Parikh Image of Pushdown Automata

We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with $n$ states and $p \geq 2n + 4$ stack symbols that accepts one single long word for which every equivalent context-free grammar needs $\Omega(n^2(p-2n-4))$ variables. This family shows that the classical algorithm for converting a PDA to an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh's theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.Comment: 17 pages, 2 figure

Hypercomplex quantum mechanics

The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a more general type. In particular, detailed discussion is given of the simplest generalization of the complex Hilbert space, that of the quaternion Hilbert module.Comment: Plain Tex, 11 page

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems

Small ball probability, Inverse theorems, and applications

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n$$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.Comment: 47 page

The new mythologies and premature aging in the youth culture

Comparative studies of aging men in a variety of preliterate traditional societies suggest that older men, across cultures, are relatively mild and uncompetitive, as compared to younger men from the same communities. Older men are more interested in receiving than in producing, more interested in communion than in agency; their sense of pleasure and security is based on food, religion, and the assurance of love. The counterculture gives priority to the same themes, and thereby seems to sponsor a premature senescence, in the psychological sense. Various contemporary myths stemming from affluence and consumerism that have led to the new geriatrics are examined, particularly the myth of the all-including, omnipotential self, which is seen as a translation of socialist, collectivist ideals into the domain of personality. The effects of the new psychic collectivism on ego development in the adolescent and postadolescent periods are also considered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45279/1/10964_2005_Article_BF02214091.pd

Nonaminoglycoside compounds induce readthrough of nonsense mutations

Large numbers of genetic disorders are caused by nonsense mutations for which compound-induced readthrough of premature termination codons (PTCs) might be exploited as a potential treatment strategy. We have successfully developed a sensitive and quantitative high-throughput screening (HTS) assay, protein transcription/translation (PTT)–enzyme-linked immunosorbent assay (ELISA), for identifying novel PTC-readthrough compounds using ataxia-telangiectasia (A-T) as a genetic disease model. This HTS PTT-ELISA assay is based on a coupled PTT that uses plasmid templates containing prototypic A-T mutated (ATM) mutations for HTS. The assay is luciferase independent. We screened ∼34,000 compounds and identified 12 low-molecular-mass nonaminoglycosides with potential PTC-readthrough activity. From these, two leading compounds consistently induced functional ATM protein in ATM-deficient cells containing disease-causing nonsense mutations, as demonstrated by direct measurement of ATM protein, restored ATM kinase activity, and colony survival assays for cellular radiosensitivity. The two compounds also demonstrated readthrough activity in mdx mouse myotube cells carrying a nonsense mutation and induced significant amounts of dystrophin protein

Exploring new physics frontiers through numerical relativity

The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology