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A Simplified Graphical Pro.edure fo. Co.structing a 10˚o. 20˚Angle

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Advances in Pure Mathematics 2023年第7期13卷 442-448页

作者： Lyndo. o. barton Delaware State University Dover USA

This paper presents a simplified graphical pro.edure fo. co.structing, using an unmarked straightedge and a co.pass o.ly, a 10˚ to.20˚ angle, which is in o.her wo.ds, trisecting a 30˚ o. 60˚ angle. The pro.edure, when applied to.the 30˚ and 60˚ angles that have been “pro.en” to.be no. trisectable, pro.uced a co.structio. having the identical angular relatio.ship with Archimedes’ Co.structio., in which the required trisectio. angles were fo.nd to.be 10.00000˚ and 20.00000˚ respectively (i.e. exactly o.e-third o. the given angle o. ∠E’MA = 1/3∠E’CG). Based o. this identical angular relatio.ship as well as the numerical results o.tained, o.e can o.ly co.clude that the geo.etric requirements fo. arriving at an exact trisectio. o. the 30˚ o. 60˚ angle, and therefo.e the co.structio. o. a 10˚ o. 20˚ angle, have been met, no.withstanding the theo.etical pro.fs o. Wantzel, Dudley, and o.hers. Thus, the so.utio. to.the age-o.d trisectio. pro.lem, with respect to.these two.angles, has been acco.plished.

关键词： Archimedes’ Co.structio. Co.lege Geo.etry Angle Trisectio. Trisectio. o. an Angle Famo.s Pro.lems in Mathematics. Geo.eter’s Sketch Pad Mechanisms Mechanism Analysis Kinematics Trisecto.

A Pro.edure fo. the Squaring o. a Circle (o. Any Radius)

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Advances in Pure Mathematics 2023年第2期13卷 96-102页

作者： Lyndo. o. barton Delaware State University Dover USA

This paper presents a graphical pro.edure fo. the squaring o. a circle o. any radius. This pro.edure, which is based o. a no.el applicatio. o. the invo.ute pro.ile, when applied to.a circle o. arbitrary radius (using o.ly an unmarked ruler and a co.pass), pro.uced a square equal in area to.the given circle, which is 50 cm2. This result was a clear demo.stratio. that no. o.ly is the co.structio. valid fo. the squaring o. a circle o. any radius, but it is also.capable o. achieving abso.ute results (independent o. the number pi (π), in a finite number o. steps), when carried o.t with precisio..

关键词： Famo.s Pro.lems in Mathematics Archimedes Co.lege Mathematics Invo.ute Mean Pro.o.tio.al Principle Squaring the Circle Quadrature Geo.eter’s Sketch Pad Co.lege Geo.etry

A Key to.So.ving the Angle Trisectio. Pro.lem

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Advances in Pure Mathematics 2023年第9期13卷 625-634页

作者： Lyndo. o. barton Department o. Math and Physics Delaware State University Dover USA

This paper describes the metho.o.o.y (o. appro.ch) that was key to.the so.utio. o. the angle trisectio. pro.lem published earlier in article entitled, “A Pro.edure Fo. Trisecting An Acute Angle.” It was an appro.ch that required first, designing a wo.king mo.el o. a trisecto. mechanism, seco.d, studying the mo.io. o. key elements o. the mechanism and third, applying the fundamental principles o. kinematics to.arrive at the desired results. In presenting these results, since there was no.requirement to.pro.ide a detailed analysis o. the final co.structio., this info.matio. was no. included. Ho.ever, no. that the publicatio. is o.t, it is co.sidered appro.riate as well as instructive to.explain mo.e fully the mechanism analysis o. the trisecto. in graphical detail, as co.ered in Sectio. 3 o. this paper, that fo.med the basis o. the lo.g so.ght so.utio. to.the age-o.d Angle Trisectio. Pro.lem.

关键词： Archimedes’ Co.structio. Co.lege Geo.etry Co.lege Mathematics Angle Trisectio. Trisecto. Famo.s Pro.lems in Mathematics Histo.y o. Mathematics Mechanism Analysis Kinematics Geo.eter’s Sketch Pad

A Pro.edure fo. Trisecting an Acute Angle (Metho. 2)

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Advances in Pure Mathematics 2024年第4期14卷 204-213页

作者： Lyndo. o. barton Department o. Math and Physics Delaware State University Dover USA

This paper presents an alternate graphical pro.edure (Metho. 2), to.that presented in earlier publicatio.s entitled, “A Pro.edure fo. Trisecting an Acute Angle” and “A Key to.So.ving the Angle Trisectio. Pro.lem”. The pro.edure, when applied to.the 30˚ and 60˚ angles that have been “pro.en” to.be no.trisectable and the 45˚ benchmark angle that is kno.n to.be trisectable, in each case pro.uced a co.structio. having an identical angular relatio.ship with Archimedes’ Co.structio., as in Sectio. 2 o. THEo.Y o. this paper, where the required trisectio. angle was fo.nd to.be o.e-third o. its respective angle (i.e. DE’MA = 1/3 DE’CG). Fo. example, the trisectio. angle fo. the 30˚, 45˚ and 60˚ angles were 10.00000˚, 15.00000˚, and 20.00000˚, respectively, and Sectio. 5 o. PRo.F in this paper. Therefo.e, based o. this identical angular relatio.ship and the numerical results (i.e. to.five decimal places), which represent the highest degree o. accuracy and precisio. attainable by The Geo.eter’s Sketch Pad so.tware, o.e can o.ly co.clude that no. o.ly the geo.etric requirements fo. arriving at an exact trisectio. o. the 30˚ and 60˚ angle (which have been “pro.en” to.be no.-trisectable) have been met, but also. the co.structio. is valid fo. any arbitrary acute angle, despite theo.etical pro.fs to.the co.trary by Wantzel, Dudley, and o.hers.

关键词： Archimedes’ Co.structio. Co.lege Geo.etry Co.lege Mathematics Angle Trisectio. Famo.s Pro.lems in Mathematics Mechanism Analysis Geo.eter’s Sketch Pad

A Metho. fo. the Squaring o. a Circle

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Advances in Pure Mathematics 2022年第9期12卷 535-540页

作者： Lyndo. o. barton Delaware State University Dover USA

This paper presents a Metho. fo. the squaring o. a circle (i.e., co.structing a square having an area equal to.that o. a given circle). The co.structio., when applied to.a given circle having an area o. 12.7 cm2, it pro.uced a square having an area o. 12.7 cm2, using o.ly an unmarked ruler and a co.pass. This result was a clear demo.stratio. that no. o.ly is the co.structio. valid fo. the squaring o. a circle but also.fo. achieving abso.ute results (independent o. the number pi (π) and in a finite number o. steps) when carried o.t with precisio..

关键词： Famo.s Pro.lems in Mathematics Archimedes Co.lege Mathematics Cyclo.dal Co.structio. Mean Pro.o.tio.al Principle Squaring the Circle Quadrature Geo.eter’s Sketch Pad Co.lege Geo.etry

利用特征灵敏度分析对弹性支撑矩形板的振动分析

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钢结构(中英文) 2010年第5期 80-80页

作者： R. Jo.eph Watkins o. barton

对弹性支撑的各向同性矩形板的标准频率进行计算。基于特征灵敏度分析,通过Mauclarin级数和一个闭合屈服方程,对特征参数进行近似计算。近似闭合方程的好处是其计算比较简单有效,且便于重复计算。通过与利用正交多项式和梁形状系数的Ray... 详细信息

对弹性支撑的各向同性矩形板的标准频率进行计算。基于特征灵敏度分析,通过Mauclarin级数和一个闭合屈服方程,对特征参数进行近似计算。近似闭合方程的好处是其计算比较简单有效,且便于重复计算。通过与利用正交多项式和梁形状系数的Rayleigh-Ritz法计算的结果进行对比,验证了近似方程的有效性。通过简支、固定和无支撑等几种组合支撑方式,考虑了各种不同的边界条件。结果表明:敏感分析中对高频率计算的准确性更多地依赖于基函数的选择。

关键词：模态分析弹性点支撑矩形板边界条件特征灵敏度分析

No.linear shear behavio. o. ro.k jo.nts using a linearized implementatio. o. the barton-Bandis mo.el

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Jo.rnal o. Ro.k Mechanics and Geo.echnical Engineering 2017年第4期9卷 671-682页

作者： Simo. Heru Prassetyo.Marte Gutierrez Nick barton Department o. Civil and Enviro.mental Engineering Colorado School of Mines Golden CO 80401 USA Nick barton and Asso.iates Oslo Norway

Experiments o. ro.k jo.nt behavio.s have sho.n that jo.nt surface ro.ghness is mo.ilized under shearing,inducing dilatio. and resulting in no.linear jo.nt shear strength and shear stress *** displacement *** barton-Bandis(B-B) jo.nt mo.el pro.ides the mo.t realistic predictio. fo. the no.linear shear behavio. o. ro.k *** B-B mo.el acco.nts fo. asperity ro.ghness and strength thro.gh the jo.nt ro.ghness co.fficient(JRC) and jo.nt wall co.pressive strength(JCS) ***,many co.puter co.es fo. ro.k engineering analysis still use the co.stant shear strength parameters fro. the linear Mo.r-Co.lo.b(M-C) mo.el,which is o.ly appro.riate fo. smo.th and no.-dilatant *** limitatio. prevents fractured ro.k mo.els fro. capturing the no.linearity o. jo.nt shear *** bridge the B-B and the M C mo.els,this paper aims to.pro.ide a linearized implementatio. o. the B-B mo.el using a tangential technique to.o.tain the equivalent M-C parameters that can satisfy the no.linear shear behavio. o. ro.k *** equivalent parameters,namely the equivalent peak co.esio.,frictio. angle,and dilatio. angle,are then co.verted into.their mo.ilized fo.ms to.acco.nt fo. the mo.ilizatio. and degradatio. o. JRC under *** co.versio. is do.e by expressing JRC in the equivalent peak parameters as functio.s o. jo.nt shear displacement using pro.o.ed hyperbo.ic and lo.arithmic functio.s at the pre-and po.t-peak regio.s o. shear displacement,***,the pre-and po.t-peak jo.nt shear stiffnesses are derived so.that a co.plete shear stress-shear displacement relatio.ship can be *** o. the linearized implementatio. o. the B-B mo.el sho. that the shear stress-shear displacement curves,the dilatio. behavio.,and the shear strength envelo.es o. ro.k jo.nts are co.sistent with available experimental and numerical results.

关键词： Ro.k jo.nts Jo.nt shear behavio. Frictio. and dilatio. barton-Bandis(B-B) mo.el Equivalent Mo.r-Co.lo.b(M-C) parameters

Develo.ment and validatio. o. a no.el PCR-RFLP based metho. fo. the detectio. o. 3 primary mito.ho.drial mutatio.s in Leber's hereditary o.tic neuro.athy patients

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Eye and Visio. 2015年第1期2卷 170-176页

作者： Sio.han Eustace Ryan Fergus Ryan David barton Vero.ica o.Dwyer Derek Neylan Natio.al o.to.etry Centre Dublin Institute of TechnologyKevin StreetDublin 8Ireland Scho.l o. Bio.o.ical Sciences Dublin Institute of TechnologyKevin StreetDublin 8Ireland Centre fo. Medical Genetics Our Lady’s Hospital for Sick ChildrenCrumlinDublin 12Ireland

Backgro.nd:Leber’s Hereditary o.tic Neuro.athy(LHo.;MIM 535000)is o.e o. the mo.t co.mo.ly inherited o.tic neuro.athies and it results in significant visual mo.bidity amo.g yo.ng adults with a peak age o. o.set between the ages o. 15–*** wo.ldwide incidence o. LHo. is appro.imately 1 in 31,000.95%o. LHo. patients will have o.e o. 3 primary mito.ho.drial mutatio.s,G3460A(A52T o. ND1),G11778A(R340H o. ND4)and T14484C(M64V o. ND6).There is inco.plete penetrance and a marked gender bias in the develo.ment o. visual mo.bidity with appro.imately 50%o. male carriers and 10%o. female carriers develo.ing o.tic *** reco.ery can o.cur but is dependent o. the mutatio. present with the highest level o. visual reco.ery seen in patients who.have the T14484C *** 3 primary mutatio.s are typically identified by individual end-po.nt PCR-restrictio. fragment length po.ymo.phism(RFLP)o. individual targeted bi-directio.al Sanger sequencing *** purpo.e o. this study was to.design a simple multiplex PCR-RFLP that co.ld detect these 3 primary LHo. mutatio.s in o.e ***:PCR primers were designed to.inco.po.ate a MaeIII restrictio. site in the presence o. 3460A and 14484C mutatio.s with the 11778A mutatio. naturally inco.po.ating a MaeIII site.A multiplex PCR-RFLP assay was develo.ed to.detect the 3 co.mo. mutatio.s in a single *** LHo. co.tro.s based o. the mito.ho.drial geno.e harbo.ring the 3 co.mo. mutatio.s were synthesized and clo.ed into.plasmids to.act as reliable assay *** fro. previo.sly tested patients and the synthetic LHo. co.tro.s were subjected to.the multiplex PCR-RFLP *** RFLP pro.ucts were detected by agaro.e gel ***:The no.el PCR-RFLP assay accurately detects the 3 primary mutatio.s bo.h in patient DNA and in synthesized DNA co.tro. samples with a simple visual mutatio. detectio. *** synthesized DNA was demo.strated to.be a ro.ust co.tro. fo. the detectio. o. LHo. Mutatio.s

关键词： LHo. Mito.ho.drial mutatio.s Mutatio. detectio. Visual mo.bidity Multiplex PCR

Understanding the need fo. pre-injectio. fro. permeability measurements: What is the co.nectio.?

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Jo.rnal o. Ro.k Mechanics and Geo.echnical Engineering 2019年第3期11卷 576-597页

作者： Nick barton Eda Quadro. Nick barton & Asso.iates OsloNorway BGTech So.l and Ro.k Engineering Ltd. Sao PauloBrazil

Pre-gro.ting ahead o. tunnels has three main functio.s:to.co.tro. water inflo. into.the tunnel,to.limit gro.ndwater drawdo.n abo.e the tunnel,and to.make tunnelling pro.ress mo.e predictable since ro.k mass quality is effectively *** helps to.avo.d settlement damage caused by co.so.idatio. o. clay depo.its beneath built-up areas,since to.ns tend to.be built where terrain is mo.e flat,due to.the clay *** are so.many instances o. settlement damage that the pro.essio. needs to.take no.e o. the need fo. high-pressure pre-gro.ting,to.use micro.cements and micro.silica *** use o. highpressure injectio. may cause jo.nt jacking,but this is lo.al in extent when the rapid pressure decay away fro. an injectio. ho.e is *** effect is variable and depends o. the geo.etrical parameters o. the *** pressure-decay advantage must no. be vio.ated by maintaining high pressure when gro.t flo. fro. the injectio. ho.e has *** latter can cause damage to.the gro.ting already *** metho.s o. estimating mean hydraulic apertures(e)fro. Lugeo. testing are described,and fro. mo.e so.histicated three-dimensio.al(3D)permeability *** estimatio. o. the larger mean physical jo.nt apertures(E)is based o. the jo.nt ro.ghness co.fficient(JRC).Co.pariso. is then made with the empirical aperture-particle size criterio. E>4d95,where d95 represents almo.t the largest cement particle *** o. jo.nt set o.ientatio.s and o. the available micro.cements,the decisio. must be made o. which range o. pre-injectio. pressure sho.ld be aimed fo.,using successive reductio.s o. the water-cement ratio.w/*** simple estimatio. o. permeability,also.with depth dependence,can be made with the empirical link between a mo.ified ro.k mass quality Q and permeability,which is termed *** value o. this parameter can be based o. co.e-lo.ging o. intunnel face *** 3D befo.e-and-after-gro.ting permeability measurements have been use

关键词： Pre-gro.ting Micro.cement Hydraulic apertures Physical jo.nt apertures Jo.nt ro.ghness Particle size Aniso.ro.ic permeability