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# If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds?

I have read this question:

The only way you can do this is to remove kinetic energy from the system. With normal matter this is done through electromagnetic interactions, which turn the kinetic energy of normal matter (protons, electrons etc.) into photons, which then escape from the system. Since these kinds of interactions do not occur for dark matter (by definition), then there is no way to get rid of kinetic energy and so the dark matter remains as a large "halo" around gravitationally clumping ordinary matter.

If dark matter only interacts with gravity, why doesn't it all clump together in a single point?

And this one:

A significant portion of the dark matter is known to be gravitationally bound to galaxies and relativistic velocities are far about the escape velocity and the stuff you propose would not remain bound.

Can dark matter be relativistic dust?

I assume that its speed with respect to the Sun will have a distribution with an rms of a few 100 km/s.

Can the Sun capture dark matter gravitationally?

In general particles created at the time of Big Bang are at non relativistic velocities at present .

What is meant by dark matter being non-relativistic and why is this?

The first one says that dark matter cannot lose kinetic energy through interactions. As far as I understand, most ordinary matter moves at non-relativistic speeds around us because these are able to lose kinetic energy through interactions and thus we see objects made of these particles in the observable universe to be mostly non-relativistic (except for example neutrinos).

Now if dark matter cannot lose kinetic energy, then shouldn't it have most of its kinetic energy from the Big Bang and travel at relativistic speeds? The second one says dark matter is definitely non-relativistic.

Question:

If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds?

• Since the big bang is not an explosion then surely dark matter would have always been "comoving". Aside from that, matter simply cannot travel at relativistic speeds WRT every inertial frame of reference, only light can do that. I might have misunderstood your question, but the more sophisticated answers below do not seem to make these points explicitly. Jul 10, 2021 at 11:20
• Why should we expect dark matter to have attributes such as kinetic energy just like any other ordinary matter?
– KP99
Jul 11, 2021 at 5:05

If dark matter consists of particles that thermally decoupled from the rest of the universe very early, then its momentum distribution with respect to the comoving reference frame was fixed at that point.

As the universe expands, then the characteristic absolute scale of this momentum distribution decreases as the scale length grows.

One way of thinking about this is that the de Broglie wavelength of a particle $$\lambda = h/p$$, gets stretched by the universal expansion just like the wavelength of light. Hence the rms $$p$$ (with respect to the comoving reference frame) decreases and the particles become non-relativistic.

Now that's an interesting question. And truth be told, as with so many things about the characteristics of dark matter, we do not know the answer yet.

First, it is not correct that "dark matter can't lose kinetic energy". We wouldn't have galaxies if that were the case. A population of dark matter can cool through gravitational three-body interactions. You know those from space probes making swing-bys on planets, typically to increase their kinetic energy, or if you travel to the inner planets, also to shed kinetic energy, at the expense or gain of the third body's kinetic energy. This is how dark matter halos cool and thus clump, which in turn stabilizes the formation of the baryonic disks that we have all come to love so dearly. This process is not as efficient in shedding kinetic energy as e.g. friction, so dark matter halos are very puffy and we should not expect dark matter discs or dark matter stars, but nonetheless, the process exists.

Indeed, as you say, from structure formation we know that dark matter is non-relativistic (aka "cold") and thus explains the structures we observe in the universe. Your question is why that is the case, and we don't know, because we don't know how dark matter came to be. Usually this is taken the other way around: we know dark matter is non-relativistic, and we use this to weed out all kinds of models and dark matter production mechanisms that are in disagreement with that.

The two most popular production mechanisms that produce non-relativistic dark matter are massive "thermal relic" particles and the "misalignment mechanism". Thermal relics are any particles that are at some point in the past in equilibrium with the rest of the universe's plasma. Protons and neutrons are a known example. WIMP dark matter is the most famous hypothetical example. By definition, thermal relics have a Maxwell-Boltzmann velocity distribution at some point. As the universe cools, that gets redshifted, thus leaving you with non-relativistic dark matter. Again, protons and neutrons do the same thing. This works well for dark matter that is more massive that a few proton masses. In turn, this rules out e.g. neutrinos as a dark matter candidate, as they are too light and would move relativistically.

The other example mechanism, misalignment, is invoked to explain why axion dark matter would not be relativistic. Axions are extremely light, but if your model prevents them from ever being in thermal equilibrium (because they interact too feebly), then their velocity distribution is set by their production mechanism. If you come up with some way to produce them close to their quantum mechanical zero point energy, then that is nonrelativistic and thus matches observation.

In summary, dark matter moving slowly is an observational fact that needs to be explained by any model attempting to describe the nature of dark matter, and indeed this constrains the possibilities for model builders.

• And of course, there might be significant quantities of dark matter that is moving relativistically - but those wouldn't form galaxies, and we would have even less of a chance of detecting them, with even fewer interactions and being far more sparse. Jul 9, 2021 at 11:15
• ...where "significant quantities" is a small mass density compared to the total dark matter density, yes
– rfl
Jul 9, 2021 at 12:02
• Thank you so much! Jul 9, 2021 at 15:53

It loses energy as the universe expands, in the same way that a gas cools as it expands. The same happened to the cosmic microwave background radiation (CMB), which is why its effective temperature is only a few Kelvin, even though the universe was about 3000 K when this light was first emitted during the Big Bang (at a moment called 'recombination').

To give a slightly more detailed answer, in an expanding universe, everything is pushed away from everything else, and the further away two objects are, the faster they move apart. This tends to reduce the relative velocities of particles. For example, if you surround yourself in a sphere of dust particles, they will gradually accelerate away from you. Thus any particles moving toward you will slow down (and it's the particles moving toward you that you will measure when you measure temperature).

• That last part sounds as if it tends to increase the relative velocities of particles (in a directions that each sees as "away from itself". Not decrease which suggests a gradual reduction in relative velocity and eventual near-unchanging distances. Can you clarify a bit? Thanks. Jul 9, 2021 at 9:35
• Minor nitpick just about your analogy: the cooling of compressed air cans is not actually the same mechanism as a gas cooling as it expands. The dominant thermal effect there is actually that in a boiling liquid, the part that remains liquid is cooled by the boiling process. A canister pressurized so that its contents are just gas actually cools a lot less than the familiar canisters do.
– Ian
Jul 9, 2021 at 12:55
• Two good points. I added a clarification and removed the gas canister analogy. The air at the output of the nozzle of a gas canister will cool as it expands into the surrounding atmosphere, but the cooling of the canister itself is presumably largely the boiling effect. A better example would be gas expanding inside an enclosure like a piston. However neither of these are perfect, because they involve cooling by doing work. Jul 10, 2021 at 17:17
• @user34722 "... in an expanding universe, everything is pushed away from everything else" This is incorrect! Matter which is gravitationally bound to other matter does not get pushed away from the matter it is bound to. Galaxies and galaxy groups have the bounding property, so they remain all together, eventually becoming a super-large black hole. Eventually these black holes will move away from each other faster than the speed of light as each such black hole becomes alone in its own observable universe.
– Buzz
Jul 13, 2021 at 1:45
• The expansion pushes things apart. Gravitational attraction can of course overcome this force, and it does if the mass density is sufficiently large, but that doesn't change the fact that such a force is acting on everything. The overall mass density of the universe is such that on intergalactic distances, the net acceleration is repulsive, as you note. This acceleration leads to cooling of the CMB and gravitationally unbound dark matter. Jul 14, 2021 at 3:29


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## Model-augmented Prioritized Experience Replay

### Youngmin Oh · Jinwoo Shin · Eunho Yang · Sung Ju Hwang

Keywords: [ reinforcement learning ] [ rl ]

[ Abstract ]
Thu 28 Apr 6:30 p.m. PDT — 8:30 p.m. PDT

Abstract: Experience replay is an essential component in off-policy model-free reinforcement learning (MfRL). Due to its effectiveness, various methods for calculating priority scores on experiences have been proposed for sampling. Since critic networks are crucial to policy learning, TD-error, directly correlated to $Q$-values, is one of the most frequently used features to compute the scores. However, critic networks often under- or overestimate $Q$-values, so it is often ineffective to learn to predict $Q$-values by sampled experiences based heavily on TD-error. Accordingly, it is valuable to find auxiliary features, which positively support TD-error in calculating the scores for efficient sampling. Motivated by this, we propose a novel experience replay method, which we call model-augmented prioritized experience replay (MaPER), that employs new learnable features driven from components in model-based RL (MbRL) to calculate the scores on experiences. The proposed MaPER brings the effect of curriculum learning for predicting $Q$-values better by the critic network with negligible memory and computational overhead compared to the vanilla PER. Indeed, our experimental results on various tasks demonstrate that MaPER can significantly improve the performance of the state-of-the-art off-policy MfRL and MbRL which includes off-policy MfRL algorithms in its policy optimization procedure.

Chat is not available.

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Electric Charge Video Lessons

Concept

# Problem: A hollow, conducting sphere with an outer radius of 0.254 m and an inner radius of 0.202 m has a uniform surface charge density of +6.63x10-6 C/m2. A charge of -0.830 μC is now introduced into the cavity inside the sphere. What is the new charge density on the outside of the sphere?

###### FREE Expert Solution

Original charge on the outer surface of the sphere:

$\overline{){{\mathbf{q}}}_{\mathbf{o}\mathbf{u}\mathbf{t}}{\mathbf{=}}{\mathbf{\sigma }}{\mathbf{A}}}$

But Area, A = 4πr02

Therefore, qout = (6.63 × 10-6)[(4π)(0.2542)] = 5.375 × 10-6C = 5.375 µC

84% (93 ratings)
###### Problem Details

A hollow, conducting sphere with an outer radius of 0.254 m and an inner radius of 0.202 m has a uniform surface charge density of +6.63x10-6 C/m2. A charge of -0.830 μC is now introduced into the cavity inside the sphere. What is the new charge density on the outside of the sphere?


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## Workshop 2: Regulatory Networks

### Organizers

Ralf Bundschuh
Departments of Physics, Chemistry&Biochemistry, Division of Hematology, The Ohio State University
Jeff Hasty
Departments of Molecular Biology and Bioengineering, University of California, San Diego
Fernand Hayot
Department of Physics, The Ohio State University

It has been over 40 years since Monod and Jacob boldly predicted that such fundamental cellular processes as differentiation and protein regulation are accomplished through signaling pathways resident at the level of the gene. This prediction laid the foundation for the ensuing progress in describing the essential regulatory mechanisms in many specific genetic systems. With the development of the field of nonlinear dynamics and the concurrent advent of available computing, mathematical models describing gene regulation began to appear regularly in the 1970s. Implicit in these studies was the realization that the "wiring" of naturally-occurring gene regulatory networks would be too complex for qualitative descriptions devoid of mathematics. Though this realization proved to be ahead of its time, mainly due to the lack of experimentally deduced regulatory pathways in the "pre-genomic" era, recent experimental advances in both sequencing and genetic engineering have made the analysis and design of gene networks amenable to quantitative analysis. These advances have reignited interest in gene regulatory models that can be used to explain and predict behavior that emerges from gene regulatory networks.

In this workshop, we will focus on recent advances in utilizing mathematical models to describe gene regulatory networks. The workshop will begin by introducing mathematical modeling techniques, including boolean representations, classical kinetic theory, stochastic simulation methods and constraints-based models. This will be followed by descriptions of specific gene regulatory networks, such as those related to the lambda and T7 life cycles, drosophilia segmentation, and xenopus oocytes differentiation. The remainder of the workshop will be devoted to the construction and analysis of synthetic gene regulatory networks, including switches, oscillators, autoregulation, and noise analysis. The workshop will highlight the utility, which an accurate mathematical description of synthetic networks provides, in describing complex naturally-occurring networks.

### Accepted Speakers

Timothy Elston
Department of Mathematics, University of North Carolina, Chapel Hill
Daniel Forger
Department of Biology, New York University
Glaxo Smith Kline
Timothy Gardner
Biomedical Engineering, Boston University
Dan Gillespie
Biochemical Computing Consultant, Paisii Hilendarski'' University of Plovdiv
Terry Hwa
Department of Physics, University of California, San Diego
Jean-Christophe Leloup
Service de Chimie Physique, Faculte des Sciences, Universite Libre de Bruxelles
Ron Milo
Complex Systems/Molecular Cell Biology, The Weizmann Institute of Science
John Reinitz
Applied Mathematics and Statistics, Stony Brook University
Michael Savageau
Department of Biomedical Engineering, University of California, Davis
Luis Serrano
Biostructure and Biocomputing, European Molecular Biology Laboratory
Michael Simpson
Materials Science and Engineering, University of Tennessee
Ron Weiss
Electrical Engineering & Molecular Biology, Princeton University
Lingchong You
Chemistry and Chemical Engineering, California Institute of Technology
Monday, November 8, 2004
Time Session
09:30 AM
10:30 AM
Dan Gillespie - Stochastic Chemical Kinetics

The time evolution of a well-stirred chemically reacting system is traditionally modeled by a set of coupled ordinary differential equations called the reaction rate equation (RRE). The resulting picture of continuous deterministic evolution is, however, valid only for infinitely large systems. That condition is usually well approximated in macroscopic chemical systems. But in biological systems formed by single living cells, the small population numbers of some reactant species can result in dynamical behavior that is noticeably discrete rather than continuous, and stochastic rather than deterministic. In that case, a more accurate mathematical modeling is obtained by using the machinery of Markov process theory, specifically, the chemical master equation (CME) and the stochastic simulation algorithm (SSA). This talk will review the theoretical foundations of stochastic chemical kinetics, and then discuss some recent efforts to (1) approximate the SSA by a faster simulation procedure, and (2) establish the formal connection between the CME/SSA description and the RRE description.

11:00 AM
12:00 PM
Daniel Forger - Towards a Biologically Rigorous Model of the Mammalian Circadian Clock

Perhaps the best understood biochemical networks are those of the circadian (near 24-hour) clock within cells. A mathematical model of the mammalian circadian clock is developed which incorporates a wide range of experimental data, and is by far the most detailed mathematical model of a circadian clock yet derived. Despite its complexity, there is enough experimental data to estimate the parameters of the model as an inverse problem. The model is accurate in its predictions with respect to mutations and can be used to understand key questions about clock structure and phase resetting.

We then investigate the behavior of an earlier circadian clock model in the presence of molecular noise. Despite a previous report, we find very accurate rhythms from this model, and study the physiological causes of this robustness. Unfortunately, this model is not detailed enough to specify individual molecular interactions, which has lead to conflicting results in the literature.

Based on an experimental estimate of the number of molecules of key proteins within the mammalian circadian clock, we can directly, without ambiguity, simulate our model of the mammalian circadian clock with stochastic molecular interactions. Amazingly, interactions with promoters on the time scale of seconds are required for accurate 24-hour timekeeping. The stochasticity of our model follows the central limit theorem. Finally we find that non-redundant gene-duplication can increase immunity to molecular noise by allowing for more interactions with promoters. This work was conducted with Charles Peskin.

Work done in collaboration with Justin Blau.

Tuesday, November 9, 2004
Time Session
09:00 AM
10:00 AM
Michael Savageau - Discovery of System Design Principles and Construction of Gene Circuits

The ability to comprehensively and quantitatively monitor dynamic changes in gene expression, together with new genome-scale informatic methods, is enabling high-throughput characterization of genetic regulatory networks. In addition, methods of genetic engineering now allow synthetic regulatory circuits to be readily built. Attention is currently being turned towards manipulating genetic regulatory circuits for therapeutic and technological applications, which increases the need to understand the functional consequences of genetic manipulations and to discover principles that can guide the design process. This issue will be addressed by comparing and contrasting what has been learned about design principles for gene circuits in their complex natural setting and how these have been put to use in designing, constructing and analyzing simple synthetic gene circuits.

02:00 PM
03:00 PM
Luis Serrano - Engineering Gene Networks to Emulate Drosophila Embryonic Pattern Formation & In Silico Biological Validation of Protein Interaction Networks

To understand in a quantitative manner how biological systems operate we need to achieve several things. First we need accurate and meaningful data of biologically relevant interactions. Second, we need to have experimental methodologies that allow us to dissect the behavior of the network in a context free environment. Third, we need computer algorithms to explore and simulate many different parameters, proposing new experiments to do. Finally we need to be able to modify and design the properties of the target network based on the previous analysis. In my presentation I will deal with the first two points: How to validate biologically meaningful interactions and how to analyze the properties of a network in an "in theory" context free environment.

Biological Validation of Protein interaction Networks:

Protein interaction networks are an important part of the post-genomic effort to integrate a parts-list view of the cell into system-level understanding. Using a set of 11 yeast genomes we show that combining comparative genomics and secondary structure information can greatly increase consensus based prediction of SH3 targets. Careful benchmarking of our method against positive and negative standards gives 83% accuracy with 26% coverage. We demonstrate the concept of an optimal divergence time, for effective comparative genomics studies, by proving that genomes of species that diverged very recently from S. cerevisiae (S. mikatae, S. bayanus and S. paradoxus), or a long time ago (S. pombe) contain less information for accurate prediction of SH3 targets. Our findings highlight several novel S. cerevisiae SH3 protein-interactions and the importance of selection of optimal divergence times in comparative genomics studies.

Engineering Gene Networks to Emulate Drosophila Embryonic Pattern Formation:

Pattern formation is essential in the development of higher eukaryotes. For example, in the Drosophila embryo, maternal morphogen gradients establish gap gene expression domain patterning along the anterior-posterior axis, through linkage with an elaborate gene network. To understand better the evolution and behaviour of such systems, it is important to establish the minimal determinants required for patterning. We have therefore engineered artificial transcription/translation networks, that generate simple patterns, crudely analogous to the Drosophila gap gene system. The Drosophila syncytium was modelled using DNA-coated paramagnetic beads, fixed by magnets in an artificial chamber, forming a gene expression network. Transient expression domain patterns were generated using various levels of network connectivity. Generally, adding more transcription repression interactions increased the 'sharpness' of the pattern while reducing overall expression levels. An accompanying computer model for our system allowed us to search for parameter sets compatible with patterning. While it is clear that the Drosophila embryo is far more complex than our simplified model, several features of interest emerge. For example, the model suggests that simple diffusion may be too rapid for Drosophila-scale patterning, implying that sublocalization or 'trapping' is required. Secondly, we find that for pattern formation to occur under the conditions of our in vitro reaction-diffusion system, the activator molecules must propagate faster than the inhibitors. Thirdly, adding controlled protease degradation to the system stabilizes pattern formation over time.

Work done in collaboration with Mark Isalan, Caroline Lemerle, Pedro Beltrao, and Luis Serrano.

Wednesday, November 10, 2004
Time Session
09:00 AM
10:00 AM
Ron Milo - Searching for Building Blocks and Design Principles in the Genetic Regulatory Network of E. coli

Little is known about the design principles of transcriptional regulation networks that control gene expression in cells. Recent advances in data collection and analysis, however, are generating unprecedented amounts of information about gene regulation networks. To understand these complex wiring diagrams, we sought to break down such networks into basic building blocks. We generalized the notion of motifs, widely used for sequence analysis, to the level of networks. We define 'network motifs' as patterns of interconnections that recur in many different parts of a network at frequencies much higher than those found in randomized networks. We found such motifs in networks from biochemistry, neurobiology, sociology and engineering. One of the best-characterized regulation networks is that of direct transcriptional interactions in Escherichia coli. We find that much of the network is composed of repeated appearances of several highly significant motifs. Each network motif has a specific function in determining gene expression, such as generating temporal expression programs and governing the responses to fluctuating external signals. The talk will present the theoretical and experimental approaches used to detect, measure and analyze functional circuits in this genetic regulatory network.

10:30 AM
11:30 AM
John Reinitz - Regulatory Networks in the Drosophila Blastoderm

The fruit fly Drosophila is a premier system for investigating how animal embryos self-organize their body plan. The blueprint for the fly's body is created by networks of genes operating in an ellipsoidal shell of cell nuclei called the blastoderm. We create predictive models of this process using systems of ordinary or partial differential equations fit to gene expression data by simulated annealing and/or Lagrangian methods. In this talk I will discuss the entire pattern formation project, from colorful fluorescently stained embryos to image processing, new optimization algorithms, and finally to new biological results. Also, although the notion of 'cis-regulatory modules' central to modern molecular biology, I will show that our understanding of the function and organization of these entities is fundamentally insufficient for understanding developmental biology. I will propose a solution to this problem through a new theoretical approach in concert with quantitative data from promoter-reporter constructs.

02:00 PM
03:00 PM
Lingchong You - Homeostasis, Oscillations, and Ecological Interactions in Re-programmed Bacterial Populations

De novo engineering of gene circuits inside cells has emerged as a powerful approach to decoding 'design principles' of biological systems. Such circuits are also of great interest for their potential applications in computation, engineering, and medicine. However, it has been challenging to realize predictable and robust circuit performance due to some major hurdles, such as noise in gene expression and cell-to-cell variation in phenotype. We address these issues by using cell-cell communication to coordinate cellular behavior across the population. To establish cell-cell communication, we take advantage of 'quorum sensing' systems that many bacteria use to detect and respond to changes in the cell density. As a prototype example, we have built and characterized a 'population control' circuit in bacterium E. coli. This circuit autonomously regulates the cell density using a negative feedback loop acting on the entire population. With the circuit, the cell density is broadcasted and detected by a quorum sensing system, which modulates the expression of a killer gene. The killer gene in turn regulates the cell density by controlling the death rate. Upon activation, the circuit will lead to a stable steady state or sustained oscillations in terms of cell density and gene expression. This circuit lays down the conceptual foundation to program interactions among multiple cell populations - essentially creating 'synthetic ecosystems' from well-characterized genetic modules.

Thursday, November 11, 2004
Time Session
09:00 AM
10:00 AM
Michael Simpson - Probing Gene Circuit Structure and Function Using Noise

The study of gene circuits is similar to many other areas of biology in as much as the principal aim is to understand the relationships between structure and function. This is true not only of chemical (e.g. the sequences of bases) or physical (e.g., 3-D structure of proteins) structure, but also the informational structure of genetic circuits and networks. New insights are emerging from the top-down analysis of the biomolecular networks from which complex cellular function emerges. Network motifs have been found that occur significantly more often than would be expected in random networks, providing a rational basis to search for the structure-function relationships in these systems. However, topology alone does not define function, which is sensitive to the specifics of kinetic parameters and the structure and function of individual gene circuits which comprise the higher order networks. Unfortunately, these parameters are usually very difficult to measure or infer. The problem is exacerbated by the fact that we wish to measure these parameters within the context of the fully functioning system of the cell, especially as intracellular molecular crowding generates kinetics that are vastly different than those found from in vitro measurements.

Concurrent with this emphasis on the informational architecture of intracellular molecular networks, a new appreciation of the role of stochastic processes in decision making in biological systems has emerged. Efforts in this direction have developed analysis and simulation techniques; described the noise consequences of gene circuit structure; and have explored how stochastic processes may play a pivotal role in gene circuit functionality. However, the use of inherent noise as a gene circuit probe has been largely ignored. Stochastic fluctuations are a broad-spectrum input excitation, and the frequency-domain structure of the resulting output spectra reveal details about the underlying gene circuit structure and parameter values. In this talk I will describe the frequency-domain processing of stochastic fluctuations by genetic circuits, the measurement of the output noise spectral densities, and the use of these spectra to infer gene circuit structure and reaction rate constants.

10:30 AM
11:30 AM
Terry Hwa - Molecular Strategies for Control and Gain in Transcriptional Regulation

Molecular Strategies for Control and Gain in Transcriptional Regulation

02:00 PM
03:00 PM
Ron Weiss - Programming Collaborative Behavior and Pattern Formation in Bacterial Communities

Cell-cell communication is a pervasive activity common to both single cell and multicellular organisms, and is used in coordinating cell behavior for a variety of tasks ranging from quorum sensing in bacteria to embryogenesis in mammalian cells. Engineering synthetic multicellular communication systems to exhibit desired functions will improve our quantitative understanding of naturally occurring cell-cell communication, and will also have biotechnology applications in areas such as biosensing, biomaterial fabrication, and tissue engineering. Here we will present theoretical and experimental results from three synthetic multicellular communication systems implemented in bacteria that have been programmed to exhibit unique coordinated cell behavior. The first system is the pulse generator where sender cells communicate to nearby receiver cells, which then respond with a transient burst of gene expression whose amplitude and duration depends on the distance from the senders. In the second system, receiver cells have been engineered to respond to cell-cell communication signals only within prespecified ranges. We will demonstrate how this system can be used to generate a variety of interesting spatial patterns. In the third system, cells have been engineered to "play" Conway's Game of Life, where cells live or die based on the density of their neighbors. This system exhibits complex global emergent behavior that arises from the interaction of cells based on simple local rules. In this talk, we will correlate experimental results from observing the behavior of these systems with our quantitative spatiotemporal models.

Friday, November 12, 2004
Time Session
09:00 AM
10:00 AM
- Balancing Molecular Fluctuations and Cellular Stability at the Level of a Gene, Cell, and Cell Community

A living cell is a noisy biochemical reactor in which low reactant concentrations lead to significant statistical fluctuations, or noise, in molecule numbers and reaction rates. This noise is often perceived as being undesirable and unpredictable. However, living systems are inherently noisy and are optimized to function in the presence of stochastic fluctuations. Some organisms can exploit noise to introduce diversity into a population. In contrast, stability against fluctuations is essential in case of a gene regulatory cascade controlling cell differentiation in a developing embryo. Stability in biological systems is often obtained by feedback regulation in the underlying regulatory network. In this talk I will address how biological systems can tune the balance between stability and noise at the level of a gene, cell, and cell community.

10:30 AM
11:30 AM
Timothy Gardner - Inferring Gene Regulatory Network Structure and Function in Microbes via Expression Profiling

We have developed a systematic methods to infer regulatory structures and properties of gene networks using microarray expression data. The methods learn first-order models of regulatory influences using RNA expression profiles for a diverse set of treatments, including exogenous compounds, environmental stresses, genetic mutations, and RNA inhibition. We have successfully applied the methods in E. coli and yeast to infer networks of 10s to 1000s of genes. The resulting network models can be used to identify transcription factor interactions, critical regulatory hubs, and to predict the mode of action of compounds and metabolites. In yeast, for example, the method was applied to a microarray data set measuring 6000 RNAs in 300 treatments. The resulting network model was used correctly identify the gene target of terbinafine, itraconazole and several other drugs. This method may be similarly applied to identify the feedback interactions between metabolic compounds and regulatory genes. These regulatory models may improve the optimization of metabolic pathways for biotechnology applications and may create new opportunities for target identification and lead optimization in drug discovery.

02:00 PM
03:00 PM
Chetan Gadgil - Stochastic Reaction Engineering Analysis of Regulatory Networks

Regulatory processes, especially those involving reactions between species that exist at very low concentrations, are inherently stochastic in nature. It is presently not clear how the structure of the reaction network model affects the results from the (numerical) calculation of the distribution of species. As a step towards analyzing the time-dependent behavior of the concentration distribution of each species in a network, we derive analytical expressions for the mean and variance of the concentration of all species in an arbitrary network where all the interactions are zero-order production reactions or first order conversion, catalytic or degradation reactions. We find the surprising, and apparently unknown, result that the time evolution of the second moments is governed by linear combinations of the eigenvalues of the matrix for the evolution of the means.

We use this theoretical framework to analyze the effect of network topology on the evolution of the mean and variance of various species in the network. In particular, we analyze the slowest time-scales for relaxation of the mean and variance for networks that are linear, and those that have positive feedback or feedforward loops. For a stochastic analysis of diffusion-reaction processes, we derive a framework that facilitates the separation of the effects of domain geometry, diffusion, and reaction rates on the distribution of species. We discuss the use of various measures to describe the 'noise' in stochastic systems, and show that the choice of the noise measure can lead to completely different conclusions for the same system.

Name Email Affiliation
Agosto, Francisco agosto.6@osu.edu Biophysics Program, The Ohio State University
Alexandridis, Roxana roxana@stat.ohio-state.edu Department of Statistics, The Ohio State University
Bazaliy, Borys Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine
Best, Janet jbest@mbi.osu.edu Mathematics, The Ohio State University
Borisyuk, Alla borisyuk@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Brewer, Daniel d.brewer@ucl.ac.uk Institute of Child Health & CoMPLEX, University College London
Buechler, Steven buechler.1@nd.edu Department of Mathematics, University of Notre Dame
Bundschuh, Ralf bundschuh@mbi.osu.edu Departments of Physics, Chemistry&Biochemistry, Division of Hematology, The Ohio State University
Carl, Joe carl.30@osu.edu IBGP, The Ohio State University
Cracium, Gheorghe craciun@math.wisc.edu Dept. of Mathematics, University of Wisconsin-Madison
DiVentura, Barbara diventur@embl.de Structural Biology & Biocomputing, EMBL
Doss, Hani Department of Statistics, The Ohio State University
Dougherty, Daniel dpdoughe@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Edwards, David Department of Mathematical Sciences, University of Delaware
Elston, Timothy telston@amath.unc.edu Department of Mathematics, University of North Carolina, Chapel Hill
Erdal, Selnur erdal.3@osu.edu Department of Biophysics, The Ohio State University
Fang, Chih gt1091b@yahoo.com Molecular Virology, Immunology & Med Genetics, The Ohio State University
Foley, Catherine foley@math.mcgill.ca Mathematics and Statistics, McGill University, Macdonald Campus
Forger, Daniel forger@umich.edu Department of Biology, New York University
Frazier, John john.frazier@wpafb.af.mil Human Effectiveness Directorate, Wright-Patterson Air Force Base (WPAFB)
Gao, Jianbo gao@ece.ufl.edu Electrical & Computer Engineering, University of Florida
Gardner, Timothy gardner@amyris.com Biomedical Engineering, Boston University
Gillespie, Dan GillespieDT@mailaps.org Biochemical Computing Consultant, Paisii Hilendarski'' University of Plovdiv
Goel, Pranay goelpra@helix.nih.gov NIDDK, Indian Institute of Science Education and Research
Gokhale, Zoee gokhale.5@osu.edu Department of Biophysics, The Ohio State University
Grotewold, Erich grotewold.1@osu.edu Department of Plant Biology, The Ohio State University
Gu, Weisong weisong@osc.edu Ohio Supercomputer Center, The Ohio State University
Guo, Yixin yixin@math.drexel.edu Department of Psychology, The Ohio State University
Hassanali, Ali hassanali@osu.edu Department of Biophysics, The Ohio State University
Hasty, Jeff hasty@ucsd.edu Departments of Molecular Biology and Bioengineering, University of California, San Diego
Hayot, Fernand hayot@mps.ohio-state.edu Department of Physics, The Ohio State University
Hsu, Jason hsu.1@osu.edu Department of Statistics, The Ohio State University
Hu, Bei Department of Mathematics, University of Notre Dame
Huang, Yifan huang.338@osu.edu Department of Statistics, The Ohio State University
Hwa, Terry hwa@matisse.ucsd.edu Department of Physics, University of California, San Diego
Isaacson, Samuel isaacson@math.utah.edu Courant Institute of Mathematical Sciences, New York University
Jayaprakash, Ciriyam jay@mps.ohio-state.edu Department of Physics, The Ohio State University
Joseph-Souriraj, Irene joseph-souriraj.1@osu.edu Biophysics Program, The Ohio State University
Joyce, Andrew ajoyce@ucsd.edu Palsson Lab, PFBH 419, University of California, San Diego
Jung, Peter Quantitative Biology Institute, Ohio University
Kannan, Dan kannan@uga.edu Department of Mathematics, University of Georgia
Kuznetsov, Alexey alexey@bu.edu Department of Mathematics, Boston University
Lauria, Mario lauria@cis.ohio-state.edu Computer Science & Engineering, The Ohio State University
Lee, Yoonkyung Department of Statistics, The Ohio State University
Lee, Chang Heong chlee@math.umn.edu Department of Mathematics, University of Minnesota
Leloup, Jean-Christophe jleloup@ulb.ac.be Service de Chimie Physique, Faculte des Sciences, Universite Libre de Bruxelles
Lim, Sookkyung limsk@math.uc.edu Department of Mathematical Sciences, University of Cincinnati
Lin, Shili lin.328@osu.edu Department of Statistics, The Ohio State University
Lipniacki, Tomasz tlipnia@ippt.gov.pl, Institute of Fundamental Technological Research
Liu, Jun jliu@stat.harvard.edu Department of Statistics, Harvard University
Liu, Liang LiuLiang@stat.ohio-state.edu Department of Statistics, The Ohio State University
Melfi, Vincent melfi@mbi.osu.edu Mathematics, Michigan State University
Milo, Ron ron.milo@weizmann.ac.il Complex Systems/Molecular Cell Biology, The Weizmann Institute of Science
Nagaraja, Haikady Department of Statistics, The Ohio State University
Palaniswamy, Saranyan palaniswamy-1@medctr.osu.edu Human Cancer Genetics Program, The Ohio State University
Pohar, Twyla pohar-2@medctr.osu.edu Division of Human Cancer Genetics, The Ohio State University
Pol, Diego dpol@mbi.osu.edu Independent Researcher, Museo Paleontologico E. Feruglio
Rassoul-Agha, Firas firas@math.ohio-state.edu Department of Mathematics, University of Utah
Ray, William ray.29@osu.edu Pediatrics, The Ohio State University
Reinitz, John reinitz@odd.bio.sunysb.edu Applied Mathematics and Statistics, Stony Brook University
Rejniak, Katarzyna rejniak@mbi.osu.edu Mathematical Biosciences Institute, The Ohio State University
Santner, Thomas tjs@stat.ohio-state.edu Department of Statistics, The Ohio State University
Savageau, Michael masavageau@ucdavis.edu Department of Biomedical Engineering, University of California, Davis
Serrano, Luis serrano@embl-heidelberg.de Biostructure and Biocomputing, European Molecular Biology Laboratory
Simpson, Michael simpsonML1@ornl.gov Materials Science and Engineering, University of Tennessee
Singer, Greg gacsinger@gmail.com Human Cancer Genetics, The Ohio State University
Smolen, Paul paul.d.smolen@uth.tmc.edu Neurobiology and Anatomy, University of Texas Medical School at Houston
Stubna, Michael stubna@mbi.osu.edu Engineering Team Leader, Pulsar Informatics
Sun, Hao sun.143@osu.edu Human Cancer Genetics, The Ohio State University
Sun, Junfeng sun@stat.ohio-state.edu Department of Statistics, The Ohio State University
Tang, Fusheng fusheng-tang@uiowa.edu Department of Mathematics, University of Iowa
Terman, David terman@math.ohio-state.edu Mathemathics Department, The Ohio State University
Thomas, Rene rthomas@dbm.ulb.ac.be Biologie Moleculaire, Universite Libre de Bruxelles
Tian, Jianjun Paul tianjj@mbi.osu.edu Mathematics, College of William and Mary
Varbanov, Alex varbanov.ar@pg.com Department of Statistics, Proctor & Gamble
Verducci, Joseph verducci.1@osu.edu Department of Statistics, The Ohio State University
Volfson, Dmitri dvolfson@ucsd.edu Institute for Nonlinear Science, University of California, San Diego
Wang, Tao wangtao@stat.ohio-state.edu Department of Statistics, The Ohio State University
Wang, Zailong zlwang@mbi.osu.edu Integrated Information Sciences, Novartis
Wechselberger, Martin wm@mbi.osu.edu Mathematical Biosciences Insitute, The Ohio State University
Weiss, Ron rweiss@princeton.edu Electrical Engineering & Molecular Biology, Princeton University
Wright, Geraldine wright.572@osu.edu School of Biology, Newcastle University
Ye, Kenny kyeblue@yahoo.com Epidemiology & Population Health, Albert Einstein College of Medicine
Yin, Lijie yin.39@osu.edu Department of Pathology, The Ohio State University
You, Lingchong you@duke.edu Chemistry and Chemical Engineering, California Institute of Technology
Zhang, Xuan zhangx@cse.ohio-state.edu Computer Science & Engineering, The Ohio State University
Zhou, Jin jzhou@mbi.osu.edu Department of Mathematics, Northern Michigan University
Zhou, Penghui zhou.154@osu.edu Department of Molecular Genetics, The Ohio State University
Zinner, Bertram zinnebe@auburn.edu Mathematics and Statistics, Auburn University
Towards a Biologically Rigorous Model of the Mammalian Circadian Clock

Perhaps the best understood biochemical networks are those of the circadian (near 24-hour) clock within cells. A mathematical model of the mammalian circadian clock is developed which incorporates a wide range of experimental data, and is by far the most detailed mathematical model of a circadian clock yet derived. Despite its complexity, there is enough experimental data to estimate the parameters of the model as an inverse problem. The model is accurate in its predictions with respect to mutations and can be used to understand key questions about clock structure and phase resetting.

We then investigate the behavior of an earlier circadian clock model in the presence of molecular noise. Despite a previous report, we find very accurate rhythms from this model, and study the physiological causes of this robustness. Unfortunately, this model is not detailed enough to specify individual molecular interactions, which has lead to conflicting results in the literature.

Based on an experimental estimate of the number of molecules of key proteins within the mammalian circadian clock, we can directly, without ambiguity, simulate our model of the mammalian circadian clock with stochastic molecular interactions. Amazingly, interactions with promoters on the time scale of seconds are required for accurate 24-hour timekeeping. The stochasticity of our model follows the central limit theorem. Finally we find that non-redundant gene-duplication can increase immunity to molecular noise by allowing for more interactions with promoters. This work was conducted with Charles Peskin.

Work done in collaboration with Justin Blau.

Stochastic Reaction Engineering Analysis of Regulatory Networks

Regulatory processes, especially those involving reactions between species that exist at very low concentrations, are inherently stochastic in nature. It is presently not clear how the structure of the reaction network model affects the results from the (numerical) calculation of the distribution of species. As a step towards analyzing the time-dependent behavior of the concentration distribution of each species in a network, we derive analytical expressions for the mean and variance of the concentration of all species in an arbitrary network where all the interactions are zero-order production reactions or first order conversion, catalytic or degradation reactions. We find the surprising, and apparently unknown, result that the time evolution of the second moments is governed by linear combinations of the eigenvalues of the matrix for the evolution of the means.

We use this theoretical framework to analyze the effect of network topology on the evolution of the mean and variance of various species in the network. In particular, we analyze the slowest time-scales for relaxation of the mean and variance for networks that are linear, and those that have positive feedback or feedforward loops. For a stochastic analysis of diffusion-reaction processes, we derive a framework that facilitates the separation of the effects of domain geometry, diffusion, and reaction rates on the distribution of species. We discuss the use of various measures to describe the 'noise' in stochastic systems, and show that the choice of the noise measure can lead to completely different conclusions for the same system.

Inferring Gene Regulatory Network Structure and Function in Microbes via Expression Profiling

We have developed a systematic methods to infer regulatory structures and properties of gene networks using microarray expression data. The methods learn first-order models of regulatory influences using RNA expression profiles for a diverse set of treatments, including exogenous compounds, environmental stresses, genetic mutations, and RNA inhibition. We have successfully applied the methods in E. coli and yeast to infer networks of 10s to 1000s of genes. The resulting network models can be used to identify transcription factor interactions, critical regulatory hubs, and to predict the mode of action of compounds and metabolites. In yeast, for example, the method was applied to a microarray data set measuring 6000 RNAs in 300 treatments. The resulting network model was used correctly identify the gene target of terbinafine, itraconazole and several other drugs. This method may be similarly applied to identify the feedback interactions between metabolic compounds and regulatory genes. These regulatory models may improve the optimization of metabolic pathways for biotechnology applications and may create new opportunities for target identification and lead optimization in drug discovery.

Stochastic Chemical Kinetics

The time evolution of a well-stirred chemically reacting system is traditionally modeled by a set of coupled ordinary differential equations called the reaction rate equation (RRE). The resulting picture of continuous deterministic evolution is, however, valid only for infinitely large systems. That condition is usually well approximated in macroscopic chemical systems. But in biological systems formed by single living cells, the small population numbers of some reactant species can result in dynamical behavior that is noticeably discrete rather than continuous, and stochastic rather than deterministic. In that case, a more accurate mathematical modeling is obtained by using the machinery of Markov process theory, specifically, the chemical master equation (CME) and the stochastic simulation algorithm (SSA). This talk will review the theoretical foundations of stochastic chemical kinetics, and then discuss some recent efforts to (1) approximate the SSA by a faster simulation procedure, and (2) establish the formal connection between the CME/SSA description and the RRE description.

Molecular Strategies for Control and Gain in Transcriptional Regulation

Molecular Strategies for Control and Gain in Transcriptional Regulation

Searching for Building Blocks and Design Principles in the Genetic Regulatory Network of E. coli

Little is known about the design principles of transcriptional regulation networks that control gene expression in cells. Recent advances in data collection and analysis, however, are generating unprecedented amounts of information about gene regulation networks. To understand these complex wiring diagrams, we sought to break down such networks into basic building blocks. We generalized the notion of motifs, widely used for sequence analysis, to the level of networks. We define 'network motifs' as patterns of interconnections that recur in many different parts of a network at frequencies much higher than those found in randomized networks. We found such motifs in networks from biochemistry, neurobiology, sociology and engineering. One of the best-characterized regulation networks is that of direct transcriptional interactions in Escherichia coli. We find that much of the network is composed of repeated appearances of several highly significant motifs. Each network motif has a specific function in determining gene expression, such as generating temporal expression programs and governing the responses to fluctuating external signals. The talk will present the theoretical and experimental approaches used to detect, measure and analyze functional circuits in this genetic regulatory network.

Regulatory Networks in the Drosophila Blastoderm

The fruit fly Drosophila is a premier system for investigating how animal embryos self-organize their body plan. The blueprint for the fly's body is created by networks of genes operating in an ellipsoidal shell of cell nuclei called the blastoderm. We create predictive models of this process using systems of ordinary or partial differential equations fit to gene expression data by simulated annealing and/or Lagrangian methods. In this talk I will discuss the entire pattern formation project, from colorful fluorescently stained embryos to image processing, new optimization algorithms, and finally to new biological results. Also, although the notion of 'cis-regulatory modules' central to modern molecular biology, I will show that our understanding of the function and organization of these entities is fundamentally insufficient for understanding developmental biology. I will propose a solution to this problem through a new theoretical approach in concert with quantitative data from promoter-reporter constructs.

Discovery of System Design Principles and Construction of Gene Circuits

The ability to comprehensively and quantitatively monitor dynamic changes in gene expression, together with new genome-scale informatic methods, is enabling high-throughput characterization of genetic regulatory networks. In addition, methods of genetic engineering now allow synthetic regulatory circuits to be readily built. Attention is currently being turned towards manipulating genetic regulatory circuits for therapeutic and technological applications, which increases the need to understand the functional consequences of genetic manipulations and to discover principles that can guide the design process. This issue will be addressed by comparing and contrasting what has been learned about design principles for gene circuits in their complex natural setting and how these have been put to use in designing, constructing and analyzing simple synthetic gene circuits.

Engineering Gene Networks to Emulate Drosophila Embryonic Pattern Formation & In Silico Biological Validation of Protein Interaction Networks

To understand in a quantitative manner how biological systems operate we need to achieve several things. First we need accurate and meaningful data of biologically relevant interactions. Second, we need to have experimental methodologies that allow us to dissect the behavior of the network in a context free environment. Third, we need computer algorithms to explore and simulate many different parameters, proposing new experiments to do. Finally we need to be able to modify and design the properties of the target network based on the previous analysis. In my presentation I will deal with the first two points: How to validate biologically meaningful interactions and how to analyze the properties of a network in an "in theory" context free environment.

Biological Validation of Protein interaction Networks:

Protein interaction networks are an important part of the post-genomic effort to integrate a parts-list view of the cell into system-level understanding. Using a set of 11 yeast genomes we show that combining comparative genomics and secondary structure information can greatly increase consensus based prediction of SH3 targets. Careful benchmarking of our method against positive and negative standards gives 83% accuracy with 26% coverage. We demonstrate the concept of an optimal divergence time, for effective comparative genomics studies, by proving that genomes of species that diverged very recently from S. cerevisiae (S. mikatae, S. bayanus and S. paradoxus), or a long time ago (S. pombe) contain less information for accurate prediction of SH3 targets. Our findings highlight several novel S. cerevisiae SH3 protein-interactions and the importance of selection of optimal divergence times in comparative genomics studies.

Engineering Gene Networks to Emulate Drosophila Embryonic Pattern Formation:

Pattern formation is essential in the development of higher eukaryotes. For example, in the Drosophila embryo, maternal morphogen gradients establish gap gene expression domain patterning along the anterior-posterior axis, through linkage with an elaborate gene network. To understand better the evolution and behaviour of such systems, it is important to establish the minimal determinants required for patterning. We have therefore engineered artificial transcription/translation networks, that generate simple patterns, crudely analogous to the Drosophila gap gene system. The Drosophila syncytium was modelled using DNA-coated paramagnetic beads, fixed by magnets in an artificial chamber, forming a gene expression network. Transient expression domain patterns were generated using various levels of network connectivity. Generally, adding more transcription repression interactions increased the 'sharpness' of the pattern while reducing overall expression levels. An accompanying computer model for our system allowed us to search for parameter sets compatible with patterning. While it is clear that the Drosophila embryo is far more complex than our simplified model, several features of interest emerge. For example, the model suggests that simple diffusion may be too rapid for Drosophila-scale patterning, implying that sublocalization or 'trapping' is required. Secondly, we find that for pattern formation to occur under the conditions of our in vitro reaction-diffusion system, the activator molecules must propagate faster than the inhibitors. Thirdly, adding controlled protease degradation to the system stabilizes pattern formation over time.

Work done in collaboration with Mark Isalan, Caroline Lemerle, Pedro Beltrao, and Luis Serrano.

Probing Gene Circuit Structure and Function Using Noise

The study of gene circuits is similar to many other areas of biology in as much as the principal aim is to understand the relationships between structure and function. This is true not only of chemical (e.g. the sequences of bases) or physical (e.g., 3-D structure of proteins) structure, but also the informational structure of genetic circuits and networks. New insights are emerging from the top-down analysis of the biomolecular networks from which complex cellular function emerges. Network motifs have been found that occur significantly more often than would be expected in random networks, providing a rational basis to search for the structure-function relationships in these systems. However, topology alone does not define function, which is sensitive to the specifics of kinetic parameters and the structure and function of individual gene circuits which comprise the higher order networks. Unfortunately, these parameters are usually very difficult to measure or infer. The problem is exacerbated by the fact that we wish to measure these parameters within the context of the fully functioning system of the cell, especially as intracellular molecular crowding generates kinetics that are vastly different than those found from in vitro measurements.

Concurrent with this emphasis on the informational architecture of intracellular molecular networks, a new appreciation of the role of stochastic processes in decision making in biological systems has emerged. Efforts in this direction have developed analysis and simulation techniques; described the noise consequences of gene circuit structure; and have explored how stochastic processes may play a pivotal role in gene circuit functionality. However, the use of inherent noise as a gene circuit probe has been largely ignored. Stochastic fluctuations are a broad-spectrum input excitation, and the frequency-domain structure of the resulting output spectra reveal details about the underlying gene circuit structure and parameter values. In this talk I will describe the frequency-domain processing of stochastic fluctuations by genetic circuits, the measurement of the output noise spectral densities, and the use of these spectra to infer gene circuit structure and reaction rate constants.

Programming Collaborative Behavior and Pattern Formation in Bacterial Communities

Cell-cell communication is a pervasive activity common to both single cell and multicellular organisms, and is used in coordinating cell behavior for a variety of tasks ranging from quorum sensing in bacteria to embryogenesis in mammalian cells. Engineering synthetic multicellular communication systems to exhibit desired functions will improve our quantitative understanding of naturally occurring cell-cell communication, and will also have biotechnology applications in areas such as biosensing, biomaterial fabrication, and tissue engineering. Here we will present theoretical and experimental results from three synthetic multicellular communication systems implemented in bacteria that have been programmed to exhibit unique coordinated cell behavior. The first system is the pulse generator where sender cells communicate to nearby receiver cells, which then respond with a transient burst of gene expression whose amplitude and duration depends on the distance from the senders. In the second system, receiver cells have been engineered to respond to cell-cell communication signals only within prespecified ranges. We will demonstrate how this system can be used to generate a variety of interesting spatial patterns. In the third system, cells have been engineered to "play" Conway's Game of Life, where cells live or die based on the density of their neighbors. This system exhibits complex global emergent behavior that arises from the interaction of cells based on simple local rules. In this talk, we will correlate experimental results from observing the behavior of these systems with our quantitative spatiotemporal models.

Homeostasis, Oscillations, and Ecological Interactions in Re-programmed Bacterial Populations

De novo engineering of gene circuits inside cells has emerged as a powerful approach to decoding 'design principles' of biological systems. Such circuits are also of great interest for their potential applications in computation, engineering, and medicine. However, it has been challenging to realize predictable and robust circuit performance due to some major hurdles, such as noise in gene expression and cell-to-cell variation in phenotype. We address these issues by using cell-cell communication to coordinate cellular behavior across the population. To establish cell-cell communication, we take advantage of 'quorum sensing' systems that many bacteria use to detect and respond to changes in the cell density. As a prototype example, we have built and characterized a 'population control' circuit in bacterium E. coli. This circuit autonomously regulates the cell density using a negative feedback loop acting on the entire population. With the circuit, the cell density is broadcasted and detected by a quorum sensing system, which modulates the expression of a killer gene. The killer gene in turn regulates the cell density by controlling the death rate. Upon activation, the circuit will lead to a stable steady state or sustained oscillations in terms of cell density and gene expression. This circuit lays down the conceptual foundation to program interactions among multiple cell populations - essentially creating 'synthetic ecosystems' from well-characterized genetic modules.

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# Skybox rotation/movement problem (again)

This topic is 4858 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

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Hi, I'm having the same (?) problem as I've seen others have with skyboxes. That is that I cannot get my skybox to stay still. I guess It's an easy fix, but I don't know what to do. I have a fixed camera, and is only moving/rotating one mesh. Unfortunately the skybox moves just like the single mesh does... This is what I do in my render: 1.) I call clear on my device with D3DCLEAR_TARGET|D3DCLEAR_ZBUFFER 2.) I then call BeginScene. 3.) I then sets the view transform using my camera's matrix. 4.) Copies the matrix used in 1, and changes it to: matView._41 = 0; matView._42 = 0; matView._43 = 0; I then set the view matrix again, disable the Z buffer and lightning, renders the skybox, enables lightning, and z buffer and set the original view transform matrix. 5.) Sets the world transform using my mesh matrix. 6.) Renders the mesh. 7.) Finally I call EndScene So the question is: What have I missed? It feels like I've tried all sorts of things but so far without luck. Any ideas anyone? Thanks, Kristoffer

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Don't kill me if I'm wrong, but I think you need to undo the world transforms between rendering the mesh and the skybox, because:

..
View Transform: Sky
Render: Sky
World Transform: Mesh
Render: Mesh
..

(next pass)

..
View Transform: Sky
Render: Sky (with world transform still in effect)
..

So it looks like to me that you still have the world transforms for the mesh set when you render the skybox, thus the skybox is drawn with the same translations + rotations.

I'm 99% sure this is the problem. But then again, I'm often wrong. Good luck!

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If you want to move one mesh but not the others in your scene, you have to make a copy of the world matrix first then set it back to that after transforming and rendering the rotating mesh. For example:

	D3DXMATRIX matEarth, matScale, matRotateY, matOld;	D3DXMatrixIdentity(&matOld);	// Calculate the earths position	D3DXMatrixRotationY(&matRotateY, timeGetTime()/1000.0f);	D3DXMatrixScaling(&matScale, 2.0f, 2.0f, 2.0f);		D3DXMatrixMultiply(&matEarth, &matScale, &matRotateY);	//Render the earth	m_pD3DDevice->SetTransform(D3DTS_WORLD, &matEarth);	m_dwTotalPolygons += m_pEarth->Render();	m_pD3DDevice->SetTransform(D3DTS_WORLD, &matOld); // Reset the world.so everything doesn't rotate together.

Hope that helps. Basically after step 6 you need to set the world matrix back to how it was before you did the transform. Oh by the way in mine I totally reset the world matrix but if you want to get its current state use pD3dDevice->GetTransform(D3DTS_WORLD, &matOld);

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Quote:
 Original post by xegothIf you want to move one mesh but not the others in your scene, you have to make a copy of the world matrix first then set it back to that after transforming and rendering the rotating mesh.

That never seems quite to work for me. Ah oh well. The many problems with VB6...

But yeah, that *should* work.

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Thanks guy!,

I'll test your tips tonight :)

/Kristoffer

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1. Set world matrix to identity
2. Save view matrix
3. Change view matrix as you said e.g.
matView._41 = 0.0f; matView._42 = 0.0f; matView._43 = 0.0f;
4. Turn off z buffer and lighting
5. Render skybox
6. Turn back on z buffer and lighting
7. Set previously saved view matrix

This is if you want to render the skbox first. If you are fill rate limited you may want to render it last. If you do you need to still have z tests but not z writes.

Make sure each mesh can be rendered in any order so it does not affect future renderings. One good way of doing this is to use state blocks. With these blocks you can store the required changes and restore them after rendering.

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Thanks Trip99,

I had missed setting the world matrix to identity. So now everything works just fine :)

/Kristoffer

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text:
Equation of a Line: General Form

Interactive practice questions

What is the gradient of the line $10x+9y-7=0$10x+9y7=0?

$\frac{10}{9}$109

A

$\frac{9}{10}$910

B

$-\frac{9}{10}$910

C

$-\frac{10}{9}$109

D

$\frac{10}{9}$109

A

$\frac{9}{10}$910

B

$-\frac{9}{10}$910

C

$-\frac{10}{9}$109

D
Easy
Less than a minute
Express $y=\frac{-x}{-10}$y=x10 in general form.
Express $y-x=2$yx=2 in general form.


text:
Browse Questions

# Which of the following groups is an auxochrome?

$(a)\;-N(CH_3)_2 \\ (b)\;-OH \\ (c)\;-COOOH \\ (d)\;All$
Can you answer this question?

Groups like $-OH,-COOH, -SO_3H, -NHR,-NR_2,-NH_2$ are anxochromic
Hence d is the correct answer.
answered Mar 7, 2014 by


text:
Stochastic Systems Group
 Home Research Group Members Programs Demos Calendar Publications Mission Statement Alumni

## Multitarget-Multisensor Data Association Using the Tree-Reweighted Max-Product Algorithm Lei Chen SSG Doctoral Student

Data association is a fundamental problem in multitarget-multisensor tracking. It entails selecting the most probable association from a very large set of possibilities. With $M$ sensors and $N$ targets in the range of each sensor, there are $(N !)^{M - 1}$ different configurations, which renders infeasible a solution by direct computation even in modestly-sized applications. (For example, for $M = 7$ and $N = 5$, there are slightly less than $3$ trillion possible configurations.) We describe an iterative method for solving the optimal data association in a distributed fashion; the work exploits the framework of graphical models. Our basic idea is to treat the measurement assignment for each sensor as a random variable, which is in turn represented as a node in an underlying graph. Neighboring nodes are coupled by the targets visible to both sensors. Thus we transform the data association problem to that of computing the maximum a posteriori (MAP) configuration in a graphical model, to which efficient techniques (e.g., the max-product/min-sum algorithm) can be applied. We use the tree-reweighted max-product algorithm instead of the regular version to get the exact MAP estimates of the association variables on any graph topology (subject to some necessary conditions). For acyclic graphs, our algorithm can solve the data association problem directly and recursively with complexity of $\mathcal{O}\left( (N !)^2 M \right)$. On graphs with cycles, the algorithm may require more iterations to converge. However, for the data association problem the coupling matrices are inherently of low rank, and experiments show that the algorithm still converges very fast in this case.

Joint work with Martin J. Wainwright, Mujdat Cetin, and Alan S. Willsky.

Problems with this site should be emailed to jonesb@mit.edu


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# 16 Conclusion

## Prerequisites

Like the introduction, this concluding chapter contains few code chunks. But its prerequisites are demanding. It assumes that you have:

• read through and attempted the exercises in all the chapters of Part I (Foundations);
• considered how you can use geocomputation to solve real-world problems, at work and beyond, after engaging with Part III (Applications).

## 16.1 Introduction

The aim of this chapter is to synthesize the contents, with reference to recurring themes/concepts, and to inspire future directions of application and development. Section 16.2 discusses the wide range of options for handling geographic data in R. Choice is a key feature of open source software; the section provides guidance on choosing between the various options. Section 16.3 describes gaps in the book’s contents and explains why some areas of research were deliberately omitted, while others were emphasized. This discussion leads to the question (which is answered in Section 16.5): having read this book, where to go next? Section 16.6 returns to the wider issues raised in Chapter 1. In it we consider geocomputation as part of a wider ‘open source approach’ that ensures methods are publicly accessible, reproducible and supported by collaborative communities. This final section of the book also provides some pointers on how to get involved.

## 16.2 Package choice

A characteristic of R is that there are often multiple ways to achieve the same result. The code chunk below illustrates this by using three functions, covered in Chapters 3 and 5, to combine the 16 regions of New Zealand into a single geometry:

library(spData)
nz_u1 = sf::st_union(nz)
nz_u2 = aggregate(nz["Population"], list(rep(1, nrow(nz))), sum)
nz_u3 = dplyr::summarise(nz, t = sum(Population))
identical(nz_u1, nz_u2$geometry) #> [1] TRUE identical(nz_u1, nz_u3$geom)
#> [1] TRUE

Although the classes, attributes and column names of the resulting objects nz_u1 to nz_u3 differ, their geometries are identical. This is verified using the base R function identical().100 Which to use? It depends: the former only processes the geometry data contained in nz so is faster, while the other options performed attribute operations, which may be useful for subsequent steps.

The wider point is that there are often multiple options to choose from when working with geographic data in R, even within a single package. The range of options grows further when more R packages are considered: you could achieve the same result using the older sp package, for example. We recommend using sf and the other packages showcased in this book, for reasons outlined in Chapter 2, but it’s worth being aware of alternatives and being able to justify your choice of software.

A common (and sometimes controversial) choice is between tidyverse and base R approaches. We cover both and encourage you to try both before deciding which is more appropriate for different tasks. The following code chunk, described in Chapter 3, shows how attribute data subsetting works in each approach, using the base R operator [ and the select() function from the tidyverse package dplyr. The syntax differs but the results are (in essence) the same:

library(dplyr)                          # attach tidyverse package
nz_name1 = nz["Name"]                   # base R approach
nz_name2 = nz |> select(Name)          # tidyverse approach
identical(nz_name1$Name, nz_name2$Name) # check results
#> [1] TRUE

Again the question arises: which to use? Again the answer is: it depends. Each approach has advantages: the pipe syntax is popular and appealing to some, while base R is more stable, and is well known to others. Choosing between them is therefore largely a matter of preference. However, if you do choose to use tidyverse functions to handle geographic data, beware of a number of pitfalls (see the supplementary article tidyverse-pitfalls on the website that supports this book).

While commonly needed operators/functions were covered in depth — such as the base R [ subsetting operator and the dplyr function filter() — there are many other functions for working with geographic data, from other packages, that have not been mentioned. Chapter 1 mentions 20+ influential packages for working with geographic data, and only a handful of these are demonstrated in subsequent chapters. There are hundreds more. As of mid-2022, there are about 200 packages mentioned in the Spatial Task View; more packages and countless functions for geographic data are developed each year, making it impractical to cover them all in a single book.

The rate of evolution in R’s spatial ecosystem may seem overwhelming, but there are strategies to deal with the wide range of options. Our advice is to start by learning one approach in depth but to have a general understand of the breadth of options available. This advice applies equally to solving geographic problems in R (Section 16.5 covers developments in other languages) as it does to other fields of knowledge and application.

Of course, some packages perform much better than others, making package selection an important decision. From this diversity, we have focused on packages that are future-proof (they will work long into the future), high performance (relative to other R packages) and complementary. But there is still overlap in the packages we have used, as illustrated by the diversity of packages for making maps, for example (see Chapter 9).

Package overlap is not necessarily a bad thing. It can increase resilience, performance (partly driven by friendly competition and mutual learning between developers) and choice, a key feature of open source software. In this context the decision to use a particular approach, such as the sf/tidyverse/raster ecosystem advocated in this book should be made with knowledge of alternatives. The sp/rgdal/rgeos ecosystem that sf is designed to supersede, for example, can do many of the things covered in this book and, due to its age, is built on by many other packages.101 Although best known for point pattern analysis, the spatstat package also supports raster and other vector geometries . At the time of writing (October 2018) 69 packages depend on it, making it more than a package: spatstat is an alternative R-spatial ecosystem.

It is also being aware of promising alternatives that are under development. The package stars, for example, provides a new class system for working with spatiotemporal data. If you are interested in this topic, you can check for updates on the package’s source code and the broader SpatioTemporal Task View. The same principle applies to other domains: it is important to justify software choices and review software decisions based on up-to-date information.

## 16.3 Gaps and overlaps

There are a number of gaps in, and some overlaps between, the topics covered in this book. We have been selective, emphasizing some topics while omitting others. We have tried to emphasize topics that are most commonly needed in real-world applications such as geographic data operations, projections, data read/write and visualization. These topics appear repeatedly in the chapters, a substantial area of overlap designed to consolidate these essential skills for geocomputation.

On the other hand, we have omitted topics that are less commonly used, or which are covered in-depth elsewhere. Statistical topics including point pattern analysis, spatial interpolation (kriging) and spatial epidemiology, for example, are only mentioned with reference to other topics such as the machine learning techniques covered in Chapter 12 (if at all). There is already excellent material on these methods, including statistically orientated chapters in Bivand, Pebesma, and Gómez-Rubio (2013) and a book on point pattern analysis by Baddeley, Rubak, and Turner (2015). Other topics which received limited attention were remote sensing and using R alongside (rather than as a bridge to) dedicated GIS software. There are many resources on these topics, including Wegmann, Leutner, and Dech (2016) and the GIS-related teaching materials available from Marburg University.

Instead of covering spatial statistical modeling and inference techniques, we focused on machine learning (see Chapters 12 and 15). Again, the reason was that there are already excellent resources on these topics, especially with ecological use cases, including A. Zuur et al. (2009), A. F. Zuur et al. (2017) and freely available teaching material and code on Geostatistics & Open-source Statistical Computing by David Rossiter, hosted at css.cornell.edu/faculty/dgr2 and the R for Geographic Data Science project by Stefano De Sabbata at the University of Leicester for an introduction to R for geographic data science. There are also excellent resources on spatial statistics using Bayesian modeling, a powerful framework for modeling and uncertainty estimation .

We have largely omitted geocomputation on ‘big data’, by which we mean datasets that do not fit on consumer hardware or which cannot realistically be processed on a single CPU. This decision is justified by the fact that the majority of geographic datasets that are needed for common research or policy applications do fit on consumer hardware, even if that may mean increasing the amount of RAM on your computer (or temporarily ‘renting’ compute power available on platforms such as GitHub Codespaces), and the fact that learning to solve problems on small datasets is a prerequisite to solving problems on huge datasets. Analysis of ‘big data’ often involves extracting a small amount of data from a database for a specific statistical analysis. Spatial databases, covered in Chapter 10, can help with the analysis of datasets that do not fit in memory. ‘Earth observation cloud back-ends’ can be accessed from R with the openeo package, as described on the openeo.org website. We omitted detailed coverage of software for geographic analysis of big data with software such as Apache Sedona because the hardware and time costs of setting-up such systems are high, relative to their relative niche use cases.

## 16.4 Getting help?

Geocomputation is a large field and it is highly likely that you will encounter challenges preventing you from achieving an outcome that you are aiming towards. In many cases you may just ‘get stuck’ at a particular point in data analysis workflows, with a cryptic error message or an unexpected result providing little clues as to what is going on. This section provides pointers to help you overcome such problems, by clearly defining the problem, searching for existing knowledge on solutions and, if those approaches to not solve the problem, through the art of asking good questions.

When you get stuck at a particular point, it is worth first taking a step back and working out which approach is most likely to solve the issue. Trying each of the following steps, in order (or skipping steps if you have already tried them), provides a structured approach to problem solving:

1. Define exactly what you are trying to achieve, starting from first principles (and often a sketch, as outlined below)
2. Diagnose exactly where in your code the unexpected results arise, by running and exploring the outputs of individual lines of code and their individual components (you can can run individual parts of a complex command by selecting them with a cursor and pressing Ctrl+Enter in RStudio, for example)
3. Read the documentation of the function that has been diagnosed as the ‘point of failure’ in the previous step. Simply understanding the required inputs to functions, and running the examples that are often provided at the bottom of help pages, can help solve a surprisingly large proportion of issues (run the command ?terra::rast and scroll down to the examples that are worth reproducing when getting started with the function, for example)
4. If reading R’s inbuilt documentation, as outlined in the previous step, does not help solve the problem, it is probably time to do a broader search online to see if others have written about the issue you’re seeing. See a list of places to search for help below for places to search
5. If all the previous steps above fail, and you cannot find a solution from your online searches, it may be time to compose a question with a reproducible example and post it in an appropriate place

Steps 1 to 3 outlined above are fairly self-explanatory but, due to the vastness of the internet and multitude of search options, it is worth considering effective search strategies before deciding to compose a question.

### 16.4.1 Searching for solutions online

A logical place to start for many issues is search engines. ‘Googling it’ can in some cases result in the discovery of blog posts, forum messages and other online content about the precise issue you’re having. Simply typing in a clear description of the problem/question is a valid approach here but it is important to be specific (e.g. with reference to function and package names and input dataset sources if the problem is dataset specific). You can also make online searches more effective by including additional detail:

• Use quote marks to maximise the chances that ‘hits’ relate to the exact issue you’re having by reducing the number of results returned
• Set time restraints, for example only returning content created within the last year can be useful when searching for help on an evolving package.
• Make use of additional search engine features, for example restricting searches to content hosted on CRAN with site:r-project.org

### 16.4.2 Places to search for (and ask) for help

• R’s Special Interest Group on Geographic data email list (R-SIG-GEO)
• The GIS Stackexchange website at gis.stackexchange.com
• The large and general purposes programming Q&A site stackoverflow.com
• Online forums associated with a particular entity, such as the RStudio Community, the rOpenSci Discuss web forum and forums associated with particular software tools such as the Stan forum
• Software development platforms such as GitHub, which hosts issue trackers for the majority of R-spatial packages and also, increasingly, inbuilt discussion pages such as that created to encourage discussion (not just bug reporting) around the sfnetworks package (see luukvdmeer/sfnetworks/discussions)

### 16.4.3 How to ask a good question with a reproducible example

In terms asking a good question, a clearly stated questions supported by an accessible and fully reproducible example is key. It is also helpful, after showing the code that ‘did not work’ from the user’s perspective, to explain what you would like to see. A very useful tool for creating reproducible examples is the reprex package.

### 16.4.4 Defining and sketching the problem

The best starting point when developing a new geocomputational methodology or approach is often a pen and paper (or equivalent digital sketching tools such as Excalidraw and tldraw which allow collaborative sketching and rapid sharing of ideas): during the most creative early stages of methodological development work software of any kind can slow down your thoughts and direct your thinking away from important abstract thoughts. Framing the question with mathematics is also highly recommended, with reference to a minimal example that you can sketch ‘before and after’ versions of numerically. If you have the skills and if the problem warrants it, describing the approach algebraically can in some cases help develop effective implementations.

## 16.6 The open source approach

This is a technical book so it makes sense for the next steps, outlined in the previous section, to also be technical. However, there are wider issues worth considering in this final section, which returns to our definition of geocomputation. One of the elements of the term introduced in Chapter 1 was that geographic methods should have a positive impact. Of course, how to define and measure ‘positive’ is a subjective, philosophical question, beyond the scope of this book. Regardless of your worldview, consideration of the impacts of geocomputational work is a useful exercise: the potential for positive impacts can provide a powerful motivation for future learning and, conversely, new methods can open-up many possible fields of application. These considerations lead to the conclusion that geocomputation is part of a wider ‘open source approach’.

Section 1.1 presented other terms that mean roughly the same thing as geocomputation, including geographic data science (GDS) and ‘GIScience’. Both capture the essence of working with geographic data, but geocomputation has advantages: it concisely captures the ‘computational’ way of working with geographic data advocated in this book — implemented in code and therefore encouraging reproducibility — and builds on desirable ingredients of its early definition :

• The creative use of geographic data
• Application to real-world problems
• Building ‘scientific’ tools
• Reproducibility

We added the final ingredient: reproducibility was barely mentioned in early work on geocomputation, yet a strong case can be made for it being a vital component of the first two ingredients. Reproducibility

• encourages creativity by shifting the focus away from the basics (which are readily available through shared code) and towards applications;
• discourages people from ‘reinventing the wheel’: there is no need to re-do what others have done if their methods can be used by others; and
• makes research more conducive to real world applications, by enabling anyone in any sector to apply your methods in new areas.

If reproducibility is the defining asset of geocomputation (or command-line GIS) it is worth considering what makes it reproducible. This brings us to the ‘open source approach’, which has three main components:

• A command-line interface (CLI), encouraging scripts recording geographic work to be shared and reproduced
• Open source software, which can be inspected and potentially improved by anyone in the world
• An active developer community, which collaborates and self-organizes to build complementary and modular tools

Like the term geocomputation, the open source approach is more than a technical entity. It is a community composed of people interacting daily with shared aims: to produce high performance tools, free from commercial or legal restrictions, that are accessible for anyone to use. The open source approach to working with geographic data has advantages that transcend the technicalities of how the software works, encouraging learning, collaboration and an efficient division of labor.

There are many ways to engage in this community, especially with the emergence of code hosting sites, such as GitHub, which encourage communication and collaboration. A good place to start is simply browsing through some of the source code, ‘issues’ and ‘commits’ in a geographic package of interest. A quick glance at the r-spatial/sf GitHub repository, which hosts the code underlying the sf package, shows that 40+ people have contributed to the codebase and documentation. Dozens more people have contributed by asking question and by contributing to ‘upstream’ packages that sf uses. More than 600 issues have been closed on its issue tracker, representing a huge amount of work to make sf faster, more stable and user-friendly. This example, from just one package out of dozens, shows the scale of the intellectual operation underway to make R a highly effective and continuously evolving language for geocomputation.

It is instructive to watch the incessant development activity happen in public fora such as GitHub, but it is even more rewarding to become an active participant. This is one of the greatest features of the open source approach: it encourages people to get involved. This book itself is a result of the open source approach: it was motivated by the amazing developments in R’s geographic capabilities over the last two decades, but made practically possible by dialogue and code sharing on platforms for collaboration. We hope that in addition to disseminating useful methods for working with geographic data, this book inspires you to take a more open source approach. Whether it’s raising a constructive issue alerting developers to problems in their package; making the work done by you and the organizations you work for open; or simply helping other people by passing on the knowledge you’ve learned, getting involved can be a rewarding experience.


text:
PL EN

Preferencje
Język
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo

## Annales Polonici Mathematici

1997 | 66 | 1 | 223-238
Tytuł artykułu

### $𝒞^∞$-vectors and boundedness

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The following two questions as well as their relationship are studied: (i) Is a closed linear operator in a Banach space bounded if its $𝒞^∞$-vectors coincide with analytic (or semianalytic) ones? (ii) When are the domains of two successive powers of the operator in question equal? The affirmative answer to the first question is established in case of paranormal operators. All these investigations are illustrated in the context of weighted shifts.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
223-238
Opis fizyczny
Daty
wydano
1997
otrzymano
1995-10-19
poprawiono
1996-05-20
Twórcy
autor
• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
autor
• Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland
Bibliografia
• [1] P. R. Chernoff, Some remarks on quasi-analytic vectors, Trans. Amer. Math. Soc. 167 (1972), 105-113.
• [2] P. R. Chernoff, A semibounded closed symmetric operator whose square has trivial domain, Proc. Amer. Math. Soc. 89 (1983), 289-290.
• [3] J. Daneš, On local spectral radius, Časopis Pěst. Mat. 112 (1987), 177-187.
• [4] W. G. Faris, Selfadjoint Operators, Lecture Notes in Math. 433, Springer, Berlin, 1975.
• [5] T. Furuta, On the class of paranormal operators, Proc. Japan Acad. 43 (1967), 594-598.
• [6] M. Hasegawa, On quasi-analytic vectors for dissipative operators, Proc. Amer. Math. Soc. 29 (1971), 81-84.
• [7] W. Mlak, The Schrödinger type couples related to weighted shifts, Univ. Iagel. Acta Math. 27 (1988), 297-301.
• [8] E. Nelson, Analytic vectors, Ann. of Math. 70 (1959), 572-615.
• [9] A. E. Nussbaum, Quasi-analytic vectors, Ark. Mat. 6 (1967), 179-191.
• [10] A. E. Nussbaum, A note on quasi-analytic vectors, Studia Math. 33 (1969), 305-309.
• [11] Y. Okazaki, Boundedness of closed linear operator T satisfying R(T)⊂ D(T), Proc. Japan Acad. 62 (1986), 294-296.
• [12] S. Ôta, Closed linear operators with domain containing their range, Proc. Edinburgh Math. Soc. 27 (1984), 229-233.
• [13] S. Ôta, Unbounded nilpotents and idempotents, J. Math. Anal. Appl. 132 (1988), 300-308.
• [14] K. Schmüdgen, On domains of powers of closed symmetric operators, J. Operator Theory 9 (1983), 53-75.
• [15] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag, Berlin, 1990.
• [16] B. Simon, The theory of semi-analytic vectors: a new proof of a theorem of Masson and McClary, Indiana Univ. Math. J. 20 (1971), 1145-1151.
• [17] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part I, Exposition. Math. 1 (1983), 71-73.
• [18] J. Stochel and F. H. Szafraniec, Boundedness of linear and related nonlinear maps, Part II, Exposition. Math. 2 (1984), 283-287.
• [19] J. Stochel and F. H. Szafraniec, Bounded vectors and formally normal operators, in: Dilation Theory, Toeplitz Operators, and Other Topics, Proc. 7th Internat. Conf. on Oper. Theory, Timişoara and Herculane (Romania), June 7-17, 1982, C. Apostol, C. M. Pearcy, B. Sz.-Nagy and D. Voiculescu (eds.), Oper. Theory Adv. Appl. 11, Birkhäuser, Basel, 1983, 363-370.
• [20] J. Stochel and F. H. Szafraniec, The normal part of an unbounded operator, Nederl. Akad. Wetensch. Proc. Ser. A 92 (1989), 495-503 = Indag. Math. 51 (1989), 495-503.
• [21] J. Stochel and F. H. Szafraniec, A few assorted questions about unbounded subnormal operators, Univ. Iagel. Acta Math. 28 (1991), 163-170.
• [22] F. H. Szafraniec, On the boundedness condition involved in dilation theory, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 877-881.
• [23] F. H. Szafraniec, Boundedness of the shift operator related to positive definite forms: an application to moment problems, Ark. Mat. 19 (1981), 251-259.
• [24] F. H. Szafraniec, Kato-Protter type inequalities, bounded vectors and the exponential function, Ann. Polon. Math. 51 (1990), 303-312.
Typ dokumentu
Bibliografia
Identyfikatory


text:
On a characterization of an abelian variety in the classification theory of algebraic varieties
Compositio Mathematica, Volume 41 (1980) no. 3, p. 355-359
@article{CM_1980__41_3_355_0,
author = {Kawamata, Yujiro and Viehweg, Eckart},
title = {On a characterization of an abelian variety in the classification theory of algebraic varieties},
journal = {Compositio Mathematica},
publisher = {Sijthoff et Noordhoff International Publishers},
volume = {41},
number = {3},
year = {1980},
pages = {355-359},
zbl = {0417.14033},
mrnumber = {589087},
language = {en},
url = {http://www.numdam.org/item/CM_1980__41_3_355_0}
}

Kawamata, Yujiro; Viehweg, Eckart. On a characterization of an abelian variety in the classification theory of algebraic varieties. Compositio Mathematica, Volume 41 (1980) no. 3, pp. 355-359. http://www.numdam.org/item/CM_1980__41_3_355_0/

[1] K. Ueno: Classification theory of algebraic varieties and compact complex spaces, Lecture Note in Math. 439, 1975, Springer. | MR 506253 | Zbl 0299.14007

[2] K. Ueno: Classification of algebraic varieties, II-Algebraic threefolds of parabolic type-, Intl. Symp. on Algebraic Geometry, Kyoto, 1977, 693-708. | MR 578882 | Zbl 0407.14011


text:
# In a PID $R$, given $a,b\in R$, what are the ideals $\langle a\rangle,\langle b\rangle$ also generated by?

Let $$R$$ be a principal ideal domain and let $$a$$ and $$b$$ be two nonunit elements in $$R$$ then ideal generated by $$a$$ and $$b$$ is also generated by

1. $$a+b$$

2. $$ab$$

3. $$\gcd(a,b)$$

4. $$\text{lcm}(a,b)$$

I am not sure but I think that answer is 3.

Let $$I=\langle a,b \rangle$$ $$=\{ax+by:x,y\in R\}$$. Let $$d=gcd(a,b)$$. Since $$R$$ is a PID we get $$d=ax+by$$ for some $$x,y\in R$$. So $$\langle d \rangle \subseteq I$$ as $$d$$ is a divisor of $$a$$ and $$b$$ clearly $$a,b \in \langle d \rangle$$ So $$I \subseteq \langle d \rangle$$. Hence $$I=\langle d \rangle$$. Is it correct?

• I like it, it's correct. – Mars Nov 19 '20 at 5:14

Yes, it is correct. In fact, some authors go a step further and define the gcd of $$a$$ and $$b$$ of a ring to be a generator of the smallest principal ideal containing both $$a$$ and $$b$$.