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# Application of smart solid lipid nanoparticles to enhance the efficacy of 5-fluorouracil in the treatment of colorectal cancer

## Abstract

5-Fluorouracil (5-FU) is a standard treatment option for colorectal cancer (CRC) but its rapid metabolism and systemic instability (short half-life) has hindered its therapeutic efficacy. The objective of this study was to develop a novel drug delivery system, solid lipid nanoparticle (SLN), capable of delivering high payload of 5-FU to treat CRC. The rational was to improve 5FU-nanocarrier compatibility and therapeutic efficacy. The SLN-loaded 5-FU was developed by utilizing a Strategic and unique Method to Advance and Refine the Treatment (SMART) of CRC through hot and cold homogenization approach. The SLN was made of unique PEGylated lipids and combination of the surfactants. Cytotoxicity studies, clonogenic assay, flow cytometry and confocal imaging were conducted to evaluate the effectiveness and cellular uptake of 5FU-SLN4 in HCT-116 cancer cells. Pharmacokinetic (PK) parameters and receptor expressions were determined while tumor efficacy studies were conducted on mouse bearing subcutaneous HCT-116 cancer. Among the all the formulations, 5FU-SLN4 was the most effective with particle size of was 263 ± 3 nm, zeta potential was 0.1 ± 0.02 and entrapment efficiency of 81 ± 10%. The IC50 value of 5FU-SLN4 (7.4 ± 0.02 µM) was 2.3 fold low compared with 5-FU (17.7 ± 0.03 µM). For tumor efficacy studies, 5FU-SLN4 significantly inhibited tumor growth in comparison to 5-FU while area-under plasma concentration-time curve (AUC) of 5FU-SLN4 was 3.6 fold high compared with 5-FU. HER2 receptors expression were markedly reduced in 5-FU-SLN4 treated mice compared with 5FU and liver and kidney tissues showed no toxicity at dose of 20 mg/kg. 5FU-SLN4 was highly cytotoxic against HCT-116 cells and significantly inhibited subcutaneous tumor growth in mice compared with 5-FU. This emphasizes the significance of developing a smart nano-delivery system to optimize the delivery efficiency of anticancer drugs to tumors.

## Introduction

CRC is the third most diagnosed cancer in individuals in US, and a third leading cause of cancer related deaths. CRC is the second highest cause of cancer deaths when men and women are combined1. According to many sources, CRC is expected to cause more than 50,000 deaths during 20191,2.

The present treatment of CRC largely depends on radiotherapy, surgery, targeted therapy and chemotherapy; however, surgery is the preferred option if CRC is at stages one through to four with or without chemotherapy and/or radiotherapy3. Based on the stage of CRC, two or more treatment options are combined to achieve the desired therapeutic effect. Further, reports show that 5-year survival rate of individuals with metastatic cancer is less than those without metastatic cancer1. The present treatment strategies, examples, targeted therapy and chemotherapy largely affect CRC cells mainly through cell senescence and apoptosis.

The existing therapeutic agents for the treatment of CRC are diverse. They include cytotoxic agents such as 5-FU, oxaliplatin, capecitabine and irinotecan, and targeted therapies such as bevacizumab and anti-epidermal growth factor receptor anti-bodies (examples are cetuximab and panitumumab).

While 5-FU is often used alone or in combination with other anticancer agents such as irinotecan and folinic acid (FOLFIRI) or folinic acid and oxaliplatin (FOLFOX)4, it’s antimetabolite mimics uracil and thymine which eventually incorporates into DNA and RNA synthesis leading to cytotoxicity and cell death5. 5-FU is a fluorinated pyrimidine that inhibits DNA and RNA activity through inhibition of the rate-limiting enzyme (thymidylate synthetase) in pyrimidine nucleotide synthesis6.

Although 5-FU shows inhibitory activity on DNA synthesis, its therapeutic efficacy is limited due to its hydrophilic nature and rapid metabolism by dihydropyrimidine dehydrogenase. Therefore, continuous administration of high doses of 5-FU must be applied to attain the required therapeutic concentration which causes severe toxic effects on the gastrointestinal tract, hematological, neural, cardiac and dermatological reactions7. Hence, there is a need to develop a smart delivery system capable of protecting and delivering a high payload of 5-FU to the desired site with improved therapeutic efficacy.

Nanotechnology has become one of the fastest growing areas with potential applications in drug therapy such as liposomes, micelles, polymeric nanoparticles, and dendrimers have been widely used to convey and overcome drugs with aqueous solubility or systemic stability related issues. Among all the nanodelivery systems, lipid-based nanocarriers are commonly considered the least toxic for in vivo applications and a remarkable progress has been achieved in DNA/RNA in delivery8.

While delivery approaches such as liposomal and polymeric nanoparticles have had setbacks, investigations of solid lipid nanoparticle (SLN) as substitute drug carrier system has proven to be a novel, due to its various advantages such as the ability to incorporate of both hydrophilic and lipophilic drugs, ease of scale-up and enhanced physical stability9,10. By virtue of nanoparticle’s size effect and improved physicochemical properties, the SLN can extravasate into the tumor through the tumor’s leaky vasculature and increase localized tumor drug exposure11.

The current study aimed to develop a smart delivery system capable of delivery a high payload of 5-FU, a hydrophilic drug, into optimized SLNs (5FU-SLNs) with special emphasis on protection and prolonging systemic circulation of 5-FU and, to improve its therapeutic outcome. In this context, the use of smart PEGylated 5FU-SLN4 was developed and optimized to obtain a high entrapment efficiency that was highly cytotoxic against HCT-116 cells, improved 5-FU pharmacokinetic parameters and significantly inhibited subcutaneous tumor growth in mice compared with 5-FU. These improved efficacy of 5-FU emphasized the significance of developing a smart nano-delivery system to optimize the delivery efficiency of anticancer drugs to tumors.

## Results

CRC mortality has slowly decline largely due to cancer prevention and early detection, however; cancer prevention and early detection may not be the only key factors for the management or treatment of patients with CRC. In 2019, 145,600 estimated new cases of CRC will occur and an estimated 51,020 deaths is expected to occur for the year12. While prevention through screening is the best approach to combat the development of CRC, many patients present with advanced disease that will require surgery and systemic therapy to improve survival. With respect to systemic therapy, the median survival of patients with metastatic CRC has improved over the past three decades due to the introduction of 5-FU.

5-FU is commonly used alone or in combination with other anticancer drugs as first—line treatment for patients who have developed advanced CRC. The anticancer agent exerts its effect by interrupting RNA processing and blocking normal DNA synthesis. However, the response rate of 5-FU is less than 15% and due to dihydropyrimindine dehydrogenase (DPD) 80% of 5-FU is rapidly broken down limiting the therapeutic efficacy of the anti-cancer agent13,14,15.

Pharmaceutical nano drug systems have become idea for the transport of anti-cancer drugs and the reduction of unwanted distribution and side effects to healthy cells. These nanoparticles provide protection for anti-cancer drugs from first—pass metabolism and enzymatic degradation. In addition, nano drug systems generally provide enhancement properties of the drug, for example, prolong systemic circulation, increase drug entrapment and loading capacity. For many of those reasons, we decided to develop a novel SLN delivery system capable of high payload of 5-FU and improve 5-FU therapeutic efficacy on HCT-116 colon cancer.

### Characterization of PEGylated 5FU-SLNs

PEGylated SLNs are carrier systems that have been developed to encapsulate, protect and deliver active drug components. Six different batches of loaded SLNs were prepared by cold homogenization method. Cold homogenization was used to overcome temperature induced drug degradation, drug distribution into aqueous phase and crystallization modification as the method is also the preferred technique for hydrophilic drugs16,17. The active drug molecules were dispersed into the lipid melt fabricating a drug enriched core, a type of SLN as illustrated in Fig. 1. All batches were used in initial in-vitro studies indicated in Fig. 2A. Table 1 presents the drug, lipid, and surfactant composition for each batch of 5FU-SLN. With the chosen lipid compositions, glycerol monostearate, Compritol 888 ATO and Precirol ATO 5 (2–3%) the particle size and dispersity stayed in range, 77–288 nm. 5FU-SLN1 had a glycerol monostearate content of 2% and an observed higher particle size with decreased dispersity compared to other formulations with 3% lipid content. The choice of surfactants and its concentration has a great impact on SLN size. Lipid and surfactant compositions were kept below 5% to prevent particle growth and maintain small particle size. Again, higher concentrations of surfactants may lead to toxicity as reported in literature that ionic emulsifiers are more likely to produce smaller size in comparison to non-ionic emulsifiers even at the equal concentrations18. With the knowledge given formulations 5FU-SLN5 and 5FU-SLN6 both contain a non-ionic emulsifier as well as Lecithin, a highly recommended co-emulsifier to help decrease the particle size. The results showed smaller particle size for those formulations in comparison to those that were prepared with frequently used Tween 80 only. The surfactant concentrations did not drastically change/effect zeta potential. The results suggest that the surfactant concentrations are sufficient for the development of SLNs as the particle size range is expected. It could be seen from Table 1 the mean particle size, zeta potential and polydispersity index (PI) for each formulation (Supplementary Figures S7 and S8). The PI is used to measure the degree of non-uniformity of a size distribution of nanoparticles. With the use of the cold homogenization technique, wide dispersion typically occurs as indicated by the PI values shown in Table 1. Although all formulations had reasonable particle sizes, ranging from 77 to 288 nm, 5FU-SLN4 with particle size of 263 ± 3 nm with transmission electronic microscopy image (TEM) as shown in Supplementary Figure S1 was chosen for this study. The reason for selecting 5FU-SLN4 was due to the desired cytotoxicity activity displayed by 5FU-SLN4 in Fig. 2A–C, Supplementary figure S6 as well the high entrapment efficiency (EE) of 81 ± 10% (Table 1 and Supplementary Table S1).

### In vitro cell viability studies

To evaluate the efficacy of PEGylated 5FU-SLNs, in vitro cytotoxicity of 5-FU and various PEGylated 5FU-SLN formulations studies were performed against HCT-116 cells for 48 h at different concentrations. Figure 2A shows the effect of increasing concentration of 5-FU along with the six batches (5FU-SLN1, 5FU-SLN2, 5FU-SLN3, 5FU-SLN4.5FU-SLN5 and 5FU-SLN6) to assess the growth inhibition of HCT-116 cancer cells. As shown in Fig. 2, PEGylated 5FU-SLN4 was most effective against HCT-116 cell growth at all concentrations in comparison to the free 5-FU (Fig. 2B) and the five 5FU-SLN formulations (Fig. 2A). To determine the extent of HCT-116 cells sensitivity to 5-FU and 5FU-SLN4, half maximum inhibitory concentrations (IC50) of 5-FU and PEGylated 5FU-SLN4 against HCT-116 cells were measured. Figure 2C shows that IC50 value (17.7 ± 0.03 μM) of 5-FU was 2.4 fold-high compared with that of 5FU-SLN4 (7.4 ± 0.02 μM) suggesting that 5FU-SLN4 was significantly effective and would require significantly less concentration to treat HCT-116 cells compared with 5-FU.

### In vitro drug release kinetics from PEGylated SLNs

In Fig. 3, the cumulative in-vitro drug release of 5-FU during a 24 h period of incubation at 37 °C was presented. In this case, the dialysis bag method was used to study in vitro evaluation of 5-FU release profile from 5FU-SLN4. The formulation was designed with the expectation that 5-FU was encapsulated and retained within the SLN4. The release of free 5-FU was rapid and almost complete from the dialysis bag with about 80% released just within 2 h and about 90% released within the first 3 h. The free 5-FU release continued until it reached 95% within the first 4 h. This suggests that any free 5-FU placed in the dialysis bag would rapidly diffuse out in a significant amount within a short period of time. In addition, its release profile could be used to differentiate the release pattern of 5-FU from SLN4 as long as sink condition was maintained and diffusion of 5-FU to the receiving medium depended on concentration gradient (Fig. 3A). In comparison to 5-FU release from SLN4, about 40% of 5-FU was released within the first 2 h followed by a gradual or slow release of additional 20% over a period of 6 h and reached steady state at time 8 h. This implies that most of the 5-FU remained entrapped in SLN4 under the study conditions (Fig. 3A).

To determine the 5-FU diffusion mechanism, in vitro release kinetics of 5-FU from SLN4 was modeled as shown in Fig. 3B,C for zero-order kinetics, Fig. 3D for Higuchi model and Fig. 2E,F for first-order kinetics. The best-fit release mechanism was determined based on the comparison of R2 values of the various kinetic models (Fig. 3B,D,E)19. The R2 value (0.92) of Higuchi model (Fig. 3D) was found to be the highest compared with R2 value (0.72) of zero-order (Fig. 3B) and R2 value (0.61) of first-order (Fig. 3E) based on the 24 h. This suggests that 5-FU released from SLN4 followed Higuchi diffusion kinetics and implies that 5-FU release come from a homogeneous delivery system and diffuses out of the delivery system over a period of time19,20. Figure 3C (R2 = 0.97) and Fig. 3F (R2 = 0.97) were not considered in determining the best-fit model of 5-FU release because the graphs covered only the first 2 h (first six data points).

### Flow cytometry and confocal imaging

To further assess PEGylated 5FU-SLN4 cell uptake by HCT-116 cells, flow cytometry analysis was performed. Supplementary Figure S2 shows cellular uptake of SLN as they were tagged with rhodamine (Rho-SLN4) and incubated for 3 h at 37 °C. As shown in Supplementary Figure S2 (A and B), flow cytometry data revealed significant Rho-SLN cellular uptake. To further confirm the uptake of PEGylated 5FU-SLN4, HCT-116 cells were treated for 24 h at 37 °C with FITC-labeled SLNs. Confocal images Supplementary Figure S2C of HCT-116 cells showed uptake of FITC-SLN as Hoechst dye was used to counterstain nuclei. The merged image indicated that the solid lipid nanoparticle localized within the cell nuclei.

### Clonogenic studies

Clonogenic survival assay was conducted to evaluate in vitro cell survival. The primary goal was to determine the proliferative ability of HCT-116 cell lines after treatment with 5-FU and 5FU-SLN4. As shown in Supplementary Figure S3A, as concentrations of both 5-FU and 5FU-SLN4 increases, survival percent decreases. More importantly as 5FU-SLN4 concentration increased from 3 to 25 μM, we observed a significantly decreased colony formation of 5FU-SLN4 treated HCT-116 cells as shown in Supplementary Figure S3B. In contrast, colony formation especially in 6 μM 5-FU treated HCT-116 cells was a remarkably higher than that treated with 5FU-SLN4. The findings seem to suggest that a significant DNA damage might have occurred in 5FU-SLN4 treated cells than that 5-FU and was most likely due to the HCT-116 cells inability to repair DNA and reproduce.

### Western blot analysis

Utilizing the inhibitory concentrations (IC50) from viability experiments, we performed protein expression analysis in HCT-116 cells treated with 5-FU and PEGylated 5FU-SLN4 for 48 h. In Fig. 4A protein analysis showed greater dose-dependent reduction of EGFR and AKT phosphorylation with treatment of PEGylated 5FU-SLN4. As studies have shown, EGFR is highly expressed in advanced CRC with associated poor survival21. More importantly, the results indicate that, our developed formulation, PEGylated 5FU-SLN4 appeared to be more effective in targeting EGFR as seen in the reduction of protein expression compared with that of 5-FU treated cells (Fig. 4B,C). Our novel formulation, PEGylated 5FU-SLN4 reduced the expression of both the total and phosphorylated EGFR as well as some of the crucial molecules in the EGFR signaling pathways, activated AKT and STAT3. Such inhibitory effects of 5FU-SLN4 may negatively impacts the signaling through the key transduction pathways activated by EGFR such as RAS, phosphatidylinositol 3-kinase (PI3K), Src and signal transducer and activator of transcription 3 (STAT3) involved in the control of cell proliferation, survival and differentiation22. Blocking the EGFR-induced activation of the RAS-Raf-MEK-MAP kinase cascade or PI3K-AKT results in the inhibition of transcription factors regulating gene expression, while inactivation of STAT3 prevents dimerization and translocation to the nucleus where they act as transcription factors23.

### Acute toxicity

Clinical usefulness of 5-FU has been precluded because of its hepatotoxic and nephrotoxic side effects24. In the present study, careful examination of the H&E-stained sections of the liver and kidney of mice treated with 5-FU and 5FU-SLN4 (dose: 20 mg/kg) revealed normal histological (hepatic and renal) architecture compared with that of the control mice (Supplementary Figure S4). While other reports indicate loss of normal hepatic and renal architecture of mouse treated with 5-FU (80 mg/kg, dose)25,26, we did not observe any abnormality in hepatocytes and nephrocytes architecture of mice treated with either 5-FU or 5FU-SLN4 (dose: 20 mg/kg). More importantly, at a dose of 20 mg/kg 5FU-SLN4 significant CRC tumor growth suppression was observed, suggesting that the dose was tolerable and more effective than the 5-FU in the mice.

### Pharmacokinetics

5-FU and 5FU-SLN4 was intraperitoneally dosed (20 mg/kg) with a single bolus administration in mice to evaluate and compare the pharmacokinetic parameters of 5-FU and 5FU-SLN4 (at 5-FU equivalent dose). As shown in Table 2, the half-life (t1/2) of 5FU-SLN4 (2.13 ± 0.04 μg/mL * h) was found to be threefold high compared with 5-FU (0.68 ± 0.02 μg/mL * h) with p-value of 0.0001, while area under curve (AUC) of 5FU-SLN4 (54.7 ± 3.2 (μg/mL * h)) was significantly higher than 5-FU (15.5 ± 1.9 μg/mL * h) with p-value of 0.0001. As expected, the plasma clearance () of 5-FU (12.5 ± 1.3 mL/h) was remarkably high compared with 5FU-SLN4 (4.1 ± 0.8 mL/h) with p-value of 0.0007. In comparison with literature on pharmacokinetic profiles of similar 5-FU loaded SLNs formulations, we found that AUC and t1/2 values of our formulation 5FU-SLN4 were significantly greater than in literature27,28,29. In addition, CL value of 5FU-SLN4 was notably lower than that found in literature27,29. Put together, the results suggest that SLN4 may have prolonged plasma circulation of 5-FU and the observed favorable pharmacokinetics of 5-FU was mostly due to the unique ability of SLN4 to protect and deliver high payload of 5-FU. However, more studies will have to be conducted to elucidate the mechanisms underlying the 5FU-SLN4’s ability to increase the systemic exposure of 5-FU when administered intraperitoneally.

### Tumor efficacy studies

The tumor-inhibition study was conducted over a period of 30 days and was concluded when the normalized mean tumor volume of control (untreated) animals reached 375 ± 12 mm3. From Fig. 5, it was clear that 20 mg/kg 5FU-SLN4 was more effective in controlling the tumor growth compared to the other treated groups such as 20 mg/kg 5-FU, 5 mg/kg 5FU-SLN4 and 10 mg/kg 5FU-SLN4. However, the normalized mean tumor volumes of 20 mg/kg 5FU (300 ± 50 mm3), 5 mg/kg 5FU-SLN4 (328 ± 23 mm3) and 10 mg/kg 5FU-SLN4 (325 ± 21 mm3) treated groups were significantly less than the control (untreated or normal saline) tumor (375 ± 12 mm3). Although 5 mg/kg 5FU-SLN4 and 10 mg/kg 5FU-SLN4 treatment groups were included in the tumor efficacy study, our final goal was to investigate the efficacy of 20 mg/kg 5-FU and 20 mg/kg 5FU-SLN4 in controlling tumor growth. At the end of the study (30 days), mean tumor volume (200 ± 8 mm3) of 20 mg/kg 5FU-SLN4 treated group was significantly lower than the mean tumor volume (300 ± 50 mm3) of 20 mg/kg 5-FU treated group (p < 0.0001).

Recently, there have been several reports focused on the tumor doubling time (TDT) of cancer30,31,32 which sought to establish relationship between TDT and cancer prognosis which states that the longer TDT the better the prognosis. In this study, mean tumor volume of 20 mg/kg 5FU-SLN4 treated group took 30 days to double while mean tumor volume of 20 mg/kg 5-FU treated group doubled within 21 days. This suggests that 20 mg/kg 5FU-SLN4 might be more effective in suppressing tumor growth compared with 20 mg/kg 5FU. This was expected, given the short half-life of 5-FU (Table 2) which is most likely due to rapid metabolism and the fact that it was administered without any protection. While SLN4 acted as an efficient vehicle or delivery system, it may have afforded 5-FU excellent protection, leading to prolong systemic circulation which may lead to more 5FU-SLN4 to accumulate in tumor site. However, biodistribution studies need to be performed to conclusively determine whether 5FU-SLN4 indeed accumulate to a greater extent in the tumors relative to other tissues. There were no significant weight changes in animals among all the treatment groups during study period, indicating minimum or total lack of apparent toxicity of the administered formulations (Supplementary Fig. S4).

The in vivo evaluations in this study employed a subcutaneous xenograft versus an orthotopic model employed in the study by Tseng et al.33. For future investigations in orthotopic models, the surface of SLN4 delivery system will be grafted with ligand specific to receptors expressed on CRC cells.

### Immunohistochemistry (IHC)

Epidermal growth factor receptor (EGFR) and human EGFR 2 (HER2) and vascular endothelial growth factor receptor (VEGFR) are tyrosine kinase receptors highly expressed in several solid tumors including CRC34,35,36. These receptors are frequently reported to harbor aberrant activities that lead to the proliferation, survival, migration and differentiation required for CRC pathogenesis. We therefore evaluated the impact of our novel formulated 5FU-SLN4 on the expression of EGFR, HER2 and VEGFR in the mice xenograft tumors derived from HCT-116 cells (Fig. 6). Immunohistochemistry staining of the tumor tissues indicated intense expression of EGFR in the control untreated (Fig. 6a) as well as 5-FU and 5FU-SLN4 treated tissues (Fig. 6b,c). Based on our previous observation on the inhibitory effect of the compounds on EGFR signaling from the western blotting, the only plausible explanation to this increased EGFR expression is a possible compensatory protective response to overcome inhibition of downstream signaling in order to overcome growth suppression and to increase survival37.

The control untreated tissue also showed intense HER2 immunoreactivity indicating relatively high expression of the protein (Fig. 6d). Treatment with 5-FU and 5FU-SLN4 caused a significant decrease in HER2 expression as shown in Fig. 6. Specifically, 5-FU-SLN4 was more effective at reducing HER2 expression compared to 5-FU as shown by relatively low immunoreactivity to HER2 in 5FU-SLN4 treated tumor tissues (Fig. 6e,f). Although in general, only trace to weak immunostaining was observed in the tumor tissue for VEGFR (Fig. 6g), immunoreactivity was markedly reduced in both 5-FU and 5FU-SLN4 treated tumors (Fig. 6h,i). Significant differences in both HER2 and VEGFR immunoreactivity intensities between the control untreated tissues (Fig. 6d,g) and 5-FU and 5-FU-SLN4 treated tumor (p < 0.0001 and p < 0.001 respectively, Table 3) were observed when the stained sections were further analyzed as shown in Table 3.

## Conclusion

The development of novel delivery system capable of delivering high payload of chemotherapeutic agents across the tumor microenvironment to target cancer cells and enhance therapeutic response has been a challenge or daunting task. 5-FU has shown efficacy in the treatment of cancers but suffers from physico-chemical limitations that restrict its use as a free 5-FU; however, SLN4 developed in this project appeared to be robust and versatile, and seemed to have the desired requisite of novel drug delivery that could be utilized to overcome the limitations encountered by 5-FU and other free anticancer agents. Most importantly, 5FU-SLN4 activity in mouse bearing HCT-116 subcutaneous tumor significantly inhibited the HER2 expression. In literature, HER2 overexpression or amplification has been implicated as one of the drivers in CRC progression34,35. But further investigation of 5FU-SLN4’s biodistribution in vivo needs to be carried out to fully determine its pattern of distribution in tissues relative to tumor tissue.

Finally, the 5FU-SLN4 developed in this project revealed an enhanced in vitro activity, which translated into to a favorable therapeutic efficacy in vivo compared to the free 5-FU.

## Materials and methods

### Materials

Compritol 888 ATO (glyceryl behenate) and Precirol ATO 5 (glyceryl palmitostearate) were obtained from Gattefosse (Saint Priest, France). Tween 80 (polysorbate 80), Lecithin, glycerol monostearate and 5-FU were purchased from Sigma-Aldrich (St. Louis, Missouri, USA). Trevigen HT PARP/Apoptosis Assay kit was purchased from Sigma-Aldrich (St. Louis, Missouri, USA). Colorectal carcinoma cell lines, HCT -116, were obtained from American Type Culture Collection (ATCC). All other chemicals used were of an analytical reagent grade.

### Preparation of PEGylated 5-FU SLN suspension

Different 5FU-SLNs were prepared by means of cold emulsification method using high-shear mixer based on a reported method with slight modifications38 as illustrated in Fig. 1. A combination of PEGylated lipids and surfactants were used in the optimization of 5FU-SLN. The following lipids formed the matrix: glycerol monostearate, mPEG2000-DSPE, Compritol or Precirol (2–3% w/v ratio) and the formulations were stabilized with Tween 80, Lecithin and Poloxamer (1.5–4% w/v ratio).

Lipids with different amounts and 5-FU (0.15% w/v) in a glass vial were heated to 70–80 °C to melt. For the cold homogenization technique, the 5FU-containing lipid melt is cooled, the solid lipid ground to lipid microparticles (approximately 50–100 μm) and these lipid microparticles were dispersed in a cold surfactant solution (water, Tween 80, Lecithin or Poloxamer) to yield a pre-suspension. This pre-suspension was homogenized (1,200 rpm (3,820 rcf), 3–5 cycles) on ice-bath (below room temperature) causing lipid microparticles to break directly to solid lipid nanoparticles. This process avoids or minimizes the melting of the lipid and therefore minimizing loss of hydrophilic drugs to the water phase39. The freshly prepared formulations were used to characterize all the various SLN formulations and, for in vitro and in vivo studies. The selected SLN formulation (5FU-SLN4) was lyophilized with 5% w/v mannitol and stored at 4 °C for future use.

### Characterization of SLN formulations

#### Transmission electron microscopy (TEM)

The morphological examination of the PEGylated 5FU-SLN4 was performed using high resolution TEM. The 3 μl samples were stained with 50 μl of ammonium molybdate solution (1% w/v) after adjusting the 5FU-SLN4 suspension to pH of 7.0 with 5 N sodium hydroxide. The stained sample was then placed on copper grids, allowed to dry and viewed by Tecnai F-20 transmission electron microscope (Philips Co. Japan).

#### Particle size and zeta potential

All analyzed samples were diluted with deionized water for both particle size and zeta potential measurements using a Zeta Potential/Particle Sizer (NICOMP 380 ZLS). All measurements were performed in triplicate. The NICOMP 380 ZLS measures the nanoparticles based on principles of dynamic light scattering (DLS).

#### HPLC analysis

5-FU analysis was performed according to method described by Ciccolini and colleagues with minor modification40. Briefly, 5-FU analysis was performed using a chromatographic system, which consisted of a HPLC (Waters Corporation, Milford, MA) equipped with an auto-sampler, photo diode array (2998 UV/Vis) detector and pumps. Separation was performed using a reverse phase column (ZORBEX SB—C18 4.6 × 250 mm, 5 μm).

A flow rate of 1.0 mL/min and injection volume 20 μl at ambient temperature were maintained while detection was performed at wavelength of 268 nm41. Prior to analysis, reverse phase column was equilibrated with mobile phase made up of methanol 5 mM sodium phosphate buffer in ratio of 595, and pH adjusted to 542. An isocratic elution was performed throughout the entire analysis including internal standards.

A calibration curve was prepared using 5-FU standard solutions with concentration range of 3–100 µg/mL (r = 0.9). A plot of the peak areas as a function of 5-FU concentration was plotted and the linear equation of the calibration curve given as y = mx + c was determined, where y is the peak area, m is the slope, x is the concentration of 5-FU and c is the y—intercept. Supernatants from controls were spiked with aliquots of 0.5 μg/ml of 5-FU.

### Encapsulation efficiency (%)

The entrapment efficiency (E.E) corresponds to the amount of drug incorporated in the nanoparticle. Entrapment efficiency analysis was performed by adding 1 mL of loaded SLN to 1 mL of methanol. The mixture was placed into the chamber of the ultrafiltration tube and centrifuged at 6000 rpm for 6 min. After centrifugation, the non-entrapped drug amount in the supernatant was collected and ran on HPLC analysis to estimate the amount of 5-FU present in the sample. The E.E was calculated using the following equation:

$$E.E = \frac{{{\text{Total amount of drug}} - {\text{Free drug}}}}{{\text{Toal amount of drug}}} \times 100$$
(1)

### In vitro release of 5-FU

The cumulative release of 5-FU from the solid lipid nanoparticles was performed at 37 ± 1 °C. Twenty milligrams (20 mg) of lyophilized 5FU-SLN4 was suspended in 1.0 ml of PBS (pH = 7.4), transferred into a dialysis bag (MWCO = 3,500) and incubated in 5 ml of PBS pre-equilibrated to temperature (37 ± 1 °C). One milliliter (1.0 ml) was collected at selected time points and replaced with 1.0 ml of fresh PBS. Collected samples were analyzed by HPLC–UV/Vis spectrophotometry to determine the amount of 5-FU released at different time points43.

### Mathematical models to determine 5-FU release kinetics

The release kinetics of 5-FU from the heat sensitive liposomes was investigated to predict the possible mechanism of release using mathematical models. The release order was determined using zero order (Eq. 2) and first order kinetic model as shown below.

$$C={C}_{o}+{K}_{o}t$$
(2)
$$Log C=Log {C}_{o}+\frac{{K}_{1}t}{2.303}$$
(3)

where Co is the initial amount of drug, C is the % cumulative 5-FU released (zero order) or first order (Eq. 3) at time “t” and Ko is zero order release constant and K1 is the first order release constant44.

A Higuchi model was used determine whether the release mechanism follows Fickian diffusion as shown below20:

$$C= {C}_{o}+{K}_{H}{t}^{1/2}$$
(4)

where C is the % cumulative 5-FU release at time, t and KH is the Higuchi constant.

### Cytotoxicity of PEGylated 5FU-SLNs against HCT-116 cells

Cell Viability: Cell viability study was performed using 96-well plates and evaluated by Alamar Blue assay. The HCT-116 cells were seeded at a density of 1 × 10cells/well. The cells were cultured in Dulbecco's Modified Eagle Medium (DMEM)/F-12 medium enriched with L-glutamine, HEPES, 10% fetal bovine serum (FBS), and 1% penicillin streptomycin. Cells were maintained at 37 °C and 5% CO2. After cells were grown to 70% confluence, they were treated with freshly prepared treatments at different concentrations of 5-FU-SLN and 5-FU equivalents. Cells were treated for a 48-h period followed by termination, stained with 0.05% resazurin, gently mixed and incubated in the dark for 2–3 h before measurement of the fluorescence with excitation at 560 nm and emission at 590 nm using GloMax Explorer Microplate Reader (Promega Biotech, Madison, WI).

### Cellular uptake of FITC-labeled PEGylated SLNs

#### Confocal imaging

HCT-116 cancer cells were grown in 6-well plates (with cover slips) at a cell density of 2 × 103, for 24 h at 37 °C45. The cells were then treated with fluorescein isothiocyanate (FITC)-labeled SLN4 in growth media. After 3 h, FITC-SLN4 was removed and the cells gently washed twice with PBS. Next, 5 μg/ml of Hoechst dye was added for nuclear staining, the cells were fixed using 4% paraformaldehyde, then mounted and imaged using Leica SP2 Multiphoton system45.

#### Flow cytometry

To determine 5-FU-loaded solid lipid nanoparticle uptake by the cells, HCT-116 cells were grown in 6-well plates at a cell density of 1 × 105 and cultured in growth media until 70% confluence45. Cells were treated with the rhodamine-labeled solid lipid nanoparticles (Rho-SLN) for 24 h at 37 °C. After treatment, the adherent cells were washed three times with PBS, detached from the culture plate with 0.25% trypsin- ethylene diamine tetraacetic acid (EDTA) solution, and centrifuged at 6,000 rpm (3820 rcf) for 5 min45. Finally, the cells were re-suspended in 500 μl PBS, fixed with 4% paraformaldehyde, and kept on ice until analysis using a Becton Dickinson (BD) Fluorescence-Activated Cell Sorting Canto Analyzer and a BD Fluorescence-Activated Cell Sorting Aria Cell Sorter (BD Biosciences)45.

### Clonogenic study

For colony assay, HCT-116 cells were seeded into T-25cm2 culture flask at a density of 5 × 105 cells and cultured in DMEM/F12 medium supplemented with 2 mM L-glutamine, 10 mM HEPES, 10% FBS, and 1% penicillin/streptomycin45. After the cells reached 70% confluency, they were exposed to different concentrations of free 5-FU and 5FU-SLN445. Treatment was repeated after 24 h for 48 h exposure after which the experiment was terminated, cells harvested, and then re-plated onto 6-well plates at a density of 1,000 cells per well and incubated with growth medium for 7 days45. After the control cells reached 75% confluence, the experiment was terminated by fixing and staining the plates with 0.5% crystal violet solution45. The stained colonies (fifty per colony) were counted using a Jenco Stereomicroscope.

### Western blot analysis

Western blot analysis were conducted by seeding (1 × 106) HCT-116 cells into 75 cm2 flask. The cells were treated with 3, 6 and 12 µM of 5FU-SLN4. After a 48 h period, treatment was terminated through trypsinization, cell pellets were collected after centrifugation and washed 3 times. The cell pellet was collected in microcentrifuge tubes and lysed with RIPA buffer supplemented with 1% Protease inhibitor cocktail (Millipore Sigma, St Louis, MO). Samples were centrifuged for 20 min at 14,000 rpm (3,820 rcf) and the supernatant stored at − 20 °C until further used. Lysates containing 100 μg of protein were mixed with SDS-PAGE sample buffer, placed in boiling water bath for 5 min and separated by Sodium dodecyl sulfate polyacrylamide Gel Electrophoresis (SDS-PAGE) and proteins were transferred onto nitrocellulose membranes (Bio-Rad, CA) overnight46. After blocking, membranes were incubated with primary antibodies at 4 °C overnight and then horseradish-peroxidase (HRP)-conjugated anti-mouse secondary antibodies (1:1,000; Cell Signaling, MA, USA) were added for 1 h at RT46. Chemiluminescence images were captured using the ChemiDoc Imaging System (Bio-Rad, CA).

#### Cutting of membranes

To enhance the detection of multiple proteins, membranes were cut into individual strips for better detection with different antibodies prior to imaging.

Western Blot bands were quantified by densitometric analysis using Image Lab software (Bio-Rad Laboratories) and GraphPad Prism version 6. The primary and secondary antibodies were: EGFR (1:1,000, sc-373746; Santa Cruz Dallas, TX), pEGFR (1:1,000, 2234) (Cell Signaling, MA), STAT3 (1:1,000, 9139; Cell Signaling, MA), pSTAT3 (1:1,000, 9134; Cell Signaling, MA), AKT (1:1,000, 9272; Cell Signaling, MA), pAKT (1:1,000, 9271; Cell Signaling, MA). Anti-Beta-Actin (1:1,000, sc-517582; Santa Cruz Dallas, TX) served as loading control for protein normalization. The experiments were performed at least two times.

### Animal studies

NOD/Scid mice were obtained from Jackson Laboratory (Bar Harbor, ME) and housed in a virus-free, indoor, light and temperature-controlled environment provided with adequate food and water. All procedures with animals were in accordance with the National Institutes of Health Guide for the Care and Use of Laboratory Animals and approved by the Florida A&M University Animal Care and Use Committee. Colorectal HCT-116 cancer cells were cultured in DMEM/F12 medium supplemented with 10% FBS and 1% penicillin-streptomycin and incubated in 5% CO2 at 37 °C47.

Mice were inoculated with HCT-116 cells (5.7 × 106 in 150 μl PBS) and injected subcutaneously in the lower flanks of NOD/Scid mice48,49. Prior to the tumor appearance within two weeks of inoculation, mice were grouped into 6 groups with 5 mice in a group [control, 5-FU, 5FU-SLN4 (5 mg/kg), 5FU-SLN4 (10 mg/kg) and 5FU-SLN4 (20 mg/kg)]. The mice were administered with selected formulations for 2 weeks, every other day via intraperitoneal (i.p.) injection50. The control group received normal saline throughout the study. Tumor measurements were taken every other day using a digital Vernier caliper throughout the experiment. Mice were euthanized when tumor size reached 2.0 cm.

#### Pharmacokinetic studies

A group of mice were injected intraperitoneally with a single dose of 20 mg/kg of 5-FU and 5FU-SLN4 (at 5-FU equivalent dose). Blood samples were taken from the tail vein of mice at different time points (5 min, 10 min, 15 min, 1, 2, 4, 8, 12 and 24 h). The samples were placed in heparinized centrifuge tubes and centrifuged at 12,000 rpm for 8–10 min. The plasma was then stored at − 20 °C until analysis.

#### Assay validation and recovery

A calibration curve was prepared using the peak areas of 5-FU against the respective concentrations of 5-FU. Linear regression analysis of the calibration data was performed using the equation y = mx + c, where y is the peak area, x is the concentration of 5-FU, and m and c are the slope and intercept, respectively51. The recovery of 5-FU from plasma or tissue samples was determined by comparing the peak areas from the samples containing known added 5-FU with those from 5-FU standard solutions51.

### Immunohistochemistry

Mice were euthanized and tissue samples (tumor, liver and kidney) were excised from mice immediately, washed with PBS and placed in 10% buffered formalin for 24 h and transferred to 70% ethanol for histopathological analysis. Histology was performed by HistoWiz Inc. (histowiz.com) using a Standard Operating Procedure and fully automated workflow. Samples were processed, embedded in paraffin, and sectioned at 4 μm52,53,54. Immunohistochemistry was performed on a Bond Rx autostainer (Leica Biosystems) with enzyme treatment (1:1,000) using standard protocols. Antibodies used were rat monoclonal F4/80 primary antibody (eBioscience, 14-4801, and 1:200) and rabbit anti-rat secondary (Vector, 1:100)52,53,54. Bond Polymer Refine Detection (Leica Biosystems) was used according to manufacturer’s protocol. After staining, sections were dehydrated and film coverslipped using a Tissue-Tek Prisma and Coverslipper (Sakura). Whole slide scanning (40 ×) was performed on an Aperio AT2 (Leica Biosystems).

### Statistical analysis

Pharmacokinetic parameters were estimated using software Pharmacokinetic Solutions 2.0 (PK Solver). The difference between 5-FU and 5FU-SLN4 treatment groups were analyzed using paired Student’s t-test and considered significant at p < 0.05. All experiments were performed at least in triplicate and analyzed using GraphPad Prism software (GraphPad Software, Inc., La Jolla, CA, USA).

## Data availability

All data generated or analyzed during this study are included in this published article and its Supplementary Information Files.

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## Acknowledgements

This research was supported by the National Cancer Institute (NCI) of the National Institutes of Health (NIH) under Award Numbers U54MD007582, U54CA233396-01 and 5P20CA192990-04. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.

## Author information

Authors

### Contributions

E.A., J.T. and T.S designed experiments, collected, analyzed, interpreted the data and wrote the main manuscript text. K.A. contributed to design of experiments. E.N. contributed to the work and performed western blot analysis. F.A., B.H. and S.K. provided data analysis and interpretation. E.A., J.T. and S.K. conceived the concept development and design, provided materials, analysis of results and manuscript write-up. All authors reviewed the revised manuscript and provided valuable insight and edits.

### Corresponding author

Correspondence to Edward Agyare.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Smith, T., Affram, K., Nottingham, E.L. et al. Application of smart solid lipid nanoparticles to enhance the efficacy of 5-fluorouracil in the treatment of colorectal cancer. Sci Rep 10, 16989 (2020). https://doi.org/10.1038/s41598-020-73218-6

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# no-cycles condition

Let $X$ be a metric space and let $f\colon X\to X$ be a homeomorphism. Suppose $\mathcal{F}=\{\Lambda_{1},\dots,\Lambda_{k}\}$ is a family of compact invariant sets for $f$. Define a relation $\rightarrow$ on $\mathcal{F}$ by $\Lambda_{i}\rightarrow\Lambda_{j}$ if

 $W^{u}(\Lambda_{i})\cap W^{s}(\Lambda_{j})-\bigcup_{l=1}^{k}\Lambda_{l}\neq\emptyset,$

that is, if the unstable set of $\Lambda_{i}$ intersects the stable set of $\Lambda_{j}$ outside the union of the $\Lambda_{l}$’s.

A cycle for $\mathcal{F}$ is a sequence $\{n_{i}:i=1,\dots,j\}$ such that

 $\Lambda_{n_{i}}\rightarrow\Lambda_{n_{i+1}}$

for $1\leq i and

 $\Lambda_{n_{j}}\rightarrow\Lambda_{n_{1}}.$

With some abuse of notation, we can write this as

 $\Lambda_{n_{1}}\rightarrow\Lambda_{n_{2}}\rightarrow\cdots\rightarrow\Lambda_{% n_{j}}\rightarrow\Lambda_{n_{1}}.$

If $\mathcal{F}$ has no cycles, then we say that it satisfies the no-cycles condition.

Title no-cycles condition NocyclesCondition 2013-03-22 14:30:53 2013-03-22 14:30:53 Koro (127) Koro (127) 5 Koro (127) Definition msc 37-00 msc 37C75 no-cycles no-cycle no cycles condition cycle

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# How can I get a product-of-sums from this sum-of-products?

I have this function:

$$f(A,B,C,D) = A'B' + CD' + ABC + A'B'CD' + ABCD'$$

I used a Karnaugh map to minimize the function to:

$$Minimum SOP = A'B' + C D' + A B C$$

How can I turn this into a product-of-sums?

My attempt was to first find the inverse:

$$(A'B' + C D' + A B C)'$$ $$= (A'B')'(C D')'(A B C)'$$ $$= (A + B)(C' + D)(A' + B' + C')$$ $$= (AC' + AD + BC' + CD)(A' + B' + C')$$ $$= A'(AC' + AD + BC') + B'(AC' + AD + BC') + C'(AC' + AD + BC')$$

I am stuck here because I see that I end up with this:

$$= A'AC' + A'AD + A'BC' + ...$$

How can this be correct?

Doesn't $A'A$ mean $A \wedge \neg A$ (which doesn't even make sense to me)?

If I made a mistake, where did I make it?

If I'm on the right track, what is my next step?

-
Here's a basic tutorial: meta.math.stackexchange.com/questions/5020/… –  Sammy Black Dec 1 '13 at 9:36
For product of sum, use K-map to find those $A,B,C,D$ which gives $f(A,B,C,D) = 0$ first –  peterwhy Dec 1 '13 at 9:40
@peterwhy And then what? –  Søren Dec 1 '13 at 9:44
Then you have found a sum of product for the negative of $f$. Use de Morgan's laws to find a product of sum for $f$. –  peterwhy Dec 1 '13 at 9:47

$$\begin{array}{r|c|c|c|c} CD|AB&00&01&11&10 \\\hline 00&1&0&0&0\\\hline 01&1&0&0&0\\\hline 11&1&0&1&0\\\hline 10&1&1&1&1 \end{array}$$

From this K-map, those $0$'s (the negative of $f$) can be written as a sum of product:

$$[f(A, B, C, D)]' = BC' + AC' + A'BD + AB'D$$

Then, using de Morgan's laws, the negative of the whole formula is

\begin{align*} f(A,B,C,D) =& (BC' + AC' + A'BD + AB'D)'\\ =&(BC')'(AC')'(A'BD)'(AB'D)'\\ =&(B'+C)(A'+C)(A+B'+D')(A'+B+D') \end{align*}

In short, you should focus on the $0$'s when finding a product of sum form.

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How did you group the zeros? Two groups of 4? –  Søren Dec 1 '13 at 9:59
Also, is there a way to turn my minimum SOP into POS, similar to the steps I wrote in my posted question? –  Søren Dec 1 '13 at 9:59
First question: two groups of 4, two groups of 2. Second question: to continue with your steps above, eliminate those terms with $AA'$ (e.g. $A'AC$), since $AA' = 0$ alone and a further $0C = 0$. But still, it is difficult to be sure your terms after grouping are minimum. –  peterwhy Dec 1 '13 at 10:04
Thanks--------! –  Søren Dec 1 '13 at 10:08
By the way, the expansion in your question is missing the $CD$ term on the fifth/last line. –  peterwhy Dec 1 '13 at 10:16


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My Math Forum Dice probability

 December 1st, 2009, 11:26 AM #1 Newbie   Joined: Dec 2009 Posts: 1 Thanks: 0 Dice probability I am trying to find a general solution to calculate a certain number with x number of dice. As in, what is the probability to roll a 10 with 3 6-sided dice? Or to roll a 17 with 4 6-sided dice? Could anyone please help me on this?
 December 1st, 2009, 11:49 AM #2 Member   Joined: Oct 2009 Posts: 64 Thanks: 0 Re: Dice probability It's very easy to write the recursion relation. Let C(N,k) be the number of ways to roll k dice so that the total is N. Then C(N,k+1) = C(N-1,k) + C(N-2,k) + ... + C(N-6,k) And C(N, 1) = 1 if 1<=N<=6 and 0 otherwise. That is, you look at the outcome of the first die, then count the number of ways to roll the remaining part of N on the remaining k dice. Add the counts for each possible roll of the first die, and you have the C(N,k+1) as desired. If you want to write a closed formula, you'll need to go a bit further.
 December 1st, 2009, 12:15 PM #3 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: Dice probability Wikipedia states that the probability of rolling k with n s-sided dice is: $F_{s,n}(k)=\frac{1}{s^n}\sum_{i=0}^{\left \lfloor \frac{k-n}{s} \right \rfloor} (-1)^i {n \choose i} {k-si-1 \choose n-1}$ So the probability of rolling 10 with 3 six-sided dice is $\frac{1}{6^3}\left [ {3\choose 0}{9\choose 2}-{3\choose 1}{3\choose 2}\right ]= \frac{27}{216} = \frac{9}{73}$. And the probability of rolling 17 with 4 six-sided dice is $\frac{1}{6^4}\left [ {4 \choose 0}{16 \choose 3}-{4 \choose 1}{10 \choose 3}+{4 \choose 2}{4 \choose 3} \right ]= \frac{560-480+24}{1296} = \frac{13}{162}$ If you want to solve the problem for all possible k at once, you can look instead at the coefficients of the expansion of (x+x²+x³+...+x^s)^n. For example, if you tell Wolfram Alpha to expand (x+x^2+x^3+x^4+x^5+x^6)^3, which represents three 6-sided dice, you get: $x^{18}+3 x^{17}+6 x^{16}+10 x^{15}+15 x^{14}+21 x^{13}+25 x^{12}+27 x^{11}+\\27 x^{10}+25 x^9+21 x^8+15 x^7+10 x^6+6 x^5+3 x^4+x^3$ which means that there is 1 way to get 18, 3 ways to get 17, 6 ways to get 16, 10 ways to get 15... etc.

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## KANO KIT DIY Computer: A Computer So Simple, Anyone Can Assemble It

The Raspberry Pi computing platform definitely has some interesting applications. This project is no exception, because it will allow users of all ages and experience to assemble a DIY computer. A Raspberry Pi is used as the brains of the operation.

Read more @ Technabob

## Kindleberry Pi: Hack Your Kindle Into Raspberry Pi Display

If you’ve got a Kindle you’re planning on upgrading, and you’re wondering what to do with your old eReader, then check out what Gef Tremblay did with his old Kindle. He hacked it into something he calls a Kindleberry. With the mod, his Kindle serves as a display for his Raspberry Pi computer.

Read more @ Technabob

## Happy Pi Day

Today is Pi Day.3/14, the first three digits  of the transcendental constant pi.

Here is last year’s post on Pi Day.

π is pretty important in maths and number theory. It’s also used a lot in real life to calculate circumferences, volumes and everything related to circles, wheels and anything circular.

I’m actually more interested in the fact that it’s irrational and transcendental, meaning that there is no repetition in the digits, no period to be found, no matter how far you go and that it’s not a solution of a non-zero polynomial equation with rational coefficients. I had a poster in my room before, π @ 1 million digits. I think that I might have posted the 1 million digits in a post last year. It took up a few MBs. Be warned.

## Pi at 1 Million

I’ve been meaning to do this for a while. I imagine that 1 million numbers should be easily contained in one single post.

Be warned. π at 1 million decimals in a txt file is about 1MB! Look at π.

On your home computer, you can calculate π at 128 million in about 15 minutes. π at 1 billion takes a bit more than 3 hours. That’s a lot of decimals. For now, 1 million is fine and dandy. I’m actually using the first 2000 decimals in a piece of art that I’m creating.

π at 1 million decimals.

## Pi Day

It’s Pi Day, so happy Pi Day.

Happy π Day!

What’s Pi Day?

Pi Day and Pi Approximation Day are two holidays held to celebrate the mathematical constant π (pi). Pi Day is observed on March 14 (3/14 in American date format), due to π being equal to roughly 3.14. Sometimes it is celebrated on March 14 at 1:59 p.m. (commonly known as Pi Minute). If π is truncated to seven decimal places, it becomes 3.1415926, making March 14 at 1:59:26 p.m., Pi Second (or sometimes March 14, 1592 at 6:53:58 a.m.). Pi Approximation Day may be observed on any of several dates, most often July 22 (22/7 (European date format) is a popular approximation of π). March 14 also happens to be Albert Einstein‘s birthday.

What’s Pi? It’s one of the most important constants in mathematics.

 Binary 11.00100100001111110110… Decimal 3.14159265358979323846… Duodecimal 3.184809493B91864… Hexadecimal 3.243F6A8885A308D31319… Continued fraction $3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \ddots}}}}$ Note that this continued fraction is not periodic.

Pi or π is one of the most important mathematical constants, approximately equal to 3.14159. It represents the ratio of any circle‘s circumference to its diameter in Euclidean geometry, which is the same as the ratio of a circle’s area to the square of its radius. Many formulas from mathematics, science, and engineering include π.

It is an irrational number, which means that it cannot be expressed as a fraction m/n, where m and n are integers. Consequently its decimal representation never ends or repeats. Beyond being irrational, it is a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it. Throughout the history of mathematics, much effort has been made to determine π more accurately and understand its nature; fascination with the number has even carried over into culture at large.

The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter “περίμετρος”, probably by William Jones in 1706, and popularized by Leonhard Euler some years later. The constant is occasionally also referred to as the circular constant, Archimedes‘ constant (not to be confused with an Archimedes number), or Ludolph‘s number.

*


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# Optimal suspicion functions for Tardos traitor tracing schemes

J. Oosterwijk, B. Skoric, J.M. Doumen

17 Citaten (Scopus)

## Samenvatting

We investigate alternative suspicion functions for Tardos traitor tracing schemes. In the simple decoder approach (computation of a score for every user independently) we derive suspicion functions that optimize a performance indicator related to the sufficient code length $\ell$ in the limit of large coalition size $c$. Our results hold for the Restricted-Digit Model as well as the Combined-Digit Model. The scores depend on information that is usually not available to the tracer -- the attack strategy or the tallies of the symbols received by the colluders. We discuss how such results can be used in realistic contexts. We study several combinations of coalition attack strategy vs. suspicion function optimized against some attack (another attack or the same). In many of these combinations the usual scaling $\ell \propto c^2$ is replaced by a lower power of $c$, e.g. $c^{3/2}$. We find that the interleaving strategy is an especially powerful attack, and the suspicion function tailored against interleaving is effective against all considered attacks.
Originele taal-2 Engels s.n. 15 Gepubliceerd - 2013

### Publicatie series

Naam Cryptology ePrint Archive 2013/154

## Vingerafdruk

Duik in de onderzoeksthema's van 'Optimal suspicion functions for Tardos traitor tracing schemes'. Samen vormen ze een unieke vingerafdruk.


text:
Journal article Open Access

# Influence of Reinforcement Oxides on Structural and Mechanical Properties of Glass-Ceramics: A Review Article

Amr Ali; Sabreen Abdallah Abdelwahab; Khaled Abdelwahed; Ibrahim Ahmed; Ahmed I. Ali

### Citation Style Language JSON Export

{
"DOI": "10.35940/ijitee.E8670.0310521",
"container_title": "International Journal of Innovative Technology and Exploring Engineering (IJITEE)",
"language": "eng",
"title": "Influence of Reinforcement Oxides on Structural and Mechanical Properties of Glass-Ceramics: A Review Article",
"issued": {
"date-parts": [
[
2021,
3,
30
]
]
},
"abstract": "
This review studied the mechanical behaviors of Glass ceramics (GC) based on the Al2O3/SiO2 system. Glass ceramics are great interest due to their wide variety of applications, which have the ability to fulfil the recent demands of advanced mechanical, optical and biomedical applications. Glass-ceramics are typically heat-stable and have greater mechanical features than glasses. In addition, mechanical properties can be customized to provide variable volume fractions of crystalline phases by regulating nucleation and growth of the crystalline phases. The distribution of these crystalline phases in the glass matrix increases the consistency of the material and, in comparison, effectively limits the growth of cracks. The crystallization process resulted in substantial improvements in micro-hardness and density values such as sodium calcium phosphate (Na4Ca(PO3)6 and calcium pyrophosphate (β-Ca2P2O7) had sufficient properties for bone grafts and dental applications. This article outlines recent developments in the field of doping Oxides as reinforced with SiO2-Al2O3-based Glass- ceramics, to enhance the mechanical properties of Glass-ceramics combination. The research focused on the mechanical and the tribological behaviour of Biomedical, Electronics applications and selection of fabrication methods.
",
"author": [
{
"family": "Amr Ali"
},
{
"family": "Sabreen Abdallah Abdelwahab"
},
{
"family": "Khaled Abdelwahed"
},
{
"family": "Ibrahim Ahmed"
},
{
"family": "Ahmed I. Ali"
}
],
"page": "117-127",
"volume": "10",
"type": "article-journal",
"issue": "5",
"id": "5830302"
}
20
13
views


text:
# Stack Snippets and non-runnable code

Since the introduction of Stack Snippets, I've noticed them showing up frequently with non-runnable code. (I spend most of my time on the tag, which is a server-side js variant. I imagine that folks see the same. Sure enough - it's mentioned in this question for as well.)

I can see value in Stack Snippets for js variants, for the code tidy-up within the editor and "copy to answer" function. But the "Run" button is prominent & useless.

Is there a way that I can edit a question to leave the snippet in place, but disable the "Run" button?

• Personally I find the 'snippet' feature a bit of a white elephant which serves little or no purpose. I follow the Android tag and anybody who programs for Android knows it's impossible to run anything without a full app context which involves multiple code files...snippets are useless. I rip out the snippet-related stuff on anything that has Android-specific code in it and wish it was disabled for any question with the Android tag on it. I suspect that should also apply in other scenarios. – Squonk May 30 '15 at 16:59
• Even worse, it's reasonably commonly seen in newbie questions in python, bash, lisp etc. – tripleee May 31 '15 at 10:41
• For the whole "other languages" can of worms, ideone.com already does it for a big ole pile o f'em. Start with a clone of that? – tripleee May 31 '15 at 10:43
• @Squonk snippets don't even support Java, much less the Android framework. My guess is that new users are just hitting the wrong buttons while creating questions. Perhaps we just need to make it more difficult to accidentally create snippets. – Bryan Herbst Jun 1 '15 at 18:39
• I've seen em in java – an earwig Jun 2 '15 at 0:11
• I see it in c++ from time to time -.- – Lightness Races in Orbit Jun 2 '15 at 0:18
• A newcomer who sees a button labeled "code snippet" won't necessarily anticipate what that button does. Would it help if it were labeled "Interactive Code Snippet" or some such? It would make it at least slightly apparent that it's a feature, not just a way to post code. – rutter Jun 2 '15 at 0:28
• For everyone complaining about snippets being misused, here's a feature request for you: meta.stackoverflow.com/questions/292421/… – BoltClock Jun 2 '15 at 11:14

If it's not a stack snippet (HTML+CSS+JS), don't use the stack snippet feature.

Copying a bog-standard code-sample to the answer is easy enough, the copy-to-answer button is not valuable enough to balance blatant misuse of the feature.

• To extrapolate that pov, just use jsfiddle if you've got (HTML+CSS+JS). The whole feature is not valuable enough when it only supports those languages, and not in server variations. Point is, many first timers are using the feature inappropriately, and it would be more effective to fix the code than fix the users. – Mogsdad May 30 '15 at 15:41
• @Mogsdad: Well, what language and platform should SE work on first? How should it detect that you the snippet is not used for client-side web-development, and what it's used for instead? And are you sure that's sufficiently more important than all the rest they could work on instead? – Deduplicator May 30 '15 at 16:18
• No need to get snippety! (hehe) As proposed, detection not needed - if it's not able to run client-side, it won't pass the "run first" rule. Far simpler. Partial support for JS variants - heck, for the whole C-like gaggle - could improve user experience more than full run-in-browser support for the next language stack. I don't know - hence the discussion. WRT relative importance - that's not for me (or you, I assume) to judge, nor is it relevant to the discussion. – Mogsdad May 30 '15 at 16:34
• Well, I'm still missing what non-runnable js-only runnable-snippets have over standard code-blocks... – Deduplicator May 30 '15 at 16:36
• For the most part, the code being posted in a question doesn't work in the first place - so the "run" button doesn't help there, either. But the separation of HTML/CSS/JS helps clarify, the ability to include additional code that supports but isn't central to the question helps, clicking "copy to answer" saves a few clicks... – Mogsdad May 30 '15 at 16:43
• I think it's nice to have a replacement for jsFiddle and CodePen and others that were often linked too. It would be nice if Runnable Snippets could run other kinds of snippets too, but HTML+CSS+JavaScript is as easy as combining them and feeding them to an iframe, so I think the next step would be excessively harder to build. Personally I'm glad with this feature, and I think it's acceptable that it works only for, basically, client-side web questions. The normal [code] button is there for other pieces of code, which is by the way also misused by new users.. – GolezTrol May 31 '15 at 7:08
• I posted such suggestion in a bit milder form. – PM 77-1 Jun 1 '15 at 18:41
• HTML+CSS+JS You can add that 3 languages are not required, a snippet can only display HTML to show a tag. Or HTML+CSS when the question is about stylesheet. Or HTML+JS when the question is about jQuery. – A.L Jun 2 '15 at 11:09
• Or any combination of whatever is necessary to reproduce the problem. I don't think every possibility needs to be mentioned. – BoltClock Jun 2 '15 at 11:10
• that was nice advice, but we need some shortcut on the editor. Not stack snippet and have language options. – Adi Prasetyo Oct 30 '17 at 5:25

Easily done, but not documented. (Until now, that is!)

Turns out there's zero tolerance on the syntax of the begin comment, therefore you can disable snippet controls by making that comment line invalid. A whitespace violation is enough, but I'm proposing using runnable: false in the hopes that it turns into a feature that only removes the "run" button, leaving the still-useful "copy snippet to answer" button.

<!-- begin snippet: js hide: false runnable: false -->

Other 'typos' that can disable the snippet widget:

• remove space after open or before close comment markers

<!-- begin snippet: js hide: false-->
^

• any extraneous character

<!-- begin snippet: js hide: false x -->
^

• I found this out a while back and it made my life so much easier – user4639281 May 31 '15 at 6:55
• I posted a bug report some time ago that also concerns the handling of snippet delimiters: meta.stackoverflow.com/questions/284254/… – BoltClock Jun 2 '15 at 11:12

I really like the runnable snippets

I think it's nice to have a replacement for jsFiddle and CodePen and others that were so often linked too. It would be nice if Runnable Snippets could run other kinds of snippets too, but HTML+CSS+JavaScript is as easy as combining them and feeding them to an iframe, so it was the obvious choice to start with.

It's going to be hard to build it for other languages

I think the next step would be excessively harder to build. You'd need a completely different UI to select from a multitude of languages, and you require a big server capacity and complex configuration to run all that stuff. Because once there is a Node.js tool, PHP users will want one too. And eventually you'll be running COBOL code somewhere.

But let's still keep the current one

Personally I'm glad with this feature, and I think it's acceptable that it works only for, basically, client-side web questions, and removing this feature because it is misused doesn't seem the right way to go. After all, new users make mistakes with the normal code snippet button too. That's what new users do, and often it's simple to fix. A small checkbox to mark it as 'Non runnable' could make life easier for those who edit the questions.

And add some features to normal code snippets

But maybe there can be some modification to the snippets inserted by the normal {code sample} button. They could have a 'copy snippet to answer' button as well, and maybe also a possibility to select a language in case the automatic detection failed.

Doing so gives no a richer experience for normal snippets, and an additional simulator for client-side web code, which may in the future be expanded with other languages as well.

• Note that there's already a way to select a language for highlighting if auto-detection didn't work: <!-- language: lang-whatever --> on a line separated by double newlines before the codeblock. – Nathan Tuggy Jun 1 '15 at 18:33
• Of course, but it would be nice if there was some GUI element that helps you with this, so you don't have to find out by trial and error. – GolezTrol Jun 1 '15 at 20:21


text:
My Math Forum Constant function

 Algebra Pre-Algebra and Basic Algebra Math Forum

 June 28th, 2019, 09:08 PM #1 Member   Joined: Aug 2018 From: RomÃ¢nia Posts: 88 Thanks: 6 Constant function Hello all, What are the extreme values of the function $\displaystyle f(x)=C$ where $\displaystyle C$ is a real constant? All the best, Integrator Last edited by Integrator; June 28th, 2019 at 09:19 PM.
 June 28th, 2019, 10:58 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,969 Thanks: 2218 The function's only value is $C$, which is the function's absolute maximum value and its absolute minimum value. Thanks from Integrator
June 29th, 2019, 09:29 PM   #3
Member

Joined: Aug 2018
From: RomÃ¢nia

Posts: 88
Thanks: 6

Quote:
 Originally Posted by skipjack The function's only value is $C$, which is the function's absolute maximum value and its absolute minimum value.
Hello,

I'm sorry, but I do not understand!
It is true that from $\displaystyle f'(x)=0$ it follows that we have extremes, but $\displaystyle f''(x)=0$ does not show that there is any global minimum or maximum global...
If the function $\displaystyle f(x)=C$ has both the global minimum and the global maximum equal to $\displaystyle C$, then this means that the function $\displaystyle f(x)=C$ is both increasing and decreasing $\displaystyle \forall x\in \mathbb R$?!? Unbelievable and I do not know why mathematicians have reached this conclusion...
The function $\displaystyle f(x)=C$ has local extremes? I say it has no local extremes, and that's because $\displaystyle f''(x)=0$. Thank you very much!

All the best,

Integrator

Last edited by skipjack; July 1st, 2019 at 05:00 AM.

June 30th, 2019, 08:25 AM   #4
Senior Member

Joined: Jun 2019
From: USA

Posts: 196
Thanks: 78

Quote:
 Originally Posted by Integrator It is true that from $\displaystyle f'(x)=0$ it follows that we have extremes , but $\displaystyle f''(x)=0$ does not show that there is any global minimum or maximum global... If the function $\displaystyle f(x)=C$ has both the global minimum and the global maximum equal to $\displaystyle C$ , then this means that the function $\displaystyle f(x)=C$ is both increasing and decreasing $\displaystyle \forall x\in \mathbb R$?!?[B]
You're confusing the simplified rules (almost always true) with the proper rules (more generally true, but worded more carefully).

$\displaystyle f'(x)= 0$ means the function is not changing locally (zero slope). This is a requirement for local minima/maxima, but not sufficient. (Look at $\displaystyle x^3\ @\ x=0$).

$\displaystyle f''(x)= 0$ means the slope is not changing locally (flat). If $\displaystyle f'(a) = 0\text{ and }f''(a) < 0$, then the function is increasing for $x<a$ and decreasing for $x>a$ (in the neighbourhood of a, anyway). This means it MUST be a local maximum. If $\displaystyle f'(a) = 0\text{ and }f''(a) > 0$, it MUST be a local minimum.

In contrast, $\displaystyle f'(x)=0\text{ and }f''(x)=0$ very often indicates neither a local minimum nor maximum, but this is not always the case. $\displaystyle f(x) = x^4\text{ and }f(x) = C$ both prove this.

Quote:
 Originally Posted by Integrator If the function f(x)=C has both the global minimum and the global maximum equal to C , then this means that the function f(x)=C is both increasing and decreasing âˆ€xâˆˆR?
I think both not decreasing and not increasing might be more accurate.

Would it help if we called $C$ the non-localised global minimum and the non-localised global maximum of $\displaystyle f(x)=C$?

Last edited by skipjack; July 1st, 2019 at 05:13 AM.

June 30th, 2019, 08:45 PM   #5
Member

Joined: Aug 2018
From: RomÃ¢nia

Posts: 88
Thanks: 6

Quote:
Originally Posted by DarnItJimImAnEngineer
You're confusing the simplified rules (almost always true) with the proper rules (more generally true, but worded more carefully).

$\displaystyle f'(x)= 0$ means the function is not changing locally (zero slope). This is a requirement for local minima/maxima, but not sufficient. (Look at $\displaystyle x^3\ @\ x=0$).

$\displaystyle f''(x)= 0$ means the slope is not changing locally (flat). If $\displaystyle f'(a) = 0\text{ and }f''(a) < 0$, then the function is increasing for $x<a$ and decreasing for $x>a$ (in the neighbourhood of a, anyway). This means it MUST be a local maximum. If $\displaystyle f'(a) = 0\text{ and }f''(a) > 0$, it MUST be a local minimum.

In contrast, $\displaystyle f'(x)=0\text{ and }f''(x)=0$ very often indicates neither a local minimum nor maximum, but this is not always the case. $\displaystyle f(x) = x^4\text{ and }f(x) = C$ both prove this.

Quote:
 Originally Posted by Integrator If the function f(x)=C has both the global minimum and the global maximum equal to C , then this means that the function f(x)=C is both increasing and decreasing âˆ€xâˆˆR?
I think both not decreasing and not increasing might be more accurate.

Would it help if we called $C$ the non-localised global minimum and the non-localised global maximum of $\displaystyle f(x)=C$?
Hello,

I think I do not confuse anything! In the case of $\displaystyle f(x)=x^4$, it is clear that the function has a global minimum because $\displaystyle f''(x)=12x^2>0$ $\displaystyle \forall x\neq 0$. In the case of $\displaystyle f(x)=C$, it follows that $\displaystyle f''(x)=0$ and that tells us nothing about the fact that the function $\displaystyle f(x)=C$ is increasing or is decreasing and so it seems incorrect to say that $\displaystyle f(x)=C$ is at the same time increasing and decreasing, which would mean that the function $\displaystyle f(x)=C$ would have the global minimum equal to the global maximum and equal with $\displaystyle C$....

All the best,

Integrator

Last edited by skipjack; July 1st, 2019 at 05:12 AM.

 June 30th, 2019, 09:09 PM #6 Member     Joined: Oct 2018 From: USA Posts: 90 Thanks: 61 Math Focus: Algebraic Geometry From an Analysis standpoint: $\displaystyle f: \mathbb{R} \mapsto \{C\}$ max$\{C\}$ = min$\{C\} = C$

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# DSFT of some basic filters

Used in ECE438

Plot of the frequency response of the average filter:

$h(k,l)=\frac{1}{16}\left[ \begin{array}{ccc}1& 2 & 1\\ 2 &4 &2 \\ 1 & 2 & 1 \end{array} \right]$

Plot of the frequency response of the filter:

$h(k,l)=\frac{1}{9}\left[\begin{array}{ccc}-1& -1 & -1\\ -1 &8 &-1 \\ -1 & -1 & -1 \end{array} \right]$

Plot of the frequency response of the following filter, with $\lambda = 0.5$.:

$h(k,l)=\frac{1}{9}\left[\begin{array}{ccc}-\lambda & -\lambda & -\lambda\\ -\lambda &9+8 \lambda & -\lambda \\ -\lambda & -\lambda & -\lambda \end{array} \right]$


text:
InertForm - Maple Programming Help

Home : Support : Online Help : Programming : Evaluation : Preventing Evaluation : InertForm Package : InertForm/Parse

InertForm

 Parse
 parse an expression to an inert form

 Calling Sequence Parse( expr, opts )

Parameters

 expr - string implicitmultiply - (optional) true or false lastread - (optional) unevaluated name names - (optional) set or the string "all", as described below corenames - (optional) set

Description

 • The Parse command accepts a string containing a Maple-syntax math expression, and returns an inert-form expression.
 • By parsing to inert form you can avoid automatic simplification that is normally done on mathematical expressions.  For example, parsing "1+1" using the parse command will give you the answer 2, because Maple automatically simplifies expressions where multiple integer constants are added.  Using this parser, the same "1+1" will give %+(1,1), where the terms are all present in a form that can be easily manipulated in a predictable way.  For more information on automatic simplification, refer to the Maple Expressions chapter of the Maple Programming Guide.
 • The inert form rewrites the following (normally infix) operators: *, +, ^, /.   It also prefixes all function calls with a percent character.
 • Note that the unary negation operator is not rewritten.  Parse of "-x" will return -x, and Parse of "a-b" will return %+(a,-b). Similarly Parse of "-a+b" will return %+(-a,b), so there is no ambiguity, and an inert minus is not necessary, except in the case of -0, which will be parsed to %*(0,-1).
 • If a string contains multiple expressions separated by a semicolon, only the first expression is parsed.  The lastread option can be used to find the character position of the last character included in the parsed result.  Pass an unevaluated name, lastread='n', and the designated name will be assigned the integer position.  This option should not be used in combination with implicitmultiply because some transformations are done to the string prior to true parsing, the offset returned will be unreliable.

Implicit Multiplication

Adding the optional parameter, implicitmultiply, or implicitmultiply=true will cause the parser to interpret some situations as implied multiplication.

 • An explicit space, such as in the expressions "2 x", or "x y", or "x 2", or "(x+4) y", is interpreted by Parse as multiplication.
 • A number followed by a variable or open parenthesis with no space, such as "2x", or "2(x-4)" is interpreted as multiplication between the number and subsequent expression.
 • Two back-to-back expressions surrounded by parentheses, such as "(x-1)(x+1)" is interpreted as multiplication between expressions.
 • Multi-letter variable names, if not otherwise specified by either the names or corenames options are interpreted as single letter variables multiplied together.  Thus "xy" is parsed as "x*y", but "norm(x)" is also interpreted as "n*o*r*m*x".

The names option lets you specify which additional variable and function names should be accepted as a single entity.  The additional names should be combined in set brackets, { } and denoted as strings.  Calling Parse("norm(x)",implicitmultiply,names={"norm"}) will now be interpreted as a single function call, %norm(x) as opposed to multiplication of single-letter variables as in the previous paragraph.

names="all" can be used to specify that all multi-letter names should be interpreted as single entities instead of a multiplication of single-letter variables.

The corenames option works in conjunction with the names option to build a complete list of recognized multi-letter names.  The union of corenames and names constitutes the full set of recognized names.  corenames is predefined to include standard special functions like sin, cos, log, and Pi.  For a complete definition, execute Describe(InertForm:-Parse).

By setting corenames={} you can override and eliminate all known names.

There is some ambiguity in identifying which names should be kept in the expression.  Consider the case where expr = phi2, and names = {phi,phi2}.  This could be interpreted as %*(phi,2) or as phi2.  This command will return the latter.  It matches the longest names first.

When implicitmultiply is enabled, the input should be restricted to mathematical expressions.  Constructs like procedures and modules are not supported, and will likely trigger a syntax error.

Package Usage

 • This function is part of the InertForm package, so it can be used in the short form Parse(..) only after executing the command with(InertForm). However, it can always be accessed through the long form of the command by using InertForm[Parse](..).

Examples

 > $\mathrm{InertForm}:-\mathrm{Parse}\left("\left(1+2\right)/6"\right)$
 ${\mathrm{%/}}{}\left({\mathrm{%+}}{}\left({1}{,}{2}\right){,}{6}\right)$ (1)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("cx2",'\mathrm{implicitmultiply}'\right)$
 ${\mathrm{%*}}{}\left({c}{,}{x}{,}{2}\right)$ (2)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("cx2",'\mathrm{implicitmultiply}',\mathrm{names}="all"\right)$
 ${\mathrm{cx2}}$ (3)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("cx2",'\mathrm{implicitmultiply}',\mathrm{names}=\left\{\mathrm{x2}\right\}\right)$
 ${\mathrm{%*}}{}\left({c}{,}{\mathrm{x2}}\right)$ (4)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("phi1phi2",'\mathrm{implicitmultiply}',\mathrm{names}=\left\{\mathrm{φ},\mathrm{φ1},\mathrm{φ2}\right\}\right)$
 ${\mathrm{%*}}{}\left({\mathrm{φ1}}{,}{\mathrm{φ2}}\right)$ (5)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("abcd",'\mathrm{implicitmultiply}',\mathrm{names}=\left\{\mathrm{ab},\mathrm{cd},\mathrm{abc}\right\}\right)$
 ${\mathrm{%*}}{}\left({\mathrm{abc}}{,}{d}\right)$ (6)
 > $\mathrm{InertForm}:-\mathrm{Parse}\left("varx^2",'\mathrm{implicitmultiply}',\mathrm{names}=\left\{\mathrm{var}\right\}\right)$
 ${\mathrm{%*}}{}\left({\mathrm{var}}{,}{\mathrm{%^}}{}\left({x}{,}{2}\right)\right)$ (7)

Compatibility

 • The InertForm[Parse] command was introduced in Maple 18.
 • For more information on Maple 18 changes, see Updates in Maple 18.


text:
## Fluid Mechanics - Torque at hinge of closed vessel

The closed vessel contains water with an air pressure of 10 psi at the water surface. One side of the vessel contains a spout closed by a 6-inch diameter circular gate hinged along one side. Horizontal axis of the hinge is located 10 ft below the water surface. Determine the minimum torque that needs to be applied at the hinge to hold the gate shut.

Okay.
Here's what I am thinking, and please let me know if I am doing this correctly.

First, I'm trying to find the resultant Force = F. Here's where I am slightly confused. Do I take this body as a curved surface or not? because the gate is located at a spout which is circular. If not, then what I did was just

A = pi * r^2 = 9*pi in^2 = pi/16 ft^2
hc (vertical distance from fluid surface to centroid of area) = 10 ft
specific weight = 62.4 lb/ft^3

F = sp wt * hc * A = 122.52 lb

Torque (T) = F * radius = F * (3/12)ft = 30.63 lb-ft

Is that right?
Attached Thumbnails

Recognitions:
Homework Help
 Calculate the velocity of efflux from the curved pipe opening using Bernoulli's Theorem. This water-flow will form a force at the circular gate trying to push it outwards, therefore equal and opposite torque is needed to counteract the torque due to water flow. The force due to water flow is given by $dAv^2$ where d is the density of water .The torque will be the moment of this force about the hinge of the circular gate. BJ


text:
# Euclid's Proof of infinite prime numbers

I think this should probably be obvious, but I having trouble understanding part of the proof: If $N=p_1p_2\cdots p_n+1$, then why is it necessarily true that any given $p$ does not divide $N$?

-
Because it leaves a remainder of $1$. – Brian M. Scott Jan 21 '13 at 23:50
Dividing by any of the given $p_k$ leaves a remainder of $1$. – Henry T. Horton Jan 21 '13 at 23:50
Oh alright I see thanks – user58437 Jan 21 '13 at 23:51
You can also use $\, N = p_1 p_2\cdots p_k +\, p_{k+1}\cdots p_n,\ \$ or, more simply, note that $N(N+1)$ has more prime factors than $N.\ \$ – Math Gems Jan 21 '13 at 23:54
Caveat: Note that this does not generate all prime numbers, only creates a new one. For example, $2 \cdot 3 + 1 = 7$, but the next prime is $5$. – George V. Williams Jan 22 '13 at 1:09

Because the rest of division for all $p_{i}$ is $1$. For example we can suppose that there are only $3$ prime numbers: $(2,3,5)$. So you can define a new number $N$ as $2*3*5+1=31$. So it results that $31=2*15+1=6*5+1=3*10+1$. Then $N$ is a new prime number!


text:
# Tag Info

## Hot answers tagged fitting

23

The following is based on the fact that the determinant of a matrix is equal to zero when two rows are the same. Thus, if you plug any of the points in, you get a true statement. SeedRandom[3]; pts = RandomReal[{-1, 1}, {5, 2}]; row[{x_, y_}] := {1, x, y, x*y, x^2, y^2}; eq = Det[Prepend[row /@ pts, row[{x, y}]]] == 0 (* Out: ...

10

The general equation of ellipse (here) is given by: ellipse[x_, y_] = a x^2 + b x y + c y^2 + d x + e y + f == 0; solving using 5 pintos result in: SeedRandom[3]; pts = RandomReal[{-1, 1}, {5, 2}]; sol = Solve[ellipse @@@ pts]; ellipse[x, y] /. sol[[1]] // Simplify (*a (-0.275185 + 1. x^2 + x (0.189022 + 0.566953 y) + 0.1281 y + 0.397124 y^2) == ...

9

Major update, version 2.0 What changed: Is a different, more clever way of solving the problem. Overcomes most of the issues of previous versions. Problem summary: I will rephrase the problem in simplest terms. There exests a named curve called limacon (french, pronounced [ˈlɪməsɒn], means snail), described by a simple equation $r=a+b\cos{\theta}$ in ...

5

Use NonlinearModelFit and put in the best guess parameters from your Manipulate. data = {0.316228, -0.316228, 0.316228, -0.316228, 0.316228, -0.316228, 0.316228, -0.316228, 0.316228, -0.316228, 0.316228} eqn[x_] := a Sin[b x + c] nlm = NonlinearModelFit[data, eqn[x], {{a, 0.4}, {c, -1}, {b, 3.1}}, x] Plot[nlm[x] /. fit, {x, 1, 11}, Epilog -> ...

4

Why not try to fit some functions to your three datasets? Using Fit we can fit e.g. polynomials of degree 10 as follows: FitPolynomial[data_] := Fit[data, Table[x^n, {n, 0, 10}], x]; {f1, f2, f3} = FitPolynomial /@ {yy1, yy2, yy3}; We can then combine the three functions f1, f2, and f3 into a single function f with PieceWise: f = Piecewise @ { {f1, 0 ...

3

Here is a Fourier Basis approach: ClearAll[FourierBasis2D]; FourierBasis2D[{numx_, numy_}, {λx_, λy_}, x_, y_] := N[With[{ωn = 2 π/λx, ωm = 2 π/λy}, Flatten[ {1}~Join~ Table[ {Cos[ n ωn x] Cos[m ωm y], Cos[ n ωn x] Sin[ m ωm y], Sin[ n ωn x] Cos[m ωm y], Sin[ n ωn x] Sin[ m ωm y]}, {n, numx}, {m, ...

3

Just an extended comment to start. I'll try to follow up with some code later today or over the weekend. This sounds like a perfect job for a Laguerre Filter and most likely an adaptive one, e.g., Laguerre Filters – An Introduction. You can find lots of info on this online. The Laguerre Filter smooths a data set based on Laguerre polynomials. Its first ...

2

Here's a rather ham-fisted approach using GaussianFilter: First, a filtering function: filter[data_, threshold_, r_] := Module[{datatemp = Transpose@data, pos = Flatten@Position[data[[All, 2]], x_ /; x >= threshold]}, datatemp[[2, pos]] = GaussianFilter[datatemp[[2, pos]], r]; Transpose@datatemp ] This function applies a Gaussian filter to all ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
Chapter 24

A 23-year-old woman presents to the office for consultation regarding her antiseizure medications. Seven years ago, this otherwise healthy young woman had a generalized tonic-clonic seizure (GTCS) at home. She was rushed to the emergency department, at which time she was alert but complained of headache. A consulting neurologist placed her on levetiracetam, 500 mg bid. Four days later, EEG showed rare right temporal sharp waves. MRI was normal. One year after this episode, a repeat EEG was unchanged, and levetiracetam was gradually increased to 1000 mg bid. The patient had no significant adverse effects from this dosage. At age 21, she had a second GTCS while in college; further discussion with her roommate at that time revealed a history of two recent episodes of 1–2 minutes of altered consciousness with lip smacking (complex partial seizures). A repeat EEG showed occasional right temporal spikes. Lamotrigine was gradually added to the regimen to a dosage of 200 mg bid. Since then, the patient has been seizure-free for almost 2 years but now comes to the office for a medication review. Gradual discontinuation of levetiracetam is planned if the patient continues to do well for another year, although risk of recurrent seizures is always present when medications are withdrawn.

*This case description includes the answers to implied questions regarding therapy.

Approximately 1% of the world's population has epilepsy, the second most common neurologic disorder after stroke. Although standard therapy permits control of seizures in 80% of these patients, millions (500,000 people in the USA alone) have uncontrolled epilepsy. Epilepsy is a heterogeneous symptom complex—a chronic disorder characterized by recurrent seizures. Seizures are finite episodes of brain dysfunction resulting from abnormal discharge of cerebral neurons. The causes of seizures are many and include the full range of neurologic diseases—from infection to neoplasm and head injury. In some subgroups, heredity has proved to be a predominant factor. Single gene defects, usually of an autosomal dominant nature involving genes coding voltage-gated ion channels or GABAA receptors, have been shown to account for a small number of familial generalized epilepsies. Commonly, one family shows multiple epilepsy syndromes including, for example, febrile seizures, absence attacks, and juvenile myoclonic epilepsy.

The antiseizure drugs described in this chapter are also used in patients with febrile seizures or with seizures occurring as part of an acute illness such as meningitis. The term "epilepsy" is not usually applied to such patients unless chronic seizures develop later. Seizures are occasionally caused by an acute underlying toxic or metabolic disorder, in which case appropriate therapy should be directed toward the specific abnormality, eg, hypocalcemia. In most cases of epilepsy, however, the choice of medication depends on the empiric seizure classification.

### Drug Development for Epilepsy

For a long time it was assumed that a single antiepileptic drug (AED) could be developed for the treatment of all forms of epilepsy. However, the ...

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