arXiv subjects
Publications and source records attributed to Inder Jeet Taneja.
In this paper we have established relations of the genetic tables with magic and bimagic squares. Connections with Hamming distances, binomial coefficients are established. The idea of Gray code is applied. Shannon entropy of magic squares of order 4x4, 8x8 and 16x16 are also calculated. Some comparison is also made. Symmetry among restriction enzymes having four letters is also studied.
In this paper we have considered a single inequality having 11 known divergence measures. This inequality include measures like: Jeffryes-Kullback-Leiber J-divergence, Jensen-Shannon divergence (Burbea-Rao, 1982), arithmetic-geometric mean divergence (Taneja, 1995), Hellinger discrimination, symmetric chi-square divergence, triangular discrimination, etc. All these measures are well-known in the literature on Information theory and Statistics. This sequence of 11 measures also include measures due to Kumar and Johnson (2005) and Jain and Srivastava (2007). Three measures arising due to some mean divergences also appears in this inequality. Based on non-negative differences arising due to this single inequality of 11 measures, we have put more than 40 divergence measures in nested or sequential form. Idea of reverse inequalities is also introduced.
Eve (2003), studied seven means from geometrical point of view. These means are \textit{Harmonic, Geometric, Arithmetic, Heronian, Contra-harmonic, Root-mean square and Centroidal mean}. Some of these means are particular cases of Gini's (1938) mean of order r and s. In this paper we have established some proportionality relations having these means. Some inequalities among some of differences arising due to seven means inequalities are also established.
In 1938, Gini studied a mean having two parameters. Later, many authors studied properties of this mean. It contains as particular cases the famous means such as harmonic, geometric, arithmetic, etc. Also it contains, the power mean of order r and Lehmer mean as particular cases. In this paper we have considered inequalities arising due to Gini-Mean and Heron's mean, and improved them based on the results recently studied by the author (Taneja, 2011).
In 1938, Gini studied a mean having two parameters. Later, many authors studied properties of this mean. In particular, it contains the famous means as harmonic, geometric, arithmetic, etc. Here we considered a sequence of inequalities arising due to particular values of each parameter of Gini's mean. This sequence generates many nonnegative differences. Not all of them are convex. We have studied here convexity of these differences and again established new sequences of inequalities of these differences. Considering in terms of probability distributions these differences, we have made connections with some of well known divergence measures.
There are three classical divergence measures exist in the literature on information theory and statistics. These are namely, Jeffryes-Kullback-Leiber J-divergence. Sibson-Burbea-Rao Jensen-Shannon divegernce and Taneja arithemtic-geometric mean divergence. These three measures bear an interesting relationship among each other and are based on logarithmic expressions. The divergence measures like Hellinger discrimination, symmetric chi-square divergence, and triangular discrimination are also known in the literature and are not based on logarithmic expressions. Past years Dragomir et al., Kumar and Johnson and Jain and Srivastava studied different kind of divergence measures. In this paper, we have presented some more new divergence measures and obtained inequalities relating these new measures and also made connections with previous ones. The idea of exponential divergence is also introduced.
Recently, Taneja studied two one parameter generalizations of J-divergence, Jensen-Shannon divergence and Arithmetic-Geometric divergence. These two generalizations in particular contain measures like: Hellinger discrimination, symmetric chi-square divergence, and triangular discrimination. These measures are well known in the literature of Statistics and Information theory. In this paper our aim is to prove metric space properties for square root of these two symmetric generalized divergence measures.
In this paper we have considered two one parametric generalizations. These two generalizations have in articular the well known measures such as: J-divergence, Jensen-Shannon divergence and Arithmetic-Geometric mean divergence. These three measures are with logarithmic expressions. Also, we have particular cases the measures such as: Hellinger discrimination, symmetric chi-square divergence, and triangular discrimination. These three measures are also well-known in the literature of statistics, and are without logarithmic expressions. Still, we have one more non logarithmic measure as particular case calling it d-divergence. These seven measures bear an interesting inequality. Based on this inequality, we have considered different difference of divergence measures and established a sequence of inequalities among themselves.
There are three classical divergence measures exist in the literature on information theory and statistics. These are namely, Jeffryes-Kullback-Leiber J-divergence. Sibson-Burbea-Rao Jensen-Shannon divegernce and Taneja Arithmetic-Geometric divergence. These three measures bear an interesting relationship among each other. The divergence measures like Hellinger discrimination, symmetric chi-square divergence, and triangular discrimination are also known in the literature. In this paper, we have considered generalized symmetric divergence measures having the measures given above as particular cases. Bounds on the probability of error are obtained in terms of generalized symmetric divergence measures. Study of bounds on probability of error is extended for the difference of divergence measures.
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, root square mean, etc. Considering the difference of these means, we can establish. some inequalities among them. Interestingly, the difference of mean considered is convex functions. Applying some properties, upper bounds on the probability of error are established in this paper. It is also shown that the results obtained are sharper than obtained directly applying known inequalities.
In this paper we shall consider some famous means such as arithmetic, harmonic, geometric, logarithmic means, etc. Inequalities involving logarithmic mean with differences among other means are presented
In this paper we have produced different kinds of bimagic squares based on bimagic squares of order 8x8, 16x16, 25x25, 49x49, etc. A different technique is applied to produce bimagic square of order 16x16, 25x25, 49x49, etc. The bimagic square of order 8x8 used is the already known in the literature. The work is neither based on any programming language nor on mathematical results. Just simple combinations are used to produce these bimagic squares. Moreover, in each case we have used consecutive numbers starting from 1
In this short paper we have produced different kinds of upside down magic squares based on a palindromic day 11.02.2011. In this day appear only the algorisms 0, 1 and 2. Some of the magic squares are bimagic and some are palindromic. Magic sums of the magic squares of order 3x3, 4x4 and 5x5 satisfies the Pythagoras theorem. Bimagic squares of order 9x9 are produced with 4, 6 and 8 digits. The bimagic square of order 9x9 with 8 digits is of palindromic numbers. We have given bimagic squares of order 16x16 and 25x25, where the magic sum S1 in both the cases is same. In order to make these magic squares upside down, i.e., 180 degrees rotation, we have used the numbers in the digital form. All these magic square are only with three digits, 0, 1 and 2 appearing in the day 11.02.2011.
In this short note we shall give connection between the most perfect "Khajuraho" magic square of order 4x4 discovered in 10th century and the "Lo-Shu" magic square of order 3x3 with the day October 1, 2010, i.e., 01.10.2010. The day has only three digits 0, 1 and 2. Here we have given an equivalent version of Khajuraho magic square using only three digits 0, 1 and 2. If we write the above date in two parts, 0110 2010, interestingly, the sum of new magic square is the first part, i.e., 0110, and the numbers appearing in the magic square are from the second part. An equivalent version of "Lo-Shu" magic square of order 3x3 is also given
In this short note we have given an equation based on the date 11.09.2001 and presented some magic squares. The magic squares presented are of order 3x3, 4x4, 5x5, 9x9, 16x16 and 25x25. While the magic square of higher order 9x9, 16x16 and 25x25 are bimagic. These magic squares are based on the digits, 1-6-9, 0-1-2, 0-1-2-9 and 0-1-2-6-9. The interesting fact in all these magic squares is that if we gave them a rotation of 180 degrees, they remain again the magic squares. In order to have this rotation, we have used the numbers in digital forms. Moreover, the day of submission of this work (20.10.2010) has only the digits 0, 1 and 2. Using only these three digits, we have presented magic squares of order 3x3 and 9x9.
In this short note we have produced different kinds of bimagic squares using only the digits 0, 1, 2, 5 and 8. The universal bimagic squares presented are of order 8x8, 9x9, 16x16 and 25x25. In order to bring universal bimagic square of order 8x8, we used only the digits 2 and 5. For the order 9x9, we used only the digits 2, 5 and 8. For the universal bimagic square of order 16x16 we used the digits 1, 2, 5 and 8 and finally for the order 25x25, we used five digits 0, 1, 2, 5 and 8. In order to produce these universal bimagic squares we have used the digits in the digital form.
In this short note we have produced for the first time in the history different kinds of universal bimagic squares. This we have made using only the digits 0,1 and 2. The universal bimagic squares of order 8x8 and 16x16 are with the digits 0 and 1. The universal bimagic square of order 9x9 is with the digits 0, 1 and 2. It is interesting to note that the day 0ctober 10 have only the digits 0 and 1 if we consider it as 10.10.10. If we consider the date as 10.10.2010, then this has the digits 0, 1 and 2.
In this short note we have produced different kind of magic squares using digital letter having only the algorisms: 0, 1, 2, 5, and 8. The interesting fact in considering these five digits is that the day 8th May 2010 also have these ones (08.05.2010). Moreover, the magic squares presented have some interesting properties, such as: they remains the same if we rotate them by 180 degree, or see in the mirror, or see on the other side of the paper, etc. Two palindrome semi-magic squares of order 3x3 are also given. Still, we have considered other dates having four digits.