Mehmet &#x15e;eng&#xf6;n&#xfc;l

On the Zweier Sequence Spaces of Fuzzy Numbers

International Journal of Mathematics and Mathematical Sciences

Retracted: Partial Sums of Functions of Bounded Turning

Assume that <svg     style="vertical-align:-0.0pt;width:11.075px;" id="M2" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#xA;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g>

		
	
</svg> is an Abelian ring. In this paper, we characterize the structure of <svg     style="vertical-align:-0.0pt;width:11.075px;" id="M3" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D445"/></g>

		
	
</svg> whenever <svg     style="vertical-align:-0.0pt;width:11.075px;" id="M4" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D445"/></g>

		
	
</svg> is <svg     style="vertical-align:-0.16302pt;width:10.175px;" id="M5" height="8.1499996" version="1.1" viewBox="0 0 10.175 8.1499996" width="10.175"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.888)"><use xlink:href="#x1D70B"/></g>

		
	
</svg>-regular. It is also proved that an Abelian <svg     style="vertical-align:-0.16302pt;width:10.175px;" id="M6" height="8.1499996" version="1.1" viewBox="0 0 10.175 8.1499996" width="10.175"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.888)"><use xlink:href="#x1D70B"/></g>

		
	
</svg>-regular ring is isomorphic to the subdirect sum of some metadivision rings.

Jianwei Lu

Liguo He

On the Structures of Abelian <svg style="vertical-align:-0.16302pt" height="8.1499996" version="1.1" viewBox="0 0 10.175 8.1499996" width="10.175" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.888)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" /></g></svg>-Regular Rings

We provide a new proof for the description of holomorphic and biholomorphic flows on multiply connected domains in the complex plane. In contrast to the original proof of Heins (1941) we do this by the means of operator theory and by utilizing the techniques of universal coverings of the underlying domains of holomorphic flows and their liftings on the corresponding universal coverings.

F. Jafari

Z. S&#x142;odkowski

T. Tonev

A New Proof for the Description of Holomorphic Flows on Multiply Connected Domains

Let <svg     style="vertical-align:-0.1092pt;width:11.075px;" id="M1" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D445" d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117&#xA;t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" /></g>

		
	
</svg> be a commutative ring with identity and let <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M2" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g>

		
	
</svg> be an infinite unitary <svg     style="vertical-align:-0.1092pt;width:11.075px;" id="M3" height="11.3125" version="1.1" viewBox="0 0 11.075 11.3125" width="11.075"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D445"/></g>

		
	
</svg>-module. (Unless indicated otherwise, all rings are commutative with identity 1<i>&#x2260;</i>0 and all modules are unitary.) Then <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M4" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g>

		
	
</svg> is called a <i  >J&#xf3;nsson module</i> provided every proper submodule of <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M5" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g>

		
	
</svg> has smaller cardinality than <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M6" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g>

		
	
</svg>. Dually, <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M7" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g>

		
	
</svg> is said to be <i  >homomorphically smaller</i> (HS for short) if <svg     style="vertical-align:-2.22495pt;width:94px;" id="M8" height="14.85" version="1.1" viewBox="0 0 94 14.85" width="94"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x7C" d="M162 -163h-61v866h61v-866z" /></g><g transform="matrix(.017,-0,0,-.017,4.533,12.012)"><use xlink:href="#x1D440"/></g><g transform="matrix(.017,-0,0,-.017,21.889,12.012)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,28.892,12.012)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.017,-0,0,-.017,43.749,12.012)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,52.929,12.012)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,67.633,12.012)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,72.103,12.012)"><use xlink:href="#x1D440"/></g><g transform="matrix(.017,-0,0,-.017,89.459,12.012)"><use xlink:href="#x7C"/></g>

		
	
</svg> for every nonzero submodule <svg     style="vertical-align:-0.1092pt;width:14.9875px;" id="M9" height="11.3125" version="1.1" viewBox="0 0 14.9875 11.3125" width="14.9875"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D441"/></g>

		
	
</svg> of <svg     style="vertical-align:-0.0pt;width:17.475px;" id="M10" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g>

		
	
</svg>. In this survey paper, we bring the reader up to speed on current research on these structures by presenting the principal results on J&#xf3;nsson and HS modules. We conclude the paper with several open problems.

Greg Oman

J&#xf3;nsson and HS Modules over Commutative Rings

We prove that the Burkill-Cesari integral is a value on a subspace of <svg     style="vertical-align:-0.23206pt;width:24.5625px;" id="M1" height="11.75" version="1.1" viewBox="0 0 24.5625 11.75" width="24.5625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D434" d="M716 28l-8 -28h-254l8 28q55 5 66 17.5t5 47.5l-30 145h-232l-74 -138q-21 -43 -11 -54.5t72 -17.5l-6 -28h-235l8 28q53 7 73.5 21t53.5 70l319 540l33 8q6 -43 33 -176l80 -381q10 -49 26.5 -63t72.5 -19zM495 281l-49 264h-2l-149 -264h200z" /></g><g transform="matrix(.017,-0,0,-.017,12.523,11.4)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g>

		
	
</svg> and then discuss its continuity with respect to both the <svg     style="vertical-align:-0.1638pt;width:23.225px;" id="M2" height="11.375" version="1.1" viewBox="0 0 23.225 11.375" width="23.225"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path id="x1D435" d="M594 511q0 -122 -171 -157l1 -2q158 -33 158 -159q0 -52 -34.5 -95t-90.5 -65q-76 -33 -217 -33h-223l8 28q63 5 79.5 19t26.5 72l83 426q9 48 -2.5 60t-77.5 17l6 28h259q195 0 195 -139zM499 509q0 59 -37 83t-91 24q-36 0 -51 -9q-17 -9 -22 -44l-35 -195h62&#xA;q82 0 128 37t46 104zM481 199q0 71 -48 102.5t-121 31.5h-56l-37 -201q-11 -58 7.5 -77t80.5 -19q76 0 125 44.5t49 118.5z" /></g><g transform="matrix(.017,-0,0,-.017,10.551,11.112)"><path id="x1D449" d="M730 650l-8 -28q-52 -4 -72 -18t-64 -77q-79 -113 -321 -539h-33l-119 541q-13 59 -29.5 73t-66.5 20l7 28h245l-8 -28l-28 -5q-33 -6 -40.5 -15.5t-0.5 -38.5l102 -450h2q191 320 246 430q21 42 15.5 56t-43.5 19l-31 4l7 28h240z" /></g>

		
	
</svg> and the Lipschitz norm. We provide an example of value on a subspace of <svg     style="vertical-align:-0.23206pt;width:24.5625px;" id="M3" height="11.75" version="1.1" viewBox="0 0 24.5625 11.75" width="24.5625"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D434"/></g><g transform="matrix(.017,-0,0,-.017,12.523,11.4)"><use xlink:href="#x1D436"/></g>

		
	
</svg> strictly containing <svg     style="vertical-align:-3.50804pt;width:37.525002px;" id="M4" height="15.8375" version="1.1" viewBox="0 0 37.525002 15.8375" width="37.525002"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D45D" d="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36&#xA;t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" /></g><g transform="matrix(.017,-0,0,-.017,10.143,11.4)"><path id="x1D441" d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153&#xA;l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" /></g><g transform="matrix(.017,-0,0,-.017,25,11.4)"><use xlink:href="#x1D434"/></g>

		
	
</svg> as well as an existence result of a Lipschitz continuous value, different from Aumann and Shapley&#x2019;s one, on a subspace of <svg     style="vertical-align:-3.3907pt;width:37.325001px;" id="M5" height="15.6875" version="1.1" viewBox="0 0 37.325001 15.6875" width="37.325001"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D434"/></g><g transform="matrix(.017,-0,0,-.017,12.523,11.4)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,24.512,15.475)"><path id="x221E" d="M983 225q0 -112 -67 -174.5t-150 -62.5q-91 0 -154.5 43.5t-113.5 129.5q-49 -85 -104 -129t-138 -44q-98 0 -158.5 66t-60.5 154q0 59 21 106.5t54.5 75.5t70.5 43t73 15q90 0 152.5 -43.5t112.5 -128.5q48 84 104.5 128t140.5 44q93 0 155 -65t62 -158zM478 196&#xA;q-27 49 -47 80t-50 67t-64 54t-73 18q-48 0 -81.5 -47t-33.5 -128q0 -96 37.5 -157.5t99.5 -61.5q68 0 117.5 47t94.5 128zM889 204q0 91 -35.5 151t-99.5 60q-68 0 -119 -47t-95 -127q27 -49 47 -80.5t50 -67.5t65 -54t74 -18q113 0 113 183z" /></g>

		
	
</svg>.

F. Centrone

A. Martellotti

The Burkill-Cesari Integral on Spaces of Absolutely Continuous Games

In this paper, some new fractional integral inequalities are established.

Jessada Tariboon

Sotiris K. Ntouyas

Weerawat Sudsutad

Some New Riemann-Liouville Fractional Integral Inequalities

Variety of intervention programs for controlling the obesity epidemic has been done worldwide. However, it is still not yet available a scientific tool to measure the effectiveness of those programs. This is due to the difficulty in parameterizing the human interaction and transition process of obesity. A dynamical model for simulating the interaction between healthy people, overweight people, and obese people in a randomly mixed population is discussed in here. Two scenarios of intervention programs were implemented in the model, dietary program for overweight people with healthy life campaign and treatment program for obese people. Assuming all control rates are constant, disease free equilibrium point, endemic equilibrium point, and basic reproductive ratio (<svg     style="vertical-align:-3.39066pt;width:22.5px;" id="M1" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path id="x211B" d="M852 669l7 -19q-33 -13 -66 -38q83 -53 83 -139q0 -65 -49 -111q-49 -47 -131 -50v-2q54 -40 54 -84q0 -36 -38 -97t-38 -77q0 -34 32 -34q52 0 145 101l17 -16q-100 -115 -176 -115q-38 0 -65.5 23.5t-27.5 58.5q-1 28 20.5 61t43.5 63t20 51q0 48 -33 48q-30 0 -47 11&#xA;q-81 -166 -177 -242.5t-219 -76.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 152 52q60 53 145 203q75 133 131 214q54 75 121 124q-76 48 -192 48q-150 0 -250 -75&#xA;q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 245h20q-8 -107 -69 -191q-62 -84 -158 -84q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41zM814 486q0 64 -48 102q-7 -6 -14 -14t-14.5 -18l-10.5 -14q-4 -5 -35 -55l-6 -11&#xA;q-6 -11 -10 -20t-10 -20l-6 -11l-17 -36q-4 -7 -7.5 -14.5l-6 -13t-3.5 -7.5q34 0 60 -17q57 6 92 50q36 43 36 99z" /></g>

			<g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g>

		
	
</svg>) as the epidemic indicator were shown analytically. We find that the disease free equilibrium point is locally asymptotical stable if and only if <svg     style="vertical-align:-3.39066pt;width:50.087502px;" id="M2" height="16.025" version="1.1" viewBox="0 0 50.087502 16.025" width="50.087502"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.737)"><use xlink:href="#x211B"/></g>

			<g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><use xlink:href="#x30"/></g>

			<g transform="matrix(.017,-0,0,-.017,27.162,11.737)"><path id="x3C" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,41.849,11.737)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g>

		
	
</svg>. From sensitivity analysis of <svg     style="vertical-align:-3.39066pt;width:22.5px;" id="M3" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.737)"><use xlink:href="#x211B"/></g>

			<g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><use xlink:href="#x30"/></g>

		
	
</svg>, we obtain that larger rate of dietary program and treatment program will reduce <svg     style="vertical-align:-3.39066pt;width:22.5px;" id="M4" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.737)"><use xlink:href="#x211B"/></g>

			<g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><use xlink:href="#x30"/></g>

		
	
</svg> significantly. With control rates are continuous in time, an optimal control approach was applied into the model to find the best way to minimize the number of overweight and obese people. Some numerical analysis and simulations for optimal control of the intervention were shown to support the analytical results.

D. Aldila

N. Rarasati

N. Nuraini

E. Soewono

Optimal Control Problem of Treatment for Obesity in a Closed Population

A Hilbert <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M3" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-quad module of finite type has a multistructure of
Hilbert <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M4" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-bimodules with two finite bases. We will construct a <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M5" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-algebra
from a Hilbert <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M6" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-quad module of finite type and prove its universality subject
to certain relations among generators. Some examples of the <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M7" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-algebras from
Hilbert <svg     style="vertical-align:-0.23206pt;width:19.7125px;" id="M8" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125"  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g>

			<g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g>

		
	
</svg>-quad modules of finite type will be presented.

Kengo Matsumoto

<svg style="vertical-align:-0.23206pt" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.688)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g></svg>-Algebras Associated with Hilbert <svg style="vertical-align:-0.23206pt" height="14.0375" version="1.1" viewBox="0 0 19.7125 14.0375" width="19.7125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D436"/></g><g transform="matrix(.012,-0,0,-.012,12.05,5.525)"><use xlink:href="#x2217"/></g></svg>-Quad Modulesof Finite Type

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<p><em>International Journal of Mathematics and Mathematical Sciences</em> publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.</p>


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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences