Studies in Experimental Design and Data Analysis with Special Reference to Computational Problems

An experimenter, designing or analysing an experiment, frequently finds that not all can be done according to the text-book. Considerations outside his control may have to be taken into account that affect the design; he may not be interested in all his treatments equally but perhaps in some unusual contrasts between them. Again, unforseen circumstances and happenings can afterwards force him to discard some of his results, so upsetting the balance or orthogonality of the design. The aim of this thesis is to help an experimenter in such a situation. The first part is concerned with analysing a block experiment that is in general unbalanced and non-orthogonal. 'Two different methods, one iterative, one non-iterative, are derived for obtaining the analysis, each with its own advantages. The non-iterative method basically is derived from the actual design and produces matrices, which can then operate on any suitable data supplied. The iterative method, however, found in appendix A, is applied directly to the data from the start, to produce the treatment effects directly. Although the iterative method is easier to apply and can also be used with a wider class of design than can the non-iterative method, the inter-block analysis and the analysis of the dual become easier using the non-iterative method. Certain contrasts are related to the design in special ways, and, if known, make the analysis of the design easier. The implications are discussed in chapter 2, which is also concerned with finding the contribution to the sum of squares for these and other, more general, contrasts of interest. The dual of a design is defined as that design formed from the original design when treatments and blocks classifications are interchanged, i.e. treatments become blocks and vice-versa in the new design. It is useful for studying block differences eliminating those due to treatments, which may, for example, be required if the blocking system arose from the possible residual effects of treatments from some previous experiment on the same material. The second part of the thesis, in chapter 3> is concerned with the analysis of the dual. It is shown that there is no need to start again from the beginning when analysing the dual if the original design had already been analysed, because the analysis can provide information about that of the dual. The method is especially easy when the non-iterative method of analysing block designs, discussed in chapter 2, has been used for the original design. The experimenter will often be more interested in some contrasts between treatments than in others and a design can be selected to give more precise information about these contrasts. The construction of such designs is discussed in the third part of the thesis. Various measures can be used to judge which design is best as regards contrasts of interest. Algorithms for finding the optimal design according to these measures are derived and discussed in chapter 4. Listings and flowcharts of a program to carry out the non-iterative analysis of chapter 2 and of a program to construct optimal block designs appear in appendices B and C

On the extension of Householder's method for weighted Moore-Penrose inverse

This note discusses how to extend and use the well-known method of Householder (1964) [6] for finding weighted Moore-Penrose inverse. The discussions cover both theoretical and numerical aspects. An approach for accelerating the initial phase of convergence will be contributed. Some tests will also be employed to check the validity and superiority of the results over the Schulz method. (C) 2014 Elsevier Inc. All rights reserved.Soleimani, F.; Soleymani, F.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2014). On the extension of Householder's method for weighted Moore-Penrose inverse. Applied Mathematics and Computation. 231:407-413. doi:10.1016/j.amc.2014.01.021S40741323

The judgement of God in the person, work and teachings of Jesus: a critical and exegetical study in the synoptic gospels

The topic to which I shall address myself in the forthcoming pages is "The Judgment of God in the Person, Work and Teaching of Jesus in the Synoptics: A Critical and Exegetical Study." This is a vast subject, and lest we lose ourselves in its vastness, it would be well as we begin to delineate the nature of our study, to examine our motives and outline the principles by which we shall be governed.1) This is first of all a study of the judgment of God. Everything that is said will be oriented frankly and deliberately to this great theme. I make no apology for thus limiting my study. It is my conviction that the full understanding of this subject is the greatest single need in the field of modern theology. In the theology of the modern world, especially of America, there is a prevailing tendency to overstress the love of God and to ignore His wrath and the other darker aspects of His nature. In the words of Bishop Temple, "this seems to involve a conception of God as so genially tolerant as to be morally indifferent, and converts the belief in immortality from a moral stiumlant to a moral narcotic." That we have is only half a God, a God stripped of that half of His nature against which His love strikes fire. This is certainly not the God whom I find revealed in scripture. I feel, with Bishop Temple, that there is a great need to completely rethink this subject of the nature of God. With only half a God, there can be only half a Gospel, and with only half a Gospel there can be only partial salvation, which is no salvation at all. The lukewarm theology of partial salvation is the curse of the modern generation. As a remedy for this I have undertaken this study of the judgment of God. I am convinced that here, in the judgment of God, is the very centrality and fulness of the nature of God. Here is the very essence of the Holy Spirit. Here is the very heart of the Incarnation. Here is the very life of a vital theology.2) Secondly, this is a study of the person, work and teaching of Jesus in the Synoptic Gospels. I shall make much use of the Old Testament, of extra-Testamental literature and of the various books of the New Testament, but my emphasis will be fully and frankly on the Synoptics. I do so because it is my conviction that Jesus Christ has given the fullest and most authoritative revelation of the nature of God, and it is in the Synoptics that we find the most accurate and detailed account of this revelation.I shall concentrate on the Synoptics for another important reason. I do so because the Synoptics have fallen into great disuse as a source of Christian theology. All too much of modern Christian theology is written from an historic standpoint, After a brief nod is given to the New Testament evidence, the theologian outlines the development of the doctrine through the ages, and often makes no attempt to distinguish between the teaching of Jesus and that of later New Testament writers and theologians. I fully recognìze the value of the historic approach, but I am concerned with the danger of confusing the history of Christian theology with Christian theology itself. Too often historical studies are the history of error, and the more the theologian piles the history of one error on top of another, the more he tends to perpetuate the error. The tendency for the historical theologian is to read into the Synoptics the beliefs of earlier thinkers, or to read back into the mind of Jesus beliefs of later thinkers found in the historical study. The tendency for the reader of historical theology is to become an eclectic and make up his own doctrine out of a combination of what he feels are the best elements of all the various approaches to Christian truth. Thus error is given a semblance of truth, and so perpetuated., The one great corrective of such tendencies is to go to the primary source of Christian theology, the life and words of Jesus Christ.Another unfortunate trend, illustrative of the point I have been making, is the tendency of the modern theologian. to derive his theology entirely or primarily from the writings of the Apostle Paul. A brief glance at the scriptural indexes of some of the greatest and most recent works on theology will indicate the correctness of this observations If there be any truth in the idea that Christianity is the revelation of God in Jesus Christ, then certainly even the Apostle Paul is a secondary source. Out of my study of the Synoptics, there has grown the conviction that one of the reasons for this modern neglect of the Synoptics is the belief, expressed or unexpressed, that the Synoptics are not a scholastically sound basis for any theology. Much of this feeling comes as the result of a generation of Synoptic skepticism, led by the German school Form-Criticsm. Out of my study has also come the conviction that such skepticism is unfounded on clear, honest scholastic grounds. It will therefore be one of the major burdens of this dissertation to demonstrate that a Synoptic theology is critically tenable, and logically inescapable.3) In the third place, this is a critical and exegetical study. My method will be that of the critical exegete. I use this method because it is my conviction that critical scholarship is here to stay. Any scholar who attempts to exegete the Synoptic Gospels without taking into account the discoveries in the fields of textual, source, historic, literary end form criticism is living in a fool's paradise and doing an injustice to the very scripture he claims to revere. In operating upon the word of God, he is deliberately choosing blunt, time-worn tools instead of the keen, sharp tools, the latest methods, which God has placed in his hands. only result can be less than the best.I shall take full cognizance of the successes and failures of the Farm-Critical movement. There is no other movement in modern history that has wielded a greater influence, for good or ill, upon Synoptic exegesis in particular and Christian theology in general. It is my conviction that this movement has spent its force and that the forces of exegetical common sense are bringing us back to a recognition of the authenticity of the Synoptic material. It is also my conviction that this "Form-Critical revolution" has taught us that never again dare we allow the exegetical obscurantism of the "literal word" to dominate the field of Synoptic scholarship. The last generation of extreme critical scholars tore the Synoptics apart. It is the task of this generation to put them together again, but along the new, resiliently strong lines of positive critical scholarship.4) In the fourth and final place, this will be an exhaustive study. It will be long and technical. I do this deliberately for two reasons. I do so because so many of my exegetical positions are new, or at least not adequately developed by competent scholars, that I feel I must Prove my case every step of the way. I do so also as a protest against a generation of exegetical"short-cutting", of exegetical "declamation" that has substituted assertion for evidence, assumption for proof, and has produced a Synoptic chaos that is an affront to the modern Christian intelligence. If the Gospel of Jesus Christ, rather than the opinions of countless scholars, is to survive, then it is the duty of the Christian scholar to base his Synoptic exegesis upon factual evidence, logically and fully demonstrated