It is known that the binary Goldbach problem is one of the open problems on linear equations in primes, and it has the Goldbach-Linnik problem, that is, representation of an even integer in the form of two odd primes and powers of two, as its approximate problem. The theme of my research is on linear equations in primes and powers of two. Precisely, there are two cases: one pair of linear equations in primes and powers of two, and one class of pairs of linear equations in primes and powers of two, in this thesis. In 2002, D.R. Heath-Brown and P.C. Puchta obtained that every sufficiently large even integer is the sum of two odd primes and k powers of two. Here k = 13, or = 7 under the generalized Riemann hypothesis. In 2010, B. Green and T. Tao obtained that every pair of linear equations in four prime variables with coefficients matrix A = (a_ij)s×t with s ≤ t, satisfying nondegenerate condition, that is, A has full rank and the only elements of the row-space of A over Q with two or fewer nonzero entries is the zero vector, is solvable. The restriction on the coefficient matrix means that they excluded the case of the binary Goldbach problem. Motivated by the above results, it is obtained that for every pair of sufficiently large positive even integers B1, B2, the simultaneous equation {█({B1 = p1 + p2 + 2v1 + 2v2 + · · · + 2vk ,@B2 = p3 + p4 + 2v1 + 2v2 + · · · + 2vk ,)┤ (1) is solvable, where p1, · · · , p4 are odd primes, each vi is a positive integer, and the positive integer k ≥ 63 or ≥ 31 under the generalized Riemann hypothesis. Note that, in 1989, M.C. Liu and K.M. Tsang have obtained that subject to some natural conditions on the coefficients, every pair of linear equations in five prime variables is solvable. Therefore one class of pairs of linear equations in four prime variables with special coefficient matrix and powers of two is considered. Indeed, it is deduced that every pair of integers B1 and B2 satisfying B1 ≡ 0 (mod 2), 3BB1 > e^(eB^48 ), B2 ≡ ∑_1^4▒= 1^(a_i ) (mod 2) and |B2| < BB1, where B = max1≤j≤4(2, |aj|), can be represented as {█(B1 = 〖p1〗_1 + p2 + 2^(v_1 ) + 2^(v_2 )+ · · · + 2^(v_k )@B2 = a1p1 + a2p2 + a3p3 + a4p4 + 2^(v_1 )+ 2^(v_2 )+ · · · + 2^(v_k ) )┤ (2) with k being a positive integer. Here p1, · · · p4 are odd primes, each 〖v 〗_iis a positive integer and the integral coefficients ai (i = 1, 2, 3, 4) satisfy {█((〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) = 1,@〖a 〗_1 〖a 〗_2< 0, 〖a 〗_3 〖a 〗_4<0,)┤ Moreover it is calculated that the positive integer k ≥ g(〖a 〗_1- 〖a 〗_2, 〖a 〗_3, 〖a 〗_4) where g(〖a 〗_21- 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = [(log⁡〖G(〖a 〗_21, …, 〖a 〗_24 〗)-log⁡〖F (〖a 〗_21, …, 〖a 〗_24)〗)/log0.975805-84.0285], (3) G(〖a 〗_21, 〖a 〗_22, 〖a 〗_23, 〖a 〗_24) = (min(1/(|a_24 |), 1/(|a_23 |)) - (〖|a〗_(21 )- a_22 |)/(|〖a_23 a〗_24 |) 〖(3B)〗^(-1) ×〖(3B)〗^(-1) (1-0.000001)- 〖(3B)〗^(-1-4), with B = max1≤j≤4(2, |a2j|), and F(a_21, …, a_24) = √(f(a_21)f〖(a〗_22 )) with f(a_2i) = {█(4414.15h (a_21-1)+5.088331 if a_21≠1@59.8411 if a_21=1,)┤ for i = 1, 2, and h(n) =∏_(p|n,p>2)▒(p-1)/(p-2). This result, if without the powers of two, can make up some of the cases excluded in Green and Tao’s paper. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Prime numbers play an integral part in many upper level mathematics courses, most notably in Number Theory. Can a course or section on prime numbers be introduced at the secondary (high school) level? This report outlines a possible course in a manner suitable for grade level instruction. These topics include: an extended section on the complete number system, a brief history of primes, their cardinality, and both the Fundamental Theorem of Arithmetic and Prime Number Theorem, the applications of primes, and the impact of primes within perfect numbers will all be explored. A brief discussion on questions that still remain relating to prime numbers will conclude this report. / text

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

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No description available.

Feature-based attention refers to the phenomenon that attending to a feature value (e.g., a specific shade of red) enhances the detection of similar feature values (e.g., the same shade of red or other shades of red similar to the attended shade) relative to different feature values (e.g., green) that belong to a different object, and that this facilitation effect can be found across the visual field. In previous studies, the participants’ task was primarily the detection or discrimination of simple features such as orientation, colour or motion. The experiments reported in this thesis investigated whether feature-based attention could also influence the speed and/or accuracy of discriminating alphanumeric stimuli such as letters and numbers. In three experiments, participants saw displays that consisted of a series of stimulus patterns at a central location followed by the appearance of an alphanumeric stimulus at one of two peripheral locations. Experiment 1 tested whether paying attention to a specific orientation in a central stimulus would affect the speed and/or accuracy of identifying a peripheral letter whose principal axis was either the same as or different from the attended orientation of the central stimulus. Experiment 2 changed the peripheral stimulus from a letter to a number. In Experiment 3, a peripheral stimulus occurred randomly on 50% of the trials instead of on 100% of the trials. The results showed that attending to a specific orientation of a central stimulus could affect the processing efficiency of both letters and numbers at a peripheral location when the alphanumeric stimulus occurred on every trial (Experiments 1 and 2), but not when it appeared on 50% of the trials. These results suggest that feature-based attention could influence the identification of alphanumeric stimuli. However, the effect may be quite short-lived.

No description available.

The ring of symmetric functions Λ, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the symmetric group. One may define a coproduct on Λ by the plethystic addition on alphabets. In this way the ring of symmetric functions becomes a Hopf algebra. The Littlewood–Richardson numbers may be viewed as the structure constants for the co-product in the Schur basis. The first part of this thesis, inspired by the umbral calculus of Gian-Carlo Rota, is a study of the co-algebra maps of Λ. The Macdonald polynomials are a somewhat mysterious qt-deformation of the Schur functions. The second part of this thesis contains a proof a generating function identity for the Macdonald polynomials which was originally conjectured by Kawanaka.

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2008/04/12). Includes bibliographical references.

Habilitationsschrift--Freiburg i.B. / Includes indexes. Includes bibliographical references (p. [335]-346).