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Hypermethylation of CpG islands in the promoter regions is an important mechanism to silence the expression of many important genes in cancer. The hypermethylation status is passed to the daughter cells through the methylation of the newly synthesized DNA strand by 5-cytosine DNA methyltransferase (DNMT). We report herein that (−)-epigallocatechin-3-gallate (EGCG), the major polyphenol from green tea, can inhibit DNMT activity and reactivate methylation-silenced genes in cancer cells. With nuclear extracts as the enzyme source and polydeoxyinosine-deoxycytosine as the substrate, EGCG dose-dependently inhibited DNMT activity, showing competitive inhibition with a Ki of 6.89 μM. Studies with structural analogues of EGCG suggest the importance of D and B ring structures in the inhibitory activity. Molecular modeling studies also support this conclusion, and suggest that EGCG can form hydrogen bonds with Pro1223, Glu1265, Cys1225, Ser1229, and Arg1309 in the catalytic pocket of DNMT. Treatment of human esophageal cancer KYSE 510 cells with 5–50 μM of EGCG for 12–144 h caused a concentration- and time-dependent reversal of hypermethylation of p16INK4a, retinoic acid receptor β (RARβ), O6-methylguanine methyltransferase (MGMT), and human mutL homologue 1 (hMLH1) genes as determined by the appearance of the unmethylation-specific bands in PCR. This was accompanied by the expression of mRNA of these genes as determined by reverse transcription-PCR. The re-expression of RARβ and hMLH1 proteins by EGCG was demonstrated by Western blot. Reactivation of some methylation-silenced genes by EGCG was also demonstrated in human colon cancer HT-29 cells, esophageal cancer KYSE 150 cells, and prostate cancer PC3 cells. The results demonstrate for the first time the inhibition of DNA methylation by a commonly consumed dietary constituent and suggest the potential use of EGCG for the prevention or reversal of related gene-silencing in the prevention of carcinogenesis.

Green tea is a popular beverage worldwide. It has been shown to prevent carcinogenesis in animal models for different organ sites, and many mechanisms have been proposed for this activity (reviewed in Ref. 1). EGCG3, the major polyphenol in green tea, has many interesting activities and is believed to be a key active ingredient (1). In previous studies, we have shown that EGCG is methylated by catechol-O-methyltransferase to form MeEGCG and DiMeEGCG both in vitro and in vivo(2, 3). EGCG is also a potent inhibitor of catechol-O-methyltransferase activity (3). Catechol-O-methyltransferase and DNMT belong to the same superfamily of S-adenosylmethionine-dependent methyltransferases. Both enzymes have a common core structure at the catalytic site (4). It is possible that EGCG may also inhibit DNMT by binding to a similar catalytic site.

Hypermethylation of DNA is a key epigenetic mechanism for the silencing of many genes, including those for cell cycle regulation, receptors, DNA repair, and apoptosis (5, 6, 7, 8). In this aberrant methylation, the cytosine of the CpG island in or near the promoter region of the newly synthesized DNA strand is methylated by DNMT. The methylated CpG island has higher binding affinity to methyl-CpG binding domain proteins that recruit transcriptional corepressors, such as histone deacetylases and Sin 3A, resulting in chromosome condensation and transcription repression (9, 10). The inhibition of DNMT, especially DNMT1, would block the hypermethylation of the newly synthesized DNA strand, resulting in the reversal of the hypermethylation and the re-expression of the silenced genes (11, 12, 13). Indeed, this point has been demonstrated by studies with DNMT inhibitors, DAC (also known as 5-aza-2′-deoxycytidine) and zebularine. These compounds have been shown to inhibit cancer cell growth, induce cancer cell apoptosis, and reduce tumor volume in mice (14, 15, 16, 17). There is high potential for developing this group of inhibitors for cancer therapy. However, side effects and toxicity are serious concerns. There is a great need for the development of effective and nontoxic inhibitors of DNMT. We hypothesize that EGCG can be used for this purpose.

In the present study, we tested this hypothesis by examining the inhibition of DNMT by EGCG and the effects of EGCG treatment on four methylation-silenced genes in human esophageal cancer cell line KYSE 510: the tumor suppressor p16INK4a, retinoic acid receptor β (RARβ), O6-methylguanine methyltransferase (MGMT), and the DNA mismatch repair gene human mutL homologue 1 (hMLH1). In our previous work, these genes have been shown to be frequently silenced by DNA hypermethylation in human esophageal cancer, and some of these genes can be partially reactivated by treatment with the DNMT inhibitor, DAC (18). The results on the inhibition of DNMT and reactivation of methylation-silenced genes by EGCG are reported herein.

### Cell Lines and Cell Culture.

The human esophageal squamous cell carcinoma cell lines KYSE 510 and KYSE 150 were gifts from Dr. Yutaka Shimada (Kyoto University, Kyoto, Japan). The cells were maintained in 5% CO2 atmosphere at 37°C in RPMI 1640 and Ham F12 mixed (1:1) medium containing 5% fetal bovine serum. To determine the dose-dependent changes, KYSE 510 cells were treated with 5, 10, 20, or 50 μM of EGCG (Unilever-Bestfoods, Englewood Cliffs, NJ) or 8.7 μM of DAC for 6 days. EGCG or DAC was added, in new culture medium, to the cells on days 1, 3, and 5. For the time course study, cells were treated with 20 μM of EGCG for 12, 24, 48, 72, or 144 h. Human colon cancer cell line HT-29 and prostate cancer cell line PC3 were obtained from American Type Culture Collection (Manassas, VA), and were grown in McCoy’s 5A and RPMI 1640 containing 10% fetal bovine serum, respectively. These cells were treated with EGCG for 6 days as described above.

### DNA Methyltransferase Assay.

Cultured KYSE 510 cells were harvested, and nuclear extracts were prepared with the nuclear extraction reagent (Pierce, Rockford, IL). The DNMT assay was performed according to published methods (19, 20). In brief, the nuclear extracts (4 μg of protein) were incubated for 1.5 or 2 h at 37°C with 0.66 μM of poly(dI-dC)·poly(dI-dC) and 10 μM of S-adenosyl-L-[methyl-3H]methionine (2 μCi; Amersham, Piscataway, NJ) in a total volume of 40 μl of a pH 7.4 buffer, containing 20 mM Tris-HCl, 25% glycerol (v/v), 10 mM EDTA, 0.2 mM phenylmethylsulfonyl fluoride, and 20 mM 2-mercaptoethanol. All of the incubations contained 0.02% DMSO, which was used to dissolve EGCG and other inhibitors. Reactions were initiated by the addition of nuclear extracts and stopped by mixing with 300 μl of a solution containing 1% SDS, 3% 4-aminosalicylate, 5% butanol, 2.0 mM EDTA, 125 mM NaCl, 0.25 mg/ml carrier salmon testes DNA (Life Technologies, Inc., Gaithersburg, MD), and 1 mg/ml protease K (Sigma, St. Louis, MO). To remove proteins, the resulting mixture was vortexed with 300 μl of a solution containing 8% phenol, 12% m-cresol, and 0.1% 8-hydroxyquinoline (Sigma), and centrifuged. The methylated DNA template was recovered from the aqueous phase by ethanol precipitation and then washed three times with 70% ethanol. The radioactivity in the pellets was counted in a scintillation counter. All of the assays were performed in duplicate. Background levels were determined in incubations without the template DNA.

### In Silico Molecular Modeling Studies of DNMT1.

The protein sequence of DNMT1 (accession no. NP_001370) was retrieved from the National Center for Biotechnology Information Reference Sequence (RefSeq) Collection. A structural model of the catalytic domain of DNMT1 was constructed using the Insight II Homology Module (Accelrys, Inc., San Diego, CA) from the published crystal structure of HhaI Mtase (RCSB Protein Data Bank;4 PDB ID = 5MHT) as the modeling template. The quality of the model was confirmed by the WHATIF-Check program.5 Initial structures of EGCG and the five analogues were built using the Insight II Builder Module and subsequently energy minimized to yield a stable conformation. Each ligand was docked into the putative DNMT1 binding pocket using GOLD (Genetic Optimization for Ligand Docking; Ref. 21), which accommodates ligand flexibility and partial enzyme flexibility to satisfy ligand-enzyme binding requirements. After these docking studies to elucidate stable ligand-binding modes, values of the ligand-enzyme binding energy (ΔEbinding) were calculated for each of the six ligands using molecular mechanics procedures. Residues within 5.0 Å of the cytosine nucleotide were defined as the “active site” pocket and were permitted full relaxation, while holding all of the other residues fixed. We define ΔEbinding as the difference between the potential energy of the ligand-enzyme (Ecomplex) and the sum of potential energies of the ligand (Eligand) and enzyme (Eenzyme).

${\Delta}E_{\mathrm{binding}}\ {=}\ \mathrm{E}_{\mathrm{complex}}\ {-}\ (\mathrm{E}_{\mathrm{ligand}}\ {+}\ \mathrm{E}_{\mathrm{enzyme}})$

A favorable (more negative) binding energy is taken as evidence that the ligand possesses high affinity for DNMT1.

### Bisulfite Modification and Methylation-Specific PCR.

DNA was extracted from the cells using the DNeasy tissue kit (Qiagen, Valencia, CA), and was denatured with 0.2 M NaOH at 37°C for 30 min. The denatured DNA was treated with 3.3 M sodium bisulfite and 0.66 mm hydroquinone according to Herman et al.(22), with slight modifications (18). The modified DNA was purified using the Qiaquick gel extraction kit (Qiagen) and eluted with water. NaOH was added to the modified DNA for a final concentration of 0.3 M and maintained for 5 min at room temperature to desulfonate the modified DNA. The resulting DNA was repurified with Qiaquick gel and used immediately or stored at −20°C.

Methylation-specific PCR was carried out using a nested two-stage PCR approach (23). Stage I PCR was performed on bisulfite-modified DNA to amplify a 208-, 289-, 425-, or 185-bp fragment of the CpG-rich promoter regions of the p16INK4a, MGMT, RARβ, or hMLH1 genes, respectively. The primers recognize the bisulfite-modified template but do not discriminate between methylated and unmethylated alleles. The stage I PCR products were diluted and subjected to a stage II PCR with primer sets specific to methylated or unmethylated template. The primers were designed according to the literature (23, 24, 25, 26, 27). The chromosomal locations of all of the genes tested, primer sequences for both the methylated and unmethylated templates, annealing temperatures used, and expected PCR product sizes are summarized in Table 1. Placental DNA and placental DNA treated with SssI methyltransferase (New England Biolabs, Beverly, MA) were used as positive controls for unmethylated alleles and methylated alleles, respectively. Amplification was carried out using a 9700 Perkin-Elmer thermal cycler (Applied Biosystems, Foster City, CA).

### Reverse Transcription-PCR.

Total RNA was isolated from cells by the Tri reagent (Sigma). Reverse transcription of RNA was performed using the Advantage RT-for-PCR kit (Clontech, Palo Alto, CA). PCR was carried out using a 9700 Perkin-Elmer thermal cycler. The primers used in the PCR reactions for the genes analyzed (13, 28, 29) are summarized in Table 1. The PCR reactions were run in an approximate linear range (for example, 25–28 cycles for p16INK4a, 30–35 cycles for RARβ, 20–25 cycles for hMLH1, and 30–35 cycles for MGMT). The amplified products were subjected to electrophoresis on a 2% agarose gel. The gel was then stained with ethidium bromide and photographed. Negative controls for PCR were run under the same conditions without RNA or reverse transcriptase. Expression of the glyceraldehyde-3-phosphate dehydrogenase gene (G3PDH) was used to examine the integrity of RNA in each sample and to standardize the amount of cDNA added to each of the PCR reactions.

### Western Blot.

The protein samples in SDS-containing lysis buffer were loaded on 4–15% acrylamide gel (40 μg/lane). After electrophoresis, the proteins were blotted onto a nitrocellulose membrane. After incubation in blocking buffer (5% nonfat milk), the membranes were incubated with the primary antibodies (rabbit antihuman RARβ and hMLH1 polyclonal antibodies; Santa Cruz Biotechnology, Santa Cruz, CA) overnight at 4°C. After washing with Tris-buffered saline containing 0.1% Tween 20, the membrane was then incubated with sheep antirabbit horseradish peroxidase-labeled secondary antibody and visualized using the electrochemiluminescense detection kit (Amersham).

### Inhibition of DNA Methyltransferase Activity by EGCG.

With nuclear extracts from the KYSE 510 cells as the enzyme source and poly(dI-dC)·poly(dI-dC) as the substrates for DNMT, under our assay conditions, the production of methylated poly(dI-dC)·poly(dI-dC) was linear for 2 h. A dose-dependent inhibition by EGCG was observed, with an IC50 of ∼20 μM (Fig. 1,A). The structural analogues of EGCG: ECG, EGC, EC, and methylated EGCG all had inhibitory activities, with potency in the rank order of EGCG > ECG, MeEGCG > EGC, and DiMeEGCG > EC. Kinetic studies indicated that EGCG increases the Km of DNMT without affecting the Vmax, suggesting a competitive inhibition with a Ki of 6.89 μM (Fig. 1 B). The same conclusion was reached when this experiment was repeated.

### Molecular Modeling of the Interaction between EGCG and DNMT.

To investigate the possible inhibition mechanism of EGCG to DNMT1, we used information available from the X-ray crystal structure of HhaI Mtase (30) to model the three-dimensional structure of the DNMT1 catalytic domain. This structural model revealed a substantial interaction region with hemimethylated DNA and a cytosine active pocket for subsequent methylation, which was folded by evolutionarily conserved sequence motifs identified from multiple sequence alignments (Fig. 2,A). Docking of EGCG into the putative cytosine pocket indicated that the gallate moiety of this compound formed potential hydrogen bonds with two catalytically important residues, Glu1265 and Pro1223, which were the same residues that appear to stabilize the flipped cytosine through hydrogen bonding. Interestingly, the D ring was oriented at approximately the same position as the pyrimidyl ring of cytosine in the structural model of DNMT1. Furthermore, Ser1229 and Cys1225 are positioned to form hydrogen bonds with the hydroxy groups in the A and B rings of EGCG, respectively, thereby contributing to high-affinity DNMT1 binding. The interactions are depicted in Fig. 2 B. These findings suggest that EGCG exerts its inhibitory effect on DNMT1 function by blocking entry of the key nucleotide cytosine into its active site and, thus, prevents methylation. A consensus-binding mode was obtained from 25 independent docking procedures for EGCG and each of the five analogues bound to DNMT1. Accordingly, calculated values of Δ Ebinding (shown in parentheses, in units of kcal/mol) for each ligand in its consensus-binding mode yielded the following order of binding affinity to DNMT1: EGCG (−49.75) > ECG (−46.53)> Me-EGCG (−44.91) > diMe-EGCG (−41.64) > EGC (−40.79) > EC (−38.32). These predicted results are generally consistent with the experimental data.

### Reversal of Hypermethylation and Reactivation of RARβ, MGMT, p16INK4a, and hMLH1 by EGCG.

In KYSE 510 cells, p16INK4a, RARβ, MGMT, and hMLH1 had hypermethylation status with the loss of the respective mRNA expression. After treating the cells with 5, 10, 20, or 50 μM of EGCG for 6 days, the methylation-specific bands of these four genes still existed. The unmethylation-specific bands of these four genes, however, appeared after treatment with 20 or 50 μM of EGCG for 6 days (Fig. 3,A). Corresponding to the appearance of the unmethylation-specific band was the re-expression of mRNA of all four of the genes. The unmethylation-specific bands and mRNA expression were both observed with 20 and 50 μM of EGCG. For hMLH1, the unmethylation-specific band was observed even with 10 μM of EGCG in comparison to the faint bands observed with 0 and 5 μM EGCG (Fig. 3,A). The reversal of hypermethylation and re-expression of these four genes by EGCG was similar to that produced by the classical DNMT inhibitor, DAC. After treating the cells with 20 μM of EGCG for 0, 12, 24, 48, 72, and 144 h, strong unmethylation-specific bands for all four of the genes began to appear at 48 h (Fig. 3 A). The mRNA expression of these four genes was also observed at 48 h and, depending on the genes, higher levels were observed at either 72 or 144 h. Again, a faint hMLH1 mRNA band was observed at 0, 12, and 24 h. Similar results were obtained in two repeated experiments.

Using the reactivation of RARβ mRNA as an assay, ECG was found to be slightly less active than EGCG; EGC, MeEGCG, and EC had lower activities than EGCG; and DiMeEGCG had no activity when the cells were treated with 20 μM of the test agents for 6 days (Fig. 3 B). In another experiment, demethylation and reactivation of RARβ was observed with DiMeEGCG at 50 μM, but not at 20 μM (data not shown).

Western blot analysis indicated that the RARβ protein expression was increased in the cells treated with 20 and 50 μM of EGCG for 6 days (Fig. 4). A time-dependent increase of hMLH1 protein expression level was observed in the cells treated with 20 μM of EGCG for 3 and 6 days. Because of the lack of suitable antibodies, we were unable to accurately analyze the re-expression of MGMT and p16INK4a proteins.

To determine whether the reactivation of methylation-silenced genes by EGCG is a general phenomenon that also occurs in other cell lines, we studied the effects of EGCG treatment on the methylation status and mRNA levels of p16INK4a or RARβ in three other human cancer cell lines (Fig. 3 C). The appearance of unmethylation-specific bands and reactivation of the mRNA of p16INK4a in HT-29 cells, and RARβ in KYSE 150 and PC3 cells, were observed after treating the cells with 20 μM EGCG for 6 days. In the KYSE 150 cells, a faint RARβ mRNA band was also observed in cells without EGCG treatment, but the band was much more intense after EGCG treatment. These results were reproduced in three experiments with the three cell lines. The results, although still preliminary in a quantitative sense, demonstrate that the reactivation of methylation-silenced genes does occur in different cell lines, suggesting it is a general phenomenon.

### Inhibition of Cell Growth and Other Effects.

After treatment with EGCG for 6 days, significant inhibition of cell growth was not observed at 5 or 10 μM, but 36 and 54% inhibition were observed at 20 and 50 μM of EGCG, respectively (Fig. 5,A). Many damaged cells and some polynuclear cells were observed with 50 μM of EGCG, and to a lesser extent with 20 μM of EGCG, in this 6-day experiment (Fig. 5,C). In experiments in which cells were treated for 2 days with EGCG, growth inhibition and signs of toxicity were not apparent at 5, 10, or 20 μM; but at 50 μM, some damaged cells with vesicles were observed (Fig. 5,C). After treatment of the cells with 50 μM of EGCG for 2 days, the colony formation (measured after an additional 7 days of culturing) was inhibited by 55%, but significant inhibition was not observed with EGCG at 20 μM or lower concentrations (Fig. 5 B). Preliminary experiments with RT-PCR also indicated that EGCG treatment did not affect the mRNA expression level of DNMT1, DNMT3a, DNMT3b, and MBD2 (methyl-CpG binding domain protein 2), which has been reported to have DNA demethylase activity (31).

The present study clearly demonstrates that EGCG inhibits DNMT activity, and causes CpG demethylation and reactivation of methylation-silenced genes. To our knowledge, this is the first demonstration of such an activity of a commonly consumed dietary constituent. Kinetic studies indicate that EGCG inhibits nuclear DNMT activity competitively with a Ki of 6.89 μM. We believe that most of the activity is because of the dominant form of DNMT, DNMT1, although DNMT 3a and 3b may also be present in smaller quantities in the nuclear extracts (14). Molecular modeling studies indicate that EGCG is well accommodated in a hydrophilic pocket of DNMT1. As depicted in Fig. 2 B, EGCG is effectively tethered within the DNMT1 binding site by at least four hydrogen bonds involving Ser1229 (2.5Å), Glu1265 (3.1Å), Pro1223 (3.0 Å), and Cys1225 (3.1Å). An additional hydrogen bond is possible with the protonated side chain of Arg1309, although the N… O distance (3.8Å) is at the outer limit of the typical hydrogen bond and, thus, would imply a weak interaction. This hydrogen-bonding scheme is consistent with the observed reduction in DNMT1 inhibitory activity for the five EGCG structural analogues. The absence of the B-ring 3′-hydroxy group in ECG abolishes the hydrogen bond with Cys1225. Likewise, the absence of the 4″-hydroxy group in MeEGCG eliminates the hydrogen bond with Pro1223. Addition of another methyl group at the B ring as in DiMeEGCG additionally interferes with the binding. The importance of galloyl moiety in EGCG-like compounds is evident from interactions with the two conserved residues, Glu1265 and Pro1223, which are associated with the catalytic activity of DNMT1. This observation concurs with the experimental results for EGC and EC, two poor DNMT1 inhibitors in which the galloyl moiety is conspicuously absent.

The activities of these compounds to reactivate RARβ mRNA expression are also correlated with their ability to inhibit DNMT activity. With EGCG, demethylation and reactivation of all four of the genes are observable after 48 h of treatment, and 20 μM is the effective concentration that activates all four of the genes. All of the results are consistent with our hypothesis that reactivation of methylation-silenced genes is due to the inhibition of DNMT by EGCG. More work is needed to determine the quantitative reactivation of the different genes as well as the extents of the demethylation in different cell lines under different treatment conditions. It would be interesting to determine whether the different hypermethylated genes respond similarly or differently to the EGCG treatment. The reactivated p16INK4a is expected to regain its function in cell cycle regulation, and re-expressing RARβ is expected to regain responsiveness to retinoic acids. It is not clear whether the presently observed growth -inhibitory effect after 2 or 6 days of treatment with EGCG are related to the reactivation of these genes. The functional importance of the reactivation of those genes remains to be determined.

Hypermethylation of DNA is a key epigenetic mechanism for the silencing of many genes including those for tumor suppressors, DNA repair enzymes, and receptors (5, 6, 7, 8). In theory, this epigenetic process is reversible if the newly synthesized DNA strands are not methylated. Therefore, DNMT inhibitors such as DAC and zebularine have been actively explored as cancer therapeutic agents, especially when they are used in combination with a histone deacetylase inhibitor, such as trichostatin A (15, 26, 32, 33, 34, 35, 36). It has also been suggested that methylation is not the initial event in triggering gene silencing in cancer; rather, the methylation of the promoter CpG islands is a consequence of prior gene inactivation, and it is a mechanism for locking the chromatin in a repressed state (5, 13, 37). Many tumor suppressor and receptor genes have been reported to be hypermethylated and transcriptionally silenced during the development of different types of cancers. It is likely that inhibition of DNMT and histone deacetylase can also prevent the hypermethylation and silencing of these key genes and, therefore, this inhibition would contribute to the prevention of carcinogenesis. The prevention of intestinal tumorigenesis by Dnmt1 deficiency and by DAC has been demonstrated in the Min mice, which carry a mutated Apc gene (38, 39).

Oral administration of green tea has been shown to inhibit tumorigenesis in different organs, and multiple mechanisms may be involved (1). It remains to be determined whether and to what extent this cancer-preventive activity is due to the inhibition of DNMT by EGCG. The presently observed effective dose of EGCG, Ki of 6.89 μM, or IC50 of 20 μM, is achievable in the oral cavity (in the saliva) after drinking green tea, and perhaps in the stomach, esophagus, and intestines where there is direct contact between EGCG and the epithelial cells. This effective concentration, however, is higher than those in the internal organs, which depend on the systemic bioavailability of EGCG (1). Therefore, the extent of DNMT inhibition in vivo would depend on the bioavailability of EGCG in a particular organ site. Although inhibition of DNMT is expected to prevent hypermethylation, severe inhibition of DNMT activity, as suggested by recent genetic studies, may cause DNA hypomethylation, genomic instability, and early development of cancers such as T-cell lymphomas and sarcomas (40, 41). There is no evidence for such adverse effects due to regular consumption of tea; however, it could be a concern if large doses of EGCG are given to humans.

On the basis of this analysis, for the practical application of tea in the prevention of cancer, it may be desirable to use doses to produce levels of EGCG in the tissue that would cause a moderate inhibition of DNMT. The possible synergy generated between EGCG with histone deacetylase inhibitors or other agents will hold great promise. For example, butyrate, which is a histone deacetylase inhibitor produced in the colon (42), may work synergistically with EGCG in the prevention of colon cancer. The application of this concept in animal models or humans remains to be demonstrated.

Grant support: NIH Grant CA88961 and CA65871 (C. S. Y.).

The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

M. Z. F. and Y. W. contributed equally to this work.

Requests for reprints: Chung S. Yang, Department of Chemical Biology, Ernest Mario School of Pharmacy, Rutgers University, 164 Frelinghuysen Road, Piscataway, NJ 08854-8020. Phone: (732) 445-5360; Fax: (732) 445-0687; E-mail: csyang@rci.rutgers.edu

3

The abbreviations used are: EGCG, (−)-epigallocatechin-3-gallate; MeEGCG, 4″-O-methyl-EGCG; DiMeEGCG, 4′, 4″-O-dimethyl-EGCG; DNMT, 5-cytosine DNA methyltransferase; DAC, 2′-deoxy-5-azacytidine; RARβ, retinoic acid receptor β; MGMT, O6-methylguanine methyltransferase; hMLH1, human mutL homolog 1; poly(dI-dC)·poly(dI-dC), polydeoxyinosine-deoxycytosine; ECG, (−)-epicatechin gallate; EGC, (−)-epigallocatechin; EC, (−)-epicatechin; RT-PCR, reverse transcription-PCR.

4

5

Fig. 1.

Inhibition of 5-cytosine DNA methyltransferase activity by EGCG. A, inhibition of DNMT by different concentrations of EGCG and 20 or 50 μM of other compounds. The reaction mixture contained nuclear extracts (4 μg protein), poly(dI-dC)·poly(dI-dC; 0.66 μM), and S-adenosyl-l-[methyl-3H]methionine (10μM, 2.0 μCi) in 40 μl. The incubation time was 2 h. The results are mean of 6–10 determinations, and are analyzed by one-way ANOVA (SigmaStat 1.01) using Student-Newman-Keuls pairwise test; bars, ±SD. Bars identified by different letters signified that they are significantly different (P < 0.05). B, kinetic studies on the inhibition of DNMT by EGCG. The reaction mixture contained 10 μM of S-adenosyl-l-[methyl-3H]methionine, and different concentrations of poly(dI-dC)·poly(dI-dC), and was incubated for 1.5 h. Each data point represents the mean ± SD of 4 determinations, and the data were analyzed by GraphPad Prism 4. In the absence of EGCG, the mean ± 95% confidence interval of the Vmax was 1.161 ± 0.089 pmol/min/mg and of the Km was 0.246 ± 0.070 μM. In the presence of 20 μM EGCG, the Vmax was 1.235 ± 0.205 pmol/min/mg, and Km was 0.897 ± 0.210 μM, showing a competitive inhibition with a Ki of 6.89 μM.

Fig. 1.

Inhibition of 5-cytosine DNA methyltransferase activity by EGCG. A, inhibition of DNMT by different concentrations of EGCG and 20 or 50 μM of other compounds. The reaction mixture contained nuclear extracts (4 μg protein), poly(dI-dC)·poly(dI-dC; 0.66 μM), and S-adenosyl-l-[methyl-3H]methionine (10μM, 2.0 μCi) in 40 μl. The incubation time was 2 h. The results are mean of 6–10 determinations, and are analyzed by one-way ANOVA (SigmaStat 1.01) using Student-Newman-Keuls pairwise test; bars, ±SD. Bars identified by different letters signified that they are significantly different (P < 0.05). B, kinetic studies on the inhibition of DNMT by EGCG. The reaction mixture contained 10 μM of S-adenosyl-l-[methyl-3H]methionine, and different concentrations of poly(dI-dC)·poly(dI-dC), and was incubated for 1.5 h. Each data point represents the mean ± SD of 4 determinations, and the data were analyzed by GraphPad Prism 4. In the absence of EGCG, the mean ± 95% confidence interval of the Vmax was 1.161 ± 0.089 pmol/min/mg and of the Km was 0.246 ± 0.070 μM. In the presence of 20 μM EGCG, the Vmax was 1.235 ± 0.205 pmol/min/mg, and Km was 0.897 ± 0.210 μM, showing a competitive inhibition with a Ki of 6.89 μM.

Close modal
Fig. 2.

Molecular modeling of the interaction between EGCG and DNMT. A, binding orientation of EGCG in DNMT1. The close-up view of the consensus orientation for EGCG. The protein is depicted in ribbon representation and colored by secondary structures (i.e., helix, strand, and loop). Both EGCG and ligand contact residues are represented in stick form and colored by atom type with carbon in gray, oxygen in red, and sulfur in yellow. B, hydrogen-binding network of EGCG in DNMT1. ---- represents hydrogen bond. Figure depicts all potential H-bonding X-H… Y interactions for which the X… Y distance is <4.0 Å.

Fig. 2.

Molecular modeling of the interaction between EGCG and DNMT. A, binding orientation of EGCG in DNMT1. The close-up view of the consensus orientation for EGCG. The protein is depicted in ribbon representation and colored by secondary structures (i.e., helix, strand, and loop). Both EGCG and ligand contact residues are represented in stick form and colored by atom type with carbon in gray, oxygen in red, and sulfur in yellow. B, hydrogen-binding network of EGCG in DNMT1. ---- represents hydrogen bond. Figure depicts all potential H-bonding X-H… Y interactions for which the X… Y distance is <4.0 Å.

Close modal
Fig. 3.

Alterations of methylation status and mRNA expression levels of RARβ, MGMT, p16, or hMLH1 genes after treatment with EGCG or other polyphenols. For each gene, the top panel presents the unmethylation-specific bands (UMSB), the middle panel presents the methylation-specific bands (MSB), and the bottom panel presents the mRNA expression levels (mRNA). A, left: KYSE 510 cells were treated with 5, 10, 20, or 50 μm of EGCG or 8.7 μM of DAC for 6 days as described in “Materials and Methods;” right: the cells were treated for 12, 24, 48, 72, and 144 h with 20 μM of EGCG added in new culture medium at 0, 48, and 96 h. B, RARβ mRNA levels of KYSE 510 cells treated with 20 μM of different polyphenols for 6 days. The RARβ mRNA level was determined with RT-PCR with G3DPHas an internal control. The band intensity was determined with densitometry. The values presented are mean (n = 3); bars, ±SD. C, HT-29, KYSE150, and PC3 cells were treated with 5 or 20 μM of EGCG for 6 days under conditions described in “Materials and Methods.”

Fig. 3.

Alterations of methylation status and mRNA expression levels of RARβ, MGMT, p16, or hMLH1 genes after treatment with EGCG or other polyphenols. For each gene, the top panel presents the unmethylation-specific bands (UMSB), the middle panel presents the methylation-specific bands (MSB), and the bottom panel presents the mRNA expression levels (mRNA). A, left: KYSE 510 cells were treated with 5, 10, 20, or 50 μm of EGCG or 8.7 μM of DAC for 6 days as described in “Materials and Methods;” right: the cells were treated for 12, 24, 48, 72, and 144 h with 20 μM of EGCG added in new culture medium at 0, 48, and 96 h. B, RARβ mRNA levels of KYSE 510 cells treated with 20 μM of different polyphenols for 6 days. The RARβ mRNA level was determined with RT-PCR with G3DPHas an internal control. The band intensity was determined with densitometry. The values presented are mean (n = 3); bars, ±SD. C, HT-29, KYSE150, and PC3 cells were treated with 5 or 20 μM of EGCG for 6 days under conditions described in “Materials and Methods.”

Close modal
Fig. 4.

Re-expression of RARβ and hMLH1 proteins after EGCG treatment. KYSE 510 cells were treated with 20 and 50 μM EGCG for 6 days, or 20 μM for 3 and 6 days; protein levels were determined with Western blot analysis using β-actin as an internal control.

Fig. 4.

Re-expression of RARβ and hMLH1 proteins after EGCG treatment. KYSE 510 cells were treated with 20 and 50 μM EGCG for 6 days, or 20 μM for 3 and 6 days; protein levels were determined with Western blot analysis using β-actin as an internal control.

Close modal
Fig. 5.

Effects of EGCG treatment on cell growth, colony formation, and morphological change. A, KYSE 510 cells were treated with EGCG, and after 6 days, its effect on cell growth was determined by counting living cell numbers by the trypan blue exclusion assay. B, after the cells were treated with EGCG for 2 days, the cells were washed, cultured with new medium (without EGCG) for another 7 days, and colony number was analyzed. C, morphological changes (magnification, ×400) after 2 and 6 days of EGCG treatment are shown. The values presented are mean (n = 3); bars, ±SD. ∗ indicates statistically significant (P < 0.05).

Fig. 5.

Effects of EGCG treatment on cell growth, colony formation, and morphological change. A, KYSE 510 cells were treated with EGCG, and after 6 days, its effect on cell growth was determined by counting living cell numbers by the trypan blue exclusion assay. B, after the cells were treated with EGCG for 2 days, the cells were washed, cultured with new medium (without EGCG) for another 7 days, and colony number was analyzed. C, morphological changes (magnification, ×400) after 2 and 6 days of EGCG treatment are shown. The values presented are mean (n = 3); bars, ±SD. ∗ indicates statistically significant (P < 0.05).

Close modal
Table 1

List of primers, annealing temperatures, and product sizes for methylation-specific PCR and RT-PCR

GeneChromosome locationForward primer (5′–3′)Reverse primer (5′–3′)Annealing temperature (°C)Product size (bp)
Methylation-specific PCRa
N: TTAAG(C/T)TTTGTGAGAATTTTG CCTATAATTAATCCAAATAATCATTTA CC 53 425
RARβ 3p24 U: TTGAGAATGTGAGTGATTTGA AACCAATCCAACCAAAACAA 62 146
M: TCGAGAACGCGAGCGATTCG GACCAATCCAACCGAAACGA 62 146
N: GGATATGTTG GGATAGTT CCAAAAACCCCAAACCC 52 289
MGMT 10q26 U: TTTGTGTTTTGATGTTTGTAGGTTTTTGT AACTCCACACTCTTCCAAAAACAAAACA 59 93
M: TTTCGACGTTCGTAGGTTTTCGC GCACTCTTCCGAAAACGAAACG 59 81
N: GGAGAGGGGGAGAGTAGGT CTACAAACCCTCTACCCACCT 60 208
P16 9p21 U: TGGGGAGTAGTATGGAGTTGGTGGT CAACCCCAAACCACAACCATAA 62 81
M: CGGGGAGTAGTATGGAGTCGGCGGC GACCCCGAACCGCGACCGTAA 62 81
N: GGTATTTTTGTTTTTATTGGTTGGAT AATACCAATCAAATTTCTCAACTCCT 55 185
hMLH1 3p22 U: TAAAAATGAATTAATAGGAAGAGT CTCTATAAATTACTAAATCTCTTCA 60 117
M: TAAAAACGAATTAATAGGAAGAGC CTCTATAAATTACTAAATCTCTTCG 60 117
RT-PCR
RARβ  GACTGTATGGATGTTCTGTCAG ATTTGTCCTGGCAGACGAAGCA 60 256
MGMT  AAAATGGACAAGGATTGTGAAA CATCCGATGCAGTGTTACACG 57 568
P16  CGGAAGGTCCCTCAGACATC TCATGAAGTCGACAGCTTCCG 60 385
hMLH1  CAGCGGCCAGCTAATGCTAT AATCCTCAAAGGACTGCAGTT 60 195
DNMT1  ACCAAGCAAGAAGTGAAGCC GCTTCCTGCAGAAGAACCTG 60 336
DNMT3a  CACACAGAAGCATATCCAGGAGTG AGTGGAAACCAAATACCC 60 550
DNMT3b  AATGTGAATCCAGCCAGGAAAGGC ACTGGATTACACTCCAGGAACCGT 60 191
MBD2  AACCCTGCTGTTTGGCTTAAC CGTACTTGCTGTACTCGCTCTTC 60 101
G3DPH  TTGCAACTGTTTTAGGACTTT AGCATTGGGAAATGTTCAAGG 60 983
GeneChromosome locationForward primer (5′–3′)Reverse primer (5′–3′)Annealing temperature (°C)Product size (bp)
Methylation-specific PCRa
N: TTAAG(C/T)TTTGTGAGAATTTTG CCTATAATTAATCCAAATAATCATTTA CC 53 425
RARβ 3p24 U: TTGAGAATGTGAGTGATTTGA AACCAATCCAACCAAAACAA 62 146
M: TCGAGAACGCGAGCGATTCG GACCAATCCAACCGAAACGA 62 146
N: GGATATGTTG GGATAGTT CCAAAAACCCCAAACCC 52 289
MGMT 10q26 U: TTTGTGTTTTGATGTTTGTAGGTTTTTGT AACTCCACACTCTTCCAAAAACAAAACA 59 93
M: TTTCGACGTTCGTAGGTTTTCGC GCACTCTTCCGAAAACGAAACG 59 81
N: GGAGAGGGGGAGAGTAGGT CTACAAACCCTCTACCCACCT 60 208
P16 9p21 U: TGGGGAGTAGTATGGAGTTGGTGGT CAACCCCAAACCACAACCATAA 62 81
M: CGGGGAGTAGTATGGAGTCGGCGGC GACCCCGAACCGCGACCGTAA 62 81
N: GGTATTTTTGTTTTTATTGGTTGGAT AATACCAATCAAATTTCTCAACTCCT 55 185
hMLH1 3p22 U: TAAAAATGAATTAATAGGAAGAGT CTCTATAAATTACTAAATCTCTTCA 60 117
M: TAAAAACGAATTAATAGGAAGAGC CTCTATAAATTACTAAATCTCTTCG 60 117
RT-PCR
RARβ  GACTGTATGGATGTTCTGTCAG ATTTGTCCTGGCAGACGAAGCA 60 256
MGMT  AAAATGGACAAGGATTGTGAAA CATCCGATGCAGTGTTACACG 57 568
P16  CGGAAGGTCCCTCAGACATC TCATGAAGTCGACAGCTTCCG 60 385
hMLH1  CAGCGGCCAGCTAATGCTAT AATCCTCAAAGGACTGCAGTT 60 195
DNMT1  ACCAAGCAAGAAGTGAAGCC GCTTCCTGCAGAAGAACCTG 60 336
DNMT3a  CACACAGAAGCATATCCAGGAGTG AGTGGAAACCAAATACCC 60 550
DNMT3b  AATGTGAATCCAGCCAGGAAAGGC ACTGGATTACACTCCAGGAACCGT 60 191
MBD2  AACCCTGCTGTTTGGCTTAAC CGTACTTGCTGTACTCGCTCTTC 60 101
G3DPH  TTGCAACTGTTTTAGGACTTT AGCATTGGGAAATGTTCAAGG 60 983
a

N, nest primers; U, unmethylation-specific primers; and M, methylation-specific primers.

We thank Drs. Jungil Hong, Joshua Lambert, Yan Nie, and Janelle Landau for helpful discussions. We also thank Sarah Philipps for assistance in the preparation of this manuscript.

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text:
# Homework Help: Help polynomial

1. Apr 1, 2004

### expscv

if p(x ) = x^6 + x^4 + x^2 + 1

show that the solutions of the equation p(x ) = 0 are among the solutions of the equation x^8 - 1 = 0

hence factorse p(x ) fully over R

2. Apr 1, 2004

### Chen

Excuse my ignorance but I don't see how p(x) can ever intersect with the X axis? Its minimum point is (0, 1)...

3. Apr 1, 2004

### expscv

this has complexy number i involved~

4. Apr 1, 2004

### expscv

answer = $$(x^2+1)(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1)$$ but i dont see how

Last edited: Apr 1, 2004
5. Apr 1, 2004

### Chen

$$p_{(x)} = x^6 + x^4 + x^2 + 1 = x^4(x^2 + 1) + x^2 + 1 = (x^2 + 1)(x^4 + 1) = 0$$

So either (1) $$x^4 = -1$$ or (2) $$x^2 = -1$$

Now what are the solutions for $$x^8 = 1$$?
$$x^4 = \pm 1$$
If $$x^4 = -1$$ we see that it will include the two solutions of equation (1). If $$x^4 = 1$$ then we can say that $$x^2 = \pm 1$$ and again, if $$x^2 = -1$$ we see it will include the two solutions of equation (2).

6. Apr 1, 2004

### rattis

did you post this in maths section as well?

format: plain


text:
## Files in this item

FilesDescriptionFormat

application/pdf

9717317.pdf (3MB)
(no description provided)PDF

## Description

 Title: Julia Sets and Symbolic Dynamics of Certain Rational and Entire Functions Author(s): Niamsup, Piyapong Doctoral Committee Chair(s): Palmore, Julian Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics Abstract: A complex analytic map f always decomposes the complex plane into two dis-joint subsets, the Julia set J(f) on which the family of iterates of f, $\{f\sp{k}\}\sbsp{k=0}{\infty}$, fails to be a normal family, and its complement, the Fatou set F(f). The dynamics of f on its Fatou set is relatively tame while the dynamics on its Julia set is always complicated. We study the Julia sets and Symbolic dynamics of certain rational and entire functions. We also estimate the size of some Julia sets in terms of their Hausdorff dimension. Issue Date: 1997 Type: Text Language: English Description: 77 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997. URI: http://hdl.handle.net/2142/86942 Other Identifier(s): (MiAaPQ)AAI9717317 Date Available in IDEALS: 2015-09-28 Date Deposited: 1997


text:
# Q2 2014 GDP a Strong 4.6%

Q2 2014 real GDP was revised to 4.6%, a strong showing not seen since Q4 2011.  Q2 Gross Domestic Product also cancels out the dismal Q1 -2.1% real GDP contraction for the year.  Growth was across the board.  Investment showed large growth.  Personal consumption expenditures increased and were a large component of GDP. Changes in private inventories was a large GDP contribution, but so were exports.  Overall Q2 GDP was a big bounce back from Q1.

As a reminder, GDP is made up of: $Y=C+I+G+{\left(X-M\right)}$ where Y=GDP, C=Consumption, I=Investment, G=Government Spending, (X-M)=Net Exports, X=Exports, M=Imports*.  GDP in this overview, unless explicitly stated otherwise, refers to real GDP.  Real GDP is in chained 2009 dollars.

This below table shows the percentage point spread breakdown of Q1 from Q2 2014 GDP major components and their spread.

Comparison of Q1 2014 and Q2 2014 GDP Components

Component

Q1 2014

Q2 2014

GDP -–2.1 +4.6 +6.7
C +0.83 +1.75 +0.92
I -1.13 +2.87 +4.00
G -0.15 +0.31 +0.46
X -1.30 +1.43 +2.73
M –0.36 –1.77 -1.41

Consumer spending, C in our GDP equation was 38% of GDP.  Durable goods were almost an entire GDP percentage point and Motor vehicles & parts was 0.45 percentage points of GDP.  Below is a percentage change graph in real consumer spending going back to 2000.

Graphed below is PCE with the quarterly annualized percentage change breakdown of durable goods (red or bright red), nondurable goods (blue) versus services (maroon).

Imports and Exports, M & X subtracted -0.34 percentage points from Q2 GDP as exports grew yet imports grew more.   This is a huge improvement from Q1 as shown in the below graph.

Government spending, G was all state and local spending increases.  Below is the graph of government spending showing the drag on economic growth is over.

Investment, I is made up of fixed investment and changes to private inventories.  Investment was the big factor for Q2 GDP, contributing 2.87 percentage points or 63.4% of GDP.  The change in private inventories alone gave a 1.42 percentage point contribution to Q2 GDP after a contraction in Q1.  Changes in private inventories alone was 30.9% of Q2 GDP.  Below are the change in real private inventories and the next graph is the change in that value from the previous quarter.

Fixed investment is residential and nonresidential and added 1.45 percentage points to GDP.  Nonresidential was 1.18 percentage points of GDP, with equipment being 0.63 percentage points.  Nonresidential structures by themselves added 0.35 percentage points.

Part of fixed investment is Residential fixed investment.  Residential added 0.27 percentage points to Q2 GDP.  One can see the housing bubble collapse in the below graph and also how there is no meteoric recovery in terms of economic growth.

Nominal GDP:  In current dollars, not adjusted for prices, of the U.S. output,was \$17,328.2 billion, a 6.8% annualized increase from Q1.  In Q1, current dollar GDP contracted -0.8%.

Real final sales of domestic product is GDP - inventories change. This figures gives a feel for real demand in the economy.  This is because while private inventories represent economic activity, the stuff is sitting on the shelf, it's not demanded or sold . Real final sales increased 3.2%.  Q1 real final sales were -1.0%.

Gross domestic purchases are what U.S. consumers bought no matter whether it was made in Ohio or China. &nbsp ; It's defined as GDP plus imports and minus exports or using our above equation: $P=Y-X+M$ where P = Real gross domestic purchases.  Real gross domestic purchases increased 4.8%.  Q1 was -0.4%.   Exports are subtracted off because they are outta here, not available for purchase by Americans, but imports, as well a know all too well, are available for purchase at your local Walmart.   When gross domestic purchases exceed GDP, that's actually bad news, it means America is buying imports instead of goods made domestically.

Below are the revised percentage changes of Q2 2014 GDP components, from Q1.  There is a difference between percentage change and percentage point change.  Point change adds up to the total GDP percentage change and is reported above. The below is the individual quarterly percentage change, against themselves, of each component which makes up overall GDP.  Additionally these changes are seasonally adjusted and reported by the BEA in annualized format.

Q2 2014 GDP Component Percentage Change

(annualized)

Component Percentage Change from Q2
GDP +4.6%
C +2.5%
I +19.1%
G +1.7%
X +11.1%
M +11.3%

Other overviews on gross domestic product can be found here,

## Forum Categories:

### 4.6% will be hard to match...

a lot of the 2nd quarter gains were just rebounds from the winter depressed 1st quarter...the 14.1% annualized rate of increase in durables goods consumption, which accounted for .99% of the GDP increase, will be particularly hard to match...almost half of that was an increase in consumption of motor vehicles and parts, which grew at a 19.1% annual rate and added .45% to GDP all by itself; in addition, real outlays for durable household equipment and furniture grew at a 12.9% rate, while real consumption of recreational goods and vehicles rose at a 13.3% rate, as all durables consumption benefited from a negative 1.9% deflator...

investment in equipment was also growing at an unsustainable rate of 11.2%, which added .63% to 2nd quarter growth..  investment in industrial equipment grew at a 27.3% rate and investment in information processing equipment grew at a 26.6% rate...blips in growth like that cannot be sustained...

rjs

### not sustained GDP

I just started back with our analysis,but since jobs are nowhere, wages flat, the same lies about "worker shortage" as the middle class has disappeared, I doubt any of this can be sustained.

Thanks for the additional detail figures. I saw autos & parts and wondered who exactly is buying up these cars? Wages are flat, rents are high....

I believe I saw a host of downgrades yesterday on Q3 estimates.


text:
# tem_tracking_test.f90 Source File

## Source Code

! Copyright (c) 2013-2016 Kannan Masilamani <kannan.masilamani@uni-siegen.de>
! Copyright (c) 2013, 2019 Harald Klimach <harald.klimach@uni-siegen.de>
! Copyright (c) 2013 Simon Zimny <s.zimny@grs-sim.de>
! Copyright (c) 2016 Jiaxing Qi <jiaxing.qi@uni-siegen.de>
! Copyright (c) 2016 Tobias Schneider <tobias1.schneider@student.uni-siegen.de>
! Copyright (c) 2017 Peter Vitt <peter.vitt2@uni-siegen.de>
!
! Redistribution and use in source and binary forms, with or without
! modification, are permitted provided that the following conditions are met:
!
! 1. Redistributions of source code must retain the above copyright notice, this
! list of conditions and the following disclaimer.
!
! 2. Redistributions in binary form must reproduce the above copyright notice,
! this list of conditions and the following disclaimer in the documentation
! and/or other materials provided with the distribution.
!
! THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
! AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
! IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
! DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
! FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
! DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
! SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
! CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
! OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
! OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
program tem_tracking_test
use, intrinsic :: iso_c_binding, only: C_NEW_LINE
use mpi
! incude treelm modules
use env_module,             only: solSpecLen
use tem_tracking_module,    only: tem_tracking_type, &

! include aotus modules
use aotus_module,           only: flu_State, close_config,      &
&                               open_config_chunk

!mpi!nprocs = 1

implicit none

character, parameter :: nl = C_NEW_LINE
type(tem_tracking_type) :: track
type(flu_State) :: conf
integer :: iError

character(len=solSpecLen), parameter :: sysConf =            &
&    'tracking = {'                                  // nl &
& // '  label = "point",'                            // nl &
& // '  variable = { "density", "velocity" },'       // nl &
& // '  folder = "tracking_",'                       // nl &
& // '  shapce = { kind = "all" },'                  // nl &
& // '  output = { format = "ascii" },'              // nl &
& // '  time = { min = 1, max = 10, interval = 1 },' // nl &
& // '}'

call MPI_Init(iError)

call open_config_chunk( L     = conf,         &
&                     chunk = trim(sysConf) )

call tem_load_tracking( me   = track, &
&                     conf = conf   )

call close_config( L = conf )
write(*,*) 'PASSED'

call MPI_Finalize(iError)

end program tem_tracking_test


text:
## ccsoccerox 2 years ago Given the triangle below, what is sec ∡B? a. 4 sqrt33/33 b. 7/4 c. 4/7 d. 7 sqrt33/33

1. ccsoccerox

2. WATTAGWAN

Sec B = 1/ Cos B and cosB = A/7 so sec B = 7/A where A = $\sqrt{7^2 - 4^2}$

3. WATTAGWAN

Seriously? you can do the rest with a calculator.


text:
# Verify is tautology by using logical equivalence

Verify is tautology by using logical equivalence:

$$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q$$

by showing what laws are using, step by step explanation needed.

Logical equivalence: de morgan's laws, commutativity, associativity, double negative, distribution, idempotent, absorption, identity, universal bound, negation

Rules of inference: modus ponens, modus tollens, generalisation, specialisation, conjunction, ellimination, transivity, proof by division into cases, contradiction

What I got so far:

I need to prove

$$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ \text{tautology}$$ Since $$p → r$$ same as $$∼ p ∨ r$$: $$((p ∨ q) ∧ (p → r) ∧ (∼ r)) → q ≡ ((p ∨ q) ∧ (∼ p ∨ r ) ∧ (∼ r)) → q$$

I am stuck here

• Please include your work or at the very least what does one have to do to verify something is a tautology? For instance, I can tell that the premise tautologically implies $q$ without making use of the equivalences. – Alvin Lepik Aug 12 '18 at 15:29
• @AlvinLepik please check my solution I got so far. – icefish Aug 12 '18 at 17:52

Proceed by $$(p\lor q) \land (\neg p\lor r)\land\neg r \rightarrow q \equiv \neg (p\lor q) \lor \neg (\neg p\lor r)\lor r \lor q\\ \equiv (\neg p\land\neg q) \lor (p\land\neg r)\lor r\lor q$$ View the latter formula grouped as follows $$\left [(\neg p\land \neg q) \lor q\right ] \lor \left [(p\land\neg r)\lor r\right ]\equiv$$ Use distributivity $$\equiv \left [(\neg p\lor q) \land (\neg q\lor q)\right ] \lor \left [(p\lor r)\land (\neg r\lor r)\right ] \equiv \neg p\lor q \lor p\lor r \equiv t$$
Next use absorption, since $~ (\neg p\vee r)\land \lnot r \equiv \neg p$


text:
Landolt C charts are considered the gold standard of alternative eye charts since 1988.

\$65.00

### Product Variations

 Chart Distance Choose an option2 Meter2.5 Meter3 Meter4 MeterReset Variations
SKU: N/A Categories: ,

## Description

Can be wall-mounted.

Landolt C charts are considered the gold standard of alternative eye charts since 1988. The chart was developed as an alternative to Snellen chart optotypes, which were not equally difficult. Each chart features only one carefully calibrated optotype placed in different positions (up, down, left, right, and 45 degree positions). The C used in the chart creates an equality throughout testing, giving the patient optimal opportunity. Patients must gesture or say which direction the C is facing, making testing simple and accurate.


text:
# Center align title, authors and affiliation on first page

So I have a Latex document in which I want to center align title, authors, and affiliation on first page.

Right now, it is left aligned by default.

\begin{document}
\mainmatter              % start of the contribution
\title*{My Title}
\titlerunning{My Title}
\author{AuthorA \and AuthorB}
\tocauthor{AuthorA, AuthorB}
\institute{Department of ABC,\\XYZ Institute, City, Country\\
Email: \{a, b\}@pqr.ijk.mn.st}
\maketitle              % typeset the title of the contribution
\renewcommand{\abstractname}{Abstract.}
\input{abstract}
\input{intro}
\input{design}
\input{evaluation}
\input{related_work}
\input{conclusion}
\input{acknowledgments}

My class file is at https://pastebin.com/wLzFsSc8.

UPDATE: I have got the title, author, and affiliation center aligned. However, I am now facing an issue. The page numbers have been shifted to the bottom center of each page. Earlier, they were on top left in even pages and top right in odd pages. I want it to remain the same. I am using fancyhdr package.

How do I solve this ?

\usepackage{afterpage}
\usepackage{fancyhdr}

\pagestyle{fancy}
\fancypagestyle{plain}
\fancyhf{}

\begin{document}

\afterpage{\cfoot{\thepage}}

\thispagestyle{plain}
\begin{center}
\Large
\textbf{Title} \\
\vspace{0.4cm}
\small{Author 1 and Author 2} \\
\vspace{0.4cm}
Department of ABC,\\XYZ Institute, City, Country\\
Email: \{a, b\}@pqr.ijk.mn.st}
\input{abstract}
\end{center}

• You can build your own titlepage using the titlepage environment (don't use \maketitle in it) – Skillmon Oct 12 '17 at 15:20
• – Skillmon Oct 12 '17 at 15:27
• @Skillmon Can you please tell me how to center align ? I am new to latex and I can't figure out. – pikaraider Oct 12 '17 at 15:31

Minimal approach, change it to your liking. The page break was introduced by the titlepage environment. If you don't want this break, you can use a center-environment instead and drop the \centering.

\documentclass[]{article}

\usepackage{blindtext}% for the abstract

\begin{document}
\begin{center}
\mbox{}\vfill
{\huge My Title}
\vfill
{\Large AuthorA and AuthorB}
\vfill
Department of ABC,\\
XYZ Institute, City, Country\\
Email: \{a, b\}@pqr.ijk.mn.st
\vfill
\end{center}
\blindtext% abstract
\end{document}

• This centers the title, author and affiliation but it pushes the abstract to the next page. What I currently have is abstract on the first page, I want it to remain that way. How do I go make the abstract and other content remain on the first page as earlier ? – pikaraider Oct 12 '17 at 16:08
• Could you please help ? The abstract and other content on the first page is getting shifted to the next page when I apply your code. – pikaraider Oct 13 '17 at 10:07
• That depends on the way your abstract is formatted. Please update your question containing everything which is necessary to reproduce your issue BUT nothing more (so without any unrelated package and stuff). – Skillmon Oct 13 '17 at 10:23


text:
# Steps of data analysis using logistic regression

I am a very beginner of statistic. Recently a project require me to analyse data using logistic regression & SPSS within a specific time frame. Although I have read few books, but still very blur on how to start off. Can someone guide me through? What is the 1st ste and what next?

Anyway, I have started some. Once entered the data into SPSS, I have done crosstab (categorical IV), descriptive (continuous IV) and spearman correlation.

Then, I proceed to test for nonlinearity by transforming into Ln which give me some problems. I have re-coded all zero cells to a small value (0.0001) to enable the Ln transformation. Then, I re-test the nonlinearity.

Question:

1) The only solution for violation is to transform the variable from continuous to categorical? I got one violation.

2) One Exp(B) is extremely large (15203.835). What does this means? Why?

3) There is one interaction has Exp(B) = 0.00. Why?

Many thanks.

-
why exactly are you testing for non-linearity? And what are you trying to model? What is IV? –  mpiktas Jan 15 '11 at 18:28
IV stands for independent variables. I thought the first step is testing assumption before we start the regression analysis? –  lcl23 Jan 16 '11 at 1:47
could you be more precise, your questions are for the results of non-linearity test, or for logistical regression? Also it would help if we knew more about your model, what is dependent variable, etc. –  mpiktas Jan 16 '11 at 7:32
it is the result of nonlinearity test using Ln transformation. My research question is to know the existence(yes/no) of management committee in public company. Predictors involved comprised of categorical & continuous variable, such as board size, auditor, etc.... –  lcl23 Jan 16 '11 at 10:28
I am perplexed about your recode of zero cells. Have you tried a sensitivity analysis? What happen if you use for example 0.01 or 0.00000001? Isn't to possible to make a conditional analysis after discarding observations with zeroes? –  glassy Jan 16 '11 at 13:37

Generally large beta coefficients signal multi-collinearity. You should look for marginals that are zero in your cross-tabulations. You should also pay attention to mpiktas's comment. Testing for linearity (and transforming to categorical) is not generally needed if you have been setting up your data correctly.

-
crosstab result is ok, with no zero values. –  lcl23 Jan 16 '11 at 1:52
crosstab result is ok, with no zero values. The problematic continuous variable gives me this result: B=9.62, SE=7.0, Wald=1.89, Sig=0.16, Exp(B)=15203.83. So, it is not sig., thus no issue of multicollinearity right? But the Exp(B) is so large which I think something wrong here. Another interaction (X*LnX) is sig. at 0.05, thus multicollinearity exists. From what I read from the book, the only solution is to transform this continuous variable to categorical? –  lcl23 Jan 16 '11 at 2:00
@lcl23, what is $B$? If it is a coefficient from logistic regression, why are you interested in $\exp(B)$? –  mpiktas Jan 16 '11 at 7:33
yes, B is the coefficient. Exp(B) is the odd ratio. Does this extremely large odd ratio indicate something wrong? or I can just ignore it? –  lcl23 Jan 16 '11 at 10:55
I suppose it's possible that it means something but more likely it means something went wrong. Multi-colinearity will not necessarily show up in the cross-tabulations. You need to run some diagnostics. There should be documentation in your software about how to get a condition number for the XX' matrix or perhaps it will tell you how to get a report on variance inflation factors. –  DWin Jan 16 '11 at 14:51
show 1 more comment

The large value of "B" would be a coefficient of a variable usually called "X" in your model. Usually, "X" has a real world meaning (could be income, could be a measured volume of something, etc.). So the job is to interpret this "B" in terms of "X". The usual definition (in ordinary least squares) is that a ONE UNIT increase in "X" corresponds to a "B" increase in "Y" (where "Y" is the dependent variable which you are modeling). It is similar (but not exact) when interpreting for a logistic regression: a ONE UNIT increase in "X" corresponds to a B increase in the log-odds (which is "Y" in this case). Therefore EXP(B) tells you the proportional increase in the odds for a ONE UNIT increase in "X". So the question which may help is "What does a ONE UNIT increase in X mean in the real world?" This may make the apparent extreme value seem more sensible. Another thing to ask is what is the range of X values in your data? for if this is much smaller than 1 then EXP(B) is effectively extrapolating well beyond what you have observed in the "X space" (something which is generally not recommended because the relationship may be different).

Put more succinctly the numerical value of your betas is related to the scale at which you measure your X variables (or independent variables).

The easiest mathematical way to see this is the form of the simple ordinary least squares estimate which is

B=(standard deviation of Y) / (standard deviation of X) * (correlation between Y and X)

This result is not exact in logistic regression, but it does approximately happen.

I would also recommend that replacing the observed proportions with (r_i+1)/(n_i+2) is not a bad way to go before plugging them into the logistic function, because it can help guard against creating extreme logit values (which are only an artifact of your choice of "small number"). Extreme logit values can create outliers and influential points in the regression, and this can make your regression coefficients highly sensitive to these observations, or to be more precise, to your particular choice of "small number".

This has been called "Laplace's rule of succession" (add 1 success and 1 failure to what was observed) and it effectively "pulls" the proportion back towards 1/2, with less "pulling" the greater n_i is. In Bayesian language it corresponds to the prior information that it is possible for both of the binary outcomes to occur.

-
"Another thing to ask is what is the range of X values in your data?" This variable ranges from 2 to 6. –  lcl23 Jan 16 '11 at 13:40


text:
## 13 February, 2016

### Linear Algebra

While linear algebra is, strictly speaking, a subject of algebra and not of analysis or
diﬀerential equations (which are, along with probability theory, the basic mathematical
tools used for studying quantitative ﬁnance), the theory of vector spaces is so fundamen-
tal to the modern formulation of either, that it is necessary to study it before proceeding
too far. Indeed, the theory has proven itself to be perhaps the most inﬂuential in unifying
signiﬁcant branches of modern mathematics, much as algebra did for classical mathemat-
ics centuries ago. We thus begin Part I: Review of Undergraduate Mathematics with the
subject of linear algebra, which we divide into two further sub-parts, one covering the
general ﬁnite-dimensional theory including that of vector spaces, inner product spaces,
dual spaces, and linear transformations on these spaces, and the other a more detailed
analysis of linear operators, focusing in particular on general canonical representations
and spectral analysis of speciﬁc types of commonly encountered operators (e.g., normal
operators). The inﬁnite-dimensional theory (i.e., functional analysis) will be postponed
to a later part, and any reference to linear algebra here implies the ﬁnite-dimensional
case, unless the contrary is stated.
We develop the theory in close accordance to the classic textbook [1]. The reader is
assumed to have studied linear algebra previously; accordingly, and also in keeping with
the main objective of quantitative ﬁnance, several elements of the theory are omitted,
many concepts implicitly implied, and the proofs of many theorems left as exercises. The
development of the theory will be mostly pedagogical; but at the end we shall summarize
many of the results into one of the most remarkable theorems in all of linear algebra: the
invertible matrix theorem (essentially a very long list of surprising equivalent conditions
for a matrix to be, among many other things, invertible, hence also a linear operator on a
ﬁnite-dimensional vector or inner produce space to be invertible). An analogous theorem
holds for linear transformations between spaces not necessarily of the same dimension,
but of course with necessarily fewer equivalences.
1. Linear Systems, Matrices and Elementary Row Operations
Before reviewing ﬁnite dimensional vector spaces and their transformations, it is ﬁrst
helpful to review linear systems of equations and matrices, both as a source of examples
later, but mainly in view of the practicality that almost all concrete calculations aris-
ing from vector spaces require one to solve a system of linear equations, the standard
algorithm by which matrices help facilitate in executing. More importantly than that,
matrices provide a concrete realization of linear transformations on ﬁnite-dimensional
vector spaces and make much of the corresponding theory of these operators possible, as
we shall see.
Theorem 1.1. Let A and B be m×n matrices. If the linear systems Ax = b and By = c
are equivalent, then x = y
Remark. In words, if two systems of linear equations are equivalent (whether they are
homogeneous or inhomogeneous), then the solution sets are the same.
Proof. . . .
1
Theorem 1.2. Let A, B, and C be matrices with dimensions such that the products
(AB)C and A(BC) are deﬁned. Then A(BC) = (AB)C; that is, matrix multiplication
is associative.
Proof. . . .
Theorem 1.3. Elementary row operations are invertible and their inverses are of the
same type. Proof. ... Theorem I.4 Let A be an m × n matrix, e an elementary row
operation, I the m × n identity matrix (i.e., I
ij
= δ
ij
), and deﬁne E := e(I). Then
e(A) = e(I)A = EA.
Corollary. A is row-equivalent to B if and only if B = P A, where P = E
1
. . . E
k
for
some ﬁnite sequence of elementary matrices.
Proof. . . .
Theorem 1.4. Let A be an m × n matrix. There exists a unique row-reduced (echelon)
matrix R that is row-equivalent to A.
Proof. . . .
Theorem 1.5. Let A and B be m × n matrices. If A is row-equivalent to B, then the
corresponding homogeneous linear systems Ax = 0 and By = 0 are equivalent and x = y.
Remarks. Applying Theorem 1.5, we see then that Rx = 0 has the same solution set
as Ax = 0, where R is the row-reduced (echelon) form of A obtained by applying a
particular sequence of elementary operations (cf. Theorem 1.4). If we apply this same
sequence of elementary operations to the column vector b in the inhomogeneous system
Ax = b, then A R and b P b and we see from Theorem 1.1 (with B = R and
c = P b) that Ax = b and Rx = P b have the same solution set. In particular, we have a
robust way of solving both homogeneous systems and inhomogeneous systems of linear
equations. This method is colloquially known as Gaussian elimination.
As we will see in later sections when we discuss linear transformations, the converse also
holds: if Ax = 0 and Bx = 0 have the same solution set, then A is row-equivalent to B.
Proof. . . .
We conclude our discussion of m × n systems before moving onto n × n systems with
one of the most fundamental theorems in linear algebra (it’s importance being revealed
when we discuss linear transformations).
Theorem 1.6 (Fundamental Theorem of Homogeneous Linear Systems). Let A be an
m × n matrix with m < n. Then the corresponding homogeneous linear system Ax = 0
has a non-trivial solution x 6= 0.
Proof. . . .
We now move on to discuss n × n matrices and their corresponding systems of linear
equations. The theory that follows is simpler and more extensive than the theory for
m × n matrices and their systems. The reason for this is largely to do with the concept
of invertability (together with ﬁnite dimensionality), which only makes sense for n × n
matrices because of the deﬁnition of matrix multiplication given earlier.
Theorem 1.7. Let A be an invertible n × n matrix. Then A
1
is an invertible n × n
matrix and
(A
1
)
1
= A.
Furthermore, if A = A
1
. . . A
k
and each of the A
j
(j = 1, 2, . . . , k) are invertible, then so
is A and
A
1
= A
1
k
. . . A
1
1
.
Remark. The converse to the second assertion is also true and will be proved as a corollary
to Theorem 1.8.
Proof. . . .
Theorem 1.8 (Invertible Matrix Theorem Part I). Let A be an n × n matrix. Then the
following are equivalent:
(1) A is invertible;
(2) A is row-equivalent to the identity matrix I; item A is the product of P and I,
P being a product of elementary matrices: A = P I = E
1
. . . E
k
I;
(3) The corresponding homogeneous system Ax = 0 has only the trivial solution
x = 0;
(4) The corresponding inhomogeneous system Ax = b has only the solution x = A
1
b
for every n × 1 column matrix b.
Corollaries. The following corollaries follow from this theorem:
(1) If BA = I = AC, then B = C = A
1
. In particular, for n × n matrices, either
A is invertible or it has no right or left inverse (equivalently, an n × n matrix
cannot have nonequal left and right inverses);
(2) If P = E
1
. . . E
k
, then P A = I = A
1
= P = P I. In particular, if A is reduced
to I by a sequence of elementary operations, then the result of the corresponding
sequence applied to I is A
1
;
(3) If A = A
1
. . . A
k
is invertible, then each A
j
is invertible (j = 1, . . . , k). This is the
converse to Theorem 1.7. In particular, invertability is preserved under products
when each factor is invertible, and destroyed if just one factor is non-invertible;
(4) For m×n matrices A and B, A B is equivalent to the existence of an invertible
m × m matrix P such that B = P A holds. This is essentially a weaker statement
than that of Theorem 1.4.
Proof. . . .
2. Structure of Vector Spaces: Bases, Dimension, and Coordinates
Deﬁnition (Vector Space). A vector (or linear) space (X, F, +, ·) consists of a set X,
whose elements we refer to as vectors, an algebraic ﬁeld F, whose elements we refer to
as scalars, a binary operator + : X × X X, which we call addition, and a binary
operator · : F × X X, which we call scalar multiplication, where the operations obey
the following vector space axioms for arbitrary x, y, z X and c, d F:
(1) (Closure) x + c · y X
(2) (Associativity) (x + y) + z = x + (y + z)
(3) (Commutativity) x + y = y + x
(4) (Additive Identity 0 X) x + 0 = x
(5) (Additive Inverse x X) x + (x) = 0
(6) (Compatibility with Scalar Multiplication and Field Multiplication) (cd) · x =
c · (d · x)
(7) (Compatibility with Field Identity) 1 · x = x
(8) (Distribution over Vector Addition) c · (x + y) = c · x + c · y
(9) (Distribution over Scalar Addition) (c + d) · x = c · x + d · x
A subset S X is called a (vector or linear) subspace of X if (S, F, +, ·) is itself a vector
space. Because of axiom #6, we will not explicitly use the notation · for scalar multipli-
cation; whether the multiplication refers to scalar multiplication or ﬁeld multiplication
is now obvious. Moreover, we use the same notation 1 and 0 for the special identity
elements in both F and X; again, to which one is being referred should be obvious from
the context.
It is customary to only refer to the set of vectors X when talking about a vector space;
in other words, we do not ordinarily explicitly state the ﬁeld and binary operations +
and ·, even though technically all of these must be speciﬁed in order for there to be a
vector space. This practice is observed throughout mathematics, e.g. when talking about
measure spaces or topological spaces, we do not usually explicitly reference the under-
lying sigma algebras, topologies, measures, etc. (except in cases where that particular
component of the space is what is speciﬁcally under consideration or subject to change
- it will often be the case in linear algebra where the ﬁeld must be explicitly stated as
certain results depend on whether one is working on R or C, or more generally, a ﬁeld
that is algebraically complete or of a particular characteristic).
The above axioms imply all of the usual properties one is accustomed to using when
working with vectors on the Euclidean vector space R
n
. For instance, one can prove
simple facts such as (1)x = x and 0x = 0 and thus if cx = 0 then x = 0.
Finally, it is clear that in evaluating whether a given subset S of X is itself a vector
space, one only needs to verify that S is closed under the vector space operations (i.e.,
s + cr S for every s, r S and c F). Deﬁnition II.2 (Linear Span / Independence)
Let X = {x
1
, . . . , x
n
} X be a collection of vectors in X. We call the set
S
X
:= {x X : x = c
1
x
1
+ . . . + c
n
x
n
, c
j
F (j = 1, . . . n)}
the (linear) span of X . If the equation
c
1
x
1
+ . . . + c
n
x
n
= 0
has only the trivial solution x
j
= 0 (j = 1, . . . , n), then X is said to be a linearly
independent subset of X, or simply independent. Otherwise X is said to be dependent.
It is clear that the span of any set S := S
X
is actually a subspace of X, regardless of
whether the spanning set X is independent or dependent. If X is independent, then X
is called a minimal spanning set or basis of S and |X | (cardinality of X ) is called the
dimension of S. Every vector in S can be uniquely expressed as a linear combination of
vectors in X . If X is dependent, then this representation is not unique; however, any
such spanning set can be reduced to a linearly independent set by eliminating certain
dependent vectors without aﬀecting the set’s span, and hence a minimally spanning set or
basis. Conversely, every subspace S in X has a basis (note that the basis of a vector space
is far from unique; it is the number of elements that make-up a basis which is unique,
i.e. a set that simultaneously spans S and is linearly independent and necessarily has
cardinality equal to the dimension of S). We record these important facts in the following
fundamental theorem:
Theorem 2.1. Fundamental Basis and Dimension Theorem . . .
Proof. . . .
3. Linear Transformations, Field Isomorphism, and Duality
Deﬁnition (Linear Transformation). If X and Y are two vector spaces over the same
ﬁeld F, then a map T : X Y is said to be a linear transformation from X to Y if
T preserves the vector space structure; in particular, for each x, y X and c F the
following holds:
T (cx + y) = cT x + T y.
If Y = X, then T is called a (linear) operator on X.
If x =
P
n
j=1
c
j
x
j
is a linear combination of vectors in X, then successive application of
the deﬁning property for T implies
y = T x = T
n
X
j=1
c
j
x
j
!
=
n
X
j=1
c
j
T x
j
,
so that T also preserves linear combinations. As we shall see, this fact allows us to
characterize linear transformations based soley on from their actions on a ﬁnite set of
vectors in X; equivalently, every ”rule” that assigns a vector in Y to each vector in such
a set can be extended to a unique linear transformation T .
Theorem 3.1. . . .
Proof. . . .
Theorem 3.2 (Rank-Nullity Theorem). . . .
Proof. . . .
Theorem 3.3. . . .
Proof. . . .
Theorem 3.4. . . .
Proof. . . .
Theorem 3.5. . . .
Proof. . . .
Theorem 3.6 (Invertible Linear Transformation Theorem). . . .
Proof. . . .
Theorem 3.7 (Riesz Representation Theorem for the dual space X
). . . .
Proof. . . .
Theorem 3.8 (Subspace Annihilator Theorem). . . .
Corollaries. Theorem 3.8 implies the following corollaries:
(1) . . .
(2) . . .
(3) . . .
(4) . . .
Proof. . . .
Theorem 3.9 (Riesz Representation Theorem for the double dual space X
∗∗
). . . .
Corollaries. Theorem 3.9 implies the following corollaries:
(1) . . .
(2) . . .
(3) . . .
(4) . . .
Proof. . . .
Theorem 3.10. . . .
Proof. . . .
Theorem 3.11. . . .
Corollaries. Theorem 3.11 implies the following corollaries:
(1) . . .
(2) . . .
(3) . . .
(4) . . .
Proof. . . .
Theorem 3.12. . . .
Proof. . . .
Theorem 3.13. . . .
Rank-Nullity Theorem for the Adjoint Transformation T
. . . .
Theorem 3.14 (Matrix Representation Theorem and Vector Space Field Isomorphism).
. . .
Corollaries. Theorem 3.14 implies the following corollaries:
(1) . . .
(2) . . .
(3) . . .
(4) . . .
Proof. . . .
Theorem 3.15 (Matrix Representation Theorem for the Adjoint Transformation T
).
. . .
Proof. . . .
Theorem 3.16 (Invertible Matrix Theorem Part II). . . .
Proof. . . .
4. Inner Product Spaces and Orthogonality
5. Summary
References
1. K. Hoﬀman and R. Kunze, Linear Algebra, 2
nd
Edition, Prentice-Hall, Inc. (1971).

#### 1 comment:

Taylor Martin said...

test


text:
+0

0
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The equation  describes the height (in feet) of a ball tossed up in the air at 26 feet per second from a height of 105 feet above the ground. In how many seconds will the ball hit the ground? Express your answer as a decimal rounded to the nearest tenth.

Aug 28, 2022

#1
+33343
+1

Set y = 0 and solve the resulting equation for t.  There will be two values; choose the only sensible one.

Aug 28, 2022
#2
+29
-3

I remember seeing this problem in Alcumus, or maybe a similar variation of it. However, the solution is pretty simple.

set y = 0, since we are trying to find out when the ball hits the ground, and when that happens, Y = 0.

This looks like a quadratic formula!

$$0 = -16t^2 + 26t + 105$$

Then, simply solve for this quadratic. There will be two answers, but one of them is negative, so choose the postitve answer.

$$\frac{-26 +\sqrt{26^2 - 4(-16)(105)} }{2(-16)}$$

That should be the postive answer. Sorry, it's a little messy. Solve this equation and the answer shall be the final answer!

Hope this helps!

Aug 28, 2022


text:
Second 0+ state of unbound $^{12}$O: Scaling of mirror asymmetry - Archive ouverte HAL Access content directly
Journal Articles Physical Review C Year : 2016

## Second 0+ state of unbound $^{12}$O: Scaling of mirror asymmetry

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D. Suzuki
• Function : Author
H. Iwasaki
• Function : Author
D. Beaumel
• Function : Author
M. Assié
H. Baba
• Function : Author
Y. Blumenfeld
• Function : Author
Francois de Oliveira Santos
N. de Séréville
• Function : Author
S. Franchoo
• Function : Author
J. Gibelin
S. Giron
• Function : Author
S. Grévy
• Function : Author
J. Guillot
• Function : Author
M. Hackstein
• Function : Author
F. Hammache
• Function : Author
N. Keeley
• Function : Author
V. Lapoux
F. Maréchal
• Function : Author
A. Matta
S. Michimasa
• Function : Author
L. Nalpas
• Function : Author
F. Naqvi
• Function : Author
H. Okamura
• Function : Author
H. Otsu
• Function : Author
J. Pancin
• Function : Author
D. Y. Pang
• Function : Author
L. Perrot
• Function : Author
C. M. Petrache
• Function : Author
A. Ramus
• Function : Author
W. Rother
• Function : Author
P. Roussel-Chomaz
• Function : Author
• PersonId : 903877
H. Sakurai
J.-A. Scarpaci
O. Sorlin
• Function : Author
P. C. Srivastava
• Function : Author
I. Stefan
C. Stodel
Y. Tanimura
• Function : Author
S. Terashima
• Function : Author

#### Abstract

The unbound O12 nucleus was studied via the two-neutron transfer (p,t) reaction in inverse kinematics using a radioactive O14 beam at 51 MeV/u. Excitation energy spectra and differential cross sections were deduced by the missing mass method using MUST2 telescopes. We achieved much higher statistics compared to the previous experiments of O12, which allowed accurate determination of resonance energy and unambiguous spin and parity assignment. The O12 resonance previously reported using the same reaction was confirmed at an excitation energy of 1.62±0.03(stat.)±0.10(syst.). MeV and assigned spin and parity of 0+ from a distorted-wave Born approximation analysis of the differential cross sections. Mirror symmetry of O12 with respect to its neutron-rich partner Be12 is discussed from the energy difference of the second 0+ states. In addition, from systematics of known 0+ states, a distinct correlation is revealed between the mirror energy difference and the binding energy after carrying out a scaling with the mass and the charge. We show that the mirror energy difference of the observed 0+ state of O12 is highly deviated from the systematic trend of deeply bound nuclei and in line with the scaling relation found for weakly bound nuclei with a substantial 2s1/2 component. The importance of the scaling of mirror asymmetry is discussed in the context of ab initio calculations near the drip lines and universality of few-body quantum systems.

#### Domains

Physics [physics] Nuclear Experiment [nucl-ex]

### Dates and versions

in2p3-01278370 , version 1 (17-03-2020)

### Identifiers

• HAL Id : in2p3-01278370 , version 1
• DOI :
• INSPIRE :

### Cite

D. Suzuki, H. Iwasaki, D. Beaumel, M. Assié, H. Baba, et al.. Second 0+ state of unbound $^{12}$O: Scaling of mirror asymmetry. Physical Review C, 2016, 93 (2), pp.024316. ⟨10.1103/PhysRevC.93.024316⟩. ⟨in2p3-01278370⟩

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Question

A pack of paper costs $1.75, including tax. Mr. Valentino wants to purchase packs of paper for his class and has a$25 budget. Write and solve an inequality to solve for the number of packs of paper Mr. Valentino can purchase, and describe the graph of the solution

Answers

1. trucchi
Answer:
The inequality will be x ≤ 14.28 and Mr. Valentino can purchase a maximum of 14 packs of paper.
Step-by-step explanation:

Reply
2. thienhuong
The inequality will be x ≤ 14.28 and Mr. Valentino can purchase a maximum of 14 packs of paper.

### What is inequality?

A difference between two values indicates whether one is smaller, larger, or basically not similar to the other.
A mathematical phrase in which the sides are not equal is referred to as being unequal. In essence, a comparison of any two values reveals whether one is less than, larger than, or equal to the value on the opposite side of the equation.
In other words, inequality is just the opposite of equality for example 2 =2 then it is equal but if I say 3 =6 then it is wrong the correct expression is 3 < 6.
Let’s say there is x pack of paper.
Cost of x pack of paper 1.75x
Now Mr. Valentino has only \$25
So,
1.75x ≤  25 ⇒  x ≤ 14.28
It means Mr. Valentino can purchase 14 packs of paper.
For more about inequality,
brainly.com/question/20383699
#SPJ2

Reply


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# What is the math behind median filter's noise reduction property?

I am interested in understanding the mathematical reason for why does applying a median filter on an image (or signal) result in reduction of noise.

• The effectiveness of a median filter would depend on the type of noise. Noise with a median value of 0, e.g. additive white Gaussian noise (AWGN) would be cut out quite well given sufficient samples. If the noise is some other form of interference (multipath fading), a median filter won't really help. – Nick T Mar 24 '14 at 21:39

Intuition: The intuition is this: Your noise is some event or events that are rare, and that when compared to other events, look like outliers that shouldn't really be there.

For example, if you are measuring the speeds of every car on the highway as they pass by you and plot them, you will see that they are usually in the range of say, $50$ mph to $70$ mph. However as you are inspecting your data for your boss, you see that you recorded a speed of $1,000,000$ mph. Not only does this value not make physical sense for the speed of an actual car on a highway, but it also sticks out wildly from the rest of your data. Chalking this event up to some strange measurement error, you remove it, and give the rest of your data to your boss.

However as you continue your measurements day in and day out, you notice that every now and then, you get those wild measurements of speed. For example, over the span of 1 hour, you measure 1000 cars, and their speeds are nicely between $50$ and $70$ mph, however 3 of those have speeds of $23,424$ mph, $12,000,121$ mph, and $192,212,121,329,982,321,912$ mph, breaking not only local state laws, but also those of theoretical physics.

You get tired of continuously having to go in, and remove those errant data points caused by your cheapo-radar by hand. Afterall, your boss is really only interested in the statistics of the speeds, not so much every actual value. He likes to make nice histograms for his bosses.

Those errant and large numbers are a kind of 'noise' you reckon - 'noise' caused by your cheapo-radar that you bought from a shady pawn shop. Is the noise additive white gaussian noise? (AWGN). Yes and no - It's spectrum is wideband and white, but it is temporally rare, sparse, and very localized. It is better referred to as 'salt and pepper' noise, (especially in the image processing domain).

Thus, what you can do, is run your data through a median filter. Your median filter will take a block of say, $5$ speed points, (points 1 to 5), find the median, and spit that value as the 'average' speed. Then it will take the next 5 points, (points 2 to 6), take that median, and spit out this as the average, etc etc.

What happens when you come across one of your faster-than-light speeds?. Let us say that your 5 speeds were [45, 65, 50, 999999, 75]. If you took the normal average, your 'average' speed here will be something quite large. However if you take the median, your 'average' will be 65. Which best approximates the average that you are really trying to measure? The median metric.

Thus, if you filter your data with a median filter, you will be sure to remove those outliers - and you have thus faithfully 'de-noised' your signal. In contrast, if you tried to remove your noise via traditional filtering, (nothing but a moving weighted sum), you will instead 'smear' the error across your data, and not get rid of it.

Math: The math is this: The median measurement is what is referred to as an order statistic. That is, it returns the value of your data, along some point, after it has been ordered. The max and min are also both order statistics - they return the extreme points of your data after it has been ordered. Taking the median also returns the value of your ordered data, but right from the middle.

But why are they different from mean filters? Well, mean-filters compute an average using all the data. If you notice from max, min, and median, you are getting an answer without using all the data. In fact, all the median does is order your data, and pick the value in the middle. It never 'touches' the outliers, like those large speeds that you measured.

This is why median - an order statistic - is able to 'remove' outlier noise for you. Outlier noise segregates itself in front of the median, and the median never comes near it or considers it, while still giving you a nice estimate of central tendency.

Assuming independent random variables with normal distributions, the probability that a value will fall beyond, say, 2 standard deviations will be about 0.01.

If you have a median filter of width 3, that triplet must contain two outliers on the same side of the mean in order for an outlier to come through. This event has a probability of $2 \cdot 0.005^2 =$ 0.00005.

As the median filter increases in width, the probability of an outlier making it through decreases exponentially.


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Convert number to date in Base (or compare to a date at least)? [closed]

The SQL command "TO_DATE(<number>)" returns an error that it's not supported. (I'm using a Calc spreadsheet as the back end because I just need to run some queries off of it, not actually have any relational database.)

Of course, in the Calc file the dates are displayed as dates, not numbers—but actually they are timestamps and so include fractions in the underlying number storage. This wouldn't be a problem, but I need to split the day at NOON, and count anything after noon as being the "next day" for purposes of my query.

I thought I could get around this by changing that part of the query to ROUND("Log"."DATE") as "Date", which at least doesn't throw an error and does seem to round correctly (i.e. anything PM becomes rounded to the next day). Two problems with this: (1) The rounded date is then shown in the query results as a number, like 41875. (2) I can't work out how to compare the resulting number to a user defined date parameter.

(1) isn't a big issue because I can include "Date" in the query twice—once rounded for filter criteria, and the other unaltered for display purposes. (I would have to do this anyways as I'm counting afternoons as part of the next day for filtering purposes, but still need to display the date that was actually entered in the spreadsheet.)

(2) is a problem. I think it would be solved if I can map the rounded date number back to an actual date, but possibly the actual issue is how to ensure the parameter entered by the user is understood as a date and properly converted to a number. (Incidentally, does Base store dates and timestamps the same way as Calc? Such that 28 Aug 14 = 41876?)

I'm giving all the extra data on what I'm actually trying to do, because it's possible there is a different approach that would be a lot better. ;)

The actual FINAL end result that I need in terms of functionality, is to have the end user fire up Base, click "generate weekly report", choose the date range (or at least one date—the range will always be from noon Wednesday to the next Wednesday at noon), and then be able to print the query results in a nice tabular form.

edit retag reopen merge delete

Closed for the following reason the question is answered, right answer was accepted by Alex Kemp close date 2020-08-15 17:35:39.915021

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Based in the LO function list and limitations of the old HSQLDB version that LO still is using,consider something like this to display a date in human-readable form:

CONCAT( CONCAT( CONCAT( MONTH( "num" ), '/' ), CONCAT( DAY( "num" ), '/' ) ), YEAR( "num" ) )

There may be other ways too. The HSQLDB version embedded in LO at 4.3.5 still only allows concatation of two strings at a time. That is the reason for all the CONCAT statements.

For date comparison, a little easier: DATEDIFF( 'dd', "date1", "date2" ) See HSQLDB reference

As for the noon to noon issue, sounded like the data is just dates so unclear how time of day would factor. If it is contained in the data, I would just add twelve hours to every date value and use the resulting date, which is the practical effect.

EDIT to fix date comparison reference; thx Wildcard

more

Thanks, the concat thing is sort of helpful.

Datedif only works in Calc, though—I tried it in a Base query and it said it's not supported.

The dates are strictly speaking timestamps, but for user purposes here I don't actually care about time, just AM/PM. Regardless, dates and times are stored identically in Calc. It is stored as an integer of the number of days from the hard-coded base date, or with a decimal added for times. 41876 = 28 Aug 14. 41876.75 = 6 PM on that day.

( 2015-02-01 07:00:49 +0200 )edit

Fixed comparison function reference, thx. Also consider TIMESTAMPDIFF from HSQLDB documentation.

( 2015-02-09 03:13:41 +0200 )edit

Hi Is it not possible to do the rounding calculation in the spreadsheet? You could filter on this new column ...

( 2015-02-10 16:01:26 +0200 )edit

Hi pierre, I just thought I'd mention that what you suggested here (rounding the date in a separate column) is what I ended up doing. I came up with the idea on my own, but thought you'd like to know that yes, that is a good solution. :)

( 2015-03-09 09:24:05 +0200 )edit

Hi @Wildcard, nice of you, thank you :)

( 2015-03-09 09:37:16 +0200 )edit