Comparing the strength of diagonally non-recursive functions in the absence of Σ02 induction

We prove that the statement there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f does not imply weak K\ onig\u27s lemma over RCA0+BΣ02. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally non-recursive function may fail in the absence of IΣ02

Reducibilities in recursive function theory.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D.Bibliography: leaves 102-103.Ph.D

Comparing the degrees of enumerability and the closed Medvedev degrees

We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees

Lower bounds on the redundancy in computations from random oracles via betting strategies with restricted wagers

The Kučera–Gács theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n+h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy nlog⁡n while Gács used a more elaborate coding procedure which achieves redundancy View the MathML source. A similar bound is implicit in the later proof by Merkle and Mihailović. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that ∑n2−g(n)=∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies ∑n2−g(n)<∞. Moreover, the class of such reals is comeager and includes a View the MathML source real as well as all weakly 2-generic reals. On the other hand, it has been recently shown that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that ∑n2−g(n)<∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log⁡n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop

Photonic processing of microwave signals

In this paper, recent advances in the implementation of photonic tuneable transversal filters for RF signal processing developed by the Optical Communications Group are described. After a brief introduction to the field and the basic concepts, limiting factors and application of these filters are described in the field. This is based on a distinction between filters based on wavelength tuneable optical taps and others based on the tuneability of the dispersive elements that provide the time delay between samples. Recently developed approaches have also been discussed

Diagrammatic many-body methods for anharmonic molecular vibrational properties

Diagrammatic many-body methods for computing the energies and other properties of anharmonic vibrations have been developed based on the Dyson equation formalism for the single-particle vibrational Green's function and the many-body perturbation theory for the total zero-point energy. Unlike similar methods based on the vibrational self-consistent field (VSCF) approximation, these XVSCF and XVMP2 methods are guaranteed to be size-consistent at the formalism level, meaning that they are applicable not only to small molecules but also to larger systems including condensed phases. The XVSCF method, initially developed by Keçeli and Hirata, is extended to calculate anharmonic corrections to geometries as well as vibrational frequencies and energies, and rendered identical to the VSCF method in the thermodynamic limit despite orders of magnitude lower computational cost. When XVSCF is formulated in terms of the Dyson equation, it is additionally revealed to be an approximation to the self-consistent phonon (SCP) method which is commonly used in solid-state physics. Furthermore, the development of XVSCF in terms of Green's functions enables the formulation of the concept of Dyson coordinates and Dyson geometries, conceived as anharmonic generalizations of the normal coordinates and equilibrium geometries of the harmonic approximation, which represent a formally exact effectively harmonic treatment of molecular and crystal vibrations, similar to the concept of Dyson orbitals from the field of electronic structure theory. Many-body perturbation theory based on XVSCF is referred to as XVMP2 and is showed to be both more efficient and more powerful than standard VMP2 methods. XVMP2 inherits the computational efficiency and manifest size-consistency of XVSCF, and additionally, through the Dyson-equation formalism, it is able to directly compute vibrational fundamental, overtone, and combination frequencies directly even in the presence of anharmonic resonance. This makes XVMP2 a rare example of a perturbative method which can defeat strong correlation. The XVSCF and XVMP2 methods are formulated in both deterministic algorithms which rely on the computation of a large number of anharmonic force constants, and stochastic algorithms which require no stored representation of the PES. This is a significant advance because the computation and storage of the PES is a significant bottleneck in terms of accuracy and computational cost. The Monte Carlo XVSCF and Monte Carlo XVMP2 methods, as they are called, are uncommon among stochastic methods in that they can compute anharmonic frequencies directly, without noisy, small differences between large total vibrational energies and without sign problems that plague other forms of quantum Monte Carlo such as DMC

Diagonal operators, qq-Whittaker functions and rook theory

We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}_q$. We solve this problem in the case where the operator is diagonalizable. The solution leads us to a new class of polynomials $b_{\mu\nu}(q)$ indexed by pairs of integer partitions. These polynomials have several interesting specializations and can be expressed as positive sums over semistandard tableaux. We present a new correspondence between set partitions and semistandard tableaux. A close analysis of this correspondence reveals the existence of several new set partition statistics which generate the polynomials $b_{\mu\nu}(q)$; each such statistic arises from a Mahonian statistic on multiset permutations. The polynomials $b_{\mu\nu}(q)$ are also given a description in terms of coefficients in the monomial expansion of $q$-Whittaker symmetric functions which are specializations of Macdonald polynomials. We express the Touchard--Riordan generating polynomial for chord diagrams by number of crossings in terms of $q$-Whittaker functions. We also introduce a class of $q$-Stirling numbers defined in terms of the polynomials $b_{\mu\nu}(q)$ and present connections with $q$-rook theory in the spirit of Garsia and Remmel.Comment: 41 pages, 10 figure