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| <?xml version="1.0"?> | |
| <pdf> | |
| <section line_height="9.96" font="KAAPYP+CMR10" letter_ratio="0.02" year_ratio="0.0" cap_ratio="0.0" name_ratio="0.2767857142857143" | |
| word_count="112" lateness="0.06666666666666667" reference_score="4.6"> | |
| <line x_offset="0.0" y_offset="117.01" spacing="0.5">Abstract.</line> | |
| <line x_offset="57.0" y_offset="116.52" spacing="-9.47">Theory is presented for the distributions of local process intensity | |
| and</line> | |
| <line x_offset="0.0" y_offset="103.56" spacing="3.0">local average pore dimensions in random fibrous materials. For complete | |
| partitioning</line> | |
| <line x_offset="0.0" y_offset="90.6" spacing="3.0">of the network into contiguous square zones, the variance of local process | |
| intensity</line> | |
| <line x_offset="0.0" y_offset="77.64" spacing="3.0">is shown to be proportional to the mean process intensity and inversely | |
| proportional</line> | |
| <line x_offset="0.0" y_offset="64.68" spacing="3.0">to the zone size. The coefficient of variation of local average pore area is | |
| shown</line> | |
| <line x_offset="0.0" y_offset="51.72" spacing="3.0">to be approximately double that of the local average pore diameter with | |
| both</line> | |
| <line x_offset="0.0" y_offset="38.76" spacing="3.0">properties being inversely proportional to the square root of zone size and | |
| mean process</line> | |
| <line x_offset="0.0" y_offset="25.92" spacing="2.88">intensity. The results have relevance to heterogenous near-planar fibrous | |
| materials</line> | |
| <line x_offset="0.0" y_offset="12.96" spacing="3.0">including paper, nonwoven textiles, nanofibrous composites and electrospun | |
| polymer</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.0">fibre networks.</line> | |
| <component x="143.76" y="396.75" width="373.2" height="126.48" page="1" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.07" year_ratio="0.0" cap_ratio="0.05" | |
| name_ratio="0.24596774193548387" word_count="496" lateness="0.13333333333333333" reference_score="6.54"> | |
| <line x_offset="195.12" y_offset="622.16" spacing="0.55">i.e.</line> | |
| <line x_offset="0.0" y_offset="621.6" spacing="-11.41">The global average pore size of thin, near-planar, heterogeneous fibrous | |
| materials</line> | |
| <line x_offset="0.0" y_offset="605.64" spacing="3.99">and its distribution have been widely studied using statistical geometry and | |
| simulation.</line> | |
| <line x_offset="0.0" y_offset="589.68" spacing="3.99">Typically the context of these studies has been materials with widespread | |
| application</line> | |
| <line x_offset="0.0" y_offset="573.84" spacing="3.87">in society and industry such as paper [1, 2], nonwoven textiles [3], and | |
| fibrous filter</line> | |
| <line x_offset="0.0" y_offset="557.88" spacing="3.99">media [4, 5]. Interest in the structural characteristics of fibrous | |
| materials has increased</line> | |
| <line x_offset="0.0" y_offset="541.92" spacing="3.99">in recent years as researchers seek to develop materials for future | |
| applications such</line> | |
| <line x_offset="0.0" y_offset="525.96" spacing="3.99">as carbon nanotube 'buckypaper' [6, 7], nanofibrous composites [8, 9] and | |
| electrospun</line> | |
| <line x_offset="0.0" y_offset="510.0" spacing="3.99">fibrous networks for application as scaffolds in tissue engineering [10, | |
| 11].</line> | |
| <line x_offset="23.4" y_offset="494.04" spacing="3.99">In a seminal paper, Miles [12] provides several properties of the polygons | |
| generated</line> | |
| <line x_offset="0.0" y_offset="478.2" spacing="3.87">by the stochastic division of a plane by a Poisson process of straight lines | |
| with infinite</line> | |
| <line x_offset="0.0" y_offset="462.24" spacing="3.99">length. A graphical representation of such a process is given in Figure 1a. | |
| The process</line> | |
| <line x_offset="356.88" y_offset="446.29" spacing="3.99">τ</line> | |
| <line x_offset="0.0" y_offset="446.28" spacing="-11.96">intensity is characterized by the expected line length per unit area, | |
| ¯. The following</line> | |
| <line x_offset="0.0" y_offset="430.32" spacing="3.99">results of Miles are utilized in the present study:</line> | |
| <line x_offset="46.8" y_offset="408.36" spacing="9.99">The expected area of polygons is</line> | |
| <line x_offset="35.04" y_offset="399.88" spacing="-11.97">•</line> | |
| <line x_offset="143.64" y_offset="397.57" spacing="-9.64">π</line> | |
| <line x_offset="157.92" y_offset="389.53" spacing="-3.92">.</line> | |
| <line x_offset="406.32" y_offset="389.52" spacing="-11.96">(1)</line> | |
| <line x_offset="118.56" y_offset="389.52" spacing="-11.97">a¯ =</line> | |
| <line x_offset="147.96" y_offset="385.73" spacing="-4.18">2</line> | |
| <line x_offset="141.6" y_offset="381.25" spacing="-7.48">τ</line> | |
| <line x_offset="142.2" y_offset="381.24" spacing="-11.96">¯</line> | |
| <line x_offset="46.8" y_offset="369.0" spacing="0.27">The distribution of diameters of the largest circle inscribed within</line> | |
| <line x_offset="35.04" y_offset="360.52" spacing="-11.97">•</line> | |
| <line x_offset="46.8" y_offset="353.04" spacing="-4.48">polygons is exponential with mean</line> | |
| <line x_offset="141.84" y_offset="339.0" spacing="2.07">1</line> | |
| <line x_offset="120.6" y_offset="334.08" spacing="-7.05">¯</line> | |
| <line x_offset="118.56" y_offset="330.97" spacing="-8.85">d</line> | |
| <line x_offset="406.32" y_offset="330.96" spacing="-11.96">(2)</line> | |
| <line x_offset="127.92" y_offset="330.96" spacing="-11.97">=</line> | |
| <line x_offset="141.6" y_offset="322.81" spacing="-3.81">τ</line> | |
| <line x_offset="142.2" y_offset="322.8" spacing="-11.96">¯</line> | |
| <line x_offset="81.6" y_offset="312.0" spacing="-1.17">¯</line> | |
| <line x_offset="46.8" y_offset="308.89" spacing="-8.85">a</line> | |
| <line x_offset="79.56" y_offset="308.89" spacing="-11.96">d</line> | |
| <line x_offset="46.92" y_offset="308.88" spacing="-11.96">¯ and are independent of the width of lines, for any probability | |
| density</line> | |
| <line x_offset="35.04" y_offset="300.4" spacing="-11.97">•</line> | |
| <line x_offset="46.8" y_offset="292.92" spacing="-4.48">of line widths.</line> | |
| <line x_offset="0.0" y_offset="270.96" spacing="9.99">Miles showed also that the expected number of sides per polygon is 4 and | |
| that the</line> | |
| <line x_offset="194.16" y_offset="264.07" spacing="0.92">2</line> | |
| <line x_offset="189.0" y_offset="260.81" spacing="-4.71">π</line> | |
| <line x_offset="119.04" y_offset="255.01" spacing="-6.16">P</line> | |
| <line x_offset="0.0" y_offset="255.0" spacing="-11.96">fraction of triangles is (3) = (2</line> | |
| <line x_offset="199.44" y_offset="255.0" spacing="-11.97">)</line> | |
| <line x_offset="222.72" y_offset="255.0" spacing="-11.97">0 355; Tanner [13] obtained the fraction of.</line> | |
| <line x_offset="191.52" y_offset="251.93" spacing="-4.9">6</line> | |
| <line x_offset="175.32" y_offset="246.52" spacing="-15.04">−</line> | |
| <line x_offset="208.68" y_offset="246.52" spacing="-20.44">≈</line> | |
| <line x_offset="77.16" y_offset="239.17" spacing="-4.6">P</line> | |
| <line x_offset="123.24" y_offset="239.17" spacing="-11.96">.</line> | |
| <line x_offset="0.0" y_offset="239.16" spacing="-11.96">quadrilaterals, (4) 0 381. The fractions of polygons with more than 4 | |
| sides are not</line> | |
| <line x_offset="104.76" y_offset="230.68" spacing="-11.97">≈</line> | |
| <line x_offset="0.0" y_offset="223.2" spacing="-4.48">known analytically, but have been obtained by Monte Carlo methods [14, 15]. | |
| We note</line> | |
| <line x_offset="0.0" y_offset="207.24" spacing="3.99">that very similar results were obtained by Piekaar and Clarenburg [4] from | |
| simulations</line> | |
| <line x_offset="0.0" y_offset="191.28" spacing="3.99">of networks of fibres with finite length. Crain and Miles [14] observed the | |
| probability of</line> | |
| <line x_offset="0.0" y_offset="175.33" spacing="3.99">n</line> | |
| <line x_offset="350.04" y_offset="175.33" spacing="-11.96">n</line> | |
| <line x_offset="6.96" y_offset="175.32" spacing="-11.96">-sided polygons was well approximated by the Poisson variable (</line> | |
| <line x_offset="373.56" y_offset="175.32" spacing="-11.97">3) with mean</line> | |
| <line x_offset="360.72" y_offset="166.84" spacing="-11.97">−</line> | |
| <line x_offset="4.56" y_offset="159.37" spacing="-4.48">n</line> | |
| <line x_offset="0.0" y_offset="159.36" spacing="-11.96">(¯ 3) = 1. This result was used by Dodson and Sampson [16] to | |
| approximate</line> | |
| <line x_offset="15.72" y_offset="150.88" spacing="-11.97">−</line> | |
| <line x_offset="0.0" y_offset="143.52" spacing="-4.6">the probability density functions for polygon areas and perimeters, assuming | |
| regular</line> | |
| <line x_offset="0.0" y_offset="127.56" spacing="3.99">polygons. The coefficient of variation of polygon area was close to 2 and | |
| that of polygon</line> | |
| <line x_offset="74.04" y_offset="112.96" spacing="-5.85">√</line> | |
| <line x_offset="84.0" y_offset="111.6" spacing="-10.6">3 2; despite the assumption of regularity, these are close to the | |
| analytic/</line> | |
| <line x_offset="0.0" y_offset="111.6" spacing="-11.97">perimeter was</line> | |
| <line x_offset="0.0" y_offset="95.64" spacing="3.99">results of Miles [12].</line> | |
| <line x_offset="23.4" y_offset="79.68" spacing="3.99">Now, each intersection between lines represents one of the vertices of four | |
| polygons</line> | |
| <line x_offset="0.0" y_offset="63.72" spacing="3.99">and for a Poisson process of infinite lines, Miles [12] gives the expected | |
| number of</line> | |
| <line x_offset="160.68" y_offset="53.21" spacing="2.54">2</line> | |
| <line x_offset="186.0" y_offset="48.44" spacing="-6.09">i.e.</line> | |
| <line x_offset="154.32" y_offset="47.89" spacing="-11.41">τ /π</line> | |
| <line x_offset="212.64" y_offset="47.89" spacing="-11.96">/a</line> | |
| <line x_offset="0.0" y_offset="47.88" spacing="-11.96">intersections per unit area as ¯ ,</line> | |
| <line x_offset="206.76" y_offset="47.88" spacing="-11.97">1 ¯. For the case of networks lines with finite</line> | |
| <line x_offset="0.0" y_offset="31.92" spacing="3.99">length and constant width (Figure 1b), the expected number of intersections | |
| per unit</line> | |
| <line x_offset="0.0" y_offset="15.96" spacing="3.99">area was derived by Kallmes and Corte [17] for infinite networks; in | |
| practical contexts,</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.99">this can be considered satisfied of samples are large relative to the | |
| dimensions of voids.</line> | |
| <component x="72.0" y="39.72" width="444.82" height="633.57" page="2" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="5.7" font="ECYBFT+Myriad-Roman" letter_ratio="0.0" year_ratio="0.0" cap_ratio="0.0" name_ratio="0" | |
| word_count="2" lateness="0.2" reference_score="5.33"> | |
| <line x_offset="0.0" y_offset="0.0" spacing="0.0">ab</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="0.0">c</line> | |
| <component x="203.85" y="697.14" width="130.19" height="5.7" page="3" page_width="612.0" page_height="792.0"/> | |
| <component x="456.22" y="697.14" width="3.54" height="5.7" page="3" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.02" year_ratio="0.0" cap_ratio="0.0" | |
| name_ratio="0.32142857142857145" word_count="28" lateness="0.26666666666666666" reference_score="5.68"> | |
| <line x_offset="0.0" y_offset="31.92" spacing="0.0">of infinite lines. From this we obtain approximate probability density | |
| functions for the</line> | |
| <line x_offset="0.0" y_offset="15.96" spacing="3.99">distribution of local averages of pore area and pore diameter in planar | |
| stochastic fibrous</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.99">materials.</line> | |
| <component x="72.0" y="657.24" width="444.76" height="43.89" page="4" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.08" year_ratio="0.0" cap_ratio="0.04" | |
| name_ratio="0.1740139211136891" word_count="431" lateness="0.26666666666666666" reference_score="7.78"> | |
| <line x_offset="0.0" y_offset="563.41" spacing="0.0">We seek the variance of local process intensity for a planar Poisson line | |
| process of mean</line> | |
| <line x_offset="51.0" y_offset="547.58" spacing="3.87">τ</line> | |
| <line x_offset="322.56" y_offset="547.58" spacing="-11.96">x</line> | |
| <line x_offset="0.0" y_offset="547.57" spacing="-11.96">intensity, ¯, partitioned into contiguous square zones of side, . We | |
| denote the local</line> | |
| <line x_offset="186.0" y_offset="531.62" spacing="3.99">τ</line> | |
| <line x_offset="0.0" y_offset="531.61" spacing="-11.96">process intensity within such zones, .</line> | |
| <line x_offset="438.36" y_offset="515.66" spacing="3.99">τ</line> | |
| <line x_offset="23.4" y_offset="515.65" spacing="-11.96">Before proceeding, it is helpful to obtain estimates of the likely range | |
| of ¯</line> | |
| <line x_offset="186.24" y_offset="506.4" spacing="-27.59">(</line> | |
| <line x_offset="0.0" y_offset="499.69" spacing="-5.26">encountered in real materials. If the expected mass per unit area, or | |
| 'areal density', of a</line> | |
| <line x_offset="128.64" y_offset="489.06" spacing="2.66">2</line> | |
| <line x_offset="432.12" y_offset="489.06" spacing="-7.97">1</line> | |
| <line x_offset="85.08" y_offset="486.97" spacing="-9.88">¯</line> | |
| <line x_offset="83.4" y_offset="483.74" spacing="-8.73">β</line> | |
| <line x_offset="388.56" y_offset="483.74" spacing="-11.96">δ</line> | |
| <line x_offset="0.0" y_offset="483.73" spacing="-11.96">fibre network is, (kg m ), and the linear density of the constituent | |
| fibres is (kg m ),</line> | |
| <line x_offset="122.04" y_offset="483.44" spacing="-10.84">−</line> | |
| <line x_offset="425.52" y_offset="483.44" spacing="-11.13">−</line> | |
| <line x_offset="206.52" y_offset="471.01" spacing="0.46">¯</line> | |
| <line x_offset="182.64" y_offset="467.78" spacing="-8.73">τ β/δ</line> | |
| <line x_offset="0.0" y_offset="467.77" spacing="-11.96">then the expected process intensity, ¯ = . For materials such as | |
| paper and nonwoven</line> | |
| <line x_offset="319.8" y_offset="457.26" spacing="2.54">2</line> | |
| <line x_offset="388.8" y_offset="451.94" spacing="-6.64">τ</line> | |
| <line x_offset="0.0" y_offset="451.93" spacing="-11.96">textiles, with mass per unit area between 20 and 100 g m we expect ¯ | |
| to be of</line> | |
| <line x_offset="313.2" y_offset="451.64" spacing="-10.84">−</line> | |
| <line x_offset="43.68" y_offset="441.3" spacing="2.37">2</line> | |
| <line x_offset="76.44" y_offset="441.3" spacing="-7.97">1</line> | |
| <line x_offset="0.0" y_offset="435.97" spacing="-6.64">order 10 mm ; for electrospun networks of polymer fibres with mass per unit | |
| area</line> | |
| <line x_offset="69.84" y_offset="435.68" spacing="-10.84">−</line> | |
| <line x_offset="79.2" y_offset="425.34" spacing="2.37">2</line> | |
| <line x_offset="404.04" y_offset="425.34" spacing="-7.97">3</line> | |
| <line x_offset="436.8" y_offset="425.34" spacing="-7.97">1</line> | |
| <line x_offset="234.36" y_offset="420.02" spacing="-6.64">µ</line> | |
| <line x_offset="308.76" y_offset="420.02" spacing="-11.96">τ</line> | |
| <line x_offset="0.0" y_offset="420.01" spacing="-11.96">around 10 g m and fibre diameter around 1 m we expect ¯ to be of | |
| order 10 mm .</line> | |
| <line x_offset="72.6" y_offset="419.72" spacing="-10.84">−</line> | |
| <line x_offset="430.2" y_offset="419.72" spacing="-11.13">−</line> | |
| <line x_offset="315.0" y_offset="404.06" spacing="3.7">l</line> | |
| <line x_offset="23.4" y_offset="404.05" spacing="-11.96">From Coleman [35] the probability density of the length, , of random | |
| secants in a</line> | |
| <line x_offset="318.48" y_offset="403.26" spacing="-7.18">1</line> | |
| <line x_offset="0.0" y_offset="388.09" spacing="3.2">unit square is</line> | |
| <line x_offset="128.64" y_offset="373.5" spacing="6.62">2 l</line> | |
| <line x_offset="136.8" y_offset="372.92" spacing="-5.4">1</line> | |
| <line x_offset="231.24" y_offset="367.71" spacing="-6.74">< l</line> | |
| <line x_offset="211.32" y_offset="367.69" spacing="-11.96">if 0</line> | |
| <line x_offset="267.84" y_offset="367.69" spacing="-11.97">1</line> | |
| <line x_offset="247.2" y_offset="366.9" spacing="-7.18">1</line> | |
| <line x_offset="132.24" y_offset="364.5" spacing="-5.57">π</line> | |
| <line x_offset="255.24" y_offset="359.22" spacing="-15.16">≤</line> | |
| <line x_offset="187.92" y_offset="357.54" spacing="-6.29">2 l</line> | |
| <line x_offset="147.6" y_offset="357.42" spacing="-7.85">4</line> | |
| <line x_offset="196.08" y_offset="356.96" spacing="-5.52">1</line> | |
| <line x_offset="267.84" y_offset="353.1" spacing="-16.58">√</line> | |
| <line x_offset="111.84" y_offset="352.08" spacing="-35.83"></line> | |
| <line x_offset="231.24" y_offset="351.75" spacing="-11.62">< l</line> | |
| <line x_offset="211.32" y_offset="351.73" spacing="-11.96">if 1</line> | |
| <line x_offset="277.8" y_offset="351.73" spacing="-11.97">2</line> | |
| <line x_offset="247.2" y_offset="350.94" spacing="-7.18">1</line> | |
| <line x_offset="155.88" y_offset="349.28" spacing="-4.32">2</line> | |
| <line x_offset="71.76" y_offset="349.22" spacing="-11.9">f l</line> | |
| <line x_offset="429.72" y_offset="349.21" spacing="-11.96">(3)</line> | |
| <line x_offset="78.84" y_offset="349.21" spacing="-11.97">( ) =</line> | |
| <line x_offset="191.4" y_offset="348.66" spacing="-7.42">π</line> | |
| <line x_offset="86.88" y_offset="348.42" spacing="-7.73">1</line> | |
| <line x_offset="128.64" y_offset="346.02" spacing="-5.57">π l</line> | |
| <line x_offset="153.24" y_offset="346.02" spacing="-7.97">l</line> | |
| <line x_offset="166.68" y_offset="346.02" spacing="-7.97">1</line> | |
| <line x_offset="137.76" y_offset="345.44" spacing="-5.4">1</line> | |
| <line x_offset="143.28" y_offset="344.58" spacing="-19.58">√</line> | |
| <line x_offset="155.76" y_offset="344.0" spacing="-5.4">1</line> | |
| <line x_offset="255.24" y_offset="343.26" spacing="-19.7">≤</line> | |
| <line x_offset="174.72" y_offset="343.26" spacing="-20.44">−</line> | |
| <line x_offset="160.08" y_offset="340.4" spacing="-8.27">−</line> | |
| <line x_offset="111.84" y_offset="337.8" spacing="-37.84"></line> | |
| <line x_offset="127.44" y_offset="330.37" spacing="-4.54">0</line> | |
| <line x_offset="211.32" y_offset="330.37" spacing="-11.97">otherwise.</line> | |
| <line x_offset="111.84" y_offset="316.2" spacing="-22.67"></line> | |
| <line x_offset="111.84" y_offset="312.72" spacing="-33.37"></line> | |
| <line x_offset="57.84" y_offset="310.93" spacing="-10.18">¯</line> | |
| <line x_offset="197.64" y_offset="309.18" spacing="-18.69">√</line> | |
| <line x_offset="117.6" y_offset="309.18" spacing="-20.44">√</line> | |
| <line x_offset="225.48" y_offset="307.82" spacing="-10.6">/ π</line> | |
| <line x_offset="277.2" y_offset="307.82" spacing="-11.96">.</line> | |
| <line x_offset="57.84" y_offset="307.82" spacing="-11.96">l</line> | |
| <line x_offset="127.56" y_offset="307.81" spacing="-11.96">2 + 3 log(1 + 2)</line> | |
| <line x_offset="231.36" y_offset="307.81" spacing="-11.97">(3 ) 0 946.</line> | |
| <line x_offset="0.0" y_offset="307.81" spacing="-11.97">with mean = 4 1</line> | |
| <line x_offset="61.32" y_offset="307.02" spacing="-7.18">1</line> | |
| <line x_offset="258.72" y_offset="299.34" spacing="-12.76">≈</line> | |
| <line x_offset="105.72" y_offset="299.34" spacing="-20.44">−</line> | |
| <line x_offset="218.04" y_offset="292.32" spacing="-29.83">(</line> | |
| <line x_offset="91.68" y_offset="292.32" spacing="-36.85">(</line> | |
| <line x_offset="323.76" y_offset="291.86" spacing="-11.5">x</line> | |
| <line x_offset="23.4" y_offset="291.85" spacing="-11.96">The expected total length of lines in a square zone of side is</line> | |
| <line x_offset="110.4" y_offset="275.94" spacing="7.94">2</line> | |
| <line x_offset="73.08" y_offset="273.01" spacing="-9.04">¯</line> | |
| <line x_offset="71.76" y_offset="269.9" spacing="-8.85">L τ x .</line> | |
| <line x_offset="83.04" y_offset="269.89" spacing="-11.96">= ¯</line> | |
| <line x_offset="429.72" y_offset="269.89" spacing="-11.97">(4)</line> | |
| <line x_offset="400.08" y_offset="251.17" spacing="6.75">¯ ¯</line> | |
| <line x_offset="258.36" y_offset="248.07" spacing="-8.85">x l</line> | |
| <line x_offset="400.08" y_offset="248.07" spacing="-11.96">l</line> | |
| <line x_offset="424.56" y_offset="248.07" spacing="-11.96">l x</line> | |
| <line x_offset="0.0" y_offset="248.05" spacing="-11.96">We denote the length of secants in a square of side , , with expected | |
| length, = .</line> | |
| <line x_offset="275.16" y_offset="247.26" spacing="-7.18">x</line> | |
| <line x_offset="403.56" y_offset="247.26" spacing="-7.97">x</line> | |
| <line x_offset="428.04" y_offset="247.26" spacing="-7.97">1</line> | |
| <line x_offset="338.52" y_offset="232.1" spacing="3.2">x</line> | |
| <line x_offset="0.0" y_offset="232.09" spacing="-11.96">It follows that the expected number of secants in a square of side | |
| is</line> | |
| <line x_offset="101.04" y_offset="216.01" spacing="4.11">¯</line> | |
| <line x_offset="99.72" y_offset="213.03" spacing="-8.97">L</line> | |
| <line x_offset="126.24" y_offset="213.01" spacing="-11.96">τ x¯</line> | |
| <line x_offset="75.12" y_offset="207.97" spacing="-6.93">¯</line> | |
| <line x_offset="150.24" y_offset="204.99" spacing="-8.97">.</line> | |
| <line x_offset="71.76" y_offset="204.98" spacing="-11.96">N</line> | |
| <line x_offset="112.56" y_offset="204.97" spacing="-11.96">=</line> | |
| <line x_offset="429.72" y_offset="204.97" spacing="-11.97">(5)</line> | |
| <line x_offset="85.68" y_offset="204.97" spacing="-11.97">=</line> | |
| <line x_offset="129.48" y_offset="199.45" spacing="-6.45">¯</line> | |
| <line x_offset="129.6" y_offset="196.35" spacing="-8.85">l</line> | |
| <line x_offset="133.08" y_offset="195.54" spacing="-7.17">1</line> | |
| <line x_offset="99.24" y_offset="195.51" spacing="-11.92">l</line> | |
| <line x_offset="102.72" y_offset="194.7" spacing="-7.17">x</line> | |
| <line x_offset="313.92" y_offset="175.1" spacing="7.64">x N</line> | |
| <line x_offset="338.4" y_offset="152.88" spacing="-14.62">(</line> | |
| <line x_offset="429.72" y_offset="126.01" spacing="14.9">(6)</line> | |
| <line x_offset="429.72" y_offset="66.61" spacing="47.43">(7)</line> | |
| <line x_offset="429.72" y_offset="14.05" spacing="40.59">(8)</line> | |
| <line x_offset="23.4" y_offset="175.09" spacing="-173.01">If the number of lines contained in a given square of side is , then the | |
| total line</line> | |
| <line x_offset="0.0" y_offset="159.13" spacing="3.99">length in that square is</line> | |
| <line x_offset="102.48" y_offset="145.93" spacing="7.44">(</line> | |
| <line x_offset="100.32" y_offset="141.9" spacing="-3.94">N</line> | |
| <line x_offset="71.76" y_offset="126.03" spacing="3.92">L</line> | |
| <line x_offset="114.72" y_offset="126.03" spacing="-11.96">l</line> | |
| <line x_offset="136.44" y_offset="126.03" spacing="-11.96">,</line> | |
| <line x_offset="83.04" y_offset="126.01" spacing="-11.96">=</line> | |
| <line x_offset="118.2" y_offset="125.22" spacing="-7.18">x,i</line> | |
| <line x_offset="97.2" y_offset="113.1" spacing="4.15">i=0</line> | |
| <line x_offset="95.4" y_offset="112.08" spacing="-35.82">(</line> | |
| <line x_offset="72.72" y_offset="103.8" spacing="-28.57">(</line> | |
| <line x_offset="0.0" y_offset="95.65" spacing="-3.82">such that the local process intensity is</line> | |
| <line x_offset="96.84" y_offset="74.67" spacing="9.03">L</line> | |
| <line x_offset="115.44" y_offset="66.63" spacing="-3.92">.</line> | |
| <line x_offset="71.76" y_offset="66.63" spacing="-11.95">τ</line> | |
| <line x_offset="81.48" y_offset="66.61" spacing="-11.96">=</line> | |
| <line x_offset="101.64" y_offset="62.94" spacing="-4.3">2</line> | |
| <line x_offset="95.04" y_offset="58.47" spacing="-7.48">x</line> | |
| <line x_offset="97.8" y_offset="52.56" spacing="-30.94">(</line> | |
| <line x_offset="72.0" y_offset="41.4" spacing="-25.69">(</line> | |
| <line x_offset="154.08" y_offset="14.07" spacing="15.38">,</line> | |
| <line x_offset="0.0" y_offset="43.09" spacing="-41.0">It follows that the variance of local process intensity is given by</line> | |
| <line x_offset="123.24" y_offset="27.54" spacing="7.58">2</line> | |
| <line x_offset="132.6" y_offset="22.23" spacing="-6.64">L</line> | |
| <line x_offset="116.16" y_offset="22.23" spacing="-11.96">σ</line> | |
| <line x_offset="128.04" y_offset="22.21" spacing="-11.96">( )</line> | |
| <line x_offset="122.76" y_offset="20.22" spacing="-5.98">x</line> | |
| <line x_offset="124.92" y_offset="5.91" spacing="2.36">x</line> | |
| <line x_offset="131.52" y_offset="10.38" spacing="-12.45">4</line> | |
| <line x_offset="133.56" y_offset="0.0" spacing="-26.46">(</line> | |
| <component x="72.0" y="32.03" width="444.81" height="575.38" page="4" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.16" year_ratio="0.0" cap_ratio="0.09" | |
| name_ratio="0.10554089709762533" word_count="379" lateness="0.3333333333333333" reference_score="10.03"> | |
| <line x_offset="103.8" y_offset="477.48" spacing="0.0">x</line> | |
| <line x_offset="0.0" y_offset="477.47" spacing="-11.96">where the subscript is included to denote that the variance depends on the | |
| zone size.</line> | |
| <line x_offset="87.48" y_offset="466.84" spacing="2.66">2</line> | |
| <line x_offset="135.48" y_offset="466.84" spacing="-7.97">2 2</line> | |
| <line x_offset="80.4" y_offset="461.52" spacing="-6.64">σ L x σ L</line> | |
| <line x_offset="219.36" y_offset="461.52" spacing="-11.96">L</line> | |
| <line x_offset="0.0" y_offset="461.51" spacing="-11.96">Note also that ( ) =</line> | |
| <line x_offset="154.08" y_offset="461.51" spacing="-11.97">( ), where</line> | |
| <line x_offset="237.12" y_offset="461.51" spacing="-11.97">is the total line length in a unit square</line> | |
| <line x_offset="166.56" y_offset="460.72" spacing="-7.18">1</line> | |
| <line x_offset="227.28" y_offset="460.72" spacing="-7.97">1</line> | |
| <line x_offset="87.0" y_offset="459.52" spacing="-6.77">x</line> | |
| <line x_offset="56.88" y_offset="445.56" spacing="2.0">N</line> | |
| <line x_offset="0.0" y_offset="445.55" spacing="-11.96">containing secants. Thus,</line> | |
| <line x_offset="97.92" y_offset="439.29" spacing="-30.59">(</line> | |
| <line x_offset="159.6" y_offset="439.29" spacing="-36.85">(</line> | |
| <line x_offset="220.32" y_offset="439.29" spacing="-36.85">(</line> | |
| <line x_offset="123.24" y_offset="429.88" spacing="1.45">2</line> | |
| <line x_offset="116.16" y_offset="424.56" spacing="-6.64">σ L</line> | |
| <line x_offset="127.92" y_offset="424.55" spacing="-11.96">( )</line> | |
| <line x_offset="140.4" y_offset="423.76" spacing="-7.18">1</line> | |
| <line x_offset="59.88" y_offset="423.33" spacing="-36.42">(</line> | |
| <line x_offset="78.84" y_offset="422.44" spacing="-7.07">2</line> | |
| <line x_offset="158.76" y_offset="416.52" spacing="-6.04">.</line> | |
| <line x_offset="71.76" y_offset="416.52" spacing="-11.95">σ τ</line> | |
| <line x_offset="429.72" y_offset="416.51" spacing="-11.96">(9)</line> | |
| <line x_offset="83.64" y_offset="416.51" spacing="-11.97">( ) =</line> | |
| <line x_offset="78.36" y_offset="414.52" spacing="-5.98">x</line> | |
| <line x_offset="133.8" y_offset="412.72" spacing="-6.17">2</line> | |
| <line x_offset="127.2" y_offset="408.36" spacing="-7.6">x</line> | |
| <line x_offset="133.44" y_offset="402.33" spacing="-30.82">(</line> | |
| <line x_offset="88.44" y_offset="391.29" spacing="-25.81">(</line> | |
| <line x_offset="31.08" y_offset="389.64" spacing="-10.3">L</line> | |
| <line x_offset="366.6" y_offset="389.64" spacing="-11.96">N</line> | |
| <line x_offset="0.0" y_offset="389.63" spacing="-11.96">Now, is a random variable obtained as the sum of the lengths of</line> | |
| <line x_offset="382.32" y_offset="389.63" spacing="-11.97">independent</line> | |
| <line x_offset="39.0" y_offset="388.84" spacing="-7.18">1</line> | |
| <line x_offset="338.64" y_offset="384.59" spacing="-1.51">(</line> | |
| <line x_offset="336.6" y_offset="380.56" spacing="-3.94">N</line> | |
| <line x_offset="272.52" y_offset="374.23" spacing="-4.53">i.e.</line> | |
| <line x_offset="352.8" y_offset="373.68" spacing="-11.41">l</line> | |
| <line x_offset="408.48" y_offset="373.68" spacing="-11.96">l</line> | |
| <line x_offset="293.4" y_offset="373.68" spacing="-11.96">L</line> | |
| <line x_offset="366.24" y_offset="373.67" spacing="-11.96">, where</line> | |
| <line x_offset="426.48" y_offset="373.67" spacing="-11.97">is a</line> | |
| <line x_offset="0.0" y_offset="373.67" spacing="-11.97">and identically distributed secants in a unit square,</line> | |
| <line x_offset="310.44" y_offset="373.67" spacing="-11.97">=</line> | |
| <line x_offset="356.28" y_offset="372.88" spacing="-7.18">1,i</line> | |
| <line x_offset="411.96" y_offset="372.88" spacing="-7.97">1,i</line> | |
| <line x_offset="301.32" y_offset="372.88" spacing="-7.97">1</line> | |
| <line x_offset="336.6" y_offset="371.2" spacing="-6.29">i=0</line> | |
| <line x_offset="32.16" y_offset="367.41" spacing="-33.06">(</line> | |
| <line x_offset="369.48" y_offset="367.41" spacing="-36.85">(</line> | |
| <line x_offset="404.76" y_offset="357.84" spacing="-2.38">N</line> | |
| <line x_offset="0.0" y_offset="357.83" spacing="-11.96">continuous random variable with probability density given by Equation (3) | |
| and takes</line> | |
| <line x_offset="323.88" y_offset="357.45" spacing="-36.47">(</line> | |
| <line x_offset="294.36" y_offset="351.57" spacing="-30.97">(</line> | |
| <line x_offset="215.28" y_offset="341.88" spacing="-2.26">L</line> | |
| <line x_offset="0.0" y_offset="341.87" spacing="-11.96">integer values. The mean and variance of</line> | |
| <line x_offset="231.96" y_offset="341.87" spacing="-11.97">are given by [36]</line> | |
| <line x_offset="223.2" y_offset="341.08" spacing="-7.18">1</line> | |
| <line x_offset="407.64" y_offset="335.61" spacing="-31.38">(</line> | |
| <line x_offset="133.68" y_offset="323.03" spacing="0.62">¯</line> | |
| <line x_offset="73.08" y_offset="322.91" spacing="-11.85">¯</line> | |
| <line x_offset="124.44" y_offset="322.91" spacing="-11.97">¯</line> | |
| <line x_offset="71.76" y_offset="319.92" spacing="-8.97">L</line> | |
| <line x_offset="121.08" y_offset="319.92" spacing="-11.96">N l</line> | |
| <line x_offset="108.72" y_offset="319.91" spacing="-11.96">=</line> | |
| <line x_offset="423.96" y_offset="319.91" spacing="-11.97">(10)</line> | |
| <line x_offset="216.36" y_offset="319.65" spacing="-36.59">(</line> | |
| <line x_offset="79.68" y_offset="319.12" spacing="-7.43">1</line> | |
| <line x_offset="137.16" y_offset="319.12" spacing="-7.97">1</line> | |
| <line x_offset="185.76" y_offset="309.52" spacing="1.63">2</line> | |
| <line x_offset="199.56" y_offset="306.88" spacing="-5.33">2</line> | |
| <line x_offset="78.84" y_offset="306.88" spacing="-7.97">2</line> | |
| <line x_offset="140.76" y_offset="306.88" spacing="-7.97">2</line> | |
| <line x_offset="124.44" y_offset="304.07" spacing="-9.16">¯</line> | |
| <line x_offset="177.24" y_offset="300.96" spacing="-8.85">l σ N .</line> | |
| <line x_offset="71.76" y_offset="300.96" spacing="-11.96">σ L N σ l</line> | |
| <line x_offset="204.24" y_offset="300.95" spacing="-11.96">( )</line> | |
| <line x_offset="423.96" y_offset="300.95" spacing="-11.97">(11)</line> | |
| <line x_offset="83.52" y_offset="300.95" spacing="-11.97">( ) =</line> | |
| <line x_offset="145.56" y_offset="300.95" spacing="-11.97">( ) +</line> | |
| <line x_offset="180.96" y_offset="300.16" spacing="-7.18">1</line> | |
| <line x_offset="96.0" y_offset="300.16" spacing="-7.97">1</line> | |
| <line x_offset="153.6" y_offset="300.16" spacing="-7.97">1</line> | |
| <line x_offset="313.2" y_offset="284.44" spacing="7.75">2</line> | |
| <line x_offset="356.76" y_offset="282.11" spacing="-9.64">¯</line> | |
| <line x_offset="85.92" y_offset="279.12" spacing="-8.97">N</line> | |
| <line x_offset="306.12" y_offset="279.12" spacing="-11.96">σ N N</line> | |
| <line x_offset="0.0" y_offset="279.11" spacing="-11.96">We assume that</line> | |
| <line x_offset="100.44" y_offset="279.11" spacing="-11.97">is a Poisson random variable, such that ( ) = and we have</line> | |
| <line x_offset="211.8" y_offset="278.85" spacing="-36.59">(</line> | |
| <line x_offset="89.04" y_offset="278.85" spacing="-36.85">(</line> | |
| <line x_offset="192.84" y_offset="263.92" spacing="6.97">2</line> | |
| <line x_offset="78.84" y_offset="261.28" spacing="-5.33">2</line> | |
| <line x_offset="147.96" y_offset="261.28" spacing="-7.97">2</line> | |
| <line x_offset="124.44" y_offset="258.35" spacing="-9.04">¯</line> | |
| <line x_offset="88.92" y_offset="256.89" spacing="-35.39">(</line> | |
| <line x_offset="325.44" y_offset="256.89" spacing="-36.85">(</line> | |
| <line x_offset="184.44" y_offset="255.36" spacing="-10.42">l</line> | |
| <line x_offset="71.76" y_offset="255.36" spacing="-11.96">σ L N σ l</line> | |
| <line x_offset="423.96" y_offset="255.35" spacing="-11.96">(12)</line> | |
| <line x_offset="83.52" y_offset="255.35" spacing="-11.97">( ) =</line> | |
| <line x_offset="152.64" y_offset="255.35" spacing="-11.97">( ) +</line> | |
| <line x_offset="188.16" y_offset="254.56" spacing="-7.18">1</line> | |
| <line x_offset="96.0" y_offset="254.56" spacing="-7.97">1</line> | |
| <line x_offset="160.68" y_offset="254.56" spacing="-7.97">1</line> | |
| <line x_offset="197.64" y_offset="243.45" spacing="-25.74">(</line> | |
| <line x_offset="133.68" y_offset="243.45" spacing="-36.85">(</line> | |
| <line x_offset="106.32" y_offset="236.44" spacing="-0.95">2</line> | |
| <line x_offset="36.96" y_offset="233.32" spacing="-4.85">2</line> | |
| <line x_offset="89.04" y_offset="233.13" spacing="-36.66">(</line> | |
| <line x_offset="78.48" y_offset="233.08" spacing="-7.91">2</line> | |
| <line x_offset="29.88" y_offset="228.0" spacing="-6.88">σ l</line> | |
| <line x_offset="74.76" y_offset="228.0" spacing="-11.95">l</line> | |
| <line x_offset="97.92" y_offset="228.0" spacing="-11.96">l</line> | |
| <line x_offset="0.0" y_offset="227.99" spacing="-11.96">Now, ( ) =</line> | |
| <line x_offset="111.12" y_offset="227.99" spacing="-11.97">. So</line> | |
| <line x_offset="49.8" y_offset="227.2" spacing="-7.18">1</line> | |
| <line x_offset="101.64" y_offset="227.2" spacing="-7.97">1</line> | |
| <line x_offset="78.24" y_offset="226.0" spacing="-6.77">1</line> | |
| <line x_offset="85.92" y_offset="219.51" spacing="-13.96">−</line> | |
| <line x_offset="78.84" y_offset="211.96" spacing="-0.41">2</line> | |
| <line x_offset="137.4" y_offset="211.12" spacing="-7.13">2</line> | |
| <line x_offset="124.44" y_offset="209.03" spacing="-9.88">¯</line> | |
| <line x_offset="150.0" y_offset="206.04" spacing="-8.97">.</line> | |
| <line x_offset="133.68" y_offset="206.04" spacing="-11.96">l</line> | |
| <line x_offset="71.76" y_offset="206.04" spacing="-11.95">σ L N</line> | |
| <line x_offset="423.96" y_offset="206.03" spacing="-11.96">(13)</line> | |
| <line x_offset="83.52" y_offset="206.03" spacing="-11.97">( ) =</line> | |
| <line x_offset="96.0" y_offset="205.24" spacing="-7.18">1</line> | |
| <line x_offset="137.16" y_offset="204.16" spacing="-6.89">1</line> | |
| <line x_offset="60.24" y_offset="189.28" spacing="6.91">2</line> | |
| <line x_offset="56.52" y_offset="184.08" spacing="-6.76">l</line> | |
| <line x_offset="0.0" y_offset="184.07" spacing="-11.96">We obtain</line> | |
| <line x_offset="69.0" y_offset="184.07" spacing="-11.97">as the second moment of the probability density given by Equation | |
| (3):</line> | |
| <line x_offset="89.04" y_offset="183.81" spacing="-36.59">(</line> | |
| <line x_offset="60.0" y_offset="182.2" spacing="-6.35">1</line> | |
| <line x_offset="107.88" y_offset="168.78" spacing="2.29">√</line> | |
| <line x_offset="114.96" y_offset="167.8" spacing="-6.99">2</line> | |
| <line x_offset="190.8" y_offset="161.75" spacing="-5.92">3</line> | |
| <line x_offset="125.4" y_offset="159.64" spacing="-5.86">2</line> | |
| <line x_offset="75.48" y_offset="158.8" spacing="-7.13">2</line> | |
| <line x_offset="202.32" y_offset="153.72" spacing="-6.88">.</line> | |
| <line x_offset="71.76" y_offset="153.72" spacing="-11.96">l</line> | |
| <line x_offset="121.68" y_offset="153.72" spacing="-11.96">l f l</line> | |
| <line x_offset="165.12" y_offset="153.72" spacing="-11.96">l</line> | |
| <line x_offset="423.96" y_offset="153.71" spacing="-11.96">(14)</line> | |
| <line x_offset="83.52" y_offset="153.71" spacing="-11.97">=</line> | |
| <line x_offset="139.2" y_offset="153.71" spacing="-11.97">( ) d =</line> | |
| <line x_offset="147.24" y_offset="152.92" spacing="-7.18">1</line> | |
| <line x_offset="168.6" y_offset="152.92" spacing="-7.97">1</line> | |
| <line x_offset="75.24" y_offset="151.84" spacing="-6.89">1</line> | |
| <line x_offset="125.16" y_offset="151.72" spacing="-7.85">1</line> | |
| <line x_offset="190.2" y_offset="145.56" spacing="-5.8">π</line> | |
| <line x_offset="96.0" y_offset="144.81" spacing="-36.1">(</line> | |
| <line x_offset="102.6" y_offset="143.92" spacing="-7.07">0</line> | |
| <line x_offset="170.28" y_offset="131.44" spacing="4.51">2</line> | |
| <line x_offset="254.28" y_offset="131.44" spacing="-7.97">2</line> | |
| <line x_offset="250.44" y_offset="129.23" spacing="-9.76">¯</line> | |
| <line x_offset="163.2" y_offset="126.0" spacing="-8.73">σ l</line> | |
| <line x_offset="218.52" y_offset="126.0" spacing="-11.96">/π l</line> | |
| <line x_offset="0.0" y_offset="125.99" spacing="-11.96">For completeness, we note that ( ) = (3 )</line> | |
| <line x_offset="275.04" y_offset="125.99" spacing="-11.97">0 0593..</line> | |
| <line x_offset="183.0" y_offset="125.2" spacing="-7.18">1</line> | |
| <line x_offset="254.04" y_offset="124.12" spacing="-6.89">1</line> | |
| <line x_offset="238.68" y_offset="117.51" spacing="-13.84">− ≈</line> | |
| <line x_offset="423.96" y_offset="66.23" spacing="39.32">(15)</line> | |
| <line x_offset="423.96" y_offset="8.27" spacing="45.99">(16)</line> | |
| <line x_offset="23.4" y_offset="110.15" spacing="-113.85">Substituting Equations (13), (14) and (5) in Equation (9) yields our | |
| final expression</line> | |
| <line x_offset="336.24" y_offset="94.2" spacing="3.99">x</line> | |
| <line x_offset="78.84" y_offset="72.16" spacing="14.07">2</line> | |
| <line x_offset="71.76" y_offset="66.24" spacing="-6.04">σ τ</line> | |
| <line x_offset="83.64" y_offset="66.23" spacing="-11.96">( ) =</line> | |
| <line x_offset="78.36" y_offset="64.24" spacing="-5.98">x</line> | |
| <line x_offset="88.44" y_offset="41.01" spacing="-13.62">(</line> | |
| <line x_offset="102.48" y_offset="38.27" spacing="-9.22">=</line> | |
| <line x_offset="0.0" y_offset="94.19" spacing="-67.89">for the variance of local process intensity for square zones of side | |
| :</line> | |
| <line x_offset="128.16" y_offset="74.28" spacing="7.95">N</line> | |
| <line x_offset="116.88" y_offset="74.27" spacing="-11.96">3</line> | |
| <line x_offset="138.36" y_offset="46.31" spacing="15.99">τ¯</line> | |
| <line x_offset="138.24" y_offset="30.0" spacing="4.35">x</line> | |
| <line x_offset="134.4" y_offset="62.44" spacing="-40.41">2</line> | |
| <line x_offset="116.28" y_offset="58.08" spacing="-7.6">π x</line> | |
| <line x_offset="122.16" y_offset="46.31" spacing="-0.2">3</line> | |
| <line x_offset="129.12" y_offset="28.12" spacing="10.22">1</line> | |
| <line x_offset="128.04" y_offset="8.28" spacing="7.88">.</line> | |
| <line x_offset="116.28" y_offset="28.92" spacing="-32.6">π l</line> | |
| <line x_offset="116.4" y_offset="16.32" spacing="0.64">τ</line> | |
| <line x_offset="117.0" y_offset="16.31" spacing="-11.96">¯</line> | |
| <line x_offset="116.28" y_offset="0.0" spacing="4.35">x</line> | |
| <component x="72.0" y="211.69" width="444.82" height="489.44" page="5" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.04" year_ratio="0.0" cap_ratio="0.09" | |
| name_ratio="0.2773722627737226" word_count="137" lateness="0.3333333333333333" reference_score="8.33"> | |
| <line x_offset="438.36" y_offset="127.45" spacing="0.0">τ</line> | |
| <line x_offset="0.0" y_offset="127.44" spacing="-11.96">From the Central Limit Theorem, we expect the distribution of local | |
| process intensity,</line> | |
| <line x_offset="0.0" y_offset="111.48" spacing="3.99">to be well approximated by a Gaussian distribution if the expected number of | |
| secants</line> | |
| <line x_offset="438.48" y_offset="102.23" spacing="-27.59">(</line> | |
| <line x_offset="120.72" y_offset="95.65" spacing="-5.38">x</line> | |
| <line x_offset="438.12" y_offset="95.65" spacing="-11.96">x</line> | |
| <line x_offset="0.0" y_offset="95.64" spacing="-11.96">in a square zone of side is sufficiently large. For low intensity processes | |
| and at small</line> | |
| <line x_offset="0.0" y_offset="79.68" spacing="3.99">we anticipate that the distribution of local process intensity will exhibit a | |
| positive skew</line> | |
| <line x_offset="282.84" y_offset="63.73" spacing="3.99">N</line> | |
| <line x_offset="0.0" y_offset="63.72" spacing="-11.96">as a consequence of the underlying Poisson process for .</line> | |
| <line x_offset="192.12" y_offset="47.77" spacing="3.99">τ</line> | |
| <line x_offset="23.4" y_offset="47.76" spacing="-11.96">Derivation of the distribution of</line> | |
| <line x_offset="202.56" y_offset="47.76" spacing="-11.97">has proved intractable, so here we estimate the</line> | |
| <line x_offset="285.72" y_offset="41.51" spacing="-30.59">(</line> | |
| <line x_offset="0.0" y_offset="31.8" spacing="-2.26">skewness of the distribution by considering a Poisson process of secants in a | |
| unit square.</line> | |
| <line x_offset="192.36" y_offset="22.55" spacing="-27.59">(</line> | |
| <line x_offset="135.12" y_offset="15.85" spacing="-5.26">τ x</line> | |
| <line x_offset="0.0" y_offset="15.84" spacing="-11.96">The influence of changing ¯ or is therefore captured entirely by | |
| varying the expected</line> | |
| <line x_offset="200.16" y_offset="3.0" spacing="0.87">¯</line> | |
| <line x_offset="196.8" y_offset="0.01" spacing="-8.97">N</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="-11.96">number of secants in the unit square, . From Equations (5) and (15), the mean | |
| and</line> | |
| <component x="72.0" y="34.56" width="445.12" height="139.41" page="5" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="7.3" font="LRQCZG+TimesNewRomanPSMT" letter_ratio="1.0" year_ratio="0.0" cap_ratio="0.0" name_ratio="0" | |
| word_count="1" lateness="0.4" reference_score="2.47"> | |
| <line x_offset="0.0" y_offset="0.0" spacing="0.0">0.4</line> | |
| <component x="196.39" y="693.86" width="10.02" height="7.3" page="6" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.09" year_ratio="0.0" cap_ratio="0.04" | |
| name_ratio="0.18269230769230768" word_count="312" lateness="0.4666666666666667" reference_score="9.24"> | |
| <line x_offset="255.24" y_offset="436.69" spacing="0.0">γ τ</line> | |
| <line x_offset="23.4" y_offset="436.68" spacing="-11.96">The skewness of the local process intensity, ( ), of the simulation data | |
| arising</line> | |
| <line x_offset="261.36" y_offset="435.89" spacing="-7.18">1</line> | |
| <line x_offset="103.68" y_offset="420.73" spacing="3.2">N</line> | |
| <line x_offset="0.0" y_offset="420.72" spacing="-11.96">from different input</line> | |
| <line x_offset="117.72" y_offset="420.72" spacing="-11.97">is plotted against that obtained using Equation (18) in Figure | |
| 2.</line> | |
| <line x_offset="270.84" y_offset="411.47" spacing="-27.59">(</line> | |
| <line x_offset="180.6" y_offset="404.77" spacing="-5.26">N</line> | |
| <line x_offset="195.84" y_offset="404.76" spacing="-11.96">input to the simulation and the broken line has</line> | |
| <line x_offset="0.0" y_offset="404.76" spacing="-11.97">Data labels represent the value of</line> | |
| <line x_offset="0.0" y_offset="388.8" spacing="3.99">unit gradient. A linear regression on the data has gradient 0.9994 with | |
| coefficient of</line> | |
| <line x_offset="84.36" y_offset="378.29" spacing="2.54">2</line> | |
| <line x_offset="78.72" y_offset="372.97" spacing="-6.64">r</line> | |
| <line x_offset="110.64" y_offset="372.97" spacing="-11.96">.</line> | |
| <line x_offset="0.0" y_offset="372.96" spacing="-11.96">determination, = 0 9997.</line> | |
| <line x_offset="23.4" y_offset="357.0" spacing="3.99">Histograms of the local process intensity arising from the simulations are | |
| plotted in</line> | |
| <line x_offset="66.36" y_offset="344.04" spacing="0.99">¯</line> | |
| <line x_offset="63.0" y_offset="341.05" spacing="-8.97">N ,</line> | |
| <line x_offset="0.0" y_offset="341.04" spacing="-11.96">Figure 3 for = 10 20 and 50. To approximate the probability density of the | |
| data, the</line> | |
| <line x_offset="0.0" y_offset="325.08" spacing="3.99">heights of the bars are given by the frequency divided by the bin width. The | |
| solid lines</line> | |
| <line x_offset="0.0" y_offset="309.12" spacing="3.99">represent the probability densities of skew-normal distributions fitted by a | |
| least-squares</line> | |
| <line x_offset="0.0" y_offset="293.16" spacing="3.99">method to the cumulative data. The probability density of the skew-normal | |
| distribution</line> | |
| <line x_offset="0.0" y_offset="277.32" spacing="3.87">is [38]</line> | |
| <line x_offset="145.56" y_offset="262.03" spacing="9.32">2</line> | |
| <line x_offset="126.36" y_offset="260.93" spacing="-3.29">τ m(</line> | |
| <line x_offset="123.48" y_offset="259.51" spacing="-4.55">(</line> | |
| <line x_offset="142.68" y_offset="259.51" spacing="-5.98">)</line> | |
| <line x_offset="206.4" y_offset="257.28" spacing="-3.53">(</line> | |
| <line x_offset="130.44" y_offset="255.33" spacing="-5.37">−</line> | |
| <line x_offset="182.4" y_offset="255.29" spacing="-7.92">α (m τ)</line> | |
| <line x_offset="138.0" y_offset="253.03" spacing="-3.72">2</line> | |
| <line x_offset="134.52" y_offset="252.77" spacing="-3.97">s</line> | |
| <line x_offset="130.92" y_offset="251.35" spacing="-4.55">2</line> | |
| <line x_offset="199.56" y_offset="249.67" spacing="-9.45">−</line> | |
| <line x_offset="115.68" y_offset="249.31" spacing="-10.77">−</line> | |
| <line x_offset="110.28" y_offset="248.53" spacing="-11.18">e</line> | |
| <line x_offset="153.48" y_offset="248.52" spacing="-11.96">erfc</line> | |
| <line x_offset="190.92" y_offset="245.47" spacing="-8.08">√</line> | |
| <line x_offset="198.0" y_offset="244.49" spacing="-6.99">2s</line> | |
| <line x_offset="215.28" y_offset="236.63" spacing="-28.98">(</line> | |
| <line x_offset="174.0" y_offset="236.63" spacing="-36.85">(</line> | |
| <line x_offset="71.76" y_offset="236.05" spacing="-11.38">g τ</line> | |
| <line x_offset="231.48" y_offset="236.05" spacing="-11.95">,</line> | |
| <line x_offset="77.88" y_offset="236.04" spacing="-11.96">( ) =</line> | |
| <line x_offset="423.96" y_offset="236.04" spacing="-11.97">(19)</line> | |
| <line x_offset="150.12" y_offset="227.92" spacing="-12.33">√</line> | |
| <line x_offset="168.0" y_offset="226.57" spacing="-10.6">π s</line> | |
| <line x_offset="160.08" y_offset="226.56" spacing="-11.96">2</line> | |
| <line x_offset="82.68" y_offset="210.83" spacing="-21.11">(</line> | |
| <line x_offset="57.48" y_offset="206.05" spacing="-7.18">ζ</line> | |
| <line x_offset="0.0" y_offset="206.04" spacing="-11.96">where erfc( ) is the complementary error function. The mean, variance and | |
| skewness</line> | |
| <line x_offset="0.0" y_offset="190.08" spacing="3.99">are given by</line> | |
| <line x_offset="155.04" y_offset="171.4" spacing="-1.77">√</line> | |
| <line x_offset="172.8" y_offset="169.93" spacing="-10.48">α s</line> | |
| <line x_offset="165.0" y_offset="169.92" spacing="-11.96">2</line> | |
| <line x_offset="114.84" y_offset="161.89" spacing="-3.93">m</line> | |
| <line x_offset="71.76" y_offset="161.88" spacing="-11.96">τ¯</line> | |
| <line x_offset="102.36" y_offset="161.88" spacing="-11.97">= +</line> | |
| <line x_offset="197.52" y_offset="156.77" spacing="-2.86">2</line> | |
| <line x_offset="159.72" y_offset="153.76" spacing="-17.44">√</line> | |
| <line x_offset="140.64" y_offset="152.44" spacing="-19.12">√</line> | |
| <line x_offset="150.6" y_offset="152.41" spacing="-11.92">π</line> | |
| <line x_offset="189.96" y_offset="152.41" spacing="-11.96">α</line> | |
| <line x_offset="169.68" y_offset="152.4" spacing="-11.96">1 +</line> | |
| <line x_offset="178.08" y_offset="141.41" spacing="3.02">2</line> | |
| <line x_offset="170.52" y_offset="136.09" spacing="-6.64">α</line> | |
| <line x_offset="162.6" y_offset="136.08" spacing="-11.96">2</line> | |
| <line x_offset="215.52" y_offset="133.85" spacing="-5.74">2</line> | |
| <line x_offset="78.84" y_offset="133.85" spacing="-7.97">2</line> | |
| <line x_offset="210.0" y_offset="127.93" spacing="-6.04">s</line> | |
| <line x_offset="71.76" y_offset="127.93" spacing="-11.95">σ τ</line> | |
| <line x_offset="125.64" y_offset="127.92" spacing="-11.96">1</line> | |
| <line x_offset="83.52" y_offset="127.92" spacing="-11.97">( ) =</line> | |
| <line x_offset="188.76" y_offset="124.25" spacing="-4.3">2</line> | |
| <line x_offset="147.24" y_offset="119.77" spacing="-7.48">π</line> | |
| <line x_offset="181.2" y_offset="119.77" spacing="-11.96">α</line> | |
| <line x_offset="156.36" y_offset="119.76" spacing="-11.96">(1 + )</line> | |
| <line x_offset="199.2" y_offset="119.63" spacing="-36.71">(</line> | |
| <line x_offset="116.76" y_offset="119.63" spacing="-36.85">(</line> | |
| <line x_offset="134.16" y_offset="119.44" spacing="-20.26">−</line> | |
| <line x_offset="189.72" y_offset="107.09" spacing="4.39">3</line> | |
| <line x_offset="125.64" y_offset="103.12" spacing="-16.48">√</line> | |
| <line x_offset="88.32" y_offset="102.71" spacing="-36.43">(</line> | |
| <line x_offset="168.48" y_offset="101.77" spacing="-11.02">π α</line> | |
| <line x_offset="135.6" y_offset="101.76" spacing="-11.96">2 (4 )</line> | |
| <line x_offset="71.76" y_offset="93.73" spacing="-3.93">γ τ</line> | |
| <line x_offset="82.56" y_offset="93.72" spacing="-11.96">( ) =</line> | |
| <line x_offset="156.6" y_offset="93.28" spacing="-20.01">−</line> | |
| <line x_offset="77.88" y_offset="92.93" spacing="-7.61">1</line> | |
| <line x_offset="198.72" y_offset="92.59" spacing="-5.64">3</line> | |
| <line x_offset="188.28" y_offset="86.57" spacing="-1.95">2</line> | |
| <line x_offset="198.72" y_offset="86.23" spacing="-5.64">2</line> | |
| <line x_offset="120.6" y_offset="82.09" spacing="-7.82">π π</line> | |
| <line x_offset="180.72" y_offset="82.09" spacing="-11.96">α</line> | |
| <line x_offset="116.04" y_offset="82.08" spacing="-11.96">( + (</line> | |
| <line x_offset="168.24" y_offset="82.08" spacing="-11.97">2) )</line> | |
| <line x_offset="156.36" y_offset="73.6" spacing="-11.97">−</line> | |
| <line x_offset="87.36" y_offset="68.51" spacing="-31.75">(</line> | |
| <line x_offset="0.0" y_offset="63.84" spacing="-7.3">respectively.</line> | |
| <line x_offset="23.4" y_offset="47.88" spacing="3.99">As anticipated from the skewness values plotted in Figure 2, the | |
| distributions</line> | |
| <line x_offset="384.48" y_offset="34.92" spacing="0.99">¯</line> | |
| <line x_offset="381.12" y_offset="31.93" spacing="-8.97">N</line> | |
| <line x_offset="0.0" y_offset="31.92" spacing="-11.96">are increasingly well approximated by a Gaussian probability density | |
| as</line> | |
| <line x_offset="396.48" y_offset="31.92" spacing="-11.97">increases.</line> | |
| <line x_offset="37.8" y_offset="18.96" spacing="0.99">¯</line> | |
| <line x_offset="34.44" y_offset="15.97" spacing="-8.97">N</line> | |
| <line x_offset="0.0" y_offset="15.96" spacing="-11.96">When is greater than about 50, the skewness is negligible and the Gaussian | |
| can be</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.99">assumed to describe the distribution of local process intensity well.</line> | |
| <component x="72.0" y="58.44" width="444.82" height="448.65" page="7" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.08" year_ratio="0.0" cap_ratio="0.03" | |
| name_ratio="0.16666666666666666" word_count="372" lateness="0.5333333333333333" reference_score="9.47"> | |
| <line x_offset="0.0" y_offset="415.2" spacing="0.0">We proceed assuming the distribution of local process intensity to be Gaussian | |
| with</line> | |
| <line x_offset="137.52" y_offset="399.92" spacing="4.42">cf.</line> | |
| <line x_offset="238.32" y_offset="399.92" spacing="-10.86">i.e.</line> | |
| <line x_offset="31.2" y_offset="399.37" spacing="-11.41">τ</line> | |
| <line x_offset="110.16" y_offset="399.37" spacing="-11.96">τ /x</line> | |
| <line x_offset="0.0" y_offset="399.36" spacing="-11.96">mean ¯ and variance ¯ ( Equation (16)),</line> | |
| <line x_offset="187.2" y_offset="383.83" spacing="9.56">2</line> | |
| <line x_offset="170.52" y_offset="382.8" spacing="-3.29">(</line> | |
| <line x_offset="161.88" y_offset="382.73" spacing="-4.15">x τ τ</line> | |
| <line x_offset="167.4" y_offset="381.31" spacing="-4.55">( ¯)</line> | |
| <line x_offset="130.56" y_offset="378.97" spacing="-9.62">x</line> | |
| <line x_offset="174.36" y_offset="377.13" spacing="-5.48">−</line> | |
| <line x_offset="177.0" y_offset="375.65" spacing="-2.74">τ</line> | |
| <line x_offset="172.08" y_offset="374.23" spacing="-4.55">2 ¯</line> | |
| <line x_offset="154.08" y_offset="371.23" spacing="-8.13">−</line> | |
| <line x_offset="71.76" y_offset="370.93" spacing="-11.66">g τ</line> | |
| <line x_offset="148.68" y_offset="370.93" spacing="-11.96">e</line> | |
| <line x_offset="200.88" y_offset="370.93" spacing="-11.96">.</line> | |
| <line x_offset="77.88" y_offset="370.92" spacing="-11.96">( ) =</line> | |
| <line x_offset="423.96" y_offset="370.92" spacing="-11.97">(20)</line> | |
| <line x_offset="109.08" y_offset="363.23" spacing="-29.15">(</line> | |
| <line x_offset="130.08" y_offset="362.65" spacing="-11.38">π τ</line> | |
| <line x_offset="122.16" y_offset="362.64" spacing="-11.96">2 ¯</line> | |
| <line x_offset="160.56" y_offset="349.13" spacing="5.54">0</line> | |
| <line x_offset="274.32" y_offset="346.97" spacing="-5.81">6</line> | |
| <line x_offset="82.68" y_offset="345.71" spacing="-35.58">(</line> | |
| <line x_offset="95.64" y_offset="341.65" spacing="-7.9">τ x ></line> | |
| <line x_offset="175.8" y_offset="341.65" spacing="-11.96">g τ τ <</line> | |
| <line x_offset="0.0" y_offset="341.64" spacing="-11.96">We note that for ¯</line> | |
| <line x_offset="131.76" y_offset="341.64" spacing="-11.97">20,</line> | |
| <line x_offset="181.92" y_offset="341.64" spacing="-11.97">( ) d</line> | |
| <line x_offset="233.28" y_offset="341.64" spacing="-11.97">4 10 ; accordingly, truncation of the</line> | |
| <line x_offset="267.72" y_offset="341.35" spacing="-10.84">−</line> | |
| <line x_offset="242.88" y_offset="333.16" spacing="-12.26">×</line> | |
| <line x_offset="158.28" y_offset="332.83" spacing="-10.79">−∞</line> | |
| <line x_offset="152.64" y_offset="326.03" spacing="-30.04">(</line> | |
| <line x_offset="137.4" y_offset="325.69" spacing="-11.62">τ <</line> | |
| <line x_offset="0.0" y_offset="325.68" spacing="-11.96">distribution such that 0</line> | |
| <line x_offset="175.32" y_offset="325.68" spacing="-11.97">is unnecessary for practical purposes.</line> | |
| <line x_offset="124.68" y_offset="317.2" spacing="-11.97">≤ ∞</line> | |
| <line x_offset="186.72" y_offset="316.43" spacing="-36.07">( (</line> | |
| <line x_offset="23.4" y_offset="309.72" spacing="-5.26">From Equation (1), we expect the local average pore area to be</line> | |
| <line x_offset="137.52" y_offset="300.47" spacing="-27.59">(</line> | |
| <line x_offset="96.84" y_offset="294.61" spacing="-6.1">π</line> | |
| <line x_offset="114.96" y_offset="286.57" spacing="-3.91">.</line> | |
| <line x_offset="71.76" y_offset="286.57" spacing="-11.96">a</line> | |
| <line x_offset="423.96" y_offset="286.56" spacing="-11.96">(21)</line> | |
| <line x_offset="81.24" y_offset="286.56" spacing="-11.97">=</line> | |
| <line x_offset="101.16" y_offset="282.77" spacing="-4.18">2</line> | |
| <line x_offset="94.8" y_offset="278.29" spacing="-7.48">τ</line> | |
| <line x_offset="0.0" y_offset="263.04" spacing="3.29">Inevitably, when a network is partitioned into contiguous square zones, some | |
| polygons</line> | |
| <line x_offset="71.52" y_offset="261.35" spacing="-35.15">(</line> | |
| <line x_offset="95.04" y_offset="253.07" spacing="-28.57">(</line> | |
| <line x_offset="0.0" y_offset="247.08" spacing="-5.98">intersect the perimeter of the zone. The expected number of polygons | |
| intersecting the</line> | |
| <line x_offset="398.04" y_offset="234.36" spacing="0.75">¯</line> | |
| <line x_offset="184.68" y_offset="231.13" spacing="-8.73">x</line> | |
| <line x_offset="327.0" y_offset="231.13" spacing="-11.96">n</line> | |
| <line x_offset="383.52" y_offset="231.13" spacing="-11.96">x/d</line> | |
| <line x_offset="429.72" y_offset="231.13" spacing="-11.96">x τ</line> | |
| <line x_offset="0.0" y_offset="231.12" spacing="-11.96">perimeter of a square zone of side can be approximated as</line> | |
| <line x_offset="361.2" y_offset="231.12" spacing="-11.97">= 4</line> | |
| <line x_offset="407.4" y_offset="231.12" spacing="-11.97">= 4 ¯</line> | |
| <line x_offset="333.96" y_offset="230.33" spacing="-7.18">perim</line> | |
| <line x_offset="361.92" y_offset="220.61" spacing="1.75">2</line> | |
| <line x_offset="410.64" y_offset="220.61" spacing="-7.97">2 2</line> | |
| <line x_offset="306.96" y_offset="215.17" spacing="-6.52">n</line> | |
| <line x_offset="355.32" y_offset="215.17" spacing="-11.96">x /a</line> | |
| <line x_offset="404.04" y_offset="215.17" spacing="-11.96">x τ /π</line> | |
| <line x_offset="0.0" y_offset="215.16" spacing="-11.96">and the expected number of polygons in the square is</line> | |
| <line x_offset="338.16" y_offset="215.16" spacing="-11.97">= ¯ = ¯ .</line> | |
| <line x_offset="313.92" y_offset="214.37" spacing="-7.18">area</line> | |
| <line x_offset="0.0" y_offset="199.32" spacing="3.08">So, the expected fraction of polygons intersecting the perimeter of the | |
| square is</line> | |
| <line x_offset="326.16" y_offset="188.69" spacing="2.66">1</line> | |
| <line x_offset="78.72" y_offset="183.37" spacing="-6.64">n /n</line> | |
| <line x_offset="166.08" y_offset="183.37" spacing="-11.96">π/ τ x</line> | |
| <line x_offset="252.48" y_offset="183.37" spacing="-11.96">τ</line> | |
| <line x_offset="360.96" y_offset="183.37" spacing="-11.96">x</line> | |
| <line x_offset="0.0" y_offset="183.36" spacing="-11.96">approximately</line> | |
| <line x_offset="142.8" y_offset="183.36" spacing="-11.97">= 4 (¯ ). When ¯ = 100 mm and = 1 mm this</line> | |
| <line x_offset="319.56" y_offset="183.07" spacing="-10.84">−</line> | |
| <line x_offset="85.68" y_offset="182.57" spacing="-7.47">perim area</line> | |
| <line x_offset="63.84" y_offset="167.41" spacing="3.2">.</line> | |
| <line x_offset="114.12" y_offset="167.41" spacing="-11.96">n ></line> | |
| <line x_offset="0.0" y_offset="167.4" spacing="-11.96">fraction is 0 125 and</line> | |
| <line x_offset="157.8" y_offset="167.4" spacing="-11.97">3000 so it is reasonable to assume that Equation (21)</line> | |
| <line x_offset="121.08" y_offset="166.61" spacing="-7.18">area</line> | |
| <line x_offset="0.0" y_offset="151.44" spacing="3.2">provides a good measure of the local average polygon area.</line> | |
| <line x_offset="23.4" y_offset="135.48" spacing="3.99">The probability density of local average pore area is obtained by a simple | |
| variable</line> | |
| <line x_offset="0.0" y_offset="119.52" spacing="3.99">transform of Equation (20):</line> | |
| <line x_offset="122.28" y_offset="100.33" spacing="7.23">τ</line> | |
| <line x_offset="115.8" y_offset="100.32" spacing="-11.96">d˜</line> | |
| <line x_offset="137.88" y_offset="92.29" spacing="-3.93">g</line> | |
| <line x_offset="160.44" y_offset="92.29" spacing="-11.96">π/a</line> | |
| <line x_offset="144.0" y_offset="92.28" spacing="-11.96">(</line> | |
| <line x_offset="173.52" y_offset="92.28" spacing="-11.97">˜)</line> | |
| <line x_offset="71.76" y_offset="92.28" spacing="-11.97">p a(˜) =</line> | |
| <line x_offset="129.84" y_offset="84.35" spacing="-28.91">(</line> | |
| <line x_offset="110.64" y_offset="84.35" spacing="-36.85">(</line> | |
| <line x_offset="122.4" y_offset="84.13" spacing="-11.74">a</line> | |
| <line x_offset="115.92" y_offset="84.12" spacing="-11.96">d˜</line> | |
| <line x_offset="148.56" y_offset="77.87" spacing="-30.59">(</line> | |
| <line x_offset="129.84" y_offset="77.27" spacing="-36.25">(</line> | |
| <line x_offset="110.64" y_offset="77.27" spacing="-36.85">(</line> | |
| <line x_offset="208.2" y_offset="71.47" spacing="-0.18">2</line> | |
| <line x_offset="179.64" y_offset="70.37" spacing="-3.13">π a τ</line> | |
| <line x_offset="165.24" y_offset="70.37" spacing="-4.23">x</line> | |
| <line x_offset="129.84" y_offset="70.07" spacing="-36.54">(</line> | |
| <line x_offset="110.64" y_offset="70.07" spacing="-36.85">(</line> | |
| <line x_offset="189.96" y_offset="69.81" spacing="-7.06">√</line> | |
| <line x_offset="173.64" y_offset="69.09" spacing="-6.6">√</line> | |
| <line x_offset="196.08" y_offset="68.95" spacing="-5.83">˜ ¯)</line> | |
| <line x_offset="170.76" y_offset="68.95" spacing="-5.98">(</line> | |
| <line x_offset="132.0" y_offset="66.61" spacing="-9.62">x</line> | |
| <line x_offset="184.2" y_offset="64.77" spacing="-5.48">−</line> | |
| <line x_offset="186.6" y_offset="63.17" spacing="-2.62">a τ</line> | |
| <line x_offset="129.84" y_offset="62.87" spacing="-36.54">(</line> | |
| <line x_offset="110.64" y_offset="62.87" spacing="-36.85">(</line> | |
| <line x_offset="181.68" y_offset="61.75" spacing="-4.86">2 ˜ ¯</line> | |
| <line x_offset="157.44" y_offset="58.75" spacing="-8.13">−</line> | |
| <line x_offset="221.88" y_offset="58.45" spacing="-11.66">.</line> | |
| <line x_offset="152.04" y_offset="58.45" spacing="-11.96">e</line> | |
| <line x_offset="423.96" y_offset="58.44" spacing="-11.96">(22)</line> | |
| <line x_offset="96.12" y_offset="58.44" spacing="-11.97">=</line> | |
| <line x_offset="135.84" y_offset="54.77" spacing="-4.3">3</line> | |
| <line x_offset="108.6" y_offset="50.75" spacing="-32.82">(</line> | |
| <line x_offset="129.72" y_offset="50.29" spacing="-11.5">a τ</line> | |
| <line x_offset="121.8" y_offset="50.28" spacing="-11.96">8 ˜ ¯</line> | |
| <line x_offset="23.4" y_offset="31.8" spacing="6.51">The probability density given by Equation (22) is plotted for a range of | |
| process</line> | |
| <line x_offset="59.04" y_offset="15.85" spacing="3.99">τ</line> | |
| <line x_offset="200.88" y_offset="15.85" spacing="-11.96">x</line> | |
| <line x_offset="0.0" y_offset="15.84" spacing="-11.96">intensities, ¯ and scales of inspection, in Figure 4. As anticipated, | |
| the distribution</line> | |
| <line x_offset="291.96" y_offset="0.01" spacing="3.87">τ</line> | |
| <line x_offset="329.76" y_offset="0.01" spacing="-11.96">x</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="-11.96">exhibits a positive skew and narrows with increasing ¯ and . A | |
| consequence of</line> | |
| <component x="72.0" y="38.88" width="444.82" height="427.17" page="8" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="36.85" font="IWMANX+CMEX10" letter_ratio="0.13" year_ratio="0.0" cap_ratio="0.1" | |
| name_ratio="0.11210762331838565" word_count="446" lateness="0.6" reference_score="11.66"> | |
| <line x_offset="0.0" y_offset="636.13" spacing="0.0">neglecting skewness and using the approximation given by Equation (18) for | |
| the variance</line> | |
| <line x_offset="0.0" y_offset="620.17" spacing="3.99">of local average process intensity is that the expected local average pore | |
| area obtained as</line> | |
| <line x_offset="228.36" y_offset="609.54" spacing="2.66">2</line> | |
| <line x_offset="363.0" y_offset="609.54" spacing="-7.97">1</line> | |
| <line x_offset="30.12" y_offset="606.08" spacing="-7.67">∞</line> | |
| <line x_offset="0.0" y_offset="604.22" spacing="-10.1">a</line> | |
| <line x_offset="209.04" y_offset="604.22" spacing="-11.96">π/τ</line> | |
| <line x_offset="294.72" y_offset="604.22" spacing="-11.96">τ</line> | |
| <line x_offset="394.56" y_offset="604.22" spacing="-11.96">x</line> | |
| <line x_offset="0.12" y_offset="604.21" spacing="-11.96">¯ =</line> | |
| <line x_offset="41.04" y_offset="604.21" spacing="-11.97">a p a a˜ (˜) d˜ is slightly greater than ¯ though | |
| for ¯ 200 mm and</line> | |
| <line x_offset="417.36" y_offset="604.21" spacing="-11.97">1 mm</line> | |
| <line x_offset="356.4" y_offset="603.92" spacing="-10.84">−</line> | |
| <line x_offset="27.72" y_offset="601.02" spacing="-5.07">0</line> | |
| <line x_offset="304.56" y_offset="595.74" spacing="-15.16">≥</line> | |
| <line x_offset="404.64" y_offset="595.74" spacing="-20.44">≥</line> | |
| <line x_offset="22.08" y_offset="588.6" spacing="-29.71">(</line> | |
| <line x_offset="0.0" y_offset="588.25" spacing="-11.62">the error is less than 2 %. The variance of local average pore area, is | |
| given by</line> | |
| <line x_offset="126.48" y_offset="570.92" spacing="6.2">∞</line> | |
| <line x_offset="78.84" y_offset="568.38" spacing="-5.43">2</line> | |
| <line x_offset="171.48" y_offset="568.38" spacing="-7.97">2</line> | |
| <line x_offset="71.76" y_offset="562.47" spacing="-6.04">σ a</line> | |
| <line x_offset="140.04" y_offset="562.47" spacing="-11.96">a a p a a .</line> | |
| <line x_offset="83.52" y_offset="562.45" spacing="-11.96">(˜) =</line> | |
| <line x_offset="135.48" y_offset="562.45" spacing="-11.97">(˜ ¯) (˜) d˜</line> | |
| <line x_offset="423.96" y_offset="562.45" spacing="-11.97">(23)</line> | |
| <line x_offset="148.8" y_offset="553.98" spacing="-11.97">−</line> | |
| <line x_offset="114.6" y_offset="553.44" spacing="-36.31">(</line> | |
| <line x_offset="121.2" y_offset="552.66" spacing="-7.19">0</line> | |
| <line x_offset="0.0" y_offset="536.29" spacing="4.4">It has not been possible to obtain a closed form solution to this integral | |
| though an</line> | |
| <line x_offset="0.0" y_offset="520.45" spacing="3.87">analytic estimate can be obtained through consideration of Equations (5) and | |
| (21).</line> | |
| <line x_offset="0.0" y_offset="504.49" spacing="3.99">From Equation (5) we expect the local average process intensity to be</line> | |
| <line x_offset="107.64" y_offset="486.01" spacing="6.51">¯</line> | |
| <line x_offset="95.04" y_offset="482.9" spacing="-8.85">N l</line> | |
| <line x_offset="111.24" y_offset="482.1" spacing="-7.17">1</line> | |
| <line x_offset="121.08" y_offset="474.75" spacing="-4.6">.</line> | |
| <line x_offset="71.76" y_offset="474.75" spacing="-11.95">τ</line> | |
| <line x_offset="423.96" y_offset="474.73" spacing="-11.96">(24)</line> | |
| <line x_offset="81.48" y_offset="474.73" spacing="-11.97">=</line> | |
| <line x_offset="102.24" y_offset="466.59" spacing="-3.81">x</line> | |
| <line x_offset="98.04" y_offset="460.68" spacing="-30.94">(</line> | |
| <line x_offset="0.0" y_offset="450.61" spacing="-1.9">Substituting in Equation (21), we obtain</line> | |
| <line x_offset="72.0" y_offset="449.52" spacing="-35.75">(</line> | |
| <line x_offset="110.52" y_offset="436.5" spacing="5.05">2</line> | |
| <line x_offset="94.8" y_offset="431.19" spacing="-6.64">π x</line> | |
| <line x_offset="124.44" y_offset="431.17" spacing="-11.96">1</line> | |
| <line x_offset="144.0" y_offset="423.15" spacing="-3.93">,</line> | |
| <line x_offset="71.76" y_offset="423.15" spacing="-11.96">a</line> | |
| <line x_offset="423.96" y_offset="423.13" spacing="-11.96">(25)</line> | |
| <line x_offset="71.88" y_offset="423.13" spacing="-11.97">˜ =</line> | |
| <line x_offset="104.52" y_offset="419.58" spacing="-4.42">2</line> | |
| <line x_offset="100.68" y_offset="417.61" spacing="-10.0">¯</line> | |
| <line x_offset="130.2" y_offset="417.18" spacing="-7.54">2</line> | |
| <line x_offset="100.8" y_offset="414.51" spacing="-9.28">l</line> | |
| <line x_offset="119.64" y_offset="412.71" spacing="-10.16">N</line> | |
| <line x_offset="104.28" y_offset="412.62" spacing="-7.89">1</line> | |
| <line x_offset="0.0" y_offset="396.25" spacing="4.4">such that</line> | |
| <line x_offset="122.64" y_offset="390.6" spacing="-31.19">(</line> | |
| <line x_offset="155.04" y_offset="385.14" spacing="-2.51">2</line> | |
| <line x_offset="140.28" y_offset="383.22" spacing="-6.05">2</line> | |
| <line x_offset="124.68" y_offset="377.91" spacing="-6.64">π x</line> | |
| <line x_offset="78.84" y_offset="375.66" spacing="-5.73">2</line> | |
| <line x_offset="170.88" y_offset="375.66" spacing="-7.97">2</line> | |
| <line x_offset="202.56" y_offset="375.66" spacing="-7.97">2</line> | |
| <line x_offset="71.76" y_offset="369.75" spacing="-6.04">σ a</line> | |
| <line x_offset="163.8" y_offset="369.75" spacing="-11.95">σ /N .</line> | |
| <line x_offset="83.64" y_offset="369.73" spacing="-11.96">(˜) =</line> | |
| <line x_offset="175.68" y_offset="369.73" spacing="-11.97">(1 )</line> | |
| <line x_offset="423.96" y_offset="369.73" spacing="-11.97">(26)</line> | |
| <line x_offset="78.36" y_offset="367.86" spacing="-6.1">x</line> | |
| <line x_offset="170.4" y_offset="367.86" spacing="-7.97">x</line> | |
| <line x_offset="134.4" y_offset="366.3" spacing="-6.41">2</line> | |
| <line x_offset="130.56" y_offset="364.33" spacing="-10.0">¯</line> | |
| <line x_offset="114.6" y_offset="361.44" spacing="-33.95">(</line> | |
| <line x_offset="146.28" y_offset="361.44" spacing="-36.85">(</line> | |
| <line x_offset="130.68" y_offset="361.22" spacing="-11.74">l</line> | |
| <line x_offset="134.16" y_offset="359.22" spacing="-5.97">1</line> | |
| <line x_offset="195.0" y_offset="347.52" spacing="-25.14">(</line> | |
| <line x_offset="333.0" y_offset="344.94" spacing="-5.39">2</line> | |
| <line x_offset="364.68" y_offset="344.94" spacing="-7.97">2</line> | |
| <line x_offset="54.84" y_offset="339.62" spacing="-6.64">N</line> | |
| <line x_offset="236.88" y_offset="339.62" spacing="-11.96">P N > σ /N</line> | |
| <line x_offset="23.4" y_offset="339.61" spacing="-11.96">Now, is a Poisson variable and since ( = 0) 0, (1 ) is undefined.</line> | |
| <line x_offset="332.52" y_offset="337.62" spacing="-5.98">x</line> | |
| <line x_offset="114.24" y_offset="326.65" spacing="-1.0">¯</line> | |
| <line x_offset="110.88" y_offset="323.67" spacing="-8.97">N</line> | |
| <line x_offset="272.88" y_offset="323.67" spacing="-11.96">P N</line> | |
| <line x_offset="0.0" y_offset="323.65" spacing="-11.96">Typically, we expect</line> | |
| <line x_offset="126.12" y_offset="323.65" spacing="-11.97">to be sufficiently large that ( = 0) is negligible such that</line> | |
| <line x_offset="57.84" y_offset="317.4" spacing="-30.59">(</line> | |
| <line x_offset="253.56" y_offset="317.4" spacing="-36.85">(</line> | |
| <line x_offset="357.0" y_offset="317.4" spacing="-36.85">(</line> | |
| <line x_offset="9.24" y_offset="307.7" spacing="-2.26">< /N</line> | |
| <line x_offset="0.0" y_offset="307.69" spacing="-11.96">0 1</line> | |
| <line x_offset="60.0" y_offset="307.69" spacing="-11.97">1. A convenient approximation to the discrete Poisson probability | |
| function</line> | |
| <line x_offset="289.56" y_offset="301.44" spacing="-30.59">(</line> | |
| <line x_offset="47.4" y_offset="299.22" spacing="-18.22">≤</line> | |
| <line x_offset="423.96" y_offset="246.37" spacing="40.88">(27)</line> | |
| <line x_offset="423.96" y_offset="147.49" spacing="86.91">(28)</line> | |
| <line x_offset="339.84" y_offset="81.37" spacing="54.15">4)</line> | |
| <line x_offset="423.96" y_offset="58.57" spacing="10.83">(29)</line> | |
| <line x_offset="423.96" y_offset="8.41" spacing="38.19">(30)</line> | |
| <line x_offset="0.0" y_offset="291.85" spacing="-295.41">is the probability density of a Gamma distributed continuous random | |
| variable with</line> | |
| <line x_offset="36.36" y_offset="285.6" spacing="-30.59">(</line> | |
| <line x_offset="0.0" y_offset="275.89" spacing="-2.26">variance equal to the mean. This probability density is given by</line> | |
| <line x_offset="128.16" y_offset="263.77" spacing="6.36">(</line> | |
| <line x_offset="149.04" y_offset="261.78" spacing="-5.98">¯</line> | |
| <line x_offset="146.64" y_offset="259.74" spacing="-5.93">N 1</line> | |
| <line x_offset="126.0" y_offset="259.74" spacing="-7.97">N</line> | |
| <line x_offset="114.0" y_offset="254.42" spacing="-6.64">e N</line> | |
| <line x_offset="154.32" y_offset="254.12" spacing="-10.82">−</line> | |
| <line x_offset="119.4" y_offset="254.12" spacing="-11.13">−</line> | |
| <line x_offset="174.6" y_offset="246.39" spacing="-4.22">,</line> | |
| <line x_offset="71.76" y_offset="246.39" spacing="-11.96">q N</line> | |
| <line x_offset="77.4" y_offset="246.37" spacing="-11.96">( ) =</line> | |
| <line x_offset="141.48" y_offset="240.97" spacing="-6.57">¯</line> | |
| <line x_offset="138.12" y_offset="237.87" spacing="-8.85">N</line> | |
| <line x_offset="126.24" y_offset="237.85" spacing="-11.96">Γ( )</line> | |
| <line x_offset="139.08" y_offset="232.2" spacing="-31.19">(</line> | |
| <line x_offset="84.84" y_offset="224.16" spacing="-28.81">(</line> | |
| <line x_offset="230.04" y_offset="221.58" spacing="-5.39">2</line> | |
| <line x_offset="185.52" y_offset="216.27" spacing="-6.64">ν /N</line> | |
| <line x_offset="0.0" y_offset="216.25" spacing="-11.96">such that the probability density of = 1</line> | |
| <line x_offset="238.68" y_offset="216.25" spacing="-11.97">is given by</line> | |
| <line x_offset="222.48" y_offset="194.04" spacing="-14.63">(</line> | |
| <line x_offset="122.4" y_offset="193.47" spacing="-11.38">N</line> | |
| <line x_offset="115.92" y_offset="193.45" spacing="-11.96">d</line> | |
| <line x_offset="71.76" y_offset="185.43" spacing="-3.93">r ν</line> | |
| <line x_offset="142.2" y_offset="185.43" spacing="-11.96">q ν</line> | |
| <line x_offset="77.4" y_offset="185.41" spacing="-11.96">( ) =</line> | |
| <line x_offset="147.84" y_offset="185.41" spacing="-11.97">( )</line> | |
| <line x_offset="110.76" y_offset="181.08" spacing="-32.51">(</line> | |
| <line x_offset="134.16" y_offset="181.08" spacing="-36.85">(</line> | |
| <line x_offset="124.44" y_offset="177.27" spacing="-8.14">ν</line> | |
| <line x_offset="117.96" y_offset="177.25" spacing="-11.96">d</line> | |
| <line x_offset="110.76" y_offset="173.88" spacing="-33.47">(</line> | |
| <line x_offset="134.16" y_offset="173.88" spacing="-36.85">(</line> | |
| <line x_offset="125.4" y_offset="171.24" spacing="-34.21">(</line> | |
| <line x_offset="110.76" y_offset="166.8" spacing="-32.41">(</line> | |
| <line x_offset="134.16" y_offset="166.8" spacing="-36.85">(</line> | |
| <line x_offset="174.24" y_offset="162.9" spacing="-4.07">¯</line> | |
| <line x_offset="130.32" y_offset="161.0" spacing="-9.23">√</line> | |
| <line x_offset="137.4" y_offset="160.86" spacing="-7.83">ν (1+N/2)</line> | |
| <line x_offset="121.92" y_offset="160.86" spacing="-7.97">1/</line> | |
| <line x_offset="110.76" y_offset="159.6" spacing="-35.58">(</line> | |
| <line x_offset="134.16" y_offset="159.6" spacing="-36.85">(</line> | |
| <line x_offset="144.6" y_offset="155.55" spacing="-7.9">ν</line> | |
| <line x_offset="109.92" y_offset="155.55" spacing="-11.96">e</line> | |
| <line x_offset="151.08" y_offset="155.24" spacing="-10.82">−</line> | |
| <line x_offset="115.32" y_offset="155.24" spacing="-11.13">−</line> | |
| <line x_offset="110.76" y_offset="152.4" spacing="-34.0">(</line> | |
| <line x_offset="134.16" y_offset="152.4" spacing="-36.85">(</line> | |
| <line x_offset="96.24" y_offset="147.49" spacing="-7.06">=</line> | |
| <line x_offset="156.48" y_offset="141.97" spacing="-6.45">¯</line> | |
| <line x_offset="153.12" y_offset="138.99" spacing="-8.97">N</line> | |
| <line x_offset="133.32" y_offset="138.97" spacing="-11.96">2 Γ( )</line> | |
| <line x_offset="189.36" y_offset="121.93" spacing="5.07">¯</line> | |
| <line x_offset="229.56" y_offset="121.93" spacing="-11.97">¯</line> | |
| <line x_offset="139.92" y_offset="118.83" spacing="-8.85">ν / N N</line> | |
| <line x_offset="0.0" y_offset="34.93" spacing="71.92">Such that</line> | |
| <line x_offset="0.0" y_offset="118.81" spacing="-95.85">The distribution has mean ¯ = 1 (</line> | |
| <line x_offset="211.2" y_offset="118.81" spacing="-11.97">1)(</line> | |
| <line x_offset="251.4" y_offset="118.81" spacing="-11.97">2) and variance</line> | |
| <line x_offset="199.2" y_offset="110.34" spacing="-11.97">− −</line> | |
| <line x_offset="176.04" y_offset="103.32" spacing="-29.83">(</line> | |
| <line x_offset="261.84" y_offset="103.32" spacing="-36.85">(</line> | |
| <line x_offset="253.92" y_offset="100.93" spacing="-9.58">¯</line> | |
| <line x_offset="242.76" y_offset="97.93" spacing="-8.97">4N</line> | |
| <line x_offset="275.88" y_offset="97.93" spacing="-11.97">10</line> | |
| <line x_offset="78.84" y_offset="95.82" spacing="-5.86">2</line> | |
| <line x_offset="122.04" y_offset="95.82" spacing="-7.97">2</line> | |
| <line x_offset="153.6" y_offset="95.82" spacing="-7.97">2</line> | |
| <line x_offset="71.76" y_offset="89.91" spacing="-6.04">σ ν σ /N</line> | |
| <line x_offset="145.92" y_offset="67.68" spacing="-14.62">(</line> | |
| <line x_offset="78.84" y_offset="14.34" spacing="45.37">2</line> | |
| <line x_offset="78.36" y_offset="6.42" spacing="-0.05">x</line> | |
| <line x_offset="71.76" y_offset="8.43" spacing="-13.96">σ a</line> | |
| <line x_offset="83.64" y_offset="8.41" spacing="-11.96">(˜)</line> | |
| <line x_offset="83.52" y_offset="89.89" spacing="-93.45">( ) = (1 ) =</line> | |
| <line x_offset="263.88" y_offset="89.46" spacing="-20.01">−</line> | |
| <line x_offset="220.32" y_offset="85.86" spacing="-4.37">2</line> | |
| <line x_offset="265.2" y_offset="85.86" spacing="-7.97">2</line> | |
| <line x_offset="187.92" y_offset="84.37" spacing="-10.48">¯</line> | |
| <line x_offset="232.92" y_offset="84.37" spacing="-11.97">¯</line> | |
| <line x_offset="277.8" y_offset="84.37" spacing="-11.97">¯</line> | |
| <line x_offset="318.0" y_offset="84.37" spacing="-11.97">¯</line> | |
| <line x_offset="209.88" y_offset="81.37" spacing="-8.97">1) (</line> | |
| <line x_offset="254.76" y_offset="81.37" spacing="-11.97">2) (</line> | |
| <line x_offset="299.64" y_offset="81.37" spacing="-11.97">3)(</line> | |
| <line x_offset="197.88" y_offset="72.9" spacing="-11.97">− − − −</line> | |
| <line x_offset="288.72" y_offset="61.69" spacing="-0.76">¯</line> | |
| <line x_offset="285.36" y_offset="58.59" spacing="-8.85">N</line> | |
| <line x_offset="267.48" y_offset="58.57" spacing="-11.96">for</line> | |
| <line x_offset="314.52" y_offset="58.57" spacing="-11.97">20</line> | |
| <line x_offset="299.28" y_offset="50.1" spacing="-11.97">(</line> | |
| <line x_offset="184.56" y_offset="81.39" spacing="-43.24">N N N N</line> | |
| <line x_offset="180.0" y_offset="81.37" spacing="-11.96">(</line> | |
| <line x_offset="191.52" y_offset="8.43" spacing="60.99">.</line> | |
| <line x_offset="190.56" y_offset="54.66" spacing="-54.21">5</line> | |
| <line x_offset="184.8" y_offset="66.73" spacing="-24.04">4</line> | |
| <line x_offset="183.36" y_offset="53.17" spacing="1.59">¯</line> | |
| <line x_offset="180.0" y_offset="50.19" spacing="-8.97">N</line> | |
| <line x_offset="166.2" y_offset="50.1" spacing="-20.36">≈</line> | |
| <line x_offset="177.0" y_offset="23.82" spacing="18.31">2</line> | |
| <line x_offset="168.24" y_offset="0.0" spacing="-13.02">(</line> | |
| <component x="72.0" y="53.03" width="444.84" height="648.1" page="9" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.09" year_ratio="0.0" cap_ratio="0.09" | |
| name_ratio="0.19883040935672514" word_count="171" lateness="0.6666666666666666" reference_score="11.38"> | |
| <line x_offset="88.32" y_offset="263.76" spacing="0.0">¯</line> | |
| <line x_offset="84.96" y_offset="260.77" spacing="-8.97">N</line> | |
| <line x_offset="0.0" y_offset="260.76" spacing="-11.96">Substituting for</line> | |
| <line x_offset="99.48" y_offset="260.76" spacing="-11.97">from Equation (5) yields</line> | |
| <line x_offset="130.92" y_offset="247.25" spacing="5.54">2</line> | |
| <line x_offset="137.52" y_offset="245.04" spacing="-9.76">¯</line> | |
| <line x_offset="137.52" y_offset="241.93" spacing="-8.85">l</line> | |
| <line x_offset="123.84" y_offset="241.93" spacing="-11.96">π</line> | |
| <line x_offset="115.92" y_offset="241.92" spacing="-11.96">4</line> | |
| <line x_offset="141.0" y_offset="241.13" spacing="-7.18">1</line> | |
| <line x_offset="78.84" y_offset="239.81" spacing="-6.65">2</line> | |
| <line x_offset="154.8" y_offset="233.89" spacing="-6.04">.</line> | |
| <line x_offset="71.76" y_offset="233.89" spacing="-11.96">σ a</line> | |
| <line x_offset="423.96" y_offset="233.88" spacing="-11.96">(31)</line> | |
| <line x_offset="83.52" y_offset="233.88" spacing="-11.97">(˜)</line> | |
| <line x_offset="135.96" y_offset="230.09" spacing="-4.18">5</line> | |
| <line x_offset="120.96" y_offset="225.61" spacing="-7.48">x τ</line> | |
| <line x_offset="130.2" y_offset="225.6" spacing="-11.96">¯</line> | |
| <line x_offset="102.12" y_offset="225.4" spacing="-20.25">≈</line> | |
| <line x_offset="73.2" y_offset="215.57" spacing="1.87">2</line> | |
| <line x_offset="30.84" y_offset="210.25" spacing="-6.64">a π/τ</line> | |
| <line x_offset="0.0" y_offset="210.24" spacing="-11.96">Since ¯ = ¯ , it follows that the coefficient of variation of | |
| local average pore area is</line> | |
| <line x_offset="0.0" y_offset="194.28" spacing="3.99">approximated by</line> | |
| <line x_offset="145.92" y_offset="176.4" spacing="5.91">¯</line> | |
| <line x_offset="146.04" y_offset="173.17" spacing="-8.73">l</line> | |
| <line x_offset="149.52" y_offset="172.37" spacing="-7.17">1</line> | |
| <line x_offset="71.76" y_offset="165.13" spacing="-4.72">CV a</line> | |
| <line x_offset="90.36" y_offset="165.12" spacing="-11.96">(˜) 2</line> | |
| <line x_offset="129.48" y_offset="159.83" spacing="-31.55">(</line> | |
| <line x_offset="142.56" y_offset="156.97" spacing="-9.1">x τ</line> | |
| <line x_offset="151.8" y_offset="156.96" spacing="-11.96">¯</line> | |
| <line x_offset="108.96" y_offset="156.64" spacing="-20.13">≈</line> | |
| <line x_offset="132.36" y_offset="145.08" spacing="-0.4">2</line> | |
| <line x_offset="156.84" y_offset="137.05" spacing="-3.93">.</line> | |
| <line x_offset="423.96" y_offset="137.04" spacing="-11.96">(32)</line> | |
| <line x_offset="122.76" y_offset="128.92" spacing="-12.33">√</line> | |
| <line x_offset="108.96" y_offset="128.56" spacing="-20.08">≈</line> | |
| <line x_offset="132.72" y_offset="127.93" spacing="-11.32">x τ</line> | |
| <line x_offset="141.96" y_offset="127.92" spacing="-11.96">¯</line> | |
| <line x_offset="0.0" y_offset="111.6" spacing="4.35">We observe that the influence of zone size and process intensity is coupled | |
| such that the</line> | |
| <line x_offset="345.84" y_offset="95.65" spacing="3.99">x τ</line> | |
| <line x_offset="0.0" y_offset="95.64" spacing="-11.96">coefficient of variation depends only on the dimensionless product, ¯. | |
| The coefficient</line> | |
| <line x_offset="0.0" y_offset="79.68" spacing="3.99">of variation of local average pore area, as calculated via numerical | |
| integration of</line> | |
| <line x_offset="0.0" y_offset="63.84" spacing="3.87">Equation (23) is plotted against mean process intensity in Figure 5. The | |
| solid lines</line> | |
| <line x_offset="0.0" y_offset="47.88" spacing="3.99">represent the approximation given by Equation (32). We note that Schweers | |
| and</line> | |
| <line x_offset="370.56" y_offset="15.96" spacing="19.95">CV a(˜) plotted</line> | |
| <line x_offset="0.0" y_offset="31.92" spacing="-27.93">Lo¨ffler [31] report a coefficient of variation of local flow velocity | |
| through a porous</line> | |
| <line x_offset="0.0" y_offset="15.96" spacing="3.99">nonwoven filter of 0.3 at the 0.5 mm scale, consistent with the values | |
| of</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.99">in Figure 5.</line> | |
| <component x="72.0" y="215.04" width="444.82" height="275.73" page="10" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.03" year_ratio="0.0" cap_ratio="0.0" name_ratio="0.2734375" | |
| word_count="128" lateness="0.6666666666666666" reference_score="8.75"> | |
| <line x_offset="0.0" y_offset="127.56" spacing="0.0">Although the voids generated by random fiber processes are irregular convex | |
| polygons,</line> | |
| <line x_offset="0.0" y_offset="111.6" spacing="3.99">it is often convenient to characterize their dimensions by an equivalent | |
| diameter, rather</line> | |
| <line x_offset="0.0" y_offset="95.64" spacing="3.99">than by area. A good candidate for such a measure is the equivalent diameter | |
| determined</line> | |
| <line x_offset="0.0" y_offset="79.68" spacing="3.99">from the hydraulic radius and defined as the ratio of the area of a polygon | |
| to its</line> | |
| <line x_offset="0.0" y_offset="63.72" spacing="3.99">perimeter. Despite the established utility of this measure, to calculate it | |
| for our system</line> | |
| <line x_offset="0.0" y_offset="47.88" spacing="3.87">we would require knowledge of the joint probability density of polygon | |
| perimeter and</line> | |
| <line x_offset="0.0" y_offset="31.92" spacing="3.99">area, which is unknown. Two alternative measures have been employed | |
| previously: the</line> | |
| <line x_offset="0.0" y_offset="15.96" spacing="3.99">diameter of the largest circle that can be inscribed within a polygon [5, 12] | |
| and the</line> | |
| <line x_offset="0.0" y_offset="0.0" spacing="3.99">diameter of a circle with the same area as a polygon [1, 20]. The expected | |
| diameter</line> | |
| <component x="72.0" y="31.56" width="444.77" height="139.53" page="10" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.15" year_ratio="0.0" cap_ratio="0.11" | |
| name_ratio="0.10526315789473684" word_count="57" lateness="0.7333333333333333" reference_score="12.78"> | |
| <line x_offset="147.48" y_offset="137.79" spacing="0.0">x</line> | |
| <line x_offset="116.04" y_offset="137.77" spacing="-11.96">1</line> | |
| <line x_offset="73.8" y_offset="132.85" spacing="-7.05">˜</line> | |
| <line x_offset="71.76" y_offset="129.63" spacing="-8.73">d</line> | |
| <line x_offset="126.72" y_offset="129.61" spacing="-11.96">=</line> | |
| <line x_offset="102.12" y_offset="129.61" spacing="-11.97">=</line> | |
| <line x_offset="115.8" y_offset="121.47" spacing="-3.81">τ</line> | |
| <line x_offset="140.28" y_offset="119.31" spacing="-9.8">N l</line> | |
| <line x_offset="156.48" y_offset="118.5" spacing="-7.17">1</line> | |
| <line x_offset="122.4" y_offset="109.38" spacing="1.15">2</line> | |
| <line x_offset="115.8" y_offset="104.07" spacing="-6.64">x</line> | |
| <line x_offset="78.84" y_offset="101.94" spacing="-5.85">2</line> | |
| <line x_offset="137.4" y_offset="101.94" spacing="-7.97">2</line> | |
| <line x_offset="90.24" y_offset="99.13" spacing="-9.16">˜</line> | |
| <line x_offset="143.28" y_offset="97.08" spacing="-34.79">(</line> | |
| <line x_offset="116.04" y_offset="96.24" spacing="-36.01">(</line> | |
| <line x_offset="71.76" y_offset="96.03" spacing="-11.74">σ d</line> | |
| <line x_offset="130.32" y_offset="96.03" spacing="-11.96">σ /N</line> | |
| <line x_offset="83.64" y_offset="96.01" spacing="-11.96">( ) =</line> | |
| <line x_offset="142.32" y_offset="96.01" spacing="-11.97">(1 )</line> | |
| <line x_offset="78.36" y_offset="94.02" spacing="-5.98">x</line> | |
| <line x_offset="136.92" y_offset="94.02" spacing="-7.97">x</line> | |
| <line x_offset="120.96" y_offset="92.94" spacing="-6.89">2</line> | |
| <line x_offset="117.24" y_offset="87.75" spacing="-6.76">l</line> | |
| <line x_offset="120.72" y_offset="85.86" spacing="-6.09">1</line> | |
| <line x_offset="161.52" y_offset="73.8" spacing="-24.78">(</line> | |
| <line x_offset="0.0" y_offset="63.25" spacing="-1.42">Again approximating the Poisson distribution for</line> | |
| <line x_offset="136.92" y_offset="30.25" spacing="21.03">1</line> | |
| <line x_offset="156.6" y_offset="22.23" spacing="-3.93">,</line> | |
| <line x_offset="142.8" y_offset="19.38" spacing="-5.13">3</line> | |
| <line x_offset="118.44" y_offset="13.74" spacing="-14.8">≈</line> | |
| <line x_offset="132.24" y_offset="10.95" spacing="-9.16">N</line> | |
| <line x_offset="0.0" y_offset="47.29" spacing="-48.31">variance equal to the mean, we obtain</line> | |
| <line x_offset="78.84" y_offset="28.14" spacing="11.18">2</line> | |
| <line x_offset="78.36" y_offset="20.22" spacing="-0.05">x</line> | |
| <line x_offset="71.76" y_offset="22.23" spacing="-13.96">σ /N</line> | |
| <line x_offset="83.64" y_offset="22.21" spacing="-11.96">(1 )</line> | |
| <line x_offset="102.84" y_offset="0.0" spacing="-14.63">(</line> | |
| <component x="72.0" y="40.31" width="261.7" height="149.74" page="11" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="36.85" font="IWMANX+CMEX10" letter_ratio="0.33" year_ratio="0.0" cap_ratio="0.0" name_ratio="0" word_count="2" | |
| lateness="0.7333333333333333" reference_score="9.9"> | |
| <line x_offset="0.0" y_offset="22.23" spacing="2.67">N</line> | |
| <line x_offset="3.0" y_offset="0.0" spacing="-14.62">(</line> | |
| <component x="339.36" y="81.35" width="9.64" height="36.85" page="11" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="11.97" font="IKTHOC+CMR12" letter_ratio="0.0" year_ratio="0.0" cap_ratio="0.0" name_ratio="0.2" word_count="5" | |
| lateness="0.7333333333333333" reference_score="9.35"> | |
| <line x_offset="0.0" y_offset="0.0" spacing="0.0">by a gamma distribution with</line> | |
| <component x="355.8" y="103.56" width="160.81" height="11.97" page="11" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="4.02" font="FJIWZS+Times-Roman" letter_ratio="1.0" year_ratio="0.0" cap_ratio="0.0" name_ratio="0" | |
| word_count="1" lateness="0.8" reference_score="5.15"> | |
| <line x_offset="0.0" y_offset="0.0" spacing="0.0">0.12</line> | |
| <component x="197.21" y="698.64" width="10.05" height="4.02" page="12" page_width="612.0" page_height="792.0"/> | |
| </section> | |
| <section line_height="9.96" font="KAAPYP+CMR10" letter_ratio="0.29" year_ratio="0.02" cap_ratio="0.22" | |
| name_ratio="0.1544811320754717" word_count="848" lateness="0.9333333333333333" reference_score="19.89"> | |
| <line x_offset="344.28" y_offset="311.29" spacing="0.5">Consolidation of the</line> | |
| <line x_offset="5.04" y_offset="310.8" spacing="-9.47">[1] H. Corte and E.H. Lloyd. Fluid flow through paper and sheet structure. | |
| In</line> | |
| <line x_offset="30.48" y_offset="298.33" spacing="3.5">Paper Web</line> | |
| <line x_offset="91.8" y_offset="298.29" spacing="-9.14">Trans. IIIrd Fund. Res. Symp.</line> | |
| <line x_offset="231.72" y_offset="297.84" spacing="-9.51">(F. Bolam, ed.), pp981-1009, BPBMA, London,</line> | |
| <line x_offset="30.48" y_offset="284.88" spacing="3.0">1966.</line> | |
| <line x_offset="5.04" y_offset="271.92" spacing="3.0">[2] C.T.J. Dodson and W.W. Sampson. The effect of paper formation and | |
| grammage on its pore size</line> | |
| <line x_offset="165.84" y_offset="259.45" spacing="3.5">22</line> | |
| <line x_offset="87.48" y_offset="259.41" spacing="-9.14">J. Pulp Pap. Sci.</line> | |
| <line x_offset="30.48" y_offset="258.96" spacing="-9.51">distribution.</line> | |
| <line x_offset="177.36" y_offset="258.96" spacing="-9.96">(5):J165-J169, 1996.</line> | |
| <line x_offset="5.04" y_offset="246.0" spacing="3.0">[3] M.S. Abdel-Ghani and G.A. Davies. Simulation of non-woven fibre mats and | |
| the application to</line> | |
| <line x_offset="78.72" y_offset="233.61" spacing="3.22">Chem. Eng. Sci.</line> | |
| <line x_offset="30.48" y_offset="233.16" spacing="-9.51">coalescers.</line> | |
| <line x_offset="153.48" y_offset="233.16" spacing="-9.96">40(1):117-129, 1985.</line> | |
| <line x_offset="416.76" y_offset="220.65" spacing="3.34">Chem.</line> | |
| <line x_offset="5.04" y_offset="220.2" spacing="-9.51">[4] H.W. Piekaar and L.A. Clarenburg. Aerosol filters-Pore size | |
| distribution in fibrous filters.</line> | |
| <line x_offset="73.56" y_offset="207.73" spacing="3.5">22</line> | |
| <line x_offset="30.48" y_offset="207.69" spacing="-9.14">Eng. Sci.</line> | |
| <line x_offset="85.08" y_offset="207.24" spacing="-9.51">():1399-1408, 1967.</line> | |
| <line x_offset="361.92" y_offset="194.73" spacing="3.34">Appl. Math. Model.</line> | |
| <line x_offset="5.04" y_offset="194.28" spacing="-9.51">[5] J. Castro and M. Ostoja-Starzewski. Particle seiving in a random fibre | |
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