Simple Mathematical Properties Research Articles

Uncovering ecological state dynamics with hidden Markov models.

Ecological systems can often be characterised by changes among a finite set of underlying states pertaining to individuals, populations, communities or entire ecosystems through time. Owing to the inherent difficulty of empirical field studies, ecological state dynamics operating at any level of this hierarchy can often be unobservable or ‘hidden’. Ecologists must therefore often contend with incomplete or indirect observations that are somehow related to these underlying processes. By formally disentangling state and observation processes based on simple yet powerful mathematical properties that can be used to describe many ecological phenomena, hidden Markov models (HMMs) can facilitate inferences about complex system state dynamics that might otherwise be intractable. However, HMMs have only recently begun to gain traction within the broader ecological community. We provide a gentle introduction to HMMs, establish some common terminology, review the immense scope of HMMs for applied ecological research and provide a tutorial on implementation and interpretation. By illustrating how practitioners can use a simple conceptual template to customise HMMs for their specific systems of interest, revealing methodological links between existing applications, and highlighting some practical considerations and limitations of these approaches, our goal is to help establish HMMs as a fundamental inferential tool for ecologists.

Wallenius Bayes

This paper introduces a new event model appropriate for classifying (binary) data generated by a “destructive choice” process, such as certain human behavior. In such a process, making a choice removes that choice from future consideration yet does not influence the relative probability of other choices in the choice set. The proposed Wallenius event model is based on a somewhat forgotten non-central hypergeometric distribution introduced by Wallenius (Biased sampling: the non-central hypergeometric probability distribution. Ph.D. thesis, Stanford University, 1963). We discuss its relationship with models of how human choice behavior is generated, highlighting a key (simple) mathematical property. We use this background to describe specifically why traditional multivariate Bernoulli naive Bayes and multinomial naive Bayes each are suboptimal for such data. We then present an implementation of naive Bayes based on the Wallenius event model, and show experimentally that for data where we would expect the features to be generated via destructive choice behavior Wallenius Bayes indeed outperforms the traditional versions of naive Bayes for prediction based on these features. Furthermore, we also show that it is competitive with non-naive methods (in particular, support-vector machines). In contrast, we also show that Wallenius Bayes underperforms when the data generating process is not based on destructive choice.

Biological Relevance of Network Architecture.

Mathematical representations of brain networks in neuroscience through the use of graph theory may be very useful for the understanding of neurological diseases and disorders and such an explanatory power is currently under intense investigation. Graph metrics are expected to vary across subjects and are likely to reflect behavioural and cognitive performances. The challenge is to set up a framework that can explain how behaviour, cognition, memory, and other brain properties can emerge through the combined interactions of neurons, ensembles of neurons, and larger-scale brain regions that make information transfer possible. "Hidden" graph theoretic properties in the construction of brain networks may limit or enhance brain functionality and may be representative of aspects of human psychology. As theorems emerge from simple mathematical properties of graphs, similarly, cognition and behaviour may emerge from the molecular, cellular and brain region substrate interactions. In this review report, we identify some studies in the current literature that have used graph theoretical metrics to extract neurobiological conclusions, we briefly discuss the link with the human connectome project as an effort to integrate human data that may aid the study of emergent patterns and we suggest a way to start categorizing diseases according to their brain network pathologies as these are measured by graph theory.

Data-driven outbreak forecasting with a simple nonlinear growth model

Recent events have thrown the spotlight on infectious disease outbreak response. We developed a data-driven method, EpiGro, which can be applied to cumulative case reports to estimate the order of magnitude of the duration, peak and ultimate size of an ongoing outbreak. It is based on a surprisingly simple mathematical property of many epidemiological data sets, does not require knowledge or estimation of disease transmission parameters, is robust to noise and to small data sets, and runs quickly due to its mathematical simplicity. Using data from historic and ongoing epidemics, we present the model. We also provide modeling considerations that justify this approach and discuss its limitations. In the absence of other information or in conjunction with other models, EpiGro may be useful to public health responders.

A physical basis for sensory perception

It is argued that the process of perception takes origin within physics itself. A simple, physical model of a biological sensory receptor unit, a unit which mediates perception at its most elemental level, is developed. This model will be not just a detector of sensory signals (like a light meter or sound level meter), but will transduce these signals to the level of consciousness. The properties of this physical model of the sensory receptor unit are drawn from classical physics. Because of its simplicity, the receptor model allows for perception of only discrete quantities of incident signal energy. My primary goal in presenting this reduced model of perception is to teach concepts without the need for detailed anatomy or physiology. Using the simple mathematical properties of the receptor model, we are able to derive a number of the empirical equations of sensory science. Since the idea has been advanced that the process of perception, at a fundamental level, belongs to physics whose validity is universal, it is suggested that the “laws” of perception of the world manifested by organisms anywhere within the universe will be similar to the laws we observe here on earth.

Creation of matter in the universe and groups of type E7

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

Grid cell hexagonal patterns formed by fast self‐organized learning within entorhinal cortex

Grid cells in the dorsal segment of the medial entorhinal cortex (dMEC) show remarkable hexagonal activity patterns, at multiple spatial scales, during spatial navigation. It has previously been shown how a self-organizing map can convert firing patterns across entorhinal grid cells into hippocampal place cells that are capable of representing much larger spatial scales. Can grid cell firing fields also arise during navigation through learning within a self-organizing map? This article describes a simple and general mathematical property of the trigonometry of spatial navigation which favors hexagonal patterns. The article also develops a neural model that can learn to exploit this trigonometric relationship. This GRIDSmap self-organizing map model converts path integration signals into hexagonal grid cell patterns of multiple scales. GRIDSmap creates only grid cell firing patterns with the observed hexagonal structure, predicts how these hexagonal patterns can be learned from experience, and can process biologically plausible neural input and output signals during navigation. These results support an emerging unified computational framework based on a hierarchy of self-organizing maps for explaining how entorhinal-hippocampal interactions support spatial navigation.

Efficient algebraic representation of quantum circuits

A novel and powerful method for representing the unitary matrix of a quantum circuit as an algebraic equation is presented. The method forms a product of sum of Kronecker products (POSOK’s) equation whereby simple mathematical properties can be used to manipulate the algebraic equation to reduce the quantum circuit and prove known circuit equivalences. The method is simple enough to be done by hand or simple automation. The benefits of such a method apply to quantum simulation where the quantum circuit can be easily parsed and reduction techniques can reduce simulation time. Also, having an algebraic expression that can be manipulated through a set of mathematical rules has applications in quantum logic synthesis and quantum computational theory.

Efficient and Accurate Construction of Genetic Linkage Maps from the Minimum Spanning Tree of a Graph

Genetic linkage maps are cornerstones of a wide spectrum of biotechnology applications, including map-assisted breeding, association genetics, and map-assisted gene cloning. During the past several years, the adoption of high-throughput genotyping technologies has been paralleled by a substantial increase in the density and diversity of genetic markers. New genetic mapping algorithms are needed in order to efficiently process these large datasets and accurately construct high-density genetic maps. In this paper, we introduce a novel algorithm to order markers on a genetic linkage map. Our method is based on a simple yet fundamental mathematical property that we prove under rather general assumptions. The validity of this property allows one to determine efficiently the correct order of markers by computing the minimum spanning tree of an associated graph. Our empirical studies obtained on genotyping data for three mapping populations of barley (Hordeum vulgare), as well as extensive simulations on synthetic data, show that our algorithm consistently outperforms the best available methods in the literature, particularly when the input data are noisy or incomplete. The software implementing our algorithm is available in the public domain as a web tool under the name MSTmap.

Autocorrelation Values of New Generalized Cyclotomic Sequences of Order Two and Length pq

Pseudo-random sequences are used extensively for their high speed and security level and less errors. As a branch, the cyclotomic sequences and the generalized ones are studied widely because of their simple mathematical structures and excellent pseudo-random properties. In 1998, Ding and Helleseth introduced a new generalized cyclotomy which includes the classical cyclotomy as a special case. In this paper, based on the generalized cyclotomy, new generalized cyclotomic sequences with order two and length pq are constructed. An equivalent definition of the sequences is deduced so that the autocorrelation values of these sequences can be determined conveniently. The construction contributes to the understanding of the periodic autocorrelation structure of cyclotomically-constructed binary sequences, and the autocorrelation function takes on only a few values.

The (mis)behavior of markets: a fractal view of risk, ruin, and reward

This international bestseller, which foreshadowed a market crash, explains why it could happen again if we don’t act now. Fractal geometry is the mathematics of roughness: how to reduce the outline of a jagged leaf or static in a computer connection to a few simple mathematical properties. With his fractal tools, Mandelbrot has got to the bottom of how financial markets really work. He finds they have a shifting sense of time and wild behaviour that makes them volatile, dangerous - and beautiful. In his models, the complex gyrations of the FTSE 100 and exchange rates can be reduced to straightforward formulae that yield a much more accurate description of the risks involved.

Efficient Index Set Generation for Compiling HPF Array Statements on Distributed-Memory Machines

In languages such as High Performance Fortran (HPF), array statements are used to express data parallelism. In compiling array statements for distributed-memory machines, efficient enumeration of local index sets and commmunication sets is important. A method based on a virtual processor approach has been proposed for efficient index set enumeration for array statements involving arrays distributed using block-cyclic distributions. The virtual processor approach is based on viewing a block-cyclic distribution as a block (or cyclic) distribution on a set of virtual processors, which are cyclically (or block-wise) mapped to the physical processors. The key idea of the method is to first develop closed forms in terms of simple regular sections for the index sets for arrays distributed using block or cyclic distributions. These closed forms are then used with the virtual processor approach to give an efficient solution for arrays with the block-cyclic distribution. HPF supports a two-level mapping of arrays to processors. Arrays are first aligned with a template at an offset and a stride and the template is then distributed among the processors using a regular data distribution. The introduction of a nonunit stride in the alignment creates “holes” in the distributed arrays which leads to memory wastage. In this paper, using simple mathematical properties of regular sections, we extend the virtual processor approach to address the memory allocation and index set enumeration problems for array statements involving arrays mapped using the two-level mapping. We develop a methodology for translating the closed forms for block and cyclically distributed arrays mapped using a one-level mapping to closed forms for arrays mapped using the two-level mapping. Using these closed forms, the virtual processor approach is extended to handle array statements involving arrays mapped using two-level mappings. Performance results on the Cray T3D are presented to demonstrate the efficacy of the extensions and identify various trade-offs associated with the proposed method.

Detecting number of folds by a simple mathematical property

Based on a simple mathematical property, a simple method is proposed for detecting the number of folds n contained in a given rotationaly symmetric shape. Skills to overcome the difficulty caused by sampling and rounding errors are also discussed. Experimental results are included to show the feasibility of this new method. As for shape orientation, it is just a by-product of the proposed method because no extra computation load is needed to get the orientation.

A compact representation of hierarchical relations using decimal notations

A decimal notation satisfies many simple mathematical properties. and it is a useful tool in the analysis of trees. A practical method is presented, that compresses the decimal codes while maintaining the fast determination of relations (e.g., ancestor, descendant, brother, etc.). A special node. called a kernel node,including many common subcodes of the other codes, is defined, and a compact data structure is presented using the kerne! nodes. Let n(m) be the number of the total (kernel) nodes. It is theoretically proved that encoding a decimal code is a constant time, that the worst-case time complexity of compressing the decimal codes is O(n+m 2), and that the size of the data structure is proportional to m. From the experimental results of some hierarchical semantic primitives for natural language processing, it is shown that the ratio m/n becomes an extremely small value, ranging from 0.047 to 0.13.

Storing a tree structure by using decimal notation

A decimal notation satisfies many simple mathematical properties, and it is a useful tool in the analysis of trees. A practical method is presented, that compresses the decimal codes while maintaining the fast determination of relations(e.g., ancestor, descendant, brother, etc.). A special node, called a kernel node , including many common subcodes of the other codes, is defined, and a compact data structure is presented using the kernel nodes. Let n ( m ) be the number of the total(kernel) nodes. It is theoretically proved that encoding a decimal code is a constant time, that the worst-case time complexity of compressing the decimal codes is O( n + m 2 ), and that the size of the data structure is proportional to m. From the experimental results of some hierarchical semantic primitives for natural language processing, it is shown that the ratio m/n becomes an extremely small value, ranging from 0.047 to 0.13.

A probability measure on parton and string states

The states obtained in hard high energy processes can, in a complementary way, be described in terms of partonic QCD language or in terms of a string-like confined force-field spanned by the partons. We define a surface characteristic of a confined force-field which breaks up into hadrons. The negative exponential of this surface multiplied by the appropriate phase-space defines a measure on the QCD states which is everywhere finite and infrared stable. It reproduces the results of perturbative QCD in the kinematical regions where such results are relevant. It has simple mathematical properties, in particular it is factorizable and in that way generates a stochastic process.

Quantitative analysis of dendritic branching. I. Simple formulae for the quantitative analysis of dendritic branching

Dendritic branching of neurons may be quantitatively studied using applications of graph theory. Dendritic ramifications may be considered as 'forests' of planted, stemmed, binary 'trees' with simple mathematical properties. Two fundamental numbers, that of dendritic tips and that of dendritic stems, usually suffice to quantify the dendritic branching of one neuron as, in most cases, they allow the computation of all other numbers of dendritic points and segments.

Valence Electron Studies with Gaussian-Based Model Potentials and Gaussian Basis Functions. I. General Discussion and Applications to the Lowest s and p States of Li and Na

A new model potential for one-valence-electron atoms is introduced. It consists [Eq. (6)] of a core Coulomb potential modified by the addition of a Gaussian screened Coulomb potential. This potential has the desired features outlined for a model potential, and it has particularly convenient and simple mathematical properties when used with Gaussian basis functions. Since the smooth valence orbitals are sought, Gaussian functions are a good basis set because their deficiencies in nuclear regions do not enter the problem. The potential is calibrated to experimental energies for the 2s and 2p states of Li and 3s and 3p states of Na, using extended basis sets. The model Hamiltonians so defined are used with a variety of more modest basis sets to determine pseudowavefunctions. Based on comparisons with ab initio orbitals, energy analyses, radial density calculations, and overlap and expectation value calculations, the conclusion is that good valence pseudowavefunctions can be obtained by this approach with relatively small basis sets. The approach looks promising for valence-only molecular studies.