Useful links of research groups
Robert Langer George Whitesides Charles Lieber Joseph Wang Xi Yin Howard Stone Kaustav Banerjee LiYi Wei Zhenan Bao David Weitz David Quere
Proposal idea for a new experiment
Releasing test on reared recruits of sea urchins \label{releasingtestonrearedrecruitsofseaurchins} Context In Sardinia (Italy, western Mediterranean) a variety of management tools are used, according to regional laws, for sea urchin fishery, including size limits, closed seasons, gear restrictions and marine reserves (Pais et al. 2007). Despite these regional decrees, the resource is drastically decreasing in several areas with consequences for fishery and may be for the ecological functions of populations and coastal ecosystems. Experiments around the world have repeatedly demonstrated strong effects due to the removal of sea urchins from the environment (Andrew et al. 2002), but few studies are carried out on the effects of the enhancement and recovery of populations for a sustainable fishery. Enhancement of sea urchin populations is divided into three categories: reseeding, habitat enhancement, and transplantation in wild populations (Andrew et al. 2002). About this study, we would like to investigate new issues concerning the reseeding of sea urchin recruits produced and grown in captivity for approximatively sixteen months (diameter without spines 10 20mm). The theoretical basis of stock enhancement by reseeding is the belief that populations are recruitment limited (i.e. due to the limited abundance by processes acting on sea urchins before they settle). The premise is that we can raise large numbers of larvae or juveniles and by releasing them into the marine environment, they compensate for the lack of these stages in nature and thereby increase stock size in the late juvenile and early adult stages (Saito 1992, Kitada 1999). Consequently, for reseeding to make sense, mortality of reared recruits have to be no higher than of the wild ones and populations receiving the outplanted organisms have to be not near to the carrying capacity of the environment. In this sense, both of the evaluation of environment carrying capacity and the development of a production process that optimizes the times of the larval growth in according with the lowest possible mortality are the keypoints for a successful reseeding. The active reseeding (or restocking) is based on the production of larvae and postlarvae under controlled laboratory conditions until those animals are potentially ready to be released. At this porpouse, we used optimal diets and we tried to minimize the environmental stress constraining the variation of the physicalchemical conditions to optimize the production and growing of the organisms (Brundu et al. 2016). However, it is still unknown if, through this procedure, mortality of reared recruits is higher respect to the wild ones and, consequently, additional methods of acclimatization should be added before the releasing. Actually, many studies have shown that vertebrate organisms grown in captivity are generally less resistance to the environmental variations than the wild ones (McGinnity et al 2009) and their antipredator behaviour is different, since they are more vulnerable to predators (Meager et al 2011). In Japanese sea urchin hatcheries, reared individuals with a diameter of about 5mm were directly placed into small mesh cages for intermediate culture in tanks on land or suspended in the sea and no acclimatizing stages were done before reseeding (Tegner, 1989). Previous studies on the effects of sea urchins reseeding (Paracentrotus lividus and Strongylocentrotus franciscanus ) did not carry out acclimatizing stage of animals before their release (JuinioMeñez et al., 1998; Couvray et al., 2015). Nevertheless, it is possible that intermediate cages or tanks cultures between laboratory and natural environment, as carried out in Japanese sea urchin hatcheries, could influence positively the successful of reseeding. Furthermore, additional methods of acclimatization, to be added before the releasing, could improve the successful of sea urchin restocking, in terms of survival. In this sense, it may be reasonable to check differences in resistance and environmental perceptions among reared and wild sea urchins. Objective and hypothesis of work The aim of this study is to evaluate the success of the restocking in a controlled environment with reared recruits (10

SMMP  Stochastic Methods for Molecular Properties
Possible titles:

Stochastic Methods for Molecular Properties (SMMP)

Stochastic Methods for Chiroptical Properties (SMCP or ChiroStoch)

Deterministic methods need large Hilbert spaces for effective expansions of the manyelectron wave function
This is however largely redundant \cite{Ivanic_2001}

Stochastic algorithms are highly parallelizable in the number of walkers.
I will develop my skills in parallel programming techniques by developing this project.
Research questions:
 Understanding chiroptical properties for large chemical systems
Objectives of the project:

The calculation of molecular properties with high accuracy and for systems of relevant size
 High accuracy means coupled cluster (CC) wave functions
 Systems for which CC is an option are limited by its polynomial scaling
 It is possible to reduce the scaling, e.g. by means of local approaches to the electron correlation problem,
but this has been proven to not be as effective for molecular properties as for energies

Devise the appropriate stochastic approach to the solution of response equations
 We want the stochastic approach because it's (supposedly) embarassingly parallel.
This can enable the study of the response properties for larger systems and provide benchmark
results for lower level calculations.
 Low scaling (or better parallelization options) + (perhaps?) controllable error of stochastic methods
are the key advantages. Find relevant literature on both!
\cite{Booth_2014}, \cite{Coccia_2012}

The creation of the appropriate software toolbox with good scalability.
 The toolbox will be freely available under the appropriate open source license (GPL most likely!)
General background on quantum chemistry:
 "Frontiers in electronic structure theory" \cite{Sherrill_2010}
 Quantum chemistry as an effective complement to experiment \cite{Lee_1995}, \cite{Helgaker_2004}, \cite{Tajti_2004}, \cite{Helgaker_2000}, \cite{Goddard_1985}
State of the art:

QMC:
 Recent reviews on QMC approaches: \cite{Dubeck__2016}, \cite{Toulouse_2016}, \cite{Austin_2012}, \cite{Needs_2009}, \cite{Assaraf_2007}, \cite{Towler_2006} and \cite{Foulkes_2001}
 DMC (Wick rotation of the Schrodinger equation \cite{Wick_1954} and isomorphism with a classical diffusion problem)
 SelfHealing QMC
 Auxiliary Field QMC
 Fermion QMC
 FCIQMC
 Stochastic Coupled Cluster: Thom introduces the FCIQMClike stochastic algorithm for solving the CC equations \cite{Thom_2010}
The methods leverages the stochastic sampling strategies of the FCI wave function in a discrete Fock space first proposed by Alavi _et al._\cite{Booth_2009}
 Linked Coupled Cluster Monte Carlo: the stochastic algorithm for the solution of the CC equations in the linked (termbyterm sizeextensive) form \cite{Franklin_2016}
 Initiator approximation: the same group proposes the initiator approximation for the CCMC algorithm \cite{Spencer_2016}
 Stochastic MøllerPlesset \cite{Thom_2007}
 Local approaches to QMC \cite{Manten_2003}, \cite{Williamson_2001}

Properties by QMC
 Dynamic polarizabilities \cite{Caffarel_1993}, \cite{Mella_2001}
 Large systems \cite{Filippi_2012}, \cite{Valsson_2010}
 Forces: correlated sampling \cite{Filippi_2000} and space warp coordinate transformation \cite{Umrigar_2009}
Work by Assaraf and Caffarel on improved estimators \cite{Assaraf_2000} and \cite{Assaraf_2003}
 Static electric properties (dipole and quadrupole moments, static polarizabilities): polarizabilities by finite differences for ethyne \cite{Coccia_2012}
polarizability of the hydrogen atom by modified sampling \cite{Li_2007}

Chiroptical properties:
 Eliel "Stereochemistry of organic compounds" (find appropriate ref)
 Barron's book \cite{Barron_2004} Rosenfeld's formulation \cite{Rosenfeld_1929}
 OR and ECD review by Pecul and Ruud \cite{Pecul_2005}
 Berova's book \cite{Berova_2011}
 Daniel's reviews \cite{Crawford_2005}, \cite{Crawford_2007a}, \cite{Crawford_2012}
 Octant rule \cite{Snatzke_1979} and its failure \cite{Rinderspacher_2004}
 Chiroptical properties by DFT \cite{Cheeseman_2000}, \cite{Furche_2000}, \cite{Grimme_2001} \cite{Stephens_2001} \cite{Stephens_2002} \cite{Grimme_2002} \cite{Autschbach_2002}
\cite{Autschbach_2002b} \cite{Autschbach_2003}
 Chiroptical properties by CC \cite{Tam_2004} \cite{Crawford_2005} \cite{Ruud_2002} \cite{Ruud_2003}
 Computational studies available on a variety of molecules \cite{Tam_2006} \cite{Kowalczyk_2006} \cite{Crawford_2005} \cite{Tam_2007} \cite{Crawford_2007a} \cite{Crawford_2007b}
\cite{Wiberg_2008}\cite{Crawford_2008} \cite{Crawford_2009} \cite{Pedersen_2009} \cite{Pedersen_2009b} \cite{Lambert_2012} \cite{Rinderspacher_2004} \cite{Wilson_2005}
\cite{Furche_2000} \cite{Pulm_1997} \cite{Kondru_1998} \cite{Grimme_1998} \cite{Polavarapu_1999} \cite{Ribe_2000} \cite{Polavarapu_2002} \cite{Diedrich_2003} \cite{Polavarapu_2003} \cite{Norman_2004} \cite{Diedrich2004xo} \cite{Stephens_2005} \cite{Wiberg_2005a} \cite{Wiberg_2005b} \cite{Wiberg_2006} \cite{Autschbach_2009} \cite{Pritchard_2010} \cite{Mach_2011} \cite{Mach_2014}
 Ultrasensitive Cavity RingDown Polarimetry (CRDP) \cite{Wilson_2005} \cite{M_ller_2002} \cite{M_ller_2000}
 Local approaches to the correlation problem in response theory \cite{Russ_2004} \cite{Russ_2008}

Response theory:
 Most recent review on wave functionbased response theory \cite{Helgaker_2012}
 Foundational work \cite{Olsen_1985}, \cite{Christiansen_1998}, \cite{Paw_owski_2015}, \cite{Coriani_2016}
 CC response theory
 Local approaches to CC (really a lot of literature...)
 Local approaches to CC response theory \cite{Friedrich_2015}, \cite{McAlexander_2012}, \cite{McAlexander_2016}
Problems to address:
 The fermion sign problem. How do FCIQMC and CCMC avoid it?
The sign problem is NPhard \cite{Troyer_2005} thus not solvable in polynomial time
TODO:
 Size of the systems investigated by stochastic methods in Fock space?
 Properties by stochastic methods? VMC/DMC/FCIQMC?
 Local correlation approaches for molecular properties?
 L. Guidoni might have published some FCIQMC calculations on large molecular systems.
The determination of coronary artery wave speed
using information theory
Kim H. Parker
and
1 collaborator
A method is described for estimating the wave speed of a coronary artery from simultaneous measurements of pressure and velocity. It involves finding the wave speed that minimises the distance between the probability density functions of the measured pressure and velocity and the probability density functions of the pressure and velocity obtained from the separation of the waves into their forward and backward components using the assumed wave speed.
King Chicken Theorem
In graph theory, directed graphs can be used to understand tournaments and theorems such as the king chicken theorem. First to understand the king chicken theorem, we will go over some terminology. A tournament is a directed graph that contains edges that have specific orientation. Tournament graphs are also used to show relationships between players and who beat who in a tournament. Complete graphs most often show this by using arrows. Any edge that points from \(i\) to \(j\) has directed orientation.
Detailed Reviewer Responses
Anisha Keshavan
and
3 collaborators
We would like to thank the reviewers for their insightful comments. The major points that have been addressed are as follows:
It was not our intention to give the impression that one needs to scan human calibration phantoms at each site to properly power a multisite study with nonstandardized parameters, which is very costly. The statistical model which takes MRI bias into account has been emphasized instead. The bias that was measured and validated via calibration served to corroborate the scaling assumption of the statistical model. For other researchers planning multisite studies, the statistical model we proposed with the biases we reported should help plan and power a study.
Our measurements have been compared with other harmonization efforts, specifically \cite{cannon2014, jovicich2013brain} and \cite{Schnack_2004}.
The scanning parameters of our consortium have been better specified.
The independence assumption between the unobserved effect and the scaling factor for a particular site have been addressed. Specifically, we emphasized that this assumption could hold for MS patients based on our experiment. The need to validate this assumption for other situations by scanning human phantoms was recommended, and the equation of variance without the independence assumption has been provided for the readers.
Blog Post 10
1 $\underline{\text{TreesA Branch of Discrete Mathematics}}$ Trees provide poets with inspiration as they sway through the breeze and their leaves, bursting with color, rustle in the wind. It is no wonder, then, that mathematicians coined the term “tree” in describing special classes of structured graphs. One author, Joe Malkevitch, makes it his goal to “convince you (readers) that mathematical trees are no less lovely than their biological counterparts.”
In discrete mathematics, and more specifically graph theory, a tree is a connected graph with no cycles. When the graph is not connected, naturally we call this a forest. In addition, a vertex of degree 1 is called a leaf. These kind of mathematical structures were first studied by mathematician Arthur Cayley. In 1889 Cayley published a formula stating that for n ≥ 1, the number of trees with n vertices is n^{n − 2}.
A few other properties of trees include the following:
Given two vertices, x and y, there is a unique path from x to y
If we remove any edge of a tree, the graph is no longer connected
If a tree has n vertices, then it has n − 1 edges
The concept of mathematical trees has applications in various fields including science, the enumeration of saturated hydrocarbons, the study of electrical circuits, and many more (Harary, 1994, p. 4).
Functional consequence of SNPs on the Tuberculosis drug metabolising enzyme, human arylamine Nacetyltransferase 1
Abstract
Background
The human arylamine Nacetyltransferase 1 (NAT1) plays a vital role in determining the duration of action and pharmacokinetics of aminecontaining drugs such as paraaminosalicylic acid used in clinical therapy, as well as influencing the balance between detoxification and metabolic activation of these drugs. Single nucleotide polymorphisms (SNPs) in this enzyme are continuously being detected and show interethnic and interindividual variation. Administrating tuberculosis (TB) treatment in the absence of genotypic information for drug metabolizing enzymes can limit the successful eradication of the disease from a patient. Recent studies have shown that loss of Hbonds affects protein function.
Results: In this study, the eects of 11 novel nonsynonymous SNPs (nsSNPs) on the structure and function of NAT1 was tested computationally using SIFT and POLYPHEN2 algorithms and structural analyses methods including loss of hydrogenbonding, stability calculation, solvent accessibility and sequence conservation. Four out of 11 nsSNPs (Q210P, D229H, V231G and V235A) were predicted to aect protein function using both algorithms. Two of these four SNPs showed a loss of 24 hydrogen bonds and in most cases a destabilized protein structure. Another two SNPs (F202V, N245I) were predicted to aect protein function using both algorithms but without any loss of hydrogenbonds. Three additional nsSNPs (T240S, S259R, T193S) were predicted to be benign with either a loss of three hydrogen bonds or no loss of hydrogenbonds. The remaining two nsSNPs (E264K and R242M) showed conflicting results between SIFT and POLYPHEN2 and both cases showed stable Gibbs free energy. No correlation could be identified between the predicted functional eects from SIFT and POLYPHEN2, and the stability calculations and the hydrogenbonding analyses. However, the structural effects of modifying an amino acid together with the conficting results from both algorithms warrant experimental testing to resolve the consequences of these 11 novel nsSNPs on NAT1.
Conclusion: The nsSNPs that aect protein function and/or have a destabilized structure provides a prioritized list of SNPs that will be tested in the laboratory by creating a SNP construct that will be cloned into an expression vector. These ndings will inform a strategy of incorporating genotypic data (i.e, functional SNP alleles) with phenotypic information (slow or fast acetylators) to better prescribe effective tuberculosis treatment.
Tournament Graphs
Tournament graphs are used in discrete mathematics to represent a winning vertex in a graph. A tournament is a complete graph in which every pair of vertices are connected by a directed edge. These types of graphs are referred to as tournaments because each of the \(n\) players competes against the other \(n1\) players where ties are not allowed and the winner can be represented on a graph. These graphs are created by assigning every player a vertex and if player \(1\) beats player \(2\), then a directed edge can be drawn with the arrow pointing from \(1\) to \(2.\) Tournaments graphs create Hamiltonian paths that go through each vertex. The Hamiltonian path theorem states that for every tournament there is a Hamiltonian path for \(n\ge1,\) for any tournament consisting of \(n\) vertices in which there is always a sequence of vertices \(v_1,v_2,...,v_n\) such that \(v_1\rightarrow v_2\rightarrow...\rightarrow v_n.\)
Autopledge
Bacon
and
2 collaborators
Even in following good coding practices, arbitrary code execution bugs can still exist. By leveraging pledge(2) system calls and a static analysis framework, we attempt to mitigate these bugs by automatically inserting pledge statements. Although an algorithm was devised to do this, time limitations prevented its full implementation.
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Supervised Learning: Classification and Regression
Naets
and
2 collaborators
Regression
\label{RegSection}
Contactless Remote Induction of Shear Waves in Soft Tissues Using a Transcranial Magnetic Stimulation Device
This study presents the first observation of shear wave induced remotely within soft tissues. It was performed through the combination of a transcranial magnetic stimulation device and a permanent magnet. A physical model based on Maxwell and Navier equations was developed. Experiments were performed on a cryogel phantom and a chicken breast sample. Using an ultrafast ultrasound scanner, shear waves of respective amplitude of 5 and 0.5 micrometers were observed. Experimental and numerical results were in good agreement. This study constitutes the framework of an alternative shear wave elastography method.
Direct measurement of \(\alpha_{\rm QED}(m_{\rm Z}^{2})\) at the FCCee
Patrick Janot
and
7 collaborators
When the measurements from the FCCee become available, an improved determination of the standardmodel “input” parameters will be needed to fully exploit the new precision data towards either constraining or fitting the parameters of beyondthestandardmodel theories. Among these input parameters is the electromagnetic coupling constant estimated at the Z mass scale, $(m^2_{\rm Z})$. The measurement of the muon forwardbackward asymmetry at the FCCee, just below and just above the Z pole, can be used to make a direct determination of $(m^2_{\rm Z})$ with an accuracy deemed adequate for an optimal use of the FCCee precision data.
MATH stuff
Complex derivative Here we provide a definition for the ’complex’ derivative of a realvalued function f : ℂn → ℝ with respect to its complex variables. The notation f : ℂn → ℝ means “f is a mapping (or function) from the set of column vectors of size n with complex components (denoted ℂn) into the set of real numbers (denoted ℝ).” The complex derivative of x = a + jb ∈ ℂ, a, b ∈ ℝ, is defined as {dx} = {da} + j{db}. Example 1. Given x = a + jb ∈ ℂ, a, b ∈ ℝ, What is Dx? SOLUTION: We have x = = = . \nonumber Applying the definition of the complex derivative yields {dx} &=& {da} + j{db} \nonumber\\ &=& {2} + j{2} \nonumber\\ &=& {} + j{} \nonumber\\ &=& {x}. \nonumber Example 2. Given x = a + jb ∈ ℂ, a, b ∈ ℝ, What is Dx²? SOLUTION: We have x^2 = x^*x = (ajb)(a+jb) = a^2 + b^2. \nonumber Applying the definition of the complex derivative yields {dx} &=& {da} + j{db} \nonumber\\ &=& 2a + j2b \nonumber\\ &=& 2x. \nonumber Suppose f : ℂn → ℝ is a realvalued function and $x \in {}{}f$. The derivative Df(x) is a 1 × n matrix (a _row_ vector), defined by Df(x) = \left[ {\partial x_1}(x), \dots, {\partial x_n}(x) \right]. Example 3. Given x = [x₁, …, xn]T ∈ ℂn with xi = ai + jbi ∈ ℂ, ai, bi ∈ ℝ, What is D∥x∥ℓ₂²? SOLUTION: We have \x\_{\ell_2}^2 &=& ^n x_i^2 = ^n x_i^*x_i \nonumber\\ &=& ^n (a_i +jb_i)^*(a_i +jb_i) \nonumber\\ &=& ^n (a_i jb_i)(a_i +jb_i) \nonumber\\ &=& ^n (a_i^2 +b_i^2). \nonumber We first look at the first element of Equation [eqn:derivative] with f(x)=∥x∥ℓ₂². Applying the definition of the complex derivative gives ^2}{\partial x_1} &=& ^2}{\partial a_1} + j^2}{\partial b_1} \nonumber\\ &=& {\partial a_1} \left(^n (a_i^2 +b_i^2)\right) + j{\partial b_1} \left(^n (a_i^2 +b_i^2)\right) \nonumber\\ &=& 2a_1 + j2b_1 \nonumber\\ &=& 2x_1. \nonumber Therefore, it follows that Df(x) &=& \left[ ^2}{\partial x_1}, \dots, ^2}{\partial x_n} \right] \nonumber\\ &=& \left[2x_1, \ldots, 2x_n \right] \nonumber\\ &=& 2x^T. \nonumber Example 4. Suppose A ∈ ℂm × n, and x = [x₁, …, xn]T ∈ ℂn with xi = ai + jbi ∈ ℂ, ai, bi ∈ ℝ. What is D(Ax)? SOLUTION: Since f(x)=Ax : ℂn → ℂm, we have D(Ax) = \left[ {\partial x_1}, \dots, {\partial x_n} \right]. \nonumber Since Ax ∈ ℂm, we express it as Ax = \left[ {c} (Ax)_1 \\ \vdots \\ (Ax)_m \right] = \left[ {c} ^n A_{1i}x_i \\ \vdots \\ ^n A_{mi}x_i \right], \nonumber and it follows that {\partial x_1} = \left[ {c} {\partial x_1} \\ \vdots \\ {\partial x_1} \right] = \left[ {c} A_{11} \\ \vdots \\ A_{m1} \right]. \nonumber Using the expression above, we write the derivative of Ax as D(Ax) = \left[ {ccc} {\partial x_1} & \cdots & {\partial x_n} \\ \vdots & \ddots & \vdots \\ {\partial x_1} & \cdots & {\partial x_n} \right] = \left[ {ccc} A_{11} & \cdots & A_{1n} \\ \vdots & \ddots & \vdots \\ A_{m1} & \cdots & A_{mn} \right] = A. \nonumber
Blog Post 9
1 $\underline{\text{An Introduction to Graph Theory}}$ The concepts of graph theory go all the way back to the eighteenth century when, in 1736, Euler published what is believed to be the first paper on the very subject. It contained a famous problem known as the Königsberg Bridges Problem. This is a puzzle which considers the question of whether or not a person, starting from their home, could pass over each of the seven bridges that crossed the Pregal River exactly once before returning home. Euler was able to reconstruct the problem in such a way that allowed for him to lay the foundation of graph theory. To solve the puzzle, Euler replaced the land masses with vertices and let edges represent the bridges. The mathematical structure that became of his work is now called a graph.
Adjacency Matrix
The adjacency matrix is used in discrete mathematics to represent the number of ways in which we can walk from one vertex to another within a graph. Any graph can be shown in an adjacency matrix where both the rows and columns are labeled with our graph vertices. We denote each entry as \(\left(i,j\right)\) which counts the number of adjacent edges between the \(i^{th}\) row and \(j^{th}\) column. We also say that \(a_{i,j}\) represents the number in row \(i\), column \(j.\) The adjacency matrix is made up of graph vertices that are either a \(0\) or \(1.\) To decide which entry to write in the matrix, we use a \(0\) if vertex \(i\) is not adjacent to vertex \(j\) and we use a \(1\) if vertex \(i\) is adjacent to vertex \(j.\)
Matrices in Graph Theory
Some of the most important matrices that are used in number theory are known as the adjacency matrix and the transition matrix. An adjacency matrix is given by the vertices of that matrix and is labeled with a \(0\) or \(1\) depending on its adjacency. The way we label such a vertex with its adjacency is by \(\left(i,j\right)\), where \(i\) is the row while \(j\) is the column. Adjacency matrices can also be used to find the number of walks between vertices. To show this we raise our matrix to the \(L\), where \(L\) is the length of the walk and read off the matrix as \(\left(i,j\right)\).
Assignment 2
FAT POINTS Proof by contradiction. Assume that the line segment between points \(P,Q\) has maximum pairwise distance, and that \(Q\) does not lie on a vertex. Let the point on the boundary of our hull found by extending the line \(PQ\) be denoted \(Q'\). This boundary segment is defined between two vertices on our convex hull which we refer to as \(A\) and \(B\). See \(\) for clarification. Q Lies on the Interior of the Hull Clearly \(PQ'>PQ\). In the following section we show that there is always a linesegment longer than \(PQ'\). By the transitive property of inequality this segment must also be longer than \(PQ\), a contradiction. Q Lies on a Hull Edge
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Blog Post 8
1 $\underline{\text{Parity}}$
Parity, in terms of mathematics, describes the classification of an integer as either even or odd. An even number is defined as an integer that is divisible by 2 while an odd number is one that is not. A more formal definition states that an even number is an integer n of the form n = 2k where k is an integer. On the other hand, an odd number is an integer of the form n = 2k + 1. In set notation we see:
\[ \text{Even} \hspace{0.5mm} = \hspace{0.5mm} {2k : k \in \mathbb{Z}} \]
\[ \text{Odd} \hspace{0.5mm} = \hspace{0.5mm} {2k+1 : k \in \mathbb{Z}} \]
In number theory, the idea of parity allows us to solve some mathematical problems simply by making note of odd and even numbers. In the same way, the impossibility of some mathematical constructions can be proven. For example, consider the following question:
The impact of boreal wildfires on carbon and nitrogen dynamics: the interplay between biotic and abiotic processes
PURPOSE AND AIMS Wildfires are a natural phenomenon but human activities are altering both the driving factors (climate) and the vulnerability (landuse factors) of ecosystems, increasing both frequency and severity of fire impacts. This is an issue of concern given that wildfires play a major role in the global carbon cycle by affecting carbon and nitrogen storage in ecosystems. Yet, our knowledge of early postfire carbon (C) and nitrogen (N) (hereafter abbreviated as CN) dynamics has been severely limited by the lack of crossscale (from soil to plant to ecosystem) and crosslandscape (wetlands to uplands, managed and unmanaged land) studies. Understanding the mechanisms causing variability in CN dynamics (e.g., CN accumulation) , in heterogeneous landscapes, is critical for predicting changes in C and N storage with more frequent disturbance. Given this immediate research need, I propose an ambitious research program to investigate the impact of wildfires on the C and N cycle in the boreal landscape, capitalizing on a recent standreplacing wildfire in Sweden. With an array of paired pre and postfire data, which is rare in wildfire ecosystem research, I aim to address whether predisturbance and initial postdisturbance conditions can be used to formulate predictions of postdisturbance ecosystem development. I will employ a novel multidiciplinary framework, which integrates ecological process, like plant community development, into the biogeochemical processes. This much needed integration makes it possible to improve and add new mechanisms to current ecosystem models and to answer under what conditions is the system is most vulnerable to change under frequent and severe wildfires. Three questionbased work packages are described below as the basis of this wildfire research program: 1. CN LOSSES. CN losses. Where in the landscape do the largest C and N losses occur, and what factors control losses? How large are CN combustion losses relative to C transformed into charcoal and hydrologicallyexported CN following fire? 2. CN POOL DEVELOPMENT. What is the relative importance of abiotic (e.g. soil moisture, temperature) and biotic (e.g. plant traits) factors in generating variation in postfire recovery rate of C and N pools at different spatial scales? 3. VEGETATION DEVELOPMENT. What controls species and trait assembly postfire? What is the role of nichebased processes (abiotic effects: environmental filtering, and biotic effects: legacy effects, regeneration traits) in contrast to neutral processes (stochasticity, priority effects)?