We shall study the problem of minimizing a functional involving the curl of vector fields in a three-dimensional, bounded multiconnected domain with prescribed tangential component on the boundary. The paper is an extension of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M1" height="11.6412pt" version="1.1" viewBox="-0.0498162 -11.5914 12.262 11.6412" width="12.262pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-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></svg> minimization problem of the curl of vector fields. We shall prove the existence and the estimate of minimizers of more general functional which contains <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M2" height="10.5908pt" version="1.1" viewBox="-0.0498162 -10.541 13.2667 10.5908" width="13.2667pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-77"/></g><g transform="matrix(.0091,0,0,-0.0091,7.185,-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></svg> norm of the curl of vector fields.

Junichi Aramaki

Variational Problem Involving Operator Curl in a Multiconnected Domain

Galerkin method is presented to solve singularly perturbed differential-difference equations with delay and advanced shifts using fitting factor. In the numerical treatment of such type of problems, Taylor&#x2019;s approximation is used to tackle the terms containing small shifts. A fitting factor in the Galerkin scheme is introduced which takes care of the rapid changes that occur in the boundary layer. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the tridiagonal system of the fitted Galerkin method. The method is analysed for convergence. Several numerical examples are solved and compared to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shifts on the boundary layer solution.

D. Kumara Swamy

K. Phaneendra

Y. N. Reddy

Solution of Singularly Perturbed Differential-Difference Equations with Mixed Shifts Using Galerkin Method with Exponential Fitting

Some Ostrowski type inequalities for functions whose first derivatives are logarithmically preinvex are established.

Badreddine Meftah

Ostrowski Inequalities for Functions Whose First Derivatives Are Logarithmically Preinvex

We prove new generalization of Hadamard, Ostrowski, and Simpson inequalities in the framework of GA-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M2" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 4.9929 6.1673" width="4.9929pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-116"/></g></svg>-convex functions and Hadamard fractional integral.

&#x130;mdat &#x130;&#x15f;can

Mustafa Aydin

Some New Generalized Integral Inequalities for GA-<svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2063904pt" height="6.1673pt" version="1.1" viewBox="-0.0498162 -5.96091 4.9929 6.1673" width="4.9929pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z" /></g></svg>-Convex Functions via Hadamard Fractional Integrals

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.

Oleg Yaremko

Natalia Yaremko

Matrix Fourier Transforms for Consistent Mathematical Models

We first compute James’ sectional category (secat) of the Ganea map <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M1" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 12.3455 9.39034" width="12.3455pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-104" d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.072,3.132)"><path id="g50-108" d="M485 418C485 434 469 451 441 451C389 451 312 387 255 335C221 304 194 278 159 242H157L259 678C264 698 264 710 255 710C242 710 183 681 96 673L90 643L128 642C165 641 171 640 162 602L24 -5L31 -12C51 -4 81 4 110 9L145 181C154 195 180 220 205 242C233 170 263 106 291 53C318 1 335 -12 360 -12C383 -12 425 0 483 82L463 105C437 79 411 57 400 57C388 57 378 68 355 110C329 156 288 244 268 298C299 333 355 376 405 376C426 376 440 373 448 368C453 365 465 368 468 373C477 386 485 405 485 418Z"/></g></svg> of any map <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M2" height="9.14241pt" version="1.1" viewBox="-0.0498162 -5.96091 10.5422 9.14241" width="10.5422pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-231" d="M211 91L201 114C175 93 136 64 124 64C117 64 117 73 121 93C150 233 183 368 203 440L200 448L119 426L44 86C38 59 35 38 35 24C35 -5 46 -12 60 -12C95 -12 178 49 211 91Z"/></g><g transform="matrix(.0091,0,0,-0.0091,2.873,3.132)"><path id="g50-89" d="M782 650H541L536 618L564 614C597 609 604 600 584 574C535 508 486 451 437 395C394 473 361 533 343 575C331 605 335 610 375 615L400 618L407 650H156L147 618C222 612 230 608 267 537L375 330C277 214 186 117 163 96C108 43 97 38 27 32L18 0H265L274 32L251 35C204 41 202 47 224 80C264 138 327 213 397 293L504 80C520 48 516 42 472 36L440 32L433 0H707L716 32C627 38 622 43 581 121L455 361L629 548C685 606 696 610 775 618L782 650Z"/></g></svg> in terms of the sectional category of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.1815pt" id="M3" height="9.14241pt" version="1.1" viewBox="-0.0498162 -5.96091 10.5422 9.14241" width="10.5422pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-231"/></g><g transform="matrix(.0091,0,0,-0.0091,2.873,3.132)"><use xlink:href="#g50-89"/></g></svg>: we show that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42942pt" id="M4" height="11.2041pt" version="1.1" viewBox="-0.0498162 -7.77468 40.0441 11.2041" width="40.0441pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g121-113" d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><path id="g121-99" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><path id="g121-97" d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z"/></g><g transform="matrix(.013,0,0,-0.013,16.004,0)"><path id="g121-95" d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"/></g><g transform="matrix(.013,0,0,-0.013,21.463,0)"><path id="g121-114" d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z"/></g><g transform="matrix(.013,0,0,-0.013,27.603,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.0091,0,0,-0.0091,34.675,3.132)"><use xlink:href="#g50-108"/></g></svg> is the integer part of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.18147pt" id="M5" height="12.4698pt" version="1.1" viewBox="-0.0498162 -9.28833 78.944 12.4698" width="78.944pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-113"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,16.004,0)"><use xlink:href="#g121-95"/></g><g transform="matrix(.013,0,0,-0.013,21.463,0)"><use xlink:href="#g121-114"/></g><g transform="matrix(.013,0,0,-0.013,27.603,0)"><use xlink:href="#g113-231"/></g><g transform="matrix(.0091,0,0,-0.0091,30.476,3.132)"><use xlink:href="#g50-89"/></g><g transform="matrix(.013,0,0,-0.013,38.035,0)"><path id="g113-48" d="M368 703H309L44 -163H104L368 703Z"/></g><g transform="matrix(.013,0,0,-0.013,43.396,0)"><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,-0.013,47.894,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g><g transform="matrix(.013,0,0,-0.013,57.339,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,67.875,0)"><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><g transform="matrix(.013,0,0,-0.013,74.115,0)"><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>. Next we compute the relative category (relcat) of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42943pt" id="M6" height="9.39034pt" version="1.1" viewBox="-0.0498162 -5.96091 12.3455 9.39034" width="12.3455pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.0091,0,0,-0.0091,7.072,3.132)"><use xlink:href="#g50-108"/></g></svg>. In order to do this, we introduce the relative category of order <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M7" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.66314 9.49473" width="6.66314pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-108"/></g></svg> (<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29115pt" id="M8" height="12.5273pt" version="1.1" viewBox="-0.0498162 -9.23615 33.7324 12.5273" width="33.7324pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g121-112" d="M181 342V451C133 431 89 419 40 411V388C98 381 102 377 102 311V104C102 38 95 32 33 26V0H263V26C186 32 181 38 181 104V287C203 343 235 372 261 372C277 372 289 366 304 352C310 346 318 345 330 350C349 359 362 379 362 399C362 422 338 449 304 449C256 449 213 393 183 342H181Z"/></g><g transform="matrix(.013,0,0,-0.013,4.653,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.126,0)"><path id="g121-106" d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z"/></g><g transform="matrix(.013,0,0,-0.013,13.415,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,18.953,0)"><use xlink:href="#g121-95"/></g><g transform="matrix(.013,0,0,-0.013,24.412,0)"><use xlink:href="#g121-114"/></g><g transform="matrix(.0091,0,0,-0.0091,28.377,3.132)"><use xlink:href="#g50-108"/></g></svg>) of a map and show that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.429399pt" id="M9" height="12.7373pt" version="1.1" viewBox="-0.0498162 -9.3079 43.0062 12.7373" width="43.0062pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g190-115" d="M181 342V451C133 431 89 419 40 411V388C98 381 102 377 102 311V104C102 38 95 32 33 26V0H263V26C186 32 181 38 181 104V287C203 343 235 372 261 372C277 372 289 366 304 352C310 346 318 345 330 350C349 359 362 379 362 399C362 422 338 449 304 449C256 449 213 393 183 342H181Z"/></g><g transform="matrix(.013,0,0,-0.013,4.653,0)"><path id="g190-102" d="M380 106C343 72 306 56 265 56C195 56 116 112 115 248C235 252 361 262 377 265C396 269 400 277 400 297C400 374 333 449 250 449H249C198 449 144 421 103 376S37 269 37 201C37 88 109 -12 232 -12C263 -12 332 6 395 84L380 106ZM225 412C281 412 315 364 314 312C314 297 308 292 290 292C232 290 176 289 120 289C135 370 180 412 225 412Z"/></g><g transform="matrix(.013,0,0,-0.013,10.126,0)"><path id="g190-109" d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z"/></g><g transform="matrix(.013,0,0,-0.013,13.415,0)"><path id="g190-100" d="M390 111C344 68 312 56 269 56C212 56 118 102 118 241C118 346 175 401 241 401C277 401 312 388 342 360C350 352 355 349 361 349C372 349 394 371 394 392C394 403 391 411 378 422C362 436 329 449 288 449H287C250 449 190 432 138 392C71 341 37 274 37 197C37 90 112 -12 238 -12C297 -12 363 32 407 90L390 111Z"/></g><g transform="matrix(.013,0,0,-0.013,18.953,0)"><path id="g190-98" d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z"/></g><g transform="matrix(.013,0,0,-0.013,24.412,0)"><path id="g190-117" d="M298 36L289 62C276 55 253 45 228 45C202 45 169 60 169 141V397H276C289 405 292 426 282 437H169V574L155 576L90 509V437H45L17 408L21 397H90V107C90 28 125 -12 188 -12C198 -12 213 -8 230 1L298 36Z"/></g><g transform="matrix(.013,0,0,-0.013,30.661,0)"><use xlink:href="#g113-104"/></g><g transform="matrix(.0091,0,0,-0.0091,37.733,3.132)"><use xlink:href="#g50-108"/></g></svg> is the integer part of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2911pt" id="M10" height="12.599pt" version="1.1" viewBox="-0.0498162 -9.3079 84.8782 12.599" width="84.8782pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g190-115"/></g><g transform="matrix(.013,0,0,-0.013,4.653,0)"><use xlink:href="#g190-102"/></g><g transform="matrix(.013,0,0,-0.013,10.126,0)"><use xlink:href="#g190-109"/></g><g transform="matrix(.013,0,0,-0.013,13.415,0)"><use xlink:href="#g190-100"/></g><g transform="matrix(.013,0,0,-0.013,18.953,0)"><use xlink:href="#g190-98"/></g><g transform="matrix(.013,0,0,-0.013,24.412,0)"><use xlink:href="#g190-117"/></g><g transform="matrix(.0091,0,0,-0.0091,28.485,3.132)"><use xlink:href="#g50-108"/></g><g transform="matrix(.013,0,0,-0.013,33.633,0)"><use xlink:href="#g113-231"/></g><g transform="matrix(.0091,0,0,-0.0091,36.506,3.132)"><use xlink:href="#g50-89"/></g><g transform="matrix(.013,0,0,-0.013,44.064,0)"><use xlink:href="#g113-48"/></g><g transform="matrix(.013,0,0,-0.013,49.426,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,53.924,0)"><use xlink:href="#g113-108"/></g><g transform="matrix(.013,0,0,-0.013,63.368,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.013,0,0,-0.013,73.905,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,80.145,0)"><use xlink:href="#g113-42"/></g></svg>. Then we establish some inequalities linking secat and relcat of any order: we show that <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.2911pt" id="M11" height="12.599pt" version="1.1" viewBox="-0.0498162 -9.3079 190.072 12.599" width="190.072pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-113"/></g><g transform="matrix(.013,0,0,-0.013,4.875,0)"><use xlink:href="#g121-99"/></g><g transform="matrix(.013,0,0,-0.013,10.466,0)"><use xlink:href="#g121-97"/></g><g transform="matrix(.013,0,0,-0.013,16.004,0)"><use xlink:href="#g121-95"/></g><g transform="matrix(.013,0,0,-0.013,21.463,0)"><use xlink:href="#g121-114"/></g><g transform="matrix(.013,0,0,-0.013,27.603,0)"><use xlink:href="#g113-231"/></g><g transform="matrix(.0091,0,0,-0.0091,30.476,3.132)"><use xlink:href="#g50-89"/></g><g transform="matrix(.013,0,0,-0.013,41.667,0)"><path id="g117-95" d="M531 103V159L115 342L531 526V582L57 367V317L531 103ZM531 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xlink:href="#g117-95"/></g><g transform="matrix(.013,0,0,-0.013,127.948,0)"><use xlink:href="#g190-115"/></g><g transform="matrix(.013,0,0,-0.013,132.601,0)"><use xlink:href="#g190-102"/></g><g transform="matrix(.013,0,0,-0.013,138.074,0)"><use xlink:href="#g190-109"/></g><g transform="matrix(.013,0,0,-0.013,141.363,0)"><use xlink:href="#g190-100"/></g><g transform="matrix(.013,0,0,-0.013,146.901,0)"><use xlink:href="#g190-98"/></g><g transform="matrix(.013,0,0,-0.013,152.36,0)"><use xlink:href="#g190-117"/></g><g transform="matrix(.0091,0,0,-0.0091,156.433,3.132)"><use xlink:href="#g50-108"/></g><g transform="matrix(.013,0,0,-0.013,161.581,0)"><use xlink:href="#g113-231"/></g><g transform="matrix(.0091,0,0,-0.0091,164.454,3.132)"><use xlink:href="#g50-89"/></g><g transform="matrix(.013,0,0,-0.013,174.918,0)"><use xlink:href="#g117-36"/></g><g transform="matrix(.013,0,0,-0.013,185.454,0)"><use xlink:href="#g113-50"/></g></svg>. We give examples that show that these inequalities may be strict.

Jean-Paul Doeraene

Sectional Category of the Ganea Fibrations and Higher Relative Category

We study the different types of Finsler space with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M1" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 29.2766 12.7178" width="29.2766pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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,-0.013,4.498,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.013,0,0,-0.013,11.883,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.013,0,0,-0.013,17.026,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g><g transform="matrix(.013,0,0,-0.013,24.58,0)"><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>-metrics which have nonholonomic frames as an application for classical mechanics and dynamics in physics using gauge transformation which helps to derive unified field theory. Further, we set up the application of Finsler geometry to geometrize the electromagnetic field completely.

Mallikarjuna Yallappa Kumbar

Sachin Jangir

Chowdari Kondasandra Chowdappa

Narasimhamurthy Senajji Kampalappa

Geometric Framework for Unified Field Theory Using Finsler Gauge Transformation

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

M. Rahimi

A. Assari

Ergodicity Space for Measure-Preserving Transformations

We study some ratios related to hyper-Horadam numbers such as <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.1275pt" id="M1" height="16.6936pt" version="1.1" viewBox="-0.0498162 -12.5661 55.145 16.6936" width="55.145pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-88" d="M962 650H739L732 622L760 619C812 613 819 604 798 552C760 457 671 267 606 131H604L511 638H480L237 131H233L183 554C177 607 183 614 226 619L248 622L257 650H24L17 622C88 615 92 611 103 524L173 -11H203L450 491H453L543 -11H575L839 529C882 609 886 613 953 622L962 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,12.669,-5.741)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.0091,0,0,-0.0091,15.927,-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 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width="39.6721pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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(.013,0,0,-0.013,10.158,0)"><path id="g117-148" d="M901 255C861 290 790 361 716 442L694 427L796 280H69V230H796L694 83L716 68C790 149 861 220 901 255Z"/></g><g transform="matrix(.013,0,0,-0.013,26.192,0)"><path id="g113-30" d="M983 225C983 348 891 448 766 448C646 448 579 377 521 276C458 383 382 448 256 448C165 448 37 373 37 208C37 98 119 -12 256 -12C375 -12 440 60 498 161C561 53 639 -12 766 -12C865 -12 983 64 983 225ZM478 196C428 106 365 21 266 21C183 21 129 114 129 240C129 353 183 415 244 415C352 415 420 301 478 196ZM889 204C889 66 838 21 776 21C666 21 598 135 540 241C590 331 655 415 754 415C842 415 889 320 889 204Z"/></g></svg> by using a symmetric algorithm obtained by the recurrence relation <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.053101pt" id="M3" height="16.3113pt" version="1.1" viewBox="-0.0498162 -12.2582 83.2423 16.3113" width="83.2423pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-98" d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z"/></g><g 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xlink:href="#g54-33"/></g><g transform="matrix(.0091,0,0,-0.0091,43.395,-5.741)"><use xlink:href="#g50-50"/></g><g transform="matrix(.0091,0,0,-0.0091,32.615,3.784)"><use xlink:href="#g50-111"/></g><g transform="matrix(.013,0,0,-0.013,51.285,0)"><path id="g117-36" d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"/></g><g transform="matrix(.013,0,0,-0.013,61.821,0)"><use xlink:href="#g113-98"/></g><g transform="matrix(.0091,0,0,-0.0091,68.401,-5.741)"><use xlink:href="#g50-108"/></g><g transform="matrix(.0091,0,0,-0.0091,67.814,3.784)"><use xlink:href="#g50-111"/></g><g transform="matrix(.0091,0,0,-0.0091,72.473,3.784)"><use xlink:href="#g54-33"/></g><g transform="matrix(.0091,0,0,-0.0091,78.034,3.784)"><use xlink:href="#g50-50"/></g></svg>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.943501pt" id="M4" height="16.2017pt" version="1.1" viewBox="-0.0498162 -12.2582 23.7475 16.2017" width="23.7475pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-88"/></g><g transform="matrix(.0091,0,0,-0.0091,12.669,-5.741)"><use xlink:href="#g50-41"/></g><g transform="matrix(.0091,0,0,-0.0091,15.927,-5.741)"><use xlink:href="#g50-115"/></g><g transform="matrix(.0091,0,0,-0.0091,19.814,-5.741)"><use xlink:href="#g50-42"/></g><g transform="matrix(.0091,0,0,-0.0091,10.777,3.784)"><use xlink:href="#g50-111"/></g></svg> is the <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>th hyper-Horadam number. Also, we give some special cases of these ratios such as the golden ratio and silver ratio.

Mustafa Bah&#x15f;i

S&#xfc;leyman Solak

A Symmetric Algorithm for Golden Ratio in Hyper-Horadam Numbers

We study the existence and uniqueness of mild solutions for neutral stochastic integrodifferential equations with Poisson jumps under global and local Carath&#xe9;odory conditions on the coefficients by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial value. Finally, an example is provided to illustrate the effectiveness of the obtained results.

Diem Dang Huan

The Neutral Stochastic Integrodifferential Equations with Jumps

Given a finite metabelian <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M1" height="8.39864pt" version="1.1" viewBox="-0.0498162 -8.34882 6.36303 8.39864" width="6.36303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><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></svg>-group <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2064009pt" id="M2" height="8.8423pt" version="1.1" viewBox="-0.0498162 -8.6359 9.02496 8.8423" width="9.02496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-72" d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"/></g></svg>, the object of this paper is to discuss some cases under which <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42947pt" id="M3" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 76.4453 12.7178" width="76.4453pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-69" d="M743 375C743 570 607 650 407 650H136L130 622C214 612 220 606 204 526L125 122C110 43 105 36 24 27L17 0H250C382 0 482 23 571 76C680 141 743 248 743 375ZM644 379C644 188 520 36 305 36C277 36 249 38 229 46C205 55 202 71 212 126L293 553C300 588 306 600 317 606C330 613 356 617 390 617C547 617 644 537 644 379Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.893,3.132)"><path id="g50-54" d="M158 548H390L417 615L410 623H122L83 318C105 326 143 337 185 337C296 337 350 275 350 188C350 116 308 42 225 42C164 42 122 74 100 93C90 101 82 99 72 92C60 82 51 68 50 59C48 46 52 38 66 24C82 9 125 -12 172 -12C225 -11 292 15 346 59C408 108 437 166 437 226C437 309 371 397 242 397C214 397 170 382 133 369L158 548Z"/></g><g transform="matrix(.013,0,0,-0.013,14.839,0)"><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,-0.013,19.337,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.013,0,0,-0.013,28.23,0)"><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><g transform="matrix(.013,0,0,-0.013,36.36,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,47.622,0)"><path id="g113-225" d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z"/></g><g transform="matrix(.0091,0,0,-0.0091,53.29,3.132)"><use xlink:href="#g50-54"/></g><g transform="matrix(.013,0,0,-0.013,58.237,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,62.735,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.013,0,0,-0.013,71.627,0)"><use xlink:href="#g113-42"/></g></svg>. Further, some examples of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04981995pt" id="M4" height="8.39864pt" version="1.1" viewBox="-0.0498162 -8.34882 6.36303 8.39864" width="6.36303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-51"/></g></svg> groups of class <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M5" height="8.55521pt" version="1.1" viewBox="-0.0498162 -8.34882 6.36303 8.55521" width="6.36303pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-52" d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"/></g></svg>, for which <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.18147pt" id="M6" height="12.4698pt" version="1.1" viewBox="-0.0498162 -9.28833 63.1738 12.4698" width="63.1738pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.0091,0,0,-0.0091,9.893,3.132)"><path id="g50-53" d="M462 177V227H365V632H320C217 496 116 350 21 208V177H284V109C284 43 280 38 190 31V0H451V31C369 38 365 43 365 108V177H462ZM284 227H88C155 336 218 430 282 519H284V227Z"/></g><g transform="matrix(.013,0,0,-0.013,14.839,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,19.337,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.013,0,0,-0.013,28.23,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,44.032,0)"><path id="g117-145" d="M-161 592L-215 609L-427 -83L-372 -100L-161 592Z"/></g><g transform="matrix(.013,0,0,-0.013,36.36,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,47.651,0)"><path id="g113-124" d="M293 -169V-141C218 -131 209 -90 209 -44C209 19 222 85 222 152C222 207 198 255 139 269V273C199 286 222 334 222 388C222 454 209 523 209 588C209 632 218 671 293 681V709C234 709 148 695 148 577C147 513 155 438 155 372C155 337 152 291 64 285V256C152 250 155 204 155 169C156 105 148 31 148 -41C148 -157 234 -169 293 -169Z"/></g><g transform="matrix(.013,0,0,-0.013,52.162,0)"><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><g transform="matrix(.013,0,0,-0.013,58.402,0)"><path id="g113-126" d="M283 255V284C195 290 192 337 192 375C192 436 200 508 200 580C200 696 116 709 54 709V681C127 671 139 634 139 588C138 524 125 452 125 387C125 333 149 286 209 272V268C148 253 125 206 125 151C126 87 139 16 139 -47C139 -92 127 -131 54 -141V-169C115 -169 200 -152 200 -41C200 32 192 104 192 164C192 202 195 249 283 255Z"/></g></svg> but <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.29107pt" id="M7" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 63.1738 12.5794" width="63.1738pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.0091,0,0,-0.0091,9.893,3.132)"><use xlink:href="#g50-54"/></g><g transform="matrix(.013,0,0,-0.013,14.839,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,19.337,0)"><use xlink:href="#g113-72"/></g><g transform="matrix(.013,0,0,-0.013,28.23,0)"><use xlink:href="#g113-42"/></g><g transform="matrix(.013,0,0,-0.013,36.36,0)"><use xlink:href="#g117-34"/></g><g transform="matrix(.013,0,0,-0.013,47.622,0)"><use xlink:href="#g113-124"/></g><g transform="matrix(.013,0,0,-0.013,52.133,0)"><use xlink:href="#g113-50"/></g><g transform="matrix(.013,0,0,-0.013,58.373,0)"><use xlink:href="#g113-126"/></g></svg>, are discussed. 

Shalini Gupta

The Fifth Dimension Subgroup for Metabelian 2 Groups

We investigate the estimation of a multivariate continuous-discrete conditional density. We develop an adaptive estimator based on wavelet methods. We prove its good theoretical performance by determining sharp rates of convergence under the <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.47421pt" id="M1" height="14.1623pt" version="1.1" viewBox="-0.0498162 -8.68809 13.7626 14.1623" width="13.7626pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g197-23" d="M571 0V44H236V662H70V0H571ZM192 44H114V618H192V44Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.683,3.132)"><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></svg> risk with <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.42938pt" id="M2" height="11.7782pt" version="1.1" viewBox="-0.0498162 -8.34882 29.0242 11.7782" width="29.0242pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g><g transform="matrix(.013,0,0,-0.013,11.342,0)"><path id="g117-94" d="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z"/></g><g transform="matrix(.013,0,0,-0.013,22.605,0)"><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> for a wide class of unknown conditional densities. A simulation study illustrates the good practical performance of our estimator.

Christophe Chesneau

Hassan Doosti

A Note on the Adaptive Estimation of a Conditional Continuous-Discrete Multivariate Density by Wavelet Methods

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Chinese Journal of Mathematics