formulasearchengine - User contributions [en]

Auslander–Reiten theory

2014-06-19T15:24:48Z

138.38.106.89: /* External links */

A pocket knife performs a [http://Sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=%0D%0A%0D%0AFor+those+of+us+of+wish+to+spend+time+outdoors%2C+whether+or+not+it+be+camping+or+looking%2C+you+most+likely+will+notice+the+significance+of+getting+the+very+best+pocket+knife+potential.+Now+when+you+go+to+an+open+air+retailer+like+REI+%2C+there+may+be+most+likely+lots+of+of+various+pocket+knives+to+choose+from.+Some+of+these+knives+are+made+just+that+will+help+you+reduce+free+threads+in+your+garments+and+that%27s+about+it.%0D%0A%0D%0AThe+age+that+you+just+introduce+your+baby+to+a+pocket+knife+is+completely+up+to+you.+Trying+again+at+how+young+I+was%2C+I+find+it+a+bit+shocking.+My+father+obviously+thought+that+I+was+prepared.+I+certainly+was+not+one+of+the+kids+mentioned+above+and+I+think+my+father+knew+that.+If+I+had+been+in+his+place+right+this+moment+I+might+be+asking+myself+the+very+questions+posed+above.+I%27d+personally+wait+till+the+age+of+eight.+Two+years+shy+of+being+an+adolescent%2C+however+still+younger+enough+to+know+and+benefit+from+the+magic+of+being+entrusted+with+your+personal+private+%E2%80%9CExcalibur.%E2%80%9D%0D%0A%0D%0ABlades+are+often+either+constituted+of+chrome+steel+or+carbon+metal.+Knife+aficionados+typically+have+clear+preferences+of+1+over+the+opposite.+If+you%27re+buying+a+present+for+a+knife-lover%2C+be+sure+to+first+find+out+about+his+or+her+chosen+blade+materials.+If+you%27re+buying+your+first-time+knife%2C+hold+just+a+few+facts+in+mind.+Stainless-steel+knives+are+much+less+durable+and+boring+extra+quickly+than+carbon+steel+knives.++Here+is+more+info+on+%5Bhttp%3A%2F%2Fwww.thebestpocketknifereviews.com%2F+pocket+knife+Reviews%5D+look+at+our+own+web+site.+Nonetheless%2C+carbon+metal+knives+are+vulnerable+to+rust+and+stain.+Generally%2C+stainless-steel+knives+are+higher+for+many+who+could+also+be+using+their+knife+sparingly+or+in+moist+circumstances.+Carbon+metal+are+greatest+for+individuals+who+count+on+heavy+use+conditions.+Handle+Materials%0D%0A%0D%0AFolding+Pocket+Knives%2C+or+guide+folders+as+many+call+them%2C+are+extraordinarily+well-liked+knives+for+a+variety+of+reasons.+As+a+result+of+they+will+fit+snugly+in+a+pocket+or+carry+simply+in+a+pack%2C+many+individuals+have+a+number+of+folding+pocket+knives.+Among+the+many+fine+quality+manufacturers+of+folding+knives+that+we+supply+are+Benchmade+knives%2C+Spyderco+Knives%2C+and+SOG+knives.+One+of+the+vital+widespread+folding+knives+of+all+time+is+the+Benchmade+Griptillian+series+which+comes+in+a+wide+range+of+blade+shapes+and+handle+colours.+Spyderco+has+a+new+sequence+of+Value+Folders+including+the+Tenacious%2C+Persistence%2C+Resilience+and+Formidable+which+are+nicely+constructed+folder+knives+with+G10+handles+and+chrome+steel+blades.%0D%0A%0D%0AWhile+%5Bhttp%3A%2F%2Fwww.wonderhowto.com%2Fsearch%2FI%2Backnowledge%2F+I+acknowledge%5D+that+lots+of+people+use+knives+for+self+protection%2C+therefore+its+point+out+in+this+article%2C+I+do+not+sometimes+advocate+a+knife+as+a+self+defense+instrument+in+untrained+fingers.+For+that+reason%2C+I+will+not+be+recommending+any+knives+specific+to+this+use+on+this+article.+For+a+self+protection+advice%2C+I+suggest+pepper+spray%2C+as+it+will+probably+shortly+and+non-lethally+disable+a+number+of+attackers+from+a+distance+and+is+tough+to+show+and+use+towards+you.+I%27ll+yield+self+protection+knife+recommendations+to+people+who+have+the+appropriate+coaching+in+this+enviornment.+A+Quick+Word+About+Options%0D%0A%0D%0AOne+other+safety+concern+is+that+it%27s+possible+to+place+a+retractable+in+your+pocket+with+a+bit+of+bit+of+the+blade+peeking+out.+I%27ve+gotten+various+sharp+pokes+over+the+years+from+this+occurring.+These+sorts+of+things+actually+can%27t+happen+with+folding+knives.+They%27re+either+open+or+closed+and+there+is+not+any+mistaking+which+one+it+is.+Say+whats+up+to+your+pint-sized+champ.+The+Spyderco+Bug+Knife+is+the+Mighty+Mouse+of+the+pocket+knife+world.+When+it+is+closed+up%2C+it%27s+smaller+than+your+pinky+and+the+blade+is+only+about+1+half%E2%80%B3.+But+don%27t+confuse+its+adorableness+for+a+scarcity+of+ability%2C+this+mini+knife+is+constructed+to+be+a+purposeful+and+lasting+everyday+carry+tool.+Made+within+the+USA%2C+the+Bug+Knife+comes+with+a+thumb+hole+opener+within+the+leaf-formed+blade.+Hyperlink%0D%0A%0D%0AA+knife+ought+to+fulfill+the+minimum+criteria+to+be+referred+to+as+as+%E2%80%9CGreatest+On+a+regular+basis+Carry+Knife%E2%80%9D.+You+need+to+search+for+some+fundamental+characteristics+when+buying+a+finest+edc+knife+to+ensure+it+isn%27t+a+cheap+imitation.+Obviously%2C+you+don%27t+want+to+buy+a+kitchen+knife+and+use+it+for+carrying+round+and+doing+day-to-day+tasks.+One+of+the+best+everyday+carry+knife+must+be+manufactured+from+sturdy+prime+quality+stainless+steel+and+sturdy.+Some+corporations+supply+life+lengthy+warranties+and+other+offer+limited+warranties.+Pocket+knives+with+the+lengthy+warranties+are+the+best+to+select.+As+well+as%2C+always+go+together+with+the+perfect+deals+available.+Test+if+you+will+get+a+better+deal+for+that+brand+elsewhere.+href%3D%22-20%2F%22%3EKershaw+Ken+Onion+Tactical+Blur+Folding+Knife+with+Velocity+Secure+Fish+Knife+SHOP%0D%0A%0D%0ATradition.+Within+the+colonial+days%2C+pretty+much+every+man+carried+a+knife+with+him.Through+the+World+Wars%2C+pocket+knives+were+standard+issue+for+all+GIs.+When+my+dad+gave+me+my+first+pocket+knife%2C+he+gave+me+the+identical+sort+of+knife+that+he+and+my+grandfather+carry.+Discuss+to+your+grandfather%2C+discover+out+what+model+his+knife+is%2C+carry+it+too%2C+give+it+to+your+son+at+some+point.+The+first+thought+lots+of+people+have+is+%E2%80%9Cis+it+legal%3F%E2%80%9D+For+almost+all+over+the+place+in+the+USA%2C+it+is+authorized+to+hold+a+%3C+three%E2%80%9D+folding+knife%2C+even+hidden+in+your+pocket.+To+look+into+the+laws+in+your+particular+state%2C+consult+this+guide+My+6+year+outdated+son+Scotty+whittling+together+with+his+pocket+knife.+It%27s+my+outdated+Kamp-King+four+blade+Scout+Knife.+Kershaw+Ken+Onion+Blur+Pocket+Knife+Professionals+MAGNUM+Alice+Pocket+Knife%0D%0A%0D%0AOkay%2C+you%27ve+gotten+your+stone.+Now%2C+the+right+way+to+sharpen+your+knife%3F+There+are+a+number+of+steps.+In+the+beginning%2C+start+with+the+rougher+grit.+Put+mineral+oil+in+your+sharpener+%28or+soak+it%29%2C+place+the+knife+blade+flat+on+the+stone+and+raise+it+to+a+10-15+diploma+angle+depending+in+your+blade.+The+bottom+line+is+to+maintain+a+constant+angle+%28it+takes+some+apply%29.+Once+you%27re+finished+with+one+facet%2C+sharpen+the+other+aspect.+Then+sharpen+each+side+with+alternating+strokes.+The+subsequent+step+is+to+modify+over+to+the+advantageous+grit+and+repeat+the+entire+course+of.+Carried+out+%E2%80%93+your+pocket+knife+is+sharp+and+able+to+use.&create=Create crucial function] in any man’s life. They serve loads of makes use of, It is not simply the size that distinguishes this knife from the others of its kind. Buck 301 Stockman Folding Pocket Knife can outlast generations to come. Those that have bought knives by Buck Knives 50 years in the [http://thebestpocketknifereviews.com/ best tactical Pocket knives] past can vouch that knives made by this firm do serve them nicely for several a long time of use. Owning a three piece pocket knife for the price of one is simply an excellent deal for most people. Want to personal a knife that outlives its owner? Blade Steel – There are dozens of different types of blade steel out there and possibly hundreds of thousands of opinions on which is finest in your knife. The metal is made up of various quantities of carbon and iron but there are different alternate options out there like ceramic knives. Every different sort has their benefits [http://en.wikipedia.org/wiki/Blade boot knives] but it largely comes down to energy and hardness. A [http://bitcoinaddict.com/webid/item.php?id=90907&mode=1 harder knife] holds an edge higher, but if your knife is simply too onerous, it is going to be less tough; which means should you hit or drop the blade it may break. Perform over aesthetics as always. However EDC pocket knives might be like pocket jewellery, whereas nonetheless being completely purposeful, let your taste and desire information you. A non-threatening look will enable your instrument to be carried in an workplace or in a public house with out upsetting your delicate coworkers. However, if you don’t interact with society or just don’t care about individuals’s opinion on this, there are plenty of tactical and aggressive-wanting designs on the market. Though these tactical knives are undoubtedly a visible treat for the eyes with awe-inspiring manufacturing, we want the traditional/timeless/tasteful look, which is usually defined as a “gentleman’s EDC pocket knife”. I have been carrying a pocket folding knife for fairly awhile, and if you realize me, I've lost a number of. I lastly acquired an affordable mini knife to carry in my back pocket, but the constant friction from sitting is carrying out my pants. So I have been looking [http://www.thebestpocketknifereviews.com/ best pocket knife under 30] for a larger knife with a clip. Just from looking knife stores, I like the Mini Griptillian by Benchmade. But being a Benchmade knife, the couldn't justify the associated fee as I'm unsure if I'll lose that one too. Finally, I discovered a [http://crimsonfists.cba.pl/index.php?site=profile&id=772 slim knife] at REI for below $20-the Gerber Evo Jr. The 1 st one is the CKRT M1614DSFG made by the Columbia River Knife and Instrument Company. I’ve been utilizing CKRT knives for fairly a while now they usually actually do produce some prime quality knives. The CKRT M1614DSFG is no exception to the rule and gives one of [http://www.victorinox.com/ch knife making supplies] the stable grips I’ve ever seen in a tactical folding knife. It has an actual hilt upon full extension and the knife itself has a particularly sharp edge. Another factor about CKRT knives is that they all include a patented auto locking system that provides you with security towards the blade closing upon your fingers. If you're like me, who believes that each man ought to have minimal one folding knife and in case you are trying to find out best folding knife that may used in day by day life then you are at proper place. Few months in the past I started my mission to discover a excellent folding knife and ended up banging my head on each single knife evaluation which claimed that their knife is the very best among the all. Finally I got here up with a few possibly greatest folding knives that have a number of options added to them and supplies nice value to our cash. Finally I wish to deal with the serrations on the blade of this folding knife. I’m not a fan of the Veff serrations. I've to agree with several different reviewers I’ve seen that the serrations are just too skinny and fragile. Too many people have reported too many problems with them bending and chipping for me to order a knife with them on it. I’[http://sss.Chaoslab.ru/tracker/mim_plugs/newticket?reporter=anonymous&summary=AttributeError%3A+%27Environment%27+object+has+no+attribute+%27get_db_cnx%27&description=In+terms+of+using+a+knife+every+individual+might+have+distinct+provisions+and+practical+activity.+Whether+or+not+a+person+go+on+mountain+climbing+or+could+want+to+use+it+as+self-safety+if+ever+they%27re+misfortune+and+mugged+by+these+robbers%2C+they+may+need+the+right+gadget+for+every+circumstances.+Some+could+use+it+for+camping+and+others+on+household+chores.+Electricians+and+plumbers+might+also+use+it+as+a+component+for+his+or+her+job.+Hunters+may+additionally+want+to+have+this+for+excoriating+animals+on+the+sphere.%0D%0A%0D%0AThe+Previous+Timer+Knife+is+the+all+time+traditional+knife%2C+the+perfect+made+American+pocket+knife+at+this+time+with+a+historical+past+that+goes+again+almost+one+hundred+ten+years.+Not+all+pocket+knives+are+created+equal.+They+all+are+very+unique+and+have+their+very+personal+features+and+enchantment.++If+you+cherished+this+post+and+also+you+wish+to+be+given+guidance+with+regards+to+%5Bhttp%3A%2F%2Fwww.Thebestpocketknifereviews.com%2F+Pocket+Knife+reviews%5D+kindly+pay+a+visit+to+the+web-page.+%5Bhttp%3A%2F%2Fwww.google.de%2Fsearch%3Fq%3DSelecting+Selecting%5D+a+pocket+knife+can+be+a+bit+confusing+and+perhaps+even+overwhelming+with+all+the+different+options+accessible+to+choose+from.+There%27s+a+wide+variety+of+kinds%2C+handles%2C+blades%2C+and+more+to+choose+from.+To+study+out+extra+about+pocket+knives+and+extensive+selection+in+the+market%2C+please+visit++for+professional+ideas+and+beneficial+perception+earlier+than+buying.+A+knife+is+a+matter+of+preference%2C+but+these+are+five+of+our+favorite+pocket+knives+that+consistently+get+rave+reviews.+Tip+up+or+tip+down+pocket+clip%0D%0A%0D%0AAt+3.75+ounces%2C+the+Spyderco+Para-Navy+2+makes+an+awesome+daily+carry+knife+as+well+as+a+common+use+knife%2C+and+it+does+this+because+of+the+great+stability+between+weight%2C+robustness+and+performance.+The+tip+on+the+blade+is+somewhat+bit+thin+for+me+to+suggest+it+as+a+survival+knife%2C+but+it+would+easily+withstand+any+camping+trip+or+each+day+carry+perform+you+may+consider.+searching+knife+%2C+the+knife+has+a+greater+size+%2C+stiffness+%2C+good+supplies+and+A+slender%2C+straight+blade+with+a+central+level+is+best+for+peeling+spuds+and+whittling+kindling.+The+just+about+infinite+variety+in+brand+names+of+knives+is+confusing+at+greatest%2C+and+misleading+at+worst.+Opening+a+letter.+Sure%2C+you+can+use+your+finger%2C+however+using+a+knife+is+simply+more+manly.+Boker+Plus+Credit+score+Card+Knife%0D%0A%0D%0APocket+knives+are+excellent+companions+as+a+result+of+they%27re+transportable+and+make+themselves+helpful+in+lots+of+conditions%2C+whether+or+not+it%27s+tenting+exterior+or+let%27s+simply+say+you+are+lost+in+a+desert+island.+When+it%27s+a+must+to+select+one%2C+make+sure+that+the+knife+meets+your+wants+and+is+in+response+to+your+preferences.+There+are+a+zillion+selections+today.+There+are+traditional+knives%2C+single+knives%2C+Swiss+knives%2C+tactical+knives+%2C+multi-device+knives+and+and+so+forth.+Selecting+out+a+knife+is+a+private+thing.+It%27s+like+choosing+out+a+pockets.+All+of+it+comes+down+to+personal+taste.+Men+have+a+bond+with+their+instruments.+Choose+something+that+feels+good+in+your+hand%2C+and+that+you%27re+going+to+take+pleasure+in+and+need+to+carry+around.+Home+%3E%3E+Make+Your+Own+Emergency+Kit+%3E%3E+The+Greatest+Pocket+Knife+Fastened-Blade+Knives%0D%0A%0D%0ANow+this+has+been+a+huge+debate+for+nevertheless+lengthy+pocket+knives+have+been+round.+The+giants+Victorinox+are+always+competitors+for+being+the+best+however+then+once+more+there+are+a+whole+lot+of+various+corporations.+I+private+love+CRKT+and+Gerber%2C+however+then+once+more+everyone+has+their+own+private+choice.+I+will+leave+the+difficult+choice+up+to+you+on+what+one+of+the+best+pocket+knife+model+is%2C+but+then+again+most+manufacturers+have+no+less+than+just+a+few+wonderful+knives+so+with+any+alternative+you+may+be+nice%21%0D%0A%0D%0AThis+is+a+knife+that+can+take+some+abuse.+From+the+stable+RotoLock+mechanism+to+the+prime+quality+Elmax+PM+one+hundred+seventy+steel+on+the+4.5mm+thick+blade%2C+you+may+trust+this+knife+to+come+back+by+way+of+for+you.+The+handles+are+CNC+machined+perpendicular+to+the+blades+size+to+make+sure+a+pleasant+stable+grip+on+the+knife.+Timothy+Dahl+of+Charles+%26+Hudson+%2C+also+reviewing+the+unique+model%2C+%E2%80%9Cabsolutely+liked+this+knife.+The+appear+and+feel+is+nice+and+the+design+is+exclusive.%E2%80%9D+7.one+hundred+twenty+five%22+three%22+3.2+oz+CPM-S30V+%24%24%24%24+4.9+Buck+112FG+Ranger+with+Finger+Grooves%2C+Lockback+Folding+Knife+3.375%22+5+oz+154CM+%24%24%24+four.8+Kershaw+Black+Shallot+Knife+with+Tungsten+DLC+Black+coating+on+Blade+%26+Deal+with+4.625%22+1.875%22+1.5+oz+420HC+Stainless+Metal+%24+4.8+Kershaw+Ken+Onion+Tactical+Blur+Folding+Knife+with+Speed+Secure%0D%0A%0D%0AEarlier+than+you+let+your+youngster+run+off+into+the+woods+unsupervised+it%27s+essential+examine+him+to+that+kid+we+all+knew+growing+up.+Is+your+child+that+baby%3F+If+so%2C+then+it+would+be+best+to+wait+a+handful+of+extra+years+or+extra+before+permitting+them+to+have+a+pocket+knife.+It+boils+down+as+to+whether+or+not+you%27re+feeling+your+baby+is+mature%2C+respectful+and+intelligent+sufficient+to+care+for+a+potentially+dangerous+instrument.+Take+a+while+and+observe+your+child%2C+especially+how+he+or+she+reacts+round+other+youngsters.+Peer+stress+can+lead+to+unlucky+resolution+making.%0D%0A%0D%0AKummerfeldt+says+the+neatest%2C+safest+place+to+stash+your+knife+is+in+a+straightforward-entry+spot+in+your+backpack.+You+are+asking+for+trouble+by+sporting+a+fixed-blade+knife+in+your+belt.+If+you+fall%2C+the+knife+may+rotate+inward+and+you+could+land+right+on+the+blade.+had+quite+a+bit+SOG+fashion+knife+participate+in+tactical+battlefield+for+the+soldiers.+electrical+meeting+shall+choose+its+Victorinox+knife.+Victorinox+is+an+effective+If+you+happen+to+use+a+knife+as+ornament+or+souvenir%2C+you+possibly+can+choose+its+SOG+knife.+SOG+has+a+very+small+product+line%3B+you+may+fairly+put+on+in+hand+everybody+may+be+assured+of+the+most+effective+choices.+Choosing+knife+depends+on+each+individual%2C+each+type+and+finance.+Hope+you+may+choose+up+a+knife+best.+Use+a+Q-tip+or+toothbrush+to+scrub+the+knife+after+using+it.%0D%0A%0D%0AThe+Flash+II+has+a+army+background%2C+but+unlike+the+Civilian%2C+it+has+uses+that+go+above+and+past+the+rippin%27+and+the+tearin%27.+It%27s+meant+to+be+a+partial+utility+knife+that+can+employ+the+serrated+lower+end+for+easy+sawing+procedures+whereas+the+graceful+higher+portion+can+handle+scraping+or+slashing.+The+blade+is+3.5+inches+of+AUS-8+metal+that+carries+a+bolt+for+easy+one-handed+opening.+It+is+spring+loaded%2C+so+it+snaps+to+attention+with+a+minimal+of+effort.+The+ergonomic+deal+with+means+it%27s+arduous+to+drop+and+almost+unattainable+to+have+pulled+out+of+your+grip.+An+awesome+folding+knife+for+the+survivalist+that+wants+just+a+little+bit+of+every+thing.++Buy&create=Create ve additionally] seen them first hand on my buddy’s My Tighe and his had been bent and chipped. I feel they give the impression of being really cool but I might take standard serrations that work over these.

Auslander–Reiten theory

2014-01-28T10:48:54Z

138.38.106.89: Undid revision 591721034 by Noix07 (talk)

In [[complex analysis]], a branch of [[mathematics]], the
'''Hadamard three-lines theorem''', named after the French mathematician [[Jacques Hadamard]], is a result about the behaviour of [[holomorphic function]]s defined in regions bounded by parallel lines in the complex plane.

==Statement==
Let ''f''(''z'') be a bounded function of ''z = x + iy'' defined on the strip

:<math>\{x+iy:a\le x \le b\},</math>

holomorphic in the interior of the strip and continuous on the whole strip. Then, if

:<math> M(x)=\sup_y |f(x+iy)|, \,</math>

log ''M''(''x'') is a convex function on [''a'', ''b''].

In other words, if <math>x=t a + (1-t) b</math> with <math>0\le t \le 1</math>, then

:<math>M(x)\le M(a)^t M(b)^{1-t}. \, </math>

==Proof==
Define <math> F(z)</math> by

:<math> F(z)=f(z) M(a)^{{z-b\over b-a}}M(b)^{{z-a\over a-b}}.</math>

Thus <math>|F(z)|\le 1</math> on the edges of the strip. The result follows once it is shown that the inequality also holds in the interior of the strip.

After an affine transformation in the coordinate ''z'', it can be assumed that ''a'' = 0 and ''b'' = 1.
The function

:<math>\displaystyle{F_n(z) = F(z) e^{z^2/n} e^{-1/n}}</math>

tends to 0 as |''z''| tends to infinity and satisfies |''F''''n''| ≤ 1 on the boundary of the strip. The [[maximum modulus principle]] can therefore be applied to ''F''''n'' in the strip. So |''F''''n''(''z'')| ≤ 1. Since ''F''''n''(''z'') tends to ''F''(''z'') as ''n'' tends to infinity. it follows that |''F''(''z'')| ≤ 1.

==Applications==
The three-line theorem can be used to prove the [[Hadamard three-circle theorem]] for a bounded continuous function <math>g(z)</math> on an
[[annulus (mathematics)|annulus]] <math>\{z: r \le |z| \le R\}</math>, holomorphic in the interior. Indeed applying the theorem to

:<math> f(z) = g(e^{z}), \, </math>

shows that, if

:<math>m(s)=\sup_{|z|=e^s} |g(z)|, \, </math>

then <math>\log\, m(s)</math> is a convex function of ''s''.

The three-line theorem also holds for functions with values in a [[Banach space]] and plays an important role in [[Interpolation space#Complex interpolation|complex interpolation theory]]. It can be used to prove [[Hölder's inequality]] for measurable functions

:<math> \int |gh|\le \left(\int |g|^p\right)^{1\over p} \cdot \left(\int |h|^q\right)^{1\over q},</math>

where <math>{1\over p} + {1\over q} =1</math>, by considering the function

:<math> f(z) = \int |g|^{pz} |h|^{q(1-z)}.</math>

==See also==
*[[Riesz–Thorin theorem]]

==References==
*{{citation|first=Jacques|last=Hadamard|authorlink=Jacques Hadamard|title=Sur les fonctions entières|journal=Bull.Soc. Math. France|volume=24|year=1896|pages=186–187|url=http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1896__24_/BSMF_1896__24__184_1/BSMF_1896__24__184_1.pdf}} (the original announcement of the theorem)
*{{citation|title=Methods of modern mathematical physics, Volume 2: Fourier analysis, self-adjointness|first= Michael|last= Reed|authorlink1=Michael C. Reed|first2=Barry|last2= Simon|authorlink2=Barry Simon|publisher=Elsevier|year=1975|isbn=0-12-585002-6|pages=33–34}}
*{{citation|title=Complex made simple|volume= 97|series= Graduate studies in mathematics|first= David C.|last= Ullrich|publisher=[[American Mathematical Society]]|year= 2008|isbn=0-8218-4479-2|pages=386–387}}

[[Category:Theorems in complex analysis]]

Multisymplectic integrator

2013-11-25T14:00:28Z

138.38.106.110: /* Preissman box scheme */

{{multiple issues|
{{external links|date=June 2013}}
{{refimprove|date=June 2013}}
{{notability|date=June 2013}}
}}
{{Financial risk types}}
'''Five risks''' are the five types of [[financial risk]] faced by long-term investors as opposed to those faced by short-term [[Trader (finance)|traders]], who make frequent purchases and have shorter holding periods. Estimating the value of assets and [[portfolio]]s may be enhanced by evaluating them through the lens of the five risks. Some of these risks are derived from traditional [[Modern Portfolio Theory]] concepts, but certain assumptions have been tweaked to provide a way of understanding how investments respond to various changes in the overall [[economy]].

==Market risk==
[[Market risk]] has traditionally been measured through the [[beta (finance)|beta]] coefficient of the [[Capital Asset Pricing Model]]. This coefficient represents the variability of asset (or portfolio) returns as they relate to the market (or some other benchmark). When market returns go up or down, beta indicates how much (and in which direction) the comparative asset returns tend to move up or down, based on historical values. Many investors use beta in their valuations of asset prices by calculating a [[risk premium]] that is higher for investments with higher betas—or greater risk.

Traditional calculations of [[beta (finance)|beta]] do not distinguish between returns variability on days when the benchmark has positive returns and days when it has negative returns. This can lead investors to under- or over-estimate the risk involved with an investment when up-market and down-market variability are asymmetrical. An alternative measure, “[[downside beta]],” has been proposed<ref name="Five Types of Risk">{{cite journal|last=Chong|first=James T., PhD|coauthors=Jennings, William P., PhD; Phillips, Michael G., PhD|title=Five Types of Risk and a Fistful of Dollars: Practical Risk Analysis for Investors|journal=Journal of Financial Service Professionals|date=May 2012|pages=68–76|url=http://www.cfpionline.org/wp-content/uploads/2012/07/Chong-Jennings-Phillips-JFSP-2012-Five-kinds-of-risk-and-a-fistful-of-dollars.pdf|accessdate=26 June 2013}}</ref> as a more precise way of measuring the real market risk investors face. Positive up-market returns variability indicates the “risk” of higher returns than the market on up-market days, which undermines traditional beta as a measure of [[financial risk]] for the rational investor. Because downside beta only measures returns variability on days when the market has negative returns, it is a more accurate proxy for risk. Furthermore, assets with high up-market returns variability and low downside beta (low market risk) are mistaken to be more risky by traditional beta measures.

==Momentum risk==
Momentum risk is a [[financial risk|risk]] associated with a decline in the popularity of an asset (or industry, or [[asset class]]). While momentum-based strategies have been shown to be effective for up to six months,<ref name="The 52-Week High and Momentum Investing">{{cite journal|last=George|first=T.J.|coauthors=Hwang, C.Y.|title=The 52-Week High and Momentum Investing|journal=Journal of Finance|year=2004|volume=59|pages=2145–2176}}</ref> asset prices tend to revert to their economic values over longer time periods. One measure of momentum risk is a price-to-high-price trailing ratio that takes into account the [[market price]] and the trailing year’s high price. This value will always be between 0 and 1 because when a new high price is recorded, it becomes the new denominator in the ratio. This measure allows investors to eliminate from consideration assets that have fallen substantially in value in the last year, thereby mitigating momentum risk.

==Intrinsic value risk==
The intrinsic value of a stock relates to the company’s long-term [[cash-flow]] potential. Traditional measures of intrinsic value have been based on cash-flow analyses and investigations of [[financial statement]]s which lead to [[forecasting]] that is subject to [[statistical error]]. These measures, however, do little to estimate the “systematic” or [[economic risk]] associated with asset value. To address this deficiency the Eta® model has been proposed as a more comprehensive proxy for intrinsic value risk. The Eta® model is an [[econometric model]] that explains, on average, over 90% of [[variance]] in asset prices. It measures how assets relate to the most influential 18 economic factors in the [[United States economy]]. Among these factors are the [[unemployment rate]], bond yields, and the [[consumer price index]]. A good measure of intrinsic value risk within the Eta® model is the MacroRisk Index, which summarizes the [[absolute value]]s of each of the economic factors’ impact on an investment’s price [[volatility]].<ref name="Five Types of Risk" />

==Residual risk==
Investors also face [[financial risk|risk]] that is not associated with general economic conditions. This is called the residual risk (also known as [[value at risk]]), and it is defined as the amount of asset value that remains unexplained by the general [[economy]], based on a regression of the asset values and the Eta® model. The Residual Risk Index measures residual risk as a percentage of [[asset price]].<ref name="Five Types of Risk" />

==Attribution instability risk==
The final type of [[financial risk]] that long-term investors face is attribution instability risk. This refers to the risk that an investment changes its relationship to the overall [[economy]] as measured by [[econometric model]]s. <math>R^2</math> is a statistical measure of how well an econometric model fits the price data for a particular investment. Attribution instability risk is calculated as <math>1- R^2</math>, and it indicates the level of risk that an asset will not behave in a manner consistent with econometric predictions.<ref name="Five Types of Risk" />

==See also==
*[[Beta (finance)|Beta]]
*[[Capital asset pricing model]]
*[[Cost of capital]]
*[[Downside beta]]
*[[Downside risk]]
*[[Dual-beta]]
*[[Financial risk]]
*[[Macro risk]]
*[[Upside beta]]
*[[Upside risk]]

==References==
{{Reflist}}

==External links==
*[http://www.macrorisk.com MacroRisk Analytics] “Patented and proprietary macro risk measurements and tools for investors since 1999”.
*[http://www.finra.org/ FINRA] "Investor Protection. Market Integrity."
*[http://www.getsmarteraboutmoney.ca/en/Pages/default.aspx Get Smarter About Money] "Investor Education Fund".
*[http://www.wiley.com/WileyCDA/WileyTitle/productCd-1118300181.html Investment Risk and Uncertainty: Advanced Risk Awareness Techniques for the Intelligent Investor]
*[http://www.overstock.com/Books-Movies-Music-Games/Fear-and-Greed-Investment-Risks-and-Opportunities-in-a-Turbulent-World-Hardcover/6818723/product.html Fear and Greed: Investment Risks and Opportunities in a Turbulent World]

{{DEFAULTSORT:Five risks}}
[[Category:Jargon]]
[[Category:Finance|Terminology, Financial]]
[[Category:Economics terminology|*Finance]]
[[Category:Technical terminology|*Finance]]

Steinitz exchange lemma

2013-03-11T15:48:39Z

138.38.106.89:

In [[scholastic logic]], a '''syncategorematic term''' (''syncategorema'') is a word that cannot serve as the subject or the predicate of a proposition, and thus cannot stand for any of Aristotle's categories, but can be used with other terms to form a proposition. Words such as 'all', 'and', 'if' are examples of such terms.<ref>Grant, p. 120.</ref>

The distinction between categorematic and syncategorematic terms was established in ancient Greek grammar. Words that designate self-sufficient entities (i.e., nouns or adjectives) were called categorematic, and those that do not stand by themselves were dubbed syncategorematic, (i.e., prepositions, logical connectives, etc.). [[Priscian]] in his ''Institutiones grammaticae'' <ref>Priscian, ''Institutiones grammaticae'', II, 15</ref> translates the word as ''consignificantia''. Scholastics retained the difference, which became a dissertable topic after the 13th century revival of logic. [[William of Sherwood]], a representative of terminism, wrote a treatise called ''Syncategoremata''. Later his pupil, [[Peter of Spain]], produced a similar work entitled ''Syncategoreumata''.<ref>[http://plato.stanford.edu/entries/peter-spain/#4 Peter of Spain], ''Stanford Encyclopedia of Philosophy'' online</ref>

In [[propositional calculus]], a '''syncategorematic term''' is a term that has no individual meaning (a term with an individual meaning is called [[categorematic]]). Whether a term is syncategorematic or not is determined by the way it is defined or introduced in the language.

In the common definition of propositional logic, examples of syncategorematic terms are the logical connectives. Let us take the connective <math>\land</math> for instance, its semantic rule is:

<math>\lVert \phi \land \psi \rVert = 1</math> iff <math>\lVert \phi \rVert = \lVert \psi \rVert = 1</math>

So its meaning is defined when it occurs in combination with two formulas <math>\phi</math> and <math>\psi</math>. But it has no meaning when taken in isolation, i.e. <math>\lVert \land \rVert</math> is not defined.

We could however define the <math>\land</math> in a different manner, e.g., using [[lambda calculus|λ-abstraction]]: <math>(\lambda b.(\lambda v.b(v)(b)))</math>, which expects a pair of Boolean-valued arguments, i.e., arguments that are either ''TRUE'' or ''FALSE'', defined as <math>(\lambda x.(\lambda y.x))</math> and <math>(\lambda x.(\lambda y.y))</math> respectively. This is an expression of [[type system|type]] <math>\langle \langle t, t \rangle, t \rangle</math>. Its meaning is thus a binary function from pairs of entities of type [[truth-value]] to an entity of type truth-value. Under this definition it would be non-syncategorematic, or categorematic. Note that while this definition would formally define the <math>\land</math> function, it requires the use of <math>\lambda</math>-abstraction, in which case the <math>\lambda</math> itself is introduced syncategorematically, thus simply moving the issue up another level of abstraction.

==Notes==
<References/>

==References==
* Grant, Edward, ''God and Reason in the Middle Ages'', Cambridge University Press (July 30, 2001), ISBN 978-0-521-00337-7.

{{DEFAULTSORT:Syncategorematic Term}}
[[Category:Logical calculi]]
[[Category:Term logic]]
[[Category:Propositional calculus]]