Open Access. Powered by Scholars. Published by Universities.®

Physical Sciences and Mathematics Commons

Open Access. Powered by Scholars. Published by Universities.®

Articles 10291 - 10317 of 10317

Full-Text Articles in Physical Sciences and Mathematics

A Comparison Of Adsorption Isotherms Of Crosslinked Poly(N-Vinylpyrrolidone)/Basic Brown 1 Binding System, Dursun Saraydin, Erdener Karadağ Jan 1996

A Comparison Of Adsorption Isotherms Of Crosslinked Poly(N-Vinylpyrrolidone)/Basic Brown 1 Binding System, Dursun Saraydin, Erdener Karadağ

Turkish Journal of Chemistry

In this study, the binding of a basic dye such as basic brown 1 [BB 1] onto a crosslinked poly(N-vinylpyrrolidone) [CPVP] was investigated. Adsorption of the azo dye onto the CPVP was studied by the batch adsorption technique at 25^{\circ}C. In adsorption experiments a Langmuir type (L) adsorption was found with respect to the Giles classification system. Binding parameters such as initial binding constant (K_{i}), equilibrium constant {\it (K)}, monolayer coverage (n), site-size (u), and maximum fractional occupancy (\hat{\theta}), and thermodynamic parameters such as heat of adsorption (\Delta H), free energy of adsorption (\Delta G) and entropy of adsorption (\Delta …


The Facile Synthesis Of Cyclopentanoid Antibiotic Kjellmanianone, Ayhan S. Demi̇r, Cihangir Tanyeli̇, Volkan Gülbeyaz, Hülya Akgül Jan 1996

The Facile Synthesis Of Cyclopentanoid Antibiotic Kjellmanianone, Ayhan S. Demi̇r, Cihangir Tanyeli̇, Volkan Gülbeyaz, Hülya Akgül

Turkish Journal of Chemistry

The synthesis of cyclopentanoid antibiotic Kjellmanianone is described in this study. The C-acylation of lithium enolates of 3-Methoxycyclopentenone by N-carbomethoxyimidazole or methylcyanoformate \beta-ketoester, which when oxidized with Mn(III) acetate produces kjellmanianone in good yield.


Oxidative Coupling Of Methane On Lnba_{2}Cu_{3}O_{7-X} Superconductors, N. T. Shamilov, V. P. Vislovskii, E. A. Mamedov, S. M. Aliyev, A. I. Suleimanov, E. G. Ismailov Jan 1996

Oxidative Coupling Of Methane On Lnba_{2}Cu_{3}O_{7-X} Superconductors, N. T. Shamilov, V. P. Vislovskii, E. A. Mamedov, S. M. Aliyev, A. I. Suleimanov, E. G. Ismailov

Turkish Journal of Chemistry

The superconducting oxides LnBa_{2}Cu_{3}O_{7-x} (Ln=Y, Gd, Er, Yb, Lu) showed poor selectivity in the oxidative coupling of methane. The addition of 0.5 mol. %Li_{2}O to these materials considerably enhanced the yield of C_{2} hydrocarbons. This effect was related to both lower redox property and higher basicity of Li-doped catalysts as compared to those of undoped ones.


An Amperometric Biosensor For The Determination Of Organophosphorus Pesticides, Azmi Telefoncu Jan 1996

An Amperometric Biosensor For The Determination Of Organophosphorus Pesticides, Azmi Telefoncu

Turkish Journal of Chemistry

Organophosphorus pesticides are inhibitors of cholinesterase and this enzyme is used as a monitoring system for pesticides containing organophosphorus compounds. The decrease in activity of cholinesterase is monitored by the choline sensor and is correlated to the concentration of pesticide present in solution. According to this observation a biosensor for pesticides was determined. This biosensor consists of a Clark oxygen electrode, modified with cellulose acetate and dialysis membranes. Between the two membranes there are two immobilized enzymes: acetyl-cholinesterase and choline oxidase.


Deacidification Of High Acid Olive Oil By Immobilized Lipase, Ayşe Makasçi, Kadir Arisoy, Azmi Telefoncu Jan 1996

Deacidification Of High Acid Olive Oil By Immobilized Lipase, Ayşe Makasçi, Kadir Arisoy, Azmi Telefoncu

Turkish Journal of Chemistry

The acidity reduction of high acid oils prior to refining could be an attractive application for industry. In this study the enzymatic deacidification degummed and dewaxed olive oil with a high free fatty acid content, catalysed by an immobilized lipase was optimzed (Novozym 435). The optimum amount of substrate for synthesizing trigliycerides was the necessary stoichiometric amount of fatty acid and glycerol. The mixing was performed with a mechanical stirrer and the dehydration was performed by vacuum pump or by bubbling dry nitrogen. The dehydration rate was faster than the deacidification rate. The four parameters chosen for this experimental plan: …


A New Flavanone From Teucrium Alyssifolium, G. Topcu, C. Eris, S. Kurucu A. Ulubelen Jan 1996

A New Flavanone From Teucrium Alyssifolium, G. Topcu, C. Eris, S. Kurucu A. Ulubelen

Turkish Journal of Chemistry

From the aerial parts of {\it Teucrium alyssifolium} in addition to previously obtained {\it neo-}clerodane diterpenoids, a new flavanone alysifolinone and three flavones cirsiliol, cirsilineol and luteolin-6,7,3', 4'-tetramethyl ether, were isolated. The structure of the compounds was established by spectral data, and by TLC comparison for the known compounds. Keywords: Teucrium alyssifolium, Lamiaceae, flavanone, alysifolinone, flovones.


A Comparison Of Normal(-Cn) And Reversed (C-18) Phase Chromatographic Behaviour Of Polycyclic Aromatic Hydrocarbons, S. Fatma Aygün, Mustafa Özci̇mder Jan 1996

A Comparison Of Normal(-Cn) And Reversed (C-18) Phase Chromatographic Behaviour Of Polycyclic Aromatic Hydrocarbons, S. Fatma Aygün, Mustafa Özci̇mder

Turkish Journal of Chemistry

Because of the well-known carcinogenic and mutagenic character of polycyclic aromatic hydrocarbons (PAHs), still new methods have been tried for their qualitative and quantitative determinations in various smaples. Liquid chromatography is one of the most widely applied techniques. In this study we used liquid chromatography in normal and reversed phase modes for the determinations of PAHs and tried to find a relation between the retention times and various parameters, such as molecular shape, water solubility and number of carbon atoms.


Adsorption Of Monovalent Cationic Dyes On Some Silicates, Oğuz İnel, Ayşegül Aşkin (Börekçi̇) Jan 1996

Adsorption Of Monovalent Cationic Dyes On Some Silicates, Oğuz İnel, Ayşegül Aşkin (Börekçi̇)

Turkish Journal of Chemistry

Adsorption of two cationic dyes, methylene blue (MB) and malachite green (MG), was studied on the bentonite, sepiolite and zeolite samples by the batch equilibration technique using a spectrophotometer. In addition, cation exchange capacities (CEC) of samples were determined by NH_4^{+} saturation using Kjeldhal distillation. The adsorption data was found to conform with the Langmuir equation within the concentration range studied, and Langmuir constants were determined for each of the samples. It was observed that the amount of dyes adsorbed was in excess of CEC in the case of bentonite and sepiolite. The adsorption capacities of the samples for both …


Investigation Of Adsorption Isotherms Used For Wool Dyeing By Aqueous Extraction Of Cehri Fruit (Fructus Rhamni Petiolari) And Dyeing Of Wool And Silk Accompanied By Various Mordants, A. Haluk Aydin, Fuat Güzel, Zeki Tez Jan 1996

Investigation Of Adsorption Isotherms Used For Wool Dyeing By Aqueous Extraction Of Cehri Fruit (Fructus Rhamni Petiolari) And Dyeing Of Wool And Silk Accompanied By Various Mordants, A. Haluk Aydin, Fuat Güzel, Zeki Tez

Turkish Journal of Chemistry

In this study, adsorption isotherms were obtained by processing wool with dyestuff solutions of different concentrations extracted from cehri fruit with water. These isotherms follow the Nernst equation; the Nernst constant was determined and interpreted. In addition, wool and silk fibres were dyed with the same dyestuff solution and various mordants. Colours having a high fastness which could be used on an industrial scale were produced.


The Effect Of Examination Stress On The Levels Of Zn^{2+}, Total Protein, Albumin In Serum And Carbonic Anhydrase Isoenzymes In Erythrocytes, Nazan Demi̇r, Yaşar Demi̇r, Ali Yildirim, Ö. İrfan Küfrevi̇oğlu, Ebubekir Bakan Jan 1996

The Effect Of Examination Stress On The Levels Of Zn^{2+}, Total Protein, Albumin In Serum And Carbonic Anhydrase Isoenzymes In Erythrocytes, Nazan Demi̇r, Yaşar Demi̇r, Ali Yildirim, Ö. İrfan Küfrevi̇oğlu, Ebubekir Bakan

Turkish Journal of Chemistry

In this study, the levels of Zn^{2+}, total protein and albumin in the serums of students with temporarily examination stress without any physical illnes were determined and the relationship between these values was evaluated. Furthermore, the change in the activity of carbonic anhydrase metaloenzyme with Zn^{2+} and in the rates of carbonic anhydrase I and carbonic anhydrase II were also investigated. With this aim, blood samples were taken from 100 students just before examination and from 20 control subjects. Zn^{2+} levels were determined enzymatically, while total protein and albumin assays were done with an autoanalyzer. Carbonic anhydrase activity was measured …


Synthesis Of 2,4-Dihydro-4-(2-Phenylethyl)-5-(Isomeric Pyridyl)-3h-1,2,4-Triazole-3-Thiones And Their Derivatives., Rashid Iqbal, Khosrow Zamani, Nasim H. Rama Jan 1996

Synthesis Of 2,4-Dihydro-4-(2-Phenylethyl)-5-(Isomeric Pyridyl)-3h-1,2,4-Triazole-3-Thiones And Their Derivatives., Rashid Iqbal, Khosrow Zamani, Nasim H. Rama

Turkish Journal of Chemistry

Intramolecular cyclization of the three isomeric 1,4-disubstituted thiosemicarbazides 3(a-c) in aqueous sodium hydroxide was attempted to obtain 2,4-dihydro-4-(2-phenylethyl)-5-(isomeric pyridyl)- 3H-1,2,4-triazole-3-thiones 4(a-c). Each of the three isolated triazoles was separately treated with benzyl chloride and with chloroacetic acid to furnish the corresponding benzylthio 5(a-c) and carboxymethylthio derivatives 6(a-c), respectively


Latex Supported Manganese(Iii) Porphyrin Catalysts For Oxidations Of Alkenes By Hydrogen Peroxide And {\It Tert-} Butyl Hydroperoxide In The Aqueous Medium, Hayrettin Türk Jan 1996

Latex Supported Manganese(Iii) Porphyrin Catalysts For Oxidations Of Alkenes By Hydrogen Peroxide And {\It Tert-} Butyl Hydroperoxide In The Aqueous Medium, Hayrettin Türk

Turkish Journal of Chemistry

Cationic polymer latexes were prepared by emulsion copolymerization of vinylbenzyl chloride and divinylbenzene followed by quaternization with trimethylamine and used as catalyst supports for 5,10,15,20-tetrakis(2, 6-dichloro-3-sulfonatophenyl)porphinatomanganese(III) chloride and 5,10,15,20-tetra(4-sul- fonatophenyl)porphinatomanganese(III) chloride. The supported catalysts and water soluble analogues were used for oxidation of styrene, 1-decene and {\it cis}-cyclooctene by hydrogen peroxide and {\it tert}- butyl hydroperoxide in the aqueous phase. The oxidation of 1-decene and {\it cis}-cyclooctene failed. In the oxidation of styrene, the porphyrin catalysts in the latex were more active than in solution. The relative molar ratios of oxidant: styrene: Mn porphyrin were 1250: 750: 1, and about …


Structural Fusion Of Condensed Benzenoid Systems, Lemi Türker Jan 1996

Structural Fusion Of Condensed Benzenoid Systems, Lemi Türker

Turkish Journal of Chemistry

On the basis of ring topologies, some theorems for the structural fusion of condensed benzenoids are presented. It has been proved that the oddness or evenness of the fused system is dictated by the degree of fusion.


The Sterols Of Cyclamen Coum, Nurettin Yayli, Cemalettin Baltaci Jan 1996

The Sterols Of Cyclamen Coum, Nurettin Yayli, Cemalettin Baltaci

Turkish Journal of Chemistry

The structures of one major and three minor constituents of sterol fraction of the Cyclamen coum have been established as stigmasterol [(22E)- (24S)- 24-ethylcholesta-\Delta ^{5,22}-dien-\beta-ol] ({\bf 1}), Poriferasterol [22E)-(24R)-24-ethylcholesta-\Delta^{5,22}-dien-3\beta- ol] ({\bf 2}), stigma-or poriferasta-\Delta^{3,5,22} triene [(22E)-24 -ethylcholesta-\Delta^{3,5,22}-trien] ({\bf 3}) stigma-or poriferosta- \Delta^{3,5,7,22}-tetraene [(22E)-24-ethylcholesta-\Delta^{3,5,7,22}- tetraene] ({\bf 4}) by spectroscopic means (NMR, EI-MS and GC-MS).


An Unusual Zinc-Promoted Reductive Retro-Wagner-Meerwein Rearrangement, Yavuz Taşkesenli̇gi̇l, Metin Balci Jan 1996

An Unusual Zinc-Promoted Reductive Retro-Wagner-Meerwein Rearrangement, Yavuz Taşkesenli̇gi̇l, Metin Balci

Turkish Journal of Chemistry

Bromination and dehydrobromination of exocyclic olefin {\bf 8} resulted in the formation of bromomethylene compound {\bf 9} and Wagner-Meerwein rearrangement product {\bf 10}. The dehydrobromination of {\bf 10} with potassium {\it tert}-butoxide provided the corresponding bromomethyl compound {\bf 11} in high yield. Reaction of {\bf 11} with Zn-Cu couple in methanol afforded the methyl derivative {\bf 12} and the parent olefin {\bf 8}. The unusual formation mechanism of {\bf 8} has been discussed in term of a reductive retro-Wagner-Meerwein rearrangement.


A Novel And Stereospecific Synthesis Of Conduritol-E Via Cyclohexa-3,5-Diene-1,2-Diol, Ahmet Maras, Hasan Secen, Yasar Sütbeyaz, Metin Balci Jan 1996

A Novel And Stereospecific Synthesis Of Conduritol-E Via Cyclohexa-3,5-Diene-1,2-Diol, Ahmet Maras, Hasan Secen, Yasar Sütbeyaz, Metin Balci

Turkish Journal of Chemistry

Conduritol-E was synthesized starting from cyclohexa-3,5-diene-1,2-diol {\bf 1a} in six steps. Acetylation of the diol {\bf 1a} followed by bromination gave (2\alpha/1\beta, \,\, 3\beta, \,\, 4\beta)-1,2-dibromo-2,3- diacetoxy- 5-cyclohexene {\bf 3} as main product. KMnO_{4}-hydoxylation of the dibromide {\bf 3} followed by acetylation, Zn-elimination and hydrolysis afforded conduritol-E {\bf 5a}.


Study Of Interaction And Solubility Parameters Of Alkyl Methacrylate Polymers By Inverse Gas Chromatography, İsmet Kaya, Eyüp Özdemi̇r, Mehmet Coşkun Jan 1996

Study Of Interaction And Solubility Parameters Of Alkyl Methacrylate Polymers By Inverse Gas Chromatography, İsmet Kaya, Eyüp Özdemi̇r, Mehmet Coşkun

Turkish Journal of Chemistry

Flory-Huggins interaction parameters (\chi) and partial molar free energies of mixing (\DeltaG^{\infty}_1) have been determined by inverse gas chromatography (i.g.c.) at temperatures 413-443 K for alkanes, acetates, ketones, aromatics, THF and methanol to estimate the polymer solubility parameters (\delta_2) of PMMA, PEMA, PİBMA and Pt-BMA. Furthermore, weight fraction activity coefficients (\alpha_1/W_1)^{\infty} of probes were calculated at infinite dilution at 413-443 K.


The Epoxidation Of Oil Obtained From The Eastern Black Sea Fishes, Eyüp Özcan, Bahri Ülküseven Jan 1996

The Epoxidation Of Oil Obtained From The Eastern Black Sea Fishes, Eyüp Özcan, Bahri Ülküseven

Turkish Journal of Chemistry

In this study we have investigated the suitable reaction conditions for the epoxidation of the semi-refined fish oil obtained from the fishes of eastern Black Sea, with peracetic acid. Various oxidation mixtures to reach the highest epoxyd-oxygene amount have been tested mean while the effective factors were established. In order to establish some specifications of the epoxy-product. Iodine-, hydroxyl-indices and epoxyd-oxygene percents were determined. We've also tried to explain how some reaction conditions (mole ratio of acetic acid to double bond and temperature) influenced the epoxy-product.


Synthesis Of Optically Active Camphorsulphonylbenzimidazoles, Mashooda Hasan, Humaira Masood, Naeema Khan, M. Zia-Ul-Haq Jan 1996

Synthesis Of Optically Active Camphorsulphonylbenzimidazoles, Mashooda Hasan, Humaira Masood, Naeema Khan, M. Zia-Ul-Haq

Turkish Journal of Chemistry

The preparation of optically active camphorsulphonylbenzimidazole {(\bf 1)}, a mixture of regioisomeric 5-nitro- {\bf (2A)} and 6-nitro-{\bf (2B)} camphorsulphonylbenzimidazoles, 5-nitro-{\bf (3A)} and 6-nitro-{\bf (3B)} 2- methyl camphorsulphonylbenzimidazoles is described. In this study the structure of the purified camphorsulphonylbenzimidazoles {\bf 1, 2B, 3A} and {\bf 3B} was confirmed by their mass and ^{1}H NMR spectral analysis. The specific rotation of each pure compound is also reported.


Synthesis, Characterization And Formation Kinetics Of Sn(Ii) And (Iv)-Schiff Base Complexes, Yakup Baran, Birgül Erk Jan 1996

Synthesis, Characterization And Formation Kinetics Of Sn(Ii) And (Iv)-Schiff Base Complexes, Yakup Baran, Birgül Erk

Turkish Journal of Chemistry

The synthesis and characterization of Sn(II) and Sn(IV) complexes of nitrogen and oxygen ligands, {\it p}-dimethyl aminophenylenesalicyladimine ( DPS), N,N'phenylenebis(salicylaldimine)(PBS) and N, N'-triethylenetetramine bis (salicylaldimine)(TTS) are reported. The structure of free ligands and their tin complexes were determined by elemental analysis, ir and nmr measurements. The kinetics of formation of the complexes were studied by stopped-flow technique at several temperatures with \mu=0.100 M (NaClO_{ 4}). Activation parameters \Delta H^{\ast} and \Delta S^{\ast} were determined.


Voltammetric Determination Of Promethazine By Platinum And Glassy Carbon Electrodes, Bengi Uslu, İnci Bi̇ryol, Sibel A. Özkan, Zühre Şentürk Jan 1996

Voltammetric Determination Of Promethazine By Platinum And Glassy Carbon Electrodes, Bengi Uslu, İnci Bi̇ryol, Sibel A. Özkan, Zühre Şentürk

Turkish Journal of Chemistry

The electrochemical oxidation of promethazine was studied by linear sweep and cyclic voltammetry on platinum and glassy carbon electrodes. The dependence of current intensities and potentials on pH, concentration, scan rate and nature of the buffer was examined. Linear calibration plots were obtained in phosphate buffer (pH 7.0) and acetate buffer (pH 4.7) for platinum and glassy carbon electrode, respectively.


Chemistry Of Aerosol In Uludağ National Park, Serpil Y. Karakaş, Semra G. Tuncel Jan 1996

Chemistry Of Aerosol In Uludağ National Park, Serpil Y. Karakaş, Semra G. Tuncel

Turkish Journal of Chemistry

Chemical composition of aerosol samples which were collected daily during September 1993 - April 1994 was examined by determining the concentrations and variations of major ions in aerosol samples, determining the binary correlations and the sources of them. Sulfate made up the 30% of aerosol mass with an 2.14 \mug m^{-3} average concentration. Measured chemical parameters in Uludağ National Park were comparable with other rural area data expect Ca^{2+} which was higher in our station with an average concentration of 1.61 \mug m^{-3}. Ion balance showed a deficiency of acid forming anions (SO^{2-}_4 and NO^-_3 for about 30% on the …


Catalysis Of Autoxidation Of 2,6-Di-Tert-Butylphenol In Water By Metallophthalocyanine Tetrasulfonates Bound To Polyelectrolyte Supports, Yasemin Tura Kopkalli, Hayrettin Türk Jan 1996

Catalysis Of Autoxidation Of 2,6-Di-Tert-Butylphenol In Water By Metallophthalocyanine Tetrasulfonates Bound To Polyelectrolyte Supports, Yasemin Tura Kopkalli, Hayrettin Türk

Turkish Journal of Chemistry

2,6-and 2,10-Ionenes and soluble poly(vinylbenzyl)trimethylammonium chloride polyelectrolytes were synthesized and used as catalyst supports for cobalt-, iron- and copper-phthalocyanine tetrasulfonates on the autoxidation of 2,6-di-{\it tert}-butylphenol in water by molecular oxygen. Polyelectrolyte supported cobalt-phthalocyanine tetrasulfonate showed the highest activity in comparison with the supported analogous of iron- and copper-phthalocyanine tetrasulfonates. The supported metallophthalocyanine tetrasulfonates had better catalytic activities than the unsupported analogous for the reaction. The reaction yielded 3,3',5,5'-tetra-tert-butyl- 4,4'-diphenoquinone as the only oxidation product of 2,6-di-tert-butylphenol. The extent of the reaction was affected by time, temperature, pH, the types and concentrations of supports and metallophthalocyanine tetrasulfonates.


Chelate Ring-Closure Kinetics Of Cr(Co)_5 (Dppm) Studied By ^{31}P-Nmr Spectroscopy, Saim Özkar, Ceyhan Kayran, Ayşin Tekkaya Jan 1996

Chelate Ring-Closure Kinetics Of Cr(Co)_5 (Dppm) Studied By ^{31}P-Nmr Spectroscopy, Saim Özkar, Ceyhan Kayran, Ayşin Tekkaya

Turkish Journal of Chemistry

^{31}P-NMR spectroscopy was used to study the kinetics of the chelate ring-closure reaction in CR(CO)_5 (DPPM) where DPPM is bis (diphenylphosphino) methane. DPPM is potentially a bidentate ligand, but in this complex it is coordinated to the transition metal in a monodentate fashion and therefore can undergo chelation. The thermal chelate ring-closure reaction was carried out in an NMR sample tube at a constant temperature and the course of the reaction was quantitatively followed by taking ^{31}P-NMR spectra in appropriate time intervals. Kinetic results have shown that the chelate ring-closure reaction of Cr(CO)_5 DPPM is of zero-order. Activation parameters were …


The Concept Of Molecular Similarity And The Angle Of Total \Pi-Electron Energy, Lemi Türker Jan 1996

The Concept Of Molecular Similarity And The Angle Of Total \Pi-Electron Energy, Lemi Türker

Turkish Journal of Chemistry

The concept of molecular similarity is developed and a novel similarity index based on the cosine of the angle of total \Pi-electron energy is proposed to interrelate structural variations to operant properties of molecules isoconjugate with alternant hydrocarbons.


Polyelectrolytes In Immunology: Fundametals And Perspectives, Mehmet I. Mustafaev Jan 1996

Polyelectrolytes In Immunology: Fundametals And Perspectives, Mehmet I. Mustafaev

Turkish Journal of Chemistry

Physico-chemical criteria for the construction of artifical immunomodulators and immunogens(and vaccines) on the basis of synthetic polyelectrolyte(PE) and polyelectrolyte complexes(PEC) with low molecular weight ligands and protein (and polysaccharide) antigens are considered. The role of electric charge, chain length, conformation, hydrophobicity, polydispersity and supramolecular structure in the immunomodulating activity of PE(PEC) and the conditions facilitating the transformation of their suppressor capacity into immunostimulating one in the same polymerhomologues series are discussed. A structure-function relationship of immunogenicity of PE(PEC), a structural model of an artificial polymer- subunit immunogen and some practical applications of them as valuable tools in theoretical immunological studies …


On The \Ell_{P} Norms Of Almost Cauchy-Toeplitz Matrices, D. Bozkurt Jan 1996

On The \Ell_{P} Norms Of Almost Cauchy-Toeplitz Matrices, D. Bozkurt

Turkish Journal of Mathematics

In this study, we have given the definition of almost Cauchy-Toeplitz matrix. i.e. its elements are t_{ij}= a(i=j) and t_{ij}=1/(i-j)\, (i\neq j) such that a is a real number. We have found a lower and upper bounds for the \ell_{p} norm of this matrix. Furthermore, we have done the proof of the conjecture that were given by myself for the spectral norm of this matrix.