By Gerard 'T Hooft

At the fiftieth anniversary of Yang–Mills thought, this helpful quantity seems again on the advancements and achievements in uncomplicated particle physics that ensued from that stunning concept.

During the final 5 many years, Yang-Mills idea, that's undeniably crucial cornerstone of theoretical physics, has elevated generally. it's been investigated from many views, and lots of new and unforeseen good points were exposed from this concept. In contemporary a long time, except excessive power physics, the speculation has been actively utilized in different branches of physics, equivalent to statistical physics, condensed subject physics, nonlinear structures, and so forth. This makes the idea an vital subject for all who're concerned about physics.

An overseas staff of specialists, every one of whom has left his mark at the advancements of this awesome thought, give a contribution essays or extra unique technical debts to this quantity. those articles spotlight the hot discoveries from the respective authors’ views. the prestigious individuals are: S Adler, F A Bais, C Becchi, M Creutz, A De Rújula, B S DeWitt, F Englert, L D Faddeev, P Hasenfratz, R Jackiw, A Polyakov, V N Popov, R Stora, P van Baal, P van Nieuwenhuizen, S Weinberg, F Wilczek, E Witten, C N Yang. incorporated in every one article are introductory and explanatory feedback through the editor, G ’t Hooft, who's himself an incredible participant within the improvement of Yang-Mills concept.

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**Example text**

I I Pnoor'. We only prove the homogeneous estimates (ii) and (iv), the proofs of (i) and (iii) being similar. Also, we assume I ( @, the necessary modifications to treat the cds€ Q: oo are obvious. 2). StBp 1. Proof of (ii). );dr); llJ\',. " aj(t):2z"illT(t)(f-t(l{l"di0))ll2r. ,,o,r,,o^ ;; . 40 2. , ) 'rL-/ 1 22" il * : clle a; u ", which is the desired estimate. Sr:pp 2. Proof of (iv). 6)) eoo(t) , . t, y . ',v,"o'' o,ovt)' JJI ' : 2ll' > 1. It follows that llt ll ui@atll. ,-. lltp(Z) "ll,t ",I f s" J llbift)ll2"p1dt I ^ r4 dt : : c |f (/-!

There exists a constant C independent of 1s) lwo ="(lt,l') (l tr"t'*t,t') Qi Qi j such that rorau uenL(s,1. Qi Indeed, suppose first l[ > 3. 18) follows by using Holder's inequality. Suppose now from Sobolev's embedding that ' llullr,"

10) Let$(r):lrl-n tt@(tl(r) where the funct'ion H lo with0