ONE SILVESTRU SEVER DRAGOMIR 1;2
|
|
- Rosalind Preston
- 5 years ago
- Views:
Transcription
1 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE SILVESTRU SEVER DRAGOMIR ; Abstract. In this paper we introuce some -ivergence measures that are relate to the Jensen s ivergence introuce by Burbea an Rao in 98. We establish their joint convexity an provie some inequalities between these measures an a combination o Csiszár s -ivergence, -mipoint ivergence an -integral ivergence measures.. Introuction Let (; A) be a measurable space satisying jaj > an be a - nite measure on (; A) : Let P be the set o all probability measures on (; A) which are absolutely continuous with respect to : For P; Q P, let p = P Q an q = enote the Raon-Nikoym erivatives o P an Q with respect to : Two probability measures P; Q P are sai to be orthogonal an we enote this by Q? P i P (q = g) = Q (p = g) = : Let : [; )! ( ; ] be a convex unction that is continuous at ; i.e., () = lim u# (u) : In 963, I. Csiszár [4] introuce the concept o -ivergence as ollows. De nition. Let P; Q P. Then (.) I (Q; P ) = (x) ; is calle the -ivergence o the probability istributions Q an P: Remark. Observe that, the integran in the ormula (.) is une ne when = : The way to overcome this problem is to postulate or as above that (.) = lim u ; x : u# u We now give some examples o -ivergences that are well-known an oten use in the literature (see also [3]). 99 Mathematics Subject Classi cation. 94A7, 6D5. Key wors an phrases. Jensen ivergence, -ivergence measures, HH -ivergence measures, -ivergence.
2 S. S. DRAGOMIR.. The Class o -Divergences. The -ivergences o this class, which is generate by the unction ; [; ); e ne by have the orm (.3) I (Q; P ) = (u) = ju j ; u [; ) p q p = p jq pj : From this class only the parameter = provies a istance in the topological sense, namely the total variation istance V (Q; P ) = R jq pj : The most prominent special case o this class is, however, Karl Pearson s -ivergence q (Q; P ) = p that is obtaine or = :.. Dichotomy Class. From this class, generate by the unction : [; )! R 8 u ln u or = ; >< (u) = ( ) [u + u ] or Rn ; g ; >: u + u ln u or = ; only the parameter = (u) = (p u ) provies a istance, namely, the Hellinger istance H (Q; P ) = ( p p q p) : Another important ivergence is the Kullback-Leibler ivergence obtaine or = ; q KL (Q; P ) = q ln : p.3. Matsushita s Divergences. The elements o this class, which is generate by the unction ' ; (; ] given by ' (u) := j u j ; u [; ); are prototypes o metric ivergences, proviing the istances I ' (Q; P ) :.4. Puri-Vincze Divergences. This class is generate by the unctions ; [; ) given by j uj (u) := ; u [; ): (u + ) It has been shown in [6] that this class provies the istances [I (Q; P )] :
3 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 3.5. Divergences o Arimoto-type. This class is generate by the unctions 8 h i ( + u ) ( + u) or (; ) n g ; >< (u) := ( + u) ln + u ln u ( + u) ln ( + u) or = ; >: j uj or = : It has been shown in [8] that this class provies the istances [I (Q; P ) )]min(; or (; ) an V (Q; P ) or = : For continuous convex on [; ) we obtain the -conjugate unction o by (u) = u ; u (; ) u an () = lim (u) : u# It is also known that i is continuous convex on [; ) then so is : The ollowing two theorems contain the most basic properties o -ivergences. For their proos we reer the reaer to Chapter o [7] (see also [3]). Theorem (Uniqueness an Symmetry Theorem). Let ; be continuous convex on [; ): We have I (Q; P ) = I (Q; P ) ; or all P; Q P i an only i there exists a constant c R such that or any u [; ): (u) = (u) + c (u ) ; Theorem (Range o Values Theorem). Let : [; )! R be a continuous convex unction on [; ): For any P; Q P, we have the ouble inequality (.4) () I (Q; P ) () + () : (i) I P = Q; then the equality hols in the rst part o (.4). I is strictly convex at ; then the equality hols in the rst part o (.4) i an only i P = Q; (ii) I Q? P; then the equality hols in the secon part o (.4). I () + () < ; then equality hols in the secon part o (.4) i an only i Q? P: The ollowing result is a re nement o the secon inequality in Theorem (see [3, Theorem 3]). Theorem 3. Let be a continuous convex unction on [; ) with () = ( is normalise) an () + () < : Then (.5) I (Q; P ) [ () + ()] V (Q; P ) or any Q; P P. For other inequalities or -ivergence see [], [7]-[].
4 4 S. S. DRAGOMIR. Some Preliminary Facts For a unction e ne on an interval I o the real line R, by ollowing the paper by Burbea & Rao [], we consier the J -ivergence between the elements t; s I given by J (t; s) := [ (t) + (s)] : As important examples o such ivergences, we can consier [], 8 < ( ) ( t+s ) ; 6= ; J (t; s) := : t ln (t) + s ln (s) () ln t+s ; = : I is convex on I; then J (t; s) or all (t; s) I I: The ollowing result concerning the joint convexity o J also hols: Theorem 4 (Burbea-Rao, 98 []). Let be a C unction on an interval I: Then J is convex (concave) on I I; i an only i is convex (concave) an is concave (convex) on I: We e ne the Hermite-Haamar trapezoi an mi-point ivergences (.) T (t; s) := [ (t) + (s)] (( an (.) M (t; s) := or all (t; s) I I: We observe that ) ) (( ) ) (.3) J (t; s) = T (t; s) + M (t; s) or all (t; s) I I: I is convex on I; then by Hermite-Haamar inequalities (a) + (b) a + b (( ) a + b) or all a; b I; we have the ollowing unamental acts (.4) T (t; s) an M (t; s) or all (t; s) I I: Using Bullen s inequality, see or instance [, p. ], a + b (( ) a + b) we also have (a) + (b) (( (.5) M (t; s) T (t; s) : Let us recall the ollowing special means: ) a + b)
5 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 5 a) The arithmetic mean A (a; b) := a + b ; a; b > ; b) The geometric mean c) The harmonic mean ) The ientric mean 8 >< I (a; b) := >: G (a; b) := p ab; a; b ; H (a; b) := a + ; a; b > ; b b b e a a b a i b 6= a a i b = a ; a; b > e) The logarithmic mean 8 b a >< i b 6= a L (a; b) := ln b ln a ; a; b > >: a i b = a ) The p-logarithmic mean 8 b p+ a >< p+ p i b 6= a; p Rn ; g L p (a; b) := (p + ) (b a) ; a; b > : >: a i b = a I we put L (a; b) := I (a; b) an L (a; b) := L (a; b) ; then it is well known that the unction R 3p 7! L p (a; b) is monotonic increasing on R. We observe that or p Rn ; g we have an [( ) a + b] p = L p p (a; b) ; [( ) a + b] ln [( ) a + b] = ln I (a; b) : = L (a; b) Using these notations we can e ne the ollowing ivergences or (t; s) I n I n where I is an interval o positive numbers: an or all p Rn ; g ; an T p (t; s) := A (t p ; s p ) L p p (t; s) M p (t; s) := L p p (t; s) A p (t; s) T (t; s) := H (t; s) L (t; s) M (t; s) := L (t; s) A (t; s)
6 6 S. S. DRAGOMIR or p = an an T (t; s) := ln M (t; s) := ln G (t; s) I (t; s) I (t; s) A (t; s) or p = : Since the unction () = p ; > is convex or p ( have ; ) [ (; ); then we (.6) T p (t; s) ; M p (t; s) or all (t; s) I I: For p (; ) the unction () = p ; > an or p = ; the unction () = ln are concave, then we have or p [; ) that (.7) T p (t; s) ; M p (t; s) or all (t; s) I I: Finally or p = we have both T (t; s) = M (t; s) = or all (t; s) I I: We nee the ollowing convexity result that is a consequence o Burbea-Rao s theorem above: Lemma. Let be a C unction on an interval I: Then T an M are convex (concave) on I I; i an only i is convex (concave) an is concave (convex) on I: Proo. I T an M are convex on I I then the sum T + M = J is convex on I I, which, by Burbea-Rao theorem implies that is convex an is concave on I: Now, i is convex an is concave on I; then by the same theorem we have that the unction J : I I! R J (t; s) := [ (t) + (s)] is convex. Let t; s; u; v I. We e ne ' () := J (( ) (t; s) + (u; v)) = J ((( ) t + u; ( ) s + v)) or [; ] : = [ (( ) t + u) + (( ) s + v)] ( ) t + u + ( ) s + v = [ (( ) t + u) + (( ) s + v)] ( ) + u + v
7 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 7 Let ; [; ] an ; with + = : By the convexity o J we have ' ( + ) = J (( ) (t; s) + ( + ) (u; v)) = J (( + ) (t; s) + ( + ) (u; v)) = J ( ( ) (t; s) + ( ) (t; s) + (u; v) + (u; v)) = J ( [( ) (t; s) + (u; v)] + [( ) (t; s) + (u; v)]) J (( ) (t; s) + (u; v)) + J (( ) (t; s) + (u; v)) = ' ( ) + ' ( ) ; which proves that ' is convex on [; ] or all t; s; u; v I: Applying the Hermite-Haamar inequality or ' we get (.8) an since an [' () + ' ()] ' () = [ (t) + (s)] ' () = [ (u) + (v)] ' () ; u + v ' () = (( ) t + u) + (( ( ) + u + v ; hence by (.8) we get [ (t) + (s)] (( ) t + u) + ( ) + u + v ) s + v) + u + v [ (u) + (v)] : Re-arranging this inequality, we get (t) + (u) (( + (s) + (v) (( u + v + (( ) t + u) ) s + v) ) s + v) ( ) + u + v ;
8 8 S. S. DRAGOMIR which is equivalent to [T (t; u) + T (s; v)] T ; u + v = T (t; u) + (s; v) ; or all (t; u) ; (s; v) I I, which shows that T is Jensen s convex on I I: Since T is continuous on I I, hence T is convex in the usual sense on I I: Now, i we use the secon Hermite-Haamar inequality or ' on [; ] ; we have (.9) Since ' = hence by (.9) we have t + u ' () ' + (( ) t + u) + s + v : (( ( ) + u + v t + u s + v + + ) s + v) u + v + u + v ; which is equivalent to t + u (( ) t + u) + s + v (( ) s + v) ( ) + u + v + u + v that can be written as [M (t; u) + M (s; v)] M ; u + v = M (t; u) + (s; v) or all (t; u) ; (s; v) I I, which shows that M is Jensen s convex on I I: Since M is continuous on I I, hence M is convex in the usual sense on I I: The ollowing reverses o the Hermite-Haamar inequality hol:
9 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 9 Lemma (Dragomir, [] an []). Let h : [a; b]! R be a convex unction on [a; b] : Then a + b a + b (.) h + h (b a) 8 an (.) The constant 8 We also have: h (a) + h (b) b a b 8 [h (b) h + (a)] (b a) 8 b a + b h + a b a a h () a + b h (b a) a + b h () h 8 [h (b) h + (a)] (b a) : is best possible in all inequalities rom (.) an (.). Lemma 3. Let be a C convex unction on an interval I: I I is the interior o I; then or all (t; s) I I we have (.) M (t; s) T (t; s) 8 C (t; s) where (.3) C (t; s) := [ (t) (s)] (t s) : Proo. Since or b 6= a b a then rom (.) we get (t) + (s) or all (t; s) I I: Remark. I b a (t) t = (( t) a + tb) t; (( ) ) t 8 [ (t) (s)] (t s) = in (t) an t I = sup (t) t I are nite, then C (t; s) ( ) jt sj an by (.) we get the simpler upper boun M (t; s) T (t; s) ( ) jt sj : 8 Moreover, i t; s [a; b] I an since is increasing on I; then we have the inequalities (.4) M (t; s) T (t; s) 8 [ (b) (a)] jt sj :
10 S. S. DRAGOMIR Since J (t; s) = T (t; s) + M (t; s) ; hence J (t; s) 4 [ (b) (a)] jt sj : Corollary. With the assumptions o Lemma 3 an i the erivative is Lipschitzian with the constant K > ; namely then we have the inequality j (t) (s)j K jt sj or all t; s I; (.5) M (t; s) T (t; s) 8 K (t s) : 3. Main Results Let P; Q; W P an : (; )! R. We e ne the ollowing -ivergence (3.) J (P; Q; W ) := J ; (x) = (x) + (x) + : I we consier the mi-point ivergence measure M e ne by + M (Q; P; W ) := (x) or any Q; P; W P, then rom (3.) we get (3.) J (P; Q; W ) = [I (P; W ) + I (Q; W )] M (Q; P; W ) : We can also consier the integral ivergence measure ( t) + t A (Q; P; W ) := t (x) : We introuce the relate -ivergences (3.3) T (P; Q; W ) := an (3.4) We observe that M (P; Q; W ) := T ; (x) = [I (P; W ) + I (Q; W )] A (Q; P; W ) M ; (x) = A (Q; P; W ) M (Q; P; W ) : J (P; Q; W ) = T (P; Q; W ) + M (P; Q; W ) : I is convex on (; ) then by the Hermite-Haamar an Bullen s inequalities we have the positivity properties M (P; Q; W ) T (P; Q; W )
11 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE an or P; Q; W P. We have the ollowing result: J (P; Q; W ) Theorem 5. Let be a C unction on an interval (; ) : I is convex on (; ) an is concave on (; ) ; then or all W P, the mappings are convex. P P 3 (P; Q) 7! J (P; Q; W ) ; M (P; Q; W ) ; T (P; Q; W ) Proo. Let (P ; Q ) ; (P ; Q ) P P an ; with + =. We have J ( (P ; Q ; W ) + (P ; Q ; W )) = J (P + P ; Q + Q ; W ) p (x) + p (x) = J ; q (x) + q (x) (x) = J p (x) + p (x) ; q (x) + q (x) (x) p (x) = J ; q (x) p (x) + ; q (x) (x) p (x) J ; q (x) p (x) + J ; q (x) (x) p (x) = J ; q (x) p (x) (x) + J ; q (x) (x) = J (P ; Q ; W ) + J (P ; Q ; W ) ; which proves the convexity o P P 3 (P; Q) 7! J (P; Q; W ) or all W P. The convexity o the other two mappings ollows in a similar way an we omit the etails. Theorem 6. Let be a C (; ) ; then or all W P unction on an interval (; ) : I is convex on (3.5) M (P; Q; W ) T (P; Q; W ) 8 (Q; P; W ) where (3.6) (Q; P; W ) := Proo. From the inequality (.) we have + 8 or all x : ( ) (x) : ( t) + t t
12 S. S. DRAGOMIR I we multiply by > an integrate on we get [I (P; W ) + I (P; W )] A (Q; P; W ) 8 = 8 which implies the esire inequality. ( ) (x) ; (x) Corollary. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; namely then j (s) (t)j K js tj or all t; s (; ) ; (3.7) M (P; Q; W ) T (P; Q; W ) 8 K (Q; P; W ) ; where (3.8) (Q; P; W ) := ( ) (x) : Remark 3. I there exists < r < < R < such that the ollowing conition hols ((r,r)) r ; R or -a.e. x ; then (3.9) M (P; Q; W ) T (P; Q; W ) 8 [ (R) (r)] (Q; P ) where (Q; P ) := Moreover, i is twice i erentiable an j j (x) : (3.) k k [r;r]; := sup j (t)j < t[r;r] then (3.) M (P; Q; W ) T (P; Q; W ) 8 k k [r;r]; (Q; P; W ) : We also have: Theorem 7. Let be a C unction on an interval (; ) : I is convex on (; ) an is concave on (; ) ; then or all W P, (3.) J (P; Q; W ) [ (P; Q; W ) + (Q; P; W )] ; where (P; Q; W ) := + ( ) (x) :
13 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 3 Proo. It is well known that i the unction o two inepenent variables F : D R R! R is convex on the convex omain D an has an on D then or all (t; s) ; (u; v) D we have the (t; (t; s) (t u) + F (t; s) F (u; (u; (t u) (u; (s v) : Now, i we take F : (; ) (; )! R given by an observe that an F (t; s) = [ (t) + (t; = = (t) (s) an since F is convex on (; ) (; ) ; then by (3.3) we get (t) (t u) + (3.4) (s) (s v) u + v [ (t) + (s)] [ (u) + (v)] + u + v (u) (t u) + u + v (v) (s v) : I we take u = v = in (3.4), then we have (t) (t ) + (3.5) (s) [ (t) + (s)] (s ) or all (t; s) (; ) (; ) : I we take t = p(x) q(x) an s = in (3.5) then we obtain + (3.6) :
14 4 S. S. DRAGOMIR By multiplying this inequality with > we get ( ) + + ( ) or all x : Corollary 3. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.7) J (P; Q; W ) Proo. We have that an similarly 4 K j j + (P; Q; W ) + j j (x) K + j j (x) = K j j (x) = K j j j j (x) = K j j (x) (P; Q; W ) K j j Finally, by the use o (3.) we get the esire result. (x) : (x) : Remark 4. I there exist < r < < R < such that the ollowing conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.8) J (P; Q; W ) 4 k k [r;r]; j j Since an + ; max R ; rg R r; (x) :
15 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 5 hence by (3.8) we get the simpler boun (3.9) J (P; Q; W ) k k [r;r]; (R r) max R ; rg : We also have: Theorem 8. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.) T (P; Q; W ) 6 K j j + (x) : Proo. Let (x; y) ; (u; v) (; )(; ) : I we take F : (; )(; )! R given by then an F (t; s) (t; (t) + (s) = (t) (t; (( ) ) ( ) (( ) ) ( ) [ (t) (( ) )] = (s) = (( ) ) [ (s) (( ) )] an since F is convex on (; ) (; ) ; then by (3.) we get (3.) (t u) + (s v) ( ) [ (t) (( ) )] (t) + (s) (u) + (v) + (t u) + (s v) [ (s) (( ) )] or all (t; s) ; (u; v) (; ) (; ) : (( (( ) ) ) u + v) ( ) [ (u) (( ) u + v)] [ (v) (( ) u + v)]
16 6 S. S. DRAGOMIR I we take u = v = in (3.), then we have (3.) (t ) + (s ) ( ) [ (t) (( ) )] (t) + (s) [ (s) (( ) )] (( ) ) or all (u; v) (; ) (; ) : I we take t = p(x) q(x) an s = (3.3) + p(x) ( ) + q(x) in (3.) then we get ( ) + ( ) + ( ) + : Since is Lipschitzian with the constant K >, hence p(x) + q(x) ( ) + ( ) ( ) + + ( ) + K ( ) + K ( ) = 6 K + : I we multiply this inequality by > an integrate, then we get the esire result (3.). Corollary 4. I there exist < r < < R < such that the conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.4) T (P; Q; W ) 3 k k [r;r]; (R r) max R ; rg : Finally, we also have:
17 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 7 Theorem 9. With the assumptions o Theorem 6 an i is Lipschitzian with the constant K > ; then (3.5) M (P; Q; W ) 8 K j j + (x) : Proo. Let (t; s) ; (u; v) (; ) (; ) : I we take F : (; ) (; )! R given by F (t; s) = (( ) ) (t; s) = ( ) (( ) = ( ) (( ) ) (t; = = (( ) ) (( ) ) an since F is convex on (; ) (; ) ; then by (3.) we get (3.6) (t u) ( ) (( ) ) + (s v) (( ) ) (( ) ) u + v (( ) u + v) + u + v (t u) ( ) (( ) u + v) u + v + (s v) (( ) u + v) : I we take u = v = in (3.6), then we have (3.7) (t ) ( ) + (s ) or all (t; s) (; ) (; ) : (( ) ) (( ) ) (( ) )
18 8 S. S. DRAGOMIR I we take t = p(x) (3.8) ( ) an s = q(x) ( ) + in (3.7) then we get ( ) + + ( ) ( ) ( ) ( ) + + K ( ) + K ( ) : Since hence ( ) = 8 ; ( ) + 8 K + + or all x : I we multiply this inequality by > an integrate, then we get the esire result (3.). Corollary 5. I there exist < r < < R < such that the conition (r; R) hols an i is twice i erentiable an (3.) is vali, then (3.9) M (P; Q; W ) 4 k k [r;r]; (R r) max R ; rg :
19 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE 9 4. Some Examples The Dichotomy Class o -ivergences are generate by the unctions : [; )! R e ne as 8 u ln u or = ; >< (u) = ( ) [u + u ] or Rn ; g ; >: u + u ln u or = : Observe that 8 >< u or = ; (u) = u or Rn ; g ; >: u or = : In this amily o unctions only the unctions with [; ) are both convex an with concave on (; ) : We have I (P; W ) = (x) an M (Q; P; W ) = 8 >< = >: 8 >< = >: R ( ) w (x) p (x) (x) ; (; ) ; R ln p(x) (x) ; = ; + h R ( ) R (x) h q(x)+p(x) i w (x) (x) i ; (; ) h i h i q(x)+p(x) ln q(x)+p(x) (x) ; = : We also have [( t) a + tb] ln [( t) a + tb] t = 4 (b + a) ln I a ; b = A (a; b) ln I a ; b : Thereore ( A (Q; P; W ) := 8 h R q(x) >< ( ) L ; p(x) = >: R A q(x) ; p(x) ln I t i t) + t (x) q(x) (x) ; (; ) ; p(x) (x) ; = :
20 S. S. DRAGOMIR We have J (P; Q; W ) = [I (P; W ) + I (Q; W )] M (Q; P; W ) ; T (P; Q; W ) = [I (P; W ) + I (Q; W )] A (Q; P; W ) an M (P; Q; W ) = A (Q; P; W ) M (Q; P; W ) : Accoring to Theorem 5, or all [; ), the mappings P P 3 (P; Q) 7! J (P; Q; W ) ; M (P; Q; W ) ; T (P; Q; W ) are convex or all W P. I < r < < R, then kk [r;r]; = sup (t) = or [; ): t[r;r] r I there exists < r < < R < such that the ollowing conition hols ((r,r)) then by (3.9), (3.4) an (3.9) we get r ; R or -a.e. x ; (4.) J (P; Q; W ) k k [r;r]; (R r) max R ; rg ; (4.) T (P; Q; W ) 3 an (R r) max R ; rg r (4.3) M (P; Q; W ) (R r) max R 4 r ; rg ; or all [; ) an W P. The intereste reaer may apply the above general results or other particular ivergences o interest generate by the convex unctions provie in the introuction. We omit the etails. Reerences [] J. Burbea an C. R. Rao, On the convexity o some ivergence measures base on entropy unctions, IEEE Tran. In. Theor., Vol. IT-8, No. 3, 98, [] P. Cerone an S. S. Dragomir, Approximation o the integral mean ivergence an - ivergence via mean results. Math. Comput. Moelling 4 (5), no. -, 7 9. [3] P. Cerone, S. S. Dragomir an F. Österreicher, Bouns on extene -ivergences or a variety o classes, Kybernetika (Prague) 4 (4), no. 6, Preprint, RGMIA Res. Rep. Coll. 6(3), No., Article 5. [ONLINE: [4] I. Csiszár, Eine inormationstheoretische Ungleichung un ihre Anwenung au en Beweis er Ergoizität von Marko schen Ketten. (German) Magyar Tu. Aka. Mat. Kutató Int. Közl. 8 (963) [5] I. Csiszár, Inormation-type measures o i erence o probability istributions an inirect observations, Stuia Math. Hungarica, (967), [6] I. Csiszár, On topological properties o -ivergences, Stuia Math. Hungarica, (967), [7] S. S. Dragomir, Some inequalities or (m; M)-convex mappings an applications or the Csiszár -ivergence in inormation theory. Math. J. Ibaraki Univ. 33 (), [8] S. S. Dragomir, Some inequalities or two Csiszár ivergences an applications. Mat. Bilten No. 5 (), 73 9.
21 SOME -DIVERGENCE MEASURES RELATED TO JENSEN S ONE [9] S. S. Dragomir, An upper boun or the Csiszár -ivergence in terms o the variational istance an applications. Panamer. Math. J. (), no. 4, [] S. S. Dragomir, An inequality improving the rst Hermite-Haamar inequality or convex unctions e ne on linear spaces an applications or semi-inner proucts, J. Inequal. Pure an Appl. Math., 3 () (), Art. 3. [] S. S. Dragomir, An inequality improving the secon Hermite-Haamar inequality or convex unctions e ne on linear spaces an applications or semi-inner proucts, J. Inequal. Pure an Appl. Math., 3 (3) (), Art. 35. [] S. S. Dragomir, Upper an lower bouns or Csiszár -ivergence in terms o Hellinger iscrimination an applications. Nonlinear Anal. Forum 7 (), no., 3 [3] S. S. Dragomir, Bouns or -ivergences uner likelihoo ratio constraints. Appl. Math. 48 (3), no. 3, 5 3. [4] S. S. Dragomir, New inequalities or Csiszár ivergence an applications. Acta Math. Vietnam. 8 (3), no., [5] S. S. Dragomir, A generalize -ivergence or probability vectors an applications. Panamer. Math. J. 3 (3), no. 4, [6] S. S. Dragomir, Some inequalities or the Csiszár '-ivergence when ' is an L-Lipschitzian unction an applications. Ital. J. Pure Appl. Math. No. 5 (4), [7] S. S. Dragomir, A converse inequality or the Csiszár -ivergence. Tamsui Ox. J. Math. Sci. (4), no., [8] S. S. Dragomir, Some general ivergence measures or probability istributions. Acta Math. Hungar. 9 (5), no. 4, [9] S. S. Dragomir, Bouns or the normalize Jensen unctional, Bull. Austral. Math. Soc. 74(3)(6), [] S. S. Dragomir, A re nement o Jensen s inequality with applications or -ivergence measures. Taiwanese J. Math. 4 (), no., [] S. S. Dragomir, A generalization o -ivergence measure to convex unctions e ne on linear spaces. Commun. Math. Anal. 5 (3), no., 4. [] S. S. Dragomir an C. E. M. Pearce, Selecte Topics on Hermite- Haamar Inequalities an Applications, RGMIA Monographs,. [Online [3] S. Kullback an R. A. Leibler, On inormation an su ciency, Ann. Math. Stat., (95), [4] J. Lin an S. K. M. Wong, A new irecte ivergence measure an its characterization, Int. J. General Systems, 7 (99), [5] H. Shioya an T. Da-Te, A generalisation o Lin ivergence an the erivative o a new inormation ivergence, Elec. an Comm. in Japan, 78 (7) (995), [6] P. Kaka, F. Österreicher an I. Vincze, On powers o -ivergence e ning a istance, Stuia Sci. Math. Hungar., 6 (99), [7] F. Liese an I. Vaja, Convex Statistical Distances, Teubuer Texte zur Mathematik, Ban 95, Leipzig, 987. [8] F. Österreicher an I. Vaja, A new class o metric ivergences on probability spaces an its applicability in statistics. Ann. Inst. Statist. Math. 55 (3), no. 3, Mathematics, College o Engineering & Science, Victoria University, PO Box 448, Melbourne City, MC 8, Australia. aress: sever.ragomir@vu.eu.au URL: DST-NRF Centre o Excellence in the Mathematical, an Statistical Sciences, School o Computer Science, & Applie Mathematics, University o the Witwatersran,, Private Bag 3, Johannesburg 5, South Arica
CONVEX FUNCTIONS AND MATRICES. Silvestru Sever Dragomir
Korean J. Math. 6 (018), No. 3, pp. 349 371 https://doi.org/10.11568/kjm.018.6.3.349 INEQUALITIES FOR QUANTUM f-divergence OF CONVEX FUNCTIONS AND MATRICES Silvestru Sever Dragomir Abstract. Some inequalities
More informationA New Quantum f-divergence for Trace Class Operators in Hilbert Spaces
Entropy 04, 6, 5853-5875; doi:0.3390/e65853 OPEN ACCESS entropy ISSN 099-4300 www.mdpi.com/journal/entropy Article A New Quantum f-divergence for Trace Class Operators in Hilbert Spaces Silvestru Sever
More informationTHE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS
THE HERMITE-HADAMARD TYPE INEQUALITIES FOR OPERATOR CONVEX FUNCTIONS S.S. DRAGOMIR Abstract. Some Hermite-Hadamard s type inequalities or operator convex unctions o seladjoint operators in Hilbert spaces
More informationFURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM
FURTHER BOUNDS FOR THE ESTIMATION ERROR VARIANCE OF A CONTINUOUS STREAM WITH STATIONARY VARIOGRAM N. S. BARNETT, S. S. DRAGOMIR, AND I. S. GOMM Abstract. In this paper we establish an upper boun for the
More informationPROPERTIES. Ferdinand Österreicher Institute of Mathematics, University of Salzburg, Austria
CSISZÁR S f-divergences - BASIC PROPERTIES Ferdinand Österreicher Institute of Mathematics, University of Salzburg, Austria Abstract In this talk basic general properties of f-divergences, including their
More informationTHEOREM AND METRIZABILITY
f-divergences - REPRESENTATION THEOREM AND METRIZABILITY Ferdinand Österreicher Institute of Mathematics, University of Salzburg, Austria Abstract In this talk we are first going to state the so-called
More informationINEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES
INEQUALITIES OF LIPSCHITZ TYPE FOR POWER SERIES OF OPERATORS IN HILBERT SPACES S.S. DRAGOMIR ; Abstract. Let (z) := P anzn be a power series with complex coe - cients convergent on the open disk D (; R)
More informationarxiv:math/ v1 [math.st] 19 Jan 2005
ON A DIFFERENCE OF JENSEN INEQUALITY AND ITS APPLICATIONS TO MEAN DIVERGENCE MEASURES INDER JEET TANEJA arxiv:math/05030v [math.st] 9 Jan 005 Let Abstract. In this paper we have considered a difference
More informationCalculus and optimization
Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function
More informationarxiv: v2 [cs.it] 26 Sep 2011
Sequences o Inequalities Among New Divergence Measures arxiv:1010.041v [cs.it] 6 Sep 011 Inder Jeet Taneja Departamento de Matemática Universidade Federal de Santa Catarina 88.040-900 Florianópolis SC
More informationResearch Article On Some Improvements of the Jensen Inequality with Some Applications
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 323615, 15 pages doi:10.1155/2009/323615 Research Article On Some Improvements of the Jensen Inequality with
More informationON A CLASS OF PERIMETER-TYPE DISTANCES OF PROBABILITY DISTRIBUTIONS
KYBERNETIKA VOLUME 32 (1996), NUMBER 4, PAGES 389-393 ON A CLASS OF PERIMETER-TYPE DISTANCES OF PROBABILITY DISTRIBUTIONS FERDINAND OSTERREICHER The class If, p G (l,oo], of /-divergences investigated
More informationSome New Information Inequalities Involving f-divergences
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 12, No 2 Sofia 2012 Some New Information Inequalities Involving f-divergences Amit Srivastava Department of Mathematics, Jaypee
More informationSome spaces of sequences of interval numbers defined by a modulus function
Global Journal o athematical Analysis, 2 1 2014 11-16 c Science Publishing Corporation wwwsciencepubcocom/inexphp/gja oi: 1014419/gjmav2i12005 Research Paper Some spaces o sequences o interval numbers
More informationNON-AUTONOMOUS INHOMOGENEOUS BOUNDARY CAUCHY PROBLEMS
25-Ouja International Conerence on Nonlinear Analysis. Electronic Journal o Dierential Equations, Conerence 14, 26, pp. 191 25. ISSN: 172-6691. URL: http://eje.math.tstate.eu or http://eje.math.unt.eu
More informationInequalities of Jensen Type for h-convex Functions on Linear Spaces
Mathematica Moravica Vol. 9-205, 07 2 Inequalities of Jensen Type for h-convex Functions on Linear Spaces Silvestru Sever Dragomir Abstract. Some inequalities of Jensen type for h-convex functions defined
More informationGLOBAL SOLUTIONS FOR 2D COUPLED BURGERS-COMPLEX-GINZBURG-LANDAU EQUATIONS
Electronic Journal of Differential Equations, Vol. 015 015), No. 99, pp. 1 14. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu ftp eje.math.txstate.eu GLOBAL SOLUTIONS FOR D COUPLED
More informationSEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES
SEMI-INNER PRODUCTS AND THE NUMERICAL RADIUS OF BOUNDED LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. The main aim of this paper is to establish some connections that exist between the numerical
More informationA SYMMETRIC INFORMATION DIVERGENCE MEASURE OF CSISZAR'S F DIVERGENCE CLASS
Journal of the Applied Mathematics, Statistics Informatics (JAMSI), 7 (2011), No. 1 A SYMMETRIC INFORMATION DIVERGENCE MEASURE OF CSISZAR'S F DIVERGENCE CLASS K.C. JAIN AND R. MATHUR Abstract Information
More informationLower bounds on Locality Sensitive Hashing
Lower bouns on Locality Sensitive Hashing Rajeev Motwani Assaf Naor Rina Panigrahy Abstract Given a metric space (X, X ), c 1, r > 0, an p, q [0, 1], a istribution over mappings H : X N is calle a (r,
More informationSOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS
International Journal o Analysis and Applications ISSN 2291-8639 Volume 15, Number 2 2017, 179-187 DOI: 10.28924/2291-8639-15-2017-179 SOME CHARACTERIZATIONS OF HARMONIC CONVEX FUNCTIONS MUHAMMAD ASLAM
More informationWUCHEN LI AND STANLEY OSHER
CONSTRAINED DYNAMICAL OPTIMAL TRANSPORT AND ITS LAGRANGIAN FORMULATION WUCHEN LI AND STANLEY OSHER Abstract. We propose ynamical optimal transport (OT) problems constraine in a parameterize probability
More informationWEAK AND STRONG CONVERGENCE OF AN ITERATIVE METHOD FOR NONEXPANSIVE MAPPINGS IN HILBERT SPACES
Applicable Analysis and Discrete Mathematics available online at http://pemath.et.b.ac.yu Appl. Anal. Discrete Math. 2 (2008), 197 204. doi:10.2298/aadm0802197m WEAK AND STRONG CONVERGENCE OF AN ITERATIVE
More informationSOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR PREINVEX FUNCTIONS
Banach J. Math. Anal. 9 15), no., uncorrected galleyproo DOI: 1.1535/bjma/9-- ISSN: 1735-8787 electronic) www.emis.de/journals/bjma/ SOME HERMITE HADAMARD TYPE INEQUALITIES FOR THE PRODUCT OF TWO OPERATOR
More informationA View on Extension of Utility-Based on Links with Information Measures
Communications of the Korean Statistical Society 2009, Vol. 16, No. 5, 813 820 A View on Extension of Utility-Based on Links with Information Measures A.R. Hoseinzadeh a, G.R. Mohtashami Borzadaran 1,b,
More informationREVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE. 1. Introduction We look at the problem of reversibility for operators of the form
REVERSIBILITY FOR DIFFUSIONS VIA QUASI-INVARIANCE OMAR RIVASPLATA, JAN RYCHTÁŘ, AND BYRON SCHMULAND Abstract. Why is the rift coefficient b associate with a reversible iffusion on R given by a graient?
More informationExistence of equilibria in articulated bearings in presence of cavity
J. Math. Anal. Appl. 335 2007) 841 859 www.elsevier.com/locate/jmaa Existence of equilibria in articulate bearings in presence of cavity G. Buscaglia a, I. Ciuperca b, I. Hafii c,m.jai c, a Centro Atómico
More informationEC5555 Economics Masters Refresher Course in Mathematics September 2013
EC5555 Economics Masters Reresher Course in Mathematics September 3 Lecture 5 Unconstraine Optimization an Quaratic Forms Francesco Feri We consier the unconstraine optimization or the case o unctions
More informationJensen-Shannon Divergence and Hilbert space embedding
Jensen-Shannon Divergence and Hilbert space embedding Bent Fuglede and Flemming Topsøe University of Copenhagen, Department of Mathematics Consider the set M+ 1 (A) of probability distributions where A
More informationSOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION
Volume 29), Issue, Article 4, 7 pp. SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION R. K. RAINA / GANPATI VIHAR, OPPOSITE SECTOR 5 UDAIPUR 332, RAJASTHAN,
More informationOn some Hermite Hadamard type inequalities for (s, QC) convex functions
Wu and Qi SpringerPlus 65:49 DOI.86/s464-6-676-9 RESEARCH Open Access On some Hermite Hadamard type ineualities for s, QC convex functions Ying Wu and Feng Qi,3* *Correspondence: ifeng68@gmail.com; ifeng68@hotmail.com
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationINVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN INTEGRAL OVERDETERMINATION CONDITION
Electronic Journal of Differential Equations, Vol. 216 (216), No. 138, pp. 1 7. ISSN: 172-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu INVERSE PROBLEM OF A HYPERBOLIC EQUATION WITH AN
More informationOn the Chi square and higher-order Chi distances for approximating f-divergences
c 2013 Frank Nielsen, Sony Computer Science Laboratories, Inc. 1/17 On the Chi square and higher-order Chi distances for approximating f-divergences Frank Nielsen 1 Richard Nock 2 www.informationgeometry.org
More informationHermite-Hadamard Type Integral Inequalities for Log-η- Convex Functions
Mathematics and Computer Science 20 (): 8-92 http://www.sciencepublishinggroup.com/j/mcs doi: 0.8/j.mcs.2000.3 Hermite-Hadamard Type Integral Inequalities for Log-η- Convex Functions Mohsen Rostamian Delavar,
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationComputing Exact Confidence Coefficients of Simultaneous Confidence Intervals for Multinomial Proportions and their Functions
Working Paper 2013:5 Department of Statistics Computing Exact Confience Coefficients of Simultaneous Confience Intervals for Multinomial Proportions an their Functions Shaobo Jin Working Paper 2013:5
More informationTWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES. S.S. Dragomir and J.J.
RGMIA Research Report Collection, Vol. 2, No. 1, 1999 http://sci.vu.edu.au/ rgmia TWO MAPPINGS RELATED TO SEMI-INNER PRODUCTS AND THEIR APPLICATIONS IN GEOMETRY OF NORMED LINEAR SPACES S.S. Dragomir and
More informationSome Hermite-Hadamard type integral inequalities for operator AG-preinvex functions
Acta Univ. Sapientiae, Mathematica, 8, (16 31 33 DOI: 1.1515/ausm-16-1 Some Hermite-Hadamard type integral inequalities or operator AG-preinvex unctions Ali Taghavi Department o Mathematics, Faculty o
More informationConservation Laws. Chapter Conservation of Energy
20 Chapter 3 Conservation Laws In orer to check the physical consistency of the above set of equations governing Maxwell-Lorentz electroynamics [(2.10) an (2.12) or (1.65) an (1.68)], we examine the action
More informationRelative Entropy and Score Function: New Information Estimation Relationships through Arbitrary Additive Perturbation
Relative Entropy an Score Function: New Information Estimation Relationships through Arbitrary Aitive Perturbation Dongning Guo Department of Electrical Engineering & Computer Science Northwestern University
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION 2 (Rates of change) A.J.Hobson
JUST THE MATHS UNIT NUMBER 10.2 DIFFERENTIATION 2 (Rates of change) by A.J.Hobson 10.2.1 Introuction 10.2.2 Average rates of change 10.2.3 Instantaneous rates of change 10.2.4 Derivatives 10.2.5 Exercises
More informationDiffusion Limit for the Vlasov-Maxwell-Fokker-Planck System
IAENG International Journal o Applie Mathematics, 40:3, IJAM_40_3_05 Diusion Limit or the Vlasov-Maxwell-Fokker-Planck System Najoua El Ghani Abstract In this paper we stuy the iusion limit o the Vlasov-Maxwell-Fokker-Planck
More informationINEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES
INEQUALITIES FOR THE NORM AND THE NUMERICAL RADIUS OF LINEAR OPERATORS IN HILBERT SPACES S.S. DRAGOMIR Abstract. In this paper various inequalities between the operator norm its numerical radius are provided.
More informationTight Bounds for Symmetric Divergence Measures and a Refined Bound for Lossless Source Coding
APPEARS IN THE IEEE TRANSACTIONS ON INFORMATION THEORY, FEBRUARY 015 1 Tight Bounds for Symmetric Divergence Measures and a Refined Bound for Lossless Source Coding Igal Sason Abstract Tight bounds for
More informationOn Picard value problem of some difference polynomials
Arab J Math 018 7:7 37 https://doiorg/101007/s40065-017-0189-x Arabian Journal o Mathematics Zinelâabidine Latreuch Benharrat Belaïdi On Picard value problem o some dierence polynomials Received: 4 April
More information1. Filling an initially porous tube under a constant head imposed at x =0
Notes on Moving Bounary problems, Voller U o M, volle00@umn.eu. Filling an initially porous tube uner a constant hea impose at x =0 Governing equation is base on calculating the water volume lux by the
More informationLogarithmic spurious regressions
Logarithmic spurious regressions Robert M. e Jong Michigan State University February 5, 22 Abstract Spurious regressions, i.e. regressions in which an integrate process is regresse on another integrate
More informationA PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks
A PAC-Bayesian Approach to Spectrally-Normalize Margin Bouns for Neural Networks Behnam Neyshabur, Srinah Bhojanapalli, Davi McAllester, Nathan Srebro Toyota Technological Institute at Chicago {bneyshabur,
More informationRobust Forward Algorithms via PAC-Bayes and Laplace Distributions. ω Q. Pr (y(ω x) < 0) = Pr A k
A Proof of Lemma 2 B Proof of Lemma 3 Proof: Since the support of LL istributions is R, two such istributions are equivalent absolutely continuous with respect to each other an the ivergence is well-efine
More informationInequalities for the numerical radius, the norm and the maximum of the real part of bounded linear operators in Hilbert spaces
Available online at www.sciencedirect.com Linear Algebra its Applications 48 008 980 994 www.elsevier.com/locate/laa Inequalities for the numerical radius, the norm the maximum of the real part of bounded
More informationMath 180, Exam 2, Fall 2012 Problem 1 Solution. (a) The derivative is computed using the Chain Rule twice. 1 2 x x
. Fin erivatives of the following functions: (a) f() = tan ( 2 + ) ( ) 2 (b) f() = ln 2 + (c) f() = sin() Solution: Math 80, Eam 2, Fall 202 Problem Solution (a) The erivative is compute using the Chain
More informationA Weak First Digit Law for a Class of Sequences
International Mathematical Forum, Vol. 11, 2016, no. 15, 67-702 HIKARI Lt, www.m-hikari.com http://x.oi.org/10.1288/imf.2016.6562 A Weak First Digit Law for a Class of Sequences M. A. Nyblom School of
More informationSOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES
SOME INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS OF TWO OPERATORS IN HILBERT SPACES SEVER S DRAGOMIR Abstract Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More information(1) T^- f"f 2 (x)dx < ( ) \ [f 2 (a) + f(a)f(b) + f(b)},
BULL. AUSTRAL. MATH. SOC. VOL. 53 (1996) [229-233] 26A51, 2615 A CONTINUOUS ANALOGUE AN AN EXTENSION OF RAO'S FORMULA C.E.M. PEARCE AN J. PECARIC A continuous analogue is derived for Rado's comparison
More informationA new proof of the sharpness of the phase transition for Bernoulli percolation on Z d
A new proof of the sharpness of the phase transition for Bernoulli percolation on Z Hugo Duminil-Copin an Vincent Tassion October 8, 205 Abstract We provie a new proof of the sharpness of the phase transition
More informationLecture 10: October 30, 2017
Information an Coing Theory Autumn 2017 Lecturer: Mahur Tulsiani Lecture 10: October 30, 2017 1 I-Projections an applications In this lecture, we will talk more about fining the istribution in a set Π
More informationNested Inequalities Among Divergence Measures
Appl Math Inf Sci 7, No, 49-7 0 49 Applied Mathematics & Information Sciences An International Journal c 0 NSP Natural Sciences Publishing Cor Nested Inequalities Among Divergence Measures Inder J Taneja
More informationOn permutation-invariance of limit theorems
On permutation-invariance of limit theorems I. Beres an R. Tichy Abstract By a classical principle of probability theory, sufficiently thin subsequences of general sequences of ranom variables behave lie
More informationarxiv: v1 [math.mg] 10 Apr 2018
ON THE VOLUME BOUND IN THE DVORETZKY ROGERS LEMMA FERENC FODOR, MÁRTON NASZÓDI, AND TAMÁS ZARNÓCZ arxiv:1804.03444v1 [math.mg] 10 Apr 2018 Abstract. The classical Dvoretzky Rogers lemma provies a eterministic
More informationON TAUBERIAN CONDITIONS FOR (C, 1) SUMMABILITY OF INTEGRALS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 54, No. 2, 213, Pages 59 65 Publishe online: December 8, 213 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS Abstract. We investigate some Tauberian
More informationThe group of isometries of the French rail ways metric
Stu. Univ. Babeş-Bolyai Math. 58(2013), No. 4, 445 450 The group of isometries of the French rail ways metric Vasile Bulgărean To the memory of Professor Mircea-Eugen Craioveanu (1942-2012) Abstract. In
More informationASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVE FOR VISCOUS CONSERVATION LAW IN TWO SPACE DIMENSIONS
TANSACTIONS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 35, Number 3, Pages 13 115 S -9947(999-4 Article electronically publishe on September, 1999 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVE FO VISCOUS
More informationAn extension of Alexandrov s theorem on second derivatives of convex functions
Avances in Mathematics 228 (211 2258 2267 www.elsevier.com/locate/aim An extension of Alexanrov s theorem on secon erivatives of convex functions Joseph H.G. Fu 1 Department of Mathematics, University
More informationA nonlinear inverse problem of the Korteweg-de Vries equation
Bull. Math. Sci. https://oi.org/0.007/s3373-08-025- A nonlinear inverse problem of the Korteweg-e Vries equation Shengqi Lu Miaochao Chen 2 Qilin Liu 3 Receive: 0 March 207 / Revise: 30 April 208 / Accepte:
More informationCHAPTER 4. INTEGRATION 68. Previously, we chose an antiderivative which is correct for the given integrand 1/x 2. However, 6= 1 dx x x 2 if x =0.
CHAPTER 4. INTEGRATION 68 Previously, we chose an antierivative which is correct for the given integran /. However, recall 6 if 0. That is F 0 () f() oesn t hol for apple apple. We have to be sure the
More informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics INEQUALITIES FOR GENERAL INTEGRAL MEANS GHEORGHE TOADER AND JOZSEF SÁNDOR Department of Mathematics Technical University Cluj-Napoca, Romania. EMail:
More informationSome Weak p-hyponormal Classes of Weighted Composition Operators
Filomat :9 (07), 64 656 https://oi.org/0.98/fil70964e Publishe by Faculty of Sciences an Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Some Weak p-hyponormal Classes
More informationOn the Cauchy Problem for Von Neumann-Landau Wave Equation
Journal of Applie Mathematics an Physics 4 4-3 Publishe Online December 4 in SciRes http://wwwscirporg/journal/jamp http://xoiorg/436/jamp4343 On the Cauchy Problem for Von Neumann-anau Wave Equation Chuangye
More informationAgmon Kolmogorov Inequalities on l 2 (Z d )
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Publishe by Canaian Center of Science an Eucation Agmon Kolmogorov Inequalities on l (Z ) Arman Sahovic Mathematics Department,
More informationOn the Chi square and higher-order Chi distances for approximating f-divergences
On the Chi square and higher-order Chi distances for approximating f-divergences Frank Nielsen, Senior Member, IEEE and Richard Nock, Nonmember Abstract We report closed-form formula for calculating the
More informationLecture 2: Correlated Topic Model
Probabilistic Moels for Unsupervise Learning Spring 203 Lecture 2: Correlate Topic Moel Inference for Correlate Topic Moel Yuan Yuan First of all, let us make some claims about the parameters an variables
More informationContinuous Solutions of a Functional Equation Involving the Harmonic and Arithmetic Means
Continuous Solutions o a Functional Equation Involving the Harmonic and Arithmetic Means Rebecca Whitehead and Bruce Ebanks aculty advisor) Department o Mathematics and Statistics Mississippi State University
More informationLiterature on Bregman divergences
Literature on Bregman divergences Lecture series at Univ. Hawai i at Mānoa Peter Harremoës February 26, 2016 Information divergence was introduced by Kullback and Leibler [25] and later Kullback started
More informationShifted Independent Component Analysis
Downloae rom orbit.tu.k on: Dec 06, 2017 Shite Inepenent Component Analysis Mørup, Morten; Masen, Kristoer Hougaar; Hansen, Lars Kai Publishe in: 7th International Conerence on Inepenent Component Analysis
More informationConvergence of generalized entropy minimizers in sequences of convex problems
Proceedings IEEE ISIT 206, Barcelona, Spain, 2609 263 Convergence of generalized entropy minimizers in sequences of convex problems Imre Csiszár A Rényi Institute of Mathematics Hungarian Academy of Sciences
More informationHermite-Hadamard Type Inequalities for GA-convex Functions on the Co-ordinates with Applications
Proceedings of the Pakistan Academy of Sciences 52 (4): 367 379 (2015) Copyright Pakistan Academy of Sciences ISSN: 0377-2969 (print), 2306-1448 (online) Pakistan Academy of Sciences Research Article Hermite-Hadamard
More informationtopics about f-divergence
topics about f-divergence Presented by Liqun Chen Mar 16th, 2018 1 Outline 1 f-gan: Training Generative Neural Samplers using Variational Experiments 2 f-gans in an Information Geometric Nutshell Experiments
More informationBalancing Expected and Worst-Case Utility in Contracting Models with Asymmetric Information and Pooling
Balancing Expecte an Worst-Case Utility in Contracting Moels with Asymmetric Information an Pooling R.B.O. erkkamp & W. van en Heuvel & A.P.M. Wagelmans Econometric Institute Report EI2018-01 9th January
More informationA Sketch of Menshikov s Theorem
A Sketch of Menshikov s Theorem Thomas Bao March 14, 2010 Abstract Let Λ be an infinite, locally finite oriente multi-graph with C Λ finite an strongly connecte, an let p
More informationSCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN
SCHUR-CONVEXITY AND SCHUR-GEOMETRICALLY CONCAVITY OF GINI MEAN HUAN-NAN SHI Abstract. The monotonicity and the Schur-convexity with parameters s, t) in R 2 for fixed x, y) and the Schur-convexity and the
More informationSINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
Communications on Stochastic Analysis Vol. 2, No. 2 (28) 289-36 Serials Publications www.serialspublications.com SINGULAR PERTURBATION AND STATIONARY SOLUTIONS OF PARABOLIC EQUATIONS IN GAUSS-SOBOLEV SPACES
More informationImproving Estimation Accuracy in Nonrandomized Response Questioning Methods by Multiple Answers
International Journal of Statistics an Probability; Vol 6, No 5; September 207 ISSN 927-7032 E-ISSN 927-7040 Publishe by Canaian Center of Science an Eucation Improving Estimation Accuracy in Nonranomize
More informationA note on asymptotic formulae for one-dimensional network flow problems Carlos F. Daganzo and Karen R. Smilowitz
A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae
More informationMany problems in physics, engineering, and chemistry fall in a general class of equations of the form. d dx. d dx
Math 53 Notes on turm-liouville equations Many problems in physics, engineering, an chemistry fall in a general class of equations of the form w(x)p(x) u ] + (q(x) λ) u = w(x) on an interval a, b], plus
More informationA LIMIT THEOREM FOR RANDOM FIELDS WITH A SINGULARITY IN THE SPECTRUM
Teor Imov r. ta Matem. Statist. Theor. Probability an Math. Statist. Vip. 81, 1 No. 81, 1, Pages 147 158 S 94-911)816- Article electronically publishe on January, 11 UDC 519.1 A LIMIT THEOREM FOR RANDOM
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationON MÜNTZ RATIONAL APPROXIMATION IN MULTIVARIABLES
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXVIII 1995 FASC. 1 O MÜTZ RATIOAL APPROXIMATIO I MULTIVARIABLES BY S. P. Z H O U EDMOTO ALBERTA The present paper shows that or any s sequences o real
More informationarxiv: v1 [math.ca] 13 Feb 2014
arxiv:1.379v1 math.ca 13 Feb 1 SOME NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR hconvex FUNCTIONS ON THE CO-ORDINATES VIA FRACTIONAL INTEGRALS. ERHAN SET, M. ZEKI SARIKAYA, AND HATICE ÖGÜLMÜŞ Abstract.
More informationProduct and Quotient Rules and Higher-Order Derivatives. The Product Rule
330_003.q 11/3/0 :3 PM Page 119 SECTION.3 Prouct an Quotient Rules an Higher-Orer Derivatives 119 Section.3 Prouct an Quotient Rules an Higher-Orer Derivatives Fin the erivative o a unction using the Prouct
More informationSTATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION
Journal o athematical Analysis ISSN: 227-342, URL: http://www.ilirias.com Volume 4 Issue 2203), Pages 6-22. STATISTICALLY CONVERGENT TRIPLE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTION AAR JYOTI DUTTA, AYHAN
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More informationWELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL PRESSURE IN SOBOLEV SPACES
Electronic Journal of Differential Equations, Vol. 017 (017), No. 38, pp. 1 7. ISSN: 107-6691. URL: http://eje.math.txstate.eu or http://eje.math.unt.eu WELL-POSEDNESS OF A POROUS MEDIUM FLOW WITH FRACTIONAL
More informationVUMAT for Fabric Reinforced Composites
VUMAT or Fabric Reinorce Composites. Introuction This ocument escribes a constitutive mo or abric reinorce composites that was introuce in Abaqus/Exicit 6.8. The mo has been imemente as a built-in VUMAT
More informationMANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec56
MANAGEMENT SCIENCE oi 0.287/mnsc.080.0970ec pp. ec ec56 e-companion ONLY AVAILABLE IN ELECTRONIC FORM inorms 2009 INFORMS Electronic Companion Eects o E-Waste Regulation on New Prouct Introuction by Erica
More informationIERCU. Institute of Economic Research, Chuo University 50th Anniversary Special Issues. Discussion Paper No.210
IERCU Institute of Economic Research, Chuo University 50th Anniversary Special Issues Discussion Paper No.210 Discrete an Continuous Dynamics in Nonlinear Monopolies Akio Matsumoto Chuo University Ferenc
More informationAlgebrability of the set of non-convergent Fourier series
STUDIA MATHEMATICA 75 () (2006) Algebrability of the set of non-convergent Fourier series by Richar M. Aron (Kent, OH), Davi Pérez-García (Mari) an Juan B. Seoane-Sepúlvea (Kent, OH) Abstract. We show
More informationLecture Introduction. 2 Examples of Measure Concentration. 3 The Johnson-Lindenstrauss Lemma. CS-621 Theory Gems November 28, 2012
CS-6 Theory Gems November 8, 0 Lecture Lecturer: Alesaner Mąry Scribes: Alhussein Fawzi, Dorina Thanou Introuction Toay, we will briefly iscuss an important technique in probability theory measure concentration
More information