Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-H&#xf6;lder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

Jes&#xfa;s &#xc1;vila

V&#xed;ctor Mar&#xed;n

H&#xe9;ctor Pinedo

Isomorphism Theorems for Groupoids and Some Applications

Ahmed Mohammed Ali

Asmaa Salah Aziz

A Relation between <i>D-</i>Index and Wiener Index for <i>r</i>-Regular Graphs

A method of finding exact solutions of the modified Veselov&#x2013;Novikov (mVN) equation is constructed by Moutard transformations, and a geometric interpretation of these transformations is obtained. An exact solution of the mVN equation is found on the example of a higher order Enneper surface, and given transformations are applied in the game theory via Kazakh proverbs in terms of trees.

Damir Kurmanbayev

Exact Solution of Modified Veselov&#x2013;Novikov Equation and Some Applications in the Game Theory

In this paper, we present a continuous mathematical model <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.150599pt" id="M1" height="12.6916pt" version="1.1" viewBox="-0.0498162 -10.541 67.1851 12.6916" width="67.1851pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-81" d="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z"/></g><g transform="matrix(.013,0,0,-0.013,7.504,0)"><path id="g113-78" d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"/></g><g transform="matrix(.013,0,0,-0.013,19.659,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.013,0,0,-0.013,30.582,0)"><path id="g113-85" d="M620 675H597C578 656 570 650 541 650H144C112 650 104 653 94 675H72C59 618 42 552 23 493L53 491C71 534 88 564 105 585C124 608 144 615 238 615H290L197 121C182 40 174 34 88 28L82 0H361L367 28C275 34 266 38 281 121L374 615H441C522 615 543 608 553 583C562 560 566 531 565 493L597 494C603 551 612 629 620 675Z"/></g><g transform="matrix(.0091,0,0,-0.0091,38.867,-5.741)"><path id="g50-115" d="M397 380C406 395 404 411 396 425S369 451 350 451C302 451 239 372 192 294H189L199 338C214 407 207 451 180 451C152 451 83 405 30 345L48 318C87 354 117 372 125 372S135 362 127 324L55 -5L61 -12C87 -5 117 3 139 5C154 87 168 162 179 207C198 250 240 310 260 332C281 355 297 366 307 366C321 366 333 360 347 346C351 342 360 342 369 348C379 355 389 366 397 380Z"/></g><g transform="matrix(.013,0,0,-0.013,43.266,0)"><use xlink:href="#g113-85"/></g><g transform="matrix(.0091,0,0,-0.0091,51.551,-5.741)"><path id="g50-113" d="M573 302C573 402 527 451 414 451C386 451 343 446 313 437L330 513L320 522C295 508 261 484 243 463L230 415C194 400 159 383 126 359L131 330C159 344 187 357 222 368L109 -147C96 -204 80 -214 18 -223L13 -255L256 -244L259 -212L236 -210C184 -205 180 -195 191 -141L219 -1C240 -10 268 -12 284 -12C352 4 431 48 484 104C543 166 573 240 573 302ZM481 290C481 165 381 37 305 37C280 37 249 56 235 71L302 395C328 399 353 402 372 402C427 402 481 378 481 290Z"/></g><g transform="matrix(.013,0,0,-0.013,57.502,0)"><path id="g113-82" d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z"/></g></svg> of alcohol drinking with the influence of private and public addiction treatment centers. We study the dynamical behavior of this model and we discuss the basic properties of the system and determine its basic reproduction number <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M2" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 13.1624 11.927" width="13.1624pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.0091,0,0,-0.0091,8.086,3.132)"><path id="g50-49" d="M245 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style="vertical-align:-0.04979992pt" id="M4" height="11.6412pt" version="1.1" viewBox="-0.0498162 -11.5914 13.0841 11.6412" width="13.0841pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-70" d="M593 650H142L136 622C219 616 224 611 209 531L130 120C114 39 108 34 23 28L17 0H501C518 32 557 130 568 158L540 169C519 132 495 93 469 70C439 43 409 35 332 35C272 35 241 37 225 50C211 61 211 89 222 149L253 320H350C441 320 443 312 441 239H471L510 434H481C455 366 448 358 360 358H260L303 572C311 613 312 615 348 615H431C509 615 522 603 532 581C544 555 546 531 545 491L574 493C583 555 592 631 593 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,8.008,-5.741)"><use xlink:href="#g50-49"/></g></svg> when <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M5" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 34.3491 11.927" width="34.3491pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-83"/></g><g transform="matrix(.0091,0,0,-0.0091,8.086,3.132)"><use xlink:href="#g50-49"/></g><g transform="matrix(.013,0,0,-0.013,16.664,0)"><path id="g117-93" d="M531 71V127L115 310L531 494V550L57 335V285L531 71ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,27.927,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg>.</span> When <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M6" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 34.3491 11.927" width="34.3491pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-83"/></g><g transform="matrix(.0091,0,0,-0.0091,8.086,3.132)"><use xlink:href="#g50-49"/></g><g transform="matrix(.013,0,0,-0.013,16.664,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 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306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g></svg> exists and the system is locally as well as globally asymptotically stable at alcohol present equilibrium <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04990005pt" id="M8" height="10.1524pt" version="1.1" viewBox="-0.0498162 -10.1025 14.2167 10.1524" width="14.2167pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-70"/></g><g transform="matrix(.0091,0,0,-0.0091,8.008,-5.741)"><use xlink:href="#g50-43"/></g></svg>.</span>

Bouchaib Khajji

Abderrahim Labzai

Omar Balatif

Mostafa Rachik

Mathematical Modeling and Analysis of an Alcohol Drinking Model with the Influence of Alcohol Treatment Centers

In this paper, we give an explicit lower bound for the class number of real quadratic field <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2681pt" id="M2" height="14.4435pt" version="1.1" viewBox="-0.0498162 -12.1754 36.323 14.4435" width="36.323pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g197-28"/></g><g transform="matrix(.013,0,0,-0.013,9.945,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,14.443,-1.934)"><use xlink:href="#g119-37"/></g><rect height="0.65243" width="7.24205" x="24.4123" y="-11.4731"/><g transform="matrix(.013,0,0,-0.013,24.412,0)"><path id="g113-101" d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z"/></g><g transform="matrix(.013,0,0,-0.013,31.654,0)"><use xlink:href="#g113-42"/></g></svg>,</span> where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3499002pt" id="M3" height="11.9413pt" version="1.1" viewBox="-0.0498162 -11.5914 53.4959 11.9413" width="53.4959pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-101"/></g><g transform="matrix(.013,0,0,-0.013,10.847,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,22.11,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.0091,0,0,-0.0091,28.636,-5.741)"><use xlink:href="#g50-51"/></g><g transform="matrix(.013,0,0,-0.013,36.488,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.013,0,0,-0.013,47.025,0)"><use xlink:href="#g113-53"/></g></svg> is a square-free integer, using <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.26807pt" id="M4" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 26.1802 11.5564" width="26.1802pt"><g transform="matrix(.013,0,0,-0.013,2.175,0)"><path id="g113-246" d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z"/></g><g transform="matrix(.013,0,0,-0.013,10.47,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,14.968,0)"><use xlink:href="#g113-111"/></g><g transform="matrix(.013,0,0,-0.013,21.495,0)"><use xlink:href="#g113-42"/></g></svg> which is the number of odd prime divisors of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M5" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 6.6501 6.1673" width="6.6501pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-111"/></g></svg>.</span>

Hasan Sankari

Ahmad Issa

Lower Bound for the Class Number of <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-2.2681pt" height="14.7403pt" version="1.1" viewBox="-0.0498162 -12.4722 60.3118 14.7403" width="60.3118pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-28" d="M703 -12C678 -16 653 -21 626 -21C591 -21 567 -1 542 24C650 81 715 195 715 332C715 529 588 676 385 676C179 676 45 536 45 332C45 139 185 -14 381 -14C422 -14 460 -8 494 4C525 -31 561 -65 626 -65C654 -65 681 -60 709 -56L703 -12ZM591 548C648 488 671 412 671 332C671 255 647 175 591 119V548ZM547 79C536 70 523 63 510 58C478 91 441 122 384 122C329 122 290 96 257 56C240 63 226 70 213 79V584C251 616 334 632 381 632C434 632 504 619 547 584V79ZM169 119C116 175 89 255 89 332C89 412 112 488 169 548V119ZM460 41C432 34 406 30 381 30C358 30 333 34 305 41C326 64 348 78 381 78C416 78 439 62 460 41Z" /></g><g transform="matrix(.013,0,0,-0.013,9.945,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" /></g><g transform="matrix(.013,0,0,-0.013,14.443,-2.231)"><path id="g119-37" d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z" /></g><rect height="0.65243" width="31.2309" x="24.4123" y="-11.7699" /><g transform="matrix(.013,0,0,-0.013,24.412,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z" /></g><g transform="matrix(.0091,0,0,-0.0091,30.938,-4.176)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z" /></g><g transform="matrix(.013,0,0,-0.013,38.79,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z" /></g><g transform="matrix(.013,0,0,-0.013,49.327,0)"><path id="g113-53" d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z" /></g><g transform="matrix(.013,0,0,-0.013,55.643,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" /></g></svg>

Paul Antony Otieno

Jean Baptiste Gatsinzi

Vitalis Onyango-Otieno

Rational Cohomology Algebra of Mapping Spaces between Complex Grassmannians

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.15061pt" id="M1" height="12.107pt" version="1.1" viewBox="-0.0498162 -9.95639 14.6956 12.107" width="14.6956pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-82" d="M699 368C699 549 574 666 407 666C186 666 23 488 23 277C23 113 129 -3 288 -13L307 -26C431 -111 501 -139 533 -147C559 -154 613 -163 658 -164L666 -141C597 -111 507 -66 430 -11L416 -1C580 42 699 190 699 368ZM601 371C601 227 518 54 381 22L354 40L278 24C175 47 120 145 120 269C120 451 235 631 398 631C540 631 601 521 601 371Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.386,-5.741)"><path id="g50-111" d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z"/></g></svg>,</span> where <i>Q</i> is an alphabet of size <i>q</i> and <i>n</i> is a positive integer. A 2-frameproof code is a code <i>C</i> such that any 2 codewords in <i>C</i> cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a <i>q</i>-ary 2-frameproof code of length 4. Our new upper bound shows that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.429399pt" id="M2" height="15.0208pt" version="1.1" viewBox="-0.0498162 -11.5914 94.0948 15.0208" width="94.0948pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path id="g113-68" d="M645 631C614 643 545 666 457 666C215 666 23 519 23 283C23 90 158 -16 337 -16C412 -16 489 2 522 10C543 39 590 127 606 167L580 181C519 89 459 18 348 18C201 18 122 136 122 287C122 464 244 632 435 632C544 632 602 595 608 472L639 475C643 526 645 581 645 631Z"/></g><g transform="matrix(.013,0,0,-0.013,12.103,0)"><use xlink:href="#g113-9"/></g><g transform="matrix(.013,0,0,-0.013,19.154,0)"><path id="g117-93" d="M531 71V127L115 310L531 494V550L57 335V285L531 71ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,30.417,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.013,0,0,-0.013,36.657,0)"><path id="g113-114" d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"/></g><g transform="matrix(.0091,0,0,-0.0091,43.04,-5.741)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.013,0,0,-0.013,50.892,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,61.429,0)"><use xlink:href="#g113-51"/></g><g transform="matrix(.013,0,0,-0.013,67.669,0)"><use xlink:href="#g113-114"/></g><g transform="matrix(.013,0,0,-0.013,76.957,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,87.494,0)"><path id="g113-50" d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"/></g></svg> when <i>q</i> is odd and <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M3" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 33.9567 11.7782" width="33.9567pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-114"/></g><g transform="matrix(.013,0,0,-0.013,10.015,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g><g transform="matrix(.013,0,0,-0.013,21.278,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,27.518,0)"><path id="g113-49" d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"/></g></svg>.</span>

Penying Rochanakul

New Bounds on 2-Frameproof Codes of Length 4

The aim of this paper is to provide an approximation of the value-at-risk of the multivariate copula associated with financial loss and profit function. A higher dimensional extension of the Taylor&#x2013;Young formula is used for this estimation in a Euclidean space. Moreover, a time-varying and conditional copula is used for the modeling of the VaR.

Vini Yves Bernadin Loyara

Remi Guillaume Bagr&#xe9;

Diakarya Barro

Estimation of the Value at Risk Using the Stochastic Approach of Taylor Formula

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International Journal of Mathematics and Mathematical Sciences