| Date post: | 08-Jul-2018 |
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of 20
8/19/2019 f(x) = k ^ 1:x
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f(x) = xth root of a For a=2x y
0 Infinity1 22 1
3 0.66666666674 0.55 0.46 0.33333333337 0.28571428578 0.259 0.2222222222
10 0.2
11 0.181818181812 0.166666666713 0.153846153814 0.142857142915 0.133333333316 0.12517 0.117647058818 0.111111111119 0.1052631579
20 0.121 0.095238095222 0.090909090923 0.086956521724 0.083333333325 0.0826 0.076923076927 0.0740740741
28 0.071428571429 0.068965517230 0.066666666731 0.06451612932 0.062533 0.060606060634 0.058823529435 0.0571428571
0 50 1000
0.5
1
1.5
2
2.5
8/19/2019 f(x) = k ^ 1:x
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36 0.055555555637 0.054054054138 0.052631578939 0.051282051340 0.05
41 0.048780487842 0.047619047643 0.046511627944 0.045454545545 0.044444444446 0.043478260947 0.042553191548 0.0416666667
49 0.040816326550 0.0451 0.039215686352 0.038461538553 0.037735849154 0.03703703755 0.036363636456 0.035714285757 0.0350877193
58 0.034482758659 0.033898305160 0.033333333361 0.032786885262 0.032258064563 0.031746031764 0.0312565 0.0307692308
66 0.030303030367 0.029850746368 0.029411764769 0.028985507270 0.028571428671 0.028169014172 0.027777777873 0.0273972603
8/19/2019 f(x) = k ^ 1:x
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74 0.02702702775 0.026666666776 0.026315789577 0.02597402678 0.0256410256
79 0.025316455780 0.02581 0.02469135882 0.024390243983 0.024096385584 0.023809523885 0.023529411886 0.023255814
87 0.022988505788 0.022727272789 0.022471910190 0.022222222291 0.02197802292 0.021739130493 0.021505376394 0.021276595795 0.0210526316
96 0.020833333397 0.020618556798 0.020408163399 0.0202020202
100 0.02101 0.0198019802102 0.0196078431103 0.0194174757
104 0.0192307692105 0.019047619106 0.0188679245107 0.0186915888108 0.0185185185109 0.0183486239110 0.0181818182111 0.018018018
8/19/2019 f(x) = k ^ 1:x
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112 0.0178571429113 0.017699115114 0.0175438596115 0.0173913043116 0.0172413793
117 0.0170940171118 0.0169491525119 0.0168067227120 0.0166666667121 0.0165289256122 0.0163934426123 0.0162601626124 0.0161290323
125 0.016126 0.0158730159127 0.0157480315128 0.015625129 0.015503876130 0.0153846154131 0.0152671756132 0.0151515152133 0.015037594
134 0.0149253731135 0.0148148148136 0.0147058824137 0.0145985401138 0.0144927536139 0.0143884892140 0.0142857143141 0.0141843972
142 0.014084507143 0.013986014144 0.0138888889145 0.0137931034146 0.0136986301147 0.0136054422148 0.0135135135149 0.0134228188
8/19/2019 f(x) = k ^ 1:x
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150 0.0133333333151 0.0132450331152 0.0131578947153 0.0130718954154 0.012987013
155 0.0129032258156 0.0128205128157 0.0127388535158 0.0126582278159 0.0125786164160 0.0125161 0.0124223602162 0.012345679
163 0.0122699387164 0.012195122165 0.0121212121166 0.0120481928167 0.0119760479168 0.0119047619169 0.0118343195170 0.0117647059171 0.0116959064
172 0.011627907173 0.0115606936174 0.0114942529175 0.0114285714176 0.0113636364177 0.011299435178 0.0112359551179 0.0111731844
180 0.0111111111181 0.0110497238182 0.010989011183 0.0109289617184 0.0108695652185 0.0108108108186 0.0107526882187 0.0106951872
8/19/2019 f(x) = k ^ 1:x
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188 0.0106382979189 0.0105820106190 0.0105263158191 0.0104712042192 0.0104166667
193 0.0103626943194 0.0103092784195 0.0102564103196 0.0102040816197 0.0101522843198 0.0101010101199 0.0100502513200 0.01
201 0.0099502488202 0.0099009901203 0.0098522167204 0.0098039216205 0.0097560976206 0.0097087379207 0.0096618357208 0.0096153846209 0.009569378
210 0.0095238095211 0.009478673212 0.0094339623213 0.0093896714214 0.0093457944215 0.0093023256216 0.0092592593217 0.0092165899
218 0.0091743119219 0.0091324201220 0.0090909091221 0.0090497738222 0.009009009223 0.0089686099224 0.0089285714225 0.0088888889
8/19/2019 f(x) = k ^ 1:x
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226 0.0088495575227 0.0088105727228 0.0087719298229 0.0087336245230 0.0086956522
231 0.0086580087232 0.0086206897233 0.008583691234 0.0085470085235 0.0085106383236 0.0084745763237 0.0084388186238 0.0084033613
239 0.0083682008240 0.0083333333241 0.0082987552242 0.0082644628243 0.0082304527244 0.0081967213245 0.0081632653246 0.0081300813247 0.008097166
248 0.0080645161249 0.0080321285250 0.008251 0.0079681275252 0.0079365079253 0.0079051383254 0.0078740157255 0.0078431373
256 0.0078125257 0.0077821012258 0.007751938259 0.0077220077260 0.0076923077261 0.0076628352262 0.0076335878263 0.0076045627
8/19/2019 f(x) = k ^ 1:x
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264 0.0075757576265 0.0075471698266 0.007518797267 0.0074906367268 0.0074626866
269 0.0074349442270 0.0074074074271 0.0073800738272 0.0073529412273 0.0073260073274 0.0072992701275 0.0072727273276 0.0072463768
277 0.0072202166278 0.0071942446279 0.0071684588280 0.0071428571281 0.0071174377282 0.0070921986283 0.0070671378284 0.0070422535285 0.0070175439
286 0.006993007287 0.0069686411288 0.0069444444289 0.0069204152290 0.0068965517291 0.0068728522292 0.0068493151293 0.0068259386
294 0.0068027211295 0.006779661296 0.0067567568297 0.0067340067298 0.0067114094299 0.0066889632300 0.0066666667301 0.0066445183
8/19/2019 f(x) = k ^ 1:x
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302 0.0066225166303 0.0066006601304 0.0065789474305 0.006557377306 0.0065359477
307 0.006514658308 0.0064935065309 0.0064724919310 0.0064516129311 0.0064308682312 0.0064102564313 0.0063897764314 0.0063694268
315 0.0063492063316 0.0063291139317 0.0063091483318 0.0062893082319 0.0062695925320 0.00625321 0.0062305296322 0.0062111801323 0.0061919505
324 0.0061728395325 0.0061538462326 0.0061349693327 0.006116208328 0.006097561329 0.0060790274330 0.0060606061331 0.0060422961
332 0.0060240964333 0.006006006334 0.005988024335 0.0059701493336 0.005952381337 0.0059347181338 0.0059171598339 0.005899705
8/19/2019 f(x) = k ^ 1:x
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340 0.0058823529341 0.0058651026342 0.0058479532343 0.0058309038344 0.0058139535
345 0.0057971014346 0.0057803468347 0.0057636888348 0.0057471264349 0.005730659350 0.0057142857351 0.0056980057352 0.0056818182
353 0.0056657224354 0.0056497175355 0.0056338028356 0.0056179775357 0.0056022409358 0.0055865922359 0.0055710306360 0.0055555556361 0.0055401662
362 0.0055248619363 0.0055096419364 0.0054945055365 0.0054794521366 0.0054644809367 0.0054495913368 0.0054347826369 0.0054200542
370 0.0054054054371 0.0053908356372 0.0053763441373 0.0053619303374 0.0053475936375 0.0053333333376 0.0053191489377 0.0053050398
8/19/2019 f(x) = k ^ 1:x
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378 0.0052910053379 0.0052770449380 0.0052631579381 0.0052493438382 0.0052356021
383 0.0052219321384 0.0052083333385 0.0051948052386 0.0051813472387 0.0051679587388 0.0051546392389 0.0051413882390 0.0051282051
391 0.0051150895392 0.0051020408393 0.0050890585394 0.0050761421395 0.0050632911396 0.0050505051397 0.0050377834398 0.0050251256399 0.0050125313
400 0.005401 0.0049875312402 0.0049751244403 0.0049627792404 0.004950495405 0.0049382716406 0.0049261084407 0.0049140049
408 0.0049019608409 0.0048899756410 0.0048780488411 0.00486618412 0.0048543689413 0.004842615414 0.0048309179415 0.0048192771
8/19/2019 f(x) = k ^ 1:x
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416 0.0048076923417 0.0047961631418 0.004784689419 0.0047732697420 0.0047619048
421 0.0047505938422 0.0047393365423 0.0047281324424 0.0047169811425 0.0047058824426 0.0046948357427 0.0046838407428 0.0046728972
429 0.0046620047430 0.0046511628431 0.0046403712432 0.0046296296433 0.0046189376434 0.0046082949435 0.0045977011436 0.004587156437 0.004576659
438 0.00456621
8/19/2019 f(x) = k ^ 1:x
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150 200 250 300 350 400 450 500
f(x) = 2^(1/x)
y Infnit
8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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For a=3x y
0 infinity1 32 1.7320508076
3 1.44224957034 1.3160740135 1.24573093966 1.20093695527 1.16993081288 1.14720269049 1.1298309639
10 1.116123174
11 1.105031503412 1.095872691113 1.088182243514 1.081633400415 1.075989624716 1.071075483117 1.066758117118 1.062935070419 1.0595260647
20 1.056467308521 1.053707472122 1.051204786623 1.048924917624 1.046839381725 1.044924351126 1.043159740127 1.0415284982
28 1.040016057729 1.038609897830 1.037299197331 1.036074555632 1.034927767133 1.033851637234 1.032839831335 1.0318867499
0 50 1000
0.5
1
1.5
2
2.5
3
3.5
8/19/2019 f(x) = k ^ 1:x
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36 1.03098742537 1.030137433638 1.029332825139 1.028570059440 1.0278459557
41 1.027157647142 1.026502543643 1.025878299244 1.025282783745 1.024714058646 1.024170355847 1.023650059648 1.0231516905
49 1.022673890950 1.022215413351 1.021775108852 1.021351917953 1.020944861954 1.020553035555 1.020175599556 1.019811775657 1.0194608405
58 1.019122121259 1.018794990760 1.018478864461 1.018173196362 1.017877475763 1.01759122564 1.017313996365 1.0170453698
66 1.016784951367 1.016532370268 1.016287277969 1.016049346470 1.015818266271 1.015593745672 1.015375509373 1.0151632971
8/19/2019 f(x) = k ^ 1:x
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74 1.014956862975 1.01475597476 1.014560409877 1.014369961478 1.0141844307
79 1.014003629680 1.013827379681 1.01365551182 1.013487862383 1.013324280184 1.013164618285 1.013008737486 1.0128565048
87 1.012707793988 1.012562483989 1.012420459490 1.012281610391 1.012145831392 1.012013021593 1.011883084894 1.011755928995 1.0116314654
96 1.011509609797 1.011390280798 1.011273400799 1.011158895
100 1.0110466919101 1.0109367228102 1.0108289216103 1.0107232248
104 1.0106195713105 1.0105179025106 1.0104181619107 1.0103202952108 1.0102242501109 1.0101299761110 1.0100374248111 1.0099465493
8/19/2019 f(x) = k ^ 1:x
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112 1.0098573046113 1.0097696471114 1.0096835348115 1.0095989273116 1.0095157855
117 1.0094340716118 1.009353749119 1.0092747827120 1.0091971385121 1.0091207836122 1.0090456859123 1.0089718148124 1.0088991405
125 1.0088276341126 1.0087572676127 1.0086880141128 1.0086198473129 1.0085527418130 1.0084866731131 1.0084216173132 1.0083575513133 1.0082944527
134 1.0082322997135 1.0081710713136 1.0081107469137 1.0080513066138 1.0079927313139 1.0079350021140 1.0078781008141 1.0078220098
142 1.0077667119143 1.0077121903144 1.0076584289145 1.0076054118146 1.0075531237147 1.0075015497148 1.0074506752149 1.0074004861
8/19/2019 f(x) = k ^ 1:x
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150 1.0073509686151 1.0073021094152 1.0072538954153 1.0072063139154 1.0071593525
155 1.0071129993156 1.0070672424157 1.0070220704158 1.0069774722159 1.006933437160 1.0068899541161 1.0068470132162 1.0068046042
163 1.0067627173164 1.0067213429165 1.0066804718166 1.0066400946167 1.0066002027168 1.0065607872169 1.0065218396170 1.0064833518171 1.0064453155
172 1.0064077229173 1.0063705663174 1.0063338382175 1.0062975311176 1.0062616379177 1.0062261515178 1.0061910651179 1.0061563719
180 1.0061220653181 1.006088139182 1.0060545866183 1.006021402184 1.0059885792185 1.0059561123186 1.0059239956187 1.0058922233
8/19/2019 f(x) = k ^ 1:x
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188 1.00586079189 1.0058296903190 1.005798919191 1.0057684707192 1.0057383406
193 1.0057085235194 1.0056790148195 1.0056498095196 1.0056209031197 1.0055922909198 1.0055639686199 1.0055359317200 1.005508176
201 1.0054806971202 1.0054534911203 1.0054265539204 1.0053998814205 1.0053734699206 1.0053473155207 1.0053214144208 1.0052957631209 1.0052703579
210 1.0052451952211 1.0052202717212 1.0051955839213 1.0051711285214 1.0051469023215 1.005122902216 1.0050991245217 1.0050755667
218 1.0050522256219 1.0050290981220 1.0050061815221 1.0049834727222 1.0049609691223 1.0049386677224 1.004916566225 1.0048946612
8/19/2019 f(x) = k ^ 1:x
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226 1.0048729507227 1.004851432228 1.0048301025229 1.0048089597230 1.0047880012
231 1.0047672246232 1.0047466275233 1.0047262077234 1.0047059627235 1.0046858905236 1.0046659888237 1.0046462554238 1.0046266882
239 1.0046072852240 1.0045880442241 1.0045689632242 1.0045500403243 1.0045312735244 1.0045126609245 1.0044942006246 1.0044758906247 1.0044577293
248 1.0044397147249 1.0044218452250 1.0044041189251 1.0043865342252 1.0043690893253 1.0043517827254 1.0043346126255 1.0043175775
256 1.0043006757257 1.0042839058258 1.0042672661259 1.0042507552260 1.0042343716261 1.0042181138262 1.0042019803263 1.0041859698
8/19/2019 f(x) = k ^ 1:x
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264 1.0041700809265 1.0041543121266 1.0041386621267 1.0041231296268 1.0041077132
269 1.0040924117270 1.0040772238271 1.0040621481272 1.0040471836273 1.0040323289274 1.0040175828275 1.0040029442276 1.0039884119
277 1.0039739848278 1.0039596616279 1.0039454413280 1.0039313228281 1.0039173049282 1.0039033867283 1.003889567284 1.0038758449285 1.0038622192
286 1.003848689287 1.0038352532288 1.0038219109289 1.0038086612290 1.003795503291 1.0037824354292 1.0037694575293 1.0037565683
294 1.0037437669295 1.0037310526296 1.0037184243297 1.0037058811298 1.0036934224299 1.0036810471300 1.0036687544301 1.0036565436
8/19/2019 f(x) = k ^ 1:x
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302 1.0036444138303 1.0036323642304 1.0036203941305 1.0036085025306 1.0035966889
307 1.0035849523308 1.0035732921309 1.0035617074310 1.0035501977311 1.0035387621312 1.0035273999313 1.0035161105314 1.0035048931
315 1.003493747316 1.0034826716317 1.0034716662318 1.0034607302319 1.0034498628320 1.0034390635321 1.0034283315322 1.0034176664323 1.0034070673
324 1.0033965339325 1.0033860653326 1.0033756611327 1.0033653206328 1.0033550433329 1.0033448286330 1.0033346758331 1.0033245846
332 1.0033145542333 1.0033045841334 1.0032946739335 1.0032848229336 1.0032750307337 1.0032652966338 1.0032556203339 1.0032460011
8/19/2019 f(x) = k ^ 1:x
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340 1.0032364386341 1.0032269323342 1.0032174817343 1.0032080862344 1.0031987455
345 1.003189459346 1.0031802262347 1.0031710468348 1.0031619202349 1.003152846350 1.0031438237351 1.0031348529352 1.0031259332
353 1.0031170641354 1.0031082452355 1.003099476356 1.0030907561357 1.0030820852358 1.0030734628359 1.0030648886360 1.003056362361 1.0030478827
362 1.0030394504363 1.0030310645364 1.0030227249365 1.003014431366 1.0030061825367 1.002997979368 1.0029898201369 1.0029817055
370 1.0029736349371 1.0029656078372 1.002957624373 1.002949683374 1.0029417846375 1.0029339283376 1.0029261139377 1.002918341
8/19/2019 f(x) = k ^ 1:x
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378 1.0029106093379 1.0029029185380 1.0028952682381 1.0028876581382 1.0028800879
383 1.0028725573384 1.002865066385 1.0028576136386 1.0028501999387 1.0028428246388 1.0028354874389 1.0028281879390 1.0028209259
391 1.0028137011392 1.0028065133393 1.002799362394 1.0027922471395 1.0027851683396 1.0027781253397 1.0027711178398 1.0027641456399 1.0027572084
400 1.0027503059401 1.0027434379402 1.0027366041403 1.0027298042404 1.0027230381405 1.0027163054406 1.002709606407 1.0027029395
408 1.0026963057409 1.0026897044410 1.0026831353411 1.0026765983412 1.002670093413 1.0026636193414 1.0026571769415 1.0026507656
8/19/2019 f(x) = k ^ 1:x
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416 1.0026443852417 1.0026380354418 1.002631716419 1.0026254268420 1.0026191676
421 1.0026129382422 1.0026067383423 1.0026005678424 1.0025944264425 1.002588314426 1.0025822303427 1.0025761752428 1.0025701483
429 1.0025641496430 1.0025581789431 1.0025522359432 1.0025463204433 1.0025404323434 1.0025345713435 1.0025287374436 1.0025229302437 1.0025171496
438 1.0025113955
8/19/2019 f(x) = k ^ 1:x
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150 200 250 300 350 400 450 500
f(x) = 3^(1/x)
y infnity
8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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8/19/2019 f(x) = k ^ 1:x
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![· T U V W X Y Z T [ X] ^ _ ` a b ` c d _ ` e b f g _ ` o b _ t Y b f m r l s Y b k j k t q _ b f k ` p c a _ f [ Z U X u v p Y w = x P012H 2> E / 7 P2 /K](https://static.documents.pub/doc/80x56/5bbfb14a09d3f2af1f8be2e7/-t-u-v-w-x-y-z-t-x-a-b-c-d-e-b-f-g-o-b-t-y-b-f-m-r-l-s-y.jpg)
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![arXiv:1507.07595v2 [math.OC] 6 Jan 2016 · Lkx−yk, ∀x,y ∈ Rd, and their average f is µ-strongly convex, i.e., k∇f(x)−∇f(y)k ≥ µkx−yk, ∀x,y ∈ Rd.We call κ =](https://static.documents.pub/doc/80x56/5ecd6799c47c1a7409122a82/arxiv150707595v2-mathoc-6-jan-2016-lkxayk-axy-a-rd-and-their-average.jpg)
