Synthesis of 6-arylspiro[tetrahydropyran-3,1′-cyclobutane]-2,4-diones

Methyl 1-(2-bromo-2-methyl-1-oxopropyl)-, 1-(1-bromocyclopentylcarbonyl)-, and 1-(1-bromocyclohexylcarbonyl) cyclobutanecarboxylates reacted with zinc and aromatic aldehydes to give, respectively, 7-aryl-2,2-dimethyl-6-oxaspiro[3.5]nonane-5,9-diones, 11-aryl-12-oxadispiro[3.1.4.3]tridecane-5,13-diones, and 12-aryl-13-oxadispiro[3.1.5.3]tetradecane-5,14-diones.     

Reaction of Zinc Enolates Derived from Substituted 2,2-Dibromobutyrophenones with Ethyl 5,5-Dimethyl-2-oxo-2,5-dihydrofuran-3-carboxylate

Zinc enolates derived from substituted 2,2-dibromobutyrophenones react with ethyl 5,5-dimethyl-2-oxo-2,5-dihydrofuran-3-carboxylate to give ethyl 6-aroyl-6-ethyl-4,4-dimethyl-2-oxo-3-oxabicyclo[3.1.0]-hexane-1-carboxylates mostly as a single geometric isomer.     

Cyclopropanation of Alkyl and Aryl 3-Aryl-2-cyanopropenoates with Bromine-containing Zinc Enolates Obtained from 1-Aryl-2,2-dibromoalkanones and Zinc

Zinc enolates prepared from 1-aryl-2,2-dibromoalkanones and zinc react with alkyl and aryl 3-aryl-2-cyanopropenoates affording alkyl and aryl 2-alkyl-3-aryl-2-aroyl-1-cyanocyclopropane-1-carboxylates.     

Reformatsky reaction of methyl 1-bromocycloalkane-1-carboxylates with phenyl-and benzoylhydrazones derived from aromatic aldehydes

Reformatsky reagents obtained from methyl 1-bromocyclohexane-and 1-bromocyclopentane-1-carboxylates reacted with aromatic aldehyde phenyl-and benzoylhydrazones to give 3-aryl-2-phenylamino-2-azaspiro[3.5]nonan-, 3-aryl-2-phenylamino-2-azaspiro[3.4]octan-, 3-aryl-2-benzoylamino-2-azaspiro[3.5]-nonan-, and 3-aryl-2-benzoylamino-2-azaspiro[3.4]octan-1-ones, respectively. The reaction of furan-2-carbaldehyde phenylhydrazone with methyl 1-bromocyclohexane-1-carboxylate and zinc led to the formation of 4-(2-furyl)-2-phenyl-2,3-diazaspiro[4.5]decan-1-one.     

Cyclopropanation of N-substituted 3-aryl-2-cyanoprop-2-enamides and derivatives of 5,5-Dimethyl-2-oxo-2,5-dihydrofuran-3-carboxylic acid and 2-oxochromene-3-carboxylic acid with bromine-containing zinc enolates

Zinc enolates derived from 1-aryl-2,2-dibromoalkanones react with N-substituted 3-aryl-2-cyanoprop-2-enamides and 5,5-dimethyl-2-oxo-2,5-dihydrofuran-3-carboxylic and 2-oxochromene-3-carboxylic acid derivatives to give, respectively, N-substituted 2-alkyl-3-aryl-2-aroyl-1-cyanocyclopropane-1-carboxamides, 6-(4-bromobenzoyl)-4,4,6-trimethyl-2-oxo-3-oxabicyclo[3.1.0]hexane-1-carboxylic acid ethyl ester and morpholide, and 1-alkyl-1-aroyl-2-oxo-1a,7b-dihydrocyclopropla[c]chromene-1a-carboxylic acids as a single geometric isomer. Treatment of 1-alkyl-1-aroyl-2-oxo-1a,7b-dihydrocyclopropa[c]chromene-1a-carboxylic acids with carboxylic acid anhydrides leads to the formation of the corresponding 9c-alkyl-1-aryl-3,4-dioxo-9b,9c-dihydro-2,5-dioxacyclopenta[2,3]cyclopropa[1,2-a]naphthalen-1-yl carboxylates.     

Reactions of Methyl Esters of 1-(α-Bromoisobutyryl)cyclohexanecarboxylic or 3-(1-Bromocyclohexyl)-2,2-dimethyl-3-oxopropanoic Acids with Zinc and Arylglyoxal

Methyl esters of 1-(-bromoisobutyryl)cyclohexanecarboxylic or 3-(1-bromocyclohexyl)-2,2-dimethyl-3-oxopropanoic acids react with zinc and arylglyoxals to form 3-aroyl-4,4-dimethyl-2-oxaspiro[5.5]undecane-1,5-diones or 1-aroyl-4,4-dimethyl-2-oxaspiro[5.5]undecane-3,5-diones, respectively. The former products react with phenylhydrazine, yielding 3-[aryl(2-phenylhydrazono)methyl]-4,4-dimethyl-2-oxaspiro[5.5]undecane-1,5-diones.     

Reaction of Zinc and Sodium Enolates of 3-Alkyl-6-aryl-5,5-dimethyl-2,3,5,6-tetrahydropyrane-2,4-diones with Acyl Chlorides or Benzyl Bromides

Zinc enolates of 3-alkyl-6-aryl-5,5-dimethyl-2,3,5,6-tetrahydropyrane-2,4-diones react with acyl chlorides to form O-acylation products, 4-acyloxy-3-alkyl-6-aryl-5,5-dimethyl-5,6-dihydropyrane-2-ones. Sodium enolates of these pyranediones react in DMSO with substituted benzyl bromides to give mainly C-alkylation products, 3-alkyl-6-aryl-3-(4-R-benzyl)-5,5-dimethyl-2,3,5,6-tetrahydropyrane-2,4-diones, as single geometric isomers. In some cases, O-alkylation products, 4-alkoxy-3-alkyl-6-aryl-5,5-dimethyl-5,6-dihydropyrane-2-ones, are formed as by-products (10-15%).     

Synthesis of Alkyl 4-(1-Alkyl-2-aryl-2-oxoethyl)-5,5-dimethyl-2-oxotetrahydrofuran-3-carboxylates

Zinc enolates derived from 1-aryl-2-bromoalkanones react with alkyl 5,5-dimethyl-2-oxo-2,5-dihydrofuran-3-carboxylates to give alkyl 4-(1-alkyl-2-aryl-2-oxoethyl)-5,5-dimethyl-2-oxotetrahydrofuran-3-carboxylates. Reactions of the latter with amines, such as p-toluidine, cyclohexylamine, and piperidine, lead to the corresponding carboxamides.     

On Hölder maps of cubes

A map of metric spaces f: X → Y satisfying the inequality $$ \left| {f(x) - f(y)} \right| \leqslant C\left| {x - y} \right|^\alpha $$ for some C and α and all x, y ∈ X is called a Hölder map with exponent α. V. I. Arnold posed the following problem: Does there exist a Höldermap from the square onto the cube with exponent 2/3? The firstmain theorem of this paper gives a general method for constructing Höldermaps of compact metric spaces. This construction yields, in particular, a dimension-raising map f: I ⁿ → I ^m with Hölder exponent arbitrarily close to m/n for m > n > 1 and a map I ¹ → I ^m with Hölder exponent 1/m. The second main theorem states the nonexistence of a regular fractal map f: I ⁿ → I ^m with Hölder exponent n/m from the n-cube onto the m-cube for m < 2n.