Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided.

Rui Huang

Xiaodong Wu

Ruihe Wang

Hui Li

A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space

We introduce the notion of Demazure descent data on a triangulated category <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1881pt;width:13.675px;" id="M1" height="12.0625" version="1.1" viewBox="0 0 13.675 12.0625" width="13.675">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.738)"><path id="x1D49E" d="M449 645l-12 -19q-55 31 -136 31q-52 0 -103 -17t-85.5 -54.5t-34.5 -84.5q0 -60 41.5 -96.5t109.5 -43.5q81 125 216 216q137 92 240 92q46 0 71 -19.5t25 -55.5q0 -60 -67 -123q-66 -63 -177 -105q-109 -42 -225 -42q-69 -100 -69 -199q0 -49 31.5 -79.5t85.5 -30.5&#xA;q79 0 141 50q61 49 61 108q0 35 -20 54.5t-51 19.5q-53 0 -103 -53q-47 -52 -55 -131l-22 2q0 91 58 158t139 67q55 0 86.5 -26.5t31.5 -76.5q0 -81 -96 -148q-78 -54 -174 -54q-86 0 -141 46q-56 47 -56 127q0 81 48 171q-72 12 -121 55t-49 115q0 75 71 131q72 57 185 57&#xA;q91 0 156 -42zM331 355l1 -1q122 0 244 62q73 37 118.5 85.5t45.5 90.5q0 40 -50 40q-73 0 -174 -79q-102 -80 -185 -198z"></path></g>

		
	
</svg> and define the descent category for such data. We illustrate the definition by our basic example. Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.15048pt;width:12.4375px;" id="M2" height="11.75" version="1.1" viewBox="0 0 12.4375 11.75" width="12.4375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z"></path></g>

		
	
</svg> be a reductive algebraic group with a Borel subgroup <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:10.6125px;" id="M3" height="11.175" version="1.1" viewBox="0 0 10.6125 11.175" width="10.6125">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><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"></path></g>

		
	
</svg>. Demazure functors form Demazure descent data on <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.72116pt;width:80.737503px;" id="M4" height="20.8375" version="1.1" viewBox="0 0 80.737503 20.8375" width="80.737503">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><path id="x1D437" d="M780 372q0 -191 -178 -297q-128 -75 -335 -75h-250l7 27q61 7 77 22t28 76l77 398q11 62 0 76.5t-77 22.5l6 28h292q164 0 258.5 -67.5t94.5 -210.5zM678 375q0 119 -73 180.5t-201 61.5q-62 0 -82 -13q-17 -7 -25 -53l-80 -427q-8 -47 -4 -62t26 -20q24 -6 80 -6&#xA;q166 0 262.5 95t96.5 244z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,13.712,8.537)"><path id="x1D44F" d="M452 333q0 -82 -46 -164t-116 -128q-81 -53 -158 -53q-26 0 -51 13t-37 32q-26 41 -6 134l91 431q7 40 2.5 47t-34.5 7h-33l2 25q36 4 72.5 13t58.5 15.5t28 6.5q10 0 4 -29l-91 -391h2q65 77 130 116.5t105 39.5q36 0 56.5 -32t20.5 -83zM365 316q0 76 -35 76&#xA;q-31 0 -93.5 -49t-107.5 -110q-12 -37 -17 -70q-9 -65 12 -96.5t59 -31.5q31 0 56 14q53 29 89.5 105t36.5 162z"></path></g>

			<g transform="matrix(.017,-0,0,-.017,19.912,16.7)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z"></path></g><g transform="matrix(.017,-0,0,-.017,25.794,16.7)"><path id="x52" d="M631 18l-4 -26q-24 0 -35 1q-57 3 -92.5 23t-69.5 68q-21 30 -101 160q-13 22 -30.5 31t-54.5 9h-32v-159q0 -63 14 -77.5t74 -19.5v-28h-262v28q62 5 76 19.5t14 77.5v402q0 62 -13 76t-73 19v28h248q109 0 159 -34q67 -42 67 -131q0 -115 -127 -172q27 -52 89 -144&#xA;q49 -71 81 -104q31 -36 72 -47zM212 316h48q73 0 108 29q55 42 55 124q0 76 -42 112t-104 36q-41 0 -54 -10q-11 -7 -11 -44v-247z"></path></g><g transform="matrix(.017,-0,0,-.017,36.401,16.7)"><path id="x65" d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3&#xA;q14 0 19 4.5t5 15.5q1 41 -23.5 70.5t-65.5 29.5z"></path></g><g transform="matrix(.017,-0,0,-.017,43.626,16.7)"><path id="x70" d="M169 380l92 53q26 16 49 16q80 0 128.5 -56t48.5 -133q0 -108 -72.5 -180.5t-176.5 -91.5q-38 1 -69 18v-131q0 -57 13 -70t71 -18v-27h-236v27q49 5 61 17t12 61v452q0 48 -9.5 58.5t-56.5 16.5v24q66 11 145 43v-79zM169 346v-270q37 -39 94 -39q63 0 101.5 50&#xA;t38.5 134q0 78 -38.5 123t-95.5 45q-45 0 -100 -43z"></path></g><g transform="matrix(.017,-0,0,-.017,52.533,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,58.415,16.7)"><use xlink:href="#x1D435"></use></g><g transform="matrix(.017,-0,0,-.017,68.903,16.7)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z"></path></g><g transform="matrix(.017,-0,0,-.017,74.784,16.7)"><use xlink:href="#x29"></use></g>

		
	
</svg> and the descent category is equivalent to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.72116pt;width:82.550003px;" id="M5" height="20.8375" version="1.1" viewBox="0 0 82.550003 20.8375" width="82.550003">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,16.7)"><use xlink:href="#x1D437"></use></g>

			<g transform="matrix(.012,-0,0,-.012,13.712,8.537)"><use xlink:href="#x1D44F"></use></g>

			<g transform="matrix(.017,-0,0,-.017,19.912,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,25.794,16.7)"><use xlink:href="#x52"></use></g><g transform="matrix(.017,-0,0,-.017,36.401,16.7)"><use xlink:href="#x65"></use></g><g transform="matrix(.017,-0,0,-.017,43.626,16.7)"><use xlink:href="#x70"></use></g><g transform="matrix(.017,-0,0,-.017,52.533,16.7)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,58.415,16.7)"><use xlink:href="#x1D43A"></use></g><g transform="matrix(.017,-0,0,-.017,70.722,16.7)"><use xlink:href="#x29"></use></g><g transform="matrix(.017,-0,0,-.017,76.603,16.7)"><use xlink:href="#x29"></use></g>

		
	
</svg>.

Sergey Arkhipov

Tina Kanstrup

Demazure Descent and Representations of Reductive Groups

Let <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.55815pt;width:14.1375px;" id="M1" height="15.4125" version="1.1" viewBox="0 0 14.1375 15.4125" width="14.1375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><path id="x210C" d="M626 695l19 -13q-96 -135 -219 -135q-71 0 -133 50q-41 -33 -41 -91q0 -78 57 -139q161 94 224 127q113 -26 176.5 -100t63.5 -179q0 -75 -22 -120q-22 -46 -67 -92q-40 -42 -127 -104q-87 -63 -117 -103h-13q-60 62 -132 62q-69 0 -114 -56l-20 9q60 136 185 136&#xA;q71 0 129 -64q105 66 145 150q36 75 36 139q0 91 -52 156q-51 66 -133 66q-63 0 -143 -54l44 -56q35 -46 35 -82q0 -44 -38 -93q-40 -49 -139 -131q-23 18 -61 37q-43 24 -56 28q-33 -31 -46 -53l-24 15l104 147l76 -45q55 -33 59 -33q17 15 23 26t7 36q0 36 -79 121&#xA;q-32 36 -55 86t-23 97q0 73 75 145q31 31 61 57q27 24 66 53q58 -53 131 -53q88 0 138 53z"></path></g>

		
	
</svg> be a class of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3135pt;width:34.724998px;" id="M2" height="8.7124996" version="1.1" viewBox="0 0 34.724998 8.7124996" width="34.724998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,8.438)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z"></path></g><g transform="matrix(.017,-0,0,-.017,12.37,8.438)"><path id="xD7" d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z"></path></g><g transform="matrix(.017,-0,0,-.017,26.122,8.438)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> Hankel matrices <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.13504pt;width:23.9375px;" id="M3" height="15.7" version="1.1" viewBox="0 0 23.9375 15.7" width="23.9375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path id="x1D407" d="M759 25v-25h-339v25q54 6 71.5 20.5t17.5 52.5v228h-241v-228q0 -40 17 -54q16 -13 72 -19v-25h-336v25q54 8 69 20q16 13 16 53v479q0 41 -16.5 55t-68.5 19v25h336v-25q-54 -4 -71.5 -18.5t-17.5 -55.5v-204h241v204q0 41 -17.5 55.5t-71.5 18.5v25h339v-25&#xA;q-51 -4 -69.5 -19t-18.5 -55v-479q0 -40 16 -53q15 -12 72 -20z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,14.538,15.637)"><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"></path></g>

		
	
</svg> whose entries, depending on a given matrix <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0pt;width:12.4px;" id="M4" height="11.85" version="1.1" viewBox="0 0 12.4 11.85" width="12.4">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.788)"><path id="x1D400" d="M689 25v-25h-322v25l32 3q46 4 46 33q0 31 -46 136h-225l-13 -34q-31 -78 -31 -97q0 -36 81 -41v-25h-202v25q36 6 53 28q18 22 46 92l221 545h28l222 -522q36 -87 58 -117q21 -26 52 -26zM189 236h195l-101 243z"></path></g>

		
	
</svg>, are linear forms in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables with coefficients in a finite field <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.74141pt;width:18.862499px;" id="M6" height="18.3125" version="1.1" viewBox="0 0 18.862499 18.3125" width="18.862499">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D53D" d="M613 503l-29 -3q-13 55 -28 80q-13 21 -35.5 28.5t-78.5 7.5h-76q-22 0 -27.5 -5t-5.5 -27v-238h103q66 0 82.5 12t25.5 62h29v-193h-29q-9 54 -25.5 67t-82.5 13h-103v-181q0 -64 15 -78.5t84 -19.5v-28h-394v28q62 5 76 19.5t14 77.5v400q0 63 -14 77.5t-76 19.5v28&#xA;h565q1 -40 10 -147zM290 131v390q0 62 -7.5 79.5t-30.5 17.5q-24 0 -31.5 -17.5t-7.5 -79.5v-390q0 -62 7.5 -79.5t31.5 -17.5q23 0 30.5 17.5t7.5 79.5z"></path></g>

			<g transform="matrix(.012,-0,0,-.012,12.337,15.188)"><path id="x1D45E" d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387&#xA;q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z"></path></g>

		
	
</svg>. For every matrix in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.55815pt;width:14.1375px;" id="M7" height="15.4125" version="1.1" viewBox="0 0 14.1375 15.4125" width="14.1375">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,11.875)"><use xlink:href="#x210C"></use></g>

		
	
</svg>, it is shown that the varieties specified by the leading minors of orders from 1 to <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:34.349998px;" id="M8" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><path id="x2212" d="M535 230h-483v50h483v-50z"></path></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><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"></path></g>

		
	
</svg> have the same number <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.29482pt;width:27.762501px;" id="M9" height="20.2125" version="1.1" viewBox="0 0 27.762501 20.2125" width="27.762501">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D45E"></use></g>

			<g transform="matrix(.012,-0,0,-.012,8.413,7.613)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.012,-0,0,-.012,14.386,7.613)"><use xlink:href="#x2212"></use></g><g transform="matrix(.012,-0,0,-.012,21.37,7.613)"><use xlink:href="#x31"></use></g>

		
	
</svg> of points in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.74141pt;width:19px;" id="M10" height="20.75" version="1.1" viewBox="0 0 19 20.75" width="19">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,13.55)"><use xlink:href="#x1D53D"></use></g>

			<g transform="matrix(.012,-0,0,-.012,12.337,5.388)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.012,-0,0,-.012,12.337,17.638)"><use xlink:href="#x1D45E"></use></g>

		
	
</svg>. Further properties are derived, which show that sets of varieties, tied to a given Hankel matrix, resemble a set of hyperplanes as regards the number of points of their intersections.

Edoardo Ballico

Michele Elia

On Determinantal Varieties of Hankel Matrices

We establish new results concerning endomorphisms of a finite chain, if the cardinality of the image of such endomorphism is no more than some fixed number. The semiring of all such endomorphisms can be seen as a simplex whose vertices are the constant endomorphisms. We explore the properties of these simplices.

Ivan Trendafilov

Simplices in the Endomorphism Semiring of a Finite Chain

We discuss several partial solutions to the so-called &#x201c;coquecigrue problem&#x201d; of Loday; these solutions parallel, but also generalize in several directions, the classical Lie group-Lie algebra correspondence. Our study highlights some clear similarities between the split and nonsplit cases and leads us to a general unifying scheme that provides an answer to the problem of the algebraic structure of a coquecigrue.

Juan Monterde

Fausto Ongay

On Integral Manifolds for Leibniz Algebras

We present a method to iteratively construct new bent functions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.51414pt;width:34.349998px;" id="M3" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><use xlink:href="#x32"></use></g>

		
	
</svg> variables from a bent function of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M4" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables and its cyclic shift permutations using minterms of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.10033pt;width:8.6625004px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"></use></g>

		
	
</svg> variables and minterms of 2 variables. In addition, we provide the number of bent functions of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.51414pt;width:34.349998px;" id="M6" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998">
	
		
			<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D45B"></use></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><use xlink:href="#x2B"></use></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><use xlink:href="#x32"></use></g>

		
	
</svg> variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.

Joan-Josep Climent

Francisco J. Garc&#xed;a

Ver&#xf3;nica Requena

A Construction of Bent Functions of <svg style="vertical-align:-0.51414pt" height="11.125" version="1.1" viewBox="0 0 34.349998 11.125" width="34.349998" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.017,-0,0,-.017,12.37,10.862)"><path id="x2B" d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" /></g><g transform="matrix(.017,-0,0,-.017,26.122,10.862)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g></svg> Variables from a Bent Function of <svg style="vertical-align:-0.10033pt" height="7.9499998" version="1.1" viewBox="0 0 8.6625004 7.9499998" width="8.6625004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D45B"/></g></svg> Variables and Its Cyclic Shifts

Based on the definition of vertex coalgebra introduced by Hubbard, 2009, we
prove that this notion can be reformulated using coskew symmetry, coassociator and
cocommutator formulas without restrictions on the grading. We also prove that a
vertex coalgebra can be defined in terms of dual versions of the axioms of Lie conformal
algebra and differential algebra.

Florencia Orosz Hunziker

Jos&#xe9; I. Liberati

Vertex Coalgebras, Coassociator, and Cocommutator Formulas

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Algebra