On the ultimate complexity of factorials

AbstractIt has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n!) by straight-line programs. Formally, we say that a sequence of integers an is ultimately f(n)-computable, if there exists a nonzero integer sequence mn such that for any n, anmn can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f(n). Shub and Smale [12] showed that if n! is ultimately hard to compute, then the algebraic version of NP≠P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound (exp(clognloglogn)) for the ultimate complexity of n! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well

Exponential Sums and Congruences with Factorials

We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials $n!m!$ and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials $n!m!$ with $\max\{n,m\}<p^{1/2+\epsilon}$ are uniformly distributed modulo $p$, and that any residue class modulo $p$ is representable in the form $m!n!+n_1! + ... +n_{49}!$ with $\max \{m,n, n_1, >..., n_{49}\} < p^{8775/8794+ \epsilon}$.Comment: 21 page

Fast and Accurate Bilateral Filtering using Gauss-Polynomial Decomposition

The bilateral filter is a versatile non-linear filter that has found diverse applications in image processing, computer vision, computer graphics, and computational photography. A widely-used form of the filter is the Gaussian bilateral filter in which both the spatial and range kernels are Gaussian. A direct implementation of this filter requires $O(\sigma^2)$ operations per pixel, where $\sigma$ is the standard deviation of the spatial Gaussian. In this paper, we propose an accurate approximation algorithm that can cut down the computational complexity to $O(1)$ per pixel for any arbitrary $\sigma$ (constant-time implementation). This is based on the observation that the range kernel operates via the translations of a fixed Gaussian over the range space, and that these translated Gaussians can be accurately approximated using the so-called Gauss-polynomials. The overall algorithm emerging from this approximation involves a series of spatial Gaussian filtering, which can be implemented in constant-time using separability and recursion. We present some preliminary results to demonstrate that the proposed algorithm compares favorably with some of the existing fast algorithms in terms of speed and accuracy.Comment: To appear in the IEEE International Conference on Image Processing (ICIP 2015

A Computational Framework for High-Throughput Isotopic Natural Abundance Correction of Omics-Level Ultra-High Resolution FT-MS Datasets

New metabolomics applications of ultra-high resolution and accuracy mass spectrometry can provide thousands of detectable isotopologues, with the number of potentially detectable isotopologues increasing exponentially with the number of stable isotopes used in newer isotope tracing methods like stable isotope-resolved metabolomics (SIRM) experiments. This huge increase in usable data requires software capable of correcting the large number of isotopologue peaks resulting from SIRM experiments in a timely manner. We describe the design of a new algorithm and software system capable of handling these high volumes of data, while including quality control methods for maintaining data quality. We validate this new algorithm against a previous single isotope correction algorithm in a two-step cross-validation. Next, we demonstrate the algorithm and correct for the effects of natural abundance for both 13C and 15N isotopes on a set of raw isotopologue intensities of UDP-N-acetyl-D-glucosamine derived from a 13C/15N-tracing experiment. Finally, we demonstrate the algorithm on a full omics-level dataset

Cellular signaling networks function as generalized Wiener-Kolmogorov filters to suppress noise

Cellular signaling involves the transmission of environmental information through cascades of stochastic biochemical reactions, inevitably introducing noise that compromises signal fidelity. Each stage of the cascade often takes the form of a kinase-phosphatase push-pull network, a basic unit of signaling pathways whose malfunction is linked with a host of cancers. We show this ubiquitous enzymatic network motif effectively behaves as a Wiener-Kolmogorov (WK) optimal noise filter. Using concepts from umbral calculus, we generalize the linear WK theory, originally introduced in the context of communication and control engineering, to take nonlinear signal transduction and discrete molecule populations into account. This allows us to derive rigorous constraints for efficient noise reduction in this biochemical system. Our mathematical formalism yields bounds on filter performance in cases important to cellular function---like ultrasensitive response to stimuli. We highlight features of the system relevant for optimizing filter efficiency, encoded in a single, measurable, dimensionless parameter. Our theory, which describes noise control in a large class of signal transduction networks, is also useful both for the design of synthetic biochemical signaling pathways, and the manipulation of pathways through experimental probes like oscillatory input.Comment: 15 pages, 5 figures; to appear in Phys. Rev.

How do middle managers deal with uncertainty in the strategy process?

Uncertainty is an unavoidable dimension of organizational life, and it is also challenging even for the most competent managers (Pich, Loch, & Meyer, 2002). This thesis examined middle managers’ uncertainty responses during the formulation and implementation of the strategy process in three studies. The first study was qualitative and intended to expose the sources of uncertainty, managerial responses, and variables, which are essential for middle managers to cope with uncertainty in the strategy process. Study 1, involving interviews with 22 middle managers in civilian and military organizations, resulted in six sources, including a new classification--International instability and disasters--and five different responses: Collaborative responses, Emotional responses, Cognitive responses, Value-based responses, and Bureaucratic responses. In the second study, we conducted a Multidimensional scaling technique (N=70) to create a taxonomy of perception of managerial uncertainty responses in order to understand the nature of uncertainty in any organization and to help to build new theories. Results indicated six clusters: Protection by Support, Protection by Structure, Protection by Scapegoats, Certainty of Change, Development by Debate, and Development by Change. In the third study, we examined the effects of individual responses to uncertainty and organizational factors on managerial responses using quantitative analysis (N=310). The results showed that organizational-level knowledge sharing has positive effects on managers’ desire of change. This study added a new source of uncertainty and five different managerial responses to uncertainty, and revealed that individual-level cognitive uncertainty and desire of change result in bureaucratic and collaborative responses. Additionally, we contend that managers respond to uncertainty variously, from suppressing to collaboration, either to protect themselves or to act towards a constructive change in the organizations.A incerteza é uma dimensão inevitável da vida organizacional, e também é um desafio até mesmo para os gerentes mais competentes (Pich, Loch & Meyer, 2002). Esta dissertação examinou as respostas de incerteza do gerente intermediário durante a formulação e implementação do processo de estratégia em três estudos. O primeiro estudo foi qualitativo e pretendia expor as fontes de incerteza, respostas gerenciais e variáveis, que são essenciais para os gerentes de nível médio lidarem com a incerteza no processo de estratégia. O estudo 1, com entrevistas a 22 gerentes de nível médio em organizações civis e militares, resulta em seis fontes, incluindo uma nova classificação; Instabilidade internacional e desastres e cinco respostas diferentes; Respostas colaborativas, respostas emocionais, respostas cognitivas, respostas baseadas em valores e respostas burocráticas. No segundo estudo, conduzimos uma técnica de dimensionamento multidimensional (N = 70) para criar uma taxonomia da percepção das respostas gerenciais de incerteza para entender a natureza da incerteza em qualquer organização e para ajudar a construir novas teorias. Os resultados indicaram cinco clusters; Proteção por Suporte, Proteção por Estrutura, Proteção por Bodes Expiatórios, Certeza de Mudança, Desenvolvimento por Debate e Desenvolvimento por Mudança. No terceiro estudo, examinamos os efeitos das respostas individuais à incerteza e fatores organizacionais sobre respostas gerenciais por meio de análise quantitativa (N = 310). Os resultados mostraram que o compartilhamento do conhecimento no nível organizacional tem efeitos positivos no desejo de mudança dos gestores. Este estudo adicionou uma nova fonte de incerteza e cinco respostas gerenciais diferentes à incerteza e revelou que a incerteza cognitiva de nível individual e o desejo de mudança resultam em respostas burocráticas e colaborativas. Além disso, afirmamos que os gerentes reagem à incerteza, desde a supressão até a colaboração, seja para se protegerem ou para agir em prol de uma mudança construtiva nas organizações