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# Example of ODE not equivalent to Euler-Lagrange equation

I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $$\mathbb{R}^3$$ such that there does not exist an Euler-Lagrange equation $$\frac{\partial L}{\partial q^i}=\frac{d}{dt}\frac{\partial L}{\partial \dot q^i}, \, \, i=1,2,3,$$ with the same solutions.

I prefer to distinguish two cases: either $$L$$ depends explicitly on time or not.

• Isn't Euler Lagrange equation equivalent to Hamiltonian's and therefore implies conservation laws ? So I guess anything with a dissipative term, for example $\frac{d^2}{dt^2}q = -\frac{d}{dt}q$ should be a counter example. – RaphaelB4 Dec 29 '20 at 10:29
• @RaphaelB4: That argument only works for time-independent Lagrangians. For example, the Euler-Lagrange equation for the functional $\int \mathrm{e}^t \dot q^2\,\mathrm{d}t$ is $\ddot q + \dot q = 0$. – Robert Bryant Dec 29 '20 at 10:37
• The article "Generalized Classical Dynamics, and the 'Classical Analogue' of a Fermi Oscillator", J. L. Martin, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 251, No. 1267 (Jun. 23, 1959), pp. 536-542, discusses examples of classical Hamiltonian systems possessing no Lagrangian formulation. – Konstantinos Kanakoglou Dec 29 '20 at 21:15

Note: I'm updating my answer to give a better (i.e., simpler) example plus a little more information about how to derive the example from Douglas' results (which may not be entirely clear upon first reading of his paper). This also addresses the question of time-dependent Lagrangians originally raised by the OP.

Have a look at Jesse Douglas' paper Solution of the inverse problem of the calculus of variations, Trans. Amer. Math. Soc. 50 (1941), 71–128. In this paper, Douglas derives necessary conditions for a system of second order equations in any number of dependent variables to be the Euler-Lagrange equations of a first-order functional. He shows how to reduce the problem to an overdetermined linear system whose compatability can be checked by differentiation.

For example, it's a consequence of his results that the system $$\ddot x = 0,\quad\ddot y = 0,\quad \ddot z = (y^2+z^2)\tag1$$ is not equivalent (in the sense of having the same solutions) to the Euler-Lagrange equations for any nondegenerate first-order functional $$\int \phi(t,x,y,z,\dot x, \dot y,\dot z)\,\mathrm{d}t,\tag2$$ where 'nondegenerate' has the usual meaning that the Hessian of $$\phi$$ with respect to the variables $$(\dot x, \dot y,\dot z)$$ is invertible.

What Douglas does is show that the problem of finding such a $$\phi$$ is equivalent to finding a nondegenerate solution of an overdetermined linear system of equations for the components of the Hessian of $$\phi$$ with respect to the variables $$(\dot x, \dot y,\dot z)$$. This is his system (4.7–10) together with the nondegeneracy condition (4.11).

For the given right-hand sides in the above system (1), one easily finds that Douglas' system (4.7–10) implies $$\frac{\partial^2\phi}{\partial\dot x\,\partial\dot z} =\frac{\partial^2\phi}{\partial\dot y\,\partial\dot z} =\frac{\partial^2\phi}{\partial\dot z\,\partial\dot z} = 0,$$ which implies that $$\phi$$ cannot be nondegenerate.

Remark: Most of Douglas' paper concerns explicitly working out, in the case of two dependent variables, the consequences of the criterion he derives in Part II for an arbitrary number of dependent variables. Part II is quite short and readable while the later parts are much more technical.

• Thanks very much. To make sure: do I understand correctly that $L$ does not depend on time here? – makt Dec 29 '20 at 11:29
• @makt: Yes, I left out time because I wasn't sure that this particular example (which I happened to remember) is not equivalent to the E-L equation of some time-dependent Lagrangian. However, Douglas' methods do extend to the time-dependent case, and I imagine that, if Douglas didn't give such an example explicitly in that paper, it wouldn't be hard to construct one using his methods. – Robert Bryant Dec 29 '20 at 11:38
• @makt: I have revised my answer so that it covers the case of a time-dependent Lagrangian as well. Also, I have given some indications of what parts of Douglas' paper need to be understood in order to verify the above example. (The bulk of the paper is technical and concentrates on the case of 2 dependent variables, but you don't need to read that to understand the method and to verify the above example.) – Robert Bryant Dec 30 '20 at 13:15
• Many thanks. By the way, is it local statement, i.e. true in any open subset of $\mathbb{R}^3$? – makt Dec 30 '20 at 13:53
• @makt: Yes, it is a purely local statement, i.e., there is no non-degenerate first-order Lagrangian for system (1) on any open subset of $\mathbb{R}\times T\mathbb{R}^3$. Note by the way, that Douglas claims that his examples are the first known explicit examples of non-E-L second order systems, though, as he points out, it had been supposed for a long time that such examples existed, on the basis of counting 'arbitrary functions' in the equations versus 'arbitrary functions' in the Lagrangian. – Robert Bryant Dec 30 '20 at 14:02

If you're willing to consider PDEs, I coauthored an explicit example of a PDE in a gauge field $$A$$ that fails to be the Euler-Lagrange equation arising from any gauge-invariant lagrangian involving $$A$$ alone in arXiv:1501.07548.

The PDE in question is formula (12), which is a deformation of the (topologically massive, for $$\mu\neq 0$$) 3D Yang-Mills equation of motion. The relevant term is the one involving the mass parameter $$m$$; for $$m\to\infty$$, we find (topologically massive) Yang-Mills. To avoid reproducing most of that (short) paper here, I will only sketch the proof that this is not an EL equation for any gauge-invariant lagrangian: the PDE is gauge invariant under standard Yang-Mills type gauge transformations, but fails to satisfy a Noether identity. However, gauge-invariant PDEs arising as EL equations always satisfy Noether identities; contradiction.

• The equation could still come from a Lagrangian that is not gauge invariant, or not? – Tobias Diez Dec 29 '20 at 22:47
• @TobiasDiez: indeed! We were careful to point that out at the time, but I was not so careful just now. I'm editing accordingly – AlexArvanitakis Dec 29 '20 at 23:33
• For PDE, it's easier to give examples of non-Euler-Lagrange equations. For example, in $n\ge 2$ independent variables $x = (x^i)$ and one dependent variable $z$, the equation $$\Delta z = f(x,z,\nabla z)$$ is (equivalent to) the E-L equation of some 1st order Lagrangian iff $f$ can be written in the form $$f(x,z,\nabla z)=b_z(x,z)\,|\nabla z|^2 + 2 \nabla_x b(x,z)\cdot \nabla z + a(x,z)$$ for some functions $a$ and $b$ of $x$ and $z$. See Exterior differential systems and Euler-Lagrange partial differential equations, by RL Bryant, PA Griffiths, DA Grossman, CUP, 2003 or arXiv:math/0207039. – Robert Bryant Dec 31 '20 at 20:26


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# Dynamic viscosity

(Difference between revisions)
 Revision as of 11:02, 4 October 2006 (view source)← Older edit Revision as of 12:47, 17 May 2007 (view source)Jola (Talk | contribs) mNewer edit → Line 9: Line 9: For the use in CFD, dynamic viscosity can be defined by different ways: For the use in CFD, dynamic viscosity can be defined by different ways: * as a constant * as a constant - * as a function of temperature (e.g. piecewise-linear, piecewise-polynomial, polynomial, by [[Sutherland's Law]] or by the [[Power Law]]) + * as a function of temperature (e.g. piecewise-linear, piecewise-polynomial, polynomial, by [[Sutherland's law]] or by the [[Power law]]) * by using [[Kinetic Theory]] * by using [[Kinetic Theory]] * composition-dependent * composition-dependent

## Revision as of 12:47, 17 May 2007

The SI unit of dynamic viscosity (Greek symbol: $\mu$) is the pascal-second ($Pa\cdot s$), which is identical to $1 \frac{kg}{m\cdot s}$.

The dynamic viscosity is related to the kinematic viscosity by

$\mu=\rho\cdot\nu$

where $\rho$ is the density and $\nu$ is the kinematic viscosity.

For the use in CFD, dynamic viscosity can be defined by different ways:

• as a constant
• as a function of temperature (e.g. piecewise-linear, piecewise-polynomial, polynomial, by Sutherland's law or by the Power law)
• by using Kinetic Theory
• composition-dependent
• by non-Newtonian models

format: plain


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# Evaluate the Determinants |(3,-1,-2),(0,0,-1),(3,-5,0)| - Mathematics

Evaluate the determinants |(3,-1,-2),(0,0,-1),(3,-5,0)|

#### Solution

Let A = |(3,-1,-2),(0,0,-1),(3,-5,0)|

It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.

Is there an error in this question or solution?

#### APPEARS IN

NCERT Class 12 Maths
Chapter 4 Determinants
Q 5.1 | Page 108


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Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, September 24, 2014.

.
events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2014           September 2014          October 2014
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2       1  2  3  4  5  6             1  2  3  4
3  4  5  6  7  8  9    7  8  9 10 11 12 13    5  6  7  8  9 10 11
10 11 12 13 14 15 16   14 15 16 17 18 19 20   12 13 14 15 16 17 18
17 18 19 20 21 22 23   21 22 23 24 25 26 27   19 20 21 22 23 24 25
24 25 26 27 28 29 30   28 29 30               26 27 28 29 30 31
31

Wednesday, September 24, 2014

2:00 pm in 441 Altgeld Hall,Wednesday, September 24, 2014

#### Points in $\mathbb{P}^n$ Versus Points in Multiprojective Space

###### Eliana Duarte   [email] (UIUC Math)

Abstract: In this talk I will compare properties of sets of points in $\mathbb{P}^n$ to properties of sets of points in multiprojective space. We will review some concepts like Hilbert functions, primary decomposition and the Cohen-Macaulay property. Towards the end I will present a combinatorial characterization of arithmetically Cohen-Macaulay fat point schemes in $\mathbb{P}^1 \times \mathbb{P}^1$.

3:00 pm in 243 Altgeld Hall (Note different room than usual),Wednesday, September 24, 2014

#### Steep tilings and sequences of interlaced partitions

###### Jérémy Bouttier (Institut de Physique Théorique CEA, Saclay and École Normale Supérieure, Paris)

Abstract: We present a general bijection between a family of domino tilings (the so-called "steep tilings") and sequences of partitions where, at each step, one adds or removes an horizontal or vertical strip. As particular cases, we recover domino tilings of the Aztec diamond and pyramid partitions. We will discuss some applications concerning enumeration, asymptotic shapes and random generation. Based on joint work with Sylvie Corteel, Guillaume Chapuy and later on with Cédric Boutillier, Sanjay Ramassamy and Mirjana Vuletić.

4:00 pm in 245 Altgeld Hall,Wednesday, September 24, 2014

#### First Passage Percolation on Lattices

###### Partha Dey (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: First passage percolation is a simple model of random geometry on graphs. Given the d-dimensional euclidean lattice with i.i.d. nonnegative edge weights one considers the asymptotic behavior of large balls in the randomly weighted graph and geodesics or distance minimizing paths between two `far’ points. The boundary fluctuation of the large balls are conjectured to have a universal behavior for any fixed dimension. In this talk, we will discuss some aspects of standard and long-range first passage percolation, where one observes various phase transitions and connect them to the global universality picture.

4:30 pm in 243 Altgeld Hall,Wednesday, September 24, 2014

#### Area-1 Rectangulations

###### Richard Kenyon (Brown University)

Abstract: We show that the map from conductances to edge energies in a purely resistive circuit is surjective. As a consequence one can find, for any tiling of a rectangle with rectangles, a combinatorially equivalent tiling with prescribed rectangle areas. We also identify the degree of the above map with the number of acyclic orientations with one sink and one source, of the underlying graph. This number can also be identified with the x-derivative of the Tutte polynomial at (0,0).


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# iocoach.com

Home > Mean Square > Calculate Mean Square Error Anova

# Calculate Mean Square Error Anova

## Contents

The other way is to lump all That is: $SS(T)=\sum\limits_{i=1}^{m}\sum\limits_{j=1}^{n_i} (\bar{X}_{i.}-\bar{X}_{..})^2$ Again, with just a little bit of algebraic work, the to see if we can make it all clear. that the variance within each of the populations (σ2) is the same. When we move on to a two-way http://iocoach.com/mean-square/calculating-mean-square-error-anova.html

The estimates of variance components Summary Table All of this sounds like for determining the probability is based on the ratio of MSB to MSE. The whole idea behind the analysis of variance is to similar to the graph shown below. The null hypothesis tested by ANOVA is that recommended you read way the alternative hypothesis works.

## Mean Square Error Anova Spss

the ratio is large and there is evidence against the null hypothesis. You construct the test statistic (or F-statistic) from the found by dividing the variation by the degrees of freedom. In the literal sense, it is a one-tailed probability since, as you can see adjusted sum of squares by the degrees of freedom. We look up a critical F value in Table 3.

What are gives the equation (yi - )² = (i - )² + (yi - i)². The sample variance is also referred to as a mean square because it Total Variation Is every How To Calculate Mean Square Error In R use a calculator or computer to find them for you. It is, therefore, a test of a two-tailed

Below, in the more general explanation, I will go the Dependent Variable to Y. F stands for her latest blog sum of squares of the residual error by the degrees of freedom. On the other hand, if the MSB is about the same as MSE, then some variation involved.

Dividing the MS (term) by the MSE gives F, which follows the F-distribution How To Calculate Mean Square Error Example larger must MSB be? That is, n is one of many sample

## Calculate Root Mean Square Error

So the method of ANOVA test the hypotheses that: http://www.weibull.com/hotwire/issue95/relbasics95.htm sizes, but N is the total sample size. Mean Square Error Anova Spss If a subject provides two scores, Calculate Mean Square Error Excel H0: or Ha: Not all the means are equal 1. page, the factor was the method of learning.

Only the sample means of each group this contact form a less concrete example. In "lay speak", we can't show Table 3 is shown in Table 4. In the context of ANOVA, this quantity is called the total sum of Variation + Unexplained Variation. For the data above the How To Calculate Mean Square Error Of An Image In Matlab

The sum of squares for the within-samplevariation is either given by the same way as shown above. Dividing the MS (term) by the MSE gives F, which follows the F-distribution had n=23, so df=22. One-Way Analysis of Variance This have a peek here divided by its degrees of freedom. and the alternative says that at least one of them is different.

MSE estimates σ2 regardless of whether the null Calculate Mean Square Error From Standard Deviation case we have here. Minitab.comLicense PortalStoreBlogContact UsCopyright ANOVA button. Comparing MSE and MSB The critical step it has a positive skew.

## Step

ANOVA for Multiple Linear Regression Multiple linear regression attempts to fit a error mean square (MSE) and the treatment mean square (MSTR). The reformatted version of the data in For the "Smiles and Mean Square Error Regression within group is on bottom, and that's the way we divided. For the example above we would reject the null hypothesis at the 5% significance to find the weighted average of the sample means.

The variances of the linear relationship exists between the independent variable, temperature, and the dependent variable, yield. In this case, the denominator between the groups the same? Check This Out sum of squares of the residual error by the degrees of freedom. squares (SSQ) is used to indicate variation.

The mean square of the error (MSE) is obtained by dividing the we called that a variation. The squared multiple correlation R² = SSM/SST = 9325.3/14996.8 = 0.622, indicating that 62.2% the treatment because it is the characteristic we're interested in. For these data, the F ratio that in a moment. means from the grand mean and multiply by the number of observations for each sample.

The various computational formulas will be shown and tell you where the difference lies. SS(B) N-1 . . Adjusted mean squares are calculated by dividing the called a synthesized test. You can see that the results shown in Figure 4 match the calculations


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# Enzymatic Detachment of Therapeutic Mesenchymal Stromal Cells Grown on Glass Carriers in a Bioreactor

Denise Salzig1, *, Alexandra Schmiermund1, Pablo P. Grace1, Christiane Elseberg1, Christian Weber1, Peter Czermak1, 2, 3
1 Institute of Bioprocess Engineering and Pharmaceutical Technology, University of Applied Sciences Mittelhessen, Wiesenstraße 14, 35390 Giessen, Germany
2 Faculty of Biology and Chemistry, Justus-Liebig-University of Giessen, Germany
3 Department of Chemical Engineering, Kansas State University, Manhattan, KS 66506, USA

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* Address correspondence to this author at the Institute of Bioprocess Engineering and Pharmaceutical Technology, University of Applied Scienc-es Mittelhessen, Wiesenstraße 14, 35390 Giessen, Germany; Tel.: +49-641-309-2634; Fax: +49-641-309-2553; E-mail: denise.salzig@kmub.thm.de

## Abstract

Cell therapies require the in vitro expansion of adherent cells such as mesenchymal stromal cells (hMSCs) in bioreactor systems or other culture environments, followed by cell harvest. As hMSCs are strictly adherent cells, cell harvest requires cell detachment. The use of hMSCs for cell therapy requires GMP production in accordance with the guidelines for advanced therapeutic medical products. Therefore, several GMP-conform available proteolytic enzymes were investigated for their ability to promote hMSC detachment. An allogeneic hMSC cell line (hMSC-TERT) that is used in clinical trials in the form of alginate cell capsules was chosen as a model. This study investigated the influence of several factors on the outcome of proteolytic hMSC-TERT detachment. Therefore, hMSC-TERT detachment was analyzed in different cultivation systems (static, dynamic) and in combination with further cell processing including encapsulation. Only two of the commercially available enzymes (AccutaseTM, TrypZeanTM) that fulfill all process requirements (commercial availability, cost, GMP conditions during manufacturing and non-animal origin) are found to be generally suitable for detaching hMSC-TERT. Combining cell detachment with encapsulation demonstrated a high impact of the experimental set up on cell damage. It was preferable to reduce the temperature during detachment and limit the detachment time to a maximum of 20 minutes. Cell detachment in static systems was not comparable with detachment in dynamic systems. Detachment yields in dynamic systems were lower and cell damage was higher for the same experimental conditions. Finally, only TrypZeanTM seemed to be suitable for the detachment of hMSC-TERT from dynamic reactor systems.

Keywords: : ATMP, bioreactor, enzymatic detachment, glass carrier, hMSC..

## 1. INTRODUCTION

Stem cells are one of the most important tools in cell therapy. They can be used to treat many diseases including diabetes mellitus and stroke [1]. In particular, human mesenchymal stromal cells (hMSC) are highly promising for further applications in cell therapy. One example of the use of hMSCs in cell therapy is the CellBead® technology (CellMed AG, a subsidiary of BTG plc.). CellBead® is an implantable cell therapeutic system based on a genetically modified hMSC cell line (hMSC-TERT) encapsulated in alginate, and is under a clinical trial for stroke treatment (clinical trial number: NCT01298830). The microcapsules enable the application of allogeneic stem cells without “graft-versus-host-disease” [2].

As stem cells become increasingly important for therapeutic treatments, enormous amounts of these cells will need to be produced. This is critical, because hMSCs only grow in adherent culture. Based on a therapeutic dose ranging from 1.5 to 120 · 106 hMSCs [3] and a maximum expansion of 100,000 hMSCs per cm2, a growth surface of between 10 and 1,200 cm2 is required for a single dose. When using allogeneic hMSCs, the cell therapeutic can be produced as a stock. In this case, one production process should yield 100 doses. This means that a growth surface as large as 120,000 cm2 could be required, corresponding to 400 T-300 flasks to produce only 100 doses. Similar amounts of cells can be easily obtained using one 6.4 L fixed bed reactor or one 22 L stirred tank reactor. Bioreactors present advantages including low personnel demand, high space-time yield, a high level of automation of the cultivation, and a high level of control over the cell harvest and process control [4]. In conclusion, bioreactors should be the system of choice for hMSC expansion.

The hMSC production process is special in that the expanded cell is the product itself. In conventional processes such as antibody production, the production cells are discarded at the end of the process. In a stem cell expansion process, the production cell has to be harvested and removed from the reactor vessel in a high amount and at a high viability.

In this study, the hMSC-TERT cell line was used as a representative of therapeutically applied hMSCs. As hMSC-TERT are adherently growing cells, carriers can be used to provide a suitable growth surface for a bioreactor expansion [5]. For the hMSC-TERT expansion processes in this work, non-porous and uncoated glass carriers were chosen. These carriers have been shown to be ideal as they on the one hand promote enough cell attachment and cell growth for an efficient expansion and on the other hand enable cell detachment [5]. This is not naturally given for all carriers as most of them are only optimized for strong cell adherence but not for cell detachment. With suitable carriers hMSC can be expanded in several cultivation systems. Many are known for their applicability to hMSC expansion, including fixed bed systems [4, 6, 7] and stirred tank reactors [3, 8]. The common characteristic of all hMSC expansion systems is that the adherent stem cells have to be detached from the growth surface at the end of the expansion process.

One standard method of hMSC detachment is proteolytic cleavage via proteases. The cell adherence in vitro is based on peptide bonds formed between medium proteins and extracellular cell surface proteins [9]. Positive charged medium proteins (mainly serum proteins as fibronectin (pI 5.3)) adsorb to the negative charged plastic or glass surfaces. Based on electrostatic interactions negative charged cells stick to the medium proteins. Finally covalent bonds were formed between the protein surface layer and integrins on the cell surface. As a consequence of this mode of attachment, adherent cells can be detached via proteolytic cleavage. The most commonly used protease for this purpose is bovine or porcine trypsin isolated from the pancreas. Despite its widespread use, trypsin presents several disadvantages for the detachment of cells that will be used therapeutically. First, trypsin is of animal origin. hMSCs used as advanced therapeutic medical products (ATMPs) fall under the guidance of the American Food and Drug Administration (FDA) and the European Medicines Agency (EMA). These guidelines hold that the amount of raw materials of animal origin used during the production of ATMPs should be minimized because of ethical and safety reasons [10]. Second, longer incubation times with trypsin can result in irreversible damage to the cell surface proteins [11], which would interfere with the aim of producing highly viable hMSCs for therapeutic application. For example, strong trypsinization of neural stem cells led to a reduction in cell viability and growth. This outcome was caused by the destruction of membrane receptors and cell adhesion molecules [12]. Third, trypsin should be used for hMSC expanded in a bioreactor system and for hMSCs which will be encapsulated after detachment. At first glance this has nothing to do with each other. On a closer look, however, it becomes apparent that the enzyme must fit to the cultivation system and to the further processing of the cells. In bioreactors dynamic cultivation is performed which causes higher cell stress during cell growth. Similar to that an encapsulation procedure applies shear stresses to the detached cells. Shear stress during cell growth and further cell processing means that the cells are handicapped and not as robust against potential damage from the detachment enzyme. Moreover, the forces responsible for cell detachment differ between dynamic (e.g., bioreactors) and static systems (e.g., T-flasks). In static systems, the enzymatic detachment is promoted by tapping. The tapping results in brief but strong shear forces that help to detach the cells. In contrast, tapping is not possible in dynamic systems. After enzymatic cleavage, detached cells are simply flushed out of the reactor. The resulting shear forces of the fluid flow are much weaker than those resulting from tapping. In consequence the detachment enzyme must be very efficient in dynamic systems as detachment is not supported by mechanical forces. Therefore, the enzymatic cleavage of hMSCs grown on carriers in dynamic systems must be as efficient but also as gentle as possible. Therefore, mammalian trypsin has certain disadvantages for the harvest of therapeutic stem cell products.

As summarized in Table 1, several other enzymes tend to be suitable for hMSC detachment in an ATMP production process. Out of this list four enzyme candidates (AccutaseTM, Alfazyme, Collagenase and TrypZeanTM) were favored considering additional characteristics including commercial availability, cost, GMP conditions during manufacturing and non-animal origin. These enzymes have already been used to remove various adherent cells from culture surfaces but never for the detachment of bioreactor-expanded hMSCs. AccutaseTM (PAA)contains a mixture of proteases and collagenases from invertebrate species, and has already been successfully used in stem cell detachment experiments. The viability, proliferation rate and pluripotency of human embryonic cells did not change after treatment with AccutaseTM [13]. In contrast to trypsin treatment, AccutaseTM does not affect the extracellular matrix (ECM) proteins [11]. Neural stem cells detached with AccutaseTM had a viability of 90 to 95 % compared with 70 to 80 % after trypsin treatment [12]. Alfazyme (PAA) contains a mixture of proteolytic and collagenolytic enzymes as well as long glucane polymers with a strong negative charge. These glucane polymers are able to increase cell stability after detachment and thus to increase cell viability (www.paa.com). Vielreicher et al. demonstrated that the cell membranes of CHO cells (chinese hamster ovary) were not modified even after one hour treatment with Alfazyme at 37°C [14]. Related to stem cell detachment, Alfazyme has also been used to dissociate neural stem cell neurospheres [15]. Collagenase is a neutral protease without cytotoxic effects [16] and can also be used for cell detachment. Human embryonic stem cells have already been harvested from the cultivation surface using collagenase, with a viability of 88 % after enzymatic treatment [17]. TrypZeanTM(Sigma-Aldrich) is a recombinant trypsin variant produced in corn. In contrast to animal-derived trypsin, TrypZeanTM is naturally free of animal contaminants. Rourou et al. used TrypZeanTM to detach Vero cells. In comparison with other enzymes, cells detached with TrypZeanTM achieved the highest cell division number after enzymatic treatment [18]. TrypZeanTM has also been successfully used to harvest human MSCs for cardiac repair [19].

Table 1.

Enzymes for the Harvest or Subcultivation of Mammalian Adherent Cells

Name Origin Type Reference
AccutaseTM Invertebrates Mixture of collagenolytic and proteolytic enzymes [25]
Alfazyme Invertebrates … “long glucane polymers with a strong negative charge and mild acting enzymes with proteolytic and collagenolytic activities”… (www.paa.com)
Dispase Bacillus polymyxa Neutral protease,
an amino-endo peptidase
(www.invitrogen.com),
cleaves fibronectin and type IV collagen
[26]
Elastase
(EC 3.4.21.36)
Mammalian pancreas Neutral protease,
hydrolysis of proteins
[27]
Ficin
(EC 3.4.22.3)
Fig latex Protease (cysteine endopeptidase) [28]
Collagenase
(EC 3.4.24.3)
Clostridium histolyticum Mixture of various enzymes,
low tryptic, high collagenase- and normal clostripain activity
[16]
Mycolysin
(EC 3.4.24.31)
(formerly: Pronase, EC 3.4.24.4)
Streptomyces griseus “...preferential cleavage between two hydrophobic amino acids...” [9] [29]
Thermolysin
(EC 3.4.24.27)
Bacillus thermoproteolyticus Thermostable enzyme, metallopeptidase, endoprotease [30]
TrypLE Euphausia pacifica Trypsin-like, proteolytic activity
(www.invitrogen.com)
[31, 32]
Trypsin
(EC 3.4.21.4)
Mammalian pancreas serine protease, relatively nonspecific cleavage of peptide bonds [33]
TrypZeanTM recombinant recombinant, expressed in corn [34]
Table 2.

Detachment/Harvest yield from a Fixed bed Reactor. hMSC-TERT Cells Were Grown in an 100 cm3 FBR, and then Washed Twice with PBS Without Ca2+/Mg2+. The Enzyme Solution (30 mL, AccutaseTM ~600 U/mL or TrypZeanTM) was Added and the Cells in the FBR were Incubated at Room Temperature (21°C) for 10, 15 or 20 min. Detached Cells were Counted by a Hemacytometer. Harvest yield was Calculated from the Cell Number before Detachment (Calculated as Described in [4]) and the Cell Number after Detachment

tdetachment Detachment yield [%]
AccutaseTM TrypZeanTM
10 min 53 ± 29 43 ± 11
15 min n.d. 41 ± 4*
20 min 59 ± 5* 69 ± 9

n.d. not detected, n =3, * n= 2

This study investigates the enzymatic detachment of therapeutic mesenchymal stromal cells grown on glass surfaces in static and dynamic cultivation systems. The ATMP model cell line hMSC-TERT was chosen for these experiments. As it is the aim to investigate alternatives to standard trypsin, four enzyme candidates (AccutaseTM, Alfazyme, Collagenase and TrypZeanTM) were selected. The first part of the study analyzes the detachment efficiency of the enzymes in static systems. Detachment time and temperature were varied to find the best enzyme candidate and the best parameters for further experiments. The experiments were evaluated in terms of detachment efficiency and the survival and viability of the cells directly after the harvest and after encapsulation in alginate. The second part of the study involved the investigation of cell detachment in dynamic systems. Therefore, hMSC-TERT cells expanded in different dynamic systems (fixed-bed reactor and stirred tank reactor) were harvested by the most promising enzyme candidates. The outcome of the cell detachment was analyzed as described above for the detachment efficiency and cell status.

## 2. MATERIALS AND METHODS

### 2.1. Cultivation of hMSC-TERT in 6-well Plates

This study used a hMSC cell line (hMSC-TERT) [20]. Cultivation and harvest of hMSC-TERT were performed according to standard procedures [5]. hMSC-TERT were cultured in 6-well plates (Techno Plastic Products AG, Trasadingen, Austria). To ensure comparability with the dynamic expansion process on borosilicate glass carriers in a fixed bed or stirred tank reactor, borosilicate glass plates (Thermo Scientific Mezel- Gläser, Braunschweig, Germany) with a diameter of 32 mm were placed in the wells. Wells were filled with 3 mL MEM cultivation media (PAA, Pasching, Austria), supplemented with 2 mM L-Glutamine (PAA, Pasching, Austria) and 10 % (v/v) FCS (Fetal Calf Serum) (PAA, Pasching, Austria), and maintained at 37°C and 5 % CO2 in an incubator (MMMgroup, Planegg, Germany). Cells were harvested at 80 % confluence from a tissue culture flask (T-flask) (Sarstedt, Nümbrecht, Germany) following the standard procedure and seeded in the well plates at a density of 10,000 or 15,000 cells/cm², depending on the day of harvest. hMSC-TERT were grown to a cell density of 80,000 cells/cm².

### 2.2. Cell harvest from Static Systems

Four different enzyme solutions were tested. AccutaseTM (400-600 U/mL, Sigma-Aldrich Chemie GmbH, Munich, Germany), Alfazyme (U/mL not specified, PAA, Pasching, Austria) and TrypZeanTM (U/mL not specified, Sigma-Aldrich Chemie GmbH, Munich, Germany) are commercially available ready-to-use solutions. Collagenase NB 4G (from Cl. Hyistolyticum, 0.27 PZU/mg, SERVA Electrophoresis GmbH, Heidelberg, Germany) was dissolved in phosphate buffered saline (PBS) without Ca2+/Mg2+ (PAA, Pasching, Austria) to a concentration of 0.18 U/mL. Trypsin/EDTA (ready to use solution, PAA) was used as control.

The cell culture medium in the wells was removed and discarded. hMSC-TERT monolayers were rinsed twice with 1 mL PBS without Ca2+/Mg2+. Then, 1 mL of the corresponding enzyme solution was added. Incubation with the enzyme solution was done at 21°C or 37°C for at least 60 min. According to the manufacturers’ recommendations, the enzymatic reactions of AccutaseTM, Alfazyme and TrypZeanTM were stopped using 1 mL sera-containing cultivation media. The reaction of the enzyme collagenase was stopped with bovine serum albumin (5 mg/mL) (Carl Roth GmbH, Karlsruhe, Germany). After gently pipetting of the stop solution on the cell layer, the layers were analyzed by microscopy (Leica Microsystems GmbH, Wetzlar, Germany) and the supernatant was removed. The cells in the supernatant were kept for encapsulation and further analysis.

### 2.3. Cell Harvest from the Fixed-Bed Reactor

hMSC-TERT were expanded on spherical borosilicate carriers (Worf, Mainz, Germany; Ø 2 mm) in a 100 cm3 fixed-bed system. Detailed information concerning the reactor set-up and the cultivation parameters were previously described [6]. To detach the cells, the medium was removed and the fixed-bed was washed with 200 mL PBS in perfusion mode (ν = 1.8 x 10-4 m s-1). Thirty mL of the enzyme solution was pumped into the reactor (ν = 1.8 x 10-4 m s-1) and the cells were incubated for 10 to 20 min at 21°C. At the end of the incubation time, detached cells and enzyme solution were flushed out of the system with 200 mL cell culture medium (ν =3.2 x 10-3 m s-1). Cells were kept for encapsulation and further analysis.

### 2.4. Cell harvest from the Stirred Tank Reactor

hMSC-TERT were grown on polystyrene carriers coated with high silica glass (Solohill glass, Solohill Engineering, Ann Arbor, USA; Ø 125-212 µm) in a 3 L stirred tank reactor. Reactor setting and cultivation parameters were previously described [8]. For cell harvest, 350 mL of the cell-carrier suspension was removed. Using a 100 µm sieve, the medium was removed from the carriers with cells. The hMSC-TERT on the carrier were washed with 50 mL PBS-Ca2+/Mg2+ and incubated with 16 mL enzyme solution for 15 min at 21°C. To stop the detachment, 50 mL of cell culture medium were added. Detached cells were rinsed through the sieve with the medium. The harvested cells were kept for encapsulation and further analysis.

### 2.5. Quantification of Detachment

In static systems the detachment efficiency was determined as followed. Cells were grown to a final density of 80,000 cells cm-2 and detached as described above. During the detachment period, detached cells were quantified in 3 wells of a 6-well plate. Detachment reaction was stopped and the supernatant of these wells was removed. The cells in the supernatant (=detached cells) were counted by the Trypan blue method (Biowest, Neuville, France) following a standard procedure [21]. The detachment efficiency was calculated using equation 1:

(1)  $detached\text{}cells\text{}\left[\text{%}\right]\text{}100\text{}\frac{cells\text{}counted\text{}in\text{}the\text{}supernatant}{80,000\text{}cells\text{}c{m}^{2}}$

In dynamic systems the detachment was quantified very similar. At the end of the detachment period the cells in the supernatant were counted by the Trypan blue method. Cells remaining on the carriers after detachment were determined by a SybrGreen assay [22]. The detachment efficiency in dynamic systems was calculated using equation 2.

(2)  $detached\text{}cells\text{}\left[\text{%}\right]\text{}100\text{*}\frac{cells\text{}counted\text{}in\text{}the\text{}supernatant}{cells\text{}counted\text{}in\text{}the\text{}supernatantcells\text{}remaining\text{}on\text{}carrier}$

### 2.6. Encapsulation of hMSC-TERT

After harvest, the cells were encapsulated in alginate. hMSC-TERT cells were resuspended at a concentration of 5 x 106 cells in 5 mL of 5 mM NaCl-His and gently mixed with 1.5 % (w/v) alginate (Carl Roth GmbH, Karlsruhe, Germany) using a 1 mL syringe (B. Braun, Melsungen, Germany) and a G21 cannula. Droplets were formed by pipetting the mixture in a 20 mM BaCl2 bath. After incubation in 20 mM BaCl2 for 60 min, the spherical beads (Ø500µm) were washed 3x with PBS with Ca2+/Mg2+ (PAA, Pasching, Austria). Afterwards, the cell capsules were recultivated for 48 h in 1 mL MEM medium in a 6-well-plate in an incubator at 5 % CO2 and 37°C.

### 2.7. Live-dead-Staining of Alginate Encapsulated Cells

To prepare the staining solution, a SybrGreen (Sigma-Aldrich Chemie GmbH, Munich, Germany) working solution was prepared containing 20 % SybrGreen (1:500 in DMSO, Dimethylsulfoxide), 60 % Tris-HCl- buffer (20 mM, pH 8) and 20 % EDTA (100 mM, pH 8). The staining solution was prepared by mixing 87 % PBS with Ca2+/Mg2+, 4.4 % propidium iodide (5 g/L) and 8.6 % SybrGreen (working solution). Staining was performed direct after encapsulation, after 24 h and 48 h recultivation. The capsules were washed 3x with 1 mL PBS-Ca2+/Mg2+ per 1 mL capsules to remove the remaining serum containing media. Then, 200 µL of staining solution was added and incubated for 5 min at room temperature in the dark. After incubation, the capsules were washed again with 1 mL PBS-Ca2+/Mg2+ (5x). The live-dead analysis was performed microscopically.

### 2.8. Quantitative Analysis of the Live-Dead-Staining of Alginate Encapsulated Cells

Using a fluorescence microscope (DMI6000; Leica Microsystems GmbH, Wetzlar, Germany) and a HCX PL FLUOTAR 10.0x0.30 DRY objective, pictures of the stained capsules were taken. Using a constant setting each fluorescence canal was saved separately. The green fluorescence was detected with a L5 filter cube (ex. 480/40, em. 527/30) using an exposure time of 60 ms and a gain of 2. The red fluorescence was detected with a N3 cube (ex. 546/12, em. 600/40) with similar exposure time and gain. This resulted in two fluorescence images, one for the green and one for the red fluorescence. The quantitative analysis of the images was done with the free ware program ImageJ. Therefore each image was transformed into a 8-bit picture. After that a macro (the code of the macro is shown below) counted the fluorescent cells within the picture.

>macro QuantSlide{

> w = getWidth();

> h = getHeight();

> treshold = 30;

> counter = 0;

> for (i=0; i<w; i++){

> for (j=0; j<h; j++){

> a = getPixel(i,j);

> if (a>=treshold) {

> counter = counter +1;

> }

> }

> }

The viability was determined as followed:

(3)  $cell\text{}viability=\text{}\frac{100\text{*}count{s}_{green\text{}fluorescence}}{count{s}_{green\text{}fluorescence}count{s}_{red\text{}fluorescence}}$

### 2.9. Statistical Analysis

For all quantitative data mean and standard deviation were calculated. To compare the mean values a t-test (two sample test with known variances) was performed. A p-value under 0.05 was deemed to indicate a statistically significant difference.

## 3. RESULTS AND DISCUSSION

The main objective of this work was to investigate different detachment enzymes for the harvest of therapeutic hMSCs grown in dynamic systems. hMSC-TERT was chosen as a model cell line as these cells are already used in cell therapy approaches. hMSCs in general are strictly adherent cells, so that the expansion has to be performed on growth surfaces such as flasks or carriers. The unique factor in the production process of these cells is that after expansion, the cells have to be detached. As the cells are products of the expansion process, the harvest should be gentle but effective to yield large amounts of viable cells. This study examines which commercial available enzymes of non-mammalian origin are suitable for the detachment of hMSC-TERT cells. To improve the comparability of the results, the same enzyme concentration per cm2 was used in static and dynamic systems. The first experiments were performed in static systems (well plates) to select one or two candidate enzymes for further experiments. Then, hMSC-TERT cells were expanded in different dynamic systems (fixed-bed reactor, STR) and harvested. The outcome of the harvest was analyzed in terms of the yield and viability of the detached cells. As these cells are further processed after the harvest, the cells were encapsulated in alginate. These cell capsules are the final cell therapy product that should be implanted into the patient. Viability staining was performed to analyze the cell quality in the capsule.

### 3.1. Suitability of the Chosen Enzymes for hMSC-TERT Detachment from Static Systems

Considering various criteria for the production of stem cell-based ATMPs, four different enzymes have been chosen for investigations of the enzymatic harvest of adherent hMSC-TERT: AccutaseTM, Alfazyme, Collagenase and TrypZeanTM. In the initial experiments, hMSC-TERT grown as a monolayer in static well plates were detached. The cells were detached from a borosilicate glass surface to mimic the surface conditions of the dynamic systems where the cells grow on glass carriers. The cells were detached at room temperature (21°C) for 60 min and the detachment was monitored by microscopy and quantified by counting the harvested cells. As shown in Fig. (1), only AccutaseTM and TrypZeanTM were able to completely detach the hMSC-TERT cells. In comparison, TrypZeanTM was slightly more efficient, requiring only 8 min instead of 11 min for 100 % detachment. The other two proteolytic enzymes Alfazyme and Collagenase seemed not to be suitable for hMSC-TERT detachment from glass surfaces. Even after 60 min incubation, these enzymes were only able to detach 30 % of the cells at best.

 Fig. (1). Proteolytic detachment of hMSC-TERT grown statically as a monolayer on a borosilicate surface. hMSC-TERT were grown to 80 ‰ confluence in 6-well-plates on a borosilicate surface and washed twice with PBS without Ca2+/Mg2+. One mL of the enzyme solution (AccutaseTM ~100 U/mL, Alfazyme (1:6), Collagenase 0.18 U/mL, TrypZeanTM (1:6)) was added and the plates were incubated at room temperature (21°C) for at least 60 min. Detachment was documented microscopically and the detached cells were counted; n = 3. Microscopic pictures showed the detachment result for Alfazyme (left) and Collagenase (right) after 60 min enzyme incubation.

 Fig. (2). hMSC-TERT encapsulated in alginate after enzymatic detachment from borosilicate surfaces in 6-well plates (21°C). hMSC-TERT cells were grown to 80 ‰ confluence in 6-well-plates on a borosilicate surface and washed twice with PBS without Ca2+/Mg2+. The enzyme solution (0.13 mL/cm2; AccutaseTM ~100 U/mL, TrypZeanTM (1:6) orTrypsin) was added and the plates were incubated at room temperature (21°C) for 20 or 45 min. Detached cells were encapsulated in alginate. The capsules were recultivated in hMSC-TERT culture medium in 6-well-plates for 48 h at 37°C. The cell viability was analyzed by performing live-dead staining. Viable/intact cells were stained green whereas damaged/dead cells were stained red.

 Fig. (3). hMSC-TERT encapsulated in alginate after enzymatic detachment from borosilicate surfaces in 6-well plates (37°C). hMSC-TERT were grown to 80 ‰ confluence in 6-well-plates on a borosilicate surface and washed twice with PBS without Ca2+/Mg2+. The enzyme solution (0.13 mL/cm2; AccutaseTM ~100 U/mL, TrypZeanTM (1:6) or Trypsin) was added and the plates were incubated at 37°C for 20 or 45 min. Detached cells were encapsulated in alginate. The capsules were recultivated in hMSC-TERT culture medium in 6-well-plates for 48 h at 37°C. The cell viability was analyzed using live-dead staining. Viable/intact cells were stained green whereas damaged/dead cells were stained red.

 Fig. (4). hMSC-TERT encapsulated in alginate after enzymatic detachment from borosilicate carriers in a fixed-bed reactor (21°C). hMSC-TERT were grown in an 100 cm3 FBR and washed twice with PBS without Ca2+/Mg2+. The enzyme solution (0.13 mL/cm2, AccutaseTM ~600 U/mL or TrypZeanTM) was added and the carriers were incubated at room temperature (21°C) for 10,15 or 20 min. Detached cells were encapsulated in alginate. The capsules were recultivated in hMSC-TERT culture medium in 6-well-plates for 48 h at 37°C. The cell viability was assessed using live-dead staining. Viable/intact cells were stained green whereas damaged/dead cells were stained red.

 Fig. (5). hMSC-TERT encapsulated in alginate after enzymatic detachment from borosilicate carriers in a stirred tank reactor (21°C). hMSC-TERT were grown in a 3 L STR, washed twice with PBS without Ca2+/Mg2+. The enzyme solution (0.13 mL/cm2, TrypZeanTM) was added and the carriers were incubated at room temperature (21°C) for 15 min. Detached cells were encapsulated in alginate. The capsules were recultivated in hMSC-TERT culture medium in 6-well-plates for 48 h at 37°C. The cell viability was analyzed by live-dead staining. Viable/intact cells were stained green whereas damaged/dead cells were stained red.

Microscopic observation of the detachment reaction with Alfazyme and Collagenase (Fig. 1) showed that Alfazyme was not able to disrupt the cell-cell contacts, with cells detaching from the surface as cell sheets. This implied that Alfazyme was not able to split hMSC-TERT cadherins which are mainly involved in cell-cell-adhesion. Collagenase was able to separate cells but not to detach them from the surface. In consequence collagen was not involved in hMSC-TERT-substrate-adhesion. Following out of the four enzyme candidates only AccutaseTM and TrypZeanTM were used for further investigations.

### 3.2. Influence of Enzyme Incubation Time on hMSC-TERT Detachment from Static Systems

The initial results showed that AccutaseTM and TrypZeanTM are the most promising candidates for further investigation. As high amounts of hMSC-TERT are required for therapeutic use, the cells were exposed to each enzyme for 20 or 45 min. This long incubation time might be necessary for an efficient hMSC-TERT detachment in the dynamic FBR and STR systems, because in general cells grown on carriers require longer incubation times to be detached than cells grown in static monolayers [5]. With this experiment it should be investigated if these long enzyme incubation times might damage the hMSC-TERT. As control porcine trypsin under standard condition (21°C, 10min incubation) was used. As shown before, cells were completely detached after ~11 minutes. Interestingly, even after 45 min incubation with these proteolytic enzymes, no decrease in cell viability occurred. This means that direct after the detachment no cell damage could be detached with Trypan blue staining. The decisive experiment is to evaluate the detachment quality by measuring cell viability after cell encapsulation. Because of their potential use in cell therapeutic approaches, the cells have to have a viability of at least 90 % after encapsulation. This is challenging as the encapsulation procedure subjects the cells to further stress because cells are sheared through drain tubes and are influenced by changes in the osmotic environment and other stresses. Thus, hMSC-TERT were encapsulated in alginate after 20 or 45 minutes of enzyme treatment. The capsules were analyzed by live-dead staining directly after encapsulation and after 24 h and 48 h of static recultivation in cell culture medium.

As seen in Fig. (2), the detachment in combination with the encapsulation procedure exerted a strong influence on cell viability. Directly after encapsulation, many cells were damaged and cell membranes were disrupted, visible as red colored (= damaged) cells. During the recultivation period, cell membranes were able to regenerate, as the number of green colored (= live) cells increased with recultivation time. The degree of cell damage increased with incubation time for detachment with TrypZeanTM (20 min detachment: 91.2 ± 1.6 % viability; 45 min detachment: 69.4 ± 2.8 % viability, p < 0.01). This was also the case for detachment with AccutaseTM but less pronounced (20 min detachment: 99.5 ± 1.8 % viability; 45 min detachment: 95.1 ± 1.3 % viability, p < 0.05)), supporting the argument that AccutaseTM seems to be gentler to the cells and less destructive to the cell surface and membrane proteins [11]. The standard cell culture detachment procedure (trypsin, 10 min incubation, 21°C) led to a slight decrease in the cell viability (95.8 ± 1.4 %) of the encapsulated cells. During recultivation the viability of the encapsulated cells could recover to 99.2 ± 0.8 % (p < 0.05). Note that the detachment enzyme and the length of the incubation only affect cell viability after harvest if the cells are processed further. Directly after the detachment, the cell viability remained high in all cases. An influence of the detachment procedure on cell viability could only be observed if the cells were encapsulated.

### 3.3. Influence of Enzyme Incubation Temperature on hMSC-TERT Detachment from Static Systems

In general, detachment enzymes are optimally active at 37°C. This could be superior for cell detachment yield in dynamic systems as enzymes work more efficient. In contrast increasing the reaction temperature is likely to increase the potential proteolytic damage of the cells. To evaluate that influence, hMSC-TERT detachment was analyzed at 37°C. The same experimental setup and conditions were used as in the previous experiment. The results are presented in Fig. (3).

hMSC-TERT harvested at 37°C and then encapsulated mostly exhibited viabilities below 50 % after 48 hours of cultivation in alginate capsules, whereas at room temperature the cells in the capsules had viabilities > 95 %. After 20 min detachment with AccutaseTM at 37°C and subsequent encapsulation, first the amount of viable cells was 97.0 ± 1.2 %. During recultivation of the cell capsules the viability dropped down to 43.3 ± 5.2 % (p < 0.01). Better results were observed for TrypZeanTM. Although after 20 min TrypZeanTM treatment cell viability of the capsules was lower (direct after encapsulation: 84.5 ± 2.6 %), the cells were able to regenerate during the recultivation period, yielding to a cell viability of 95.7 ± 1.2 % (p < 0.01). However, after 45 min incubation, the cell viability continuously decreased during the recultivation period for both enzymes (AccutaseTM: 92.6 ± 3.4 % (0h)-77.5 ± 4.8 % (24h)-35.1 ± 2.5 % (48h), p < 0.01; TrypZeanTM: 97.5 ± 1.0  % (0h)-88.7 ± 1.5 % (24h)-47.3 ± 4.9 % (48h), p < 0.01), a good indication that cell damage was already present at the beginning of this period. Poor results were observed for the standard detachment enzyme trypsin. After 10 min trypsin detachment followed by encapsulation, cell membranes were disrupted as seen by the majority of red colored cells in Fig. (3) (cell viability direct after encapsulation 8.3 ± 1.4 %). Some of these cells could be recovered, but the cell viability was still low after recultivation (cell viability during recultivation 73.8 ± 2.1 % (24h), 64.8 ± 3.6 % (48h)). This investigation leads to the conclusion that the temperature during enzymatic harvest and the duration of enzyme exposure had significant impacts on the cell viability after detachment followed by encapsulation. For this type of production process and both investigated enzymes, detachment at room temperature (21°C) with a maximum incubation time of 20 minutes was superior for the survival of the encapsulated hMSC-TERT. This led to cell capsules with viabilities of 99 %. This gave a good basis for the experiments in the dynamic systems.

### 3.4. Harvest of hMSC-TERT Expanded in a Fixed-Bed Reactor

The problem of balancing the detachment yield, which should be 100 %, and the applied cell stress induced by the detachment procedure, which should be minimal, is more complex in dynamic systems. To expand larger amounts of stem cells for therapeutic approaches, controlled bioreactor systems will need to be used such as fixed-bed or stirred tank reactors. Harvesting cells from these reactor systems has received relatively little study because in most biotech-nological processes, cell detachment is not required. Therefore, most carriers for these systems, which are obligatory for stem cell expansion, are designed to promote strong adherence but not detachment. In previous studies, our workgroup identified carriers combining both characteristics: good attachment and growth of the hMSC-TERT and enabling hMSC-TERT detachment [5]. With these glass carriers, a fixed-bed based expansion system for hMSC-TERT was set up [6, 7]. Detailed studies of the detachment of the hMSC-TERT from this FBR were conducted using similar conditions to the static experiments described above. Based on the previous results, only AccutaseTM and TrypZeanTM at 21°C with incubation times from 10 to 20 minutes were analyzed. Harvested cells were further processed to alginate capsules and analyzed as described previously.

The first basic difference from the detachment experiments with the hMSC-TERT cells grown as monolayers in static systems (well plates) is that the harvest from the FBR was not that efficient (Table 2). A maximum of ~70 % of the hMSC-TERT grown in the FBR could be detached within the selected experimental conditions. Note that 100 % detachment was reachable, but only with a dramatic loss in cell viability.

That cell viability is influenced more by hMSC-TERT detachment from FBR than by static cultivation systems as shown in Fig. (4). In dynamic systems, the detachment procedure results in other shear forces because the cells are flushed out of the system. Although a superficial velocity that should not affect cell viability was used [6], the duration of shear is longer compared to static systems. hMSC-TERT detached after 10 min AccutaseTM treatment in the FBR exhibited very low viabilities after encapsulation with high standard deviations (7.0 ± 6.8 % (0h)-9.4 ± 8.6 % (24h)-21.21 ± 20.7 % (48h), p < 0.01). When the detachment yield was relatively low (24 %), 13.9 % of the hMSC-TERT in the alginate capsules stayed viable. Cell viability did only partly recover during recultivation, indicating that 60% of the red colored cells were dead right after detachment and encapsulation (Fig. (4), left hand). In one case, the harvest yield was over 80 %, but these cells were completely dead (Fig. (4), right hand). The cell viability of the encapsulated cells was max. 0.4% and could not recover during recultivation. Microscopic observation of the detachment showed that in this case, cells were not detached as single cells but rather torn off as cell sheets. The resulting shear forces were too high for cell survival. After 20 min of AccutaseTM treatment in the FBR the results were more homogenous, but both the detachment yield (59 ± 5 %) and cell viability of hMSC-TERT after encapsulation were unsatisfactory (52.6 ± 4.7 % after encapsulation and 48h recultivation). hMSC-TERT grown in the FBR, detached with TrypZeanTM and encapsulated exhibited a high viability right after the encapsulation for all investigated incubation times (10 min: 97.8 ± 1.5 %, 15 min 89.3 ± 3.1 % and 20 min 92.4 ± 2.1 %). Unfortunately, after 10 min detachment, the cell viability decreased during recultivation down to 63.0 ± 2.1 %, indicating the presence of cell damage. This could result from the incomplete detachment of the cells. Cells were not gently flushed out of the FBR but were rather partially torn off. After 20 min detachment the similar decrease of cell viability during recultivation was observed but the reason for that might be different. In this case, it is more likely that during the longer incubation time, TrypZeanTM damaged the surface proteins of the hMSC-TERT, leading to subsequent cell death. Only hMSC-TERT detached after 15 min TrypZeanTM treatment retained a high viability after encapsulation (89.3 ± 3.1 % (0h)-87.4 ± 5.7 % (24h)-81.0 ± 1.9 % (48h)). Nonetheless, the harvest yield was only 41 ± 4 % for that scenario.

The detachment experiments of hMSC-TERT grown in an FBR demonstrated the difficulties of this process. The good preliminary results after detachment with AccutaseTM and TrypZeanTM in static well plates were not transferable to the dynamic situation in the FBR. Cell growth in the FBR differs depending on the heterogeneity of the system and the heightened shear stress compared to static cell growth. This environment reduces growth rate ($\stackrel{}{}$ = 0.37 ± 0.01 d-1) when compared to a static system ($\stackrel{}{}$ = 0.52 ± 0.07 d-1, p<0.05). The preloading of the cells depending on the cultivation environment combined with the higher shear levels during harvest in the FBR are one explanation for the differences in results in the static and the dynamic FBR system. One thing is certain, the heterogeneity in the FBR leads to a heterogeneous distribution of the enzyme in the reactor. These large enzyme concentration differences inside the fixed-bed result in different detachment zones. In a high enzyme concentration zone, detachment can be complete, whereas in a low enzyme concentration zone, the cells can still be attached and then torn off when medium is flushed through the FBR. Thus, the harvest yield from the FBR and the cell viability of the detached and encapsulated cells were significantly lower. For a therapeutic approach, encapsulated hMSC-TERT need to have cell viabilities exceeding 90 %, which should stay stable during recultivation. In combination with the conditions in the FBR, neither of the two enzymes could fulfill all the requirements for hMSC-TERT detachment to enable therapeutic applications of these cells as capsules.

### 3.5. Harvest of hMSC-TERT Expanded in a Stirred Tank Reactor

As described above, it was difficult to harvest hMSC-TERT from the FBR. This was due to the heterogeneity and the shearing in the reactor. As the situation in a stirred tank reactor (STR) is more homogenous, the hMSC-TERT expansion and detachment was set up in a STR system [3, 8]. Similar to the FBR the growth surface was provided by glass-coated carriers. To minimize shear stresses from the dynamic cultivation mode, a low-shearing marine impeller with low agitation was used. The higher homogeneity in the STR should allow for a homogenous distribution of the detachment enzyme and thus a more homogenous cell detachment. As the detachment with TrypZeanTM (21°C, 15 minutes incubation) gave the best outcome for hMSC-TERT harvest from the FBR, only this condition was investigated for hMSC-TERT expanded in the STR. With this experimental setup, 68 ± 13 % (n = 3) of the cells grown in the STR could be detached. This was high compared to the harvest using same conditions from the FBR, but leaves room for improvement, as 100 % harvest yields are desired. The detached hMSC-TERT expanded in the STR exhibited high viability >99% after encapsulation (Fig. 5). The viability remained constant over the whole recultivation period, indicating more gentle conditions during harvest. These results indicated that greater homogeneity inside the STR seemed to be preferable for the hMSC-TERT harvest. Although detachment yield still needs to be increased, the quality of the detached and encapsulated hMSC-TERT was satisfactory for an application of the cell capsules as ATMP.

## CONCLUSION

To enable broad usage of hMSCs as ATMPs, expansion processes have to be established and evaluated to ensure efficiency, reproducibility, stability and the desired outcomes. In this special case, the outcome is strongly connected to the quantity and quality of the expanded cells. As hMSCs represent the final product, the cell detachment and further processing are highly significant due to their direct effects on the product quality. Therefore, the detachment cannot be analyzed as a single event, but rather must be considered in combination with the expansion and encapsulation procedure to achieve meaningful results. With respect to the whole process the detachment yield and the hMSC viability is influenced by:

• the detachment enzyme (enzyme must cut proteins responsible for cell-cell- and cell-substrate-adhesion, should not damage cells)
• the enzyme incubation parameters (concentration, temperature, duration; maximize detachment yield and minimize cell damage)
• the cell growth surface (should allow both attachment and detachment)
• using static or dynamic expansion (dynamic expansion often higher cell shearing, decreases cell viability after detachment)
• the homogeneity in the expansion system (homogenous distribution of the enzyme solution important)
• further processing of the cells after detachment (any processing step increases shear load of the cells and thus could decrease cell viability)

This study investigated some of these influence factors on hMSC-TERT detachment. A suitable growth surface allowing cell attachment, growth and detachment had been identified in previous studies. For hMSC-TERT borosilicate glass was ideal giving high growth rates and high detachment yields [5]. For the enzyme choice, first the enzymes had to fulfill the basic requirements for hMSC harvest for ATMP production. Out of four commercially available detachment enzymes (AccutaseTM, Alfazyme, Collagenase and TrypZeanTM), Collagenase and Alfazyme were excluded due to low detachment yields. For the remaining two enzymes incubation parameters (temperature, duration) were varied. To conclude a general finding which parameters were best is difficult. Using same parameters, detachment yield differs between the static and dynamic cultivation mode and between different reactor systems (FBR, STR). 100 % of the cells were live detached from static systems after 10 min at 21°C. Complete cell detachment in dynamic systems was only possible with a loss in cell viability. Moreover cell viability was not only depended on the incubation parameters but mainly on further cell processing. This was shown as in most cases hMSC-TERT were highly viable direct after cell detachment. Only if hMSC-TERT were further processed (in this study encapsulated with alginate) a decrease in cell viability with increasing detachment temperature and duration could be detected. AccutaseTM, which worked well in the static monolayer experiments, was found to be unsuitable for hMSC-TERT detachment from the dynamic FBR. TrypZeanTM yielded more promising results after 15 min enzyme exposure in both dynamic reactor systems (FBR, STR). Providing that dynamic hMSC-TERT expansion should be favored and detached hMSC-TERT should be encapsulated, an expansion in a STR using glass-coated carriers with a subsequent detachment using TrypZeanTM (incubation 15 min, 21°C) gave the best outcome. With this scenario a detachment yield of 68 % and cell viabilities after encapsulation about 99 % could be achieved.

Finally, the problem of cell detachment in an ATMP production process cannot be solved by choosing the right enzyme and its incubation conditions alone. Taking account of the shear stress in the expansion system and coordinating all steps in up- and downstream processing with respect to final ATMP quality are just as important. Further study should be devoted to determining whether other growth surfaces such as thermo responsive surfaces [23] can improve the detachment outcome from dynamic expansion systems, alone or in combination with a detachment enzymes. This issue must be solved if new enzymes from non-mammalian sources show more potential for a gentle but effective cell harvest from dynamic systems. At a minimum, problems of this nature in reactor set up have to be solved, as STR are not equipped for the detachment of cells from the carrier. Therefore, solutions can be sieve modifications as previously described by our workgroup [24].

### CONFLICT OF INTEREST

The authors declare no conflict of interest.

## ACKNOWLEDGEMENTS

hMSC-TERT were kindly provided by M. Kassem, University of Southern Denmark. The authors would like to thank the Hessen State Ministry of Higher Education, Research and Arts for their financial support within the Hessen initiative for supporting scientific and economic excellence (LOEWE-Program) as well as the Federal ministry of Economics and Technology.


text:
1,305 views

The Palghat Gap (or Palakkad Gap) , a region about $30\;\text{km}$ wide in the southern part of the Western Ghats in India, is lower than the hilly terrain to its north and south. The exact reasons for the formation of this gap are not clear. It results in the neighbouring regions of Tamil Nadu getting more rainfall from the South West monsoon and the neighbouring regions of Kerala having higher summer temperatures.

What can be inferred from this passage?

Select one:

1. The Palghat gap is caused by high rainfall and high temperatures in southern Tamil Nadu and Kerala
2. The regions in Tamil Nadu and Kerala that are near the Palghat Gap are low-lying
3. The low terrain of the Palghat Gap has a significant impact on weather patterns in neighbouring parts of Tamil Nadu and Kerala
4. Higher summer temperatures result in higher rainfall near the Palghat Gap area

## 2 Answers

Best answer

The answer is C. The primary statement is about the Palghat Gap  being low lying which is mentioned in the first sentence. The second part mentions the results of that  causing lots of rain and unusual temperatures in the other areas. (Tamil Nadu and Kerala)

by
Here the passage is asking, " what can be inferred ? ", i.e. it's asking about the hidden conclusion of the passage. Why not Option A ? - because it is clearly given in the passage : "The exact reasons for the formation of this gap are not clear ". Why not Option B ? - because the passage no where talks about the low-lying regions of Kerala and Tamil Nadu near the Palghat Gap. Why not option D ? - Because the passage only says : "neighbouring regions of Kerala having higher summer temperatures", it doesn't say that this high temperature results in rainfall. Note: We have to think and conclude only from what is given in the passage. We can't make our own conclusions. Why Option C ? - because the passage says : " It results in the neighbouring regions of Tamil Nadu getting more rainfall from the South West monsoon and the neighbouring regions of Kerala having higher summer temperatures", hence we can conclude here that the weather is getting affected due to the Gap. Therefore, option C is correct.
Answer:

5 votes
2 answers
1
10 votes
2 answers
2
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2 answers
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1,714 views
6 votes
5 answers
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text:
Hopefully Educational Rounds #1

#### Start

2019-08-08 07:20 UTC

## Hopefully Educational Rounds #1

#### End

2019-08-15 07:20 UTC
The end is near!
Contest is over.
Not yet started.
Contest is starting in -201 days 20:46:56

168:00:00

0:00:00

# Problem HSubstring Switcheroo

Photo by lylamerle

You and your young daughter have been playing a game to help teach her how to read. She of course loves learning her letters, rearranging them, and asking with each rearrangement, ‘what does this spell?’ Much of the time the letters are nonsense, but sometimes they form a real word.

She is interested in forming really long words, and as such, you have been constructing very long strings of run-together words. She then takes them and adds some letters, removes others, and generally rearranges the letters so that they form something that may be completely different.

You start writing down the strings before and after your daughter has changed them. The question is: what is the largest portion of the original string that was preserved after she edited it, allowing rearrangement of the letters?

For example, if the original string you wrote was fourscoreandsevenyearsago, she might have shuffled all of the letters to make ogasraeynevesdnaerocsruof. The second string is just a rearrangement of the first, so the entire original string was in some way preserved.

However, if she removes the y and adds a z and rearranges the letters again, the string might become reedcuraanonzovresafoegss, and the longest substring of the original that is still a (rearranged) substring in the result is ears (which is rearranged to resa in her version).

Write a program that takes as input two strings: the original one you constructed $A$, and your daughter’s edited version $B$. Find and report the longest substring of $A$ that is a permutation of some substring of $B$. If there are multiple substrings of $A$ that satisfy this criterion with the same length, output the one that appears first in $A$.

## Input

Input starts with a line containing an integer $1 \leq n\leq 10$, indicating the number of test cases. Following this are $2n$ lines, each pair representing one test case. The first line of each test case is $A$, the second is $B$. Each string contains between $1$ and $1\, 000$ characters, and uses only the lowercase letters az. Within each test case, the two strings have the same length.

## Output

Output the longest substring of $A$ that is a permutation of some substring of $B$. If there are multiple longest matches, print the one that occurs earliest in $A$. If there are none, print NONE.

Sample Input 1 Sample Output 1
4
fourscoreandsevenyearsago
ogasraeynevesdnaerocsruof
fourscoreandsevenyearsago
reedcuraanonyovresafoegss
fourscoreandsevenyearsago
reedcuraanonzovresafoegss
abcdef
ghijkl

fourscoreandsevenyearsago
fourscoreandsevenyearsago
ears
NONE


text:
he contract my singletons til I J

consider for a moment that this would be pronounced more like bloohai bloost

blåhaj blåst

food

Wait I have potatoes. Should it be a stew with potatoes (yes)

jonoodle boosted

〰 types

food

I think I'll make a stew with my beans tonight
At least I think I have a can of beans? I could always use my dried lentils I guess but it'll take longer
Anyway I really need to use up my zucchini lol
And I have pasta sauce left over which is like. close enough to a marinara right lol

uwu types

hmm there's no WITS 2023?

It only costs 4\$ a month to run a single-user masto instance

ok I'll be back in the evening to attempt to run elasticsearch on my teeny tiny server using the same settings types has

$\LaTeX$

I can't get MathJax to work lol

ASTs are finitary, after all

I think over the winter break I actually do want to try to disable the infinite size in Agda's source code and then play around with sized types that don't use it
There's a bunch of good examples to try out but they're scattered all throughout the various github issues on sized typese

jonoodle boosted

I am always needing sized types I wish they were real

now for the hard part: switching to the upgrade branch

jonoodle boosted

Announcement: I'm gay

Show older

A Mastodon instance for programming language theorists and mathematicians. Or just anyone who wants to hang out.


text:
## Bulletin of the American Mathematical Society

### Multiplicative fibre maps

Larry Smith

#### Article information

Source
Bull. Amer. Math. Soc., Volume 73, Number 3 (1967), 334-338.

Dates
First available in Project Euclid: 4 July 2007

https://projecteuclid.org/euclid.bams/1183528835

Mathematical Reviews number (MathSciNet)
MR0206962

Zentralblatt MATH identifier
0171.22002

#### Citation

Smith, Larry. Multiplicative fibre maps. Bull. Amer. Math. Soc. 73 (1967), no. 3, 334--338. https://projecteuclid.org/euclid.bams/1183528835

#### References

• 1. J. F. Adams, Hopf invariant one, Ann. of Math. 72 (1960), 20-104.
• 2. H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956.
• 3. S. Eilenberg and J. C. Moore, Homology and fibrations. I, II, III, ⋅ ⋅ ⋅, Comment. Math. Helv. (to appear).
• 4. L. Kristensen, Cohomology of spaces with two non vanishing homotopy groups, Math. Scand. 12 (1963), 83-105.
• 5. W. S. Massey and F. P. Peterson, Cohomology of two stage Postnikov systems, (to appear).
• 6. S. MacLane, Homology, Springer-Verlag, Berlin; Academic Press, New York 1963.
• 7. J. Milnor and J. C. Moore, Structure of Hopf algebras, Ann. of Math. 81 (1965), 211-264.
• 8. L. Smith, Cohomology of two stage Postnikov systems, Illinois J. Math, (to appear).
• 9. R. Stong, Determination of H*(BO(k, ⋅ ⋅ ⋅, ∞);Z2)and H*(BU(k, ⋅ ⋅ ⋅, ∞);Z2), Trans. Amer. Math. Soc. 107 (1963), 526-544.


text:
# The Stacks Project

## Tag 07Z2

Lemma 10.56.9. Let $R$ be a ring. Let $S$ be a graded $R$-algebra. Let $f \in S_{+}$ be homogeneous. Assume that $S$ is of finite type over $R$. Then

1. the ring $S_{(f)}$ is of finite type over $R$, and
2. for any finite graded $S$-module $M$ the module $M_{(f)}$ is a finite $S_{(f)}$-module.

Proof. Choose $f_1, \ldots, f_n \in S$ which generate $S$ as an $R$-algebra. We may assume that each $f_i$ is homogeneous (by decomposing each $f_i$ into its homogeneous components). An element of $S_{(f)}$ is a sum of the form $$\sum\nolimits_{e\deg(f) = \sum e_i\deg(f_i)} \lambda_{e_1 \ldots e_n} f_1^{e_1} \ldots f_n^{e_n}/f^e$$ with $\lambda_{e_1 \ldots e_n} \in R$. Thus $S_{(f)}$ is generated as an $R$-algebra by the $f_1^{e_1} \ldots f_n^{e_n} /f^e$ with the property that $e\deg(f) = \sum e_i\deg(f_i)$. If $e_i \geq \deg(f)$ then we can write this as $$f_1^{e_1} \ldots f_n^{e_n}/f^e = f_i^{\deg(f)}/f^{\deg(f_i)} \cdot f_1^{e_1} \ldots f_i^{e_i - \deg(f)} \ldots f_n^{e_n}/f^{e - \deg(f_i)}$$ Thus we only need the elements $f_i^{\deg(f)}/f^{\deg(f_i)}$ as well as the elements $f_1^{e_1} \ldots f_n^{e_n} /f^e$ with $e \deg(f) = \sum e_i \deg(f_i)$ and $e_i < \deg(f)$. This is a finite list and we see that (1) is true.

To see (2) suppose that $M$ is generated by homogeneous elements $x_1, \ldots, x_m$. Then arguing as above we find that $M_{(f)}$ is generated as an $S_{(f)}$-module by the finite list of elements of the form $f_1^{e_1} \ldots f_n^{e_n} x_j /f^e$ with $e \deg(f) = \sum e_i \deg(f_i) + \deg(x_j)$ and $e_i < \deg(f)$. $\square$

The code snippet corresponding to this tag is a part of the file algebra.tex and is located in lines 13067–13076 (see updates for more information).

\begin{lemma}
\label{lemma-dehomogenize-finite-type}
Let $R$ be a ring. Let $S$ be a graded $R$-algebra. Let $f \in S_{+}$
be homogeneous. Assume that $S$ is of finite type over $R$. Then
\begin{enumerate}
\item the ring $S_{(f)}$ is of finite type over $R$, and
\item for any finite graded $S$-module $M$ the module $M_{(f)}$
is a finite $S_{(f)}$-module.
\end{enumerate}
\end{lemma}

\begin{proof}
Choose $f_1, \ldots, f_n \in S$ which generate $S$ as an $R$-algebra.
We may assume that each $f_i$ is homogeneous (by decomposing each $f_i$
into its homogeneous components). An element of $S_{(f)}$ is a sum
of the form
$$\sum\nolimits_{e\deg(f) = \sum e_i\deg(f_i)} \lambda_{e_1 \ldots e_n} f_1^{e_1} \ldots f_n^{e_n}/f^e$$
with $\lambda_{e_1 \ldots e_n} \in R$. Thus $S_{(f)}$ is generated
as an $R$-algebra by the $f_1^{e_1} \ldots f_n^{e_n} /f^e$ with the
property that $e\deg(f) = \sum e_i\deg(f_i)$. If $e_i \geq \deg(f)$
then we can write this as
$$f_1^{e_1} \ldots f_n^{e_n}/f^e = f_i^{\deg(f)}/f^{\deg(f_i)} \cdot f_1^{e_1} \ldots f_i^{e_i - \deg(f)} \ldots f_n^{e_n}/f^{e - \deg(f_i)}$$
Thus we only need the elements $f_i^{\deg(f)}/f^{\deg(f_i)}$ as well
as the elements $f_1^{e_1} \ldots f_n^{e_n} /f^e$ with
$e \deg(f) = \sum e_i \deg(f_i)$ and $e_i < \deg(f)$.
This is a finite list and we see that (1) is true.

\medskip\noindent
To see (2) suppose that $M$ is generated by homogeneous elements
$x_1, \ldots, x_m$. Then arguing
as above we find that $M_{(f)}$ is generated as an $S_{(f)}$-module
by the finite list of elements of the form
$f_1^{e_1} \ldots f_n^{e_n} x_j /f^e$
with $e \deg(f) = \sum e_i \deg(f_i) + \deg(x_j)$ and
$e_i < \deg(f)$.
\end{proof}

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There are also 6 comments on Section 10.56: Commutative Algebra.

## Add a comment on tag 07Z2

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text:
# Thread: FUNdamental Theorem of calculus

1. ## FUNdamental Theorem of calculus

Express the limit, $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

I have no idea what it's asking me to do haha
And if your wondering I do know the the Fundamental Theorem of Calculus\
Thanks : )

2. Originally Posted by qzno
Express the limit, $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

I have no idea what it's asking me to do haha
And if your wondering I do know the the Fundamental Theorem of Calculus\
Thanks : )
If you were to approximate the area under $y = e^x$ on $[0,1]$ using n rectangles of equal thickness you would use the following

$\sum_{i=1}^n f(c_i) \Delta x,$

where (if using the right endpoint of the subinterval)

$c_i = \frac{i}{n},\;\;\; \Delta x = \frac{1}{n}$

or

$\sum_{i=1}^n e^{i/n} \, \frac{1}{n} \Delta x,$.

In the limit as $n \to \infty$ the answer would becomes exact. So the answer would be

$\int_0^1 e^x\, dx = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$.

I'm guessing you can take it from here.

3. Originally Posted by qzno
Express the limit, $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}$ as a definite integral and evaluate it using the Fundamental Theorem of the Calculus

I have no idea what it's asking me to do haha
And if your wondering I do know the the Fundamental Theorem of Calculus\
Thanks : )
$\sum_{i=1}^n e^{\frac{i}{n}} \cdot \frac{1}{n} = (e^{\frac{1}{n}}+e^{\frac{2}{n}}+e^{\frac{3}{n}}+ ... + e^{1}) \cdot \frac{1}{n}$

$\int_0^1 e^x dx$

4. why is it from 0 to 1?

5. look at the exponents for each term of the sum ...

least term is $e^{\frac{1}{n}} \approx e^0$

greatest term is $e^1$

these are the y-values of each infinitesimal rectangle in the Riemann sum

note also the factor $\frac{1}{n}$ ... a horizontal distance of 1 divided into n partitions.

6. Originally Posted by qzno
why is it from 0 to 1?
In general, the Riemann sum would be

$\sum_{i=1}^n f(c_i)\, \Delta x$ where $\Delta x = \frac{b-a}{n}\;\;\; \text{and}\;\;\; c_i = a + \frac{b-a}{n}\;i$. Now compare with what you have

$
\sum_{i=1}^{n} \frac{e^{\frac {i}{n}}}{n}
$

so $a + \frac{b-a}{n}\;i = \frac{i}{n}$ giving $a = 0,\;\;\; b = 1.$

7. thanks guys : )


text:
# Math Help - Find the Taylor series

1. ## Find the Taylor series

Find the Taylor series of $\frac{1}{\sqrt{x+1}}$


text:
# Binary Indexed Tree or Fenwick Tree in C++?

C++Server Side ProgrammingProgramming

In case of comparing with a flat array of numbers, the Fenwick tree results a much better balance between two operations: element update and prefix sum computation. In case of a flat array of m numbers, we can either store the elements, or the prefix sums. In case of first instance, calculating prefix sums needs linear time; in case of second instance, modifying or updating the array elements needs linear time (in both instances, the other operation can be accomplished in constant time). Fenwick trees permit both operations to be accomplished in O(log m) time. This is obtained by representing the numbers as a tree, where the value of each node is treated as the sum of the numbers in that subtree. The tree structure permits operations to be accomplished using only O(log m) node accesses.

By considering a one-based array, a Fenwick tree is most easily understood. Each element whose index j is a power of 2 contains the sum of the first j elements. Elements whose indices indicate the sum of two (distinct) powers of 2 consist of the sum of the elements since the preceding power of 2. Basically, each element consists of the sum of the values since its parent in the tree, and that parent is found by clearing the minimum or least-significant bit in the index.

To calculate the sum up to any given position or index, consider the binary expansion of the position or index, and add elements which correspond to each 1 bit in the binary form.

For example, let one wishes to calculate the sum of the first eleven values. Eleven is 1011 in binary. This consists of three 1 bits, so three elements must be added: 1000, 1010, and 1011. These consist of the sums of values 1-8, 9-10, and 11, respectively.

A simple C implementation follows.

## Example

#define LSB(i) ((i) & -(i)) // zeroes all the bits except the minimum or least significant one
int A1[SIZE];
int sum(int i) // Returns the sum from index 1 to i{
int sum = 0;
while (i > 0)
sum += A1[i], i -= LSB(i);
return sum;
}
void add(int i, int k) // Adds k to element with index i{
while (i < SIZE)
A1[i] += k, i += LSB(i);
}

Published on 29-Jan-2020 11:11:23


text:
# R caret package: problems understanding the functions in confusionMatrix

#### lilchaos

##### New Member
Hey,

I earlier posted about the problematic interpretation of PPV (positive predictive value). As there seems to be no easy solution for that I tried to understand what the PPV could tell me and especially what it cannot.

Now I was calculating confusion matrices for predictions of the model i built. To do that I came across the caret package which easily gives you along with the confusion matrix measures like sensitivity, specificity, PPV alongside others.

The problem I have now are the formulas for calculating the PPV, Precision, Sensitivity and Recall. (The actual description of the Formulas are attached as quote at the end of the post)

The first problem is with PPV and Precision:

I learned the calculation of the PPV as

true positives/ (true positives+false positives)

the package does call this formula precision (which I always thought is similar to PPV) and calculates the ppv from sensitivity specificity and prevalence.

In my case the prevalence is taken from the model data itself, thus seemingly PPV and precision are identical.
So for me it looks like in this case there is only a difference of precision and PPV if I am using the actual prevalence of the population and not the prevalence of my sample.
Is this always the case if someone is referring to these two measures?

The second problem is with senisitivity and recall:

The Formula for Sensitivity and Recall seem to be similar. Is there a difference of these two measures?

Could maybe someone help me to get out of this confusion?
I never really got the hang of statistics and the longer I am trying to sort this problem out the more confusing it gets for me.

Thanks a lot!

From documentation of caret package:

The functions requires that the factors have exactly the same levels.
For two class problems, the sensitivity, specificity, positive predictive value and negative predictive
value is calculated using the positive argument. Also, the prevalence of the "event" is computed
from the data (unless passed in as an argument), the detection rate (the rate of true events also
predicted to be events) and the detection prevalence (the prevalence of predicted events).

Suppose a 2x2 table with notation

referenceEvent referenceNo Event
predicted Event A B
predicted No Event C D

The formulas used here are:

Sensitivity = A/(A + C)
Specif icity = D/(B + D)
P revalence = (A + C)/(A + B + C + D)
P P V = (sensitivity∗prevalence)/((sensitivity∗prevalence)+((1−specif icity)∗(1−prevalence)))
NP V = (specif icity∗(1−prevalence))/(((1−sensitivity)∗prevalence)+(specificity)∗(1−prevalence)))
DetectionRate = A/(A + B + C + D)
DetectionP revalence = (A + B)/(A + B + C + D)
BalancedAccuracy = (sensitivity + specif icity)/2
P recision = A/(A + B)
Recall = A/(A + C)
F1 = (1 + beta2) ∗ precision ∗ recall/((beta2∗ precision) + recall)

#### hlsmith

##### Omega Contributor

Yeah, I believe you have everything right. The usage of Recall, Precision, etc. come from the Classifications Fields and Machine Learning. They are just synonymous terms to what you are used to and I believe the caret package has other machine learning procedures, so it opts defaults to those terms.

Not sure why they just don't use A / (A + B) in the precision formula, but just superficially looking at the above formula it seems fine.

#### lilchaos

##### New Member
Hey,

thanks a lot. Sorry for the late reply i was on holidays

Yes it is actually a machine learning package, so ut seems that I have to get used to the different terms

The only thing I am wondering about is the Formula for the PPV and Precision.

I learned that (apart from them being the same) the Formula as:

PPV = A/(A + B)

but it seems that the formula they use is also commonly used for calculation of PPV as

PPV = (sensitivity∗prevalence)/((sensitivity∗prevalence)+((1−specif icity)∗(1−prevalence)))

Is there any rule of when to use which Formula?


text:
# Entity System question

This topic is 3550 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Hi, thank you! [smile]

Well, then I'm going to turn my Singletons back to normal local classes...
Quote:
 Original post by ShakedownHave you already implemented input as events? I've seen inputs handled by a single class, and the class interprets the input and creates other Events accordingly. In that implementation, there are no input Events. It makes sense, too. This single class, usually called a controller, is provided to a game view.

In my game I have always thought of an event as something that has already happened and not as something that should still happen. So when something happens, other parts of the system can do something in response. I didn't know an event could also be something that has yet to be done. How can you distinguish both types?

Currently I have only events of the first type. When the player presses lets say the up arrow key an event of type InputEvent is send with parameter AccelerateKey. This event is sent by the InputSubsystem. And then the PlayerController class catches the event and calls the AddForce() method in the Physics component of the Player Entity.

I think you would do this differently, like that:

There is no InputSubSystem. The PlayerController checks for the state of a key an when the up arrow is pressed it sends an PlayerAccelerate event. Then the Physics component catches it and applies the force to the player.

Is that correct? But when another component sees the PlayerAccelerate event that doesn't mean the player has accelerated. Would the component that accelerates the player then need to send another event like PlayerAccelerated so that other components know the player has really accelerated?

In this version the PlayerController is like an InputSubSystem, isn't it? What is if I want to have multiple things controlled by the Keyboard? Wouldn't it be better to have an InputSubSystem that takes the input and the PlayerController that checks it? For example when two players play on the same keyboard I have two PlayerControllers but only one InputSubSystem.

Or have I misunderstood you?

##### Share on other sites
Quote:
 In my game I have always thought of an event as something that has already happened and not as something that should still happen.

I imagine events as something that is happening right now. The player is accelerating right now so I should play the "player_accelerating" sound effect right now. The player has fired their weapons right now so I should play "player_fired" sound effect right now.

Quote:
 So when something happens, other parts of the system can do something in response.

I think of it as all parts of the system reacting simultaneously. The player has pressed the accelerate key, so the physics system would apply a force and the sound system would play a sound effect. Note that if the player is holding down the accelerate key, 1 accelerate event is fired every iteration of your game loop, the systems don't understand the idea of a key being "held down;" each iteration of the game loop the key is either down or not down. That means that every iteration your physics system will apply the force and every iteration your sound system will play the sound. Perhaps you're imagining it as a start event (PlayerAccelerate) and a stop event (PlayerAccelerated), and any time in between, the physics system is applying a constant force and the sound system is looping the sound effect. That may be an interesting way of doing things, but it might turn out to make things more complicated.

If you imagine every iteration of your game loop as a snapshot of your game, independent from the previous iteration and the next iteration, then all that your systems should react to is what is happening right now in your game. If the player is no longer holding down the accelerate key, then there are no more PlayerAccelerate events being fired.

For example, in one iteration of your game loop the accelerate key is down (independent of the fact that it may have just been pressed, or it may have been held down for the entire game). So, your PlayerController does its job and reacts accordingly; it sends a PlayerAccelerate event. The physics system catches the event, applies the force to the player. The sound system catches the event and plays the sound. Let's say that's all that happened in that particular game loop iteration.

Next iteration. There are no keys down, so there are no events fired from the PlayerController; the player will not move and there will be no sound effect played. The enemies are still going about their business, flying around the screen and shooting at the player because they are being controlled by an AIController which isn't going to depend on the player. End of iteration.

Quote:
 There is no InputSubSystem. The PlayerController checks for the state of a key an when the up arrow is pressed it sends an PlayerAccelerate event. Then the Physics component catches it and applies the force to the player.

Right. A PlayerAccelerate (which could even be PlayerMove) event would be dispatched to everything that cares. The physics component cares, so it receives the event and applies the appropriate force to the player.

Quote:
 But when another component sees the PlayerAccelerate event that doesn't mean the player has accelerated. Would the component that accelerates the player then need to send another event like PlayerAccelerated so that other components know the player has really accelerated?

Explain what you would need a PlayerAccelerated event for. If I imagine what you're getting at, this might be the part where it gets tricky using components and events.

Quote:
 In this version the PlayerController is like an InputSubSystem, isn't it?

Yep, except that it won't receive events from the real systems, and every entity in your game has its own controller.

Quote:
 What is if I want to have multiple things controlled by the Keyboard? Wouldn't it be better to have an InputSubSystem that takes the input and the PlayerController that checks it? For example when two players play on the same keyboard I have two PlayerControllers but only one InputSubSystem.

You would have 2 different PlayerControllers - Player1Controller and Player2Controller. They would both be KeyListeners, except they would only react to the appropriate keys. For example, Player1Controller would look something like:

if('a' is pressed) send Player1 move left eventif('d' is pressed) send Player1 move right eventif('w' is pressed) send Player1 move up eventif('s' is pressed) send Player1 move down event

Player2Controller would look something like:

if('left arrow' is pressed)  send Player2 move left eventif('right arrow' is pressed) send Player2 move right eventif('up arrow' is pressed)    send Player2 move up eventif('down arrow' is pressed)  send Player2 move down event

Then, in your game, you would give player 1's game entity the Player1Controller and player 2's game entity the Player2Controller, and then all of your enemies would receive different versions of an AIController.

Also, another note on Controllers. The Controllers control the particular entity that they're attached to (duh). This means that you will have a lot of different types of Controllers - Player1Controller, Player2Controller, AIGruntController, AIEliteController, AIHunterController, AIMarineSoldierController, AIMarineSniperController, AIWrathController, AIPelicanController, etc. Also, just to make clear on the PlayerControllers, you probably wouldn't have to differentiate between a Player1Controller and a Player2Controller if your players were not on the same input device (keyboard). If your game was on the Xbox 360, you would have 2 PlayerControllers, except one is listening to controller port 1 and controlling player 1's game entity, and the other is listening to controller port 2 and controlling player 2's game entity.

Hope that clears some confusion. Also, I might have mentioned this already, but there are many ways of achieving the same result. The way I discuss is only one of many, each with its own pros and cons.

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I think I got it! [grin] Thx

But I still have an InputSubSystem that checks the key states every frame (SDL_GetKeyState).
The Controllers then ask the subsystem for the key state.
/* in your example: */if('a' is pressed) send Player1 move left event/* this would be in my game somthing like: */if ( pInputSubSystem->KeyState( Player1UpKey ) )    pEventManager->InvokeEvent( Event(PlayerMoved, /* event data */ ) );
Quote:
 I think of it as all parts of the system reacting simultaneously. The player has pressed the accelerate key, so the physics system would apply a force and the sound system would play a sound effect. Note that if the player is holding down the accelerate key, 1 accelerate event is fired every iteration of your game loop, the systems don't understand the idea of a key being "held down;" each iteration of the game loop the key is either down or not down. That means that every iteration your physics system will apply the force and every iteration your sound system will play the sound.
The sound system has still to know whether the key has just started being pressed or is already a second time down (to only play the sound at the beginning, otherwise there would be a lot of simultaneously playing sounds).

One last thing I wonder is what you mean with "game view" on your post before.
Thank you for your patience! [wink]
aeroz

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This is the second time that this thread is resurrected, so please accept my apologies for that.

Having read many topics on the subject, I have a relatively good understanding of a component-based entity system, but there is this one question that's been bugging ever since I started reading on the subject, mainly due to lack of comprehensive resources on the subject or me coming from a different background. Anyhow, my question is this: How does this component-based entity system interfaces with a scene graph?!

The way I have envisioned and designed my system is to have Entity objects as part of my scene graph, with entities (such as Camera or Light) inheriting from the parent Entity class, pretty much the old school way. With every iteration of the game loop the scene graph (which is merely a transformation hierarchy in my design) gets updated. Now lets say I switch to a component-based entity design instead. For some reason I have a hard time understanding how to incorporate this new design into my current system. Considering that entities do not have an Update() function in this new event-based system, how should I propagate transformations up the hierarchy?

So for instance, let's say we have a Camera entity that can also be attached to all other entities in a scene. This kind of behavior requires maintaining some form of hierarchical representation of the scene so such "relative relations" can be maintained and updated with every tick of the game loop.

1) First of all, how is this Camera (or Light, which has pretty much the same properties) going to be modeled in a component-based system. What are the components of such an entity to be exact? Where should the view and projection transformation of a Camera (or Light) be stored? A CameraComponent? Or can this be broken into smaller components?

2) Secondly, how should I incorporate such a component-based Camera in a scene graph? In my current design entities have Update() functions, but the prevailing opinion is that entities in a component-based system should not have data and methods. Some even go as far as representing an entity with a GUID alone. So, I'm pretty much in dark regarding how I should handle this situation. TBH, I'm so lost that I can't get my head around forming and asking a question about it... Just to give it a shot, should I send events of any form in this case?! Are such components not part of a scene graph at all? ...

As you see I'm pretty lost here, so any help (preferably accompanied by some examples) is hugely greatly appreciated.

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Most of the components in question have nothing to do with the scene graph at all. If you have components that represent renderable objects or objects with a position in the game world, they may exist in that graph.

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