ROOT PROBLEM FOR CONVENIENT MAPS

In this paper we study when the minimal number of roots of the so-called convenient maps horn two-dimensional CW complexes into closed surfaces is zero We present several necessary and sufficient conditions for such a map to be root free Among these conditions we have the existence of specific fittings for the homomorphism induced by the map on the fundamental groups, existence of the so-called mutation of a specific homomorphism also induced by the map, and existence of particular solutions of specific systems of equations on free groups over specific subgroups

Wecken type problems for self-maps of the Klein bottle

We consider various problems regarding roots and coincidence points for maps into the Klein bottle . The root problem where the target is and the domain is a compact surface with non-positive Euler characteristic is studied. Results similar to those when the target is the torus are obtained. The Wecken property for coincidences from to is established, and we also obtain the following 1-parameter result. Families which are coincidence free but any homotopy between and , , creates a coincidence with . This is done for any pair of maps such that the Nielsen coincidence number is zero. Finally, we exhibit one such family where is the constant map and if we allow for homotopies of , then we can find a coincidence free pair of homotopies.

The Brezis-Nirenberg type problem involving the square root of the Laplacian

We establish existence and non-existence results to the Brezis-Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

Minimal Nielsen Root Classes and Roots of Liftings

Given a continuous map f : K -> M from a 2-dimensional CW complex into a closed surface, the Nielsen root number N(f) and the minimal number of roots mu(f) of f satisfy N(f) <= mu(f). But, there is a number mu(C)(f) associated to each Nielsen root class of f, and an important problem is to know when mu(f) = mu(C)(f)N(f). In addition to investigate this problem, we determine a relationship between mu(f) and mu((f) over tilde), when (f) over tilde f is a lifting of f through a covering space, and we find a connection between this problems, with which we answer several questions related to them when the range of the maps is the projective plane.

Root Water Extraction under Combined Water and Osmotic Stress

Using a numerical implicit model for root water extraction by a single root in a symmetric radial flow problem, based on the Richards equation and the combined convection-dispersion equation, we investigated some aspects of the response of root water uptake to combined water and osmotic stress. The model implicitly incorporates the effect of simultaneous pressure head and osmotic head on root water uptake, and does not require additional assumptions (additive or multiplicative) to derive the combined effect of water and salt stress. Simulation results showed that relative transpiration equals relative matric flux potential, which is defined as the matric flux potential calculated with an osmotic pressure head-dependent lower bound of integration, divided by the matric flux potential at the onset of limiting hydraulic conditions. In the falling rate phase, the osmotic head near the root surface was shown to increase in time due to decreasing root water extraction rates, causing a more gradual decline of relative transpiration than with water stress alone. Results furthermore show that osmotic stress effects on uptake depend on pressure head or water content, allowing a refinement of the approach in which fixed reduction factors based on the electrical conductivity of the saturated soil solution extract are used. One of the consequences is that osmotic stress is predicted to occur in situations not predicted by the saturation extract analysis approach. It is also shown that this way of combining salinity and water as stressors yields results that are different from a purely multiplicative approach. An analytical steady state solution is presented to calculate the solute content at the root surface, and compared with the outputs of the numerical model. Using the analytical solution, a method has been developed to estimate relative transpiration as a function of system parameters, which are often already used in vadose zone models: potential transpiration rate, root length density, minimum root surface pressure head, and soil theta-h and K-h functions.

Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

We discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation-(u' / root 1 - u'(2))' = f(t, u). Depending on the behaviour of f = f(t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

Root-locus practical sketching rules for fractional-order system

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.

Positive radial solutions of the Dirichlet problem for the Minkowski-Curvature equation in a ball

We study the existence and multiplicity of positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation { -div(del upsilon/root 1-vertical bar del upsilon vertical bar(2)) in B-R, upsilon=0 on partial derivative B-R,B- where B-R is a ball in R-N (N >= 2). According to the behaviour off = f (r, s) near s = 0, we prove the existence of either one, two or three positive solutions. All results are obtained by reduction to an equivalent non-singular one-dimensional problem, to which variational methods can be applied in a standard way.

Natural Arabidopsis brx loss-of-function alleles confer root adaptation to acidic soil.

Soil acidification is a major agricultural problem that negatively affects crop yield. Root systems counteract detrimental passive proton influx from acidic soil through increased proton pumping into the apoplast, which is presumably also required for cell elongation and stimulated by auxin. Here, we found an unexpected impact of extracellular pH on auxin activity and cell proliferation rate in the root meristem of two Arabidopsis mutants with impaired auxin perception, axr3 and brx. Surprisingly, neutral to slightly alkaline media rescued their severely reduced root (meristem) growth by stimulating auxin signaling, independent of auxin uptake. The finding that proton pumps are hyperactive in brx roots could explain this phenomenon and is consistent with more robust growth and increased fitness of brx mutants on overly acidic media or soil. Interestingly, the original brx allele was isolated from a natural stock center accession collected from acidic soil. Our discovery of a novel brx allele in accessions recently collected from another acidic sampling site demonstrates the existence of independently maintained brx loss-of-function alleles in nature and supports the notion that they are advantageous in acidic soil pH conditions, a finding that might be exploited for crop breeding.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Root parametrized differential equations

The present thesis is about the inverse problem in differential Galois Theory. Given a differential field, the inverse  problem asks which linear algebraic groups can be realized as differential Galois groups of Picard-Vessiot extensions of this field.   In this thesis we will concentrate on the realization of the classical groups as differential Galois groups. We introduce a method for a very general realization of these groups. This means that we present for the classical groups of Lie rank $l$ explicit linear differential equations where the coefficients are differential polynomials in $l$ differential indeterminates over an algebraically closed field of constants $C$, i.e. our differential ground field is purely differential transcendental over the constants.   For the groups of type $A_l$, $B_l$, $C_l$, $D_l$ and $G_2$ we managed to do these realizations at the same time in terms of Abhyankar's program 'Nice Equations for Nice Groups'. Here the choice of the defining matrix is important. We found out that an educated choice of $l$ negative roots for the parametrization together with the positive simple roots leads to a nice differential equation and at the same time defines a sufficiently general element of the Lie algebra. Unfortunately for the groups of type $F_4$ and $E_6$ the linear differential equations for such elements are of enormous length. Therefore we keep in the case of $F_4$ and $E_6$ the defining matrix differential equation which has also an easy and nice shape.   The basic idea for the realization is the application of an upper and lower bound criterion for the differential Galois group to our parameter equations and to show that both bounds coincide. An upper and lower bound criterion can be found in literature. Here we will only use the upper bound, since for the application of the lower bound criterion an important condition has to be satisfied. If the differential ground field is $C_1$, e.g., $C(z)$ with standard derivation, this condition is automatically satisfied. Since our differential ground field is purely differential transcendental over $C$, we have no information whether this condition holds or not.   The main part of this thesis is the development of an alternative lower bound criterion and its application. We introduce the specialization bound. It states that the differential Galois group of a specialization of the parameter equation is contained in the differential Galois group of the parameter equation. Thus for its application we need a differential equation over $C(z)$ with given differential Galois group. A modification of a result from Mitschi and Singer yields such an equation over $C(z)$ up to differential conjugation, i.e. up to transformation to the required shape. The transformation of their equation to a specialization of our parameter equation is done for each of the above groups in the respective transformation lemma.

Soil injections of carbohydrates improve fine root growth of established urban trees

Four established mature tree species (Aesculus hippocastanum L., Betula pendula Roth., Primus avium L. and Quercus rohur L.) commonly planted in UK urban landscapes were subjected to soil injections of the carbohydrate sucrose at 25, 50 and 70g per litre of water. Fine root dry weight was recorded at month 5 following soil injections. Soil injections of sucrose significantly increased fine root dry weight compared to controls, however; growth responses were influenced by species and the concentration of sucrose applied. Results indicate soil injections of sucrose ≥ 50g litre of water may be able to improve root growth of established mature trees. Such a response is desirable as root damage following construction is a frequent problem encountered by established trees growing in UK towns and cities.

Linear vs. log-linear unit-root specification: an application of mis-specification encompassing

The objective of this paper is to apply the mis-specification (M-S) encompassing perspective to the problem of choosing between linear and log-linear unit-root models. A simple M-S encompassing test, based on an auxiliary regression stemming from the conditional second moment, is proposed and its empirical size and power are investigated using Monte Carlo simulations. It is shown that by focusing on the conditional process the sampling distributions of the relevant statistics are well behaved under both the null and alternative hypotheses. The proposed M-S encompassing test is illustrated using US total disposable income quarterly data.

Epidemics in networks of spatially correlated three-dimensional root-branching structures

Using digitized images of the three-dimensional, branching structures for root systems of bean seedlings, together with analytical and numerical methods that map a common susceptible-infected- recovered (`SIR`) epidemiological model onto the bond percolation problem, we show how the spatially correlated branching structures of plant roots affect transmission efficiencies, and hence the invasion criterion, for a soil-borne pathogen as it spreads through ensembles of morphologically complex hosts. We conclude that the inherent heterogeneities in transmissibilities arising from correlations in the degrees of overlap between neighbouring plants render a population of root systems less susceptible to epidemic invasion than a corresponding homogeneous system. Several components of morphological complexity are analysed that contribute to disorder and heterogeneities in the transmissibility of infection. Anisotropy in root shape is shown to increase resilience to epidemic invasion, while increasing the degree of branching enhances the spread of epidemics in the population of roots. Some extension of the methods for other epidemiological systems are discussed.

Projektarbete i praktiken : Hur kan oförutsedda problem vid projektöverlämningen minimeras

Projektarbete förekommer på många företag och trots att det ofta finns tydligt beskrivna rutiner för hur projektarbetet skall ske händer det att parterna inte är helt överens då projektet ska avslutas. Ibland är det ändringar under projektets gång som ligger till grund för kommande tvister, men i grunden är det oftast kommunikationen som brister mellan parterna. Genom att intervjua några olika företag och analyserat hur de arbetar i projekt har vi i detta arbete kunnat se en röd tråd som består av den viktiga kommunikationen för att skapa och bibehålla sunda relationer. För att hantera de förändringar som kan ske under projektresans gång ser vi i vår analys av de företag som ingått i vår studie att det är av fördel att ha ett nära samarbete med kunder eller leverantörer och helst starta samarbetet tidigt för att möjliggöra justeringar som annars i ett senare skede blir svårare att genomföra. Ett tillvägagångssätt att arbeta med ett projekts interna problem kan vara plattformar som möjliggör en delning av information mellan projektets olika medlemmar. Angreppsvinkeln är då att det ska bli lättare för att samtliga projektdeltagare att kunna ta del av informationen med en förhoppning att minimera missförstånd inom projektet.