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chargaff's rules for the pairing of nitrogen bases is

Attached to each sugar is one of four bases--adenine (A), cytosine (C), guanine (G), or thymine (T). Chargaff's rules state that DNA from any species of any organism should have a 1:1 stoichiometric ratio (base pair rule) of pyrimidine and purine bases and, more specifically, that the amount of guanine should be equal to cytosine and the amount of adenine should be equal to thymine.This pattern is found in both strands of the DNA. DNA polymerase is an enzyme that joins individual nucleotides to produce a new strand of DNA. The structure of DNA consists of two strands of nucleotides that are paired together to form a ladder-like structure. A with T: the purine adenine (A) always pairs with the pyrimidine thymine (T) ; C with G: the pyrimidine cytosine (C) always pairs with the purine guanine (G) . 13) Chargaff's base-pairing rules helped Watson and Crick build the DNA model immensely. These observations became the basis for Chargaff's rules, also known as the base pairing rules. The 4 DNA Bases and Their Strict Pairing Rules. A=t g=c. Adenine (A), Thymine (T), Guanine (G), and Cytosine (C). the shape (structure) of the nitrogen bases. The base pairing rules for DNA are governed by the complementary base pairs: adenine (A) with thymine (T) in an A-T pairing and cytosine (C) with guanine (G) in a C-G pairing. A and T, and G and C needs to be paired. In RNA, there is no binding between nitrogen-containing bases. This helped Watson and Crick assemble the nitrogen bases accurately. Chargaff's rule states for every Adenine there's a thymine and for every cytosine there' a guanine The DNA molecule is made up of very long chains of the 4 bases: A, C, G and T. In 1950, Erwin Chargaff published a paper stating that in DNA of any given species, the ratio of adenine to thymine is equal, as is the ratio of cytosine to guanine. Chem. Elucidate the importance of Chargaffs rule in the structure of DNA molecules. The DNA of all the living beings is composed of just four bases i.e. Base pairing of the nitrogen bases takes place according to Chargaff's rules. A. \mathrm{A}-\mathrm{C}, \mathrm{T}-\mathrm{G} c. A-T, G-C d. \mathrm{A}-\mathrm{A}, \mathrm{G}-\mathrm{G… Chargaff's rules state that DNA from any cell of any organisms should have a 1:1 ratio (base Pair Rule) of pyrimidine and purine bases and, more specifically, that the amount of guanine should be equal to cytosine and the amount of adenine should be equal to thymine. During replication, DNA may be lost from the tips of chromosomes, which are called telomeres. A pairs with T and G pairs with C. C. A pairs with G and C pairs with T. D. A pairs with C and G pairs with T. E. T = C and G = A. A segment of DNA has 120 adenine and 120 cytosine bases. Purine (Adenine and Guanine) always base pairs with a pyrimidine ( Cytosine, Uracil, and Thymine). a. A-G, T-C b. The base pairing rules for DNA are often called Chargaff’s rules of DNA base pairing. Discover the nitrogenous bases of each nucleotide and how they combine into specific combinations to form the structure of DNA. A = C and G = T. B. He developed the fact that Adenine only pairs with Thymine, and Cytosine only pairs with Guanine. Also, the ratio of bases vary from species to species. The two strands are held together by hydrogen bonds between the bases, with adenine forming a base pair with thymine, and cytosine forming a base pair with guanine. The number of adenine components equaled the thymine bases and the number of guanines were equal to the cytosines. This pattern is found in both strands of the DNA. the color of the nitrogen bases. 14) Base-pairing simply means the pairing rules of the nitrogen "bases". This is known as Chargaff's ratios and it was a crucial clue that helped solve the structure of DNA. The discovery of its double-helix structure in 1953 catapulted James Watson and Francis Crick a Nobel Prize, and even among non-science nerds, DNA is widely known for playing a major part in the innumerable traits that are passed from parents to offspring. Base Pairing The rules of base pairing (or nucleotide pairing) are: . The various juxtapositions of these 4 bases give rise to the genetic codes of all the biota on the planet. A nitrogenous base is an organic molecule that contains the element nitrogen and acts as a base in chemical reactions. The total number of nucleotides present in the segment is. In case of double stranded DNA, Purine=Pyrimidine. Nucleotides are full of groups that can participate in hydrogen bonds. Each new DNA molecule has one original strand and one new strand. Chargaff rule: The rule that in DNA there is always equality in quantity between the bases A and T and between the bases G and C. (A is adenine, T is thymine, G is guanine, and C is cytosine. 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# Classes containing boolean closure of CFLs

So I asked a similar question on MathOverflow, but I now realize I asked the wrong question and asked it in the wrong place.

Anyway, I'm searching for a class of languages containing the boolean closure of CFLs that has a 'pumping-type' condition on it. What I've found so far are:

• MCFGs (multiple context-free languages), which have a nice pumping lemma and contain CFLs (though it's not obvious if they contain the boolean closure of CFLs, since they're not boolean-closed themselves),
• ET0L languages (which contain CFLs, have a 'pumping-type' lemma, and are (full) AFLs themselves--but I don't think they necessarily contain the boolean closure of CFLs),
• Mildly CSLs (which apparently have pumping-type lemmas?),
• Indexed languages (which have a 'shrinking' lemma), etc.

Now, I'm certainly no specialist in formal language theory, so I don't know too much about these classes. Anyway, does such a class exist, and, if so, where can I find papers/etc. on it?

This is not an answer to your question per se, but might explain why finding sensible classes of languages more or less fitting your requirements will be hard: recall that the emptiness of intersection of two context-free languages is undecidable, thus the full boolean closure of CFLs is not a very interesting class... (In fact, any R.E. language $L$ can be expressed as $h(L_1\cap L_2)$ for $L_1$ and $L_2$ deterministic context-free and $h$ an homomorphism.)

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# Math Help - Polynomials

1. ## Polynomials

Why is it that the most general form of the remainder in a polynomial is given by R(x) = ax +b?

2. Originally Posted by xwrathbringerx
Why is it that the most general form of the remainder in a polynomial is given by R(x) = ax +b?
Please rephrase this question, at the moment it makes no sense

RonL

3. Sori

The question was : "Explain why the most general form of R(x) i.e. the remainder in the polynomial P(x) = (x+1)(x-4) is given by R(x) = ax+b."

4. Originally Posted by xwrathbringerx
Sori

The question was : "Explain why the most general form of R(x) i.e. the remainder in the polynomial P(x) = (x+1)(x-4) is given by R(x) = ax+b."
No there is still not enough context, we have established that you are talking about second degree polynomials, but remainder for what?

RonL

5. Oops really sori it seems I've been reading the question wrong.

Explain why the most general form of R(x) is given by R(x) = ax+b.

WHEN the polynomial P(x) is divided by (x+1)(x-4) where the quotient is Q(x) and the remainder is R(x).

Sori once more

6. Originally Posted by xwrathbringerx
Oops really sori it seems I've been reading the question wrong.

Explain why the most general form of R(x) is given by R(x) = ax+b.

WHEN the polynomial P(x) is divided by (x+1)(x-4) where the quotient is Q(x) and the remainder is R(x).

Sori once more
If the remainder were a polynomial of degree greater $n$ than $1$ then it can be rewriten in the form:

$R(x)=(x+1)(x-4)P_{n-2} (x) +Q_{n-1}(x)$

Where $P_{n-2}$ and $Q_{n-1}$ are polynomials of degree $n-2$ and no more than $n-1$ respectivly, so it would not have been the remainder.


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Environ Eng Res > Volume 25(1); 2020 > Article
Chollom, Rathilal, Swalaha, Bakare, and Tetteh: Comparison of response surface methods for the optimization of an upflow anaerobic sludge blanket for the treatment of slaughterhouse wastewater

### Abstract

This study was aimed at using the Central Composite Design (CCD) and Box-Behnken Design (BBD) to compare the efficiency and to elucidate the main interacting parameters in the upflow anaerobic sludge blanket (UASB) reactor, namely: Organic Loading Rate (OLR), Hydraulic Retention Times (HRT) and pH at a constant temperature of 35°C. Optimum HRT (15 h), OLR (3.5 kg.m−3.d−1) and pH (7) resulted in biogas production of 5,800 mL/d and COD removal of 80.8%. BBD produced a higher desirability efficiency of 94% as compared to the CCD which was 92%. The regression quadratic models developed with high R2 values of 0.961 and 0.978 for both CCD and BBD, respectively, demonstrated that the interaction models could be used to pilot the design space. BBD model developed was more reliable with a higher prediction of biogas production (5,955.4 ± 225.3 mL/d) and COD removal (81.5 ± 1.014%), much close to the experimental results at a 95% confidence level. CCD model predictions was greater in terms of COD removal (82.6 ± 1.06% > 80.8%) and biogas production (4,636.31 mL/d ± 439.81 < 5,800 mL/d) which was less than the experimental results. Therefore, RSM can be adapted for optimizing various wastewater treatment processes.

### 1. Introduction

Wastewater treatment processes are complex, therefore, the incorporation of multiple variables during the treatment process for the optimum performance of the system is necessary. Each of the chosen variables is expected to be operated at an optimum range to maximise the benefit of the process, therefore, optimization becomes very important [1, 2]. In fact, the feasibility of the conventional techniques, such as the one-factor-at-time (OFAT) is very challenging in environmental processes and wastewater treatment systems for acquiring the best conditions to maximise the desired response or treatability performance [1, 3]. Conventionally, the OFAT method which involves changing one independent variable while holding others at fixed level is extremely time consuming and expensive to run high number of experiments. Subsequently, optimization using multivariable techniques is encouraged, as they are economical, faster, and effective to optimize multivariable simultaneously. Furthermore, the multivariable approach offers the advantage of interactive effects among the factors [4].
Consequently, due to the powerful efficiency of the multivariable technique, response surface methodology (RSM) has been encouraged for optimization of various process systems. The RSM is a resourceful tool which is made up of mathematical and statistical techniques that are based on the fit of a polynomial equation to the experimental data in order to predict the behaviour of the system [5]. It has been employed for the optimization of various wastewater treatment processes to improve the water quality [3, 6].
A study by Bashir et al. [7] reported that out of 3,190 articles published within the year 2000 to 2013 (data extracted from Scopus), about 352 documents were related to the application of RSM for process optimization in different subject areas of research. Some of which were engineering (15.6%), biochemistry (11.2%), agricultural and biological science (10.6%), chemical engineering (9.5%) and environmental science (6%) [7, 8]. The studies carried out in the optimization of operational factors in different types of wastewater treatment processes by various authors includes textile dye wastewater [3] landfill leachate [6], oil refinery wastewater [1] and the sugar refinery wastewaters [9].
The RSM has been shown to be advantageous over the traditional techniques of optimization with respect to the number of experiments and multifactor interactions [1]. RSM, as a collection of a mathematical and statistical tool, is usually used in designing of experiments as well as for the evaluation of the most influential factors on chosen responses. This tool describes the interactional effects between a set of data. Again, it can be used to generate models for the prediction of the responses. To expand the application of RSM, there is the need to evaluate which of the design types fit for the number of process parameters. Some of the design types of RSM include Doehlert Design (DD), Central Composite Design (CCD), the Box-Behnken Design (BBD) and a three-level full factorial design [3, 6]. Each of the design types offers both advantages and disadvantages.
To increase the bioremediation efficiency, more especially in UASB systems, it is crucial to know the optimal operating conditions. Since some of the influential parameters in biological systems have interactional effects which can inhibit a shift in microbial community. Bezerra et al. [3] reported that using minimum number of experimental runs, a correlation can be developed to evaluate the relationship between the response of interest and the input variables. Ibrahim, et al [4] used BBD to model and optimize external parameters of biological system for the removal of COD and phosphorus. The results showed that at optimum temperature of 45.33°C and 60 d, desirability performance of 99.8% was obtained for the removal of COD and phosphorus, which correspond to 95% confidence at index R2 of 0.955 and 0.91, respectively. Amr et al [10] conducted a study on three independent factors (ozone dosage, COD concentration, and reaction time) using RSM to evaluate the treatment of semi-aerobic stabilized leachate. The obtained optimum conditions were 70 g/m3 ozone, 250 mg/L COD, and 60 min reaction time for 26.7, 7.1, and 92% removal for COD, NH3–N, and colour, respectively.
The use of RSM as an optimization tool is seen to be increasing as earlier indicated by Bashir et al. [7], little information for its application in biological wastewater treatment systems exist. Biological wastewater treatment systems such as the upflow anaerobic sludge blanket (UASB) are preferred for the treatment of wastewaters with high organics such as those from the slaughterhouses due to their robustness. The incomplete removal of organic matters and pathogens is one of the disadvantages of the UASB reactor [11, 12]. The performance of the UASB is easily affected by both environmental and operational factors. Therefore, to improve the efficiency of a UASB processes, it is expected that the operational parameters are operated at an optimum. The optimization of UASB processes has been carried out mostly with the use of OFAT which does not consider the interactional effects on the response. The use of RSM for the optimization of both input variables either independently or in combination will be necessary.
The information with regards to the comparative modelling and optimization of biogas production and COD removal efficiency in an UASB process is limited. A few studies have attempted to model and optimize UASB processes using one of the optimization tools, but very little or none have looked at the comparison of both optimization tools [1, 9, 13, 14]. The present study therefore focused on simulating and comparing the CCD and BBD response surface designs to evaluate their desirability efficiency in a biological wastewater treatment process. In this study, the effects of the HRT, pH and Organic Loading Rate (OLR) as independent parameters of UASB for the breaking down of the organic and inorganic components such as the COD to generate the biogas was investigated. Statistically designed RSM were used to optimize the aforementioned parameters at different levels and understand their interactive effects on the responses (COD and biogas) at maximum target. These two parameters were chosen due to the fact that the treatment efficiency is mainly influenced by them.

### 2.1. Synthetic Wastewater

A 25 L synthetic wastewater was prepared before each experimental run. The wastewater was characterized according to specification by Lettinga and Pol [15] with a Biological Oxygen Demand (BOD)/COD ratio of 0.40–0.53 comparable to slaughterhouse wastewater. The synthetic wastewater was continuously fed into a 5 L UASB at the flowrate predetermined by a peristatic pump (Flexflo A-100E, Blue-White Industries Ltd.).

### 2.2. Seed sludge and Experimental Apparatus

The UASB reactor used for the experiment was constructed from plexiglas glass and its effective working volume was 4.5 L. The UASB was continuously fed with synthetic wastewater at the bottom of the reactor and the effluent was collected at the top. It was operated at a constant mesophilic temperature of 35 ± 1°C. Fig. S7 shows a schematic diagram for the experimental set-up. The reactor was started with a TSS of 19.4 g/L and VSS of 13.8 g/L digested seed sludge collected from a local wastewater treatment plant treating slaughterhouse wastewater in South Africa. The pH of the reactor was maintained by the addition of 1 M HCl and 1 M NaOH. Table S3 shows the characteristics of the wastewater that was used for the experiments. The performance of the reactor was calculated using Eq. (1).
##### (1)
$Reactor efficiency=(1-CODoutCODin)×100$
Where, CODout = Effluent and CODin = Influent
The reactor was started with an OLR of 1.54 kg.m−3.d−1 at a HRT of 24 h for the first three days of the start-up where it was operated in a batch mode, thereafter, the continuous mode was used. The OLR was thereafter increased in a stepwise manner according to the chosen HRT (6, 12 and 18 h) and it was maintained to enable the microorganisms to adapt to the new loadings until the system had attained stability. Upon HRT change, the reactor was run for a period of 50 d before changing to the next.

### 2.3. Analytical Methods

The water displacement technique was used to measure biogas that evolved from the anaerobic digestion (AD) process [14]. COD, which is measured based on the quantity of oxidant (Cr+6) consumed and expressed in terms of its oxygen equivalence, was determined using close refluxing according to the standard method 5220D [16, 17]. Other parameters were determined as shown in Table S3.

### 2.4. Experimental Design and Procedure

Design Expert software (10.0.3) was used to perform the statistical design of experiments and data analysis. Three independent effective variables were selected: HRT, pH and OLR. The OLR is a measure of the capability of the microorganism in the AD process to breakdown organic compounds present in the effluent.

#### 2.4.1. CCD

The CCD is a RSM design type which estimates the second order polynomial on a wider range of the design space and processes the input data to give out the interactions between the response and the process variables. The CCD is a five-level fractional factorial design, which comprises of a two level factorial design, central designs and two axial designs. The centre point is replicated to enhance the measurement of reproductively and the model lack of fit. It also has rotatability and orthogonality properties, which helps in the curvature description when represented in 3D plots. There were 30 experimental runs derived from the design matrix, with two centre points and an alpha value of 1.89. The total number of the experiments (N) depended on the replication number of the centre points. The range and levels of the variables in coded and actual units are given in Table 1.

#### 2.4.2. BBD

The BBD is a spherical, rotatable, or nearly rotatable second-order design. It is based on a three-level incomplete factorial design, which consists of the centre point and middle points like the edge of a cube. Although, BBD can be derived from a cube, it can be represented spherically, making the vertices of the cube not covered by the design. It can be considered as three interlocking factorial designs along with centre points. The BBD is said to be a more economical and viable tool than the CCD, because its design matrix are usually generated with a fewer number of experimental runs [3, 6].
As shown in Table 1, three levels were chosen for the independent variables (operating variables). The CCD and BBD experimental design were conducted for the three factors and the two responses (COD removal and biogas production). Other parameters that were monitored alongside were the alkalinity, Volatile Fatty Acids (VFA), Total Suspended Solids (TSS) and Volatile Suspended Solids (VSS). The parameters were chosen to indicate the performance of the reactors in terms of operational stability and efficiency.
Input variables were coded according to Eq. (2) [1].
##### (2)
$Xn=xn-xoΔxn$
Where xn, xo and Δxn represent the coded level, the real value, the centre point value and the variable step change, respectively.
A second order polynomial Eq. (3) was used to describe the effect of the chosen parameters and their interactions on the responses using the BBD for the experimental design.
##### (3)
$Yi=βo+Σi=1kβiXi+Σi=1kβiiXi2+Σj=1k-1βijXiXj+C$
Where βo, βi, βii and βij are constant, linear, quadratic, and cross factor interaction coefficients, respectively; Xi and Xj represent the independent variables; Yi is the predicted response; and k and c are the number of factors and the residual terms, respectively. The significance of the model equations, individual parameters and factor interaction were evaluated by the analysis of variance (ANOVA) at a confidence intervals (CI) of 95% (ρ = 0.05). Two-dimensional (2D) contour plots and three-dimensional (3D) surface responses were obtained after applying the developed quadratic models.

### 3. Results and Discussion

Treatment efficiency for the UASB was observed to be above 65% of the contaminants removal. VFA, alkalinity, TSS and VSS, respectively were reduced to 64.9 ± 2.5, 92 ± 5.5, 35 ± 7.8 and 125 ± 8.6 mg/L. A significant desludging of 5 ± 1.5 mg TSS/L was carried out to reduce the UASB sludge volume intermittently. RSM was employed to reduce the number of experiments by selecting a small number of experiments, which were best representatives of the test domain to investigate the responses (COD and biogas).
Tables S4 and S5 illustrate the experimental matrices, which have 16 factorial points, 8 axial points and 6 centres points. They were then evaluated using multiple regression analysis of the CCD and BBD adapted from the RSM. The ANOVA was employed to evaluate the significant effect of the regression models statistically. The regression models mathematically explained the relationship between the responses and the independent variables on the system. The visual representation of the responses is graphically presented in 3D surface and 2D contour plots. The regression models developed using CCD and BBD together with the experimental data are expressed in coded and actual values as depicted equations 47 for CCD and 8–11 for BBD. The combination of the factors at different levels showed effective responses of the models, with the models predicted value and experimental value difference being less than 5%. The experimental data were well fitted to the quadratic polynomial function.
From the equations for both the CCD (Eq. (4)-(7)) and BBD (Eq. (8)-(11)), the models developed showed a good fit to the quadratic design space with high regression coefficients. Quadratic models had high R2 (0.9606; 0.9782), adjusted R2 (0.9429; 0.9614) and predicted R2 (0.8499; 0.8945) values for the CCD and BBD, respectively. All the models were significant with an insignificant lack of fit (LOF) at the 95% confidence level. The BBD based models had superlative values for the R2 as compared with the CCD. The response models developed and their respective predicted values were then evaluated to validate the experimental results for the treatment of the wastewater using UASB. In addition, the models derived by the BBD seem to be simpler than the CCD, which had only 9 terms, whereas the CCD had 10. According to Zolgharnein et al. [6], the simplicity of the BBD makes it stand out from a theoretical statistical standpoint for better prediction. The response models, due to their complexity, validity, accuracy and operational interactions were then compared (section 3.1) with respect to their design’s performance.
##### (4)
$CCD-COD model actual=169.37-2.701HRT-13.081OLR-19.971pH+1.14(HRT×OLR)+0.575(HRT×pH)+2.133(OLR×pH)-0.041(HRT2)-0.101(OLR2)+0.778 (pH2)-0.168 (HRT×OLR×pH)$
##### (5)
$CCD-COD model coded=82.443+0.551A+0.307B-1.448C-0.786AB-3.114AC+0.401BC-A2-1.238B2+0.778C2-3.533ABC$
##### (6)
$CCD-Biogas model actual=21304.49+479.896HRT+21.593OLR-5719.225pH-19.675(HRT×OLR)-64.836(HRT×pH)-19.268(OLR×pH)+2.019(HRT2)+17.462(OLR2)+470.692 (pH2)+1.399 (HRT×OLR×pH)$
##### (7)
$CCD-Biogas model coded=4168.13-16.85A-16.85B+76.37C-207.43AB-334.43AC-8.65BC+72.71A2+213.9B2+470.69C2+29.39A$
##### (8)
$BBD-COD model actual=161.29-0.856HRT-3.389OLR-17.363pH-0.00694(HRT×OLR)+0.102(HRT×pH)+0.4739(OLR×pH)+0.00429(HRT2)+0.00776(OLR2)+0.85 (pH2)$
##### (9)
$BBD-COD model coded=82.443+0.551A+0.307B-1.448C-0.786AB-3.114AC+0.401BC-A2-1.238B2+0.778C2-3.533ABC$
##### (10)
$BBD-Biogas model actual=-6994.49+1055.50HRT-1089.921OLR+2640.099pH-16.22(HRT×OLR)-107.61(HRT×pH)+144.1(OLR×pH)-7.431(HRT2)+17.648(OLR2)-163.65 (pH2)$
##### (11)
$BBD-Biogas model coded=4976.81+110.91A-162.5B-5.67C-340.63AB-645.63AC+504.35BC-267.52A2+216.2B2-163.65C2$

### 3.1. The Analysis of Variance (ANOVA) and Significance of the Models Using CCD and BBD

The ANOVA is essential in determining the significance and adequacy of the model. From model equations for both the CCD and BBD, the models developed were expressed in terms of their coded and actual values. The relationship and interactions between the HRT (A), OLR (B) and pH(C) and the responses were established by applying the multiple regression analysis. With the ANOVA, the statistical values of the model terms were determined. Tables 2 and 3 depict the ANOVA summary of the CCD and BBD for the COD based models, respectively. Likewise Tables 4 and 5 show the biogas production using CCD and BBD. In terms of the CCD, the coefficient of determination (R2) for COD and biogas were 0.9617 and 0.9653, respectively. In addition, the R2 of the COD and biogas-based models using the BBD were 0.9752 and 0.9802, respectively. In both occurrences the R2 values were found to be greater than 0.8 indicating a good fit of the models. Each of the model terms evaluated by the F-test showed 5% significance level (p < 0.05) with a p-value (probability) of 95% confidence level. The CCD and BBD F-values for COD (47.70; 3.37) and biogas (3.31; 9.02) implied that there was less than 2% chance that an F-value that large could occur due to error. Hence, the significant terms with p-values less than 0.05 were considered as significant terms, whilst those with p ≥ 0.05 were considered as limited terms. The high hierarchal were of the model terms with p-values greater than or equal to 0.05, which had limited influence on the response were also excluded. The adequate precision measured the signal to noise ratio.
The adequate precision ratio and the low values of the coefficient of variation (CV) validated the reliability and good precision of the models as shown in Table 2Table 3Table 45. A precision greater 4 indicated that the model was significant while the CV should be ≥ 0.1. The ANOVA results prove that the response surface models for predicting the COD and biogas yield are considerably coherent. This is due to the high R2 values, which satisfied the adjustment of the models to suit the experimental data. This demonstrated a good agreement between the experimental data and the predicted results (Table S4 and S5). Furthermore, the ANOVA results showed a desirable and coherent agreement with the adjusted R2. Therefore, the use of the quadratic models could be used to optimize the system under the same given condition instead of the conventional method.

### 3.2. Diagnostic Checking of the Fitted Models Using CCD and BBD

To authenticate the normality assumptions, the normal probability plot of residuals was used. Such that, if the residuals plot approaches a straight line, then the normality assumption is satisfied. According to Zolgharnein et al. [6], although some points are anticipated to scatter evenly, if the data points falls along the straight line then it is acceptably significant. Fig. S3 and S4 show that the normal probability vs the studentized residuals plots for both CCD and BBD, respectively. The plots for the CCD and BBD responses suggest that the random scattering of the point is distributed along the line. However, some of the points were evenly distributed, hence indicating a constant variance of the response. Fig. S4 and S5 presented the predicted vs experimental value plots, which demonstrated that the predicted values depended on the main effect of the factor levels. Hence, it is important to identify these influential factors and their levels to ascertain the significance of the model. As depicted in Fig. S5 (a) and (b), the removal efficiency of the COD ranged from 72% to 88% and for biogas production from 4,200 mL/d to 5,700 mL/d, respectively for the CCD. In Fig. S6(a) and (b), COD removal of 78% to 88% and biogas production of 3,800 mL/d to 5,800 mL/d was obtained by using the BBD.

### 3.3. The Three-dimensional (3D surface) Plots of the Models

To graphically analyse the interactive relationships among the factors and the response, the Design Expert 10.0.3 generated the 3D surface. The visual representation of the response surface in 3D surface plots assisted the investigation and optimization of the UASB, by determining the maximum region of interest on the curvature of the surface. The precise location and optimum point identified (Fig. 1 and 2) indicated the true representation of the regression Eq. (4)-(11). The 3D surface showed the model’s variation with two factor levels (HRT and OLR) whereas the other factor level (pH) was kept constant. In addition, the plots depict the sensitivity of the responses due to the change of factor levels with the degree of their interactions. Fig. 1(a) and 1(b) illustrated the maximum COD removal (72–88%) and biogas produced (3,500–5,500 mL/d) at a pH of 7 using CCD. While, Fig. 2(a) and 2(b) also depicts response surface plots at a pH of 7 for maximum COD removal (77–90%) and biogas production (3,714–5,600 mL/d) using BBD. All the 3D surface plots depict the effect of HRT and OLR on COD removal and biogas production at alkaline medium. The optimization of the process variables was aimed at finding the levels of the factors to obtain a maximum desirability for all the responses within the design space.

### 3.4. Numerical Evaluation of the Effect of pH, OLR and HRT on the Response

Table S1 and Fig. 3 shows some of the CCD suggested optimal conditions using the numerical analysis. Likewise, Table S2 and Fig. 6 do the same for the BBD response model. Comparatively, high desirability performance of the BBD (94%) was obtained as compared to the use of the CCD at 92%. However, almost the same optimum conditions such as HRT (15 h) and pH (7) were obtained by using the CCD and BBD, except the OLR that differed as 3.5 kg.m−3.d−1 and 3 kg.m−3.d−1, respectively.
A series of mechanism occurred in the reactor to enhance the microbial community in breaking down the organic matter. These mechanisms enhanced the biodegradability to generate more biogas during the digestion process, which includes the acidogenesis and methanogenesis that occurred at different pH levels. The control of the pH was very critical to detect the abnormalities in the system in respect of acidity and alkalinity, which might inhibit the system. From the results, it was found that high OLR (10 kg.m−3.d−1) at low HRT (8 h) could result in high VFA, which might have inhibited the biogas production and decrease the degradability of the COD. Thus, within the pH range of 6.5–7.2, the COD removal and biogas production increased, whilst below 6.5, the efficiency of the system was inhibited. This agree with the report by Lettinga and Pol [15 Lettinga and Pol [15] and Torkian et al. [18 Torkian et al. [18]. They reported that a decrease in pH below 6.5 could hinder the methanogenic microorganism activities thus declining biogas production. At a lower pH, an accumulation of VFA could occur and a minimal buffering capacity of the UASB is induced thereby inhibiting the methanogenic activity.
The OLR rate describes the liquid flowrate and contaminant concentration and can be defined as the mass of the pollutant introduced in a unit volume of the reactor per unit time. As such, this parameter integrates reactor characteristics, operational characteristics, and bacterial mass and activity into the volume of media Ruiz et al. [19]. plays a significant role on the performance of the UASB reactor. With an increase in the OLR, an increase in the biogas production was experienced. However, at an OLR above 5 kg.m−3.d−1 and at a shorter HRT of less than 8 h was found to be detrimental to the methanogens. Thus, the high OLR (10 kg.m−3.d−1) resulted in poor performance of the system, due to overload of the substrate, which supersedes the microorganism. At a lower HRT, there is a reduction in the contact time for the breaking down of the organic components, therefore impeding the performance of methanogens present in the system such that the conversion of the high COD or the acetates into biogas is limited. Operating at an optimal region of the OLR (3–5 kg.m−3.d−1) was found suitable to prevent the washout of the organic matter in the effluent and increased the treatable desirability of the system. At higher OLR, a possible washout of organic matter could occur.
The HRT is considered an important operating paramter, such that its effectiveness is the controlling factor with respect to the entire UASB performance. The observation of a long HRT (18 h) could lead to a siutuation in the reactor whereby the microorganism would compete over the available the substrate in other to withstand adjust to the discomfort. This phenomenon can affect the growth rate of the microbial community in the reactor, even though a higher yield of biogas and degradability could be achieved as seen with a HRT of 8–15 h.
A confirmation test (Fig. 3) was carried out with the selected optimum conditions in Tables S1 and S2 with highest desirability through the numerical optimization analysis ramp plots (Fig. S1 and S2) provided in the supplementary section. It was found that the BBD based model predicted values agree with the experimental results with less than 2% disparities. Whereas, the CCD based model predictions was greater in terms of COD removal (82.6 ± 1.06% > 80.8%) and biogas production (4,636.31 mL/d ± 439.81 < 5,800 mL/d) which was less than the experimental results. This made the BBD to be more superior over the CCD.

### 4. Conclusions

Optimization of multiple variables is a major concern in wastewater treatment processes to the obtainability of the desirable discharge effluent limits. This research paper was aimed at comparing the applicability of CCD and BBD for the optimization of HRT, pH and OLR in determining their desirability efficiency in a UASB for the treatment of wastewaters. The two response surface designs investigated produced almost the same optimum conditions: HRT (15 h) and pH (7), except the OLR that differed as 3.5 kg.m−3.d−1 and 3 kg.m−3.d−1 for CCD and BBD, respectively. Both designs were found suitable in predicting high treatability performance of the UASB above 90%. Based on the confirmation test, the BBD model prediction (biogas 5,955.4 ± 225.3 mL/d; COD 81.5 ± 1.014%) was found to be in good agreement with the experimental results (biogas 5,800 mL/d; COD 80.8%). Conversely, the BBD produced a high desirability efficiency of 94% as opposed to the CCD of 92%. The models obtained were found to be predictive, adequate and significant at 95% confidence level. RSM is an effective and economically viable alternative technique that can be adapted for optimizing various wastewater treatment processes to favourably maximise the output. Operating the UASB at HRT (8 to 15 h), OLR (3 to 5 kg.m−3.d−1) and pH (6.5 to 7.2) were found to be workable conditions to maximise the outputs. Furthermore, the UASB is an effective and reliable process for the treatment and generation of biogas from wastewaters. In addition, this area can be of economic interest to stakeholders for protecting the environment and producing green energy as well as reducing the footprint of greenhouse gases.

### Acknowledgments

The authors wish to thank Durban University and South Africa National Research Foundation (NRF) for their support.

### References

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2. Dasgupta J, Sikder J, Chakraborty S, Curcio S, Drioli ERemediation of textile effluents by membrane based treatment techniques: A state of the art review. J Environ Manage. 2015;147:55–72.

3. Bezerra MA, Santelli RE, Oliveira EP, Villar LS, Escaleira LAResponse surface methodology (RSM) as a tool for optimization in analytical chemistry. Talanta. 2008;76:965–977.

4. Ibrahim S, Wahab NA, Anuar AN, Bob MParameter optimisation of aerobic granular sludge at high temperature using response surface methodology. Int J Electr Comput Eng. 2017;7:1522–1529.

5. Najib T, Solgi M, Farazmand A, Heydarian SM, Nasernejad BOptimization of sulfate removal by sulfate reducing bacteria using response surface methodology and heavy metal removal in a sulfidogenic UASB reactor. J Environ Chem Eng. 2017;5:3256–3265.

6. Zolgharnein J, Shahmoradi A, Ghasemi JBComparative study of Box-Behnken, central composite, and Doehlert matrix for multivariate optimization of Pb(II) adsorption onto Robinia tree leaves. J Chemometr. 2013;27:12–20.

7. Bashir MJ, Amr SSA, Aziz SQ, Aun NC, Sethupathi SWastewater treatment processes optimization using response surface methodology (RSM) compared with conventional methods: Review and comparative study. Middle-East J Sci Res. 2015;3:244–252.

8. Bashir MJ, Aziz HA, Aziz SQ, Amr SAAn overview of wastewater treatment processes optimization using response surface methodology (RSM). In : The 4th International Engineering Conference–Towards engineering of 21st century; 15–16 January 2012; Gaza.

9. Pambi R, Musonge PApplication of response surface methodology (RSM) in the treatment of final effluent from the sugar industry using Chitosan. In : WIT Transactions on Ecology and the Environment; 27–29 June 2016; Venice. p. 209–219.

10. Amr SSA, Aziz HA, Bashir MJApplication of response surface methodology (RSM) for optimization of semi-aerobic landfill leachate treatment using ozone. Appl Water Sci. 2014;4:231–239.

11. Chong S, Sen TK, Kayaalp A, Ang HMThe performance enhancements of upflow anaerobic sludge blanket (UASB) reactors for domestic sludge treatment – A state-of-the-art review. Water Res. 2012;46:3434–3470.

12. Bustillo-Lecompte CF, Mehrvar MTreatment of an actual slaughterhouse wastewater by integration of biological and advanced oxidation processes: Modeling, optimization, and cost-effectiveness analysis. J Environ Manage. 2016;182:651–666.

13. Phanduang O, Lunprom S, Salakkam A, Reungsang AAnaerobic solid-state fermentation of bio-hydrogen from microalgal Chlorella sp. biomass. Int J Hydrog Energ. 2017;42:9650–9659.

14. Jha P, Kana EBG, Schmidt SCan artificial neural network and response surface methodology reliably predict hydrogen production and COD removal in an UASB bioreactor? Int J Hydrog Energ. 2017;42:18875–18883.

15. Lettinga G, Pol LHUASB-process design for various types of wastewaters. Water Sci Technol. 1991;4:87–107.

16. Rezaee S, Zinatizadeh AA, Asadi AHigh rate CNP removal from a milk processing wastewater in a single ultrasound augmented up-flow anaerobic/aerobic/anoxic bioreactor. Ultrason Sonochem. 2015;23:289–301.

17. Chollom M, Rathilal S, Swalaha F, Bakare B, Tetteh EStudy of the start-up of an upflow laboratory-scale anaerobic sludge blanket for the treatment of slaughterhouse wastewater. In : 4th Water and Society internatioanl conference; 4–6 June 2017; Seville, Spain.

18. Torkian A, Eqbali A, Hashemian SJThe effect of organic loading rate on the performance of UASB reactor treating slaughterhouse effluent. Resour Conserv Recycl. 2003;40:1–11.

19. Ruiz I, Veiga M, Santiago PD, Blàzquez RTreatment of slaughterhouse wastewater in a UASB reactor and an anaerobic filter. Bioresour Technol. 1997;60:251–258.

##### Fig. 1
CCD -response surface plot effects of HRT and OLR at constant pH (7) on (a) COD removal (b) biogas production on 3D surface plot.
##### Fig. 2
BBD -response surface plot effects of HRT and OLR at constant pH (7) on (a) COD removal (b) on biogas production on 3D surface plot.
##### Fig. 3
Confirming the models prediction at HRT (15 h), OLR (3.5 kg.m−3·d−1) and pH (6.5).
##### Table 1
Input variables Levels (X)
−1 0 1
X1: HRT (h) 6 12 18
X2: OLR (kg.m−3·d−1) 3 7 10
X3: pH 6 7 8
##### Table 2
ANOVA Results and Adequacy of the Quadratic Models for COD Using CCD
Source Sum of squares df Mean square F-value p-value
COD-Model 540.39 10 54.04 47.7 < 0.0001 significant
A-HRT 8.28 1 8.28 7.31 0.0141
B-OLR 2.58 1 2.58 2.27 0.148
C-pH 57.24 1 57.24 50.52 < 0.0001
AB 9.88 1 9.88 8.72 0.0082
AC 155.13 1 155.13 136.94 < 0.0001
BC 2.57 1 2.57 2.27 0.1487
A2 26.46 1 26.46 23.35 0.0001
B2 18.57 1 18.57 16.39 0.0007
C2 7.32 1 7.32 6.46 0.0199
ABC 199.71 1 199.71 176.29 < 0.0001
Residual 21.52 19 1.13
Lack of Fit 0.5219 4 0.1305 0.0932 0.9831 not significant
Pure Error 21 15 1.4
Cor Total 561.91 29

Std. Dev. 1.06 R2 0.9617 Adjusted R2 0.9415 Predicted R2 0.8955 Adeq Precision 23.8754 Mean 80.68 C.V. % 1.32
##### Table 3
ANOVA Results and Adequacy of the Quadratic Models for COD Using BBD
Source Sum of squares df Mean square F-value p-value
COD-Model 31.21 9 3.47 3.37 0.0114 significant
A-HRT 2.19 1 2.19 2.13 0.1601
B-OLR 0.366 1 0.366 0.3553 0.5578
C-pH 10.84 1 10.84 10.52 0.0041
AB 0.0851 1 0.0851 0.0826 0.7767
AC 1.49 1 1.49 1.44 0.2439
BC 10.96 1 10.96 10.64 0.0039
A2 0.1188 1 0.1188 0.1153 0.7377
B2 0.045 1 0.045 0.0436 0.8366
C2 3.59 1 3.59 3.49 0.0766
Residual 20.6 20 1.03
Lack of Fit 8.29 3 2.76 3.82 0.0294 significant
Pure Error 12.31 17 0.7243
Cor Total 51.82 29

Std. Dev. 1.01 R2 0.9752 Adjusted R2 0.943 Predicted R2 0.8826 Adeq Precision 9.62 Mean 79.87 C.V. % 1.27
##### Table 4
ANOVA Results and Adequacy of the Quadratic Models for Biogas Using CCD
Source Sum of squares df Mean square F-value p-value
Biogas-Model 6.40E + 06 10 6.40E + 05 3.31 0.012 significant
A-HRT 1.05E + 05 1 1.05E + 05 0.5404 0.4713
B-OLR 7,758.16 1 7,758.16 0.0401 0.8434
C-pH 1.59E + 05 1 1.59E + 05 0.8234 0.3755
AB 6.88E + 05 1 6.88E + 05 3.56 0.0746
AC 1.79E + 06 1 1.79E + 06 9.25 0.0067
BC 1,197.02 1 1,197.02 0.0062 0.9381
A2 64,001.6 1 64,001.59 0.3309 0.5719
B2 5.54E + 05 1 5.54E + 05 2.86 0.107
C2 2.68E + 06 1 2.68E + 06 13.86 0.0014
ABC 13,824.3 1 13,824.26 0.0715 0.7921
Residual 3.68E + 06 19 1.93E + 05
Lack of Fit 2.94E + 05 4 73,456.79 0.3258 0.8563 not significant
Pure Error 3.38E + 06 15 2.25E + 05
Cor Total 1.01E + 07 29

Std. Dev. 439.82 R2 0.9653 Adjusted R2 0.9433 Predicted R2 0.8417 Adeq Precision 5.481 Mean 4,857.68 C.V. % 9.05
##### Table 5
ANOVA Results and Adequacy of the Quadratic Models for Biogas Using BBD
Source Sum of squares df Mean square F-value p-value
Biogas-Model 4.13E + 06 9 4.58E + 05 9.02 < 0.0001 significant
A-HRT 98,404.39 1 98,404.39 1.94 0.1793
B-OLR 2.11E + 05 1 2.11E + 05 4.16 0.0549
C-pH 257.03 1 257.03 0.0051 0.944
AB 4.64E+05 1 4.64E + 05 9.14 0.0067
AC 1.67E + 06 1 1.67E + 06 32.83 < 0.0001
BC 1.02E + 06 1 1.02E + 06 20.03 0.0002
A2 3.55E + 05 1 3.55E + 05 7 0.0155
B2 2.32E + 05 1 2.32E + 05 4.57 0.0451
C2 1.33E + 05 1 1.33E + 05 2.62 0.1213
Residual 1.02E + 06 20 50,795.58
Lack of Fit 1.50E + 05 3 49,990.4 0.9814 0.4248 not significant
Pure Error 8.66E + 05 17 50,937.67
Cor Total 5.14E + 06 29

Std. Dev. 225.38 R2 0.9802 Adjusted R2 0.9713 Predicted R2 0.8345 Adeq Precision 14.656 Mean 4,919.48 C.V. % 4.58
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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
1 초 128 MB 11 3 3 27.273%

## 문제

One of the most fundamental data structure problems is the dictionary problem: given a set D of words you want to be able to quickly determine if any given query string q is present in the dictionary D or not. Hashing is a well-known solution for the problem. The idea is to create a function h : Σ → [0..n − 1] from all strings to the integer range 0, 1, .., n − 1, i.e. you describe a fast deterministic program which takes a string as input and outputs an integer between 0 and n−1. Next you allocate an empty hash table T of size n and for each word w in D, you set T[h(w)] = w. Thus, given a query string q, you only need to calculate h(q) and see if T[h(q)] equals q, to determine if q is in the dictionary. Seems simple enough, but aren’t we forgetting something? Of course, what if two words in D map to the same location in the table? This phenomenon, called collision, happens fairly often (remember the Birthday paradox: in a class of 24 pupils there is more than 50% chance that two of them share birthday). On average you will only be able to put roughly √n-sized dictionaries into the table without getting collisions, quite poor space usage!

A stronger variant is Cuckoo Hashing1. The idea is to use two hash functions h1 and h2. Thus each string maps to two positions in the table. A query string q is now handled as follows: you compute both h1(q) and h2(q), and if T[h1(q)] = q, or T[h2(q)] = q, you conclude that q is in D. The name “Cuckoo Hashing” stems from the process of creating the table. Initially you have an empty table. You iterate over the words d in D, and insert them one by one. If T[h1(d)] is free, you set T[h1(d)] = d. Otherwise if T[h2(d)] is free, you set T[h2(d)] = d. If both are occupied however, just like the cuckoo with other birds’ eggs, you evict the word r in T[h1(d)] and set T[h1(d)] = d. Next you put r back into the table in its alternative place (and if that entry was already occupied you evict that word and move it to its alternative place, and so on). Of course, we may end up in an infinite loop here, in which case we need to rebuild the table with other choices of hash functions. The good news is that this will not happen with great probability even if D contains up to n/2 words!

1Cuckoo Hashing was suggested by the danes R. Pagh and F. F. Rödler in 2001

## 입력

aOn the first line of input is a single positive integer 1 ≤ t ≤ 50 specifying the number of test cases to follow. Each test case begins with two positive integers 1 ≤ m ≤ n ≤ 10000 on a line of itself, m telling the number of words in the dictionary and n the size of the hash table in the test case. Next follow m lines of which the i:th describes the i:th word di in the dictionary D by two non-negative integers h1(di) and h2(di) less than n giving the two hash function values of the word di. The two values may be identical.

## 출력

For each test case there should be exactly one line of output either containing the string “successful hashing” if it is possible to insert all words in the given order into the table, or the string “rehash necessary” if it is impossible.

## 예제 입력

2
3 3
0 1
1 2
2 0
5 6
2 3
3 1
1 2
5 1
2 5

## 예제 출력

successful hashing
rehash necessary


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# Numerically solve differential equation that has no solution

I am not really experienced in the field of numerical differential equations. I am only studying existence and uniqueness results about PDEs at the moment. Now I was wondering what would happen, if one tries to solve a PDE numerically, although the PDE 1) has no solution or 2) the solution is not unique?

The Navier-Stokes-Equation in 3 dimensions comes to mind. How do physicists approach this equation? Do they work with numerical solutions although there is no proof, that there is a solution at all?

Best,

Luke

What happens depends strongly on the software to compute the solution, but in general if you don't have a proof of existence and uniqueness as well as a proof of convergence to the unique solution, you are on very shaky ground.

Let's consider $y' = f$ with $y(0) = 0$. This can be solved by an ODE solver, under some assumptions on $f$ (say, continuity). Now let's say $f$ is a delta function at a random location in the domain. A numerical solver will never find it, and the answer will always be wrong.

The next situation is non-uniqueness. I wrote a few solvers for a PDE which had many solutions. What we attempted was to compute them all, and then select a subset of the solutions based on some property we thought was relevant. (To be fair, what actually happened is we got into fights about what was relevant, and eventually we just modeled the phenomenon with a different PDE that did have a unique solution. I can honestly say that non-uniqueness was one of the biggest headaches I've had solving PDEs.)

I've never attempted to solve an equation which known to not have a solution, but I cannot imagine it turning out well. The best you can hope for is an error message, but you probably won't get it. My suspicion is that your numerical method will compute the solution to an equation which is in some sense "close" to the original, and then you'll be in a truly devastating situation where you have a bunch of garbage which you think is a solution.

Hopefully someone else can offer insight into Navier-Stokes; I've never worked on it so I can't help you there.

• Thank you for your comment! I read on Wikipedia, that solutions to the Navier-Stokes-Equations are used in physics, although it has not been proven that they exist. If several different solution calculators come to a (in some sense) close solution, is that a hint, that the solution might exist and be unique? Is that enough for applications? – Luke May 31 '17 at 6:48
• Sure it's a hint. That's probably part of why people generally believe that solutions to Navier-Stokes do exist. – user14717 May 31 '17 at 14:29
• Does that mean when I use a PDE solver such as gPROMS or COMSOL to solve multiphysics problem, just because the numerical solver converges, the results may not be unique! Doesn't the fact that you add boundary conditions and physical constraints increase the likelihood that the results are unique? – SPIL Oct 31 '17 at 11:12
• I'm not sure "likelihood" is the correct way of understanding uniqueness. As PW Anderson says, computation should come after understanding, not before. As to the behavior of COMSOL and gPROMS, I cannot say what their behavior will be. – user14717 Oct 31 '17 at 18:18


text:
## Calculus: Early Transcendentals (2nd Edition)

$g'(y) = 18y^5-52y^3+8y$
$g(y) = (3y^4-y^2)(y^2-4)$ a) Using Product Rule: $g'(y) = (12y^3 -2y)(y^2-4)+(3y^4-y^2)(2y) = 12y^5-2y^3-48y^3+8y +6y^5 - 2y^3 = 18y^5-52y^3+8y$ b) Expansion then Power Rule: $g(y) = 3y^6-12y^4-y^4+4y^2 = 3y^6 - 13y^4 + 4y^2$ $g'(y) = 18y^5-52y^3+8y$ Answers to Part A and Part B match


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# Arctic regions in higher dimensional zonotopes

Same way as the two dimensional tilings by rhombi come from minimal surfaces in a $D$ dimensional cubical lattice as mentioned in this answer, one can consider higher dimensional zonotopes tiled by rhombic polytopes. I've seen these usually denoted as $D\to d$ tilings, or $d$-dimensional, codimension $D-d$ tilings. We can call a region in such a tiling effectively $D'$ dimensional, if it looks locally like a $D'\to d$ tiling.

In "An n-dimensional generalization of the rhombus tiling" by J. Linde, C. Moore, and M.G. Nordahl, it is conjectured that the octahedron inscribed inside a rhombic dodecahedron plays the same role for $4\to 3$ tilings as the arctic circle does for $3\to 2$ tilings. Outside this octahedron, the tiling is most likely frozen, i.e. locally periodic and with vanishing entropy, while inside the octahedron we have the only region which is effectively $4$ dimensional. However it is also conjectured that the entropy inside the arctic octahedron approaches a uniform distribution, unlike the 2-dimensional case where the entropy peaks at the center of the arctic ellipse.

What is the status of these conjectures? Is it expected that for all zonotope tilings ($d>2$) the arctic regions are polyhedral? Is the entropy expected to be constant inside the arctic region? Is there a conceptual reason why the behaviour changes so drastically from planar tilings to higher dimensions?

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## 1 Answer

Basically nothing is known beyond what you've said, and it doesn't look like that's going to change anytime soon.

There is at least one piece of relevant numerical work which I know of: J. Linde, the first author in the paper you mentioned, posted some graphs on his website which suggest that the height function isn't quite flat inside the octahedron. This means that the entropy isn't likely to be constant in there, either.

There are some heuristic reasons why we should expect flat facets on the arctic regions for these problems when d>3. For instance, similar problems have been studied with torus boundary conditions and a slightly different lattice, and the height function there is extremely flat. See http://arxiv.org/abs/1005.4636.

Other than that, the only other work I know about these higher dimensional tilings is numerical work on solid partitions, counted by volume. MacMahon made some guesses about how many solid partitions there are of a given volume, which Knuth proved to be incorrect in the sixties. It seems that MacMahon might have guessed an asymptotically correct answer, however, according to the results of this project. It uses massively parallel computation to count solid partitions by volume:

http://boltzmann.wikidot.com/solid-partitions

and their project has generated a paper,

http://arxiv.org/pdf/1105.6231v3

which I just found and haven't read properly yet. There's also some approximate counting work due to Mustonen and Rajesh, which reaches the same conclusion; it's cited in the above paper.

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### Praxis Raw Score Calculator

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Praxis Raw Score Converter! free convert online with more formats like file, document, video, audio Details: HOW TO CONVERT A PRAXIS II RAW SCORE INTO A PRAXIS II UNOFFICIAL SCORE › Get more: Slp praxis raw score conversionView Convert. Listen to credible commentators. Example: The Praxis 5081 raw score I described above is 92/130. 2013 praxis raw score conversion table 2017 Golden Education World Book Document ID 7389880f Golden Education World Book Praxis Raw Score Conversion Table 2017 Description Of. Apply the formula to every raw score to get your scaled score. Next, divide 35 by 56, and you get 62. How to Calculate Your Praxis Score. How to Calculate Scores on a Practice Praxis Test Bizfluent. The number of correct answers you give is your raw score. Praxis raw scores are calculated only from selected response questions-questions that are multiple choice or numeric entry. The actual exam is harder that the exams in this book or any other test prep exams that I used.

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# relationship between two planes (or their (N,d) vectors)

I'm reading this article in computer vision and I just can't get my head around eqution(6). The scenario is as follows: We have the pose of a camera in world coordinate system as $T_{w,c}$ so that a 3D point defined in local camera coordinate system $p_c$ could be transformed to the world coordinate system using $p_w=T_{w,c}p_c$. Then a plane in camera coordinate system, represented by a 4D vector of its normal and diatnce to the origin $\pi_c=[n^T,d]$. could be transformed to world coordinates using $\pi_w=T_{c,w}^{-T}\pi_c$

I have no idea how we got to this final equation for planes from the info given above. Any help on this will be appreciated.

A general equation of a plane is $Ax+By+Cz+D=0$, which can be written in vector form as $[A,B,C]\cdot[x,y,z]=-D$. From this equation we can find that the point $$\mathbf x_0 = {-D\over A^2+B^2+C^2}[A,B,C]$$ lies on the plane. Now, for any point $\mathbf x$ on the plane, we have $[A,B,C]\cdot(\mathbf x-\mathbf x_0)=-D-(-D)=0$, but $\mathbf x-\mathbf x_0$ is parallel to the plane, so $[A,B,C]$ is normal to it. This also means that $$d=\|\mathbf x_0\| = \left\| {-D\over A^2+B^2+C^2}[A,B,C] \right\| = {|D| \over \sqrt{A^2+B^2+C^2}}$$ is the distance of the plane from the origin. Without the absolute value, you have $d=-D/\sqrt{A^2+B^2+C^2}$ as the signed distance from the origin, i.e., it is the distance you need to move along $[A,B,C]$ to reach the plane, which is negative if this vector points away from the plane.
If we take $\mathbf n$ to be a unit normal to the plane, we then have $\mathbf n\cdot\mathbf x = d$ as the vector equation of the plane, which we can rewrite as $[\mathbf n,-d]\cdot[\mathbf x,1]=0$. Switching to column vectors, we get $\mathbf\pi=[\mathbf n^T,-d]^T$ (note the change of sign from what you have in your question). This vector is covariant, which means that if points transform as $\mathbf p_w=T_{w,c}\mathbf p_c$, it transforms as $\mathbf\pi_w = T_{w,c}^{-T}\mathbf\pi_c$. I’ll leave that to you to verify.
• @Musaab It was not at all clear to me from the way you phrased your question that it was this just final step that you didn’t understand. It’s almost trivial: $\mathbf\pi_c^T\mathbf p_c=0 \Leftrightarrow \mathbf\pi_c^TT_{w,c}^{-1}T_{w,c}\mathbf p_c=0 \Leftrightarrow (T_{w,c}^{-T}\mathbf\pi_c)^T(T_{w,c}\mathbf p_c)=0 \Leftrightarrow (T_{w,c}^{-T}\mathbf\pi_c)^T\mathbf p_w=0$, so $\mathbf\pi_w=T_{w,c}^{-T}\mathbf\pi_c$. – amd Nov 9 '17 at 19:40


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# Best Mathematics Text Book (for 1st year university)

1. Jun 10, 2018

### Nathan Elward

I'm going to Uni in October and want to get a head with maths for physics NOW so its not so daunting when I am there.

What textbooks/resources are an excelent introduction to physical mathematics?

Thanks

Nathan

2. Jun 10, 2018

### Staff: Mentor

I think calculus and linear algebra are the keywords here. You can have a look on the books on OpenStax , e.g. Calculus I or higher in case you already know the content. However, I would concentrate on linear algebra, as I think it's the bigger step from school to university. Matrices and vector spaces are usually taught only very basically at school, if at all, whereas in physics you use them all the time. So it might be helpful to get used to the concepts and terms.

However, there is always a risk to learn wrong, i.e. to interpret the subtleties of a subject in a wrong way: e.g. the difference between a linear transformation or vector and its representation by coordinates, the concept of continuity, i.e. confusing the $\varepsilon - \delta$ definition.

3. Jun 10, 2018

### Nathan Elward

Thanks. Openstax is a decent resource I hadn't heard of before. I'll definitely be using it.

4. Jun 12, 2018

### MidgetDwarf

Alonso and Finn University Physics. A book forgotten in the US, but used in other parts of the world. Concise explanations and diagrams are meaningful. Ie., the pages are not cluttered with pictures like most of todays physics book. Diagrams are drawn clearly and well explained. Everything is derived from first principle. Problems can be challenging. Derivations are clear. The real gem is part 2 and part 3. Topics are introduced that are usually covered in upper division courses. This book is below Klepner and Kolenkow, but can prepare you for upper division books of the that ilk. Calculus is highly emphasized at the beginning. Majority of intro physics books in mechanics, do not use calculus extensively.
https://www.amazon.com/Fundamental-...id=1528785589&sr=8-3&keywords=alonso+and+finn

Also purchase another introductory book to reference. Maybe Giancoli Physics for Scienties and Engineers or Serway Physics. Always choose the older editions. Much cheaper, and practically the same.

Edwin E. Moise : Calculus. Much easier than Spivak/Apostol/Courant. Everything is motivated carefully. Great precise, clear, and informative writing. Explains things like the well ordering principle, the power and motivation behind the Mean Value Theorem. Fields and their axioms are explained. Explains what a parabolic sector is, and how Archimedes approached the area of a curve. Logirithmic functions and their integrals are constructed and well defined.
Explains sequences and series very well.Mathematically correct derivation of arc length, most modern calculus book that have a proof of this are incorrect. Great place to learn to think mathematically. Great balance between theory and application. Reminds of a gentler Courant Calculus.
https://www.amazon.com/Calculus-com...qid=1528785912&sr=1-1&keywords=calculus+moise

Stewart Calculus. It's more towards engineers and very generic. Use it as a problems book.
https://www.amazon.com/Calculus-7th...d=1528786244&sr=1-3&keywords=calculus+stewart

Thomas Calculus 3rd ed. Mathwonk wrote a great summary of the books strengths. More of an applied approach, but a goody. I learned Calculus from this book.
https://www.amazon.com/Calculus-Ana...528786304&sr=1-2&keywords=thomas+calculus+3rd

I recommend that you read Moise as the main book, and supplement it with Thomas. If Moise is too hard, you can use Thomas, and later do Moise.

This should be enough for now.

5. Jun 14, 2018

### FourEyedRaven

I would say Apostol's Calculus, vols 1 and 2. It has sequences, series, single variable calculus, vector algebra, linear algebra, vector calculus, multivariable calculus, differential equations, (and some probability and numerical analysis). The book is excelent at ballancing between intuition and formalism. It is not overly formal or pedantic, and contains many computational exercises. But it is a rigorous textbook with clear explanations of the material.

Be careful with the edition, though. There are many bad reviews that criticize the international edition.

6. Jun 14, 2018

### vela

Staff Emeritus
I'd concentrate on first-year calculus and differential equations. It's more important at this point to learn how to do calculations efficiently than get bogged down in knowing how to prove everything. Don't discount getting better at algebra and trig.

The OpenStax physics book, University Physics, isn't great. My students largely don't care for it. You'd be better off buying an old edition of Young and Freedman.


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# Is a projective module of constant finite rank finitely generated?

If $R$ is a (commutative) ring and $P$ is a projective $R$-module, then every localization of $P$ at a prime of $R$ is free by Kaplansky's theorem, and has a well-defined rank. If these ranks are all finite, must $P$ be finitely generated?

The answer is no: one can take $R=k[x_1,x_2,\dots]/(x_1^2-x_1,x_2^2-x_2,\dots)$ and $P=\bigoplus_{n=1}^\infty R/\bigl(1-x_1x_2\cdots x_{n-1}(1-x_n)\bigr)$; then $P$ is projective, not finitely generated, and at each prime at most one of its summands is nonzero. At each prime $\mathfrak{p}$ of $R$, then, $P_\mathfrak{p}$ is a free rank-$1$ $R_\mathfrak{p}$-module, unless $\mathfrak{p}=(1-x_1,1-x_2,\dots)$, for which $P_\mathfrak{p}=0$.

Or here's another example: let $R$ be the ring of continuous functions $[0,1]\to\mathbb{R}$, and let $P\subseteq R$ be the ideal consisting of those functions that vanish on a neighborhood of zero. Then again $P_\mathfrak{p}$ is free of rank 1 at every prime $\mathfrak{p}$ of $R$, except for those $\mathfrak{p}$ consisting of functions vanishing at $0$; at such $\mathfrak{p}$ we have $P_\mathfrak{p}=0$.

Such examples must be reasonably complicated due to this result of Bass's:

Proposition 4.2. Suppose $R$ has only finitely many primes minimal above $0$... If $P$ is a locally finitely generated projective $R$-module, then $P$ itself is finitely generated.

("Big Projective Modules are Free", 1963)

However, I'm wondering if it's possible to remove the conditions on $R$ if we assume not only that $P$ is projective, but that its rank is a continuous function $\mathrm{Spec}(R)\to \mathbb{N}$. Since continuity of the rank function is equivalent to the rank being locally constant, we can work locally on affine opens where the rank is constant. Thus the question reduces to:

If $P$ is a projective $R$-module of constant finite rank, is $P$ finitely generated?

• Yes, according to Exercises I.2.13 and I.2.14 of Weibel's K-Book. The hints there say: use the determinant to reduce to constant rank one, and use the trace to show finite generation in the rank one case (much like showing quasicompactness of affine schemes). – Matthias Wendt Aug 16 '14 at 9:20
• Thanks for the reference; that's exactly what I'm looking for! But the hint to use the trace is a little fuzzy: in the exercise where he introduces the properties of traces, he also talks about an integer-valued rank function, which makes it sound like $P$ is implicitly assumed to be finitely generated. – Owen Biesel Aug 18 '14 at 10:14
• I think the rank statement is not so relevant because we are assuming constant rank one anyway. The argument probably should go: the image of $\check{P}\otimes P\to R$ is an ideal, and constant rank one implies that this ideal can not be contained in any prime ideal. Then you write $1=\sum f_i(x_i)$, so that the $x_i$ are the generators. It does not seem that finite generation is used in this argument (but I could be missing something). – Matthias Wendt Aug 18 '14 at 12:52
• I was concerned that Weibel's exercises about the trace were only valid in the finitely-generated case, but the relevant statement (that for each prime $\mathfrak{p}$, we have $\tau_P\subset\mathfrak{p}$ iff $P_\mathfrak{p}=0$) is in fact true for general projective $P$. (That also explains why the supports of the non-constant-rank projective modules in the question were open—their complements are the closed subschemes cut out by their trace ideals!) The rest is just checking that the generators are what you think they are. If you expand your comment into an answer, I'll accept it. – Owen Biesel Aug 18 '14 at 13:31

For any commutative ring $R$ with 1, any projective $R$-module of constant finite rank is finitely generated. This is the content of Exercises I.2.13 and I.2.14 of Weibel's $K$-book. The argument goes as follows (all the relevant hints are in Weibel's book), I hope I did not introduce any tacit finiteness assumptions.
1) In the constant rank one case, we have the dual $\check{P}=\operatorname{Hom}_R(P,R)$ and the evaluation map $\operatorname{ev}:\check{P}\otimes_R P\to R$ whose image $\tau_P=\operatorname{im}\left(\operatorname{ev}:\check{P}\otimes_RP\to R\right)$ is called the trace. By a result of Kaplansky, $P_{\mathfrak{p}}$ is free for any prime ideal $\mathfrak{p}\subseteq R$, and we can choose a generator $x\in P_{\mathfrak{p}}$. Then $\check{x}\otimes x\mapsto 1$ under $\check{P_{\mathfrak{p}}}\otimes_{R_{\mathfrak{p}}}P_{\mathfrak{p}}\to R_{\mathfrak{p}}$. In particular, $\tau_P\not\subseteq\mathfrak{p}$ since otherwise $P\otimes_R R/\mathfrak{p}=0$ contradicts the constant rank one assumption (as pointed out in the comments of Owen Biesel). Since $\tau_P$ is an ideal, it contains $1$ and we can write $1=\sum f_i(x_i)$. For each $\mathfrak{p}$, some $x_i$ will be a generator of $P_{\mathfrak{p}}$, so the $x_i$ are generators for $P$.
2) If $P$ has constant rank $r$, then $\bigwedge^r P$ has constant rank one and by Step 1 has finitely many generators. These are of the form $\sum a_i y_{1i}\wedge\cdots\wedge y_{ri}$ and the (finitely many) $y_{ij}$ appearing generate $P$.
• There actually is a bit of finiteness I think you've introduced. To conclude that $\tau_P\nsubseteq\mathfrak{p}$, you need to know that $\mathrm{Hom}_R(P,R)_\mathfrak{p} \cong \mathrm{Hom}_{R_\mathfrak{p}}(P_\mathfrak{p},R_\mathfrak{p})$, but generally this requires $P$ to be finitely presented. However... – Owen Biesel Aug 19 '14 at 9:56
• ... if it were the case that $\tau_P\subseteq\mathfrak{p}$, then we find (e.g. using a dual basis for $P$) that $P=\mathfrak{p}P$, so $P\otimes_R \kappa(\mathfrak{p})=0$. But this is impossible, since $P_\mathfrak{p}$ is assumed to be free of rank 1, so $P\otimes_R \kappa(\mathfrak{p})\cong \kappa(\mathfrak{p})$. – Owen Biesel Aug 19 '14 at 9:59
• Dear Mathias, your sentence "Since $\tau_P$ is an ideal, it contains $1$ " is a bit strange, to put it mildly... – Georges Elencwajg Sep 29 at 18:47
• @GeorgesElencwajg: True. I probably should have written that $\tau_P\subseteq R$ is an $R$-submodule not contained in any prime ideal and therefore contains 1. – Matthias Wendt Oct 1 at 7:41