The Dimensions of Individual Strings and Sequences

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0,1]. Sequences that are random (in the sense of Martin-Lof) have dimension 1, while sequences that are decidable, \Sigma^0_1, or \Pi^0_1 have dimension 0. It is shown that for every \Delta^0_2-computable real number \alpha in [0,1] there is a \Delta^0_2 sequence S such that \dim(S) = \alpha. A discrete version of constructive dimension is also developed using termgales, which are supergale-like functions that bet on the terminations of (finite, binary) strings as well as on their successive bits. This discrete dimension is used to assign each individual string w a dimension, which is a nonnegative real number dim(w). The dimension of a sequence is shown to be the limit infimum of the dimensions of its prefixes. The Kolmogorov complexity of a string is proven to be the product of its length and its dimension. This gives a new characterization of algorithmic information and a new proof of Mayordomo's recent theorem stating that the dimension of a sequence is the limit infimum of the average Kolmogorov complexity of its first n bits. Every sequence that is random relative to any computable sequence of coin-toss biases that converge to a real number \beta in (0,1) is shown to have dimension \H(\beta), the binary entropy of \beta.Comment: 31 page

A Race to the Stars and Beyond: How the Soviet Union’s Success in the Space Race Helped Serve as a Projection of Communist Power

In the modern era, the notion of space travel is generally one of greater acceptance and ease than in times previously. Moreover, a greater number of nations (and now even private entities) have the technological capabilities to launch manned and unmanned missions into Earth’s Orbit and beyond. 70 years ago, this ability did not exist and humanity was simply an imprisoned species on this planet. The course of humanity’s then-present and the collective future was forever altered when, in 1957, the Soviet Union successfully launched the world’s first satellite into space, setting off a decades-long completion with the United States to cosmically outperform the other. In the context of the Cold War, the ensuring Space Race was more than friendly completion, rather it was a race to determine who’s military and civil society could produce the most powerful interstellar technologies, which in turn demonstrated the combative readiness of either side. This paper seeks to examine the Soviet Union’s success during the Space Race (and subsequently, the global Arms Race) and its place within the larger East versus West conflict which occurred in the earlier years of the Cold War. By utilizing academic literature and primary Soviet sources, this paper will analyze how the Space Race allowed the Soviet Union to promote the successes of a Communist government and how such a leadership style served as a positive determinant for advancements in space and the Soviet Union’s premier place in many suc

Constraints on RG Flow for Four Dimensional Quantum Field Theories

The response of four dimensional quantum field theories to a Weyl rescaling of the metric in the presence of local couplings and which involve $a$, the coefficient of the Euler density in the energy momentum tensor trace on curved space, is reconsidered. Previous consistency conditions for the anomalous terms, which implicitly define a metric $G$ on the space of couplings and give rise to gradient flow like equations for $a$, are derived taking into account the role of lower dimension operators. The results for infinitesimal Weyl rescaling are integrated to finite rescalings $e^{2\sigma}$ to a form which involves running couplings $g_\sigma$ and which interpolates between IR and UV fixed points. The results are also restricted to flat space where they give rise to broken conformal Ward identities. Expressions for the three loop Yukawa $\beta$-functions for a general scalar/fermion theory are obtained and the three loop contribution to the metric $G$ for this theory are also calculated. These results are used to check the gradient flow equations to higher order than previously. It is shown that these are only valid when $\beta \to B$, a modified $\beta$-function, and that the equations provide strong constraints on the detailed form of the three loop Yukawa $\beta$-function. ${\cal N}=1$ supersymmetric Wess-Zumino theories are also considered as a special case. It is shown that the metric for the complex couplings in such theories may be restricted to a hermitian form.Comment: 86 pages, version 2, various corrections, section 3 significantly revised, version 3 further minor corrections, as to be published, version 4, some corrections and additional material in sections 2,