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# Chapter 10 Statistical modelling of spatial data

Spatial data almost always (and everywhere) has the property that it is spatially structured: observations done closeby in space tend to be more similar than observations done at larger distance from each other. This phenomenon, in the geography domain attributed to Waldo Tobler (as in “Waldo Tobler’s first law of geography”) was already noted by (Fisher and others 1937) and was a motivation for developing randomized block design in agricultural experiments: allocating treatments randomly to blocks avoids that spatial structure gets mixed up (or: confounds) with a signal caused by the treatment.

The often heard argument that spatially structured data means that the data is spatially correlated, which would exclude estimation methods that assume independent observations is false. Correlation is a property of two random variables, and there are different ways in which spatial data can be approached with random variables: either the observation locations are random (leading to design-based inference) or the observed values are random (leading to model-based inference). The next section points out the difference between these two.

## 10.1 Design-based and model-based inference

Statistical inference means the action of estimating parameters about a population from sample data. Suppose we denote the variable of interest with $$z(s)$$, where $$z$$ is the attribute value measured at location $$s$$, and we are interested in estimating the mean value of $$z(s)$$ over a domain $$D$$, $z(s)=\frac{1}{|D|} \int_{ u \in D} z(u)du,$ with $$|D|$$ the area of $$D$$, from sample data $$z(s_1),...,z(s_n)$$.

Then, there are two possibilities to proceed: model-based, or design-based. A model-based approach considers $$z(s)$$ to be a realisation of a superpopulation $$Z(s)$$ (using capital letters to indicate random variables), and could for instance postulate a model for its spatial variability in the form of $Z(s) = m + e(s), \ \ \mbox{E}(e(s)) = 0, \ \ \mbox{Cov(e(s))} = \Sigma(\theta)$ with $$m$$ a constant mean and $$e(s)$$ a residual with mean zero and covariance matrix $$\Sigma(\theta)$$. This would require choosing the covariance function $$\Sigma()$$ and estimating its parameters $$\theta$$ form $$z(s)$$, and then computing a block kriging prediction $$\hat{Z}(D)$$ (section 12.5). This approach makes no assumptions about how $$z(s)$$ was sample spatially, but of course it should allow for choosing the covariance function and estimating its parameters; inference is conditional to the validity of the postulated model.

Rather than assuming a superpopulation model, the design-based approach (De Gruijter and Ter Braak 1990; D. J. Brus 2021a; Breidt, Opsomer, and others 2017) assumes randomness in the locations, which is justified (only) when using random sampling. It requires that the sample data were obtained by probability sampling, meaning that some form of spatial random sampling was used where all elements of $$z(s)$$ had a known and positive probability of being included in the sample obtained. The random process is that of sampling: $$z(s_1)$$ is a realisation of the random process $$z(S_1)$$, the first observation taken over repeated random sampling. Design-based estimaters only need these inclusion probabilities to estimate mean values with standard errors. This means that for instance given a simple random sample, the unweighted sample mean is used to estimate the population mean, and no model parameters need to be fit.

The misconception here, as explained in D. J. Brus (2021a), is that this is only the case when working under model-based approaches: $$Z(s_1)$$ and $$Z(s_2)$$ may well be correlated (“model-dependent”), but although in a particular random sampling (realisation) $$z(s_1)$$ and $$z(s_2)$$ may be close in space, the corresponding random variables $$z(S_1)$$ and $$z(S_2)$$ considered over repeated random sampling are not close together, and are design-independent. Both situations can co-exist without contradiction, and are a consequence of choosing to work under one inference framework or the other.

The choice whether to work under a design-based or model-based framework depends on the purpose of the study and the data collection process. The model-based framework lends itself best for cases:

• where predictions are required for individual locations, or for areas too small to be sampled
• when the available data were not collected using a known random sampling scheme (i.e., the inclusion probabilities are unknown, or are zero over particular areas or/and times)

Design-based approaches are most suitable when:

• observations were collected using a spatial random sampling process
• aggregated properties of the entire sample region (or sub-region) are needed
• estimates are required that are not sensitive to potential model misspecification, e.g. when needed for regulatory or legal purposes

In case a sampling procedure is to be planned (De Gruijter et al. 2006), some form of spatial random sampling is definitely worth considering since it opens up the possibility of following both inference frameworks.

## 10.2 Predictive models with coordinates

In data science projects, coordinates may be seen as features in a larger set of predictors (or features, or covariates) and treated accordingly. There are some catches with doing so.

As usual when working with predictors, it is good to choose predictive methods that are not sensitive to shifts in origin or shifts in unit (scale). Assuming a two-dimensional problem, predictive models should also not be sensitive to arbitrary rotations of the x- and y- or latitude and longitude axes. For projected (2D, Cartesian) coordinates this can be assured e.g. by using polynomials of order $$n$$ as $$(x+y)^n$$, rather than $$(x)^n + (y)^n$$; for a second order polynomial this involves including the term $$xy$$, so that an ellipsoidal-shape trend surface does not have to be aligned with the $$x-$$ or $$y-$$axis. For a GAM model with spline components, one would use a spline in two dimensions rather than two independent splines in $$x$$ and $$y$$. An exception to this “rule” is when e.g. a pure latitude effect is desired, for instance to account for solar energy influx.

When the area covered by the data is large, the difference between using ellipsoidal coordinates and projected coordinates will automatically become larger, and hence choosing one of both will have an effect on predictive modelling. For very large extents, e.g. global models, polynomials or splines in latitude and longitude will not make sense as they ignore the circular nature of longitude and the coordinate singularities at the poles. Here, spherical harmonics, base functions that are continuous on the sphere with increasing spatial frequencies can replace polynomials or be used as spline base functions.

In many cases, the spatial coordinates over which samples were collected also define the space over which predictions are made, setting them apart from other features. Many simple predictive approaches, including most machine learning methods, assume sample data to be independent. When samples are collected by spatially random sampling over the spatial target area, this assumption may be justified when working under a design-based context (D. J. Brus 2021b). This context however treats the coordinate space as the variable over which we randomize, which affords predicting values for a new randomly chosen location but rules out making predictions for fixed locations; this implies that averages over areas over which samples were collected can be obtained, but not spatial interpolations. In case predictions for fixed locations are required, or in case data were not collected by spatial random sampling, a model-based approach (as taken in chapter 12) is needed and typically some form of spatial and/or temporal autocorrelation of residuals must be assumed.

A common case is where sample data are collected opportunistically (“whatever could be found”), and are then used in a predictive framework that does not weigh them. This has a consequence that the resulting model may be biased towards over-represented areas (in predictor space and/or in spatial coordinates space), and that simple (random) cross validation statistics may be over-optimistic when taken as performance measures for spatial prediction (Meyer and Pebesma 2020). Adaptive cross validation measures, e.g. spatial cross validation may help getting more relevant measures for predictive performance.

There is a large number of papers and books available on analysing and statistical modelling of spatial and spatiotemporal data, and a very large number of R packages help doing so. Several CRAN task views try to maintain an overview of the R packages, e.g. on:

• spatial data (R. Bivand 2021a)
• spatiotemporal data (E. Pebesma 2021a)
• tracking data (Joo et al. 2021), see also (Joo et al. 2020)

The introductions to subsequent chapters contain more pointers to relevant literature references. Introductions to using the integrated nested Laplace approximation (INLA) for analysing spatial data are given in Blangiardo et al. (2013), Blangiardo and Cameletti (2015), and Virgilio Gómez-Rubio (2020). Krainski et al. (2018) combine the INLA approach with stochastic partial differential equations. Spatiotemporal Bayesian modelling of change of support problems is presented in Raim et al. (2021) and Raim et al. (2020).

### References

Bivand, Roger. 2021a. CRAN Task View: Analysis of Spatial Data. https://cran.r-project.org/web/views/Spatial.html.

Blangiardo, Marta, and Michela Cameletti. 2015. Spatial and Spatio-Temporal Bayesian Models with R-Inla. John Wiley & Sons.

Blangiardo, Marta, Michela Cameletti, Gianluca Baio, and Håvard Rue. 2013. “Spatial and Spatio-Temporal Models with R-Inla.” Spatial and Spatio-Temporal Epidemiology 4: 33–49. https://doi.org/https://doi.org/10.1016/j.sste.2012.12.001.

Breidt, F Jay, Jean D Opsomer, and others. 2017. “Model-Assisted Survey Estimation with Modern Prediction Techniques.” Statistical Science 32 (2): 190–205.

Brus, Dick J. 2021a. “Statistical Approaches for Spatial Sample Survey: Persistent Misconceptions and New Developments.” European Journal of Soil Science 72 (2): 686–703. https://doi.org/https://doi.org/10.1111/ejss.12988.

Brus, Dick J. 2021b. “Statistical Approaches for Spatial Sample Survey: Persistent Misconceptions and New Developments.” European Journal of Soil Science 72 (2): 686–703. https://doi.org/https://doi.org/10.1111/ejss.12988.

De Gruijter, Jaap, Dick J Brus, Marc FP Bierkens, and Martin Knotters. 2006. Sampling for Natural Resource Monitoring. Springer Science & Business Media.

De Gruijter, JJ, and CJF Ter Braak. 1990. “Model-Free Estimation from Spatial Samples: A Reappraisal of Classical Sampling Theory.” Mathematical Geology 22 (4): 407–15.

Fisher, Ronald Aylmer, and others. 1937. The Design of Experiments. The Design of Experiments. 2nd Ed. Oliver & Boyd, Edinburgh & London.

Gómez-Rubio, Virgilio. 2020. Bayesian Inference with Inla. CRC Press.

Joo, Rocío, Matthew E. Boone, Thomas A. Clay, Samantha C. Patrick, Susana Clusella-Trullas, and Mathieu Basille. 2020. “Navigating Through the R Packages for Movement.” Journal of Animal Ecology 89 (1): 248–67. https://doi.org/https://doi.org/10.1111/1365-2656.13116.

Joo, Rocío, Matthew E. Boone, Michael Sumner, and Mathieu Basille. 2021. CRAN Task View: Processing and Analysis of Tracking Data. https://cran.r-project.org/web/views/SpatioTemporal.html.

Krainski, Elias T, Virgilio Gómez-Rubio, Haakon Bakka, Amanda Lenzi, Daniela Castro-Camilo, Daniel Simpson, Finn Lindgren, and Håvard Rue. 2018. Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and Inla. CRC Press.

Meyer, Hanna, and Edzer Pebesma. 2020. “Predicting into Unknown Space? Estimating the Area of Applicability of Spatial Prediction Models.” https://arxiv.org/abs/2005.07939.

Pebesma, Edzer. 2021a. CRAN Task View: Handling and Analyzing Spatio-Temporal Data. https://cran.r-project.org/web/views/SpatioTemporal.html.

Raim, A.M., S.H. Holan, J.R. Bradley, and C.K. Wikle. 2021. “Spatio-Temporal Change of Support Modeling with R.” Computational Statistics 36: 749–80. https://doi.org/ https://doi.org/10.1007/s00180-020-01029-4.

Raim, Andrew M., Scott H. Holan, Jonathan R. Bradley, and Christopher K. Wikle. 2020. Stcos: Space-Time Change of Support. https://github.com/holans/ST-COS.


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Search: MSC category 11R11 ( Quadratic extensions )

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1. CJM 2011 (vol 63 pp. 1220)

Baake, Michael; Scharlau, Rudolf; Zeiner, Peter
 Similar Sublattices of Planar Lattices The similar sublattices of a planar lattice can be classified via its multiplier ring. The latter is the ring of rational integers in the generic case, and an order in an imaginary quadratic field otherwise. Several classes of examples are discussed, with special emphasis on concrete results. In particular, we derive Dirichlet series generating functions for the number of distinct similar sublattices of a given index, and relate them to zeta functions of orders in imaginary quadratic fields. Categories:11H06, 11R11, 52C05, 82D25

2. CJM 2007 (vol 59 pp. 553)

Dasgupta, Samit
 Computations of Elliptic Units for Real Quadratic Fields Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an earlier joint article with Henri Darmon, we presented the definition of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$ and each $\tau \in K$. We conjectured that the $p$-adic number $u(\alpha, \tau)$ lies in a specific ring class extension of $K$ depending on $\tau$, and proposed a Shimura reciprocity law" describing the permutation action of Galois on the set of $u(\alpha, \tau)$. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha, \tau)$, and implement this algorithm with the modular unit $\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$ and $11$, and all real quadratic fields $K$ with discriminant $D < 500$ such that $2$ splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha, \tau)$, instead of only up to a root of unity. Categories:11R37, 11R11, 11Y40

3. CJM 2004 (vol 56 pp. 55)

Harper, Malcolm
 $\mathbb{Z}[\sqrt{14}]$ is Euclidean We provide the first unconditional proof that the ring $\mathbb{Z} [\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then: $$the~ring~of~integers~of~K~is~a~Euclidean~domain~if~and~only~if~it~is~a~principal~ideal~domain.$$ The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property. Categories:11R04, 11R11

4. CJM 2000 (vol 52 pp. 369)

Granville, Andrew; Mollin, R. A.; Williams, H. C.
 An Upper Bound on the Least Inert Prime in a Real Quadratic Field It is shown by a combination of analytic and computational techniques that for any positive fundamental discriminant $D > 3705$, there is always at least one prime $p < \sqrt{D}/2$ such that the Kronecker symbol $\left(D/p\right) = -1$. Categories:11R11, 11Y40

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• Book Details

Volume: 1572014; 356 pp
MSC: Primary 81; 46; 34; 47;

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators.

Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrödinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.

This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.

Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.

This new edition has additions and improvements throughout the book to make the presentation more student friendly.

The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints).

Zentralblatt MATH

The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.

Mathematical Reviews

Graduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

• Part 0. Preliminaries
• Chapter 0. A first look at Banach and Hilbert spaces
• Part 1. Mathematical foundations of quantum mechanics
• Chapter 1. Hilbert spaces
• Chapter 2. Self-adjointness and spectrum
• Chapter 3. The spectral theorem
• Chapter 4. Applications of the spectral theorem
• Chapter 5. Quantum dynamics
• Chapter 6. Perturbation theory for self-adjoint operators
• Part 2. Schrödinger operators
• Chapter 7. The free Schrödinger operator
• Chapter 8. Algebraic methods
• Chapter 9. One-dimensional Schrödinger operators
• Chapter 10. One-particle Schrödinger operators
• Chapter 11. Atomic Schrödinger operators
• Chapter 12. Scattering theory
• Part 3. Appendix
• Appendix A. Almost everything about Lebesgue integration

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Volume: 1572014; 356 pp
MSC: Primary 81; 46; 34; 47;

Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrödinger operators.

Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrödinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory.

This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics.

Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature.

This new edition has additions and improvements throughout the book to make the presentation more student friendly.

The book is written in a very clear and compact style. It is well suited for self-study and includes numerous exercises (many with hints).

Zentralblatt MATH

The author presents this material in a very clear and detailed way and supplements it by numerous exercises. This makes the book a nice introduction to this exciting field of mathematics.

Mathematical Reviews

Graduate students and research mathematicians interested in spectral theory and quantum mechanics, with an emphasis on Schrödinger operators.

• Part 0. Preliminaries
• Chapter 0. A first look at Banach and Hilbert spaces
• Part 1. Mathematical foundations of quantum mechanics
• Chapter 1. Hilbert spaces
• Chapter 2. Self-adjointness and spectrum
• Chapter 3. The spectral theorem
• Chapter 4. Applications of the spectral theorem
• Chapter 5. Quantum dynamics
• Chapter 6. Perturbation theory for self-adjoint operators
• Part 2. Schrödinger operators
• Chapter 7. The free Schrödinger operator
• Chapter 8. Algebraic methods
• Chapter 9. One-dimensional Schrödinger operators
• Chapter 10. One-particle Schrödinger operators
• Chapter 11. Atomic Schrödinger operators
• Chapter 12. Scattering theory
• Part 3. Appendix
• Appendix A. Almost everything about Lebesgue integration
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1

M39a9am3R 3176
5 years ago

The transparencies are working fine, it's just that whenever I click the button it will kill me (at least right now I know how to make a reset button). Can anyone find my problem?

local nameshowing=true
local plr=script.Parent.Parent.Parent.Parent
script.Parent.MouseButton1Click:connect(function()
local shouldhidename = true
if nameshowing == true then
shouldhidename = true
else shouldhidename=false
end
if shouldhidename==true then
if plr.Character ~=nil then
nameshowing = false
end
end
else
nameshowing=true
end
end)

Any help will be appreciated, thanks for your time.

### Locked by M39a9am3R, adark, SanityMan, and TheMyrco

This question has been locked to preserve its current state and prevent spam and unwanted comments and answers.

4

Why did you make a script inside another thing? You need to create a LocalScript in some storage service, then clone it in the playergui and execute the function in your mouse script, kinda like this:

_G.hideBillboard = function()
local player = game.Players.LocalPlayer
local weld = Instance.new("Weld")
end

_G.unhideBillboard = function()
-- Opposite of hideBillBoard()
end

After you can call those method outside everything:

function onMouseClick()
-- I suppose that you have a boolean called "hidden" that checks if the game needs to hide/unhide the billboard
if hidden then
_G.unhideBillboard()
else
_G.hideBillboard()
end

0
The script is inside the button inside a screengui, inside the startergui, which is inside the player so there are four parents. I think my problem here is the welds. But does it need to be inside a localscript in order for it to work? M39a9am3R 3176 — 5y
0
I edited the message alessandro112 171 — 5y
0
Just moved my code to a LocalScript, thank you. M39a9am3R 3176 — 5y
0
No problem ;) alessandro112 171 — 5y
-1

Would it have been possible to rename your humanoid to nothing so the name doesn't show up? Just wondering.

0
No, it is not possible. Since your character is connected to your player I guess that makes your character's name ROBLOX locked, since I've tried changing it in two ways in studio one being script and the other being through properties with no change. M39a9am3R 3176 — 5y
0
Ok thanks! Mind checking my most recent question? Champion121212 22 — 5y


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# IEEE Transactions on Image Processing

## Filter Results

Displaying Results 1 - 25 of 41

Publication Year: 2006, Page(s):c1 - c4
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• ### IEEE Transactions on Image Processing publication information

Publication Year: 2006, Page(s): c2
| PDF (36 KB)
• ### Error-Resilient Video Communications Over CDMA Networks With a Bandwidth Constraint

Publication Year: 2006, Page(s):3241 - 3252
Cited by:  Papers (14)
| | PDF (493 KB) | HTML

We present an adaptive video transmission scheme for use in a code-division multiple-access network, which incorporates efficient bandwidth allocation among source coding, channel coding, and spreading under a fixed total bandwidth constraint. We derive the statistics of the received signal, as well as a theoretical bound on the packet drop rate at the receiver. Based on these results, a bandwidth... View full abstract»

• ### On the Use of Context-Weighting in Lossless Bilevel Image Compression

Publication Year: 2006, Page(s):3253 - 3260
Cited by:  Papers (5)
| | PDF (2674 KB) | HTML

We present a context-weighting algorithm that adaptively weights in real-time three-context models based on their relative accuracy. It can automatically select the better model over different regions of an image, producing better probability estimates than using either one of these models exclusively. Combined with the previously proposed block arithmetic coder for image compression (BACIC), the ... View full abstract»

• ### Reversing Demosaicking and Compression in Color Filter Array Image Processing: Performance Analysis and Modeling

Publication Year: 2006, Page(s):3261 - 3278
Cited by:  Papers (16)  |  Patents (3)
| | PDF (3253 KB) | HTML

In the conventional processing chain of single-sensor digital still cameras (DSCs), the images are captured with color filter arrays (CFAs) and the CFA samples are demosaicked into a full color image before compression. To avoid additional data redundancy created by the demosaicking process, an alternative processing chain has been proposed to move the compression process before the demosaicking. ... View full abstract»

• ### Precompression Quality-Control Algorithm for JPEG 2000

Publication Year: 2006, Page(s):3279 - 3293
Cited by:  Papers (10)
| | PDF (987 KB) | HTML

In this paper, a precompression quality-control algorithm is proposed. It can greatly reduce computational power of the embedded block coding (EBC) and memory requirement to buffer bit streams. By using the propagation property and the randomness property of the EBC algorithm, rate and distortion of coding passes is approximately predicted. Thus, the truncation points are chosen before actual codi... View full abstract»

• ### Quadratic Weighted Median Filters for Edge Enhancement of Noisy Images

Publication Year: 2006, Page(s):3294 - 3310
Cited by:  Papers (17)
| | PDF (9373 KB) | HTML

Quadratic Volterra filters are effective in image sharpening applications. The linear combination of polynomial terms, however, yields poor performance in noisy environments. Weighted median (WM) filters, in contrast, are well known for their outlier suppression and detail preservation properties. The WM sample selection methodology is naturally extended to the quadratic sample case, yielding a fi... View full abstract»

• ### The Multidimensional Phase Unwrapping Integral and Applications to Microwave Tomographical Image Reconstruction

Publication Year: 2006, Page(s):3311 - 3324
Cited by:  Papers (15)  |  Patents (2)
| | PDF (2286 KB) | HTML

Spatial unwrapping of the phase component of time varying electromagnetic fields has important implications in a range of disciplines including synthetic aperture radar (SAR) interferometry, MRI, optical confocal microscopy, and microwave tomography. This paper presents a fundamental framework based on the phase unwrapping integral, especially in the complex case where phase singularities are encl... View full abstract»

• ### An Improved Observation Model for Super-Resolution Under Affine Motion

Publication Year: 2006, Page(s):3325 - 3337
Cited by:  Papers (25)
| | PDF (5115 KB) | HTML

Super-resolution (SR) techniques make use of subpixel shifts between frames in an image sequence to yield higher resolution images. We propose an original observation model devoted to the case of nonisometric inter-frame motion as required, for instance, in the context of airborne imaging sensors. First, we describe how the main observation models used in the SR literature deal with motion, and we... View full abstract»

• ### Fast IIR Isotropic 2-D Complex Gabor Filters With Boundary Initialization

Publication Year: 2006, Page(s):3338 - 3348
Cited by:  Papers (8)
| | PDF (2156 KB) | HTML

Gabor filters are widely applied in image analysis and computer vision applications. This paper describes a fast algorithm for isotropic complex Gabor filtering that outperforms existing implementations. The main computational improvement arises from the decomposition of Gabor filtering into more efficient Gaussian filtering and sinusoidal modulations. Appropriate filter initial conditions are der... View full abstract»

• ### Face Verification Across Age Progression

Publication Year: 2006, Page(s):3349 - 3361
Cited by:  Papers (102)
| | PDF (2453 KB) | HTML

Human faces undergo considerable amounts of variations with aging. While face recognition systems have been proven to be sensitive to factors such as illumination and pose, their sensitivity to facial aging effects is yet to be studied. How does age progression affect the similarity between a pair of face images of an individual? What is the confidence associated with establishing the identity bet... View full abstract»

• ### Translation-Invariant Contourlet Transform and Its Application to Image Denoising

Publication Year: 2006, Page(s):3362 - 3374
Cited by:  Papers (87)
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Most subsampled filter banks lack the feature of translation invariance, which is an important characteristic in denoising applications. In this paper, we study and develop new methods to convert a general multichannel, multidimensional filter bank to a corresponding translation-invariant (TI) framework. In particular, we propose a generalized algorithme agrave trous, which is an extension of the ... View full abstract»

• ### Fast Splitting$alpha$-Rooting Method of Image Enhancement: Tensor Representation

Publication Year: 2006, Page(s):3375 - 3384
Cited by:  Papers (18)
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In the tensor representation, a two-dimensional (2-D) image is represented uniquely by a set of one-dimensional (1-D) signals, so-called splitting-signals, that carry the spectral information of the image at frequency-points of specific sets that cover the whole domain of frequencies. The image enhancement is thus reduced to processing splitting-signals and such process requires a modification of ... View full abstract»

• ### Multiresolution MAP Despeckling of SAR Images Based on Locally Adaptive Generalized Gaussian pdf Modeling

Publication Year: 2006, Page(s):3385 - 3399
Cited by:  Papers (63)
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In this paper, a new despeckling method based on undecimated wavelet decomposition and maximum a posteriori (MAP) estimation is proposed. Such a method relies on the assumption that the probability density function (pdf) of each wavelet coefficient is generalized Gaussian (GG). The major novelty of the proposed approach is that the parameters of the GG pdf are taken to be space-varying within each... View full abstract»

• ### Ground Target Recognition Using Rectangle Estimation

Publication Year: 2006, Page(s):3400 - 3408
Cited by:  Papers (14)
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We propose a ground target recognition method based on 3-D laser radar data. The method handles general 3-D scattered data. It is based on the fact that man-made objects of complex shape can be decomposed to a set of rectangles. The ground target recognition method consists of four steps; 3-D size and orientation estimation, target segmentation into parts of approximately rectangular shape, identi... View full abstract»

• ### The Bayesian Operating Point of the Canny Edge Detector

Publication Year: 2006, Page(s):3409 - 3416
Cited by:  Papers (16)
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We have investigated the operating point of the Canny edge detector which minimizes the Bayes risk of misclassification. By considering each of the sequential stages which constitute the Canny algorithm, we conclude that the linear filtering stage of Canny, without postprocessing, performs very poorly by any standard in pattern recognition and achieves error rates which are almost indistinguishabl... View full abstract»

• ### Recognition of Dynamic Video Contents With Global Probabilistic Models of Visual Motion

Publication Year: 2006, Page(s):3417 - 3430
Cited by:  Papers (21)
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The exploitation of video data requires methods able to extract high-level information from the images. Video summarization, video retrieval, or video surveillance are examples of applications. In this paper, we tackle the challenging problem of recognizing dynamic video contents from low-level motion features. We adopt a statistical approach involving modeling, (supervised) learning, and classifi... View full abstract»

• ### Unsupervised Variational Image Segmentation/Classification Using a Weibull Observation Model

Publication Year: 2006, Page(s):3431 - 3439
Cited by:  Papers (33)
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Studies have shown that the Weibull distribution can model accurately a wide variety of images. Its parameters index a family of distributions which includes the exponential and approximations of the Gaussian and the Raleigh models widely used in image segmentation. This study investigates the Weibull distribution in unsupervised image segmentation and classification by a variational method. The d... View full abstract»

• ### A Statistical Evaluation of Recent Full Reference Image Quality Assessment Algorithms

Publication Year: 2006, Page(s):3440 - 3451
Cited by:  Papers (828)  |  Patents (10)
| | PDF (1234 KB) | HTML

Measurement of visual quality is of fundamental importance for numerous image and video processing applications, where the goal of quality assessment (QA) algorithms is to automatically assess the quality of images or videos in agreement with human quality judgments. Over the years, many researchers have taken different approaches to the problem and have contributed significant research in this ar... View full abstract»

• ### Perceptual Image Hashing Via Feature Points: Performance Evaluation and Tradeoffs

Publication Year: 2006, Page(s):3452 - 3465
Cited by:  Papers (125)  |  Patents (7)
| | PDF (4361 KB) | HTML

We propose an image hashing paradigm using visually significant feature points. The feature points should be largely invariant under perceptually insignificant distortions. To satisfy this, we propose an iterative feature detector to extract significant geometry preserving feature points. We apply probabilistic quantization on the derived features to introduce randomness, which, in turn, reduces v... View full abstract»

• ### Structure From Planar Motion

Publication Year: 2006, Page(s):3466 - 3477
Cited by:  Papers (6)
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Planar motion is arguably the most dominant type of motion in surveillance videos. The constraints on motion lead to a simplified factorization method for structure from planar motion when using a stationary perspective camera. Compared with methods for general motion , our approach has two major advantages: a measurement matrix that fully exploits the motion constraints is formed such that the ne... View full abstract»

• ### On the Orientability of Shapes

Publication Year: 2006, Page(s):3478 - 3487
Cited by:  Papers (35)
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The orientation of a shape is a useful quantity, and has been shown to affect performance of object recognition in the human visual system. Shape orientation has also been used in computer vision to provide a properly oriented frame of reference, which can aid recognition. However, for certain shapes, the standard moment-based method of orientation estimation fails. We introduce as a new shape fea... View full abstract»

• ### A Wavelet-Based Two-Stage Near-Lossless Coder

Publication Year: 2006, Page(s):3488 - 3500
Cited by:  Papers (23)
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In this paper, we present a two-stage near-lossless compression scheme. It belongs to the class of "lossy plus residual coding" and consists of a wavelet-based lossy layer followed by arithmetic coding of the quantized residual to guarantee a given Linfin error bound in the pixel domain. We focus on the selection of the optimum bit rate for the lossy layer to achieve the minimum total b... View full abstract»

• ### An Optimal Nonorthogonal Separation of the Anisotropic Gaussian Convolution Filter

Publication Year: 2006, Page(s):3501 - 3513
Cited by:  Papers (18)
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We give an analytical and geometrical treatment of what it means to separate a Gaussian kernel along arbitrary axes in Ropfn, and we present a separation scheme that allows us to efficiently implement anisotropic Gaussian convolution filters for data of arbitrary dimensionality. Based on our previous analysis we show that this scheme is optimal with regard to the number of memory access... View full abstract»

• ### Embedding Motion in Model-Based Stochastic Tracking

Publication Year: 2006, Page(s):3514 - 3530
Cited by:  Papers (27)  |  Patents (1)
| | PDF (2094 KB) | HTML

Particle filtering is now established as one of the most popular methods for visual tracking. Within this framework, there are two important considerations. The first one refers to the generic assumption that the observations are temporally independent given the sequence of object states. The second consideration, often made in the literature, uses the transition prior as the proposal distribution... View full abstract»

## Aims & Scope

IEEE Transactions on Image Processing focuses on signal-processing aspects of image processing, imaging systems, and image scanning, display, and printing.

Full Aims & Scope

## Meet Our Editors

Editor-in-Chief

Gaurav Sharma
University of Rochester
Rochester, NY, USA
E-mail: gaurav.sharma@rochester.edu
Phone: +1 585-275-7313


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# Insufficiency of Newton's third law to solve multiple body problems

In The Variational Principles of Mechanics Lanczos describes what he calls 'vectorial mechanics': the process of solving mechanical problems by recourse to the immediate consequences of Newton's laws, exemplified by decomposing the forces that act on particles and applying Newton's differential equation $$(F=\frac{dp}{dt})$$ to derive solutions. Lanczos contrasts such a process with 'variational mechanics', where one focuses on the analysis and synthesis of work functions and kinetic energies rather than forces and momenta. When describing the vectorial mechanics approach to problems involving multiple bodies, Lanczos remarks

This force-analysis [the standard approach of vectorial mechanics] sometimes becomes cumbersome. The unknown nature of the interaction forces makes it necessary to introduce additional postulates, and Newton thought that the principle "action equals reaction", stated as his third law of motion, would take care of all dynamical problems. This, however, is not the case, and even for the dynamics of a rigid body the additional hypothesis that the inner forces of the body are of the nature of central forces had to be made. In more complicated situations the Newtonian approach fails to give a unique answer to the problem. (p. 3-4)

Here are my related questions:

-What are examples of the failure of Newton's third law to solve problems involving multiple bodies and requiring the additional assumption of central forces?

-What are examples of 'more complicated situations' where Newton's third law fails to give a unique answer to multiple body problems?

-Assuming a variational mechanics approach is capable of solving a multiple body problem, is the motion that the variational mechanics approach predicts always unique? (Not asking whether the Lagrangian is always unique, but is the prediction for the motion of an object [the position of the object at each instant of time] always unique [of course allowing for the typical symmetries imposed on classical physics equations, e.g. position and time shifts]).

Note: I am aware that very similar questions were asked in this post in 2013, but the two answers there don't seem to be sufficiently explained or addressing to the questions.


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Home > Mean Square > Normalized Mean Square Error Wiki

# Normalized Mean Square Error Wiki

## Contents

The result for S n − 1 2 {\displaystyle S_{n-1}^{2}} follows easily from the χ n − 1 2 {\displaystyle \chi _{n-1}^{2}} variance that is 2 n − 2 {\displaystyle 2n-2} more stack exchange communities company blog Stack Exchange Inbox Reputation and Badges sign up log in tour help Tour Start here for a quick overview of the site Help Center Detailed It is an average.sqrt(sum(Dates-Scores).^2)./Dates Thus, you have written what could be described as a "normalized sum of the squared errors", but it is NOT an RMSE. That case could be due to time and/or space shifting. http://dlldesigner.com/mean-square/normalized-root-mean-square-error-wiki.php

## Mean Square Error Formula

An internet search however only shows strange definitions like $$\frac{ \sum_i (x_i-y_i)^2}{N\sum_i (x_i)^2} \quad\text{or} \quad \frac{N \sum_i (x_i-y_i)^2}{\sum_i x_i \sum_i y_i}$$ Is my interpretation not the standard definition? Compared to the similar Mean Absolute Error, RMSE amplifies and severely punishes large errors. $$\textrm{RMSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}$$ **MATLAB code:** RMSE = sqrt(mean((y-y_pred).^2)); **R code:** RMSE p.229. ^ DeGroot, Morris H. (1980).

• Some experts have argued that RMSD is less reliable than Relative Absolute Error.[4] In experimental psychology, the RMSD is used to assess how well mathematical or computational models of behavior explain
• Mean squared error is the negative of the expected value of one specific utility function, the quadratic utility function, which may not be the appropriate utility function to use under a
• For this reason, the NMSE generally shows the most striking differences among models.

Your version of NMSE I'd interpret as "normalized square error" ? –Evan Sep 10 '13 at 2:01 @Evan, The 1/N in the numerator and denominator cancel each other. –Mark For this reason, the NMSE generally shows the most striking differences among models. Play games and win prizes! Mean Square Error Definition Variance Further information: Sample variance The usual estimator for the variance is the corrected sample variance: S n − 1 2 = 1 n − 1 ∑ i = 1 n

See also James–Stein estimator Hodges' estimator Mean percentage error Mean square weighted deviation Mean squared displacement Mean squared prediction error Minimum mean squared error estimator Mean square quantization error Mean square Root Mean Square Error Formula On the other hand, high NMSE values do not necessarily mean that a model is completely wrong. The mean absolute error used the same scale as the data being measured. Also in regression analysis, "mean squared error", often referred to as mean squared prediction error or "out-of-sample mean squared error", can refer to the mean value of the squared deviations of

Probability and Statistics (2nd ed.). Mean Square Error Calculator Join the conversation ERROR The requested URL could not be retrieved The following error was encountered while trying to retrieve the URL: http://0.0.0.9/ Connection to 0.0.0.9 failed. Image Analyst (view profile) 0 questions 20,721 answers 6,534 accepted answers Reputation: 34,810 Vote0 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/4064#answer_205645 Answer by Image Analyst Image Analyst (view profile) 0 questions Measuring air density - where is my huge error coming from?

## Root Mean Square Error Formula

Please help improve this article by adding citations to reliable sources. How can I then find microcontrollers that fit? Mean Square Error Formula In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the Root Mean Square Error Interpretation Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.

It is defined as: Contrary to the bias, in the NMSE the deviations (absolute values) are summed instead of the differences. http://dlldesigner.com/mean-square/normalized-mean-square-error.php Moreover, it must be pointed out that differences on peaks have a higher weight on NMSE than differences on other values. Criticism The use of mean squared error without question has been criticized by the decision theorist James Berger. International Journal of Forecasting. 8 (1): 69–80. Root Mean Square Error Example

Related Content 3 Answers John D'Errico (view profile) 4 questions 1,877 answers 683 accepted answers Reputation: 4,318 Vote5 Link Direct link to this answer: https://www.mathworks.com/matlabcentral/answers/4064#answer_12671 Answer by John D'Errico John D'Errico The same data filtering for FAa calculation is applied for NMSE calculation. Apply Today MATLAB Academy New to MATLAB? weblink These all summarize performance in ways that disregard the direction of over- or under- prediction; a measure that does place emphasis on this is the mean signed difference.

But how r dates and scores related? 1 Comment Show all comments Enne Hekma Enne Hekma (view profile) 0 questions 0 answers 0 accepted answers Reputation: 0 on 9 Jan 2016 Mean Absolute Error The bootstrap technique has to be used. square error is like (y(i) - x(i))^2.

## Not the answer you're looking for?

When normalising by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity.[3] This is analogous to the coefficient of For example, when measuring the average difference between two time series x 1 , t {\displaystyle x_{1,t}} and x 2 , t {\displaystyle x_{2,t}} , the formula becomes RMSD = ∑ This also is a known, computed quantity, and it varies by sample and by out-of-sample test space. Root Mean Square Error Excel Note that, although the MSE (as defined in the present article) is not an unbiased estimator of the error variance, it is consistent, given the consistency of the predictor.

so that ( n − 1 ) S n − 1 2 σ 2 ∼ χ n − 1 2 {\displaystyle {\frac {(n-1)S_{n-1}^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2}} . This is known as a scale-dependent accuracy measure and therefore cannot be used to make comparisons between series using different scales.[1] The mean absolute error is a common measure of forecast Further, while the corrected sample variance is the best unbiased estimator (minimum mean square error among unbiased estimators) of variance for Gaussian distributions, if the distribution is not Gaussian then even check over here Mathematical Statistics with Applications (7 ed.).

ISBN0-387-96098-8. Suppose the sample units were chosen with replacement. NMSE

The NMSE (Normalised Mean Square Error) is an estimator of the overall deviations between predicted and measured values. This property, undesirable in many applications, has led researchers to use alternatives such as the mean absolute error, or those based on the median.

In simulation of energy consumption of buildings, the RMSE and CV(RMSE) are used to calibrate models to measured building performance.[7] In X-ray crystallography, RMSD (and RMSZ) is used to measure the What to do with my pre-teen daughter who has been out of control since a severe accident? This article needs additional citations for verification. The difference is that a mean divides by the number of elements.

I have always assumed that $$MSE(x,y)=\frac 1N \sum_i (x_i-y_i)^2$$ and $$NMSE(x,y)=MSE(x,y)/MSE(x,0) = \frac{\| x-y\|_2^2}{\| x\|_2^2}$$ where $y$ is the approximation to $x$. Related Content Join the 15-year community celebration. Reload the page to see its updated state. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply.

Unbiased estimators may not produce estimates with the smallest total variation (as measured by MSE): the MSE of S n − 1 2 {\displaystyle S_{n-1}^{2}} is larger than that of S The minimum excess kurtosis is γ 2 = − 2 {\displaystyle \gamma _{2}=-2} ,[a] which is achieved by a Bernoulli distribution with p=1/2 (a coin flip), and the MSE is minimized This gives a simple relation between NMSE and relative $\ell^2$ error. C V ( R M S D ) = R M S D y ¯ {\displaystyle \mathrm {CV(RMSD)} ={\frac {\mathrm {RMSD} }{\bar {y}}}} Applications In meteorology, to see how effectively a

Wiki (Beta) » Root Mean Squared Error # Root Mean Squared Error (RMSE) The square root of the mean/average of the square of all of the error. Let say x is a 1xN input and y is a 1xN output. The denominator is the sample size reduced by the number of model parameters estimated from the same data, (n-p) for p regressors or (n-p-1) if an intercept is used.[3] For more If we define S a 2 = n − 1 a S n − 1 2 = 1 a ∑ i = 1 n ( X i − X ¯ )

Hard to compute real numbers What is this strange almost symmetrical location in Nevada?


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## California Governor signs agreement with China national government to cooperate on green tech

##### 06 June 2017

In China, California Governor Edmund G. Brown Jr. signed an agreement with China’s Minister of Science and Technology to deepen cooperation on the development of green technology. Governor Brown also met with China President Xi Jinping for approximately 45 minutes at the Great Hall of the People in Beijing. The Governor and President discussed the importance of expanding cooperation on green technology, innovation and trade.

The agreement on behalf of California builds on two sub-national pacts on green technology that Brown signed earlier this week with Sichuan and Jiangsu provinces.

Specifically, the Memorandum of Understanding (MoU)—explicitly not a contract or a treaty—expands cooperation on the advancement of low-carbon, renewable energy and energy efficiency technologies such as zero-emission vehicles, energy storage, grid modernization and low-carbon urban development. Under the agreement, California and China will also deepen their partnership and coordination on greenhouse gas emission and air pollution reduction programs, including emissions trading systems and carbon capture.

Governor Brown also spoke at a Clean Energy Ministerial forum focused on electric vehicles; met with Hebei Province’s Governor, Xu Qin; met with China’s Minister of Commerce, Zhong Shan; and spoke at a Clean Energy Ministerial event organized by the Energy Foundation on the economic transition to clean energy.

Governor Brown is cohosting the Under2 Clean Energy Forum with China’s Ministry of Science and Technology and Sichuan Province. The forum builds on the event Governor Brown launched last year in San Francisco during the Clean Energy Ministerial.

The Under2 Coalition—an international pact among cities, states and countries committed to limiting the increase in global average temperature to below 2 degrees Celsius—now includes 170 jurisdictions on six continents. These members collectively represent more than 1.18 billion people and $27.5 trillion GDP—equivalent to 16% of the global population and 37% of the global economy. Among the coalition members are 10 US states and eight US cities that together account for nearly one-third of US population and GDP. ### Comments The Trump White House is completely disconnected from the people and forces the states to do the bidding of the American people. Jerry Brown is doing the work Trump should be doing for the people; but, his only concern is the welfare of the rich people in the U.S., not the declining middle-class. For two very different reasons, those two are very recently on the same wavelength: a) China is choking. b) California is drying up. Could China supply fresh water for fresh air? Apropos fresh water; the oceans have been trumped. There's supposedly more garbage floating in the oceans than fish swimming in them. I.o.w. what is the difference between garbage and Trump? California's water storage is back to normal and there are$30 billion in funds for new water programs. California learns its lessons.

The comments to this entry are closed.


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# roots of minimal and characteristic polynomial

Why is it, that for the matrix

$A \in Mat(n\times n, \mathbb{C})$

the characteristic polynomial $\chi_A(t)$ and the minimal polynomial $\mu_A(t)$ have same roots?

Since $\chi_A(t) = \mu_A(t) \cdot p(t)$ it should be easy to follow, that $\chi_A(t)$ has roots where $\mu_A(t)$ has roots.

But why can't $\chi_A(t)$ have roots where $\mu_A(t)$ hasn't?

-
Actually, in general, the characteristic polynomial and the minimal polynomial have the same irreducible factors. – lhf Jan 22 '12 at 12:52

First, show the following easy claim: Let $f(x)\in\mathbb{C}[x]$ be any polynomial. If $\lambda$ is an eigenvalue of $A$ associated with the eigenvector $v$, then $f(\lambda)$ is an eigenvalue of $f(A)$ associated with the eigenvector $v$. (prove by calculating $f(A)v$, using distributivity of multiplication over addition).
Now let $\lambda$ be a root of $\chi_A(t)$. Thus $\lambda$ is an eigenvalue and so has an associated eigenvector $v\neq 0$. Using the claim, we get $0=\mu_A(A)\cdot v=\mu_A(\lambda)\cdot v$ (since $\mu_A(A)=0$). Since $v\neq 0$, we get $\mu_A(\lambda)=0$.


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# Continuous shear stress alters metabolism, mass-transport, and growth in electroactive biofilms independent of surface substrate transport

## Abstract

Electroactive bacteria such as Geobacter sulfurreducens and Shewanella onedensis produce electrical current during their respiration; this has been exploited in bioelectrochemical systems. These bacteria form thicker biofilms and stay more active than soluble-respiring bacteria biofilms because their electron acceptor is always accessible. In bioelectrochemical systems such as microbial fuel cells, corrosion-resistant metals uptake current from the bacteria, producing power. While beneficial for engineering applications, collecting current using corrosion resistant metals induces pH stress in the biofilm, unlike the naturally occurring process where a reduced metal combines with protons released during respiration. To reduce pH stress, some bioelectrochemical systems use forced convection to enhance mass transport of both nutrients and byproducts; however, biofilms’ small pore size limits convective transport, thus, reducing pH stress in these systems remains a challenge. Understanding how convection is necessary but not sufficient for maintaining biofilm health requires decoupling mass transport from momentum transport (i.e. fluidic shear stress). In this study we use a rotating disc electrode to emulate a practical bioelectrochemical system, while decoupling mass transport from shear stress. This is the first study to isolate the metabolic and structural changes in electroactive biofilms due to shear stress. We find that increased shear stress reduces biofilm development time while increasing its metabolic rate. Furthermore, we find biofilm health is negatively affected by higher metabolic rates over long-term growth due to the biofilm’s memory of the fluid flow conditions during the initial biofilm development phases. These results not only provide guidelines for improving performance of bioelectrochemical systems, but also reveal features of biofilm behavior. Results of this study suggest that optimized reactors may initiate operation at high shear to decrease development time before decreasing shear for steady-state operation. Furthermore, this biofilm memory discovered will help explain the presence of channels within biofilms observed in other studies.

## Introduction

Bacteria exist in biofilms more than in planktonic states to (1) protect bacteria from predation and chemical attack and (2) to allow bacteria to manipulate their environment1. Electrochemically active biofilms2,3,4 not only have potential to bridge the energy-water nexus, producing power from wastewater treatment5,6, but also to be used as a tool to study biofilm development because they produce a measurable current as part of their metabolism2. Though bioelectrochemical systems employ flow to improve nutrient delivery and metabolic waste removal6,7,8,9, which is known to influence biofilm behavior, the independent influence of shear on electrochemically active biofilms remains largely unexplored6,7,8,9,10,11. Short-term exposures of electroactive biofilms to shear showed expected improvements in salt transport12 and unexpected resilience to high shear (635 s−1)13 despite forming thicker biofilms than aerobic bacteria5. Long-term exposures to shear showed that high shear (200 s−1) selects for high-current producing bacteria compared to low shear (80 s−1)14. That fluid shear affects bacteria biofilms fixed to a surface and that biofilms have channels15 is not well understood because pore sizes limit convection16,17,18,19 and thus the influence of shear. Electroactive bacteria biofilms remain more metabolically active20,21 than soluble respiring bacteria and might not be limited by respiration in high shear unlike aerobic biofilms22. Using the analytic equations for fluid flow and mass flux that exist for a rotating disk electrode, the absolute concentration of nutrients is reduced in a way that decouples the influence of mass flux and shear stress. We use a pure culture of a high-current producing bacteria to reduce the effect shear has on selection of species with high adhesion strength, exopolymeric substance production, or current production. However, there will still be phenotypic selection within the strain induced by time and possibly shear23. In doing so, we find high metabolic rate increases with increased shear that may improve the performance of bioelectrochemical systems during startup. We find two dimensionless groups that account for initial, exponential and maximum growth though not decay. We demonstrate that while these biofilms are affected by their environment, their porosity does not change at steady-state so they are still susceptible to decay from pH stress5. This paper has been published as a BioRxiv pre-print24, is extended from A. Jones’ dissertation25, and has been presented in part at previous meetings26.

We used a rotating disk electrode that has analytic equations for flow and mass flux that can be varied independently, and through scaling laws, have been used to simulate pipe flow, conduit flow and spray jets27. The property of separation, specifically, makes rotating disk electrodes useful for studying electrochemical reactions; however, when it has been used to study electroactive biofilms in the past, this particular property was not used28. We benchmarked our initial flux and shear to that of a practical bioelectrochemical system6, and varied the shear by two orders of magnitude. We used a pure culture of Geobacter sulfurreducens, as it has been shown to produce the highest current5 in a bioelectrochemical system and is the most abundant species in mixed-culture bioelectrochemical systems when selected through time, shear, or voltage methods.

## Results and Discussion

We find that maximum current and rates of current increase follow increasing shear stress for continuous growth of G. sulfurreducens PCA with fixed mass flux to the surface of a rotating disk, Fig. 1a. The rise in current follows a logistic growth like pattern from slow initial rise, exponential increase to maximum followed by a decay. Increasing shear stress from 0.01 Pa to 1 Pa reduced the initial doubling time of the current from 0.26 days to 0.14 days, as shown in Table 1 and Supplementary Figure S1. The near immediate current response is found in the literature29, but here we also found a dependence on shear stress. We propose that the rapid development of measurable current is also independent of the flux of bacteria, which was not held constant. If the variable flux of bacteria caused the rapid current development, it would be reflected in the initial average open circuit potential (OCP) which is used to describe active bacteria colonization of an electrode30. However, the OCP shows a decay rate (Table 1, Fig. 1b) that is inconsistent with the trend in current rise. We assume that all the bacteria had the same initial electrochemical activity, as they were all taken from mid-log planktonic growth phase21 and seeded at 600 RPM for 10 minutes prior to the start of each experiment. However, the variability shown in Fig. 1b demonstrates that further study of the influence of bacteria on electrode potential is needed30.

We used dimensional analysis to estimate that the time to maximum current is decreased for increased shear rates from 9.8 days to 3.8 days, as shown in Fig. 1c and Table 1. The electrochemical signal of the biofilm displayed similar characteristics to optical measurements of planktonic cell growth31 and optical measurements of biofilm growth32. Assuming that current is a proxy for metabolic rate2, the initial lag phase, where little current is found, was seen in the 0.01 Pa case (Fig. 1a). A rapid growth phase can be seen in all three shear rates tested, a stable growth phase can be seen in the 0.1 Pa case, and a decay phase can be seen in the 1.0 Pa case. Thus, we assumed that the dimensionless current should scale with dimensionless time according to a logistic growth model.

Our results indicate a shear dependence on maximum current not described in the literature28 when both mass and shear were coupled (Fig. 2a.) for the same species. It is accepted that varying concentration (mass flux) changes metabolic (current) output for all bacteria, and varying electrode potential33 and electric fields34 can improve power output and select for electroactive biofilms. From this hypothesis, previous studies have grown the biofilm under one shear stress condition and varied shear to see the impact on current28. A common technique for determining the influence of shear, substrate convection, and substrate diffusion is the use of the product of the Schmidt and Reynolds number, where the Schmidt number is a ratio of viscous diffusion to mass diffusion, Sc = v/D, to scale mass transport and the Reynolds number is a ratio of the inertial to viscous forces, Re = uD/v, to scale shearing forces. Combining this product and the scaling of current from Fig. 1c, we find different metabolic efficiencies and transport regimes for biofilms subjected to shear from different starting conditions determined by a parameter α, as $$i/{i}_{\mathrm{lim}}\sim {{\rm{Sc}}}^{\alpha }{\rm{Re}}$$. The maximum current of biofilms initially grown under shearing or static conditions, 136 µA and 101 µA respectively, becomes the same when scaled and occurs at the same scaled transport condition, Fig. 2b. Furthermore, these points coincide whether using the diffusivity of acetate (substrate) or protons (toxic waste byproduct), Fig. 2b. However, biofilms grown in static conditions show a decrease in dimensionless current with increasing transport. Our dimensionless current is equivalent to the Coulombic efficiency, so we hypothesize that the biofilm is adapted to maximum utilization at its initial growth condition (shear stress of 0 Pa) and any increase does not improve substrate utilization. This adaptation is likely the higher porosity found for biofilms not grown under shear35. For both, protons or acetate, biofilms in this work are strictly dependent on mass transport, α = 1. However, biofilms initially grown in static conditions are dependent on both diffusivity and viscosity for acetate transport α = −1 and weakly dependent on viscosity α = 1.1 for proton transport. Since no hysteresis in current output was found for the biofilms grown in static conditions and then sheared indicating no loss of biomass28, this dependence on viscosity should be further investigated.

High metabolic rates described appear to come at the cost of sustained growth and stable metabolic activity. The highest shear stress, 1.0 Pa, produced the highest current density (see Table 1) yet immediately declined (Fig. 1a). This decline in current was not present for the lower shear stress cases. While it has been shown that higher shear induces higher metabolic rates in non-electroactive biofilms36, previous experiments were run for half the duration of the present study. As a result, they did not show the trade-offs found here-in for the highest shear36. We caution that our dimensional model, Fig. 1c, does not describe the decay shown in the high shear stress case, though it does help determine when it may be necessary to run experiments for longer durations1.

Measured electron diffusion current followed similar trends as the electrical current with shear (Supplementary Fig. S6.). Since electron diffusion current is dependent on electron transport protein concentration, bacteria concentration, and reaction rate37, and high current production5 and/or high metabolic activity32 leads to decreased pH in the biofilm with a concomitant loss of electron transport protein activity5, we hypothesize that the decline is due to either the death of bacteria or the loss of electron transport protein activity. Nyquist plots, Supplementary Fig. S3, of the electrochemical impedance spectroscopy, a measure of the complex resistance to charge transfer, similarly showed an increase in resistance upon a decrease in current (both electron diffusion and absolute), which could either imply cell death or loss of electron transport protein activity. To interpret this further impedance spectra measured at 340 mV were fit using a simple equivalent circuit model38. The charge transfer resistance shows a nonlinear decay for the lower shear cases and a nonlinear increase for the highest shear. The pseudocapacitance shows an oscillation about a mean time. These two features do not explain the overall trends in current or the full impedance. However, using an apparent diffusivity calculation for the substrate acetate following Tribollet et al.39, shows steady exponential decay for the highest shear, and a crossover for the lowest contrast with the electron diffusivity (Supplementary Fig. S6). Taken together, there are likely internal regulations of substrate and electron transport similar to those found in static conditions on pH and turnover conditions that must be further investigated with continuous structural measurements to produce the external results shown here.

While biofilm health is negatively affected by higher shear stress over long-term growth, the biofilm retains memory of initial fluid flow conditions and does not change structure to counteract such stress, Table 2. The independence of thickness and shear for the two higher shear stress cases agrees with previous studies on membrane aerated biofilms10. Since the lowest shear stress case has not reached maximal current, we conclude that biofilm thickness, surface roughness, and porosity are independent of shear for the conditions tested once the biofilm is fully developed (Fig. 1d).

The fluid streamline pattern (Fig. 1f) is imprinted on the biofilm interior at a height of 6 µm, but not at the surface of the biofilm after seven days of growth. The stability of this interior structure is quantitatively seen in stable open circuit potential30 and redox potential37, indicating a stable surface attachment of electrically respiring bacteria/proteins, Supplementary Figures S4S5. The mechanisms of this imprinting were not studied, though it may be due to the transport of quorum sensing chemicals during biofilm formation. The memory of a complex substructure from initial growth conditions of a biofilm has been seen before15, yet this is the first time to the authors’ knowledge that it has been shown under controlled initial conditions.

This analysis does not account for additional coupling of growth factors. Imposing shear stress and electric field gradients34 on mixed species biofilm in long-durations has been shown to select for electroactive bacteria. When decoupled from the impact of mass transport, long-time exposure to shear may show genetic adaptation in structure and current generation40. Further tradeoffs between EPS production and current generation may be affected by shear22. We propose that the linear relationship of dimensionless current to dimensionless biofilm growth rate, Fig. 1c, would be improved with data showing transport through the biofilm. This could be in the form of pH monitoring, acetate utilization, or continuous structural monitoring as described. In comparing with the results Babauta and Beyenal28, electrode potential was not fixed as the present study used the recommendation for maximum current production from Soussan et al.41. The coincident dimensionless current warrants further study as potential has a strong effect on current production34,41.

## Conclusion

This study shows that electroactive biofilms increase metabolic activity in response to increasing shear stress independent of mass transport. This occurs both at early times and at long-times. Using a dimensionless model, we show initial growth conditions can shift the system from diffusion dependent to convection dependent mass transport by comparing with results from the literature. However, this increased metabolism comes at the cost of decreased viability, likely due to an inability to transport metabolic byproducts. This result is consistent with the existing literature suggesting that shear stress does not induce changes in biofilm structure in both single and multi-species biofilms10,42. Continuous optical measurements and metabolic substrate monitoring may further improve understanding of the coupled phenomena and refine the fit between dimensionless growth and substrate utilization. These results are promising for optimizing bioelectrochemical system startup times as dynamic control of shear stress should be considered an independent tool for biofilms to maintain reactor stability similar to variation of potential33, batch feeding43, and removal of “old” biomass44.

## Methods

### Practical Benchmark

We benchmarked this study to an upflow microbial fuel cell by He et al.6. An upflow microbial fuel cell is designed in a similar manner to an upflow anaerobic sludge blanket bioreactor where organic laden fluid is flown upwards over a biofilm-coated porous media, having a smaller footprint than continuous stirred tank reactors and subsequently higher shear45. Matching two fluid flow systems, using only the dimensionless Reynolds number that is a ratio of momentum fluxes, produces vastly different results between the two systems46. Instead, both the mass flux and the momentum flux must be matched. The dimensionless shear stress for the upflow microbial fuel cell is found using Supplementary Eq. 5, $$\mathop{\tau }\limits^{ \sim }=0.056$$. This value is matched to the shear of rotating disc setups which for our system results in a rotation rate of 739 rpm, a shear stress of 1.048 Pa and shear rate of 1046 s−1. Using the data from He et al.6, and Supplementary Eq. 6 the flux can be found $$N=5.35\text{e-6}\,{\rm{g}}\,{\rm{C}}{\rm{O}}{\rm{D}}\,{{\rm{c}}{\rm{m}}}^{-2}{{\rm{s}}}^{-1}.$$ We varied the dimensional shear stress over two orders of magnitude keeping the mass flux fixed resulting in the concentrations, rotation rates, and dimensionless shear stress in Table 3.

The current will be normalized to the mass transfer limiting current for a rotating disk,

$$\begin{array}{c}\frac{i}{{i}_{lim}}=\frac{i}{nFAN}=i{({n}_{e}FAG^{\prime} (0){\nu }^{-\frac{1}{6}}{\omega }^{\frac{1}{2}}{c}_{0}{D}_{0}^{\frac{2}{3}})}^{-1}\end{array}$$
(1)

To correlate this with the flux through the biofilm, the biofilm growth time is considered as

$$\mathop{t}\limits^{ \sim }=tY{q}_{m},$$
(2)

where Y is the biomass yield of the substrate and qm is the maximum substrate utilization rate47. A plot of $$\mathop{i}\limits^{ \sim }$$ versus $$\mathop{\tau }\limits^{ \sim }$$ at constant mass flux results in curve that is similar to that of logistic growth48 with parameters that may relate to physical mechanisms.

### Media

The media was 0.59 g potassium dihydrogenphosphate, 0.38 g potassium chloride, 2.19 g sodium hydrogen carbonate, 0.36 g sodium chloride, 0.20 g ammonium chloride, 0.04 g calcium chloride dihydrate, and 0.10 magnesium chloride hexahydrate (MilliporeSigma, Darmstadt, GER) in 1 L of deionized water based on the media used in the literature28,49. The resulting conductivity was 2.10 mS.cm−1 at 30 °C. To this media was added 14.97 mM, 32.24 mM, or 69.47 mM of sodium acetate. This resulted in conductivity between 5.5 to 6.7 mS.cm−1. Acetate concentration was held constant by removing old media and adding gas sparged new media using a peristaltic pump at 0.07 mL. min−1.

### Electrodes

The working electrode was a 0.5 cm diameter glassy carbon exchange disk (Pine Instruments, Durham, NC, USA). The electrode was soaked in 1 M hydrochloric acid and 1 M sodium hydroxide for 24 hours after wiping with ethanol and roughened using 5 µm grit silicon carbide polishing paper to clean from previous experiments. A single junction Ag/AgCl (3.8 M KCl) reference electrode (MilliporeSigma, Darmstadt, GER) was used because the conductivity of the media is too low to sustain a double-junction. While this may cause leakage of the 3.8 M sodium solution into the electrolyte, the abiotic controls do not show any significant deviation. A 0.64 cm diameter graphite electrode behind a glass frit was used as the counter electrode (Pine Instruments, Durham, NC, USA).

### Apparatus

The reactor was a water jacketed inverted 500 mL Erlenmeyer flask, with 4 ports for electrodes and gas (Pine Instruments, Durham, NC, USA). The reactor was maintained at a constant temperature of 30 °C using a water jacket (Julabo E5, Julabo USA, Allentown, PA, USA). The system was purged and blanketed at 50 sccm with ultra-high purity 80% nitrogen and 20% carbon dioxide gas (Airgas Inc, Radnor Twp, PA, USA). The current was measured with a Gamry Reference 600 or a Gamry Reference 3000 (Gamry Instruments, Philadelphia, PA, USA) with a resolution of 600 mA and a bandwidth of 10 MHz which is much lower than the sampling rate and current measured.

### Bacteria

Geobacter sulfurreducens PCA (ATCC, 51573), was grown anaerobically from a frozen stock in a media matching Coppi et al. with 4 mM L-cystine and 40 mM fumarate49 at 30 °C. This was transferred to a 100 mL flask and grown for two days. The cells were centrifuged twice at 6000 rpm for 10 min, rinsed in between with the media described earlier. Optical density was measured at 600 nm (UV-1800, Shimadzu, Nakagyo-ku, Kyoto, JPN) after suspending the cells in the full 100 mL. Cell density was then calculated using a disposable hemocytometer (Incyto, Co., Ltd, Chonan-si, Chungnam-do, KOR). Cell volume was calculated assuming the bacteria were cylindrical and measuring them using a 100 X oil immersion objective.

### Procedure

The media was brought to 30 °C and internal resistance of the media was measured. All conditions were run in biological triplicate. A roughly 24 hr cycle included 30 min open circuit potential measurement, followed by two cycles of cyclic voltammetry, chronoamperometry for 22 hours, and electrochemical impedance spectroscopy. Chronoamperometry was conducted at −0.156 V vs Ag/AgCl based on Soussan et al.41. Open circuit potential was measured at a sampling rate of 0.016 Hz for 30 min as this appeared sufficient to reach steady-state. Cyclic voltammetry had an equilibration time of 5 s, a step size of 2 mV and a scan rate of 1 mV.s−1 from −0.755–0.045 V. following the method of Marsili et al.38. The electron diffusion current was measured as the maximal current generated during a slow (2 mV.s−1) cyclic voltammetry scan. Electrochemical impedance of the biofilm is measured since it may be related to the structure of the biofilm and is a key characteristic of microbial fuel cell performance16,50,51. Electrochemical impedance spectroscopy was conducted using two sets of parameters 1000000 Hz to 0.10 Hz at Edc = −0.340 V, Eac = 5 mV and 1000000 Hz to 0.01 Hz at Edc = −0.157 V, Eac = 10 mV following both Babauta and Beyanal and Marsili et al.28,38. We fit the impedance spectra to the equivalent circuit in Marsili et al.38, where the solution resistance is in series with the charge transfer resistance and a constant phase element in parallel. However, there is no widely accepted model for biofilm impedance. We further fit the impedance data by calculating an apparent diffusivity for acetate following the method of Tribollet et al.39 which holds when the system is mass transport limited.

### Optical Analysis

The biofilm density and thickness are necessary to compare the results to previous research on bioreactors44. After 7 days, the bacteria on disk were dyed using 4′,6-diamidino-2-phenylindole a DNA stain (NucBlue, Molecular Probes, Inc, Eugene, OR, USA) and Alexa Fluor® 594 -Concanavalin A, Conjugate (Molecular Probes, Inc, Eugene, OR, USA) that binds to glycocalyx18. After 10 minutes, the cells were fixed for 24 hrs using 4% SEM grade glutaraldehyde (MilliporeSigma, Darmstadt, GER) solution in their native buffer. The biofilm was then rinsed in an ethanol series 30, 50, 70, 90, 100, 100. These were imaged using confocal microscopy on four 50 µm2 regions at the perimeter of the disk where the shear was calculated. The sample was then critical point dried and sputter coated. The images were processed in MATLAB where a threshold was set and the images were converted to black and white. Porosity was calculated layer by layer.

## Data Availability

All data available upon request.

## References

1. 1.

Bryers, J. D. Medical biofilms. Biotechnology and Bioengineering 100, 1–19, https://doi.org/10.1002/bit.21838 (2008).

2. 2.

Koch, C. & Harnisch, F. Is there a Specific Ecological Niche for Electroactive Microorganisms? Chem Electro Chem 3, 1282–1295, https://doi.org/10.1002/celc.201600079 (2016).

3. 3.

Lovley, D. & Phillips, E. Rapid assay for microbially reducible ferric iron in aquatic sediments PMCID: PMC203906 PMID: 16347384. Appl. Environ. Microbiol. 53, 1536–1540 (1987).

4. 4.

Myers, C. & Nealson, K. Bacterial manganese reduction and growth with manganese oxide as the sole electron acceptor. Science 240, 1319–1321, https://doi.org/10.1126/science.240.4857.1319 (1988).

5. 5.

Popat, S. C. & Torres, C. I. Critical transport rates that limit the performance of microbial electrochemistry technologies. Bioresource Technology 215, 265–273, https://doi.org/10.1016/j.biortech.2016.04.136 (2016).

6. 6.

He, Z., Minteer, S. & Angenent, L. T. Electricity generation from artificial wastewater using an upflow microbial fuel cell. Environmental Science and Technology 39, 5262–5267, https://doi.org/10.1021/es0502876 (2005).

7. 7.

He, W. et al. The effect of flow modes and electrode combinations on the performance of a multiple module microbial fuel cell installed at wastewater treatment plant. Water Research 105, 351–360, https://doi.org/10.1016/j.watres.2016.09.008 (2016).

8. 8.

Li, J., Ge, Z. & He, Z. A fluidized bed membrane bioelectrochemical reactor for energy-efficient wastewater treatment. Bioresource Technology 167, 310–315, https://doi.org/10.1016/j.biortech.2014.06.034 (2014).

9. 9.

Deng, Q., Li, X., Zuo, J., Ling, A. & Logan, B. E. Power generation using an activated carbon fiber felt cathode in an upflow microbial fuel cell. Journal of Power Sources 195, 1130–1135, https://doi.org/10.1016/j.jpowsour.2009.08.092 (2010).

10. 10.

Casey, E., Glennon, B. & Hamer, G. Biofilm development in a membrane-aerated biofilm reactor: Effect of flow velocity on performance. Biotechnology and Bioengineering 67, 476–486, https://doi.org/10.1002/(SICI)1097-0290(20000220)67 (2000).

11. 11.

Shen, Y., Wang, M., Chang, I. S. & Ng, H. Y. Effect of shear rate on the response of microbial fuel cell toxicity sensor to Cu(II). Bioresource Technology 136, 707–710, https://doi.org/10.1016/j.biortech.2013.02.069 (2013).

12. 12.

Shen, L. et al. Anodic concentration loss and impedance characteristics in rotating disk electrode microbial fuel cells. Bioprocess and Biosystems Engineering 39, 1627–1634, https://doi.org/10.1007/s00449-016-1638-1 (2016).

13. 13.

Babauta, J. T., Beasley, C. A. & Beyenal, H. Investigation of Electron Transfer by Geobacter sulfurreducensBiofilms by using an Electrochemical Quartz Crystal Microbalance. ChemElectroChem 1, 2007–2016, https://doi.org/10.1002/celc.201402127 (2014).

14. 14.

Pham, H. T. et al. High shear enrichment improves the performance of the anodophilic microbial consortium in a microbial fuel cell. Microbial Biotechnology 1, 487–496, https://doi.org/10.1111/j.1751-7915.2008.00049.x (2008).

15. 15.

Wilking, J. N. et al. Liquid transport facilitated by channels in Bacillus subtilis biofilms. Proceedings of the National Academy of Sciences 110, https://doi.org/10.1073/pnas.1216376110 (2013).

16. 16.

Bonanni, S. P., Bradley, D. F., Schrott, G. D. & Busalmen, J. Limitations for Current Production in Geobacter sulfurreducens Biofilms. Chem Sus Chem 6, 711–720, https://doi.org/10.1002/cssc.201200671 (2013).

17. 17.

Renslow, R., Babauta, J. T., Majors, P. & Beyenal, H. Diffusion in biofilms respiring on electrodes. Energy Environ. Sci. 6, 595–607, https://doi.org/10.1039/C2EE23394K (2013).

18. 18.

Wey, V. A. et al. Anisotropic nutrient transport in three-dimensional single species bacterial biofilms. Biotechnology and Bioengineering 109, 1280–1292, https://doi.org/10.1002/bit.24390 (2011).

19. 19.

Stoodley, P., Dirk, D. & Lewandowski, Z. Liquid Flow in Biofilm Systems PMCID: PMC201713 PMID: 16349345. Appl Environ Microbiol 60, 2711–2716 (1994).

20. 20.

Jana, P., Katuri, K., Kavanagh, P., Kumar, A. & Leech, D. Charge transport in films of Geobacter sulfurreducens on graphite electrodes as a function of film thickness. Physical Chemistry Chemical Physics 16, 9039–9046, https://doi.org/10.1039/C4CP01023J (2014).

21. 21.

Reguera, G. et al. Biofilm and nanowire production leads to increased current in Geobacter sulfurreducens fuel cells. Appl. Environ. Microbiol. 72, 7345–7348, https://doi.org/10.1128/AEM.01444-06 (2006).

22. 22.

Thomen, P. et al. Bacterial biofilm under flow: First a physical struggle to stay, then a matter of breathing. Plos One 12, 1–24, https://doi.org/10.1371/journal.pone.0175197 (2017).

23. 23.

Wang, Q., Jones, A.-A. D., Gralnick, J. A., Lin, L. & Buie, C. R. Microfluidic Dielectrophoresis Illuminates the Relationship between Microbial Cell Envelope Polarizability and Electrochemical Activity. Science Advance, https://doi.org/10.1126/sciadv.aat5664 (Accepted).

24. 24.

Jones, A.-A. D. & Buie, C. R. Shear stress mediates metabolism and growth in electroactive biofilms. bioRxiv, 1–10, https://doi.org/10.1101/391029 (2018).

25. 25.

Jones, A.-A. D. Exoelectrogenic Biofilm Growth in Shearing Flows Doctor of Philosophy thesis, Massachusetts Institute of Technology, (2018).

26. 26.

Jones, A.-A. D. & Buie, C. R. The impact of shearing flows on electroactive biofilm formation, structure, and current generation in 69th Annual Meeting of the APS Division of Fluid Dynamics Vol. 61, http://meetings.aps.org/link/BAPS.2016.DFD.H20.5 (2016).

27. 27.

Liu, G., Tree, D. & High, M. Relationships between Rotating Disk Corrosion Measurements and Corrosion in Pipe Flow. Corrosion 50, 584–593, https://doi.org/10.5006/1.3293529 (1994).

28. 28.

Babauta, J. T. & Beyenal, H. Mass transfer studies of Geobacter sulfurreducensbiofilms on rotating disk electrodes. Biotechnology and Bioengineering 111, 285–294, https://doi.org/10.1002/bit.25105 (2013).

29. 29.

Marsili, E., Sun, J. & Bond, D. R. Voltammetry and Growth Physiology of Geobacter sulfurreducens Biofilms as a Function of Growth Stage and Imposed Electrode Potential. Electroanalysis 22, 865–874, https://doi.org/10.1002/elan.200800007 (2010).

30. 30.

McLean, J. S. et al. Quantification of Electron Transfer Rates to a Solid Phase Electron Acceptor through the Stages of Biofilm Formation from Single Cells to Multicellular Communities. Environmental Science and Technology 44, 2721–2727, https://doi.org/10.1021/es903043p (2010).

31. 31.

Ausubel, F. M. et al. Current Protocols in Molecular Biology. John Wiley & Sons, Inc, 1–4648 (2003).

32. 32.

Garrett, T., Bhakoo, M. & Zhang, Z. Bacterial adhesion and biofilms on surfaces. Progress in Natural Science 18, 8, https://doi.org/10.1016/j.pnsc.2008.04.001 (2008).

33. 33.

Gardel, E. J., Nielsen, M. E., Grisdela, P. T. & Girguis, P. R. Duty Cycling Influences Current Generation in Multi-Anode Environmental Microbial Fuel Cells. Environmental Science & Technology 46, 5222–5229, https://doi.org/10.1021/es204622m (2012).

34. 34.

Du, Q., Mu, Q., Cheng, T., Li, N. & Wang, X. Real-Time Imaging Revealed That Exoelectrogens from Wastewater Are Selected at the Center of a Gradient Electric Field. Environmental Science & Technology 52, 8939–8946, https://doi.org/10.1021/acs.est.8b01468 (2018).

35. 35.

Liu, Y. & Tay, J. H. The essential role of hydrodynamic shear force in the formation of biofilm and granular sludge. Water Research 36, 1653–1665, https://doi.org/10.1016/S0043-1354(01)00379-7 (2002).

36. 36.

Trulear, M. & Characklis, W. G. Dynamics of Biofilm Processes. Journal (Water Pollution Control Federation) 54, 1288–1301, https://doi.org/10.2307/25041684 (1982).

37. 37.

Richter, H. et al. Cyclic voltammetry of biofilms of wild type and mutant Geobacter sulfurreducens on fuel cell anodes indicates possible roles of OmcB, OmcZ, type IV pili, and protons in extracellular electron transfer. Energy Environ Sci 2, 506–516, https://doi.org/10.1039/B816647A (2009).

38. 38.

Marsili, E., Rollefson, J. B., Baron, D. B., Hozalski, R. M. & Bond, D. R. Microbial Biofilm Voltammetry: Direct Electrochemical Characterization of Catalytic Electrode-Attached Biofilms. Appl Environ Microb 74, https://doi.org/10.1128/AEM.00177-08 (2008).

39. 39.

Tribollet, B., Newman, J. & Smyrl, W. H. Determination of the Diffusion Coefficient from Impedance Data in the Low Frequency Range. Journal of The Electrochemical Society 135, 134–134, https://doi.org/10.1149/1.2095539 (1988).

40. 40.

Yi, H. et al. Selection of a variant of Geobacter sulfurreducens with enhanced capacity for current production in microbial fuel cells. Biosensors and Bioelectronics 24, 3498–3503, https://doi.org/10.1016/j.bios.2009.05.004 (2009).

41. 41.

Soussan, L., Erable, B., Delia, M.-L. & Bergel, A. The open circuit potential of Geobacter sulfurreducens bioanodes depends on the electrochemical adaptation of the strain. Electrochemistry Communications 33, 35–38, https://doi.org/10.1016/j.elecom.2013.04.013 (2013).

42. 42.

Taylor, S. W. & Jaffé, P. R. Biofilm Growth and the related changes in the pysical properties of a porous medium 1. Experimental Investigation. Water Resources Research 26, 2153–2159, https://doi.org/10.1029/WR026i009p02161 (1990).

43. 43.

Ye, Z., Ellis, M. W., Nain, A. S. & Behkam, B. Effect of electrode sub-micron surface feature size on current generation of Shewanella oneidensis in microbial fuel cells. 347, 270–276, https://doi.org/10.1016/j.jpowsour.2017.02.032 (2017).

44. 44.

Characklis, W. G., Trulear, M. G., Bryers, J. & Zelver, N. Dynamics of Biofilm Processes: Methods. Water Research 16, 1207–1216, https://doi.org/10.1016/0043-1354(82)90139-7 (1982).

45. 45.

Hammer, J., Mark, J., Hammer, S. & Mark, J. Water and wastewater technology. 7 edn, (Pearson Prentice Hall, 2012).

46. 46.

Becerra, H., Retamoso, C. & Macdonald, D. D. The corrosion of carbon steel in oil-in-water emulsions under controlled hydrodynamic conditions. Corrosion Science 42, 561–575, https://doi.org/10.1016/S0010-938x(99)00068-2 (2000).

47. 47.

Abraham, E. N., Rothermich, M., Sharma, M. & Lovley, D. Growth of Geobacter sulfurreducens under nutrient‐limiting conditions in continuous culture. Method Enzymol 7, 641–648, https://doi.org/10.1111/j.1462-2920.2005.00731.x (2005).

48. 48.

Zwietering, M. H. et al. Modeling of the Bacterial Growth Curve. Applied and Environmental Microbiology 56 (1990).

49. 49.

Coppi, M., Leang, C., Sandler, S. & Lovley, D. R. Development of a Genetic System for Geobacter sulfurreducens. Applied and Environmental Microbiology 67, 3180–3187, https://doi.org/10.1128/AEM.67.7.3180-3187.2001 (2001).

50. 50.

Sarah, S.-G. & Snider, R. A, G.-E. & Tender, L. M. On the electrical conductivity of microbial nanowires and biofilms PMCID: PMC184525 PMID: 16348228. Energy Environ Sci 4, 4366–4379, https://doi.org/10.1039/C1EE01753E (2011).

51. 51.

Bond, D. R., Sarah, S.-G., Tender, L. M. & Torres, C. I. On Electron Transport through Geobacter Biofilms. ChemSusChem 5, 1099–1105, https://doi.org/10.1002/cssc.201100748 (2012).

## Acknowledgements

A.-A.J. would like to acknowledge the financial support of the Alfred P. Sloan Foundation’s Minority PhD Program. We must also acknowledge the resources and instruction of the W.M. Keck Biological Imaging Facility at the Whitehead Institute.

## Author information

A.-A.J. and C.R.B. conceived of the experiments. A.-A.J. performed the experiments and carried out all the data analysis. A.-A.J. organized and wrote the first draft of the manuscript. Both authors discussed and interpreted the results and contributed to drafts of this final manuscript.

Correspondence to Cullen R. Buie.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

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Reprints and Permissions

Jones, A.D., Buie, C.R. Continuous shear stress alters metabolism, mass-transport, and growth in electroactive biofilms independent of surface substrate transport. Sci Rep 9, 2602 (2019). https://doi.org/10.1038/s41598-019-39267-2

• Accepted:

• Published:

• ### Strategies for improving the electroactivity and specific metabolic functionality of microorganisms for various microbial electrochemical technologies

• P. Chiranjeevi
•  & Sunil A. Patil


text:
## Chemistry and Chemical Reactivity (9th Edition)

a) Enthalpies of formation: $H_2O(g): -241.83, CO(g): -110.525$ Enthalpy change: $-110.525- -241.83=+131.58\ kJ/mol$ b) Since $\Delta H_r\gt0$, it's an endothermic reaction. c) Number of moles of carbon: $1\times10^6\ g\div 12.011\ g/mol=8.33\times 10^4\ mol$ Heat required: $131.58\ kJ/mol\times 8.33\times 10^4\ mol=10.95\times10^6\ kJ$


text:
# If P\left(A\right)=0.4, P\left(B\right)=0.5, and P\left(A\cap B\

If $$\displaystyle{P}{\left({A}\right)}={0.4}$$, $$\displaystyle{P}{\left({B}\right)}={0.5}$$, and $$\displaystyle{P}{\left({A}\cap{B}\right)}={0.3}$$, find a. $$\displaystyle{P}{\left({A}\cup{B}\right)}$$, b. $$\displaystyle{P}{\left({A}\cap{B}'\right)}$$, and c. $$\displaystyle{P}{\left({A}'\cup{B}'\right)}$$.

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Explanation 1
Step 1
PROBABILITY RULES
Complement rule
$$\displaystyle{P}{\left({A}'\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}$$
General addition rule for any two events:
$$\displaystyle{P}{\left({A}\cup{B}\right)}={P}{\left({A}\right)}+{P}{\left({B}\right)}-{P}{\left({A}\cap{B}\right)}$$
Step 2
SOLUTION
$$\displaystyle{P}{\left({A}\right)}={0.4}$$
$$\displaystyle{P}{\left({B}\right)}={0.5}$$
$$\displaystyle{P}{\left({A}\cap{B}\right)}={0.3}$$
(a) Use the general addition rule for any two events:
$$\displaystyle{P}{\left({A}\cup{B}\right)}={P}{\left({A}\right)}+{P}{\left({B}\right)}-{P}{\left({A}\cap{B}\right)}$$
$$\displaystyle={0.4}+{0.5}-{0.3}$$
$$\displaystyle={0.6}$$
Explanation 2
Step 1
Probability rules, used n the exercise:
Complement Rule:
$$\displaystyle{\left({A}'\right)}={P}{\left(\neg{A}\right)}={1}-{P}{\left({A}\right)}$$
$$\displaystyle{\left({A}\cup{B}\right)}={P}{\left({A}\right)}+{P}{\left({B}\right)}-{P}{\left({A}\cap{B}\right)}$$
Rule of Complementary events B and B':
$$\displaystyle{P}{\left({A}\right)}={P}{\left({A}\cap{B}\right)}+{P}{\left({A}\cap{B}'\right)}$$
De Morgan's Rules
$$\displaystyle{P}{\left({\left({A}\cup{B}\right)}'\right)}={P}{\left({A}'\cap{B}'\right)}$$
$$\displaystyle{P}{\left({\left({A}\cap{B}\right)}'\right)}={P}{\left({A}'\cap{B}'\right)}$$
Step 2
Solution:
$$\displaystyle{P}{\left({A}\right)}={0.4}$$
$$\displaystyle{P}{\left({B}\right)}={0.5}$$
$$\displaystyle{P}{\left({A}\cap{B}\right)}={0.3}$$
(a)
$$\begin{matrix} \\ &{\left(Addition Rule\right) } \\P \left(A \cup B\right) & {=} & P\left(A\right)-P\left(A \cap B\right)\end{matrix}$$
$$\displaystyle={0.4}+{0.5}-{0.3}$$
$$\displaystyle={0.6}$$

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Crom1970

Explanation 1
Step 3
(b) The event of A occurring but not B is the event A without all outcomes that occur for A and B have in common.
$$\displaystyle{P}{\left({A}\cap{B}'\right)}={P}{\left({A}\right)}–{P}{\left({A}\cap{B}\right)}={0.4}-{0.3}={0.1}$$
Explanation 2
(b)
$$\begin{matrix} \\ &{\left(Rule\ of\ Complementary\ events \right) } \\P \left(A \cap B'\right) & {=} & P\left(A\right)-P\left(A \cap B\right)\end{matrix}$$
$$\displaystyle{0.4}-{0.3}$$
$$\displaystyle={0.1}$$

Don Sumner

Explanation 1
Step 4
(c) The event of A not occurring or B not occurring is the event that A and B cannot occur together.
Use the complement rule:
$$P\left(A'\cup B'\right)= P\left(\left(A \cap B\right)'\right)=1- P\left(A \cap B\right)=1-0.3 = 0.7$$
Explanation 2
(c)
$$\begin{matrix} \\ &{\left(De Morgan's Rule (2) \right) } \\P \left(A' \cup B'\right) & {=} & P\left(\left(A \cap B\right)'\right)\\ &{\left(Complement Rule \right) } & \\ & = & 1-P\left(A \cap B\right) \end{matrix}$$
$$=1-0.3$$
$$=0.7$$