1)
   There exists a pair \({(C\,,\varphi)}\) , where C is a compact connected Riemann surface and

   $\varphi\,:\, C\,\longrightarrow\, X$

   a holomorphic map, such that the pull back \({\varphi^* {\it TX}}\) is not semistable.
    
2. (2)
   The variety X admits an étale covering by an abelian variety.
    
3. (3)
   The dimension dim X ≤ 1.
    

We prove that the following classes are among those that are of the above type.

• All X with a finite fundamental group.
• All X such that there is a nonconstant morphism from \({{\mathbb C}{\mathbb P}^1}\) to X.
• All X such that the canonical line bundle K _X is either positive or negative or \({c_1(K_X)\,\in\,H^2(X,\, {\mathbb Q})}\) vanishes.
• All X with \({{\rm dim}_{\mathbb C} X\, =\,2}\).

     

Leibniz’s Ontological Proof of the Existence of God and the Problem of »Impossible Objects«

The core idea of the ontological proof is to show that the concept of existence is somehow contained in the concept of God, and that therefore God’s existence can be logically derived—without any further assumptions about the external world—from the very idea, or definition, of God. Now, G.W. Leibniz has argued repeatedly that the traditional versions of the ontological proof are not fully conclusive, because they rest on the tacit assumption that the concept of God is possible, i.e. free from contradiction. A complete proof will rather have to consist of two parts. First, a proof of premise

1. (1)
   God is possible.
    

Second, a demonstration of the “remarkable proposition”

1. (2)
   If God is possible, then God exists.
    

The present contribution investigates an interesting paper in which Leibniz tries to prove proposition (2). It will be argued that the underlying idea of God as a necessary being has to be interpreted with the help of a distinguished predicate letter ‘E’ (denoting the concept of existence) as follows:

1. (3)
   \(g=_{\mathrm{df}} \,\upiota x\square E(x)\).
    

Proposition (2) which Leibniz considered as “the best fruit of the entire logic” can then be formalized as follows:

1. (4)
   \(\diamondsuit E(\upiota x\square E(x)) \rightarrow \, E(\upiota x\square E(x))\).
    

At first sight, Leibniz’s proof appears to be formally correct; but a closer examination reveals an ambiguity in his use of the modal notions. According to (4), the possibility of the necessary being has to be understood in the sense of something which possibly exists. However, in other places of his proof, Leibniz interprets the assumption that the necessary being is impossible in the diverging sense of something which involves a contradiction. Furthermore, Leibniz believes that an »impossible thing«, y, is such that contradictory propositions like \(\hbox {F}(y)\) and \(\lnot F(y)\) might both be true of y. It will be argued that the latter assumption is incompatible with Leibniz’s general views about logic and that the crucial proof is better reinterpreted as dealing with the necessity, possibility, and impossibility of concepts rather than of objects. In this case, the counterpart of (2) turns out to be a theorem of Leibniz’s second order logic of concepts; but in order to obtain a full demonstration of the existence of God, the counterpart of (1), i.e. the self-consistency of the concept of a necessary being, remains to be proven.