2,4-Bis(2-fluoro­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one

The title compound, C20H19F2NO, exists in a twin-chair conformation with an equatorial orientation of the two 2-fluoro­phenyl groups on both sides of the secondary amine group. The benzene rings are orientated at an angle of 25.68 (4)° with respect to one another and the F atoms point upwards (towards the carbonyl group). The crystal is stabilized by an inter­molecular N—H⋯π inter­action

2,6-Bis(4-meth­oxy­phen­yl)-1,3-dimethyl­piperidin-4-one O-benzyl­oxime

The central ring of the title compound, C28H32N2O3, exists in a chair conformation with an equatorial disposition of all the alkyl and aryl groups on the heterocycle. The para-anisyl groups on both sides of the secondary amino group are oriented at an angle of 54.75 (4)° with respect to each other. The oxime derivative exists as an E isomer with the methyl substitution on one of the active methyl­ene centers of the mol­ecule. The crystal packing features weak C—H⋯O inter­actions

2,6-Bis(4-chloro­phen­yl)-1,3-dimethyl­piperidin-4-one O-benzyl­oxime

The piperidin-4-one ring in the title compound, C26H26Cl2N2O, exists in a chair conformation with equatorial orientations of the methyl and 4-chlorophenyl groups. The C atom bonded to the oxime group is statistically planar (bond-angle sum = 360.0°) although the C—C=N bond angles are very different [117.83 (15) and 127.59 (15)°]. The dihedral angle between the chloro­phenyl rings is 54.75 (4)°. In the crystal, mol­ecules inter­act via van der Waals forces

2,4-Bis(3-methoxy­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one

In the crystal structure, the title compound, C22H25NO3, exists in a twin-chair conformation with equatorial orientations of the meta-methoxy­phenyl groups on both sides of the secondary amino group. The title compound is a positional isomer of 2,4-bis­(2-methoxy­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one and 2,4-bis­(4-methoxy­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one, which both also exhibit twin-chair conformations with equatorial dispositions of the anisyl rings on both sides of the secondary amino group. In the title compound, the meta-methoxy­phenyl rings are orientated at an angle of 25.02 (3)° with respect to each other, whereas in the ortho and para isomers, the anisyl rings are orientated at dihedral angles of 33.86 (3) and 37.43 (4)°, respectively. The crystal packing is dominated by van der Waals inter­actions and by an inter­molecular N—H⋯O hydrogen bond, whereas in the ortho isomer, an inter­molecular N—H⋯π inter­action (H⋯Cg = 2.75 Å) is found

2,4-Bis(4-bromo­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one

The title compound, C20H19Br2NO, shows a chair–chair conformation for the aza­bicycle with an equatorial disposition of the 4-bromo­phenyl groups [dihedral angle between the aromatic rings = 16.48 (3)°]. In the crystal, a short Br⋯Br contact [3.520 (4) Å] occurs and the structure is further stabilized by N—H⋯O hydrogen bonds and C—H⋯O inter­actions

2,4-Bis(4-chloro­phen­yl)-3-aza­bicyclo­[3.3.1]nonan-9-one

In the mol­ecular structure of the title compound, C20H19Cl2NO, the mol­ecule exists in a twin-chair conformation with equatorial dispositions of the 4-chloro­phenyl groups on both sides of the secondary amino group; the dihedral angle between the aromatic ring planes is 31.33 (3)°. The crystal structure is stabilized by N—H⋯O inter­actions, leading to chains of molecules

2,4,6,8-Tetra­kis(4-chloro­phen­yl)-3,7-diaza­bicyclo­[3.3.1]nonan-9-one O-benzyl­oxime acetone monosolvate

In the title compound, C38H31Cl4N3O·C3H6O, the 3,7-diaza-bicycle exists in a chair–boat conformation. The 4-chloro­phenyl groups attached to the chair form are equatorially oriented at an angle of 18.15 (3)° with respect to each other, whereas the 4-chloro­phenyl groups attached to the boat form are oriented at an angle of 32.64 (3)°. In the crystal, mol­ecules are linked by N—H⋯π and C—H⋯O inter­actions

2,4-Bis(2-eth­oxy­phen­yl)-7-methyl-3-aza­bicyclo­[3.3.1]nonan-9-one

The crystal structure of the title compound, C25H31NO3, exists in a twin-chair conformation with an equatorial orientation of the ortho-eth­oxy­phenyl groups. According to Cremer and Pople [Cremer & Pople (1975 ▶), J. Am. Chem. Soc. 97, 1354–1358], both the piperidone and cyclo­hexa­none rings are significantly puckered with total puckering amplitutdes Q T of 0.5889 (18) and 0.554 (2) Å, respectively. The ortho-eth­oxy­phenyl groups are located on either side of the secondary amino group and make a dihedral angle of 12.41 (4)° with respect to each other. The methyl group on the cyclo­hexa­none part occupies an exocyclic equatorial disposition. The crystal packing is stabilized by weak van der Waals inter­actions

A Comparative Study of Latin Square Design Under Fuzzy Environments Using Trapezoidal Fuzzy Numbers

This paper deals with the problem of Latin Square Design (LSD) test using Trapezoidal Fuzzy Numbers (Tfns.).  The proposed test is analysed under various types of trapezoidal fuzzy models such as Alpha Cut Interval, Membership Function, Ranking Function, Total Integral Value and Graded Mean Integration Representation.  Finally a comparative view of the conclusions obtained from various test is given.  Moreover, two numerical examples having different conclusions have been given for a concrete comparative study.   Keywords: LSD, Trapezoidal Fuzzy Numbers, Alpha Cut, Membership Function, Ranking Function, Total Integral Value, Graded Mean Integration Representation.   AMS Mathematics Subject Classification (2010): 62A86, 62F03, 97K8

One-Factor ANOVA Model Using Trapezoidal Fuzzy Numbers Through Alpha Cut Interval Method

Most of our traditional tools in descriptive and inferential statistics is based on crispness (preciseness) of data, measurements, random variable, hypotheses, and so on.  By crisp we mean dichotomous that is, yes-or-no type rather than more-or-less type.  But there are many situations in which the above assumptions are rather non-realistic such that we need some new tools to characterize and analyze the problem.  By introducing fuzzy set theory, different branches of mathematics are recently studied.  But probability and statistics attracted more attention in this regard because of their random nature.  Mathematical statistics does not have methods to analyze the problems in which random variables are vague (fuzzy). In this regard, a simple and new technique for testing the hypotheses under the fuzzy environments is proposed.  Here, the employed data are in terms of trapezoidal fuzzy numbers (TFN) which have been transformed into interval data using  interval method and on the grounds of the transformed fuzzy data, the one-factor ANOVA test is executed and decisions are concluded.  This concept has been illustrated by giving two numerical examples. Keywords: Fuzzy set, , Trapezoidal fuzzy number (TFN), Test of hypotheses, One-factor ANOVA model, Upper level data, Lower level data