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Abstract:

In this paper, the author has opened a new horizon in the theory of quadratic equations. The author proved that the value of x which satisfies the quadratic equation cannot be the only criteria to designate as the root or roots of an equation. The author has developed a new mathematical concept of the dimension of a number. By introducing the concept of the dimension of number the author structured the general form of a quadratic equation into two forms: 1) Pure quadratic equation and 2) Pseudo quadratic equation. First of all the author defined the pure and pseudo quadratic equations. In the case of pure quadratic equation ax^2+bx+c=0 , the root of the equation will be a two-dimensional number having one root only while in the case of pseudo quadratic equation ax^2+bx+c=0, the root of the equation will be a one-dimensional number having two roots only. The author proved that all pseudo quadratic equation is factorizable but all factorizable quadratic equation is not a pseudo quadratic equation. The author begs to differ from the conventional theorem: "A quadratic equation has two and only two roots." By introducing the concept that any quadratic surd is a two-dimensional number, the author developed a new theorem: “In a quadratic equation with rational coefficients, irrational roots cannot occur in conjugate pairs” and proved it. Any form of quadratic equation ax^2+bx+c=0, can be solved by the application of the ‘Theory of Dynamics of Numbers’ even if the discriminant b^2-4ac<0 in real number only without introducing the concept of an imaginary number. Therefore, the question of imaginary roots does not arise in the method of solution of any quadratic equation

Keywords:

Dimension of Numbers,Dynamics of Numbers,,Quadratic Equation,Rectangular Bhattacharra’s Coordinates,significance of roots of a Quadratic Equation,

Refference:

I. Bhattacharyya, Prabir Chandra. : “AN INTRODUCTION TO THEORY OF DYNAMICS OF NUMBERS: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-1, January (2022). pp 37-53

II. Bhattacharyya, Prabir Chandra. : “A NOVEL CONCEPT IN THEORY OF QUADRATIC EQUATION”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-3, March (2022) pp 41-63

III. Bhattacharyya, Prabir Chandra. : “AN INTRODUCTION TO RECTANGULAR BHATTACHARYYA’S CO-ORDINATES: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, November (2021). pp 76-86.

IV. Boyer, C. B. & Merzbach, U. C. (2011). A history of mathematics. New York: John Wiley & Sons.

V. Cajori, F., (1919). A History of Mathematics 2nd ed., New York: The Macmillan Company.

VI. Dutta, B.B. ( 1929). The Bhakshali Mathematics, Calcutta, West Bengal: Bulletin of the Calcutta Mathematical Society.

VII. Datta, B. B., & Singh, A. N. (1938). History of Hindu Mathematics, A source book. Mumbai, Maharashtra: Asia Publishing House.

VIII. Gandz, S. (1937). The origin and development of the quadratic equations in Babylonian, Greek, and Early Arabic algebra. History of Science Society, 3, 405-557.

IX. Gandz, S. (1940). Studies in Babylonian mathematics III: Isoperimetric problems and the origin of the quadratic equations. Isis, 3(1), 103-115.

X. Hardy G. H. and Wright E. M. “An Introduction to the Theory of Numbers”. Sixth Edition. P. 52.

XI. Katz, V. J. (1997), Algebra and its teaching: An historical survey. Journal of Mathematical Behavior, 16(l), 25-36.

XII. Katz, V., J. (1998). A history of mathematics (2nd edition). Harlow, England: Addison Wesley Longman Inc.

XIII. Katz Victor, (2007). The Mathematics of Egypt, Mesopotamia, China, India and Islam: A source book 1st ed., New Jersey, USA: Princeton University Press.

XIV. Kennedy, P. A., Warshauer, M. L. & Curtin, E. (1991). Factoring by grouping: Making the connection. Mathematics and Computer Education, 25(2), 118-123.

XV. Ling, W. & Needham, J., (1955). Horner’s method in Chinese Mathematics: Its root in the root extraction procedures of the Han Dynasty, England: T’oung Pao.
XVI. Nataraj, M. S., & Thomas, M. O. J. (2006). Expansion of binomials and factorisation of quadratic expressions: Exploring a vedic method. Australian Senior Mathematics Journal, 20(2), 8-17.

XVII. Rosen, Frederic (Ed. and Trans). (1831). The algebra of Mohumed Ben Muss. London: Oriental Translation Fund; reprinted Hildesheim: Olms, 1986, and Fuat Sezgin, Ed., Islamic Mathematics and Astronomy, Vol. 1. Frankfurt am Main: Institute for the History of Arabic-Islamic Science 1997.

XVIII. Smith, D. (1951). History of mathematics, Vol. 1. New York: Dover. Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover. Stols, H. G. (2004).

XIX. Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover.

XX. Thapar, R., (2000). Cultural pasts: Essays in early Indian History, New Delhi: Oxford University Press.
XXI. Yong, L. L. (1970). The geometrical basis of the ancient Chinese square-root method. The History of Science Society, 61(1), 92-102.

XXII. http://en. wikipedia.org/wiki/Shridhara

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Abstract:

                  Breast cancer is a serious trouble and one of the greatest causes of death for women throughout the world. Computer-aided diagnosis (CAD) techniques can help the doctor make more credible decisions. We have determined the possibility of knowledge transfer from natural to histopathological [IX][XII] images by employing a pre-trained network ResNet-50.This pre-trained network has been utilized as a feature generator and extracted features are used to train support vector machine (SVM), random forest, decision tree, and K nearest neighbor(KNN) classifiers[X]. We altered the softmax layer to support the vector machine classifier, random forest classifier, decision tree classifier, and k-nearest neighbor classifier, to evaluate the classifier performance of each algorithm. These approaches are applied for breast cancer classification and evaluate the performance and behavior of different classifiers on a publicly available dataset named Bttheeak-HIS dataset. In order to increase the efficiency of the ResNet[III] model, we preprocessed the data before feeding it to the network. Here we have applied to sharpen filter and data augmentation techniques, which are very popular and effective image pre-processing techniques used in deep models.

Keywords:

Machine learning,Support Vector Machine (SVM),K-Nearest Neighbor (KNN),RESNET (Residual Network) model,Random Forest.[VII],

Refference:

I. Altman, N. S. (1992). “An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression” (Pdf). The American Statistician.
II. Araújo, Teresa, Guilherme Aresta, Eduardo Castro, José Rouco, Paulo Aguiar, Catarina Eloy, António Polónia, and Aurélio Campilho. “Classification of breast cancer histology images using Convolutional Neural Networks.” PloS one 12, no. 6 (2017): e0177544.
III. “Automatic white blood cell classification using pre-trained deep learning models: ResNet and Inception,” Proc. SPIE 10696, Tenth International Conference on Machine Vision (ICMV 2017), 1069612 (13 April 2018);
IV. Balestriero, R. Neural Decision Trees. Arxiv E-Prints, 2017.
V. Bayramoglu, Neslihan, Juho Kannala, and Janne Heikkilä. “Deep learning for magnification independent breast cancer histopathology image classification.” In Pattern Recognition (ICPR), 2016 23rd International Conference on, pp. 2440- 2445. IEEE, 2016.
VI. B. E. Bejnordi, G. Zuidhof, M. Balkenhol et al., “Contextaware stacked convolutional neural networks for classifcation of breast carcinomas in whole-slide histopathology images,” Journal of Medical Imaging, vol. 4, no. 04, p. 1, 2017.
VII. Diaz-Uriarte R, Alvarez De Andres S: Gene Selection And Classification Of Microarray Data Using Random Forest. Bmc Bioinformatics 2006, 7:3.

VIII. George, Yasmeen Mourice, Hala Helmy Zayed, Mohamed Ismail Roushdy, and Bassant Mohamed Elbagoury. “Remote computer-aided breast cancer detection and diagnosis system based on cytological images.” IEEE Systems Journal 8, no. 3 (2014): 949-964.

IX. Gupta, Vibha, and Arnav Bhavsar. “Breast Cancer Histopathological Image Classification: Is Magnification Important?.” In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 17- 24. 2017.
X. Hall P, Park Bu, Samworth Rj (2008). “Choice Of Neighbor Order In Nearest-Neighbor Classification”. Annals Of Statistics. 36 (5): 2135–2152.
XI. Hammer B, Gersmann K: A Note On The Universal Approximation Capability Of Support Vector Machines. Neural Processing Letters 2003, 17:43-53.
XII. Han, Zhongyi, Benzheng Wei, Yuanjie Zheng, Yilong Yin, Kejian Li, and Shuo Li. “Breast cancer multi-classification from histopathological images with structured deep learning model.” Scientific reports 7, no. 1 (2017): 4172
XIII. Kahya, Mohammed Abdulrazaq, Waleed Al-Hayani, and Zakariya Yahya Algamal. “Classification of breast cancer histopathology images based on adaptive sparse support vector machine.” Journal of Applied Mathematics and Bioinformatics 7.1 (2017): 49
XIV. Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun,”Deep Residual Learning for Image Recognition,” arXiv:1512.03385 [cs.CV], 10 Dec 2015;
XV. Kowal, Marek, Paweł Filipczuk, Andrzej Obuchowicz, Józef Korbicz, and Roman Monczak. “Computer-aided diagnosis of breast cancer based on fine needle biopsy microscopic images.” computers in biology and medicine 43, no. 10 (2013): 1563-1572.
XVI. Mehdi Habibzadeh, Mahboobeh Jannesari, Zahra Rezaei, Hossein Baharvand, Mehdi Totonchi,
XVII. Pan, S.J. And Yang, Q., 2010. A Survey On Transfer Learning. Ieee Transactions On Knowledge And Data Engineering, 22(10), Pp.1345–1359.
XVIII. Prinzie, A., Van Den Poel, D. (2008). “Random Forests For Multiclass Classification: Random Multinomial Logit”. Expert Systems With Applications
XIX. Qicheng Lao, Thomas Fevens,”Cell Phenotype Classification using Deep Residual Network and its Variants,” International Journal of Pattern Recognition and Artificial Intelligence © World Scientific Publishing Company, 01/17/19;
XX. Rawat, W. And Wang, Z., 2017. Deep Convolutional Neural Networks For Image Classification: A Comprehensive Review. Neural Computation, 29(9), Pp.2352–2449.

XXI. Rokach, Lior; Maimon, O. (2008). Data Mining With Decision Trees: Theory And Applications. World Scientific Pub Co Inc.
XXII. Shalev-Shwartz, Shai; Ben-David, Shai (2014). “18. Decision Trees”. Understanding Machine Learning. Cambridge University Press.
XXIII. Scornet, Erwan (2015). “Random Forests And Kernel Methods”.

XXIV. Simonyan, K. And Zisserman, A., 2014. Very Deep Convolutional Networks For Large-Scale Image Recognition. Arxiv Preprint Arxiv:1409.1556.
XXV. Spanhol, Fabio A., Luiz S. Oliveira, Caroline Petitjean, and Laurent Heutte. “A dataset for breast cancer histopathological image classification.” IEEE Transactions on Biomedical Engineering 63, no. 7 (2016): 1455-1462.
XXVI. Spanhol, Fabio Alexandre, Luiz S. Oliveira, Caroline Petitjean, and Laurent Heutte. “Breast cancer histopathological image classification using convolutional neural networks.” In Neural Networks (IJCNN), 2016 International Joint Conference on, pp. 2560-2567. IEEE, 2016.
XXVII. Spanhol, Fabio A., Luiz S. Oliveira, Paulo R. Cavalin, Caroline Petitjean, and Laurent Heutte. “Deep features for breast cancer histopathological image classification.” In Systems, Man, and Cybernetics (SMC), 2017 IEEE International Conference on, pp. 1868-1873. IEEE, 2017.
XXVIII. Tang, Y., 2013. Deep Learning Using Linear Support Vector Machines. Arxiv Preprint Arxiv: 1306.0239.
XXIX. T. Araujo, G. Aresta, E. Castro et al., “Classifcation of breast cancer histology images using convolutional neural networks,” PLoS ONE, vol. 12, no. 6, Article ID e0177544, 2017.
XXX. Veta, Mitko, Josien PW Pluim, Paul J. Van Diest, and Max A. Viergever. “Breast cancer histopathology image analysis: A review.” IEEE Transactions on Biomedical Engineering 61, no. 5 (2014): 1400-1411
XXXI. Yosinski, J., Clune, J., Bengio, Y. And Lipson, H., 2014. How Transferable Are Features In Deep Neural Networks?. In Advances In Neural Information Processing Systems (Pp. 3320–3328).
XXXII. Zeiler, M.D. And Fergus, R., 2014, September. Visualizing And Understanding Convolutional Networks. In European Conference On Computer Vision (Pp. 818–833). Springer, Cham.
XXXIII. Zhang, Yungang, Bailing Zhang, Frans Coenen, and Wenjin Lu. “Breast cancer diagnosis from biopsy images with highly reliable random subspace classifier ensembles.” Machine vision and applications 24, no. 7 (2013): 1405-1420.

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Abstract:

Information communication and technology are the most important skills for 21^st-century learning and help promote other skills, including life and career skills and learning and innovation skills. This kind of learning allows the learner to connect as a learning network without barriers or borders. The growth of online education has taken our education system to another level. Now anyone can learn from anywhere, anytime as per convenience. In different platforms, questions link are shared with a submission time. Although, learners are taking up unfair means to clear the test provided online in which students usually search up the topic, use different means to get the answers, and get good marks. Hence, teachers cannot get an idea of who is good in the class and who needs extra attention. So, our idea is to make such a platform where the teacher will be taking the test just like our offline classes. In this platform, the teacher will be discussing every question after the students submit the answer in a time duration which will also be proctored and at the same time, the teacher will get the top performer and their submission time. This way we can assure minimal malpractice and identify the students who need more explanation for the questions. This will clear their doubts and the teacher understands the actual performance ratio.

Keywords:

Skill test platform,MongoDB,MERN Stack, MongoDB,student's performance,

Refference:

I. C. Ying-ying, “The Design and Implementation of Online Examination System with Characteristics of Cloud Service”, Beijing University of Posts and Telecommunications. 2013.
II. H-R. Ouyang, H-F. Wei, H-X. Li, A-Q. Pan, Y. Huang, “Checking Causal Consistency of MongoDB”, Journal of Computer Science and Technology. Vol. 37, pp: 128–146, 2022.
III. M. Radoev, “A Comparison between Characteristics of NoSQL Databases and Traditional Databases,” Comput. Sci. Inf. Technol. vol. 5, no. 5, pp: 149–153, 2017.
IV. M.Yagci, M. Unal, “Designing and implementing an adaptive online examination system”, Procedia – Social and Behavioral Sciences. Vol. 116, pp: 3079-3083, 2014.
V. W. Schultz, T. Avitabile, A. Cabral, “Tunable consistency in MongoDB.”, Proc. VLDB Endow. Vol. 12, No. 12, pp 2071-2081, 2019.
VI. Z. Yong-Sheng, F. Xiu-Mei and B. Ai-Qin, “The Research and Design of Online Examination System,” 2015 7th International Conference on Information Technology in Medicine and Education (ITME). pp.: 687-691, 2015.

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