Let L(f;x,y;r), A(f;x,y;r) be the mean values of a function f(x,y) of\r\ntwo real variables on the perimeter and on the interior, respectively, of a\r\ncircle of center (x,y) and radius r. The limits
\r\n\r\nlimr\u21920 L(f;x,y;r) - f(x,y)/r2 = f'(x,y)
\r\n\r\nlimr\u21920 A(f;x,y;r) - f(x,y)/r2 = f'D(x,y)
\r\n\r\nare called Mean Value Derivatives of f(x,y). This paper is concerned with\r\nthe investigation of functions with mean value derivatives. These derivatives\r\nare essentially generalizations of LaPlace's operator, and, as such, were\r\ninvestigated by Blaschke and Privaloff. In addition Zaremba has investigated\r\nanother form of generalized LaPlacian, and Plancherel has investigated a\r\ngeneralization of Beltrami's parameter. Many of the results obtained for these\r\nlast two operators hold true for mean value derivatives.
\r\n\r\nChapter I contains some results relating to the mean value derivative\r\nas given by eqn. (1) while Chapter II is a similar treatment of eqn.(2.). Most of\r\nthe results given in these two chapters are known for at least one of the four\r\noperators, i.e. those of Blaschke, Privaloff, Zaremba, and Plancherel. Chapter\r\nIII discusses briefly uniform mean value derivatives. Chapter IV is devoted to\r\nthe use of potential theory in the subject and Chapter V to higher derivatives.\r\nChapter VI is concerned with further problems on the subject and Chapter VII\r\ncontains a summary of the results of the authors mentioned above. The principle\r\nnew results obtained are as follows:
\r\n\r\n(1) If f'0(x,y) exists then so does f'(x,y). This is a generalization of a result\r\ndue to Kozakiewicz, who assumed continuity of f. Tb.is assumption is not\r\nnecessary.
\r\n\r\n(2) If (i)f(x,y)is continuous, (ii) f'(x,y) exists and is bounded, (iii) f'(x,y)=o almost\r\neverywhere on a domain D, then f(x,y) is harmonic on D.
\r\n\r\n(3) If f(x,y) is a logarithmic potential function for which the density of the mass\r\ndistribution exists at a point P then f'(P) exists.
\r\n\r\n(4) Expansions in powers of r2 are obtained for the means L(f;x,y;r), A(f;x,y;r) in\r\nwhich the coefficients involve the higher mean value derivatives of\r\nin a manner analogous to Taylor's Theorem.
", "doi": "10.7907/vzy3-vc51", "publication_date": "1947", "thesis_type": "phd", "thesis_year": "1947" }, { "id": "thesis:4601", "collection": "thesis", "collection_id": "4601", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-11182008-135603", "primary_object_url": { "basename": "Hayes_wd_1943.pdf", "content": "final", "filesize": 1952056, "license": "other", "mime_type": "application/pdf", "url": "/4601/1/Hayes_wd_1943.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Some Investigations in the Buckling of Thin Rods", "author": [ { "family_name": "Hayes", "given_name": "Wallace Dean", "clpid": "Hayes-Wallace-Dean" } ], "thesis_advisor": [ { "family_name": "Bateman", "given_name": "Harry", "clpid": "Bateman-H" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "GALCIT" }, { "literal": "div_eng" } ], "abstract": "The general case of a non-uniform, straight rod under varying axial load is investigated, and several methods of solution are indicated or described. The case of a uniform rod with constant axial load is investigated by means of its deflection curve, and the direct determination of the stability with general end restraints is made possible by the use of a graph. The correlation between the end fixities of a rod and its behavior as a beam is given.\r\n", "doi": "10.7907/2G6H-SH82", "publication_date": "1943", "thesis_type": "engd", "thesis_year": "1943" }, { "id": "thesis:14022", "collection": "thesis", "collection_id": "14022", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:12092020-212456873", "primary_object_url": { "basename": "Gould_MJ_1941.pdf", "content": "final", "filesize": 36624883, "license": "other", "mime_type": "application/pdf", "url": "/14022/1/Gould_MJ_1941.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "The Ground Roll Phenomenon of Applied Seismology", "author": [ { "family_name": "Gould", "given_name": "Martin James", "clpid": "Gould-Martin-James" } ], "thesis_advisor": [ { "family_name": "Gutenberg", "given_name": "Beno", "clpid": "Gutenberg-B" }, { "family_name": "Bateman", "given_name": "Harry", "clpid": "Bateman-H" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_gps" } ], "abstract": "Since the preceding series of investigations are of a somewhat different character than the one to follow, it is desirable at this point to summarize the results so far obtained:
\r\n\r\n1. Gravitational waves in a viscous incompressible medium are far too slow to account for the velocity of first arrival of the ground roll.
\r\n\r\n2. The importance of Bateman's secondary Rayleigh wave cannot be known without the solution of the complex problem of the partition of energy at the source of the Rayleigh waves.
\r\n\r\n3. The theory of the propagation of Love waves on the surface of a layered visco-elastic medium indicates the possibility of velocities less than the shear wave velocity obtained without viscosity. The rapid damping of very short waves is also indicated. The possibility of obtaining a damping of the order of the ground roll damping has been demonstrated.
\r\n\r\n4. An examination of seismic data on the velocity of the ground roll has indicated that the observations are not necessarily inconsistent with the theory of dispersion of Rayleigh waves in a layered elastic medium, but, that there may be other causes of the observed dispersion.
\r\n\r\nCertain anomalous ground roll velocity variation near Fresno; California is possibly explanable on the hypothesis of dispersion in a layered elastic medium. For small values of L/H, the velocity of the ground roll agrees roughly with the velocity of Rayleigh waves, but these same velocities are associated with an initial anomalous forward cycle of the particle motion. Theoretically thereis an inherent disadvantage in obtaining dispersion data on the ground roll by seismic means, because of the factor of possible resonance making the interpretation of the results difficult. The three component ground roll data suggest the possible co-existence of Love waves and Rayleigh waves.
", "doi": "10.7907/77xt-pm46", "publication_date": "1941", "thesis_type": "phd", "thesis_year": "1941" }, { "id": "thesis:16433", "collection": "thesis", "collection_id": "16433", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292024-201308777", "primary_object_url": { "basename": "Hall_NA_1938.pdf", "content": "final", "filesize": 32054763, "license": "other", "mime_type": "application/pdf", "url": "/16433/1/Hall_NA_1938.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Part I. A Formal Expansion Theory for Functions of One or More Variables. Part II. The Number of Representations Function for Binary Quadratic Forms", "author": [ { "family_name": "Hall", "given_name": "Newman Arnold", "clpid": "Hall-Newman-Arnold" } ], "thesis_advisor": [ { "family_name": "Bateman", "given_name": "Harry", "clpid": "Bateman-H" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "No abstract.", "doi": "10.7907/k80p-sp52", "publication_date": "1938", "thesis_type": "phd", "thesis_year": "1938" }, { "id": "thesis:803", "collection": "thesis", "collection_id": "803", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-02282005-140719", "primary_object_url": { "basename": "Hicks_hc_1928.pdf", "content": "final", "filesize": 411960, "license": "other", "mime_type": "application/pdf", "url": "/803/1/Hicks_hc_1928.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Lagrangian Functions Which Determine a Symmetrical Tensor by Schrodinger's Rule", "author": [ { "family_name": "Hicks", "given_name": "Hervey Crandall", "clpid": "Hicks-Hervey-Crandall" } ], "thesis_advisor": [ { "family_name": "Bateman", "given_name": "Harry", "clpid": "Bateman-H" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The choice of a Lagrangian function to be used in a variational principle may be limited by the condition that the tensor derived from it by Schrodinger\u2019s rule shall be symmetrical. To meet this condition the function must satisfy a certain set of partial differential equations. Particular and general solutions of these equations are found in various cases\u2014according as the function is restricted to depend (A) only on the components of a vector, (B) only on their first derivatives, or (C) on both; and according to the number of dimensions of the vector. Methods of obtaining such solutions, and of proving their independence or of finding the relations between them, are discussed.\r\n", "doi": "10.7907/EKPF-YT05", "publication_date": "1928", "thesis_type": "phd", "thesis_year": "1928" }, { "id": "thesis:4823", "collection": "thesis", "collection_id": "4823", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-12072004-142200", "primary_object_url": { "basename": "Robertson_hw_1925.pdf", "content": "final", "filesize": 1094163, "license": "other", "mime_type": "application/pdf", "url": "/4823/1/Robertson_hw_1925.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "On the Dynamical Space-Times Which Contain a Conformal Euclidean 3-Space", "author": [ { "family_name": "Robertson", "given_name": "Howard Percy", "clpid": "Robertson-Howard-Percy" } ], "thesis_advisor": [ { "family_name": "Bateman", "given_name": "Harry", "clpid": "Bateman-H" }, { "family_name": "Epstein", "given_name": "Paul Sophus", "clpid": "Epstein-P-S" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "No abstract.", "doi": "10.7907/4RWV-4C94", "publication_date": "1925", "thesis_type": "phd", "thesis_year": "1925" } ]