Some characterizations of boundedness in <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.18147pt" id="M1" height="12.4698pt" version="1.1" viewBox="-0.0498162 -9.28833 35.3035 12.4698" width="35.3035pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-79" d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,10.062,3.132)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 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270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2681pt" id="M2" height="12.8091pt" version="1.1" viewBox="-0.0498162 -10.541 36.0238 12.8091" width="36.0238pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0091,0,0,-0.0091,10.91,-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,-0.013,16.861,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,21.359,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.013,0,0,-0.013,31.318,0)"><use xlink:href="#g113-42"/></g></svg>  <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="22.001pt" style="vertical-align:-3.42947pt" id="M3" height="12.7178pt" version="1.1" viewBox="-0.0498162 -9.28833 22.001 12.7178"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><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,-0.013,14.37,0)"><path id="g117-91" d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="18.973pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="25.5831838 -9.28833 18.973 12.7178"><g transform="matrix(.013,0,0,-0.013,25.633,0)"><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,-0.013,36.975,0)"><use xlink:href="#g117-91"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="17.758pt" style="vertical-align:-3.42947pt" height="12.7178pt" version="1.1" viewBox="48.1881838 -9.28833 17.758 12.7178"><g transform="matrix(.013,0,0,-0.013,48.238,0)"><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><g transform="matrix(.013,0,0,-0.013,61.498,0)"><use xlink:href="#g113-42"/></g></svg></span> will be described, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.18147pt" id="M4" height="12.4698pt" version="1.1" viewBox="-0.0498162 -9.28833 35.3035 12.4698" width="35.3035pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0091,0,0,-0.0091,10.062,3.132)"><use xlink:href="#g50-43"/></g><g transform="matrix(.013,0,0,-0.013,16.141,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,20.639,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.013,0,0,-0.013,30.597,0)"><use xlink:href="#g113-42"/></g></svg> denote the Smirnov class and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.2681pt" id="M5" height="12.8091pt" version="1.1" viewBox="-0.0498162 -10.541 36.0238 12.8091" width="36.0238pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-79"/></g><g transform="matrix(.0091,0,0,-0.0091,10.91,-5.741)"><use xlink:href="#g50-113"/></g><g transform="matrix(.013,0,0,-0.013,16.861,0)"><use xlink:href="#g113-41"/></g><g transform="matrix(.013,0,0,-0.013,21.359,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.013,0,0,-0.013,31.318,0)"><use xlink:href="#g113-42"/></g></svg> the Privalov class on the upper half plane <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="21.221pt" style="vertical-align:-2.26807pt" id="M6" height="11.5564pt" version="1.1" viewBox="-0.0498162 -9.28833 21.221 11.5564"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-69"/></g><g transform="matrix(.013,0,0,-0.013,13.59,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="21.482pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="24.803183800000003 -9.28833 21.482 11.5564"><g transform="matrix(.013,0,0,-0.013,24.853,0)"><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,-0.013,29.364,0)"><path id="g113-123" d="M475 445L452 468C423 436 401 426 385 426S352 427 305 436C268 443 216 448 185 448C156 448 128 425 102 382C89 360 77 338 64 304L89 291C117 340 143 380 190 380C214 380 268 376 295 373C320 370 344 371 362 374L70 101C32 65 23 36 23 18C23 5 27 -3 31 -8C36 -8 39 -6 42 -1C52 15 67 31 80 40S110 49 143 33C195 8 257 -20 290 -20C339 -20 382 11 439 129L416 148C366 74 334 55 305 55C278 55 236 68 203 81C169 94 141 100 122 96C205 170 348 308 411 376L475 445Z"/></g><g transform="matrix(.013,0,0,-0.013,39.471,0)"><path id="g117-173" d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="53.089pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="49.9171838 -9.28833 53.089 11.5564"><g transform="matrix(.013,0,0,-0.013,49.967,0)"><path id="g13-65" d="M625 186C574 85 512 32 429 32C293 32 194 153 194 337C194 485 274 621 419 621C496 621 573 586 606 468L643 475C636 537 629 583 620 638C593 644 519 665 438 665C199 665 42 523 42 317C42 158 152 -15 418 -15C486 -15 578 5 604 11C621 46 647 122 660 171L625 186Z"/></g><g transform="matrix(.013,0,0,-0.013,62.387,0)"><path id="g113-9" d="M162 -163V703H101V-163H162Z"/></g><g transform="matrix(.013,0,0,-0.013,69.438,0)"><path id="g204-42" d="M77 28C79 11 88 2 105 0H420C436 1 447 13 448 28V37C447 53 436 62 420 65H299V546H420C436 548 445 557 448 574V583C447 599 436 608 420 611H105C88 608 79 599 77 583V574C79 557 88 548 105 546H226V65H105C88 63 79 54 77 37V28Z"/></g><g transform="matrix(.013,0,0,-0.013,76.263,0)"><path id="g204-78" d="M-6 28C-3 11 7 2 22 0H140C156 2 165 11 168 28V37C165 54 156 63 140 65H111V244C111 301 136 372 196 372C220 372 232 344 232 288V65H214C197 63 188 54 186 37V28C188 11 197 2 214 0H321C336 2 346 11 349 28V37C346 54 337 63 321 65H292V244C292 301 319 372 377 372C401 372 413 344 413 288V65H395C378 63 369 54 367 37V28C369 11 378 2 395 0H502C517 2 527 11 530 28V37C527 54 517 63 502 65H473V292C473 389 443 437 383 437C342 437 307 420 278 385C263 417 234 437 202 437C168 437 138 425 111 401C111 417 100 430 83 431H22C7 429 -3 420 -6 403V394C-3 377 7 368 22 366H51V65H22C7 63 -3 54 -6 37V28Z"/></g><g transform="matrix(.013,0,0,-0.013,85.318,0)"><use xlink:href="#g113-123"/></g><g transform="matrix(.013,0,0,-0.013,95.425,0)"><path id="g117-92" d="M512 230V281L75 514V456L453 256V254L75 55V-3L512 230Z"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="10.751pt" style="vertical-align:-2.26807pt" height="11.5564pt" version="1.1" viewBox="106.63818380000001 -9.28833 10.751 11.5564"><g transform="matrix(.013,0,0,-0.013,106.688,0)"><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><g transform="matrix(.013,0,0,-0.013,112.928,0)"><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></span>, respectively.

Yasuo Iida

Bounded Subsets of Smirnov and Privalov Classes on the Upper Half Plane

We investigate the structure of symmetric solutions of the matrix equation <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="30.343pt" style="vertical-align:-0.0498209pt" id="M2" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 30.343 8.68572"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.013,0,0,-0.013,9.135,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.013,0,0,-0.013,22.712,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="7.813pt" style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="33.9251838 -8.6359 7.813 8.68572"><g transform="matrix(.013,0,0,-0.013,33.975,0)"><use xlink:href="#g113-67"/></g></svg></span>, where <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M3" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 9.2729 8.68572" width="9.2729pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-66"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M4" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 7.94191 8.68572" width="7.94191pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-67"/></g></svg> are <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 10.3951 6.1673" width="10.3951pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g></svg>-by-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M6" 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)"><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></svg> matrices over a principal ideal domain <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1672497pt" id="M7" height="8.55523pt" version="1.1" viewBox="-0.0498162 -8.38798 8.33339 8.55523" width="8.33339pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g121-80" d="M631 18C609 24 585 35 559 65C534 91 514 117 478 169C448 214 406 281 389 313C462 346 516 399 516 485C516 545 490 590 449 616C412 641 363 650 290 650H42V622C120 615 128 612 128 527V125C128 40 120 34 38 28V0H300V28C221 34 212 40 212 125V284H244C295 284 312 272 329 244C359 195 395 133 430 84C475 19 516 -3 592 -7C603 -8 615 -8 627 -8L631 18ZM212 316V563C212 591 215 602 223 607C231 613 248 617 277 617C352 617 423 577 423 469C423 415 407 375 368 345C343 324 310 316 260 316H212Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M8" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 10.0819 8.68572" width="10.0819pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-89"/></g></svg> is unknown <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M9" 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>-by-<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063904pt" id="M10" 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> matrix over <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1672497pt" id="M11" height="8.55523pt" version="1.1" viewBox="-0.0498162 -8.38798 8.33339 8.55523" width="8.33339pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-80"/></g></svg>.</span> We prove that matrix equation <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="30.343pt" style="vertical-align:-0.0498209pt" id="M12" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 30.343 8.68572"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.013,0,0,-0.013,9.135,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.013,0,0,-0.013,22.712,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="7.813pt" style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="33.9251838 -8.6359 7.813 8.68572"><g transform="matrix(.013,0,0,-0.013,33.975,0)"><use xlink:href="#g113-67"/></g></svg></span> over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1672497pt" id="M13" height="8.55523pt" version="1.1" viewBox="-0.0498162 -8.38798 8.33339 8.55523" width="8.33339pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-80"/></g></svg> has a symmetric solution if and only if equation <span class="inline_break"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="30.343pt" style="vertical-align:-0.0498209pt" id="M14" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 30.343 8.68572"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.013,0,0,-0.013,9.135,0)"><use xlink:href="#g113-89"/></g><g transform="matrix(.013,0,0,-0.013,22.712,0)"><use xlink:href="#g117-34"/></g></svg><span class="irelop"></span><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="7.813pt" style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="33.9251838 -8.6359 7.813 8.68572"><g transform="matrix(.013,0,0,-0.013,33.975,0)"><use xlink:href="#g113-67"/></g></svg></span> has a solution over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.1672497pt" id="M15" height="8.55523pt" version="1.1" viewBox="-0.0498162 -8.38798 8.33339 8.55523" width="8.33339pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g121-80"/></g></svg> and the matrix <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.04979992pt" id="M16" height="11.97pt" version="1.1" viewBox="-0.0498162 -11.9202 23.5048 11.97" width="23.5048pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-66"/></g><g transform="matrix(.013,0,0,-0.013,9.135,0)"><use xlink:href="#g113-67"/></g><g transform="matrix(.0091,0,0,-0.0091,16.948,-5.741)"><path id="g50-85" d="M655 676H628C608 656 600 650 571 650H156C118 650 114 651 102 676H77C63 615 45 548 24 485L60 486C81 531 97 562 114 580C134 603 148 611 252 611H308L217 127C201 41 196 38 105 32L98 0H384L393 32C300 38 288 41 304 127L393 611H466C553 611 571 604 582 580C592 559 597 530 595 485L630 489C636 548 645 624 655 676Z"/></g></svg> is symmetric. If symmetric solution exists we propose the method for its construction. 

V. M. Prokip

The Structure of Symmetric Solutions of the Matrix Equation <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 42.0208 8.68572" width="42.0208pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z" /></g><g transform="matrix(.013,0,0,-0.013,9.135,0)"><path id="g113-89" d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z" /></g><g transform="matrix(.013,0,0,-0.013,22.712,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" /></g><g transform="matrix(.013,0,0,-0.013,33.975,0)"><path id="g113-67" d="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z" /></g></svg> over a Principal Ideal Domain

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

B. Z. Shavarovskii

Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices

The aim of this paper is to present a new improved semilocal and local convergence analysis for two-step secant method to approximate a locally unique solution of a nonlinear equation in Banach spaces. This study is important because starting points play an important role in the convergence of an iterative method. We have used a combination of Lipschitz and center-Lipschitz conditions on the Fr&#xe9;chet derivative instead of only Lipschitz condition. A comparison is established on different types of center conditions and the influence of our approach is shown through the numerical examples. In comparison to some earlier study, it gives an improved domain of convergence along with the precise error bounds. Finally, some numerical examples including nonlinear elliptic differential equations and integral equations validate the efficacy of our approach.

Abhimanyu Kumar

D. K. Gupta

Shwetabh Srivastava

Influence of the Center Condition on the Two-Step Secant Method

Qing Zou

Comment on &#x201c;Some Congruence Properties of a Restricted Bipartition Function <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-3.29107pt" height="12.5794pt" version="1.1" viewBox="-0.0498162 -9.28833 28.1489 12.5794" width="28.1489pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-100" d="M383 397C383 424 344 448 299 448C244 448 172 409 132 375C66 319 23 227 23 146C23 42 74 -12 146 -12C208 -12 298 30 359 103L343 124C315 95 248 48 192 48C145 48 111 85 111 163C111 228 129 294 151 330C171 363 201 401 241 401C275 401 302 384 325 356C332 347 339 344 348 348C373 360 383 381 383 397Z" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="406" vert-adv-y="406" XplusCharwidth="5.278" /></g><g transform="matrix(.0091,0,0,-0.0091,4.173,3.132)"><path id="g50-79" d="M870 650H622L613 619C694 615 708 606 708 559C708 531 702 477 686 396L636 136H633L281 650H127L120 619C180 616 198 611 212 592C227 572 232 558 220 503L166 254C148 169 139 131 126 89C112 43 85 35 33 32L24 0H280L286 32C208 37 189 44 189 93C189 124 198 179 212 250L267 526H271L622 -8H655L732 396C747 477 762 532 770 556C785 600 798 616 862 619L870 650Z" /><glyph.data ascent="3443" descent="-2856" horiz-adv-x="888" vert-adv-y="888" XplusCharwidth="12.25377336" /></g><g transform="matrix(.013,0,0,-0.013,12.453,0)"><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" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" XplusCharwidth="16.951" /></g><g transform="matrix(.013,0,0,-0.013,16.951,0)"><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" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="502" vert-adv-y="502" XplusCharwidth="23.477" /></g><g transform="matrix(.013,0,0,-0.013,23.478,0)"><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" /><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346" XplusCharwidth="27.976000000000003" /></g></svg>&#x201d;

We introduce the concept of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.291101pt" id="M1" height="11.927pt" version="1.1" viewBox="-0.0498162 -8.6359 14.462 11.927" width="14.462pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-75" d="M429 650H175L169 622C251 614 259 612 244 532L174 152C163 91 152 37 135 -6C109 -72 67 -101 23 -113L33 -144C45 -143 73 -134 102 -119C186 -76 229 -1 254 133L327 532C341 609 347 613 423 622L429 650Z"/></g><g transform="matrix(.0091,0,0,-0.0091,5.031,3.132)"><path id="g50-98" d="M490 97L476 124C442 96 405 70 398 70C392 70 390 78 396 114C419 243 448 379 463 432L457 436C446 436 431 439 418 442C393 447 368 451 343 451C281 451 204 418 155 381C74 320 24 206 24 107C24 23 59 -12 88 -12C118 -12 155 5 191 34C236 70 290 122 328 177H330L312 84C296 0 311 -12 331 -12C355 -12 425 24 490 97ZM374 387C371 367 360 299 347 264C323 202 187 53 142 53C128 53 113 73 113 120C113 224 157 332 221 380C241 395 274 403 303 403C330 403 360 395 374 387Z"/></g><g transform="matrix(.0091,0,0,-0.0091,9.71,3.132)"><path id="g185-40" d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"/></g></svg>-distance (an analogue of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M2" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-metric), <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.47418pt" id="M3" height="14.4363pt" version="1.1" viewBox="-0.0498162 -8.96212 13.0971 14.4363" width="13.0971pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-243" d="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z"/></g><g transform="matrix(.0091,0,0,-0.0091,7.02,3.132)"><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>-proximal contraction, and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-5.47418pt" id="M4" height="14.4363pt" version="1.1" viewBox="-0.0498162 -8.96212 13.0971 14.4363" width="13.0971pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-243"/></g><g transform="matrix(.0091,0,0,-0.0091,7.02,3.132)"><use xlink:href="#g50-113"/></g></svg>-proximal cyclic contraction for non-self-mappings in Hausdorff uniform spaces. We investigate the existence and uniqueness of best proximity points for these modified contractive mappings. The results obtained extended and generalised some fixed and best proximity points results in literature. Examples are given to validate the main results.

Victoria Olisama

Johnson Olaleru

Hudson Akewe

Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces

The purpose of this paper is to introduce new concepts of <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 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>-admissible Geraghty type generalized <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M4" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g></svg>-contraction and to prove that some fixed point results for such mappings are in the perspective of partial <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2063999pt" id="M5" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-99"/></g></svg>-metric space. As an application, we inaugurate new fixed point results for Geraghty type generalized graphic <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.0498209pt" id="M6" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-71"/></g></svg>-contraction defined on partial metric space endowed with a directed graph. On the other hand, one more application to the existence and uniqueness of a solution for the first-order periodic boundary value problem is also provided. Our findings encompass various generalizations of the Banach contraction principle on metric space, partial metric space, and partial <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 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g113-99"/></g></svg>-metric space. Moreover, some examples are presented to illustrate the usability of the new theory.

Deepak Singh

Varsha Chauhan

R. Wangkeeree

Geraghty Type Generalized <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.0498209pt" height="8.68572pt" version="1.1" viewBox="-0.0498162 -8.6359 8.02022 8.68572" width="8.02022pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z" /></g></svg>-Contractions and Related Applications in Partial <svg  xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"   style="vertical-align:-0.2063999pt" height="9.49473pt" version="1.1" viewBox="-0.0498162 -9.28833 6.21949 9.49473" width="6.21949pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z" /></g></svg>-Metric Spaces

For Banach algebras <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455397pt" id="M1" height="9.09021pt" version="1.1" viewBox="-0.0498162 -8.84467 11.2563 9.09021" width="11.2563pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-2" d="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z"/></g></svg> and <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455397pt" id="M2" height="9.25986pt" version="1.1" viewBox="-0.0498162 -9.01432 12.496 9.25986" width="12.496pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-3" d="M895 660C863 648 835 636 808 620C736 668 639 687 564 687C395 687 153 589 153 403C153 307 221 260 300 260C447 260 534 401 534 561H514C506 396 415 290 305 290C234 290 194 335 194 396C194 555 383 657 566 657C626 657 716 643 776 601C669 531 592 432 501 271C389 71 313 16 189 16C118 16 69 60 69 95H71C78 84 93 83 101 83C129 83 144 106 144 131C144 160 121 179 93 179C60 179 34 150 34 107C34 41 100 -15 207 -15C365 -15 509 76 621 311C626 300 647 292 668 292C675 292 701 294 717 298C752 278 763 243 763 203C763 140 734 78 685 49C666 38 647 34 624 34C596 34 574 55 574 82C574 127 623 171 679 176L675 199C586 184 537 132 537 67C537 32 567 -11 631 -11C672 -11 710 0 743 22C804 63 839 127 839 186C839 234 813 289 749 316V318C837 329 902 398 902 463C902 524 873 566 830 601C850 615 874 628 902 641L895 660ZM841 485C841 420 802 346 717 338C697 349 675 352 662 352C655 352 644 351 639 348C691 449 736 524 801 580C825 555 841 524 841 485Z"/></g></svg>, we show that if <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3499298pt" id="M3" height="9.36425pt" version="1.1" viewBox="-0.0498162 -9.01432 63.0336 9.36425" width="63.0336pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-22" d="M805 658H719L589 415C537 319 349 15 237 15C214 15 208 28 208 43C208 59 219 86 247 130L433 417C464 464 471 506 471 541C471 638 384 687 292 687C164 687 36 597 36 474C36 406 83 360 151 360C238 360 296 442 309 509H289C265 440 225 390 150 390C103 390 77 426 77 474C77 582 175 652 283 652C340 652 404 619 404 568C404 541 397 512 368 467L223 240C170 155 153 111 153 73C153 18 186 -15 234 -15C290 -15 367 41 439 138L442 137C427 106 425 83 425 66C425 18 463 -12 511 -12C575 -12 638 42 714 134L693 147C621 63 565 18 525 18C506 18 492 28 492 53C492 60 493 77 509 108L805 658Z"/></g><g transform="matrix(.013,0,0,-0.013,14.578,0)"><path id="g117-34" d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"/></g><g transform="matrix(.013,0,0,-0.013,25.841,0)"><use xlink:href="#g198-2"/></g><g transform="matrix(.013,0,0,-0.013,39.861,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"/></g><g transform="matrix(.013,0,0,-0.013,50.398,0)"><use xlink:href="#g198-3"/></g></svg> is unital and commutative, each bi-Jordan homomorphism from <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455397pt" id="M4" height="9.25986pt" version="1.1" viewBox="-0.0498162 -9.01432 11.0867 9.25986" width="11.0867pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><use xlink:href="#g198-22"/></g></svg> into a semisimple commutative Banach algebra <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2455397pt" id="M5" height="9.25986pt" version="1.1" viewBox="-0.0498162 -9.01432 11.6478 9.25986" width="11.6478pt"><g transform="matrix(.013,0,0,-0.013,0,0)"><path id="g198-5" d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z"/></g></svg> is a bihomomorphism.

Abbas Zivari-Kazempour

A Characterization of Bi-Jordan Homomorphisms on Banach Algebras

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International Journal of Analysis