On mappings preserving orthogonality of non-singular vectors

Letq be a regular quadratic form on a vector space (V,F) and letf be the bilinear form associated withq. Then, \(\dot V: = \{ z \in V|q(z) \ne 0\} \) is the set of non-singular vectors ofV, and forx, y ∈ \(\dot V\) , ?(x, y) ?f(x, y) ²/(q(x) · q(y)) is theq-measure of (x, y), where ?(x,y)=0 means thatx, y are orthogonal. For an arbitrary mapping \(\sigma :\dot V \to \dot V\) we consider the functional equations $$\begin{gathered} (I)\sphericalangle (x,y) = 0 \Leftrightarrow \sphericalangle (x^\sigma ,y^\sigma ) = 0\forall x,y \in \dot V, \hfill \\ (II)\sphericalangle (x,y) = \sphericalangle (x^\sigma ,y^\sigma )\forall x,y \in \dot V, \hfill \\ (III)f(x,y)^2 = f(x^\sigma ,y^\sigma )^2 \forall x,y \in \dot V, \hfill \\ \end{gathered} $$ and we state conditions on (V,F,q) such thatσ is induced by a mapping of a well-known type. In case of dimV ∈N?{0, 1, 2} ∧ ∣F∣ > 3, each of the assumptions (I), (II), (III) implies that there exist aρ-linear injectionξ :V →V and a fixed λ ∈F?{0} such thatF x ^σ =F x ^ξ ?x ∈ \(\dot V\) andf(x ^ξ,y ^ξ)=λ · (f(x, y))^ρ ?x, y ∈V. Moreover, (II) implies ρ =id _F ∧q(x ^ξ) = λ ·q(x) ?x ∈ \(\dot V\) , and (III) implies ρ=id _F ∧ λ ∈ {1,?1} ∧x ^σ ∈ {x ^ξ, ?x ^ξ} ?x ∈ \(\dot V\) . Other results obtained in this paper include the cases dimV = 2 resp. dimV ?N resp. ∣F∣ = 3.     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.     

Über die Glattheit von Quotientenabbildungen

Let G be a semisimple algebraic group acting on a factorial Gorenstein algebra S. Let X:=Spec S, Y:=Spec S^G and π:X→Y be the quotient map. The main results are:

1. Let x be a smooth point of X whose orbit has maximal dimension and such that π(x) is a smooth point of Y. Then π is smooth at x.
2. Let S be positively graded and let χ_S(t) be its generating function which is a rational function. Then: deg χ_S≦deg \(X_{S^G } \) .

     

Analytische Fortsetzbarkeit von zufälligen Potenzreihen bezüglich abhängiger Zufallsvariabler

Рассматриваются слу чайная величина \(\mathfrak{X} = (X_n (\omega ))\) , удовлетворяющая усл овиюE(X _n ⁴ )≦M, и соответствующ ий случайный степенн ой ряд \(f_x (z;\omega ) = \mathop \sum \limits_{n = 0}^\infty a_n X_n (\omega )z^n\) . Устанавливаются тео ремы непродолжимост и почти наверное:

1. дляf _x при условиях с лабой мультипликати вности на \(\mathfrak{X}\) ,
2. для \(f_{\tilde x}\) , где \(\mathop \mathfrak{X}\limits^ \sim = (\mathop X\limits^ \sim _n )\) есть подп оследовательность в \(\mathfrak{X}\) ,
3. для по крайней мере од ного из рядовf _x′ илиf _x″ , где \(\mathfrak{X}'\) и \(\mathfrak{X}''\) — некоторые п ерестановки \(\mathfrak{X}\) , выбираемые универс ально, т. е. независимо от коэффициентовa _n .

     

On optimal backward perturbation bounds for the linear least squares problem

Consider the linear least squares problem min _x ‖b?Ax‖₂ whereA is anm×n (m<n) matrix, andb is anm-dimensional vector. Lety be ann-dimensional vector, and let ηls(y) be the optimal backward perturbation bound defined by $$\eta _{LS} (y) = \inf \{ ||F||_F :y is a solution to \mathop {min}\limits_x ||b - (A + F)x||_2 \} .$$ . An explicit expression of ηls(y) (y≠0) has been given in [8]. However, if we define the optimal backward perturbation bounds ηmls(y) by $$\eta _{MLS} (y) = \inf \{ ||F||_F :y is the minimum 2 - norm solution to \mathop {min}\limits_x ||b - (A + F)x||_2 \} ,$$ , it may well be asked: How to derive an explicit expression of ηmls(y)? This note gives an answer. The main result is: Ifb≠0 andy≠0, then ηmls(y)=ηls (y).     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

On some incidence structures

In this paper,suggested by André's papers ([2), [3]), we construct geometrical structures (X,?,//}) where X is a finite set of points, ? is a set of lines, and // is an equivalence relation on ?. These constructions are made starting with a finite and not empty set X and a permutation group G which is 2-transitive on X and such that the stabilizer of two distinct points of X is different from the identical subgroup. We look for conditions such that the structure (X, ?) is a (3,q)-Steiner system. We remember that a (3,q)-Steiner system is a pair (X,B), where X is a set of elements (called points), B is a system of subsets of X (called blocks), such that:

1. every block contains q points exactly;
2. given three distinct points x,y,z of X, there is exactly one subset of X belonging to B and containing x,y,z.

At the end we construct such a system with the help of a nearskewfield (according to Zassenhaus [7], [8]).     

Green’s relations and regularity for semigroups of transformations that preserve double direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let $T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.$ Then $T_{E^{*}}(X)Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

TE*(X)={a ? TX:"x,y ? X,(x,y) ? E?(xa,ya) ? E}.T_{E^*}(X)=\{\alpha\in T_X:\forall x,y\in X,(x,y)\in E\Leftrightarrow(x\alpha,y\alpha)\in E\}.      9. Les F-reguliers a gauche      A. Batbedat 《Semigroup Forum》1985,31(1):69-86 The concept of left F-regular semigroups was first defined by Batbedat at the Oberwolfach meeting in 1981. It generalizes the notion of F-regular semigroup introduced by Edwards [4], itself a generalization of the F-inverse semigroups defined by McFadden/O’Carroll [6] In the present paper we generalize the results of [4] and [6] by defining two preorders and ? on a monoid S with a distinguished band E, as follows: iff x=ay for some a∈E xδy iff x=yb for some b∈E When S is regular orthodox and E=E(S), is the preorder of [1] p. 29 and is the order of [1] p. 31 (the order of [4]): in fact is the natural partial order introduced by Nambooripad [7]. In b), we define the relation Σ on S: xΣy iff exe=eye for some e∈E Then we consider the congruence σ generated by Σ. DEFINITION. S is left FE-monoid if each σ-class contain a greatest element with respect to . PARTICULAR CASES. When S is regular, S is left FE-regular. When S is regular orthodox and E=E(S), S is left F-regular. We describe the structure of left F-regular semigroups like in [1], [2], [4] and [6]. Note that every left F-regular semigroup is a gammasemigroup [3] Particular Cases (gamma morphism) and applications (congruences).       10. Transformations of polar Grassmannians preserving certain intersecting relations      Wen Liu  Mark Pankov  Kaishun Wang 《Journal of Algebraic Combinatorics》2014,40(2):633-646 Let Π be a polar space of rank n≥3. Denote by \({\mathcal{G}}_{k}(\varPi)\) the polar Grassmannian formed by singular subspaces of Π whose projective dimension is equal to k. Suppose that k is an integer not greater than n?2 and consider the relation \({\mathfrak{R}}_{i,j}\) , 0≤i≤j≤k+1, formed by all pairs \((X,Y)\in{\mathcal{G}}_{k}(\varPi)\times{\mathcal{G}}_{k}(\varPi)\) such that dim p (X ⊥∩Y)=k?i and dim p (X∩Y)=k?j (X ⊥ consists of all points of Π collinear to every point of X). We show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{1,1}\) is induced by an automorphism of Π, except the case where Π is a polar space of type D n with lines containing precisely three points. If k=n?t?1, where t is an integer satisfying n≥2t≥4, we show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{0,t}\) is induced by an automorphism of Π.      11. The canonical embedding of an unramified morphism in an ��tale morphism      David Rydh 《Mathematische Zeitschrift》2011,268(3-4):707-723 We show that every unramified morphism ${X\to Y}$ has a canonical and universal factorization ${X\hookrightarrow E_{X/Y}\to Y}$ where the first morphism is a closed embedding and the second is étale (but not separated).      12. Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances      Yimin Xiao  Xinghua Zheng 《Probability Theory and Related Fields》2013,156(1-2):1-26 Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dimH R = dimP R = 2 almost surely, where dimH and dimP denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X t for arbitrarily large t > 0 is given in terms of dimH A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.      13. Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below      Luigi Ambrosio  Nicola Gigli  Giuseppe Savaré 《Inventiones Mathematicae》2014,195(2):289-391 This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$ . Our main results are: A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d). The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures . The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$ . A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.      14. The adjunction morphism for retgular differential forms and relative duality      REINHOLD HÜBL 《Compositio Mathematica》1997,106(1):87-123       15. A rotation theorem in the class of bounded univalent functions      V. V. Goryainov 《Mathematical Notes》1975,18(5):967-971 Let SM, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class SM.      16. Condensed cyclic and bridged-ring systems. VII. Acid-catalysed cyclisation of methyl substituted benzylcyclohexanols. Factors influencing the nature of cyclisation products      Jyotirmoy Chakravarty  Rupak Dasgupta  Jayanta K Ray  Usha R Ghatak 《Proceedings Mathematical Sciences》1977,86(3):317-325 The gross structures of the cyclised products from the acid-catalysed cyclisations of 2-benzyl-1, 3-dimethylcyclohexanol (6) and 1-benzyl-3, 5-dimethylcyclohexanol (11) revealing the influence of the structure of the benzylcyclohexanol derivative, and of the cyclisation reagent, have been evaluated. Polyphosphoric acid and aluminium chloride catalysed cyclisations of (6) result in the formation of predominantly 1, 4a-dimethyl-1, 2, 3, 4, 4a, 9a-hexahydrofluorene (7) and 4, 9-dimethyl-7, 8-benzobicyclo [3.3.1] non-7-ene (9) respectively. Under the same conditions, (11) produced cyclised products consisting mostly the benzobicyclo [3.3.1] non-7-ene derivative (12), characterised through 1,3-dimethyl-7,8-benzobicyclo [3.3.1] non-6-oxo-7-ene (14) by oxidation with chromium trioxide. Phosphorus pentoxide induced cyclisation of (6), followed by oxidation gave a mixture of the bridged-ring ketone (10) and the 9-oxohydrofluorene (8) in a ratio ofca. 3 : 2, whereas 2-benzyl-5-methylcyclohexanol (19) resulted in mostly 2-methyl-7,8-benzobicyclo [3.3.1] non-6-oxo-7-ene (19).      17. On the conservativity of the axiom of choice over set theory      Timothy J. Carlson 《Archive for Mathematical Logic》2011,50(7-8):777-790 We show that for various set theories T including ZF, T + AC is conservative over T for sentences of the form ${\forall x \exists ! y}$ A(x, y) where A(x, y) is a ??0 formula.      18. Power Majorization and Majorization of Sequences      G. D. Allen 《Results in Mathematics》1988,14(3-4):211-222 Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x1 ≥ x2 ≥ … ≥ xn and y1 ≥ y2 >- … ≥ yn and Σxi = Σyi. We say that \(\bar x\) is power majorized by \(\bar y\) if Σxi p ≤ Σyi p for all real p ? [0, 1] and Σxi p ≥ Σyi p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .      19. Scalar-type spectral operators and holomorphic semigroups      Ralph de Laubenfels 《Semigroup Forum》1986,33(1):257-263 We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold. A generates a uniformly bounded holomorphic semigroup {e?zA}Re(z)≥0. If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖FN‖} N=1 ∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {FN(s)x} N=1 ∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation. The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e?zA}Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.      20. Markov type and threshold embeddings      Jian Ding  James R. Lee  Yuval Peres 《Geometric And Functional Analysis》2013,23(4):1207-1229 For two metric spaces X and Y, say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps ${\{\varphi_{\tau} : X \to Y : \tau > 0\}}$ such that for every ${x,y \in X}$ , $$d_X(x, y) \geq \tau \implies d_Y(\varphi_\tau (x),\varphi_\tau (y)) \geq \|{\varphi}_\tau\|_{\rm Lip}\tau/K,$$ where ${\|{\varphi}_{\tau}\|_{\rm Lip}}$ denotes the Lipschitz constant of ${\varphi_{\tau}}$ . We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset ${X \subseteq L_1}$ threshold-embeds into Hilbert space if and only if X has Markov type 2.

Les F-reguliers a gauche

1. The concept of left F-regular semigroups was first defined by Batbedat at the Oberwolfach meeting in 1981. It generalizes the notion of F-regular semigroup introduced by Edwards [4], itself a generalization of the F-inverse semigroups defined by McFadden/O’Carroll [6]
2. In the present paper we generalize the results of [4] and [6] by defining two preorders and ? on a monoid S with a distinguished band E, as follows: iff x=ay for some a∈E xδy iff x=yb for some b∈E
3. When S is regular orthodox and E=E(S), is the preorder of [1] p. 29 and is the order of [1] p. 31 (the order of [4]): in fact is the natural partial order introduced by Nambooripad [7].
4. In b), we define the relation Σ on S: xΣy iff exe=eye for some e∈E Then we consider the congruence σ generated by Σ.
5. DEFINITION. S is left F_E-monoid if each σ-class contain a greatest element with respect to .
6. PARTICULAR CASES. When S is regular, S is left F_E-regular. When S is regular orthodox and E=E(S), S is left F-regular.
7. We describe the structure of left F-regular semigroups like in [1], [2], [4] and [6]. Note that every left F-regular semigroup is a gammasemigroup [3]
8. Particular Cases (gamma morphism) and applications (congruences).

     

Transformations of polar Grassmannians preserving certain intersecting relations

Let Π be a polar space of rank n≥3. Denote by \({\mathcal{G}}_{k}(\varPi)\) the polar Grassmannian formed by singular subspaces of Π whose projective dimension is equal to k. Suppose that k is an integer not greater than n?2 and consider the relation \({\mathfrak{R}}_{i,j}\) , 0≤i≤j≤k+1, formed by all pairs \((X,Y)\in{\mathcal{G}}_{k}(\varPi)\times{\mathcal{G}}_{k}(\varPi)\) such that dim _p (X ^⊥∩Y)=k?i and dim _p (X∩Y)=k?j (X ^⊥ consists of all points of Π collinear to every point of X). We show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{1,1}\) is induced by an automorphism of Π, except the case where Π is a polar space of type D _n with lines containing precisely three points. If k=n?t?1, where t is an integer satisfying n≥2t≥4, we show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{0,t}\) is induced by an automorphism of Π.     

The canonical embedding of an unramified morphism in an ��tale morphism

We show that every unramified morphism ${X\to Y}$ has a canonical and universal factorization ${X\hookrightarrow E_{X/Y}\to Y}$ where the first morphism is a closed embedding and the second is étale (but not separated).     

Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances

Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E _d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dim_H R = dim_P R = 2 almost surely, where dim_H and dim_P denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X _t for arbitrarily large t > 0 is given in terms of dim_H A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.     

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$ . Our main results are:

• A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).
• The equivalence of the heat flow in $L^{2}(X,\mathfrak {m})$ generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional $\mathrm {Ent}_{\mathfrak {m}}$ in the space of probability measures .
• The proof of density in energy of Lipschitz functions in the Sobolev space $W^{1,2}(X,\mathsf {d},\mathfrak {m})$ .
• A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.

Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality.     

A rotation theorem in the class of bounded univalent functions

Let S_M, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class S_M.     

Condensed cyclic and bridged-ring systems. VII. Acid-catalysed cyclisation of methyl substituted benzylcyclohexanols. Factors influencing the nature of cyclisation products

The gross structures of the cyclised products from the acid-catalysed cyclisations of 2-benzyl-1, 3-dimethylcyclohexanol (6) and 1-benzyl-3, 5-dimethylcyclohexanol (11) revealing the influence of the structure of the benzylcyclohexanol derivative, and of the cyclisation reagent, have been evaluated. Polyphosphoric acid and aluminium chloride catalysed cyclisations of (6) result in the formation of predominantly 1, 4a-dimethyl-1, 2, 3, 4, 4a, 9a-hexahydrofluorene (7) and 4, 9-dimethyl-7, 8-benzobicyclo [3.3.1] non-7-ene (9) respectively. Under the same conditions, (11) produced cyclised products consisting mostly the benzobicyclo [3.3.1] non-7-ene derivative (12), characterised through 1,3-dimethyl-7,8-benzobicyclo [3.3.1] non-6-oxo-7-ene (14) by oxidation with chromium trioxide. Phosphorus pentoxide induced cyclisation of (6), followed by oxidation gave a mixture of the bridged-ring ketone (10) and the 9-oxohydrofluorene (8) in a ratio ofca. 3 : 2, whereas 2-benzyl-5-methylcyclohexanol (19) resulted in mostly 2-methyl-7,8-benzobicyclo [3.3.1] non-6-oxo-7-ene (19).     

On the conservativity of the axiom of choice over set theory

We show that for various set theories T including ZF, T + AC is conservative over T for sentences of the form ${\forall x \exists ! y}$ A(x, y) where A(x, y) is a ??₀ formula.     

Power Majorization and Majorization of Sequences

Let \(\bar x\) , \(\bar y\ \in\ R_n\) be vectors which satisfy x₁ ≥ x₂ ≥ … ≥ x_n and y₁ ≥ y₂ >- … ≥ y_n and Σx_i = Σy_i. We say that \(\bar x\) is power majorized by \(\bar y\) if Σx_i ^p ≤ Σy_i ^p for all real p ? [0, 1] and Σx_i ^p ≥ Σy_i ^p for p ∈ [0, 1]. In this paper we give a classification of functions ? (which includes all possible positive polynomials) for which \(\bar\phi(\bar x) \leq \bar\phi(\bar y)\) (see definition below) when \(\bar x\) is power majorized \(\bar y\) . We also answer a question posed by Clausing by showing that there are vectors \(\bar x\) , \(\bar y\ \in\ R^n\) of any dimension n ≥ 4 for which there is a convex function ? such that \(\bar x\) is power majorized by \(\bar y\) and \(\bar\phi(\bar x)\ >\ \bar\phi(\bar y)\) .     

Scalar-type spectral operators and holomorphic semigroups

We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold.

1. A generates a uniformly bounded holomorphic semigroup {e?^zA}_Re(z)≥0.
2. If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖F_N‖} _N=1 ^∞ is uniformly bounded on [0,∞) and, for all x in X, the sequence {F_N(s)x} _N=1 ^∞ converges pointwise on [0, ∞) to a vector-valued function of bounded variation.

The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e^?zA}_Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E[0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X.     

Markov type and threshold embeddings

For two metric spaces X and Y, say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps ${\{\varphi_{\tau} : X \to Y : \tau > 0\}}$ such that for every ${x,y \in X}$ , $$d_X(x, y) \geq \tau \implies d_Y(\varphi_\tau (x),\varphi_\tau (y)) \geq \|{\varphi}_\tau\|_{\rm Lip}\tau/K,$$ where ${\|{\varphi}_{\tau}\|_{\rm Lip}}$ denotes the Lipschitz constant of ${\varphi_{\tau}}$ . We show that if a metric space X threshold-embeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien’s theorem. For instance, a subset ${X \subseteq L_1}$ threshold-embeds into Hilbert space if and only if X has Markov type 2.