This thesis is concerned with the problem of exhibiting all the rational integer solutions satisfying certain diophantine equations. The thesis consists of two sections. The first section indicates how some types of diophantine equations completely reducible in a single quadratic field may he solved completely, and also the complete solution is obtained for an interesting class of cubic diophantine equations. In the past there have been many isolated\r\ninvestigations on the separate equations of the class considered here and at most only partial integral solutions have been given. In this thesis the complete solutions for these equations are deduced from a single multiplicative equation in a quadratic field.
\r\n\r\nIn the second section Diophantine equations completely reducible in two or more different quadratic fields are considered. These equations are solved by operating on multiplicative equations in biquadratic fields.
\r\n\r\nThe success of our method depends fundamentally upon the\r\ncomplete solution of the simple multiplicative equation xy= zw.
", "doi": "10.7907/dchd-cd06", "publication_date": "1944", "thesis_type": "phd", "thesis_year": "1944" }, { "id": "thesis:16920", "collection": "thesis", "collection_id": "16920", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:12122024-224942949", "primary_object_url": { "basename": "Levit_RJ_1939.pdf", "content": "final", "filesize": 9437152, "license": "other", "mime_type": "application/pdf", "url": "/16920/1/Levit_RJ_1939.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Postulate Sets for Certain Mathematical Systems and their Complete Existential Theory", "author": [ { "family_name": "Levit", "given_name": "Robert Jules", "clpid": "Levit-Robert-Jules" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "No abstract.", "doi": "10.7907/e99q-m874", "publication_date": "1939", "thesis_type": "masters", "thesis_year": "1939" }, { "id": "thesis:1457", "collection": "thesis", "collection_id": "1457", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-04222008-093535", "type": "thesis", "title": "Many-Valued Logics", "author": [ { "family_name": "Webb", "given_name": "Donald Loomis", "clpid": "Webb-Donald-Loomis" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "No abstract.\r\n", "doi": "10.7907/N5WZ-W144", "publication_date": "1936", "thesis_type": "phd", "thesis_year": "1936" }, { "id": "thesis:5773", "collection": "thesis", "collection_id": "5773", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05052010-102644606", "primary_object_url": { "basename": "Poole_ar_1935.pdf", "content": "final", "filesize": 1211324, "license": "other", "mime_type": "application/pdf", "url": "/5773/1/Poole_ar_1935.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Finite Ova", "author": [ { "family_name": "Poole", "given_name": "Albert Roberts", "clpid": "Poole-Albert-Roberts" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis systems consisting of a finite number of elements and one binary commutative associative rule of combination are considered. Such systems are called ova. The distinctness of ova is first discussed. The elements of ova are then classified according to their behavior when raised to powers. A necessary and sufficient condition that an ovum have no associate elements is found, and ova having no associate elements are discussed in detail. A necessary and sufficient condition that an ovum be a finite Abelian group is also found. All the distinct ova of orders 2,3,4, have been computed and are listed in the course of the paper. There are 3 distinct ova of order 2, 12 of order 3, and 56 of order 4. All concepts introduced in the discussion are illustrated in these ova.", "doi": "10.7907/H8RN-2G38", "publication_date": "1935", "thesis_type": "phd", "thesis_year": "1935" }, { "id": "thesis:8171", "collection": "thesis", "collection_id": "8171", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:03262014-091535570", "primary_object_url": { "basename": "Worth_cr_1933.pdf", "content": "final", "filesize": 7213125, "license": "other", "mime_type": "application/pdf", "url": "/8171/1/Worth_cr_1933.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "The Subvarieties of a Field", "author": [ { "family_name": "Worth", "given_name": "Carleton Russell", "clpid": "Worth-Carleton-Russell" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Number systems which satisfy part but not all\r\nof the postulates for a field are called subvarieties\r\nof a field. The purpose of this paper is the determination\r\nof as great as possible a number of such\r\nvarieties by suitable definitions of the class of\r\nelements and of the two operations involved.
\r\n\r\nTwo postulate systems are considered. The first\r\ngives rise to 284 varieties, instances of all of which\r\nare given for infinite classes of elements, and of all\r\nexcept three for finite classes.
\r\n\r\nOf the 8192 combinations of postulates arising\r\nfrom the second system, not more than 1146 can be\r\nconsistent. Instances are given of 1054 of these.\r\nAs the postulates of this system are not independent,\r\nno conclusion has been reached regarding the remaining\r\ncases.
", "doi": "10.7907/0MK5-ZQ13", "publication_date": "1933", "thesis_type": "phd", "thesis_year": "1933" }, { "id": "thesis:1373", "collection": "thesis", "collection_id": "1373", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-04132005-154104", "type": "thesis", "title": "Arithmetized Trigonometrical Expansions of Doubly Periodic Functions of the Third Kind", "author": [ { "family_name": "Elder", "given_name": "John Dyer", "clpid": "Elder-John-Dyer" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We present a method of expanding the members of a certain class of functions, doubly periodic of the third kind, due to Appell. An extension of the details is given, and the method is used to obtain the expansions of those numbers of the class considered whose complexity, in a technical sense, is within an arbitrary chosen range. The details in the expansion of any member of the class are discussed and some information is obtained as to the general form of the result.\r\n", "doi": "10.7907/HY0G-JX06", "publication_date": "1929", "thesis_type": "phd", "thesis_year": "1929" }, { "id": "thesis:2747", "collection": "thesis", "collection_id": "2747", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06282004-092506", "type": "thesis", "title": "The Gravitational Field of a Body with Rotational Symmetry in Einstein's Theory of Gravitation", "author": [ { "family_name": "Zhou", "given_name": "Peiyuan", "clpid": "Zhou-Peiyuan" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Einstein's set of field equations in vaccuo
\r\n\r\nG\u03bc\u03c5 = 0
\r\n\r\nis reduced to such a form that simple problems like the sphere (Schwarzschild's solution), the infinite plane and the infinite cylinder can be solved. The fundamental quadratic differential forms for the latter two cases are respectively
\r\n\r\nds2 = - [(1+4\u03c0\u03c3z)-1dz2 = - [(1+4\u03c0\u03c3z)-1dz + d\u03c12 + \u03c12d\u03c62] + 1+4\u03c0\u03c3z)dt2,
\r\n\r\nds2 = - c24\u03c1-2[(1+4mlog\u03c1)-1d\u03c12 + \u03c12d\u03c62] - dz2 + (1+4mlog\u03c1)dt2,
\r\n\r\nwhere \u03c3 is the surface density of matter on the plane, z=0; m the linear density of matter on the cylinder, \u03c1=const.; (\u03c1,z,\u03c6) the cylindrical coordinates; c4 an indeterminate constant and the velocity of light is unity. Setting g44 = the Newtonian potential + const., we can get the solution of the general gravitational problem for a body whose mass is distributed symmetrically about an axis provided we can solve
\r\n\r\n2\u03b4/\u03b4\u03c8[(1-2M\u03c8)\u03b4n/\u03b4\u03c8] + \u03b42/\u03b4\u03b82e2n = 0 (M = mass of the body).
\r\n\r\nThe gravitational field of an oblate spheroidal homoeoid is characterized by
\r\n\r\nds2 = - \u03c8-4(1-2M\u03c8)-1d\u03c82 - \u03c8-2dE2 - \u03c8-2cos2Ed\u03c82 + (1-2M\u03c8)dt2,
\r\n\r\nwhere \u03c8 = k-1cot-1(sinh\u03b7), M = mass of the homoeoid whose equation is c2\u03c12a2z2 = a2c2, k2 = a2-c2 and E,\u03b7 are related to the cylindrical coordinates (\u03c1,z,\u03c6) by \u03c1+iz = kcos(E+i\u03b7). Analogous expressions for a prolate spheroidal homoeoid are obtainable. The oblateness of the homoeoid causes a slight increase in the advance of the perihelion of a planet's orbit derived from Schwarzschild's solution.
\r\n", "doi": "10.7907/GH13-VA35", "publication_date": "1928", "thesis_type": "phd", "thesis_year": "1928" }, { "id": "thesis:863", "collection": "thesis", "collection_id": "863", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-03042005-135853", "primary_object_url": { "basename": "Ward_m_1928.pdf", "content": "final", "filesize": 2913329, "license": "other", "mime_type": "application/pdf", "url": "/863/1/Ward_m_1928.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "The Foundations of General Arithmetic", "author": [ { "family_name": "Ward", "given_name": "Morgan", "clpid": "Ward-Morgan" } ], "thesis_advisor": [ { "family_name": "Bell", "given_name": "Eric Temple", "clpid": "Bell-E-T" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "No abstract.", "doi": "10.7907/XDSZ-KF26", "publication_date": "1928", "thesis_type": "phd", "thesis_year": "1928" } ]